meyer user's guide 3

User’s Guide Sixth Edition

MeyerFracturing

Simulators

Meyer & Associates, Inc.

Information in this document is subject to change without notice. No part of this document or theassociated software may be reproduced or transmitted in any form or by any means, electronic ormechanical, for any purpose, other than as permitted in the license agreement, without the expressedwritten permission of Meyer & Associates, Inc. (Meyer).

Copyright © 1997, 1999, 2003-2008 Meyer & Associates, Inc. All rights reserved. Printed in theUnited States of America.

MFrac, MF2d, MAcid, MPwri, MView, MAcq, MinFrac, MProd, MNpv, MFast, MFrac-Lite, andMWell are trademarks of Meyer & Associates, Inc.

Microsoft, MS-DOS, Windows and Word are registered trademarks of the Microsoft Corporation.

All names of companies, wells, persons or products contained in this documentation are part of ficti-tious scenarios and are used solely to document the use of a Meyer & Associates, Inc. product.

The databases provided with the software are based on sound engineering practices, but because ofvariable well conditions and other information which must be relied upon, Meyer & Associates, Inc.and its database suppliers make, no warranty, express or implied, as to the accuracy of their data or ofany calculations or opinions expressed therein or derived therefrom. You agree that Meyer & Associ-ates, Inc. and its database suppliers shall not be liable for any loss or damage whether do to negli-gence or otherwise arising out of or concerning such data, calculations or opinions.

Software License Agreement

License Grant

This License Agreement permits you to use one copy of the Meyer software program(s) included inthe package. Meyer grants to the end user, a nonexclusive, nontransferable license, with no right tosublicense or prepare derivative works from the program(s) in connection with your computers.

Warranty

Meyer hereby warrants that it has the right to license the program(s). Meyer agrees to defend, indem-nify and hold harmless the end user against claims or alleged claims of infringement of any patent,copyright or other intellectual property rights.

The licensed software is provided “as-is.” All warranties and representations of any kind with regardto the licensed software are hereby disclaimed, including the implied warranties of merchantabilityand fitness for a particular purpose. Under no circumstances will Meyer & Associates, Inc. be liablefor any consequential, incidental, special or exemplary damages even if appraised of the likelihood ofsuch damages occurring.

“The chessboard is the world, the pieces are the phenomena of theuniverse, the rules of the game are what we call the laws of nature.”

Thomas Huxley

Table of Contents

Introduction___________________________________________xxxi

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiProgram Descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxii

MFrac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiiMView . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiiiMinFrac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiiiMProd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiiiMNpv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxivMFast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxivMPwri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxivMFrac-Lite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxivMWell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

About this Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxvWhat’s in this Guide?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

Technical Support Documentation . . . . . . . . . . . . . . . . . . . . . . . . xxxviiHow to use this Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix

Limited Experience with Meyer Software and Fracture Design . . . xxxixGeneral Knowledge of Meyer Software and Fracturing . . . . . . . . . xxxixExperienced with Meyer Software and Fracturing . . . . . . . . . . . . . . . .xl

Symbols and Conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl

Chapter 1Getting Started _________________________________________ 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2 System & Hardware Requirements . . . . . . . . . . . . . . . . . . . . . .11.3 Installing the Meyer Software . . . . . . . . . . . . . . . . . . . . . . . . . .2

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Before Installing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2To Install the Meyer Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Program Maintenance - Modify, Repair or Remove . . . . . . . . . . . . . . . . . . 4Database Installation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 General Application Information . . . . . . . . . . . . . . . . . . . . . . . 5Application Installation Directories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Application Data Folder Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Starting Folder for Open/Save-as Dialogs. . . . . . . . . . . . . . . . . . . . . . . . . . 6Application File Name Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Application File Extension Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Long File Name Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Working Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Read Only Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Last Opened Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Starting the Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Connecting the Hardware Security Key . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Quick Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Program Check List. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 Customer Support. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Updating Your Hardware Key - MKey Utility . . . . . . . . . . . . . . . . . . . . . . . 11

1.8 Windows Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.9 Meyer Program Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Button Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11File Management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

File Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Opening a File. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Creating a New File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Saving a File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Copying Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Selecting a Printer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Defining the System Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Setting the Input and Output Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Getting Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Accessing Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Error Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Data Entry Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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Run-Time Error Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21File Version Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Working with Spreadsheets and Dialogs. . . . . . . . . . . . . . . . . . . . . . . . . . 23Spreadsheet Keyboard Commands . . . . . . . . . . . . . . . . . . . . . . . . . . 23Spreadsheet Mouse Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Freezing Spreadsheet Panes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Spreadsheet Options Dialog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Spreadsheet Speed Buttons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Dialog and Spreadsheet Column Sizing . . . . . . . . . . . . . . . . . . . . . . . 29Spreadsheets With Movable Columns . . . . . . . . . . . . . . . . . . . . . . . . 30

Working with Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Arranging Plot Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Moving Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Zooming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Printing Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Plot Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Configuring Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Plot Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Axis Title Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Run Menu for Other Meyer Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 64Simulation Data Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Run Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Display the following data windows when the simulator is run . . . . . . 67Scale plots based on the last run while calculating. . . . . . . . . . . . . . . 67Beep after each time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Disable MIN MAX error checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Show template before running simulator. . . . . . . . . . . . . . . . . . . . . . . 67

Generating Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Viewing Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Report Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exporting Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Exporting to Exodus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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2.2 Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73General Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Reservoir Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Real-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Net Present Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Fluid Loss Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Treatment Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Treatment Design Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Wellbore Hydraulics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Fracture Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Fracture Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Fracture Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Flowback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Simulate to Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Fracture Fluid Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Propagation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Fracture Initiation Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Fracture Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Wall Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Tip Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Proppant Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Proppant Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Proppant Ramp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Proppant Flowback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Perforation Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Proppant Transport Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Proppant Settling Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Wellbore-Proppant Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Fracture-Proppant Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Wellbore Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

General Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Deviation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Casing/Tubing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Restrictions Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114BHTP References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117


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Active . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Zone Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Perforation and Fracture Intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . 118Zone Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Treatment Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Auto Design - Treatment Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . 130Input - General Treatment Schedule. . . . . . . . . . . . . . . . . . . . . . . . . 139Acid Frac Treatment Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Real-Time/Replay Treatment Schedule . . . . . . . . . . . . . . . . . . . . . . 148Graphical Treatment Scheduling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Foam Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Rock Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Rock Property Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Insert from Database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Log File Importing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Fluid Loss Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Constant Fluid Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Harmonic and Dynamic Fluid Loss Models. . . . . . . . . . . . . . . . . . . . 178Time Dependent Fluid Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Pressure Dependent Fluid Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Fluid Type Dependent Fluid Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Proppant Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Minimum Number of Proppant Layers to Prevent Bridging. . . . . . . . 186Minimum Concentration/Area for Propped Frac . . . . . . . . . . . . . . . . 187Embedment Concentration/Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Closure Pressure on Proppant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Non-Darcy Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Acid Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Conductivity Damage Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Minimum Conductivity for Etched Length . . . . . . . . . . . . . . . . . . . . . 192Acid Fracture Closure Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Rock Embedment Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192In-situ Acid Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Carbonate Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Fraction of Non-Reactive Fines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

2.4 Run/Performing Calculations. . . . . . . . . . . . . . . . . . . . . . . . .194Calculation Menu Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Stop Menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Simulate Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Simulation Data Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195


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2.5 Plots - Graphical Presentation. . . . . . . . . . . . . . . . . . . . . . . 196Viewing Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

To Create a Plot:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Plot Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Fracture Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Leakoff/Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Wellbore Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Proppant Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Acid Transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Net Present Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Input Treatment Schedule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Multilayer Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Multilayer Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Multilayer Legends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Composite Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Multi-Axes Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Three-Dimensional Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

2.6 Generating Reports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Viewing a Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Explanation of the Report Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

2.7 Program Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Fluid Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Fluid Code and Name. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Shear Rate - Viscosity at . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Rheology Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Friction Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Proppant Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Proppant Database Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Non-Darcy Database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Non-Darcy Database Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Acid Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Description of the Acid Database Parameters . . . . . . . . . . . . . . . . . 225

Casing and Tubing Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Rock Properties Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

2.8 Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231


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Proppant Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Beta Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Proppant Property Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

2.9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236

Chapter 3MView _________________________________________________ 239

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .239Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

3.2 Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241Creating a Parameter List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Using Parameter List Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244Data Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Importing Real-Time Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Importing a Replay Data File. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Importing an ASCII Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Importing an MView Acquired Data File . . . . . . . . . . . . . . . . . . . . . . 246Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Importing an MFrac or Mpwri Data File. . . . . . . . . . . . . . . . . . . . . . . 249Setup Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Editing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Save Data as a Text File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252Merge Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

3.4 Real-Time Menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254Acquisition Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Making a Modem Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265Troubleshooting Modem Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Real-Time Data Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Raw Data View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267Translated Data View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Digital Data View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268


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Configuring the Real-Time Data Window . . . . . . . . . . . . . . . . . . . . . 269Add Log Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Recover Real-Time Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

3.5 Simulation Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Sending Data To MFrac and/or MinFrac . . . . . . . . . . . . . . . . . . . . . . . . . 272

3.6 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274Building Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275Viewing Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Graphically Editing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Graphical Edit Menu Bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Chapter 4MinFrac________________________________________________ 285

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Fracture Closure Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288Fracture Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Total Leakoff Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Fracture Geometry Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Pressure During Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Determining Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294Step Rate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Step Down Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Horner Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Derivative Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297After Closure Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Permeability and Reservoir Pressure . . . . . . . . . . . . . . . . . . . . . . . . 298Diagnostic Plots and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

4.2 Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302General Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Graphical Technique - ASCII text file . . . . . . . . . . . . . . . . . . . . . . . . 304


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Graphical Technique - from MView. . . . . . . . . . . . . . . . . . . . . . . . . . 305User Specified Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Graphical Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305User Specified Pumping Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Fracture Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308Fracture Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308Wall Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309Tip Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310Proppant Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

4.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Base Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Young's Modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Total Leakoff Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318Total Fracture Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Ellipsoidal Aspect Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Flow Behavior Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Consistency Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Spurt Loss Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Total Vertical Depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320Wellbore Fluid Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320Flowback Time (after ISIP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320Flowback Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Leakoff Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320Average Reservoir Fluid Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 321Total Reservoir Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Equivalent Reservoir Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Equivalent Reservoir Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Frac Fluid Leakoff Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Filter Cake Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Closure Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323Injection Rate (2-wings) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324Pumping Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324Closure Time (after ISIP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324Closure Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

History Match Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Import Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

Selecting a Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326Edit Imported Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328


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4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329Select Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Analysis Wizard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Select Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334Wizard Window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Step Rate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348Select Ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349Select Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350Pressure Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352Diagnostic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

Step Down Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355Select Ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356Select Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357Pressure Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Diagnostic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Horner Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363Select Ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364Select Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367Select Ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368Select Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369History Match . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

After Closure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Select Ranges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Select TC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Select Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

4.5 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388Simulation Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Base Data Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389History Match Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392Manage Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

4.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393


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Chapter 5MProd _________________________________________________ 397

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3975.2 Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399

General Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401Simulation Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404Fluid Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405Internal PVT Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406Production Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Fracture Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Non-Darcy Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Permeability Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408NPV/Multi-Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

5.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .411Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411Formation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Reservoir Drainage Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413Dimensionless Reservoir Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . 413Dimensionless Well Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414Total Pay Zone Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414Initial Reservoir Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415Total Reservoir Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415Equivalent Reservoir Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 416Equivalent Reservoir Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Equivalent Reservoir Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Gas Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Bubble Point Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Oil API . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Reservoir Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

Fracture Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Calculate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419Total Pay Zone Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419Effective Propped Pay Zone Height . . . . . . . . . . . . . . . . . . . . . . . . . 419Total Propped Fracture Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Propped Fracture Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Fracture Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Fracture Width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420


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Average Fracture Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Dimensionless Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421Beta Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Variable Fracture Conductivity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422Conductivity Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422Fracture Position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422Fracture Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423Dimensionless Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

History Match Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423No Fracture Case - Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Fracture Case - Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

NPV/Multi-Case Fracture Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 425Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Calculate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Propped Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Effective Propped Pay Zone Height . . . . . . . . . . . . . . . . . . . . . . . . . 427Fracture Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Fracture Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Average Fracture Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Dimensionless Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Beta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428Maximum Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428Slurry Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428Liquid Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428Total Proppant Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Gas PVT Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429Proppant Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430Design Optimization Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Total Proppant Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432Pay Zone Proppant Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432Proppant Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Production Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433Measured Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435Well Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Wellbore Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436Formation Volume Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436Wellbore Skin Factor (base) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437Wellbore Skin Factor (stimulated). . . . . . . . . . . . . . . . . . . . . . . . . . . 437Dimensionless Wellbore Storage Factor. . . . . . . . . . . . . . . . . . . . . . 437Wellbore Skin Factor (base - prefrac) . . . . . . . . . . . . . . . . . . . . . . . . 438Wellbore Skin Factor (stimulated). . . . . . . . . . . . . . . . . . . . . . . . . . . 438Fracture Skin Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Inverse Fracture Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439


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5.4 Run/Performing Calculations. . . . . . . . . . . . . . . . . . . . . . . . .4405.5 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .440

Plot Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441Viewing Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

5.6 Generating Reports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .442Viewing a Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442Explanation of the Report Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442Production Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

5.7 Program Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443Non-Darcy Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Non-Darcy Database Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

5.8 Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .445Proppant Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

5.9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .445

Chapter 6MNpv __________________________________________________ 447

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4476.2 Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .448

Fluid Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449Revenue/Unit Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450Unit Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450Partner Share Option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450MProd Output File with NPV/Multi-Case Data. . . . . . . . . . . . . . . . . . . . . 451

6.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .452Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

6.4 Economic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .453Frac Fluid Unit Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454Proppant Unit Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455Hydraulic Power Unit Cost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455


xviii Table of Contents

Fixed Equipment Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456Miscellaneous Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456Currency Escalation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Unit Revenue for Produced Oil or Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 457Unit Revenue Escalation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458Share of Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458Share of Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

6.5 Variable Unit Revenue Table . . . . . . . . . . . . . . . . . . . . . . . . 4586.6 Variable Share% Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4606.7 Variable Unit Cost Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4626.8 Run/Performing Calculations. . . . . . . . . . . . . . . . . . . . . . . . 4636.9 Plots - Graphical Presentation. . . . . . . . . . . . . . . . . . . . . . . 464

Plot Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464Viewing Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

6.10 Generating Reports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465Viewing a Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465Explanation of the Report Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

Treatment Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466Net Present Value Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

6.11 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

Chapter 7MFast __________________________________________________ 469

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

7.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471Fracture Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472Wall Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473Tip Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474


Table of Contents xix

Proppant Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476Base Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Young's Modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478Fracture Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480Total Pay Zone Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Total Fracture Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Ellipsoidal Aspect Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Injection Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Flow Behavior Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Consistency Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Total Leakoff Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Spurt Loss Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Input Total Volume Injected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Input Fracture Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Maximum Proppant Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . 483

7.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .484Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

7.4 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .487

Chapter 8MPwri__________________________________________________ 489

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

8.2 Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .491General Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

Reservoir Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492Thermal and Poro-Elastic Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . 493Fluid Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494Fluid Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Fracture Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495


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8.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496Treatment Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

General Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496Stage Tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Thermal/Poro-elastic Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498Zone Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Initial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Coefficient of Thermal Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Layer Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500Biot’s Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

Thermal/Water Front Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500Injected Fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Reservoir Lithology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501In-situ Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Reservoir Half-Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Drainage Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

Fluid Loss Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502Constant Fluid Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502Dynamic Fluid Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504Time Dependent Fluid Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510Pressure Dependent Fluid Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

Internal/External Cake Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511Internal Cake Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512External Cake Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

8.4 Run/Performing Calculations. . . . . . . . . . . . . . . . . . . . . . . . 5218.5 Plots - Graphical Presentation. . . . . . . . . . . . . . . . . . . . . . . 522

Water/Thermal Front Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

8.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

Chapter 9MFrac-Lite_____________________________________________ 527

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5279.2 Options and Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

General Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528


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Fracture Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529Proppant Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

9.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .531Zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Zone Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532Rock Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533Fluid Loss Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Constant Fluid Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534Harmonic or Dynamic Fluid Loss Models . . . . . . . . . . . . . . . . . . . . . 534

Chapter 10MWell__________________________________________________ 535

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .535Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

10.2 Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .537General Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538Real-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538Treatment Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Treatment Design Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Wellbore Hydraulics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Wellbore Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541Fluid Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

Proppant Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542Proppant Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543Wellbore-Proppant Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

10.3 Data Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .545Wellbore Hydraulics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545Zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

Zone Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546Perforation and Fracture Intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . 546Zone Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546


xxii Table of Contents

Appendix AHydraulic Fracturing Theory_________________________ 551

A.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551A.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552During Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552After Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553Width-Opening Pressure Elasticity Condition . . . . . . . . . . . . . . . . . . . . . 553Fracture Propagation Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

A.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554A.4 Parametric Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 555A.5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563A.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

Appendix BMultilayer Fracturing _________________________________ 567

B.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567B.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

B.3 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571Momentum Eqns. (i=1,…,n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

B.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574B.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575


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Appendix CMultiple Fractures ____________________________________ 577

C.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .577C.2 Far Field - Multiple Fractures . . . . . . . . . . . . . . . . . . . . . . . .578

Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578Interaction Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579Fluid Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579Width-Opening Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

C.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .580Net Pressure for Multiple Fractures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

Viscous Dominated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581Toughness Dominated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582Constant Critical Stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

Net Pressure Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582Viscous Dominated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583Toughness Dominated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583Constant Critical Stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

C.4 Near Wellbore - Multiple Fractures . . . . . . . . . . . . . . . . . . . .584Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

Laminar Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

Width-Opening Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586Near Wellbore Pressure Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

GDK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587PKN Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

General Near Wellbore Dissipation Function. . . . . . . . . . . . . . . . . . . . . . 587

C.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589


Appendix DFluid Loss_____________________________________________ 591

D.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591D.2 Leakoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

C - Total Leakoff Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592CI - Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592CII - Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592CIII - Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

D.3 Spurt Loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

Appendix EWellbore Friction Factor _____________________________ 597

E.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597E.2 Friction Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597E.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

Appendix FMinifrac Methodology ________________________________ 601

F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601F.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602Width-Opening Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603Fracture Propagation Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605Fluid Loss During Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606Mass Conservation After Shut-In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608Minifrac Closure Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

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Dimensionless Net Pressure Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614Pressure Decline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

F.3 Minifrac Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . .619F.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .621F.5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .623F.6 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .626

Appendix GProduction Model Theory ____________________________ 629

G.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .629G.2 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .630

Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630Pseudopressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632Trilinear Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

Constant Flow Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633Constant Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

Pseudo-Radial Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634Productivity Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

Constant Flow Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634Constant Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

Desuperposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635Method of Images - Generate Closed Systems. . . . . . . . . . . . . . . . . . . . 636

No Fracture: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638Fracture: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

G.3 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .639G.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .641


xxvi Table of Contents

Appendix HNet Present Value Theory____________________________ 643

H.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643H.2 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Fracture Net Present Value (NPV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645Discount Well Revenue (DWR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646Discounted Return on Investment (DROI). . . . . . . . . . . . . . . . . . . . . . . . 647

Appendix ITSO & Frac-Pack Methodology______________________ 649

I.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649I.2 Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

Design Criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

I.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654I.4 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 659I.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

Appendix JProduced Water Reinjection Fracturing ____________ 661

J.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661J.2 Thermal and Water Front Equations . . . . . . . . . . . . . . . . . . 661J.3 Thermoelastic and Poroelastic Stresses . . . . . . . . . . . . . . 664

Thermoelastic Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664Poroelastic Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666


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J.4 Governing Fluid Loss Equations. . . . . . . . . . . . . . . . . . . . . .667Carter’s Solution - Linear Fluid loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668Dimensionless Pressure Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669The uniform fracture flux solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670The infinite-conductivity solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670Dimensionless Rate Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671Linear Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

General Dimensionless Rate Solution . . . . . . . . . . . . . . . . . . . . . . . 674

J.5 Fracture Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .676External Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676Internal Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

J.6 Internal and External Filter Cakes . . . . . . . . . . . . . . . . . . . . .681Internal Filtration Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

Transition Volume Loss per Unit Area . . . . . . . . . . . . . . . . . . . . . . . 685Internal Cake Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686Internal Cake Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688Internal Cake Build and Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

External Cake Filtration Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690Filter Cake Coefficient, Thickness, and Other Relationships . . . . . . 691Filter Cake Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693Filter Cake Build Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694Filter Cake Erosion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

J.7 MPwri Input Dialog Nomenclature. . . . . . . . . . . . . . . . . . . . .696J.1 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .697

Appendix KAfter-Closure Analysis _______________________________ 699

K.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .699K.2 Superposition or Duhamel’s Theorem: General Solution .700K.3 Impulse Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .701K.4 Linear Solution - Background . . . . . . . . . . . . . . . . . . . . . . . .705

Constant Velocity Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 705


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Constant Pressure Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . 706Time Dependent Velocity Boundary Condition . . . . . . . . . . . . . . . . . . . . 708

Variable Injection Rate followed by a Shut-in . . . . . . . . . . . . . . . . . . 709Constant Injection Rate followed by a Shut-in . . . . . . . . . . . . . . . . . 711Apparent Closure Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712

Linear Solution Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

K.5 Radial Solution - Infinite-acting time period . . . . . . . . . . . 714Horner Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715Nolte’s After-Closure Radial Time Function . . . . . . . . . . . . . . . . . . . . . . 717

K.6 Summary and Implementation of After Closure Analysis 718Impulse Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718Horner Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719Nolte After Closure Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720Graphical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

General Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721Permeability and Reservoir Pressure . . . . . . . . . . . . . . . . . . . . . . . . 721Diagnostic Plots and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

K.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

Appendix LPseudosteady State Analysis of Finite Conductivity Vertical Fractures ____________________________________ 727

L.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727Dimensionless Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

L.2 Pseudosteady Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 729Dimensionless Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729Effective Wellbore Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

L.3 Pseudosteady Fractured System Model. . . . . . . . . . . . . . . 732Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732

Reservoir Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733Fracture Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

Inverse Dimensionless Productivity Index. . . . . . . . . . . . . . . . . . . . . . . . 734


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No Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734Finite Conductivity Vertical Fracture System . . . . . . . . . . . . . . . . . . 738Square Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739Rectangular Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

L.4 Pseudosteady Fracture Solutions. . . . . . . . . . . . . . . . . . . . .744Slot Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

Uniform Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747Effective Wellbore Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749

L.5 Pseudosteady Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .750Constant Finite Conductivity Fracture . . . . . . . . . . . . . . . . . . . . . . . . 750Non-Uniform Fracture Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . 754Tail-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757Over Flush. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758

L.6 Infinite Fracture Conductivity . . . . . . . . . . . . . . . . . . . . . . . .761Vertical Fracture in an Infinite System. . . . . . . . . . . . . . . . . . . . . . . . . . . 762

Uniform Flux Vertical Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762Vertical Fracture in a Rectangular Closed Reservoir. . . . . . . . . . . . . . . . 764Effective Wellbore Radius - Infinite Conductivity . . . . . . . . . . . . . . . . . . . 772

L.7 Shape Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .774L.8 Fracture Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .777

External Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778Internal Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778

L.9 Radial Flow Differential Equations . . . . . . . . . . . . . . . . . . . .779Derivation of Radial Diffusivity Equation . . . . . . . . . . . . . . . . . . . . . . . . . 780Linearization of Radial Diffusivity Equation . . . . . . . . . . . . . . . . . . . . . . . 781

Pseudo-Pressure and Pseudo-Time Functions . . . . . . . . . . . . . . . . 781Improved Pseudo-Pressure and Pseudo-Time Functions . . . . . . . . 786Pseudopressure Relationships and Limits . . . . . . . . . . . . . . . . . . . . 789

L.10 Dimensionless Rate & Pressure Solutions . . . . . . . . . . . .793Infinite or Infinitely Acting System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794

Constant Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794Constant Flowing Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

Closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802Constant Mass Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802Constant Flowing Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803


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Numerical Solution -Total Mass and Mass Rate. . . . . . . . . . . . . . . . 814

L.11 Real Gas Potential and Related Equations. . . . . . . . . . . . 821Agarwal Pseudopressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821General Pseudopressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822

L.12 Forchheimer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823Effective Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827

L.13 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

Subject Index______________________________831


Introduction

OverviewThis User's Guide is designed to explain the installation and use of the suite ofhydraulic fracture design and analysis software developed by Meyer & Associates,Inc. The suite of computer programs include MFrac, MView, MPwri, MinFrac,MFast, MProd, MNpv, MFrac-Lite, and MWell. The Meyer software is designed foruse with Windows Vista, Windows 2003, Windows XP SP1, and Windows 2000.

This guide explains the available options and basic procedures used for running theMeyer programs. A number of example files are included with the software. Theseexamples demonstrate the utility of many of the program features, as well as, themanipulation of data. In addition to the examples and the explanation of programoptions, supplementary technical information is contained in the appendices.

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The figure below shows an overall simulation flowchart for the Meyer intergratedsuite of software.

Program DescriptionsA brief description of each of the programs in the Meyer suite of software is givenbelow. Please refer to the corresponding chapter for specific information.

MFracMFrac is a comprehensive design and evaluation simulator containing a variety ofoptions including three-dimensional fracture geometry, auto design features, andintegrated acid fracturing solutions. Fully coupled proppant transport and heattransfer routines, together with a flexible user interface and object oriented devel-opment approach, permit use of the program for fracture design, as well as treat-


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ment analysis. MFrac is the calculation engine for real-time and replay fracturesimulation. When operating in this manner, the program works in conjunction withour real-time data acquisition and display program, MView.

MViewThis program provides a data handling system and display module for real-timehydraulic fracturing and minifrac analysis. MView was intended primarily for usewith the MFrac and MinFrac hydraulic fracturing design and analysis simulators;however, its flexible structure also makes it functional as a general data handlingsystem. This program was created as a separate utility in order to provide a simplesystem that would be reliable and operate in a straight-forward independent man-ner.

MinFracThe MinFrac program has been specifically designed for use as an analysis tool forinjection tests and minifrac analysis. The program provides a means of examiningrate and pressure data during and after a period of injection. This includes, pump-in/shut-in tests for the determination of stress, step rate interpretation and conven-tional or “unconventional” pressure decline analysis. MinFrac, like MFrac, can alsocommunicate dynamically in real-time with MView to share data. This allows datainterpretation during real-time acquisition.

The primary purpose of MinFrac is to calculate closure pressure, fracture effi-ciency, individual fracture geometry leakoff coefficients and near wellbore effects.

MProdMProd is a single phase analytical production simulator developed by Meyer &Associates, Inc. primarily for hydraulic fracturing applications. The program isused to assess the production benefit for a variety of treatment scenarios with com-parison of unfractured wells. Mprod includes an objective methodology for deter-mining unknown or uncertain parameters through regression analysis of simulatedand measured data by history matching. A Fracture Design Optimization optionenables the user to determine the optimum fracture design (length, width, conduc-tivity) that will maximize production for a given amount of proppant mass.

As part of a treatment optimization methodology, MProd is integrated and fullycompatible with MFrac. Output produced by MFrac (propped fracture characteris-tics) can be used by this program as input. Once calculations are performed usingMProd, the results are available for use by MNpv.


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MNpvMNpv provides a tool for forecasting fractured well net present value (NPV) orreturn on investment (ROI). Simply stated, fracture treatment optimization is amethodology used for maximizing well profitability. This process requires a com-parison of the cost penalties and revenue benefits associated with any proposedtreatment scenario. The program has been designed for use with MProd to automat-ically determine and compare various NPV fracture scenarios in order to identify anoptimum design.

MFastMFast is a two-dimensional analytical hydraulic fracturing simulator for aiding thefracturing engineer in designing 2D fractures. The simulator illustrates the impor-tance of various parameters and provides a fast first order solution to fracturegeometry, net pressure, fracture efficiency and treatment design. This simulatorprovides the capability to compare the fracture geometries for Geerstma-deKlerk(GDK), Perkins-Kern (PKN) and ellipsoidal type two dimensional models. SinceMFast was developed from analytical solutions it has the inherent limitations ofsteady state injection, constant mechanical properties, time independent fluid rheol-ogy and single layer properties.

MPwriMPwri is a highly specialized simulator for predicting the pressure and geometry ofhydraulic fractures associated with waterflooding. The program was specificallydesigned for evaluating the effects of injecting large fluid volumes over long peri-ods and for fracture efficiencies approaching zero. Thermal and poro-elastic effectsare also include.

MPwri has options for conventional (diffusion controlled) and ellipsoidal (non-dif-fusion) fluid loss. At early times, fluid loss from the fracture is generally diffusioncontrolled, but at large times the fluid loss is governed by steady-state or pseu-dosteady-state leakoff. This fluid loss option has a marked effect on fracture geom-etry with larger leakoff rates at later times as compared to diffusion alone.

MFrac-LiteMFrac-Lite is a three-dimensional hydraulic fracturing simulator similar to MFracbut with a limited number of MFrac features and capabilities (i.e., a lite version).This simplified three-dimensional simulator provides ease of use with less input


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data and fewer options to choose from for applications which do not require someof the advanced features in MFrac.

MFrac-Lite uses the same numerical routines as MFrac but without some of themore advanced and user specified options. MFrac-Lite has similar real-time capa-bilities as MFrac and is designed to be compatible with like features in MFrac. Thissimulator is designed for those who do not need the full functionality of MFrac.

MWellMWell is a wellbore hydraulics simulator for calculating surface or bottomholepressures, gravitational head, restrictions, transport times and hydraulic powerrequirements in the wellbore. Near wellbore and perforation pressure losses arealso calculated to determine the bottomhole treating pressure in the formation.

MWell was designed for real-time analysis to calculate BHTP’s, from surface con-ditions but can also be used as a design tool for determining wellbore pressure char-acteristics prior to the treatment. MWell is essentially a subset of the MFracsimulator without the fracture simulation. MWell however does provide the capabil-ity to simulate a time dependent formation pressures with a user specified table thatfor inputting the minimum horizontal stress and a time dependent net pressure(pressure above or below the reference minimum stress). If the formation is notfracture the reference pressure should be the reservoir pressure.

About this GuideThis guide assumes the user is familiar with basic Windows operations and termi-nology. When practical, screen displays are shown to demonstrate an operation orresult. It is important to realize that depending on the options selected, the input oroutput displayed in the program may vary from the examples shown. All of thescreens depicted in this manual correspond to the Windows convention establishedby the Microsoft Corporation.

What’s in this Guide?This guide begins by covering topics that are general and common to all programs.To find information specific to an individual program, refer to the chapter for theprogram in question. Each program chapter is organized to correspond to the typi-cal steps needed to begin working with the software. Selecting options, enteringdata, performing calculations and viewing the output are covered. To assist you infinding answers to your questions quickly, an index is provided at the end of thisguide.


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Chapter 1, “Getting Started,” covers the hardware and system requirements to runthe software, as well as, installation procedures and program initiation techniques.Program basics, such as, opening and saving files, as well as, entering and editingvalues in data fields are discussed. This chapter outlines the hardware and systemrequirements for the Meyer suite of software. Specific instructions for installingand running the software are also given.

Chapter 2, “MFrac, Hydraulic Fracturing Simulator,” provides detailed instruc-tions for using the MFrac simulator. This chapter begins with a discussion of theprogram’s options and their influence on the modeling methodology used. Each ofthe data input screens is described with logical steps for data entry. In addition, theuse and maintenance of program databases, performing calculations and viewingprogram output are discussed. Application to replay and real-time simulation is alsopresented.

Chapter 3, “MView, Acquired Data Visualization,” describes how to use MViewto setup, connect and transfer replay or real-time fracture treatment data. Thisincludes the procedures for acquiring data from a service company, either via aserial cable or remotely using a modem. This chapter also outlines how to send datato MFrac and MinFrac for use as simulation input (e.g., rate, pressure, volume,proppant concentration, …). The program’s graphical capability for viewing andanalyzing fracturing data is covered and directions for specifying plot data and con-figuring plot preferences are given. In addition, instructions for graphical editing ofdata are discussed. This includes the application of shifting and statistical curve fit-ting functions.

Chapter 4, “MinFrac, Minifrac Analysis,” begins by explaining the proceduresinvolved in importing, formatting and editing injection test data. The options asso-ciated with the analysis of minifrac data and the techniques used in performing ananalysis are described. An Analysis Wizard is provided for the systematic methodof selecting and performing minifrac analyses. The MinFrac analyses include: StepRate, Step Down, Horner and Regression with history matching. Various timefunctions and pressure derivatives are provided for analyzing rate-pressure data.

Chapter 5, “MProd, Analytical Production Simulator,” explains the function anduse of MProd. It describes the available options and the basic procedures needed torun the program. This includes importing MFrac output data for forecasting frac-tured well performance. Examples are given to demonstrate MProd’s role in treat-ment optimization, as well as, its use as a standalone production forecast tool forfractured and unfractured reservoirs.

Chapter 6, “MNpv, Economic Analysis,” outlines economic analysis as part of atreatment optimization method. All of the procedures and options available inMNpv are covered in this chapter. A description of all required input is also given.


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The use of MProd output data is discussed and examples are given to demonstratethe basic program functions.

Chapter 7, “MFast, An Analytical 2D Fracturing Simulator,” begins by explain-ing the basic 2D fracture models and the associated input data. This simulator pro-vides the capability to compare the fracture geometries for Geerstma-deKlerk(GDK), Perkins-Kern (PKN) and ellipsoidal type two dimensional models. SinceMFast was developed from analytical solutions it has the inherent limitations ofsteady state injection, constant mechanical properties, time independent fluid rheol-ogy and single layer properties. The main advantage of this program is the ease ofuse and ability to provide solutions quickly. It is also a great tool for the first timeuser in that it demonstrates parametric effects how various input parameters affectthe fracture propagation process.

Chapter 8, “MPwri, Produced Water Re-Injection Fracturing Simulator,” explainsthe function and use of our highly specialized simulator. Mpwri includes thermal-and poro-elastic stresses for large injection volumes, low fluid efficiencies frac-tures, and thermal and water front tracking in each layer. All of the procedures andoptions available in MPwri are covered in this chapter. A description of all MPwrirequired input is also given. The use of MPwri output data is discussed and exam-ples are given to demonstrate the basic program functions.

Chapter 9, “MFrac-Lite, Hydraulic Fracturing Simulator - Lite Version,” providesdetailed instructions for using the MFrac-Lite simulator. This chapter begins with acomparison of the MFrac-Lite and MFrac features. Only the options and data inputscreens that are different than those in MFrac are presented in the chapter. Theseinclude the Options, Zones, Rock Properties, and Fluid Loss Data dialogs. Thereader is referred to the MFrac chapter for a description of all other comparable dia-logs.

Chapter 10, “MWell, A Wellbore Hydraulics Simulator,” begins by describing thethe utility and application of the software to calculate surface or bottomhole pres-sures, gravitational head, restrictions, transport times and hydraulic power require-ments in the wellbore. The Main Menu, Options and Input data are discussed indetail. The reader is referred to the MFrac chapter for a description of all othercomparable dialogs.

Technical Support DocumentationAdditional technical support documentation and theory may be found in the Manu-als directory on both the CD and the install directory. The Meyer Appendices existat the end of the mhelp.chm file and in pdf format within the Manuals directory.


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Appendix A, “Hydraulic Fracturing Theory,” provides a technical reference forthe analytical and numerical methods used for fracture simulation. Appendix A is asupplement to the technical presentations provided elsewhere in this guide. It isprovided as a source of information for users who want to develop a better under-standing of fracture modeling concepts.

Appendix B, “Multilayer Fracturing,” explains the methodology used in the pro-gram for simulating the propagation of multiple fractures from multilayer perfo-rated intervals. The general concept and theory are discussed.

Appendix C, “Multiple Fractures,” documents the simulation methods used whenmodeling far field and near wellbore multiple fractures in a single perforated inter-val or layer. Multiple parallel and dendritic fractures which may or may not interactare presented.

Appendix D, “Fluid Loss,” outlines the numerical procedures used to model fluiddiffusion or leakoff to the reservoir. The theory behind various fluid loss options isgiven.

Appendix E, “Wellbore Friction Factor,” presents the correlations and methodsused to simulate frictional pressure dissipation in the wellbore. The correlations aredeveloped in terms of the Fanning friction factor.

Appendix F, “Minifrac Methodology,” describes some of the fundamental rela-tionships associated with the interpretation of minifrac pressure records and thebasic theory implemented in the MinFrac software.

Appendix G, “Production Model Theory,” contains a description of the basic the-ory behind the computational methods used in MProd.

Appendix H, “Net Present Value Theory,” provides a brief overview of the meth-ods used for conducting economic analyses of fracture treatments. The concepts offracture Net Present Value (NPV) and Discounted Return On Investment (DROI)are presented.

Appendix I, “TSO & Frac-Pack Methodology,” presents the basic methodologiesof TSO and frac-packs. A discussion of the design criteria, procedures, differencesand benefits of each are given.

Appendix J, “Produced Water Reinjection Theory or Waterflood Theory” pre-sents the solution methodology and governing equations for our hydraulic fractur-ing produced water reinjection simulator. A summary of the governing waterfront,thermal front, thermal- and poro-elastic stresses, and ellipsoidal fluid loss equationsare presented. The theory building internal and external skins and cakes due to par-ticulate matter in the injected fluid is also presented.


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Appendix K, “After-Closure Analysis” presents the solution methodology fordetermining formation permeability after hydraulic fracturing is formulated in thisreport. A summary of the governing linear and radial flow equations are presentedalong with the graphical method to determine permeability and reservoir pressurefrom the infinite-acting time period. This appendix sets forth the methodology anddocumentation of the governing equations for after-closure analysis as originallypresented by Gu et al. (1993) and Nolte (1997).

Appendix L, “Pseudosteady State Analysis of Finite Conductivity Vertical Frac-tures” presents the solution methodology for pseudosteady behavior of a well witha finite conductivity vertical fracture in rectangular shaped formations based on anew reservoir/fracture domain resistivity concept. The formulation encompasses atransformed resistivity domain that utilizes an equivalent or effective wellboreradius. The resulting pseudosteady solution is presented in the form of the dimen-sionless productivity index ( ).

How to use this GuideThe Meyer software has been designed to minimize the time required to becomefamiliar with its use. Depending on your level of experience, we recommend one ofthe following methods to familiarize yourself with the software:

Limited Experience with Meyer Software and Frac-ture DesignIf you are just learning fracture design and analysis, we recommend you read thisguide from beginning to end to familiarize yourself with the program features,menus and options. This approach will provide a thorough understanding of theprograms and prepare a base from which more advanced applications can beexplored. We recommend starting with the simple analytical 2D fracturing simula-tor MFast.

General Knowledge of Meyer Software and Frac-turing Review the contents of this guide for installation and a basic understanding of thegeneral operations and options. Working through each of the examples providedwith the software will demonstrate the general operation of each program. Many ofthe operational details can be picked up as needed using the on-line Help feature.

JD


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Experienced with Meyer Software and FracturingBrowse the contents of this guide, exploring those topics you are least familiar within more detail. To optimize your time and sample the program features quickly, werecommend working through the demo examples provided with the software. Theseexample files demonstrate many of the program’s primary features. Secondaryoptions and preferences can then be investigated as time permits.

Symbols and ConventionsThroughout this User’s Guide, special fonts and/or icons are used to emphasizespecific steps, instructions and procedures in the program as shown below:

ALL CAPS Represent directories, file names, and commands.

Italics Used to emphasize certain points of information.

Keycap Bold italics are used to indicate a specific action or command to betaken. For example: “Click OK to proceed to the next screen.”

MView Program names are written as shown.

Menu⏐Command To avoid repeating the phrase “Click the File menu and choose theOpen command,” we use the File⏐Open convention.

Emphasizes specific information to be entered or a point of inter-est.

To Step-by-step instructions normally start with a bold To.


Chapter 1

Getting StartedThe Meyer Software

1.1 IntroductionThe Meyer Software is a powerful suite of engineering programs for the design,analysis and monitoring of hydraulic fractures.

This chapter outlines the hardware and system requirements for the Meyer suite ofsoftware. Specific instructions for installing and running the software are alsogiven.

After reading this chapter and installing the software, you will be ready to beginusing the programs. This guide assumes you have a working knowledge of the Win-dows terminology and procedures. If you are unfamiliar with the Windows operat-ing system, we recommend reading the relevant sections concerning Windows,menus and using a mouse found in your “Microsoft Windows User's Guide.”

1.2 System & Hardware RequirementsTable 1.1 shows the minimum system and hardware requirements for satisfactoryinstallation and performance of the Meyer Software. Additional resources, such as

1 Meyer & Associates, Inc.

2 Getting Started: The Meyer Software

additional RAM and a faster processor will greatly improve the performance of thesoftware.

1.3 Installing the Meyer SoftwareYou must install the application from the Meyer CD onto your hard disk; you can-not run the program from the CD. Installation instructions are available in the ViewReadme file on the CD.

Before InstallingBefore installing the software, you should first do the following:

1. Close any open applications.

2. Close any virus-detection application.

3. Make sure your computer meets the minimum system requirements listedabove in Table 1.1.

4. Determine the drive where the software will be installed.

5. Check the amount of space available on the selected drive.

Table 1.1: Minimum System Requirements.

Computer: IBM or fully compatible PC using a Pentium 400 MHz or more recent-processor. AMD Athlon systems are also supported.A CD or DVD drive (to install the software).USB or Parallel port (to install the security key).A Meyer & Associates, Inc. authorized software protection key.

Operating System: Windows Vista, Windows 2003, Windows XP SP1, and Windows 2000.

Memory: 128 MB RAM; more is highly recommended.

Disk Drive: A hard disk with at least 250 MB of free disk space is recommended (80MB are required for the executables with additional space required forthe data and database files).

Mouse: A Windows compatible pointing device (e.g. mouse, trackball).

Optional Equipment:

Windows supported printers.Modem (for remote real-time).


1.3 Installing the Meyer Software 3

6. Verify that you have the necessary access rights (administrative privileges) tocreate directories, files and to install device drivers on the designated drive.

To Install the Meyer ApplicationsThe software can be installed from a CD drive. Figure 1.1 shows the Install Soft-ware Setup menu.

Figure 1.1: Install Software Setup Menu

The Meyer Setup application installs all the main applications and components.Please read the directions below, and then follow the instructions in the Setup pro-gram to complete the installation.

1. Insert the Meyer CD into the CD-ROM drive. After inserting the CD, a menuscreen as shown in Figure 1.1 will appear. If the Meyer Install Setup Menudoes not display, click the Start button on the Windows taskbar, and click Run.Type D:\Setup (where D is the letter corresponding to the CD-ROM drive) inthe Open box.

2. Click Install Software. If you want to view the Readme HTML file, click onthe View Readme button. The Readme file contains FAQ’s, installation tips,new features and other information that may be important prior to installationor other information that was not available at press time.

3. Click the Next > button and follow the instructions.

4. Read the Meyer license agreement, and click the Next > button. If you don’taccept the license agreement, the Setup application will terminate withoutinstalling the software.

5. Type in the Customer Information User Name and Organization. Also selectone of the Install this application for options and click the Next > button.

6. Destination Folder. Click Next > to install this folder, or click Change... toinstall to a different directory. Choose a directory where the software will be



installed. It is recommended that a directory name that does not exist be speci-fied. Setup will then create the new directory.

7. Setup Type. Choose the setup type option that best suits your needs:

• Complete. All program features will be installed (requires the most diskspace).

• Custom. Choose which program features you want installed and wherethey will be installed. Recommended for advanced users. Click on the iconin the list to change how the feature is installed. If you are not authorizedto use all of the Meyer application software, you may wish to excludethese components by clicking “This feature will not be available”. Somecomponents have sub-components. To change the sub-component selec-tions, highlight the component in the list and click the appropriate desiredfeature.

8. Click the Next > button, and follow the installation instructions in the SetupWizard. Click the Install > button to begin the installation. When the installa-tion is completed, click the Finish > button to return to the Install SoftwareSetup Menu.

Program Maintenance - Modify, Repair or RemoveAfter installing the Meyer software, you can Modify, Repair or Remove the soft-ware. Simply follow the steps above for starting the Setup program. The ProgramMaintenance menu will then be displayed as shown in Figure 1.2 giving you thefollowing options:

1. Modify. Change which program features are installed.

2. Repair. Repair installation errors in the Meyer Software.

3. Remove. Remove the Meyer Software from your computer.

4. Click the Next > button, and follow the instructions in the Wizard.

To change selections in the Setup Wizard, click the < Back button.


1.4 General Application Information 5

Figure 1.2: Install - Program Maintenance Menu

These options give you the capability to free up disk space, update components torefresh their configurations settings and to repair or reinstall any components whichyou have inadvertently deleted.

Database InstallationThe Meyer Installer will preserve the current user databases when you install thenew version. There is no chance of them being overwritten or lost. If you have morethan one version of a user database on your computer, the installer will pick themost-recently-modified one and copy it in the appropriate default directory for youruse. No user interaction is necessary.

1.4 General Application InformationThis section presents general information on the application installation directories,file name extensions, application directories, long file name support, applicationfolder support and other support information.

Application Installation DirectoriesThe applications store the user databases, plots, etc. in the user profile. Technically,we store this information in the “roaming profile per-user application data folder.”



The location of this folder depends on the version of Windows and the version ofthe Meyer Software you are running.

For Meyer 2008:

• The English application data folder is “Meyer 2008 (en).”

• The Russian application data folder is “Meyer 2008 (ru).”

Both of these folders may be found under the “Meyer & Associates, Inc” folder,which in turn may be found under the Windows application data folder.

1. The Windows application data folder is determined by the %AppData% envi-ronmental variable, some example locations are shown below:

• For a user called Bruce on Windows XP, our folder might be C:\Docu-ments and Settings\Bruce\Application Data\Meyer & Associates, Inc\Meyer 2008 (en)

• For a user called Bruce on Windows Vista, the application data foldermight be C:\Users\Bruce\AppData\Roaming\Meyer & Associates, Inc\Meyer 2008 (en)

Application Data Folder SupportAll working files, user databases, plot configuration files are stored in a sub-folderof the user profile’s Application Data folder. In previous versions of the software,this was stored in the application folder. By having these files in the user profile, ifmore than one person uses a computer, each user will have their own copies ofthese files. Locked down environment is also supported.

Starting Folder for Open/Save-as DialogsWhen selecting a file name for opening or saving files associated with the inputfile, its useful to keep them in the same folder. The Open/Save-as dialogs for thefollowing tasks now start in the folder of the currently opened input file. A list ofsome of these dialog tasks is: 1) selecting a report bitmap, 2) exporting a plot; 3)exporting a report; 4) exporting an Exodus file (MFrac), 5) importing a log file(MFrac), 6) selecting a data-set file (MView), 7) exporting data as text (MView), 8)merging data files (MView), 9) selecting an MFrac output file (MProd) and 10)selecting an MProd output file (MNpv).


1.4 General Application Information 7

Application File Name Extensions

GeneralTable 1.2 provides a list and description of general file extensions.

Table 1.2: General File Extensions.

Extension Description

adb Acid Database (generic)

adtAcquired data text file orMAcq output data file

aie Acid Import/export file

axt Axis template file

bin Default plot configuration file

csv Comma separated value file

dat Data file

dbs Database Files (pipe and rock)

emf Exodus export file

fdb Fluid database file (vendor)

fie Fluid import/export file

fpc Fracture plot configuration file

key MKey file

las Log data file

pdb Proppant database

pie Proppant import/export file

stp MAcq setup file

txt Text file

usr User Database (acid, fluid, andproppant)

vhd MView Import header file

vtp MView Plot template file

wrl Virtual Reality Modeling Lan-guage file



Application File Extension SummaryTable 1.3 provides a summary of the Meyer application specific file extensions.

Long File Name SupportAll of the Meyer applications use long file names for input, output, database, unitand configuration files. Following is a list of general Windows limitations for longfile names:

1. The path and file name cannot exceed 255 characters (i.e., WindowsMAX_PATH).

2. The following characters not allowed in a path or file name are < > : “ / \ |.These characters are reserved for Windows.

3. The application must not use reserved words, such as aux, con, and prn, as file-names or directory names.

4. Long file name support when displaying an opened file in the title bar, MostRecent File list, and Window menu.

Working FilesThe Meyer applications use temporary working files not only for input files, butalso for output and other configuration files associated with the input files. This

Table 1.3: Application File Extension Summary.

Application Main Data File Plot Template File Units File

MFrac mfrac mtp mfu

MFrac-Lite mfrac-lite mtp mfu

MWell mwell mtp mfu

MPwri mpwri mtp mfu

MView mview vtp mvu

MinFrac minfrac mwz mmu

MProd mprod ptp mpu

MNpv mnpv ntp mnu

MFast mfast atp mtu


1.5 Starting the Software 9

allows you to open and run read-only files because the original (including the out-put) files are only changed when you save them. This also allows you to create andrun untitled projects without having to save to a file first.

Read Only FilesRead Only input files can be opened. However, during the save operation you willbe prompted to use the file Save As command.

Last Opened FilesThe Meyer applications remember the last 6 files opened. Now any file that doesnot exist in the Most Recent Used file (MRU) list that is selected/opened will bedeleted from the list.

1.5 Starting the SoftwareThe Meyer programs can be started using one of several methods. A program canbe started from the Windows desktop by double-clicking the associated icon orusing the Start menu. A program can also be started by “Running” the main execut-able module (e.g., MFrac.exe) from the desktop File or Start menus. It is also possi-ble to start an executable file from the Windows File Manager or Explorer.

Connecting the Hardware Security KeyThe security key must be attached to the parallel (printer) port or USB port. If youhave a USB key it must be plugged into the USB port. If you are using securitykeys for other software, some experimentation with the order and compatibilitymay be required.

If you are using a network security key, it is not necessary to have an individualsecurity key on your PC. Your network administrator should install the networksecurity key and configure your PC to access the network security key server.

Before starting a Meyer application, make sure the hardware security key is con-nected to your PC. If your PC is on a network with a network security key server,check with your network administrator to ensure that your PC is properly con-figured.

WARNING; Do not connect the parallel security key to a serial port, as this candamage the key and/or your PC.



1.6 Quick StartProgram Check List

To ensure trouble-free processing and access to the Meyer software, please checkthe following:

1. Sufficient disk space is available. A minimum of 80 MB is required for pro-gram installation. Additional free space is needed for data, output and data-bases files. A minimum of 250 MB free space is recommended.

2. The software protection key can be either a USB or Parallel port device. TheUSB key is connected to the USB port and the parallel port key to the Parallelprinter port. Do Not connect the key to the serial port, as this can damage thekey or your PC.

3. The software protection key is firmly in place to ensure a good connection. Ifthe key is loose, the program may not be able to access it.

4. The PC system date is set to the current date and time (i.e., today's date).

1.7 Customer SupportIf a question cannot be resolved by referring to the User’s Guide or on-line help,any user with a current maintenance contract can obtain Technical Support duringregular business hours (8:30 A.M. - 5:00 P.M. U.S. Eastern Standard Time) at thefollowing numbers;

Telephone: (724) 224-1440

FAX: (724) 224-1442

Information concerning our products, updates and upcoming events can beaccessed by visiting our home page on the Internet. To send data files or correspondwith us via email, our address is

Home Page - http://www.mfrac.com

Email - [email protected]

When requesting Technical Support, please include the program and version num-ber listed in the About Box.


1.8 Windows Fundamentals 11

Updating Your Hardware Key - MKey UtilityAll Meyer software requires a hardware protection device to run. The utility pro-gram, MKey, allows Meyer technical support to access your key remotely as neces-sary. MKey is also used to upgrade your device to run new software versions.

1.8 Windows FundamentalsIf you are unfamiliar with the graphical environment of the Windows operating sys-tem, we suggest reading the first few chapters of the “Microsoft Windows User'sGuide.” Some of the fundamental techniques can be acquired by using any of theon-line Windows Tutorials that are available.

1.9 Meyer Program BasicsThis section covers the essential features of data management for the Meyer pro-grams. The procedures used to open, save and print files, define system units andobtain on-line help are discussed. The options and procedures discussed in the fol-lowing sections are found under the File, Units, and Help menus of a program.

Button ConventionsThe Meyer software buttons are used to control many of the program functions.The conventions used for these buttons are summarized in Table 1.4.

In general, you should not need to use MKey unless specifically instructed to doso by Meyer or for software upgrades. In such cases, you will be given system-atic instructions on how to use MKey.

Table 1.4: Button Conventions.

Name Function

Browse The Browse button can be used to specify the path in the Directories dia-log box.

Cancel Selecting Cancel at any time during an editing session closes the activedialog box without accepting any changes made during the session.

Clear All Use this button to clear all selected plot choices in a dialog box.

Config. Plot This button is found in the plot frames. Use it to access the Plot Configu-ration screen.



File ManagementThe following section describes the File menu commands available within theMeyer programs.

File ExtensionsWhenever a file is saved in a Meyer application, it is saved with a specific exten-sion, such as *.mfrac (used for MFrac files); however, during simulation other filesmay be created with the same base name but different extensions. These files gener-ally contain results of the simulations and plot preferences.

Copy Use this button to send a copy of the plot to the clipboard. This providesa way to copy the plot as a Windows bitmap.

Done This button is used to close the current screen.

Help

The Help button is used to access the corresponding help information forthe dialog box that is open. Once Help is accessed, additional contextsensitive topics may be available and will be shown in green. Clickingone of these topics will display additional information related to a partic-ular parameter.

Load Units Use this button to apply a template of a units configuration.

OK The OK button is used to close and accept the contents of a screen keep-ing all changes which have occurred during the active editing session.

Print Found on the Plot frame, this button is used to send the active plot to theWindows specified plotter or printer.

Printer Setup This button, found in the Select Printer dialog box, is used to access theWindows printer properties options.

Save Units The Save Units button is used to save a template of the current units con-figuration. A saved configuration may then be used as a template at alater time.

Select All This button can be used to choose all available plot choices in a dialogbox.

Zoom-Out

Zooming-in on a plot is accomplished by dragging a box around the areayou want to zoom with the left mouse button. For a framed plot, this but-ton provides a way of quickly returning to the starting magnification. Onnon-framed plots, (i.e., normal ones) you can zoom-out by clicking theright mouse button to reveal a shortcut menu containing a zoom-outcommand.

Table 1.4: Button Conventions.


1.9 Meyer Program Basics 13

Opening a FileWhen starting a Meyer program, the file last accessed during the previous session isautomatically opened. If you do not want to work with this file, other data files canbe opened quickly and easily using standard Windows procedures.

A list of the last six data files accessed is always maintained at the bottom of theFile menu. You can select from this list to access one of these files. Otherwise, toopen a file, choose File⏐Open. The program checks to see if data has been modi-fied since the last time the file was saved. If it has, the screen shown in Figure 1.3 isdisplayed prompting you to save before terminating the files editing session. If nochanges have been made, the file Open dialog box, as shown in Figure 1.4, is pre-sented.

Figure 1.3: Program File Close Message.

Figure 1.4: Program File Open Dialog Box.

The File⏐Open dialog box alphabetically lists the available files matching yourselection criteria. The files in the default data directory are automatically shownfirst.



To Open a File, Use One of the Following Methods:

1. Type in the complete name of the file in the File Name box and press ENTER.

2. Click the Files box anywhere but on a filename, type the first letter of the file-name and press ENTER repeatedly until the desired filename is highlighted.

3. Use the TAB key to move to the Files selection box, next use the ↓ arrow key tohighlight the desired file and press ENTER.

4. Double-click on the filename.

If the file is not listed, it is possible that:

• the file is in a different sub-directory,

• the file is on a different drive, or

• the file is of a different type.

To change the directory or drive, type in the correct path in the directory field orscroll to and click on the directory you want.

Creating a New FileTo create a new file choose File⏐New. The program clears the application screen,title bar and re-initializes the program input/output data. When creating a new file,it is necessary to specify a file name using the Save As command as describedbelow.

Saving a FileTo save a file, choose either File⏐Save or File⏐Save As. The Save command storeschanges made to the current active file and overwrites the previously saved data.By default, the Save command saves a file under its original name to the drive anddirectory last selected. To save the file in a different directory or to another name,select Save As. After selecting Save As, a screen will appear as shown in Figure1.5.



Figure 1.5: Save As File Dialog Box.

Enter the new file name in the field provided or choose an existing file to overwrite.The file may be stored in any existing directory by making a choice from the Direc-tories selection box. To accept the new name and directory select OK.

Copying FilesTo make more than one copy of a file, select File⏐Save As using the proceduredescribed in the previous section. Enter the new file name in the space provided.This command is useful for saving several similar input files to different drives, orunder a different name on the same drive.

When copying a file, the default data directory is automatically displayed first. If a“Save As” file name already exists, the program prompts you to replace the file.You can choose Yes to replace the existing file or No to select a new name. To copya file, enter a new name in the File Name field and press ENTER or click OK.

Selecting a PrinterBefore a report or plot can be printed, the printer hardware must be specified andset up properly to work with Windows. First, make sure that the correct printerdriver(s) are loaded by accessing the Control Panel from the Windows Desktop.Refer to the “Windows User's Guide” if you are unfamiliar with this procedure.From the Desktop, select the Printers icon to display the list of available printers. Ifyour printer is not among the list displayed, follow the instructions given in the



Windows Guide to install the necessary driver. After making this verification orinstallation return to the application and specify the printer for output.

To Select and Set Up a Printer:

Select Printer Setup from the File Menu. The screen shown in Figure 1.6 will thenappear.

Choose from the list of printers displayed and click OK. Only printers installedunder Windows are displayed. Make sure the port specified is correct. If it is not,refer to your Windows User's Guide for instructions or access the Control Panelfrom the Desktop and select the Connect button to change the setting.

Select the Properties button to configure your specific printer.

Figure 1.6: Selected Printer Dialog Box.

When the Properties button found in the Select Printer dialog shown in Figure 1.6is used the dialog box displayed corresponds to the printer device selected. Allprinters have varying printing capabilities; however, most printers allow you toselect the paper size and source, as well as the page orientation and number of cop-ies. The example shown in Figure 1.7 is for a HP Laser Jet 4000 Series PCL printer.



Figure 1.7: Example Printer Specific Setup Screen.

Defining the System UnitsThe Units menu is used to define the units that are applied to the program dialogboxes and output displays. All of the Meyer programs use a flexible system of unitsthat are user defined for each variable. This flexibility makes it possible to custom-ize the units system to suit personal preferences or change the units to correspondwith data reports supplied by service companies.

The Meyer software also allows for mixing and matching of input and output unitsets. To create a custom unit template, make selections from the categories avail-able and save the modified units system under a new name by clicking the Savebutton found at the bottom of the Units dialog box. To apply a saved units template,use the Load button also found on the Units screen. English and metric default tem-plates are provided.

The units are not associated with data files. All changes to the units are kept for alldata files and work sessions. In general, once the units are set to your preferences, itshould not be necessary to enter the units box again.

To access the Units menu, click the menu name. The dialog box shown in Figure1.8 will be displayed. To list the parameters in alphabetical order (or reverse order)



click on the triangular sort marker in the right hand corner of the parameters col-umn.

Figure 1.8: Meyer Units System.

The Units screen is divided into the following areas:

• Measurement variables, which lists all the variables currently used in a pro-gram,

• Input and Output unit group categories to the left and right of the screen, and

• Command buttons.



Setting the Input and Output UnitsTo select the input and output units system, you may want to start with a predefinedunits template. To do this, click on the Load button and select a template.

To view the units list, move the scroll box up or down until your choice appears onthe list. The corresponding input and output unit categories will scroll simulta-neously. To select a unit, click the unit to highlight the item. Click again on the newunit you want. If you prefer using the keyboard, use the TAB key to scroll to aparameter and then use the directional ↑ or ↓ arrows to scroll to the unit itemdesired. When all the parameters are set to their desired unit click OK or pressENTER.

Unit templates can be applied or changed at any time during an active program ses-sion. Simply open the Units dialog box and make the desired changes. All units dis-played or contained in any open plot will be automatically converted.

Getting HelpThe Meyer User’s Guide (this document) can be used as a standalone reference, orfor context sensitive help while using the Meyer applications. It is available inCHM (Microsoft HTML Help) and PDF formats.

Accessing HelpWhen information is needed quickly, use one of the following methods to displaythe help file:

Help through the MenuFrom the menu bar, choose Help⏐Contents and select the desired subject from thelist of Help topics provided. Additional capabilities are also offered, including anindex, search, and favorites tab. A PDF version of the User’s Guide is available bychoosing Help⏐More Documentation⏐Meyer User’s Guide (PDF). A PDF vieweris required to view the PDF version of the Meyer User’s Guide.

Getting Help using Help ButtonsCommand buttons found within dialog boxes can also be used to access help.Clicking a Help button directs the help function to information about the current

For English language versions of the software, the units default to the Englishoilfield units; however, for the Russian version of the software, the units defaultto metric units.



dialog box and its components. Pressing the F1 key also accesses context sensitivehelp depending on what dialog or window is currently opened.

Error CheckingThe error checking routines contained in the Meyer programs provide an extra levelof protection when working within the Windows environment. This protection isrequired because Windows, unlike other environments such as DOS, does notrequire sequential display and input for the data screens. The Windows’ user inter-face allows multiple entry points into the data input dialogs.

Extensive error checking procedures contained in the programs check that anyentered parameter is within defined limits; and also verifies the existence of datarelative to the options that are set. The objective is to avoid taking a step in the pro-gram that will cause a problem during the calculations or during any subsequentprocess (e.g., plotting, etc.).

Data Entry ErrorsWhen entering data in any dialog box, the program checks the data after it isentered to ensure that it falls within the limits set by the program. If it does not, amessage similar to the one shown in Figure 1.9 is displayed. The example showncorresponds to a case where Young's modulus has been entered incorrectly.

Figure 1.9: Example Error Checking Message for Young's Modulus.

When the program posts an error checking message, you must respond by eitherclicking the Change button to return to the data entry dialog to correct the problemor the Ignore button to continue.

When an Error Message is Displayed During Data Entry:

1. Select either the Change or Ignore button located at the bottom of the messagescreen.



2. If the Ignore button is selected, the program will attempt to use the dataentered. When the Change button is used the program returns to the data fieldwhich caused the error and allows you to re-enter a value.

3. When re-entering a value, make sure that it is within the acceptable limits asposted by the error message tolerance range.

The programs use general solutions that are open to very broad ranges of data input.The ranges limited by the program are typically adequate for handling most cases.It is important to realize that even though the programs do allow you to override thelimits, the out of range values may cause a problem with the calculations.

When a dialog box is closed by selecting the OK, Next Page or Previous Pagebuttons, the program again checks to make sure that the data is within the rangesrequired and that all rows containing data are complete. This process is repeated foreach dialog box that is opened. The program provides a message when an out ofbounds error occurs.

Run-Time Error CheckingThe next level of error checking occurs once the data has been entered and the Runcommand has been selected. Prior to performing calculations, the program onceagain checks the data relative to the options selected. Whenever the programencounters a problem, whether it is out of range or missing data, the program postsa message and terminates the simulation (see Figure 1.10). Before continuing withthe calculations, it is necessary to correct the error indicated. After the correctionshave been made, return to the Run menu and initiate the simulation again. Ifanother data error is found the program will display another message requiring thecorrection process to be repeated. This procedure is repeated until all errors are cor-rected. The program will then automatically proceed with the simulation.

Figure 1.10: Example Run-Time Error Checking Message.

In order to run the calculations when the Ignore button is used, disable the min/max checking from the Run Options dialog.



When a Run-Time Error Checking Message is Displayed:

1. Select the OK button located at the bottom of the message screen.

2. Return to the appropriate dialog box to correct or add data as required.

3. Return to the Run menu and re-initiate the calculations.

For those instances when the Ignore button has been used during data entry it isnecessary to disable the min/max error checking in order to continue with the cal-culations. This is accomplished by selecting the appropriate check box from theRun Options dialog (e.g., See “Run Options” on page 66). If this is not done theprogram will continuously detect an out of bounds error and not permit the simula-tion to continue.

Once again, the error checking has been designed to cover as much of the programstructure as possible; however, to prevent the possible loss of data, a recoverymechanism has been implemented in the program. As a “last ditch effort,” if a prob-lem occurs with one of the data sets or other Windows applications that results in aprogram “crash”, the next time the program is started, an option to recover the datais provided. After a crash the message shown in Figure 1.11 is displayed. SelectYes to recover your data. If the program crashes while recovering a data file, youshould select No the next time the recover message appears.

Figure 1.11: Recover Unsaved Data File Message.

File Version CheckingThere is a system in place that compares the data file version for compatibility withdata files from previous versions of the program. If a message like the one shown inFigure 1.12 appears, choose OK to upgrade the file to the current version. Note thatafter doing this, the file will not be compatible with the previous version of the pro-gram.



Figure 1.12: Program Version Checking Message.

Working with Spreadsheets and DialogsSpreadsheet controls are used throughout the software to input tabular data. Thissection describes the special functionality of these controls and the use of thespreadsheet speed buttons.

Spreadsheet Keyboard CommandsTable 1.5 lists the action keys used to edit data and move the active cell within aselection.

Table 1.5: Action Keys.

Key Description

ENTERWhen in edit mode; accepts the current entry. The next cell is selectedaccording to the user preference specified in the spreadsheet options dia-log (Tools⏐Options...).

SHIFT+ENTER

When in edit mode; accepts the current entry. The next cell is selectedaccording to the user preference specified in the spreadsheet options dia-log, but in the opposite direction.

TAB When in edit mode; accepts the current entry and moves active cell hori-zontally to next cell in selection.

SHIFT+ TABWhen in edit mode; accepts the current entry. When a range is selected,accepts the current entry and moves active cell horizontally to previouscell in selection.

F2 Enters edit mode.

DEL Clears current selection.

ESC Cancels current data entry or editing operation.

CTRL + A Selects all rows and columns that contain data.



Table 1.6 lists the movement keys used to move the active cell within a spreadsheetand to display different sections in the spreadsheet.

CTRL + C Copies selection to the clipboard.

CTRL + - Deletes/removes the selected rows. (Speed button shortcut)

CTRL + D Fill down. (Speed button shortcut)

CTRL + + Inserts rows. (Speed button shortcut)

CTRL + O Opens/imports a file into the spreadsheet. (Speed button shortcut)

CTRL + P Prints the spreadsheet. (Speed button shortcut)

CTRL + S Saves/exports the contents of the spreadsheet. (Speed button shortcut)

CTRL + T Opens the linear transformation dialog based on the current selection.(Speed button shortcut)

CTRL + V Pastes from the clipboard to the current selection.

CTRL + Z When in edit mode, will undo a current active cell input.

Table 1.6: Movement Keys.

Key Description

Up Arrow Moves active cell up one row.

Down Arrow Moves active cell down one row.

Left Arrow Move active cell left one column.

Right Arrow Moves active cell right one column.

Page Up Moves up one screen.

Page Down Moves down one screen.

Home Goes to first column of current row.

End Goes to last column of current row that contains data.

CTRL Home Goes to first row, first column.

CTRL End Goes to last row and last column that contains data.

Table 1.5: Action Keys.



Table 1.7 lists the keys used to modify the action of the movement keys.

Spreadsheet Mouse ActionsTable 1.8 lists the mouse actions used in the spreadsheet.

Freezing Spreadsheet PanesThe rows of a spreadsheet can be frozen or locked to the top of the screen whilescrolling. This is accomplished by right clicking on a highlighted row and selectingFreeze Panes from the right-click menu. All rows above the selected line willbecome ‘frozen,’ and the other lines will scroll normally (See Figure 1.13). Theother way to freeze panes is to left click and drag the pane freezing bar to thedesired location.

Table 1.7: Modifying keys.

Key Description

SHIFT + any movement key

Extends the current selection.

Table 1.8: Mouse Actions.

Action Description

Left Click Moves the active cell to the pointer position.

Left Click in Rowor Column Headings

Selects entire row or column.

Left Click in TopLeft Corner Selects entire spreadsheet.

Left Double-Click Invokes in-cell editing.

Left Click and DragSelects a range. The previous selected ranges are deselected. Youcan also freeze spreadsheet panes by dragging the pane freezing bar(located above the first spreadsheet row) to the desired location.

SHIFT + Left Clickand Drag Extends the current selection.

Right Click Opens the right-click menu.



Figure 1.13: Freezing Spreadsheet Panes

Spreadsheet Options DialogThe spreadsheet options dialog is opened by choosing Tools⏐Options... from themain application menu and clicking the Spreadsheet Options tab. It can also beopened by right clicking on a spreadsheet field and choosing Options... from theright click menu. All of the options within the Spreadsheet Options dialog are glo-bal and apply to all of the Meyer applications.

A bold face font within the Spreadsheet Options Dialog identifies a non-defaultvalue.



Move selection after enter (direction)

This option determines what happens when the ENTER key is pressed while editinga spreadsheet cell. The Enter key direction can be set to move the cursor Down,Right, Up, Left, or to simply toggle a spreadsheet cell in and out of edit mode if thecheckbox is left unchecked.

Display ellipsis for truncated strings

When set to ‘Yes’ (default), an ellipse will appear at the end of spreadsheet text thatdoes not fit entirely in a cell. If this option is set to ‘No’, the text will be truncated.

Active cell border style

This allows the user to specify the visual cue that is used to identify the currentactive spreadsheet cell.

Alternate background color

When set to ‘User Defined,’ the background of every other spreadsheet row will befilled using the specified color.

Frozen rows background color

When set to ‘User Defined’ the background color of frozen cells will be filled usingthe specified color. This allows frozen rows to stand out better from non-frozenrows.

Spreadsheet Speed ButtonsAt the top of dialog boxes that contain spreadsheet controls is a group of small but-tons. These are the spreadsheet speed buttons. If there is more than one spreadsheetcontrol in a dialog box then these buttons affect the one that was most recentlyactive. Table 1.9 lists the speed buttons used with spreadsheets.

Table 1.9: Spreadsheet Speed Buttons.

Button Description

Imports spreadsheet file into spreadsheet control. The file should be in ExcelBIFF 4 format with the XLS file extension. It is recommend that only filesexported from the same control are imported.

Exports spreadsheet control into a spreadsheet file. The file should havethe.XLS extension and will be in Excel BIFF 4 format.

Prints the active spreadsheet to the currently selected printer (PrinterSetup... in the File Menu).



Data Shift provides an easy way to shift numerical data. Data shift can be selected

by clicking on . To use Data Shift, highlight the numbers you want to adjust,select the Data Shift icon and add, subtract, or multiply by a fraction. This is a con-venient way to calibrate stress, stress gradient, or Young's modulus from actualminifrac results.

Figure 1.14 shows the data shift (linear transformation) screen. The New Values arecalculated from the Old Values by a simple linear transformation ( ). Atable of new and old column values is displayed. Selecting OK will replace the oldvalues with the new values.

Cut operation removes the selection and copies it to the Clipboard. Similar tothe copy command below.

Copies current selection to the Clipboard. Copied selections can then bepasted in another selection in this or other spreadsheet controls or in Excel.

Pastes from the Clipboard to the current selection. The Clipboard selectioncan come from other spreadsheet controls or Excel.

Inserts rows at the current selection.

Deletes rows of the current selection.

Fills down the top cells of the selection to the cells below in a column.

Linear transformation provides an easy way to shift and multiply numericaldata.

Table 1.9: Spreadsheet Speed Buttons.

y mx b+=



Figure 1.14: Data Shift Screen.

Dialog and Spreadsheet Column SizingAll dialogs can be moved and scaled. To resize a dialog, grab the gripper bar on thelower right corner of the dialog box by holding down on the left mouse button anddragging. The dialog size and position is preserved and will be restored the nexttime you access the dialog. The default dialog and spreadsheet layout can be resetby clicking Window⏐Restore Default Layout.

The column widths in a spreadsheet can be resized by 1) placing the cursor on thecolumn separation line in the header, 2) holding down on the left mouse button, and3) dragging the column to a given width. To adjust the spreadsheet within the dia-log to have equal column widths, highlight the columns to be resized (see Figure1.15) and resize as discussed above.



Figure 1.15: Spreadsheet Column Sizing to Equal Widths.

Spreadsheets With Movable ColumnsCertain spreadsheets allow the columns to be rearranged. The simulation data win-dows in all of the applications support this feature. To move a spreadsheet columngraphically, click and drag the column header to a new location. Columns may alsobe rearranged or hidden via the Spreadsheet Column Configuration dialog accessi-ble from the context menu of spreadsheets that support the feature (Figure 1.16).



Figure 1.16: Spreadsheet Column Setup Menu.

Clicking Column Setup... will open up the Spreadsheet Column Configuration dia-log (Figure 1.17).

Figure 1.17: Spreadsheet Column Configuration Dialog.

Columns may be hidden by unchecking the check box and made visible by check-ing the check box associated with the column. The columns may also be rearrangedusing the Move Up and Move Down buttons.



Working with PlotsAll of the Meyer programs use plots as a graphical method to display information.These plots can be moved, zoomed, printed, exported and configured as describedin the following sections.

Arranging Plot WindowsAll plots are in separate windows that are located within the main program window.There can be any number of plot windows open at any given time; however, onlyone document window is active at a time. This is indicated by a highlighted title barand its foreground position in front of all other opened plots. When plot windowsare open, their position may be adjusted by accessing commands located under theWindows menu as shown in Figure 1.18.

Figure 1.18: Window Command Menu.

Plot windows can be “cascaded” from the upper left or “tiled” so that each plot win-dow is fully visible. This sub-menu also has a list of all the plot windows that areopen. Selecting from the list moves the associated window to the foreground desig-



nating it as the active window. Any opened report windows will also be listed in theWindow menu and will be affected by the Window commands.

After the desired plots are displayed, they can be positioned and arranged accordingto your preferences. Plots are treated much the same as any window on the Win-dows desktop. Plots can be moved, sized and arranged relative to each other forcomparison. Figure 1.19 illustrates the Tile arrangement for displaying plots. Fig-ure 1.20 illustrates the Cascade form of displaying all the plots in a single window.

Moving PlotsMouse

1. Click the title bar of the plot to move and hold the left mouse button downwhile dragging the plot to its new position.

2. Release the mouse button. To cancel the move, press ESC before you releasethe mouse button.

Keyboard

1. Select the plot to move.

2. Press ALT + SPACEBAR to open the Control menu.

3. From the Control menu choose Move. The pointer changes to a four-headedarrow.

4. Use the arrow keys to move the plot to its new location.

5. Press ENTER.

To Arrange Opened Plots:

1. Access the available commands from the Window menu (ALT + W).

2. Select either Tile (ALT + T), Cascade (ALT + C), or Arrange Icons (ALT + A) toaffect the arrangement of the plots (Figure 1.19 and Figure 1.20).

To Close a Plot:

1. In Window 3.x, select the Control menu of the plot you want to close. Click thecontrol box or press ALT + SPACEBAR.

2. From the Control menu choose Close (C).



To Close All Opened Plots:

1. Select Close All (ALT + L) from the Window menu.

Figure 1.19: The Tile Arrangement.



Figure 1.20: The Cascade Arrangement.

ZoomingAll of the Meyer plots have the ability to zoom in on a specific area. You may zoomin on a zoomed area as many times as you like. Use the Zoom Out command toreturn to the previous magnification. Use the Zoom Out 100% (Zoom 100%) toreturn to the original magnification.

To Zoom in on a Plot:

1. Position the cursor cross hair at one corner of the plot area.

2. Hold the left mouse button down and drag a box around the area to be zoomed.

3. Release the mouse button when the new visible plot area has been defined.

4. To return to the previous magnification, select Zoom Out from the Plot menuor the right mouse button menu, or simply press F5.



5. To return to the original magnification select Zoom 100% from the Plot menuor the right mouse button menu, or simply press F6.

Printing PlotsPlot windows may be printed using any Windows compatible printer. To print, youmust first select and configure the printer hardware for operation with Windows.This can be done with the Printer Setup command in the File menu, which allowsyou to select a printer and set its properties.

To Print a Plot on a Windows Compatible Output Device:

1. Open the desired plots and configure them according to your preferences.

2. Select the Print command from the File menu. The Print Graphics dialogappears listing the available print options (Figure 1.21). Select the plot optionand page orientation.

3. Click OK to accept the selections and complete the printing process.

4. While the Print Manager is sending the information to the printer, a message isdisplayed. During this time the printing process may be aborted by clicking theCancel button or pressing ENTER.

Figure 1.21: Program Print Graphics Dialog Box.

When zooming in on a plot with both left and right axes, both axes will bezoomed to display exactly what was in the selected area. To change the left andright scales independently, use the Scales section of the Plot Configuration box.



The progress of any printing operation can be monitored and controlled with theWindows’ Print Manager. Refer to the Windows User's Guide for complete instruc-tions for using the Print Manager.

On each printout, the program adds a footer (small identifier) in the lower left cor-ner of each plot that contains the file name and date. An option to hide the footer isalso available.

Plot MenuAll Meyer programs have a Plot menu that is used to create and manipulate plots.The menu commands used to manipulate plots are common to all programs.

If a plot is displayed, the Plot menu can be accessed from the Main Plot menu asshown in Figure 1.22. Figure 1.23 shows the resulting Plot Menu screen by clickingon the right mouse button.

Figure 1.22: Plot Menu Screen from Main Menu.



Figure 1.23: Plot Menu Screen from Right Mouse Button Click.

Copying to the ClipboardTo copy a plot to the clipboard, select the Copy to Clipboard command from thePlot menu. This will place a bitmap of the current selected plot on the clipboard.This bitmap can be pasted into any other Windows program. For best results, try tosize the plot window to the size of the desired bitmap before copying the picture.This will eliminate the need to scale the bitmap after it has been pasted.

Exporting PlotsMeyer plots can be exported for use in another program, such as a word processor,by saving a enhanced metafile (*.emf), bitmap (*.bmp), JPEG (*.jpg) or PortableNetwork Graphics (*.png) file. Figure 1.24 shows the Exporting Plot Menu.



Figure 1.24: Exporting Plot Menu.

It is recommended that metafiles be used whenever possible, since they producebetter quality output generally. A metafile is stored as a series of vectors, which canbe scaled to any size, but a bitmap is a fixed size and does not scale well. PortableNetwork Graphics (*.png) is also scalable. Most figures in this Guide are png files.

To save a plot as a metafile, select Enhanced MetaFile (*.emf) from the Save Fileas Type menu. Then enter a file name for the metafile. This will create a standardwindows metafile of the currently active plot, which can be imported into manygraphics packages and word processors. For example, to insert the metafile intoMicrosoft Word, choose Picture from the Insert menu in Word. Then specify themetafile that was saved from the Meyer program. This will insert the picture intoyour document. Note that the picture can be stretched without losing quality.

Default Plot AttributesThe default plot attributes for all plots can be configured by selecting the DefaultPlot Attributes under the Plot Menu from the Main Menu (see Figure 1.22). Thismenu is comprised of the Colors, Fonts and Layout attributes as shown in Figure1.25.



Figure 1.25: Default Plot Attributes Menu.

ColorsThe Colors command allows you to specify which colors are used in the plots. Thebackground, text, frame and grid colors are user specified. By default, all plots usethe colors selected in this box; however, a plot can be configured to have a differentset of colors if desired. For more information on configuring colors, see the PlotConfiguration section below.

General Colors

The default background, text, frame and grid colors are user specified under theGeneral Colors Menu.

To select the default background color click on the color box. A color menu willthen be displayed as shown in Figure 1.26. Select the color you want and press theOK button.



Figure 1.26: Default Color Menu.

Curve Attributes

To change the default curve attributes click on the Curve Attributes color box. Thiswill bring up the Line & Curve Attributes dialog box shown in Figure 1.27.

Figure 1.27: Default Line & Curve Attributes Selection Screen.

The attributes of a specific line or curve can be modified by clicking on the buttonrepresenting the curve. This will bring up the dialog box shown in Figure 1.27.From here the Line Color, Marker Style, Line Style and Line Width can bechanged. Setting the Marker Style to Off turns off the markers for an individualcurve. Likewise, setting the Line Style to Off allows you to plot only the markers



without lines connecting them. Choosing a Line Width greater than one will take alonger time to draw the plot.

FontsTo change the title font, select the font button next to the corresponding title. Tochange the font used to draw the legend, see the Legend section below.

The Meyer programs support all True Type fonts installed and available in Win-dows. Clicking on the Font folder will display the font selection screen shown inFigure 1.28. This will allow you to modify the font and its size.

Figure 1.28: Fonts Selection Screen.

To change the Main Title, X-axis Label, Y-axis Labels, Scale or Legend fonts clickon the appropriate selection box. Figure 1.29 displays the default Plot Font Selec-tion screen. Since font size is always scaled based on the window size, it is impossi-ble to select an exact point size; however, you may modify the ratio of the font sizeto window size with the Font Size list box.



Figure 1.29: Default Plot Font Selection Screen.

LayoutFigure 1.30 shows the Default Layout folder which allows you to specify somebasic attributes (Plot Layout, Mouse Coordinates, and Left and Right Axes) used todraw plots. All of these attributes may be overridden in any specific plot by choos-ing the Plot Attributes button in the configuration screen as discussed below.

Figure 1.30: Layout Selection Dialog Box.



Plot Layout

The different options for determining how the plot is drawn in the window are:

1. Standard plot layout: The plot is always drawn such that the frame has a con-stant aspect ratio, regardless of the size of the window.

2. Maximize plot area: The plot is drawn such that no empty space is wasted. Theaspect ratio of the plot will change as the plot window is sized

3. Allow graphical arrangement: The elements of the plot can be arrangedgraphically. By using the Switch Mouse to Arrange Mode command, the ele-ments of the plot can be manipulated with the mouse. The aspect ratio of theplot can be user defined.

Mouse Coordinates

The mouse coordinates option allows the coordinates of the current mouse positionto be continuously updated on the screen. When using the graphical plot layoutoption, the coordinates may be positioned anywhere in the window. The coordi-nates are drawn using the same font as the labels on the axes. To view mouse coor-dinates of multi-axes plots place the mouse cursor on the existing mousecoordinates, click the right mouse button and select the desired curve. Figure 1.31illustrates the mouse button selection dialog box.



Figure 1.31: Mouse Coordinate Selection.

Left and Right Axes

There are four options for how the curves of an axis (left or right) are drawn on theplot. These options are 1) default, 2) like curves share an axis, 3) all curves drawnon one axis, and 4) each curve has its own axis. Figure 1.32 illustrates the case ofall curves drawn on one axis. Figure 1.33 shows a plot where each curve has itsown axis. When they each have their own scale, there are labels for each curvealong the axis. The order of the curves in the legend is the same order of the labelson the axis. This option is most useful when there are different kinds of parameterson the same axis, for example rate and concentration.



Figure 1.32: Axes Label – All Curves on One Axis with Maximize Area.



Figure 1.33: Axes Label – Each Curve has Its Own Axis.

ConfigurationThis will bring up the Plot Configuration dialog box described in detail belowunder Configuring Plots.

Add Text BlockA block of text may be added to any plot by selecting Add a Text Block from thePlot menu or the right mouse button menu as illustrated in Figure 1.34. Then clickon the plot at the desired location to create a new text block (clicking and draggingallows you to specify the text block width). To edit an existing text block, click on itonce to select it, and again to enter edit mode. Once in edit mode, pressing ENTERwill go to the next line, and CTRL+Z will undo the last change. Press ESCAPE orclick elsewhere in the plot to exit edit mode.



Figure 1.34: Adding a Text Block to a Plot.

By double clicking on the block, or selecting configuration from the right mousebutton menu (Figure 1.34), the attributes of the text block, such as colors, frame,and font, can be configured. Figure 1.35 shows the text block configuration dialogbox. A text block can be deleted by using the right mouse button menu, or pressingDELETE when a text block is selected. The text blocks can also be moved and sizedwith the mouse.



Figure 1.35: Text Block Configuration.

Zoom OutThis will return the currently selected plot to the previous magnification. For moreinformation on zooming, see the Zoom section above.

Zoom 100%This will return the currently selected plot to the original magnification. For moreinformation on zooming, see the Zoom section above.

Configuring PlotsThe attributes of an open plot may be re-configured by either double-clicking theactive plot in the main plot area or by selecting the Configuration command fromthe Plot menu. This action displays the configuration dialog shown in Figure 1.36.

All of these commands are easily accessible at any time by clicking the rightmouse button within a plot window.



Figure 1.36: Plot Configuration Screen.

The Plot Configuration screen provides a method for modifying the text, colors andscales as described in the sections below.

When finished editing the configuration, choose one of the buttons in Table 1.10 toclose the configuration box.

Plot LabelsTo change a plot label, edit the appropriate text in the Plot Labels section at the topof the box. To change the text on the plot legend, see the Legend section below.

MarkersTurn on the Markers checkbox to see markers on the curve. Fill in the Draw mark-ers every ith point to specify how often to draw markers. For example, a value of

Table 1.10: Buttons For Closing Plot Configuration Box.

OK Changes only the current plot in the Plot Configuration screen.

Change Default Applies the changes to this plot and all future plots of the same typethat are plotted, even in different data files.

Cancel Cancels all changes.



two will specify to draw a marker at every other data point. To turn off markers forall curves of the plot, turn this option off. To turn off markers for only selectedcurves of the plot, turn this option on and modify the marker type for the curves onwhich markers are not wanted.

General Sub CategoriesThe sub categories of the General section are shown in Figure 1.37. This menu iscomprised of the Colors, Fonts and Layout attributes.

Figure 1.37: Individual Plot Attributes.

General ColorsOne set of colors, the default colors, is used for all plots. To change these defaultcolors, check the Apply to defaults check box. However, individual plots may havetheir own color scheme.

To use colors other than the defaults, turn off the Use Defaults check box. Thenconfigure the individual colors by clicking on them, which brings up a palette.Choose the desired color from the palette and click OK. Double-click on the colorsin the palette for quick selection. The definition of the different colors is describedin Table 1.11.

Curve AttributesThe attributes of a specific curve can be modified by un-checking the Use Defaultsbox and clicking on the box representing the curve. This will bring up the dialog



screen shown in Figure 1.38. From here the Line Color, Marker Style, Line Styleand Line Width can be changed. Setting the Marker Style to Off turns off the mark-ers for an individual curve. Likewise, setting the Line Style to Off allows you toplot only the markers without lines connecting them. Choosing a Line Widthgreater than one will slow down the plot drawing process. Clicking Line Color willbring up the color selection dialog as shown in Figure 1.26.

To return to the curve defaults, select Use Defaults. From this box, the attributes ofany curve can be changed.

Figure 1.38: Line & Curve Attributes Selection Screen.

FontsTo change the title font, select the font button next to the corresponding title. Tochange the legend font, see the Legend section below.

The Meyer programs support all True Type fonts installed and available in Win-dows. Clicking on the Font button will display the font selection screen shown in

Table 1.11: Plot General Colors.

Area Definition

BackgroundColor All the area inside and outside of the plot frame.

Text Color All of the labels, but not the legends (the legends are drawn with thecurve colors).

Frame Color The rectangle that separates the curves from the labels.

Grid Color The grid drawn inside the frame.



Figure 1.39. This will allow you to modify the font and its size. Since font size isalways scaled based on the window size, it is impossible to select an exact pointsize; however, you may modify the ratio of the font size to window size with theFont Size list box.

Figure 1.39: Plot Font Selection Screen.

LayoutSee “Layout” on page 43.

ScalesOn certain plots, it may be desirable to have the X and Y-axes use the same scale.To force them to use the same scale, click on the Scales button in the Plot Control.This will bring up the scales dialog box as shown in Figure 1.40. Then check X-Yscales have same Aspect Ratio. To turn off this feature, leave the checkboxunchecked. Only use this feature when the X and Y scales are in the same unit, forexample the width contour plot in MFrac. In this plot, the Y scale represents depthand the X scale represents fracture length.



Figure 1.40: X-Y Aspect Ratio Selection Screen.

Axes

The X and Y axis may be plotted on a linear or log scale. Select Linear or Log asdesired.

The Upper, Lower and Increment may be specified for each axis. The Upper andLower define the range of the scale and the Increment defines how often gridlines,tick marks and numeric labels are drawn. The Increment must be less than theUpper minus the Lower (for positive numbers).

To have the program automatically pick the appropriate settings, check the Autoscale box. This is the default. To manually specify the parameters, turn off theAuto scale box. Note that the X, Y Left and Y Right, and Z axes may be configuredindividually; however the Y Left and Y Right axes share Auto scale and Linear/Log buttons.

To temporarily zoom in on a section of the plot, the Zoom feature as describedabove may be easier to use than manually specifying the scale.



LegendThe plot legend is used to reference a curve that corresponds or represents a givenparameter. To modify the legend’s attributes, click on the Legend button in the PlotAttributes tree view. The Legend Control dialog box for the vertical orientation isshown in Figure 1.41.

Figure 1.41: Legend Control Dialog Box - Vertical.

To hide the Legend, make sure the Show Legend checkbox is not checked. Toeliminate showing a given curve, enter the Line & Curve Attributes and set bothLine Style and Marker Style to Off.

The legends Show Units option has three radial button choices: 1) Yes - to showthe units, 2) No - do not show units, and 3) Automatic - the code will display unitswhen more than one parameter with different units is present on a given axes. Threeoptions are available for under

The legend Orientation also has three radial button choices: 1) Vertical - the legendwill be placed in the vertical position, 2) Horizontal - the legends will be orientedhorizontally, and 3) Automatic - the legend will be place vertically for a single vari-able on each axes and in the Horizontal position for multiple variables with theoption that each has it’s own axis



There are five fixed locations for the vertical legend orientation as illustrated inFigure 1.41, the four corners of the plot and along the right side of the plot. TheLegend Control dialog box for the Horizontal Orientation is shown in Figure 1.42.There are three locations for the Horizontal legend orientation: 1) Across the Top,2) Position with Mouse, and 3) Across the Bottom.

To allow movement of the legend, anywhere on the screen, select Position withMouse. If Allow graphical arrangement is checked in the Layout dialog box, thelegend can be arranged anywhere on the plot and the Legend Control dialog boxwill be dimmed. Choose the desired location with the corresponding radio button.

Figure 1.42: Legend Control Dialog Box - Horizontal.

To modify the legend font, click on the Fonts button in the tree view. This font canbe modified like the label fonts, described above.

To change the legend text, click on the corresponding button at the bottom of thescreen. The left and right buttons allow you to change the text for the curves on theleft and right scales, respectively. After clicking on either of these buttons, a dialogbox will be displayed that contains all of the legends. Modify them as desired andclick OK. To revert to the original legend text, click on the Default button.



For help with the Multilayer Legend Names section see “Multilayer Legends” onpage 200.

GridlinesTo draw a grid on the plot, click the Gridlines section. The grid is drawn at inter-vals specified by the Increment section of the X and Y-axes. Figure 1.43 shows thegrid selection screen.

Figure 1.43: Grid Selection Screen.

Color FillWith contour plots, it is possible to show a shaded, color filled or a plain contourplot. To change this option, click on the Contours button in the Plot Attributes treeview. This will bring up the contour selection screen as shown in Figure 1.44. Thenchoose Shaded, Color Filled Contour, or Plain Contour.



Figure 1.44: Color Fill Contour Selection Screen.

Shading ColorsThe shading colors are dependent on the Color Fill Contour Selection shown in Fig-ure 1.44. Figure 1.45 shows the Shaded Contour Color Selection Screen.

Figure 1.45: Shading Colors Selection Screen.

Figure 1.46 shows the Color Filled and Plain Contour Selection Screen. To changea Filled Area Color, click on the color or curve you wish to change. This will bringup a dialog box for interactively choosing the color values for Red, Green and Blue.The values must be in the range of 0 to 255.



Figure 1.46: Filled Area Colors Selection Screen.

To return to the defaults for the Filled Area Colors, select Area Color from theDefault section. To make all the Filled Area Colors be shades of a certain color,click on the Shading button and select a base color.

Chart TypeFor chart plots, it is possible to change the chart type. Click on the Chart button inthe Plot Attributes tree view. This brings up the chart dialog as shown in Figure1.47. Then choose the desired type of plot from the list. The number displayed inthe legend can also be configured. Select either Stage Number, Percentage (percentof total) or Unit Values (the value for the stage).



Figure 1.47: Chart Type Selection Screen.

Plot TemplatesThe current state of all the plots on the screen can be saved into a template file. Thistemplate can then be recalled later to view the same plots again, even for a differentsimulation. This makes it easy to recall groups of plots that are always viewedtogether. Any template that is saved can be added directly on the Template menufor easy access.

The template contains the current plot default attributes, identifies opened plots andrelative window positions, and the plot configuration for each of these open plots.Any options associated with these plots (e.g., plotted versus time or volume) arealso saved.

To Save a TemplateOnce the windows are arranged properly, select Save Template from the Templatepop-up menu that is under the Plot menu as shown in Figure 1.48.



Figure 1.48: Plot Template Dialog Box.

Then specify a descriptive name for the template as illustrated in Figure 1.49. If youdo not want the template to appear in the template menu, uncheck the Add to Tem-plate Menu check box. Click on the OK button and then specify a file name for thetemplate.

Figure 1.49: Save Plot Template Descriptive Name.



To Recall a TemplateIf the template is listed on the Template menu, just select it. Otherwise, select LoadTemplate from the Template menu. Then specify the template file. The plots savedin the template will appear.

Organize Template MenuThe Organize Template Menu screen allows for user customization of the Templatemenu. Any template that has been saved may be added to the Template menu foreasy access as shown in Figure 1.50. Within this screen, the templates and the orderin which they appear in the menu can be configured.

Figure 1.50: Plot Template Menu Organization Screen.

Figure 1.50 shows the current configuration of the menu, the template descriptionand file location.

To add a template Click on the Add button and select the desired template file. Todelete a Template select the desired template in the list. Then click on the Deletebutton. To Move a Template Up in the list select the desired template and click onthe Up button as many times as needed. To Move a Template Down in the list selectthe desired template and click on the Down button as many times as needed.

Click on the OK button after the menu has been configured.



Axis Title TemplatesThe Axis Title Templates feature allows you to save and recall the main captionand X, Y-Left, and Y-Right axes titles.

You can Save (F12) or Load (Shift+F12) a template to any plot by selecting AxisTitle Templates from the right mouse button menu as illustrated in Figure 1.51.Figure 1.52 shows the Save Template selection.

Figure 1.51: Axis Title Templates - Right Mouse Button.



Figure 1.52: Axis Title Templates - Save As.

Run Menu for Other Meyer Applications Figure 1.53 shows the Run Menu for other Meyer Applications. Simply click on theMeyer Application you wish to execute.

Figure 1.53: Run Menu for Other Meyer Applications.



Simulation Data WindowsSimulation Data Windows (Figure 1.54) are available in all of the applications thatcontain a simulator. An application can be identified as having a simulator if thereis a Run item in the main application menu. The Simulation Data Windows pro-vide a read only view of the simulation output data while the simulator is running,and after it has finished running. The data columns within these windows can berearranged or hidden as desired (See “Spreadsheets With Movable Columns” onpage 30)

Figure 1.54: Simulation Data Windows.

Right clicking on any of the Simulation Data Window spreadsheets will display acontext menu providing additional options. To reset the Simulation Data Windowcolumns to their original positions and original visibility, select Restore DefaultColumn Layout from the spreadsheet context menu.



To open a Simulation Data Window at any time, the Show Simulation Data Win-dows sub menu under Plots provides a full list of available windows and a userinterface for opening or closing more than one window at a time.

The Simulation Data Windows are magnetic, when a data window is moved orsized too close to another magnetic window, the two windows will ‘snap’ together.To override this feature, hold the CTRL key while sizing or moving the SimulationData Windows.

When the simulation is started, various Simulation Data Windows are openedaccording to the current Run Options described in the next section.

Run OptionsThe Run Options dialog box is shown in Figure 1.55 and is available in all applica-tions that contain simulators. To open the Run Options dialog, choose Options...from the Run menu. These options are used during the program execution (Run-ning) and provide flexibility for auto scaling of plots, beeping after each iterationand min./max. error checking.

Figure 1.55: Run Options Dialog Box.

Simulation Data Windows are magnetic; they will snap together if they get closeto one another. Holding the CTRL key overrides this feature.



Display the following data windows when the sim-ulator is runWhen the simulator is run, the selected windows are automatically opened. Thispreference is ignored if the ‘Show template before running simulation’ option ischecked.

Scale plots based on the last run while calculatingWhen running a simulation, all open plots are automatically updated each time step.Since most output parameters are growing during the simulation, this causes theplot scales to change. Use this option to prevent the scale from changing so often.During a simulation, if the option is on, the plots will automatically use the scale ofthe last simulation. If an output parameter goes beyond the scale of the last simula-tion, the plots will then automatically rescale. It is advised to leave this option on.

Beep after each time step If desired, MFrac can provide audio feedback on the progress of the calculations bygenerating a system beep for each time step. This may be desirable for complicatedsimulations when the calculation times are long. To use this feature click the Beepafter each time step check box.

Disable MIN MAX error checkingThis will disable error checking. Error checking ensures that all input parametersare within the minimum and maximum allowable range when starting a simulation.Use this option if you have elected to enter a value that is out of range, by clickingon the Ignore button. Otherwise, it is advisable to keep this option off.

Show template before running simulatorThe current state of all the plots on the screen can be saved into a template file. Thistemplate can then be recalled later to view the same plots again, even for a differentsimulation.

Any template that is saved can be shown during running by selecting the Templatefile using the browse button (The button containing an ellipse; see Figure 1.55).

See “Plot Templates” on page 60 for additional information.



Generating ReportsAll of the Meyer programs can generate reports which can be viewed on the screenor exported to a file.

Viewing ReportsTo view a report, choose the View Report command from the Report menu andselect the report type. A report window will then appear. Use the arrow keys or thescroll bars to scroll around the report. The appearance of the report can be config-ured as described below.

Report ConfigurationTo configure the appearance of on-screen reports, select Report Configurationfrom the Report menu. This will bring up the Report Configuration dialog boxshown in Figure 1.56.

Figure 1.56: Report Configuration Dialog Box.



To include a bitmap, such as a company logo, at the top of the report, click on theBitmap at top of Report checkbox. Then click on the Select File button to selectthe bitmap file. The bitmap can be left, center or right justified.

There are four different levels of text in reports: the Main Title, Title, Subtitle andMain Text. For each, the justification and font may be configured. Choose thedesired justification with the Left, Center and Right radio buttons. The Main Textjustification is not configurable. To select the font for a text level, click on theappropriate Select Font button. From the Font dialog box, select the desired Font,Font style and Size.

The table justification is used to configure how tables are displayed. Select Left,Center or Right.

To make report windows automatically maximized when they come up, check theAutomatically maximize report windows check box.

Exporting ReportsThere are three options for exporting a report, either as a text, HTML or RTF file. Atext file is formatted by spaces and tabs. An RTF file contains all of the formattingthat is in a report window; however, an RTF file can only be read by word proces-sors that support RTF files.

Save Report as a Text FileTo save a report as a text file, select the text File command from the Report menu.After selecting desired kind of report, enter a file name for the text file.

Save Report as an HTML FileTo save a report as an HTML file, select the HTML command from the Reportmenu. After selecting desired kind of report, enter a file name for the HTML file.

Save Report as an RTF FileTo save a report as an RTF file, select the Save Report as an RTF File commandfrom the Report menu. After selecting desired kind of report, enter a file name forthe RTF file.


Chapter 2

MFracA Three Dimensional Hydraulic

Fracturing Simulator

2.1 IntroductionMFrac is a three-dimensional hydraulic fracturing simulator that is designed to beused as an everyday tool. MFrac accounts for the coupled parameters affecting frac-ture propagation and proppant transport. MFrac is not a fully 3-D model. It is how-ever formulated between a pseudo-3D and full 3-D type model with an applicablehalf-length to half-height aspect ratio greater than about 1/3 (Meyer15). MFrac alsohas options for 2-D type fracture models.

This chapter covers the available menu options and basic procedures required torun MFrac. Please refer to the Meyer Appendices and listed references for specificdetails regarding the governing equations, modeling techniques, methodology andnumerical procedures. Example files are provided with the software to demonstrateMFrac’s features, utility and general data entry procedures.

An outline of the basic steps for using MFrac is shown in Table 2.1.

Table 2.1: MFrac Basic Steps.

Step Program Area

1. Open an existing MFrac data file (*.mfrac) or create a newdata file. File Menu

2. Specify Units (optional) Units Menu

3. Select Program Options Data Menu

4. For a real-time or replay case, start MView and import theacquired data. MView

5. Input Required Data Data Menu


72 MFrac: A Three Dimensional Hydraulic Fracturing Simulator

MenuThe MFrac menu bar is shown in Figure 2.1. Generally, the menus are accessedfrom left to right with the exception of the Units and Database menus.

Figure 2.1: MFrac Main Menu.

A description of each menu item is described in the following chapters or sectionsas given in Table 2.2:

6. Run Simulation Run Menu

7. View Plots during or after the simulation Plot Menu

8. Generate Report Report Menu

Table 2.2: Description and Location of Menu Items

Main Menu Item Description Location

File File Management Chapter 1

Options Model Options Section 2.2

Data

DescriptionWellbore HydraulicsZonesTreatment ScheduleFoam ScheduleRock propertiesFluid loss DataProppant CriteriaAcid dataHeat Transfer

Section 2.3

Run Start Simulation Section 2.4

Plots Graphical Presentation Section 2.5

Table 2.1: MFrac Basic Steps.


2.2 Options 73

Exporting to ExodusAfter the simulation has completed, the propped fracture characteristics may beexported for use with T.T. & Associates Exodus reservoir simulator. To do this,select the Export to Exodus Reservoir Simulator command from the File menu. Ifthere is more than one active zone, MFrac will ask which zone to export. Make aselection by choosing the desired zone in the list box and pressing OK. Then MFracwill ask for the file name. Specify the desired file name and press OK. Use this filename when importing data into Exodus.

For more information on the Exodus reservoir simulator, contact T.T. & Associates.

2.2 OptionsThe Options screen is the first input dialog box under the Data menu in MFrac. It isused to establish the primary model options in the program. Each option relates to aspecific aspect of the fracture and proppant/acid modeling approach.

To access the Options screen, select Options from the Data menu by clicking themenu name. The dialog box displayed in Figure 2.2 will then be presented.

Reports Generating Reports Section 2.6

Databases

Fluid DatabaseProppantNon-DarcyAcidCasingTubingCoil TubingRock Properties

Section 2.7

Table 2.2: Description and Location of Menu Items



Figure 2.2: Data Options Screen.

The Options screen determines what information is needed for a particular type ofanalysis. The specific data displayed in a screen or the existence of a data screenitself varies depending on the options selected. This “smart-menu” approach, mini-mizes data input and prevents unnecessary or misleading data entry. Simply decidethe relevant options for a specific simulation and the program will only displaythose menus and input fields necessary. Any time the options are changed the inputdata screens will be updated to enable new input or hide data that is not needed.This hierarchy methodology is used throughout MFrac.

The selections made in the Data Options screen set the scope for all data enteredinto the MFrac program. These options establish the input data required and specifythe nature of the calculations to be performed.

To select an option, click the radio button adjacent to the option preference. A blackdiamond will then appear in the center of the button selected. Continuing, select aradio button within the next option section or use the TAB button to move sequen-tially through the choices. Once within a section, the current selection for thatoption is highlighted with a dotted rectangle. The option choice may be changed byusing either the mouse or the arrow keys.

General Options The General Options screen allows the user to specify the type of analysis to be per-formed. The choices available for each of the General Options are summarized asfollows:


2.2 Options 75

Simulation MethodDesign ModeThis option is the traditional methodology used for hydraulic fracturing design. Theprogram flexibility allows for running in standard mode based on a given inputfracture length, slurry volume or treatment schedule. Depending on other optionsspecified, the program uses the formation and treatment data in the calculation ofassociated fracture and proppant transport characteristics. Design Mode refers tothe fact that the design engineer must design (and optimize) the fracture treatment.

Replay/Real-TimeThe Replay/Real-Time option is required for replaying or performing real-timefracture analysis using the data collected during a treatment. This procedurerequires the use of MView as the real-time or replay data handler. Please refer toChapter 3 for instructions on the use of MView.

With respect to MFrac, there is essentially no difference in the procedures used forperforming real-time or replay simulations. The difference between these methodsonly involves the source data input which is handled by MView. Either method per-mits pressures matching, geometry prediction and proppant transport simulation.

Reservoir CouplingThe mechanisms which control fluid loss from a propagating fracture are discussedin the Appendices. Typically, for general fracturing applications where the leakoffdistance perpendicular to the fracture face is small compared to the fracture lengthCarters one-dimensional fluid loss model is adequate. However, for cases withlarge fluid loss volumes (e.g., produced water reinjection and large scale frac-packs), the fluid loss behavior is two-dimensional or ellipsoidal. The ellipsoid fluidloss option should be used for cases when the leakoff distance becomes greater thana small fraction of the fracture length.

Linear (Conventional)This is the classic linear fluid loss model as proposed by Carter and assumes thatthe fluid loss is one-dimensional. This option can be used for most applications andis the most common fluid loss model used for propagating fractures.

Ellipsoidal (Koning)This option allows for ellipsoidal (2D) fluid loss from the fracture and generallyresults in lower fracture efficiencies in high permeability formations when largevolumes of fluid are injected. This option requires selection of either the Harmonic



or Dynamic Fluid Loss Model. The Ellipsoidal model should be selected for pro-duced water reinjection and high permeability large volume frac-packs.

Real-Time The Real-Time options are only available if the Replay/Real-Time radio button isclicked On in the Simulation Method dialog. If MView Concentration is selectedthe proppant concentration will be taken from the replay/real-time data as sent toMFrac from MView. If the Input Concentration button is selected the proppantconcentration used by MFrac will be taken from the values specified in the Treat-ment Schedule. Generally, the MView Concentration is desirable unless the actualproppant concentration injected is not available.

The Synchronize Well Solution radio button is used to synchronize the numeri-cally calculated time steps for wellbore events with the replay/real-time data.

Synchronizing the wellbore solution with the incoming real-time or replay dataenables for very refined calculations of the wellbore and near-wellbore frictionalpressure losses. Since the fracture net pressure is not as dependent on the instanta-neous rate changes, this provides the capability to run the fracture model with agreater time step while still simulating the effects of rate changes on frictionallosses in the wellbore and near well region.

Net Present ValueThe Net Present Value option can only be activated if the Simulation Method isDesign Mode.

When the NPV option is turned On, MFrac automatically sets the TreatmentDesign to Auto Design and the Treatment Type to Proppant. For this option, amaximum fracture length is specified in the treatment schedule. MFrac then auto-matically calculates the proppant distribution and fracture conductivity for a num-ber of incremental fracture lengths up to the maximum value specified. Thepurpose of this type of analysis is to optimize the design length and conductivity forpropped fractures. This process is accomplished by coupling our analytical produc-tion simulator MProd to forecast productivity for each subdivision of the fracturelength. MProd, in turn, produces output used by MNpv to perform Net PresentValue economic optimization calculations.

Turning NPV Off enables the simulator to perform in standard Design, Replay/Real-time or Auto Design mode. This is the general mode of operation unless an NPVanalysis to optimize fracture length is desired.


2.2 Options 77

Fluid Loss ModelThe rate of fluid loss to the formation is governed by the total leakoff coefficient .The three types of flow resistance mechanisms making up are: 1) - leakoff

viscosity and relative permeability effects, 2) - reservoir viscosity and com-

pressibility effects, and 3) - wall building effects.

This option determines which fluid leakoff model is used. The fluid loss modeloptions include specifying the total leakoff coefficient (Constant Model) or the

coefficient and the corresponding components which comprise and (Har-monic or Dynamic Models). A detailed description of the components characteriz-ing the Harmonic and Dynamic models is given in Appendix D of the Appendicesand in the Fluid Loss Data section of this chapter.

If Constant is selected, the total leakoff coefficient, , is entered in the Fluid LossData screen. The total leakoff and spurt loss coefficients are then input as a functionof depth to characterize fluid loss in the fracture at different intervals.

When either the Harmonic or Dynamic models are chosen, the filter cake coeffi-cient ( ) and reservoir diffusivity parameters are input in the Fluid Loss Data

screen for each layer. The and coefficients are then calculated from theinput reservoir data and fracture propagation characteristics. The total leakoff coef-ficient is then calculated internally as a function of differential pressure.

The weighting of the individual leakoff coefficients for the Harmonic andDynamic models is given in Appendix D.

When the Fluid Type Dependent check box under Fluid Loss Model option ischecked, different total leakoff coefficients (C and Spurt Loss) for each fluid can beentered for the Constant Leakoff model. When the Harmonic or Dynamic Leakoffmodel is chosen the user can input different wall building coefficients ( ), fil-trate viscosities, and spurt loss coefficient for each fluid.

This option is useful when large volumes of 2% KCl or treated fluids are in thewellbore prior to pumping the main fracturing treatment. The option is also helpfulfor modeling leakoff during acid fracturing treatments with alternating pad/acidstages.

If the Include Fluid Loss History check box under Fluid Loss Model option ischecked, the simulator will remember the fluid loss history if the fracture closesand then re-opens. This option should be selected to include the effect of the mini-

CC CI

CII

CIII

CIII

CI CII

C

CIII

CI CII

CIII



frac on the main fracture treatment and when multiple open/close cycles are gener-ated. If this option is checked, the model assumes that the filter cake, viscosity, andcompressibility effects from the previous fracture remain upon re-opening.

Treatment TypeThis selection determines the type of fracture treatment. The Treatment Type canbe either a propped (Proppant) or acid (Acid) fracture. In addition, the treatmentcan accommodate an optional foam schedule by checking the Foam box. WhenFoam is checked, MFrac will include compressibility effects.

Treatment Design OptionsThe treatment design options are only available if the Simulation Method is inDesign Mode and the Treatment Type selected is Proppant with no Foam. Thedefault setting is Input for all other cases.

In MFrac, either the pumping schedule can be input manually or determined auto-matically. When Auto Design is chosen, the desired design fracture length or totalslurry volume is input in the treatment schedule dialog box. Depending on theProppant Transport Methodology selected, specific criteria for controlling theproppant scheduling will also be required.

When Input is chosen, the pumping parameters must be entered into the TreatmentSchedule screen. The exact data input required will depend on selections made forother options (e.g., ramped proppant scheduling, user specified proppant settling,acid fracturing, etc.).

Wellbore Hydraulics ModelThis option determines the wellbore hydraulics model to be used in calculating fric-tional pressure losses in the wellbore. Surface and bottomhole pressures, gravita-tional head, restrictions, transport times and hydraulic power requirements are alsocalculated. The near wellbore and perforation pressure losses are calculated sepa-rately below the BHP reference point for each fracture and coupled to the wellbore.The available wellbore model options are listed below:

When the Net Present Value Option is On the Treatment Schedule Option isautomatically set to Auto Design. For Replay/Real-Time this option is automat-ically set to Input.


2.2 Options 79

NoneWhen this option is selected, wellbore hydraulics calculations are still performed;however, the frictional pressure loss is assumed to be zero. The wellbore hydraulicsoutput data is also not displayed or written to file.

EmpiricalThe Empirical option is an internal correlation for calculating the frictional pres-sure loss of Newtonian and non-Newtonian fluids. This option provides a combinedcorrelation that is applicable for a variety of fluids ranging from linear systems tohighly non-Newtonian and viscoelastic fluids that exhibit drag reduction due to slipor shear thinning during turbulent flow. Three distinct types of behavior are possi-ble with the combined correlation used in MFrac. These behaviors are illustrated inFigure 2.3 and summarized in the explicit expressions for the Fanning friction fac-tor outlined in Table 2.3

Figure 2.3: Pipe Friction Empirical Correlations.

Table 2.3: Fanning Friction Factors

Maximum Drag Reduction, P.S. Virk1 (Predicts Minimum Friction)

1f

----- 19 Res f( )log 32.4–=



When a value for the Relative Pipe Roughness is entered into one of the WellboreHydraulics dialog boxes, the expression for friction factor based on Prandtl’s “Uni-versal” Law is modified. See Appendix E for additional information.

To include the effects of proppant concentration on friction, the program has a builtin correlation for slurry rheology. The relationship used, originally described byKeck, et al., is also presented in Appendix E.

User DatabaseWhen User Database is selected, the information specified in the fluid database isused for calculating the frictional pressure loss in the tubing, annulus, and casing.This data can be edited and plotted by accessing the database. The information inthe database does not represent proppant-laden fluid. Consequently, if the proppantconcentration wellbore option is selected in the proppant option screen, the frictionfactor will be adjusted for proppant concentration in a manner similar to the methoddescribed in Appendix E for the Empirical option.

Fracture SolutionThese options provide control and flexibility for the time dependent discretizationmethodology used in the program. To enable time step size control for capturingvarious time dependent events, the user can specify the number of fracture Itera-tions and the Maximum Time Step. MFrac will also automatically adjust the timestep size as necessary to control local and global errors. For Replay/Real-Timeanalysis, the data Restart Time can also be specified.

The base time step used for discretization in the numerical simulation will be theminimum of the values calculated from either the number of Iterations or MaxTime Step input.

IterationsThe value for the number of Iterations determines the target number of time stepsto be used for the fracture propagation solution. The total or estimated simulationtime is then divided by the number of iterations to determine the time step size.

Transitional Flow, Keck, et al.2

No Drag Reduction, Prandtl, et al.3

(Predicts Maximum Friction)


1f

----- A Res f( )log B+=

1f

----- 4 Res f( )log 0.4–=


2.2 Options 81

For example, if the number of iterations is 100 and the pump time is 100 minutes,the average time step would be one (1) minute. The actual time step may varydepending on other numerical considerations. For most simulations, a value of 20to 30 iterations is sufficient.

Generally, the number of iterations is most effectively used for design and autodesign. For Replay/Real-Time, the Max Time Step constraint may be more applica-ble.

The number of time steps should be increased for cases with order-of-magnitudechanges in the injection rate or fluid rheology properties (e.g., pad/acid). It shouldalso be increased if the fracture encounters numerous layers in a given time step(especially with diverse properties) or when the injection times are very large (e.g.,years as in water flooding). The maximum time step can also be specified to mini-mize the time step.

If the Alpha Plots accessed from the Fracture Characteristics Plot dialog boxappear erratic, the number of iterations specified may be too small. Generally, thesolution will not be very sensitive to the number of time steps because of the solu-tion technique used. If the governing equations cannot be solved within a specifictolerance, an alternate solution method is used for that time step. The number ofactual fracture solution iterations will always be slightly greater than the valuespecified. The larger this value is, the longer the program will take to run.

Max Time StepThe Max Time Step can be used to control the program time step. This is especiallyuseful when performing real-time or replay simulations. To simulate events thatoccur over a very narrow range of time (e.g., rate changes or pressure spikes) thetime step size must be small enough to capture the event. If the time step is toolarge, significant rate and pressure changes may be missed. Also, the smaller theMax Time Step the longer it will take the program to run.

The Real-Time option of synchronizing the wellbore solution to the input dataenables time refinement for wellbore and near-wellbore pressure losses. This isvery useful for history matching pressure changes due to rate. Also, as stated in theReal-Time options section, the fracture net pressure is a weaker function of ratethan frictional pressure dissipation. This provides the capability to limit the fracturetime step while still maintaining the time refinement necessary to accurately modelwellbore friction.

Restart TimeThe Restart Time is used to start or restart a simulation at a time other than the firstentry point in the data file for real-time or replay simulations. This option is nor-



mally used when earlier data is not relevant or multiple injection cycles (i.e., mini-frac) are pumped and only the later time cycle data (i.e., main frac) is to beanalyzed. Consequently, this option provides the flexibility to restart a simulation atthe beginning or middle of any injection cycle. Enter the time in the replay/real-time data at which the simulation should begin.

Heat TransferWhen turned On, the heat transfer solution can be used to predict the heat-up of thefracturing fluid in the wellbore and the exchange of heat transfer in the fracture tothe reservoir during fracture propagation. The heat transfer solution accounts forconservation of energy and results in simulating the effects of temperature on fluidrheology. The fluid rheology properties as a function of time and temperature arespecified in the Fluid Database.

If heat transfer is turned Off, the in-situ Fluid Temperature is required. The fluidrheological properties are then calculated from the fluid database as a function oftime based on this temperature.

Fracture Options This group of options is accessed by clicking the Fracture tab found on the DataOptions screen. The Fracture Options provide choices for the fracture geometrymodel and constitutive relationships that affect the fracture solution methodology(see Figure 2.4). The choices are as follows:

Figure 2.4: Fracture Options.


2.2 Options 83

Fracture GeometryThis section describes the five geometry models used in MFrac. See Appendix Afor a detailed description of each model. The parametric relationships for the 2-Dtype models is also discussed.

Horizontal EllipsoidalThe Horizontal Ellipsoidal fracture is an ellipsoidal two-dimensional planar-typefracture geometry model for propagation in the horizontal plane. It is similar to theVertical Ellipsoidal model but with a number of different constraints and constitu-tive relationships unique for horizontal propagation (e.g., proppant transport, for-mation properties, stress orientation, etc.).

This geometry model is similar to a radial or penny shaped fracture propagating in ahorizontal plane; however, rather than restricting the model to radial growth, anEllipsoidal Aspect Ratio can be specified in the Zones dialog box. This value rep-resents the ratio of the major to minor radii. When the Ellipsoidal Aspect Ratio isequal to unity, the geometry reduces to the radial solution illustrated in Figure 2.5.

Figure 2.5: Horizontal Ellipsoidal Fracture Geometry with an Ellipsoidal Aspect Ratio of Unity.

Vertical EllipsoidalThis model produces a vertical ellipsoidal shaped fracture geometry that intersectsthe wellbore along the fracture height. This fracture also has an ellipsoidal widthprofile in the vertical and lateral planes. The Ellipsoidal Aspect Ratio is equal tothe ratio of the total length (tip to tip, 2L) to the total fracture height ( ).λ 2L H⁄=



Figure 2.6 shows a vertical ellipsoidal fracture and illustrates the height, width, andlength coordinates.

Figure 2.6: Vertical Ellipsoidal Fracture Geometry.

This model is similar to the Horizontal Ellipsoidal model in appearance only. Themajor differences are related to orientation, intersection effects, influence of in-situstresses, proppant transport and a number of other constitutive relationships.

GDKThe GDK option invokes a constant height fracture geometry model with a verti-cally constant fracture width. That is a vertically unbounded geometry with slip atthe upper and lower extremities as shown in Figure 2.7. This model is characterizedby a decreasing net pressure with time. This option is most applicable for fractureswith length to height aspect ratios less than unity or for fractureswhich display slip at the upper and lower boundaries. Slip may be an appropriateassumption when rock interfaces are “weak” due to rock competence or when nor-mal stresses may be low (e.g., at shallow depths). The GDK model typically pre-dicts greater wellbore widths and shorter fracture lengths than the PKN model forfractures with aspect ratios greater than unity. The major difference between thePKN and GDK models is in the width-opening pressure relationship as shownbelow:

λ 2L H⁄= 1<

GDK - Ww ΔpL E⁄∝

PKN - Ww ΔpH E⁄∝


2.2 Options 85

where is the maximum wellbore width, is the half-length, is the height

and is the fracture net pressure.

Figure 2.7: GDK Geometry.

PKNThe PKN model, like the GDK model, is also a constant height model but differs byhaving an elliptically shaped width in the vertical plane (vertically bounded geome-try). Unlike the GDK model, the PKN net pressure increases with time for a con-stant injection rate. The PKN fracture model is most applicable when the totalfracture length is greater than the total fracture height ( ). The majordifference between the GDK and PKN model is in the width-opening pressure rela-tionship given above. Figure 2.8 shows the PKN fracture geometry profile.

Ww L H

Δp

λ 2L H⁄= 1>



Figure 2.8: PKN Geometry.

3-DimensionalThis is a 3-dimensional planar fracture model with both lateral and vertical frac-ture propagation. For large length to height aspect ratios, the model approaches thePKN constant height type geometry. When no confining stress, toughness or mod-uli contrast are entered, the model approaches a vertical radial-type geometry. Thismodel produces the most realistic geometries and is applicable for all length toheight aspect ratios. Figure 2.9 shows a typical 3-Dimensional fracture geometryprofile. As illustrated, the model assumes a bounded geometry at the leading edge(perimeter).

The GDK vertically constant width and PKN elliptical width profiles do not con-trol either models characteristic behavior. It’s all in the width-opening pressureconstitutive relationship. See Meyer4 for additional comments on model differ-ences.


2.2 Options 87

Figure 2.9: Three-Dimensional Fracture Geometry.

To use a 3-D model effectively, the formation should be characterized sufficientlyto adequately describe the rock and fluid loss properties. To better characterize theformation, the model currently allows up to one thousand (1000) layers for the rockand reservoir properties.

A detailed explanation, formulation and solution methodology for our three-dimen-sional fracturing simulator (MFrac) is presented in the appendices.

FlowbackThis option provides the user with the capability to allow fluid and proppant toflowback (negative injection rates and/or cross-flow) from the fracture. To simulatecross-flow between multilayer fractures, this option must be On. If this option isOff, cross-flow during closure and flowback will not be permitted.

This option must be clicked on to simulate flowback, since a negative rate must beinput into the treatment schedule.

Simulate to ClosureWhen Simulate to Closure is On, the program will simulate closure after pumping.Simulate to closure assumes the spatial fracture compliance factors used during clo-sure remain constant and equal to the values at the end of pumping. This method isless rigorous than the alternative (i.e., when this option is Off) when closure is sim-ulated with a treatment schedule rate of zero.



When Simulate to Closure is Off the program will not simulate fracture propagationor lateral proppant transport during closure, unless a zero rate is input in the treat-ment schedule. The proppant transport solution during closure can accommodateboth simulate to closure and a zero rate stage (shut-in) in the treatment schedule.

To maximize compliance precision and propagation accuracy during closure, turnthis option Off and enter a zero rate stage in the Treatment Schedule. Keep in mindthat Simulate to Closure may affect the pressure decline behavior, especially forfractures that grow into multi-stress layers and close with multiple inflection points.

Simulate to closure may not be an appropriate choice for multilayer fractures whichexhibit cross-flow.

Fracture Fluid GradientThis option is only used for the three-dimensional model and allows the user toInclude or Exclude the effect of the fluid gradient on fracture pressure. WhenInclude is checked, the pressure distribution within the fracture will include thehydrostatic pressure changes as a function of depth (fracture height). The pressuredistribution in turn will affect the fracture propagation.

Normally this option has little effect on the fracture geometry, except for caseswhen the hydrostatic head difference in the fracture is of the same order of magni-tude or greater than the fracture net pressure. This can happen in soft formations(low Young’s modulus) when the fracture height is large.

Propagation ParametersThe options for Propagation Parameters are Default Growth, No Growth DuringShut-in, and Positive Growth Only. This option is used to override the propagationrate calculated by the mass conservation, continuity and momentum equations.Fracture propagation will be set to zero during closure if no growth during shut-inis selected. If the Positive Growth Only option is selected the fracture will not beallowed to recede.

These options may be helpful for interference closure, proppant effects, flowbackand history matching pressure declines. Low permeability rocks fractured with highefficiency fluids may continue to propagate even after the pumps have shut down.To model the pressure decline the user may want to select Default Growth or Posi-tive Growth Only.

When fracturing wells with large stress contrasts between the reservoir rock and thebounding layers, the upper and/or lower zones may close before the reservoir rock


2.2 Options 89

closes. This will effect the fracture compliance changing the slope of the pressuredecline.

Fracture Initiation IntervalThis option is used to select the manner in which the program initiates the fracture.MFrac, like other fracture models, is a fracture propagation simulator which doesnot simulate the initial formation break down process. Normally for conditionswhere the final fracture geometry is much greater than the initiation geometry, thisinitial boundary condition becomes insignificant in the solution. The choices forfracture initiation are explained below.

Perforated IntervalWhen this option is chosen, the program uses the entire Perforated Interval as theinitial fracture height. The effective closure pressure is then calculated as the mini-mum fracture pressure necessary to keep the fracture open over the perforated inter-val. This initiation pressure may not be (and usually is not) equal to the minimumhorizontal stress in this interval.

Min. Stress IntervalThe Min. Stress Interval option invokes a routine that uses an initial fracture heightequal to ten (10) percent of the total perforation interval. For this option, the pro-gram examines the formation stress profile and identifies the portion of the perfo-rated interval that contains the zone with the minimum closure pressure required tokeep the fracture open over this limited depth of the perforated interval. This maynot be the interval which contains the lowest minimum horizontal stress. Thisnumerically selected depth interval is then used as the location for fracture initia-tion.

Fracture Friction ModelNormally, laminar flow exists in the fracture and this option may not be needed(i.e., select Off). For this case, the classical solution for fluid flow in a rectangularslot (as modified for an ellipsoidal fracture width) is used and the Darcy frictionfactor takes the form

where is the Reynolds number ( and )

Deviations from laminar flow affect the frictional dissipation in the fracture and,therefore, on the pressure predicted by a model. Turbulent flow in the fracture may

fD 24 Re⁄=

Re Re υw ν⁄= dp dx⁄ 1 2⁄ fDρυ2 w⁄–=



also occur when very low viscosity fluids (e.g., gas) at high rates are pumped. Toaccount for these phenomena and improve the ability to predict non-laminar fric-tional pressure loss in a fracture, the following friction factor expression is usedwhen the Fracture Friction Model is turned On:

Irregularities along the fracture face (e.g., tortuosity, bifuraction and wall rough-ness) that interrupt and disturb fluid flow can also result in greater energy dissipa-tion. These effects can be modeled by increasing the coefficient or modifying thewall roughness factor discussed below.

The approximations for the a and b coefficients in Table 2.4 have been developedempirically in accordance with Prandtl's Universal Law of the Wall (seeSchlichting3).

Wall RoughnessWhen this option is turned Off, the Darcy friction factor inside the fracture is usedwithout modification as determined from the selections made in the Fracture Fric-tion Model option. This selection assumes that the fracture surface is a smooth pla-nar feature without roughness.

To include the effects of roughness (or waviness) on the frictional dissipation, turnthis option On. This will result in an increase in the frictional pressure drop andfracture width, as well as, a decrease in fracture length. If this option is used, thefriction factor defined in the Fracture Friction Model option will be modified usinga Friction Factor Multiplier. The relationship used is defined in the expressionshown below:

Table 2.4: Typical a and b Friction Coefficients.

Laminar flow Re < 750; a=24; b=1

Transitional flow 750 < Re < 2000; a=0.5; b=0.44

Turbulent flow 2000 < Re < 30,000; a=0.13; b=0.25

Turbulent flow 30,000 < Re < 106; a=0.08; b=0.20

Turbulent flow Re > 106; a=0.035; b=0.14

MFrac does not use apparent viscosity. The above formulation is based on apower-law Reynolds number to provide the user with insight on the effect ofthese energy dissipation mechanisms.

fD a Reb⁄=

a


2.2 Options 91

where

An empirically derived correlation5-8 for determining the Friction Factor Multi-plier is shown in Figure 2.10.

Figure 2.10: Friction Factor Multiplier Empirical Correlation.

Tip EffectsThe observed field pressures for some treatments may at times be much higher thanthe simulated pressure. This discrepancy in measured pressure can be minimized ina number of ways. Typically, the friction factor multiplier, fracture toughness, near-wellbore effects, confining stress or rock/reservoir properties are modified to obtain

= modified Darcy friction factor= Darcy friction factor= friction factor multiplier

fD ′ Mf fD=

fD ′

fD

Mf



a match; however, if the pressure discrepancy is due to excess pressure, an over-pressure function can be applied at the tip. In MFrac, excess pressure can be appliedusing any of these three mechanisms: 1) Fracture Toughness, 2) Critical Stress, or3) Tip Over-pressure.

Over-pressure, as it is incorporated in MFrac, accounts for the extra pressurerequired at the fracture leading edge for propagation to occur. This extra resistanceat the fracture perimeter (tip) requires additional pressure (energy) to propagate thefracture. As a result, when this option is used, higher pressure must be applied atthe inlet (surface or BHTP) to compensate for losses that occur in the fracture.

Tip effects, in general, remain an area of some controversy and considerable dis-cussion. Plausible explanations for these effects have been proposed. The possibili-ties include tip friction due to flow resistance, rock properties effects (e.g.,toughness as a function of stress at the leading edge or poroelasticity), or it may bea consequence of fracture geometry (e.g., complex geometry and/or multiple frac-tures).

In this version of MFrac, tip effects represent a flow resistance at the tip. Regard-less of whether you believe this flow resistance is due to viscosity effects or someother phenomena related to the tip region (e.g., tip geometry), the general effect onpressure is typically the same (i.e., resistance is resistance). It is important to note,however, that this type of resistance differs from fracture toughness in its classicalapplication in that over-pressure varies with injection rate and time, fracture tough-ness does not.

The range on the over-pressure factor allowed by MFrac is between 0 and 1.0. Ifthis option is disabled, Off, a default value of zero is used. Usually, the Tip Effectoption is never suggested unless all other reasonable measures have been taken andthe measured injection pressures are still well above the theoretical values predictedby the simulator.

When reasonable values have been implemented for wall roughness, friction factormultiplier, toughness and other formation properties, a value between 0.1 to 0.4may be justifiable. The larger the over-pressure factor the greater the increase inthe net pressure. If you are having difficulty relating the over-pressure factor topressure, one approach is to use the MinFrac program to automatically regress onthe tip factor to determine an appropriate value. This best fit value from matchingthe net pressure in a minifrac analysis is a good place to start.

Many engineers mistake near wellbore pressure loss for excess net pressure. Keepin mind that when the injection rate changes suddenly, the near wellbore pressureloss also changes instantly; whereas, the fracture net pressure cannot because of


2.2 Options 93

storage (i.e., if the rate drops suddenly and the BHTP follows, this is not excesspressure but frictional dissipation in the near wellbore region).

Figure 2.11: Fracture Tip Width Reduction due to Non-Linear Elastic Effects.

Proppant Options This group of options is accessed by clicking the Proppant tab found on the DataOptions screen. The proppant options specify the proppant transport methodologyto be employed. Figure 2.12 illustrates the proppant options available. Theseoptions are discussed below.

The phenomena of tip over-pressure has been referred to as “dilatancy” by someresearchers. It is not clear whether these researchers are referring to rock dila-tancy or fluid dilatancy. Fluid dilatancy refers to a shear-thickening fluid. Rockdilatancy describes volumetric expansion of a material that is rapidly approach-ing failure and is usually associated with the micro-cracking process. There hasbeen no published explanation on the effects of rock dilatancy on net pressure ina crack, and to our knowledge, no correlations exist. The desired effect (i.e., anincrease in pressure) can be achieved due to viscosity effects (i.e., fluid dila-tancy) or as a result of stress dependent rock properties that may or may not berelated to rock dilatancy. This is commonly referred to as nonlinear elasticdeformation. Figure 2.11 illustrates one possibility.



Figure 2.12: Proppant Options.

Proppant SolutionThe Proppant Solution may be turned On or Off. If Off is chosen, no proppanttransport calculations will be performed and all other options related to the prop-pant transport will be dimmed. If Off is selected, there will be no coupling betweenthe fracture propagation and proppant transport solutions. Consequently, the frac-ture may propagate backward (negative growth) during closure depending on theR.O.C. of energy and mass conservation. If the proppant transport solution is Onand the Proppant Transport Methodology is not on conventional, negative growthwill not be allowed (this assumes proppant interaction).

When the Proppant Solution option is turned On, proppant transport calculationswill be performed.

If Input is chosen for the Treatment Schedule Options, MFrac calculates the geom-etry and proppant distribution based on the input treatment schedule.

When Auto Design is selected, either the desired fracture length or designed slurryvolume must be entered in the Treatment Schedule screen. The program then auto-matically designs a treatment schedule to satisfy the design criteria. For AutoDesign, the Proppant Type, Initial Proppant Concentration, Incremental ProppantConcentration, Final Proppant Concentration, Maximum Proppant Concentrationand Proppant Damage Factor must be entered. Depending on the Proppant Trans-port Methodology option selected, the program will internally calculate therequired PAD and proppant schedule to prevent (or create) a screen-out andachieve, if possible, the optimum treatment specified.


2.2 Options 95

Proppant RampThe ramp option controls the ability to ramp the proppant concentration between aspecified range. When this option is On, the concentration of proppant will beramped linearly from an initial value (From) to a final (To) value for each fluidstage in the Treatment Schedule dialog box. This results in a linear proppant rampwith liquid volume.

When this option is turned Off, a uniform proppant concentration is assumed foreach stage. The Treatment Schedule screen will then permit only one entry valuefor concentration.

Proppant FlowbackThe Proppant Flowback may be turned On or Off. This option when selected Onsimulates the flow of proppant back towards the wellbore.

Perforation ErosionThis option allows for perforation erosion during the treatment. If this option isturned On the perforation frictional pressure loss will decrease as the mass of prop-pant through each perforation increases.

Limited entry designs require a certain differential pressure across the perforationsto ensure that each zone accepts a proportionate amount of fluid and proppant. Dur-ing the limited entry treatment, perforations are exposed to a proppant and liquidslurry. The effect of proppant is to increase the discharge coefficient, , and thehydraulic diameter of the perforation. The increase in the discharge coefficient canbe described as a rounding of the perforation.

Proppant Transport MethodologyThe selection determines how the proppant transport model is linked or coupled tothe fracture propagation calculations. When the Net Present Value option is On,this option is set to Conventional automatically. The proppant transport methodol-ogy options are: 1) Conventional, 2) Conventional (link proppant), 3) TipScreen-Out (TSO), and 4) Frac Pack. With the exception of Conventional, all ofthe other options couple the proppant transport and fracture propagation solutions.The TSO and Frac-Pack options are normally used with the Auto Design option.However, they are handled the same as the Conventional (link proppant) optionwhen the treatment schedule is Input. The above proppant transport choices are dis-cussed below. Please refer to Appendix I for a detailed description of TSOs andFrac-Packs.

CD



ConventionalThis option disables the link between the proppant transport and fracture propaga-tion solutions. When this option is selected, the proppant transport calculations willnot affect the fracture propagation behavior. Consequently, for a bridge-out orscreen-out the condition of an increasing net pressure will not occur. The proppantcalculations will be performed after each fracture iteration but the results will notbe coupled.

Conventional (Link Proppant)This option links or couples the proppant transport solution with the fracture propa-gation solution to simulate the effects of slurry transport on fracture pressure distri-bution and propagation. When this option is used, the program will calculateincrementally (i.e., at each time step) a unit of fracture propagation followed by thecorresponding proppant transport time step. If a bridge-out or screen-out conditionoccurs or exists it will affect the pressure distribution in the fracture and subsequentfracture propagation behavior. The resulting change in fracture geometry in turninfluences the overall proppant transport.

When the Auto Design option is used, the choice of Conventional (Link Prop-pant) will cause the program to calculate an optimum proppant transport schedulewithout screening-out. To simulate a TSO or Frac-Pack please refer to one of theoptions described below.

Tip Screen-Out (TSO)This is a specialized form of Meyer & Associates’ proprietary, linked proppantsolution methodology used in MFrac (see Appendix I). The TSO option was specif-ically developed to automatically design a TSO and output the corresponding treat-ment schedule. This option also couples the proppant transport solution with thefracture solution as does the Conventional (Link Proppant) option when the Treat-ment Schedule Option is on Input. It only differs from the Conventional (LinkProppant) option when the Auto Design feature is used.

Therefore, the following discussion will be directed toward using the TSO option asit pertains to the Auto Design capabilities.

For this option, based on user specified criteria (e.g., target Frac Length, Initial andMax. Inlet Concentration, Average Concentration/Area and Max. BHP), the pro-gram will automatically design a pumping schedule that optimizes a tip screen-outcondition (TSO). Once a screen-out occurs, the net pressure will continue risingduring the remaining part of the treatment to meet the desired design criteria. Dur-ing this type of simulation, the linked solution methodology permits continuousproppant transport calculations as the slurry is concentrating in the fracture due toleakoff.


2.2 Options 97

The program always satisfies the specified length and then, if possible, the averageconcentration per unit area. User specified constraints are used as absolute limitsand maintain the following priority: Maximum BHP has first priority, Max. Prop-pant Concentration (lbm/ft2) in the fracture is second and Maximum Inlet Con-centration is last. If any of the specified criteria are not met, it means that a higherpriority constraint was imposed.

The result is a proppant distribution that approaches a uniform concentration perunit volume throughout the fracture based on the target concentration per unit areaspecified.

Frac-PackThis is a more aggressive variation of the TSO solution technique described above.Once again, it only differs from the Conventional (Link Proppant) option whenthe automatic design feature is used.

Like the TSO option, it is also based on the user specified criteria, Frac Length, Ini-tial and Max. Inlet Concentration, Average Concentration/Area and MaximumBHP. When this option is used, the program will automatically determine a pump-ing schedule to first produce a tip screen-out and then pack the fracture. Unlike theTip Screen-out option described above, the Frac-Pack method prevents excessivewidth growth (ballooning) by decreasing the injection rate after the maximum inletconcentration is attained.

The basic premise of this methodology is to control excessive ballooning by match-ing the injection rate to the leakoff rate once the desired geometry is achieved.Please refer to Appendix I for a comparison of the TSO and Frac-Pack Methodolo-gies.

Proppant Settling OptionsFour correlations are available for simulating proppant transport and settling 9-12:

Empirical - lowest settling velocity.

Convective Transport - medium settling velocity.

Cluster Settling - highest settling velocity.

User Specified - user defined settling velocity.

A description of each method is as follows:



Empirical This single particle settling velocity correlation is based on the work of Bird, et al.9Extension of this work to non-Newtonian fluids for all flow regimes from StokesLaw to Newtonian flow results in the friction factor equations listed below bySlatery11. The terminology upper and lower bound refers to the coefficient usedin these equations. Wall and slurry concentration (bulk viscosity) effects areincluded in this option to account for hindered settling. These effects are based onmodified density and viscosity correlations.

where

and

Convective TransportThe characteristic dimension that has the most significant influence on the particlesettling velocity for classical (Empirical) proppant transport solutions is the particlediameter. Experimental results have demonstrated for certain fluid/proppant combi-nations that this approach, based on modified forms of Stokes Law, significantlyunderestimates vertical settling velocity when large fluid bulk gradients exist.These conditions may occur when large pad volumes, high proppant concentra-tions, or significant stage density contrasts exist.

To account for bulk flow effects, a convective transport correlation is included. Asimplified form of the equation is shown below:

= friction factor= non-Newtonian Reynolds number= 1.1= 0.9= 7.5

Lower Bound12 ( )

Empirical9,11

Upper Bound12 ( )

X

f 24Rea--------- 24

Reb---------X2 c+⎝ ⎠

⎛ ⎞ Re 500<;=

f 0.44 500 Re 2.0x105<<;=

fReabc

X 3n ′-1 n′ 2+3n′

--------------n ′

= X 1≤

X 1=

X 1 0.8 1 n′–( )0.7+= X 1≥


2.2 Options 99

where

The vertical velocity determined from the relationship shown above is corrected forhindered settling using a power law model substitution for apparent viscosity andeffective density approach. The single particle velocity, calculated for an individualstage using the Empirical method described above, is superimposed over the bulksettling solution to characterize the complete settling history of the treatment.

Cluster SettlingLike the convective transport methodology described above, Cluster Settlingincludes the effects of bulk density gradients and can, therefore, be used to simulategravity induced flow. The formulations developed for this option are given below.The cluster settling rates produced may be higher than convective transport becauseof the differences in the rheology and density corrections used. These correctionsresult in an apparent viscosity that is lower than predicted by convection and aneffective density that is higher. This has the result of reducing the drag or walleffects that are calculated; hence higher settling velocity.

In addition, with Cluster Settling, only the bulk flow is considered. No single par-ticle, Stokes-type settling velocity is included. Consequently, there is no internalparticle settling assumed within an individual cluster.

This correlation may be most applicable when base fluid viscosity is low and prop-pant concentrations are relatively high (e.g., when you are pumping a bankingfluid). The cluster settling formulation is:

where

= vertical (bulk) fluid velocity= fracture width= apparent slurry viscosity

= potential gradient due to gravity, density, and pressure

= vertical (bulk) fluid velocity

vsw2

12μa------------

z∂∂Ψ=

vs

wμa

z∂∂Ψ

vsgΔρdeq

2

18μa-------------------=

vs



and

A power law correlation for apparent viscosity is used for proppant settling.

User Specified SettlingThis option allows the user to input the proppant settling rates manually. Whenselected an additional column or data box will be displayed in the Treatment Sched-ule screen that will allow you to enter a Proppant Settling Rate for each stage. Thevalue entered will be used as a constant rate of settling for that stage.

Wellbore-Proppant EffectsThis option controls the methodology used to simulate the effects of proppant con-centration on pipe friction. The options are as follows:

NoneFor this selection, proppant has no effect on the friction factors used in the wellborehydraulics calculations.

EmpiricalThis option includes the effects of proppant concentration on pipe friction as origi-nally described by Keck, et al.2 This correlation uses an expression for relativeslurry viscosity to account for the effects of proppant on increased friction. Therelationship is shown below:

= particle diameter= equivalent or characteristic diameter= apparent slurry viscosity= proppant void fraction= packed proppant void fraction= density difference of particulates and fluid= gravitational acceleration= fracture width

ddeq

μa

φφs

Δρgw

deq φ φs⁄( )αdw 1 φ φs⁄( )

αd–[ ]d+=

μr 1 0.75 e1.5n′ 1–( )e 1 n′–( )γ 1000⁄–[ ] 1.25φ1 1.5φ–-------------------+⎝ ⎠

⎛ ⎞ 2=


2.2 Options 101

where

For laminar flow, the friction factor multiplier, M, for proppant-laden fluids is equalto the value of . For proppant-laden fluids in turbulent flow, the expressionshown below is used to estimate the effect of proppant on friction:

and

where

User SpecifiedFor some slurry systems, adequate characterization of the frictional dissipation isnot possible with the empirical correlation contained in MFrac. If this occurs, thefriction factor multiplier as a function of proppant concentration can be specified intabular form.

Fracture-Proppant EffectsThis option controls the effect of proppant concentration on frictional pressurelosses in the fracture. The options are summarized below:

= relative slurry viscosity= power-law behavior index for base fluid= Newtonian shear rate= proppant void or particle volume fraction

= friction factor multiplier= base density= relative slurry density; = slurry density= relative slurry viscosity= friction factor of base fluid= friction factor of slurry

μr

n'γφ

μr

Mf μr0.55ρr

0.45=

fs Mf fb=

Mf

ρb

ρr ρr ρs ρb⁄=ρs

μr

fb

fs



NoneFor this selection, proppant has no effect on the friction factor calculated in thefracture.

EmpiricalThis option includes the effects of proppant concentration on fracture friction basedon the empirical expressions shown below. The procedure involves a correction tothe base viscosity to produce a relative viscosity term. Once this is done, the fric-tion factor is calculated based on the Fracture Friction Model selected and theFriction Factor Multiplier (if the Wall Roughness option is activated). The gen-eral correlation as a function of proppant void fraction is:

where

User SpecifiedFor some slurry systems, adequate characterization may not be possible with theempirical correlation contained in MFrac. If this occurs, the friction factor multi-plier as a function of proppant concentration can be input in tabular form. The val-ues entered are then used to determine the relative viscosity term based on thesolids fraction present. Like in the Wellbore Proppant Effects, the friction factor iscalculated based on the Fracture Friction Model and friction factor multiplier spec-ified (if the Wall Roughness is enabled).

2.3 Data InputOnce the Options are selected the scope of a simulation is set. Data may then beentered by accessing the various dialog boxes from the Data menu. As previouslystated, the option screens determine what information is needed for a particulartype of analysis. Consequently, the specific data displayed in a screen or the exist-ence of the data screen will vary depending on the options selected. This approachminimizes data input and prevents unnecessary or misleading data entry. Simplydecide what options are relevant to your simulation and the program will automati-cally display only those menus and input fields necessary. Any time an option ischanged, the input screens will vary to enable new input or hide data that is notneeded. This methodology is used throughout MFrac.

= relative slurry viscosity= exponent coefficient= proppant void or particle volume fraction

μr 1 φ–( )αμ=

μr

αμ

φ


2.3 Data Input 103

The following sections pertain to the Data menu items found within the main menu.Each Data menu item is covered in detail along with a description of the data dia-logs and their associated variables. When pertinent, the conditions or case sensitiveoptions for a data screen are noted and an example of the resulting dialog shown.All of the different data screens available within MFrac and the variables containedwithin them are presented.

DescriptionThe Data Description screen shown in Figure 2.13 provides a location for enteringinformation about a simulation. Space is provided for entering the CompanyName, Well Name, Well Location and Simulation Date. In addition, a Comments,section is included so that descriptive information can be entered. All informationcontained in this dialog is optional.

Figure 2.13: Data Description Dialog Box.

To enter data, position the cursor at the desired location using the mouse or TABkey. Any text can be selected by using the mouse or the keyboard. The scroll bar tothe right of the Comments section can be used to view all the data entered.

Wellbore HydraulicsMFrac offers an integrated wellbore hydraulics module that couples the fracturewith the wellbore to provide additional simulation capability. An energy balance



approach is used to calculate the pressure changes due to potential energy, kineticenergy, frictional dissipation and restrictions in the wellbore.

This general solution permits calculations of surface pressure, BHP in the wellbore,Frac-Pack screen pressure drop, hydrostatic head, frictional loss and hydraulicpower requirements for a treatment design. The flexibility of the model providesthe capability to history match measured pressures during real-time or replay treat-ment analysis.

The capability to model tapered deviated wellbores is also included. Treatmentstage movement in the wellbore is also simulated during pumping.

General DataThe General Wellbore Hydraulics screen is shown in Figure 2.14.

Figure 2.14: Wellbore Hydraulics - General Tab.

Injection DownPerforming wellbore hydraulics calculations requires that the wellbore configura-tion and perforations be described. Numerous configurations are possible. Injectioncan be simulated down tubing, casing, annulus or both. To select the configuration,choose one of the radio buttons found in the Injection Down section of the GeneralWellbore Hydraulics screen shown in Figure 2.14.

The different Injection Down configurations are:


2.3 Data Input 105

1. Casing - Injection is down the casing. This option assumes there is no tubingin the wellbore. Therefore, tubing data is ignored for this option. If you onlywant to pump down the casing/tubing annulus, check Annulus rather thanCasing.

2. Tubing - Fluid is pumped down the tubing string. When the fluid reaches theend of the tubing it will flow through the casing to the perforations.

3. Coiled Tubing - Fluid is pumped down the coiled tubing. When the fluidreaches the end of the coiled tubing it will flow through the casing to the perfo-rations.

4. Annulus - Fluid is pumped down the annulus between the casing and tubing.Both casing and tubing data must be entered.

5. Tubing and Annulus - For this option, flow will be simulated down both thetubing and annulus. Both tubing and casing data are required for this option.

Horizontal Wells and Frac-PacksSpecial wellbore configurations are also available for Horizontal Wells and Frac-Packs using screen and crossover assemblies. To use one of these configurations,enable the appropriate check box located in the General Wellbore Hydraulicsscreen.

When Horizontal Well is selected, the user can input the MD at the center of theperforations rather than TVD. This provides the capability for locating the perfora-tions on a horizontal section of the casing.

When selecting Frac-Pack Screen, the Screen O.D. and Crossover PressureLoss Coefficient must be entered. This provides the added capability of simulatingflow around the screen (pressure loss) and storage volume reduction due to thedecreased flow area. Once the frac-pack screen diameter is entered it will be dis-played on the wellbore schematic extending from the end of tubing to the end of thecasing. The Crossover Pressure Loss Coefficient is used to calculate the minorpressure loss associated with the crossover port:

where

= pressure loss= loss coefficient

Δp Kρsν

2

2-----------–=

ΔpK



Typical values for the loss coefficient are between two and ten.

Surface Line VolumeThe Surface Line Volume is the volume of fluid/slurry in the service line(s)upstream of the well entrance. Normally this volume is negligible; however, thisline volume may be significant at times (e.g., on Frac Boats). If all the treatmentparameters (rate, volume and staging) are referenced at the well entrance this valueshould be set to zero; however, if all treatment references (measurements and vol-umes) are upstream of the well entrance, this line volume should be specified.

Wellbore Volume Reference DepthThe Wellbore Volume Reference Depth is the depth used for calculating the vol-ume of the wellbore. Either the Measured Depth (MD) or True Vertical Depth(TVD) may be entered. This reference depth must be below the bottom of the tub-ing and above the bottom of the casing. This only applies if there is tubing in thewellbore.

Maximum BHTPThis is the maximum allowable BHTP at the BHTP Reference Depth. If this valueis exceeded during the simulation a warning message will be displayed. This Maxi-mum BHTP is also used as a design criteria in Auto Design mode to ensure that theoptimum design does not exceed this value.

VolumeOn the top of the General tab of the Wellbore Hydraulics dialog box, a Volume isdisplayed. This volume is calculated based on the wellbore configuration andincludes the surface line volume. This volume is also used as the wellbore/flushvolume in the Treatment Schedule.

Deviation DataThe first step in describing the wellbore configuration is to enter the well deviationsurvey. To do this, click the Deviation tab found on the Wellbore Hydraulics dialogbox, as shown in Figure 2.15. A table for the measured depth (MD), true verticaldepth (TVD), deviation angle and angle build rate will then be displayed.

The Angle Build Rate is a feature that allows the deviation angle or angle build rateto be specified. If the deviation angle is input, the angle build rate will be calcu-lated. If the angle build rate is specified the deviation angle will be calculated. This

= slurry density= velocity

ρs

v


2.3 Data Input 107

feature is only available if the Linear Segment check box is not checked (i.e., forlinear segments the build rate by definition is zero).

Another feature is the Fixed Depth option. When changing the wellbore deviationyou can fix either the MD or TVD depths. If MD is fixed the TVD values will be re-calculated based on the new wellbore configuration. If TVD is fixed all of the MDvalues in the Rock Properties, Fluid Loss and Zones dialog boxes will be updated.The corresponding reference MD’s in the Wellbore Hydraulics folders will also beupdated if TVD is fixed.

Figure 2.15: Wellbore Hydraulics - Deviation Tab.

The schematic of the wellbore profile for this input data is shown in Figure 2.16.



Figure 2.16: Wellbore Cross Section

If the Linear Segments check box is selected, the Wellbore Hydraulics dialog box,as shown in Figure 2.17 will be displayed. Typically, if linear segments is chosenthe MD and TVD should be entered and the Angle calculated.


2.3 Data Input 109

Figure 2.17: Wellbore Hydraulics - Deviation Tab (Linear Segments).

The wellbore profile for this input data and linear segments checked is shown inFigure 2.18.



Figure 2.18: Wellbore Cross Section - Linear Segments

When you are finished entering the deviation data or want to view the data graphi-cally to check its validity, click on the plot button. A dialog box will open with aselection of three plot types. The first is the cross-section that plots the deviation’sTVD versus horizontal displacement. The second plots TVD versus MD. The lastplots the deviation angle versus MD. Choose the plot type. Click the OK button andthe appropriate plot dialog box will open.

Casing/Tubing DataTo enter pipe data, choose either the Casing and/or Tubing tabs found on the Well-bore Hydraulics screen, depending upon your application. The program uses theinformation entered to calculate the pipe volume and/or flush volume. The valuecalculated is displayed in the upper left corner of the General Wellbore Hydraulicsscreen and also in the Wellbore section of the Treatment Schedule. Make sure that


2.3 Data Input 111

the volume calculated is correct if you are planning to evaluate under- or over-dis-placement conditions. This is especially important for real-time and replay simula-tion.

For Casing, enter the Measured Depth and Section Length for the portion of thecasing you want to describe (Figure 2.19). Selecting Allow Casing Overlapenables the user to input both the measured depth and section length. If no overlapexists, do not check this box.

Figure 2.19: Wellbore Hydraulics - Casing Tab.

To specify the OD, ID and pipe Weight, scroll through the Casing Database, foundat the bottom of the Casing screen to locate the correct pipe size. When you havefound the size, click the corresponding database row and the data will automaticallybe entered in the casing data table. If the data is not present in the database, it caneither be entered directly in the table or it can be added to the database.

In MFrac, the entry of Casing and Tubing data is handled in a similar manner. ForTubing data, however, it is only necessary to enter the Measured Depth, as nooverlap in sizes is allowed (Figure 2.20). Like the Casing data, once the depths



have been specified, fill in the table with the correct pipe sizes from the TubingDatabase. As each row of data is entered, it is displayed in the wellbore configura-tion diagram located on the left side of the Wellbore Hydraulics screen. The pipecomponents and their positions are drawn with the same relative scaling.

Figure 2.20: Wellbore Hydraulics - Tubing Tab.

For both the Casing and Tubing, a Relative Pipe Roughness and Friction LossMultiplier can be entered independently for each string of pipe. The Friction LossMultiplier can be used to simulate additional pressure losses, due to collars, joints,roughness, restrictions, etc. This parameter modifies the calculated frictional pres-sure loss. When using this feature the calculated frictional pressure loss is multi-plied by the Friction Loss Multiplier. For example, if the simulated pressure loss is1000 psi and the friction multiplier is 0.75 the actual pressure loss reported by theprogram would be 750 psi.

The Tubing dialog changes when coiled tubing is specified. Only one set of tubingparameters is applicable, and the option to calculate friction loss (based on the reelcore diameter) or to use a user-defined reel friction loss multiplier is available. Thelength of coil on the reel is calculated by subtracting the Measured Depth from the


2.3 Data Input 113

Total Coiled Tubing Length. An example of the coiled tubing tab is shown in Fig-ure 2.21.

Figure 2.21: Wellbore Hydraulics - Coiled Tubing Tab.

The Total Coiled Tubing Length is the total length of tubing on the reel and in thewell. The Measured Depth is the length of tubing in the well. Length of Coil on theReel is difference between the Total Coiled Tubing Length and the Measure Depth.

The Reel Core Mean Diameter is required input if the User-defined Coiled-TubingReel Friction-Loss Multiplier is unchecked. The friction factor multiplier for thecoil on the reel is then calculated from a modified McCann’s correlation (SPE36345) as given by

where is the fanning friction factor for the coiled-tubing, is the fanning fric-

tion factor for straight tubing, is the coil tubing inside diameter, and is the reelcore diameter (i.e., mean diameter), and the coefficients and arerepresentative values. The reel friction multiplier ( , relative to straight tubing) isgiven by

fct fst 1 a d D⁄( )b+( )=

fct fst

d Da 1.5= b 0.1=

fm



If the User-defined Coiled-Tubing Reel Friction-Loss Multiplier is checked, theuser must enter the Reeled-Tubing Friction Loss Multiplier for the coil. Thisparameter is applied as an additional frictional pressure loss multiplier for the tub-ing on the reel.

In general, the frictional loss multiplier for the tubing on the reel is an additionalmultiplier to the straight tubing friction factor times the straight tubing multiplier.

Pipe roughness is also included in the frictional pressure loss calculation when theWellbore Hydraulics Model is selected as Empirical. When a value is entered forthe Relative Pipe Roughness, the expression used for friction factor is modified inaccordance with Prandtl’s “Universal” law expression. Refer to Appendix E foraddition information regarding the effect of wall roughness on the friction factor.

Restrictions DataThe last component included in the wellbore configuration is the description of anyrestrictions that may exist in the tubing. Restriction data is optional and can beentered by clicking the Restrictions tab found on the Wellbore Hydraulics dialogbox as shown in Figure 2.22. The table that is displayed requires the measureddepth, inside diameter and optional Friction Loss Multiplier for each restriction.This feature can be used to simulate the effects of various tool or pipe configura-tions.

fm 1 a d D⁄( )b+=


2.3 Data Input 115

Figure 2.22: Wellbore Hydraulics - Restrictions Tab.

BHTP ReferencesFigure 2.23 shows the BHTP reference depth table. Three (3) BHTP referencedepths can be specified for reporting BHTP in the wellbore. Either the MeasuredDepth (MD) or True Vertical Depth (TVD) may be entered. If the reference depthis above the tubing and you are pumping down both tubing and casing, the BHTP inthe tubing will be reported.



Figure 2.23: Wellbore Hydraulics - BHTP References Tab.

ProfileAfter configuring the wellbore components, click the Profile tab on the WellboreHydraulics dialog box to view a graphical representation of the components andwellbore deviation (see Figure 2.24). Clicking on the Plot button will open a dialogbox to allow reconfiguring, zooming and printing of the plot.


2.3 Data Input 117

Figure 2.24: Wellbore Hydraulics - Profile Tab.

ZonesThe Zones dialog box is used to specify the number and location of the perforatedintervals and corresponding Zone Data (Figure 2.25). A maximum of ten differentperforated intervals or limited entry type fractures can be specified. The methodol-ogy and governing equations for multilayer or limited entry fracturing is discussedin Appendix B.



Figure 2.25: Zones Dialog Screen.

The type of data required to define an interval depends on whether the well and/orthe fracture is horizontal or vertical. A well is assumed to be a “vertical well”unless the Horizontal check box in the General Wellbore Hydraulics screen ischecked.

Active Any zone that is defined in the program can be enabled or disabled for use in simu-lation of multilayer fractures by double-clicking the left column to display or cleara check mark. A zone is Active when the check mark is displayed. Each Activezone represents the possibility of creating a multilayer fracture in that zone. If onlyone zone is active, the fracture will initiate in that zone.

Zone Name To assist in keeping track of the data depth intervals, an optional Zone name can beentered in the second column of the table. This name is only used to help organizethe input and output data.

Perforation and Fracture IntervalsFor vertical wells with vertical fractures, regardless of whether the well is deviatedor not, the perforation data is entered relative to the true vertical depth (i.e., Top of


2.3 Data Input 119

Perfs TVD, Bottom of Perfs TVD) or measured depths (i.e., Top of Perfs MD, Bot-tom of Perfs MD).

If a horizontal well is specified in the General Wellbore Hydraulics screen, the cen-ter of the perforated measured depth (Center of Perfs MD) is input and the true ver-tical center of the perforated depth (Center of Perfs TVD) is calculated. The Centerof Perfs TVD is dimmed and cannot be edited. This same convention is used whenthe Horizontal Ellipsoidal fracture model is specified, even if the well is vertical.

When either the Vertical Ellipsoidal or 3-D geometry options are used in combina-tion with a horizontal well, it is also necessary to enter the TVD for the top and bot-tom of fracture initiation.

When any of the two-dimensional fracture geometry models are chosen from theGeneral Options screen, additional columns of data are required. For the PKN orGDK models, the Zones spreadsheet will contain two additional columns. In thesecolumns you must enter the top true vertical depth of the fracture (2-D Top of FracTVD) and the bottom true vertical depth of the fracture (2-D Bottom of Frac TVD).This data is used to characterize the total gross height of the fracture. If an Ellipsoi-dal fracture geometry model is chosen, the Ellipsoidal Aspect Ratio must also bespecified. This is the ratio of the major and minor ellipse axes (i.e., the ratio of thetotal length (tip to tip) to the total height of the fracture (2L/H)).

Zone Data After entering the Zones perforated depth information, open the Zone Data screenfor each interval by clicking the Edit button found in the far right column. TheZones Data screen shown in Figure 2.26 has tabs for Perforations, Pay Zone, Mul-tiple Fractures and Near Wellbore. A feature to include perforation erosion is alsoavailable. To activate the perforation erosion folder you must have PerforationErosion selected to User Specified in the Proppant Option Dialog (see Figure2.12).



Figure 2.26: Zones Data Dialog Tabs.

Figure 2.26 shows a dimmed perforation erosion table illustrating that PerforationErosion was selected to None.

PerforationsFigure 2.27 shows the perforation tab screen. Perforation erosion has been set toUser Specified. This activates the screen for inputting perforation erosion data. Theperforation data requirements are discussed below.


2.3 Data Input 121

Figure 2.27: Zone Data - Perforation Tab.

Number and Diameter of Perforations

The Number and Diameter of perforations must also be specified for each perfo-rated zone. These values are entered in the boxes provided at the top of the Perfora-tions screen. This information is used to calculate the perforation friction pressureloss.

Perforation Erosion

The perforation erosion feature is based on the work of Shah12, Cramer17, and El-Rabaa, Shah, and Lord18. This option allows for perforation erosion during thetreatment.

Limited entry designs require a certain differential pressure across the perforationsto ensure that each zone accepts a proportionate amount of fluid and proppant. Dur-ing the limited entry treatment, perforations are exposed to a slurry of proppant andfluid. The effect of the proppant is to increase the discharge coefficient, , and

the hydraulic diameter of the perforation ( ). The increase in can bedescribed as a rounding of the perforation.

CD

CD1 2/ D CD



Figure 2.28 shows the data required to model perforation erosion. To calculate Per-foration Erosion for limited entry fracturing treatments, select Intercept or FinalDischarge Coefficient from the Perforation Erosion dialog box. When Intercept isselected, the intercept is calculated. Enter an Initial Discharge Coefficient, FinalDischarge Coefficient, Perforation Erosion Rate, and Critical Proppant Mass.When Final Discharge Coefficient is chosen enter an Intercept along with the InitialDischarge Coefficient, Perforation Erosion Rate, and Critical Proppant Mass. Thefinal discharge coefficient is then calculated.

Figure 2.28: Perforation Erosion Data Screen.

The discharge coefficient for a sharp-edged perforation entrance is 0.60. For arounded perforation entrance, the discharge coefficient is 0.83. The Final DischargeCoefficient should be set to a larger value than the Initial Discharge Coefficient.Unless reliable data is available, set the initial value to 0.6 and the Final DischargeCoefficient to 0.83.

Note, if perforation erosion is not selected a 0.83 discharge coefficient (i.e.,rounded orifice entrance) is used.


2.3 Data Input 123

Typical values for the Perforation Erosion Rate and Critical Proppant Mass are0.004 in./1000 lbm and 6000 lbm.

To view a plot of the Perforation Erosion correlation, select the Plot icon. Figure2.29 shows a plot of the hydraulic perforation diameter ( ) and pressure lossratio as a function of proppant mass through each perforation. The pressure lossratio is the ratio of the perforation pressure loss after proppant has gone throughcompared to the base case of no perforation erosion. As the amount of proppantmass passes through a perforation, the hydraulic diameter increases and the pres-sure loss ratio decreases. After 6000 lbm of proppant has passed through the perfo-ration, the pressure loss ratio has dropped to less than 60% of its original value.

Figure 2.29: Hydraulic Diameter (CD1/2D) & Pressure Loss vs. Mass.

The upper and lower dashed lines are the theoretical limits for the initial and finalhydraulic perforation diameters. These limits are based on Initial and Final Dis-charge Coefficients of 0.60 and 0.83, respectively.

Pay ZoneFigure 2.30 shows the Pay Zone data screen. To determine the fracture conductiv-ity in a pay zone, a productive interval (pay zone) and average zone permeabilitymust be assigned. These values are used to determine an integrated (average) con-ductivity ( ) and a dimensionless conductivity over the pay interval.

CD1 2/ D

kfwf



Figure 2.30: Pay Zone Data Screen.

Pay Zone Permeability

This is the average pay zone permeability used to calculate an average dimension-less conductivity.

The associated Dimensionless Fracture Flow Capacity, , is calculated from

The average fracture conductivity for long term production is given by

and for short term production or reduced conductivity near the wellbore the follow-ing relationship may be more applicable

FCD

FCDkfwfkrL---------=

kfwf kf x( )wf x( ) xd0

L

∫ L⁄=


2.3 Data Input 125

where

A more detailed analysis of the effective conductivity for variable conductivityfractures is given in Appendix L.

Pay Zone Depth

A pay zone is defined by entering the TVD or MD Depth From (top) and TVD orMD Depth To (bottom) in the boxes provided. These depths do not have to conformto the perforated interval. If the fracture is not propped in this interval, the proppedconductivity and propped fracture length will be zero.

Multiple FracturesThis section allows the user to specify the number and degree of interaction of mul-tiple fractures in the specific multilayer zone. This is not the same as multilayer orlimited entry fracturing. Multiple fractures refer to fractures in the far field (notnear wellbore) which may or may not be interacting. These fractures may also beparallel or dendritic (tree like). Figure 2.31 shows the multiple fractures input datascreen.

= average fracture conductivity= proppant permeability in the fracture= reservoir permeability= propped fracture width= propped fracture half-length

If the Proppant Transport Plots show a conductive fracture (propped width andconductivity contours) and the pay zone does not appear on the screen or thepay zone plots show zero conductivity, it indicates that the fracture is not withinthe pay zone.

kfwf L 1kf x( )wf x( )------------------------- xd

0

L

∫⎝ ⎠⎛ ⎞⁄=

kfwf

kf x( )

kr

wf x( )

L



Figure 2.31: Multiple Fractures Data Screen.

The individual fracture interaction factors and degrees of interaction as given inAppendix C are

Flow Rate

where

Stiffness

where

Fluid Loss

where

Qi ΨqQt= Ψq 1 N⁄=

Ei ΨEE0= ΨE N 1–( )ΦE 1+=

Vi Vt N⁄=

Vt ΨlV0=


2.3 Data Input 127

The interaction factors and degrees of interaction are given by and , respec-tively. The degrees of interaction are the values entered into the program. Theindividual fracture properties and parameters are identified by the subscript i. Thetotal value for N fractures is given by the subscript t.

Depending on the degree of fracture interaction, the fracture net pressure can belower for multiple fractures than for a single fracture.

If near wellbore multiple fractures are causing near wellbore pressure losses, thenear wellbore pressure loss table should be used.

The reader is referred to Appendix C for additional information regarding multiplefractures and fracture interaction.

Number of Fractures

This is the number of multiple fractures (with two wings) to be modeled in a givenlayer. The default is a single two wing fracture (Number of Fractures equal toone).

Fracture Interaction

Check this box to specify the degree of stiffness and fluid loss interaction for com-peting fractures. The default is no fracture interaction.

Stiffness Interaction

This represents the per cent of stiffness interaction between the multiple fracturesystem. This parameter can range from 0 to 100%. The interaction values for noand full interaction are zero and 100%, respectively. The closer the fractures aretogether, the greater the stiffness. For multiple parallel fractures within a fraction oftheir characteristic height, the stiffness increases by a factor equal to the number offractures (i.e., full interaction). For tree like (dendritic) fractures the stiffness inter-action may be negligible (no interaction).

Fluid Loss Interaction

This represents the percent of fluid loss interaction between the multiple fracturesystem. This parameter can have a value from 0 to 100%. The interaction values forno and full interaction are zero and 100%, respectively. Depending on the reservoirproperties and vicinity of the fracture system, this value may not be the same as thedegree of the stiffness interaction.

Ψl 1 N–( )Φl N+=

Ψ Φ

Φ



Near Wellbore Pressure TableThe near wellbore pressure loss table is shown in Figure 2.32. MFrac has the capa-bility to model time and rate dependent near wellbore pressure drop for each frac-ture. This pressure drop can represent any near wellbore effect such as tortuosity,perforation erosion, near wellbore multiple fractures, etc. The methodologyemployed is explained in Appendix C.

Figure 2.32: Near Wellbore Pressure Loss Screen.

To include the near wellbore pressure drop as a function of time, fill in the spread-sheet located on the right side of the Zone Data screen. Up to fifty rows can bespecified to define the near wellbore pressure drop as a function of time and rate.

Import RT Button

When performing real-time or replay analysis using MView, MFrac automaticallyrecords “significant rate and pressure changes” and generates a near wellbore pres-sure loss relationship. After running the acquired data through MFrac, open theZone Data dialog box and choose the Import RT Button. The program will thenload the corresponding data file to fill in the Near Wellbore Pressure Table. If no


2.3 Data Input 129

significant rate/BHTP changes were encountered or if the data was not run throughMFrac, a message like the one shown in Figure 2.33 will be displayed.

Figure 2.33: Import RT Message.

The imported near wellbore pressure table includes the total near wellbore pressureloss. This table can be manually changed to incorporate small rate/pressure changesnot considered significant or indeterminate by MFrac.

Once a Near Wellbore Pressure Table has been created, choose how it will beapplied by clicking one of the radio buttons located below the spreadsheet. Theoptions are: to ignore the table completely, use the pressure drop as the total nearwellbore effects (including perforations), or to add the resulting effects to the calcu-lated perforation pressure losses (near well effects only).

The program performs a linear interpolation between successive data points for

where .If the job duration is longer than the maximum timeentered in the table, the last (final) value will be used

Treatment ScheduleThe data required for the Treatment Schedule screen varies depending upon theselections made in the General, Fracture and Proppant Options screens. The smart-menu system used by MFrac only requests the data needed to perform a specificsimulation.

Note for limited entry the imported table must be modified to account for thefractional flow rate going into each fracture.

K t( ) Δp t( ) K t( )q t( )α=K t( )



Auto Design - Treatment ScheduleWhen Auto Design is chosen for the Treatment Schedule Option, a comprehensiveTreatment Schedule screen for auto design is enabled that requires variable inputdepending on the Proppant Transport Methodology (i.e., Conventional, Tip Screen-Out (TSO) or Frac Pack) (Figure 2.34).

The treatment design will be scheduled from the Input Parameter choices of Maxi-mum Fracture Length, Total Slurry Volume. or Total Proppant Mass. The spe-cific data required will also depend on the selection made for the ProppantDistribution Style Option which allows the user to auto design a treatment sched-ule based on Maximum Proppant Concentration, a specific Concentration perUnit Area, Dimensionless Fracture Conductivity, or Fracture Conductivity inthe pay zone at closure. An option for the proppant staging profile (staging timefor increasing the proppant concentration using a power law relationship) may alsobe selected.

Frac pack designs allow for the additional options for the rate step down duringpacking, and a rate step down criteria for multi-stage step downs. An option is alsoavailable to maintain a net pressure increase during the rate step down in order topack the fracture.


2.3 Data Input 131

Figure 2.34: Auto Design Treatment Schedule Screen.

During an auto design simulation, the automatically designed treatment schedule issaved into an output file. This treatment schedule can then be imported into a tabu-lar Treatment Schedule screen (Auto Design is Off) for additional editing and sim-ulation using the Import from Output Data button. To use this capability, it isnecessary to run the automatic design case and then change the Treatment Sched-ule Option to Input. This enables the tabular type Treatment Schedule that containsthe Import from Output Data button. Click the button to fill in the TreatmentSchedule table using the data automatically generated by the previous MFrac run.



The following input parameters and their descriptions are specific to the AutoDesign Treatment Schedule. Those input parameters not found here are in the sec-tion “Input of Treatment Schedule” below.

Input Parameter MenuThe Input Parameter choices are: Maximum Fracture Length, Total Slurry Vol-ume. or Total Proppant Mass. These choices are available for all Proppant Trans-port Methodology options including NPV optimization. If a NPV design isselected, designs will be created in increments up to the maximum specified InputParameter value.

Maximum Fracture Length

For an auto design based on fracture length (fracture half-length), a treatmentschedule will be automatically developed to create a fracture length equal to theMaximum Fracture Length input into the dialog.

Total Slurry Volume

The slurry volume is required input when the Auto Design option is based on thetotal slurry volume pumped. The resulting fracture length is then calculated fromthe slurry volume injected.

Total Proppant Mass

The proppant mass is required input when the Auto Design option is based on thetotal proppant mass pumped. The resulting fracture length is then calculated fromthe proppant mass injected and value for the specific Proppant Distribution Styleselected.

Proppant Distribution StyleThe Proppant Distribution Style can have up to four options depending on the Prop-pant Transport Methodology. If Conventional is selected, the user can specify theauto design proppant scheduling methodology based on 1) The Maximum ProppantConcentration, 2) Concentration per Unit Area, 3) Dimensionless Fracture Conduc-tivity, or the 4) Fracture Conductivity. If the TSO or Frac Pack option is selected,proppant design scheduling can be based on target values for the Concentration perUnit Area, the Dimensionless Fracture Conductivity, or Fracture conductivity. (i.e.,the Maximum Proppant Concentration option will be dimmed).

If the maximum Proppant Concentration is selected for the Proppant DistributionStyle, the code will design a treatment with the last proppant stage at the final prop-pant concentration. The auto design will create a treatment schedule so that thestages do not screen- or bridge-out and that the maximum proppant concentration inthe fracture will not exceed the maximum value specified in the table.


2.3 Data Input 133

The target values for the Concentration per Unit Area, Dimensionless FractureConductivity and Fracture Conductivity are limited by the maximum proppant con-centration pumped. Following are some of the limiting scenarios and numericalresults that will prevent achieving (below or above) the target values for Concentra-tion per Unit Area and/or Conductivity based on the Proppant Transport Methodol-ogy selected in the options screen:

Conventional and Conventional (Link Proppant)

Simulated Concentration per unit Area

Simulated Concentration per unit Area is lower than the specified Target Value.Possible remedies:

• The fracture width is not large enough. Solutions a) Select a TSO or Frac Packdesign, b) increase the fluid viscosity, increase the fracture efficiency (decreasefluid loss) or c) Increase the size of the treatment.

• Increase the final proppant concentration.

Simulated Concentration per unit Area is higher than the specified Target Value.Possible remedies:

• Decrease the initial and incremental proppant concentrations.

Simulated Dimensionless Conductivity and Fracture Conductivity

Simulated dimensionless conductivity or conductivity is lower than the specifiedTarget Value. Possible remedies:

• The fracture width is too small. Solutions a) Select a TSO or Frac Pack design,b) increase the fluid viscosity, increase the fracture efficiency (decrease fluidloss) or c) increase the size of the treatment.


• Pump a higher permeability proppant.

• The formation permeability may be too high. Therefore the designed fracturelength may be too large for the formation (Dimensionless Conductivity only).

Simulated Dimensionless Conductivity or Fracture Conductivity is higher than thespecified Target Value. Possible remedies:




• The propped fracture may be a monolayer. Therefore, design on concentrationper unit area.

• Chose a proppant with a lower permeability.

• If in TSO or Frac Pack mode switch to Conventional.

TSO and Frac Pack

Simulated Concentration per unit Area

Simulated Concentration per unit Area is lower than the specified Target Value.Possible remedies:.

• Increase the final proppant concentration and or/increase the maximum allow-able BHTP.

Simulated Concentration per unit Area is higher than the specified Target Value.Possible remedies:


• If TSO is selected change the mode to Conventional. If Frac Pack is selectedchange to TSO.

Simulated Dimensionless Conductivity and Fracture Conductivity

Simulated dimensionless conductivity or fracture conductivity is lower than thespecified Target Value. Possible remedies:

• Increase the final proppant concentration and or/increase the maximum allow-able BHTP.


• Pump a higher permeability proppant.

• The formation permeability may be too high. Therefore, the designed fracturelength may be too large for the formation (dimensionless conductivity only).

Simulated dimensionless conductivity or fracture conductivity is higher than thespecified Target Value. Possible remedies:


• The propped fracture may be a monolayer. Therefore, design on concentrationper unit area.


2.3 Data Input 135

• Chose a proppant with a lower permeability.

• If TSO is selected change the mode to Conventional. If Frac Pack is selectedchange to TSO

Override Internal Concentration Staging ProfileIf this option is checked the user can specify the rate of proppant ramping or thechange in proppant concentration with time. The user will also then be asked toinput a Staging Profile Power Law Coefficient. If this box is not checked MFracwill automatically design a treatment schedule based on the minimum, maximum,and maximum concentration at the tip.

Maintain Net Pressure IncreaseThis option is only available for Frac pack designs and if Multi-Stage Step Down isselected. If this box is checked, a treatment design will be calculated to ensure thatthe fracture net pressure does not decrease during the slow down stages. This isaccomplished by designing the slow down rate to always be greater than the leakoffrate. If this option is checked, the user does not have to input the Minimum StepDown Rate.

Fluid TypeWhen clicking or using the TAB key to access a Fluid Type data box, the FluidDatabase pop-up screen is presented allowing the selection of the stage fluid type.Select the desired fluid from the list by clicking on it and the fluid code will auto-matically be entered in the Treatment Schedule. To view the fluid properties for theselected fluid type, click on the Fluid DB button. The specified fluid will be high-lighted in the Fluid Database pop-up screen. Next, press the Edit button to view thefluid properties.

Proppant TypeWhen a Proppant Type field is entered, either by clicking on it or using the TABkey, the Proppant Database pop-up screen is displayed. Choose the desired prop-pant type from the list. The proppant code will automatically be entered in theTreatment Schedule. To view the properties that correspond to this proppant code,click on the Proppant DB button. Next, press the Edit button to view the proppantproperties.

Fracture LengthThis input field is only available if the Input Parameter selected is Maximum Frac-ture Length. For an auto design based on fracture length (fracture half-length), thevalue entered will be used to automatically develop a pumping schedule. MFracautomatically determines the required pad and stage volumes to create the input



length based on the other parameters entered in the treatment schedule screen.Proppant scheduling is optimized based on the proppant type, concentrations andProppant Transport Methodology option specified.

Total Slurry VolumeThis input field is only available if the selected Input Parameter is Total Slurry Vol-ume. Enter the total slurry to be pumped for the simulation. Based on this volumeand other criteria specified in the Treatment Schedule screen, MFrac will automati-cally design the pad volume and proppant staging.

Total proppant MassThis input field is only available if the selected Input Parameter is Total ProppantMass. Enter the total proppant mass to be pumped for the simulation. Based on thismass and other criteria specified in the Treatment Schedule screen, MFrac willautomatically design the pad volume and proppant staging.

Pump RateThis is the slurry rate at which the treatment will be pumped. When performing anautomatic Frac-Pack design, the rate entered is the maximum value that will beused. If the frac pack Rate Schedule option of Single Step Down is selected, MFracwill automatically reduce the rate once the maximum proppant concentration isreached in order to match the leakoff rate and achieve a stabilized fracture pressure.

Minimum Step Down RateThis is the minimum slurry rate at which the treatment will be pumped at the end ofthe job when performing an automatic Multi-Stage Step Down rate Frac-Packdesign. This input parameter is only required if the frac pack Rate Schedule optionof Multi-Stage Step Down Step Down is selected and Maintain net PressureIncrease is not checked., MFrac will automatically reduce the rate once the maxi-mum proppant concentration is reached in order to match the leakoff rate andachieve a stabilized fracture pressure.

Initial and Incremental Prop ConcentrationThis is the initial concentration MFrac will begin automatic scheduling. This valueis also used by MFrac as the step increment for proppant scheduling. In otherwords, if 2 lb/gal is entered, the proppant concentration will begin at 2 lb/gal for thefirst stage and then proceed with subsequent 2 lb/gal stage increments (i.e., 2, 4, 6,8.... lb/gal).

When the Proppant Ramp option is turned On, this value is used as the initial rampconcentration and increment for staging.


2.3 Data Input 137

Final Proppant ConcentrationThis value indicates the final (or last stage) proppant concentration to be pumpedby MFrac. When the Proppant Ramp option is turned On, this value is used as thefinal concentration for the ramp.

Maximum Proppant Concentration (at tip)This value is the maximum volumetric concentration allowed at the fracture tip(e.g., lb/gal) during pumping. It is used as a limit to determine the inlet concentra-tion schedule based on the selection made for the Proppant Transport Methodol-ogy option (i.e., TSO).

All automatic scheduling starts with the Initial Concentration and ends with theFinal Concentration. The Max. Proppant Concentration determines how the pro-gram schedules proppant between these two points.

Target Concentration/Unit AreaThe average target concentration per unit area is required input when the ProppantDistribution Style is selected as Concentration per Unit Area. This target parametermay be used when performing a NPV optimization, conventional, automatic tipscreen-out or frac-pack design. Based on this value and the specified Frac Length(or Volume), Initial and Max. Inlet Concentrations, and Max. BHTP, MFrac willautomatically determine a pumping schedule using either the tip screen-out (TSO)or frac-pack criteria selected in the Proppant Transport Methodology option.MFrac always designs to the specified length; and then, if possible, to the averageconcentration per unit area.

The value required by the program is the average mass per unit area of fracture face(e.g., lb/ft2) achieved at closure.

Target Dimensionless ConductivityThe Target Dimensionless Conductivity may be input when the Proppant Distribu-tion Style is selected as Dimensionless Conductivity. This input parameter may beused when performing a net present value, conventional, automatic tip screen-outor frac-pack design. Based on this value and the specified Frac Length (or Volume),Initial and Max. Inlet Concentrations, and Max. BHTP, MFrac will automaticallydetermine a pumping schedule based on the selected Proppant Transport Methodol-ogy option. MFrac always designs to the specified length for NPV, Auto Design,TSO and Frac Pack (or volume for conventional); and then, if possible, to the targetdimensionless fracture conductivity in the pay zone as calculated from



where is the average conductivity in the pay zone, is the formation perme-

ability, and is the propped fracture length in the pay.

The value required by the program is the target value to be achieved at closure.

Target Fracture ConductivityThe Target Fracture Conductivity may be input when the Proppant DistributionStyle is selected as Fracture Conductivity. This input parameter may be used whenperforming a net present value, conventional, automatic tip screen-out or frac-packdesign. Based on this value and the specified Frac Length (Volume, or Mass), Ini-tial and Max. Inlet Concentrations, and Max. BHTP, MFrac will automaticallydetermine a pumping schedule based on the selected Proppant Transport Methodol-ogy option. The target fracture conductivity in the pay zone is calculated from

where is the average conductivity in the pay zone, is the variable fracture

conductivity with position in the fracture, and is the propped fracture length inthe pay.

The value required by the program is the target value to be achieved at closure.

Staging Profile Power Law CoefficientThe staging proppant profile power law coefficient is only required input if theOverride Internal Concentration Staging Profile option box is checked. Theproppant concentration as a function of time or volume (for a constant injectionrate) will then be increased (ramped) according to the following formula:

where

= profile power law coefficient= proppant concentration at time

CfDkfwfkLp---------=

kfwf k

Lp

kfwf kfwf x( ) xd0Lp∫ Lp⁄=

kfwf kfwf

Lp

c t( ) c t0( )–c tp( ) c t0( )–------------------------------

t t0–tp t0–--------------⎝ ⎠

⎛ ⎞α

=

αc t( ) t


2.3 Data Input 139

The proppant ramp will be linear for , concave upward for , and con-cave downward for . Typical values range from 0.5 to 2.0.

Maximum BHTPThe maximum allowable bottomhole treating pressure is required as a constraintwhen performing automatic tip screen-out or frac-pack designs. Based on this valueand the specified Fracture Length, Initial and Max. Inlet Concentrations, AverageConcentration/Area and target Average Concentration/unit area or target dimen-sionless conductivity, MFrac will automatically determine a pumping scheduleusing the criteria selected in the Proppant Transport Methodology options. Themaximum BHTP is used as a pressure limit to prevent excessive ballooning. If asimulation reaches this constraint before achieving the specified fracture length orproppant concentration, a message will be displayed indicating that the maximumallowable pressure has been reached. Selecting the message box Cancel button willterminate the simulation. Selecting the OK button will allow you to enter a newpressure limit.

Input - General Treatment ScheduleIf Input is selected for the Treatment Design Option, a more intuitive TreatmentSchedule is presented. Two tabs are listed under Treatment Schedule, the Generaltab and the Stages tab.

General TabThe General tab for the non-foam Treatment Schedules is shown in Figure 2.35.

= initial proppant concentration at time = final proppant concentration at time = time= time of initial concentration or pad= time of final concentration or end of pumping

c t0( ) t0

c tp( ) tp

tt0

tp

α 1= α 1>

α 1<



Figure 2.35: Input Treatment Schedule – General Tab.

The General tab contains dialog boxes for Schedule Type and Wellbore.

Schedule Type

In the Schedule Type dialog box select Surface or Bottomhole to specify whetherthe data entered (e.g., volumes, rates, etc.) represent surface or bottomhole condi-tions. If pumping from Surface, specify a Wellbore Fluid Type. If pumping fromBottomhole, specify the Flush Fluid Type.

Select Stage Friction Multipliers to enter friction multipliers for each stage.

Wellbore

The wellbore dialog box is related to the initial condition of the wellbore.

Along with the Wellbore Volume displayed from the wellbore hydraulics screen,you can specify a Recirculation Volume. You can also specify whether the well isfilled or partially filled prior to injection. To indicate a partially filled wellbore,enter a fraction (0-1) in the Fraction of Well Filled box. A value of one (1) indi-cates the wellbore is 100% filled. A value of 0.5 means that the well is 50% filled.

An initial portion of the pumping schedule can be recirculated by entering a slurryvolume in the Recirculation Volume box. This is useful for setting stages, such as


2.3 Data Input 141

in Frac-Packs. All stages with a total slurry volume less than the Recirculation Vol-ume will be recirculated. If necessary, a fraction of a stage may be recirculated.

If the stage friction multipliers box is selected the Wellbore Fluid Friction Multi-plier can be specified.

When Input is selected for the Treatment Schedule Option, a user specified Treat-ment Schedule screen is presented. The resulting screen allows specification of thesize and pumping parameters for each stage. This variety of treatment scheduleuses a spreadsheet table for the data input. A toolbar is provided with each spread-sheet screen to control functions such as cut, paste, copy, insert, delete and filldown (see Chapter 1 Working with Spreadsheets). For reference, the Wellbore Vol-ume from the wellbore hydraulics screen is displayed.

Stage TabThe Stage tab for the Input Proppant Treatment Schedule is shown in Figure 2.36.This type of treatment schedule uses a spreadsheet type of interface as shown. Usethe toolbar located at the top of the screen to control functions such as cut, paste,copy, insert, delete and fill down (see Chapter 1 Working with Spreadsheets). Whenpumping from Bottomhole, it is necessary to specify the Flush Fluid Type. This isselected in the same way as the Wellbore Fluid Type as described above. For refer-ence, the Wellbore Fluid Type (surface) or Flush Fluid Type (bottomhole) and Well-bore Volume are displayed.

Figure 2.36: Treatment Schedule – Stages Tab.



To aid in defining the Treatment Schedule, the last column of the table (the VariableColumn) displays a variety of parameters. Use the Variable Column list box tochoose the desired parameter. Depending on the options, the list will contain somesubset of Total Time, Total Liquid Volume, Total Slurry Volume, Total Mass, StageMass, Stage Slurry Volume and Stage Liquid Volume. All of these parameters, withthe exception of the mass parameters, may be modified in the last column. Whenmodifying a total parameter (such as Total Time), changes will be made in the cur-rent stage and the next stage, such that the total parameter for the next stageremains constant. For example, consider the case where Stage 1 has a stage time of10 and Stage 2 has a stage time of 20. Changing the Total Time of Stage 1 to 15 willcause both Stage 1 and Stage 2 to have stage times of 15.

The treatment schedule table has a few variations that depend on the data options. Ifthe Proppant Ramp option is enabled, additional columns for specifying the rampincrement (i.e., From, To) are included. Likewise, when the Proppant SettlingModel is User Specified, a column is added to enter the Proppant Settling Rate.

There are other features of the Input Treatment Schedule screen worthy of note.After entering any two of the first three columns, MFrac will automatically calcu-late the third. It is possible to either enter the rate and volume to calculate the time,or enter the rate and time to calculate the volume. In short, MFrac will always guar-antee that the rate, time and volume are synchronized.

The following section has a complete list of input parameters and their definitions.

Slurry Rate

This is the simulated constant rate at which the slurry will be pumped for a specificstage or volume specified. Entering a zero rate and volume is equivalent to specify-ing a shut-in period. For flowback, a negative rate must be specified. The flowbackoption should be On to enter a negative slurry rate.

Stage Liquid Volume

This is the liquid volume of the stage. In design mode, this is the second column ofthe treatment schedule table. Since proppant concentration is entered as the mass ofproppant per unit volume of liquid, the calculated pump times displayed include thevolume of the proppant, as well as the liquid (i.e., based on a given slurry rate).

Stage Slurry Volume

This is the slurry volume of the stage. In real-time/replay mode, this is the secondcolumn of the treatment schedule table.


2.3 Data Input 143

Stage Time

This is the time required to inject the stage slurry volume at the stage slurry rate.When a slurry rate and stage volume are input, the time is automatically calculated.Entering or editing the time for a stage will result in an adjustment to the stage vol-ume.

Stage Type

Figure 2.37 shows the Stage Type in the Treatment Schedule. The stage type optionwas added to clearly identify the type of stage in the treatment schedule. The fol-lowing stage types can be entered in the Treatment Schedule: None (blank), Pre-pad, Pad, Slug (proppant slug), Prop (proppant), Acid, Flush and Shut-in.

Figure 2.37: Stage Type Identifiers.

Clicking on the Stage Type box will display a pull down box with the choicesshown in Figure 2.37. Not only are the stage types cosmetically pleasing, they arefunctional. The stage type is now an identifier as to whether a given stage canscreen-out or bridge-out.



The Stage Type identifier only allows a stage type of Blank (none), Prop, Flush orFlowback to screen- or bridge-out. The only exception is if the proppant type is0000. Then the proppant will not screen-out, independent of the stage type.

Fluid Type

When clicking or using the TAB key to access a Fluid Type data box, the FluidDatabase pop-up screen is presented allowing the selection of the stage fluid type.Select the desired fluid from the list by clicking on it and the fluid code will auto-matically be entered in the Treatment Schedule. To view the fluid properties for agiven fluid type either select the corresponding spreadsheet row or wellbore fluidtype and then click on the Fluid DB button. The specified fluid will be highlightedin the Fluid Database pop-up screen. Next, press the Edit button to view the fluidproperties.

Proppant Type

When a Proppant Type field is entered, either by clicking on it or using the TABkey, the Proppant Database pop-up screen is displayed. Choose the desired prop-pant for the corresponding stage from the list. The proppant code will automaticallybe entered in the Treatment Schedule. To view the properties that correspond to aproppant code in a given row, click on the Proppant DB button. The specified prop-pant type will be highlighted in the Proppant Database pop-up screen. Next, pressthe Edit button to view the proppant properties.

Only Stage Types of Blank (none), Prop, Flush and Flowback are allowed toscreen- or bridge-out in the fracture. All other Stage Types (Pre-pad, Pad,Slug, Acid, and Shut-in) will prevent the proppant from concentrating andbridging in the fracture. The bridge-out and screen-out criteria are also disabledfor proppant type 0000. In real-time if the proppant type is set to Prop for allstages the model could screen-out due to slight meter reading fluctuations in theproppant concentration during the pad or when pumping a proppant slug. If theProppant Transport Methodology is set to Conventional, the proppant canscreen- and bridge-out but it will not stop the fracture from propagating. There-fore, to achieve a TSO the proppant Stage Type must be of type Blank (none),Prop, Flush or Flowback and the Proppant Transport Methodology must be setto an option that links the fracture and proppant solutions (i.e., choose Conven-tional (link proppant), TSO, or Frac Pack. Conventional does not link the solu-tions).

The bridge-out and screen-out criteria are disabled for Proppant Type 0000independent of the Proppant Transport Methodology option.


2.3 Data Input 145

Prop. Concentration

When the Proppant Ramp Option is Off, the value entered is a constant for the cor-responding stage. It represents the inlet concentration of proppant per unit volumeof liquid to be injected (mass/volume liquid i.e., lbm/gal liquid). When the Prop-pant Ramp Option is turned On, the inlet concentration is assumed to increase ordecrease as a linear ramp in liquid volume between the From and To values. Thisresults in a uniform increase (or decrease) in proppant concentration with increas-ing liquid volume.

Proppant Damage Factor

The reported final permeability of the proppant in the fracture is calculated from:

where

The final permeability is used to determine the fracture conductivity and dimen-sionless fracture conductivity.

Proppant Settling Rate

When the User Specified option is chosen for the Proppant Settling Model, theProppant Settling Rate must be entered in the Treatment Schedule. The valueentered will be used as a constant for the settling velocity of the associated proppantstage during the simulation.

Friction Loss Multiplier

To activate the Friction Loss Multiplier column feature, the Stage Friction Multi-plier must be checked in the General tab of the Treatment Schedule dialog box. Ifstage friction multiplier is checked, friction loss multipliers can be specified foreach stage in the Treatment Schedule. This is useful for history matching surfacetreating pressures.

The following explains how the Friction Loss Multiplier is implemented. If thecalculated pipe friction for a stage is 1000 psi and a Treatment Schedule FrictionLoss Multiplier value of 0.85 is entered, the resulting pipe friction would be 850psi. This multiplier is in addition to the Friction Loss Multiplier in the WellboreHydraulics Screen. Therefore, if the base pressure loss was 1000 psi, and the Fric-

= final (damaged) fracture permeability= proppant permeability (undamaged from database)= proppant damage factor

kf k 1 DF–( )=

kf

kDF



tion Loss Multiplier in the Treatment Schedule was 0.85 with a Wellbore Hydrau-lics multiplier of 0.5, the total pressure loss would be 425 psi (1000*0.85*0.5).

Database AccessThe Fluid, Proppant, and Acid Databases can be accessed from the TreatmentSchedule. Figure 2.38 shows accessing the Fluid Database. Once the Fluid Data-base screen is activated you can add, delete, copy, edit and plot data as if you hadselected the Database menu from the Main Menu tool bar.

Figure 2.38: Database Access from the Treatment Schedule.

Acid Frac Treatment ScheduleWhen using the Acid Fracturing option, the treatment schedule replaces the prop-pant scheduling parameters with the required acid fracturing parameters (Figure2.39). The acid parameters and a brief description of each are given below. Thegeneral treatment parameters of slurry rate, stage volume, etc. are discussed in the“Treatment Schedule Input Parameters” section.


2.3 Data Input 147

Figure 2.39: Acid Treatment Schedule Screen.

Rock/Acid SystemThis five (5) character code identifier specifies which data to use from the AcidFrac Database. The Rock/Acid System database list automatically appears whenthe cursor is placed in this column. Click on the desired rock/acid system from thelist and it will automatically be placed in the column. Moving to any other columncloses the Rock/Acid System list box.

Acid Conc. at InletThis is the acid concentration to be pumped. This value has the units of mass acid/mass liquid (i.e., lbm acid / lbm liquid, or kg acid / kg liquid).

The mass of acid in each stage is calculated and included in the output using the fol-lowing relationship:

where

= mass of acid= inlet acid concentration= stage volume

Ma CiVρ=

Ma

Ci

V



Acid Conc. at EquilibriumThis is the acid concentration below which no reaction with the formation willoccur. This value has the units of mass acid/mass liquid (i.e., lbm acid / lbm liquid,or kg acid / kg liquid).

Diffusivity MultiplierThe diffusivity of an acid stage used in a simulation is calculated from the follow-ing:

where

Since the characteristic acid diffusivity is rarely understood, the diffusivity multi-plier is provided as a means of quickly exploring the sensitivity of diffusivity with-out changing the diffusivity permanently in the Rock/Acid database.

Real-Time/Replay Treatment ScheduleWhen using the Replay/Real-Time option, the treatment schedule functions differslightly from the design table (Figure 2.40). The second column of the table is nowthe Stage Slurry Volume instead of the Stage Liquid Volume. If the Real-Timeoption MView Concentration is selected, the proppant concentration columns inthe treatment schedule will not be editable, since the concentration will be takenfrom the real-time data. If the Input Concentration option is selected, then the con-centration values must be entered in the table. In this case, the Liquid Volumes maybe accessed from the Variable Column.

The most important difference between design and replay/real-time is that the vol-ume and time values are synchronized with the real-time data. This is done auto-matically each time a value is changed. For example, when a stage slurry volume ischanged, MFrac will calculate the total slurry volume up to the end of that stage.Then it will look through the real-time data and find the total time that correspondsto that total volume. The stage time is then calculated from this total time.

= fluid density

= diffusivity used in the fracture= base diffusivity as given in the database= diffusivity multiplier

ρ

D D0 DM×=

DD0

DM


2.3 Data Input 149

The average slurry rates, proppant concentrations and proppant mass for each stageare calculated from the real-time data. These columns are grayed out, since theycannot be changed. That is for a specified slurry volume these values are fixedbased on the real-time data.

Figure 2.40: Real-Time & Replay Treatment Schedule Screen.

When a stage time is changed, MFrac uses a similar process to calculate a newstage volume. This process insures that the time and volume values are properlysynchronized with the real-time data. If a time or volume value is entered thatcauses a stage to go beyond the real-time data, then the rate column is enabled. Inthis case, the rate, volume and times are synchronized as they are in design mode.To better understand the process of synchronizing with the real-time data, explorethe Graphical Treatment Schedule as described below.

Graphical Treatment SchedulingTo see a graphical picture of the treatment schedule, click on the Graphical TS but-ton. This will display an interactive plot (Figure 2.41) of Total Slurry Volume vs.Total Time.

When doing a real-time job, as more data becomes available, the time and ratecolumns may change in order to maintain the proper synchronization. In gen-eral, the slurry volume always takes precedence over time in determining stag-ing.



This plot allows graphical manipulation of stages. The different stages are repre-sented along the Time and Volume axes by boxes of alternating colors. The alter-nating colors of the Time scale extend through the plot area. Horizontal lines extendfrom the boxes of the Volume scale to aid in visualizing the stage volumes. Whenusing the Replay/Real-time option, the data in the plot is the real-time data fromMView. The real-time rate is integrated to get the volume curve.

The right axis drop down selection box can now display any parameter sent fromMView (i.e., Rate, Surface Pressure, BHTP, Concentrations etc.).

Figure 2.41: Graphical Treatment Schedule Screen.

Another Graphical Treatment Schedule feature is graphical placement of the restarttime (i.e., the start of the first stage can now be moved like any of the other stages).


2.3 Data Input 151

Graphical Treatment MenuThe graphical treatment schedule has its own menu bar at the top of the screen.Many of the menu commands are standard Meyer menu commands described inChapter 1. The others are described below:

The File Menu

End Editing without making changes

This will end the editing session without keeping any changes made during the edit-ing session.

End Editing

This will end the editing session and keep all changes made.

The Edit Menu

Add Stage

This will add a new stage after the currently selected stage. If the currently selectedstage is last stage, the new stage will have exactly the same properties as the currentstage. If the current stage is not the last stage, it will cut the current stage’s volumein half. The new stage will have the same properties as the current stage, except itwill have one half of the slurry volume. In this case, adding a stage is essentiallysplitting the current stage into two equal stages. A dialog box will allow modifica-tion of the new stage.

Pressing the Insert key will also activate this command.

Delete Stage

This will delete the currently selected stage. The stage numbers of all the subse-quent stages will then decrease by one.

Pressing the Delete key will also activate this command.

Modify Stage

This will allow manual modification of the currently selected stage. A dialog boxthat resembles the treatment schedule table will appear; however, it will only haveone row representing the selected stage. This box functions exactly the same as thenormal treatment schedule with a few exceptions. Changing total values in the Vari-able Column has no effect on other stages. Any change in stage volume may besubtracted from the previous or next stage as selected with the Previous, Next or



Neither buttons. When Neither (the default) is selected, none of the other stage vol-umes will be affected.

Double-clicking the mouse on a stage while in Select mode will also bring up themodify stage box.

Undo all changes

This will undo all changes made during this editing session.

The ToolbarAt the top of the plot is a toolbar. In the top left of the toolbar, the current mouselocation on the X, YL (left), and YR (right) scales is displayed. The End Edit buttoncan be used to end the current editing session. The mouse can be set to Selectstages or Zoom in with the Mouse list box. The Right Axis list box is used tochange which data is plotted on the right axis. The choices are Rate & Concentra-tion, Rate, Concentration or None. The bottom portion of the toolbar displays infor-mation about the currently selected stage.

Modifying Stages GraphicallyThe real power of the graphical treatment plot is the ability to manipulate the stagevolumes and times with the mouse. It is important to understand that there are twodistinct mouse modes, Select and Zoom. The mode can be toggled with the Mouselist box or the right mouse button menu.

When in Zoom mode, the mouse functions as in any other Meyer plot, the leftmouse button can be used to define an area of the plot for zooming.

The Select mode is used for selecting and modifying the active stage. Clicking themouse on any stage will make it the active stage. To graphically change a stage,click and drag on the desired boundary (either time or volume). The stage informa-tion in the toolbar will be updated automatically while dragging the stage boundary.Also note that a boundary cannot be moved past the boundaries on either side of it.The stage before and after a boundary will be modified if the boundary is changed.To keep the stage time (or volume) after the boundary constant, hold down the Con-trol key when dragging the boundary. Then, only the stage before the modifiedboundary will change.

When working with a real-time treatment schedule, the times and volumes willautomatically be synchronized to the real-time data. If a stage goes beyond the endof the current real-time data, its box on the X axis will extend to the right edge ofthe plot. All subsequent stages will not have a box on the X scale; however, allstages will be represented on the Y axis. As more real-time data becomes available,


2.3 Data Input 153

those stages which were previously beyond the end of the data may be in the rangeof available data. In this case, the stage will get a new stage time from MFrac.

When flowback is enabled, all stages after the first stage with a negative rate stagewill not be represented on the Y axis. The volume of these stages may not bemanipulated graphically; however, the time of these stages may.

Foam ScheduleThe Foam Schedule is enabled when the Foam option in the General Option dialogbox is checked. This option is used in simulating foam treatments of nitrogen and/or carbon dioxide.

General TabFigure 2.42 shows the Foam Schedule General Tab selection screen. The Generaltab includes the same Schedule Type and Wellbore input data as the conventionaltreatment schedule. The additional features for foam are Foam Input, Quality, CO2Flow Meter, and Bottomhole Pressure and Temperature.

Figure 2.42: Schedule - General Tab.



Foam Input

The foam schedule permits entering of either bottomhole foam qualities or surfacerates. Use the radio buttons in the middle of the dialog box to select which one toinput; the other will then be calculated.

Quality

The foam schedule allows for the design of Mitchell, Both External Phase, InternalPhase (CO2), and Internal Phase (N2) foam quality treatments. These qualities areat bottomhole conditions as defined below:

Mitchell

and

Both External Phase

and l

Internal Phase CO2

and l

Internal Phase N2

and l

The foam and total volumes are defined as:

where

= foam quality (void fraction)= carbon dioxide volume= foam volume= liquid volume= nitrogen volume

αN2VN2

Vf⁄= αCO2VCO2

Vf⁄=

αN2VN2

Vt⁄= αCO2VCO2

Vt⁄=

αN2VN2

Vt⁄= αCO2VCO2

Vp+( ) Vt⁄=

αN2VN2

Vp+( ) Vt⁄= αCO2VCO2

Vt⁄=

Vf VN2VCO2

Vl+ +=

Vt Vf Vp+=

αVCO2

Vf

Vl

VN2


2.3 Data Input 155

CO2 Flow Meter and Bottomhole

The foam schedule permits entering of either bottomhole foam qualities or surfacevalues. To perform these calculations reference bottomhole pressure and tempera-ture are needed, enter these in the Bottomhole section. If using a carbon dioxidetreatment, a reference pressure and temperature where the flow is measured are alsorequired, enter these in the CO2 Flow Meter section. Densities are calculated fromthe pressure and temperature pairs using van der Waals equation.

Stages TabFigure 2.43 shows the Foam Treatment Schedule based on the General Tab infor-mation specified.

Figure 2.43: Foam Treatment Schedule - Stage Tab.

This foam schedule is very flexible. If bottomhole rates and concentrations areinput the surface values will be calculated and vise versa. All of the pertinent stagedata is now presented in one spreadsheet making it easier to manage the data.

The foam schedule can also be graphically edited as shown in Figure 2.44. Thismay make it easier to manipulate and visualize the slurry volume and rates in eachof the foam stages.

The Variable Column now has menu selections to display the N2, CO2, Foam,Slurry, Liquid, and Total stage volumes.

= proppant volume= total volume

Vp

Vt



Figure 2.44: Foam Schedule - Graphical Editing.

Rock PropertiesThe Rock Properties dialog box provides a table for entering the mechanical prop-erties of the reservoir and adjacent lithologies including in-situ stresses as a func-tion of depth (see Figure 2.45). For each layer an optional Lithology Symbol, andZone Name can be specified to help organize the rock properties table. The TVDdepth at the bottom of the zone (Depth at Bottom) is the next entry. By convention,this is the true vertical depth (TVD) at the bottom of each zone or layer. The MD atBottom is then calculated from the TVD or if the MD is specified the TVD depthwill be calculated.

Once a zone is defined, after MD at Bottom, the representative, Stress Gradient,Stress, Young's Modulus, Poisson's Ratio, Fracture Toughness, and CriticalStress are entered. Either the stress gradient or stress can be entered. If you enterone the other will be calculated. The stress gradient is defined as the stress dividedby the TVD. The Interpolate Stress Gradient column allows for an interpolatedstress gradient over the given layer (explained below).

Use the Help button located at the bottom of the Rock Properties screen to obtain adescription of these parameters or refer to the description of each property given atthe end of this section.


2.3 Data Input 157

Options to insert rock properties from our rock properties database (Insert fromDatabase) and to import mechanical rock properties (Import Log) data are alsoavailable. The file format for the Import Log can be LAS or text.

Figure 2.45: Rock Properties Dialog Box.

Depending upon the Fracture Geometry model selected and the data available, thenumber of layers or entries to the Rock Properties table may vary. Typically, as aminimum, at least three layers should be entered to describe the pay zone and thelayers above and below. Even for the two-dimensional models, the modulus in theadjacent layers will affect the formation stiffness. However, MFrac will run withjust one layer using a constant value across the entire height. This approach is moreappropriate for the two-dimensional models. A maximum of one thousand (1000)layers can be specified.

Rows in the table may be deleted using the Delete Rows icon and new rowsinserted using the Insert Rows icon.

Rock Property DataA description of the Rock Property screen parameters is as follows:

Lithology SymbolsLithology symbols can be added to each zone or layer by double clicking in the boxto the left of the zone name and selecting the appropriate symbol. This is illustratedby the highlighted box in Figure 2.45. Figure 2.46 shows a list of the available



lithology symbols. Options are also available for the foreground and backgroundcolors.

Figure 2.46: Select Lithology Symbol Screen.

ZoneAn optional zone name can be specified for each layer to help organize the rockproperties table.

TVD at BottomThe TVD depth at the bottom of the zone (TVD at Bottom) is the next entry. This isthe true vertical depth (TVD) at the bottom of each zone or layer. If the TVD isentered, the MD is calculated.

MD at BottomThe MD depth at the bottom of the zone (MD at Bottom) is the next entry. This isthe measured depth (MD) at the bottom of the zone or layer. If the MD is entered,TVD is calculated. If the value for TVD results in a MD greater than the maximumvalue specified for the wellbore deviation, a dashed mark (-) will be placed in thecolumn.


2.3 Data Input 159

Stress GradientThe Stress Gradient is defined as the stress at depth divided by the TVD (StressGradient = Stress/TVD). If the stress gradient is entered, the stress will be calcu-lated. If stress is entered, the stress gradient will be calculated. The stress isassumed to be the minimum horizontal stress for vertically oriented fractures andthe overburden stress for horizontal fractures.

StressIt is generally accepted that one of the most important parameters affecting fracturecontainment is the in-situ stress. Under ideal conditions adequate contrast will existbetween the target interval and surrounding layers. Normally, for ideal containmentto exist, the stress contrast in the adjacent rock layers must be much greater than thefracture net pressure (see Appendix A).

Several methods are regularly used in the petroleum industry to estimate the in-situstresses. These include micro-hydraulic fracturing methods, pump-in flowbacktests, well logging procedures and in some cases laboratory experiments (ASR &DSCA).

Stresses may be input to represent a constant value for a layer or a linear gradientmay be used by selecting the gradient check box in the far right column. Using thisswitch results in a linear increase or decrease in stress magnitude from the bottomof the overlying zone to the bottom of the specified zone.

For more information on the influence of stress on hydraulic fracture propagationsee SPE 15240, “Design Formulae for 2-D and 3-D Vertical Hydraulic Fractures”.4

Young’s ModulusYoung`s modulus or the modulus of elasticity is the slope (or derivative) of a stress-strain curve over the elastic portion of the curve. For linear-elastic deformation,Young’s modulus is a constant with a unique value for a particular material and in-situ conditions. The modulus represents the material’s ability to resist deformationunder load. It is, therefore, a measure of the materials stiffness. As the stiffness (E)of the rock increases, the fracture width will decrease and the length will increasefor a given set of input parameters. See Appendix A for more information regardingthe sensitivity of this parameter.

A range of Young’s modulus values for various rock types is given in Table 2.5.

Poisson’s RatioPoisson's ratio is defined as the ratio of the transverse strain to the axial strainresulting from an applied stress (see Figure 2.47).



Figure 2.47: Definition of Poisson’s Ratio.

The theoretical value for Poisson’s ratio is 1/4 for any isotropic body with strainsbelow the proportional (elastic) limit. For strains beyond the proportional limit, theratio increases and approaches the limiting plastic value of 1/2.

Typical Poisson’s ratios for rock formations are 0.25. From parametric studies,Poisson's ratio affects the fracture propagation characteristics to a very minorextent. Therefore, if in doubt, use 0.25.

Table 2.5: Ranges of Young’s Modulus.

Rock Type Range

(106 psi)

Range

(107 kPa )

Limestone-Reef Breccia 1 - 5 0.7 - 3.5

Limestone-Porous or Oolitic 2 - 7 1.4 - 5

Limestone-Med. to Fine Grained 4 - 11 3 - 7.5

Dolomite 6 - 13 4.2 - 9

Hard Dense Sandstone 4 - 7 2.8 - 5

Medium Hard Sandstone 2 - 4 1.4 - 3

Porous unconsolidated to poorly consoli-dated 0.1 - 2 0.07 - 1.4

Poisson’s ratio

υ = −ε wε l

ε w0

=wΔw ε =l

Δll

0

Poisson’s ratio = − Lateral strain

Longitudinal strain

l0

w0


2.3 Data Input 161

Poisson’s ratio is also used by logging companies to infer in-situ stresses. Thismethod assumes the rock behaves elastically and that the tectonic stresses areknown or insignificant. The typical relationship is

where

Fracture ToughnessThe definition of fracture toughness is obtained from the concept of stress intensityfactor, developed in linear elastic fracture mechanics (LEFM). Fracture toughnessis a measure of a materials resistance to fracture propagation. It is proportional tothe amount of energy that can be absorbed by the material before propagationoccurs. The basis for this relationship involves the assumption that pre-existingdefects exist which induce high stress concentrations in their vicinity. These sitesbecome points for crack initiation and propagation.

If represents the area of the “largest” defect, it can be shown that the tensile

strength, , of the rock can be approximated by

where is the fracture toughness.

In hydraulic fractures, propagation is assumed to occur once the stress intensity fac-tor reaches a critical value. This critical value, related to the propagation resistance(or energy balance) is assumed to be a material property and is given the name frac-ture toughness (or critical stress intensity factor). For a crack in the vicinity of auniform stress field, , the stress intensity is

minimum horizontal stressPoisson’s ratiovertical stress or overburdenpore or reservoir pressurecomponent of stress due to tectonicsBiot’s constant

σHminυ

1 υ–------------⎝ ⎠

⎛ ⎞ σv αp0–( ) αp0 σT+ +=

σHmin =υ =σv =p0 =σT =α =

ac

T

T KIC πac⁄=

KIC

σ

KI σ γHξ=



and for failure to occur we have

where is a geometric coefficient and is the characteristic fracture dimension.See Appendix A for more information on stress intensity factors.

Table 2.6 lists some measured values of fracture toughness. The values shown werereported by van Eekelen13, Thiercelin14 reviewed the testing procedures for deter-mining this parameter in his article, “Fracture Toughness and Hydraulic Fractur-ing”.

Setting fracture toughness to zero will result in the classical hydraulic fracturingpropagation solutions dominated by viscous pressure loss. For very low viscosityfluids, fracture toughness may be the dominate parameter controlling fracturegrowth.

Critical StressThe Critical Stress is the minimum critical stress ( ) for the fracture to propa-gate in the vicinity of a constant stress field. This parameter may also be thought ofas the apparent tensile strength since it is the critical stress that must be over comefor the crack to propagate (in a uniform stress field).

MFrac uses the maximum of or to determine the critical stress intensity

at the fracture leading edge (see Figure 2.48). If is set equal to zero, onlyfracture toughness will be considered. For illustration purposes, the above discus-sion was simplified by using a uniform stress field. See Meyer15 for a more generaldiscussion of the stress intensity factor.

Table 2.6: Fracture Toughness Ranges.

Formation Type psi-in1/2 kPa-m1/2

siltstone 950-1650 1040-1810

sandstone 400-1600 440-1760

limestone 400-950 440-1040

shale 300-1200 330-1320

σc KIC γHξ⁄=

γ Hξ

σc min

σc σc min

σc min


2.3 Data Input 163

Figure 2.48: Critical Stress.

Incorporating this parameter allows for modeling a constant critical pressure at thefracture tip. Using fracture toughness, the critical pressure decreases as the charac-teristic crack size increases.

Interpolate Stress GradientWhen entering stresses, if a linear stress is desired across the zone, click the Inter-polate Stress Gradient box located to the far right of the Rock Properties table.This places a check mark in the box and instructs the program to use the StressGradient across the zone. The stress in a layer at a true vertical depth of D isStress(D)=Stress(TVD)+(Stress Gradient)*(D-TVD). Therefore, the stress at thebottom of a layer will always be greater than the stress at the top of the layer if theinterpolate stress gradient check mark is selected.



If Interpolate Stress Gradient is not specified, the stress value for that zone isassumed constant across that layer.

Insert from DatabaseSelecting Insert from Database will bring up the screen shown in Figure 2.49. TheRock (Lithology) Database is comprised of a Zone Name, Stress Gradient,Young's Modulus, Poisson's Ratio, Fracture Toughness, and Critical Stress. Thisdatabase can be modified by the user.

Figure 2.49: Insert from Lithology Database Screen.

Pressing OK will place the selected lithology properties and icon into the rock prop-erties table.

If the Interpolate Stress Gradient option is checked, the Fracture Fluid Gradientshould also be checked to Include. This will ensure a true total liquid and stressgradient over a given fracture interval for 3D fracture propagation. Note, that ifthe fluid gradient is included in a uniform stress field (i.e. constant stress for alllayers with no stress gradient checks) and assuming all other parameters areunchanged the fracture will tend to propagate downward due to increasing frac-ture pressure with depth. The fracture width profile would also be more tearshaped downward. The inclusion of fluid and stress gradients is more importantas the ratio of the resultant fluid and stress gradient to the fracture net pressureincreases. Thus inclusion of a fluid gradient is only important for low net pres-sure cases when the hydrostatic fluid gradient in the fracture is of the order ofthe fracture net pressure.


2.3 Data Input 165

Log File ImportingThe Import Log option allows the user to import rock properties from a mechanicalproperty log (or logs). To access this import option select the Import Log button inthe Rock Properties dialog.

Upon entering the Log File Importing dialog, there are 5 tabs that control theimport process. The tabs are designed to be edited in sequential order. The basicsteps in the sequence are described in Table 2.7 below.

The functionality of each screen, and the log file importing process in general isdescribed below.

ParametersThe first tab is the parameters tab, it contains a single spreadsheet with a Parame-ters column, and a Unit Type column. It is used to determine what parameters areavailable for import or mapping purposes, and associates each parameter with aunit type. An example parameters setup is shown below (Figure 2.50).

Table 2.7: Log File Importing Basic Steps.

Step Dialog Area

1. Specify Parameters: common parameters are alreadysetup for you, just add any special ones that you will beimporting from a file.

Parameters tab

2. Add Data Sources: specify the log files and associate theircolumns to parameters. Data Sources tab

3. Select the source of the data for each of the rockproperties that will be imported: you can choose a datasource parameter, or to generate the data (or to calculatedata for stress or stress gradient from the other).

Import Properties tab

4. To generate some of the rock properties (or lithologysymbols and zone names) map values of these propertiesto a data source parameter’s values. (Optional)

Property Generation tab

5. Specify Zones: this can be done manual or automaticallyfrom the menu. Also the zone boundaries can be manuallyadjusted by dragging them.

Zones tab

6. Import the properties: click the import button to close thedialog and populate the rock properties table. Zones tab



Figure 2.50: Log File Importing - Parameters Screen.

The parameters spreadsheet is designed to be very flexible, the parameters may bein any order, have any name, and any unit type found in the drop down combo list.The parameter name is editable at any time, and the unit type is editable as long asno data source has been assigned to it yet. If the Unit type is disabled, then it hasalready been associated with a data source; if you wish to modify it, you mustremove the association in the Data Sources tab.

Once data is entered into the parameters spreadsheet, the next step is to assign adata source to some or all of the parameters via the Data Sources tab.

Data SourcesThis tab (Figure 2.51) is used to associate data found in one or more files with someor all of the parameters defined on the previous tab.


2.3 Data Input 167

Figure 2.51: Log File Importing - Data Sources Screen.

By clicking the Add... button, a file open prompt will open requesting the selectionof a data file. The default extensions of the files are *.txt, or *.las. To choose a filewith a different extension, choose All Files (*.*) from the Files of Type drop downlist. Once you open the file, Log data will appear in spreadsheet form as shownbelow (Figure 2.52).

Figure 2.52: Data Sources - Setup Screen.



The purpose of the data sources setup screen is to associate the appropriate data col-umns with a parameter. For example, if column A contains depth information placean “A” next to the parameter that represents measured depth. Clicking on the dropdown menu listed under Unit allows you to define what units the data is in. A sam-ple plot may be generated by defining the sample plot axes and clicking the ViewPlot... button.

Once the columns and units are defined, the sampling criteria can be setup based onmeasured depth, or by row number. If the sampling parameter is set to (Row Num-ber), then a minimum, and maximum row along with a row increment determinewhich rows are read. If the sampling parameter is set to Measured Depth, a mini-mum and maximum depth can be entered to quickly find (and use) the rows corre-lated with those depths. Once this is done, press OK to apply the changes.

To edit existing data sources, select the data source you wish to edit, then click theSetup... button to re-open the data sources setup screen (Figure 2.52).

To delete an existing data file, select the file you wish to delete, then click theDelete button (Note: this does not remove the file from your computer). All of theparameter associations for that file will be removed.

Once the data sources are setup, the next step is the Import Properties tab which isused to specify the data sources that will be used for the current rock propertiesimport.

Import PropertiesThe Import Properties tab (Figure 2.53) contains one spreadsheet. It is where theproperties for the current import are associated with a data source.


2.3 Data Input 169

Figure 2.53: Log File Importing - Import Properties Screen.

Each property may be associated with a previously mapped data source (from afile), set to Calculate for Stress or Stress Gradient but not both, or set to Generatewhich allows manual data entry of that particular property. Once the import proper-ties are setup, the next step is the Property Generation tab.

Property GenerationThe Property Generation tab is optional. Check the “Enable Property Generation”check box in the upper left hand corner to use this tab.

Note: If the Property Generation tab is disabled, any parameter set to Generate inthe Import Properties tab will be set to zero when imported.

The Property Generation tab (shown in Figure 2.54) is used to do the following:

• Specify the lithology symbols and zone names associated with each zone type

• Generate the rock properties that were set to Generate in the Import Proper-ties tab. These properties are generated by a table of zone types that maps zonetypes to a range of mapping parameter values. Property values are mapped todata from a file, such as gamma ray or Pe data.



Figure 2.54: Log File Importing - Property Generation Screen.

Generating Property Values

To generate property values, do the following:

1. Choose the mapping parameter from the drop-down menu. The list containsthe parameters you selected in the Data Sources tab. The parameter’s valuesare plotted in the display area.

2. Enter property values in the table for each zone type. Click on a row in thetable of zone types, then do the following:

a. Click on a cell in the first column to select the zone’s lithology symbol.(This step is optional.)

b. In the next column, enter a zone name. (This step is optional.)

c. In the Mapping Parameter column, enter the upper range of the mappingparameter for this zone type. Each row of the table must have a largervalue than the previous row because this value represents the range of val-ues that are mapped to the zone type.


2.3 Data Input 171

d. The rest of the columns in the table contain values for the generatedparameters that were selected in the Import Properties tab. Enter theproperty’s value for the zone type.

Inserting zone types from the rock database

As an alternative to entering zone types by hand, you can insert them from the rockdatabase. This method specifies values for the lithology symbols, zone name, andthe properties that are to be generated. You still need to specify a mapping parame-ter value.

To insert a zone type from the database:

1. Select the row in the zone type table where you want to insert the new zonetype. Alternatively, you can click on the zone type on the plot.

2. Click the Insert from Database button. The Rock Database window appearsas shown in Figure 2.45.

3. Click on the database entry that you wish to insert into the zone type table.

4. Click Insert or double-click to insert the new zone type into the property gen-eration table.

5. Enter the mapping parameter value for the new zone type.

Adjusting the range of mapping parameter values

When you move the cursor over the range boundaries on the plot, you can graphi-cally adjust the range of the mapping parameter values associated with the zonetypes.

Adding zone types with the plot

To add a zone type by using the plot, do the following:

1. Click the left mouse button on a zone to select it.

2. Choose the Add Entry command from the Edit menu or press the Insert key.The zone type you selected is split into two zone type and a new zone type isadded to the table.

3. Edit the zone type’s properties as described in “Generating Property Values”on page 170.



Deleting zone types with the plot

To delete a zone type by using the plot, do the following:

1. Click the left mouse button on a zone to select it.

2. Choose the Delete Entry command from the Edit menu or press the Deletekey. The zone type you selected is deleted and its row is removed from thetable.

Interpolate generated data

A check box is provided to linearly interpolate generated data between mappingparameter values. The generated value will remain constant over a given mappingparameter range if this box is not checked.

ZonesThe zones tab contains 5 plots depicting measured depth versus Stress Gradient,Stress, Young's Modulus, Poisson's Ration, and Fracture Toughness as shownbelow. If the Property Generation tab is enabled and contains data, the zones willbe initially generated according to that data.


2.3 Data Input 173

Figure 2.55: Log File Importing - Zones Screen.

This is where the zones used for the final import are defined. Zones may be auto-matically selected via the Edit⏐Auto Select Zones menu. The choices for auto-matic selection are described below.

Auto Select Zones

• From Property Generation Tab: This option selects zones based on the map-pings defined in the Property Generation tab.

• By Threshold: This option selects zones based on a threshold of one of thecurrent properties (defined in the Import Properties tab). e.g. If a stressparameter is selected, and a threshold of 100 psi is chosen, a new zone will bedefined every time the stress varies by 100 psi.

• By Depth: This option selects zones based on a depth increment along withminimum and maximum bounds.



Graphically Editing Zones

Zones may also be graphically edited similarly to the process of editing the map-ping parameters in the Property Generation tab.

Import Stress/Stress Gradient Radio Buttons

When data sources for both stress and stress gradient are available, these radio but-ton allow you to choose which one to import (the other will be calculated).

Interpolated Stress Gradient Check Box

When checked, this enables Interpolate Stress Gradient for all zones in the rockproperties table. See “Interpolate Stress Gradient” on page 163.

ImportingOnce you have finished editing the log data, press the Import button. This willimport the mechanical rock properties into the Rock Properties table as shown inFigure 2.56.

Figure 2.56: Rock Properties Table - After Importing from Stress Log.


2.3 Data Input 175

If some of the parameters are not selected in the Stress Log (as is toughness in thiscase) input this data manually using the fill down speed icon. Figure 2.57 shows aplot of imported rock properties layered to specification.

Figure 2.57: Plot of Imported Layered Data.

Fluid Loss DataTo model fluid loss from the fracture into the reservoir and surrounding layers,additional information characterizing the formation and in-situ diffusivity parame-ters is necessary. The format for the fluid loss data entry is flexible and allows any-thing from a single layer reservoir to multi-layered zones with diverse properties.The specific data required by the program depends on which fluid loss model isselected in the General Options dialog.

It is not necessary for these depths to correspond directly to the depths specified inthe Rock Properties screen, although they may. A maximum of 1000 layers is per-mitted in both the Rock Properties and Fluid Loss data screens.

Please refer to Appendix D for a detailed description of the individual leakoff coef-ficients which control fluid loss.



Constant Fluid Loss ModelWhen the Constant Fluid Loss Model is chosen, the total leakoff, , and the SpurtLoss coefficients for each layer are entered in the Fluid Loss Data screen shown inFigure 2.58. Normally, this is the best choice for modeling fluid loss and estimatingfracture efficiency when a minifrac has been performed using the same fluid type asthe main treatment. When this model is used, it is not necessary to calculate thethree individual linear flow resistance mechanisms , , and (see Appen-dix D). The diffusivity parameters of permeability, porosity, compressibility andviscosity are not required for this option because they are inherently included in thetotal coefficient.

Figure 2.58: Fluid Loss Data Dialog Box - Constant Fluid Loss Model.

The specific data required by the program when using a Constant Fluid Loss Coef-ficient Model is as follows:

ZonesAn optional zone name can be specified for each layer to help organize the fluidloss data properties table.

C

CI CII CIII


2.3 Data Input 177

Depth at Bottom

The TVD depth at the bottom of the zone (Depth at Bottom) is the next entry. Byconvention, this is the true vertical depth (TVD) at the bottom of each zone or layer.

Total Leakoff CoefficientThe total leakoff coefficient is a combination of the , , and leakoffmechanisms. These leakoff coefficients are discussed in Appendix D. The total lea-koff coefficient is used in calculating the time dependent leakoff velocity and over-all fluid loss based on mass conservation. The general diffusional leakoff velocity is

where is time and is the initial time of fluid leakoff. The total fluid loss volumeto the formation is

where is a fluid loss parameter and A is the total leakoff area (one face) for bothwings. This equation illustrates that the fluid loss volume is proportional to the lea-koff coefficient and leakoff area product.

Spurt LossSpurt loss is the “instantaneous” volume loss of fluid per unit area of fracture facethat occurs prior to the development of a filter cake. The volume of fluid loss due tospurt for both wings is

where is the spurt loss coefficient and is the leakoff area the pay zone.

For multilayer leakoff, spurt loss is calculated in each layer separately. Please referto Appendix D for additional information.

By convention, the depth entered is the true vertical depth TVD at the bottom ofthe interval. The reservoir parameters are assumed to have constant propertiesover this interval.

CI CII CIII

υ C t τ–⁄=

t τ

Vl 2 v A tdd0

A

∫0

t

∫=

πCA tΦ=

Φ

Vsp

Vsp 2ASp=

Sp A



Harmonic and Dynamic Fluid Loss ModelsWhen either the Harmonic or Dynamic fluid loss models are chosen, the filter cakecoefficient ( ) is input for each layer desired. and are calculated based onthe reservoir parameters input in the Fluid Loss dialog box shown in Figure 2.59.The parameters required for this option are described below. These properties, likethe Rock Properties, are input as a function of the TVD depth. Also like the RockProperties, an optional Zone name is permitted to assist in preparing and organizingthe data.

Figure 2.59: Fluid Loss Data Dialog Box - Harmonic/Dynamic Fluid Loss Model.

For the Harmonic and Dynamic models the total leakoff coefficient for each zoneis calculated internally by combining , , and (see Appendix D). Thisvalue is used to simulate leakoff from the fracture to each associated interval as afunction of differential pressure.

The specific data required for the Dynamic or Harmonic Fluid Loss Models is asfollows:


CIII CI CII

CI CII CIII


2.3 Data Input 179

Depth at BottomThe TVD depth at the bottom of the zone (Depth at Bottom) is the next entry. Byconvention, this is the true vertical depth (TVD) at the bottom of each zone or layer.

Reservoir PressureThe reservoir or pore pressure is used in conjunction with the minimum horizontalstress and fracture pressure to calculate the differential pressure for leakoff. Theleakoff pressure differential is

where

The pressure difference between the minimum horizontal stress and average porepressure is, therefore, a critical component in calculating the and leakoffcoefficients.

For new wells, enter the initial reservoir pore pressure for the productive interval.This value is typically obtained from either a production log or well test. Variationsin pore pressure versus depth can be inferred and entered based on gradient mea-surements and/or the fluid saturation changes within the interval (e.g., gas caps,aquifers, etc.).

When a well has been produced for some period of time, enter the average reservoirpressure as interpreted from a well test. In all cases, the value entered should be lessthan the minimum horizontal stress.

Total CompressibilityThe total reservoir compressibility is defined as the total change in the reservoirvolume per unit volume per unit pressure difference. It is the reciprocal of the un-drained bulk modulus and is typically expressed as follows:

= minimum horizontal stress= pressure in the fracture= pore or reservoir pressure= net fracture pressure, = differential leakoff pressure

Δploss pf p0– Δpf σHmin p0–( )+= =

σHmin

pf

p0

Δpf pf σHmin–Δploss

CI CII

ct Soco Swcw Sgcg cr+ + +=



where

The compressibility is used to relate the permeability and porosity with pressureand time using the expression

leakoff pressure differential is

where

PermeabilityThe reservoir permeability is the formation property that characterizes its ability totransfer a fluid through the pores when subjected to a pressure gradient. FromDarcy's law

where

= gas compressibility= oil compressibility= bulk rock compressibility= total formation compressibility= water compressibility= gas saturation= oil saturation= water saturation

= formation permeability= total formation compressibility= formation porosity= reservoir fluid viscosity= distance= pressure= time

= flow rate per unit area

cg

co

cr

ct

cw

Sg

So

Sw

t∂∂p k

ctφμ-----------⎝ ⎠

⎛ ⎞z2

2

∂

∂ p⎝ ⎠⎜ ⎟⎛ ⎞

=

kct

φμzpt

q kμ---

xddp

⎝ ⎠⎛ ⎞–=

q


2.3 Data Input 181

The permeability/mobility is used to calculate the coefficient in order to modelthe rate of fluid leakoff into the formation during injection. The values enteredshould reflect the effective permeability to the mobile portion of the reservoir fluid.An effective permeability to the frac fluid filtrate is used to derive . The coef-ficient is calculated from the permeability and filtrate viscosity.

PorosityThe equivalent reservoir porosity is the fraction of a rock’s bulk volume that isfilled with mobile hydrocarbons. The porosity is used to calculate the and leakoff coefficients used to simulate fluid loss during injection.

Reservoir ViscosityThe equivalent reservoir viscosity is the total effective viscosity of a multi-phasefluid system at reservoir conditions. This value is used in calculating the lea-koff coefficient for modeling leakoff resistance due to the viscosity and compress-ibility effects of the in-situ fluids.

Filtrate ViscosityThe filtrate viscosity is the effective leakoff viscosity of the fracturing fluid. This isthe fracturing fluid which leaks off through the fracture face. This viscosity hasbeen reduced from its original state due to the deposition of polymer on the fractureface which forms a filter cake. This parameter is used to calculate the coefficientfor modeling viscosity and relative permeability effects caused by fracturing fluidleakoff to the formation.

The effective fluid leakoff viscosity must also account for the relative permeabilityeffect of the leakoff fluid to that of the reservoir fluid. This is especially importantfor a gas reservoir. The effective leakoff viscosity, , in terms of the fluid leakoffviscosity and relative permeability is

where is the true fluid leakoff viscosity and is the relative permeability of theleakoff to the reservoir fluid.

= formation permeability= reservoir fluid viscosity= pressure gradient

kμdp dx⁄

CII

CI CI

CI CII

CII

CI

μe

μe μf kr⁄=

μf kr



Wall Building CoefficientThe wall building or filter cake coefficient is equivalent to the inverse of the frac-turing fluid leakoff resistance. A value of zero (0) represents an infinite filter cakeresistance, whereas, a value approaching infinity (e.g., >100 ft/min½) repre-sents no wall building. This coefficient is used in calculating the total leakoff coef-ficient . It reduces the fluid loss rate by increasing the resistance due to leakoff atthe fracture face.

The wall building coefficient is typically acquired by performing either a static ordynamic laboratory test to determine the relationship between volume loss andtime. The slope of this relationship is proportional to the Wall Building Coefficient(see Figure D.2 in the Meyer Appendices).

Spurt LossSpurt loss is the “instantaneous” volume loss of fluid per unit area of fracture facethat occurs prior to the development of a fracturing fluid filter cake. The volume offluid loss due to spurt for both faces of a single wing fracture is

where is the spurt loss coefficient and is the leakoff area in the pay zone.

For multilayer leakoff, spurt loss is calculated in each layer separately. Refer toAppendix D for additional information.

Time Dependent Fluid LossTo use time dependent fluid loss, open the Time Dependent Fluid Loss tab. Thiswill then display the Time Dependent Fluid Loss screen shown in Figure 2.60. Iftime dependent fluid loss is to be modeled, simply check the ‘Enable Time Depen-dent Fluid Loss’ check box as shown in Figure 2.60.

CIII

C

Vsp

Vsp 2ASp=

Sp A


2.3 Data Input 183

Figure 2.60: Time Dependent Fluid Loss Data Table.

This feature allows you to increase or decrease the leakoff coefficient and spurt lossas a function of time. This is helpful for modeling leakoff in naturally fractured res-ervoirs. While fracturing a naturally fractured formation, the pressure in the frac-ture may approach the critical pressure. When the critical pressure of the formationis reached, natural fractures open and accelerated leakoff occurs. A zero slope onthe Nolte plot may characterize this period of accelerated leakoff.

To use this feature, enter time dependent multipliers for the leakoff and spurt losscoefficients. If the simulation time is less than the minimum value in the table, thefirst multiplier will be used. If the fracture time is greater than the maximum valuein the table, the last data enter will be used.

The multipliers will be interpolated as varying linearly with the time values in thetable.

Pressure Dependent Fluid LossTo use pressure dependent fluid loss, open the Pressure Dependent Fluid Losstab. This will then display the Pressure Dependent Fluid Loss screen shown in Fig-ure 2.61. If pressure dependent fluid loss is to be modeled simply check the ‘EnablePressure Dependent Fluid Loss’ check box and enter the desired fluid loss multipli-ers.s shown in Figure 2.61.



Figure 2.61: Pressure Dependent Fluid Loss Data Table.

This feature allows you to increase or decrease the leakoff coefficient and spurt lossas a function of pressure. This is helpful for modeling leakoff in naturally fracturedreservoirs. While fracturing a naturally fractured formation, the pressure in the frac-ture may approach the critical pressure. When the critical pressure of the formationis reached, natural fractures open and accelerated leakoff occurs. A zero slope onthe Nolte plot may characterize this period of accelerated leakoff.

To use this feature, enter pressure dependent multipliers for the leakoff and spurtloss coefficients. The pressure (pressure in the fracture) is input as increasing func-tion with row number. Normally, for pressure dependent fluid loss, the multiplierwill increase as the pressure in the fracture increases. If the fracture pressure is lessthan the minimum value in the table, the first multiplier will be used. If the fracturepressure is greater than the maximum value in the table, the last data enter will beused.

The multipliers will be interpreted as varying linearly with the pressure values inthe table.

Fluid Type Dependent Fluid LossWhen this option is selected, different total leakoff coefficients and spurt loss foreach fluid can be entered for the Constant Leakoff model. When the Harmonic or


2.3 Data Input 185

Dynamic Leakoff model is chosen the user can input different fluid filtrate viscosi-ties, wall building coefficients, CIII, and spurt for each fluid.

This option is useful when large volumes of 2% KCl or treated fluids are in thewellbore prior to pumping the main fracturing treatment. The option is also helpfulfor modeling leakoff during acid fracturing treatments when alternating pad/acidstages are pumped.

Figure 2.62 illustrates the Fluid Loss Data screen (Harmonic or Dynamic Model)with Fluid Type Dependent Fluid Loss selected in the General Options dialog box.

Figure 2.62: Dynamic Model - Fluid Type Dependent Fluid Loss.

Fluid loss data for up to fifty different fluids can be stored in this table even if thefluid type is deleted from the treatment schedule. Only the fluid types currentlylisted in the Treatment Schedule will be displayed.

Proppant CriteriaThe Proppant Criteria screen is enabled when the Proppant Solution option isturned On (Figure 2.63). When this occurs, the program requires additional infor-mation to characterize the proppant transport process.



Figure 2.63: Proppant Criteria Dialog Box.

The following parameters are required input for the Proppant Criteria Dialog.

Minimum Number of Proppant Layers to Prevent BridgingThis value determines the proppant bridging criteria. It is the minimum number ofproppant layers in the fracture at which bridging occurs. One layer of proppant isdefined as a thickness equal to the average proppant diameter.

The average diameter is the value input in the database record for each proppant.This is typically determined from a sieve analysis according to API standards. InMFrac, a bridge-out is assumed to occur if the average fracture width integrated


2.3 Data Input 187

over the fracture height is less than the Minimum Number of Proppant Layers toPrevent Bridging. In other words, the fracture width must be greater than thebridging criteria in order for the proppant to pass through. Typically, a value of 1.5to 3 is used.

Minimum Concentration/Area for Propped FracThis is the minimum concentration per unit area in the fracture below which thepropped fracture is not included in the reported propped length. In other words,below this concentration after embedment is included, the fracture will not bereported as being “propped”. A typical minimum concentration per unit area rangesfrom 0 to 0.2 lbm/ft2 (1.0 kg/m2).

Embedment Concentration/AreaThis is the proppant embedment concentration per unit area, , in the fracture at

closure ( ) has units of proppant mass per unit area where

is the proppant concentration of embedment and is the

embedded width. The amount of embedment depends upon the proppant and for-mation type. The lost propped width, , due to embedment is given by

where is the proppant porosity and is the proppant

density.

Closure Pressure on ProppantThis is the effective closure pressure on the proppant during production. As the clo-sure pressure (stress) increases, the proppant pack permeability decreases. Theeffective closure pressure on the proppant is equal to the minimum horizontal stressminus the fluid pressure (i.e., bottomhole flowing pressure) in the fracture.

Using interpolation, the Closure Pressure on the Proppant is used to determinethe proppant permeability from the Proppant Database. This value is used to inter-polate the proppant permeability from the Proppant Database.

Non-Darcy EffectsThe equation to describe non-Darcy flow is a form of the Forchheimer [1901] equa-tion

ce

ce cs we⁄=

cs 1 φ–( )ρ= we

we

we 1 φ–( )ρ ce⁄= φ ρ



where is the permeability of the porous media with units of (i.e., md or ft2,

etc.) and is the non-Darcy flow factor or simply factor with units of (e.g.,cm-1, ft-1, atm-s2/gm etc.). Clearly the first term in this equation accounts for vis-cous effects and the second term for inertial or minor loss effects. If the second termon the right hand side is omitted, the equation simplifies to Darcy’s law. Thus non-Darcy flow describes the flow regime that does not obey Darcy’s law. Holdith[1976] reports that the original form of the second term on the right hand side of

Eq. by Forchheimer was which was replaced by Cornell and Katz [1953] bythe product of the fluid density, , and the factor.

The generalized correlation for the beta factor in terms of the fracture permeability and porosity is of the form

where , , and are constants. The effect of immobile water saturation, , canbe incorporated by modifying the porosity to be the effective porosity( ). A number of correlations for the beta factor (inertial coefficient)are provided in the data base.

The Non-Darcy Effects options are given below:

Darcy Only

Non-Darcy effects will not be considered. This is the same as assuming .

Input Beta Coefficient

The non-darcy beta coefficient is user specified and assumed constant. A valuemust be entered in the dialog.

User Database, Beta Coefficient

If this option is selected, a non-darcy beta coefficient correlation is selected fromthe Non-Darcy proppant Database drop down menu. The beta coefficient will thenbe calculated throughout the fracture as a function of proppant permeability andporosity.

xddp– μ

kf----υ β ρυ2( )+=

kf L2

β β L 1–

aυ2

ρ β

kf φ

β akf

bφc-----------=

a b c Sw

φe φ 1 Sw–( )=

β 0=


2.3 Data Input 189

Heat TransferAn analytical heat transfer model is included in MFrac that combines thermal con-vection in the fracture, with transient conduction and convection in the reservoir.These calculations can be linked to an option for simulating the heat exchangebetween the fluid and the wellbore in order to provide a complete thermodynamicmodel for the system. When the Heat Transfer option is used, it can predict theheat-up of the fracturing fluid within the wellbore and/or the exchange of heatbetween the fluid and the reservoir during fracture propagation. The model is usefulin high temperature reservoirs when the fracture fluid rheology is temperaturedependent. It couples the heat transfer, fluid flow and fracture propagation expres-sions to characterize the time-dependent fracture temperature profiles.

When the Heat Transfer option is turned On in the General Options screen, heatexchange calculations can be performed in the wellbore and fracture (Fluid Inlet:Surface), or only in the fracture (Fluid Inlet: Bottomhole). The radio buttonslabeled Surface and Bottomhole shown in Figure 2.64 can be used to make thisselection. When the Surface radio button is selected, the Fluid Inlet Temperaturemust be input. This is the temperature of the fracturing fluid at the surface. The pro-gram will use this value as the initial temperature of the fluid and then simulate theheat-up or cool-down of the fluid within the wellbore and fracture.

Figure 2.64: Heat Transfer Dialog Box.



The surface option uses the wellbore configuration to determine the wellbore heattransfer coefficients needed for the simulation. The wellbore data is taken from theWellbore Hydraulics screen.

To only perform heat transfer calculations in the fracture, choose the Bottomholeradio button located on the Heat Transfer dialog box. This selection will disable thesurface calculations in the simulation and enable a small spreadsheet on the HeatTransfer screen. With this option, bottomhole fluid temperature versus time can beimported from another source or entered manually.

The heat transfer coefficients and parameters used in the calculations are based onthe Heat Transfer dialog box selections. The program contains an internal databaseof values for various reservoir conditions. These conditions are determined basedon the selections for Base Fluid, Reservoir Lithology, In-situ Fluid, Average Poros-ity and Mean Formation Temperature. The properties contained in the internal data-base are as follows:

The Base Fluid category should be input to represent the base fluid of the fractur-ing fluid used. The choices are water, oil, foam and binary foam. The ReservoirLithology represents the primary rock type in the region to be fractured. Inextremely heterogeneous reservoirs or when detailed lithology is not known, selectshale, as it typically represents a good average. The database assigns dry rock val-ues based on the lithology selection and then modifies it depending on the porosityand in-situ fluid type entered. The Average Porosity and In-situ Fluid that occu-pies the pores characterize the fractured zone. The Mean Formation Temperatureis the average bottomhole static temperature for the fractured interval. For mostcases, we recommend entering the static temperature at the middle of the perfora-tions. The general idea for these selections is to characterize the bulk rock that willbe exposed to the fracturing fluid.

Acid DataThe optional acid fracturing module offered in MFrac provides a comprehensivemodel for simultaneously calculating hydraulic fracture geometry, leakoff, heattransfer and acid reaction. The modeling approach uses a numerical implementa-tion that involves control of the acidizing process by mass transfer (i.e., diffusionand convection), as well as, the rate of reaction. All of the other options in MFrac,

Frac Fluid ConductivityFrac Fluid DensityLeakoff Fluid DensityPower Law Nusselt NumberFrac Slurry Heat Capacity

Frac Fluid Heat CapacityRock ConductivityRock Heat CapacityPore Fluid ConductivityPore Fluid Heat Capacity


2.3 Data Input 191

such as fracture geometry models, wellbore hydraulics and production forecastingremain available for acid fracturing, just as they are for fractures using proppants.At the present time, the program does not allow the acid transport to be mixed withproppant transport. All of the screens related to propped fracturing, therefore, aredisabled when the acid frac option is selected.

When Acid is specified for the Treatment Options selection found in the GeneralOptions screen, the Acid Data dialog box shown in Figure 2.65 replaces the Prop-pant Criteria screen. This enables the following data to be entered:

Figure 2.65: Acid Data Dialog Box.

Conductivity Damage FactorThis allows the conductivity of the fracture simulated by the program to be adjustedfor incomplete clean-up (e.g., the deposition of insoluble residue). This may beimportant for accurately predicting the anticipated production from a design. Thefinal conductivity of the etched fracture is calculated from:

where

= final damaged conductivity in the fracture= undamaged fracture conductivity= etched width conductivity damage factor

kfwf finalkfwf 1 DF–( )×=

kfwf final

kfwf

DF



Minimum Conductivity for Etched LengthThe minimum conductivity is the cut off value below which the etched fracture isnot included in the etched fracture length. In other words, below this concentrationthe fracture will not be reported as having conductivity. Typical values range from 0to 100 md-ft (30 md·m).

Acid Fracture Closure StressThis is the effective closure pressure on the etched width during production. As theclosure pressure (stress) increases the etched width decreases. The effective closurestress on the proppant is equal to the minimum horizontal stress minus the fluidpressure in the fracture.

Rock Embedment StrengthThe rock embedment strength is defined as the force required to push a steel ballbearing into a rock up to a distance equal to the radius of the ball divided by theprojected area of the bearing.

This value is used to determine the final fracture conductivity based on the theoret-ical ideal conductivity, closure stress and embedment strength. For more informa-tion see Nierode and Kruk16.

Typical measured embedment strengths on dry carbonate rocks are shown in Table2.8.

Table 2.8: Rock Embedment Strengths.

Rock Type Rock Embedment Strength (psi)

Desert Creek B Limestone 42,000

San Andres Dolomite 50,000 to 175,000

Austin Chalk - Buda Limestone 20,000

Bloomberg Limestone 93,000

Caddo Limestone 38,000

Canyon Limestone 50,000 to 90,000

Capps Limestone 50,000 to 85,000

Cisco Limestone 40,000

Edwards Limestone 53,000


2.3 Data Input 193

Exposure to certain fluids, particularly reactive ones, can have a softening effect onsome rock types. Many values, such as the ones reported above, may be an overestimation of a rocks true embedment resistance.

In-situ Acid TemperatureThis is the temperature of the acid in the fracture. This value is used to modify thediffusivity coefficient. Currently it is used as a constant.

Carbonate Specific GravityThe specific gravity of the carbonate (rock) is used to calculate the mass and vol-ume of the etched rock dissolved. Typical values are 2.2 to 2.8 as given in Table 2.9

Indiana Limestone 45,000

Novi Limestone 106,000

Penn Limestone 48,000

Wolfcamp Limestone 63,000

Clearfork Dolomite 49,000 to 200,000

Greyburg Dolomite 75,000 to 145,000

Rodessa Hill Dolomite 170,000

San Angelo Dolomite 100,000 to 160,000

Table 2.9: Typical Rock Specific Gravities.

Carbonate Type

Average Compressive Strength (psi)

Average Specific Gravity

Average Porosity(%)

Fine grained 11660 2.71 3.4

Med. grained 18480 2.68 4.7

Porous- vugular 19320 2.44 13.9

Chalcedonic 15580 2.60 5.4

Oölitic 14420 2.67 1.6

Reef Breccia 4960 2.35 15.0

Table 2.8: Rock Embedment Strengths.



Fraction of Non-Reactive FinesThis is the fraction of insoluble material in the carbonate rock that will not reactwith the acid. It affects the equivalent etched rock porosity, which is

where

2.4 Run/Performing CalculationsOnce all of the required data relevant to the options selected have been entered, it istime to perform calculations. Up to this point, MFrac has checked the validity of thedata contained in every dialog box opened during the active session; however, sinceyou are not required to view every data screen sequentially prior to performing cal-culations, it is possible that some input parameters may not have been checked. Toavoid problems, when the calculation process is initiated, MFrac checks to ensurethat the minimum data requirements are met and that the data entered is withinacceptable limits. This extra level of error checking is designed to prevent calcula-tion errors due to missing or “bad” data.

To start the simulation, select the Run command from the Run menu. This willbring up the Simulation Data window and the Calculation menu bar. During thesimulation, the Simulation Data window and all open plots will be updated to showthe current state of the simulation.

Calculation Menu BarMost of the Calculation Menu Bar commands are also in the main menu bar and aredescribed elsewhere in this guide; however, the commands specific to the Calcula-tion Menu Bar are described below.

Reef Head 3080 1.79 36.0

Stylolitic 11530 2.73 3.9

equivalent void fractionvoid fraction created by reactionfraction of insoluble fines

Table 2.9: Typical Rock Specific Gravities.

φ e φ 1 φ–( )F+=

φ e =φ =F =


2.4 Run/Performing Calculations 195

Stop MenuThe Stop! menu item stops the current simulation. After stopping, a message boxwill be displayed to save the simulation results up to the current point. PressingAlt+S will also invoke the Stop command.

Simulate ClosureThis command is only available in the Replay/Real-time simulation mode. Select-ing this command will allow MFrac to start simulating closure immediately, ignor-ing all real-time data beyond the current time step. The utility of this option is tosimulate closure after one stops monitoring real-time data. This is especially truefor fractures which have large closure times (high efficiencies).

Simulation Data WindowsFor general information regarding Simulation Data Windows See “Simulation DataWindows” on page 65. Below is Simulation Data Window information specific toMFrac.

The Fracture Characteristics window contains a system of symbols to indicate thezone status during the simulation. The status legend may be toggled on and off byright clicking on the window then clicking the Status Legend menu item. The leg-end appears at the bottom of the window when this option is checked. Table 2.10contains an explanation of these symbols.

Table 2.10: Simulation Data Screen Symbols.

Symbol Meaning Description

Open The fracture is open and propagating.

Screened-out The fracture has screened-out at the tip. It can still takefluid and proppant.

Packed The fracture is packed all the way to the wellbore. It can-not accept fluid or proppant.

Closed The fracture is closed and is not propped. At one time thisfracture was open.

No Frac A fracture was never initiated in this zone.



2.5 Plots - Graphical PresentationMFrac provides a vast selection of plots that can be produced to illustrate the simu-lation results. These plots have all the characteristics of Meyer plots as described inChapter 1. This section describes the plotting facilities that are specific to MFrac.

The MFrac plots are grouped into different categories as described below. Plotsfrom any number of categories may be viewed at the same time. During a simula-tion, the plots will update after each time step, providing an animated view of thefracturing process. When not simulating, the plots contain the results of the lastsimulation that was saved. It is important to note that changing an input parameterdoes not affect a plot until the simulation has been run again. For more informationon running a simulation, see Section 2.4.

Viewing PlotsThe plots that are contained in MFrac are divided into categories that can beaccessed by different commands in the Plot menu. The specific plots that are avail-able will be directly controlled by the options selected in the General, Fracture andProppant Options tab dialog boxes for the last saved simulation. For example, ifHeat Transfer is disabled by “unselecting” it in the General Options screen, theseplots will not be available in the Plot menu.

To Create a Plot:1. Select Plot from the main menu. The Plot menu appears listing the available

plot groups.

2. Choose from the list of groups. The associated group selection dialog appears(see Figure 2.66). Groups that are unavailable due to the options selected willappear dimmed. To obtain these plots you must activate the option and then re-run the simulation.

3. Select the desired plots by clicking the adjacent check boxes. Use the SelectAll button to view all the plots for the group. To disable a plot, click off thecheck box or use the Clear All button.

4. If this is a Multilayer case, click on the Multilayer tab at the top of the screen.Then select the desired layers. More information on Multilayer plots is givenbelow.

5. Once the desired selections have been made, click OK to view the plots.


2.5 Plots - Graphical Presentation 197

Figure 2.66: Plot Selection Dialog Box Example.

Plot CategoriesThe plots in MFrac are grouped into different categories, each of which are accessi-ble with the Plot menu. Each category is summarized below.

Fracture CharacteristicsThese are plots of general fracture characteristics, such as length, height and width.Some of the plots may be plotted versus time or volume. The volume selection but-ton is the total slurry volume pumped.

Leakoff/RheologyThese plots display the leakoff rate and fluid rheology as a function of time.

Wellbore HydraulicsThese plots are used to display the results of the wellbore hydraulics model andreal-time pressure matching. The first section of plots are various pressures versustime. For these plots, there are check boxes for Replay/Real-Time, BH Rate, Sur-



face Rate, BH Concentration and Surface Concentration. Clicking these boxes willshow the corresponding curves on the graph. Clicking the Replay/Real-Time boxshows the corresponding real-time pressure data, if it is available. These can beused for real-time pressure matching. The Real Calc BHP from Surface plot is aplot of the calculated BHP from real-time surface data using the wellbore model.Similarly, the Real Calc Surface from BHP plot is a plot of the calculated surfacepressure from real-time bottomhole data using the wellbore model.

The last two plots correspond to the near wellbore pressures losses and BHP in thewellbore and fracture for each multilayer.

DiagnosticThe BHP Difference Plot is used for real-time pressure matching. This is the differ-ence between the measured and simulated BHP.

A number of net pressure plots versus time in linear and log-log coordinates areprovided to represent the familiar net pressure slope as given in the Nolte plots. TheNolte slope for each fracture is also plotted to give a graphical representation ofheight growth, etc.

The measured net pressure plots for real-time or replay analysis are (of course) notmeasured values but calculated from the BHTP differences and fracture net pres-sure:

where

and

= bottomhole treating pressure at the gauge depth= measured bottomhole treating pressure in frac= fracture pressure= net fracture pressure, = frictional pressure loss in wellbore and near wellbore

Δpm BHTPm σ–=

BHTPm pf pf σ–( )+–=

BHTPm pf Δpf+–=

BHTPm BHTP Δp fric– Δp grav+=

BHTPBHTPm

pf

Δpf Δpf pf σ–=Δp fric



TreatmentThese plots show the fracture treatment. This category is divided into two parts.The first section pertains to the surface and bottomhole treatment. The volume,rate, concentration and mass may be plotted versus time, bottomhole volume or sur-face volume. Bar charts of surface treatment may also be plotted.

The second section pertains to the treatment of the bottomhole and individual frac-ture(s). The volume, rate, concentration and mass are plotted versus time or bot-tomhole volume. The bottomhole stage bar plots show the total bottomholetreatment and how much went into each individual layer.

Proppant TransportA variety of plots are available to show the placement of proppant during and at theend of the simulation.

Acid TransportThe plots contained in this category illustrate the generation of etched fracturelength and conductivity for an acid fracturing stimulation.

Heat TransferThese plots depict the changes in temperature as a function of time and position inthe fracture as predicted for a particular simulation.

Net Present ValueThe conductivity, hydraulic power, liquid volume, slurry volume and proppant con-centration versus fracture length can be plotted.

Input Treatment ScheduleThis is the same plot as the one shown by pressing the Graphical Edit button in thetreatment schedule. It is important to note that this plot always corresponds to thecurrent input data. When doing a real-time case, the plot will update with each real-time time step. To graphically edit the treatment schedule, click on the Edit button.

= gravitational head in the wellbore= measured net fracture pressure= minimum horizontal stress

Δp grav

Δpm

σ



Multilayer PlotsWhen simulating multiple active layers, it is possible to see the data for each activelayer using the Multilayer Selection box. The different layers will be represented onthe plot legend as configured with the Multilayer legend option.

Multilayer SelectionAny combination of the active layers can be selected for a plot. Selecting thedesired layers is accomplished using the Multilayer dialog box, which is accessiblefrom the Plot menu, the plot shortcut menu or by clicking on the Multilayer tab in aplot dialog box. When using the Multilayer tab, the selections will be applied to allplots started with that dialog box. When using the menu options, you can choose toapply the selection to just the active plot or to all open plots.

The selection box has a list of all the available active layers. Toggle the selection ofthe layers by clicking on them. At least one layer must be selected.

For plots that have a Y axis depth scale, there is another option available when thedepth scales of different layers overlap or are significantly far apart. The choicesare summarized below:

Depth scale is not continuous - When two layers are significantly far apart, abreak in the Y axis will be drawn on the plot. This allows for a more detailed viewof each layer. When two layers are overlapping, no break is drawn and the data forone layer may be drawn on top of the other.

Each zone always has its own depth scale - Each zone will have its own depthscale, regardless of any overlapping. This is particularly useful for a horizontalwellbore where all fractures are around the same TVD. The first layer will alwaysbe at the top of the plot.

Continuous Depth Scale - The depth scale will be continuous, regardless of over-lapping layers. If the layers are far apart, the larger scale will force the fracture out-lines to be smaller.

Multilayer LegendsFor plots that do not have a depth axis, the legend contains both the name of thevariable and the name of the layer. There are various options which may be config-

For best results use Each zone always has its own depth scale for horizontalwells and Depth scale is not continuous for vertical wells.



ured in the Legend section of the Plot Configuration box. Consider an example ofa length curve for layer number two (2) named “Upper Frac”. The followingoptions, with examples in parenthesis are available:

Layer Number, Variable Name - This is the layer number, followed by the variablename (i.e., #2 Length).

Layer Name, variable Name - This is the name of the layer followed by the nameof the variable (i.e., Upper Frac Length).

Layer Name only - This is the name of the layer (i.e., Upper Frac). This is usefulfor plots of only one variable, for example, length versus time.

Variable Name Only - This is only the variable name (i.e., Length). This is useful ifonly one of the active layers is selected for plotting.

There is also an option at the top to Hide Legend when there is only one layer.This is used to hide the legend when only one layer is plotted, but show the legendwhen more than one layer is plotted.

Composite PlotsA number of composite plots are available in MFrac. Composite plots refer to theconfiguration of displaying more than one figure in a given plot.

Figure 2.67 illustrates a composite width contour plot that contains the stress, widthand width contours. Figure 2.68 shows a composite plot of the rock propertiestable. Similar plots are available for proppant transport display.



Figure 2.67: Composite Plot - Width Contours.

Figure 2.68: Composite Plot - Rock Properties Data.

Multi-Axes PlotsFollowing are a few examples of Multi-Axes Plots with various options for plotLayout and configuration.



Figure 2.69: Multi-Axes Plot - BHTP, Rate etc.

Figure 2.70: Multi-Axes Plot - Wellbore Friction & Gravity.

Three-Dimensional PlotsTo display a 3D plot select Three-Dimensional from the Plot Menu as shown inFigure 2.71.



Figure 2.71: Three-Dimensional Plot Menu Selection.

Figure 2.72 displays the dialog menu for creating Three-Dimensional Plots. Thesefeatures are mostly related to options for a propped fracture. If acid is selected someof these items will be dimmed. The dialog menu has options for Plot, Shading,Show Options, and Titles. This virtual reality plot is saved in a VRML at the loca-tion shown at the bottom of this menu. The location of the web Browser can also bechanged.

In order to view these 3D plots, a VRML plug-in must be installed with your webbrowser. You can download the free Cortona VRML plug-in at:

http://www.parallelgraphics.com/products/cortona/

After installing the 3D VRML plug-in you must register the 3D WRL file type gen-erated by MFrac/MPwri with the web browser so that the browser can find theplug-in. In the web browser (e.g., Mozilla, Internet Explorer, etc.) go to FolderOptions⏐File Types and Register the WRL Extension with a web browser (e.g.,Internet Explorer). After installing the plug-in and registering the WRL file typerestart your computer.


http://www.parallelgraphics.com/products/cortona/


Figure 2.72: Three-Dimensional Plot Dialog Menu.

To illustrate the functionality of the three-dimensional virtual reality plots a fewexamples are presented below. Figure 2.73 shows a single fracture with the MeyerLogo. Figure 2.74 shows a multilayer 3D plot with lithology and wellbore selected.Figure 2.75 shows the 3D plot from the top view. The flexibility here is to createyour own virtual reality 3D fracture.

These 3D plots can be shared with anyone on the Internet. It is not necessary tohave MFrac to view a 3D plot any VRML plug-in (e.g., Cortona) will work.



Figure 2.73: Virtual Reality 3D Plot - Single Fracture.

Figure 2.74: Virtual Reality 3D Plot - Opening View.


2.6 Generating Reports 207

Figure 2.75: Virtual Reality 3D Plot - Top View.

2.6 Generating ReportsAfter the calculations have been successfully performed, various options are avail-able for viewing the results and creating reports. Working with reports is the samein all Meyer programs as described in Chapter 1. The Report can also be saved asan HTML file.

Figure 2.76 displays the Report Menu for the Multilayer (limited entry) case.



Figure 2.76: Report Menu - Multilayer Case.

Viewing a ReportTo see the results of a simulation select View Report from the Report menu. TheReport Generation dialog shown in Figure 2.77 will then be presented with the FullReport option enabled as a default. The Full Report contains all of the simulationresults corresponding to the options that were selected in the Options dialog boxes.It contains all of the input data, as well as the calculation solutions.

When it is not necessary to view all of the data, you may select a pre-formattedSummary Report, with or without input data. These reports summarize the varioussolution tables presented in the Full Report. In many cases, average values are pre-sented together with fracture geometry and proppant transport information at theend of the treatment simulation. The summary is useful in comparing simulationsand making an overall assessment of a particular design. To acquire specific infor-mation about the development of a simulation, or to diagnose a potential problem ina design refer to the Full Report or choose Selected Sections.



Figure 2.77: Report Generation Dialog Box - MFrac.

Choosing the Selected Sections item enables the previously dimmed list con-tained in the Report Generation dialog. These sections represent the components ofthe Full Report and can be selected individually or together as a group. This capa-bility allows you to view a portion of the complete report without displaying anyundesired information. Once again, some of the sections will remain dimmeddepending on the simulation options that have been used.

If there is more than one active layer, clicking on the Multilayer tab will allow youto choose which layers to include in the report. Select the desired layers from thelist.

Once the selection(s) are made, the report may be viewed by clicking the OK but-ton.

Explanation of the Report OutputFor reports that have input data, the input data will have the same form as the inputscreens. The output data (simulation results) will be presented in a variety of tables,depending on the options. Tables relating to the wellbore will be first. Then for eachselected layer, the layer name and output tables are given. Each of the output tablesis summarized below:



Treatment Schedule PumpedThis table will have the title Surface Treatment Schedule Pumped or BottomholeTreatment Schedule Pumped, depending on the option selected in the treatmentschedule. This table represents what was actually pumped in the simulation. Inmost cases it will exactly match the input table.

Wellbore Hydraulics SolutionThis table will show Delta P Friction, Delta P Gravity, BHP, Surface Pressure andHydraulic Power as a function of time.

Fracture Treatment ScheduleThis table displays what was actually pumped into a layer.

Fracture Propagation SolutionThis table shows various fracture characteristics as a function of time.

Fracture Wellbore Hydraulics SolutionThis table has the wellbore hydraulics output parameters for a given layer as a func-tion of time.

Proppant Transport SolutionThis table shows the propped fracture characteristics as a function of length. Thestage numbers are intermixed with the rows of the table, allowing visualization ofwhere the stages are within the fracture.

Acid Transport SolutionThis table displays the acid concentrations and etched width characteristics as afunction of position.

Acid Schedule CommentsThis table shows where the different acid stages are at the end of the treatment.

Summary of Etched CharacteristicsThis table gives a summary of the etched characteristics.

Fluid Leakoff OutputThis table shows the fluid leakoff rate and rheology as a function of time.


2.7 Program Databases 211

Heat Transfer SolutionThis table shows the heat transfer solution as a function of time. The output showsboth real temperature and dimensionless temperature versus position. The mixedmean fracture temperature as a function of time is also tabulated.

Temperature vs. Position (End of Pumping)Various fracture temperatures versus position and time are tabulated.

Automated NPV DesignThe automated NPV design table lists the calculated propped fracture characteris-tics (e.g., width, conductivity, treatment volumes, etc.) as a function of the createdand propped fracture length.

Proppant Design SummaryThis table shows the propped fracture characteristics for the selected layers at theend of the job and at closure.

Sand Transport Summary TableThis table shows the proppant transport characteristics (properties of each stage) atthe end of the job and at closure.

2.7 Program DatabasesTo simplify database input, MFrac offers several program databases provided bynumerous industry suppliers. While these databases are offered as an integral partof the program, Meyer and the database suppliers make no guarantee or expressedwarranty as to their use or accuracy.

Fluid DatabaseA complete fracturing fluid database and database management system is providedwith MFrac to simplify entry of fluid rheology information. The databases con-tained in MFrac are comprised of System Databases and a User Database (seeFigure 2.78). The System Databases supplied with MFrac cannot be edited orchanged. This is to prevent users from loosing the original data provided to Meyerby various participating service companies.



Figure 2.78: Fluid Database Dialog Box.

To use the data from the System Databases, it must first be copied into the UserDatabase. To do this, select the desired System Database from the list box at the topof the screen. Next, select a fluid from the System Database list and press the Insertbutton located in the center of the dialog box (or double-click on the databasefluid). This action copies the selected fluid data from the System Database to theUser Database. Once a fluid has been inserted, its position relative to the other flu-ids in the list can be adjusted by using the Up or Down arrow buttons.

Any fluid record contained in the User Database can be edited by selecting the fluidname and clicking the Edit button or double-clicking on the fluid. A new blankrecord can be created with the Add button. Selections may be deleted from the UserDatabase by clicking the Delete button. To exit the fluid database dialog box, clickon the Done button.

When either the Add button or the Edit button is used to create or edit a fluid, thescreen shown in Figure 2.79 appears permitting the rheology and friction data to beentered, viewed, modified or plotted. For a new fluid, a blank screen is presentedproviding a template for the entry of data.



Figure 2.79: Fluid Database Edit Screen.

Fluid Code and NameEach database record contains rheological information the program requires forsimulation. The Fluid Code is a unique, seven (7) character name used to identifythe fluid data. Any character may be used for this code; however, by default certaincharacters are automatically used to denote specific service company data (e.g., B,D, H). The Fluid Name is a free format field to describe the fluid composition. TheFluid Code and Fluid Name appear in the pop-up screen presented when the FluidType field is entered in the Treatment Schedule dialog.

Specific GravityThe Specific Gravity of the fluid at bottomhole conditions must also be entered foreach fluid in the space provided. The fluid specific gravity is used in the proppanttransport solution and for 3-D fracture propagation solution (energy equation forheight growth) when the fluid gradient is clicked on.

Shear Rate - Viscosity atThe Viscosity @ Shear Rate is used to calculate the apparent viscosity of the fluidat this shear rate. This apparent viscosity, at the specified shear rate, can be dis-



played by pressing the Viscosity Plot button. The apparent viscosity as afunction of the fluid rheology parameters and shear rate is

where is the consistency index, is the flow behavior index, and is the shearrate. Clearly, as the shear rate increases the apparent viscosity decreases for and in the limit as the shear rate approaches zero the apparent viscosity goes toinfinity. Apparent viscosity is not used in the Meyer software calculations. Theapparent Viscosity Plot should only be used for relative comparisons.

Rheology DataThe Rheology Data section found in the database record is used to specify the and of the fluid system as a function of Time and Temperature. At least two (2)entries must be made for each temperature to define the properties for a range oftime. Use the Previous << and Next >> buttons to switch between temperatures.

Typical and fluid rheology parameters as a function of time and temperatureare shown in Figure 2.80 and Figure 2.81. Figure 2.82 illustrates the apparent vis-cosity at a given shear rate as a function of time and temperature.

Figure 2.80: Flow Behavior Index (n’) Plot.

μapp

μapp k′γn′ 1–=

k′ n′ γ

n′ 1<

n′

k′

n′ k′



Figure 2.81: Consistency Index (k’) Plot.

Figure 2.82: Apparent Viscosity Plot.


Friction DataIn addition to the Rheology Data section, the fluid database also provides a sectioncontaining characteristic friction data for various pipe sizes. This information hasalso been provided by the respective service companies. When available, the pres-sure loss data is entered as a function of the pumping rate and Hydraulic Diameter.This data is only used when the Wellbore Hydraulics Model Option is selected asUser Database.

Hydraulic DiameterThe hydraulic diameter is defined as:

where

For a circular pipe,

and

where

For an annulus,

and

hydraulic diametercross-sectional areawetted perimeter

pipe inside diameter

Dh4AP

-------=

Dh =A =P =

A πD2 4⁄=P πD=

Dh4 πD2 4⁄( )

πD------------------------- D= =

D =

A π D2 d2–( ) 4⁄=P πD πd+=


where

The program assumes that the data entered is specified as a function of hydraulicdiameter. Data specifically generated for an annular configuration should be inputbased on the hydraulic diameter. When simulating annular flow, MFrac calculatesthe fluid velocities and uses the appropriate friction data from the database.

Pressure Loss TableUp to ten friction tables with corresponding hydraulic diameters can be entered.The frictional pressure loss as a function of rate is input into the table. The morerate and pressure loss fields entered the better the interpolation will be. Use the Pre-vious << and Next >> buttons to access the pressure loss values for additionalhydraulic diameters. After the data is entered, it can be plotted for verification usingthe Friction Plot button which is configured in a similar manner to the rheologyplots (see Figure 2.83).

casing inside diametertubing outside diameter

DhD2 d2–D d+

------------------ D d–= =

D =d =



Figure 2.83: Frictional Pressure Loss Plot.

Once the data has been entered, it may be plotted for verification using the variousplot buttons. When plotted, a curve is generated through the data points using acubic spline fit. The spline fitting function may be disabled by clearing the checkbox found at the bottom of the plot dialog.

After creating a plot, the plot attributes can be changed by selecting the Plot Con-figuration button, located on the bottom of the screen. To generate a hard copy,click the Print button located on the tool bar. This plot can be zoomed like otherMeyer plots; however, to zoom out you must click the Zoom out button or pressF5.

Proppant DatabaseTo characterize the many different proppants available in the industry, variousproppant suppliers have provided us with their databases.



To enter the Proppant Database select the Proppant Database command from theDatabase menu. The first screen presented is the Proppant List dialog shown inFigure 2.84. Just like in the Fluid database, a User Database can be built by copyingproppants from the program’s System Databases. The System Databases containdata supplied by most of the major proppant manufacturers and suppliers. Onceproppants have been copied from the System Database, they can be repositioned byusing the Up and Down buttons. You can Edit a record, Delete a record, Copy arecord or Add a new record to the list by choosing the appropriate button. To exitthe proppant database dialog box, click on the Done button.

Figure 2.84: Proppant Database Dialog Box.

Proppant Database ParametersWhen the Edit button is selected, the proppant data screen appears as shown in Fig-ure 2.85. The screen displays a proppant’s database record. The Proppant Code isa unique, seven (7) character identifier, used in the Treatment Schedule to indicatewhich proppant data to use in the simulation. The Description of the proppant isdisplayed with the Proppant Code in the report for all proppants used.



Figure 2.85: Proppant Database Edit Screen.

For each proppant entry in the database, specific information is required. A descrip-tion of these properties is given below.

The Porosity, and Permeabilities at 0.5, 1, and 2 lbm/ft2 must be entered as a func-tion of Closure Pressure. Once entered, this information can be plotted andviewed by selecting the Plot button at the bottom of the dialog box. The ability toconfigure plots for display or printing is handled exactly the same way as it is in theFluid Database.

The following information is required:

Specific GravityThis is the ratio of the proppant density to the density of water. The specific gravityof a proppant is based on the grain density, not the bulk density of the proppant.Typical values are shown in Table 2.11.

The proppant Specific Gravity is used in the calculation of the proppant settlingvelocity, as well as the pipe frictional, gravitational and perforation pressure losses.



Average DiameterThis is a weighted average value as determined from a sieve analysis according toAPI standards. It is used to determine the proppant settling velocity, propped frac-ture width, monolayer criteria and as a reference value for bridge-out determina-tions. “Bridging-out” is assumed to occur if the average fracture width integratedover the fracture height is less than the value entered for the No. of Prop Layers toPrevent Bridging.

PermeabilityThis is the proppant pack permeability at concentrations of 0.5, 1, and 2 lbm/ft2 as afunction of Closure Pressure. The fracture permeability is assumed to vary withconcentration and closure pressure throughout the fracture. The proppant transportmodel elementally tracks the proppant displacement and calculates the effectivepropped width and height as a function of position in the fracture. The variation offracture conductivity with position is then used to calculate an equivalent dimen-sionless fracture conductivity at the start of pseudo-radial flow in the specific payzone.

PorosityThis is defined as the void fraction between sand grains (i.e., liquid volume toslurry ratio of the settled bank). It is used to calculate the final propped fracturewidth. Typical values of porosity for proppants are shown in Table 2.12.

Table 2.11: Specific Gravity of Proppants.

Proppant Type Specific Gravity

Absolute Density

(lbm/ft3)

Absolute Density

(kg/m3)

Resin Coated Sand 2.55 159.2 2550

Sand 2.65 165.4 2650

ISP-Lightweight 2.72 169.8 2720

Intermediate Strength 3.15 196.6 3150

Sintered Bauxite 3.70 231.0 3700

Table 2.12: Porosity of Proppants

Mesh Size Sphericity Porosity (fraction)

6-8 angular 0.36



Non-Darcy Database To enter the Non-Darcy Database select the Non-Darcy Database command fromthe Database menu. The first screen presented is the Non-Darcy List dialog shownin Figure 2.86. Just like in the Fluid database, a User Database can be built by copy-ing non-Darcy beta correlations from the program’s System Database. The SystemDatabase contains correlations for the beta factor as used in the petroleum industry.Once beta correlations have been copied from the System Database, they can berepositioned by using the Up and Down buttons. You can Edit a record, Delete arecord, Copy a record or Add a new record to the list by choosing the appropriatebutton. To exit the non-Darcy database dialog box, click on the Done button. (Note:This Non-Darcy database is also used in MProd.)

Figure 2.86: Non-Darcy Database Dialog Box.

10-20 angular 0.36

10-20 round 0.32

20-40 round 0.35

40-60 round 0.32




Non-Darcy Database ParametersWhen the Edit button is selected, the non-Darcy proppant data screen appears asshown in Figure 2.87. The screen displays a non-Darcy database record. The Refer-ence Code is a unique, seven (7) character identifier, used to indicate which betacorrelation to use in the simulation. The Description of the Non-Darcy Equation isdisplayed with the Reference Code in the report for correlation selected.

Figure 2.87: Non-Darcy Database Edit Screen.


where , , and are the input constants. The coefficient has units consistentwith the permeability power constant, , and the units for permeability. The powercoefficients , and are dimensionless.

kf φ

β akf

bφc-----------=

a b c ab

b c



Acid DatabaseAn acid fracturing database and database management system is provided withMFrac to simplify entry of acid information. The databases contained in MFrac arecomprised of System Databases and a User Database (see Figure 2.88). The Sys-tem Databases supplied with MFrac cannot be edited or changed. This is to preventusers from loosing the original data provided to Meyer by various participating ser-vice companies.

An Acid Fracturing Database is included to describe the rock/acid physical charac-teristics and mass transfer mechanisms. The objective of the Acid Frac Database isto simplify the manner in which specific data is entered in the program to describephysical processes. In this particular case, it is the thermodynamic and mass diffu-sion relationships used to characterize the reaction between acid and rock.

Figure 2.88: Acid Frac Database.

To use the data from the System Databases, it must first be copied into the UserDatabase. To do this, select the desired System Database from the list box at the topof the screen. Next, select an acid/rock system from the System Database list andpress the Insert button located in the center of the dialog box (or double-click onthe database fluid). This action copies the selected acid/rock data from the SystemDatabase to the User Database. Once an acid/rock has been inserted, its positionrelative to the other acids in the list can be adjusted by using the Up or Down arrowbuttons.



Any acid record contained in the User Database can be edited by selecting the acidname and clicking the Edit button or double-clicking on the acid. A new blankrecord can be created with the Add button. Selections may be deleted from the UserDatabase by clicking the Delete button. To exit the acid database dialog box, clickon the Done button.

When either the Add button or the Edit button is used to create or edit an acid/rocksystem, the screen shown in Figure 2.89 appears permitting the acid/rock data to beentered, viewed or modified. For a new acid, a blank screen is presented providinga template for the entry of data.

Figure 2.89: Acid Database Edit Screen.

Like all of the databases, the Acid Frac Database is accessible from the programDatabase menu and the Treatment Schedule.

Description of the Acid Database ParametersThe Reference Code is the identifier used in the Treatment Schedule to indicatewhich Rock/Acid System data to use in the simulation. This code must be a uniquestring of five (5) characters or less. Each Reference Code can be associated with aRock/Acid System description to define the content of the record. This informationis only used for reference and for future selection. The Reference Code and thedescription are displayed in the pop-up which appears whenever the cursor entersthe Rock/Acid System column in the Treatment Schedule.



Along with the information described above, each entry must have the followingCharacterization Parameters:

Acid Specific GravityEnter the specific gravity of the base acid extrapolated to a concentration of 100%(Note: A 100% acid concentration does not exist. This information is only used forvolumetric conservation.). For 100% HCl (hydrogen chloride) use a specific grav-ity of 1.5. This value is used to determine the specific gravity of the dilute acid mix-ture as demonstrated in the following expression:

where

Heat of ReactionThe heat generated from the reaction of the acid with the rock has an effect on thetemperature in the fracture and, therefore, on the acid spending rate. The heat ofreaction is the associated energy per unit mass of acid that reacts.

Acid Molecular WeightThis is the molecular weight of the base acid used. The molecular weight or atomicweight of HCl is 36.465 (H = 1.008, Cl = 35.457).

Dissolving Power

The governing acid transport relationship used by MFrac based on the rate of cre-ation of volume (R.O.C Volume) is as follows:

where

specific gravity of the mixturespecific gravity of acid (1.5)specific gravity of the solvent (i.e., water)concentration of the base acid

total volume loss per unit area per unit timeleakoff in the carbonateleakoff in the non-carbonate

γmixture γacidCacid γsolvent 1 C– acid( )+=

γmixture =γacid =γsolvent =Cacid =

V· l″ 2vylλl 2vyp 1 λl–( )λfl V· w

″ 2V″λl 2V· R″ λl–+ + +=

V· l″ =

2vylλl =2vyp 1 λl–( )λfl =



Because the internal acid fracturing calculations are based on the rate of creation ofmass (rather than volume), it is necessary to convert the entered volumes to anequivalent mass. The dissolving power facilitates this conversion. It is the mass ofrock dissolved divided by the mass of acid used (mass of rock/mass of acid).

Reaction Order and Reaction Rate CoefficientsWhen an acid comes in contact with rock, the rate of reaction can be described bythe following equation:

where

The Reaction Order represents the variation in reaction kinetics as a function ofacid concentration. The Reaction Rate Coefficient is a measure of the reactivity ofthe acid/rock combination at one specific temperature. Both of these values aredetermined experimentally by measuring the acid or product concentrations as afunction of time while controlling the environmental conditions of a specimen. Alog-log plot of the flux versus the concentration at the wall, , theoretically yields

a straight line with the slope equal to the reaction order, , and the intercept equalto . It is generally accepted that the reactivity (i.e., reaction rate coefficient)will increase exponentially with temperature according to the Arrhenius equationshown below:

where

leakoff due to worm holeschange in volume due to diffusion

change in volume due to the reaction products

flux or reaction rate (g-mole/cm2sec)reaction rate coefficient (units depend on n)acid concentration at the fracture surfacereaction order (dimensionless)

adjusted reaction rate coefficient at

V· w″ =

2V· ″λl =2V· R

″ λl =

V k Cs( )n=

V =K =Cs =n =

Cs

nK( )ln

KT′ KTo

′EaR------ 1

To----- 1

T---–⎝ ⎠

⎛ ⎞exp=

KT′ = T



If the reaction rate coefficient is determined for a series of temperatures, then a log-log plot of vs. will produce a straight line with slope equal to and the

inverse log of the intercept equal to . The Activation Energy, , and the Ref-

erence Temperature, , therefore, are input in order to provide a basis for adjust-ing the reaction rate for variation in temperature.

Standard reaction parameter units are used throughout MFrac. The required unit forthe reaction rate coefficient is (g-mole/cc)1-n-cm/sec. The units for the ActivationEnergy and Reference Temperature are Kcal/g-mole and degrees Kelvin, respec-tively.

DiffusivityThe ability to quantify the total mass transfer in the system is directly related to theability to characterize the diffusivity (diffusion coefficient). Normally in acid frac-turing, when laminar flow exists, the diffusivity is equal to the molecular diffusion.When turbulence is encountered diffusion typically increases. Unfortunately, diffu-sivity, as it relates to acid fracturing, has historically been difficult to measure in thelaboratory. There are currently several organizations investigating new methods forcharacterizing acid diffusion.

The mass transfer coefficient is calculated from the Sherwood number, :

where is the mass transfer coefficient, is the fracture width and is the dif-fusivity. Notice the similarity to the Nusselt number.

Just like the reactive rate coefficient, the diffusion coefficient also varies as func-tion of temperature according to the Arrhenius equation. The following formulationis used:

reaction rate coefficient at the reference temperature,

activation energy, cal/g-molegas law constant, 1.987 cal/g-mole

KTo′ = To

Ea =R =

KT′ 1 T⁄ Ea R⁄

KT0′ Ea

To

Nsh

NshKgW

D-----------=

Kg W D

DT DTo

EDR

------- 1To----- 1

T---–⎝ ⎠

⎛ ⎞exp=



where

A separate Activation Energy, , and the Reference Temperature, , as theyrelate to diffusion are required to characterize the effect of temperature on the diffu-sivity. Using this relationship, the acid transport model can be coupled to the heattransfer solution. The result is a fully coupled approach for determining mass trans-port as a function of temperature.

The required units for the Diffusivity are cm2/sec. Like the Reactive Rate Coeffi-cient, the associated Activation Energy must be input in Kcal/g-mole and the Ref-erence Temperature entered in degrees Kelvin.

Casing and Tubing DatabasesTo aid in defining the wellbore configuration, Casing, Tubing and Coiled Tubingdatabases are provided. These are tables with three columns: OD (outer diameter),Weight and ID (inner diameter). The Weight is never used by MFrac; it is only pro-vided as a reference. Many common pipe measurements have been provided. Youmay add, modify or delete rows in the table as you would in any other Meyer table.When finished editing the database, click on the OK button.

Figure 2.90 illustrates the tubing database dialog.

diffusivity at temperature diffusivity at the reference temperature, activation energy for diffusion, cal/g-molegas law constant, 1.987 cal/g-mole

DT = TDTo = To

ED =R =

ED To



Figure 2.90: Tubing Database.

Rock Properties DatabaseTo aid in inputting rock property data a Rock Properties Database has been pro-vided. Figure 2.91 shows the rock properties database table. The database tableincludes a lithology symbol, zone name, stress gradient, Young’s Modulus, Pois-son’s Ratio, fracture toughness, and critical stress.

You may add, modify or delete rows in the table as you would in any other Meyertable. When finished editing the database, click on the OK button.


2.8 Tools 231

Figure 2.91: Rock Properties Database Screen.

Lithology symbols can be added by placing the mouse cursor to the left of the ZoneName and double clicking the left mouse button. The Rock Database is accessiblefrom the Rock Properties dialog box by selecting Insert from Database icon.

2.8 ToolsThe Tools menu provides the user with options and analytical tools for calculatingor determining scientific parameters. Currently, all applications have a ToolSpreadsheet option that allows the user to customize the spreadsheet. MFrac andMProd also have a Proppant Calculator for determining the proppant permeabilityand beta factor based on proppant properties.

Proppant CalculatorThe Proppant Calculator Dialog screen displayed in Figure 2.92 allows the user tocalculate the theoretical proppant permeability and non-Darcy beta factor.



Figure 2.92: Proppant Calculator - Data Input Screen

Beta CorrelationThe equation to describe non-Darcy flow is a form of the Forchheimer [1901] equa-tion


etc.) and is the non-Darcy flow factor or simply factor with units of (e.g.,cm-1, ft-1, atm-s2/gm etc.). Clearly the first term in this equation accounts for vis-cous effects and the second term for inertial or minor loss effects. If the second termon the right hand side is omitted, the equation simplifies to Darcy’s law. Thus non-Darcy flow describes the flow regime that does not obey Darcy’s law. Holdith[1976] reports that the original form of the second term on the right hand side of

Eq. by Forchheimer was which was replaced by Cornell and Katz [1953] bythe product of the fluid density, , and the factor.


xdd p( )– μ

kf----υ β ρυ2( )+=

kf L2

β β L 1–

aυ2

ρ β

kf φ


2.8 Tools 233

where , , and are constants. The effect of immobile water saturation, , canbe incorporated by modifying the porosity to be the effective porosity( ). A number of correlations for the beta factor (inertial coefficient)are provided in the database.

Proppant Property DataFollowing is a list of the proppant property data that is required to calculate theproppant permeability and Non-Darcy beta factor:

Proppant PorosityThis is defined as the void fraction between sand grains (i.e., liquid volume toslurry ratio of the settled bank). It is used to calculate the propped fracture perme-ability. Typical values of porosity for proppants are shown in Table 2.13.

Proppant DiameterThis is a weighted average value as determined from a sieve analysis according toAPI standards. It is used in calculating the theoretical proppant permeability.

SphericitySphericity is a measure of how spherical (round) an object is. The sphericity, , isdefined as



6-8 angular 0.36

10-20 angular 0.36

10-20 round 0.32

20-40 round 0.35

40-60 round 0.32

β akf

bφc-----------=

a b c Sw

φe φ 1 Sw–( )=

Φs



where

Typical values of Sphericity for various shapes are shown in Table 2.13.


= specific surface area of a particle= particle volume= particle surface area= characteristic diameter of the particle

Table 2.14: Sphericity of Common Objects

Object Sphericity,

tetrahedron 0.671

cube 0.806

dodecahedron 0.910

half-sphere 0.840

sphere 1.0


Proppant Type SpecificGravity

AbsoluteDensity

(lbm/ft3)

AbsoluteDensity

(kg/m3)


Sand 2.65 165.4 2650




Φs6

dpas----------

6VpdpSp-----------=≡

as

Vp

Sp

dp

Φs


2.8 Tools 235

The proppant Specific Gravity is used in the calculation of the proppant density.

Concentration/Area and Propped WidthThis is the effective concentration per unit area in the fracture. The concentrationper unit area, , as a function of the proppant porosity, , proppant specific grav-

ity, , and propped fracture width, , is given by

where is the reference density of water at (i.e., or ).

The propped fracture width in terms of the concentration per unit area is

Proppant Damage FactorThe proppant damage factor is defined as

where

The permeability in the fracture is used to determine the fracture conductivityand dimensionless fracture conductivity as given by

Calculated Fracture PermeabilityThe theoretical fracture permeability, , is calculated in terms of the proppant

diameter, , porosity, , proppant sphericity, , propped fracture width, ,and

damage factor, , from the equation given below

= fracture permeability= proppant permeability (undamaged or theoretical)= proppant damage factor

Ca φ

γ wf

Ca γρo 1 φ–( )wf=

ρo 4 °C 62.43 lbm ft3⁄ 1g cm3⁄

wf Ca γρo 1 φ–( )[ ]⁄=

DF 1 kf k⁄–=

kf

kDF

kf

kf k 1 DF–( )=

kf

dp φ Φs wf

DF

kfφ3 dpΦs( )2

72λm 1 φ–( )2-------------------------------- 1

dpΦs3 1 φ–( )wf--------------------------+⎝ ⎠

⎛ ⎞2–

1 DF–( )=



where for . The permeability for open slot flow (i.e., no packing

in a slot as ) is

where (see Bird [1960]).

Beta

The Beta factor is calculated from the proppant property data and selected beta cor-relation. The generalized correlation for the beta factor in terms of the fracture per-meability and porosity is of the form


2.9 References1. Virk, P.S.: “Drag Reduction Fundamentals”, AIChE Journal, Vol. 21, No. 4,

July 1975.

2. Keck, R. et al.: “A New Method for Predicting Friction Pressures and Rheol-ogy of Proppant Laden Fracturing Fluids”, SPE Production Engineering, Feb.1992.

3. Schlichting, H., Boundary Layer Theory, McGraw-Hill, NY (1955).

4. Meyer, B.R.: “Design Formulae for 2-D and 3-D Vertical Hydraulic Fractures:Model Comparison and Parametric Studies,” paper SPE 15240, May 1986.

5. Hudson, P. J. and Matson, R.: “Fracturing Horizontal Wells,” presented at the54th Annual SPE Technical Conf., Midland, TX, Sept. 1992.

6. Huit, J.K.: “Fluid Flow in Simulated Fractures,” AIChE Journal, Vol. 2, p 259.1956.

λm 25 12⁄≈ φ 0.5<

φ 1→

kfwf

2

12------ 1 DF–( )×=

λm 3 2⁄=

kf φ

β akf

bφc-----------=

a b c Sw

φe φ 1 Sw–( )=


2.9 References 237

7. Louis, C.: “Etude des écoulements d'eau dans les roches fissurées et leursinfluence sur la stabilité des massifs rocheux,” Bull. de la Direction des Etudeset Recherches, Series A, No. 3, p. 5-132, 1968.

8. Warpinski, N.R.: “Measurment of Width and Pressure in a propagatingHydraulic Fracture,” SPEJ (Feb. 1985) pp 46-84.

9. Bird, R.B, Stewart, W.E., Lightfoot, E.N., Transport Phenomena, J. Wiley &Sons Inc., NY, 1965, page 204.

10. Meyer, B.R.: “Generalized Drag Coefficients Applicable for All FlowRegimes,” OGJ, 1986.

11. Slatery, J.C., doctoral thesis, University of Wisconsin, 1959.

12. Shah, S.N.: “Proppant Settling Correlation for Non-Newtonian Fluids UnderStatic & Dynamic Conditions,” paper SPE 9330, 1980.

13. van Eekelen, H.A.: “Hydraulic Fracture Geometry: Fracture Containment inLayered Formations,” SPEJ (June 1982) pp 341-349.

14. Thiercelin, M.: “Fracture Toughness and Hydraulic Fracturing,” Int. J. RockMech. & Geomechanics, vol 26, No3/4, pp 177-183, 1989.

15. Meyer, B.R.: “Three-Dimensional Hydraulic Fracturing Simulation on Per-sonal Computers: Theory and Comparison Studies,” paper SPE 19329, Oct.,1989.

16. Nierode, D.E. and Kruk, K.F.: “An Evaluation of Acid Fluid Loss Additives,Retarded Acids, and Acidized Fracture Conductivity,” paper SPE 4549, Sept./Oct., 1973.

17. Cramer, D.D.: “The Application of Limited-Entry Techniques in MassiveHydraulic Fracturing Treatments”, presented at the SPE Production OperationsSymposium, Oklahoma City, OK, March 1987.

18. El-Rabaa, A.M., Shah, S.N., and Lord, D.L.: “New Perforation Pressure LossCorrelations for Limited Entry Fracturing Treatments”, presented at the SPERocky Mountain Regional Meeting, Casper, WY, May 1997.


Chapter 3

MViewAcquired Data Visualization

3.1 IntroductionThis chapter is a guide to the use of MView. Developed as a data handling systemand display module for real-time hydraulic fracturing; this software is intended foruse with MFrac (a hydraulic fracturing design simulator) and MinFrac (a minifracanalysis simulator), but can also be used as a general data collection and displayprogram. This chapter explains the basic procedures for using MView.

MView is a data handling system and display module for the real-time and replayanalysis of hydraulic fracture treatments and minifrac analysis. MView can accom-modate up to two hundred (200) data channels and simultaneously allows selectionof up to two hundred (200) parameters for processing. A channel can also be speci-fied for Multi-parameters (e.g., channel C can be assigned to more than one param-eter.).

MView can process a maximum of 86,400 lines of data, but can read or import vir-tually an unlimited number of lines of data (limited only by memory) from a file. Ifthere are more than 86,400 lines of data in a file, it will be necessary to adjust theSample every box so that 86,400 lines or less of data are actually used for the anal-yses. For example, if a data file has 1,700,000 lines of data, it would be necessary toset the Sample every box to 20 or higher, or select a range of data (start row to endrow) with less than or equal to 86,400 rows of data. This is discussed below in theData Set dialog.

All data manipulation in MView is done through the use of data sets. A data set is agroup of related data that may come from a text file, an MFrac file (.mfrac), anMPwri file (.mpwri) or real-time data from the serial port.

Each data set may have up to two hundred (200) different parameters. A maximumof five data sets may be used at a time. All data sets share the same list of parameter


240 MView: Acquired Data Visualization

names, which includes unit types (length, rate) and units (ft, bpm). There are manyfunctions available for processing the data sets. Data sets may be edited or saved astext files. In addition, any number of data sets can be merged into a single file witha single time scale and any number of them can be plotted together.

The first step in using MView is to specify the parameter names, types and units byaccessing the Parameters dialog box from the main menu.

Once a group of parameters has been specified, go to the Data menu and selectData Sets. To import a data file into an MView data set, specify the file name andformat of its contents. The contents are defined by associating a parameter with achannel (or column) of data in the file.

MView provides a means of sharing the real-time or replay data set with MFrac andMinFrac for use as simulation input. This data can include the parameters: pumprate, bottomhole and surface pressure, proppant concentration, and nitrogen or CO2injection rate. Sharing a data set with MFrac and MinFrac can be done after the datasets are setup by choosing Simulation Setup from the Simulation menu.

It is important to realize that MFrac, MinFrac and MView are always in constantcommunication with one another. When these programs are open, they automati-cally share any Real-Time or Replay data set.

After data has been imported into MView, it is possible to construct compositeplots, by choosing the Build Plot command from the Plot menu. This commandwill display all of the data that has been imported and allow a choice of whichparameters to include in a plot. MView allows the construction of up to six plots forsimultaneous display.

After the plots have been built, they can be viewed by selecting the View Plot com-mand from the Plot menu. Whenever a plot is open, all the standard configurationand exporting functions of all Meyer plots are available. In addition, MView allowsmanipulation of the data contained in the plot using statistical and editing functions.This is described in Section 3.6 “Creating Plots- Graphically Editing Data.”

An outline of the basic steps for using MView is shown in Table 3.1.

It is important to realize that MFrac, MinFrac, and MView are always in con-stant communication with one another. When these programs are open, theyautomatically share any replay/real-time data set.


3.2 Parameters 241

MenuThe MView menu bar is shown in Figure 3.1. Generally, the menus are accessedfrom left to right as shown in Figure 3.1.

Figure 3.1: MView Main Menu.

3.2 ParametersThe first step in using MView is to create a list of the parameters to include in theanalysis.

Table 3.1: MView Basic Steps

Step Program Area

1. Open an existing MView data file (*.mview) orcreate a new data file File Menu

2. Specify Parameters Parameters Menu

3. Import Data into Data Sets4. Specify data files5. Associate columns of the file with parameters

Data Sets command from theData Menu

6. Start the Acquisition Toolbar (for real-timedata)

7. Specify the Acquisition Output File8. Setup the Communications Parameters9. Start Data Acquisition10. Real-Time Data Window

Real-Time⏐Acquisition Tool BarMAcq (data file)SetupMAcq -Start buttonMView - Real-Time Menu

11. Send real-time/replay data to MFrac and/orMinFrac

Simulation Menu

12. Build Plots Plot Menu

13. View Plots Plot Menu

14. Simulate in MFrac and/or MinFrac MFrac and/or MinFrac



Creating a Parameter ListGenerating a parameter list means cataloging the different parameters to beincluded in the analysis. Therefore, any parameter to be imported from either theacquired data, simulated data or any ASCII file, must be in the parameter list.MView will display data using the selected units of the parameter list.

To create a list of parameters, choose Parameters from the Main menu. This willdisplay the screen shown in Figure 3.2.

Figure 3.2: Parameters Dialog Box.

To Specify a Parameter:

1. Enter a descriptive name in the text box under the Parameters column.


3.2 Parameters 243

2. Select a corresponding Unit Type for the parameter using the list box providedto the right of each Parameter Name. If the desired unit type is not present inthe list, choose No Units.

3. After the Unit Type has been selected, a list of the available units correspond-ing to the Unit Type will be in the adjacent Output Unit list box. Choose thedesired unit.

4. Repeat this process for each Parameter Name.

To Reset the Parameters:

1. Press the Reset button located at the bottom of the Parameters screen.

2. This will set all of the Parameter Names to blank, the Unit Type to No Units,and the Output Unit list box to blank.

Using Parameter List TemplatesThe current set of parameters can be saved for future recall. This is useful whenworking with frequently used configurations in order to avoid creating the same listrepeatedly.

To Save a Parameters Template:

1. Enter a unique name in the Unit Set Description box located at the top of theParameters screen.

2. Click the Save button located in the lower left corner of the Parameters screen.This will display the Save As screen.

3. Enter a name for the file in the space provided and click OK. By default,parameter files contain the extension mvu (e.g., default.mvu).

Once saved, a set of parameters can be recalled and applied to an MView session byusing the Load Units button.

To Apply a Parameters Template:

1. Open the Parameters screen from the main menu and click the Load buttonlocated in the lower left corner. This will display the Open screen.

2. Browse to select the saved parameters file containing the extension mvu.

3. After selecting the template file, click the OK button.



3.3 DataA key aspect in the analysis of fracturing data is the manner in which data files ofvarious types are accessed, viewed and edited. This chapter outlines the data han-dling capabilities of MView and provides instructions for importing and manipulat-ing data sets.

Data SetsThe Data Sets dialog box is used to import data from files for display or use in asimulation (Figure 3.3). Up to five different data files can be imported for simulta-neous display. The first of these data sets may be setup as a real-time or replay fileto be used as input for an MFrac and/or MinFrac simulation. This file may be eithera real-time file that is transmitted to MView or it can be an acquired data set thatwas provided by a service company after the treatment was performed (i.e.,Replay). The remaining four files may be either ASCII formatted (delimited withcommas, spaces or tabs), MFrac or MPwri output files (with the extension “mfrac”or “mpwri”, e.g., Samplefile.mfrac).

To import a data file into a data set, choose the desired data set by clicking the cor-responding tab in the Data Sets box (e.g., Real-Time/Replay, Data Set A). Noticethat a Reference Name can be entered for each data set (e.g., Design). The Refer-ence Name is used as the data set’s unique identifier and is displayed on each dataset tab.

Figure 3.3: Data Sets Dialog Box.

For details on importing data files, refer to the instructions below that correspond tothe data file type.


3.3 Data 245

Importing Real-Time DataTo Import Real-Time Data:

1. Connect your computer to a data acquisition system that is capable of transmit-ting data through a serial connection (e.g., RS232). Instructions for setting-upand controlling the flow of real-time data is covered in Section 3.4, “Real-Time Menu”. The real-time data set may be set up before or after the data isflowing.

2. Open the Data Sets dialog box by selecting Data Sets from the Data menu andclick on the left-most tab. Then click on the Real-Time radio button locatednear the center of the screen (Figure 3.3).

3. Enter a descriptive Reference Name in the space provided.

4. Click on the Setup button and follow the instructions below for Setting up aText File.

Importing a Replay Data FileTo Import a Replay Data File:

1. Open the Data Sets dialog box, by selecting Data Sets from the Data menu andclick on the left-most tab. Then click on the Replay radio button located nearthe center of the screen (Figure 3.3).

2. Enter a Reference Name in the space provided.

3. Be sure to click on the Data File check box which is used to enable or disable afile setup from the active session. When the box is checked, the data set isused. When it is cleared, the data set is omitted from all plots.

4. After checking the Data File box, click on the Ellipses (Select File button). Afile browse dialog box will then be displayed. Select the desired file and clickOK.

MView and MAcq no longer support the unconventional time format of HoursMinutes Seconds format (HH MM SS) separated by spaces. The standard formatis Hours Minutes Seconds separated by colons (HH:MM:SS). If you have a datafile with time in the format (HH MM SS) separated by spaces you can import thefile into Excel or other spreadsheet program, separate the columns by colons,and save the new format. If real time data is being sent in a time format sepa-rated by spaces, in MAcq under Acquisition Setup check the Time Box to “Addthe computer’s time to the start of each data line”. As far as we know, Hallibur-ton is the only company that occasionally uses this format.



5. Click on the Setup button and follow the instructions below for Setting up aText File.

Importing an ASCII Data FileTo import an ASCII data file, click on the tab for the desired data set. Then use theabove procedure for importing a replay data file.

Importing an MView Acquired Data FileWhen MView is used to acquire data in real-time from a service company dataacquisition system (cable or modem link), MView stores a copy of the received datain a file with an ADT extension. To use the stored data to replay the treatment at alater time, follow the instructions above for Importing an ASCII data file. Note thatwhen you click the Select File button, you may choose MAcq Data Files from theList Files of Type box. This will help in choosing files with the ADT extension.

SetupOnce the data source has been defined and the Setup button has been clicked,MView will display the Data File setup screen (Figure 3.4).

The left side of the screen has the list of Parameters, the right side has a spreadsheetthat contains the data from the data file. Each column of data displayed in thespreadsheet is labeled with a column letter. The primary purpose of the Setupscreen is to associate each of these columns with a parameter name and unit type.The Parameter names are those that are entered in the Parameters screen.

Note that ASCII files and real-time data must contain data lines delimited witheither tabs, spaces, or commas. When reading a data file, MView will interpretand display these delimiters as column breaks. If the data file contains text head-ers, we recommend that you use single word phrases. This will keep the headersaligned properly with the correct column of data. If you must use more than oneword, try hyphenating the words to prevent unwanted column breaks (e.g., Sur-face-Pressure).


3.3 Data 247

Figure 3.4: Data Setup Dialog Box.

To Associate a Column of Data with a Parameter:

1. Enter the column letter in the corresponding column box to the right of theParameter name. Alternatively, click and drag the column letter from thespreadsheet to the desired column box. If there are no Parameter names dis-played, close the dialog box and return to the Parameters screen to set upparameters.

2. Choose the appropriate input unit for the parameter from the Unit box. Notethat this is the unit of the data in the spreadsheet, it may not necessarily be thesame unit as specified in the Parameters screen. Thus, it is possible to havedifferent input and output units. If the column is a time column with the formatHH:MM:SS, the input unit will automatically be set.



3. Data may be shifted in order to superimpose data trends during analysis or tosynchronize events. Any parameter may be shifted by selecting the Shift Valuecolumn and entering a value for the shift. The shift value will be added to thevalue in the data file. Graphical shifts can also be performed on a plot when itis displayed.

4. To filter out unreasonable parameter values, specify Filter Min. and Filter Max.values. To use this option move the cursor to the Filter columns and type inboth the minimum and maximum filter values. The Filter Min. and Filter Max.must be entered using the input unit selected in the Unit box. All lines of datathat contain a parameter which is out of its range will be ignored.

For ASCII files, the range of interpretable data is indicated by the Start and EndRow listed in the Row Selection portion of the screen (Figure 3.4). These valuesmay be changed as desired. Character strings, such as column headers, are ignoredand can be included in the selection of rows.

To sample the data contained in the spreadsheet, specify the sampling frequency byentering a number in the Sample every box. For example, to use every fifth datapoint, enter a 5 in this box. MView can select a maximum of 86,400 lines of data,but can read or import virtually an unlimited number of lines of data (limited onlyby memory) from a file. If there are more than 86,400 lines of data in a file, it willbe necessary to adjust the Sample every box so that 86,400 lines or less of data areactually used for the analyses. For example, if a data file has 1,700,000 lines ofdata, it would be necessary to set the Sample every box to 20 or higher, or select arange of data (start row to end row) with less than or equal to 86,400 rows of data.

When importing data it is sometimes difficult to determine which column of datacorresponds to which parameter type. To assist in this situation, MView provides aquick method of generating X-Y plots using any of the columns of a file.

To preview the data in the Setup screen, type or drag the column labels of thedesired parameters to the boxes found in the View Plot portion of the dialog box(Figure 3.4). One X parameter and two Y parameters may be included (e.g., rateand proppant concentration vs. time). Once a selection has been made, click on theView Plot button to display the plot.

When you are finished associating data columns with Parameters, click on the OKbutton. Then the data will be imported into the MView data set. If any change ismade to the data from within MView, it will not be reflected in the original data file.


3.3 Data 249

Importing an MFrac or Mpwri Data FileTo Import an MFrac File (*.mfrac) or MPwri (*.mpwri):

1. Choose one of the tabs located at the top of the screen, see Figure 3.5 (e.g.,Data Set A). If you choose the left most tab, make sure to choose the Replayradio button located near the left side the screen (Figure 3.3). Enter a Refer-ence Name in the space provided.

2. Check the Data File check box and then click the Select File button. Beforebrowsing to select the data file, choose MFrac Files from the List Files ofType box. Then select the desired file and click the OK button to return to theData Sets screen.

3. When an MFrac output file has been chosen MView automatically adds a listbox adjacent to the Select File button (Figure 3.5). This list box contains threechoices: Wellbore Hydraulics, Fracture Characteristics or Proppant Trans-port. Choose which type of data you want to import.

4. Click on the Setup button. This will bring up the MView Data Setup box (Fig-ure 3.6).

5. Associate rows in the list on the right of the screen to parameters on the leftside of the screen, similar to setting up text files.

6. Click on the OK button to import the data.

Figure 3.5: Data Sets Dialog Box - MFrac File.



Figure 3.6: Data Setup Dialog Box - MFrac Output File.

When importing an MFrac File it is possible to configure the units for previewingplots by selecting the Units button.

After data has been imported from an MFrac output file, the data in MView is inde-pendent of the file. Thus, if the simulation is run again, there will be no change inthe data in MView. However, to update the information in MView, select theRefresh button in the data sets screen. This will re-import the data from the outputfile, using the same setup as the last time the data was imported.

When importing a multilayer MFrac File, there will be more than one fracture(and/or proppant) characteristic zone. To analyze Fracture Characteristics orProppant Transport, select the desired zone #. The wellbore hydraulics resultsare contain in the Wellbore Hydraulics item list.

Re-running MFrac will automatically update the *.mfrac file. There is no longerany need to save the *.mfrac file after running before selecting the Refresh but-ton.


3.3 Data 251

Setup TemplatesIf the same basic data setup is used repeatedly from job to job, it may be saved as atemplate. Once saved, a template may be recalled and applied at any time.

To save a setup as a template, click the Save As... button located at the bottom ofthe Setup dialog box. This will present a screen for entering a file name and loca-tion for the template. Choose a directory and enter a name in the Save As dialogbox. Click the OK button. Setup template files are automatically given the exten-sion VHD (e.g., demo.vhd).

Once a template is created, it can easily be applied to another data set by choosingthe Load... button and selecting a file name with the extension VHD (e.g.,demo.vhd). The same Parameters must be defined when a template is loaded aswhen the template was saved.

Editing DataThe Edit Data command in the Data menu allows some basic spreadsheet editing ofa data set. For more advanced editing, refer to the Graphically Editing Data part ofSection 3.6.

When the Edit Data command is used from the Data menu, a box listing the currentdata sets is displayed. From the list, choose the desired data set and click the Editbutton. A spreadsheet containing the data will be presented. Real-time data may beviewed, but not edited.

The edit command allows you to view the entire data set and inspect it for inconsis-tencies or potential problems. If a problem is discovered, you may either attempt tocorrect it by re-typing the data or completely removing the row. The data that isviewed and edited is not the original ASCII or MFrac file, but rather a copy kept inmemory by MView. The original data file is never changed by MView.

To Re-Type Data:

1. Select the desired data cell by clicking on it.

2. Type the new value. Press the ENTER key when finished.

To Delete a Row of Data:

1. Select the desired row by clicking on the row number. The entire row will behighlighted.



2. Use the Delete Row button located at the bottom of the spreadsheet to removethe highlighted row. If you make a mistake and remove the wrong data use theCancel button to avoid saving the mistake. If the cancel button is used, allchanges made during the active editing session will be lost.

The edit function allows data to be viewed either with the defined shifts applied ordisabled. This choice is made by selecting either the View Data with Shifts or ViewData Without Shifts radio buttons found at the bottom of the spreadsheet.

The shifts applied to data may also be edited by clicking on the Edit Shifts at thebottom of the screen. This will provide a list of parameters and their shifts, whichcan be edited.

Save Data as a Text FileAfter a data set has been imported or transmitted from the data acquisition system,it can be exported as an ASCII text file with the parameter names and units as col-umn headers and data points as rows.

To Export a Text File:

1. Select the Save Data as Text File command from the Data menu. A list of theactive data sets will be displayed.

2. Select the desired data set and click the Save As Text button located at the bot-tom of the screen.

3. Enter a name and location for the text file and click OK.

4. Answer accordingly when MView asks if shifted or unshifted data should bewritten to the text file.

Merge Data SetsThere may be times when the data needed for an analysis exists in two separate filesand you would like to merge them into one file with a single time scale (e.g., sur-face pressure and bottomhole pressure). MView provides the capability to mergeany number of data sets into a single set of data.

To Merge Data Sets:

1. Select the Merge Data Sets command from the Data menu. A list of availabledata sets will be displayed.


3.3 Data 253

2. Select the data sets to be merged by checking the adjacent box next to eachdata set name. When finished, click the OK button to display the Merge DataSets dialog box. If the time range for each of the files is not overlapping or ifthe times are not increasing, the following message will appear “There is noparameter of time for which these files have increasing data. These files cannotbe merged.” In order to merge the data you will need to return to either theSetup or Edit Data dialog boxes to synchronize the data by applying a shift.The shifted values, when active, are always used when merging data files. Theerror message may also occur if you have used different parameters for time.

3. From this screen, select the time parameter to base the match on. Normally,there is just one time parameter; however, it is possible to have more than one.Consequently, any parameter that is defined as a Time Unit Type and has datain all selected data sets will appear in the list.

4. The program scans the selected data sets and determines the Min. and Max. val-ues for the selected match parameter and displays them in the Range section ofthe screen. A Time Step will also be suggested based on the existing data. Anyof these values may be changed. Notice that by adjusting the time step value,the function can also be used to apply filtration to data files and affect the deltatime between data points. Use the Smallest Common Range or LargestRange buttons to automatically set the Min. and Max. values. For example, ifthe range for data set A is 0-100 minutes and 50-150 minutes for Data Set Bthe smallest common range is 50-100, while the largest range is 0-150. Anydata set which does not have time values within the selected range will not becompletely interpolated. There is an option to specify what is written to thedata file when interpolation is impossible in the Preferences dialog box.

5. MView lists all of the active parameters in the Possible portion of the dialogbox. Select which of these parameters to include in the composite file by high-lighting them and clicking the Add button. Likewise, parameters may be elim-inated from the Selected list by using the Remove button.

6. After making selections, choose the OK button to display the File Save box.Enter or choose a location and name for the file and click OK. The selecteddata is written to the file along with the parameter names and units as columnheaders.

PreferencesMView allows the selection of certain preferences that provide some variation in theprogram control. These preferences are described as follows:



Ignore consecutive identical Real-Time linesSome data acquisition systems transmit and save data with duplicate data lines orrows. This can occur when the sampling rate of the data acquisition is much fasterthan the cycle time for the sensor it is connected to. When this occurs, the acquisi-tion system records duplicate sensor values until the sensor updates. These dupli-cate data points do not necessarily cause problems, but they consume thecomputer’s resources and cause MView and MFrac to use redundant data. To filterthese duplicate data segments from the incoming real-time data stream, check thisbox in the Preferences screen. To view and use all duplicate data, disable thisoption.

Ignore consecutive identical lines in data filesThis option is the same as the option described above; however, it applies only toimported files (e.g., replay files, etc.). Use this option to eliminate duplicate datalines for any file except real-time data.

Beep when filtering real-time data that is out of rangeIf this option is on, there will be a beep when the program has detected and ignoredreal-time data that fell outside of the range specified in the Filter section of the datasetup screen. If this option is used and numerous beeps occur, you might want toreconsider the filtration range defined.

Text to be inserted when a number cannot be interpolatedThis option is used to specify the string to be inserted when MView cannot interpo-late a number contained in a file during the Merge Data process. The string is a wayof marking locations in the merged file where data could not be interpolated. Ifsomething other than a number is specified, there may be problems importing themerged data file into MView.

Default data range for merged data setsThis preference determines the default range for the merge data set process. Forexample, if the range for data set A is 0-100 minutes and 50-150 minutes for DataSet B the smallest common range is 50-100, while the largest range is 0-150. Checkeither the Smallest Common Range or the Largest Range radio button.

3.4 Real-Time MenuThe Real-Time menu contains commands used to control the receipt of real-timedata from a data acquisition system as illustrated in Figure 3.7. The Add Log Entryand Recover Real-Time Data options in the menu are only enabled when real-timedata set is being received from MAcq (Meyer Data Acquisition).


3.4 Real-Time Menu 255

Figure 3.7: Real-Time Data Menu.

MView can receive real-time data either through a serial cable (Direct Connect),network, or remotely using a modem. MView is also capable of emulating real-timedata transfer internally without the use of a second computer or acquisition system.The TEST MODE-Send ASCII file instead of real-time data is included to facili-tate training and demonstration of MView in real-time mode.

In addition to receiving data, MView can also transmit the data that it receives. Thisprovides a means of sending treatment data from the field to other remote locations(e.g., someone’s office). A common example of this type of data transfer would bethe capture of data from the service company acquisition system via an MViewcable link on location. A second copy of MView could then be run remotely andlinked to the location copy of MView via modem. In this manner, it is conceivablethat the process of sending data in a daisy-chain fashion could continue repeatedlyas many times as needed. When used, this type of setup permits real-time simula-tion and analysis from virtually anywhere telephone service exists (cellular or oth-erwise).

The procedures used for setting-up various types of real-time data links are outlinedbelow.

Acquisition ToolbarTo receive data from a remote data acquisition system, either through a serial con-nection or using a modem, the Acquisition Toolbar must be activated by selectingthe Acquisition Toolbar command from the Real-Time menu.

The toolbar is used to setup a communication link, dial the modem, start and stopdata flow, and save/load a setup template file. The toolbar is shown in Figure 3.8.



Figure 3.8: Real-Time Acquisition Toolbar.

This program has been upgraded for enhanced collecting and parsing of a datastream. As illustrated, a data window is included to display the raw data as it isbeing processed. Also when the system is paused, a message will appear in theupper left hand corner of the data window as shown in Figure 3.9. When the systemis in the Paused mode, data will not be saved to the ADT file.

Figure 3.9: Real-Time Acquisition Tool Bar - Paused Mode.



To Acquire Real-Time Data:

1. Click the Setup button located on the Acquisition Toolbar and follow theinstructions in the Setup section below to make sure the serial port is properlyconfigured.

2. Click on the File ellipse button and specify an output file for the acquired data.The output is an ASCII (text) format with a default file extension of “adt”(other commonly used extensions are “dat” and “txt”).

3. Click on the Start button to enable data acquisition.

4. If transmitting by modem, click on the Dial button to dial a remote computer.

The individual components of the Acquisition Toolbar are described below:

Normally, once data acquisition has been setup, the toolbar is minimized or ends upin the background behind other open applications (e.g., MView). To bring the tool-

Table 3.2: Acquisition Toolbar Components

File Data that is received in real-time is always written to a file on disk. This filecan then be imported into MView for replay analysis after the job. It alsoprovides a backup of the raw data and is used for restarting or appendingthe data file in the event the acquisition is stopped or in the case of a systemcrash. The toolbar File Button is used to specify the name of the acquisitionfile. When it is selected, a File Save dialog box is displayed. Enter a newfilename or choose an existing file with extension ADT (*.adt) to append oroverwrite.

Setup Use the Setup button to configure the real-time serial connection.

Start Use this button to start MView data collection. Normally, this button isaccessed before the data acquisition system begins collecting and sendingdata.

Pause The Pause Button can be used to temporarily suspend data collection with-out terminating a session. No data will be received while pause is selected.

Continue When in paused mode, clicking on the Continue Button will resume datacollection.

Help Use this button to access the help file.

Dial When using a modem to transfer data, the Dial Button causes MView todial an entry from the phone book and connect to a remote site using thesetup parameters specified in the current setup. If the button is disabled,make sure to click the Start button to initialize the modem.



bar back to the foreground, select the Acquisition Toolbar command from theReal-Time menu in MView.

SetupThe Acquisition Setup button allows the user to configure the communication linksfor the input and output ports. The Acquisition Setup dialog box is shown in Figure3.10.

Figure 3.10: Acquisition Setup Dialog Box.

Time - Include Computer’s TimeTo include a column containing the computer’s time for each row of data, check thebox located at the top of the screen entitled Add the computer’s time to the startof each data line. This data will reside in the first column of the saved data file(*.ADT) in the form HH:MM:SS.

Input and Output PortsDepending on whether you want to configure data Input (reception) or Output(transmission) choose the Setup... button located on the left or right side of thescreen.

This action will either display the Communications Setup Input (Figure 3.11) orOutput dialog box (Figure 3.12). Keep in mind that data transmission (Output) isonly used to pass along the real-time data to another computer.



Figure 3.11: Communication Setup - Input Port Dialog Box.

Figure 3.12: Communication Setup - Output Port Dialog Box.



Configuring a Serial LinkTo transfer data through a serial link (Direct Connection), from one computer toanother, connect the computer to a data acquisition system that is capable of trans-mitting data through a serial connection (e.g., RS232). Use a null modem cable toconnect the two computers. Once the cable connection is made, the next step is toset the communication parameters for the transfer.

To Setup a Communication as Input or Output:

1. For input ports, it is necessary to specify if the data is ASCII Data from aSerial Port or if it comes from a Stewart & Stevenson Data Acquisition pre1997 software system. For an output port, make sure to check the Enable Out-put Port box located at the top of the screen.

2. Next, choose the Direct Connection button for the Communication Typeselection.

3. Continue with the setup by choosing the appropriate COM Port, Baud Rate,Stop Bits, Data Bits, Parity and Flow Control. A definition of each of theseparameters is given in Table 3.2. These settings should match the settings ofthe data acquisition software that will be sending data to MView.

4. A check box has been added to Ignore First Line of Data for the Input port(Figure 3.13). This may be helpful if the first line of data is normally bad. Withthe new parser algorithms, the first line of data will be filtered if it does notmeet certain standard preference rules for acceptable data.

5. Click the OK button to complete the setup.

MView requires that each line ends in either a carriage return or a carriagereturn followed by a line feed.

Table 3.3: Definition of Communication Parameters.

Parameter Definition

COM PortThe COM Port refers to the serial port used for data transfer. Nor-mally, this is a male DB-9 pin connector found on the back of thecomputer.

Baud Rate

The Baud Rate specifies the serial port transfer rate. For most data,sampling rates of 9600 is more than adequate. Specifying anextremely high baud rate may result in the creation of “garbage”characters in the data.



Setting-up a Modem ConnectionTo send data to or receive data from a remote location, MView must be configuredto use a modem. Normally, this is done to receive data from the field for real-timedisplay or simulation.

Depending on whether you want to setup a Modem Connection for data Input(reception) or Output (transmission), either the Communications Setup Input (Fig-ure 3.13) or Output dialog box (Figure 3.14) will be displayed. Keep in mind thatdata transmission (Output) is only used to pass along the real-time data to anothercomputer.

Stop Bits This value determines how many bits are used to mark the end of atransmitted character. The default is one (1).

Data BitsThis is the number of bits or size of each data package that is sentbetween two computers. Normally, 8-bits are used when Parity is setto None, while 7-bits are used with Even parity.

Parity

This value controls the parity of the currently specified COM Port.The selections are Even, Odd, None, Mark or Space. If parity is setto None and you see “garbage” on the screen try changing to Even.Odd, Mark and Space are rarely used unless required by the transmit-ting software. Make sure to use the correct Data Bits setting that cor-responds to the selected parity.

Flow Control

This setting determines how MView handles buffer overflows. Thisparameter controls what happens if the receiving computer's bufferbecomes too full to receive more data. XON/XOFF (software flow control) uses a character-based tech-nique, while hardware flow control uses an RS-232 Ready to Sendand Clear to Send methodology.

Table 3.3: Definition of Communication Parameters.



Figure 3.13: Communication Setup - Input Modem Dialog.



Figure 3.14: Communication Setup - Output Modem Dialog.

To Set-up a Modem for Transferring Data In or Out:

1. For an input port, it is necessary to specify if the data is ASCII Data from theSerial Port or if it comes from a Stewart & Stevenson Data Acquisition pre1997 software version. For an output port, make sure to check the Enable Out-put Port box located at the top of the screen

2. Select the Modem radio button as the Communication Type to ensure that themodem functions properly, a valid modem initialization string must be entered.Check the modem documentation for a valid string. Please refer to “Trouble-shooting Modem Problems” below for more information.

3. Options are available for the modem to Auto Answer and/or Redial if the con-nection is lost. The Output port also has an option Upon connection, sendall data that was acquired during off-line time (see Figure 3.14).

4. Once an initialization string has been entered, continue with the setup bychoosing the appropriate COM Port, Baud Rate, Stop Bits, Data Bits, Parityand Flow Control. For a definition of each of these parameters, refer to Table



3.2. These settings must be consistent with the system you are communicatingwith.

5. Click the OK button to complete the setup.

Simulating a Real-Time Data TransferChoosing the TEST MODE - Send ASCII File Instead of Real-Time Data option atthe top of the Input Communications Setup disables port communications com-pletely Figure 3.13). When activated, this command allows you to simulate thereal-time process by reading an ASCII file from your computer and transmitting thedata as if it were being sent by a remote data acquisition system. To do this, selectthe TEST MODE - Send ASCII File Instead of Real-Time Data option at the top ofthe Input Communications Setup screen. Select the file to replay with the SelectFile button. Specify the data sampling rate, and choose the OK button.

Saving a Real-Time SetupAs with the Data Connection and Parameters setup within MView, once the Real-Time setup has been completed, the configuration can be saved as a template usingthe Save button found in the Acquisition Setup dialog box (Figure 3.10). The savedtemplate file will have the extension STP (e.g., demo.stp). To recall a saved setup ata later date use the Load button found in this screen. It is not necessary to use thetemplates; the current configuration will be kept between work sessions.

Working with the Phone BookTo Add, Modify or Delete Phone Book Entries:

1. Click on the Phone Book button located in the Acquisition Setup dialog box(Figure 3.10). This will bring up the dialog box shown in Figure 3.15.

2. Depending on whether you want to add, revise or delete an entry, choose theAdd Entry, Edit Entry or Delete Entry buttons respectively.

3. If you are adding or revising an entry, the screen shown in Figure 3.16 will bedisplayed.

4. When finished typing the required information, click the OK button to add theinformation to the database.



Figure 3.15: Phone Book Dialog Box.

Figure 3.16: Phone Book Entry Dialog Box.

Making a Modem ConnectionAfter properly configuring the modem, click the Start button found on the Acquisi-tion Toolbar. This will initialize the modem. If the initialization is successful, thetoolbar Dial button will be enabled. If the initialization is not successful, an errormessage will be displayed.

When the Dial button is enabled, it can be used to dial a phone book entry. To con-nect, click the Dial button to present the Modem Dial Selection dialog box shown inFigure 3.17. Choose the target location from the list displayed and click the DialModem button found in the dialog box. The program will display message screensto indicate that it is dialing, connected or otherwise (e.g., busy, etc.).



Figure 3.17: Modem Dial Selection Dialog Box.

If another computer dials your computer’s modem, MView will ask if you want toanswer the phone. Answer yes to answer the phone and establish communication.

Troubleshooting Modem ProblemsIf the modem will not initialize, follow these guidelines:

1. First, try initializing the modem using no Modem Initialization String.

2. If you cannot connect without a modem initialization string try typing some-thing as simple as “ATZ”.

3. If after trying the steps listed above, the modem still does not connect, consultthe modem’s manual or contact the modem’s manufacturer for assistance.

In general, it is important to realize that the amount of data that is actually beingsent over the modem is quite minimal (usually only one line of text each second).Thus, for best results, try to simplify everything as much as possible. Setting thebaud rate lower will solve many problems; 2400 Baud should be more than suffi-cient for most cases. If this still does not work, try turning off any compression fea-tures of the modem. Also, the modem must be in verbose mode. For most modems,adding “V1” to the initialization string will put the modem in verbose mode. Ifthere are still problems, consult the modem’s manual for the proper initializationstring.

Real-Time Data WindowWhen real-time data is received, the data that is passed to MView can be displayedduring the transmission by using the Real-Time Data Window (see Figure 3.18)command from the Real-Time menu.



Figure 3.18: Real-Time Data Window Menu

Raw Data ViewIf the Raw Data View is selected, the raw data is displayed without column headers.The raw data is the data exactly as it comes over the serial port, before MViewparses it. Figure 3.19 shows a typical display of raw data.

Figure 3.19: Raw Data View.



Translated Data ViewWhen the Translated Data View is selected, the data contained in the View List isdisplayed using the Parameter name as a column header as shown in Figure 3.20.

Figure 3.20: Translated Data View.

Digital Data ViewFigure 3.21 shows a real-time digital data view. The appearance and configurationof this display is determined by the settings chosen in the Configure Real-TimeData Window dialog box.



Figure 3.21: Digital Data View.

Once the Digital Data View window is opened, its position and size can be manipu-lated like any standard window.

Configuring the Real-Time Data WindowTo configure the Real-Time Data Window, access the Configure View dialog boxby choosing the Configure Real-Time Data Window command found on the Real-Time menu.

When the Configure View dialog box (Figure 3.22) is opened, it displays the activereal-time parameters in the Possibilities section of the screen. To view a parameterit must be added to the View list.



Figure 3.22: Configure Real-Time View.

To Add a Parameter to the View List:

1. Select a parameter or multiple parameters from the Possibilities list and clickthe Add button. Double-clicking a parameter will also add it to the View list. Ifthe desired parameter does not appear in the Possibilities list, return to theSetup Data screen and add the parameter to the data set (e.g., associate theparameter with a channel).

2. The Configure View dialog box can also be used to specify whether to viewthe data with any specified shifts applied or not. To use the shifts, click on theUse Shifts check box.

3. The Selection parameter order can also be changed by using the Move Up andMove Down buttons on highlighted items.

4. The text color and decimal format of a highlighted Selection parameter can bechanged by clicking on the Properties button. This will bring up the dialog asshown in Figure 3.23. The Text Color and the number of Decimal Places canbe different for each selected parameter. To use scientific notation check theScientific check box.



Figure 3.23: Configure Real-Time View - Properties Dialog.

The fonts, for both the caption and data, can be specified independently by choos-ing either the Caption Font or Data Font buttons. The caption can also be Left,Center or Right justified. The data is right justified to ensure that the decimal placeremains in the same position for easy of reading the display. To view the headers inthe text color specified by the parameter’s properties, check the Use colors fortitles box.

Add Log EntryThe Add Log Entry menu allows the user to add text to a log file during real-timedata acquisition. If this menu option is selected, a time stamp with the current real-time data stream will be saved in a user specified data file.

Recover Real-Time DataIf for some reason (system crash) MView must be restarted during a real-time ses-sion, it is possible to recover the data that has already been recorded before MViewwas shut down. System crashes are rare, however, they can occur during powerfailures, or if you overload the system resources and create an unstable situation.During real-time applications, it is always recommended that you minimize extra-neous use of the system. To be safe, try not to run unnecessary applications (e.g.,solitaire) during real-time acquisition.

If MView was collecting data, but suddenly shut down and then started back upagain, it would only have the real-time data collected since the last (second) start.To recover the data collected during the first session, use the following steps:



To Recover from a System or Program Crash:

1. If the entire system crashed as a result of a power failure, restart Windows andMView. When MView restarts, the program will provide a message indicatingthat “a temporary file exists, do you want to recover it?” Answer yes.

2. Next, activate the Acquisition Toolbar and open the same file that you wereworking with before the crash. A message will inquire if you want to append oroverwrite the data contained in the file. Choose append to save the data alreadyacquired and click the start button on the toolbar as quickly as possible to pre-vent any further loss of data. If for some reason only MView crashed, but theAcquisition Toolbar did not, just restart MView.

3. Once the acquisition has been restarted, select the Recover Real-Time Datacommand from the MView’s Real-Time menu. This will display the file opendialog box and pre-set the file extension to *.ADT.

4. Select the name that was given to the data file from the Acquisition Toolbar(e.g., *.ADT) and choose OK.

5. This will cause the program to re-read all the data up to the point that the con-nection was lost or the crash occurred.

6. At this point you will be back to where you were when the crash occurred,minus the data lost during any Acquisition Toolbar down-time.

3.5 Simulation Setup The process of using MView to import replay or real-time data is described in Sec-tions 3.3 and 3.4. After successfully connecting to a data source, the next step is touse the data to perform replay or real-time analysis. MView can send the real-timeor replay data to MFrac and/or MinFrac for analysis.

Sending Data To MFrac and/or MinFracTo send data to MFrac and/or MinFrac make sure that, as a minimum, you havesetup a Parameter for Time and Slurry Rate. If these parameters are not configured,a warning message will be displayed. Bottomhole Pressure, Surface Pressure,Concentration, N2 Rate and CO2 Rate may also be used for simulation; however,they are optional and only used if available.

To send real-time or replay data from MView to MFrac and/or MinFrac chooseSimulation Setup from the Simulation menu to display the screen shown in Figure3.24. MView will automatically place all configured parameters with units of pres-


3.5 Simulation Setup 273

sure in the Bottomhole Pressure list box. Likewise, all rate parameters will beshown in the Rate list box, etc. When there are multiple parameters with the sameunit, select the parameter to use from the list. Other than Time and Rate, a parame-ter can be disabled by choosing Not Available from the list.

You must indicate for all rate and concentration data whether the data was mea-sured at the Surface or Bottomhole. This information is required so that MFracinitializes the data at the correct time. When using surface data, make sure that thewellbore volume is adequately described within MFrac. If the wellbore volume isincorrect, the transport time for any portion of the treatment will be wrong and thechronology of the job will be misrepresented.

Figure 3.24: Simulation Setup Dialog Box.

Additional parameters can also be sent to MFrac for display and staging by access-ing the Additional Parameters tab as shown in Figure 3.25. Checking the Sendbox will send this parameter data to MFrac



Figure 3.25: Simulation Setup Dialog Box - Additional Parameters.

Once you have selected which parameters to pass from MView to MFrac and/orMinFrac, the programs will be in constant communication with each other.

3.6 PlotsAs soon as data is available in a data set(s), it can be used to construct plots to assistin the analysis. Composite or multi-component plots may be useful for creatingparameter match plots (e.g., measured vs. simulated pressure), comparing plausiblescenarios (e.g., best case/worst case) or simply depicting data trends in order todocument the quality control of a treatment.

Before a plot can be displayed, you must first specify the data to be displayed in theplot. In MView, this process is referred to as “building a plot” and is accomplishedby choosing the Build Plot command from the Plot menu. When selected, thiscommand will display all data sets and their available parameters.

Once the desired plots are built, they can be viewed by selecting the View Plotcommand located on the Plot menu. Whenever a plot is opened, it can be config-ured (colors, line styles, fonts, etc.) and graphically edited. It is also possible toexport the produced graphics to include them in a customized report.

Any changes that are made to the data in MView (e.g., graphical editing) areautomatically updated in MFrac and MinFrac.


3.6 Plots 275

Building PlotsAfter the data sets have been imported and setup, the Build Plot dialog shown inFigure 3.26 can be accessed from the Plot menu. This screen is used to choosewhich parameters from the different data sets are to be included on a plot. Once aplot is selected, the Parameters spreadsheet reflects the available data channelsassociated with the selected Data Set or Data Sets. In order to build a valid plot, aparameter must be assigned to the X-Axis, and at least one parameter must beassigned to a Y-Axis. The plot description will be displayed in a dark red and boldfaced font along with an error message if the settings are invalid.

The first time you build a plot or create a new plot, the legends, axes names and theplot title are set to the default Parameter text for a given Data Set. If more than oneparameter is plotted on an axis, the default axis caption will be blank.If an axisparameter is changed, the legends and affected axis name will be changed to theirdefault settings. The default settings are obtained from the Parameter text precededby the Data Set Reference Name.

Figure 3.26: Build Plot Dialog Box.



To Build a Plot:

1. Use the Build Plot command from the Plot menu to access the Build Plotscreen.

2. Select a plot by highlighting one of the plot rows located near the top of thescreen or add a new plot by clicking the Add button.

3. Enter a title for the plot within the cell in the Name column. This can beaccomplished by double clicking within the cell, or by clicking the Renamebutton.

4. To add data to a plot, under Data Sets highlight the data set box from whichthe desired data set parameters are to be selected.

5. From the Parameters drop down menu associate the parameters with a givenAxis. Only one X parameter may be chosen per data set; however, multiple Yparameters may be used.

6. Once the plot parameter selections are made, the plot can be previewed byclicking the Preview Plot button.

7. Repeat the above process to build other plots. Click the OK button to finishbuilding the plots.

The Delete button under the Plots menu can be used to delete a plot.

The Clear All button under the Parameters spreadsheet can be used to clear all theparameter associations for a given data set and plot.

Viewing PlotsAfter the plots have been built, they may be viewed. To view the built plots, accessthe View Plot selection screen as shown in Figure 3.27 by choosing View Plot fromthe Plot menu. This screen will contain all of the built plots. Choose the plots todisplay by clicking the adjacent check box and then click the OK button. Theselected plot(s) will be presented using all of the data that is available. If a plot con-tains real-time data, the real-time data will appear as the data is acquired.


3.6 Plots 277

Figure 3.27: View Plot Dialog Box.

Graphically Editing Data Whenever a plot is displayed, it is possible to edit the data using MView’s GraphicalEdit feature. Graphical editing allows you to manipulate or process the data con-tained on a plot using statistical or editing functions (moving point(s), averaging,interpolating, or smoothing data). This capability provides a powerful tool for pro-cessing poor quality data and permits operations such as graphical data shifts.

To graphically edit the active plot, choose Graphically Edit Data from the Plotmenu. The plot will continue to be displayed; however, it will automatically bemaximized and the Graphically Edit toolbar and menu will be added at the top ofthe plot (Figure 3.28). The plot must remain maximized during the graphical edit-

When graphically editing the data sent to MFrac or MinFrac for use as simula-tion input, the changes that occur as a result of the editing are automaticallyreflected in MFrac and MinFrac.



ing session. To end the graphical editing session, select End Editing from the Filemenu.

Figure 3.28: Graphical Edit Screen.

The next step in the graphical edit process is to choose the curve to be edited. Makethis selection using the list box located on the left side of the toolbar. All editingoperations are only applied to the Active Curve. This selection can be changed atany time during the editing session by accessing the list box.

Because only a limited number of mouse buttons are available to work with, MViewallows toggling the function of the left mouse button as you work with the graphicalediting feature. This is accomplished by selecting the desired function from the sec-ond list box on the toolbar. The possible choices are: Select Range, Zoom, SelectSingle Point, Drag Single Point and Drag Single Point (Y only). The function foreach of these selections is defined below.

Select RangeThe Select Range mouse function is used to choose a range of data for processing.The editing functions under the Range menu will only be applied to the active range

The Graphical Edit mode requires that all X axes parameters be increasing orconstant (i.e., non-decreasing) sequences. For example, if there are time valuesof 1.0, 2.0, 1.5 and 3.0, they may not be graphically edited. In this situation,either choose a different parameter as an X axis, or manually edit the data usingthe Edit Data command in the Data menu.


3.6 Plots 279

on the active curve. This mouse mode permits the range of the active curve to bedefined by dragging a box around it. After the range is specified, a function fromthe Range menu can be selected. The arrow keys can be used to expand or contractthe range. The selected range is shown as a box on the plot and the starting andstopping values are displayed at the top of the screen.

ZoomThis mouse function allows the normal zooming of all Meyer plots as described inChapter 1. Click and drag to define a zoom area on the screen. To return to the nor-mal magnification, access the Zoom Out command in the shortcut menu (rightmouse button), choose Zoom Out from the Plot menu or press the F5 key.

Select Single PointTo work with single points rather than a range of data, choose this mouse function.Whenever the mouse is clicked on the plot area, the program will select the closestdata point on the active data curve. If it is difficult to select a specific point, tryzooming in before selecting the point. The arrow keys can be used to change theselected point along the curve. The X and Y coordinates of the selected point aredisplayed at the top of the screen.

Drag Single PointThis mouse function is used to reposition a single point by dragging it from its cur-rent location to a new location. Click the point to be moved and then hold themouse button down while dragging it to a new location. With respect to the Y coor-dinates, there are no limitations as to where the point can be dragged. However, apoint cannot be dragged beyond adjacent points on the X axis. The arrow keys canbe used to drag the currently selected point. The X and Y coordinates of theselected point are displayed at the top of the screen.

Drag Single Point (Y only)This mouse function limits the point drag function to movement relative to the Y-axis only. When data points are spaced relatively far apart, this function may pro-vide better control of the drag function, especially for only shifting in the Y direc-tion. This feature works well for time-based plots when you want to change themagnitude of the parameter, but not the time for each data point. The left and rightarrow keys may be used to change the currently selected point. The up and downarrow keys may be used to drag the point up or down. The X and Y coordinates ofthe selected point are displayed at the top of the screen.



Graphical Edit Menu BarWhen in the Graphical Edit Mode the menu bar changes to include commands forworking with data as described below.

Edit MenuWhenever a change is made to data it can be undone by selecting Undo from theEdit menu. MView tracks the operations performed on a data set so that you canundo (or redo) in series to trace forward or backwards through the editing steps.

Range MenuThe functions provided in the Range menu provide statistical operations that canbe applied to the active range of data on the active curve. This menu also lets youdelete a group of data points directly or filter the data points based on a specifiedmaximum standard deviation. The name of the active curve is in the title bar of thedialog boxes that are displayed after selecting a Range menu item. This helps youto determine if you are working on the proper curve. The specific operations con-tained in the Range menu are summarized as follows.

Set to Value

Use this command to set all data point in the range to a specific constant value. Thedialog box shown in Figure 3.29 will then be displayed. Either a constant value canbe specified or it can be selected graphically using the Graphically Choose button.

Figure 3.29: Range Menu - Set Range to a Value.

Deleting points from a curve cannot be undone.


3.6 Plots 281

Set to Average

To replace a portion of data with the statistical average, use this command. To viewthe value calculated and used as the average, choose the Show Statistics commandfound on the Range menu.

Linear Interpolation

This will linearly interpolate each point in the range. The Linear InterpolationScreen (Figure 3.30) allows you to specify an X1,Y1 and X2,Y2 or to define themgraphically. These coordinates are used to define a line. Then for each point in theselected range, the Y value will be set on this line. Thus, even though X values maybe specified that are not on the borders of the range, all points in the range will beaffected. If Graphically Choose is selected, two points on the graph, point 1 andpoint 2, must be selected. After selecting point one, the interpolation line will bedisplayed as the mouse is moved around to select point two.

Figure 3.30: Range Menu - Linear Interpolation.

Add Shift

This command can be used to shift a range of data relative to its current Y coordi-nates. When used, this function will allow you to enter a Shift Value or select itgraphically (Figure 3.31). This shift value will be added to the current data (new =old + shift.) Note that this shift is completely different from the shift that can beentered in the Data setup screen or under the Shift menu. To shift an entire curve,relative to X and Y or only X, use the commands found in the Shift menu.



Figure 3.31: Range Menu - Add Shift.

Smooth

When data is very erratic it can be smoothed with the Smooth command. Selectthis command to display the Smooth Data dialog box shown in Figure 3.32. Usethe horizontal scroll bar to adjust the level of smoothness.

Figure 3.32: Range Menu - Smooth.

Remove Points

This will delete all of the points in the selected range. Use it with caution. It willdelete all points in the selected range for all parameters in the data set, even if theseparameters are not on the current plot. This cannot be undone.

Remove Points n Standard Deviations from the Average

This command allows you to remove bad data points in the current range. Eventhough it is using the Active Curve to determine if a point is bad or not, if a point isbad, it will delete all corresponding points in all parameters of the data set. Thiscannot be undone. A dialog box will ask how many standard deviations around theaverage are acceptable. This box will interactively show the valid range as youchange the number of deviations. The number of deviations does not have to be aninteger. After clicking OK, each point in the selected range that lies beyond thisvalid range will be deleted.


3.6 Plots 283

Show Statistics

This command displays the information in Table 3.4 about the selected range.

Show Derivative

This command will show the derivative of the active curve within the selectedrange. If the active curve is on the left scale, the derivative scale is on the right andall curves on the right scale are hidden. If the active curve is on the right scale, theopposite happens.

Point MenuThe commands in the Point menu apply only to the currently selected point on theactive curve. To see or choose the currently selected point, set the mouse functionto Select Single Point, Drag Single Point or Drag Single Point (Y only).

Set Values

Choose set values to manually type in a new X and Y value for the selected point.Note that the X value cannot be before the previous point’s X value and cannot beafter the next point’s X value.

Remove Point

This will remove the currently selected point from the active curve. It will alsoremove all other points in the data set that correspond to this point. This cannot beundone.

Shift MenuThe Shift menu allows a graphical means of shifting an entire parameter of a dataset. These are the same shifts that are set in the data sets setup screen.

Table 3.4: Definition of Statistics of the Active Range.

Value Definition

Range Start The starting point on the X scale of the selected range.

Range Stop The stopping point on the X scale of the selected range.

Min The minimum value of the active curve in the selected range.

Max The maximum value of the active curve in the selected range.

Average The average value of all points of the active curve in the selected range.

StandardDeviation

The standard deviation of all points of the active curve in the selectedrange.



Shift Active Curve

This allows access to the shift value of the active curve. Options are available toturn the shift on or off, set the shift value manually, or choose the shift value graph-ically.

Shift Active Curve’s X Data

This is the same as Shift Active Curve, except it applies to the parameter that isused for the X axis.


Chapter 4

MinFracMinifrac Analysis

4.1 IntroductionThis chapter is a user’s guide for the MinFrac program. All of the available optionsand the basic procedures used for running the software are covered in this chapter.Please refer to Appendix F for specific information regarding the governing equa-tions, modeling techniques and numerical procedures used in the MinFrac AnalysisSoftware.

The term minifrac is commonly used to describe any type of injection test per-formed in a reservoir to obtain characteristic information associated with thehydraulic fracturing process. Tests such as these are usually applied, as part of thedesign optimization process, to calibrate the fracture model input data and redesignthe treatment. These tests typically involve periods of intermittent injection fol-lowed by intervals of shut-in and/or flowback. As with any well test, pressure andrate are measured throughout a minifrac and recorded for subsequent analyses.

MinFrac for Windows was developed to aid the fracturing engineer in analyzing thedata recorded during a minifrac treatment. The principles of fracturing pressureanalysis have been discussed extensively in the literature during the past severalyears1-9. The evolution of this technology has resulted in procedures that permit theinterpretation of injection and fall-off pressures in order to characterize the basicfracturability of a reservoir. This process results in the ability to approach an opti-mum treatment design.

MinFrac is a comprehensive tool that implements the latest fracture injection andpressure decline theory. With the many types of analyses and superposition deriva-tives available, MinFrac is considered a “state of the art” simulator by the petro-leum industry.


286 MinFrac: Minifrac Analysis

An outline of the basic steps for using MinFrac is shown in Table 4.1.

Table 4.1: MinFrac Basic Steps.

Step Program Area

1. Open an existing MinFrac data file (*.minfrac)or a new file File Menu


3. Data Optionsa. Generalb. Graphicalc. Fracture

Options Menu

4. Data Inputa. Fracture Model

• Description• Base Data• Leakoff Data• Closure Data (User Specified)

b. History Match Datac. Import Data Filed. Edit Imported Data

Data Menu

5. User Specified Closure –go to step 7


4.1 Introduction 287

MethodologyMinifrac analysis provides a method of estimating fracture efficiency, closure pres-sure, ISIP, net pressure, fracture dimensions and leakoff coefficients prior todesigning a full-scale fracture treatment. These types of analyses, as originally for-mulated by Nolte1-5 and modified by Castillo6 quantify the fracturing process asestimated from the measured pressure decline data.

Most minifrac analyses are based on Nolte's equations and do not account for theeffects of fluid rheology or the conservation of momentum. The measured pressuredecline data is simply used in place of solving the momentum equation. Neglectingmomentum can result in unrealistic estimations of fracture characteristics and fluidleakoff coefficients that are critical to the design of the main fracture treatment.

6. Analysisa. Analysis Wizard

• Select Analysis• Follow Wizard Steps…

b. Step Rate• Select Ranges• Select Points• Pressure Table• Diagnostic Plot

c. Step Down• Select Ranges• Select Points• Pressure Table• Diagnostic Plot

d. Horner• Select Ranges• Select Points (Horner Plot)

e. Regression• Select Ranges• Select Points (Analysis)• History Match

Analysis Menu

7. Outputa. Simulation (base data)b. Simulation (history match data)c. View Reportd. Export Reporte. Report Configuration

Output Menu

Table 4.1: MinFrac Basic Steps.



Up until 1987, only the width-opening pressure relationship and pressure declinedata were used to estimate minifrac characteristics. Lee7 improved upon this byincluding Biot's energy balance equation for two-dimensional type fractures geom-etry models.

The energy balance method does eliminate some of the anomalies in minifrac anal-ysis. However, this method does not fully account for viscous driven fractures.

Meyer and Hagel8-9 (1988, 1992) reported a new minifrac methodology that solvedthe governing conservation of mass and momentum equations for power-law typefluids using the 2-D fracture propagation equations-of-state. The solution techniquedoes not assume the fracture width is proportional to the measured pressure.Instead, the governing mass and momentum equations are coupled with the mea-sured closure time to predict fracture propagation characteristics. From the numeri-cally simulated fracture geometry, pressure, fluid efficiency and leakoff coefficient,you can determine which fracture model more closely represents the measuredpressure response and formation permeability.

The main advantage of this technique is that mass and momentum are both satis-fied. In addition, the important effects of flowback, interference closure, timedependent leakoff and fluid rheology are simulated.

The numerical results are used in conjunction with the measured pressure declinedata to history match a number of fracture characteristics such as fracture height,pay zone height, Young's modulus and spurt loss. Closure time can also be moreaccurately estimated from these parametric studies.

The equations of mass conservation, continuity, width-opening pressure, momen-tum and constitutive relationships for fracture propagation models are formulatedbased on the methodology of Meyer10-12. Refer to these references and Appendix Afor a detailed description of the model assumptions and solution technique.

DefinitionsThe following basic concepts are critical to the use of the MinFrac Program. If youare uncertain about the terminology, please review the following definitions. Formore information about minifrac theory, refer to Appendix F.

Fracture Closure PressureThe fracture closure pressure ( ) has been given many names by the petroleumindustry. For example, it is often referred to as the instantaneous shut-in pressure

pc



( ), the minimum horizontal stress ( ), the least principal stress ( ), the

frac gradient (i.e., F.G. or fracture closure pressure/depth to formation) or even thefracture propagation pressure. Each of these parameters have their own unique def-inition. Many of these definitions of course are used inappropriately and refer moreto the extension or propagation pressure than to the closure pressure.

The correct definition of fracture closure pressure as used throughout this guide isthe pressure in the fracture at the point of closure. This is more commonly referredto as the minimum horizontal stress or least principal stress.

The fracture closure pressure is an important parameter that is generally obtainedfrom the decline pressure(s) following a minifrac or stress test. Once the fractureclosure pressure is known, it can be used as a reference to determine the fractureclosure time, which in turn is used to find fluid efficiency. Fracture closure pressureis also necessary in defining net pressure during injection: net fracturing pres-sure, , is the difference between the pressure in the fracture, , and the

closure pressure (i.e., ). The fracture net pressure at the

end pumping is .

Fracture EfficiencyFracture efficiency is defined as the ratio of the fracture volume to the volume ofslurry injected. The fluid efficiency, therefore, changes as a function of volumeinjected. This change depends on the rate of creation of fracture area, as well as, theleakoff characteristics of the fracturing fluid and reservoir. The fracture efficiencyis approximately equal for all models, since it is only a function of the pressuredecline slope and closure time. For a minifrac, the fluid efficiency of interest is gen-erally the efficiency at the end of pumping. This instantaneous value provides a ref-erence point for determining the total leakoff coefficient relative to a given volumeof fluid injected and fracture geometry model.

Total Leakoff CoefficientThe leakoff coefficient obtained from a minifrac is the total leakoff coefficient ( ).The total leakoff coefficient that is calculated for each model, is a function of thefluid efficiency and the fracture area1-13. This coefficient is generally expressed inunits of . This has become a standard way of reporting the leakoff charac-teristics of a fluid in a given reservoir. The total leakoff coefficient is actually acombination of all the mechanisms acting to prevent fluid loss to the formation.These mechanisms are broken down into numerical relationships that are coupled

ISIP σHminσmin

Δpnet pfracture

Δpnet pfracture pclosure–=

Δpnet ISIP pclosure–=

C

ft min⁄



in the MinFrac program using the Harmonic fluid loss model as defined in Appen-dix D.

For additional information about the fluid leakoff models and minifrac leakoff referto Appendices D and F.

Fracture Geometry ModelsUnderstanding the types of two-dimensional (2-D) models used in the interpreta-tion of a minifrac treatment is critical to the data analysis process. Each 2-D modelsolves the fracture geometry with inherent assumptions about how hydraulic frac-tures propagate. The assumptions that are part of a numerical model solution arewhat dictates whether a particular fracture geometry model is applicable to a givenreservoir. For more information regarding the 2-D models used in MinFrac, pleaserefer to Chapter 2.

Pressure During InjectionThe magnitude of the injection pressure and the rate of change of pressure may beas important as all other dependent parameters obtained from a minifrac treatment.Rate changes and fluid property changes have predictable influences on theexpected pressure response during a minifrac as well as the propped fracture treat-ment. Deviations from the expected pressure response can be analyzed for probablecauses. For example, if bottomhole pressure is available during a minifrac by plac-ing a gauge close to the perforations, then the measured pressure, , can beexpressed as follows:

Each of these components has their own dependency on rate and fluid proper-ties. A change in either one (i.e., rate or fluid properties), therefore, should bringabout a predictable change in . If actual changes in differ from expec-tations, plausible explanations may be considered. When this occurs, the signifi-cance of the deviation should be weighed against it's influence on the desiredoutcome of the propped fracturing treatment. This process allows the engineer todecide whether the appropriate model(s) is being used; and to decide whether thesignal (i.e., ) and model combination provide a unique understanding of thecurrent fracturing process.

pgauge

pgauge Δpnet Δpperf Δpnear wellbore σmin+ + +=

Δp

pgauge pgauge

pgauge



Determining ClosureThe method chosen to identify fracture closure will depend on the test procedureand on the quality of the acquired data. This guide is not meant as a primer to theseprocedures; however, a list of the industry accepted methods is given below alongwith a brief description of each. All of these methodologies are grouped here asminifrac analyses. For additional information on each of these methods, refer to thereferences cited.

Micro-Frac TestThis type of test is used to measure the in-situ stress in discrete intervals within azone. The procedure involves isolating a narrow interval using packers in order topump a minimum volume of fluid (usually low viscosity) to break-down and ini-tiate a hydraulic fracture. The resulting can be determined by plotting theacquired pressure versus the square root of time. When performed properly, it isgenerally accepted that the is a reasonable estimate of the minimum horizon-tal stress and, therefore, is a form of closure pressure. The problems associated withusing this type of test, particularly in cased and perforated wellbores is discussed byWarpinski14,15.

Step Rate TestA step rate test is used to determine the fracture extension pressure that is typicallyconsidered the upper bound for the minimum horizontal stress or closure pressure.After break-down, fluid is pumped at increasing flow rates in a stair-step fashion.Ideally, each flow rate is maintained until a stabilized pressure is achieved. In lieuof achieving a stabilized pressure, it has been proposed that periods of equal timefor each flow rate be used. Regardless, the bottomhole pressure at the end of eachrate interval is then plotted versus rate to identify a change in slope. This change or“break” indicates the start of fracture extension that is theoretically equal to themagnitude of the closure pressure plus the fracture friction and propagation resis-tance.3

Pump-In/DeclineUnlike the discrete measurements associated with Micro-Frac Testing describedabove, un-isolated injection tests performed through the entire perforated heightprovide estimates of the state-of-stress over the total interval affected by the frac-ture. These types of tests involve larger volumes of fluid than a test designed solelyfor the purpose of determining in-situ stress. To obtain representative stress data, aswell as characteristic fluid loss, it is essential that the volume of fluid pumped cre-ate a fracture of significant dimensions. For this reason, a “pump-in” test is usuallypreceded by a step rate test to ensure that the propagation pressure is obtained and afracture is created.

ISIP

ISIP



The basic principle of this type of injection test is to create and propagate a fracturein order to monitor the natural pressure decline following shut-in. Once the pres-sure and time data is acquired, various graphical methods are used to identify theclosure event. Diagnostic plots involving numerous time and characteristic func-tions can be produced to identify fracture behavior. For high permeability wells, aHorner plot is sometimes used to identify pseudo-radial flow. The utility of this plotis based on the concept that if a semi-log straight line is observed, the flow ispseudo radial and therefore, the fracture is closed during the remaining portion ofthe test. That is, data beyond the point of pseudo-radial flow is not used to evaluateclosure time.

To “pick” the closure time, normally, a plot of pressure versus square root of time isa good place to start. For this time function, initially, pressure should decline alonga straight line indicating linear flow in the fracture. The point at which the fracturecloses should be marked by a distinct change in slope. Unfortunately, depending onthe relative relationship between the physical properties of the fracture and the res-ervoir, the change in slope may be either positive, negative or so subtle that it is notdetected. Different situations require different diagnostic plots (i.e., different X-axis functions) to interpret the closure event. When conditions result in an inabilityto identify closure (e.g., low permeability resulting in long closure times), a Pump-In/Flowback test may be required.

Pump-In/FlowbackWhen fluid loss is extremely low, as it is in many low permeability reservoirs, theincreased time required for a fracture to close due to natural pressure decline canmake the identification of closure extremely difficult. When this scenario is likely,the closure identification process may be augmented by using a Pump-In/Flowbackprocedure. This type of test uses constant rate flowback immediately following theinjection to increase the deflation rate of the fracture. Flowback is designed tomatch the order-of-magnitude rate of leakoff. The objective is to affect the pressureresponse in such a way as to develop a characteristic curvature (S-shaped) thatreverses from positive up to positive down (see Figure 4.1). This procedure usuallyinvolves trial and error to determine the proper flowback rate. The return rate istypically controlled using a manifold assembly containing operable valves (e.g.,gate valves) or an adjustable choke. A low-rate flow meter is also beneficial inmonitoring the flowback rate and may allow digital acquisition of the data for sub-sequent analysis.



Figure 4.1: Idealized Flowback Pressure.

Using flowback to close a fracture is based on an interpretation of the closure pro-cess that produces at least three distinct stages of pressure versus time behavior.Each stage represents a change in characteristic behavior, described as follows.During the first stage, fracture deflation, due to leakoff and flowback, dominatesthe pressure response producing a pressure trend that appears similar to naturaldecline (i.e., concave up with respect to time). The second stage of the pressuredecline behavior is characterized as a transition from a fracture that is fully open toone that is partially closed. It is this process that initiates the desired curvaturereversal.

The fracture continues to close and the communication between it and the wellboreis choked as a result of decreasing conductivity and corresponding flowback rate.The restriction caused by fracture closure reduces the wellbore recharge rate result-ing in an increase in the relative wellbore deflation rate. This produces a character-istic acceleration in the rate of pressure decline as the wellbore pressure is reduced.Once this point of acceleration is identified, the fracture is assumed to be closed.The identification of this point is typically made using a plot of the derivative( ) to illustrate the change of pressure with respect to time. The maximumpoint on the derivative plot represents the maximum rate-of-change in the pressuredecline and, therefore, the point at which the fracture is closed8,16.

Derivative MethodAnalyzing the derivative, as a function of time is another method of deter-mining closure. The resulting trend represents the rate-of-change in the pressure

pd td⁄

pd td⁄



with respect to time. Depending on the type of data (i.e., flowback or naturaldecline), the derivative plot can be used to identify the closure by observing a char-acteristic change in the shape of this relationship. Refer to Figure 4.2 for an exam-ple of the desired trends.

Figure 4.2: Idealized Pressure and Derivative Trends for Time and Slope.

Basic ConceptsThe fundamental methodology implemented in the MinFrac Program is discussedby Nolte1-5, Castillio6, and Meyer8.

The basic concepts essential to using MinFrac and understanding the results are dis-cussed below for each type of analysis:

Natural Decline Flowback

P

Time

tc

P

dP

dt

Time

P

dP

dt

dP

dt

P

Root Time

P

dP

dt

Root Time



Step Rate AnalysisA step rate test is used to determine the fracture extension pressure. This is typi-cally considered the upper bound for the minimum horizontal stress or closure pres-sure.

After breakdown, fluid is pumped at increasing flow rates in a stair-step fashion.Ideally, each flow rate is maintained until a stabilized pressure is achieved. In lieuof achieving a stabilized pressure, it has been proposed periods of equal time foreach flow rate can be used. Regardless, the bottomhole pressure at the end of eachrate interval is then plotted versus rate to identify a change in slope. This change or“break” indicates the start of fracture extension that is theoretically equal to themagnitude of the closure pressure plus the fracture friction and propagation resis-tance.

Step Down AnalysisThe step down analysis is used to calculate perforation and near wellbore frictionlosses. If the step down analysis is performed using surface treating pressure, thepipe friction needs to be entered for analysis credibility. This analysis is used todetermine near wellbore pressure loss effects (i.e., problems with anomaly highpressures which may cause a near wellbore screen-out).

This analysis is performed after fracture propagation has been established. Thenduring shut down the rate is decreased in a stair-step fashion for a short period oftime while the pressure stabilizes. As the injection rate decreases, the pressure willalso decrease as a result of perforation and near wellbore pressure losses. The rela-tionship between the decreasing rate and pressure results in a determination of nearwellbore pressure losses.

Horner AnalysisThe Horner plot is used to determine if pseudo-radial flow developed during pres-sure decline. If a semi-log straight line is observed and the line can be extrapolatedto a reasonable value of reservoir pressure, radial or pseudo-radial flow may beaffecting the decline behavior. This suggests that the fracture is already closed andthat data beyond the point of influence need not be considered in the evaluation ofclosure.

The Horner plot provides a lower bound of the minimum horizontal stress or clo-sure pressure.



Regression AnalysisRegression Analysis is a procedure whereby time dependent rate-pressure data dur-ing fracture propagation and shut-in is analyzed to determine information regardingfracture characteristics. The specific methodology used is given below:

1. Graphically identify the major events that occurred during the treatment cycle(e.g., Initiation, , Closure etc.). Diagnostic plots can be generated using avariety of time functions. These plots are used in the determination of closure.In addition, a statistical procedure can be invoked to automatically determineclosure.

2. Once the events, including closure time, have been identified, parameters canbe selected and history matches performed comparing the theoretical responseto the actual measured data for each fracture model contained in the program.A regression technique is used to minimize the difference between the modelresults and the measured data. This process can be repeated as many times asdesired.

3. When satisfied with the combination of history-matched responses and param-eter optimization for the fracture geometry models, calculation of the fracturegeometry and associated fluid loss parameters can be made.

The following information can be determined from a properly conducted Regres-sion Analysis:

1. Instantaneous shut-in pressure, .

2. Closure pressure, .

3. Fracture net pressure, .

4. Closure time, .

5. Fracture efficiency, .

6. Fraction of PAD, and .

7. Parametric uncertainty (history match).

8. Applicable fracture model (history match and net pressure).

9. Fracture area, , based on the best-fit geometry model.

ISIP

ISIP

pc

Δp ISIP pc–=

tc

η G tc( ) 2 G tc( )+( )⁄≅

fpad min1 η–( )2≅ fpad max

1 η–( ) 1 η+( )⁄≅

A



10. Leakoff coefficient, , if the fracture area is known.

Derivative MethodThe Derivative Method is one of the methodologies for determining inflectionpoints (i.e. fracture closure). Analyzing the derivative, as a function of timeis a method of determining closure. The resulting trend represents the rate-of-change of pressure with respect to time. Depending on the type of data (i.e., flow-back or natural decline), the derivative plot can be used to identify the closure byobserving a characteristic change in the shape of this relationship.

Nolte1-5 was the first to implement this concept. In simple terms, if one can find atime function where the rate of pressure decline with respect to a time function is aconstant during fracture closure, the closure time would be indicated by a deviationfrom the measured and theoretical pressure declines. This concept is formulatedbelow:

or

Where is the pressure, , and is a time function.

The time function Nolte purposed was the Nolte G time (i.e., ). Table4.2 lists a number of other time functions also used in MinFrac. During closure theuser may perform a minifrac analysis with a time function that gives the best fitpressure decline match with an inflection point at closure. Although this time func-tion may give the best fit, it may not be a “unique solution”.

Table 4.2: MinFrac Time Functions.

Log/Log Slopes*

Time Scale Definition Storage Fracture RadialFlow

Data Time - - -

Time - - -

C

pd td⁄

ISIP p Ψ( )– ΨΨdd ISIP p–( ) Ψ

ΨddP= =

p Ψ( ) ISIP ΨΨd

dP–=

p P ISIP p–= Ψ

P GdP dG⁄=

tData

t tData INIT–=



After Closure AnalysisThe purpose of the after closure analysis is to determine the formation permeabilityand reservoir pressure from the pressure response of a fractured (or unfractured)well during the infinite-acting time period (i.e., late time period or radial solution).Following is a brief summary of the after closure analysis. A detailed formulationand description of the governing equations is presented in Appendix K.

General EquationThe general form of the after closure pressure response is

(4-1)

where is slope, is a time function and is the straight line intercept at.

Permeability and Reservoir PressureAs discussed above, if the pressure is plotted against in Cartesian coordinates,the late time portion of the curve should follow a straight line. The permeability can be calculated from the slope of the straight line (i.e., ). The apparentreservoir pressure can be found from the intercept of the extension of thestraight line with the axis.

Delta Time 1 1/2 1/4

Nolte Time 1 1/2 1/4

Root Theta’ Time 2 1 1/2

Root Delta Time 2 1 1/2

Root Nolte Time 2 1 1/2

Nolte G Time 2 1 1/2

Data Start of Pumping Time = , Data End of Pumping Time =

Pumping Time; .

See the Meyer Appendices for additional information on these functions.

Table 4.2: MinFrac Time Functions.

Δt t tp–=

tD t tp⁄ 1–=

Θ′1 2⁄ t tp⁄( )1 2⁄ 1–=

Δt1 2⁄ t tp–( )1 2⁄=

tD1 2⁄ t tp⁄ 1–( )1 2⁄=

G f t tp ..., ,( )=

INIT EP

tp EP INIT–=

p p*– m F×=m F p*

F 0=

p Fk

m k 1 m⁄∝

p*F 0=



Diagnostic Plots and DerivativesDiagnostic plots similar to those used in the regression analysis using the Nolte Gfunction can be used to help identify radial flow (pressure transient). The generalrelationships are given below.

Taking the derivative of Eq. (4-1) with respect to the time function, we find

(4-2)

or

(4-3)

Therefore at late time (small values of ) the measured pressure data should over-lay Eq. (4-3) in Cartesian coordinates. Figure 4.3 illustrates the use of Eq. (4-3) byoverlaying the derivative function to help identify the intercept (reservoir pressure)and late time slope (permeability).

dp dF⁄ m=

p p* F dpdF-------+=

F



Figure 4.3: After Closure Analysis - Surface Pressure vs. Nolte - FR Linear Plot.

If a plot (see Figure 4.4)of net pressure vs. is generated, the pres-sure should overlay the following equation

(4-4)

Δp p p*–= F

Δp p= p*– F dpdF-------=



Figure 4.4: After Closure Analysis - Delta Pressure vs. Nolte - FR Linear Plot.

Taking the natural log of Eq. (4-4) we find

(4-5)

Therefore, the net pressure curve in log space should also overlay the derivativefunction for radial flow.

Another important derivative is the log slope. Taking the natural log of Eq. (4-1) wefind

(4-6)

where for the slope is equal to the net pressure (i.e., ).

Eq. (4-6) also illustrates that if is plotted versus , the log-log slopewill approach unity for large times. That is

as

Δp( )ln F dpdF-------⎝ ⎠

⎛ ⎞ln=

p p*–( )ln m( ) F( )ln+ln=F 1= p p*–( )ln m( )ln=

Δp( )ln F( )ln

d Δp( )lnd F( )ln

-------------------- 1→ F 0→



as shown in Figure 4.5.

Figure 4.5: After Closure Analysis - Delta Surface Pressure vs. Nolte- FR Log-Log Plot

Please refer to Appendix K for a detailed formulation and description of the gov-erning equations.

MenuThe Meyer MinFrac menu bar is shown in Figure 4.6.

Figure 4.6: MinFrac Main Menu.

4.2 OptionsThis section describes the various options available in MinFrac. The Menu layout(see Figure 4.6) and Data options are as shown in Figure 4.7. A complete descrip-tion of the options is presented in this section.


4.2 Options 303

Figure 4.7: MinFrac Data Menu.

To access the Options screen, select the Data⏐Options menu. The dialog box dis-played in Figure 4.8 will then be presented. The default menu selection of the radiobutton is Graphical Technique – Use real-time data from MView opening a newfile.

Figure 4.8: Options Screen.

The Options screen determines what information is needed for a particular type ofanalysis. The specific data displayed in a screen or the existence of a data screenitself varies depending on the options selected. The selections made in the DataOptions screen set the scope for data used in MinFrac. These options establish theGeneral technique, Graphical and Fracture options.



General OptionsThe General Options screen allows the user to specify the type of Graphical or userspecified technique used to perform the analysis. Figure 4.9 shows the GeneralOptions screen.

Figure 4.9: General Options.

To take full advantage of MinFrac’s ability to process acquired pressure and ratedata, choose one of the Graphical Technique options. Selecting the GraphicalTechnique option configures the program for graphically analyzing the data froman actual minifrac treatment. This data can be replay data from an ASCII data fileor Replay/Real-Time data sent by MView.

The third choice is to simply perform minifrac closure and geometry calculations.To use MinFrac to perform calculations only, select the User Specified Closureoption. This option is not recommended, since the user is required to specify thepertinent regression data of closure time, pressure, etc. This option is used to per-form parametric studies with different scenarios for various closure times andhence fracture efficiencies.

Graphical Technique - ASCII text fileSelect this option to import data from a text file directly into MinFrac.


4.2 Options 305

Graphical Technique - from MViewSelect this option to use the real-time or replay data from MView for the graphicalanalysis. When using this option, all of the plots will update automatically whenreal-time data is acquired. This is the recommended method.

User Specified ClosureIf the complete pressure and rate data records are not available to analyze, or if clo-sure time is to be determined by another method, select User Specified Closure.

This selection will disable all data input that corresponds to graphical analysis andsimply present data dialog boxes that require characteristic reservoir and fracturedata.

In addition to the reservoir data, specific information regarding the minifrac mustalso be entered. For User Specified Closure the following information is required:

• Injection rate

• Volume injected or injection time

• Closure time

• Closure pressure

Graphical OptionsThe Graphical Options are only fully available if a Graphical Technique option isselected in the General Options box. The Graphical Options provide choices for theUser Specified Pumping Data, Derivative, Mouse Button and Wizard steps. Figure4.10 shows the Graphical Option choices.



Figure 4.10: Graphical Options.

The Graphical option allows you to decide where specific information for the anal-ysis will be obtained. This set of options is provided for occasions when a completedata set is not available for analysis. When this occurs, the data set can be supple-mented by using the calculation facility contained in this dialog.

User Specified Pumping DataInterpretation of a minifrac treatment requires that the rate and volume of fluidinjected be defined.

These may be defined graphically (recommended if rate data is available) or manu-ally. Select the option that corresponds to how you want to enter the data. To enterall data graphically, choose Nothing (All data taken from graph).

These variables are necessary to determine fluid efficiency and the leakoff coeffi-cients for each fracture model. The program is capable of obtaining these from therate and pressure versus time graphs by using the event selections to start or end atime interval for processing. The data contained within a time interval (e.g.,between Initiation and End of Pumping) must contain sufficient data to identifythe injection period, as well as the period of pressure decline. For those instances,when data is missing from the imported file, it must be entered in this dialog or cal-culations cannot be performed. For example, when only the bottomhole pressureand time records are available, the injection rate and volume or some complemen-


4.2 Options 307

tary combination of time and volume, or rate and time must be input in order tocontinue with the analysis.

The injection time found in the calculator portion of this dialog is considered to beΔt. This is important because of the link between these values and those that may beobtained graphically (i.e., end of pumping time, ISIP time, closure time, etc.).

DerivativeThis option pertains to the three-point method used to calculate the derivative in theregression plots. However, since many data pressure values stay constant over afew time steps, a three-point method would result in many data points having aderivative of zero. To avoid this situation, consecutive points are not used.

Percentage of (MAX-MIN) used to find the derivative at each point

The Derivative Option allows you to enter the percentage of the data between theMIN and the MAX values used to calculate the derivative. For example, a value of12% indicates that for each point, the derivative is calculated using 12% of the totalrange defined between the MIN and MAX selections. Therefore, when possible,data within 6% on either side of a calculation point is used.

Use right mouse buttons to select points

The default is to select points with the left mouse button. To select points with theright mouse button, check this box. It is recommended to leave this box unchecked.

Minimize number of wizard steps

The Wizard Analysis allows the user to step through the different analyses system-atically. The wizard also has an option (see the Wizard screen) to have a descriptionof the next step visible or hidden in the regression plots. The description and proce-dure for doing the analyses is beneficial for first time users or as a refresher. Conse-quently, the only time this box should be unchecked is if you want the wizard tostep you through (very slowly) every detail of the procedure.

If the values entered in the Graphical Calculation Options dialog are inputprior to importing the actual data records and subsequently these values arespecified graphically, it will be necessary to return to this option screen andreset the option to: Nothing (All data taken from graph). This will allow theactual data values to take precedence over those input manually in this datascreen. Keep in mind that these options are global for a given interpretation.Changing them to obtain a different set of results will change the entire analy-sis.



It is recommended that this box be checked. Also, if you have to click next a num-ber of times to proceed to the next step in the regression analysis, you may have thisbox unchecked.

Fracture OptionsClicking the Fracture tab found in the Options screen will access this group ofoptions. The Fracture Options provide choices for the fracture geometry model andconstitutive relationships that affect the fracture solution methodology. Figure 4.11shows the Fracture Option choices.

Figure 4.11: Fracture Options Screen.

Fracture Friction ModelNormally, laminar flow exists in the fracture and this option may not be needed(i.e., unchecked). For this case, the classical solution for fluid flow in a rectangularslot (as modified for an ellipsoidal fracture width) is used and the Darcy frictionfactor takes the form:


fD 24 Re⁄=



4.2 Options 309

Deviations from laminar flow effect the frictional dissipation in the fracture and,therefore, on the pressure predicted by a model. Turbulent flow in the fracture mayalso occur when very low viscosity fluids (e.g., gas) are pumped at high rates. Toaccount for turbulent flow and improve the ability to predict non-laminar frictionalpressure loss in a fracture, the following friction factor expression is used when theFracture Friction Model is turned on:

Irregularities along the fracture face (e.g., tortuosity, bifuraction and wall rough-ness) that interrupt and disturb fluid flow can also result in greater energy dissipa-tion. These effects can be modeled by increasing the a coefficient or modifying thewall roughness factor discussed below.

Typical values for the a and b coefficients have been developed empirically inaccordance with Prandtl's Universal Law of the Wall17 as shown in Table 4.3.

Wall RoughnessWhen this option is turned off (not checked), the Darcy friction factor inside thefracture is used without modification as determined from the selections made in theFracture Friction Model option. This selection assumes that the fracture surface isa smooth planar feature without roughness.

To include the effects of roughness (or waviness) on the frictional dissipation, turnthis option on. This will result in an increase in the frictional pressure drop andfracture width, as well as, a decrease in fracture length. If this option is used, thefriction factor defined in the Fracture Friction Model option will be modified usinga Friction Factor Multiplier. The relationship used is defined in the expressionshown below:







fDa

Reb---------=



where



Tip EffectsThe observed field pressures for some treatments are at times much higher than thesimulated pressure. This discrepancy in measured pressure can be minimized in anumber of ways. Typically, the friction factor multiplier, fracture toughness, nearwellbore effects, confining stress or rock/reservoir properties are modified to obtain


fD′ Mf fD=

fD′

fD

Mf


4.2 Options 311

a match. However, if the pressure discrepancy is due to excess pressure, an over-pressure function can be applied at the tip. In MinFrac, excess pressure can beapplied using two mechanisms: 1) Fracture Toughness, and 2) Tip Over-pressure.

Over-pressure, as it is incorporated in MinFrac, accounts for the extra pressurerequired at the fracture leading edge for propagation to occur. This extra resistanceat the fracture perimeter (tip) requires additional pressure (energy) to propagate thefracture. As a result, when this option is used, higher pressure must be applied atthe inlet (surface or BHTP) to compensate for losses that occur in the fracture.

Tip effects, in general, remain an area of some controversy and considerable dis-cussion. Plausible explanations for these effects have been proposed. The possibili-ties include tip friction due to flow resistance, rock properties effects (e.g.,toughness as a function of stress at the leading edge or poroelasticity), or it may bea consequence of fracture geometry (e.g., complex geometry and/or multiple frac-tures).

In this version of MinFrac, tip effects represent a flow resistance at the tip. Regard-less of whether you believe this flow resistance is due to viscosity effects or someother phenomena related to the tip region (e.g., tip geometry) the general effect onpressure is typically the same (i.e., resistance is resistance). It is important to note,however, that this type of resistance differs from fracture toughness in its classicalapplication; over-pressure varies with injection rate and time, fracture toughnessdoes not.

The range of the over-pressure factor allowed by MinFrac is between 0 and 1.0. Ifthis option is disabled, a default value of zero is used. Usually, the Tip Effect optionis suggested when the measured injection pressures are well above the theoreticalvalues predicted by a classical model (i.e., Linear Elastic Fracture Mechanics).

When reasonable values have been implemented for wall roughness, friction factormultiplier, toughness and other formation properties, a value between 0.1 to 0.4may be justifiable. The larger the over-pressure factor the greater the increase inthe net pressure. If you are having difficulty relating the over-pressure factor topressure, one approach is to use MinFrac to automatically regress on the tip factorto determine an appropriate value. This best fit value from matching the net pres-sure in a minifrac analysis is a good place to start.

Many engineers mistake near wellbore pressure loss for excess net pressure. Keepin mind that when the injection rate changes suddenly, the near wellbore pressureloss also changes instantly whereas the fracture net pressure cannot because of stor-



age (i.e., if the rate drops suddenly and the BHTP follows, this is not excess pres-sure but frictional dissipation in the near wellbore region).


Proppant EffectsAnalysis of the pressure decline from a fracture containing proppant (i.e., interfer-ence closure) requires special considerations. If the fracture is partially filled withproppant, the fracture will close on the proppant before all of the fluid leaks off.The propped fracture volume corresponds to the reduction in fluid required to lea-koff before the fracture closes. The limiting case occurs if the fracture is 100%packed with proppant. This exists when the Propped Fraction entered in the Frac-ture Model Options dialog box is set to 1 (see Figure 4.11). For this scenario, thefracture closure time would be equal to 0 (see Appendix F).

The phenomena of tip over-pressure has been referred to as “dilatancy” bysome researchers. It is not clear whether these researchers are referring to rockdilatancy or fluid dilatancy. Fluid dilatancy refers to a shear-thickening fluid.Rock dilatancy describes volumetric expansion of a material that is rapidlyapproaching failure and is usually associated with the micro-cracking process.There has been no published explanation on the effects of rock dilatancy on netpressure in a crack, and to our knowledge, no correlations exist. The desiredeffect (i.e., an increase in pressure) can be achieved due to viscosity effects (i.e.,fluid dilatancy) or as a result of stress dependent rock properties that may ormay not be related to rock dilatancy. This is commonly referred to as nonlinearelastic deformation. Figure 4.13 illustrates one possibility.


4.3 Data Input 313

If there is no proppant in the fracture, the entire volume in the fracture at the end ofpumping must leakoff into the reservoir before the fracture actually closes (unless itdoes not close totally). This case corresponds to a value of 0 for the Propped Frac-tion. The fracture closure time and pressure can be obtained using the methodologygiven in Appendix F.

4.3 Data InputIn addition to the actual treatment records imported for use by the program, otherparameters characterizing the reservoir physical and mechanical properties arerequired. All analyses are performed with 2-D fracture geometry models, whichgreatly reduces the required data input.

Use the Data menu commands to input data (see Figure 4.7). The Closure Datacommand is only activated if the User Specified Closure option has been previ-ously selected. Likewise, the History Matched Data command is not enabledunless the Graphical Technique is used. This section explains the Data menu inputscreens.

The input data requirements are presented below.

DescriptionThe Data Description screen shown in Figure 4.14 provides a location for enteringmiscellaneous information about the specific analysis being performed.



Figure 4.14: Description Dialog Box - MinFrac.

Base DataThe Base Data dialog box shown in Figure 4.15 provides the information neces-sary to describe the rock properties, fluid rheology and fracture parameters. Each ofthese data items are discussed below.


4.3 Data Input 315

Figure 4.15: Base Data Dialog Box.

Young's ModulusYoung`s modulus or the modulus of elasticity is the slope (or derivative) of a stress-strain curve over the elastic portion of the curve. For linear-elastic deformation,Young’s modulus is a constant with a unique value for a particular material and in-situ conditions. The modulus represents the materials ability to resist deformationunder load. It is therefore a measure of the materials stiffness. As the stiffness (E)of the rock increases, the fracture width will decrease and the length will increasefor a given set of input parameters. See Appendix A for more information regardingthe sensitivity of this parameter.


Table 4.4: Young’s Modulus for Various Rock Types.

Rock Type Range (106 psi)

Range (107 kPa)

Limestone-Reef Breccia 1 - 5 0.5 - 3

Limestone-Porous or Oolitic 2 - 7 1 – 5

Limestone-Med. to Fine Grained 4 - 11 2.8 - 7.6



Fracture ToughnessThe definition of fracture toughness is obtained from the concept of stress intensityfactor, developed in linear elastic fracture mechanics (LEFM). Fracture toughnessis a measure of a material’s resistance to fracture propagation. It is proportional tothe amount of energy that can be absorbed by the material before propagationoccurs. The basis for this relationship involves the assumption that pre-existingdefects exist and induce high stress concentrations in their vicinity. These sitesbecome points for crack initiation and propagation. See the MFrac chapter for moreinformation.







Dolomite 6 - 13 4.14 - 9

Hard, dense Sandstone 4 - 7 2.8 - 5.2

Medium Hard Sandstone 2 - 4 1.4 - 2.8

Porous, unconsolidated to poorly consolidated 0.1 - 2 0.35 - 1.4


ac

T

T KIC πac⁄=

KIC

σ

KI σ γHξ=

σc KIC γHξ⁄=

γ Hξ


4.3 Data Input 317

Table 4.5 lists some measured values of fracture toughness. The values shown werereported by van Eekelen22. Thiercelin23 reviewed the testing procedures for deter-mining this parameter in his article, “Fracture Toughness and Hydraulic Fractur-ing.”

Setting the values of fracture toughness to zero will result in the classical hydraulicfracturing propagation solutions dominated by viscous pressure loss. For very lowviscosity fluids, fracture toughness may be the dominate parameter controllingfracture growth.

Poisson’s RatioPoisson’s ratio is defined as the ratio of the transverse strain to the axial strainresulting from an applied stress (see Figure 4.16).


Typical Poisson's ratios for rock formations are 0.25. From parametric studies,Poisson's ratio affects the fracture propagation characteristics to a very minorextent. Therefore, if in doubt, use 0.25.

Table 4.5: Fracture Toughness Values for Various Rocks.


Siltstone 950-1650 1040-1810

Sandstone 400-1600 440-1040

Limestone 400-950 440-1040

Shale 300-1200 330-1320




Poisson’s ratio is also used by logging companies to infer in-situ stresses. Thismethod assumes the rock behaves elastically and that the tectonic stresses areknown or insignificant. The typical relationship is

where

Total Leakoff HeightThis is the total or net permeable height penetrated by the fracture for leakoff. Thismay or may not be equal to the hydrocarbon pay thickness used to estimate produc-tion. The total leakoff height is also referred to as the net pay zone thickness.

= minimum horizontal stress= Poisson’s ratio= vertical stress or overburden= pore or reservoir pressure= component of stress due to tectonics= Biot’s constant

Poisson’s ratio

υ = −ε wε l

ε w0

=wΔw ε =l

Δll

0


Longitudinal strain

l0

w0

σHminυ

1 υ–------------⎝ ⎠

⎛ ⎞ σv αp0–( ) αp0 σT+ +=

σHmin

υσv

p0

σT

α


4.3 Data Input 319

Total Fracture HeightThis is considered the total fracture height for the PKN and GDK fracture models.These 2-D models have fixed fracture heights by definition (see Chapter 2). Thisparameter is one of the most difficult to estimate and is one of the most importantinput parameters used in the analysis of the minifrac treatment for the PKN model.MinFrac has an option to history match on fracture height for the PKN model. Thetotal fracture height is not used for the ellipsoidal geometry model.

Ellipsoidal Aspect RatioThis is the ratio between the length of the major and minor ellipse axes. If this valueis equal to unity (1), the model reduces to the standard radial or penny shaped solu-tion. Any value greater than one will produce an elliptical profile and correspond-ing fracture area. For example, an Ellipsoidal Aspect Ratio of two (2) results in afracture half length that equals the total height of the fracture.

Flow Behavior Index Rheological characterization of non-Newtonian fluid is required to calculate thefrictional dissipation in the fracture. Fracturing fluids are most often characterizedby the power law model. This model is typically defined as:

where is the wall shear rate, is the wall shear stress, is the consistency

index, and is the flow behavior index (dimensionless).

Consistency Index See the explanation of the flow behavior index above.

Spurt Loss CoefficientSpurt loss is the “instantaneous” volume loss of fluid per unit area of fracture facethat occurs prior to the development of a filter cake. The volume of fluid loss due tospurt for both faces of a single wing fracture is

n′( )

τw k′γn′=

γ τw k′

n′

k′( )

Vsp

Vsp 2ASp=




Total Vertical DepthEnter the true vertical depth (TVD) to the center of the perforations. This parameteris used to calculate the hydrostatic head required to adjust surface pressures to bot-tomhole pressures.

Wellbore Fluid Specific GravityThe specific gravity of the fluid occupying the wellbore during shut-in (i.e., pres-sure decline) is required to calculate the hydrostatic head. This value, along withthe Total Vertical Depth described above, is needed when bottomhole pressure isnot available.

Flowback Time (after ISIP)Because flowback data is rarely available in digital form, there is no graphicaloption associated with this event. To include flowback in the calculations, the start-ing time for flowback must be specified. Enter the shut-in time from the ISIP. Forany injection interval, this is the difference between the time flowback began andthe time of the ISIP.

Flowback RateThe flowback rate is simply the average negative return flow rate following aperiod of injection. This rate is applied in the calculations beginning at the Flow-back Time (after ISIP) described above and continues until closure.

Leakoff DataThe Leakoff Data screen shown in Figure 4.17 is provided for the additional reser-voir data needed to calculate the leakoff coefficients and permeability of the frac-tured interval. Each of the data items in this screen are described below. The readeris referred to Appendix D for additional information regarding fluid loss.

Sp A


4.3 Data Input 321

Figure 4.17: Leakoff Data Dialog Box.

Average Reservoir Fluid PressureThe average reservoir pressure is the fluid pressure in the pore spaces. This valuemust be less than the fracture closure pressure. If it is not, a warning message willbe displayed. See Chapter 2 and Appendix D for additional information.

Total Reservoir CompressibilityThe total reservoir compressibility is defined as the total change in the reservoirvolume per unit volume per unit pressure difference. It is the reciprocal of the un-drained bulk modulus and is typically expressed as follows:

where



cg

co

cr

ct

cw

Sg

So

Sw




where

Equivalent Reservoir Porosity This is the fraction of a rock’s bulk volume that is filled with mobile hydrocarbons.This value is used to calculate the CI and CII leakoff coefficients. Refer to Appen-dix D for information on the leakoff models contained in the program.

Equivalent Reservoir Viscosity The equivalent reservoir viscosity is the total effective viscosity of a multiphasefluid system at reservoir conditions. This value is used to calculate the CII leakoffcoefficient, which models the leakoff resistance due to the viscosity and compress-ibility effects of the in-situ fluids.

Frac Fluid Leakoff Viscosity Enter the effective viscosity of the fracturing fluid filtrate. This is the portion of thefracturing fluid which passes through the fracture face. Its viscosity has beenreduced from its original magnitude due to the deposition of polymer on the frac-ture face to form a filter cake, as well as, environmental consequences (i.e., stressand temperature). This parameter is used to calculate the CI coefficient for model-ing viscosity and relative permeability effects caused by the fracturing fluid.

The effective fluid leakoff viscosity must also account for the relative permeabilityeffect of the leakoff fluid to that of the reservoir fluid. This is especially important

= total formation compressibility= equivalent reservoir permeability= pressure= time= distance= equivalent reservoir viscosity= equivalent reservoir porosity

t∂∂p k

ctφμ-----------⎝ ⎠

⎛ ⎞z2

2

∂

∂ p=

ct

kptzμφ

φ( )

μr( )

μe( )


4.3 Data Input 323

for a gas reservoir. The effective leakoff viscosity, , in terms of the fluid leakoffviscosity and relative permeability is


Filter Cake Coefficient The wall building or filter cake coefficient is equivalent to the inverse of the frac-turing fluid leakoff resistance. A value of zero (0) represents an infinite filter cakeresistance, whereas, a CIII value approaching infinity (e.g., >100 ) repre-sents no wall building. This coefficient is a component in calculating the total lea-koff coefficient C. The filter cake reduces the fluid loss rate by increasing theresistance due to leakoff at the fracture face.


Closure DataWhen using MinFrac with the User Specified Closure (i.e., no imported data file),it is necessary to enter specific information regarding injection rate, volumes andclosure time, in addition to the reservoir and leakoff data described in the precedingsections. This data is entered in the Closure Data dialog box shown in Figure 4.18.The items contained in this dialog box are defined below.

μe

μe μf kr⁄=

μf kr

CIII( )

ft min⁄



Figure 4.18: Closure Data Dialog Box.

Injection Rate (2-wings)The average injection rate is required when the actual rate data is not available.Make sure the product of this value and the Pumping Time is equal to the volumepumped. The injection rate is also used to couple the closure time to the fracturepropagation solution (see Appendix F).

Pumping TimeThe total pump time is needed to calculate the volume of fluid injected during atreatment. This value is also needed to calculate certain required time functions,such as the dimensionless total time, (see Appendix F). The value required is thetotal time of injection (i.e., the difference between the start and end of pumping).

Closure Time (after ISIP)This is perhaps the single most important parameter required for the calculations.From closure, an estimation of fluid efficiency is determined. The fracture propaga-tion characteristics are also predicted by coupling the closure time with the govern-ing mass and momentum equations. Determining a definitive closure time can bedifficult for some data sets. The recommended procedure is presented in Section4.4. This value is the delta time after ISIP to closure.

Closure PressureFracture closure pressure or the minimum horizontal stress is needed to define thenet pressure during injection. The net fracturing pressure, , is the difference

θ

Δpnet


4.3 Data Input 325

between the pressure in the fracture, , and the closure pressure, ,

(i.e., ). These values can be determined by performing aminifrac analysis as discussed below.

History Match DataThe Graphical Technique can be used to process treatment data and evaluate dif-ferent model parametric effects when adequate information is available. This pro-cess, referred to as History Matching, uses regression analysis to compare andimprove the theoretical pressure response with the actual measured data.

The regression parameters can be selected from the Regression Analysis window.During the regression (i.e., History Match) the selected parameters are optimized tominimize the error between the measured and calculated pressure decline. Whenthe best fit is achieved (i.e., error is minimized), the regression is complete. Formore information on History Matching, see the Regression Analysis section.

The individual fracture model data values used in the regression can be viewed andedited by accessing the History Match Data dialog shown in Figure 4.19. Thisscreen is not updated until after the first regression is performed. The values dis-played or entered in this dialog box are the values used each time a new regressionis started (i.e., the History Match is performed). These values are also reported inthe History Match section of the report. The first time you regress on a data set, thisscreen contains the values entered in the Base Data dialog box. Each subsequentregression analysis then uses the History Match Data contained in this screen. Toreset all values to the base data press Reset to Base Data. For reference, the origi-nal Base Data values are shown on the right hand side in the dialog box.

pfracture pclosure

Δpnet pfracture pclosure–=



Figure 4.19: History Match Data Dialog Box.

For information on the individual parameters in this screen, refer to the appropriatepart of the Base Data section.

Import Data FileIf the Graphical Technique – Use data from an ASCII text file is selected, theImport Data function can be accessed by selecting the Data⏐Import Data Filemenu. When using this option, an ASCII file must be imported into MinFrac beforegraphical analyses are performed.

There are two steps to importing a data file, selecting a file and importing theparameters.

Selecting a Data FileTo Select an ASCII Data File for Importing:

1. From the Import menu, select Import Data File. The screen shown in Figure4.20 will be presented.

2. Browse to select the file to import.

3. Once a file is highlighted, use the OK button to finalize the selection. The dia-log box shown in Figure 4.21 will be presented.


4.3 Data Input 327

Figure 4.20: Import Data - File Open Dialog Box.

Figure 4.21: Import Data Format Screen.



The file should be an ASCII file with columns of data separated by commas, spacesor tabs. Text headers at the top of the file will be ignored. Any text in between rowsof data will also be ignored.

To sample the data contained in the spreadsheet, specify the sampling frequency byentering a number in the Sample every box. For example, to use every fifth datapoint, enter a 5 in this box. MinFrac can select a maximum of 16,382 lines of data,but can read or import up to 170,000 lines of data from a file. If there are more than16,382 lines of data in a file, it will be necessary to adjust the Sample every box sothat 16,382 lines or less of data are actually imported. For example, if a data file has170,000 lines of data, it would be necessary to set the Sample every box to 11 orhigher. The other option is to adjust the Data starts at row and Data ends at rowvalues to limit the number of selected rows to a maximum of 16,382.

Edit Imported DataAfter a data file has been imported successfully, the data may be viewed and edited.

To Edit Imported Data:

1. From the Main menu, select the Data⏐Edit Imported Data menu. The dialogbox shown in Figure 4.22 will be displayed containing the current data. If thisoption is dimmed, data has not yet been imported.

2. Click on a row or record to change and type the correction. Press ENTER toaccept the changes.

3. To copy a selection to the Clipboard, select the desired rows by clicking ontheir row numbers. Then click on the Copy button.

4. To delete row(s) of data, select the desired rows by clicking on the row num-bers. Then click on the Delete button.

5. When you are finished editing the file use the OK button to close the editingwindow and accept all changes. Use the Cancel button to abort and discard thechanges made.

Note any changes made here will only affect the imported data, no changes willbe made to the original ASCII file.


4.4 Analysis 329

Figure 4.22: Edit Imported Data Dialog Box.

A more powerful method to edit data is to use the graphical editing features inMView (see Chapter 3).

4.4 AnalysisThe main purpose of MinFrac is to graphically analyze time dependent rate andpressure data associated with minifrac treatments. This section summarizes theoptions and features involved in performing graphical analyses.

The fracture data found in the Data menu is used for history matching and modeldependent parameter optimization. A minifrac analysis to find fracture efficiencyand closure pressure can be performed without knowledge of the specific fracturemodel.

To use MinFrac as a graphical tool for processing minifrac data, choose one of theGraphical Technique options. After importing all the data, begin an analysis usingthe commands under the Analysis menu.

Figure 4.23 shows the Analysis menu in MinFrac. The different types of analysesavailable displayed under the Analysis menu are Analysis Wizard, Step Rate, StepDown, Horner, and Regression. The Analysis Wizard is a step by step tool for per-forming all types of different analyses.



Figure 4.23: Analysis Menu.

The analysis menu and methodology for performing minifrac analysis has evolvedover the years. A detailed description of all the analyses is presented below for con-tinuity.

Since each analysis begins with the Select Ranges menu, a general Select Rangesprocedure is presented first.

Select RangesBy default, MinFrac uses the entire range of imported time and pressure data for ananalysis. This command provides the capability to work with a limited range ofdata (i.e., when multiple injection cycles have been recorded).

Figure 4.24 shows a typical Select Ranges menu item for the Step Rate Analysis.

Figure 4.24: Select Ranges Menu.

The dialog box as shown in Figure 4.25 is only displayed if both surface and bot-tomhole pressure data are available.


4.4 Analysis 331

Figure 4.25: Select Ranges Dialog Box for Pressure.

The radio buttons in the Select Ranges dialog box are used to indicate which pres-sure to use in the analysis if both are available. After making your selections, clickthe OK button and a plot of the data will be generated. An example is shown in Fig-ure 4.26.

Figure 4.26: Select Ranges Plot.

The plot created using the Select Ranges procedure includes all of the importeddata. The purpose of this plot is to graphically define a range of data to use for anal-ysis. To make a selection slide the mouse to the left or right edge of the highlighteddata range. Then drag the highlighted edge horizontally to bracket a segment oftime on the plot. This is accomplished by using the left mouse button. Click and



hold the mouse button while dragging the vertical edge to the desired position.Release the button to fix the position of the highlighted edge. Remember the colorfilled region outlines the range used for the analysis.

Notice that the Start and Stop position is continuously displayed in the upper left-hand corner of the plot dialog (or where you graphically place the box). The abso-lute position of the selected range can be viewed and edited by choosing the RangeMenu button as shown in Figure 4.27. The Range Menu has two options: ExtendRange to the End of Data and Edit Selections. This displays a dialog box contain-ing the coordinates of the selected range (see Figure 4.28).

Figure 4.27: Range Menu Items.


4.4 Analysis 333

Figure 4.28: Range - Edit Selections Dialog Box.

The Range⏐Extend Range to End of Data menu defines the end of the range to bethe end of the data. This is useful for working with real-time data when the end timeof the data is continually increasing.

Analysis WizardThe Analysis Wizard is a systematic method for selecting and performing minifracanalyses. The Wizard allows for all the analyses under the Analysis menu (StepRate, Step Down, Horner and Regression).

This Analysis Wizard section is a “How to use the Wizard” presentation. Thedetails on performing a specific analysis are discussed in the individual analysissections presented below.

Figure 4.29 shows the Analysis⏐Analysis Wizard menu. To utilize the manymenus and features in the Wizard, systematically follow the Wizard steps.

Figure 4.29: Analysis Wizard Menu.

The purpose of the Wizard is to organize and step the user through an analysis. Thisis a great way for novice users to begin or for experienced users to set up a Wizardtemplate for performing analyses consistently.



The Wizard’s systematic menu methodology will step you through the analysis andwill not allow you to access a dependent procedure until the previous step has beenperformed. Informational help screens are provided at the top of each Wizard stepto guide you in performing a particular analysis.

The Analysis⏐Analysis Wizard menu consists of two parts: Select Analyses andWizard Window.

Select AnalysesAfter selecting the Analysis⏐Analysis Wizard menu, the Select Analyses dialogbox will be displayed as shown in Figure 4.30.

Figure 4.30: Analysis Wizard - Select Analyses Dialog Box.

The Analysis Wizard⏐Select Analyses menu is setup very similar to the MFracFluid, Proppant and Acid database dialogs. The buttons at the bottom of the screenare used to Edit, Add, Delete or move the Selected Analyses Up or Down in theMenu. The select analyses Save Template and Load Template buttons can be usedto save and load a file. This makes it easy to setup the MinFrac Wizard with stan-dardized templates.

Press the OK button to enter the Wizard Window or Cancel to exit the AnalysisWizard.

To add a selection to the Select Analyses menu press the Add button. Selecting theEdit or Add buttons will bring up the menu shown in Figure 4.31.


4.4 Analysis 335

Figure 4.31: Analysis Wizard - Edit Selections.

This Edit Selection menu consists of the Analysis Type, Regression Options andReport Options. The Regression Options are only highlighted if AnalysisType⏐Regression is selected. Likewise, the Report Options will be dimmed unlessthe Analysis Type⏐Report menu is selected.

Analysis TypeThe analysis type provides a drop down selection box as shown in Figure 4.31. TheAnalysis Types available are Step Rate, Step Down, Horner, Regression, andReport. Each of these analyses is discussed below in detail. To generate a report asone of your Wizard Window tabs select Analysis Type⏐Report.

If both the surface and bottomhole pressures are available for analysis, select theData Source drop down menu and choose either surface or bottomhole pressure.

Regression OptionsIf you choose Analysis Type⏐Regression the Regression Options will be high-lighted (see Figure 4.32). Select the Regression Options Time Axis (e.g., Nolte Gtime). The allowable choices are given in the drop down menu (see also Table 4.2).



Figure 4.32: Edit Analysis - Regression Options.

Next select the Right Axis derivative for Pressure (e.g., ISIP-GdP/dG). Then checkif you want to see the Delta Pressure Log-Log and/or Linear analyses. The RightAxis derivative for each of these checked items can also be specified via the dropdown window.

If you want to perform a history match analysis check the History Match box.

Report OptionsIf you choose Analysis Type⏐Report the Report Options will be activated asshown in Figure 4.33.


4.4 Analysis 337

Figure 4.33: Edit Analyses - Report Options.

The Report Options specify which items you want to include in the report.

Wizard WindowAfter selecting OK in the Analysis⏐Analysis Wizard⏐Select Analyses menu, theWizard Window will be displayed as shown in Figure 4.34. The number of foldertabs corresponds directly to the number of analyses selected in the Analysis Wiz-ard⏐Select Analyses menu.



Figure 4.34: Analysis Wizard - Wizard Window.

Please refer to the specific analysis in question when using the Select Points menu.The Select Points menu is discussed below for each analysis.

At the top of the Wizard Window, descriptive text informs the user of any actionsor steps to be taken. Plot configuration and point selection features are alsodescribed at the top right corner of each plot.

At the bottom of the Wizard Window is a set of selection buttons as shown in Fig-ure 4.35.

Figure 4.35: Wizard Window - Select Buttons at Bottom.


4.4 Analysis 339

The options at the bottom of the Wizard Window are; < Back, Next >,Select Analyses and a check box to Hide Description.

To hide the descriptive text above the plot check Hide Description. This willenlarge the viewable plot area as shown in Figure 4.36.

Figure 4.36: Wizard Window - Hide Description Checked.

The Select Analyses button will return you to the Analysis⏐Analysis Wiz-ard⏐Select Analyses menu (see Figure 4.30). The < Back and Next > buttons areused to navigate through each step of an analysis. After a step is completed you willbe allowed to proceed to the Next > step. The Back and Next buttons are addressedin the next section where a complete regression analysis in the Analysis Wizard ispresented.

Any time you are in the Wizard Window, each of the tabs will be displayed at thestep last viewed. The main menu Tool Bar will also change depending on the anal-ysis type.

Figure 4.37 shows the second Wizard Window tab for this example. Notice thedrop down Axes menu selection to change the right axis derivative function.



Figure 4.37: Wizard Window - Nolte G Time Plot with Axes Selected.

Figure 4.38 shows the next tab of Sqrt Delta time plot with the Select Points menuselected. From the Select Points menu the user can perform all the point, line, hide,edit etc., options in this menu. The Select Points menu for the Regression Analysisis discussed below in the Regression Analysis section.


4.4 Analysis 341

Figure 4.38: Wizard Window - Sqrt Delta Time with Select Points Menu.

Figure 4.39 shows the Report tab based on the Report Options selected. The fullreport menu features are available in the Main Tool Bar.



Figure 4.39: Wizard Window - Report.

Wizard Window ExampleFollowing is an example session using the Wizard to perform a Regression Analy-sis. The steps the Wizard takes you through are identical to the steps required to dothis analysis in the Analysis⏐Regression menu. Figure 4.40 shows the Nolte Gtime Wizard Window the first time it is opened. The first step is to Select Ranges.Descriptive text is shown above the Wizard Plot to help the user select the pumptime and pressure decline cycles.


4.4 Analysis 343

Figure 4.40: Wizard Window - Select Ranges.

Selecting the Next > button will change the Wizard Window to the plot shown inFigure 4.41. Here the Wizard asks the user to select the Min/Max Range Bar. Thiscan be done from the Select Points menu. Please refer to the Regression Analysissection describing the Select Points menu Min/Max Range Bar.



Figure 4.41: Wizard Window - Select Min/Max Range Bar.

The next step is to do the Regression Analysis to find closure by choosing theSelect Points menu. Figure 4.42 shows the Wizard Window for the SelectPoints⏐Automatically Find Points menu. The instructions at the top of the Wiz-ard Window describe the Select Points menu. The scroll bar can be used to showadditional comments.


4.4 Analysis 345

Figure 4.42: Wizard Window - Select Points.

Figure 4.43 shows the next item in the Analysis Wizard⏐ Select Analyses⏐EditAnalyses menu (see Figure 4.31). This is the Nolte G time in log coordinates.



Figure 4.43: Wizard Window - Delta Pressure Log Coordinates.

Figure 4.44 shows the Next > step of the Delta Pressure in linear coordinates withthe right axis derivative of GdP/dG.


4.4 Analysis 347

Figure 4.44: Wizard Window - Delta Pressure Linear Coordinates.

The final item selected in the Analysis Wizard⏐Select Analyses⏐Edit Selectionsmenu is to history match the decline data (see Figure 4.31). The history match Wiz-ard Window is shown in Figure 4.45. The pressure decline line as predicted by eachof the two-dimensional fracture models is shown on the plot. For each of the frac-ture geometry models (GDK, PKN and Ellipsoidal) a drop down parameter selec-tion menu is available. Choose the parameter to history match on and then press theHistory Match button.



Figure 4.45: Wizard Window - History Match.

The user has total control on which regression plots to place in the Wizard Win-dow. The Wizard allows you to create your own methodology of performing StepRate, Step Down, Horner, Regression, and Report windows. The Wizard is a tool,which will help minimize the complicated procedures by systematically performingan Analysis.

Step Rate AnalysisA step rate test provides a means for determining the fracture propagation or exten-sion pressure. Since the propagation pressure (dynamic condition) is typically onthe order of a few hundred psi (several hundred to several thousand kPa) greaterthan the closure pressure (static condition), the value determined from this type ofprocedure yields an upper bound for closure.

Figure 4.46 shows the Analysis⏐Step Rate menu. To perform a step rate analysis,systematically follow the menu.


4.4 Analysis 349

Figure 4.46: Step Rate Menu.

The Analysis⏐Step Rate menu consists of four parts: Select Ranges, Select Points,Pressure Table and Diagnostic Plot. The first time you perform a Step Rate testwith a new data file all of the menu items will be dimmed except for the SelectRanges as shown in Figure 4.47. The reason these menus are dimmed is that SelectPoints, Pressure Table and Diagnostic Plot are all dependent on Select Ranges.Consequently, this menu step-by-step methodology will not allow you to access adependent procedure until the previous step has been performed.

Figure 4.47: Step Rate Menu - Dimmed.

The menu command items for the Step Rate analysis are described in the sectionsbelow.

Select RangesUsing the graphical method described above in this section, select a range of data toanalyze by activating the Analysis⏐Step Rate⏐Select Ranges menu (Figure4.47). Figure 4.48 shows a typical Select Ranges plot.



Figure 4.48: Step Rate - Select Ranges Menu.

Select PointsTo select the data points for the step rate analysis, click on the Step Rate⏐SelectPoints menu item as shown in Figure 4.49.

Figure 4.49: Step Rate - Select Points Menu.

This will bring up a plot as shown in Figure 4.50. Once the Step Rate plot is dis-played the mouse pointer changes to a vertical bar. Slide the bar horizontally toalign it with the time values to be used in the analysis. To choose a time, click the


4.4 Analysis 351

left mouse button. Two markers will be displayed corresponding to the selection,one on rate and the other on pressure.

Figure 4.50: Step Rate - Select Points Plot.

To edit or erase the selected points click on the Select Points menu shown in Figure4.51. The Select Points⏐Edit Selections menu can be used to display and edit thecoordinates (Figure 4.52). To erase all of the data points use the SelectPoints⏐Erase All menu. You can also erase selected points by clicking the mousebutton on an existing point.

Figure 4.51: Step Rate - Select Points Edit Menu.



Figure 4.52: Step Rate - Edit Selections Dialog Box.

Pressure TableTo perform a meaningful Step Rate Analysis from the surface (or bottomhole) thefrictional pressure drop in the wellbore and across the perforations and the net frac-ture pressure should be accounted for in the calculations.

To input the wellbore friction and perforation losses, and net fracture pressuredependence on rate select the Analysis⏐Step Rate⏐Pressure Table menu (seeFigure 4.53).

Figure 4.53: Step Rate - Pressure Table Menu.

After selecting the Pressure Table menu, the table shown in Figure 4.54 will be dis-played.


4.4 Analysis 353

Figure 4.54: Step Rate - Pressure Table.

Figure 4.54 shows the step rate pressure table with the selected rate and pressurepoints in the first two columns, respectively.

Then the wellbore pressure loss (DP Fric) is entered. Since this case was with bot-tomhole pressure data the DP Fric in the wellbore was set to zero. For surface pres-sure the frictional pressure loss as function of rate should be specified.

The fracture net pressure (DP Frac) is then specified in column four. Once the frac-ture begins to propagate the net pressure may be relevant to the analysis. In thiscase a value of zero was entered for the fracture net pressure. As the net pressureincreases the calculated Extension Pressure will decrease.

Next is the ideal perforation pressure loss (DP Perf Ideal) as a function of rate. Theperforation pressure loss is calculated from the Specific Gravity of Fluid and theNumber, Discharge Coefficient, and Diameter of the perforations. Typical valuesfor the discharge coefficient are 0.60 for a sharp orifice entrance and 0.83 for arounded entrance. If no proppant has passed through the perforations select thelower discharge coefficient value of 0.60.

The extension pressure (surface or bottomhole) is then calculated from the equa-tion: Extension Pressure = Pressure – DP Fric – DP Frac – DP Perf Ideal.



Diagnostic PlotWhen you finish editing the Pressure Table, choose the Step Rate⏐Diagnostic Plotmenu (see Figure 4.55). A diagnostic plot like the one shown in Figure 4.56 will bepresented.

Figure 4.55: Step Rate Diagnostic Plot Menu.

Figure 4.56: Step Rate - Diagnostic Plot.

Two lines may be placed on this plot to determine the extension pressure by locat-ing two points per line. The active point is selected from the Select Points menu(see Figure 4.57). To position a point on the plot click the left mouse button. The


4.4 Analysis 355

points are labeled 1A, 1B, 2A and 2B. An alternative to manually selecting thepoints and line positions is to perform the selection automatically by regression.

Figure 4.57: Step Rate - Automatically Find Points.

When the Select Points⏐Automatically Find Points menu is selected, as shown inFigure 4.57, two lines are positioned by regression to a best fit of the data. Thismethod develops an intersection that many use as an upper bound to the extensionpressure. The intersection of the Y-axis at a rate of zero may also be taken as a min-imum value of the extension pressure. This point may be a better representation ofthe true extension pressure (minimum horizontal stress).

The intersection point of the two lines is generally considered the upper bound ofthe closure pressure.

Step Down AnalysisThe Step Down Analysis is used to calculate perforation and near wellbore frictionlosses.

This analysis is performed after fracture initiation and propagation has been estab-lished. During shut down the rate is decreased in a stair-step fashion for a shortperiod of time while the pressure stabilizes. As the injection rate decreases, thepressure also decreases as a result of perforation and near wellbore pressure losses.The relationship between the decreasing rate and pressure results in a mechanisticapproach for determining near wellbore losses.



If the step down analysis is performed using surface treating pressure, the pipe fric-tion needs to be entered.

Figure 4.58 shows the Analysis⏐Step Down menu. To perform a Step Down anal-ysis systematically, follow the menu steps.

Figure 4.58: Step Down Menu.

The Analysis⏐Step Down menu consists of four parts: Select Ranges, SelectPoints, Pressure Table and Diagnostic Plot. The first time you perform a Step Downanalysis with a new data file all of the menu items will be dimmed except for theSelect Ranges as shown in Figure 4.59.

The reason these menu items are dimmed is that Select Points, Pressure Table andDiagnostic Plot are all dependent on Select Ranges.

Figure 4.59: Step Down Menu - Dimmed Steps.

The menu command items for the Step Down analysis are described below.

Select RangesTo select ranges click on the Analysis⏐Step Down⏐Select Ranges menu.


4.4 Analysis 357

Using the graphical method described above in this section, select a range of data toanalyze by activating the Select Range Menu. Figure 4.60 shows a typical selectrange plot.

Figure 4.60: Step Down Analysis - Select Ranges.

Select PointsTo select the data points for the Step Down analysis, click on the Analysis⏐StepDown⏐Select Points menu shown in Figure 4.61. As illustrated for a new data file,the Pressure Table and Diagnostic Plot menus are dimmed since Select Points hasnot been performed.



Figure 4.61: Step Down - Select Points Menu.

After accessing Select Points a plot will be displayed as shown in Figure 4.62. Ifthis is the first time you entered Select Points no markers will be on the curve.

Figure 4.62: Step Down - Select Points Plot.

Once the Step Down plot is displayed the mouse pointer changes to a vertical bar.Slide the bar horizontally to align it with the time values to be used in the analysis.To choose a time, click the left mouse button. Two markers will be displayed corre-sponding to the selection: one on rate and the other on pressure. The program willautomatically find a zero rate and associated pressure point.

To edit or erase the selected points click on the Select Points menu. The SelectPoints⏐Edit Selections menu (Figure 4.63) can be used to display and edit the


4.4 Analysis 359

coordinates (Figure 4.64). To erase all of the data points select Erase All. You canalso erase a set of selected points by placing the mouse pointer on the existing pointand left clicking the mouse button.

Figure 4.63: Step Down - Select Points Edit Selections Menu.

Figure 4.64: Step Down - Edit Selections Show Picks.

Pressure TableTo perform a meaningful Step Down Analysis from the surface (or bottomhole) thewellbore losses, calculated perforation friction, and the net fracture pressure mustbe accounted for in the calculations.



To input the wellbore friction and net fracture pressure dependence on rate selectthe Analysis⏐Step Down⏐Pressure Table menu (see Figure 4.65).

Figure 4.65: Step Down - Pressure Table Menu.

After selecting the Pressure Table menu, a table will be displayed as shown by Fig-ure 4.66.

Figure 4.66: Step Down - Pressure Table.

Figure 4.66 shows the selected rate and pressure points in the first two columns.The Delta Pressure is the next column which is calculated by subtracting the Pres-sure at a given rate from the ISIP (i.e., Delta Pressure =Surface Pressure-ISIP).

The user-specified values for frictional pressure loss in the wellbore (DP Fric) andfracture net pressure (DP Frac) go in the next two columns. The change in thewellbore friction and net pressure from the ISIP (zero rate value) is then calculatedand placed in the Change DP Fric+ DP Frac column.


4.4 Analysis 361

The total near wellbore loss including perforations (DP Total NW) is then calcu-lated from the difference between Delta Pressure and Change DP Fric+ DP Frac.

The ideal perforation pressure loss (DP Perf Ideal) as a function of rate is calcu-lated from the Specific Gravity of Fluid and the Number, Discharge Coefficient,and Diameter of the perforations. Typical values for the discharge coefficient are0.60 for a sharp orifice entrance and 0.83 for a rounded entrance. If no proppantpasses through the perforations select the lower discharge coefficient value of 0.60.

For surface pressure the frictional pressure loss as a function of rate should be spec-ified. The fracture net pressure may be relevant to the analysis if it is rate/timedependent. In this case a value of zero was entered for the fracture net pressure.Entering a variable value for net pressure will change the calculated near wellborepressure loss. If the fracture net pressure is assumed to be relatively constant, it willnot effect the analysis.

The Apparent Number of Perfs is given in the last column. This is the equivalentnumber of perforations that would have to be open to match the Total Near Well-bore Pressure loss (DP Total NW).

Diagnostic PlotWhen you finish editing the Pressure Table, choose the Analysis⏐StepDown⏐Diagnostic Plot menu (see Figure 4.67). A diagnostic plot similar to theone shown in Figure 4.68 will be presented.

Figure 4.67: Step Down Diagnostic Plot Menu.



Figure 4.68: Step Down - Diagnostic Plot.

The diagnostic plot shows the Total Pressure Loss, Perforation Only and NearWellbore Only losses. The data points displayed by the markers are the calculatedtotal and near wellbore pressure losses as given in the Pressure Table (see Figure4.66). The near wellbore power coefficient (Alpha) for this case is shown to be1.21018.

The total near wellbore pressure loss is calculated from the following equation:

where

= total pressure loss= perforation (ideal) pressure loss= near wellbore pressure loss= perforation coefficient= near wellbore coefficient= injection rate= near wellbore power coefficient

ΔpTotal ΔpPerfs ΔpNW+ KPerfs Q2 KNW+× Qα×= =

ΔpTotal

ΔpPerfs

ΔpNW

KPerfs

KNW

Qα


4.4 Analysis 363

The near wellbore pressure loss and Alpha coefficient ( ) curve is calculated froma regression analysis. If the near wellbore pressure loss is much less than the perfo-ration or total pressure loss there probably is not a near wellbore problem (like thisexample). If the alpha power coefficient is near two, it may be a condition wherenot all the perforations are opened.

As discussed, the near wellbore pressure loss calculation is a combination of manyfactors, pipe and perforation friction, and fracture pressure. If the flow in the well islaminar the wellbore friction power coefficient may be near unity not 1.8 or 2.0.Consequently, many apparent near wellbore problems with a low power coefficientmay just be the uncertainty in the laminar wellbore pressure loss. The flow rate inthe wellbore may be turbulent at the high rate values but as the rate decreases tozero the Reynold’s Number will be in the laminar region.

Horner AnalysisFor some reservoirs it may be desirable to evaluate the decline data for productioneffects (i.e., high permeability reservoirs) to determine the lower bound for fractureclosure. Normally, plotting pressure versus the log of Horner time will help identifythe onset of pseudo radial flow (i.e., fracture closure). This time function is typi-cally defined as , or where is the pump time and is the

shut-in time that is equal to .

The Horner plot provides a lower bound, first estimate of closure pressure.

Figure 4.69 shows the Analysis⏐Horner menu. To perform a Horner analysis, fol-low the menu steps.

The Horner menu consists of two parts: Select Ranges and Select Points. The firsttime you perform a Horner analysis with a new data file Select Points will bedimmed as shown in Figure 4.69. Again, the reason this menu is dimmed is becauseSelect Points is dependent on Select Ranges.

Figure 4.69: Horner Menu - Dimmed Steps.

α

tp ts+( ) ts⁄ t t tp–( )⁄ tp ts

t tp–



The menu command items for the Horner analysis are described below.

Select RangesTo select the Horner range click on Analysis⏐Horner⏐Select Ranges menu (seeFigure 4.70).

Figure 4.70: Horner Menu - Select Ranges.

Using the graphical method described above in this section, select a range of data toanalyze. Figure 4.71 shows a typical select range plot.

Figure 4.71: Horner - Select Ranges.

For the Horner and Regression analyses two ranges must be selected. A range fromthe initiation to end of pumping (Pump Time) and a range from the end of pumpingto the end of the pressure decline data (beyond closure) must be specified.


4.4 Analysis 365

After the range cycles have been selected the Horner plot can be generated. Thisanalysis is only reliable if the fracture closes.

Select PointsTo display the Horner Plot, click on the Analysis⏐Horner⏐Select Points menushown in Figure 4.72.

Figure 4.72: Horner - Select Points Menu.

After accessing the Analysis⏐Horner⏐Select Points menu a plot of the data willbe displayed as shown in Figure 4.73. If this is the first time you entered SelectPoints no lines will be on the graph as illustrated. The pressure data is only plottedfrom the Select Ranges data from the end of pumping to the end of the selectedrange data (stop).

Figure 4.73: Horner - Raw Data Plot.



Straight lines may be drawn on this Plot with the use of the Select Points menu.Two points may be selected on the graph, 1A and 1B and a line will be drawnbetween them. To select the points, choose the desired point from the Select Pointsmenu. Then click on the graph with the left mouse button. The coordinates of theselected point are shown under the Select Points⏐Edit Selections menu.

To manually enter coordinates for the points, click on the Select Points⏐EditSelections menu. To click and drag the points that are already on the graph, clickon the Select Points⏐Drag Points menu. The mouse coordinates can also beshown on the plot to pin point the start of pseudo-radial flow.

Figure 4.74 shows a typical line placement after closure. The deviation of thestraight line from the Horner data (Horner time of about 0.15 and pressure of about5200 psi) signifies the start of pseudo radial flow and represents the lower boundfor the closure pressure.

Figure 4.74: Horner Plot - Lower Bound for Closure Pressure.

The Horner plot is used to determine if pseudo-radial flow developed during thetest. If a semi-log straight line is observed as shown in Figure 4.74 and the line canbe extrapolated to a reasonable value of reservoir pressure (4400 psi), radial orpseudo-radial flow may be affecting the decline behavior. This conclusion suggeststhat the fracture is already closed and that data beyond the point of influence neednot be considered in the evaluation of closure.


4.4 Analysis 367

Regression AnalysisThe main purpose of a minifrac analysis is to provide a method of estimating clo-sure pressure, near wellbore effects, fracture dimensions, fluid efficiency, and lea-koff coefficients prior to designing and pumping the main treatment.

The various methods to estimate the upper and lower bounds of closure pressure arethe Step Rate and Horner analyses as addressed above. Near wellbore pressurelosses can be determined from a Step Down analysis as presented. These analysesprovide alternate ways to help identify the fundamental characteristics of wellborefriction, perforation and near wellbore losses and closure pressure.

The following information can be determined from a properly conducted analysis:

1. Instantaneous shut-in pressure, .

2. Closure pressure, .

3. Closure time, .

4. Fracture efficiency, .

5. Fraction of PAD, and .

6. Fracture net pressure, .

7. Parametric uncertainty (history match).

8. Applicable fracture model (history match and net pressure).

9. Fracture area, , based on best-fit model.

10. Leakoff coefficient, , if the fracture area is known.

The derivative method is one of the MinFrac methodologies for determining inflec-tion points (i.e. fracture closure). As discussed in Section 4.1, the derivative plotcan be used to identify closure by observing a characteristic change in the slope.

Figure 4.75 shows the Analysis⏐Regression menu. To perform a Regression Anal-ysis, systematically follow the menu.

The Regression menu consists of three parts: Select Ranges, Select Points, and His-tory Match. The first time you perform a Regression analysis with a new data file

ISIP

pc

tc

η G tc( ) 2 G tc( )+[ ]⁄≅

fmin 1 η–( )2≅ fmax 1 η–( ) 1 η+( )⁄≅

Δp ISIP pc–=

A

C



Select Points will be dimmed as shown in Figure 4.75. The reason this menu isdimmed is because Select Points is dependent on Select Ranges.

Figure 4.75: Regression Menu - Dimmed Items.

The menu command items for the Regression analysis are described below.

Select RangesTo select the Regression range click on Analysis⏐Regression⏐Select Rangesmenu (see Figure 4.75).

Using the graphical method described above in this section, select a range of data toanalyze. Figure 4.76 shows a typical select range plot. As illustrated this is the samerange as used for the Horner Plot (see Figure 4.71).

Figure 4.76: Regression - Select Ranges.


4.4 Analysis 369

For the Regression analysis two ranges must be selected. A range from the initia-tion to end of pumping (Pump Time) and a range from the end of pumping to theend of the pressure decline data (beyond closure) must be specified.

After the ranges have been selected the regression plot can be generated.

Select PointsTo display the regression analysis select points, click on the Analysis⏐Regres-sion⏐Select Points menu shown in Figure 4.77.

Figure 4.77: Regression - Select Points.

After accessing the Analysis⏐Regression⏐Select Points menu a plot of the datawill be displayed as shown in Figure 4.78. If this is the first time you entered SelectPoints no lines will be drawn on the graph as illustrated. The pressure data is onlyplotted from the Select Ranges data from the end of pumping to the end of theselected data range (stop).



Figure 4.78: Regression - Data Time Plot.

Figure 4.78 shows the pump time cycle (shaded area) and the pressure decline cyclein data time. The solid bar at the top of the plot is the Min/Max Range Bar. Thisbar defines the range of data to be used for performing the regression analysis. Thisbar is discussed below.

Figure 4.79 shows the Main Menu bar after selecting the Analysis⏐Regres-sion⏐Select Points menu. This Main Menu bar allows you to use all of the SelectPoints and Axes menu features. The Main Select Points menu is shown in Figure4.80.

Figure 4.79: Select Points - Main Menu Bar.


4.4 Analysis 371

Figure 4.80: Select Points - Regression Main Menu.

The Select Points and Axes Main Menu Bar features are discussed below.

Select Points from Main Menu BarThe Select Points from the main menu bar is used to graphically choose specifictime events. These time events can then be dragged, hidden and edited. Lines canalso be manually or automatically placed on the plot to determine inflection points.

Shut-in/Closure/Select Line Points

This menu is used to manually specify or change the Shut-in (ISIP) and Closure(TC) times.

Straight lines may be drawn on this Plot with the use of the Select Points menu.Four points may be selected on the graph, 1A, 1B, 2A and 2B. A line will be drawnbetween 1A and 1B and between 2A and 2B. To select the points, choose thedesired point from the Select Points menu. Then click on the graph with the leftmouse button. The coordinates of the selected point are shown under the SelectPoints⏐Edit Selections menu. Points can also be chosen using the F7 and F8 func-tion keys.

Automatically Find Points

A statistical method is available for evaluating the pressure decline plot. Using lin-ear and nonlinear regression to fit the data, an estimate of the closure pressure canbe obtained. The technique is performed in the active plot space using the time



interval specified by the Min/Max Range Bar. MinFrac uses a statistical functionthat fits two discrete curves through the data between the Select Ranges data. Thefirst curve has the form of a straight line (i.e., ) and is assumed to passthrough the portion of data immediately following the minimum value of the Min/

Max Range Bar. The second curve has the form of , and isassumed to pass through the data that begins immediately after the early straight-line portion of data and before the maximum time range selected. This nonlinearregression methodology minimizes a difference function to fit both curves.

The intersection of the resulting curves represents the fracture closure pressure.This statistical technique is based on the MinFrac Methodology given in AppendixF. The basic premise is that there is some function or functions that linearizes thepressure decline (e.g., where ). This technique thuscharacterizes the deviation from a straight line on a pressure versus time plot tohelp determine when an inflection point occurs.

A Regression Analysis is in the current space coordinates of the plot (i.e., uses thesame time scale and pressure scale as the plot). Thus, if the X-axis changes, theregression results will be different. The TC that is found by the regression willalways be in the range defined by the Min/Max Range Bar.

To Use the Automatic Closure Method:

1. First define the Min/Max Range Bar by graphically arranging it on the screen.This is done by Clicking on the left or right edge of the Min/Max Range Barwith the left mouse button and while holding down the button drag the left orright edge to the desired position. Keep in mind that the Min/Max Range Bardefines the range of data that is used in the regression.

2. After the points are selected, select one of the time and pressure functionsgiven in the Axes menu. The Axes menu is discussed below.

3. To begin the regression, click the Select Points⏐Automatically Find Pointsmenu.

4. When the regression is complete, a message box similar to the one shown inFigure 4.81 will be displayed. This message is provided to update the ISIP.The theoretical or calculated ISIP is estimated from the projection of the earlytime decline data back to the y-axis. Choose Yes to use the calculated ISIP.

During and after the regression, the TC curve is displayed on the plot. To hide theselines, choose the Select Points⏐Hide Lines menu.

y mx b+=

y ax bx1 2⁄ c+ +=

P GdP dG⁄= P ISIP p–=


4.4 Analysis 373

Figure 4.81: Regression - Automatic Find Points and Update ISIP.

Figure 4.82 illustrates the Select Points⏐Automatic Find Points method for theNolte G function in linear coordinates.

Figure 4.82: Regression - Nolte G Time Linear Analysis.

Figure 4.82 shows the calculated time of closure and regression data. The regres-sion data displayed is the Pump Time, Delta TC (closure time after pumping), ISIPand Closure Pressure, BH (bottomhole) Closure Pressure, Efficiency (after spurt),Residual (difference between pressure data and line fit), and Slope 1 and 2 (of line1 and 2).

Put TC at the intersection of line 1 and line 2

This will place the time of closure (TC) at the intersection of the two lines. This isdone automatically for Automatically Find Points.

Drag/Hide Points and Lines

Numerous options are available to drag or hide points and lines on the graph.



Edit Selections

The Select Points⏐Edit Sections menu allows you to edit the data points for Shut-in, Closure and the line points (1A, 1B, 2A, and 2B). Figure 4.83 shows the EditSelection dialog box.

Figure 4.83: Regression - Edit Selections Dialog Box.

Axes - Main MenuThe Axes menu is used to vary the X and Y-axes coordinates. Figure 4.84 showsthe Axes menu after selection of the Analysis⏐Regression⏐Select Points menu.The X-axis is the time function (Time), the left Y-axis is the pressure coordinate(Pressure), and the right Y-axis is the Derivative/Rate coordinate. The regressionselect Axes⏐Time, Axes⏐Pressure and Axes⏐Derivative/Rate menus are dis-cussed below.

Figure 4.84: Regression - Axes Main Menu.

Axes Time

The Axes⏐Time menu is shown in Figure 4.85. This is the time function to be usedon the X-axis.


4.4 Analysis 375

Figure 4.85: Regression Axes - Time Menu.

The pressure data may be plotted versus a variety of time functions. As presented,Table 4.2 lists the formulas used to calculate the different time scales

Axes Pressure (Left Y- Axis)Pressure or Delta Pressure may be plotted on the left Y-axis. Delta Pressure isdefined to be . When in the Delta Pressure mode, the plot axis can beeither log-log or linear coordinates.

Figure 4.86: Regression Axes - Pressure (left axis).

Three options are available from the Axes⏐Pressure menu:

Axes⏐Pressure⏐Pressure

Pressure on the left Y-axis and the X-axis time function are in linear coordinates.

Axes⏐Pressure⏐Delta Pressure (Log-Log)

Delta Pressure on the left Y-axis and the X-axis time function are in log coordi-nates. Please refer to Appendix F regarding the Delta Pressure theory.

p tp( ) p t( )–



Axes⏐Pressure⏐Delta Pressure (Linear)

Delta Pressure on the left Y-axis and the X-axis time function are in linear coordi-nates.

Axes Derivative/Rate (Right Y- Axis)The right axis can be used to plot either the rate or a number of derivative functionsas illustrated in Figure 4.87.

Figure 4.87: Regression Axes - Derivative/Rate Menu.

Select the Axes⏐Derivative/Rate⏐None menu, to turn off the right axis.

Select the Axes⏐Derivative/Rate⏐Rate list menu, to plot the rate on the right axis.

To plot a derivative, choose one of the following derivative options from the listmenu:

• Linear [dy/dx]

• [xdP/dx]

• [ISIP-xdP/dx]

• Semi Log [dy/d(log x)]

• Semi Log [d(log y)/dx]

• Log [d(log y)/d(log x)]

For Nolte G time derivatives, the nomenclature will use the G symbol (i.e., ISIP-GdP/dG).


4.4 Analysis 377

The derivative is calculated using the formula for the option selected, where y is thecurrent pressure curve (Pressure or Delta Pressure) and x is the current time func-tion. A modified three-point method is used to calculate the derivative, as describedin the Options section. The derivative is plotted for the data defined by the Min/Max Range Bar at the top of each plot.

When using a log scale on the Y-axis, it is impossible to plot a negative derivative.To remedy this situation, check the Plot absolute value of derivative from theAxes⏐Derivative/Rate menu. Then the absolute value of the derivative will beplotted.

Example Regression AnalysisFollowing are a set of regression analyses using the different Axes options.

Figure 4.88 shows a Nolte G plot in linear coordinates. The right axis displays theISIP-GdP/dG derivative. As illustrated, this derivative helps identify closure asshown by the deviation from the measured data.

Figure 4.88: Nolte G Time Plot - Linear Coordinates.

Figure 4.89 shows the corresponding Delta Pressure plot in log coordinates with theGdP/dG derivative function. The same deviation pattern is also illustrated in thegraph.

Note the time of closure will always be slightly after the derivative curve devi-ates from the data (to the right of deviation).



Figure 4.89: Delta Pressure Nolte G Time - Log Coordinates.

Figure 4.90 shows the Delta Pressure plot in log coordinates. The slope,, is presented. From Table 4.2 the slope in log space should be

unity for the fracture and for radial flow. Values of two represent storage.Slopes greater than two represent other anomalous effects. As illustrated the slopeis near unity up to the time of closure (TC) and then drops to around . The earlytime data derivative is not critical since some water hammer is present.

Note: for the Nolte G function the GdP/dG derivative should normally be usedwith the Delta Pressure plots and the ISIP-GdP/dG derivative with the Pressureplots. This also applies to the generic plots (i.e., Pressure use ISIP-xdP/dx, andDelta Pressure uses xdP/dx). In log-log coordinates, the slope, ,may also be used.

d ylog( ) d xlog( )⁄

d ylog( ) d xlog( )⁄

1 2⁄

1 2⁄


4.4 Analysis 379

Figure 4.90: Delta Pressure - Log Slope.

Figure 4.91 shows the Delta pressure (ISIP-p(t)) in linear coordinates with the GdP/dG derivative on the right axis. As illustrated the time of closure is slightly to theright of the derivative deviation from the Delta Pressure curve.

Figure 4.91: Delta Pressure Plot - Linear Coordinates.



History MatchTo display the history match, click on the Analysis⏐Regression⏐History Matchmenu shown in Figure 4.92.

Figure 4.92: Regression - History Match.

History Matching allows for specifying a dependent parameter for each fracturegeometry model. Then regression is performed by systematically varying thedependent parameter to achieve the best match with the measured decline data.During the regression calculations, the dependent parameter is continuouslyupdated and displayed along with a graphical representation of the simulation. Toevaluate parameter sensitivity or other parametric variations different regressionscan be performed.

To History Match Data:

1. Make sure the Base Data and Leakoff Data have been entered, and the closuretime (TC) has been selected from the Analysis⏐Regression⏐Select Pointsmenu.

2. Choose the regression parameter for each model using the GDK, PKN andEllipsoidal list boxes found in the toolbar.

3. After choosing the models and dependent variables for regression, click theHistory Match button to History Match on the selected parameters (see Figure4.93).

4. View the History Match Data and make any desired changes. For successiveregressions another data parameter must be changed to get a different regres-sion solution. If nothing is changed from one regression to another, MinFracwill have already minimized the error, and therefore, cannot improve on thematch (i.e., the program will have nothing to do).


4.4 Analysis 381

5. After performing a history match, the calculation results can be viewed byselecting the Data⏐History Match menu as shown in Figure 4.94 (see alsoFigure 4.19).

Figure 4.93: Regression - History Match Solution.



Figure 4.94: History Match - Data Dialog Box.

After Closure AnalysisThe purpose of an after closure analysis is to determine the formation permeabilityand reservoir pressure from the pressure response of a fractured (or unfractured)well during the infinite-acting time period (i.e., late time period or radial solution).

The following information can be determined from a properly conducted analysis:

1. Reservoir pressure, .

2. Formation permeability, .

Since the after closure analysis plots and procedures are very similar to the discus-sion of the Regression Analysis redundant explanations of various steps and theWizard will not be reiterated.

The menu command items for the After Closure Analysis are described below.

pi

k


4.4 Analysis 383

Select RangesThe Analysis⏐After Closure⏐Select Ranges menu item, as seen in Figure 4.95 isused to specify a specific data range.

Figure 4.95: After Closure - Select Ranges Menu

Select TCThe Time of Closure (TC) is specified within the Select TC dialog box (Figure4.97), and can be opened by clicking the Analysis⏐After Closure⏐Select TCmenu item (Figure 4.96).

Figure 4.96: After Closure - Select TC Menu



Figure 4.97: After Closure - Select TC Dialog Box

Select PointsTo select the data points for the After Closure analysis, click on the Analysis⏐After Closure⏐Select Points menu item as shown in Figure 4.98.

Figure 4.98: After Closure - Select Points Menu

The Select Points plot is used to graphically select data points for the after closureanalysis. Two data points are required to generate data.


4.4 Analysis 385

Example Regression AnalysisFollowing are a set of after closure analyses using the different Axes options as dis-cussed in Appendix K.

Figure 4.99 shows a Nolte After Closure plot in linear coordinates. The right axisdisplays the derivative. As illustrated, this derivative helps identifythe correct slope and p* value as shown by the deviation of the derivative from themeasured data

Figure 4.99: After Closure Analysis - Surface Pressure vs. Nolte - FR Linear Plot.

Figure 4.100 shows the Delta Pressure plot in linear coordinates with the derivative function. The same deviation pattern is also illustrated in the graph.

p* x÷ dp dx⁄

xdP dx⁄



Figure 4.100: After Closure Analysis - Delta Surface Pressure vs. Nolte - FR Linear Plot with the xdP/dx Derivative.


4.4 Analysis 387

Figure 4.101: After Closure Analysis - Delta Surface Pressure vs. Nolte - FR Log-Log Plot with xdP/dx derivative.

Figure 4.102 shows the corresponding Delta Pressure plot in log coordinates withthe log-log derivative function. As illustrated the log-log slope asymptotes to onefor pseudo-radial behavior.



Figure 4.102: After Closure Analysis - Delta Surface Pressure vs. Nolte - FR Log-Log Plot with the Log Slope Derivative.

4.5 OutputWhen the closure time has been determined and all of the required reservoir infor-mation has been entered, calculations can be performed to determine fracture andreservoir characteristics (leakoff coefficients, etc.). The methodology used tomodel fracture characteristics and fluid leakoff behavior is outlined in Appendix F.

The Output menu is shown in Figure 4.103. There are two main Output sectionitems, simulation results and viewing the report.


4.5 Output 389

Figure 4.103: Output Menu.

Simulation CalculationsThere are two types of simulations that MinFrac can do, Simulation using BaseData and Simulation using History Match Data as selected from the Output menu.

Base Data CalculationsThe Base Calculations are accessible from the Output⏐Simulation using BaseData menu. This will run the simulation for all the models using the Base Data.When using the Graphical Technique, some of the input data will be taken from theimported data, as specified in the Options⏐Graphical calculation menu. Then asummary of the simulation results will be shown for the GDK, PKN and Ellipsoidalmodels as shown in Figure 4.104.



Figure 4.104: Output - Simulation Using Base Data.

Table 4.6 contains a description of the output data.

Table 4.6: Output Data.

Parameter Description

Vol. Fluid Inj. Volume of fluid injected into the fracture

Length Half length of the fracture

Height (wellbore) Height of the fracture

Max. Well Width Maximum width at the wellbore

Avg. Well Width Average width at the wellbore

Avg. Frac. Width Average width throughout the fracture

Net Pressure Fracture net pressure

Efficiency Fracture efficiency

Closure time - Delta Time from the ISIP to closure

% error % error in closure time

Ct Total leakoff coefficient

CI Leakoff fluid viscosity effects


4.5 Output 391

History Match CalculationsWhen using the Graphical Technique, the History Match Calculations are accessi-ble with the Output⏐Simulation using History Match Data menu. This will runthe simulation like the Base Calculations, except where applicable, the data in theHistory Match dialog box will be substituted for the Base Data. The same set ofsimulation results will be displayed.

The parameters used for history matching are the ISIP, closure time and closurepressure. Consequently, the history matched net pressures (ISIP- closure pressure)will be the same for all models. The fracture efficiencies will also be approximatelyequal for all models. A summary of the History Match results for the GDK, PKNand Ellipsoidal models is shown in Figure 4.105.

Figure 4.105: Output - Simulation Using History Match Data.

CII Reservoir compressibility and viscous effects

CIII Filter cake coefficient

Permeability Reservoir permeability




The history match solution matches the calculated net pressure (ISIP-Closure Pres-sure). Figure 4.105 shows that the net pressures are the same for all three geometrymodels. This is not so for the base data output (see Figure 4.104).

From history matched parameters, the individual leakoff coefficients for each frac-ture model can be calculated based on the leakoff area. Remember that minifracanalysis only gives insight into the fracture efficiency. A fracture model is requiredto calculate the leakoff area.

Although all fracture models will give a history-matched solution for variousdependent parameters (within reason), it is up to the design engineer to evaluate/determine if this is a reasonable solution. The model with the most reasonable his-tory matched parameters should be the model of choice.

Since all fracture models will have about the same fracture efficiencies or leakoffcoefficient and area product (based on mass conservation), the calculated leakoffcoefficient is strongly dependent on the fracture model. Consequently, do not use aleakoff coefficient calculated for a GDK model in the PKN model unless the lea-koff areas are the same.

The estimated reservoir permeability is calculated from the leakoff coefficients asdiscussed in Appendix F. Therefore, if CIII is dominant, the calculated permeabil-ity uncertainty will be high.

ReportsMinFrac can generate reports similar to the other Meyer Programs. Before generat-ing a report, MinFrac will inquire if Base Calculations and/or History Match Calcu-lations are to be included in the report. Figure 4.106 shows the View Report dialogbox.

Simulation using History Match Data is not available when using the User Speci-fied Closure option.


4.6 References 393

Figure 4.106: Output - View Report Dialog Box.

The order of the Time of Closure (TC) points can be changed using the Move Upand Move Down buttons. Only the rows marked active will be displayed in thereport.

For more report options, select Configuration... from the Output menu (See“Report Configuration” on page 68.)

Manage PointsThe Manage Points dialog (Output⏐Manage Points...) can be used to deleteunwanted points or to change the order of existing points. Any changes madewithin the Manage Points or View Report dialog will be reflected in the other dia-log boxes and maintained per minfrac file.

4.6 References 1. Nolte, K.G., Smith, M.B.: “Interpretation of Fracturing Pressures”, SPE 8297,

Sept. 1979.



2. Nolte, K. G.: “Determination of Fracture Parameters from Fracture PressureDecline,” SPE 8341 presented at the 54th Annual Technical Conference, LasVegas, Sept. 1979.

3. Nolte, K.G.: “Fracture Design Considerations Based on Pressure Analysis”,SPEPE, Feb. 1988, pp 22-30.

4. Nolte, K. G.: “A General Analysis of Fracturing Pressure Decline With Appli-cation to Three Models,” (SPE 12941) JPT (Dec. 1986), 571-582.

5. Nolte, K. G.: “Application of Fracture Design based on Pressure Analysis,”SPEPE (Feb. 1988), 31-41.

6. Castillo, J.L.: “Modified Pressure Decline Analysis Including Pressure Depen-dent Leakoff”, SPE 16417, May 1987.

7. Lee, W.S.: “Study of the Effects of Fluid Rheology on Minifrac Analysis,”SPE 16916, Sept. 1987.

8. Meyer, B.R., Hagel, M.W., “Simulated Mini-Frac Analysis”, Petroleum Soci-ety of CIM, Calgary June 1988.

9. Hagel, M. W. and Meyer, B. R.: “Utilizing Mini-Frac Data to Improve Designand Production,” CIM paper 92-40 June, 1992.

10. Meyer, B. R.: “Frac model in 3-D - 4 Parts,” Oil and Gas Journal, June 17,July 1, July 22 and July 29, 1985.

11. Meyer, B. R.: “Design Formulae for 2-D and 3-D Vertical Hydraulic Fractures:Model Comparison and Parametric Studies,” paper SPE 15240 presented at theSPE Unconventional Gas Technology Symposium, Louisville, KY, May. 18-21, 1986.

12. Meyer, B. R.: “Three-Dimensional Hydraulic Fracturing Simulation on Per-sonal Computers: Theory and Comparison Studies,” paper SPE 19329 pre-sented at the SPE Eastern Regional Meeting, Morgantown, Oct. 24-27, 1989.

13. Gidley, J. L., Holditch, S. A., Nierode, D. E. and Veatch, R. W.: “RecentAdvances in Hydraulic Fracturing,” SPE Monograph Vol. 12, Chapter 14,1989.

14. Warpinski, N.R.: “In-Situ Stress Measurements at U.S. DOE's MultiwellExperiment Site, Mesa Verde Group, Rifle, Colorado,” JPT (March 1985) pp527-536.


4.6 References 395

15. Warpinski, N.R.: “Investigation of the Accuracy and Reliability on In-SituStress Measurements Using Hydraulic Fracturing in Perforated, Cased Holes,”Proc., 24th U.S. Symposium on rock Mechanics, College Station, TX (June1983) pp 773-786.

16. Smith, M.B.: “Stimulation Design for Short, Precise Hydraulic Fractures,”SPEJ (June 1985) pp 371-379.



19. Huit, J.K.: “Fluid Flow in Simulated Fractures,” AIChE Journal, Vol. 2, pp259, 1956.


21. Warpinski, N.R.: “Measurement of Width and Pressure in a propagatingHydraulic Fracture,” SPEJ (Feb. 1985) pp 46-84.




Chapter 5

MProdAnalytical Production Simulator

5.1 IntroductionMProd is a single phase analytical production simulator developed by Meyer &Associates, Inc. Although the program was designed primarily for hydraulic frac-turing applications, it can also be used to explore the production potential of unfrac-tured reservoirs. This chapter explains the options available and basic proceduresfor running MProd. Detailed information regarding the methodology and basic the-ory is presented in Appendix G.

MProd has options for Production Simulation, History Match Production Simula-tion, and Fracture Design Optimization. Production Simulation, allows the user toinput typical production data to simulate well performance for fractured and unfrac-tured wells. The capability to compare the output (numerical simulated results)with measured data is also provided. An objective methodology for determiningunknown or uncertain parameters by regression analysis of measured data throughhistory matching is available. The procedure is implemented by parametric optimi-zation to minimize the error (standard deviation) between the measured and historymatched results. The history matching process allows the user to history match onvarious parameters depending on whether the well is unfractured or fractured. Foran unfractured well, the user has an option to history match on the followingparameters: 1) reservoir drainage area (closed system), 2) permeability, 3) reservoiraspect ratio, and/or 4) wellbore skin. For the fractured well, the user has an optionto history match on one or all of the following parameters: 1) reservoir drainagearea (closed system), 2) permeability, 3) reservoir aspect ratio, 4) fracture length,and/or dimensionless fracture conductivity. A Fracture Design Optimization fea-ture enables the user to determine the optimum fracture design (length, width, con-ductivity) that will maximize production for a given amount of proppant mass.Appendix L provides a detailed explanation of the theory and optimization method-ology,


398 MProd: Analytical Production Simulator

MProd is integrated and fully compatible with MFrac to provide full feature optimi-zation. Output produced by MFrac can be used by MProd. The numerical results ofMProd, in turn, can be imported by MNpv to perform economic analysis.

An outline of the basic steps for using MProd is shown in Table 5.1

Table 5.1: MProd Basic Steps

Step Program Area

1. Open an existing MProd data file (*.mprod) orcreate a new data file File Menu


3. Select a Simulation Optiona) Production Simulationb) History Match Productionc) Fracture Design Optimization

Data Menu

4. Input Dataa) Production Simulation Formation Data Fracture Characteristics Variable Fracture Conductivity NPV Fracture Data (if NPV) Gas PVT (if gas) Production Data Measured Data (if option is on) Well Datab) History Match Production Formation Data Fracture Characteristics History Match Parameters Gas PVT (if gas) Production Data Measured Data (if option is on) Well Datac) Fracture Design Optimization Formation Data Fracture Characteristics Proppant Data Design Optimization Well Data

Data Menu


5.2 Options 399

5.2 OptionsThis section outlines the flexibility of the MProd program and describes the param-eters that define the conditions and intended use of the software. The first step inperforming a simulation is establishing the options that will be used. This is accom-plished by accessing the Options dialog box from the main menu.

The option selections determine the scope of the MProd program. The optionsestablish the input to be entered and the nature of the calculations to be performed.The parameters selected are global for the current file you are working with. Theyremain with the file and are saved and recalled with the data.

The Options dialog box is typically the first input screen used in the MProd pro-gram. Its function is to establish the primary model options that will be employed.Each section deals with a different aspect of the modeling approach.

To access the Options screen, select Options from the main menu by clicking themenu name. The dialog box displayed in Figure 5.1 will then be presented.




Table 5.1: MProd Basic Steps




The Options screen determines what information is needed for a particular type ofanalysis. The specific data displayed in a screen or the existence of a data screenitself varies depending on the options selected. This “smart-menu” approach, mini-mizes data input and prevents unnecessary or misleading data entry. Simply decidethe relevant options for a specific simulation and the program will only displaythose menus and input fields necessary. Any time the options are changed the inputdata screens will be updated to enable new input or hide data that is not needed.This hierarchy methodology is used throughout MProd.

To Select an option, click the radio button adjacent to the option preference. Ablack dot appears in the center of the button when a choice is made. To move to thenext option, click a radio button within the next option section, or use the TAB but-ton to move sequentially through the choices. Once you are within a section thecurrent selection for that option is highlighted with a dotted rectangle. The optionchoice may be changed by using either the mouse or the arrow keys.


5.2 Options 401

An explanation of the choices available for each of the Program Options are sum-marized in the following section.


Simulation OptionsThis section describes the fundamental options available for running MProd.Depending on the Simulation Options selected different general options and dialogswill be displayed. Throughout this User’s Guide, we will identify which sectionsare applicable for the different options.

Production SimulationThis option allows the user to input typical production data to simulate well perfor-mance for fractured and unfractured wells. The fluid and formation properties areinput and the code will simulate the projected production rate or bottomhole treat-ing pressure based on the specified boundary condition. The production simulationoption also provides the capability to compare the output (numerical simulatedresults) with measured data.

Overlay Measured Data

If Production Simulation is selected the user has an option to overlay measured datafor comparison. If the Overlay Measured Data option is checked the user will beprompted in another dialog to either enter the measure rate or bottomhole flowingpressure versus time. This data will then be overlaid on the numerical simulationresults.

History Match Production Simulation DataThis option should be selected if one wishes to history match actual measure datawith simulated data. The numerical procedure uses an objective methodology fordetermining unknown or uncertain parameters (through regression). This procedureis implemented by parametric optimization to minimize the error (standard devia-tion) between the measured and history match results.

The history matching process allows the user to history match on various parame-ters depending on whether the well is unfractured or fractured. For an unfracturedwell, the user has an option to history match on the following parameters: 1) reser-voir drainage area (closed system), 2) permeability, 3) reservoir aspect ratio, and/or



4) wellbore skin. For the fractured well, the user has an option to history match onone or all of the following parameters: 1) reservoir drainage area (closed system),2) permeability, 3) reservoir aspect ratio, 4) fracture length, and/or dimensionlessfracture conductivity.

Since the history match algorithm uses a methodology of steepest decent, a mini-mum, estimated, maximum value for a parameter must be specified. The initial esti-mate for a given parameter can either be Internally Estimated or User Defined. Theuser must also select the Maximum Iterations allowed for the regression analysis.

The user must be aware that if the solution is not dependent on a given parameterthe code may select any value in the minimum/maximum range depending where itinitializes or estimates the starting point. (e.g., a short term well test may not beable to accurately determine the drainage area or aspect ratio).

Internal Estimate

Given a minimum and maximum range for a history match parameter, the code willprovide an estimate of the best parameter values to minimize the error between themeasured and history matched data. A report is then available that lists the best andworst fit parameters based on the associated error.

This option is useful if you have no idea what a reasonable history match value forthe parameters may be.

User Estimate

Given a minimum and maximum range for a history match parameter, the usermust also provide an estimate of the best parameter values to minimize the errorbetween the measured and history matched data.

This option is useful if a reasonable history match value can not be readily esti-mated.

Number of Iterations

This is the maximum number of iterations the code will preform prior to exitingwith a last best fit parameter. This is necessary in case the code has not fully con-verged within some reasonable number of iterations (i.e., the solution is non-lin-ear). Normally, 20 to 50 iterations are sufficient for a reasonable solution. If you aretrying to get a quick estimate of the history match parameters (especially a largenumber) you may want to put in fewer iterations and then refine your minimum andmaximum ranges in the history match parameters dialog.


5.2 Options 403

Fracture Design OptimizationFracture Design Optimization is not a new concept but has received great attentionin the last few years as attributed to concept of “Unified Fracture Design” as pre-sented by Econimides, Valko and others. This concept, however, was firstaddressed by Prats in 1961.

The main idea of Fracture Design optimization is to provide the user with an opti-mum fracture design (length, width, conductivity) for a given proppant mass thatwill maximize production. Appendix L provides a detailed explanation of the the-ory and optimization methodology,

Selecting this option, a pseudosteady state analysis will be preformed based on theformation data, proppant properties, and desired proppant mass pumped to optimizeproductivity. The numerical results provide the user with the optimum design char-acteristics for fracture width, length, penetration ratio, dimensionless conductivity,conductivity, concentration/area, productivity index and productivity ratio. Theseoptimum values can then be used as input into MFrac for auto scheduling the opti-mum design.

This option also allows for the addition of Optimization Diagnostic Plots option forMcGuire and Sikora Type Curves and Optimum Fracture Performance Curves. Theadditional diagnostic plots are discussed in Appendix L.

To provide a curve of optimum fracture characteristics versus proppant number ormass additional values of proppant number (mass) are used. The Number of Subdi-visions between the minimum and maximum proppant masses input into the DesignOptimization data table is used to generate a smooth curve.

Number of Sub-divisions

The number of data points used to provide a smooth curve of fracture optimizationparameters is the Number of sub-divisions plus one. If the number of subdivisionsis less than the number needed between each table sequence, the code will addadditional points.

If the Optimum Fracture Performance Curves are activated, the Number of sub-divisions is specified within that section instead.

Optimization Diagnostic PlotsOptimization Diagnostic Plots for McGuire and Sikora Type and Optimum Frac-ture Performance Curves may be selected by checking the Active check box foreach. A description of the required input is given below.



McGuire and Sikora Curves

If the McGuire and Sikora Curves are activated, the range (minimum, maximum),and number of sub-divisions for the dimensionless conductivity are required. Nor-mally 100 sub-divisions (101 data points) is sufficient to generate a smooth curveas presented in Appendix L (see also SPE 95941).

Optimum Fracture Performance Curves

The Optimum Fracture Performance Curves if activated will provide a set of diag-nostic plots versus dimensionless conductivity and proppant numbers as presentedin SPE 95941. The Fracture Design Optimization curves for each of the proppantmasses (and proppant numbers) entered in the Design Optimization Data table willbe presented as constant proppant numbers as a function of dimensionless conduc-tivity and productivity index (and productivity ratio). To generate these curves theuser must input 1) Dimensionless Conductivity range (minimum, maximum), andnumber of sub-divisions and 2) Proppant Number range (minimum, maximum),and number of sub-divisions. If the proppant minimum and maximum values arenot within the minimum and maximum values in the Design Optimization Datatable, the Design Optimization Data table minimum and maximum proppant num-bers will be used.

ModelThe Model options are only applicable when the Simulated Options of ProductionSimulation or History Match Production Simulation Data is selected. These optionsare not applicable for pseudosteady-state analyses as used with the Fracture DesignOptimization Option.

MProd provides four (4) alternatives to describe the reservoir configuration andboundary conditions for a simulation. The selections are:

No Fracture• Closed System - This case describes a square or rectangular, bounded reser-

voir that has not been hydraulically fractured. The well can be located any-where in the closed system by describing the Reservoir Drainage Area andDimensionless Well Locations found in the Formation Data dialog box. Thisoption is not available when the NPV Option is selected.

• Infinite Reservoir - This case models the performance of a well located in aninfinite (unbounded) reservoir. Options for a base and stimulated wellbore skinfactor are available in the well data dialog.


5.2 Options 405

Fractured Well• Closed System - This model simulates a finite conductivity fracture in a

closed reservoir. Reservoir boundaries and well location are specified with theReservoir Drainage Area and Dimensionless Well Locations as in the caseabove. When this model is used in combination with the NPV option, the out-put provides a comparison of the fractured and unfractured well productivity.Unfractured well skin factors can be specified and will be used in both cases.

• Infinite Reservoir - This case models the performance of a finite conductivityvertical fracture penetrating an infinite reservoir. When the NPV option is On,the productivity of the fractured well is compared to the potential of the samewell without a fracture. If Well Skin is specified, it is used for both the frac-tured and unfractured case. If production is below the bubble point, then aninfinite reservoir with a modified oil compressibility should be used.

The method of images is used to generate rectangular drainage shapes for closedsystems. Drainage area aspect ratios can be as large as 100.

Fluid TypeThe Fluid Type options are only applicable when the Simulated Options of Produc-tion Simulation or History Match Production Simulation Data is selected. Theseoptions are not applicable for pseudosteady-state analyses as used with the FractureDesign Optimization Option.

MProd is an analytical, single phase reservoir simulator capable of modeling thediffusion of a homogenous liquid or gas through porous media. To predict bothpressure and rate changes through the reservoir, it is important to accuratelydescribe the properties of the fluid. The Fluid Type option is used to specify the pri-mary fluid produced from the reservoir to direct the calculations through the appro-priate algorithms. The algorithms used will vary depending upon the fluid typeselected. Either Oil and Water, or Dry Gas can be specified.

In general, for liquid production above the bubble point, the oil or water phase isassumed to be slightly compressible. Conversely, when the fluid type is specified asgas, the gas phase is assumed to be highly compressible. These basic differencesobviously influence model behavior.

Regardless of Fluid Type, an additional option is available for selecting whetherinternal or user-specified fluid property correlations are to be used. Refer to thedescription of the Internal PVT Table option later in this section for further expla-nation.



Internal PVT TableFor liquids, if the Internal PVT Table is turned Off, the Total Reservoir Com-pressibility and the Equivalent Reservoir Viscosity must be entered in the Forma-tion Data dialog box. The values input for these parameters will be used asconstants for the simulation. If the Fluid Type is gas and the Internal PVT Table isturned Off, a table of Gas Viscosity and Z-Factor as a function of pressure must beentered. It is also necessary to enter a Reservoir Temperature in the FormationData dialog.

When the Internal PVT Table is On, internal correlations are used to generate thePVT data for the Fluid Type specified. For oil, the oil solution GOR and formationvolume factor versus pressure, are determined using the correlation originallydescribed by Vazquez and Beggs1. For these same conditions, the oil viscosity ver-sus pressure is estimated based on the work of Beggs and Robinson2. For this com-bination of options, the Gas Gravity, Oil gravity and bubble point pressure must beentered.

When the Fluid Type is gas and the Internal PVT Table is On, the Gas SpecificGravity and Reservoir Temperature are required in the Formation Data dialog box.The correlation used for gas viscosity is as originally described by Lee &Gonzalez3. If the Internal PVT Table is Off, and the Fluid Type is gas, a separateGas PVT Table must be entered. This table contains a list of the Gas Viscosity andZ-Factor as a function of pressure.

Production Boundary ConditionThe Production Boundary Condition option is only applicable when the SimulatedOptions of Production Simulation or History Match Production Simulation Data isselected. This option is not applicable for pseudosteady-state analyses as used withthe Fracture Design Optimization Option.

This option specifies whether the simulation will be based on a specified rate orpressure production. When Rate is selected, production will be simulated at con-stant rate or a series of variable rates depending on the data entered. SpecifyingPressure results in the simulation of a constant bottomhole flowing pressure orseries of variable bottomhole flowing pressures. Since the simulator uses superpo-sition, mixing constant rate and pressure production is not permitted.

The convention for rate is positive for production and negative for injection. If anegative rate is specified in the production table, the properties are assumed to bethe same as the reservoir fluid.


5.2 Options 407

Fracture Options The Fracture Options screen displayed in Figure 5.2 allows the user to specify thetype of analysis to be performed.

Figure 5.2: Data Options Screen - Fracture Tab

The choices available for each of the Fracture Options are summarized as follows:

Non-Darcy EffectsThe equation to describe non-Darcy flow is a form of the Forchheimer [1901] equa-tion

xddp– μ

kf----υ β ρυ2( )+=




etc.) and is the non-Darcy flow factor or simply factor with units of (e.g.,cm-1, ft-1, atm-s2/gm etc.). Clearly the first term in this equation accounts for vis-cous effects and the second term for inertial or minor loss effects. If the secondterm on the right hand side is omitted, the equation simplifies to Darcy’s law. Thusnon-Darcy flow describes the flow regime that does not obey Darcy’s law. Holdith[1976] reports that the original form of the second term on the right hand side of the

above equation by Forchheimer was which was replaced by Cornell and Katz[1953] by the product of the fluid density, , and the factor.



The Non-Darcy Effects options are given below:

Darcy Only

Non-Darcy effects will not be considered. This is the same as assuming .

Input Beta Coefficient

The non-darcy beta coefficient is user specified and assumed constant. A valuemust be entered in the dialog.

User Database, Beta Coefficient

If this option is selected, a non-darcy beta coefficient correlation is selected fromthe Non-Darcy proppant Database drop down menu. The beta coefficient will thenbe calculated as a function of proppant permeability and porosity.

Permeability OptionsThe fracture Permeability Options are given below:

kf L2

β β L 1–

aυ2

ρ β

kf φ

β akf

bφc-----------=

a b c Sw

φe φ 1 Sw–( )=

β 0=


5.2 Options 409

Input Fracture Conductivity

The fracture conductivity ( ) is user specified. This option is only available ifthe Darcy Only option is selected. Since for inertial flow (non-Darcy) both the frac-ture permeability and fracture width are required.

Calculate Fracture Permeability, Width, or Conductivity

This option provides the user with greatest flexibility in inputting and calculatingan unknown any combination of fracture permeability, width, or conductivity arespecified. This option is required for Non-Darcy flow. The theoretical proppant per-meability may also be determined by using the Proppant Calculator under ToolsMenu.

Proppant Property DataFollowing is a list of the proppant property data required to calculate the Non-Darcy Beta factor:

Proppant Porosity

This is defined as the void fraction between sand grains (i.e., liquid volume toslurry ratio of the settled bank). It is used to calculate the propped fracture perme-ability. Typical values of porosity for proppants are shown in Table 5.2.

NPV/Multi-CaseThe Production Boundary Condition option is only applicable when the SimulatedOption of Production Simulation is selected. This option is not applicable for His-tory Match Production Simulation Data or the Fracture Design OptimizationOptions.



6-8 angular 0.36

10-20 angular 0.36

10-20 round 0.32

20-40 round 0.35

40-60 round 0.32

kfwf



When NPV/Multi-Case is turned Off, the simulator will run in standard mode simu-lating the production response of a well and reservoir with or without a fracture,depending on the Model selected. Turning NPV/Multi-Case calculations On allowsthe simulator to calculate a series of automated runs based on a variety of fracturegeometries. The fracture characteristics may be identified by either specifying aMFrac file containing a *.mfrac extension or inputting a table of fracture datadirectly (i.e., User Specified). Please refer to the description of the NPV/Multi-Case Fracture Data Source option for more information on this topic as presentedbelow.

If NPV/Multi-Case is On, a series of fracture designs (Multi-Case) with variablepropped fracture length and fracture conductivity can be input into a table. Thenumerical results for the multi-case can then be compared in the report or graphi-cally to determine the effect of fracture length and/or conductivity on production.

When the NPV/Multi-Case option is used, the output produced can be read by theMNpv program to perform economic calculations. This procedure can be used todesign the optimum treatment.

NPV/Multi-Case Fracture Data Source

When the NPV/Multi-Case Option is On, there are three available methods forspecifying the fracture characteristics for the production simulation. They are:

1. Select MFrac File and enter or browse for the path and file to use containing anmfrac extension. The file must have been created in MFrac with the NPVoption set to On. If the NPV option was not used to create the fracture outputfile, an error message will be displayed. Also, the MProd and MFrac versionsof the software must be compatible.

2. Choose User Specified and the NPV Fracture Data dialog will be enabledfrom the Data menu. Use the import button provided in the NPV/Fracture Datadialog to read the data from any MFrac “.mfrac” file containing NPV data.Once the data is imported it can be modified.

Variable ConductivityThis option allows the user to input a spatially varying fracture conductivity. Thisoption is only available if the NPV/Multi-Case is not selected. If the option ischecked, a Variable Fracture Conductivity dialog will be activated with input forfracture height, width, permeability, conductivity, and dimensionless conductivityas function of fracture position.


5.3 Data Input 411

5.3 Data InputOnce the options are selected and the scope of a simulation is set, data may beentered by accessing the various dialogs available under the Data menu. As previ-ously stated, the options selected determine what information is needed for a partic-ular type of analysis. Therefore, the specific data displayed in a screen or theexistence of the data screen itself will vary depending upon the options chosen.This approach minimizes data input and prevents unnecessary or misleading dataentry. Simply decide what options are relevant to the simulation and MProd willonly display those menus and input fields necessary. Any time an option ischanged, the screens will vary to enable new input, or hide data that is not needed.This methodology is used throughout MProd.

The following sections pertain to the Data menu items found by selecting Datafrom the program's Main menu. Each Data menu item is covered in detail alongwith a description of the data dialogs and their associated variables. When perti-nent, the conditions or case sensitive options for a data screen are noted and anexample of the resulting dialog is shown. All of the different data screens availablein MProd and the variables contained within them are presented.

DescriptionThe Data Description screen shown in Figure 5.3 provides a location for enteringinformation about a simulation. Space is provided for entering the CompanyName, Well Name, Well Location and Simulation Date. In addition, a Commentssection is included so that descriptive information can be entered. All informationcontained in this dialog is optional.



Figure 5.3: Data Description Dialog Box - MProd.

Formation DataThe Formation Data dialog box (see Figure 5.4) provides a location for enteringmost of the reservoir properties needed to perform a simulation. As you becomefamiliar with this data screen, you will notice that many of the parameters willappear or disappear as the program options are changed. For example, changing theFluid Type from oil to gas with the Internal PVT Table option enabled, will resultin the elimination of the variables related to the oil PVT (e.g., oil gravity, bubblepoint pressure).


5.3 Data Input 413

Figure 5.4: Formation Data Dialog Box.

Each of the items found in the Formation Data dialog screen are explained below. Ifnecessary, the conditions for data entry are indicated.

Reservoir Drainage AreaThe Reservoir Drainage Area is only required when the Fractured Well/ClosedSystem or No Fracture reservoir model is enabled. This is the area of the reservoirto be produced. It usually represents, in map view, the lateral extent of the reservoirdrained by a particular well. The reservoir volume is obtained by multiplying thearea times the pay zone thickness (see Figure 5.5).

Figure 5.5: Definition of the Reservoir Drainage Area.

Dimensionless Reservoir Aspect RatioConcerning the reservoir's spatial relationship with the wellbore and fracture,MProd currently assumes that the drainage area, described above, can be approxi-mated by a rectangular shape. A rectangle is defined by entering the Dimension-



less Reservoir Aspect Ratio ( ). This is the ratio of the reservoir half-length

( ) and half-width ( ) illustrated in Figure 5.6. Entering a value of one (1) forthis parameter corresponds to a square drainage area. An aspect ratio of four (4)represents a rectangular area with a length 4 times greater than its width.

For a hydraulically fractured reservoir, the fracture azimuth is always assumed toparallel the orientation. The aspect ratio should, therefore, be greater than orequal to one (1) so that the fracture parallels the long axis of the block.

Dimensionless Well LocationIn MProd, the wellbore can be positioned anywhere within the defined drainagearea. This is accomplished by specifying two Dimensionless Well Locationsparameters titled Dimensionless Well Location - Direction and DimensionlessWell Location - Direction. Entering and values of zero centers the well onthe axis. The other positions can be achieved by using the convention summarizedin Figure 5.6. The values input must range between -1.0 and 1.0. Any fraction ofthe total reservoir dimension can be used to position the wellbore within the rectan-gular coordinate system used.

Figure 5.6: Wellbore Positioning within the Drainage Area.

Total Pay Zone HeightBy definition, the Total Pay Zone Height or net pay thickness is the portion of theformation that contains the mobile hydrocarbons. Together with the drainage radiusand effective porosity, it defines the effective reservoir drainage volume.

If the productive rock is discontinuous or made up of intermittent layers of perme-able and impermeable material, this value should be entered as the sum of the indi-vidual permeable zones. For example, if the pay zone lies between 8500 and 8600ft., but only 50 ft. of the interval is permeable, enter 50 ft. for the Total Pay ZoneHeight.

XL YH⁄

XL YH

XL

XY X Y


5.3 Data Input 415

Initial Reservoir PressureTo simulate pressure behavior in the reservoir, the initial conditions must be char-acterized. For new wells, enter the initial reservoir pore pressure for the productiveinterval. This value is typically obtained from either a production log or well test.

When a well has been produced for some period of time, enter the average reservoirpressure as interpreted from a well measurement. In all instances, the value enteredfor the Initial Reservoir Pressure should be less than the minimum horizontalstress in the pay zone.

Total Reservoir CompressibilityThe Total Reservoir Compressibility is defined as the total change in the reservoirvolume per unit volume per unit pressure difference. It is the reciprocal of the un-drained bulk modulus. This value is only required when the Fluid Type is oil orwater and can be estimated using the following relationship:

where

When the Fluid Type is gas, the compressibility is calculated internally using thefollowing relationship:

where


= gas compressibility= initial reservoir pressure


cg

co

cr

ct

cw

Sg

So

Sw

cg1p--- 1

Z---

pddZ–=

cg

p



Please refer to SPE Monograph 5, “Advances in Well Test Analysis” and AppendixD for typical oil, gas, water and rock compressibilities. This information isextremely important in order to accurately characterize the reservoir diffusivity, D,as illustrated in the following expression:

where

When possible, it is always advisable to “calibrate” the rate of diffusion by per-forming a well test or adjusting the parameters shown above in an attempt to historymatch the flow rate for some period of time.

Equivalent Reservoir PermeabilityAs shown in the above equation, the permeability is an important parameter indetermining the reservoir diffusivity. The reservoir permeability is the formationproperty that characterizes its ability to transfer a fluid through the pores when sub-jected to a pressure gradient. From Darcy's law:

where

MProd is a single-phase simulator, therefore, the reservoir permeability, or Equiva-lent Reservoir Permeability (as well as, reservoir viscosity) must match the sys-tem mobility for a multi-phase system. The total flowing mobility is

= gas Z-factor

= total reservoir compressibility= equivalent reservoir permeability= equivalent reservoir viscosity= equivalent reservoir porosity

= fluid mobility= fluid velocity

= pressure gradient

Z

D kφμct-----------=

ct

kμφ

v kμ---– xd

dp=

k μ⁄v

xddp


5.3 Data Input 417

where

Provided the distribution and relative saturations of the fluid phases remain fairlyconstant throughout the reservoir by entering appropriate “equivalent” values, themobility should be reasonably accurate for a multi-phase system.

Equivalent Reservoir PorosityThe reservoir porosity is the fraction of a rock’s bulk volume that is filled withmobile hydrocarbons. For a multi-phase system, the equivalent reservoir porositymust be used to satisfy the diffusivity relationship given in the explanation of TotalReservoir Compressibility above. Normally, this value can be determined with rea-sonable accuracy from well logs and/or core measurements.

Equivalent Reservoir ViscosityThe Equivalent Reservoir Viscosity is only required when the Internal PVT Tableis disabled and the Fluid Type is oil or water. The equivalent reservoir viscosityshould represent the total effective viscosity of a multi-phase fluid system. Asstated above, the equivalent permeability and reservoir viscosity must match thesystem mobility for a multi-phase system.

Gas Specific GravityThe gas specific gravity is only required when the Internal PVT Table is used.When this occurs, correlations will be applied to characterize the fluid behavior.

= total system mobility

= equivalent reservoir permeability= relative permeability to gas= relative permeability to oil= relative permeability to water= equivalent reservoir viscosity= gas viscosity= oil viscosity= water viscosity

kμ---

tk

kroμo-------

krgμg-------

krwμw-------+ +⎝ ⎠

⎛ ⎞=

kμ---

tkkrg

kro

krw

μμg

μo

μw



For gas production, enter the specific gravity of the dry gas. When oil is selected,enter the specific gravity of the solution gas (i.e., at stock-tank conditions).

Bubble Point PressureDuring a decrease in pressure, as is typically the case during production, the bubblepoint is the pressure at which gas begins to evolve from solution. This value is onlyrequired when oil or water is the produced fluid and the Internal PVT Table isselected.

Oil APIThe oil API gravity is required when the Internal PVT Table is used and the fluidtype is oil. The definition of API is as follows:

where is the specific gravity of the oil at 60°F and atmospheric pressure.

Reservoir TemperatureThe initial mean reservoir temperature is used to calculate the fluid properties. Thisvalue is only required when the Internal PVT Table option is selected or the fluidtype is gas.

Fracture CharacteristicsThe Fracture Characteristic Data dialog box (see Figure 5.7) provides a location forentering the propped fracture properties needed to perform a simulation when thefracture option is selected. As you become familiar with this data screen, you willnotice that many of the parameters will appear or disappear depending on theoptions selected.

°API 141.5γo

------------- 131.5–=

γo


5.3 Data Input 419

Figure 5.7: Fracture Characteristics - Data Dialog Box.

Each of the items found in the Fracture Data dialog screen are explained below. Ifnecessary, the conditions for data entry are indicated

CalculateA calculate radial option is available at the bottom of the dialog to calculate eitherthe fracture permeability, fracture width, or fracture conductivity. This gives theuser the flexibility input two of the variables and have the third calculated. The cal-culated value will then be dimmed. The Calculate option may not be availabledepending on the Fracture options selected (i.e., if Input Conductivity or Calculatefracture permeability is selected. The code will only display this feature if it is nec-essary.

Total Pay Zone HeightThe Total Pay Zone Height or net pay thickness is an input parameter from the For-mation Data dialog. It is dimmed here since it is only shown for reference

Effective Propped Pay Zone HeightAs stated, this is the effective propped height of the proppant in the pay zone ( )and is used only if the fractured well option is selected. This value maybe greater or

hp′



less than the Total Pay Height ( ). The Effective Propped Pay Zone Height, how-ever, must be less than or equal to the Total Propped Fracture Height that is usedfor the Fracture Design Optimization cases.

The effective dimensionless conductivity in the pay zone is calculated from

The reader is referred to Appendix L for additional information.

Total Propped Fracture HeightThe Total Propped Fracture Height is only required if the Fracture Design Optionis selected. The Total Propped Fracture Height along with the Effective ProppedPay Zone Height to calculate the proppant mass in the pay zone and the effectiveproppant number in the pay zone. Please refer to Appendix L for additional discus-sions.

Propped Fracture LengthWhen the NPV option is used and the User Specified NPV Fracture Data is dis-abled, the propped fracture lengths are read from the MFrac output file (“.mfrac”).If the NPV option is Off, it is necessary to specify the propped fracture length. Entera value that represents the fracture half-length which is propped or conductive. Thefracture is assumed to have two symmetrically propped wings of equal half-length.The Average Fracture Conductivity is used over the entire length.

Fracture PermeabilityThis is the average value for fracture conductivity (with damage).

Fracture WidthThis the average value for the propped fracture width.

Average Fracture ConductivityThe Average Fracture Conductivity is only input in this dialog box, if the NPVoption is Off.

hp

CfDkfwfkL

---------hp

′

hp-----•=


5.3 Data Input 421

The average fracture conductivity for slightly varying conductivities may be esti-mated using the following relationship for long term production

and for short term production or reduced conductivity near the wellbore the follow-ing relationship may be more applicable

where

For a variable fracture conductivity with position, the User should consider usingthe Variable Conductivity option for a fractured well. This option requires theUser to input fracture conductivity as a function of fracture position. The code willthen calculate the more rigorous Average Fracture Conductivity based on pseu-dosteady-state analyses (see Appendix L).

Because MProd uses an analytical approach to reservoir simulation, the AverageFracture Conductivity entered should be weighted towards the near wellbore con-ductivity when modeling early time production. Effectively, this will more accu-rately characterize a shorter, more conductive fracture. When simulating longerproduction times (pseudosteady-state), the entire propped fracture length should beused. If an analytical approach is unsatisfactory, a numerical simulator that allowsthe fracture conductivity as a function of position to be specified, must be used.Exodus, developed by T.T. & Associates, is such a simulator and is fully compati-ble with MFrac output.

Dimensionless ConductivityThe effective dimensionless conductivity in the pay zone is calculated from

= average fracture conductivity= proppant permeability in the fracture= propped fracture width= propped fracture length

kfwf kf x( )wf x( ) xd0

L

∫ L⁄=

kfwf L 1kf x( )wf x( )------------------------- xd

0

L

∫⎝ ⎠⎛ ⎞⁄=

kfwf

kf x( )

wf x( )

L



If the Average Fracture Conductivity is input, the effective Dimensionless Conduc-tivity in the pay zone will be calculated, and if the Dimensionless Conductivity isinput, the Average Fracture Conductivity will be calculated.

Beta FactorThe field will not be displayed if the Darcy-Only option is selected in the FractureOptions tab. The non-Darcy beta factor is required input if input Beta has beenselected. If the option to calculate beta from a database correlation is selected, avalue will be present but dimmed since it will be a calculated value from the prop-pant porosity and permeability.

Variable Fracture Conductivity DataIf the Variable Conductivity option is selected for a fractured well a dialogprompting the user to input Fracture Conductivity or Dimensionless Conductiv-ity as a function of Fracture Position will be available. The numerical simulatorwill then calculate the effective fracture and dimensionless conductivities.

Conductivity GradientAn option is provided at the bottom of the dialog to select a Conductivity Gradi-ent in the fracture. Selecting this option will model the conductivity as a linearlyvarying value from fracture position to fracture position within the fracture. If Con-ductivity Gradient is not selected the Fracture Conductivity (and DimensionlessConductivity) will be assumed to be piece-wise constant over a given fractureposition (see Appendix L for additional information and discussions).

Fracture PositionThe Fracture Position is the location of the proppant in the fracture measured fromthe wellbore (i.e., a fracture position of zero or a value equal to the well radius).The final entry in the table is the propped fracture length from which the Dimen-sionless Conductivity is calculated as discussed below.

CfDkfwfkL

---------hp

′

hp-----•=


5.3 Data Input 423

Fracture ConductivityThe variable Fracture Conductivity is entered into the table as a function of fractureposition. As discussed earlier in this chapter, if the Fracture Conductivity is inputthe equivalent spacial Dimensionless Conductivity will be calculated.

Dimensionless ConductivityThe Dimensionless Conductivity as a function of fracture position can be input andthe Fracture Conductivity as a function of fracture position will be calculated, TheDimensionless Conductivity as a function of position is defined as

where the final Fracture Position in the table is the Propped Fracture Length ( ).

History Match ParametersIf the History Match Production Simulation Data option is checked in the Optionsdialog, then the History Match Parameters and Measured Data screens becomeavailable.

The history matching process allows the user to history match on various parame-ters depending on whether the well is unfractured or fractured. For an unfracturedwell, the user has an option to history match on the following parameters: 1) reser-voir drainage area (closed system), 2) permeability, 3) reservoir aspect ratio, and/or4) wellbore skin. For the fractured well, the user has an option to history match onone or all of the following parameters: 1) reservoir drainage area (closed system),2) permeability, 3) reservoir aspect ratio, 4) fracture length, 5) beta factor and/or 6)dimensionless fracture conductivity depending on the Fracture Options.

History matching is a methodology of finding a set of input parameters (historymatch parameters) that will minimize the error between the measured data and thehistory match data (simulated numerical results).

To history match on any given parameter the user must specify the minimum/maxi-mum range and provide an estimate of a starting value. If the option Internal Esti-mate is selected the user will not have to enter an estimate for any of the historymatch parameters.

CfD x( )kfwf x( )

kL-----------------

hp′

hp-----•=

L



Any parameter (property) with the same minimum and maximum values will not bea history parameter. That is, the code will keep this parameter at a constant value.History matching by its very nature may result in a non-unique solutions. The codemay find parameter values that minimize the error in saddles but may not actuallyrepresent the true history match values. To ensure a pleasant experience, it may beadvantages to history match on only a few parameters at a time.

Following is a partial list of history matching scenarios for which the differentialequations will provide a non-unique solution:

1. Short time well test. The simulator will not be able to history match (converge)on the drainage area or reservoir aspect ratio unless the drainage area isextremely small.

2. High dimensionless fracture conductivities. For fracture conductivities greaterthan 100 or 1000, there is essential no difference in the solution. Consequently,if the dimensionless conductivity range is fro 0.1 to 10,000 and an estimate isat 1000, the code may converge on an infinite conductivity fracture (i.e., thenumerical solution for is no different than if ).

3. Fracture length. If the fracture length is chosen to be a value greater than themaximum reservoir drainage dimensions.

4. Permeability. Calculations may not be unique for short term data. Once pseu-dosteady state occurs the probability of solution uniqueness increases.

5. Beta. Calculations may not be unique for constant production data, since thereare an infinite combination of fracture conductivity and beta that will give thesame effective dimensionless conductivity.

6. Fluid properties. The uncertainty in fluid properties affects the reservoir mobil-ity and history matching uniqueness.

7. No fracture case. The effective estimated wellbore skin may not be very accu-rate if the flow has not reached pseudosteady state. A negative skin valueshould also be increases if the effective wellbore radius is greater than thedrainage area.

Although some of the above items are error checked in the code, the user must beresponsible for reasonable input values to obtain a unique solution. History match-ing is an engineering methodology not a trial and error exercise.

CfD 1000= CfD 10000=


5.3 Data Input 425

No Fracture Case - PropertiesFollowing is a list of the history match parameters for an unfractured well:

1. Reservoir Drainage Area (Closed system only).

2. Dimensionless Reservoir Aspect Ratio (Closed system only).

3. Equivalent Reservoir Permeability.

4. Skin Factor.

Fracture Case - PropertiesFollowing is a list of the history match parameters for a fractured well:

1. Reservoir Drainage Area (Closed system only).

2. Dimensionless Reservoir Aspect Ratio (Closed system only).

3. Fracture Length.

4. Dimensionless Fracture Conductivity.

5. Beta factor.

6. Equivalent Reservoir Permeability.

NPV/Multi-Case Fracture CharacteristicsTo perform treatment optimization studies using a Net Present Value approach,MProd provides three options for gathering MFrac data. This can be found in thesection containing the description of the NPV/Multi-Case Fracture Data Source.

When User Specified fracture data is selected for the NPV Fracture Data Source,the NPV Fracture Data category is added to the Data menu. This sub-menu pro-vides a table for defining up to 25 different fractures for simultaneous productionsimulation as shown in Figure 5.8.



Figure 5.8: NPV Fracture Characteristics Dialog.

The categories of data required to characterize each fracture in the NPV table are asfollows:

ImportUse the Import button provided in the dialog to read the proppant transport datafrom any MFrac file containing a proppant transport solution. Once the data isimported it can be modified.

As a means of simplifying the manner in which the Proppant Transport Characteris-tic table is constructed, a facility for importing MFrac output is provided. This util-ity allows preliminary viewing of MFrac data and offers a method for editing andadding data prior to its use by the program. To take advantage of this capability,Click on the Import button to browse and select an MFrac file. Since data containedin the dialog box is over-written during an import operation, caution should be usedwhen using this facility. Normally, prior to importing, we recommend starting witha blank data dialog, or using the Save As function to avoid loosing the originaldata.

CalculateA calculate radial option is available at the bottom of the dialog to calculate eitherthe fracture permeability, fracture width, or fracture conductivity. This gives theuser the flexibility input two of the variables and have the third calculated. The cal-culated value will then be dimmed. The Calculate option may not be available


5.3 Data Input 427

depending on the Fracture options selected (i.e., if Input Conductivity or Calculatefracture permeability is selected. The code will only display this feature if it is nec-essary.

Propped LengthThe Propped Length is the propped fracture half length for which the productionwill be simulated.

Effective Propped Pay Zone HeightThis is the effective propped height of the proppant in the pay zone ( ) and is usedonly if the fractured well option is selected. This value maybe greater or less thanthe Total Pay Height ( ).

Fracture PermeabilityThis is the average proppant permeability (with damage) as a function of position inthe fracture.

Fracture WidthThis is the effective average propped fracture width in the fracture as a function ofposition.

Average Fracture ConductivityLike the Average Fracture Conductivity discussed earlier in this chapter, thisshould be consistent with the average integrated value over the fracture length.

Dimensionless ConductivityThe effective dimensionless conductivity in the pay zone is calculated from

If the Average Fracture Conductivity is input, the effective Dimensionless Conduc-tivity in the pay zone will be calculated, and if the Dimensionless Conductivity isinput, the Average Fracture Conductivity will be calculated.

hp′

hp

CfDkfwfkL

---------hp

′

hp-----•=



BetaThe field will not be displayed if the Darcy-Only option is selected in the FractureOptions tab. The non-Darcy beta factor is required input if input Beta has beenselected. If the option to calculate beta from a database correlation is selected, avalue will be present but dimmed since it will be a calculated value from the prop-pant porosity and permeability.

Maximum PowerThis is the maximum power required to create the specified fracture geometry.These values are entered and included as output to be used later by MNpv in thecalculation of Net Present Value.

Slurry VolumeThis is the total slurry volume required to create the specified fracture geometry.This quantity is only included to allow the determination of Net Present Value for adesign.

Liquid VolumeThis is the total Liquid Volume (frac fluid) injected necessary to create the speci-fied fracture geometry. This Liquid Volume is only included to allow the determi-nation of fracture Net Present Value based on the cost of the fracturing fluid.

Total Proppant MassThis is the amount of proppant used to create the specified fracture geometry.These values are also used for the determination of Net Present Value.

To Enter User Specified NPV Fracture Data:

1. Make sure that the NPV option is turned on by opening the Options dialog boxfrom the Main Menu and selecting the appropriate radio button. The UserSpecified option for the NPV Fracture Data Source must also be selected.

2. Next, open the NPV Fracture Data dialog from the Data menu. The dialogboxes shown in Figure 5.8 will appear. Use the Import button provided in thedialog to read the data from any MFrac “.FD*” file containing NPV data. Oncethe data is imported it can be modified.


5.3 Data Input 429

3. Complete as many rows as necessary by entering data in each of the six col-umns. The Insert and Delete buttons can be used to edit the table as it is con-structed.

4. When finished characterizing the fractures for evaluation, click OK or the NextPage button to close the dialog box.

As a means of simplifying the manner in which a new NPV Fracture Characteristictable is constructed, a facility for importing MFrac output is provided. This utilityallows preliminary viewing of MFrac data and offers a method for editing and add-ing data prior to its use by the program. To take advantage of this capability, Clickon the Import button to browse and select an MFrac file. Since data contained in thedialog box is over-written during an import operation, caution should be used whenusing this facility. Normally, prior to importing, we recommend starting with ablank data dialog, or using the Save As function to avoid loosing the original data.

Gas PVT DataIf Gas is specified as the Fluid Type and the Internal PVT Table is Off, a table ofup to 25 values of Viscosity and Z-factor as a function of pressure must be entered.This table is used to calculate the pseudo-pressures used in the simulation (see Fig-ure 5.9).

The Gas PVT table must be constructed with increasing values of pressure andcover the expected range of pressures for the simulation. The first entry should beat standard pressure (14.696 psi) with a Z-factor equal to 1.0. The last entry shouldbe equal to or greater than the initial reservoir pressure.



Figure 5.9: Gas PVT Dialog Box.

Proppant DataProppant Data is required when the Fracture Optimization option is selected. Thedata in this screen is used in the calculation of the optimum fracture characteristicsand proppant number. The reader is referred to Appendix L for additional informa-tion.

The required proppant input data are: 1) permeability, 2) porosity, 3) specific grav-ity and 4) the fracture permeability damage factor. These parameters are discussedbelow:

PermeabilityThis is the fracture proppant pack permeability which is a function of concentrationper unit area and Closure Pressure. The effective propped fracture permeability iscalculated based on a damage factor as discussed below. The user is referred to theMFrac proppant database for typical values.


5.3 Data Input 431

PorosityThis is defined as the void fraction between sand grains (i.e., liquid volume toslurry ratio of the settled bank). It is used to calculate the final propped fracturewidth. Typical values of porosity for proppants are shown in Table 5.3.


The proppant Specific Gravity is used in the calculation of the proppant settlingvelocity, as well as the pipe frictional, gravitational and perforation pressure losses.

Proppant Damage Factor

The reported final permeability of the proppant in the fracture is calculated from:



6-8 angular 0.36

10-20 angular 0.36

10-20 round 0.32

20-40 round 0.35

40-60 round 0.32


Proppant Type SpecificGravity

AbsoluteDensity

(lbm/ft3)

AbsoluteDensity

(kg/m3)


Sand 2.65 165.4 2650




kf kf0 1 DF–( )=



where

The final permeability is used to determine the fracture conductivity and dimen-sionless fracture conductivity.

Design Optimization DataThis data is only required if the Fracture Design Optimization option is selected.

The fracture optimization methodology is based on placing a given amount of prop-pant mass in a propped hydraulic fracture in such a manor as to optimize perfor-mance. That is for a given mass of injected proppant, the simulator will calculatethe optimum fracture characteristics and dimensionless conductivity to maximizeproduction (i.e., for a given volume of proppant do we need fracture length or widthto optimize productivity).

The design optimization is based on the proppant mass in the pay zone or for agiven reservoir the Proppant Number. This table allows the user to input variousvalues for the Total Proppant Mass. The proppant mass placed in the pay zone(Pay Zone Proppant) and the effective Proppant Number are also calculated basedon the other proppant and formation properties. Entering the Proppant Numberwill result in the calculation of the Total and Pay Zone Proppant Masses. The useris referred to Appendix L for additional details.

Total Proppant MassThis is the total mass of proppant pumped into the fracture.

Pay Zone Proppant MassThis is the mass of proppant placed in the pay zone.

Proppant NumberThe proppant number is defined as twice the ratio of the propped fracture volume inthe pay zone ( ) to reservoir volume ( ) multiplied by the fracture to

reservoir permeability ratio ( ).

= final (damaged) fracture permeability= proppant permeability (undamaged from database)= proppant damage factor

kf

kf0

DF

Vprop frac Vreservoir

kf k⁄


5.3 Data Input 433

The proppant number for a rectangular shaped reservoir is given by

where is the fracture penetration and is the reservoir aspect ratio. See Appen-dix L for additional information.

Production DataThe Production Data dialog box provides a table for entering the production rate orbottomhole flowing pressure schedules as a function of time. The appearance andrequired data for the dialog box presented will depend upon the selection made forthe Production Specified option. When Rate is selected for this option, a tablecontaining the rates, durations (time) and time step for the calculations is required.The time step determines the number of iterations calculated for each correspond-ing time value. The data shown in Figure 5.10 would result in pressure calculationsperformed using a rate of 300 bpd for 30 days with calculations made every day.This would be followed by a rate of 200 bpd for the next 70 days to 100 days withcalculations performed every 10 days. The last entry corresponds to 100 bpd for900 days calculated every 100 days up to a total of 1000 days.

Figure 5.10: Production Data Dialog Table - Variable Rate.

To Enter a Series of Constant Rates or Flowing Pressures:

1. Select Rate or Pressure for the Production Specified option by opening theOptions dialog box from the Main Menu and making a selection.

Nprop2Vprop fracVreservoir

-----------------------kfk----× CfDIx

2λ= =

Ix λ



2. Next, open the Production Data dialog from the Data menu. One of the dialogboxes shown in either Figure 5.10 or Figure 5.11 will be displayed dependingon the selection in Item 1 above.

3. Fill in at least one (1), and up to 25 rows of data making sure to complete eachrow that is used. Use the Insert and Delete buttons to edit the table as it is con-structed.

4. When finished entering tabular data, click OK to close the dialog box.

Figure 5.11: Production Data Dialog Table - Variable Flowing Pressure.

When the Production Specified option is set to pressure, the data required is Bot-tomhole Flowing Pressure, Time and Time Step. The example shown in Figure5.11 would start with calculations using a flowing pressure of 2500 psi every dayfor 30 days, followed by a flowing pressure of 1500 psi to 365 days with calcula-tions performed every 3.65 days and finally a calculation every 36.5 days at 500 psito a total of 3650 days.

The principle of superposition is used to integrate the data input into a series of con-stant rate or pressure changes. For hydraulically fractured wells, the model is appli-cable for all flow regimes including linear, bilinear, trilinear and pseudo-radial.Calculations are made both for real and dimensionless time and flow rate or pres-sure. In addition, cumulative production and folds of increase (both instantaneousand average) are calculated.


5.3 Data Input 435

Measured DataThe measured data is input in a table very similar to the Production Data tableabove. The difference is that measured data is input and not the boundary condition.An overview of the measured input data is given below.

The Measured dialog box provides a table for entering the measured productionrate or bottomhole flowing pressure data as a function of time. The appearance andrequired data for the dialog box presented will depend upon the selection made forthe Production Specified option. When Rate is selected for this option, a tablecontaining the time and time dependent bottomhole pressures is required.

To Enter a Series of measured Rates or Flowing Pressures:

1. Select Rate or Pressure for the Production Specified option by opening theOptions dialog box from the Main Menu and making a selection.

2. Next, open the Measured Data dialog from the Data menu. A dialog box willbe displayed requiring input for pressure or rate versus time depending on theselection in Item 1 above.

3. Fill in and up to 200 rows of data making sure to complete each row that isused. Use the Insert and Delete buttons to edit the table as it is constructed.

4. When finished entering tabular data, click OK to close the dialog box.

When the Production Specified option is set to rate, the data required is Time andBottomhole Flowing Pressure,. When the Production Specified option is set topressure, the data required is Time and Rate.

Well DataProduction forecasting typically requires a minimum of data describing certain fea-tures of the well in order to perform a simulation. For MProd, these featuresinclude the Wellbore Radius and the permeability damage due to skin. The wellradius defines the contact area between the well and the reservoir. This value is par-ticularly important for unfractured wells, or when performing NPV analyses. Theskin damage characterizes the additional pressure drop associated with the nearwell regime. In addition to Well Skin, MProd also considers damage along the frac-ture face, Fracture Skin. Both of these parameters may have a significant effect oncalculating the folds of increase.



To include the effects of wellbore storage on the calculation of dimensionless pres-sure and avoid errors during early time, the Dimensionless Wellbore StorageCoefficient may be entered. Likewise, additional early time effects may be incor-porated by entering a value for the Inverse Fracture Diffusivity. Both of theseparameters are important in the early-time solutions and are typically insignificantfor long-time production. Generally, entering values of zero for these two parame-ters will not affect the overall results.

The required and/or optional well information entered in the Well Data dialog boxfor an unfractured well is shown in Figure 5.12. The parameters contained in thedialog are listed and described below.

Figure 5.12: Well Data Dialog - Unfractured Oil Well.

Wellbore RadiusThe Wellbore Radius is the radius of the borehole. It is used as the base case forradial flow from the reservoir to the well and in the calculation of wellbore storageeffects. When performing NPV analyses the productivity increase or folds ofincrease is based on the production ratio of the fractured well to an unfractured wellwith a known borehole radius.

Formation Volume FactorThe Formation Volume Factor is the ratio of the volume of oil plus dissolved gasat reservoir pressure and temperature divided by the volume of the oil at stock tankconditions. The formation volume factor is not required when the produced fluid isgas.


5.3 Data Input 437

Wellbore Skin Factor (base)The dimensionless pressure drop at the wellbore face as a result of damage (posi-tive value) or stimulation (negative value) is commonly referred to as Well Skin.The base Wellbore Skin Factor is the skin before the well is stimulated or simply abase skin that a stimulated well can be compared to determine a productivityincrease index.

Wellbore Skin Factor (stimulated)The dimensionless pressure drop at the wellbore face as a result of damage (posi-tive value) or stimulation (negative value) is commonly referred to as Well Skin.The stimulated wellbore skin is used to calculate the well productivity. The basewellbore skin factor for an unfractured well is used for the base productivity forcomparison with the stimulated case (i.e., the dimensionless productivity index isthe ratio of the stimulated to the base case productivity).

Dimensionless Wellbore Storage FactorThe Dimensionless Wellbore Storage Factor can be included to improve theaccuracy of early time predictions of pressure and rate when storage is present. Thisparameter is defined as:

where

The well information required for a fractured well is shown in Figure 5.13. Theadditional parameters contained in this dialog for a fractured well are listed anddescribed below.

= dimensionless wellbore storage= wellbore storage coefficient= total reservoir compressibility= pay zone height= wellbore radius= equivalent reservoir porosity

CD5.16146C2πφcthrw

2------------------------=

CD

Cct

hrw

φ



Figure 5.13: Well Data Dialog - Fractured Gas Well.

Wellbore Skin Factor (base - prefrac)The prefrac Wellbore Skin Factor is the wellbore skin before the well is fractured.The prefrac skin factor is used in the calculation of the flow resistivity in the nearwellbore region. For high conductivity fractures, the flow entering the wellbore inthe near well region is normally negligible and positive prefrac skin values willhave a minor effect when the fracture length is much greater than the wellboreradius (or effective wellbore radius). If the prefrac skin is negative, the fractureresistivity model simulates a reduced resistance in the near wellbore region result-ing in an increased productivity. This is especially true for short fractures with largenegative prefrac skins. (Please note that the storage of this variable in the input fileis shared with the base wellbore skin factor for an unfractured well).

Wellbore Skin Factor (stimulated)The productivity from a well with a stimulated wellbore skin factor is also calcu-lated which is used for comparison with the fractured well productivity. For thefractured well with a prefrac skin, the productivity increase is the ratio of the frac-tured well productivity to that of a well with the stimulated Wellbore Skin Factor asgiven by this input value (Please note that the storage of this variable in the inputfile is shared with the simulated wellbore skin factor for an unfractured well).

Fracture Skin FactorDamage to the fracture face can occur as a result of the fracture fluid leaking offand fluid loss additives, as well as mechanical damage due to fracture propagationprocess. The Fracture Skin Factor can be used to model this damage by simulatingan additional pressure drop adjacent to the fracture face. As with any type of skin, a


5.3 Data Input 439

positive value corresponds to damage, while a negative value indicates stimulation.Stimulation effects may be inferred as a result of the chemical reactions that occur,for example, during acid fracturing. The Fracture Skin factor, , used by MProdcan be expressed as:

where

Inverse Fracture DiffusivityNormally, the ratio of the diffusivity of the reservoir to the diffusivity of the frac-ture approaches zero for low permeability reservoirs and reasonable fracture con-ductivities. For these conditions, inverse fracture diffusivity is insignificant. As theconductivity of a fracture approaches the conductivity of the reservoir, however,this may significantly influence the early time production behavior of a well. Toinclude the effects of fracture diffusivity, enter the Dimensionless Inverse Frac-ture Diffusivity, , in the space provided. The definition of this parameter is as fol-lows:

where

= equivalent reservoir permeability= damaged zone permeability due to leakoff= propped fracture half length= damaged zone adjacent to the fracture

= dimensionless inverse fracture diffusivity= equivalent reservoir permeability= equivalent fracture permeability= total reservoir compressibility= total fracture compressibility= equivalent reservoir porosity= equivalent fracture porosity

Sf

Sfπ2---

ysxf----⎝ ⎠

⎛ ⎞ kkl---- 1–⎝ ⎠

⎛ ⎞=

kkl

xf

ys

cf

cfkφct

kfφfctf---------------=

cf

kkf

ct

ctf

φφf



For more information on inverse fracture diffusivity refer to Lee andBrockenbrough4.

5.4 Run/Performing CalculationsOnce all of the required data relevant to the options selected have been entered, thecalculations can be performed. Up to this point, the program has checked the valid-ity of the data contained in every dialog screen opened during the active session.Since you are not forced to view every data screen sequentially prior to performingcalculations, it is possible that some input parameters may not have been checked.To avoid problems, when the calculation process is initiated, the program checks toensure that the minimum data requirements are met and that the data entered iswithin acceptable limits. This extra level of error checking is designed to preventcalculation errors due to missing or “bad” data. Please refer to Chapter 1 for moreinformation about the general error checking processes.

To Perform Calculations:

1. Click Run from the main menu.

2. The Run⏐Options dialog can be used to specify options while the simulator isrunning. See “Run Options” on page 66.

After clicking the Run menu item, a Simulation Data window will be displayedand the menu bar will change to reflect that the simulation is running. To stop thesimulation, choose Stop! from the menu bar. Once the calculations are finished, themenu bar will return to normal. Due to the nature of the MProd calculations, if thesimulator is stopped before the calculations are finished, none of the calculated datawill be saved.

5.5 Plots MProd provides a vast selection of plots that can be produced to illustrate the simu-lation results. These plots have all the characteristics of Meyer plots as described inChapter 1. This section describes the plotting facilities that are specific to MProd.

The MProd plots are grouped into different categories as described below. Plotsfrom any number of categories may be viewed at the same time. The plots containthe results of the last simulation saved. It is important to note, that changing aninput parameter does not affect a plot until the simulation has been run again.


5.5 Plots 441

Plot CategoriesThe plots in MProd are grouped into different categories, each of which are accessi-ble with the Plot menu.

• Production (Single Case): Non Multi-Case NPV plots.

• Fracture Characteristics: These are plots of general characteristics of thefracture plotted versus propped length.

• Cumulative Production: These plots include the cumulative production andthe productivity ratio versus time or propped length.

• Net Cumulative Production: These plots are for net cumulative production.

• Flow Rate: These plots are of various flow rates versus time or proppedlength.

• History Match: These plots are for history matching (e.g. flow rate versustime, and flowing BHP versus time).

• Design Optimization: Various design optimizations plots.

Viewing PlotsThe plots that are contained in MProd are divided into categories that can beaccessed by different commands in the Plot menu. The specific plots available willbe directly controlled by the options selected for the last saved simulation. Forexample, if NPV is disabled by “unselecting” it in the Options screen, these plotswill not be available in the Plot menu.

To View a Plot:

1. Select Plot from the main menu. The Plot menu appears listing the availableplot groups.

2. Choose from the list of groups. The associated group selection dialog appears(See Figure 5.14). Groups that are unavailable due to the options selected willappear dimmed. To obtain these plots, you must activate the option and thenre-run the simulation.

3. Select the desired plots by clicking the adjacent check boxes. Use the SelectAll button to view all the plots for the group. To disable a plot, click Off thecheck box or use the Clear All button.




Figure 5.14: Example Plot Selection Dialog Box - MProd.

5.6 Generating ReportsAfter the calculations have been successfully performed, various options are avail-able for viewing the results and creating reports. Working with reports is the samein all Meyer programs as described in Chapter 1.

Viewing a ReportTo see the results of a simulation select View Report from the Report menu.

Explanation of the Report OutputThe input data will have the same form as the input screens. The output data (simu-lation results) will be presented in a set of tables.

Production SolutionThese tables, one per case (fracture length), show the flow rate, production, pres-sure and productivity ratio solutions indexed by time.


5.7 Program Database 443

5.7 Program DatabaseTo simplify database input, MProd offers a Non-darcy database for the beta factorcorrelation. While this databases is offered as an integral part of the program,Meyer and the reference sources make no guarantee or expressed warranty as totheir use or accuracy. This Non-Darcy database is also used in MFrac.

Non-Darcy Database To enter the Non-Darcy Database select the Non-Darcy Database command fromthe Database menu. The first screen presented is the Non-Darcy List dialog shownin Figure 5.15. Just like in the Fluid database, a User Database can be built by copy-ing non-Darcy beta correlations from the program’s System Database. The SystemDatabase contains correlations for the beta factor as used in the petroleum industry.Once beta correlations have been copied from the System Database, they can berepositioned by using the Up and Down buttons. You can Edit a record, Delete arecord, Copy a record or Add a new record to the list by choosing the appropriatebutton. To exit the non-Darcy database dialog box, click on the Done button.

Figure 5.15: Non-Darcy Database Dialog Box.



Non-Darcy Database ParametersWhen the Edit button is selected, the non-Darcy proppant data screen appears asshown in Figure 5.16. The screen displays a non-Darcy database record. The Refer-ence Code is a unique, seven (7) character identifier, used to indicate which betacorrelation to use in the simulation. The Description of the Non-Darcy Equation isdisplayed with the Reference Code in the report for correlation selected.

Figure 5.16: Non-Darcy Database Edit Screen.


where , , and are the input constants. The coefficient has units consistentwith the permeability power constant, , and the units for permeability. The powercoefficients , and are dimensionless.

kf φ

β akf

bφc-----------=

a b c ab

b c


5.8 Tools 445

5.8 ToolsThe Tools menu provides the user with options and analytical tools for calculatingor determining scientific parameters. Currently, all applications have a ToolSpreadsheet option that allows the user to customize the spreadsheet. MFrac andMProd also have a Proppant Calculator for determining the proppant permeabilityand beta factor based on proppant properties.

Proppant CalculatorThe Proppant Calculator allows the user to calculate the theoretical proppant per-meability and non-Darcy beta factor. Please refer to the MFrac section labeled“Proppant Calculator” on page 231 for more information.

5.9 References1. Vazquez, M. and Beggs, H.D.: “Correlations for Fluid Physical Property Pre-

diction” JPT (June 1980) 32, 968-970.

2. Beggs, H.D. and Robinson, J.R.: “Estimating the Viscosity of Crude Oil Sys-tems” JPT (Sept. 1975) 1140-1144.

3. Lee, A.L. Gonzalez, M.H. and Eakin, B.E.: “The Viscosity of Natural Gases”Trans., AIME (1966) 997-1002.

4. Lee, S.T. and Brockenbrough, J.R: “A New Approximate Analytic Solution forFinite-Conductivity Vertical Fractures,” SPEFE (Feb. 1986) 75-88.


Chapter 6

MNpvEconomic Analysis

6.1 IntroductionThis chapter is a user’s guide for MNpv, a fracturing design optimization simulator,based on the concept of Net Present Value developed by Meyer & Associates, Inc.MNpv is designed for use with MProd to automatically determine and compare theNPV of various fracture scenarios in order to identify an optimal design. UsingMNpv, treatment advantages or disadvantages can be ascertained by evaluating pre-dicted cash flow and future return on investment.

The complete process of performing an analysis is covered in this chapter. Adescription of the available options and the basic procedures required for runningMNpv are given. Detailed information about the MNpv methodology and basic the-ory is covered in Appendix H. In addition, examples are provided with the softwareto demonstrate the utility of certain features and manipulation of the data.

Hydraulic fracturing optimization is a basic requirement to maximize economicreturns on investment. Although, there are many factors which influence the resultsof a design, the principle objective of any stimulation application is to maximizewell profitability. Typically, this involves careful economic analysis of the costsand potential benefits of individual well operations. For many, forecasting NetPresent Value (NPV) has become an integral part of the preferred methodology tooptimize hydraulic fracture treatments.

The integrated capability of MFrac, MProd, and MNpv provides the ability to opti-mize the potential propped fracture length of a design based on the pumping sched-ule, fracture geometry, treatment cost and production revenue. The NPV of variousfracture lengths can be calculated and displayed as a function of producing time.Fracture characteristics, treatment cost, revenue and cumulative production plots asa function of fracture length and producing time can also be created.


448 MNpv: Economic Analysis

To perform an integrated NPV analysis, the simulators are run in sequence. First,the pumping schedules and associated fracture geometries are automatically deter-mined using MFrac’s NPV option. This step may be omitted if the data is entereddirectly in the MProd program. Next, MProd is used to generate production fore-casts for each of the different fracture geometries. Finally, the output generated byMProd is used in MNpv to perform an economic analysis.

An outline of the basic steps for using MNpv is shown in Table 6.1

6.2 OptionsThis section outlines the available options and describes the parameters that definethe conditions for MNpv. The first step before performing calculations is to estab-lish the options. This is accomplished by accessing the Options dialog box from themain menu.

The option selections determine the scope of the MNpv simulation. They establishthe kind of input to be entered and specific nature of the calculations to be per-formed. The selected options are stored in the current file.

The Options dialog box is typically the first input screen used in the MNpv pro-gram. Its function is to establish the primary model options that will be employed.There are four sections to this dialog. Each section deals with a different aspect ofthe modeling approach.

Table 6.1: MNpv Basic Steps

Step Program Area

Open an existing MNpv data file (*.mnpv) or cre-ate a new data file File Menu

Specify Units and currency symbol (optional) Units Menu

Select Program Options and an MProd output file Data Menu

Input Data:Input Economic DataInput Unit Revenue (if required)Input Share% Table (if required)

Data Menu

Run Simulation Run Menu

View Plots Plot Menu

Generate Report Report Menu


6.2 Options 449

To access the Options screen select Options from the main menu by clicking themenu name. The dialog box in Figure 6.1 will then appear.

Figure 6.1: Program Options.

To select an option, click the radio button adjacent to the option preference. A blackdiamond will appear in the center of the button when a choice is made. To move tothe next option, click on its radio button or use the TAB button to move sequentiallythrough the choices. Within a section the current selection for that option is high-lighted with a dotted rectangle. The option choice may be changed by using eitherthe mouse or the arrow keys.

An explanation of the choices available for each of the Options are summarized inthe following section.

Fluid TypeMNpv uses the production predictions from MProd to forecast the potential futurerevenue for each fracture geometry evaluated. The Fluid Type, as read in from theMProd output file, is used internally in MNpv to specify the primary fluid producedfrom the reservoir in order to direct the calculations through the appropriate algo-rithms for the purpose of calculating incremental revenue. This informational head-ing insures that the appropriate units are used and that the subsequent input screensare oriented towards the correct fluid (i.e., oil or gas).



Either Oil or Gas will appear in this dialog depending on information from theMProd output data file. The Fluid Type cannot be changed by the user.

Revenue/Unit VolumeThis option specifies the type of production revenue schedule to use. There are twochoices available.

When Fixed is selected, the Unit Revenue per volume of oil or gas produced isspecified as a constant in the Economic Data section of the Data menu. For thisselection, a Unit Revenue Escalation Rate may also be specified.

The other choice for this option is Variable. This selection enables the Unit Reve-nue Table found under the Data Menu to be used. The program will use the revenueversus time data entered in this table for all calculations.

Unit CostsThis option is used to modify the fluid and proppant unit cost for each case. If it isFixed then it will use the same fluid and proppant unit cost for each case. If it isVariable, you can enter a separate fluid and proppant unit cost for each table entryof conductivity and length.

When Fixed is selected, the Frac Fluid Unit Cost and Proppant Unit Cost arespecified as constants in the Economic Data section of the Data menu.

The other choice for this option is Variable. This selection enables the Unit CostTable found under the Data Menu to be used. The program will use the cost versusfracture length and conductivity (i.e., row specific) data entered in a table for allcalculations.

Partner Share OptionWhen ownership of a well involves more than one partner, this option may be usedto specify the percent share of revenue a partner receives for the Net Present Valuecalculations.

If Variable Percentage vs. Rate is chosen, the Share% Table is enabled. Thistable allows the entry of share percentage as a function of net flow rate. It can befound under the Data menu.


6.2 Options 451

Selecting Fixed Percentage requires entering the Share of Revenue percentage inthe Economic Data section of the Data menu. This value is used as a constant inthe economic calculations.

MProd Output File with NPV/Multi-Case DataTo perform NPV calculations, the output from MProd’s simulation must be linkedto MNpv’s file. The path to the data file is specified by using the Select buttonfound in the Options dialog. Using this button displays the dialog box shown inFigure 6.2 from which the desired data file may be selected. This is accomplishedby browsing directories to locate and select the file. Once the file is highlighted,click the OK button to complete the selection process. During this procedure, thedefault filename extension for MProd output is used. The file extension is POD(Production Output Data).

Figure 6.2: MProd Output File Selection.

Attempting to select an MProd file that does not contain the necessary NPV datawill result in a message like the one shown in Figure 6.3. This indicates that thedata file was not created with the NPV option on within MProd. Either selectanother data file or return to MProd and re-run the simulation with the NPV/Multi-Case option checked as On.



Figure 6.3: MProd Output File Selection Error.

6.3 Data InputAfter the options are selected and the scope of a simulation is set, data can beentered by accessing the categories found under the Data menu. As previouslydescribed, the options selected determine what information is needed for a particu-lar type of analysis. The specific data requirements of a screen or the existence ofthe data screen itself will vary depending upon the options chosen. This approach,minimizes data input and prevents unnecessary or misleading data entry. Simplydecide what options are relevant to the simulation and MNpv will display only thosemenus and input fields necessary. Any time an option is changed, the screens willvary to enable new input or hide data that is not needed. This methodology is usedthroughout MNpv.

The following sections pertain to the Data menu items found by selecting Datafrom the main menu. Each Data menu item is explained and a description of theassociated variables given. When pertinent, the conditions or case sensitive optionsfor a data screen are noted and an example of the resulting dialog is shown. All ofthe data screens available within MNpv and the variables contained within them arepresented.

DescriptionThe Data Description screen shown in Figure 6.4 provides a location for enteringdescriptive information about a simulation. Space is provided for entering the Com-pany Name, Well Name, Well Location and Simulation Date. In addition, a Com-ments section is included so that additional information can be entered. Thisinformation can include items such as the specific fluids, program options or anexplanation of the data used. All of the information contained in this dialog isoptional.


6.4 Economic Data 453

Figure 6.4: Data Description.

6.4 Economic DataWhen the Revenue/Unit Volume and Partner Share Option are both fixed, theEconomic Data dialog box containing all of the necessary information to performthe calculations appears as shown in Figure 6.5. Variations of this dialog may bepresented along with additional data screens when the Variable options are modi-fied. The units displayed, along with the variables contained in this dialog, are con-sistent with the Fluid Type selected and the preferences made from the Unit menu.For example, if the Fluid Type changes from oil to gas, the unit revenue parameterwill change to currency per unit volume gas (e.g., $/Mscf).



Figure 6.5: Economic Data Dialog Box.

Each of the items found in the Economic Data dialog screen are explained in thenext section. When necessary, the conditions for which the data is required are alsoindicated.

Frac Fluid Unit CostThe Frac Fluid Unit Cost is the cost per unit volume of fracturing fluid for a spe-cific treatment design or group of designs. Normally, this includes the total cost ofthe base fluid and gelling agents, as well as, miscellaneous additives such as, claystabilizers, fluid loss additives, nitrogen, CO2, surfactants, inhibitors, etc. Specialfees associated with license arrangements and material handling may be factored into this unit cost or they may be added in the Miscellaneous Cost category found inthis dialog. The value required for the fluid unit cost can usually be obtained byreferring to the pumping service companies price book or their specific job ticket.The number entered is used to determine the fluid cost as a function of fracturelength for the NPV calculations. The typical relationship between fracture penetra-tion (i.e., length) and the fluid cost is illustrated in Figure 6.6.



Figure 6.6: Treatment Cost vs. Propped Fracture Length.

Proppant Unit CostLike the fluid unit cost, the Proppant Unit Cost is used to determine the proppantcost as a function of fracture length. This is the cost per unit mass of proppant usedfor a particular treatment design. While the proppant type may have a significanteffect on the resulting fracture conductivity and productivity, the cost of the prop-pant may also dramatically influence the economic viability of a particular treat-ment design. A typical proppant cost versus propped length relationship is depictedin Figure 6.6.

Hydraulic Power Unit CostThe hydraulic power requirements for each specific pumping schedule, as deter-mined by MFrac or entered in MProd, are passed to MNpv for calculating the asso-ciated costs to be included in the economic analysis. The Hydraulic Power UnitCost is the cost per unit power required for each design. This value is used to deter-mine the hydraulic cost. This data can be obtained from the pumping service com-pany involved. Concerning standby power, mobilization costs, or overtime charges,these expenses may be factored into the unit cost or included in the Fixed Equip-ment Cost category explained below. Don’t forget that service companies usuallycharge based on the power ordered and not necessarily on what is actually used.You may want to add additional costs in the Fixed Equipment Cost or Miscella-neous Cost if extra pumping capability is ordered.

Treatment Cost vs Propped Length

0 200 400 600 800 1000 1200 1400

Propped Length (ft)0

40000

80000

120000

160000

200000

240000

Trea

tmen

t Cos

t ($)

Job costFluid costProppant cost



Fixed Equipment CostThis category may be used to represent the additional cost of equipment that is nota function of hydraulic power. The value entered is added to the variable powercost to determine the total equipment cost. Items that fall in this category are listedbelow:

Truck Mileage and Transportation - These are charges typically incurred formobilizing equipment and delivering materials.

Fixed Power Costs - This includes standby power and non-pumping service time.

Blending Charges - Other than mobilization and minimum charges, these costs aresometimes based on the average injection rate.

Pressure Multipliers (Intensifiers) - The cost of pressure intensification is nor-mally based on the hydraulic power ordered. There are usually fixed costs alsoassociated with this equipment.

Proppant or Slurry Handling Equipment - These are charges associated withany equipment related to proppant handling not included in typical blending equip-ment charges. This includes proppant transports, conveying equipment and specialproppant concentration devices.

Liquid Additive Equipment - These costs are related to liquid delivery systemsnot covered by blending charges. Examples include LFC units, as well as otherancillary upstream pumping equipment used for delivering fluids to the blender.

Nitrogen or CO2 Equipment - Special charges usually apply for the transport,preparation and pumping of energized fluids. The costs associated with the equip-ment should be included in this category.

Auxiliary Equipment - The cost of miscellaneous equipment such as, tree savers,manifolds, flowmeters, data acquisition, transport units, treating connections, andfrac support units should also be considered in the fixed cost of equipment.

Miscellaneous CostMiscellaneous costs should also be entered to compute the total cost associatedwith a treatment. This category should be used to represent potential expendituresnot related to hydraulic power, equipment or materials already included in the cate-gories described above. This includes site preparation, workover costs, tubular rent-als, specialized wellbore equipment (e.g., bridge plugs, etc....) and perforating.



Tank rentals, water hauling, license fees, logging or testing services, and consultingfees can also be included. Basically, this is a general category where any fixed costrelated to the analysis can be included.

Currency Escalation RateIn MNpv, future economic calculations are always made relative to the presentvalue of currency. To make this transformation, the time value of money must bedefined. The Currency Escalation Rate is the parameter which describes this rela-tionship. It can be thought of as the effective interest rate used for calculatingpresent worth from the future value of produced oil or gas. Another way of thinkingabout this parameter is to equate it to the change in the average investment opportu-nity rate. For example, the average interest rate that could have been achieved witha similar amount of money during the same time period if it had been invested else-where. This value is sometimes referred to as the discount rate. The present worthof a future value is defined as:

where

Unit Revenue for Produced Oil or GasWhen the Revenue/Unit Volume option is Fixed, the current unit revenue for theproduced fluid must be entered. The unit revenue escalation rate (discussed below)can then be specified to forecast the future revenue. The title and unit for this vari-able automatically change depending on the Fluid Type selected in the Optionsmenu.

If Variable is selected for the Revenue/Unit Volume option, this parameter doesnot appear in the Economic Data dialog. It is replaced by the Variable Revenue perUnit Volume Table found in the Data menu.

= present worth= future worth= currency escalation rate or interest rate= number of periods

P F1 i+( )n

------------------=

PFin



Unit Revenue Escalation RateThe Unit Revenue Escalation Rate is used to specify the percent change in theunit revenue for produced oil or gas as a function of time. If this value is positive,the future revenue of gas or oil will increase. A negative value indicates that as timeincreases, the unit revenue price will decrease.

Share of CostWhen partnerships are involved, this is your percentage of the total cost of the treat-ment for the NPV analysis. If you have no partners, your percentage or share of thecost is 100%.

Share of RevenueThis data is only required when the Partner Share Option has been Fixed from theOptions dialog box. It represents your percent share of the total revenue. If youhave no partners your share of the revenue is 100%.

If Variable is selected for the Partner Share Option, this parameter is disabled and itis replaced by the Share% Table located in the Data menu. This allows the entry ofyour share percentage as a function of net flow rate.

6.5 Variable Unit Revenue TableGiven the history of oil and gas prices and their potential future variability, fore-casting production revenue may be speculative. Nevertheless, to evaluate differenteconomic scenarios, as they influence the engineering decision, it is necessary tohave some flexibility in how future hydrocarbon price estimates are applied. MNpvprovides the ability to use a fixed revenue and an associated escalation rate. MNpvalso allows a table to be defined for increasing or decreasing revenue values.

This table is only available if Variable is selected in the Revenue/Unit Volumeoption found in the Options dialog box. For this condition, the table is used to enterthe unit revenue of oil or gas as a function of time. The total number of entries mustbe between two (2) and twenty-five (25). The table must be constructed withincreasing values of time to cover the total period of the simulation. The first entryshould be at zero producing time with a corresponding initial unit revenue. The last

When the Currency Escalation Rate for NPV calculations and the Unit RevenueEscalation Rate are the same, the unit revenue is a constant in terms of presentvalue.


6.5 Variable Unit Revenue Table 459

time entry should be equal to or greater than the maximum time of the productionsimulation.

To Enter the Unit Revenue as a Function of Time:

1. Select Variable for the Revenue/Unit Volume option by opening the Optionsdialog box from the main menu and making a selection.

2. Next, open the Variable Revenue/Unit volume Table screen from the Datamenu. The dialog boxes shown below will be displayed.

3. Fill in at least 2 and up to 25 rows of data making sure to complete each rowthat is used. Use the Insert and Delete buttons to edit the table as it is con-structed.

4. Click OK or Next Page to close the dialog box and save the changes you made.

The unit revenue of produced oil or gas is assumed to change linearly between con-secutive times. Consequently, midway between two times (Time 1 and Time 2) theunit revenue is equal to the average between the two entered values.

Assuming your net unit revenue for oil is $10.00/bbl today and will increase by$1.00 per year, the table in Figure 6.7 would be input.

Figure 6.7: Variable Revenue Table - Step in Unit Revenue.

If your net unit revenue for oil is $10/bbl today and increases linearly to $50/bbl intwenty years, you would enter the data shown in Figure 6.8.



Figure 6.8: Variable Revenue Table - Linearly Increasing Unit Revenue.

Just as the unit revenue field in the Economic Data dialog is updated correspondingto the Fluid Type selected, the unit and title for the variable revenue table alsochange.

6.6 Variable Share% TablePartnerships complicate any economic evaluation, especially when share percent-ages do not remain constant. Share of cost is usually more straight forward and inMNpv your percent share of expenditures is entered in the Economic Data dialog(see Share of Cost). If the percent share of revenue is a constant, simply applyyour percentage to the fixed Revenue/Unit Volume. When share of revenue isbased on periodic production rates, the Share% Table can be used to specify thechange in percent share of revenue of oil or gas as a function of the net flow rate.To access this table the Partner Share Option must be chosen as Variable Percent-age vs. Rate in the Options dialog screen. The net flow rate is defined as the flowrate from the fractured well minus the unfractured base case (i.e., radial produc-tion).

When using the Share% Table, the total number of entries must be between 2 and25 and the table must be constructed with increasing values of net flow rate to coverthe expected range of production. The first entry should be for a net rate of zero.The last time entry should be equal to or greater than the maximum expected rate. Ifthe actual net rate is lower than the first data entry value, the share percentage for


6.6 Variable Share% Table 461

the first value will be used. If the calculated net rate is greater than the last value inthe table, the last entered share percentage will be used.

As in the Revenue/Unit Volume Table (see Figure 6.7), the percent share of reve-nue of produced oil or gas is interpreted linearly between consecutive net rates.Consequently, midway between two rates the share percentage is equal to the aver-age between the two values.

Assuming the share percentage of revenue for oil will increase linearly from 0% ata rate of 0 bbls/day to 90% for 200 bbls/day of net flow rate increase, the data inFigure 6.9 would be entered.

Figure 6.9: Share% versus Net Flow Rate Table.

This corresponds to a Share versus Flow Rate relationship as shown in Figure 6.10.



Figure 6.10: Share% vs. Net Rate Relationship.

The unit for the net flow rate column will automatically correspond to the FluidType that is specified in the Options dialog.

6.7 Variable Unit Cost TableWhen the Variable unit cost option is selected you can enter a separate fluid andproppant unit cost for each table entry of length/conductivity and length.

When Fixed is selected, the Frac Fluid Unit Cost and Proppant Unit Cost arespecified as constants in the Economic Data section of the Data menu.

The Variable selection enables the Unit Cost Table found under the Data Menu tobe used. The program will use the cost versus fracture length and conductivity (i.e.,row specific) data entered in this table for all calculations as shown in Figure 6.11.



Figure 6.11: Variable Fluid and Proppant Unit Cost Table.

This table provides the user with the capability to optimize the NPV based on dif-ferent unit costs for the fluid and proppant types. The Frac Fluid Unit Cost andProppant Unit Cost can vary with either job sizes or for different proppants withvarious conductivities at a given fracture length.

6.8 Run/Performing CalculationsOnce all of the required data relevant to the options selected have been entered, thecalculations can be performed. Up to this point, MNpv has checked the validity ofthe data contained in every dialog screen opened during the active session. Sinceyou are not forced to view every data screen sequentially prior to performing calcu-lations, it is possible that some input parameters may not have been checked. Toavoid problems, once the calculation process is initiated, the program checks toensure that the minimum data requirements are met and, once again, that the dataentered is within acceptable limits. This extra level of error checking is designed toprevent calculation errors due to missing or “bad” data.

To start the calculations, select the Run command from the Run menu. If there areany error checking messages, correct the errors and select Run again. TheRun⏐Options dialog is available for specifying options for when the simulator isrunning. See “Run Options” on page 66.

After clicking the Run menu item, a Simulation Data window will be displayedand the menu bar will change to reflect that the simulation is running. To stop thesimulation, choose Stop! from the menu bar. Once the calculations are finished, themenu bar will return to normal. Due to the nature of the MNpv calculations, if the



simulator is stopped before the calculations are finished, none of the calculated datawill be saved.

6.9 Plots - Graphical PresentationMNpv provides a vast selection of plots that can be produced to illustrate the simu-lation results. These plots have all the characteristics of Meyer plots as described inChapter 1. This section describes the plotting facilities that are specific to MNpv.

The MNpv plots are grouped into different categories as described below. Onlyplots from the same categories may be viewed at the same time. The plots containthe results of the last simulation. It is important to note that changing an inputparameter does not affect a plot until the simulation has been run again.

Plot CategoriesThe plots in MNpv are grouped into different categories, each of which are accessi-ble with the Plot menu. The Fracture Characteristics, Cumulative Production, NetCumulative Production, and Yearly Average Flow Rate categories have plots thatuse MProd output data. Each category is summarized below:

Fracture Characteristics: These are plots of general characteristics of the fractureplotted versus propped length.

Cumulative Production: These plots include the cumulative production and theproductivity ratio versus time or propped length.

Net Cumulative Production: These plots are for net cumulative production.

Yearly Average Flow Rate: These plots are of various flow rates versus time orpropped length.

Treatment Cost & NPV: These plots include the treatment cost versus proppedlength plot and NPV; and DROI plots versus time or propped length.

Partner Share Treatment Cost & NPV: These plots are similar to those above.

Viewing PlotsThe plots in MNpv are divided into categories that can be accessed by differentcommands in the Plot menu. The specific plots that are available will be directlycontrolled by the options selected for the last run simulation.



To View a Plot:

1. Select Plot from the main menu. The Plot menu appears listing the availableplot groups.

2. Choose from the list of groups. The associated group selection dialog appears(see Figure 6.12). Groups that are unavailable due to the options selected willappear dimmed. To obtain these plots you must activate the option and then re-run the simulation.

3. Select the desired plots by clicking the adjacent check boxes. Chose theSelect All button to view all the plots for the group. To disable a plot, click offthe check box or use the Clear All button.


Figure 6.12: Plot Selection Dialog Box - NPV/Multi-Case.

6.10 Generating ReportsAfter the calculations have been successfully performed, various options are avail-able for viewing the results and creating reports. Working with reports is the samein all Meyer programs and is described in Chapter 1.

Viewing a ReportTo see the results of a simulation select View Report from the Report menu.



Explanation of the Report OutputThe input data will have the same form as the input screens. The output data (simu-lation results) will be presented in a variety of tables. Each of the output tables issummarized below.

Treatment CostThis table gives the proppant cost, the fluid cost, the total job cost and the share ofthe job cost indexed by the propped length of the fracture.

Net Present Value SolutionThese tables, one per time step, show the production, revenue and net present valuesolutions indexed by the propped fracture length.

6.11 UnitsThe Units dialog box (Figure 6.13) works the same as the other Meyer Unit dialogboxes as described in Chapter 1. However, there is an option to specify the currencyname and currency symbol.


6.11 Units 467

Figure 6.13: MNpv Units Dialog Box.


Chapter 7

MFastAnalytical 2D Fracture Simulator

7.1 IntroductionMFast is an analytical two-dimensional hydraulic fracturing simulator for design-ing 2D fractures. The simulator illustrates the importance of various parameters andprovides a fast first order solution to fracture geometry, net pressure, fracture effi-ciency and treatment design. This simulator provides the capability to compare thefracture geometries for Geerstma-deKlerk (GDK), Perkins-Kern (PKN) and ellip-soidal type two dimensional models. Since MFast was developed from analyticalsolutions it has the inherent limitations of steady state injection, constant mechani-cal properties, time independent fluid rheology and single layer properties.

The governing equations of mass, momentum and energy conservation used in thedevelopment of MFast are presented in full detail by Meyer1-4. Numerous coeffi-cients are implemented from analytical relationships based on the limiting solutionsof numerical results obtained from MFrac.

MFast is a great tool for beginners to understand the parametric effects of the vari-ous input data and their importance in fracturing. MFast is also a useful tool for themore experienced user in providing a fast first order analysis and in performingparametric studies based on field data. Utilizing this program to perform simple netpressure history matching will provide a first order estimate of the fracture geome-try, correct geometry model to use, fracture efficiency and proppant mass for aspecified inlet concentration.

An outline of the basic steps for using MFast is shown in Table 7.1.


470 MFast: Analytical 2D Fracture Simulator

MenuThe MFast menu bar is shown in Figure 7.1. Generally, the menus are accessedfrom left to right as shown in Figure 7.1.

Figure 7.1: MFast Main Menu.

7.2 DataThis Data menu describes the various data options and input data required forMFast. The Data options are shown in Figure 7.2. A complete description of therequired input data is presented in this section.

Figure 7.2: MFast Data Menu.

Table 7.1: MFast Basic Steps.

Step Program Area

1. Open an existing data file or a new file File Menu


3. Input Dataa. Optionsb. Descriptionc. Formation/Fracture Data

Options MenuBase Data Menu



6. View Report Report Menu


7.2 Data 471

OptionsTo access the Options screen, select the Data⏐Options menu. The dialog box dis-played in Figure 7.3 will then be presented.

Figure 7.3: Options Screen.

The Options screen allows the user flexibility on which features to include for aparticular type of fracture design analysis. The specific data displayed in the basedata screen varies depending on the options selected. The selections made in theOptions screen set the scope for data used in MFast.

The Options provide choices for the fracture geometry model and constitutive rela-tionships that affect the fracture solution methodology (Figure 7.3).

InputThis option determines whether the simulations will be based on specifying a fluid(slurry) volume or entering the desired fracture length. If volume is selected, theslurry volume is input and the fracture length will be calculated. If the fracturelength radio button is selected, the volume to create a fracture of this dimensionwill be calculated.



Fracture Friction ModelNormally, laminar flow exists in the fracture and this option may not be needed(i.e., unchecked). For this case, the classical solution for fluid flow in a rectangularslot (as modified for an ellipsoidal fracture width) is used and the Darcy frictionfactor takes the form:


Deviations from laminar flow effect the frictional dissipation in the fracture andtherefore the fracture pressure predicted by a model. Turbulent flow in the fracturemay also occur when very low viscosity fluids (e.g., gas) at high rates are pumped.To account for these phenomena and improve the ability to predict non-laminarfrictional pressure loss in a fracture, the following friction factor expression is usedwhen the Fracture Friction Model is turned On:

Irregularities along the fracture face (e.g., tortuosity, bifuraction and wall rough-ness) that interrupt and disturb fluid flow can also result in greater energy dissipa-tion. These effects can be modeled by increasing the a coefficient or modifying thewall roughness factor as discussed below.

Typical values for the a and b coefficients have been developed empirically inaccordance with Prandtl's Universal Law of the Wall5 as shown in Table 7.2.







fD 24 Re⁄=


fDa

Reb---------=


7.2 Data 473

Wall RoughnessWhen Wall Roughness is turned off (not checked), the Darcy friction factor insidethe fracture is used without modification as determined from the selections made inthe Fracture Friction Model option. This selection assumes that the fracture sur-face is a smooth planar feature without roughness.

To include the effects of roughness (or waviness) on the frictional dissipation, turnthis option on. This will result in an increase in the frictional pressure drop andfracture width, as well as, a decrease in fracture length. If this option is used, thefriction factor defined in the Fracture Friction Model option will be modified usinga Friction Factor Multiplier. The relationship used is defined in the expressionshown below:

where



fD′ Mf fD=

fD′

fD

Mf




Tip EffectsThe observed field pressures for some treatments are at times much higher than thesimulated pressure. This discrepancy in measured pressure can be minimized in anumber of ways. Typically, the friction factor multiplier, fracture toughness, nearwellbore effects, confining stress or rock/reservoir properties are modified to obtaina match. However, if the pressure discrepancy is due to excess pressure, an over-pressure function can be applied at the tip. In MFast, excess pressure can be appliedusing two mechanisms: 1) Fracture Toughness, and 2) Tip Over-pressure.

Over-pressure, as it is incorporated in MFast, accounts for the extra pressurerequired at the fracture leading edge for propagation to occur. This extra resistanceat the fracture perimeter (tip) requires additional pressure (energy) to propagate thefracture. As a result, when this option is used, higher pressure must be applied atthe inlet (surface or BHTP) to compensate for losses that occur in the fracture.

Tip effects, in general, remain an area of some controversy and considerable dis-cussion. Plausible explanations for these effects have been proposed. The possibili-


7.2 Data 475

ties include tip friction due to flow resistance, rock properties effects (e.g.,toughness as a function of stress at the leading edge or poroelasticity), or it may bea consequence of fracture geometry (e.g., complex geometry and/or multiple frac-tures).

In MFast, tip effects represent a flow resistance at the tip. Regardless of whetheryou believe this flow resistance is due to viscosity effects or some other phenomenarelated to the tip region (e.g., tip geometry) the general effect on pressure is typi-cally the same (i.e., resistance is resistance). It is important to note, however, thatthis type of resistance differs from fracture toughness in its classical application;over-pressure varies with injection rate and time, fracture toughness does not.

The range of the over-pressure factor allowed by MFast is between 0 and 1.0. Ifthis option is disabled, a default value of zero is used. Usually, the Tip Effect optionis suggested when the measured injection pressures are well above the theoreticalvalues predicted by a classical model (i.e., Linear Elastic Fracture Mechanics).

When reasonable values have been implemented for wall roughness, friction factormultiplier, toughness and other formation properties, a value between 0.1 to 0.4may be justifiable. The larger the over-pressure factor the greater the increase inthe net pressure. If you are having difficulty relating the over-pressure factor topressure, one approach is to use MinFrac to automatically regress on the tip factorto determine an appropriate value. This best fit value from matching the net pres-sure in a minifrac analysis is a good place to start.

Many engineers mistake near wellbore pressure loss for excess net pressure. Keepin mind that when the injection rate changes suddenly, the near wellbore pressureloss also changes instantly whereas the fracture net pressure cannot because of stor-age (i.e., if the rate drops suddenly and the BHTP follows, this is not excess pres-sure but frictional dissipation in the near wellbore region).

The phenomena of tip over-pressure has been referred to as “dilatancy” bysome researchers. It is not clear whether these researchers are referring to rockdilatancy or fluid dilatancy. Fluid dilatancy refers to a shear-thickening fluid.Rock dilatancy describes volumetric expansion of a material that is rapidlyapproaching failure and is usually associated with the micro-cracking process.There has been no published explanation on the effects of rock dilatancy on netpressure in a crack, and to our knowledge, no correlations exist. The desiredeffect (i.e., an increase in pressure) can be achieved due to viscosity effects (i.e.,fluid dilatancy) or as a result of stress dependent rock properties that may ormay not be related to rock dilatancy. This is commonly referred to as nonlinearelastic deformation. Figure 7.5 illustrates one possibility.




Proppant TypeA proppant type must be selected from the drop down list box. The proppant type isused to determine the amount of proppant which can be placed in fracture withoutscreening-out the fracture. The physical properties of each proppant type includedin the list box are contained in an internal database.

DescriptionThe Data Description screen shown in Figure 7.6 provides a location for enteringdescriptive information about the specific analysis being performed.


7.2 Data 477

Figure 7.6: Data Description Dialog Box - MFast.

Base DataThe MFast Base Data dialog box shown in Figure 7.7 provides the informationnecessary to describe the rock properties, fluid rheology and fracture parameters.Each of these data items are discussed below.



Figure 7.7: Base Data Dialog Box.

Young's ModulusYoung`s modulus or the modulus of elasticity is the slope (or derivative) of a stress-strain curve over the elastic portion of the curve. For linear-elastic deformation,Young’s modulus is a constant with a unique value for a particular material and in-situ conditions. The modulus represents the materials ability to resist deformationunder load. It is therefore a measure of the materials stiffness. As the stiffness (E)of the rock increases, the fracture width will decrease and the length will increasefor a given set of input parameters. See Appendix A for more information regardingthe sensitivity of this parameter.



Rock Type Range (106 psi)

Range (107 kPa)

Limestone-Reef Breccia 1 - 5 0.5 - 3

Limestone-Porous or Oolitic 2 - 7 1 – 5

Limestone-Med. to Fine Grained 4 - 11 2.8 - 7.6


7.2 Data 479

Fracture ToughnessThe definition of fracture toughness is obtained from the concept of stress intensityfactor, developed in linear elastic fracture mechanics (LEFM). Fracture toughnessis a measure of a material’s resistance to fracture propagation. It is proportional tothe amount of energy that can be absorbed by the material before propagationoccurs. The basis for this relationship involves the assumption that pre-existingdefects exist and induce high stress concentrations in their vicinity. These sitesbecome points for crack initiation and propagation. See the MFrac chapter for moreinformation.







Dolomite 6 - 13 4.14 - 9

Hard, dense Sandstone 4 - 7 2.8 - 5.2

Medium Hard Sandstone 2 - 4 1.4 - 2.8

Porous, unconsolidated to poorly consolidated 0.1 - 2 0.35 - 1.4


ac

T

T KIC πac⁄=

KIC

σ

KI σ γHξ=

σc KIC γHξ⁄=

γ Hξ



Table 7.4 lists some measured values of fracture toughness. The values shown werereported by van Eekelen9. Thiercelin10 reviewed the testing procedures for deter-mining this parameter in his article, “Fracture Toughness and Hydraulic Fractur-ing.”

Setting the values of fracture toughness to zero will result in the classical hydraulicfracturing propagation solutions dominated by viscous pressure loss. For very lowviscosity fluids, fracture toughness may be the dominate parameter controllingfracture growth.

Poisson’s RatioPoisson’s ratio is defined as the ratio of the transverse strain to the axial strainresulting from an applied stress (see Figure 7.8).


Typical Poisson's ratios for rock formations are 0.25. From parametric studies,Poisson's ratio affects the fracture propagation characteristics to a very minorextent. Therefore, if in doubt, use 0.25.

Table 7.4: Fracture Toughness Values for Various Rocks.


Siltstone 950-1650 1040-1810

Sandstone 400-1600 440-1040

Limestone 400-950 440-1040

Shale 300-1200 330-1320


7.2 Data 481


Poisson's ratio is also used by logging companies to infer in-situ stresses. Thismethod assumes the rock behaves elastically and that the tectonic stresses areknown or insignificant. The typical relationship is

where

Total Pay Zone HeightThis is the total pay zone or net permeable leakoff height penetrated by the fracturefor leakoff. This may or may not be equal to the hydrocarbon pay thickness used toestimate production. The total leakoff height is also referred to as the net pay zonethickness.

= minimum horizontal stress= Poisson’s ratio= vertical stress or overburden= pore or reservoir pressure= component of stress due to tectonics= Biot’s constant

Poisson’s ratio

υ = −ε wε l

ε w0

=wΔw ε =l

Δll

0


Longitudinal strain

l0

w0

σHminυ

1 υ–------------⎝ ⎠

⎛ ⎞ σv αp0–( ) αp0 σT+ +=

σHmin

υσv

p0

σT

α



Total Fracture HeightThis is considered the total fracture height for the PKN and GDK fracture models.These 2-D models have fixed fracture heights by definition (see Chapter 2). Thisparameter is one of the most difficult to estimate and is one of the most importantinput parameters for the PKN and GDK models. MinFrac has an option to historymatch on fracture height for the PKN model. The total fracture height is not usedfor the ellipsoidal geometry model.

Ellipsoidal Aspect RatioThis is the ratio between the length of the major and minor ellipse axes. If this valueis equal to unity (1), the model reduces to the standard radial or penny shaped solu-tion. Any value greater than one will produce an elliptical profile and correspond-ing fracture area. For example, an Ellipsoidal Aspect Ratio of two (2) results in afracture half length that equals the total height of the fracture.

Injection RateThe injection rate is the total constant slurry injection rate for a two wing fracture.

Flow Behavior Index Rheological characterization of non-Newtonian fluid is required to calculate thefrictional dissipation in the fracture. Fracturing fluids are most often characterizedby the power law model. This model is typically defined as:

where is the wall shear rate, is the wall shear stress, is the consistency

index, and is the flow behavior index (dimensionless).

Consistency Index See the explanation of the flow behavior index above.

Total Leakoff CoefficientThe total leak-off coefficient, C, is made up of a combination of three flow resistantmechanisms that are encountered in fluid loss from the fracture. These mechanisms

n′( )

τw k′γn′=

γ τw k′

n′

k′( )


7.2 Data 483

are: CI - fracture fluid leak-off viscosity and relative permeability effect; CII - reser-voir fluid viscosity-compressibility effects; and CIII - wall-building effects.

The total leak-off coefficient is one of the most important parameters in determin-ing the fluid efficiency and therefore the fracture geometry. Fluid loss is onlyassumed to occur over the pay zone height.

Spurt Loss CoefficientSpurt loss is the “instantaneous” volume loss of fluid per unit area of fracture facethat occurs prior to the development of a filter cake. The volume of fluid loss due tospurt for both faces of a single wing fracture is


Input Total Volume InjectedWhen the Input option is selected as Input Volume, the total slurry volume mustbe entered. This is the amount of slurry which will be used for the simulation. Fromthis total slurry volume, the program will automatically calculate the proppant massrequired based on the proppant type and maximum proppant concentration speci-fied.

Input Fracture LengthWhen the Input option is selected as Input Length, the fracture half length mustbe entered. The total amount of slurry is then automatically calculated to create thedesired input fracture length. From the total calculated slurry volume, the programwill then calculate the proppant mass required based on the proppant type and max-imum proppant concentration specified.

Maximum Proppant ConcentrationThe maximum proppant concentration is the desired or final concentration in thefracture. This value is normally the final or maximum inlet concentration. From thisthe total amount of proppant mass pumped is calculated based on this uniform con-centration at the end of pumping in the fracture.

Vsp

Vsp 2ASp=

Sp A



7.3 OutputThe Calculations menu is shown in Figure 7.9. There are two main Output sectionitems, simulation results and viewing the report. There is also an option to display2D Plots of the calculations.

RunThe calculations are accessible from the Run⏐Show Calculations menu. This willrun the simulation for all the models using the Input Data. A summary of the simu-lation results are displayed for the GDK, PKN and Ellipsoidal models as shown inFigure 7.9.

Figure 7.9: Calculations for the GDK, PKN and Ellipsoidal 2D fracture propagation models.

Table 7.5 contains a description of the output data.


7.3 Output 485

PlotThe MFast Plot dialog box is shown in Figure 7.10. It allows the user to selectwhich plots to display, and whether or not the plots are displayed versus time orvolume.


Parameter Description

Length Half length of the fracture. The fracture is assumed tohave two symmetric wings.

Height (wellbore) Total height of the fracture at the wellbore.

Max. well width Maximum width at the wellbore.

Avg. well width Average width at the wellbore.

Avg. frac. width Average width throughout the fracture.

Net pressure Fracture net pressure.

Efficiency Fracture efficiency = Fracture volume/ Total injected vol-ume.

Pumping time Time to inject a given volume or to create a given fracturelength.

Volume Total volume of slurry injected. (Fluid + Proppant).

Proppant mass Total mass of proppant pumped based on the specifiedmaximum allowable concentration.

Percent proppedPercent of the created fracture volume that will remainpropped after closure. (this will be the same for all geom-etry models).



Figure 7.10: Plot Options Dialog Box

ReportsMFast can generate reports similar to the other Meyer Programs. Figure 7.11 showsa typical MFast Report which includes a summary of the input data and calculatedresults.


7.4 References 487

Figure 7.11: MFast Output Report.

7.4 References 1. Meyer, B. R.: “Frac model in 3-D - 4 Parts,” Oil and Gas Journal, June 17,

July 1, July 22 and July 29, 1985.








7. Huit, J.K.: “Fluid Flow in Simulated Fractures,” AIChE Journal, Vol. 2, p 259.1956.





Chapter 8

MPwriProduced Water ReInjection -

Fracturing Simulator

8.1 IntroductionThis chapter is a User’s Guide for the three-dimensional (3-D) hydraulic fracturingwaterflood simulator MPwri. This simulator has many of the same features asMFrac with the exception of proppant transport, acid, foam etc. Options are avail-able for various fluid loss models, inclusion of thermal- and poro-elastic stressesbehaviors. Additional options are available for inputting internal and external frac-ture skins and cakes. The solution methodology for our Produced Water Reinjec-tion (PWRI) hydraulic fracturing simulator is formulated in the Appendices. Asummary of the governing water and thermal front equations, thermal- and poro-elastic stresses and fluid loss equations are also provided.

MPwri is a highly specialized simulator for predicting the pressure and geometry ofhydraulic fractures associated with waterflooding1,2. The program was specificallydesigned for evaluating the effects of injecting large fluid volumes over long peri-ods and for fracture efficiencies approaching zero.

MPwri has options for conventional (1D Carter type leakoff) and ellipsoidal (2D)fluid loss. At early times, fluid loss from the fracture is generally diffusion con-trolled or 1D, but at large times the fluid loss is governed by ellipsoidal and pseu-dosteady-state leakoff. The reservoir coupling with ellipsoidal fluid loss has amarked effect on fracture geometry for high permeability large injection volumescompared to 1D leakoff. Boundary conditions for the reservoir drainage areainclude, closed system, constant pressure boundary condition and pseudo-steadystate behavior.

Options are available in MPwri to modified the vertical layer and lateral stresses toaccount for thermal and poro-elastic effects.


490 MPwri: Produced Water ReInjection - Fracturing Simulator

Only features unique to MPwri will be presented in this chapter. Please refer to theMFrac Chapter for a complete description of the fracture input data. An outline ofthe basic steps for using MPwri is shown in Table 8.1.

MenuThe MPwri menu bar is shown in Figure 8.1. Generally, the menus are accessedfrom left to right with the exception of the Units and Database menus.

Figure 8.1: MPwri Main Menu.

A description of each of the menu items is described in the MFrac chapter. Theitems unique to MPwri are described in the following sections:

Options - Section 8.2

Table 8.1: MPwri Basic Steps

Step Program Area

1. Open an existing MPwri data file (*.mpwri) or create a newdata file. File menu

2. Specify units (optional) Units menu

3. Select program options Data menu

4. Input required dataWellbore hydraulicsZonesTreatment schedule Rock propertiesThermal/Poro-stressThermal frontFluid loss dataInternal/External cake properties

Data menu

5. Run simulation Run menu

6. View plots during or after the simulation Plot menu

7. Generate report Report menu


8.2 Options 491

Data Input - Section 8.3

• Thermal/Poro-elastic Stress Properties

• Thermal/Water Front Data

• Treatment Schedule

Run - Section 8.4

Plots - Section 8.5

The data sections are in the same order as in the MFrac Chapter for ease of refer-ence.

8.2 OptionsTo access the Data Options screen, select Options from the Data menu by clickingthe menu name. The dialog box displayed in Figure 8.2 will then be presented.


The Data Options screen determines what information is needed for a particulartype of analysis. The specific data displayed in a screen or the existence of a datascreen itself varies depending on the options selected. The selections made in theData Options screen set the scope for all data entered in the MPwri program. Theseoptions establish the input data required and specify the nature of the calculations tobe performed.



Figure 8.2 shows the data screen speed buttons at the top of the dialog box. All datascreens in MPwri now contain these speed buttons to make it easier to go from onedata screen to another without pressing OK or Cancel. Selecting a speed button hasthe same effect as pressing OK when exiting a dialog.

General Options The General Options screen allows the user to specify the type of analysis to be per-formed. Figure 8.2 shows the MPwri General Options screen. The Filtration Lawand Thermal Stress options are features unique to MPwri.

Reservoir CouplingThis option provides control and flexibility for the fluid loss mechanisms. Conven-tional is the standard diffusion type fluid loss model as used in MFrac. The Steady-State option is useful for long injection times when the leakoff rate is no longercontrolled by diffusion but rather by steady-state injection and production.

Linear (Conventional)The Linear or Conventional option is the standard type of fluid loss mechanismwhere the rate of fluid loss to the formation is governed by the total leakoff coeffi-cient C. This is referred to as diffusion type leakoff because the fluid loss mecha-nism is diffusion controlled. See the MFrac chapter for additional information onthe conventional leakoff mechanisms.

This option should only be used if the leakoff distance perpendicular to the fractureis much less than the fracture length

Ellipsoidal (Koning)This option should be used for long periods of produced water reinjection or water-flood injection. This model assumes an ellipsoidal (2D) fluid loss model based onthe work of Koning. Three reservoir boundary conditions are available for ellipsoi-dal fluid loss: 1) Closed system - no flow boundaries, 2) Constant pressureboundary condition (b.c.) at the initial reservoir pressure, and 3) Pseudo-SteadyState - where the average reservoir pressure is maintained at the initial reservoirpressure. The pseudo-steady state solution is based on the assumption that the res-ervoir is in a pseudosteady-state mode of injection and production. That is, the pro-duction rate is equal to the injection rate resulting in a pseudo-steady state pressurebehavior of the reservoir.

Although the closed system, constant pressure b.c. and steady-state fluid loss arenot diffusion controlled at long injection periods, the leakoff velocity at early times


8.2 Options 493

is diffusion controlled or linear (i.e. the leakoff velocity is inversely proportional tothe square root of time). This option accounts for the fluid loss behavior changingfrom the conventional diffusion leakoff to a steady-state fluid loss controlled mech-anism.

At large dimensionless times, the resulting leakoff velocity for the constant pres-sure b.c. and steady-state behavior approaches an asymptotic value. This results ina constant leakoff velocity since as time increases the fracture length asymptotes toa constant value.

When the Ellipsoidal law is specified, either the Harmonic or Dynamic Fluid LossModel must be chosen. The Constant fluid loss option will be dimmed.

The governing equations for the ellipsoidal fluid loss model are given inAppendix J.

Thermal and Poro-Elastic StressesOptions are available to Include or Exclude Thermal and/or Poro-elastic stresses.If either thermal or poro-elastic stresses are included a Thermal/Poro-Elastic Prop-erties Table must be input. This table includes the zone layer (depth), initial stress,and coefficient of thermal expansion and layer temperature (if thermal stresses areincluded) and Biot’s constant (only if poro-elastic effects are included). From thisinformation, the thermal and poro-elastic stresses can be calculated (seeAppendix J).

Normally, the fluid injected is cooler than the reservoir temperature which for largeinjection times and volumes, will result in a lowering of the minimum horizontalstresses in zones that have fluid leakoff (if only thermal stresses are included). Thisresults is a lower BHTP and more contained fracture. If the thermal front is aheadof the fracture leading edge, the modified minimum horizontal stresses due to ther-mal effects are seen by the fracture.

However, if poro-elastic stresses are included this will increase the minimum hori-zontal stress in layers with large leakoff volumes. Consequently, when thermalstresses are included one should also include poro-elastic effects.

If you exclude thermal and poro-elastic stresses, they can in some special cases(under pseudo-steady conditions when the thermal front is well ahead of the frac-ture) be modelled using modified stresses in layers where thermal and poro-elasticstresses become time independent.



Fluid TemperatureThe fracture fluid temperature is specified in this dialog box. There is no option forheat transfer in the wellbore or fracture because of the long injection periods thatresult in the fluid temperature in the wellbore and fracture being equal to the injec-tion temperature.

The thermal and water fronts are calculated based on the rate of creation of energyand mass. The fluid ahead of the thermal front is assumed to be at the reservoir tem-perature and the fluid behind the thermal front is at the fluid temperature specifiedin this dialog box.

The injection fluid temperature is also used to calculate the induced thermoelasticstresses.

Fluid Loss ModelTwo fluid loss models are available: 1) Constant and 2) Dynamic. If the reservoircoupling is ellipsoidal only the dynamic fluid loss option is available.

If the Reservoir coupling is selected as linear, the rate of fluid loss to the formationis governed by the total leakoff coefficient C. The three types of flow resistancemechanisms making up C are: 1) CI - leakoff viscosity and relative permeabilityeffects, 2) CII - reservoir viscosity and compressibility effects, and 3) CIII - wallbuilding effects.

This option determines which fluid leakoff model is used. The fluid loss modeloptions include specifying the total leakoff coefficient (Constant Model) or the CIIIcoefficient and the corresponding components which comprise CI and CII (Har-monic or Dynamic Models). A detailed description of the components characteriz-ing the Harmonic and Dynamic models is given in Appendix D and J and in theFluid Loss Data section of this chapter.

ConstantIf Constant is selected, the total leakoff coefficient, C, is entered in the Fluid LossData screen. The total leakoff and spurt loss coefficients are then input as a functionof depth to characterize fluid loss in the fracture at different intervals.

Dynamic ModelFor the Harmonic model, the three additional options available are: 1) Input FilterCake, 2) Input Fracture Skin, and 3) Calculate Fracture Skin.


8.2 Options 495

If Input Filter Cake is selected, the wall building coefficient must be input in thetable. If the Input Fracture Skin option is selected, the user will be asked to input anInternal Skin and External Skin for each layer. If the Calculate Fracture skin optionis selected, the user must enter the particulate properties given in the Internal/External Cake Properties dialog.

Please refer to Appendix J for additional information of the definition of internaland external fracture skin.

Include Fluid Loss HistoryIf the Include Fluid Loss History check box under Fluid Loss Model option ischecked, the simulator will remember the fluid loss history if the fracture closesand then re-opens. This option should be selected to include the effect of when mul-tiple open/close cycles are generated. If this option is checked, the model assumesthat the cake, skin, viscosity, and compressibility effects from the previous fractureremain upon re-opening.

Fracture Options This group of options is accessed by clicking the Fracture tab found on the DataOptions screen. The Fracture Options provide choices for the fracture geometrymodel and constitutive relationships that affect the fracture solution methodology.Figure 8.3 shows the Fracture Option choices.


See MFrac “Fracture Options” on page 82 for additional information.



8.3 Data InputThe following sections pertain to the features unique to MPwri in the Data menu.All other MPwri data menus are covered in detail along with a description of thedata dialogs and their associated variables under MFrac. When pertinent, the condi-tions or case sensitive options for a data screen are noted and an example of theresulting dialog shown. The data screens applicable for MPwri are presented below.

Treatment ScheduleThe input Treatment Design schedule for waterflood applications is given below.Two tabs are listed under Treatment Schedule, the General tab and the Stages tab.

General TabThe General tab for the Waterflood Treatment Schedule is shown in Figure 8.4.

Figure 8.4: Input Treatment Schedule – General Tab.

The General tab contains dialog boxes for Schedule Type and Wellbore.

Schedule TypeIn the Schedule Type dialog box, select Surface or Bottomhole to specify whetherthe data entered (e.g., volumes, rates, etc.) represent surface or bottomhole condi-


8.3 Data Input 497

tions. If pumping from Surface, specify a Wellbore Fluid Type. If pumping fromBottomhole, specify the Flush Fluid Type.

Select Stage Friction Multipliers to enter friction multipliers for each stage.

WellboreThe wellbore dialog box is related to the initial condition of the wellbore.

Along with the Wellbore Volume displayed from the wellbore hydraulics screen,you can specify a Recirculation Volume. You can also specify whether the well isfilled or partially filled prior to injection. To indicate a partially filled wellbore,enter a fraction (0-1) in the Fraction of Well Filled box. A value of one (1) indi-cates the wellbore is 100% filled. A value of 0.5 means that the well is 50% filled.

An initial portion of the pumping schedule can be recirculated by entering a slurryvolume in the Recirculation Volume box. This is useful for setting stages, such asin Frac-Packs. All stages with a total slurry volume less than the Recirculation Vol-ume will be recirculated. If necessary, a fraction of a stage may be recirculated.

If the Stage Friction Multiplier box is selected the Wellbore Fluid Friction Multi-plier can be specified.

Stage TabThe Stage tab for the Waterflood Treatment Schedule is shown in Figure 8.5. Thistype of treatment schedule uses a spreadsheet type of interface as shown. Use thetoolbar located at the top of the screen to control functions such as cut, paste, copy,insert, delete and fill down (see Chapter 1 Working with Spreadsheets). Whenpumping from Bottomhole, it is necessary to specify the Flush Fluid Type. This isselected in the same way as the Wellbore Fluid Type as described above. For refer-ence, the Wellbore Fluid Type (surface) or Flush Fluid Type (bottomhole) andWellbore Volume are displayed.



Figure 8.5: Treatment Schedule – Stages Tab.

To aid in defining the Treatment Schedule, the last column of the table has a Vari-able Column list box, which displays either the Total Time or Total Volumeinjected.

Thermal/Poro-elastic StressesThe Thermal/Poro-elastic Stress Properties dialog box provides a table for enteringthe Coefficient of Thermal Expansion, and formation Layer Temperature (only ifthermal stresses are included), and Biot’s constant (only if poro-elastic stresses areincluded) as a function of depth. Figure 8.6 shows the dialog layout when both thethermal and poro-elastic stress options are selected.


8.3 Data Input 499

Figure 8.6: Thermal/Poro-elastic Stress Properties Table.

Ahead of the thermal front, the stresses are equal to the initial formation stressesplus any poro-elastic effects. Behind the thermal front (and toward the wellbore),the modified stresses as a result of thermal and poro-elastic effects are seen by thefracture system.

Appendix J has a complete list of the Thermal/Poro-elastic stress equations.

Therefore, if the injection temperature is less than the reservoir temperature thethermoelastic stresses will be negative.

Zone DepthThe zone depth is the TVD at the bottom of the zone. This data is taken from theRock Properties dialog box. This depth can only be changed in the Rock Propertiesdialog.

Initial StressThe Initial Stress is the stress value input as a function of depth in the Rock Proper-ties dialog. This value can only be changed in the Rock Properties dialog.

Coefficient of Thermal ExpansionThe Coefficient of Thermal Expansion, , is used to calculate the thermal stressesin the formation as a function of depth. The magnitude of the thermal stresses will

The Thermal/Poro-elastic Stress data must be input after the Rock Propertiesdata.

α



increase as this coefficient increases. Typical coefficients of thermal expansion forrocks are 2x10-6 /°F to 5x10-6/°F (i.e., 10-6/°C to 2.5x10-6/°C).

Layer TemperatureThis is the formation layer temperature. The temperature difference for thermalstresses is calculated from where is the injection temperature and

is the formation layer temperature.

Biot’s ConstantThe change in the minimum horizontal stress is related to the change in the porepressure by Biot’s Constant, where . The general form of the poro-elas-tic equation (see Appendix J) is

where the Perkins factor, , is used to account for the magnitude of theellipsoidal pressure extent around the fracture.

Thermal/Water Front DataThe Thermal/Water Front Data screen is shown in Figure 8.7. This data is used tocalculate the thermal front, waterfront, ellipsoidal waterflood shape, oil displace-ment and leakoff characteristics. A description of the input data is discussed below.

ΔT Ti Tf–= Ti

Tf

α 0 α 1≤ ≤

Δσ3 p1 2ν–1 ν–

---------------⎝ ⎠⎛ ⎞ αfkΔp f a1 b1 h, ,( )⋅=

f a1 b1 h, ,( )


8.3 Data Input 501

Figure 8.7: Thermal/Water Front Data Screen.

Injected FluidThe injected fluid represents the properties (thermal conductivity, heat capacity,specific gravity, etc.) of the fracturing fluid. For waterflood applications, the usershould specify Water. The internal database associated with the injected fluid isused in the calculation of the thermal front.

Reservoir LithologyThe Reservoir Lithology represents the primary rock type in the region to be frac-tured.

In-situ FluidThe In-situ Fluid is the formation fluid that occupies the pores. Typically, this fluidis oil.

Reservoir Half-LengthThe Reservoir Half-Length is used in the dimensionless pressure solution for fluidleakoff. If the reservoir is of infinite extent simply put in a very large value. Thedrainage half-length/area is only used if the Reservoir Coupling option is set toEllipsoidal. The reservoir is assumed to be a square.

If the reservoir half-length, , is input, the drainage area, , will be calculated

from ( ).

xe A

A 4xe2=



Drainage AreaThe Drainage Area, , is used in the dimensionless pressure solution for fluid lea-koff. If the reservoir is of infinite extent simply put in a very large value. The drain-age half-length/area is only used if the Reservoir Coupling option is set toEllipsoidal.

The reservoir is assumed to be a square with sides of half-length where the equivalent drainage radius is defined as

( ).

Fluid Loss DataTo model fluid loss from the fracture into the reservoir and surrounding layers,additional information characterizing the formation and in-situ diffusivity parame-ters is necessary. The format for the fluid loss data entry is flexible and allows any-thing from a single layer reservoir to multi-layered zones with diverse properties.The specific data required by the program depends on which fluid loss model isselected in the General Options dialog.

It is not necessary for these depths to correspond directly to the depths specified inthe Rock Properties screen, although they may. A maximum of 1000 layers is per-mitted in both the Rock Properties and Fluid Loss data screens.

Please refer to Appendix D for a detailed description of the individual leakoff coef-ficients which control fluid loss.

Constant Fluid Loss ModelThis option is only available if the Reservoir Coupling is set to Linear. When theConstant Fluid Loss Model is chosen, the total leakoff, and CTotal coefficients foreach layer are entered in the Fluid Loss Data screen shown in Figure 8.8. There isno Spurt Loss for produced water reinjection or waterflood. When this model isused, it is not necessary to calculate the three individual linear flow resistancemechanisms CI, CII, and CIII (see Appendix D). The diffusivity parameters of per-meability, compressibility and viscosity are not required for this option becausethey are inherently included in the total coefficient.

However, the reservoir porosity and fluid saturations are required to calculate thewater and thermal fronts. The mobile porosity is also displayed (non-editable) ascalculated from

A

xe 1 2⁄ A πRe= =

Re A π⁄( )1 2⁄=


8.3 Data Input 503

where is the residual oil saturation, and is the irreducible water saturation.


The specific data required by the program when using a Constant Fluid Loss Coef-ficient Model is as follows:

ZonesAn optional zone name can be specified for each layer to help organize the fluidloss data properties table.


Total PorosityThe total reservoir porosity is the fraction of a rock’s bulk volume that is filled withhydrocarbons, water, and gas.

Hydrocarbon SaturationThe hydrocarbon saturation is the fraction of the total porosity initially filled withoil or gas.

Residual Hydrocarbon SaturationThe residual hydrocarbon saturation is the fraction of the total oil or gas initially inplace that is immobile. The Oil Displacement Factor is the fraction of oil that is dis-

By convention, the depth entered is the true vertical depth TVD at the bottom ofthe interval. The reservoir parameters are assumed to have constant propertiesover this interval.

φm φ= 1 Sor Siw––( )

Sor Siw



placed by the water. This factor is directly related to the irreducible or residual oilsaturation (i.e., Oil Displacement Factor = 1- irreducible oil saturation).

Irreducible Water SaturationThe irreducible water saturation is the fraction of the water that is immobile.

Mobile PorosityThe mobile or equivalent reservoir porosity is the fraction of a rock’s bulk volumethat is filled with mobile hydrocarbons. This porosity is calculated from

and is not editable. The mobile porosity is used to calculatethe CI and CII leakoff coefficients used to simulate fluid loss during injection. Thisvalue is also used to determine the extent of the water front (see Appendix J).

Total Leakoff CoefficientThe total leakoff coefficient is a combination of the CI, CII and CIII leakoff mecha-nisms. These leakoff coefficients are discussed in Appendix D. The total leakoffcoefficient is used in calculating the time dependent leakoff velocity and overallfluid loss based on mass conservation. The general diffusional leakoff velocity is

where is time and is the initial time of fluid leakoff. The total fluid loss volumeto the formation is

where is a fluid loss parameter and A is the total leakoff area (one face) for bothwings. This equation illustrates that the fluid loss volume is proportional to the lea-koff coefficient and leakoff area product.

For multi-layer leakoff, spurt loss is calculated in each layer separately. Please referto Appendix D for additional information.

Dynamic Fluid Loss ModelWhen the Dynamic fluid loss model is chosen three additional options are availablefor modeling fluid loss. In the Options screen the user can specify either 1) InputFilter Cake, 2) Input Fracture Skin, or 3) Calculate Fracture Skin.


υ C t τ–⁄=

t τ

Vl 2 v A tdd0

A

∫0

t

∫=

πCA tΦ=

Φ


8.3 Data Input 505

If Input Filter Cake is selected, the wall building coefficient must be input in thetable. If the Input Fracture Skin option is selected, the user will be asked to input anInternal Skin and External Skin for each layer. If the Calculate Fracture skin optionis selected, the user must enter the particulate properties given in the Internal/External Cake Properties dialog.

Figure 8.9 illustrates the input parameters when the internal and external skin areinput.

The parameters required for this option are described below. These properties, likethe Rock Properties, are input as a function of the TVD depth. Also like the RockProperties, an optional Zone name is permitted to assist in preparing and organizingthe data.

Figure 8.9: Fluid Loss Data Dialog Box - Dynamic Fluid Loss Model (Input Fracture Skin).

A detailed description of the Linear and Ellipsoidal fluid loss models are given inAppendix J.

The specific data required for the Dynamic Fluid Loss Model for the variousoptions is as follows:




Reservoir PressureThe reservoir or pore pressure is used in conjunction with the minimum horizontalstress and fracture pressure to calculate the differential pressure for leakoff. Theleakoff pressure differential is

where

The pressure difference between the minimum horizontal stress and average porepressure is, therefore, a critical component in calculating the CI and CII leakoffcoefficients.

For new wells, enter the initial reservoir pore pressure for the productive interval.This value is typically obtained from either a production log or well test. Variationsin pore pressure versus depth can be inferred and entered based on gradient mea-surements and/or the fluid saturation changes within the interval (e.g., gas caps,aquifers, etc.).

When a well has been produced for some period of time, enter the average reservoirpressure as interpreted from a well test. In all cases, the value entered should be lessthan the minimum horizontal stress.

Total CompressibilityThe total reservoir compressibility is defined as the total change in the reservoirvolume per unit volume per unit pressure difference. It is the reciprocal of the un-drained bulk modulus and is typically expressed as follows:

where

= minimum horizontal stress= pressure in the fracture= pore or reservoir pressure= net fracture pressure, ( )= differential leakoff pressure

= gas compressibility= oil compressibility= bulk rock compressibility

Δploss pf p0– Δpf σHmin p0–( )+= =

σHmin

pf

p0

Δpf pf σHmin–Δploss


cg

co

cr

8.3 Data Input 507


leakoff pressure differential is

where

PermeabilityThe reservoir permeability is the formation property that characterizes its ability totransfer a fluid through the pores when subjected to a pressure gradient. FromDarcy's law

where

= total formation compressibility= water compressibility= gas saturation= oil saturation= water saturation

= formation permeability= total formation compressibility= formation porosity= reservoir fluid viscosity= distance= pressure= time

= flow rate per unit area= formation permeability= reservoir fluid viscosity= pressure gradient

ct

cw

Sg

So

Sw

t∂∂p k

ctφμ-----------⎝ ⎠

⎛ ⎞z2

2

∂

∂ p⎝ ⎠⎜ ⎟⎛ ⎞

=

kct

φμzpt

q kμ---

xddp

⎝ ⎠⎛ ⎞–=

qkμdp dx⁄



The permeability/mobility is used to calculate the CII coefficient in order to modelthe rate of fluid leakoff into the formation during injection. The values enteredshould reflect the effective permeability to the mobile portion of the reservoir fluid.An effective permeability to the frac fluid filtrate is used to derive CI. The CI coef-ficient is calculated from the permeability and filtrate viscosity.

Total PorosityThe total reservoir porosity is the fraction of a rock’s bulk volume that is filled withhydrocarbons, water, and gas.

Hydrocarbon SaturationThe hydrocarbon saturation is the fraction of the total porosity initially filled withoil or gas.

Residual Hydrocarbon SaturationThe residual hydrocarbon saturation is the fraction of the total oil or gas initially inplace that is immobile. The Oil Displacement Factor is the fraction of oil that is dis-placed by the water. This factor is directly related to the irreducible or residual oilsaturation (i.e., Oil Displacement Factor = 1- irreducible oil saturation).

Irreducible Water SaturationThe irreducible water saturation is the fraction of the water that is immobile.

Mobile PorosityThe mobile or equivalent reservoir porosity is the fraction of a rock’s bulk volumethat is filled with mobile hydrocarbons. This porosity is calculated from

and is not editable. The mobile porosity is used to calculatethe CI and CII leakoff coefficients used to simulate fluid loss during injection. Thisvalue is also used to determine the extent of the water front (see Appendix J).

Reservoir ViscosityThe equivalent reservoir viscosity is the total effective viscosity of a multi-phasefluid system at reservoir conditions. This value is used in calculating the CII leakoffcoefficient for modeling leakoff resistance due to the viscosity and compressibilityeffects of the in-situ fluids.

Filtrate ViscosityThe filtrate viscosity is the effective leakoff viscosity of the fracturing fluid. This isthe fracturing fluid which leaks off through the fracture face. This viscosity hasbeen reduced from its original state due to the deposition of polymer on the fracture



8.3 Data Input 509

face which forms a filter cake. This parameter is used to calculate the CI coefficientfor modeling viscosity and relative permeability effects caused by fracturing fluidleakoff to the formation.

The effective fluid leakoff viscosity must also account for the relative permeabilityeffect of the leakoff fluid to that of the reservoir fluid. This is especially importantfor a gas reservoir. The effective leakoff viscosity, , in terms of the fluid leakoffviscosity and relative permeability is


Wall Building CoefficientThe wall building coefficient is only require if the Dynamic fluid loss option isselected to Input Filter Cake.

The wall building or filter cake coefficient is equivalent to the inverse of the frac-turing fluid leakoff resistance. A value of zero (0) represents an infinite filter cakeresistance, whereas, a CIII value approaching infinity (e.g., >100 ft/min½) repre-sents no wall building. This coefficient is used in calculating the total leakoff coef-ficient C. It reduces the fluid loss rate by increasing the resistance due to leakoff atthe fracture face.


Input Fracture SkinThe internal and external fracture skins are input for each layer, if the Input Frac-ture Skin option is selected. The input fracture skins are assumed to remain con-stant (note: use the Calculate Fracture Skin option for time dependent fractureskins).

The internal fracture skin for linear leakoff from Appendix J is

μe

μe μf kr⁄=

μf kr

sf internal

12 πφσ

′⁄-----------------

δ k ks⁄ 1–( )x 0=

L t( )-----------------------------------------=



The effective external skin based on the filter cake thickness at the wellbore fromAppendix J is

The reader is referred to Appendix J for a detailed outline of the governing equa-tions for ellipsoidal skin factors. the nomenclature is also presented in Appendix J.

For multi-layer leakoff, spurt loss is calculated in each layer separately. Refer toAppendix D additional information.

Time Dependent Fluid LossTo use time dependent fluid loss, open the Time Dependent Fluid Loss tab. Thiswill then display the Time Dependent Fluid Loss screen shown in Figure 8.10. Iftime dependent fluid loss is to be modeled, simply check the ‘Enable Time Depen-dent Fluid Loss’ check box as shown in Figure 8.10.

Figure 8.10: Time Dependent Fluid Loss Data Table

This feature allows you to increase or decrease the fluid loss multiplier as a func-tion of time. This is helpful for modeling leakoff in naturally fractured reservoirs.While fracturing a naturally fractured formation, the pressure in the fracture mayapproach the critical pressure. When the critical pressure of the formation is

sf external

12 πφ′⁄---------------

δc t( )L t( )------------ k μ⁄

kc μc⁄--------------

⎩ ⎭⎨ ⎬⎧ ⎫

=


8.3 Data Input 511

reached, natural fractures open and accelerated leakoff occurs. A zero slope on theNolte plot may characterize this period of accelerated leakoff.

To use this feature, enter time dependent fluid multipliers.

Pressure Dependent Fluid LossTo use pressure dependent fluid loss, open the Pressure Dependent Fluid Losstab. This will then display the Pressure Dependent Fluid Loss screen. If pressuredependent fluid loss is to be modeled simply check the ‘Enable Pressure DependentFluid Loss’ check box and enter the desired fluid loss multipliers.

Internal/External Cake PropertiesWennberg and Sharma (1987), Pang and Sharma (1994) etc., have proposed vari-ous internal and external cake filtration theories to model the injectivity decline inwater injection wells. These models are based on the initial development of aninternal skin (or cake) as a result of particulate deposition and permeability decreas-ing with increase concentration of deposited particulates. After some period of time(transition time) and external filter cake begins to develop.

The internal and external cake properties dialog is shown in Figure 8.11.



Figure 8.11: Internal and External Cake Properties.

Internal Cake PropertiesThe particle mass conservation equation (see Appendix J) is

where is the particulate concentration, is the leakoff velocity, and is the

current porosity in the reservoir at a given position.

The volume concentration of particulates per total volume deposited (porosity ofdeposited particulates), is

t∂∂ φcs( ) v cs∇• t∂

∂cs+ + 0=

cs v φ

σ

σ φ0 φ–=


8.3 Data Input 513

where is the original porosity and is the current porosity.

The concentration distribution in the formation is

where is the filtration coefficient with units of (i.e., ) and

The volume loss per unit area, , is given by

It is shown that for the suspended concentration is time independent and onlya function of position. The average distance a particle travels into the formation,

is

The critical deposition porosity at which transition takes place is

Inserting the critical deposition porosity, , into the deposited porosity equation,we find the transition volume loss per unit area to be

Wennberg and Sharma (1987) report that a reasonable guess for the critical poros-

ity, , at which transition occurs from a theoretical stand point is when the pore

space is about 50% filled, or .

A reasonable value for the deposited porosity ratio is

φ0 φ

σ y t,( ) σ 0 t,( )e λy–=

λ L 1– ft 1–

σ 0 t,( ) V″ t( )λcs 0( )=

V″

V″ t τ–( ) v tdτ

t

∫=

t τ>

δd

δd 2( )ln λ⁄=

σ* φ0 φ*–=

σ*

V″* σ*

λcs 0( )----------------=

φ*

φ* φ0 2⁄=



The theoretical maximum deposited porosity ratio is

where is the particulate porosity within the pore system.

The internal cake permeability reduction at a given point in the formation is gener-ally expressed in the following form (e.g., see Bachman (2003) or Pang (1994))

where is a constant. A more general form of this equation is

where represent the damage factor and the power coefficient.

Refer to Appendix J for additional discussions and these generalized equations.

Total Suspended Solids (TSS) ConcentrationThe total suspended solids concentration, , is the volume of the particulates per

unit total slurry volume. This value is typically in parts per million (ppm) andshould only include the particulates that build the internal and external cakes.

Filtration CoefficientThe filtration coefficient, , quantifies the distance a particulate travels and it’sdistribution in the formation. The average particulate travels a distance of

. Consequently, as the particulates can travel an unlim-

ited distance in the formation without deposition. As the filtration coefficientincreases the penetration in to the formation diminishes.

σ* φ0⁄ 1 2⁄→

σ*

φ0------

max

1 φp–( )=

φp

ksk---- 1

1 Ωσ+-----------------=

Ω

ksk---- 1

1 β σ σ*⁄( )α+----------------------------------=

β α

cs

λ

δd 2( )ln λ⁄= λ 0→


8.3 Data Input 515

Deposited Concentration Ratio at TransitionThe critical deposition porosity or Deposited Concentration ratio at Transition isdefined as



Permeability Damage FactorThe permeability damage factor, , determines the permeability damage and skinof the internal cake as given by


If there is no damage due to particulate deposition. The permeability at

transition is

Permeability Power CoefficientThe permeability power coefficient, , relates the permeability damage to the ratioof particulate distribution in the formation. For linear permeability damage withdeposition is unity. Typical values are near unity. Small values of indicate

σ*

φ0------ 1 φ*

φ0-----–=

σ* φ0⁄ 1 2⁄→

σ*

φ0------

max

1 φp–( )=

β

ksk---- 1

1 β σ σ*⁄( )α+

----------------------------------=

β α

β 0→σ σ*→

ksk----

σ σ*→

11 β+------------=

α

α α



that damage happens quickly with little deposition while large values of indicatethat substantial damage does not occur until the deposited concentration reaches thetransition or maximum value.

External Cake PropertiesThe external filter cake thickness can be calculated from conservation of mass

where is the cake thickness, is the filter cake porosity, and is the par-

ticulate concentration available for deposition in the cake, and the volume loss per

unit area after transition is .

The pressure drop across the filter cake from Darcy’s law is

The external filter cake resistance, , is defined as

where is pressure loss across the cake, is the leakoff velocity, and is the

fluid leakoff viscosity.

The pressure loss across the filter cake from Darcy’s law is

α

δc t( ) 11 φc–( )

-------------------cs

1 cs–-------------ΔV″ t( )=

δc t( ) φc cs

ΔV″ V″ t( ) V″*–=

Δpsμckc-----δcv=

μckc----- 1

1 φc–( )-------------------

cs1 cs–-------------ΔV″v=

Rs

RsΔpsvμc---------≡

Δps v μc

Δps vμδckc----- vμRs= =


8.3 Data Input 517

where the flow resistance term, , is shown to be a function of filter cake thick-

ness and permeability. The external cake resistance can also be written as

which is the same definition presented by Mayerhofer, Economides and Nolte(1991).

Consequently, the filter cake resistance as a function of the particulate concentra-tion and volume loss per unit area is

Erosion of the filter cake can also take place during produced water reinjection.Defining the erosion to build rate as

where the volume loss per unit area to build the filter cake form to is

and is the volume loss per unit area required to erode the filter cake from

to . The volume loss at the minimum filter cake thickness from is

The change in the filter cake thickness during the erosion process is

Rs

Rsδckc-----=

Rsδckc----- 1

kc----- 1

1 φc–( )-------------------

cs1 cs–-------------ΔV″ t( )

⎩ ⎭⎨ ⎬⎧ ⎫

= =

ω V″ddδ

e V″ddδ

b⁄

ΔVb″

ΔVe″

-----------= =

δmin δmax

ΔVb″ ΔVmax

″ ΔVmin″–=

ΔVe″

δmax δmin

ΔVmin″ ΔVmax

″δminδmax-----------⎝ ⎠

⎛ ⎞1 βg⁄

=

Δδe δmax δmin–( ) δV″ΔVe

″----------⎝ ⎠

⎛ ⎞ βe=



where is erosion rate power coefficient and the change in volume loss per unit

area after reaching the maximum filter cake thickness is

and is the volume loss per unit area when the filter cake reached it’s

last maximum value.

The filter cake thickness during erosion is then

or

Following is a description of the input parameters for the external cake.

Cake PermeabilityThis is the permeability of the external filter cake, . The lower the cake perme-

ability the greater the resistance and fracture skin will be.

Cake PorosityThis is the external cake porosity, , of the filter cake. The greater the porosity

the faster the filter cake builds for a given concentration and fluid loss volume perunit area.

Fractional Deposition of TSS Building CakeSince not all particulates after transition will be deposited on an external filter cake,the Fractional Deposition of TSS building the filter cake, , is defined as the frac-

tion of TSS which actually build the cake. The effective concentration building thecake is then .

βe

δV″ ΔV″ t( ) ΔV″ tmax( )–=

ΔV″ tmax( )

δc t( ) δmax Δδe–=

δmax δ– c t( ) δV″( )βe∝

kc

φc

fd

cs fdcs TSS→


8.3 Data Input 519

Maximum Filter Cake ThicknessThis input variable allows one to limit the maximum filter cake thickness, ,

and corresponding fracture skin. The Maximum Volume Loss per Unit Area aftertransition to reach the maximum filter cake thickness is given by

The filter cake resistance at the maximum thickness is

Typically one assumes that the maximum filter cake thickness (one face) should beless than 1/2 the fracture width. However, if filter cake embedment/compactionoccurs or if one assumes that the filter cake is outside the fracture control volume,the filter cake thickness may be greater than the fracture half-width. The filter cakethickness does not necessarily have to have a physical meaning with respect to thefracture width but rather as a calculated parameter for mass conservation and amechanism for fluid leakoff resistance based on the user input cake properties.

Minimum Cake Erosion ThicknessThis is the minimum cake thickness after erosion. If no erosion occurs this valuecan be set equal to the maximum cake thickness. The minimum cake thicknessmust be less than or equal to the maximum value.

The minimum filter cake resistance is then

Cake Build CoefficientNormally the filter cake is assumed to build proportional to the volume loss per unitarea (i.e., linear build rate)

A more general form of this equation for non-linear building of the filter cake is

δc max

ΔV″ max δc t( )1 cs–( )

cs------------------ 1 φc–( )=

Rs max

δc maxkc

---------------=

Rs min

δc minkc

--------------=

Δδc t( ) ΔV″ t( )∝



where is the cake build coefficient. For a linear build rate, is equal to unity.

The filter cake grows at a faster rate for smaller build coefficients. If , the

filter cake will instantly grow to the maximum value and if , the filter cake

will not grow until the volume loss is near the maximum value. Theoretically thecake build coefficient should be near unity. See Appendix J for additional informa-tion of the build coefficient.

Cake Erosion CoefficientThe change in the filter cake thickness during the erosion process is




last maximum value.


or

The erosion coefficient controls the erosion rate over the time or volume to erode.If is set to unity the filter cake erosion will be linear with the volume loss per

unit area. If , the filter cake will erode instantly and then remain at the

δc t( ) ΔV″ t( )βg∝

βg βg

βg 0→

βg 1»


″----------⎝ ⎠

⎛ ⎞ βe=

βe


ΔV″ tmax( )



βe

βe 0→



minimum value until the change in erosion volume is met. If , the filter

cake thickness will remain at nearly the maximum value until the change in erosion

volume, , is obtained.

The actual erosion to build rate, , is given below.

Erosion to Build Rate RatioErosion of the filter cake can also take place during produced water reinjection.Defining the erosion to build rate as



to .

The larger the erosion rate ratio the less time it takes to erode the filter cake fromthe maximum to the minimum thickness. If the time to erode the filter cake is thesame as the build ratio then should be set to unity. If the erosion process is very

fast then or

8.4 Run/Performing CalculationsOnce all of the required data relevant to the options selected have been entered, it istime to perform calculations.

To start the simulation, select the Run command from the Run menu. All openSimulation Data windows and plots will be updated to show the current state of thesimulation. See “Run Options” on page 66 for information about the available runoptions.

βe 1»

ΔVe″

ω

ω V″ddδ

e V″ddδ

b⁄

ΔVb″

ΔVe″

-----------= =

δmin δmax

ΔVb″ ΔVmax

″ ΔVmin″–=

ΔVe″

δmax δmin

ωω 1> ω 1»



8.5 Plots - Graphical PresentationThis section describes the plots unique to MPwri.

Water/Thermal Front PlotsFigure 8.12 illustrates the water and thermal front profiles, and the fracture lengthat the end of pumping. This plot only displays the zone with the greatest fluid lossvolume The minor to major axis aspect ratio is illustrated in Figure 8.13.

Figure 8.12: Thermal/Water Front Profiles.



Figure 8.13: Thermal/Water Front Aspect Ratio as a Function of Time.

The thermal, water and fracture major fronts as a function of time are shown in Fig-ure 8.14.



Figure 8.14: Thermal/Water and Fracture Fronts.



Figure 8.15: Thermal/Poro-Stresses and Fronts with Fracture Profile



Figure 8.16: Volume Loss versus Depth - Two Limited Entry Fractures

8.6 References1. Morales, R.H., Abou-Sayed, A.S., Jones, A.H. and Al-Saffar, A.: “Detection of

a Formation Fracture in a Waterflooding Experiment,” Journal of PetroleumTechnology, October 1986, 1113-1121.

2. Detienne, J-L, Creusot, M., Kessler, N., Sahuquet, B., and Bergerot, J-L:“Thermally Induced Fractures: A Field Proven Analytical Model,” SPE 30777,October 1996.

3. Perkins, T.K. and Gonzalez, J.A.: “The Effect of Thermoelastic Stresses onInjection Well Fracturing,” SPE Journal, February 1985, 78-88.


Chapter 9

MFrac-LiteThree Dimensional Hydraulic

Fracturing Simulator - Lite Version

9.1 IntroductionMFrac-Lite is a three-dimensional hydraulic fracturing simulator similar to MFracbut with a limited number of MFrac features and capabilities (i.e., a lite version).This simplified three-dimensional simulator provides ease of use with less inputdata and fewer options to choose from for applications which do not require someof the advanced features in MFrac.

MFrac-Lite uses the same numerical routines as MFrac but without some of themore advanced and user specified options. MFrac-Lite has similar real-time capa-bilities as MFrac and is designed to be compatible with like features in MFrac.MFrac-Lite can open *.mfrac files. Upon importing the MFrac data will be pro-cessed to produce a compatible three-layer single layer fracture (not limited entry)MFrac-Lite file (*.mfrac-lite). This MFrac file should be saved with the *.mfrac-liteextension. MFrac can also open MFrac-Lite files which are fully compatible butshould be saved as a *.mfrac file. This simulator is designed for those who do notneed the full functionality of MFrac.

A summary of the major MFrac-Lite and MFrac feature comparisons are shown inTable 9.1.

Table 9.1: MFrac-Lite and MFrac Feature Comparison.

Feature/Option MFrac MFrac-Lite

Reservoir Coupling Linear or Ellipsoidal Linear

Fluid Loss ModelOptions

Fluid Type DependentInclude History

NoneNone

Treatment Type Proppant, Acid, & Foam Proppant Only


528 MFrac-Lite: Three Dimensional Hydraulic Fracturing Simulator - Lite Version

Since MFrac-Lite is a subset of MFrac, the MFrac chapter should be referred to fora specific description of a given feature or data input parameter.

A detailed list of the MFrac-Lite features and differences between MFrac-Lite andMFrac is provided.

9.2 Options and FeaturesMFrac-Lite has many of the same capabilities as MFrac but with limited features.The differences between MFrac-Lite and MFrac are presented below by category:The major differences in MFrac-Lite and MFrac is most easily distinguished bycomparing the options and various other data input screens.

General OptionsTo access the Options screen, select Options from the Data menu by clicking themenu name. The dialog box displayed in Figure 9.1 will then be presented.

Heat Transfer On or Off option Off

Flow Back On or Off option Off

Proppant Flow Back On or Off option Off

Perforation Erosion On or Off option Off

Rock Properties 1000 layers 3 layers

Fluid Loss Data 1000 layers 3 layers

Import LAS Yes No

Limited Entry Yes, 10 multiple zones No. Single zone

Table 9.1: MFrac-Lite and MFrac Feature Comparison.


9.2 Options and Features 529

Figure 9.1: General Data Options Screen.

The Options screen determines what information is needed for a particular type ofanalysis.

The General Options not available in MFrac-Lite are:

1. Reservoir Coupling. MFrac has options for Linear and Ellipsoidal. Thedefault in MFrac-Lite is Linear only.

2. Fluid Loss Model. MFrac-Lite does not support Fluid Type Dependent or theoption to Include Fluid Loss History.

3. Treatment Type. MFrac-Lite only supports a treatment type of Proppant.MFrac also supports options for Acid and Foam.

4. Heat Transfer. There is no heat transfer option in MFrac-Lite. By default heattransfer is set to off and the user only needs to specify the fluid temperature atwhich rheological properties are evaluated.

See MFrac “General Options” on page 74 for more information.

Fracture Options This group of options is accessed by clicking the Fracture tab found on the DataOptions screen. The Fracture Options provide choices for the fracture geometrymodel and constitutive relationships that affect the fracture solution methodology(see Figure 9.2). The choices are as follows:




MFrac-Lite does not support Flowback as does MFrac.

See MFrac “Fracture Options” on page 82 for more information.

Proppant Options This group of options is accessed by clicking the Proppant tab found on the DataOptions screen. The proppant options specify the proppant transport methodologyto be employed. Figure 9.3 illustrates the proppant options available.


9.3 Data Input 531


As illustrated, options for Proppant Flowback and Perforation Erosion are notsupported in MFrac-Lite.

See MFrac “Proppant Options” on page 93 for more information.

9.3 Data InputOnce the Options are selected the scope of a simulation is set. Data may then beentered by accessing the various dialog boxes from the Data menu. The followingsections pertain to the Data menu items found within the main menu.

Only Data menus that are different than the MFrac Data menus will be covered inthis section. The reader is referred to the MFrac chapter for Data Input menus notcovered in this chapter that are common to both simulators.

ZonesThe Zones dialog box is used to specify the number and location of the perforatedintervals and corresponding Zone Data (Figure 9.4). Only one perforated intervalcan be specified. Limited entry type fractures are not supported in MFrac-Lite.




See MFrac “Zones” on page 117 for more information.

Zone DataPerforationsFigure 9.5 shows the Perforation tab screen. Perforation erosion is not supported inMFrac-Lite.

Figure 9.5: Zone Data - Perforation Tab.

See MFrac “Zone Data” on page 119 for more information.


9.3 Data Input 533

Rock PropertiesThe Rock Properties dialog box provides a table for entering the mechanical prop-erties of the reservoir and adjacent lithologies including in-situ stresses as a func-tion of depth (see Figure 9.6).

MFrac-Lite only supports three lithology layers.

MFrac-Lite does support an option to insert rock properties from our rock proper-ties database (Insert from Database) but DOES NOT support an option to importmechanical rock properties (Import Log) data as does MFrac.

Figure 9.6: Rock Properties Dialog Box.

MFrac-Lite supports up to three layers. A maximum of one thousand (1000) layerscan be specified in MFrac.

See MFrac “Rock Properties” on page 156 for more information.

Fluid Loss DataTo model fluid loss from the fracture into the reservoir and surrounding layers,additional information characterizing the formation and in-situ diffusivity parame-ters is necessary.

MFrac-Lite only supports three fluid loss zones whereas MFrac supports up to amaximum of one thousand (1000) layers. The specific data required by the programdepends on which fluid loss model is specified in the General Options Dialog.MFrac-Lite only supports Constant, Harmonic or Dynamic Fluid loss.



Constant Fluid Loss ModelWhen the Constant Fluid Loss Model is chosen, the total leakoff, , and the SpurtLoss coefficients for each layer are entered in the Fluid Loss Data screen shown inFigure 9.7.


The specific data required by the program when using a Constant Fluid Loss Coef-ficient Model is discussed in the MFrac chapter.

Harmonic or Dynamic Fluid Loss ModelsWhen either the Harmonic or Dynamic fluid loss models are chosen, the filter cakecoefficient ( ) is input for each layer desired. and are calculated based onthe reservoir parameters input in the Fluid Loss dialog box shown in Figure 9.8.

MFrac-Lite only supports three layers while MFrac supports up to one thousand.

Figure 9.8: Fluid Loss Data Dialog Box - Harmonic/Dynamic Fluid Loss Model.

See MFrac “Fluid Loss Data” on page 175 for more information.

C

CIII CI CII


Chapter 10

MWellA Wellbore Hydraulics Simulator

10.1 IntroductionMWell is a wellbore hydraulics simulator for calculating surface or bottomholepressures, gravitational head, restrictions, transport times and hydraulic powerrequirements in the wellbore. Near wellbore and perforation pressure losses arealso calculated to determine the bottomhole treating pressure in the formation.

MWell was designed for real-time analysis to calculate BHTP’s, from surface con-ditions but can also be used as a design tool for determining wellbore pressure char-acteristics prior to performing the treatment. MWell is essentially a subset of theMFrac simulator without the fracture simulation. MWell however does provide thecapability to simulate time dependent formation pressures with a user specifiedtable for inputting the minimum horizontal stress and a time dependent net pressure(pressure above or below the reference minimum stress). If the formation is notfractured the reference pressure should be the reservoir pressure.

MWell is structured in a manor similar to MFrac and uses the same databases. Thedata files *.mwell and *.mfrac or *.mfrac-lite are compatible in that the commondata is shared.

This chapter covers the available menu options and basic procedures required torun MWell.

An outline of the basic steps for using MWell is shown in Table 10.1.

Table 10.1: MWell Basic Steps.

Step Program Area

1. Open an existing MWell data file (*.mwelll) or create a newdata file. File Menu


536 MWell: A Wellbore Hydraulics Simulator

MenuThe MWell menu bar is shown in Figure 10.1. Generally, the menus are accessedfrom left to right with the exception of the Units and Database menus.

Figure 10.1: MWell Main Menu.

A description of each of the menu items is described in the following chapters orsections:

File - Chapter 10

Options - Section 10.2

Data - Section 10.3

• Wellbore Hydraulics

• Zones


3. Select Program Options Data Menu

4. For a real-time or replay case, start MView and import theacquired data. MView

5. Input Required DataWellbore HydraulicsZonesTreatment ScheduleFoam Schedule

Data Menu


7. View Plots during or after the simulation Plot Menu


Table 10.1: MWell Basic Steps.


10.2 Options 537

10.2 OptionsThe Options screen is the first input dialog box under the Data menu in MWell. It isused to establish the primary model options in the program. Each option relates to aspecific aspect of the fracture and proppant/acid modeling approach.

To access the Options screen, select Options from the Data menu by clicking themenu name. The dialog box displayed in Figure 10.2 will then be presented.

Figure 10.2: Data Options

The Options screen determines what information is needed for a particular type ofanalysis. The specific data displayed in a screen or the existence of a data screenitself varies depending on the options selected. This “smart-menu” approach, mini-mizes data input and prevents unnecessary or misleading data entry. Simply decidethe relevant options for a specific simulation and the program will only displaythose menus and input fields necessary. Any time the options are changed the inputdata screens will be updated to enable new input or hide data that is not needed.This hierarchy methodology is used throughout MWell.

The selections made in the Data Options screen set the scope for all data enteredinto the MWell program. These options establish the input data required and specifythe nature of the calculations to be performed.

To select an option, click the radio button adjacent to the option preference. A blackdiamond will then appear in the center of the button selected. Continuing, select aradio button within the next option section or use the TAB button to move sequen-tially through the choices. Once within a section, the current selection for that



option is highlighted with a dotted rectangle. The option choice may be changed byusing either the mouse or the arrow keys.


Simulation MethodDesign ModeThis option is used for determining the pressure losses, hydrostatic head, perfora-tion friction, wellbore pressures etc. for a specific design. The program flexibilityallows for running in standard mode based on a given input treatment schedule.Depending on other options specified, the program uses the formation and treat-ment data to calculate surface and bottomhole pressures. Design Mode refers to thefact that the design engineer must design (and optimize) the fracture treatmentschedule.

Replay/Real-TimeThe Replay/Real-Time option is required for replaying or performing real-timefracture analysis using the data collected during a treatment. This procedurerequires the use of MView as the real-time or replay data handler. Please refer toChapter 3 for instructions on the use of MView.

With respect to MWell, there is essentially no difference in the procedures used forperforming real-time or replay simulations. The difference between these methodsonly involves the source data input which is handled by MView.

Real-Time The Real-Time options are only available if the Replay/Real-Time radio button isclicked On in the Simulation Method dialog. If MView Concentration is selectedthe proppant concentration will be taken from the replay/real-time data as sent toMFrac from MView. If the Input Concentration button is selected the proppantconcentration used by MWell will be taken from the values specified in the Treat-ment Schedule. Generally, the MView Concentration is desirable unless the actualproppant concentration injected is not available.

The Synchronize Well Solution radio button is used to synchronize the numeri-cally calculated time steps for wellbore events with the replay/real-time data.


10.2 Options 539

Synchronizing the wellbore solution with the incoming real-time or replay dataenables for very refined calculations of the wellbore and near-wellbore frictionalpressure losses.

Treatment TypeThis selection determines the type of fracture treatment. The Treatment Type canbe either a propped (Proppant) or acid (Acid) fracture. In addition, the treatmentcan accommodate an optional foam schedule by checking the Foam box. WhenFoam is checked, MWell will include compressibility effects. Since Mwell uses thesame treatment type as MFrac, the reader is referred to the MFrac chapter for moredetailed information not duplicated below

Treatment Design OptionsThe treatment design options are only available if the Simulation Method is inDesign Mode and the Treatment Type selected is Proppant with no Foam. Thedefault setting is Input for all other cases.

In MWell, either the pumping schedule can be input manually or determined auto-matically. When Auto Design is chosen, the desired design fracture length or totalslurry volume is input in the treatment schedule dialog box. Depending on theProppant Transport Methodology selected, specific criteria for controlling theproppant scheduling will also be required.

Wellbore Hydraulics ModelThis option determines the wellbore hydraulics model to be used in calculating fric-tional pressure losses in the wellbore. Surface and bottomhole pressures, gravita-tional head, restrictions, transport times and hydraulic power requirements are alsocalculated. The near wellbore and perforation pressure losses are calculated sepa-rately below the BHP reference point for each fracture and coupled to the wellbore.The available wellbore model options are listed below:

NoneWhen this option is selected, wellbore hydraulics calculations are still performed;however, the frictional pressure loss is assumed to be zero. The wellbore hydraulicsoutput data is also not displayed or written to file.

EmpiricalThe Empirical option is an internal correlation for calculating the frictional pres-sure loss of Newtonian and non-Newtonian fluids. This option provides a combined



correlation that is applicable for a variety of fluids ranging from linear systems tohighly non-Newtonian and viscoelastic fluids that exhibit drag reduction due to slipor shear thinning during turbulent flow. Three distinct types of behavior are possi-ble with the combined correlation used in MWell. These behaviors are illustrated inFigure 10.3 and summarized in the explicit expressions for the Fanning friction fac-tor outlined in Table 10.2

Figure 10.3: Pipe Friction Empirical Correlations.

When a value for the Relative Pipe Roughness is entered into one of the WellboreHydraulics dialog boxes, the expression for friction factor based on Prandtl’s “Uni-versal” Law is modified. See Appendix E for additional information.






1f

------ 19 Res f( )log 32.4–=

1f

------ A Res f( )log B+=

1f

------ 4 Res f( )log 0.4–=


10.2 Options 541

To include the effects of proppant concentration on friction, the program has a builtin correlation for slurry rheology. The relationship used, originally described byKeck, et al., is also presented in Appendix E.

User DatabaseWhen User Database is selected, the information specified in the fluid database isused for calculating the frictional pressure loss in the tubing, annulus, and casing.This data can be edited and plotted by accessing the database. The information inthe database does not represent proppant-laden fluid. Consequently, if the proppantconcentration wellbore option is selected in the proppant option screen, the frictionfactor will be adjusted for proppant concentration in a manner similar to the methoddescribed in Appendix E for the Empirical option.

Wellbore SolutionThese options provide control and flexibility for the time dependent discretizationmethodology used in the program. To enable time step size control for capturingvarious time dependent events, the user can specify the number of wellbore solu-tion Iterations and the Maximum Time Step. For Replay/Real-Time analysis, thedata Restart Time can also be specified.

The base time step used for discretization in the numerical simulation will be theminimum of the values calculated from either the number of Iterations or MaxTime Step input.

IterationsThe value for the number of Iterations determines the target number of time stepsto be used for the fracture propagation solution. The total or estimated simulationtime is then divided by the number of iterations to determine the time step size.

For example, if the number of iterations is 100 and the pump time is 100 minutes,the average time step would be one (1) minute. The actual time step may varydepending on other numerical considerations. For most simulations, a value of 20to 30 iterations is sufficient.

Generally, the number of iterations is most effectively used in design mode. ForReplay/Real-Time, the Max Time Step constraint may be more applicable.

The number of time steps should be increased for cases with order-of-magnitudechanges in the injection rate or fluid rheology properties (e.g., pad/acid). It shouldalso be increased when the injection times are very large (e.g., years as in waterflooding). The maximum time step can also be specified to minimize the time step.



The larger this value is, the longer the program will take to run.

Max Time StepThe Max Time Step can be used to control the program time step. This is especiallyuseful when performing real-time or replay simulations. To simulate events thatoccur over a very narrow range of time (e.g., rate changes or pressure spikes) thetime step size must be small enough to capture the event. If the time step is toolarge, significant rate and pressure changes may be missed. Also, the smaller theMax Time Step the longer it will take the program to run.

The Real-Time option of synchronizing the wellbore solution to the input dataenables time refinement for wellbore and near-wellbore pressure losses. This isvery useful for history matching pressure changes due to rate. This provides thecapability to accurately model wellbore friction.

Restart TimeThe Restart Time is used to start or restart a simulation at a time other than the firstentry point in the data file for real-time or replay simulations. This option is nor-mally used when earlier data is not relevant or multiple injection cycles (i.e., mini-frac) are pumped and only the later time cycle data (i.e., main frac) is to beanalyzed. Consequently, this option provides the flexibility to restart a simulation atthe beginning or middle of any injection cycle. Enter the time in the replay/real-time data at which the simulation should begin.

Fluid TemperatureThis is the wellbore Fluid Temperature. The fluid rheological properties are thencalculated from the Fluid Database as a function of time based on this temperature.

Proppant Options This group of options is accessed by clicking the Proppant tab found on the DataOptions screen. Figure 10.4 illustrates the proppant options available. Theseoptions are discussed below.


10.2 Options 543


Proppant RampThe ramp option controls the ability to ramp the proppant concentration between aspecified range. When this option is On, the concentration of proppant will beramped linearly from an initial value (From) to a final (To) value for each fluidstage in the Treatment Schedule dialog box. This results in a linear proppant rampwith liquid volume.

When this option is turned Off, a uniform proppant concentration is assumed foreach stage. The Treatment Schedule screen will then permit only one entry valuefor concentration.

Wellbore-Proppant EffectsThis option controls the methodology used to simulate the effects of proppant con-centration on pipe friction. The options are as follows:

NoneFor this selection, proppant has no effect on the friction factors used in the wellborehydraulics calculations.

EmpiricalThis option includes the effects of proppant concentration on pipe friction as origi-nally described by Keck, et al.2 This correlation uses an expression for relative



slurry viscosity to account for the effects of proppant on increased friction. Therelationship is shown below:

where

For laminar flow, the friction factor multiplier, M, for proppant-laden fluids isequal to the value of . For proppant-laden fluids in turbulent flow, the expressionshown below is used to estimate the effect of proppant on friction:

and

where

User SpecifiedFor some slurry systems, adequate characterization of the frictional dissipation isnot possible with the empirical correlation contained in MWell. If this occurs, thefriction factor multiplier as a function of proppant concentration can be specified intabular form.



μr 1 0.75 e1.5n′ 1–( )e 1 n′–( )γ 1000⁄–[ ] 1.25φ1 1.5φ–-------------------+⎝ ⎠

⎛ ⎞ 2=

μr

n'γφ

μr

Mf μr0.55ρr

0.45=

fs Mf fb=

Mf

ρb

ρr ρr ρs ρb⁄=ρs

μr

fb

fs


10.3 Data Input 545

10.3 Data InputOnce the Options are selected the scope of a simulation is set. Data may then beentered by accessing the various dialog boxes from the Data menu. The followingsections pertain to the Data menu items found within the main menu.

Only Data menus that are different than the MFrac Data menus will be covered inthis section. The reader is referred to the MFrac chapter for Data Input menus notcovered in this chapter that are common to both simulators.

Wellbore HydraulicsMWell offers an integrated wellbore hydraulics module that couples the fracture orformation with the wellbore to provide additional simulation capability. An energybalance approach is used to calculate the pressure changes due to potential energy,kinetic energy, frictional dissipation and restrictions in the wellbore.

This general solution permits calculations of surface pressure, BHP in the wellbore,Frac-Pack screen pressure drop, hydrostatic head, frictional loss and hydraulicpower requirements for a treatment design. The flexibility of the model providesthe capability to history match measured pressures during real-time or replay treat-ment analysis.

The capability to model tapered deviated wellbores is also included. Treatmentstage movement in the wellbore is also simulated during pumping.

Since the MWell and MFrac wellbore hydraulics dialog’s are identical, the readeris referred to the MFrac chapter section on Wellbore Hydraulics for a detaileddescription of the required input data and features.

ZonesThe Zones dialog box is used to specify the number and location of the perforatedintervals and corresponding Zone Data (Figure 10.5). Only one perforated intervalcan be specified.




The type of data required to define an interval depends on whether the well and/orthe fracture is horizontal or vertical. A well is assumed to be a “vertical well”unless the Horizontal check box in the General Wellbore Hydraulics screen ischecked.

Zone Name To assist in keeping track of the data depth intervals, an optional Zone name can beentered in the second column of the table. This name is only used to help organizethe input and output data.

Perforation and Fracture IntervalsFor vertical wells with vertical fractures, regardless of whether the well is deviatedor not, the perforation data is entered relative to the true vertical depth (i.e., Top ofPerfs TVD, Bottom of Perfs TVD) or measured depths (i.e., Top of Perfs MD,Bottom of Perfs MD).

If a horizontal well is specified in the General Wellbore Hydraulics screen, the cen-ter of the perforated measured depth (Center of Perfs MD) is input and the truevertical center of the perforated depth (Center of Perfs TVD) is calculated. TheCenter of Perfs TVD is dimmed and cannot be edited.

Zone Data After entering the Zones perforated depth information, open the Zone Data screenfor each interval by clicking the Edit button found in the far right column. TheZones Data screen shown in Figure 10.6 has tabs for Perforations, Near Wellbore,and Fracture Pressure.


10.3 Data Input 547

Figure 10.6: Zones Data Dialog Tabs.

PerforationsFigure 10.6 shows the perforation tab screen. The perforation data requirements arediscussed below.

Number and Diameter of Perforations

The Number and Diameter of perforations must also be specified for each perfo-rated zone. These values are entered in the boxes provided at the top of the Perfora-tions screen. This information is used to calculate the perforation friction pressureloss.

Near Wellbore Pressure TableThe near wellbore pressure loss table is shown in Figure 10.7. MWell has the capa-bility to model time and rate dependent near wellbore pressure drop for each frac-ture. This pressure drop can represent any near wellbore effect such as tortuosity,perforation erosion, near wellbore multiple fractures, etc. The methodologyemployed is explained in Appendix C.



Figure 10.7: Near Wellbore Pressure Loss Screen.

To include the near wellbore pressure drop as a function of time, fill in the spread-sheet located on the right side of the Zone Data screen. Up to fifty rows can bespecified to define the near wellbore pressure drop as a function of time and rate.

Import RT Button

When performing real-time or replay analysis using MView, MWell automaticallyrecords “significant rate and pressure changes” and generates a near wellbore pres-sure loss relationship. After running the acquired data through MWell, open theZone Data dialog box and choose the Import RT Button. The program will thenload the corresponding data file to fill in the Near Wellbore Pressure Table. If nosignificant rate/BHTP changes were encountered or if the data was not run throughMWell, a message like the one shown in Figure 10.8 will be displayed.


10.3 Data Input 549

Figure 10.8: Import RT Message.

The imported near wellbore pressure table includes the total near wellbore pressureloss. This table can be manually changed to incorporate small rate/pressure changesnot considered significant or indeterminate by MWell.

Once a Near Wellbore Pressure Table has been created, choose how it will beapplied by clicking one of the radio buttons located below the spreadsheet. Theoptions are: to ignore the table completely, use the pressure drop as the total nearwellbore effects (including perforations), or to add the resulting effects to the calcu-lated perforation pressure losses (near well effects only).

The program performs a linear interpolation between successive data points for

(where ). If the job duration is longer than the maximumtime entered in the table, the last (final) value will be used.

Fracture Pressure TableThe fracture pressure table is shown in Figure 10.9. This table enables the user tocalculate surface pressures in Design Mode (from a given BHFP reference) and his-tory match the net, surface, and bottomhole pressures in Replay/Real-Time Model

The time dependent bottomhole formation or fracture pressure ( ) can eitherbe input as function of time and the net pressure or delta pressure ( ) above thereference minimum horizontal stress ( ) for a fractured system calculated or thedelta pressure can be calculated based on a given . The general formulationfor the bottomhole formation or fracture pressure is

K t( ) Δp t( ) K t( ) Q t( )• α=K t( )

BHFPΔp

σ

BHFP

BHFP σ Δp t( )+=



If the formation is not fractured the minimum stress reference is the reservoir pres-sure and the delta pressure (negative for production and positive for injection) is thedeviant form this value.

Figure 10.9: Fracture Pressure Dialog.

Following is a detailed description of the parameters in the Fracture Pressure Dia-log.

Minimum Stress

This is the minimum horizontal stress ( ) in the formation for a hydraulically frac-tured system. If the formation is not fractured during injection, the minimum stressrepresents the reservoir pressure.

Delta Pressure

The Delta Pressure ( ) or net pressure is the pressure above (positive) or below(negative) the reference Minimum Stress.

BHFP

The bottomhole formation or fracture pressure ( ) is the pressure in the for-mation or fracture. The bottomhole treating pressure is then calculated by addingthe near wellbore and perforation pressure loss to this value.

σ

Δp

BHFP


Appendix A

Hydraulic Fracturing Theory

A.1 IntroductionThis appendix describes the solution methodology for our hydraulic fracturing sim-ulator. The coupled rock and fluid mechanics equations governing fracture propa-gation are presented. These non-linear partial differential equations are thentransformed and solved using integral methods.

The hydraulic fracturing simulator accounts for the coupled parameters affectingfracture propagation and pressure-decline. The major fracture, rock and fluidmechanics phenomena included are: (1) multilayer unsymmetrical confining stresscontrast, (2) multilayer leakoff, (3) fracture toughness and dilatancy (tip effects),(4) variable injection rate and time dependent fluid rheology, (5) vertical and lateralrock deformation, (6) wall roughness and (7) coupled proppant transport, heattransfer and fracture propagation.

A list of parametric relationships which affect the fracture characteristics and frac-ture net pressure is also given.

A.2 Governing EquationsAn overview of the governing equations of mass conservation, continuity, width-opening pressure, momentum, fracture propagation criteria and constitutive rela-tionships to model fracture propagation are presented here for completeness. Adetailed description of these equations and the solution methodology is provided byMeyer et al.1-6 Symbols are given in the nomenclature.


552 Hydraulic Fracturing Theory:

Mass ConservationThe governing mass conservation equation for an incompressible slurry in a frac-ture is

where

The above mass conservation equations are solved numerically in MFrac by ele-mentally descritizing a fracture grid and then integrating over each element.

The above equations for performing minifrac analysis can be simplified for 2-Dtype models for fluid loss due to leakoff during and after pumping:

During Pumping

After Pumping

ContinuityThe mass continuity equation in terms of the flow rate per unit length q = v W is

q d V t V t V tf l sp

t( ) ( ) ( ) ( )τ τ − − − =∫ 0

0 (A-1)

[ ]

[ ]

V tC A t

t AdAdt

V t S A t

A t A A t

l

t

sp p

A

a

( )( , )

( )( ) ( )

( ) ( )

=−

=

=

∫∫2

2

00 τ

τ

α

α

τ

(A-2)

V t C t A t tl a c( ) ( ) ( ) ( )= π α α Φ (A-3)

V C t A t t G

t tl p p p a c

p

( ) ( ) ( ) ( , )

.

θ α α θ

θ

=

=

22

where

(A-4)


A.2 Governing Equations 553

where and is the leakoff rate per unit leakoff

area (i.e., leakoff velocity).

Momentum ConservationThe momentum equation (equation of motion) for steady flow is

where

; laminar flow

; turbulent flow

is the Darcy friction factor, Re is the Reynolds Number and ε is the relative wallroughness.

Width-Opening Pressure Elasticity ConditionThe crack-opening and opening pressure relationship is of the form:

where is a generalized influence function, is a characteristic half-height and

is the net fracture pressure .

Fracture Propagation CriteriaThe criterion for fracture propagation is based on the concept of a stress intensityfactor . The fracture will propagate when the stress intensity factor equals the

r r∇ ⋅ + + =q q W

tL2 0∂∂

(A-5)

∇ q⋅ ∂qL ∂x ∂z ∂z⁄+⁄= ∂L

r r∇ = −P f q w1

22 3ρ (A-6)

f 24 Re⁄=

f fR e ε,( )=

f

),0,()1(2),,,(),,( txPHG

tzyxtzxW W Δ−

Γ= ξν

(A-7)

ΓW Hζ

ΔP P σ–

KI



fracture toughness or critical stress intensity of the rock ( or

whichever is greater).

A.3 Solution MethodologyThe fracture propagation solution is obtained numerically by satisfying mass con-servation (Eqn. (A-1)), continuity (Eqn. (A-5)), momentum (Eqn. (A-6)), elasticityrelationship (Eqn. (A-7)) and the fracture propagation criteria ( or

whichever is greater).

The governing differential equations for fracture propagation are differentiatedwith respect to time and then simplified by the transformations

to form a set of equations in terms of the alpha parameters .

The length propagation parameter is of the form:

where accounts for the time dependent gamma parameters, non-steady injectionrates and fluid rheology, spurt loss, fracture toughness, etc. The fracture efficiencyis given by and . The geometric factor is equal to unity for thePKN and 3-D type fracture models and equal to zero for the GDK model. Addi-tional alpha parameters for 2-D type fractures are also given by Meyer3.

Equation (A-9) and the formulated constitutive relationships control the timedependent length propagation solution:

KIC KI KIC=

σI σIC=

KI KIC=

σI σIC=

α α

α α

α α

L a

ww

wH

w

w

p c

tL t

dL tdt

tA t

dA tdt

tW t

dW tdt

tH t

dH tdt

tP t

d P tdt

tC t

dC tdt

≡ ≡

≡ ≡

≡ ≡

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

;

;

; Δ

Δ

(A-8)

αζ t ζdζ dt⁄⁄=( )

α α η αη β

βλ

Lca term

H nn

=− + − +

+ + + ′′ + −

1 1 2 1

1 1 31 1

( )( )( ( )

( )( )

(A-9)

αζ

η βH αH αL⁄= βλ


A.4 Parametric Relationships 555

A.4 Parametric RelationshipsThis section describes some of the functional relationships between various fractureparameters and their effect on fracture characteristics and pressure response (seeHagel and Meyer6 for additional details).

Table A.1 shows the effect of various parameters on fracture length, width and netpressure for PKN, GDK and Penny type 2-D fracture models for viscous and tough-ness dominated fracture propagation. The viscous equations are for laminar flowwith negligible toughness and no spurt loss. The toughness equations are for negli-gible viscous dissipation. The penny shape model is referred to as the Sneddonmodel for toughness controlled propagation.

Parameters with the largest exponents have the greatest influence on the specificfracture characteristic. Therefore, more emphasis should be put on refining thesecritical parameters. A systematic approach is a good method of determining param-eters which best match the fracture characteristics and response.

The proportionality equations in Table A.1 can be used to refine input data and todetermine parameter sensitivity for 2-D type models. The parameters which affectnet pressure the most are: Young's modulus, fracture height and viscosity for thePKN model. To match the net pressure in a GDK model only the fluid rheology orYoung's modulus can be varied to get a match assuming negligible toughness. Thenet pressures for the GDK and Penny models are shown not to be a function of frac-ture height.

Table A.1 illustrates that the fracture net pressure decreases with volume (time) forthe GDK and Penny models for both viscous and toughness dominated fractures.The PKN model is the only 2-D model where the net pressure increases with time(volume). Replacing the injected volume ( ) by fracture volume ( ) demonstratesthe approximate effect on fluid efficiency.

Table A.1 shows that for fracture propagation controlled by toughness, as tough-ness increases the net pressure and width increase, and the length decreases. Thefracture characteristics are shown to be only a function of the critical energy releaserate ( ).

)(

)()(t

nn

L

tttLtL

α

⎟⎟⎠

⎞⎜⎜⎝

⎛= (A-10)

V Vf

Gcr KIC 1 υ–( ) G⁄=



Table A.2 summarizes the parametric equations for the total leakoff coefficientbased on satisfying the governing equations of mass and momentum. Since the clo-sure time for a minifrac analysis is a constant and known, the fracture efficiencyand fluid loss volume ( ) are also constant for a specific model. The proportion-ality relationships given in Table A.1 are for negligible spurt loss.

The leakoff coefficient in minifrac analysis is also shown to be strongly influencedby the pay zone and total fracture height for the GDK and PKN type models.Clearly, to obtain an accurate total leakoff coefficient, the pay zone and total frac-ture heights must be known. One of the major reasons for using a 3-D hydraulicfracturing simulator is to predict height growth as a function of time in fracturepressure analysis. A 2-D PKN type model can also be used with reasonable resultsfor well contained fracture by modifying the fracture height to match at givenpoints in time. However, to history match the entire pressure response in formationswhich display height growth, the model must account for this behavior.

Table A.2 also shows that the total leakoff coefficient is inversely proportional tothe pay zone height. Changing the fracture height and injection rate has a majoreffect on the predicted leakoff coefficient. The penny shape equations assume lea-koff over the entire fracture area.

Table A.3 lists the relationships for the fracture length, width and leakoff coeffi-cient as a function of net pressure. As illustrated, the leakoff coefficient and widthincrease and the length (radius) decreases as the net pressure increases. Table A.3also demonstrates that if the simulated net pressure is different than the measurednet pressure, a significant error in the leakoff coefficient and fracture characteristicscan occur. Compatibility in net pressure is far more important for mass conserva-tion than the fracture model used. For the PKN, GDK and Penny shape models theleakoff coefficient is proportional to net pressure raised to the 1, 1/2 and 2/3 pow-ers, respectively.

Table A.3 also illustrates the problem with assuming the measured net pressureapplies to all three models. This is evident by the fact that if momentum is satisfied,each model will predict different values for the net pressure. Therefore, the mini-frac net pressure results should also be applied to the correct model. The fractureefficiency is approximately equal for all models, since it is only a function of thepressure decline slope and closure time (see Appendix F).

Table A.5 lists a qualitative representation of the effect various parameters have onthe fracture geometry for 2-D and 3-D type models. This table is useful in deter-mining what happens if a given parameter is increased or decreased. Since theseequations do not account for coupled effects and assume no parameter interaction,they should only be used as general guidelines. Exceptions can be found in non-homogeneous formations and for treatments with time dependent parameters.

CA



The utility of Table A.1 through Table A.4 is in providing the engineer with a quickand qualitative method of establishing the effect various parameters have on frac-ture characteristics without running numerous simulations. For example, Table A.4illustrates that increasing fracture toughness generally increases fracture width andnet pressure, and decreases height growth. Whereas, increasing dilatancy, viscosityor wall roughness will tend to increase the fracture width, height and net pressure(generally), and result in a shorter length.

Generally, many of the parameters listed in Table A.1 through Table A.4 are notconstant but vary with time and space. To accurately model such variations it isnecessary to use a fracture simulator which accounts for these complex interac-tions. For well contained fractures, a 2-D model will provide acceptable results.However, for fractures which exhibit height growth the importance of a 3-D modelis realized to correctly simulate height, net pressure and compliance (width).

To determine the effect of coupling numerous parameters in Table A.1 throughTable A.4 a numerical simulator is almost essential. A hydraulic fracturing simula-tor can account for the following effects not normally modeled in fracture-pressureanalysis:

• Time dependent fluid rheology.

• Multilayer leakoff.

• Height growth through multi-stress zones.

• Temperature effects on fluid rheology.

• Fracture tip effects. Dilatancy and fracture toughness.

• Variable formation properties.

• Coupled fracture propagation, heat transfer and proppant transport.

• Wall roughness and waviness.

• Variable injection rates.

• Consistency in the fracture-pressure analysis and treatment design.

Table A.5 lists the major elements used in the numerical simulator to calculate thefracture pressure and pressure decline. They are divided into three sub-groups; res-ervoir, geomechanical and fracture fluid. Each element has been rated according toits relative influence on the pressure solution. The engineer is encouraged to makean effort to obtain the best possible values as the final accuracy of the analysis will



be affected. The items which have received a rating of minor, need only be approx-imated within a range of 25 percent.

Table A.1: Two-Dimensional Hydraulic Fracture Parametric Equations.

Viscous Dominated

PKN

Mod

elG

DK

Mod

elR

adia

l Mod

el

Toughness Dominated

PKN

Mod

elG

DK

Mod

elR

adia

l Mod

el

L EQ n ′– V2n ′ 2+

1 v2–( )k′Hn ′ 2+----------------------------------------

12n ′ 3+-----------------

∝ W 1 v2–( )E

-------------------k′Qn ′VHn ′

-----------------1

2n ′ 3+-----------------

∝ ΔP E2n ′ 2+ k′Qn ′V1 v2–( )2n ′ 2+ H3n ′ 3+

----------------------------------------------------1

2n ′ 3+-----------------

∝

L EQ n ′– V2n ′ 2+

1 v2–( )k′Hn ′ 2+----------------------------------------

12n ′ 4+-----------------

∝ W 1 v2–( )E

-------------------k′Qn ′VHn ′ 2+-----------------

12n ′ 4+-----------------

∝ ΔP En ′ 1+ k′Qn ′

1 v2–( )n ′ 1+ Vn ′--------------------------------------

1n ′ 2+--------------

∝

R EQ n ′– V2n ′ 2+

1 v2–( )k′---------------------------------

13n ′ 6+-----------------

∝ W 1 v2–( )E

-------------------k′Qn ′V2 n ′–

2-------------

23n ′ 6+-----------------

∝ ΔP En ′ 1+ k′Qn ′

1 v2–( )n ′ 1+ Vn ′--------------------------------------

1n ′ 2+--------------

∝

L E1 v2–( )

------------------- VKICH3 2/---------------------∝ W 1 v2–( )

E-------------------KICH1 2/∝ ΔP

KIC

H1 2/-----------∝

L E1 v2–( )

------------------- VKICH-------------

23---

∝ W 1 v2–( )2

E2---------------------

KIC2 VH

-------------13---

∝ ΔP 1 v2–( )E

-------------------KIC

4 HV

--------------13---

∝

L E1 v2–( )

------------------- VKIC---------

25---

∝ W 1 v2–( )4

E4---------------------KIC

4 V15---

∝ ΔP 1 v2–( )E

-------------------KIC

4

V---------

15---

∝



Table A.2: Leakoff Coefficient Parametric Equations.

Viscous Dominated C - Leakoff Coefficient

PKN Model

GDK Model

Radial Model

Toughness Dominated

PKN Model

GDK Model

Sneddon Model

C 1 η–( )η

-----------------Q4n ′ 1+4n ′ 6+-----------------

V2n ′ 1+2n ′ 3+-----------------

------------------Hn ′ 3+

2n ′ 3+-----------------

Hp------------------ 1 v2–( )

E-------------------K′

12n ′ 3+-----------------

∝

C 1 η–( )η

----------------- Qn ′ 1+n ′ 2+--------------

Vn ′

2n ′ 4+-----------------

------------------H1 2/

Hp----------- 1 v2–( )

E-------------------K′

12n ′ 4+-----------------

∝

C 1 η–( )η

-----------------Q12--- 2n ′

3n ′ 6+-----------------+

V12--- 2n ′

3n ′ 6+-----------------–

-------------------------- 1 v2–( )E

-------------------K′2

3n ′ 6+-----------------

∝

C 1 η–η

------------Q1 2/

V1 2/-----------H3 2/

Hp----------- 1 v2–( )

E-------------------KIC∝

C 1 η–η

------------Q1 2/

V1 6/-----------H2 3/

Hp----------- 1 v2–( )

E-------------------KIC

23---

∝

C 1 η–η

------------ Q1 2/

V3 10/------------- 1 v2–( )

E-------------------KIC

45---

∝



Table A.3: Leakoff Coefficient Parametric Equations

PKN

Mod

elG

DK

Mod

elSn

eddo

n M

odel

Table A.4: Qualitative Parametric Effects on Fracture Charactersitics.

PARAMETER FRACTURE CHARACTERISTICS

Two-Dimensional Models (PKN, GDK, Penny)

η → 1 η → 0

Toughness, KIC↑ W↑ L↓ W↑ L-

L ηV

ΔP1 v2–E

--------------H2-----------------------------∝ W ΔP1 v2–

E--------------H∝ C 1 η–

η------------Q1 2/

V1 2/-----------H2

Hp------ΔP 1 v2–( )

E----------------∝

L ηV

ΔP 1 v2–( )E

-------------------H

--------------------------------

12---

∝ W ηVH

-------ΔP1 v2–E

--------------

12---

∝ C 1 η–η1 2/------------Q1 2/ H1 2/

Hp----------- ΔP 1 v2–( )

E----------------

12---

∝

R ηV

ΔP 1 v2–( )E

-------------------H

--------------------------------

13---

∝ W ηV( )13---

ΔP1 v2–E

--------------

23---

∝ C 1 η–η2 3/------------Q1 2/

V1 6/-----------H2

Hp------ ΔP 1 v2–( )

E----------------

23---

∝



Inj. Rate, Q↑ W↑ L↓ H↑ W↑ L↑ H?

Leakoff Coef., C↑ W↓ L↓ H↓ W↓ L↓ H↓

Viscosity, μ↑ W↑ L↓ H↑ W↑ L- H↑

Pay Height, Hp↑ W↑ L↓ H? W↑ L↓ H?

Temperature, T↑ W↓ L↑ H↓ W↓ L- H↓

Volume, V↑ W↑ L↑ H↑ W↑ L↑ H↑

↑ increase ↓ decrease? not sure (may go either way)- not a function of that parameter

W - widthL - lengthH - heightη - efficiency

Table A.5: Summary of Major Elements for Proper Analysis.

RESERVOIR Sources Importance Rating

Porosity Logs, Cores Minor

Compressibility Tests, Calculations Medium

Reservoir Pressure Buildup Medium

Net Pay Logs, Cores Important

Gross Pay (Estimate of FracBarriers) Logs Medium

Permeability (estimated) Cores, Tests Medium

Fluid Viscosity Lab Tests, PVT Medium

Fluid Compressibility Lab Tests, PVT Minor

GEOMECHANICAL Sources Importance Rating

Poissons's Ratio Logs, Core Tests Minor

Young's Modulus Logs, Core Tests Medium

Fracture Toughness Tests, History Match Medium

Minimum Horizontal Stress Minifrac Important

Stress Contrasts Logs, Core Tests Important

FRAC FLUID Sources Importance Rating

Rheology Lab Tests Important

Density Lab Tests Minor

Table A.4: Qualitative Parametric Effects on Fracture Charactersitics.



Filter Cake Lab Tests Medium

Filtrate Viscosity Lab Tests Important

Table A.5: Summary of Major Elements for Proper Analysis.


A.5 Nomenclature 563

A.5 Nomenclature= Leakoff area (one face of the fracture)

= Total reservoir compressibility

= Total leakoff coefficient

= Leakoff viscosity control coefficient

= Reservoir compressibility and viscosity coefficient

= Wall building coefficient

= Young's modulus

= Fluid loss function, Eqn. (F-18)

= Fracture half-height

= Pay zone height

= Total wellbore height

= Permeability

= Consistency index

= Fracture half-length

= Flow behavior index

= Pressure

= Injection flow rate

= Spurt loss coefficient

= Time

= Dimensionless Nolte time,

= Fracture volume

= Fluid loss volume (no spurt loss)

= Volume loss by spurt

A

Ct

C

CI

CII

CIII

E

G θ( )

H

Hp

Hw

k

k′

L

n′

P

q

Sp

t

tD

Vf

Vl

Vsp



Greek

= Fracture width

= Average wellbore fracture width

= Fracture width as a function of position

= Lateral coordinate along fracture length

= Coordinate perpendicular to frac face

= Vertical Coordinate

= Leakoff area parameter, Eqn. (A-8)

= Leakoff parameter during pumping, Eqn. (A-8)

= Length propagation parameter, Eqn. (A-8)

= Pressure parameter, Eqn. (A-8)

= Width propagation parameter, Eqn. (A-8)

= Friction coefficients

= Width profile coefficients

= Maximum width-opening pressure coefficient

= Net fracturing pressure

= Fracture efficiency

= Efficiency excluding spurt loss

= Dimensionless time,

= Dimensionless lateral coordinate

= Time of fracture leakoff area creation

= Fluid loss parameter, Eqn. (A-3)

= Fluid loss parameter, Eqn. (A-3)

= Minimum horizontal stress

W

W 0 t,( )

W ζ t,( )

x

y

z

αa

αc

αL

αp

αw

γf Γf,

γw Γw,

γwoΓwo

,

ΔP

η

η*

θ

ξ

τ

Φ

ψ

σ


A.6 References 565

Subscripts

A.6 References1. Meyer, B. R.: “Frac model in 3-D - 4 Parts,” Oil and Gas Journal, June 17, July

1, July 22 and July 29, 1985.



4. Meyer, B. R., Cooper, G. D. and Nelson, S. G.: “Real-Time 3-D HydraulicFracturing Simulation: Theory and Field Case Studies,” paper SPE 20658 pre-sented at the SPE 65th Annual Technical Conf., New Orleans, Sept. 23-26,1990.

5. Meyer, B.R., Hagel, M.W., “Simulated Mini-frac Analysis”, Petroleum Soci-ety of CIM, Calgary June 1988.

6. Hagel, M. W. and Meyer, B. R.: “Utilizing Mini-frac Data to Improve Designand Production,” CIM paper 92-40 June, 1992.

c = Closure, leakoff parameter

D = Dimensionless

f = Fracture

l = Fluid loss

p = Pay zone, end of pumping

sp = Spurt loss


Appendix B

Multilayer Fracturing

B.1 IntroductionMultilayer or limited entry fracturing is a process whereby multiple zones are stim-ulated simultaneously. The initiation and propagation of multilayer fractures isgoverned by conservation of mass and momentum for the system of fractures. Thisprocess is controlled by the limited entry techniques employed which include thenumber of perforations, perforation spacing, near wellbore effects, fracture pres-sures, stresses, etc.

The methodology for limited entry fracturing was first presented by Elbel1 et al.where an analytical PKN fracture model was linked to an analytical wellboremodel. Elbel developed his formulation based on an analogy of Kirchoff's currentand voltage laws to those of mass and momentum conservation.

Although the solution techniques presented here are similar for solving a non-linearsystem of equations, the implementation is not limited by the fracture model, well-bore restrictions or time dependent dissipation losses. Our formulation is based onconservation of mass and momentum (formulation based on “TransportPhenomena2 “).

The methodology presented here for multilayer fracturing couples a generalhydraulic fracturing simulator with a general wellbore model. The fractures maycoalesce, interact or may remain isolated from one another. The governing equa-tions presented are not specific to any limiting constraints. Only the constitutiverelationships and boundary conditions imposed on the fracture geometry modelgovern the interaction behavior of the fracture system.

Figure B.1 shows a schematic of the multilayer fracturing technique implementedin MFrac. As illustrated, this concept is based on conservation of mass and energywith an analogy to electrical theory.


568 Multilayer Fracturing:

Symbols are given in the nomenclature.

Figure B.1: Schematic of Multilayer Fracturing.


B.2 Governing Equations 569

B.2 Governing EquationsFollowing are the governing mass and momentum conservation equations for mul-tilayer (limited entry) fracturing.

Mass ConservationMass conservation is based on an overall mass balance for fluid flow into the casingand fracture system:

Eqn. (B-1) states that the rate of mass accumulation in the casing (below the refer-ence point) is equal to the rate of mass injected into the system minus the sum ofthe rates of mass out of the control volume into the individual fracture intervals. Forpositive flow is into the fracture and for negative flow (flowback) is out ofthe fracture.

Assuming a constant fluid density in the control volume (i.e., same density in thecasing and at the entrance to each fracture layer) Eqn. (B-1) simplifies to

where

Eqn. (B-2) also applies to compressible and changing density slurries with time solong as the fluid density to each fracture layer is constant for a given time step. Eqn.(B-3) allows for wellbore storage (filling the wellbore and re-circulation).

∂ ρ∂

ρ ρ

- Q ii=1

n

iVt

Qin= ∑

(B-1)

Qi Qi

Q Qt i i=1

n

= ∑

(B-2)

Q Q Vtt in =

−

∂∂

(B-3)



Momentum ConservationThis equation is based on an overall momentum balance for fluid flow in the casingand into the fracture layers. From the steady-state macroscopic mechanical energybalance2:

where

For fluid flow of a constant density , Eqn. (B-4) simplifies to the “extended” Ber-noulli's Equation:

where is positive in the direction of the gravity vector.

Rearranging Eqn. (B-5) and gathering like terms:

where

From Eqn. (B-6) for each of the fracture layers (i=1,…,n) we have

= cross-sectional velocity= change in potential energy= mechanical work

= frictional loss

Δ Δ Φ12

10

3

1

2vv

dp W EP

P

v+ + + + =∫ ρ) )

(B-4)

v⟨ ⟩ΔΦ

W

)

E v

)

ρ

P v gz P v gz W Ev2 2

2

21 1

2

12 2ρ ρ+ − = + − + −

) )(B-5)

z

P P P Pl h1 2= + −Δ Δ (B-6)

Δ

Δ

P g z z

P W Ev v

h

l v

= −

= − + +−⎛

⎝⎜

⎞

⎠⎟

ρ

ρ

( )

( )2 1

22

12

2) )


B.3 Numerical Solution 571

where

and and are time dependent energy dissipation coefficients calculated numer-ically from losses in the casing. This also includes restrictions, velocity headchanges, etc. The terms and are the dissipation coefficients calculatedfor the total near wellbore effects (multiple parallel fractures, tortuosity, perforationlosses, etc.).

B.3 Numerical SolutionEqns. (B-2) and (B-7) represent a system of n+1 non-linear equations and n+1unknowns (i.e. and for i=1,..,n) for the n fracture layers.

Non-dimensionalizing Eqns. (B-2) and (B-7), and rewriting in the form of a zerocontour function we have:

Momentum Eqns. (i=1,…,n)

P P P P Pf i NW i l jj

i

h jj

i

01 1

= + + −= =

∑ ∑, , , ,Δ Δ Δ (B -7 )

P

f,i = +

= −

=

=

−

=

−

=

−

∑ ∑

σ

ρα

α

i f i

h j j j

l j j jj i

n

jj i

n

NW i NW i i i

PP g z z

P c Q Q

P c Q Q

j

NW i

Δ

Δ

Δ

Δ

,

,

,

, ,

( )

,

1

1

1

cj αj

cNW ,i αNW , i

P0 Qi

f P P P P Pi f i NW i l jj

i

h jj

i

= − + + −⎛

⎝⎜

⎞

⎠⎟

= =∑ ∑0 0

1 10/ /, , , ,σ σΔ Δ Δ (B-8)



Mass Conservation

where and are normalizing reference values for rate and stress.

The governing equations are non-dimensionalized to ensure that fi and fn+1 are ofthe same order. This is extremely beneficial and necessary for solving non-linearsets of equations. This follows from the statement: “There are no good, generalmethods for solving systems of more than one non-linear equation3.”

The general solution methodology utilizes the Newton-Raphson Method for non-linear system of Equations3.

The typical problem gives functional relations to be zeroed, involving variables, i = 1,2,...,N:

If we let denote the entire vector of values then in the neighborhood of ,

each of the functions can be expanded in Taylor Series

By neglecting terms of order and higher, we obtain a set of linear equations forthe corrections that move each function closer to zero simultaneously, namely

where

f Q Q Q Qn t ii

n

+=

= / - 1 0 01

/∑

(B-9)

Q0 σ0

Nxi

f x x x i Ni N( , ,...., ) , , ,....,1 2 0 1 2= = (B-10)

X xi X

fi

f X X f X fx

x Xi ii

jj

i

j( ) ( ) ( )r r r r

+ = + +=

∑∂ ∂∂

∂ ϑ ∂

1

2

(B-11)

∂X2

∂X

α ∂ βijj

N

j ix=

∑ =1

(B-12)


B.3 Numerical Solution 573

Matrix Eqn. (B-12) can be solved by LU decomposition as described in Ref. (3).The corrections are then added to the solution vector

where

and the process is iterated to convergence.

Although the above numerical procedure (solution of non-linear equations) is simi-lar to that of Ref. (1), the fracture pressures are calculated numerically for any

geometry model (not just an analytical PKN model). In general, the flow rates ,

reference pressure , frictional dissipation and gravitational head terms are

implicit. The fracture pressures are semi-implicit and utilize a “shooting

method” to better approximate , ( ), at the new time step and as a func-

tion of . Utilizing explicit values for can result in an unstable solution if thenear wellbore and casing frictional losses are negligible. The implementation of theabove numerical solution is stable for all fracture geometry models (including theGDK model whose net pressure is not a function of flow rate) and for negligibleenergy dissipation.

α∂∂

βiji

ji i

fx

f≡ ≡ −

;

x x x i ninew

iold

i= + = +∂ , ,...,1 1 (B-13)

x , i = 1,..., n x

i

n+1

==Q Q

Pi /

/0

0 0σ

Pf, i

Qi

P0

Pf, i

Pf ,i σi ΔPf,i+

Qi Pf,i



B.4 Nomenclature

Greek

= Viscous dissipation term

= Equations to minimize

= Gravitational acceleration

= Fracture layer number

= Interval number

= Number of layers

= Number of equations to solve (n+1)

= Reference pressure at

= Fracture pressure in layer i

= Reference rate

= Injection rate into layer i

= Inlet injection rate

= Total injection rate (Eqn. (B-3))

= cross-sectional velocity

= mechanical work

= Variables

= Depth to center of layer i

= Reference depth

= Matrix coefficients,

= Matrix function,

E v

)f

g

i

j

n

N

P0 z0

Pf i,

Q0

Qi

Qin

Qt

v⟨ ⟩

W

)

xi

zi

z0

αij ∂fi ∂xj⁄

βi fi–


B.5 References 575

Superscripts

Subscripts

B.5 References1. Elbel, J. L., Piggott, A. R., and Mack, M.G.: “Numerical Modeling of Multi-

layer Fracture Treatments,” paper SPE 23982 presented at the 1992 SPE Per-

= Pressure difference

= Change in potential energy

= Variable correction

= Function partial derivative

= Minimum horizontal stress in layer i

= Reference pressure/stress

= Density

new = New or current value

old = Old value

= Vector

1 = Reference point 1

2 = Reference point 2

f = Fracture

i = Fracture layer number

j = Interval number

l = Loss

h = Gravitational head

NW = Total near wellbore

perf = Perforations

ΔP

ΔΦ

∂x

∂f ∂x⁄

σi

σ0

ρ

→



mian Basin Oil and Gas Recovery Conference, Midland, Texas, March 18-20,1992.

2. Bird, R. B., Stewart, W.E., and Lightfoot, E. N.: “Transport Phenomena,”Wiley, New York, 1960.

3. Press, W.H. et al.: “Numerical Recipes..,” Cambridge University Press, NewYork, 1988.


Appendix C

Multiple Fractures

C.1 IntroductionMultiple fractures refers to the condition when more than one fracture is created inthe same zone. This is not the same as multilayer or limited entry fracturing whichinitiate fractures in different zones. Consequently, multiple zones can create multi-layer fractures with each having their own system of multiple fractures. Also, mul-tiple fractures may or may not be parallel to one another.

A definition of multiple fractures as implemented in MFrac is necessary because ofthe general misconception of what they are and how they are modeled. Multiplefractures can occur in two regions, the near wellbore and far field. Each has aunique impact on the fracture geometry and pressure response.

Near wellbore multiple parallel fractures occur near the wellbore. They have beenreferred to as tortuosity (or a result of tortuosity), multiple fractures, initiation frac-tures non-perpendicular to the minimum horizontal stress, etc. Many times thecause of “excess Net Pressure” is postulated to be multiple fractures.

Whether near wellbore pressure loss is a result of perforations, tortuosity, multiplefractures, fracture initiation or some other form of viscous dissipation, it can bemodeled as a near wellbore pressure loss function. This near wellbore pressure losshas been modeled by some using the “Multiple Parallel Fracture” approach. InMFrac, near wellbore pressure losses are entered into a table as a function of timeand rate, since they are not known a priori.

Only the far field multiple fractures are modeled as a fracture system in MFrac.These fractures may propagate parallel to one another or spread out in a dendritic(tree like or radial) pattern from the wellbore. These multiple far field fractures mayor may not interact.


578 Multiple Fractures:

Although, MFrac provides a number of options for modeling multiple fractures, wedo not believe they should be used as a general methodology for increasing netpressure. This is discussed in greater detail below. These options should be usedwith care based on sound engineering judgment.

For completeness, the governing equations for far field and near wellbore multiplefractures are presented.

C.2 Far Field - Multiple Fractures This section presents a set of constitutive equations describing multiple fractures inthe far field (away from the near wellbore). The basic premise is that there are Nmultiple (parallel or dendritic) fractures which may or may not interact. To modelthis phenomena (if it exists) we assume that the multiple fracture system consists ofa set of N similar fractures.

Governing EquationsFollowing are a set of governing momentum, conservation and constitutive equa-tions to model multiple fractures in the far field (and near wellbore region). Theseconstitutive relationships have been integrated into our general fracture simulatorMFrac.

Interaction FactorsThe governing constitutive relationships for the interaction factors and degree ofinteractions used in modeling multiple fractures are given below.

Multiple fractures can be modeled independently for each multilayer fracture. Toallow for interaction between multiple fractures, interaction factors are introducedfor the flow rate, stiffness (elastic) and fluid loss. The individual fracture interac-tion factors and degrees of interaction are

Flow Rate

where

Q Qi q T= Ψ

(C-1)


C.2 Far Field - Multiple Fractures 579

Stiffness

where

Fluid Loss

where

The interaction functions and degrees of interaction are given by and , respec-tively. The degrees of interaction are the values input into the program. The indi-vidual fracture properties and parameters are identified by the subscript . The totalvalue for fractures is given by the subscript .

The degrees of interaction for stiffness and fluid loss are functions of the formationproperties and relative position of the multiple fracture system. For non-interactingfractures, the degrees of interaction for stiffness and fluid loss are zero. If thefractures are fully interacting (competing for the same space and leakoff area), the

values are equal to unity.

Momentum ConservationThe generalized momentum equation for N multiple fractures in laminar or turbu-lent flow is

Ψ q = 1 / N

E Ei E= Ψ 0 (C-2)

Ψ ΦE EN= − +( )1 1

V V Ni T= / (C-3)

V VN N

T l

l l

== − +

ΨΨ Φ

0

1( )

Ψ Φ

Φ

iN T

Φ

Φ



where

is the Darcy friction factor, is the Reynolds number and is the flux per unitlength.

Width-Opening PressureThe crack-opening and opening pressure relationship is

where

C.3 DiscussionMultiple parallel fractures have been used as a mechanism in the industry toincrease fracture net pressure. The paradox here is: why would mother nature createmultiple far field fractures at an energy level (pressure) that is higher than what a

= generalized influence function= stiffness interaction factor= characteristic half-height= fracture width= opening pressure,

r r∇ = −P f q W1 2 3/ /ρ

(C-4)

ff f

qq Q N

kq

Wk k

nn

aq q

a a

n

a

n

==

≡ = =

=⎛

⎝⎜⎜

⎞

⎠⎟⎟ =

+⎛⎝⎜

⎞⎠⎟

−

24

1

6 2 132

1

/ Re;(Re);

Re ; ; /

;

' '

''

'

laminar flow turbulent Flow

ρμ

μ

rr r

r

Ψ Ψ

f Re q

WG

H PWE

=−

ΓΨ

Δ2 1( )ν

ξ

(C-5)

ΓW

ΨE

Hξ

W

ΔP Pf σ–


C.3 Discussion 581

single fracture requires? This, of course, assumes that multiple fractures alwaysinteract to be at a higher pressure state (which certainly is not always true).Although multiple fractures surely exist in nature, they are not the norm for generalhydraulic fracturing treatments.

The above set of equations have been fully integrated into MFrac which account forthe coupled effects of stiffness, fluid loss and flow rate interactions.

Net Pressure for Multiple FracturesTo illustrate the effect of the degree of interaction on net pressure, a simple formu-lation is presented for two-dimensional type models. This analysis clearly illus-trates, in a simple format, why net pressure may increase or decrease with anincreasing number of multiple fractures depending on the degree of interaction.

The net pressure for a single fracture from conservation of mass and momentumcan be expressed in the form:

Viscous Dominated

ΔP E QE q∝ α α

(C-6)

PKN: η α α

η α α

= =++

=++

= =++

=

1 2 22 3

12 3

0 2 12 2

12

: ;

: ;

'

'

'

'

'

'

,

,

E q

E q

nn

nn

nn

GDK & Penny: η α α

η α α

= =++

=

= =++

=−

+

1 12

0

0 2 12 2 2 2

: ;

: ;

'

'

'

'

'

'

,

,

E q

E q

nn

nn

nn



Toughness Dominated

Constant Critical Stress

The net pressure for a N multiple parallel fractures is

where

Net Pressure RatioThe ratio of the net pressure for N multiple fractures ( ) to a single fracture

( ) from Eqn. (C-7) is

PKN: α αE q= =0 0 , ;

GDK: α αE q= − = −1 3 1 3/ / ; ,

Penny: α αE q= − = −1 5 1 5/ / ; ,

All Models: α αE q= =0 0 , ;

Δ Ψ ΨP E QN E qE q∝ ( ) ( )α α

(C-7)

Ψ ΦΨ

E E

q

NN

= − +=

( )/

1 11

ΔPN

ΔP

ψ

α

α

pN

E

PPN

N

E

q

≡

=− +

ΔΔ

Φ

( )1 1

(C-8)


C.3 Discussion 583

The condition for equivalent multiple and single fracture net pressures ( ) is

where

The range of beta values is

Viscous Dominated

Toughness Dominated

Constant Critical Stress

Ψp

ΦE pNN

( )ψβ

= =−

−1 1

1 (C-9)

β α α= q E/

PKN: η β

η β

= =

= =++

1 1 2

0 12 1

: / ;

: ;'

'

nn

GDK & Penny: η β

η β

= =

= = −+

1 0

02 2

: ;

:'

'

;nn

PKN: β = → ∞ = →0 0 1,ΦE

GDK: β = =1 1, ;ΦE

All Models: β = → ∞ =0 1, ;ΦE



Figure C.1 shows as a function of the degree of interaction and number of frac-

tures for the PKN model ( and ). As illustrated, for interacting frac-tures the net pressure ratio increases. However, for non-interacting fractures the netpressure decreases. The required degree of stiffness interaction to maintain thesame net pressure as a single fracture is given by Eqn. (C-9) and represented by ahorizontal line with a value of unity.

Figure C.1: Net Pressure Ratio for Multiple Fractures (PKN Model).

The above analytical analysis can also be extended to include the degree of fluidloss interaction. This presentation illustrates that multiple fractures do not necessar-ily always increase fracture net pressure.

C.4 Near Wellbore - Multiple Fractures A set of equations is presented to illustrate the concept of multiple fractures (tortu-osity) in the near wellbore region. This formulation is non-unique and cannot quan-tify the number or physical characteristics of multiple fractures in the near wellregion. The details of this formulation are only useful in modeling phenomenarelated to increased near wellbore pressure loss or dissipation.

Figure C.2 shows a schematic of the near wellbore multiple fracture network. Hereis the length, is the height, and is the width of each near wellbore fracture

Ψp

η 1= n′ 0.5=

LT H W


C.4 Near Wellbore - Multiple Fractures 585

Figure C.2: Schematic of Near Wellbore Multiple Fractures.

Governing EquationsA set of governing momentum conservation and constitutive equations are pre-sented to model multiple fractures in the near wellbore region. This formulation isbased on a simplistic analysis to illustrate the general form of dissipation.

Momentum ConservationThe momentum equation for N multiple fractures in the near wellbore region fromEqn. (C-4) is

Laminar Flow

ΔΨ

PQ

HW LT

qn

nT=

⎛⎝⎜

⎞⎠⎟ − +μ

'

'( )2 1 (C-10)



Turbulent Flow

where

Width-Opening PressureThe crack-opening and opening pressure relationships from Eqn. (C-6) for theGDK and PKN type two-dimensional models are

where

Near Wellbore Pressure LossThe near wellbore pressure loss due to multiple fractures (or tortuosity) is found bysubstituting the constitutive relations for width-opening pressure into the momen-tum equation (Eqns. (C-1) to (C-5)). To maximize the near wellbore pressure loss,

=== flow rate into one-wing= number of multiple fractures=

ΔΨ

P fQ

HW LT

qT=

⎛⎝⎜

⎞⎠⎟ −1 2

2

3/ ρ (C-11)

μ 2 6n• ka Γf⁄

Ψq 1 N⁄QNΔPT ∇PLT–

GDK: W L PE

T

E

=Δ

Ψ '

(C-12)

PKN: W H PEE

=Δ

Ψ2 '

(C-13)

EG'

) )≡ =

2(1- E

4(1-ν ν 2


C.4 Near Wellbore - Multiple Fractures 587

it will also be assumed that the near wellbore fractures are fully interacting (com-peting for the same space):

GDK Model

PKN Model

General Near Wellbore Dissipation Function The general form of the near wellbore dissipation function is

(C-18)

where is a time dependent proportionality constant and is the powercoefficient.

This formulation for near wellbore pressure loss can be used for perforation fric-tion, tortuosity, near wellbore multiple fractures or any other near wellbore effectthat is a result of friction dissipation.

The coefficients for the near wellbore pressure loss are usually based on analyzingthe measured BHP and rate data. This technique works best if a number of substan-tial rate and BHP changes occur during the job. MFrac has a built in automatic fea-

Laminar Flow: Δ

ΔP

Q NL H

NEPT

T

n n

=⎛

⎝⎜

⎞

⎠⎟

⎛⎝⎜

⎞⎠⎟

+

μ /

' '' ( )

2

2 1

(C-14)

Turbulent Flow: Δ

ΔP f

Q NL H

NEPT

T=

⎛

⎝⎜

⎞

⎠⎟

⎛⎝⎜

⎞⎠⎟1 2 2

2 3

// '

ρ (C-15)

Laminar Flow : Δ

ΔP

Q NH

NEP

LHT

n nT=

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

+

μ /

' '' ( )

3

2 12

(C-16)

Turbulent Flow : Δ

ΔP f

Q NH

NEP

LHT

T=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟1 2

23

2 3

// '

ρ (C-17)

ΔP t( ) K t( )Q t( )α t( )=

K t( ) α t( )



ture to perform the necessary regression analysis to determine and aspiece wise continuous functions of time. Currently, the analysis regresses to findthe best fit a coefficient for the entire time cycle.

Since is a coefficient that is not readily associated with pressure loss, it is con-venient to transform Eqn. (C-18) so that it can be easily represented in tabular data.The pressure loss as a function of flow rate for a given time is

(C-19)

where

The near wellbore pressure loss methodology is summarized in Figure C.3 and Fig-ure C.4

.

Figure C.3: Near Wellbore Pressure Function.

= pressure drop at time and rate, = table pressure drop at time and rate = injection rate at time = table rate at time = simulation time

K t( ) α t( )

K t( )

ΔP ΔPiQQi-----⎝ ⎠

⎛ ⎞ αi=

ΔP t QΔPi t Qi

Q tQi tt


C.5 Conclusions 589

Figure C.4: Near Wellbore Pressure Loss Analysis.

C.5 Conclusions1. Far field multiple fractures can be modeled using the interaction factors given

in Eqns. (C-1) through (C-3). These fractures can be parallel or dendritic andmay or may not be interacting.

2. Far field multiple fractures may not result in a net pressure increase, dependingon their degree of interaction (see Figure C.1).

3. Near wellbore pressure loss is some function of rate and the total number of

near wellbore multiple fractures ( ). This resulting rela-tionship clearly matches observed field measurements and intuition. The power coefficient can range from less than unity for laminar flow to a value ofabout two for turbulent flow.

4. Clearly, Eqns. (C-14) to (C-17) cannot describe near wellbore tortuosity quan-titatively. Certainly, the number of multiple fractures or tortuous path lengthcannot be inferred.

ΔP t( ) K t( )Q t( )α t( )=α t( )


Appendix D

Fluid Loss

D.1 IntroductionFluid loss from the fracture to the formation is modeled by two mechanisms 1) lea-koff and 2) spurt loss. These leakoff and spurt loss coefficients are discussed below.

D.2 LeakoffThe rate of fluid loss to the formation is governed by the total leakoff coefficient,

. The three types of flow resistance mechanisms making up are: 1) - Lea-

koff viscosity and relative permeability effects, 2) - reservoir viscosity and

compressibility effects and 3) - wall building effects.

The fluid loss model options include specifying the total leakoff coefficient (Con-stant Model) or the coefficient and the components which comprise and CII(Harmonic or Dynamic Models). A detailed description of the components charac-terizing the Harmonic and Dynamic models is given below.

If a Constant leakoff model is selected, the total leakoff coefficient, , is entered inthe Fluid Loss Data screen. The total leakoff and spurt loss coefficients are theninput as a function of depth to characterize fluid loss in the fracture at differentintervals.

When either the Harmonic or Dynamic models are chosen, the filter cake coeffi-cient ( ) and reservoir diffusivity parameters must be input in the Fluid Loss

Data screen for each layer. The and coefficients are then calculated from theinput reservoir data and fracture propagation characteristics. The total leakoff coef-

C C CI

CII

CIII

CIII CI

C

CIII

CI CII


592 Fluid Loss:

ficient is calculated internally from the individual components as a function of dif-ferential pressure.

C - Total Leakoff CoefficientThe weighting of the individual leakoff coefficients for the Harmonic and Dynamicmodels is shown below:

CI - Coefficient

The or coefficient is used to simulate the effects of the fracturing fluid fil-trate viscosity and relative permeability. It is calculated from the following relation-ship:

where

CII - Coefficient

The reservoir fluid viscosity and compressibility effects are modeled using the

or coefficient:

= viscosity control coefficient, = differential leakoff pressure, psi= effective frac fluid filtrate permeability, darcy= porosity= effective viscosity of fracturing fluid filtrate, cp

Harmonic: )(

)(CC C C

C C C C C CI II III

I II II III I III

=+ +

(D-1)

Dynamic: ( ) ( )[ ]C

C C C

C C C C C C CI II III

I III I III II I III

=+ + +

2

42 2 2 2

(D-2)

CI Cv

C C 0.0469K p

I vf

f

= =Δ φ

μ (D-3)

CI ft min⁄ΔpKf

φμf

CII

Cc


D.2 Leakoff 593

where

CIII - Coefficient

The wall building or filter cake coefficient ( ) represent the inverse of fracfluid leakoff resistance through the filter cake. A value of zero (0.0) represents an

infinite filter cake resistance while a value of infinity (i.e., )represents no wall building effect. The wall building mechanism is calculated fromlaboratory data as follows:

where

= compressibility control coefficient, = differential leakoff pressure, psi= reservoir permeability to reservoir fluid, darcy= total formation compressibility, 1/psi= formation porosity= reservoir fluid viscosity, cp

= wall building coefficient, = cross-sectional area, cm2

= slope of volume versus square-root of time plot, ml/min1/2

For more information about the fluid leakoff models refer to SPE MonographVolume 12, Chapter 8.

C C 0.0374 p k c

II cr t

r= = Δ

φμ

(D-4)

CII ft min⁄Δpkrcφμr

CIII Cw

CIII CIII 100 ft/min1 2⁄>

C C0.0164 m

AIII w= = (D-5)

CIII ft min⁄Am


594 Fluid Loss:

D.3 Spurt LossSpurt loss is the “instantaneous” volume loss of fluid per unit area of fracture facethat occurs prior to the development of a filter cake. Generally, spurt loss occursonly in the pad volume. However, if the spurt time constant is of the same order asthe pump time, modeling spurt loss as instant mechanism (i.e., linear leakoff) canresult in inadequate modeling of the total fluid loss and fluid loss during closure(pressure decline) (See SPE Monograph Volume 12, Recent Advances in HydraulicFracturing, page 158-174).

The total fluid loss due to spurt is

where

For multilayer leakoff the spurt loss is calculated separately in each layer.

The value of the spurt loss coefficient is usually determined from the same labora-tory procedures used to measure the Wall Building Coefficient (see Figure D.1).The spurt in units of volume per unit area is equal to the vertical intercept from theprojection of the final slope, shown in Figure D.2, divided by the cross-sectionalarea of the core or filter paper used

= leakoff area (both wings)= spurt loss coefficient= leakoff volume due to spurt

V S Asp p= 2 (D-6)

ASp

Vsp


D.3 Spurt Loss 595

.

Figure D.1: Laboratory Apparatus.

Figure D.2: Laboratory Procedure for determining the Wall Building Coefficient and Spurt Loss of a Fluid.


596 Fluid Loss:


Appendix E

Wellbore Friction Factor

E.1 IntroductionThe wellbore friction factor is used to determine the energy dissipation (pressureloss) in the wellbore when the Empirical option is specified. The Empirical optionis an internal correlation for calculating the frictional pressure loss of Newtonianand non-Newtonian fluids. This option provides a combined correlation that isapplicable for a variety of fluids ranging from linear systems to highly non-Newto-nian and viscoelastic fluids that exhibit drag reduction due to slip or shear thinningduring turbulent flow.

E.2 Friction Factor ModelThree distinct types of behavior are possible with the combined correlation used inMFrac. These behaviors, summarized in the explicit expressions for the Fanningfriction factor outlined below, are given in Table E.1.

Table E.1: Fanning Friction Factors





1f

----- 19 Res f( )log 32.4–=

1f

----- A Res f( )log B+=

1f

----- 4 Res f( )log 0.4–=


598 Wellbore Friction Factor:

Through an iterative process, MFrac determines which correlation is most applica-ble in determining the friction factor. The criteria are based on the argument that

will always be greater than the Prandtl-Karman Law (lower bound) and lessthan Virk's maximum drag reduction asymptote (upper bound). Therefore, when thetransitional correlation developed by Keck, et al.2 reaches either the upper or lowerbound, it is automatically adjusted as shown in Figure E.1 to meet the above crite-ria.

Figure E.1: MFrac Pipe Friction Empirical Correlations.

When pipe roughness is included, by entering a value for the Relative Pipe Rough-ness in one of the Wellbore Hydraulics dialog boxes, the expression for friction fac-tor based on Prandtl's “Universal” Law is modified. An example of the modifiedcorrelation containing this additional parameter is shown below:

(E-1)

where

=== hydraulic diameter= fanning friction factor= flow behavior index= Reynolds number of solvent (e.g., water)= absolute pipe roughness

1 f⁄

1f

----- 4 Res f( ) 0.4– A 0.2εeRes f 1+( )log–log=

A 14.9n' 1.6– d0.13

B 53.9– n' 1.9– d0.27

dfn'Res

δ


E.2 Friction Factor Model 599

To include the effects of proppant concentration on friction, the program also has abuilt in correlation for slurry rheology. The relationship used, originally describedby Keck, et al., is

(E-2)

where

For laminar flow the Friction Factor Multiplier, , for proppant-laden fluids isequal to the value of . For proppant-laden fluids in turbulent flow, the expressionshown below is used to estimate the effect of proppant on pipe friction:

(E-3)

and

(E-4)

where

= relative pipe roughness,

The effect of pipe wall roughness is not applicable for Virk’s or Keck’s correla-tions because, by definition, they assume slip at the boundary.



εe δ d⁄

μr 1 0.75 e1.5n′ 1–( )e 1 n′–( )γ 1000⁄–[ ] 1.25φ1 1.5φ–-------------------+⎝ ⎠

⎛ ⎞ 2=

μrn'γφ

Mμr

Mf μr0.55ρr

0.45=

fs Mf fb=

Mfρbρr ρr ρs ρb⁄=ρsμrfbfs


600 Wellbore Friction Factor:

E.3 References1. Virk, P.S.: “Drag Reduction Fundamentals”, AIChE Journal, Vol. 21, No. 4,

July 1975.

2. Keck, R. et al.: “A New Method for Predicting Friction Pressures and Rheol-ogy of Proppant Laden Fracturing Fluids”, SPE Production Engineering, Feb.1992.



Appendix F

Minifrac Methodology

F.1 IntroductionMinifrac analysis provides a method of estimating fracture efficiency, closure pres-sure, dimensions and leakoff coefficients prior to designing a full scale fracturetreatment. These type of analyses, as originally formulated by Nolte1-5 quantify thefracturing process as estimated from the measured pressure decline data.

Most minifrac analyses are based on Nolte's equations and do not account for theeffects of fluid rheology or the conservation of momentum. The measured pressuredecline data is simply used in place of solving the momentum equation. Neglectingmomentum can result in unrealistic estimations of fracture characteristics and fluidleakoff coefficients that are critical to the design of the main fracture treatment.

Up until 1987, only the width-opening pressure relationship and pressure declinedata were used to estimate minifrac characteristics. Lee6 has recently improvedupon this by including Biot's energy balance equation for two-dimensional typefractures geometry models.

The energy balance method does eliminate some of the anomalies in minifrac anal-ysis. However, this method does not fully account for viscous driven fractures.

A new minifrac methodology was reported by Meyer and Hagel7-8 which solves theconservation of mass and momentum equations for power law type fluids. Themethodology utilizes 2-D fracture propagation equations-of-state. The solutiontechnique does not assume the fracture width is proportional to the measured pres-sure. Instead, the governing mass and momentum equations are coupled with themeasured closure time to predict fracture propagation characteristics. From thenumerically simulated fracture geometry's, pressures, efficiencies and leakoff coef-


602 Minifrac Methodology:

ficients, you can determine which fracture model more closely represents the mea-sure pressure response and formation permeability.

The main advantage of this technique is that mass and momentum are both satis-fied. In addition, the important effects of flowback, interference closure, timedependent leakoff and fluid rheology are simulated.

The numerical results are used in conjunction with the measured pressure declinedata to history match a number of fracture characteristics such as fracture height,pay zone height, Young's modulus and spurt loss. Closure time can also be moreaccurately estimated from these parametric studies.

F.2 Governing EquationsThe equations of mass conservation, continuity, width-opening pressure, momen-tum and constitutive relationships for fracture propagation models are formulatedbased on the methodology of Meyer9-11. Refer to these references and Appendix Afor a detailed description of the model assumptions and solution technique. A sum-mary of the formulated equations is presented below.

Conservation of Momentum The momentum equations for power law fluids in 2-D and 3-D type fracture geom-etries simplified by order-of-magnitude analysis are shown below:

x-component:

z-component:

Since the leakoff velocity perpendicular to the fracture face is an order-of-magni-tude less than the velocity of the fracture, the y-component of the momentum equa-tion reduces to .

∂∂Px

K qH

W x ta

f x

n=− ⎛

⎝⎜

⎞

⎠⎟ − ′+2 6

0 2 1

ΓΓ ( , , ) ( ) (F-1)

∂∂ γ

γPz

K qH W z ta

f z

n=− ⎛

⎝⎜

⎞

⎠⎟ − ′+2 6

0 2 1( , , ) ( ) (F-2)

∂P ∂y 0=⁄


F.2 Governing Equations 603

The z-component of the momentum equation is required for 3-D type fractures. Forradial or ellipsoidal fractures an equation similar to Eqn. (F-1) in radial coordinatesis used.

Equations (F-1) and (F-2) are generalized momentum equations used in the mini-frac analysis. The minifrac solution methodology, however, is applicable with anyset of fracture propagation relationships.

Width-Opening PressureThe width-opening pressure and width profile for GDK, PKN and radial type frac-tures are

GDK:

PKN:

Radial:

where and are the fracture widths at any position or. The maximum wellbore widths are given by and .

and

W tG

L t P t

W t W t

W( , ) ( ) ( ) ( , )

( , ) ( , )( )

0 2 1 0

0 1

0

2 12

=−

= −

γ γ

ξ ξ

Δ

and

W x tG

H P x t

W t W t

W

W

( , , ) ( ) ( , )

( , , ) ( , , )( )

0 2 1

0 0 0 1

0=−

= −

ΓΔ

Γ

γ

ξ ξ

(F-4)

and

W tG

R t P t

W t W t

W( , ) ( ) ( ) ( , )

( , ) ( , )( )

0 2 1 0

0 1

0

2 12

=−

= −

γ γ

ξ ξ

Δ

(F-5)

W ξ t,( ) W ξ 0 t, ,( ) ξ x L t( )⁄=ξ r R t( )⁄= W 0 t,( ) W 0 0 t, ,( )



Fracture Propagation SolutionThe momentum, mass conservation and width-opening equations are simplified bythe following transformations:

to form a set of equations in terms of the alpha parameters. The transformed equa-tions are then solved simultaneously to determine the fracture propagation charac-teristics.

The length, width, pressure and area propagation alpha parameters for a constantinjection rate and no spurt are

GDK:

PKN:

α α

α α

α

L w

a c

p

tL t

dL tdt

tW t

dW tdt

tA t

dA tdt

tC t

dC tdt

tP t

d P tdt

≡ ≡

≡ ≡

≡

( )( )

( , )( , )

( )( )

( )( )

( , )( , )

;

;

00

00

ΔΔ

(F-6)

α η α ηη

α αα α α α

α αα α

Lc

w L

p w L L

a L

L c

nn

nn n

=′ + + − −

′ + += ′ += = − ′ ′ +

=< < =

( )( ( ))( )

( )( )

.

1 1 2 12 1

11

1 2 2 3 0

-

where for

(F-7)

α η α ηη

α αα α

α αα α

Lc

w L

p w

a L

L c

nn

n

=′ + + − −

′ + += ′ +=

=< < =

( )( ( ))

( )

.

1 1 2 12 2

2 2

1 2 4 5 0

where for

(F-8)



Radial:

For radial shaped fractures, the radius propagation parameter is equal to .Leakoff is also only assumed to occur in the pay zone.

The leakoff coefficient alpha parameter is a measure of the time dependent fluidloss. If the total leakoff coefficient is only a function of the fluid loss, then

where is a constant which ranges from zero (0) to unity (1). If the net fracturingpressure, , is much less than the difference between the minimum horizontalstress and the reservoir pressure , is equal to zero. If the net pressure is much

greater than , is equal to 1/2 and 1 for dominant and resistance, respec-tively.

Mass ConservationThe total system mass balance for an incompressible slurry requires that the totalvolume of slurry injected minus the volume of slurry in the fracture and the volumeof fluid loss to the formation by leakoff and spurt loss be equal to zero:

α η α ηη ∂ η

α η αα α α α

α ∂ α∂

∂

αα α α

Lc

r

w L

p w L L

a r L

r p

r p

c

L a L p

nn n n

nn n

R HR H

R H

= ′ + + − −′ + + ′ + + ′ + −

= − ′ += − = − ′ ′ +

= += <

= >>

=< < = <

( )( ( ))( ) ( )( )

( ) ( )( )

( )

):

1 1 2 12 2 4 2 2 12 2 2

3 2 21

1 20 2

01 4 4 9 2 2

; ;

where (for and ; and ; 1 3 1 2 2< < = >>α α αL a L pR H

(F-9)

αR αL

αc

α ∂ αc = p p (F-10)

αp

ΔPΔPl

ΔPl CI CII

q d V t V t V tf l sp

t( ) ( ) ( ) ( )τ τ − − − =∫0

0 (F-11)



where

and is equal to 1/2 for square-root fluid loss. The equivalent leakoff coefficientis given by .

Equation (F-11) is applicable throughout the fracturing process and from shut-in toclosure. Since spurt is an instantaneous loss of fluid occurring when an element offracture leakoff area is initially created, spurt loss will not occur during closure, aslong as the fracture stops propagating.

The total leakoff area, , (one face of the fracture only) for the 2-D and radialshaped fractures is

2-D:

Radial:

where leakoff only occurs in the pay zone, .

Fluid Loss During PumpingThe total volume of fluid loss due to leakoff (excluding spurt loss) from Eqn. (F-12) is

[ ]

[ ]

V t Ct A

dAdt

V t S A t

A t A A t

leAt

sp p

a

( )( )

( ) ( )

( ) ( )

=−

=

=

∫∫2

2

00

1

τ

τ

α

α

τ

(F-12)

ατ

Ce

A t( )

A t H L tp( ) ( )= (F-13)

A t R t R t H p( ) ( ) ; ( )= ≤π2

22 (F-14)

A t H R t R t Hp p( ) ( ) ; ( )= >>2 2 (F-15)

Hp



where

and is the gamma function. For and , is equal to unity. The

leakoff scale constant is equal to one if .

The total leakoff coefficient is the instantaneous time dependent value. For aconstant leakoff coefficient independent of time and pressure, is equal to zero.

The range of values for 2-D and radial models is

GDK:

PKN:

Radial:

The above coefficients are based on a constant leakoff coefficient

( ).

( )V t

C A dd

C t A t t

let

a c

a( )

( )

( ) ( ) ( , )

=−

=

∫ ∫210 10

1ξξ

λ

λξ

π α α

α α ατ τ

Φ

(F-16)

( )

α α

β β

β α αα α ψ α α β

ψ α α θ α α α α

θ α αλ

λ

α

τ

α α

α

π

α ατ

= −

= =

=

= + +== + +

≡−

=

− −

∫

1 2

1

1

12

12

2 12

10

1

c

e c p p c

p p

c c c

a c a c c

a c a c a c

a c

a

C C t t C t t

C t C t t t

d

c c

c

a

( )( ) ( )( )

( ) ( )( )( ) ( ) ( )

( , ) ( , )( , ) ( , ) ( )

( , )

(

Γ Γ ΓΦ

Γ

+ ++ +1 1

21

2

) ( )( )

ΓΓ

αα α

c

a c

Γ αa 1 2⁄= αc 0=

βc αc 0=

C t( )

αc

Φ αa 0,( )

09411 Φ αa 0,( ) 1≤<

0.9008 Φ α( a 0 ) 1≤,<

1 Φ α( a 0 ) 1.0713 R t( );<,≤ Hp 2⁄<

0.8760 Φ αa 0,( ) 1 R t( ) Hp 2⁄»;≤≤

Φ αa 0,( )

αc 0=



Mass Conservation After Shut-InAfter pumping, the fracture is assumed stop propagating. The fluid loss volumeafter shut-in, from Eqn. (F-12) is

(F-17)

where

(F-18)

During closure, square-root fluid loss with a variable leakoff coefficient isassumed. This assumption is generally accurate enough for engineering purposesand flexible enough for modeling most leakoff situations during shut-in. Althoughother non-square-root forms of Eqn. (F-17) may be more rigorous mathematically,they result in incomplete beta functions and can only be evaluated for a limitednumber of αa ατ and αc2 values.

The effective leakoff coefficient is the time dependent value during shut-in.

is equal to at the end of pumping where is a constant. The lea-

koff alpha parameter after pumping is negative for a decreasing coefficient

with time. Typically, ranges from 0 to -1/2.

The dimensionless total time is defined as

where is the Nolte shut-in time .

ΔVι

[ ]22

21

2

),,()()(2

)(2)(

0 0

ccappp

t A el

GttAtC

dAdtAt

CtV

βθαα

τ

=

−= ∫ ∫

( )2

2

2

21

22

))((

1),,(

1 0 1

cppe

cca

c

a

a

ca

tttCC

ddG

β

ξλ

λξβθαα

α

θ ξ

α

ααα

=

−= ∫ ∫

−−+

Ce2

Ce2C tp( )( ) βc2

⁄ βc2

αc2

αc2

θ

θ ≡

= +

t tt

p

D 1

(F-19)

tD tD t tp⁄ 1–=



Eqn. (F-18) is a function of the incomplete beta function and, therefore, is only ana-lytically integratable for specific values of and . functions for various val-

ues of and are given below.

(F-20)

Although, is similar to Nolte's function, it is evaluated at

and rather than at fracture efficiency. The relationship between the Nolte

GNolte and the Meyer function is

Since is reported for a number of different and values, other

functions can easily be interpolated by fitting a cubic spline

through the analytic formulae in Eqn. (F-20). Because is nearly

linear in , the following expression is a good approximation:

αa αc2

αa αc2

{ }{ }

{ }{ }

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−−+

+−++−=+

−+−−=

−−−−++−=−

−−−−++−=+

−−−=

−−−++−=−

−++−++=+

−−+=

−−++=−

+−=

=−

−

−

−

+

)232()()1)(22(163

)1(83198),,2(

)1)(23(11516),0,2(

)21()()1(23134),,2(

)21()()1(21),,1(

))1(1(34),0,1(

)()1(12),,1(

()1(231),,(

2)1(),0,(

)1(22),,(

)21()1(),,0(

),,0(

212212

213

21

2325

21221221

21221221

2323

2122121

21221212321

21

212121

21212121

21

2121

2

2

2

θθθθθθ

θθθθ

θθθθ

θθθθθθθ

θθθθθθθ

θθθ

θθθθθθ

πθθθθθθθ

πθθθθ

πθθθθθ

αθθα

θθα

e

e

e

e

e

e

e

cc

e

Log

LogG

G

LogG

LogG

G

LogG

LogArcsinG

ArcsinG

LogArcsinG

G

LogGc

),,(2

θαα caG )( DtGαa αc2

G

G Ga Nolte( , , )α θ π04

=

(F-21)

G αa αc2

),,(2

θαα caG

),,(2

θαα caGαa



Figure F.1 shows the function for various values of with

. Generally, and bound the range of expected

values. The bounds of G are shown to differ by less than about ten

percent (see also Nolte13). Figure F.1 also illustrates that is approxi-

mately a linear function of for all values. This result can be used by plot-

ting pressure decline data as a function of for estimating closure time asdiscussed below.

Figure F.1: Fluid Loss Function - Constant Leakoff.

Figure F.2 shows as a function of for three values of . As

illustrated for an increasing leakoff coefficient , the fluid loss rate

{ }{ }G G G

Ga c c c a

c

( , , ) ( , , ) ( , , )

( , , )

α α θ α θ α θ α

α θ2 2 2

2

1 2 112

12

≅ − −

+ (F-22)

),,(2

θαα caG αa

αc20= ),0,( 2

1 θG ),0,1( θG

),0,( θαaG

),0,( θαaG

21θ aα

21θ

),,(22

1 θαcG θ 2cα

)21(2

+=cα



increases; whereas, for a decreasing leakoff coefficient with time ,

the fluid loss rate decreases. Consequently, the effect of a pressure dependent lea-koff coefficient can be a major factor in modeling fluid loss during shut-in.

Figure F.2, also illustrates that the fluid loss volume function isnot a linear function of if is not zero. This non-linear effect can result in errone-ous estimation of the closure time and minimization of an inflection point at clo-sure.

Figure F.2: Fluid Loss function - Time Dependent Leakoff.

Minifrac Closure EquationsMass conservation during shut-in (Eqn. (F-11)) requires that the fracture volume

during shut-in be equal to the fracture volume at the end of pumping

minus the volume of fluid loss after pumping due to leakoff and

flowback :

αc21 2⁄–=

),,(2

θαα caG

),,21(2

θαaG

)(tVf

)( pf tV lVΔ

)(tV fb



The flowback volume is given by

where is the flowback rate and is the time when flowback is initiated

. For a constant flowback rate:

The fracture volume at the end of pumping is

where is the fracture efficiency at the end of pumping and is the averagepumping rate.

From mass conservation the total fluid loss by leakoff (excluding spurt loss) at theend of pumping is

The ratio of fluid loss by leakoff to the fracture volume at the end of pumping fromEqns. (F-27) and (F-26) is

V t V t V t V tf f p l fb( ) ( ) ( ) ( )= − −Δ (F-23)

V t q dfb fbt

t

fb

( ) ( )= ∫ τ τ (F-24)

fbq fbt

)( cfbp ttt ≤≤

fbfbfb

fbfb

ttttqtttV

>−=

≤=

; )( ; 0)(

(F-25)

V t q tf p o p( ) = η (F-26)

η oq

V t q t V tl p o p sp p( ) ( ) ( )= − −1 η (F-27)

V tV t

l p

f p

( )( )

=− ∗

∗

1 ηη

(F-28)



where is the fracture efficiency after spurt loss. This efficiency is defined as

The ratio of fluid loss during shut-in and fluid loss volume at the end of pumpingfrom Eqns. (F-17) and (F-16) is

The fracture volume ratio during shut-in as a function of time is found by substitut-ing Eqns. (F-25), (F-26), (F-28) and (F-30) into Eqn. (F-23):

where .

The final propped fracture volume at closure is

where is the propped fracture fraction due to interference closure or proppant

in the fracture.

Substituting Eqn. (F-32) into (F-31) and rearranging results in the closure condi-tion:

where is the dimensionless closure time.

∗η

[ ]η η∗ ≡ −1 V t q tsp p o p( ) (F-29)

ΔV tV t

Gl

l p

a c

a cc

c

( )( )

( , , )

( , )=

α α θ

ψ α απ ββ

2

22

(F-30)

V tV t

qq

Gf

f p

fb

ofb

a c

c

( )( )

( )( , , )( )

= − − −− ∗

∗1 1 12

2ηθ θ

α α θ ηγ ηπ

(F-31)

pfbfbcaccc tt== θααψββγ and ),( 2

V t V tf c p f p( ) ( )= φ (F-32)

pφ

Gqqa c c c p

fb

oc fb( , , ) ( )α α θ

πγ

ηη

φη

θ θ2 2 1

11

=−

− − −⎛

⎝⎜

⎞

⎠⎟

∗

∗ (F-33)

pcc tt=θ



Equation (F-33) is the governing closure equation which couples closure time to thefracture propagation solution.

The Nolte G function at closure for no flow back or propped width from Eqn. (F-33) and (F-21) is

where can be approximated as unity for most engineering applications, if theleakoff coefficient is a constant and the fracture efficiency is low.

Assuming the fracture closes in a self similar manner, pressure decline during shut-in can be calculated from Eqn. (F-31) with

If the fracture does not close in a self similar manner, the net wellbore pressure at

the end of pumping, in Eqn. (F-34), should be replaced by an average net

fracture pressure. For non-similar closure, the bottomhole pressure may initiallydecrease faster than Eqn. (F-34) would predict; however, as time progresses pres-sure decline will follow Eqn. (F-31). Therefore, minifrac analyses based on onlymeasured pressure decline data may not satisfy Eqn. (F-31), if Eqn. (F-34) does notapply.

Dimensionless Net Pressure SlopeThe dimensionless net pressure slope is determined from mass, momentum,elasticity and constitutive relationships. Differentiating Eqn. (A-1) and the elastic-ity equation (Eqn. (A-7)) with respect to time and rearranging results in the dimen-sionless pressure slope ( ):

η*GNolte θc( )

2γc GNolte θc( )+----------------------------------------=

γc

ΔΔ

P tP t

V tV tp

f

f p

( )( )

( )( )

= (F-34)

)( ptPΔ

αp

αp



Eqn. (F-35) is a generalized net dimensionless pressure slope ( ) equation whichrepresents how various parameters affect the fracture pressure. The alpha parame-ters in Eqn. (F-35) are numerically calculated in the simulator. Eqn. (F-35) can alsobe used to illustrate fracture pressure responses for special cases.

A specialized case of Eqn. (F-35) is pressure decline during closure. If the fracture

is assumed to stop propagation (with alpha parameters ),the dimensionless pressure slope for no spurt loss is

where

Assuming the fracture closes in a self similar manner, the dimensionless net pres-

sure decline is

[ ]LH

csp

ca

p

wv

VV

tqtq

dttPd

tPtt

αβαβαα

ααη

αααηη

α

λλ )2()1(

)(

))(1()()(1

)()(

)(

21

21

−++++−

+++

⎥⎦

⎤⎢⎣

⎡+++−−=

ΔΔ

≡

ΓΓ

Φ

Φ (F-35)

αp

0,,,, =Γααααα caHL

)()1()( 21

Φ+−

−= αη

ηα tp (F-36)

[ ]

[ ])(),(

2)(),()(

)( )(

)(

),,(),( )()(

=)(

2

22

4

21

21

21

2

θααθθααθθ

θθ

θθα

θθαααα

πθ

π

ππ

π

GGG

dd

GttAtC

VV

ca

ca

caca

ll

+Φ+Φ−′

=

ΦΦ

=

+Φ=

Δ+Φ

Φ

DPΔ



where .

From Eqn. (F-37), the pressure decline is shown to be proportional to the G func-tion:

The dimensionless square root of time net pressure decline slope is

Substituting into Eqn. (F-39) and neglecting second order effects, the rate of

pressure decline slope as a function of is

where is the fracture efficiency at the end of pumping.

Equation (F-40) illustrates that if the fracture decline pressure during closure isplotted as a function of the square root of time or on a semi-log axis a straight linewill result with a constant slope equal to Eqn. (F-40). Equation (F-40) can also be

derived by differentiating Eqn. (F-37) with respect to where

(see Meyer7 et al.).

)−

Φ−=Δ θαα

ηη

ααπθ ,,(

)()(1

),(21)(

2cap

p

caD G

tt

P (F-37)

)1()()( e =ΔΔ≡Δ θθθ PPPD

p

P P Gf f a c( ) ( ) ( , , )θ θ α α θ= − ∝12

(F-38)

αθ

α θ

ΔΔ

Δ

PD

p D

D

d Pd

P

=

= −

12

122

(F-39)

pα

21θ

α πα α

ηηΔ ΦP

a c

p

pD

tt

=−4 1

( , )( )

( ) (F-40)

)( ptη

21θ

)1(2),,( 21

2−= θθαα caG



Pressure Decline

Pressure decline responses during closure for various parameters as a function

of and are shown in Figure F.3 and Figure F.4, respectively. The pres-

sure decline curves are reported for no spurt loss,

.

Figure F.3: Dimensionless Pressure Decline vs. Square-Root Total Time.

2cα

21θ 2

1

Dt

1 and , 21

21 === ca γαη



Figure F.4: Dimensionless Pressure Decline vs. Square-Root Shut-In Time.

Figure F.3 and Figure F.4 show that for a decreasing fluid loss coefficient during

shut-in , the pressure declines at a slower rate than for an increasing

coefficient, , as expected. This illustrates that the effect of a time

dependent fluid loss coefficient can have a significant impact on determining theclosure time based on an inflection point or deviation from square-root pressuredecline. This is generally true for a decreasing pressure decline profile, since aunique straight line may not fit the data.

It is also observed that the pressure plotted as a function of is approximately

linear if . Consequently, plotting the decline pressure as a function of

will generally yield a more prominent inflection point at closure than if the

data is plotted versus the square-root of shut-in time ( ). This is especially trueif closure is relatively fast (i.e., ; see Figure F.4).

)( 21

2−=cα

)( 21

2=cα

21θ

)0(2

=cα

21θ

21

DttD 1<


F.3 Minifrac Numerical Solution 619

F.3 Minifrac Numerical SolutionThe governing minifrac equations of mass and momentum conservation are solvediteratively to match the measured closure time. The numerical procedure is basedon solving the equations-of-state for a given leakoff coefficient.

The following procedure assumes the total leakoff coefficient is to be calculatedfrom a measured closure time. If the total leakoff coefficient is known, the proce-dure is non-iterative and closure time is found directly from Eqns. (F-33) and (F-18).

The purpose of the numerical solution is to minimize the object function :

where is the measured closure time and is the predicted closure time.

With this objective defined the numerical procedure is outlined below:

Step (1)

Given the equation and initial approximations to the leakoff coeffi-

cients and with .

Step (2)

Update the total leakoff coefficient using the Secant method:

A Bisection Algorithm in logarithmic C space is used for the first few iterations

with and having opposite signs.

)(Cf

f C cm

cp

cm( ) ( )= −θ θ θ (F-41)

mcθ p

cθ

0)( =Cf

0C1C 10 CC ≠

2 =iSet

)()())((

21

2111

−−

−−−− −

−−=

ii

iiiii CfCf

CCCfCC

)( 0Cf )( 1Cf



Step (3)

Solve the governing mass and momentum equations during fracture propagationbased on to determine:

a) Fracture characteristics

b) Alpha parameters

Step (4)

Solve the minifrac closure Eqn. (F-33) with Eqn. (F-18) for to

determine the predicted closure time .

Step (5)

Calculate the error, , between the predicted and measured closure timesfrom Eqn. (F-41)

If is less than a specified tolerance go to Step (6). Otherwise, add one to iand go to Step (2).

Step (6)

When the measured and predicted closure times are within the specified tolerance,the procedure is complete. The numerical procedure outlined above usually con-

verges in less than ten iterations for initial values of and

.

After convergence, the formation permeability is calculated from the total leakoffcoefficient at the end of pumping using the following relationship:

1C

...),,,( ηPWL Δ

...),,,( apwL αααα

),,(2

θαα caG

)( ip

c Cθ

)( iCf

mci

pc

mci CCf θθθ ))(()( −=

)( iCf

80 10−=C

minft 0.11 =C


F.4 Conclusions 621

In the formulation above is the leakoff pressure differential expressed in psi,

k in darcies, is the formation compressibility in 1/psi. The reservoir viscosity,

, is in centipoise. The equivalent leakoff viscosity, , is equal to zero if the

frac fluid leakoff viscosity, is equal to (if >> , = ).

Formation permeability is solved directly from Eqn. (F-41) since it is the onlyunknown. A dynamic weighting of the individual leakoff coefficients, in place ofthe harmonic weighting shown in Eqn. (F-41) can also be used to calculate perme-ability.

F.4 ConclusionsPressure analysis during and after a fracture treatment is primarily performed toestablish fracture characteristics and critical parameters governing fracture propa-gation. Minifrac analysis is a first step in providing an estimate of fracture geome-try, efficiency, leakoff and possibly height growth prior to designing the fracturetreatment.

Since fracture efficiency is virtually independent of the fracture model; using a lea-koff coefficient based on a measured net pressure or pressure decline can result in adifferent design fracture efficiency. The reason for this is that most minifrac analy-ses only determine the leakoff coefficient and area product. Satisfying momentumand matching the fracture pressure responses provides the analyst with a great dealmore information on the fracture geometry and sensitive parameters.

Consistency in solving the governing equations is essential in any type of historymatching or simulation. If a given method is used to calculate the leakoff coeffi-

1 1 1 1C t C C Cp I II III( )

= + + (F-42)

constant a0374.0

0469.0

where

=

Δ=

Δ=

III

rtlII

elI

CCkPC

PkC

μφ

μ

lPΔ

tC

rμ eμ

lμ rμ lμ rμ eμlμ



cient from measured data, a consistent method must be used to design the fracturetreatment using this data. The fracture efficiency should match the minifrac andmain treatment design at the end of the minifrac.

Due to the large number of uncertainties, history matching field data may not resultin a unique solution if only part of the data is used. Often the problem with historymatching is that unverifiable and unrealistic data is used to force the solution tomatch at some point in the pressure response. History matching throughout thestimulation thereby eliminates many of the erroneous matches and conclusionsencountered by only using a few selected measured data points. Using logs to ver-ify fracture height and fracture characteristics are very useful in establishing realis-tic input parameters.


F.5 Nomenclature 623

F.5 Nomenclature= Leakoff area (one face of the fracture)

= Total reservoir compressibility

= Equivalent leakoff coefficient

= Leakoff viscosity control coefficient

= Reservoir compressibility and viscosity coefficient

= Wall building coefficient

= Young's modulus

= Fluid loss function, Eqn. (F-18)

= Fracture half-height

= Pay zone height

= Total wellbore height

= Permeability

= Apparent consistency index

= Fracture half-length

= Flow behavior index

= Pressure

= Fracture closure pressure

= Reservoir fluid pressure

= Flowback rate

= Injection flow rate

= Radial coordinate

= Fracture radius

= Spurt loss coefficient

A

Ct

Ce

CI

CII

CIII

E

G θ( )

H

Hp

Hw

k

Ka

L

n′

P

Pc

Pr

qfb

qo

r

R

Sp



Greek

= Time

= Closure time,

= Dimensionless Nolte time,

= Flowback time,

= Fracture volume

= Fluid loss volume (no spurt loss)

= Volume loss by spurt

= Fracture width

= Average wellbore fracture width

= Fracture width as a function of position

= Lateral coordinate along fracture length

= Coordinate perpendicular to frac face

= Vertical Coordinate

= Leakoff area parameter, Eqn. (F-6)

= Leakoff parameter during pumping, Eqn. (F-6)

= Leakoff parameter during shut-in

= Length propagation parameter, Eqn. (F-6)

= Pressure parameter, Eqn. (6)

= Width propagation parameter, Eqn. (F-6)

= Leakoff scale factor during pumping

= Leakoff scale factor during shut-in

= Parameter defined in Eqn. (F-31)

t

tc

tD

tfb

Vf

Vl

W

W 0 t,( )

W ζ t,( )

x

y

z

αa

αc

αc2

αL

αp

αw

βc

βc2

γc


F.5 Nomenclature 625

Subscripts

= Friction coefficients

= Pressure profile coefficients

= Width profile coefficients

= Maximum width-opening pressure coefficient

= Radial coefficient , Eqn. (F-9)

= Pressure coefficient, Eqn. (F-10)

= Net fracturing pressure

= Leakoff pressure differential

= Fracture efficiency

= Efficiency excluding spurt loss

= Dimensionless time,

= Dimensionless closure time

= Dimensionless flowback time

= Integration indices

= Dimensionless lateral coordinate

= Time of fracture leakoff area creation

= Propped fracture volume fraction

= Fluid loss parameter, Eqn. (F-16)

= Fluid loss parameter, Eqn. (F-16)

= Minimum horizontal stress

2 = Shut-in

c = Closure, leakoff parameter

D = Dimensionless

γf Γf,

γp Γp,

γw Γw,

γwoΓwo

,

∂r

∂p

ΔP

ΔPl

η

η*

θ

θc

θfb

λ ξ,

ξ

τ

φp

Φ

ψ

σHmin



F.6 References1. Nolte, K.G., Smith, M.B.: “Interpretation of Fracturing Pressures”, SPE 8297,

Sept. 1979.

2. Nolte, K. G.: “Determination of Fracture Parameters from Fracture PressureDecline,” SPE 8341 presented at the 54th Annual Technical Conf., Las Vegas,Sept. 1979.

3. Nolte, K.G.: “Fracture Design Considerations Based on Pressure Analysis”,SPEPE, Feb. 1988, pp 22-30.

4. Nolte, K. G.: “A General Analysis of Fracturing Pressure Decline With Appli-cation to Three Models,” (SPE 12941) JPT (Dec. 1986), 571-582.

5. Nolte, K. G.: “Application of Fracture Design based on Pressure Analysis,”SPEPE (Feb. 1988), 31-41.

6. Lee, W.S.: “Study of the Effects of Fluid Rheology on Minifrac Analysis,”SPE 16916, Sept. 1987.


8. Hagel, M. W. and Meyer, B. R.: “Utilizing Mini-Frac Data to Improve Designand Production,” CIM paper 92-40 June, 1992.

9. Meyer, B. R.: “Frac model in 3-D - 4 Parts,” Oil and Gas Journal, June 17,July 1, July 22 and July 29, 1985.

10. Meyer, B. R.: “Design Formulae for 2-D and 3-D Vertical Hydraulic Frac-tures: Model Comparison and Parametric Studies,” paper SPE 15240 presentedat the SPE Unconventional Gas Technology Symposium, Louisville, KY,May. 18-21, 1986.

f = Fracture

fb = Flowback

l = Fluid loss

p = Pay zone, end of pumping

sp = Spurt loss


F.6 References 627



Appendix G

Production Model Theory

G.1 IntroductionThis appendix provides a brief overview of the computational methods used inMProd and highlights some of the program's capabilities. The equations and rela-tionships employed are listed and the variables used are defined. In addition, a listof references is furnished at the end of the chapter. Refer to references1-10 foradditional information regarding the simulation methodologies used.

Three distinct analytical models are contained in MProd for performing productionsimulations on different types of reservoirs. The basic reservoir production model(i.e., No Fracture) is based on the transient behavior of a well in a closed (finite)reservoir. The method of images is used to generate rectangular drainage shapes inclosed formations. Drainage area aspect ratios can be as large as 100.

The hydraulically fractured models employed are based on a finite conductivityvertical fracture. The analytical method used was developed by coupling the shorttime solution of Lee and Brockenbrough7 (1986) with the well known semi-logasymptotic (pseudo-radial) solution that has been publish by many authors. Thesemodels are applicable for all flow regimes from linear, bilinear, trilinear to pseudo-radial.

All of the models used in MProd allow either a rate or pressure boundary conditionat the well. Using the principle of superposition, a series of rates or pressures maybe specified. For closed systems, the concept of desuperposition has been incorpo-rated to simulate the effects of wellbore and fracture skin on the production.

Available for use with either gas or liquid, the numerical solutions have beenimproved over earlier versions (version 3.0 or before) for low conductivity frac-tures. The transition from trilinear to pseudo-radial flow has increased accuracy for


630 Production Model Theory:

very low values of ( ). The transition is now calculated in Laplacespace eliminating the need to specify time steps that are sufficiently small at thematch point. Additional information concerning the analytical techniques used iscontained in the following sections of this appendix.

G.2 Governing EquationsThis section presents the governing equations for simulating production from frac-tured and unfractured wells in closed and infinite reservoirs. When appropriate thesolution methodology is also given.

Dimensionless ParametersThe dimensionless and Laplace parameters used in the production models aredefined as follows:

FCD FCD 0.001>


G.2 Governing Equations 631

( )p

kh p pqBwD

i wf

o=

−

141 2. μ ; for liquids

( ) ( )[ ]p

kh m p m p

qTwD

i wf=

−

1424 ; for gas

( )[ ]qS p SD

wD

=1

2

t t rA

DADw w=

2

t ktc XDw

t f

=0 000264

2

.φμ

FK wK XCD

f

f

=

C Cc hrD

t w

=5 161462 2

.πφ

C C rXDfD w

f

=2

2

C K cK cF

t

f f t f

=φφ

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛= 1

lf

sLee K

KXys

aFCD

=2

bFCD

=− π

R rr

ewwa

w

s= = −

(G-1)



where is the Laplacian operator.

The true fracture skin, , is related to the Lee et. al. skin, , defined above by

.

PseudopressureThe real gas potential or pseudopressure is defined as

where is an arbitrary base pressure and is the real gas deviation factor.

The real gas pseudopressure equation can be simplified for certain pressure ranges.At low pressures is essentially constant, while at higher pressures it is directlyproportional to pressure (Earlougher2, 1977).

At low pressure (p < 2000 psi), the real gas pseudopressure based on a constant product is

where and are initial condition gas properties at .

At high pressures (p > 3000 psi), the real gas pseudopressure based on a constant product is

S

Sf SLee

Sfπ2---SLee=

dppZp

ppmp

pb∫=

)()(2)(

μ

(G-2)

bp Z

μZ

μZ

( )Z

pppm b

μ

22

)( −=

or

( )ii

wfii Z

pppm

μ

22

)(−

=

(G-3)

iμ iZ ip

p μZ( )⁄



When the fluid type is gas and the Internal PVT correlations are not used, the modelrequires that a table of μ and Z be entered as a function of pressure. Thepseudopressure function is then automatically used to calculate the dimensionlesspressure.

Trilinear Solution

The trilinear solution given by Lee et al.6, (1983) for a fractured well in Laplacespace is

Constant Flow Rate

Constant Pressure

The Ω parameter used in the equations above is defined by:

Zppppm b

μ)(2)( −

=

or

ii

wfiii Z

ppppm

μ)(2

)(−

=

(G-4)

p S bCoshS SbC Cosh SinhwD

Df

( )( )

=−

ΩΩ Ω

(G-5)

q SS p SD

wD

( )( )

=1

2 =−SbC Cosh Sinh

SbCoshDf

Ω Ω

Ω (G-6)

( )( )

Ω =+

+ ++

a S S

S S SC S

f

f

12

12

12

121

(G-7)



This analytical solution is based on transforming the equations above from Laplacespace to real time using the Stehfast inversion algorithm9 (1970).

Pseudo-Radial Flow SolutionThe nondimensional pressure drop at the wellbore, for an unfractured well, givenby Earlougher2 (1977) is

where the exponential integral, , is defined as follows:

This solution is also used for the pseudo-radial solution of the fractured well bymatching the pseudo-radial and trilinear solutions.

Productivity IncreaseThe productivity increase as defined by the ratio of the productivity indices for thefractured and unfractured wells as given by Meyer8 (1985) are given below.

Constant Flow Rate

p E twD i Dw= −0 5 0 25. ( . ) (G-8)

iE

E xe

dix

( )− = −−∞

∫μ

μμ (G-9)

JJ t

ppo

wD

wD

o

f

( ) = (instantaneous)

JJ t

p dt

p dto

wD

t

wD

to

f

( ) =∫∫

0

0

(average)

(G-10)



Constant Pressure

where f refers to the fractured well and o to the unfractured case. For unfracturedreservoirs the subscript f refers to the well with no skin. When considering a seriesof constant rate or pressure changes, the productivity parameters outlined above areequal to the equivalent values calculated by superposition.

DesuperpositionThe concept of desuperposition has been illustrated by Gringarten, Ramey and

Raghavan3 (1974) for modifying known values of to dimensionless pressuresdescribing somewhat different systems. This method is used to calculate the effectof skin and fractures in closed systems. For closed systems, the dimensionless pres-

sure, , is found from the following relationship:

In the above equation is the dimension pressure for a closed system and is the dimensionless pressure for an infinite (unbounded) reservoir. The first termon the right side of the equation is for the closed system with zero skin and zero

wellbore storage. for a single well in an infinite system with zero skin and

wellbore storage is subtracted from this dimensionless pressure. for a singlewell in an infinite system with the desired wellbore storage and skin factor is thenadded.

JJ t

qqo

D

Do

f( ) = (instantaneous)

JJ t

q dt

q dto

D

t

Do

tf( ) =

∫∫

0

0

(average)

(G-11)

Dp

Dp

p C S p C S p C S p C SD D D D D D D Dx x( , ) ( , ) ( , ) ( , )= = = − = = +0 0 0 0 (G-12)

DpxDp

xDp

xDp



Method of Images - Generate Closed SystemsThe method of images is used to determine the pressure distribution in closed rect-angular formations. The drainage shapes considered are generated by adding a reg-ular infinite pattern of image wells to the actual well in an unbounded homogenousreservoir. If each well is a producer, a closed drainage area is defined.

The dimensionless pressure drop, , at the well is given by:

where . Here it is assumed that the response of each well is given

by the line source solution and is the location of the ith image well for

.

The number of image wells to approximate the infinite series simulation is based onthe analytical and numerical studies of Larsen4 (1985).

The summation, in terms of a single well rectangle, is carried out by adding

columns on each side in the x direction and rows in the y direction, where

In the equations above the reservoir aspect ratio, F, is given by

where and are the reservoir half length and half width, respectively. Fora fractured reservoir, the aspect ratio should be greater than or equal to one (i.e., thefracture is assumed to run along the long axis).

Dp

( )p t E

x yAtD D i

i

i i

DA

A

( ) .= −− +

=

∞

∑0 541

2 2

(G-13)

222wii ryx =+

),( ii yxi 2≥

cn

rn

n t Fc DA= +2 1 4 1

2[ ( ) ]

and

])(41[2 21

ADr tFn +=

(G-14)

HL YXF =

LX HY



The well location at position within the closed rectangle is shown in Fig-ure G.1

Figure G.1: Well Location.

The dimensionless well coordinate given by and

is shown in Figure G.2.

Figure G.2: Dimensionless Well Coordinates.

The dimensionless coordinate and is the geometric center ofthe rectangle.

For long periods of production, the closed system will reach a pseudo-steady condi-tion. The dimensionless pressure at times greater than the pseudo-steady state time

is given by the following:

),( 11 YX

),( 11 DD YX LD XXX 11 =

HD YYY 11 =

01 =DX 01 =DY

DASSt



No Fracture:

Fracture:

where is the dimensionless time, A is the drainage area and is the shapefactor used.

It can be shown that the dimensionless rate solution isin the form of a decaying exponential for times greater than the pseudo-steady state

time .

p tA

r CD D ew

eA

A= +

⎛⎝⎜

⎞⎠⎟ +

⎛⎝⎜

⎞⎠⎟2 0 5 0 5

2 24562π . log . log

. (G-15)

p t f A X C FD D f A CDA= +2π ( , , , ....), (G-16)

ADt AC

))(1)(( 2 SpSSq DD =

DASSt


G.3 Nomenclature 639

G.3 Nomenclature= Reservoir drainage area, ft2

= Propped fracture width, ft

= Formation volume factor, RB/STB

= Total reservoir compressibility, psi-1

= Fracture compressibility, psi-1

= Wellbore storage coefficient, RB/psi

= Dimensionless inverse fracture diffusivity

= Dimensionless wellbore storage coefficient

= Dimensionless fracture storage coefficient

= Reservoir aspect ratio

= Dimensionless fracture conductivity

= Pay zone height, ft

= Productivity index

= Equivalent reservoir permeability, md

= Propped fracture permeability , md

= Fracture damaged zone permeability, md

= Fracture conductivity, md-ft

= Propped fracture half-length, ft

= Initial real gas pseudopressure, psi2/cp

= Flowing real gas pseudopressure, psi2/cp

= Initial reservoir pressure, psi

= Dimensionless wellbore pressure

A

w

Bo

ct

ctf

C

CF

CD

CDf

F

FCD

h

J

K

kf

kl

kfw

Xf

m pi( )

m pwf( )

pi

pwD



Superscripts

Subscripts

= Bottomhole flowing pressure, psi

= Flow rate, bpd or Mcf/d

= Dimensionless flow rate

= Wellbore radius, ft

= Apparent wellbore radius, ft

= Dimensionless apparent wellbore radius

= Laplace space variable

= Wellbore skin factor

= Fracture skin factor

= Time, hours

= Dimensionless time based on drainage area

= Dimensionless time based on fracture length

= Dimensionless time based on wellbore radius

= Reservoir temperature, °R

= Damaged zone adjacent to fracture, ft

= Real gas deviation factor

= Equivalent reservoir viscosity, cp

= Equivalent reservoir porosity

- = Time averaged

D = Dimensionless

f = Fracture

pwf

q

qD

rw

rwa

Rw

S

s

sfracture

t

tDA

tDf

tDw

T

ys

Z

μ

φ


G.4 References 641

G.4 References1. Beggs, H.D. and Robinson, J.R.: “Estimating the Viscosity of Crude Oil Sys-

tems” JPT (Sept. 1975) 1140-1144.

2. Earlougher, R.C., “Advances in Well Test Analysis,” SPE of AIME, NewYork, 1977.

3. Gringarten, A.C., Ramey, J., and Raghaven, R., “Unsteady-State Pressure dis-tributions Created by a Well with a Single Infinite Conductivity Vertical Frac-ture,” SPEJ (August 1974) 347-360.

4. Larsen, L., “A Simple Approach to Pressure Distributions in GeometricShapes,” SPEJ (Feb. 1985), Vol 25, No. 1 pp. 113-120.

5. Lee, A.L. Gonzales, M.H. and Eakin, B.E.: “The Viscosity of Natural Gases”Trans., AIME (1966) 997-1002.

6. Lee, S.T. and Brockenbrough, J.R., “A New Analytical Solution for FiniteConductivity Vertical Fractures with Real Time and Laplace Space ParameterEstimation,” SPE12013, 1983.

7. Lee, S.T. and Brockenbrough, J.R.: “A New Approximate Analytic Solutionfor Finite-Conductivity Vertical Fractures,” SPEFE (Feb. 1986) 75-88.

8. Meyer, B.R., “Frac Model in 3D”- Parts 1-4, Oil and Gas Journal, June 17, July1, July 22 and July 29, 1985.

9. Stehfast, H., “Numerical Inversion of Laplace Transforms,” Communicationsof the ACM, Jan., 1970, Vol. 13, pp. 47-49.

10. Vazquez, M. and Beggs, H.D.: “Correlations for Fluid Physical Property Pre-diction” JPT (June 1980) 32, 968-970.

I = Initial conditions

o = Unfractured

w = Wellbore


Appendix H

Net Present Value Theory

H.1 IntroductionThis appendix provides a brief overview of the methods used for conducting eco-nomic analyses of fracture treatments as applied in the MFrac, MProd and MNpvprograms. The basic methodology is outlined and the pertinent economic equationsare presented. The purpose of employing these methods is to maximizing well prof-itability by accelerating production, reducing operating costs and perhaps evenincreasing ultimate recovery.

Normally, economic analysis, as it pertains to hydraulic fracture design, involvescomparing the cost of treatments with their expected revenues to maximize thepotential return on investment. This typically includes several steps. First, produc-tion potential is evaluated as a function of fracture penetration and conductivity.These fracture characteristics have a dramatic impact on the cash flow income of awell. After generating the relationships between production and fracture character-istics, the next step is to consider the costs associated with each of the fracturegeometries under evaluation. The final step is to combine the productivity data withthe cost information to determine the maximum economic return. This basic meth-odology is consistent with the approach used in MFrac, MProd and MNpv (see Fig-ure H.1).

Prior to performing a productivity analysis, it is necessary to have an idea of plausi-ble fracture designs (or geometries) to explore. Since reservoir properties usuallycontrol fracture geometry, it makes sense to use a fracturing simulator to develop aset of characteristics for the model to evaluate. MFrac provides a convenient man-ner to obtain this data by generating automatic designs. When the NPV option is onin MFrac, a maximum fracture length and proppant concentration are specified.Pumping schedules, as well as, fracture geometry and proppant transport solutionsare automatically created for ten subdivisions of this length. The information cre-


644 Net Present Value Theory:

ated from this procedure can then be imported directly into MProd to perform agroup of production simulations.

Figure H.1: Economic Optimization Concept.

Although it is convenient to use MFrac to determine the fracture characteristics andtreatment parameters for evaluation in MProd, fracture length, conductivity andmaterial quantity information can be directly input in MProd to perform a produc-tion or economic analysis. Consult the “MFrac” and “MProd” chapters for moreinformation on their respective options.

With fracture data entered in MProd and all of the remaining reservoir parametersdescribed, production simulations can be performed to predict the relationshipbetween cumulative production, fracture contact area (e.g., frac length) and fracturedeliverability (e.g., conductivity). These results will normally show that for reason-able conductivity, the greater the fracture length, the more substantial the produc-tivity improvement. To determine whether or not the benefit of creating additionalreservoir contact area is worth the cost, the economic value of each treatment mustbe considered.


H.2 General Equations 645

The economic value of a stimulation treatment is typically assessed by using one ofthree methods. The first of these methods is predicting the time it takes for thecumulative post-frac revenue to reach the level of the initial investment. This is thetime required to pay for the treatment with revenues from the well. This criterion isusually referred to as “payout.” Particularly for low permeability wells, whereinvestments are large and payout time is long, this method does not take intoaccount the time value of money (e.g., Currency Escalation Rate or achievableinterest rate) and can, therefore, lead to false conclusions. Even if the value ofmoney remained constant over the producing time of a well, the payout or net reve-nue of each treatment design under consideration would not always be an accept-able method to determine an optimum design. For example, comparing a smallertreatment and associated fracture penetration with a larger job may show that thesmaller treatment pays out sooner because of early time production effects. Evalu-ating the long-term production decline of the two scenarios, however, may revealthat the larger treatment and contact area, once it achieves stabilized flow, declinesless rapidly and, therefore, results in a higher ultimate recovery.

To include the time value of money in the economic evaluation of fracture treat-ments, the preferred methods of either Net Present Value (NPV) or DiscountedReturn On Investment (DROI) are used. The primary difference between the twoapproaches is that DROI is sensitive to the rate of change in NPV and, therefore,can be thought of as an indicator of capital efficiency. In other words, the incremen-tal DROI decreases when the cost associated with an increase in productionincreases at a greater rate than the NPV. When capital is strictly limited, DROIindicates the more conservative optimization criteria. A decreasing DROI suggeststhat your limited capital may receive a higher rate of return if invested in anothermanner (i.e., one with a higher DROI).

H.2 General EquationsThis section presents the equations used for conducting economic calculations inMNpv. When appropriate the solution methodology is given.

Fracture Net Present Value (NPV)The present value or present worth, , of a future-value, , is

ate ( ) is defined as

(H-1)

P F

q

P F1 i+( )n

------------------=



where

This simple formula is the basis for calculating the Net Present Value of an invest-ment. Fracture Net Present Value, , is defined as the revenue from a hydrauli-cally fractured reservoir less the production from the same reservoir without ahydraulic fracture and the cost of the treatment in current dollars. This relationshipis expressed as follows:

(H-2)

or

where

Discount Well Revenue (DWR)The discounted well revenue (DWR) in terms of the net incremental cash flow(NCF)j is

(H-3)

or

= present worth= future worth= currency escalation rate or interest rate= number of periods

= total fixed and variable cost of a fracture treatment= number of periods= fracture Net Present Value= present value production revenue of a fractured reservoir= future value production revenue of an unfractured reservoir= future value production revenue of a fractured reservoir= future value production revenue of an unfractured reservoir

PFin

NPV

NPV RF R0– CF–=

NPVVF( )j

1 i+( )j-----------------

⎝ ⎠⎜ ⎟⎛ ⎞ V0( )j

1 i+( )j-----------------

⎝ ⎠⎜ ⎟⎛ ⎞

CF–j 1=

n

∑–j 1=

n

∑=

CF

nNPVRF

R0

VF

V0

DWRVF( )j

1 i+( )j-----------------

⎝ ⎠⎜ ⎟⎛ ⎞ V0( )j

1 i+( )j-----------------

⎝ ⎠⎜ ⎟⎛ ⎞

j 1=

n

∑–j 1=

n

∑=


H.2 General Equations 647

The NPV can be expressed in terms of the DWR by:

(H-4)

From these expressions it is shown that the Fracture NPV is a function of time,propped fracture length, conductivity, drainage area, reservoir properties, etc. Thismethodology, therefore, is an excellent criteria for basing the optimization strategy.

Discounted Return on Investment (DROI)The DROI also takes into account the time value of money invested and can beused as a indicator of the capital investment efficiency. The DROI is simply theratio of the Discounted Well Revenue divided by the total cost of a treatment. For afracture design this is

(H-5)

The decision on which optimization criteria to use rests with your companies busi-ness philosophy and financial position. DROI is the approach used by many opera-tors to evaluate new prospects. Typically, this involves considering the total cost ofdrilling and completing a well. When estimating the value of a hydraulic fracturetreatment you can identify a design which may result in the highest NPV; however,this does not mean that you have identified a design which makes the most financialsense relative to other investments. For that we recommend the use of DROI with“realistic” estimates of the future hydrocarbon revenue per unit volume and cur-rency escalation trends.

DWRNCF( )j

1 i+( )j------------------

j 1=

n

∑=

NPV DWR CF–=

DROI DWRCF

-------------=


Appendix I

TSO & Frac-Pack Methodology

I.1 IntroductionTip Screen-out (TSO) and Frac-Pack designs are generally performed in moderateto high-permeability reservoirs that require greater conductivity than achieved withconventional hydraulic fracturing. The implementation of TSO and Frac-Packs,over the past few years, has resulted in substantially greater fracture conductivitiesand improved proppant placement. As a consequence, these applications havegained popularity in the industry, especially in high permeability wells in the Gulfof Mexico where inadequate conductivity and formation damage have been prob-lems. Fundamentally, these techniques are similar up to the step of fully packingthe fracture.

The TSO methodology as presented by Smith1 et al. and applied to the RavensurnSouth gas field2 is used to deliberately create a proppant screen-out or bridgingcondition around the perimeter of the fracture to prevent further propagation andheight growth. Continued pumping results in “ballooning” or an increase in thefracture aperture with continued increasing fracture pressure. The increased aper-ture results in a greater propped width and increased fracture conductivity. Typi-cally, only the perimeter of the fracture is packed.

Frac-Packs differ from TSO's by packing the entire fracture with proppant from thetip to the wellbore at the settled bank concentration which greatly increases thefracture conductivity. This technique is typically performed in higher permeabilityformations that require large average conductivities to increase productivity.

The relative popularity and success of the so-called Frac-Pack technique forhydraulic fracturing has resulted in many misconceptions regarding the objectivesand procedures used for these non-conventional treatments. The Frac-Pack method-ology presented here (and compared to the classical TSO methodology) was origi-


650 TSO & Frac-Pack Methodology:

nally developed and implemented into MFrac-IIä ver. 7.1 July of 1994. Ananalytical form of this methodology was presented at the 1995 SPE annualmeeting3.

This appendix presents in a concise manner a summary of Meyer & Associates, Inc.1994 tech notes for design of TSO's and Frac-Packs. A clear definition of thedesign objectives and a step-by-step procedure that can be used as an engineeringguide for implementation are also included. A comparison is made between theclassical TSO and Frac-Pack methodologies with a presentation of results achiev-able with Frac-Packs. The following discussions will help reduce the confusion sur-rounding the design of fractures in high permeability reservoirs and offeralternatives to increase productivity.

I.2 MethodologyMFrac uses numerical, state-of-the-art, Frac-Pack and TSO methodologies todesign “fully packed” or TSO type proppant distributions. The modeling tech-niques used require that the fracture propagation and proppant transport solution belinked in such a way that each can influence the other. Normally, this means that foreach time step in the fracture propagation calculation, the proppant transport simu-lation must be assessed and coupled. This methodology differs substantially from aconventional fracture stimulation approach which by design tries to prevent prop-pant screen-outs or bridging.

In order to fully pack a fracture and achieve a desired conductivity, it is necessaryto accurately model and control the rate of creation of fracture volume. If this isaccomplished, the fracture can be filled or packed with an injection concentrationfar below the packed value.

For a Frac-Pack or TSO design the slurry treatment must be scheduled such that asthe earlier stages concentrate, due to slurry dehydration or leakoff, the later stagesfill the void created by a continuous and declining rate of fluid loss. The only oper-ational alternatives to fully pack a fracture is to either decrease the injection rate orincrease the proppant concentration to offset the decreased leakoff rate during theFrac-Pack process. Increasing the leakoff velocity (rate) during the Frac-Pack pro-cess will also enable the fracture to be fully packed. However, accomplishing thisin a diffusion controlled environment may be unrealistic. In practice, the maximumpumped concentration is normally limited by an upper constrained value far belowthe packed concentration needed for Frac-Packs. Therefore, the only practical wayto accomplish a Frac-Pack reliably is to decrease the injection rate after a pre-spec-ified design criteria is satisfied to offset the decline in fluid loss once screen-outoccurs and fracture growth has stopped. The advantage of decreasing the injectionrate also minimizes excess “ballooning” by maintaining a constant fracture pres-


I.2 Methodology 651

sure. This methodology is easily implemented in the field (by controlling pressureand decreasing rate) and can help force a TSO or enhance the rate of Frac-Packing.

Design CriteriaThe criteria for automatic TSO and Frac-Pack designs include:

• Designing to a pre-specified fracture length to optimized near wellbore con-ductivity;

• Basing the design on a maximum allowable inlet concentration;

• Designing to achieve a minimum concentration per unit area; and

• Maintaining pumping pressures below a critical maximum.

ProceduresIn terms of procedures, operations should design for a target fracture length. After aperimeter tip screen-out is achieved, fracture extension (length and height growth)will stop and the fracture width and pressure will begin to increase. The rate of fluidleakoff begins to decrease.

For a TSO, the fracture pressure is allowed to continue increasing until the mini-mum concentration per unit area is satisfied or the pressure rises to the maximumallowable value. The TSO methodology assumes a constant injection rate duringthe entire pumping schedule.

For a Frac-Pack once the fracture width (or pressure) reaches a value to satisfy theminimum concentration per unit area at the bank concentration, the fracture pres-sure (compliance) is held constant by decreasing the injection rate to match the lea-koff rate.

Because excess ballooning is permitted in a TSO, the inlet concentrations toapproach a “fully” packed fracture are not feasible.

Figure I.1 illustrates the methodologies for tip screen-out and frac-pack designs. Asshown, the behavior of the fracture length (extension) is similar for both methodswith arresting of fracture propagation after the time of tip screen-out (TSO). Theinlet proppant concentration for a TSO is shown to continually increase with time(after the initial TSO stage) until it reaches a maximum pre-specified inlet concen-tration.



The Frac-Pack schedule shows a similar behavior up to the time of the maximuminlet concentration. After reaching maximum concentration, the injection rate isdecreased to match the leakoff rate while maintaining a constant inlet concentra-tion. The leakoff rate decreases as a result of the decreased fracture propagationrate. Since no new fracture area is being created during the packing process, the lea-koff velocity will decrease with time as a result of diffusion. If leakoff is not con-trolled by diffusion or is time dependent, the leakoff rate will decline at a differentslope.

For a Frac-Pack, once the injection rate is cut to the leakoff rate, the fracture pres-sure will remain essentially constant. This mitigates the pressure dependence effecton fluid loss. If leakoff is a strong function of fracture pressure, the leakoff coeffi-cient would change more drastically for a TSO than a Frac-Pack because of thecontinued increasing net fracture pressure with time after a TSO.

Figure I.1 shows that the fracture net pressure and width both increase with timeafter a TSO. However, the Frac-Pack net pressure and width remain constant afterthe time the maximum concentration is reached. Since our 3-D model is not alumped model, the spatial compliance factors may change during the declininginjection rate period resulting in slight variations in the pressure and aperture. Thefracture volume will, however, remain constant during this period. Figure I.1 alsoshows the final concentration at the end of the job (eoj). This clearly illustrates thatthe main advantage of packing a fracture all the way back to the wellbore is toincrease the propped width and minimize excess pressure


I.2 Methodology 653

Figure I.1: Tip Screen-Out vs. Frac-Pack Methodology.

Generally, Frac-Packs are performed in formations which have higher permeabilityand lower fracture efficiencies than TSO's. Therefore, to achieve an adequatedimensionless conductivity the conductivity must beFCD kfwf( ) krLp( )⁄= kfwf( )



greater for frac-packs. This is achieved by designing for short high conductivityfractures. Frac-Packs are also most easily realized in formations conducive to lowfracture efficiencies (typically less than forty percent). The lower the efficiency theeasier and quicker it is to achieve a TSO or Frac-Pack. For fracture efficienciesgreater than fifty per cent it is difficult to perform a classical “fully packed” frac-ture. Other important considerations are the minimum allowable flow rate, prop-pant settling, time/pressure dependent leakoff, spurt loss and changing fracturecompliance. The numerical procedure developed here for 3-D (and 2-D) TSO's andFrac-Pack's automatically accounts for these effects and other time dependentparameters generally ignored in analytical solutions.

Typically TSO's are performed in moderate permeability “hard” rock countrywhere as Frac-Packs have been successfully performed in high permeability uncon-solidated “soft” rock formations. The advantage of a successful Frac-Pack is thatthe fracture will be packed at the settled bank proppant concentration and at thedynamic pumped width. The propped width for a TSO (no settling) will be equal tothe ratio of the slurry concentration in the fracture at the end of pumping divided bythe settled bank slurry concentration ( ). If 20/40 Jordan sand is

placed at a maximum proppant concentration of 12 lbm/gal (7.8 lbm/gal slurry) theTSO propped width ratio would be 0.61 (i.e., bank concentration of 30.5 lbm/gal liq(12.8 lbm/gal slurry), where or )). Thisclearly illustrates why many TSO's are ballooned to a much greater extent than nec-essary to achieve the same concentration per unit area as a Frac-Pack.

I.3 Numerical SimulationThe above methodology for Frac-Packs and TSO's was implemented in our 3-Dhydraulic fracturing simulator (MFrac) in early 1994. The code was beta tested andreleased in late summer. This methodology is applicable for all types of 2-D and 3-D type fracture geometry models. The methodology is simple and based on soundengineering principles of mass and momentum conservation. Since this methodol-ogy has been incorporated in a numerical simulator, implementation of differentboundary conditions or assumptions is possible and the effect of such changesquantified. Although all the underlying boundary conditions outlined in this meth-odology may not always be satisfied, these tools enable the design engineer toinvestigate the simplicity of this first order analysis and how substantially it devi-ates from conventional fracturing.

To illustrate the Frac-Pack methodology an automatic design was numerically sim-ulated based on the following criteria:

• Pre-specified fracture length of 100 feet;

wp weoj⁄ cs cs bank⁄=

cl cs 1 cs ρ⁄–( )⁄= cs cl 1 cl ρ⁄+( )⁄=


I.3 Numerical Simulation 655

• Maximum allowable inlet concentration of 12 lbm/gal liquid;

• Designed to achieve a concentration per unit area of 9 lbm/ft2; and

• Maintain a pumping pressure below 10000 psi.

Figure I.2 shows the inlet slurry, liquid and resulting leakoff rate as a function oftime which satisfy the pre-specified Frac-Pack criteria. Figure I.3 illustrates thesimulated automated inlet proppant concentration schedule.

Figure I.2: Automated Injection and Leakoff Rates vs. Time.



Figure I.3: Automated Inlet Proppant Concentration vs. Time.

Once the design fracture length is achieved at about 13 minutes fracture extension(length and height) stops as a result of the tip screen-out condition (Figure I.4). Thefracture continues to balloon from 13 to 17.5 minutes to a width of about 1.3 in.(Figure I.5) to meet the design concentration/area of 9 lbm/ft2 for a fully packedfracture.


I.3 Numerical Simulation 657

Figure I.4: Fracture Extension vs. Time.

Figure I.5: Fracture Net Pressure and Width vs. Time.



Once the fracture stops propagating the pressure and width continue increasing(Figure I.5). After the optimum design width is achieved the injection rate is cut tothe leakoff rate and the inlet concentration is maintained at the maximum value.This stops the fracture from ballooning, resulting in an approximate constant pres-sure throughout the remainder of the job.

Figure I.2 shows that once the fracture stops propagating the leakoff rate decreases.Also, the liquid rate decreases with increasing inlet sand concentration. After theslurry rate decreases to the leakoff rate at tcmax the liquid injection rate falls belowthe leakoff rate. The higher the maximum allowable inlet concentration the lowerthe liquid rate will be. Consequently, during this decreasing injection period, therate of fracture packing is equal to where is the settled bankporosity.

Figure I.6 shows the behavior of fracture efficiency as a function of time. After thepropagation rate diminishes at 13 minutes the efficiency rises as a result of thedecreased leakoff rate. However, once the injection rate decreases to the leakoffrate the fracture volume remains approximately constant (i.e., the compliance fac-tor may, however, change slightly with time) and the efficiency will continuedecreasing until the fracture is fully packed. The fracture efficiency at closure rep-resents the fraction of propped volume to total injected slurry volume.

Figure I.6: Fracture Efficiency vs. Time.

qs ql–( ) 1 φ–( )⁄ φ


I.4 Results and Conclusions 659

Figure I.7 shows the final fully packed fracture concentration per unit area con-tours. This profile is shown to match the desired final value of 9 lbm/ft2. The twomaximum concentration variations in the contours are a result of the two low con-fining stresses layers in each of the pay intervals.

Figure I.7: Proppant Concentration per Unit Area Contours.

Figure I.2 through Figure I.7 illustrate the Frac-Pack methodology as implementedin our 3-D hydraulic fracturing simulator. The advantage of using a numerical sim-ulator is that the leakoff rate, compliance factors, spurt loss, height growth andother typical simplifying analytical assumptions made by 2-D models are not nec-essary to solve the governing equations.

I.4 Results and ConclusionsThe methodology and procedures outlined in Figure I.1 will help the design engi-neer better understand TSO and Frac-Pack treatments. The advantage of a Frac-Pack, in controlling the pressure rise to minimize excess “ballooning” and in opti-mizing proppant placement, was also demonstrated. The application for either theTSO or Frac-Pack is more a function of the fracture efficiency than if it is of “hard”or “soft” rock. Lower fracture efficiencies (high reservoir permeability) favor theFrac-Pack while higher efficiencies (moderate permeability) favor the TSO meth-



odology. Excessive leakoff control for both the TSO and Frac-Pack may be a strongdisadvantage resulting in higher fracture efficiency jobs.

High permeability reservoirs require high conductivity fractures, hence the term“packed” is applied since the fracture must be fully packed with proppant to accom-plish an optimum conductivity. To approach a truly “packed” condition it is neces-sary to control the injection rate and inlet proppant concentration once a TSO hasoccurred and throughout the Frac-Pack process.

When classical TSO methods are applied to small scale treatments undesirable orless desirable effects may occur due to the resulting proppant distribution. Nor-mally, Frac-Packs are performed in high permeability reservoirs that require a moreaggressive approach to achieve the optimum proppant placement for full develop-ment of short high conductivity fractures.

Achieving the optimum condition described in the methodology above requires anunderstanding of the fundamental dynamic time dependent diffusion fluid loss pro-cess for a specific application. Fracture growth equilibrium can then be inferred byconsidering the material balance between injection, fluid loss and overall fractureconservation of volume (mass).

Frac-Packs are most applicable in design of hydraulic fracturing treatments whenthe target conductivity is high and more control in the spatial distribution of prop-pant is required.

I.5 References1. Smith, M. B., Miller, W.K. and Haga, J.: “Tip Screenout Fracturing: A Tech-

nique for Soft Unstable Formations,” SPEPE, May 1987, 95-103.

2. Martins, J.P. and Stewart, D.R.: “Tip Screenout Fracturing Applied to theRavenspurn South Gas Field Development,” SPE Prod. Eng., Aug. 1992, 252-258.

3. Fan, Y. and Economides, M.J.: “Fracture Dimensions in Frac&Pack Stimula-tion,” Paper SPE 30469 Presented at the 1995 Annual Technical Conferenceand Exhibition, Dallas, TX, Oct. 22-25, 1995.


Appendix J

Produced Water Reinjection Fracturing

J.1 IntroductionThe solution methodology for our Produced Water Reinjection (PWRI) hydraulicfracturing simulator is formulated in this report. A summary of the governing waterand thermal front equations, thermo- and poro-elastic stresses and fluid loss equa-tions are presented.

J.2 Thermal and Water Front EquationsThe mass and energy conservation equations are developed based on the assump-tion that there are two zones, one at the injection temperature and another at the for-mation temperature. The separation of these zones is identified by the thermal front.Both zones are assumed to be at an irreducible oil saturation. The thermal zoneextends from the wellbore to the thermal front and the second zone extends fromthe thermal front to the waterfront. The first zone is at the injection temperature andthe second is at the formation temperature.

Perkins and Gonzalez (1985) presented a simplified methodology for determiningthe cooled region and thermal fronts. The volume of the cooled region (assumingthe injected fluid is cooler than the reservoir) is determined from energy conserva-tion. The cooled volume in the reservoir is assumed to be the region where the res-ervoir is at the injected fluid temperature. We also assume conduction heat transferis negligible in the reservoir.

The governing energy equation is

(J-1)

where

Ein Eout– ΔEsystem=


662 Produced Water Reinjection Fracturing:

The governing energy equation with a reference temperature equal to the initial res-ervoir temperature of (i.e., ), and a fluid injection temperature of

is

(J-2)

The volume of the cooled region is

(J-3)

where is the residual oil saturation.

Perkins and Gonzalez approximated the cooled region as an elliptical inclusionconfocal with a line crack of length and having a volume . The major axis

( ) is parallel to the fracture length and the minor axis ( ) is perpendicular to thefracture plane. The resulting ellipsoidal equations are

(J-4)

and

or

. (J-5)

Ein Energy into the System=

Eout Energy out of the System=

ΔEsystem Change in System Energy=

Tr Eout 0=

Tf

Vi ρc( )w Tf Tr–( ) Vc ρc( )r 1 φ–( ) ρc( )wφ 1 Sor–( ) ρc( )oφSor+ +[ ] Tf Tr–[ ]=

Vcρc( )wVi

ρc( )r 1 φ–( ) ρc( )wφ 1 Sor–( ) ρc( )oφSor+ +-----------------------------------------------------------------------------------------------------------=

Sor

Lf Vc

a b

a0 Lf ξ0cosh Lfξ0( )exp ξ– 0( )exp+

2-------------------------------------------------⎝ ⎠

⎛ ⎞= =

b0 Lf hξ0sin Lfξ0( )exp ξ– 0( )exp–

2-------------------------------------------------⎝ ⎠

⎛ ⎞= =

Vc πa0b0h πLf2h hξ0 hcos ξ0sin==

VcπLf

2h4

------------- 2ξ0( )exp 2– ξ0( )exp–[ ]=


J.2 Thermal and Water Front Equations 663

Substituting into the above equation, we find

(J-6)

The solution to is

(J-7)

where

The major and minor thermal front axes are

(J-8)

and

. (J-9)

The volume of the water flooded region from mass conservation is

(J-10)

where is the irreducible water saturation.

Approximating the water flooded region as ellipsoidal in a similar manor to thethermal region, we have

(J-11)

where is

F0 2ξ0( )exp=

F02

4Vc

πLf2h

-------------F0– 1– 0=

F0

F0 γ0 γ0( )2 4++( ) 2⁄=

γ04Vc

πLf2h

-------------=

a0 Lf F0 1 F0⁄+( ) 2⁄=

b0 Lf F0 1 F0⁄–( ) 2⁄=

Vw

VwVi

φ 1 Sor Siw––( )--------------------------------------=

Siw

F12

4Vw

πLf2h

-------------F1– 1– 0=

F1



(J-12)

and

The major and minor water front axes are then given by

(J-13)

and

. (J-14)

Since , the ellipsoidal thermal front ( ) will lag the ellipsoidal

water front ( ).

J.3 Thermoelastic and Poroelastic StressesThis section summarizes the governing equations for thermoelastic and poroelasticstresses generated during Produced Water Reinjection (PWRI) fracturing. Themethodology is based on the work of Perkins & Gonzalez (1985) for ellipticallyshaped regions of finite thickness.

Thermoelastic StressesThe thermoelastic stresses are generated if the inject fluid is at a temperature differ-ent than that of the formation. The region of changed rock temperature has a sharpboundary interface which progresses outward in an ellipsoidal shape.

The thermoelastic stresses for regions of elliptical cross-section of finite heighthave been developed by Perkins and Gonzalez using numerical analysis. Theyreported the following equation for estimating the average thermal stress perpen-dicular to the fracture face in the interior of an elliptical cooled region of anyheight:

F1 γ1 γ12 4++( ) 2⁄=

γ14Vw

πLf2h

-------------=

a1 Lf F1 1 F1⁄+( )=

b1 Lf F1 1 F1⁄–( )=

Vw Vc> a0 b0,

a1 b1,


J.3 Thermoelastic and Poroelastic Stresses 665

(J-15)

or

(J-16)

where the thermoelastic coefficient is

(J-17)

Figure J.1 shows the thermoelastic coefficient for various thermal front ellipsoidalshapes.

Figure J.1: Thermoelastic Coefficient vs. Ellipsoidal Shape

1 ν–( )Δσ3 TEβΔT

-------------------------------- f a0 b0 h, ,( )=

Δσ3 TEβΔT1 ν–--------------- f a0 b0 h, ,( )⋅=

f a0 b0 h, ,( )b0 a0⁄

1 b0 a0⁄+------------------------= 1

1 b0 a0⁄+------------------------⎝ ⎠

⎛ ⎞ ⋅+

1 1 12--- 1.45 h

2b0--------⎠

⎞ 0.90.35 h

2b0--------⎝ ⎠

⎛ ⎞ 2+⎝ ⎠

⎛ ⎞ 1b0a0-----⎝ ⎠

⎛ ⎞0.774

++⎩ ⎭⎨ ⎬⎧ ⎫

⁄⎝ ⎠⎜ ⎟⎛ ⎞



A corresponding second principal thermal stress change is also reported by

Perkins & Gonzalez.

Poroelastic Stresses

The change in horizontal stress ( )as a result of a change in pore pressure

( ) is analogous to the thermal stresses equations. To accomplish this transfor-mation, the following linear coefficient of pore pressure expansion is introduced

(J-18)

which is analogous to the linear coefficient of thermal expansion. The change in theminimum horizontal stress as a result of pore pressure changes from Eq. (J-15) is

(J-19)

Introducing Biot’s constant

(J-20)

where is the bulk modulus of the material and is the bulk modulus of the

solid constituents. The bulk modulus of the material, , is defined as

(J-21)

Rearranging Eq. (5), the grain compressibility in terms of Biot’s constant and thebulk modulus of the material is

(J-22)

Then substituting Eq. (7) and (5) into (3) and rearranging we find

Δσ2

Δσ3 p

Δp

J 1 2ν–E

---------------cgr3

-------–=

1 ν–( )Δσ3 pEJΔp

-------------------------------- f a1 b1 h, ,( )=

α 1cgrcb-------– 1

kbkgr-------–= =

kb kgr

kb

kbE

3 1 2ν–( )-----------------------=

cgr1 α–

kb------------ 1 α–( )3 1 2ν–( )

E-----------------------= =


J.4 Governing Fluid Loss Equations 667

(J-23)

Placing Eq. (4) in terms of Biot’s constant we have

(J-24)

or

(J-25)

where .

Perkins and Gonzales used the same approach for calculating the poro-elastic stressas the thermal stress (i.e., they assumed the elliptical area of constant pressure vari-ations surround the fracture). Since Koning (1985) realized that this assumptionwas “clearly unrealistic”, he developed a different approach following the method-ology of Muskat. Koning’s ellipsoidal pressure distribution around the fracture forplane strain yields a limiting factor on the effective pressure differential. This factorwhich we will call the Koning factor, , is equal to 0.5 as derived by Koning.

The resulting poro-elastic equation from Eq. 10 is

(J-26)

where the Perkins factor, , is used to account for the magnitude of the

ellipsoidal pressure extent around the fracture.

J.4 Governing Fluid Loss EquationsThis section discusses the mechanisms which control fluid loss form a propagatingfracture. The classic linear (1D) fluid loss equations as proposed by Carter whichare exclusively used today are first presented. Next we present the fundamentaldimensionless pressure and rate solutions for linear and ellipsoidal. A relationshipbetween the dimensionless rate and pressure solutions is then discussed based onthe methodology of Koning. A discussion of internal and external skin is then pre-sented, This is followed by the governing equations for modeling internal and

EJ 1 2ν–( )Ecgr

3-----------– 1 2ν–( )α= =

1 ν–1 2ν–---------------⎝ ⎠

⎛ ⎞ Δσ3 pαΔp

-------------- f a1 b1 h, ,( )=

Δσ3 p1 2ν–1 ν–

---------------⎝ ⎠⎛ ⎞ αΔp f a1 b1 h, ,( )⋅=

Δp Pf pi–=

fk

Δσ3 p1 2ν–1 ν–

---------------⎝ ⎠⎛ ⎞ αfkΔp f a1 b1 h, ,( )⋅=

f a1 b1 h, ,( )



external filter cakes based on the total suspended solids in the injected fluid whichis based on the work of Pang and Sharma (1994) concludes the fluid loss section.

Carter’s Solution - Linear Fluid lossThe total fluid leakoff rate as a function of time in a hydraulic fracture is normallybased on Carter’s one-dimensional fluid loss equation

(J-27)

where is the fluid loss rate at time , is the fracture area (one face),

is the total leakoff coefficient, is the time of fracture area creation. Carter’s1D equation assumes that the fluid loss to the formation is linear and perpendicularto the fracture face (see Howard and Fast (1970, page 33)).

Meyer and Hagel (1988) presented a general equation to replace the leakoff rate ofEq. (J-27)

(J-28)

where

(J-29)

and

(J-30)

Meyer and Hagel also showed that for a constant leakoff area propagation parame-ter, , Eq. (J-29) simplifies to

q t( ) 4 Ct τ A( )–

----------------------- Ad0

A t( )

∫=

q t( ) t A t( ) Cτ A( )

q t( ) 4CA t( )t

------------------ 11 f λ( )–

----------------------- λd0

1

∫=

4CA t( )t

------------------φ′=

f λ( ) τ λ( ) t⁄=

φ′ 11 f λ( )–

----------------------- λd0

1

∫=

αa



(J-31)

and

(J-32)

where is the gamma function and . Typical values for as a

function of are given in Table J.1.

Dimensionless Pressure SolutionThe dimensionless pressure for a constant fluid loss rate is defined as

(J-33)

Gringarten (1974), as report by Earlougher (1977), presented two dimensionlesspressure solutions for a static vertical fracture in an infinite-acting system. The twogeneral solutions for elliptical leakoff are presented below.

Table J.1: Fluid Loss Integral versus Propagation Parameter

Propagation Parameter, Fluid loss Integral,

0 1

1/2

1 2

2 8/3

10 5.675463855...

f λ( ) τ λ( )t

----------- A τ( )A t( )-----------

⎩ ⎭⎨ ⎬⎧ ⎫

1 αa⁄

= =

λ1 αa⁄

=

φ′ 1

1 λ1 αa⁄

–-------------------------- λd

0

1

∫=

Γ αa 1+( )Γ 1 2⁄( )Γ αa 1 2⁄+( )

-------------------------------------------=

Γ Γ 1 2⁄( ) π= φ′αa

αa φ′

π 2⁄

pD2πkh

qμ-------------Δp=



The uniform fracture flux solution This solution assumes that the fluid loss in the fracture is a uniform leakoff rate perunit area of the fracture face so that there is a pressure drop in the fracture. Thedimensionless pressure is calculated from

(J-34)

where the dimensionless time based on the fracture half-length is defined as

(J-35)

and

(J-36)

The dimensionless pressure solution at early times (i.e., , linear or 1D

leakoff) for a static (non-propagating) uniform flux fracture in an infinite system isgiven by

(J-37)

When , Eq. (J-34) becomes

(J-38)

with less than 1% error.

The infinite-conductivity solutionThis solution assumes that the fracture has an infinite permeability and that thepressure is uniform in the fracture. The approximate solution for an infinite-con-ductivity fracture as given by Gringarten (1974) is

pD tDxf( ) πtDxf erf 12 tDxf

----------------⎝ ⎠⎛ ⎞ 1

2---Ei

1–4tDxf------------⎝ ⎠

⎛ ⎞–=

tDxfλtL2-----=

λ kcφμ---------=

tDxf 0.1<

pD πtDxf→

tDxf 10>

pD12--- tDxf 2.80907+ln( )=



(J-39)

The dimensionless pressure solution at early times (i.e., ) for an infi-

nite conductivity fracture in an infinite system is given by

(J-40)

When , Eq. (J-39) becomes

(J-41)


Dimensionless Rate Solution

Defining the fluid loss rate in terms of a dimensionless rate function, , we have

(J-42)

where is the formation permeability, is the formation height, is the reser-

voir viscosity, and is the differential pressure (fracture pressure minus the ini-tial reservoir pressure).

Linear Solution

The differential equation for the pressure distribution in the formationassuming one-dimensional (linear) fluid loss from a fracture is

(J-43)

pD tDxf( ) 12--- πtDxf erf 0.134

tDxf

-------------⎝ ⎠⎛ ⎞ erf 0.866

tDxf

-------------⎝ ⎠⎛ ⎞+

⎩ ⎭⎨ ⎬⎧ ⎫

=

0.067Ei0.018–tDxf

----------------⎝ ⎠⎛ ⎞– 0.433Ei

0.750–tDxf

----------------⎝ ⎠⎛ ⎞–

tDxf 0.01<

pD πtDxf→

tDxf 10>

pD12--- tDxf 2.2000+ln( )=

qD

q t( ) 2πkhΔpμ

--------------------qD t( )=

k h μΔp

p x t,( )

kcφμ---------

y2

2

∂

∂ p⎝ ⎠⎜ ⎟⎛ ⎞

t∂∂p=



where the reservoir viscosity ( ), permeability ( ), porosity ( ), and compress-

ibility ( ) effect the pressure transient.

The boundary and initial conditions, assuming that the reservoir is infinite and thata constant pressure exists in the fracture between the fracture and the reservoir, are

(J-44)

The solution to Eq. (J-43) as given by Holman (1977, page 102) is

(J-45)

where

(J-46)

The velocity at any position in the formation from Darcy’s law is

(J-47)

Performing the partial differentiation of Eq. (J-45) gives

(J-48)

At the fracture face, the leakoff velocity is

(J-49)

where the leakoff coefficient is given by

(J-50)

μ k φc

p y 0,( ) pi=

p 0 t,( ) pf for t 0>=

p y t,( ) pf–pi pf–

-------------------------- erf y2 λt------------⎝ ⎠

⎛ ⎞=

λ kcφμ---------=

y

v kμ---

y∂∂p

⎝ ⎠⎛ ⎞–=

y∂∂p pi pf–

πλt-------------- y2

4λt--------–⎝ ⎠

⎛ ⎞exp=

v kμ---

y∂∂p

⎝ ⎠⎛ ⎞

y 0=– k

μ--- Δp

πλt-------------

CII

t-------= = =

CIIkμ--- Δp

πλ----------- Δp kφc

πμ---------= =



and .

This is the same linear velocity formulation as presented by Howard and Fast(1970, page 35).

Equating Eq. (J-42) to Eq. (J-28) for 1D flow and substituting Eq. (J-50) for reser-voir dominated fluid loss ( ), we find

or

(J-51)

where the dimensionless time is given by

(J-52)

The dimensionless rate can be calculated from the dimensionless pressure solutionbased on the following well known relationship in Laplace space (see Lee and Bro-chenbrough (1983, Appendix A)

(J-53)

Then for linear fluid loss

(J-54)

or

The dimensionless rate in Laplace space is

Δp pf pi–=

C CII→

q t( ) 2πkhΔpμ

--------------------qD t( )= 4CA t( )t

------------------φ′=

qD t( ) 2π---φ′L t( )

πλt---------------= 1

πtD

-------------=

tDλtL2----- 1

2 π⁄ φ′---------------⎝ ⎠

⎛ ⎞ 2•=

qD s( ) 1s2pD s( )-------------------=

pD πtDxf→ π2---2 tDxf π⁄=

pD s( ) π2--- 1

s s---------=



(J-55)

Inverting Eq. (J-55), we find the dimensionless rate solution for a static fracturewith linear leakoff to the formation to be

(J-56)

Consequently, if we make the following substitution for a propagating fracture

(J-57)

then the dimensionless rate solution as a function of dimensionless pressure for lin-ear fluid loss is

(J-58)

General Dimensionless Rate SolutionKoning (1985) first showed that the solution for fractures propagating with a con-stant injection rate could be obtained by making the following substitution

(J-59)

Our first reaction to this substitution is that it’s too simple to be correct. However,Koning goes on to explain that this assumption is based partially on the work ofHaggort et al. who “observed that in an infinite reservoir the fracture length isalways proportional to the square root of time, even in the region intermediatebetween (16) and (17)”. Koning defines these conditions as follows: “Condition(16) means that the velocity of fracture propagation is much greater than the veloc-ity with which the pressure disturbance travels into the reservoir” (i.e., )

and condition (17) is “if the fracture propagates much slower than the pressure dis-turbance” (i.e., ) and Carter’s model is certainly not valid”. Koning fur-

ther goes on to state that (his) “2-D ellipsoidal solution is compared with the graphin Hagoort’s thesis on p. 132. The agreement is everywhere within 10%.”

qD s( ) 1s2pD s( )------------------- 2 π⁄

s----------= =

qD t( ) 2π---= 1

πtDxf

----------------

1πtD

------------- 2 πφ′⁄πtDxf

----------------→

qD tD( ) 1pD tD( )----------------=

qD tDxf( ) 1pD tDxf( )--------------------→

tDxf 1«

tDxf 1»



The correctness of Koning’s substitution for a constant injection rate (at low frac-ture efficiencies) becomes clear from Eq. (J-57). If the fracture propagates propor-tional to the square root of time, , then the fluid loss integral

parameter is given by . Eq. (J-57) then can be written as

(J-60)

or

(J-61)

Therefore, Eq. (J-58) simplifies to the Koning solution (Eq. (J-59)) for a fracturepropagating proportional to the square of time for 1-D and 2-D ellipsoidal flow.

van den Hoek (2000) “proceeded along the lines of Koning in order to come upwith an elliptical leak-off rate for the general fracture case”. His substitution, basedon our nomenclature, was

(J-62)

where is given by

(J-63)

This equation is similar to Eq. (J-32) for a time dependent propagation rate. van denHoek justifies this equation by the correctness of the asymptotic behavior for 1) aconstant propagation parameter (i.e., is a constant) and 2) for linear fluid loss

(i.e. Carter’s Solution). Clearly, van den Hoek’s substitution does not simplify toCarter’s 1-D solution for a general propagating fracture (i.e., only for

does it simplify to the Carter solution).

Therefore, again following along the lines of Koning, a general ellipsoidal solutionfor non-constant fracture propagation rates (from Eq. (J-57) and Eq. (J-58)) isobtained by making the substitution,

αa 1 2⁄=

φ′ π 2⁄=

1πtD

------------- 2 πφ′⁄πtDxf

----------------→ 1πtDxf

----------------

αa 1 2⁄→

=

tD tDxf→

1tD

--------- 2 π⁄tDxf

------------- φ′fσ----→

fσ

fσ1

1 λ1 αa t( )⁄

–------------------------------- λd

0

1

∫=

αa

αa 1 2⁄=



(J-64)

This is the same as van den Hoek’s Eq. (J-62) with set equal to unity.

The general dimensionless rate solution as a function of dimensionless pressure is

(J-65)

J.5 Fracture SkinThe pressure drop, , as a result of a skin effect is defined as

(J-66)

where is the fracture skin.

External SkinThe leakoff velocity through the filter cake as a function of the cake permeability( ), cake thickness ( ), and pressure drop across the cake ( ) from Darcy’s

law can be written as

(J-67)

The total leakoff rate over the fracture face is

(J-68)

The pressure drop across the cake in terms of the flow rate is

1tD

--------- 2 πφ′⁄tDxf

---------------→

fσ

qD tD( ) 1pD tD( )----------------=

Δps

Δpsqμ

2πkh-------------⎝ ⎠

⎛ ⎞ s=

s

kc δc Δps

vkcμc-----

Δpsδc

---------=

q 4hL( )kcμc-----

Δpsδc

---------=


J.5 Fracture Skin 677

(J-69)

The fracture skin for an external filter cake from Eq. (J-66) is

(J-70)

Eq. (J-70) is for a static (non-propagating) fracture. For a propagating fracture, thecake thickness varies as function of exposure time ( ). From conservation ofmass (volume) and Eq. (J-67) we have

(J-71)

The cake thickness as a function of time from Eq. (J-71) is

(J-72)

Therefore, the cake thickness as a function of exposure time is easily shown to beof the form

(J-73)

The total fluid loss rate for a propagating fracture at time is

(J-74)

where and is understood to be the filter cake thickness at the

wellbore. Please refer to Eq. (J-28) for additional details.

Δpsqμc4h---------

δcL----- 1

kc----=

qμ2πkh------------- π

2---

δcL----- k μ⁄

kc μc⁄--------------⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

sf external

π2---

δcL----- k μ⁄

kc μc⁄--------------⎝ ⎠

⎛ ⎞=

t τ–

t∂∂δc v 1 δc⁄∝ ∝

δc t( ) t∝

δc t τ–( ) δc t( ) 1 τ t⁄–=

t

q 4hL( )kcμc-----

Δpsδc t( )------------ 1

1 f λ( )–----------------------- λd

0

1

∫=

4hL( )kcμc-----

Δpsδc t( )------------φ′=

f λ( ) τ t⁄= δc t( )



The effective skin based on the filter cake thickness at the wellbore with from Eq. (J-70) is

(J-75)

Internal SkinThe pressure drop of the leakoff fluid in the formation for linear fluid loss as resultof mobility effects (see Eq. (J-69)) can be written as

(J-76)

where the subscript refers to the leakoff fluid properties, is the extent of the

damage region into the formation normal to the fracture face, and is the

leakoff fluid mobility.

The mobility ratio, , is given as

(J-77)

The internal fracture skin from Eq. (J-66) for a static fracture is

(J-78)

The internal skin for a propagating fracture based on the same methodology for theexternal skin (i.e., ) is

δc δc t( ) φ′⁄→

sf external

12 πφ′⁄---------------

δc t( )L t( )------------ k μ⁄

kc μc⁄--------------

⎩ ⎭⎨ ⎬⎧ ⎫

=

Δpsq

4h------

δsL---- μ

k---

s

μk---–⎝ ⎠

⎛ ⎞=

qμ2πkh------------- π

2---

δsL---- k μ⁄

ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

s δs

ks μs⁄

M

M k μ⁄ks μs⁄--------------=

sf internal

π2---

δsL---- k μ⁄

ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

δs δs t( ) φ′⁄→


J.5 Fracture Skin 679

(J-79)

where is the damaged distance in the reservoir at the wellbore.

Eq. (J-79) can be written as

(J-80)

where

(J-81)

and for linear fluid loss

(J-82)

It is also observed that for linear fluid loss is related to the dimensionless

pressure solution (i.e., for small ) if we make the

following substitution

(J-83)

Since the above equations were developed for linear resistance or , we

now will develop an approximate solution for an ellipsoidal internal skin.

The skin for a radial system (see Economides and Nolte (1987, pp. 1-11, eq 1-77))is

(J-84)

sf internal

12 πφ′⁄---------------

δs t( )L t( )----------- k μ⁄

ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

δs t( )

sf internal

k μ⁄ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞ f ξ( )•=

ξδs t( )L t( )-----------=

f ξ( ) 12 πφ′⁄---------------ξ=

f ξ( )

f ξ( ) pD tD( )→ πtD= tD

πtD1

2 πφ′⁄---------------ξ→

δs L 1«⁄

δs L 1»⁄

s k μ⁄ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞ δs

r′w

------ln=



where is the effective wellbore radius. The effective wellbore radius for aninfinite conductivity fracture is

(J-85)

Substituting Eq. (J-85) into Eq. (J-84) and rearranging we find

(J-86)

where

The dimensionless pressure solution for an infinite conductivity fracture when is

(J-87)

Therefore, if we make the following substitution

(J-88)

or

(J-89)

Then for a non-propagating fracture, we have when .

Similar to the methodology above we can postulate that for a propagating fracturewith ellipsoidal fluid loss that Eq. (J-89) becomes

(J-90)

r′w

r′w L 2⁄=

s k μ⁄ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞ f ξ( )•=

f ξ( ) 2ξ( )ln=

tD 10>

pD 1 2⁄ tD( ) 2.2+ln[ ]=

9tD( )1 2⁄ln=

9tD 2ξ→

πtD2 π

3----------ξ→

f ξ( ) pD tD( )= δs L 1»⁄

πtD2 π

3---------- 1

2 πφ′⁄---------------ξ⎝ ⎠

⎛ ⎞→


J.6 Internal and External Filter Cakes 681

Since the constant is near unity, an approximate solution for

the external fracture skin with is

(J-91)

where

(J-92)

and the multiplier can be approximated by

(J-93)

where is a constant.

J.6 Internal and External Filter CakesWennberg and Sharma (1987), Pang and Sharma (1994) etc., have proposed vari-ous internal and external cake filtration theories to model the injectivity decline inwater injection wells. These models are based on the initial development of aninternal skin (or cake) as a result of particulate deposition and permeability decreas-ing with increase concentration of deposited particulates. After some period of time(transition time) and external filter cake begins to develop.

Pang and Sharma (1994) best summarized their development of a model whichaccounts for both internal and external filter cakes (skin) and the concept of transi-tion time. The foundation of their model is stated as “In developing filtration mod-els, both internal and external filter cakes need to be accounted fro since both aregenerally present in the filtration process. We postulate that during some initialperiod an internal filter cake is formed. As more particles are trapped on the surfaceof the rock a point is reached where very few particles can invade the rock and anexternal filter cake begins to build. We refer to the time at which no more particlesinvade the rock, i.e., the time at which the initial layer of external filter cake is com-

pletely formed, as the transition time ( ). If we can determine the conditionsunder which particles will form internal and external filter cakes and the time

2 π3

---------- 1.181635≅

f ξ( ) pD tD( )→

s k μ⁄ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞ f ξ( )•=

πtD ψ ξ( ) 12 πφ′⁄---------------ξ⎝ ⎠

⎛ ⎞→

ψ ξ( )

ψ ξ( ) 2 π3

---------- 2 π3

---------- 1–⎝ ⎠⎛ ⎞ e aξ––≅

a

t*



required to form the initial layer of external filter cake (transition time), then theentire filtration process can be approximated by applying and internal cake filtra-tion model before the transition time and an external cake filtration model after the

transition time. Purely external filtration can be obtained in the limit as ,

and pure internal filtration can be obtained in the limit as .”

Internal Filtration Equations The governing mass conservation equations for the particles and liquid are

(J-94)

and

(J-95)

where

(J-96)

and is the volume concentration of particulates per total volume deposited

(porosity of deposited particulates), is the original porosity, and is the cur-

rent porosity.

It is common practice to simplify the above mass balance equations by substitutingthe particulate mass balance equation into the liquid equation. Differentiation Eq.(J-95) and rearranging, we can write the liquid mass balance equation in the follow-ing form

(J-97)

where from Eq. (J-96)

has been substituted.

t* 0→t* ∞→

y∂∂ vcs( )

t∂∂ φcs σ+( )+ 0=

y∂∂ v 1 c– s[ ]( )

t∂∂ φ 1 c– s[ ]( )+ 0=

σ φ0 φ–=

σφ0 φ

y∂∂v

y∂∂ vcs( )

t∂∂ φcs σ+( )+

⎩ ⎭⎨ ⎬⎧ ⎫

– 0=

t∂∂σ

t∂∂φ–=



Noting that the quantity between the brackets in Eq. (J-97) is the particulate massbalance equation which is equal to zero, we now substitute the particulate mass bal-ance Eq. (J-94) into Eq. (J-97) and find

(J-98)

Substituting Eq. (J-98) into Eq. (J-94) and rearranging, the particle mass conserva-tion equation is simplified to

(J-99)

or

(J-100)

This is the same basic mass conservation equation as reported by Wennberg andSharma (1987). However, the particle concentration here is that of the slurry andnot of the liquid. The liquid and slurry concentrations are related by

Dividing by the liquid volume , we find that

or

and (J-101)

Consequently for dilute solutions .

The form of the deposition function as first proposed by Iwasaki is

y∂∂v 0=

v y∂∂cs

t∂∂ φcs σ+( )+ 0=

t∂∂ φcs( ) v cs∇• t∂

∂cs+ + 0=

Vlcl Vscs Vl Vp+( )cs= =

Vl

cl 1 Vp Vl⁄+( )cs=

1 cl+( )cs=

cscl

1 cl+-------------= cl

cs1 cs–-------------=

cs cl≅



(J-102)

where is the filtration coefficient with units of (i.e., ). For dilute solu-

tions, the filtration coefficient is assumed to be independent of .

Eq. (J-99) can be simplified by substituting Eq. (J-102) and noting that

for all practical purposes except at very early times.

(J-103)

The solution to Eq. (J-103) is

(J-104)

where is the time the suspension reaches a position in the formation.

The deposition function is found by integrating Eq. (J-102)

(J-105)

where is the volume loss per unit area as given by

(J-106)

From Eq. (J-105), we have

(J-107)

where

(J-108)

t∂∂σ λvcs=

λ L 1– ft 1–

cs

σ φcs»

y∂∂cs λ– cs=

cs y( ) cs 0( )e λy– for t τ>=

τ y

σ

σ y t,( ) λvcs y( ) tdτ

t

∫=

V″ t τ–( )λcs 0( )e λy–=

V″

V″ t τ–( ) v tdτ

t

∫=

σ y t,( ) σ 0 t,( )e λy–=

σ 0 t,( ) V″ t( )λcs 0( )=



From Eq. (J-104), it is shown that for the suspended concentration is timeindependent and only a function of position. The average distance a particle travelsinto the formation, , can be found from Eq. (J-104)

or

(J-109)

Transition Volume Loss per Unit AreaThe transition volume loss per unit area, , is the injected volume loss per unit

area at which time, , the transition from internal filtration to external cake build-

up takes place. When a critical porosity, , is reached, particles can no longerenter into the formation and the external cake starts to develop.

The critical deposition porosity at which transition takes place is

(J-110)

Inserting the critical deposition porosity, , into Eq. (J-105), we find the transi-tion volume loss per unit area to be

(J-111)

Wennberg and Sharma (1987) report that a reasonable guess for the critical poros-

ity, , at which transition occurs from a theoretical stand point is when the pore

space is about 50% filled, or .



t τ>

δd

cs δd( ) cs 0( )eλδd–

cs 0( ) 2⁄= =

δd 2( )ln λ⁄=

V″*

t*

φ*

σ* φ0 φ*–=

σ*

V″* σ*

λcs 0( )----------------=

φ*

φ* φ0 2⁄=

σ* φ0⁄ 1 2⁄→



(J-112)

where is the particulate porosity within the pore system.

Internal Cake PermeabilityThe internal cake permeability reduction at a given point in the formation is gener-ally expressed in the following form (e.g., see Bachman (2003) or Pang (1994))

(J-113)

where is a constant.

A more general form of Eq. (J-113) is

(J-114)


The pressure drop across the internal cake (assuming linear flow for now) fromDarcy’s law is

or

(J-115)

where is the leakoff distance in the formation and is the local dam-

aged permeability.

The average or equivalent formation permeability ratio in the internal cake fromEq. (J-114) and Eq. (J-115) is

σ*

φ0------

max

1 φp–( )=

φp

ksk---- 1

1 Ωσ+-----------------=

Ω

ksk---- 1

1 β σ σ*⁄( )α+----------------------------------=

β α

dpdy------ μ

ks y t,( )-----------------v–=

Δp μks y t,( )-----------------v– yd

0

δ t( )

∫=

δ t( ) ks y t,( )



(J-116)

where the deposited particle porosity function from Eq. (J-107) is

(J-117)

Substituting Eq. (J-117) into Eq. (J-116) and integrating, we find an expression forthe average internal cake permeability

(J-118)

The limiting formation to damage permeability ratios for small and large values of, are

and

The characteristic leakoff distance at transition, , can be approximated

from mass conservation of the particles when a large fraction of the particles aredeposited in the internal cake

kks

---- 1ks k⁄----------- yd

0

δ t( )

∫⎩ ⎭⎨ ⎬⎧ ⎫

δ t( )⁄=

1 β σ σ*⁄( )α+ yd0

δ t( )

∫ δ t( )⁄=

σ y t,( )σ*

---------------- σ 0 t,( )σ*

----------------e λy–=

kks

---- 1 β σ 0 t,( )σ*

----------------⎩ ⎭⎨ ⎬⎧ ⎫α

e αλy– yd0

δ t( )

∫ δ t( )⁄+=

1 β σ 0 t,( )σ*

----------------⎩ ⎭⎨ ⎬⎧ ⎫α

1 e αλδ––( )αλδ t( )

---------------------------+=

α

k ks⁄α 0→

1 β+→

k ks⁄α ∞→

1→

δ t( ) δ*→γ



(J-119)

From Eq. (J-111) we find

or

(J-120)

The characteristic distance at transition in terms of the average distance a particletravels from Eq. (J-109) is

(J-121)

Consequently, the characteristic distance at transition (where most of the particlesare deposited) is within an order of magnitude of the average distance a particle

travels in the formation, (e.g., if , and if ,

).

Rearranging Eq. (J-118), we have

(J-122)

Internal Cake SkinThe internal filtrate skin can be calculated from Eq. (J-122) if the variation of thedeposited particulate porosity at the fracture face as a function of exposure time(position) is known. Similar to methodology used for the external cake thickness inEq. (J-75), the deposited porosity at the fracture face for linear leakoff is of theform

(J-123)

γV″*cs 0( ) σ csφ+( ) yd0

δ*

∫ σ y σ* e λy– yd0

δ*

∫=d0

δ*

∫≅=

γV″*cs 0( ) σ*

λ------ 1 e λδ*––( )→

γ 1 e λδ*––=

δ* 1 γ–( )ln–λ

-------------------------=

δ* δd⁄ 1 γ–( )ln–2ln

-------------------------=

γ 0.9= δ* δd⁄ 3.322→ γ 0.99=

δ* δd⁄ 4.605→

δ t( ) kks

---- 1–⎝ ⎠⎛ ⎞ β σ 0 t,( )

σ*----------------

⎩ ⎭⎨ ⎬⎧ ⎫α

1 e αλδ––( )αλ

---------------------------=

σ t τ–( ) σ t( ) 1 τ t⁄–( )→



The effective form of Eq. (J-122) for the entire fracture is

or

(J-124)

where

(J-125)

with the limits

The internal fracture skin for linear leakoff from Eq. (J-79) is

(J-126)

Internal Cake Build and ErosionIt is common practice to assume once the internal cake is fully developed it remainsa constant. However, once the external cake begins to developed the continuedfluid loss through the external cake may erode the deposited particulate material.

The deposited porosity in the formation from Eq. (J-105) is

1δ k ks⁄ 1–( )

frac

--------------------------------------- 1δ k ks⁄ 1–( )

x 0=

---------------------------------------- σ x( ) σ 0( )⁄[ ] α– xd0

L

∫⎩ ⎭⎨ ⎬⎧ ⎫

L t( )⁄=

φσ′

δ k ks⁄ 1–( )x 0=

----------------------------------------=

δ k ks⁄ 1–( )frac

β σ 0( )σ*

-----------⎩ ⎭⎨ ⎬⎧ ⎫α

1 e αλδ––( )αλ

--------------------------- 1φσ

′------=

δ k ks⁄ 1–( )x 0=

1φσ

′------=

φσ′ 1

1 f λ( )–[ ]α------------------------------- λd

0

1

∫=

φσ′ φ′ : α 1→( )→

φσ′ 1 : α 0→( )→

sf1

2 πφσ′⁄

-----------------δ k ks⁄ 1–( )

x 0=L t( )

----------------------------------------=



(J-127)

Assuming erosion can occur immediately after transition, the form of the depositedporosity can be written as

(J-128)

where is the erosion coefficient and

External Cake Filtration Equations The external filter cake thickness can be calculated from conservation of mass

(J-129)

where is the cake thickness, is the filter cake porosity, is the particu-

late concentration available for deposition in the cake, and is the time dependent

velocity at the leading edge of the filter cake.

From mass conservation of liquid, at the leading edge of the external cake we have

where is the incident velocity at the face of the external cake and is the inci-

dent velocity of the fluid at the cake/formation interface (note: , see

also Eq. (J-99)). Therefore, the filter cake thickness as a function of the leakoffvelocity is

(J-130)

σ y t,( ) V″ t τ–( )λcs 0( )e λy–=

σσ*------ σ

σ*------

min

σσ*------

min1–⎝ ⎠

⎛ ⎞ αeΔ– V″ V″*⁄( )exp–=

αe

ΔV″ t τ–( ) V″ V″* for V″ V″*>( )–=

δc t( ) 1 φc–( ) cs vs td0

t

∫=

δc t( ) φc cs

vs

vs 1 cs–( ) vl=

vs vl

vscs vlcl=

δc t( ) 11 φc–( )

-------------------cs

1 cs–------------- vl td

0

t

∫=

11 φc–( )

-------------------cs

1 cs–-------------V″ t( )=



where the volume loss per unit area is

(J-131)

Since the external filter cake does not start to develop until after transition,

, Eq. (J-130) must be written as

(J-132)

where for .

The pressure drop across the filter cake from Darcy’s law and Eq. (J-132) is

(J-133)

Eq. (J-133) is identical to the pressure drop equation given by Tongchun (1994) fora Newtonian fluid flowing through an external filter cake.

The effective external skin based on the filter cake thickness at the wellbore with from Eq. (J-70) is

(J-134)

Filter Cake Coefficient, Thickness, and Other RelationshipsA more general form for the filter cake thickness as a function of time (see also Eq.(J-130)) is

(J-135)

V″ t( )

V″ t( ) vl td0

t

∫=

V″ t( ) V″*>

δc t( ) 11 φc–( )

-------------------cs

1 cs–------------- V″ t( ) V″*–[ ]=

δc 0= V″ t( ) V″*<

Δpsμckc-----δcv=

μckc----- 1

1 φc–( )-------------------

cs1 cs–------------- V″ t( ) V″*–[ ]v=

δc δc t( ) φ′⁄→

sf external

12 πφ′⁄---------------

δc t( )L t( )------------ k μ⁄

kc μc⁄--------------

⎩ ⎭⎨ ⎬⎧ ⎫

=

tdd δc t( ) 1

1 φc–( )-------------------

cs1 cs–-------------vl=



Substituting Darcy’s law (Eq. (J-67)) into Eq. (J-135), we find

(J-136)

Integrating Eq. (J-136) over time, the filter cake thickness as a function of time isfound to be

(J-137)

where we have assumed that the differential pressure, mobility, porosity and partic-ulate concentration are constant.

Eq. (J-137) illustrates, that for the above simplifying assumptions, the filter cakethickness grows proportional to the square root of time. Consequently, with the aidof Darcy’s law we have

(J-138)

where is the wall building coefficient and is given by

(J-139)

The filter cake thickness in terms of the filter cake coefficient from Eq. (J-135) andEq. (J-138) is

or

(J-140)

Inversely, the filter cake coefficient as a function of the cake thickness is

tddδc t( )2

21 φc–( )

-------------------cs

1 cs–-------------=

kcμc-----Δpc

δc t( ) 11 φc–( )

-------------------cs

1 cs–-------------⎝ ⎠

⎛ ⎞ 2kcΔpcμc

----------------- t=

vlkcμc-----

Δpcδc t( )------------

CIII

t---------==

CIII

CIII 1 φc–( )1 c– s

cs-------------⋅

kcΔpc2μc

--------------=

tdd δc t( ) 1

1 φc–( )-------------------

cs1 cs–-------------

CIII

t---------=

δc t( ) 11 φc–( )

-------------------cs

1 cs–-------------2CIII t=



(J-141)

Filter Cake ResistanceThe external filter cake resistance, , is defined as

where is pressure loss across the cake, is the leakoff velocity, and is the

fluid leakoff viscosity.

The pressure loss across the filter cake from Darcy’s law (Eq. (J-67)) is

(J-142)

where the flow resistance term, , is shown to be a function of filter cake thick-

ness and permeability. The external cake resistance can also be written as

(J-143)

which is the same definition presented by Mayerhofer, Economides and Nolte(1991).

Consequently, from Eq. (J-130) the filter cake resistance as a function of the partic-ulate concentration and volume loss per unit area is

(J-144)

The total leakoff rate over the fracture face is

(J-145)

CIIIδc t( )

2 t------------

1 cs–cs

-------------⎝ ⎠⎛ ⎞ 1 φc–( )=

Rs

RsΔpsvμc---------≡

Δps v μc

Δps vμδckc----- vμRs= =

Rs

Rsδckc-----=

Rsδckc----- 1

kc---- 1

1 φc–( )-------------------

cs1 cs–-------------V″ t( )

⎩ ⎭⎨ ⎬⎧ ⎫

= =

q 4hL( )Δpsμc

--------- 1Rs-----=



The pressure drop across the cake in terms of the flow rate and flow resistance is

(J-146)

The fracture skin in terms of external filter cake resistance from Eq. (J-66) and Eq.(J-146) is

(J-147)

The external skin for a propagating fracture from Eq. (J-75) is

(J-148)

where is the flow resistance at the wellbore.

Filter Cake Build RateThe filter cake thickness build rate from Eq. (J-132) is shown to be directly propor-tional to the volume loss per unit area. This also is the conclusion stated by Perkins“according to usual filtration theories, the resistance should be directly proportionto the volume loss per unit area”.

However, a more general form of Eq. (J-132) is

(J-149)

where is the volume loss per unit area after transi-

tion and is the growth rate coefficient. For , the filter cake thickness is

proportional to the volume loss per unit area, .

Rewriting Eq. (J-149), we have

Δpsqμc4h---------

RsL-----=

qμ2πkh------------- π

2---

μcμ-----

RskL

--------⎝ ⎠⎛ ⎞

⎩ ⎭⎨ ⎬⎧ ⎫

=

sf external

π2---

μcμ-----

RskL

--------⎝ ⎠⎛ ⎞=

sf external

12 π⁄ φ′---------------

μcμ-----

RskL

--------⎝ ⎠⎛ ⎞=

Rs

δc t( ) ΔV″ t( )βg∝

ΔV″ t( ) V″ t( ) V″*–[ ]=βg βg 1=

V″ t( )



(J-150)

where

(J-151)

The filter cake resistance for a constant cake permeability, , is also of the form

(J-152)

In general, both the cake thickness and permeability may be non-linear functions ofthe volume loss as described by Eq. (J-152).

Filter Cake Erosion RateErosion of the filter cake can also take place during produced water reinjection.Defining the erosion to build rate as

(J-153)


(J-154)


to . The volume loss at the minimum filter cake thickness from Eq. (J-

150) is

(J-155)

The change in the filter cake thickness during the erosion process is

δc t( ) δmaxΔV″ t( )ΔVmax

″-----------------

βg=

δmax1

1 φc–( )-------------------

cs1 cs–-------------ΔVmax

″=

kc

Rs t( ) RmaxΔV″ t( )ΔVmax

″-----------------

βg=

ω V″ddδ

e V″ddδ

b⁄

ΔVb″

ΔVe″

----------= =

δmin δmax

ΔVb″ ΔVmax

″ ΔVmin″–=

ΔVe″

δmax δmin

ΔVmin″ ΔVmax

″δminδmax-----------⎝ ⎠

⎛ ⎞1 βg⁄

=



(J-156)



(J-157)


last maximum value.


(J-158)

or

The filter cake thickness during the build cycle after the first erosion cycle is

(J-159)

where

(J-160)


last minimum value.

J.7 MPwri Input Dialog NomenclatureFollowing is a list of some of the input data required for calculating the internal andexternal filter cakes:

1.Deposited Concentration Ratio at Transition,


″----------⎝ ⎠

⎛ ⎞ βe=

βe


ΔV″ tmax( )



δc t( ) δmaxδV″ t( ) ΔVmin

″+ΔVmax

″--------------------------------------

βg

=

δV″ ΔV″ t( ) ΔV″ tmin( )–=

ΔV″ tmin( )

σ* φ0⁄


J.1 References 697

2.Permeability Damage Equation,

3.Permeability Damage Factor, 4.Permeability Damage Coefficient,

5.Cake Build Coefficient,

6.Cake Erosion Coefficient, 7.Cake Erosion to Build Rate Ratio,

J.1 References1. Perkins, T.K. and Gonzalez, J.A.: “The Effect of Thermoelastic Stresses on the

Injection Well Fracturing,” SPE Journal, February 1985, 78-88.

2. Koning, E.J.L.: “Fractured Water Injection Wells - Analytical Modeling ofFracture Propagation,” SPE 14684, August 1985.

3. Howard, G.C. and Fast, C.R.: Hydraulic Fracturing, Monograph Vol. 2, SPE,1970, 33.

4. Meyer, B.R. and Hagel, M.W.: “Simulated Mini-Frac Analysis”, PetroleumSociety of CIM, Calgary, June 1988.

5. Gringarten, A.C., Ramey, H.J., and Raghavan, R.: “Unsteady-State PressureDistributions Created by a Well with a Single Infinite-Conductivity Fracture,”SPEJ, August 1974, 347-360.

6. Earlougher, R.C.: Advances in Well Test Analysis, Monograph Vol. 5, SPE,1977.

7. Holman, J.P.: Heat Transfer, McGraw-Hill, Inc., NY, 1977, 102.

8. Lee, S.T., and Brockenbrough, J.R.: “A New Analytical Solution for FiniteConductivity Vertical Fractures with Real Time and Laplace Space ParameterEstimation,” SPE 12013, 1983.

9. van den Hoek, P.J.: “A Simple and Accurate Description of Non-Linear FluidLeak-off in High-Permeability Fracturing,” SPE 63239, October 2000.

ksk---- 1

1 β σ σ*⁄( )α+-----------------------------------=

β α

βb, Δδ ΔV″( )βb ∝

βe, δc t( ) δ– max δV″( )–βe ∝

ω, ω dδe dV″⁄( ) dδb dV″⁄( )⁄–=



10. Economides, M.J., and Nolte, K.G.: Reservoir Simulation, Schlumberger Edu-cational Services, 1987, 1-10.

11. Wennberg, K.E. and Sharma, M.M.: “Determination of the Filtration Coeffi-cient and the Transition Time for Water Injection Wells,” SPE 38181, June1987.

12. Pang, Shutong and Sharma, M.M.: “A Model for Predicting Injectivity Declinein Water Injection Wells,”, SPE 28489, September 1994.

13. Bachman, R.C., Harding, T.G., Settari, A., and Walters, D.A.: “Coupled Simu-lation of Reservoir Flow, Geomechanics, and Formation Plugging with Appli-cation to High-Rate Produced Water Reinjection,” SPE 79695, February 2003.

14. Mayerhofer, M.J., Economides, M.J. and Nolte, K.G.: “An Experimental andFundamental Interpretation of Fracturing Filter-Cake Fluid Loss,” SPE 22873,October, 1991.

15. Tongchun, Yi and Panden, J.M.: “A Comprehensive Model of Fluid Loss inHydraulic Fracturing,” SPE Production & Facilities Journal, November, 1994.


Appendix K

After-Closure Analysis

K.1 IntroductionThe solution methodology for determining formation permeability after hydraulicfracturing is formulated in this report. A summary of the governing linear andradial flow equations are presented along with the graphical method to determinepermeability and reservoir pressure from the infinite-acting time period.

The purpose of this report is to set forth the methodology and documentation of thegoverning equations for after-closure analysis as originally presented by Gu et al.(1993) and Nolte (1997). The formulation is developed for linear and then radialflow. Although, the main focus is on the asymptotic solution for large times (i.e.,radial flow) from which the formation permeability can be determined, the linearsolution is important since it sets forth the frame work Nolte used to derive theeffective or an apparent closure time used in his radial time function.

Since Nolte (1997) provided a background formulation for his after closure analy-sis, the only intent of this report is to supplement Nolte’s original work and providea framework of understanding (possibly only for my benefit and enjoyment) of theunderlaying formulation, boundary conditions, assumptions, etc. This is of coursein lieu of just using Nolte’s equations and skipping directly to the implementationsection of the after closure analysis found in Section 6.

This report contains the following sections:

1. Abstract: After Closure Analysis Overview.

2. Superposition: Duhamel’s superposition theorem is presented. This is the foun-dation used to solve the response of a linear homogeneous system with timedependent (unsteady) boundary conditions (i.e., a pump-in followed by a shut-in).


700 After-Closure Analysis:

3. Impulse Injection: Classic impulse injection solution by Gu et. al. (1993) fromwhich all pressure responses (unfractured and fractured wells) must asymptoteto at infinite-acting times.

4. Linear Solution: Presents the formulation of Nolte’s (1997) linear fluid losstime function and apparent closure time, .

5. Radial Solution: Presents the formulation of the pressure response for the infi-nite acting time period. Solutions are presented for the Horner, , and

Nolte, , time functions.

6. Summary and Implementation: The methodology used for the implementationof the after closure analysis to determine the initial reservoir pressure and per-meability is discussed with the aid of various graphical techniques.

K.2 Superposition or Duhamel’s Theorem: General Solution

Duhamel’s theorem is the general theory of superposition used to solve theresponse of a linear homogeneous system with time dependent boundary conditionsby using the solution to a corresponding fundamental problem with steady stateboundary conditions (see Myers (1971, pg 153)).

If is the response of a linear homogeneous system (initially at zero) to a sin-gle unit step input, the response of the system to the input (in place of the unitstep) is given by either

(K-1)

where N is the number of finite discontinuities between and , or by

(K-2)

Myers also addresses the ability of Duhamels’s method to handle one non-homoge-neous boundary condition.

FL t· tc,( ) χtc

Fh t tξ,( )

FR t tc,( )

U x θ,( )

F θ( )

u x θ,( ) U x θ, τ–( )F′ τ( ) τdτ 0=

θ

∫ U x θ, τj–( ) FjΔ

j 1=

N

∑+=

τ 0= τ θ=

u x θ,( ) F τ( )θ∂

∂ U x θ, τ–( ) τdτ 0=

θ

∫=


K.3 Impulse Injection 701

The general solution in terms of a dimensionless pressure , to the radial dif-

fusivity equation at the wellbore for a constant production rate, , is (Drake (1990,pg 166))

(K-3)

where is the formation permeability, is the formation height, is the forma-tion viscosity, is initial reservoir pressure, is the pressure at any dimensionless

position , is the well skin, is the dimensionless time given by

(K-4)

where is time, is the formation compressibility, and is the wellbore radius.

The general solution for a multi-rate draw-down using Duhamel’s theorem is

(K-5)

where the system input boundary condition has been substituted

and is an arbitrary constant reference flow rate.

The general form of the net draw-down pressure for multiple constant injectionrates from Eq. (K-5) is

(K-6)

The reader is also referred to Earlougher (1977) and Drake (1990) for similar repre-sentations of the multi-rate drawdown pressure response at the wellbore.

K.3 Impulse InjectionGu et. al. (1993) first presented the theory and analysis that “the late-time pressurebehavior after fracture closure is like that of an instantaneous source solution,

pD tD( )

q

2πkhqμ

------------- pi pwf–( ) pD tD( )= S+

k h μ

pi p

rD r rw⁄= S tD

tDkt

φμctrw2

-----------------=

t ct rw

2πkhqμ

------------- pi p r t,( )–[ ] pD rD t, D tDj 1––( ) S+[ ]

qj qj 1––q

---------------------×

j 1=

N

∑=

ΔFjqj qj 1––

q----------------------=

q

pi p r t,( )– μ2πkh------------- pD rD tD, tDj 1–

–( ) S+[ ] qj qj 1––( )×

j 1=

N

∑=



whether the formation is fractured or not during the injection.” The importance ofGu’s analysis is that the pressure response at late time is independent of the fracturegeometry. Consequently, any late time fracture solution must asymptote to theinstantaneous line source solution.

The pressure response in the reservoir after an instantaneous injection or productionas given by Gu is

(K-7)

where is the elapsed time since shut-in and is the injection or produced vol-ume. In late time, when is large, the exponent in Eq. (K-7) approaches zero andthe pressure at the wellbore simplifies to

(K-8)

Eq. (K-7) can be derived using Duhamel’s theorem.

The boundary conditions for a well flowed or produced at a constant injection rate for a total time and then a shut-in are

(K-9)

The pressure solution using Duhamel’s method of superposition from Eq. (K-6)with the above boundary conditions is

(K-10)

where and is assumed positive for injection an negative for produc-tion.

The dimensionless pressure solution as given by Earlougher (1977) for an infinite-acting system is

(K-11)

Δp r Δt,( )V0μ

4πkhΔt-------------------e

φμcr2

4kΔt--------------–

=

Δt VΔt

Δp Δt( ) Vμ4πkh------------- 1

Δt-----=

q tp

t tp q; 1≤ q=

t tp q; 2> 0=

Δp r t,( ) μq2πkh------------- pD rD tD,( ) pD r t, tp–( )D–{ }×=

Δp p pi–= q

pD12---Ei

rD2

4tD--------–⎝ ⎠

⎛ ⎞– 12---E1

rD2

4tD--------–⎝ ⎠

⎛ ⎞= =


K.3 Impulse Injection 703

when . Here is the exponential integral and is the exponentialintegral of the first kind.

Substituting Eq. (K-11) into Eq. (K-10) we find

(K-12)

The pressure response for an impulse injection (i.e., as or ) is found bydifferentiating the numerator and denominator of the second term in Eq. (K-12)

(K-13)

The pressure response at large times is

(K-14)

Eq. (K-14) can also be derived from the late time approximation of Eq. (K-11)given by

(K-15)

tD rD2⁄ 25≥ Ei E1

Δp r t,( )μqtp2πkh-------------

12---– Ei

rD2

4tD--------–⎝ ⎠

⎛ ⎞ 12---Ei

rD2

4 t tp–( )D-----------------------–

⎝ ⎠⎜ ⎟⎛ ⎞

+⎩ ⎭⎨ ⎬⎧ ⎫

tp-------------------------------------------------------------------------------------×=

μV4πkh-------------

E1rD

2

4tD--------–⎝ ⎠

⎛ ⎞ E– 1rD

2

4 t tp–( )D-----------------------–

⎝ ⎠⎜ ⎟⎛ ⎞

tp------------------------------------------------------------------×=

tp 0→ tp t«

Δp r t,( ) μV2πkh-------------

tpdd E– 1

rD2

4 t tp–( )D-----------------------–

⎝ ⎠⎜ ⎟⎛ ⎞

⎩ ⎭⎨ ⎬⎧ ⎫

dtp dtp⁄--------------------------------------------------------×=

μV4πkh------------- e

rD2 4 t tp–( )D⁄–

t tp–-----------------------------×=

μV4πkh------------- e

φμcr2

4kΔt--------------–

Δt----------------×=

Δp t( ) μV4πkh------------- 1

Δt-----×=

pD12--- tD rD

2⁄( )ln 0.80907+[ ]=



when (or within 1-percent error when ).

Substituting Eq. (K-15) into Eq. (K-10), the infinite-acting period solution becomes

(K-16)

where or is recognized as the Horner time.

For a pulse injection or in the limit as we find

(K-17)

where is the slope of vs .

Gu states that “If the pressure is plotted against in Cartesian coordinates,the late time portion of the curve should follow a straight line. The permeability can be calculated from the slope of the straight line from the following expres-sion:

(K-18)

The apparent reservoir pressure ( )can be found from the interceptof the extension of the straight line with the axis.”

Gu also states that if one takes the derivative of pressure in Eq. (K-17) with respectto , we have

(K-19)

Consequently, a plot in Cartesian coordinates of and should coin-cide with the pressure curve at late time. This property can be used as a diagnosticplot to help determine and the late time slope from which the permeability canbe calculated.

tD rD2 100>⁄ tD rD

2 10>⁄

Δp r t,( ) μV2πkh-------------

12--- tD rD

2⁄( )ln 0.80907+[ ] 12--- t tp–( )D rD

2⁄( )ln 0.80907+[ ]–⎩ ⎭⎨ ⎬⎧ ⎫

tp----------------------------------------------------------------------------------------------------------------------------------------------------×=

μV4πkh-------------

t t tp–( )⁄[ ]lntp

--------------------------------× μV4πkh-------------

1 tp Δt⁄+[ ]lntp

---------------------------------×==

t t tp–( )⁄[ ]ln 1 tp Δt⁄+[ ]ln

tp Δt«

Δp t( ) μV4πkh------------- 1

Δt-----×= m 1

Δt-----×=

m p 1 Δt⁄

p 1 Δt⁄

km

k μV4πhm--------------=

p* Δp p p*–=1 Δt⁄ 0=

Δt( )ln

dp–d Δt( )ln------------------- Vμ

4πkh------------- 1

Δt-----×= m 1

Δt-----×=

p dp– d Δt( )ln⁄

p*


K.4 Linear Solution - Background 705

K.4 Linear Solution - BackgroundThe differential equation for the pressure distribution in the formationassuming one-dimensional (linear) fluid loss from a fracture is

(K-20)

where the reservoir viscosity ( ), permeability ( ), porosity ( ), and compress-

ibility ( ) effect the pressure transient.

Constant Velocity Boundary ConditionThe boundary and initial conditions, assuming that the reservoir is infinite and thatthe leakoff velocity is a constant

(K-21)

The solution for this case as given by Holman (1977, page 104) is

(K-22)

where

(K-23)

At the fracture face ( ) the pressure is

(K-24)

The velocity in terms of the changing net leakoff pressure differential is

p x t,( )

kcφμ---------

y2

2

∂

∂ p⎝ ⎠⎜ ⎟⎛ ⎞

t∂∂p=

μ k φc

v0

p y 0,( ) pi=

v 0 t,( ) v0kμ---

y∂∂p

⎝ ⎠⎛ ⎞

y 0=– for t 0>= =

p y t,( ) pi– 2v0μ

k-------- λt π⁄ y2

4λt--------–⎝ ⎠

⎛ ⎞ v0μyk

----------- 1 erf y2 λt------------–⎝ ⎠

⎛ ⎞–exp=

λ kcφμ---------=

y 0=

p 0 t,( ) pi– 2π---

v0μk

-------- πλt=



(K-25)

where and .

The total volume loss at time is

(K-26)

where the fracture area , is the fracture half-length, and is the forma-tion height.

Placing Eq. (K-24) in terms of rate and dimensionless time ,we find

(K-27)

Defining the net differential pressure in terms of a dimensionless

pressure function, , we have

(K-28)

The dimensionless pressure for linear fluid loss from Eq. (K-27) is found to be

(K-29)

Constant Pressure Boundary ConditionThe boundary and initial conditions, assuming that the reservoir is infinite and thata constant pressure exists in the fracture between the fracture and the reservoir, are

v0π2--- k

μ--- Δp

πλt------------- π

2---

CII

t-------= =

Δp p 0 t,( ) pi–= CIIkμ--- Δp

πλ-----------=

tp

Vl tc( ) 4Av0tp 4Aπ2---

CII tp( )

tp

----------------= =

2πCII tp( )A tp=

A Lh= L h

q Vl tp⁄= tD λt L2⁄=

p 0 t,( ) pi– μq2πkh------------- πtD=

Δp p 0 t,( ) pi–=

pD

Δp μq2πkh-------------pD t( )=

pD πtD=



(K-30)

The solution to Eq. (K-20) as given by Holman (1977, page 102) is

(K-31)

The velocity at any position in the formation from Darcy’s law is

(K-32)

Performing the partial differentiation of Eq. (K-31) gives

(K-33)

The leakoff velocity at the fracture face from Eq. (K-32) and Eq. (K-33) is

(K-34)

where the leakoff coefficient is given by

(K-35)

and .

This is the same linear velocity formulation as presented by Howard and Fast(1970, pg 35).

The total volume loss for a static (non-propagating) fracture with a constant pres-sure boundary condition at time is

p y 0,( ) pi=

p 0 t,( ) pf for t 0>=

p y t,( ) pf–pi pf–

-------------------------- erf y2 λt------------⎝ ⎠

⎛ ⎞=

y

v kμ---

y∂∂p

⎝ ⎠⎛ ⎞–=

y∂∂p pi pf–

πλt-------------- y2

4λt--------–⎝ ⎠

⎛ ⎞exp=

v kμ---

y∂∂p

⎝ ⎠⎛ ⎞

y 0=– k

μ--- Δp

πλt-------------

CII

t-------= = =

CIIkμ--- Δp

πλ----------- Δp kφc

πμ---------= =

Δp pf pi–=

tp



(K-36)

Rearranging Eq. (K-34) in terms of flow rate, , we have

(K-37)

Defining the fluid loss rate in terms of a dimensionless rate function, , we have

(K-38)

where

(K-39)

The general dimensionless rate solution as a function of dimensionless pressure fora non-propagating fracture is

(K-40)

Time Dependent Velocity Boundary ConditionNolte (1997) first presented a closed form solution for “the changed-boundary con-dition and an infinite-length fracture, or equivalently for linear flow from a fixedlength fracture, the constant-pressure condition for pre-closure” followed by a zero-flux condition. Nolte’s solution was based on the work of Carslaw and Jaeger. Thefinal closed form solution is

(K-41)

where is the pressure differential carried by the reservoir and the lea-

koff velocity at closure is given by .

Vl tc( ) 4A v td0

tp

∫ 4ACII1t

----- td0

tp

∫= =

8CIIA tp=

q 4vA=

q t( ) 4kAμ

---------- Δpπλt

------------- 2πkhμ

-------------Δp 2 π⁄πtD

-------------= =

qD

q t( ) 2πkhΔpμ

--------------------qD t( )=

qD 2 π⁄( ) πtD⁄=

qD tD( ) 2 π⁄pD tD( )----------------=

ΔpR vc tcπμkcφ--------- 2

π---arcsinθ 1 2⁄–

⎩ ⎭⎨ ⎬⎧ ⎫

=

ΔpR pR pi–=

vc CT tc⁄=



The dimensionless linear time function from Eq. (K-41) is

(K-42)

Following is a more detailed formulation of Nolte’s linear time function and how itcan be derived using Duhamel’s superposition method.

Variable Injection Rate followed by a Shut-inThe homogeneous pressure response solution for a non-propagating fracture with aconstant leakoff velocity from Eq. (K-24) is

(K-43)

or

(K-44)

where is the response of a linear homogeneous system (initially at zero) to asingle unit step input and .

The boundary condition for a typical fracture from initiation to closure is one ofessentially a constant pressure in the fracture during propagation up to closure andthen a zero leakoff velocity at the fracture face during shut-in.

To obtain a constant pressure in the fracture to the time of closure , the velocity atthe fracture face from Eq. (K-34) must be of the form

or (K-45)

where is the leakoff velocity at .

The time dependent Duhamel’s step function for this case is

F θ( ) 2π---arcsinθ 1 2⁄–=

vc

p pi– 2π---vc t kcφ

πμ---------⁄=

U θ( )p pi–

vc tc

------------- kcφπμ--------- 2

π--- θ= =

U θ( )

θ t tc⁄=

tc

v 1t

-----∝ vvc----

tct--- 1

θ-------= =

vc tc



(K-46)

where .

The response of the system to the input (in place of the unit step) is given by

(K-47)

where .

Substituting the change of variable into Eq. (K-47), we find

(K-48)

Simplifying, we find

(K-49)

The unsteady pressure solution from Eq. (K-44) is

(K-50)

or

F τ( ) vvc---- τ 1 2⁄– and F′ τ( ) 1– 2τ 3 2⁄–⁄= τ 1≤( );= =

F τ( ) 0 and F′ τ( ) 0= τ 1>( );=τ t tc⁄=

F τ( )

u θ( ) U θ τ–( )F′ τ( ) τdτ 0+=

1

∫ U θ τj–( ) FjΔ

j 1=

2

∑+=

1π--- θ τ–

τ3 2⁄--------------- τd

τ 0+=

1

∫– U θ( ) F 0+( ) U θ 1–( ) F 1( )Δ–Δ+=

F 1( )Δ 1–=

ξ2 τ=

u θ( ) 2π--- θ ξ2–

ξ2------------------ ξd

0+

1

∫– θ F 0+( )Δ θ 1––+⎩ ⎭⎨ ⎬⎧ ⎫

=

2π---= θ ξ2– ξ⁄ arcsinξθ 1 2⁄–+( )

0+

1θ F 0+( )Δ θ 1––+

⎩ ⎭⎨ ⎬⎧ ⎫

2π--- θ 1– θ F 0+( )Δ– arcsinθ 1 2⁄–+ θ F 0+( )Δ θ 1––+{ }=

u θ( ) 2 π⁄ arcsinθ 1 2⁄–=

p pi–

vc tc

------------- kcφπμ--------- 2 π⁄ arcsinθ 1 2⁄–=



(K-51)

where .

Placing Eq. (K-51) in terms of the dimensionless pressure ratio at closure, we have

(K-52)

where the pressure at closure ( ) from Eq. (K-51) is

(K-53)

Eq. (K-51) is the closed-form solution initially given by Nolte (1997a, Eq. (K-5),pg.5) for the constant-pressure condition for pre-closure and a zero-flux conditionafter closure.

Constant Injection Rate followed by a Shut-inThe pressure response for a constant injection rate followed by a shut-in is given bythe boundary conditions

(K-54)

The general form of the pressure solution for multiple injection rates using the prin-ciple of superposition from Eq. (K-6) is

(K-55)

The dimensionless pressure solution for linear fluid loss from Eq. (K-29) is

p pi– vc tcπμ

kctφ---------- 2


⎩ ⎭⎨ ⎬⎧ ⎫

=

CTπμ

kctφ---------- 2


⎩ ⎭⎨ ⎬⎧ ⎫

=

vc CT tc⁄=

Fp pi–

p tc( ) pi–---------------------- 2 π⁄ arcsinθ 1 2⁄–==

θ 1=

p tc( ) pi– CTπμ

kctφ----------=

q1 q t tc≤;=

q2 0 t tc>;=

pf pi– μq2πkh------------- pD t( )D pD t tc–( )D–[ ]=



(K-56)

Substituting Eq. (K-56) into Eq. (K-55) we find

(K-57)

Placing Eq. (K-57) in terms of the dimensionless pressure ratio at closure we have

(K-58)

where the pressure at closure ( ) from Eq. (K-58) is

(K-59)

The dimensionless fluid loss time function in Eq. (K-58) can also be written inthe form

(K-60)

where is the shut-in time.

Apparent Closure TimeNolte (1997a) devised an apparent closure time defined as where .This apparent closure time represents an equivalent time of exposure and if usedwith Eq. (K-60) will approximate the more rigorous linear time function given by

pD πtDπkt

ctφμL2-----------------= =

pf pi– μq2πkh------------- πkt

ctφμL2-----------------

πk t tc–( )

ctφμL2-----------------------–=

q2πhL------------- πμ

kctφ---------- t t tc––[ ]=

4v tc2π

-------------- πμkctφ---------- θ θ 1––[ ]=

FLp pi–

p tc( ) pi–---------------------- θ θ 1––==

θ 1=

p tc( ) pi–4v tc

2π-------------- πμ

kctφ----------=

FL

FL t tc,( ) 1 t tc–( ) tc⁄+ t tc–( ) tc⁄–=

1 Δt tc⁄+ Δt tc⁄–=

Δt t tc–=

χtc χ 16 π2⁄=



Eq. (K-52). Nolte also used this apparent closure time in the radial solution formu-lation based on a constant injection rate followed by a shut-in.

Implementation of the apparent closure time in Eq. (K-60) we find

(K-61)

Simplifying Eq. (K-61) for large time ( )

or

Equating Eq. () to Eq. (K-52) for , a value of that fits the infinite-acting timeperiod can be found

(K-62)

Solving for we find

. (K-63)

The maximum error between Eq. (K-61) and Eq. (K-52) with is 3.5% at. The error diminishes toward zero as increases with an error less of

than 1% for .

Linear Solution SummaryThe governing pressure response equation for after closure analysis from Eq. (K-51) is

where

χtc

FL 1 t tc–( ) χtc⁄+ t tc–( ) χtc⁄–=

θ 1»

FL θ χ⁄ χ θ⁄ 1+ 1–( ) θ χ⁄ 1 χ 2θ( )⁄ 1–+( )→ →

FL12--- χ θ⁄→

θ 1» χ

12--- χ θ⁄ 2 π⁄ arcsinθ 1 2⁄– 2 π⁄ 1 θ⁄==

χ

χ 16 π2⁄=

χ 16 π2⁄=θ 1.35= θ

θ 7>

p pi– mLFL t tc,( )=



.

The dimensionless linear function can also be set equal to since

this is the exact function from which was developed. That is

K.5 Radial Solution - Infinite-acting time periodHorner (1951) first proposed that for any pressure build up test, the bottom-holeshut-in pressure response could be expressed using the principle of superpositionfor a well producing at a rate until time , and a zero rate thereafter (see Ear-lougher (1977), pg. 45). The pressure response after shut-in is

(K-64)

During the infinite-acting time period, if there are no fractures or storage effects,the exponential integral for could be replace by the logarithmic approximation

for

(K-65)

where .

Substituting Eq. (K-65) into Eq. (K-64), the bottom-hole pressure response is foundto be

mL CTπμ

kctφ----------=

FL t tc,( ) 1 t tc–( ) χtc⁄+ t tc–( ) χtc⁄–=

χ 16 π2⁄=

FL t tc,( ) F t tc,( )

χ

FL t tc,( ) F t tc,( )→ 2π---arcsin tc t⁄=

q tp

pws pi– μ2πkh------------- pD tD tDj 1–

–( ) S+[ ] qj qj 1––( )×

j 1=

N

∑=

μ2πkh-------------q pD t( )D pD t tp–( )D–[ ]=

pD

tD 100>

pD12--- tD 0.80907+ln( )=

tD λt rw2⁄=


K.5 Radial Solution - Infinite-acting time period 715

(K-66)

where the shut-in time is given by . Horner noted that Eq. (K-66)

describes a straight line with intercept which will be equal to the initial reser-voir pressure for an infinite acting system and slope , where

(K-67)

The reservoir permeability can then be estimated from the slope

(K-68)

This is the proposed methodology by Horner for determining formation permeabil-ity. Thus a plot of vs. is commonly called the Horner plot.

To distinguish between closure and pump time, the nomenclature for Horner timewill be

(K-69)

where and is either equal to or .

Clearly for , the Horner time asymptotes to and Eq. (K-66)becomes

(K-70)

where is the total produced or injected volume. This equation is identicalto the impulse solution given by Eq. (K-14).

Following is the after closure analysis for fractured systems based on Horner’s timeand the methodology of Nolte.

Horner TimeThe general form of dimensionless pressure solutions for a static (non-propagating)fracture in an infinite system is

pws pi– m–tp Δt+

Δt----------------⎝ ⎠

⎛ ⎞ln=

Δt t tp–=

p*pi m–

m qμ4πkh-------------=

k qμ4πmh--------------=

p 1 tp Δt⁄+[ ]ln

Fh t tξ,( ) 1 tξ Δt⁄+[ ]ln=

Δt t tξ–= tξ tp tc

Δt tξ» Fh t tξ,( ) tξ Δt⁄→

Δp t( ) μq4πkh-------------

tξ

Δt-----× μV

4πkh------------- 1

Δt-----×= m 1

Δt-----×= =

V qtξ=



(K-71)

where is a constant (i.e., for a uniform flux and for an infinite con-ductivity fracture ). Eq. (K-71) is with less than 1% error of the exact

solution when ( ).

The pressure response for a constant injection period followed by a shut-in fromEq. (K-6) is

(K-72)

Substituting Eq. (K-71) into Eq. (K-72) and rearranging

(K-73)

where the injection rate and is the total volume injected.

Eq. (K-73) can also be written in terms of a Horner slope, , and time, ,

(K-74)

The Horner time based on closure is

(K-75)

where is the time after closure.

The slope in Eq. (K-74) is given by

(K-76)

For large times Eq. (K-75) becomes

pD12--- tDxf β+ln( )=

β β 2.80907=β 2.2000=

tDxf 10> tDxf λt L2⁄=

pf pi– μ2πkh-------------q pD t( )D pD t tc–( )D–[ ]=

pf pi– μ4πkh-------------V

tc---

t( )Dt tc–( )D

-------------------ln=

μ4πkh-------------V

tc--- t

t tc–-----------⎝ ⎠

⎛ ⎞ln=

q V tc⁄= V

mh Fh

pf pi– mh Fh t tc,( )×=

tc

Fh t tc,( ) tt tc–-----------⎝ ⎠

⎛ ⎞ln 1 tc Δt⁄+( )ln= =

Δt t tc–=

mh

mhμ

4πkh-------------V

tc---=


K.5 Radial Solution - Infinite-acting time period 717

(K-77)

and Eq. (K-73) simplifies to

(K-78)

Eq. (K-78) illustrates that at long times the radial solution for a fractured systemalso asymptotes to the impulse solution given by Eq. (K-14).

Nolte’s After-Closure Radial Time FunctionNolte’s derived his radial time function, , by substituting the apparent clo-

sure time, , into Eq. (K-75) and noting that for large times

and

(K-79)

The Nolte pressure response from Eq. (K-73) with the aid of Eq. () is

(K-80)

Simplifying Eq. (K-79), we find

(K-81)

where the Nolte radial time function is given by

(K-82)

and the slope

Fh t tc,( ) 1 tc Δt⁄+( )ln= tc Δt⁄→

pf pi– μV4πkh------------- 1

Δt-----×=

FR t tc,( )

χtc

1 tc Δt⁄+( )ln tc Δt⁄→

1χ--- 1 χtc Δt⁄+( )ln tc Δt⁄→

pf pi– μ4πkh-------------V

tc--- 1

χ--- 1 χtc Δt⁄+( )ln×=

μπkh---------V

tc--- 1

χ--- 1

4--- 1 χtc Δt⁄+( )ln×=

p pi– mR FR t tc,( )×=

FR t tc,( ) 14--- 1 χtc Δt⁄+( )ln=



(K-83)

where has been substituted.

The choice for (i.e., the 1/4 constant) was based on the asymptotic behav-

ior of at large times. That is

and (K-84)

or

(K-85)

for .

The good fortune is that Eq. (K-85) is a very good approximation to at alltimes.

K.6 Summary and Implementation of After Clo-sure Analysis

The purpose of the after closure analysis is to determine the formation permeabilityand reservoir pressure from the pressure response of a fractured (or unfractured)well during the infinite-acting time period (i.e., late time period or radial solution).The use of Nolte’s linear time function is only used in conjunction with it’s

relationship to the radial time function, . The methodology to determine

fracture characteristics using will not be addressed.

The governing pressure response equations for the radial or infinite-acting timeperiod of Section 5.0 are presented below for the Impulse Injection, Hornerand Nolte Analyses.

Impulse InjectionThe pressure response for a pulse injection, from Eq. (K-14)

mRμ

πkh---------V

tc--- 1

χ--- πμ

16kh------------V

tc---= =

χ 16 π2⁄=

FR t tc,( )

FL

FL12---

χtcΔt-------→ FR

14---

χtcΔt-------→

FR t tc,( ) FL2 t tc,( )→

t tc»

FR t tc,( )

FL t tc,( )

FR t tc,( )

FL t tc,( )

t tc»


K.6 Summary and Implementation of After Closure Analysis 719

(K-86)

where is the slope of vs and .

The permeability can be calculated from the slope of the straight line from thefollowing expression:

(K-87)

The apparent reservoir pressure ( ) can be found from the interceptof the extension of the straight line with the axis.”

Horner TimeThe pressure response from Eq. (K-73) for an injection followed by a shut-in periodin terms of a Horner slope and time is

(K-88)

The Horner time and slope based on a pump ( ) or closure time ( ) is

(K-89)

(K-90)

where

The permeability can be calculated from the slope of the straight line fromthe following expression:

(K-91)

The apparent reservoir pressure can be found from the intercept of the extensionof the straight line with the axis.

p pi– μV4πkh------------- 1

Δt-----×= m 1

Δt-----×=

m p 1 Δt⁄ Δt t tp–=

k m

k μV4πhm--------------=

p* Δp p p*–=1 Δt⁄ 0=

mh Fh

pf pi– mh Fh t tξ,( )×=

tξ tp= tξ tc=

Fh t tξ,( ) 1 tξ Δt⁄+( )ln=

mhμ

4πkh------------- V

tξ----=

Δt t tξ–=

k mh

k μ4πh---------- V

tξ---- 1

mh------×=

p*Fh t tξ,( ) 0=



Nolte After Closure TimeThe pressure response of a fractured well for an injection followed by a shut-inperiod in terms of the Nolte after closure time and slope from Eq. (K-81) is

(K-92)

The Nolte radial time function and slope are given by

(K-93)

and

(K-94)

where .

The Nolte after closure function can also be represented as a function of the follow-ing linear time functions

(K-95)

where

(K-96)

and

(K-97)

The permeability can be calculated from the slope of the straight line fromthe following expression:

(K-98)

The apparent reservoir pressure can be found from the intercept of the extensionof the straight line with the axis.

FR mR

p pi– mR FR t tc,( )×=

FR mR

FR t tc,( ) 14--- 1 χtc Δt⁄+( )ln=

mRμ

πkh---------V

tc--- 1

χ--- πμ

16kh------------V

tc---= =

χ 16 π2⁄=

FR t tc,( ) FL2 t tc,( ) F2 t tc,( )≅ ≅

FL t tc,( ) 1 t tc–( ) χtc⁄+ t tc–( ) χtc⁄–=

F t tc,( ) 2π---arcsin tc t⁄=

k mR

k πμ16h---------V

tc--- 1

mR-------×=

p*Fh t tξ,( ) 0=



Graphical Method

General EquationThe general form of the after closure pressure response is

(K-99)

where is slope, is a time function and is the straight line intercept at.

Permeability and Reservoir PressureAs discussed above, if the pressure is plotted against in Cartesian coordinates,the late time portion of the curve should follow a straight line. The permeability can be calculated from the slope of the straight line (i.e., ). The apparentreservoir pressure can be found from the intercept of the extension of thestraight line with the axis.

Diagnostic Plots and DerivativesDiagnostic plots similar to those used in the regression analysis using the Nolte Gfunction can be used to help identify radial flow (pressure transient). The generalrelationships are given below.

Taking the derivative of Eq. (K-99) with respect to the time function, we find

(K-100)

or

(K-101)

Therefore at late time (small values of ) the measured pressure data should over-lay Eq. (K-101) in Cartesian coordinates. Figure K.1 illustrates the use of Eq. (K-101) by overlaying the derivative function to help identify the intercept (reservoirpressure) and late time slope (permeability).

p p*– m F×=m F p*

F 0=

p Fk

m k 1 m⁄∝

p*F 0=

dp dF⁄ m=

p p* F dpdF-------+=

F



Figure K.1: After Closure Analysis - Surface Pressure vs. Nolte - FR Linear Plot.

If a plot (see Figure K.2)of net pressure vs. is generated, the pres-sure should overlay the following equation

(K-102)

Δp p p*–= F

Δp p= p*– F dpdF-------=



Figure K.2: After Closure Analysis - Delta Pressure vs. Nolte - FR Linear Plot.

Taking the natural log of Eq. (K-102) we find

(K-103)

Therefore, the net pressure curve in log space should also overlay the derivativefunction for radial flow.

Another important derivative is the log slope. Taking the natural log of Eq. (K-99)we find

(K-104)

where for the slope is equal to the net pressure (i.e., ).

Eq. (K-104) also illustrates that if is plotted versus , the log-log slopewill approach unity for large times. That is

as

Δp( )ln F dpdF-------⎝ ⎠

⎛ ⎞ln=

p p*–( )ln m( ) F( )ln+ln=F 1= p p*–( )ln m( )ln=

Δp( )ln F( )ln

d Δp( )lnd F( )ln

-------------------- 1→ F 0→



as shown in Figure K.3.

Figure K.3: After Closure Analysis - Delta Surface Pressure vs. Nolte- FR Log-Log Plot

K.7 References1. Abousieiman, Y, Cheng, A.A-D., and Gu, H: “Formation Permeability Deter-

mination from Micro or Mini-Hydraulic Fracturing.” ASME, Vol. 116, pg.104-114, June, 1994.

2. Benelkadi, S., Belhaouas, R., and Sonatrach, M.S.: “Use of After ClosureAnalysis to Improve Hydraulic Fracturing designs, Application on Algeria’sIn-Adaoi Gas Field,” SPE 80936, March 2003.

3. Drake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Science Pub-lishers, B.V., The Netherlands, 1990, 174.


5. Gu, Hongren, Elbel, J.L, Nolte, K.G., Cheng A.H-D., and Abousieiman, Y.:“Formation permeability Determination Using Impulse-Fracture Injection,“SPE 25425, March 1993.


K.7 References 725

6. Holman, J.P.: Heat Transfer, McGraw-Hill, Inc., NY, 1977, 102.

7. Howard, G.C. and Fast, C.R.: Hydraulic Fracturing, Monograph Vol. 2, SPE,1970, 33.

8. Horner, D.R.: “Pressure Build-Up in Wells,” Proc., Third World Pet. Cong.,The Hague (1951) Sec. II, 503-523. Also Reprint Series No. 9 - Pressure Anal-ysis Methods, SPE AIME, Dallas (1967), 25-43.

9. Myers, G.E.: Analytical Methods in Conduction Heat Transfer, McGraw-Hill,Inc., NY, 1971, 153-160.

10. Nolte, K.G.: “Background for After-Closure Analysis of Fracture CalibrationTests,” SPE 39407, July 1997.

11. Nolte, K.G., Maniere, J.L., and Owens, K.A.: “After-Closure Analysis of Frac-ture Calibration Tests,” SPE 38676, October 1997.


Appendix L

Pseudosteady State Analysis of FiniteConductivity Vertical Fractures

L.1 IntroductionThe solution methodology for pseudosteady behavior of a well with a finite con-ductivity vertical fracture is formulated based on a new reservoir/fracture domainresistivity concept. The formulation encompasses a transformed resistivity domainbased on an equivalent or effective wellbore radius. The resulting pseudosteadysolution is presented in the form of the dimensionless productivity index ( ).

Some of the major advantages of this pseudosteady solution for finite conductivityvertical fractures are 1) the methodology is based on fundamental principles, 2) thesolution is analytical, 3) the equations are formulated for rectangular shaped reser-voirs, and 4) the solution and concepts are easily understood and implemented. Themethodology accounts for a piece wise continuous linearly varying fracture con-ductivity including: proppant tail-ins, over-flushing, pinch zones, choked flow(external skin, fracture), and internal skin mechanisms (reservoir).

A summary of the fundamental building blocks, effective wellbore radius concept,pseudo-skin functions and fracture skin are discussed. An improvement to Gringar-ten’s dimensionless productivity solution for infinite vertical conductivity fracturesin rectangular closed reservoirs is also presented.

This report begins by presenting the dimensionless parameters and definitions,methodology, governing equations, and finally solutions to specific fundamentalfinite conductivity scenarios. Gringarten’s infinite conductivity solution for verticalfractures in a closed rectangular reservoir is then presented. Gringarten’s infiniteconductivity solution that is based on utilizing the uniform flux solution with anequivalent dimensionless fracture position ( ) is then discussed. A modi-fication of Gringarten’s infinite conductivity solution is presented that addresses

JD

xD 0.732=


728 Pseudosteady State Analysis of Finite Conductivity Vertical Fractures:

the singularities and anomalies with the assumption of constant value for . An

analytical expression for the formation shape factor ( ) is presented for rectangu-lar shaped reservoirs. A formulation for internal and external fracture skins is thenpresented.

Finally a discussion on pseudopressure and pseudotime is presented followed bysolutions to the radial diffusivity equation for undersaturated oil (liquids), real gas,and two phase behavior. A presentation on methodology for non-Darcy flow is alsoaddressed.

A summary of the “Pseudosteady-State Analysis of Finite Conductivity VerticalFractures is presented by Meyer (2005).

Dimensionless ParametersThe following dimensionless parameters will be used throughout this report. Thedimensionless pressure ( ) for a constant production rate ( ) is defined as

(L-1)

where is the formation permeability, is the formation height, is the reservoirviscosity, and is the differential pressure (the initial reservoir pres-

sure ( ) minus the flowing pressure ( )).

The dimensionless time based on the drainage area, , is defined as

(L-2)

where is the formation porosity and is the formation compressibility.

The dimensionless rate ( ) for a constant flowing pressure ( ) is defined as

(L-3)

The flow rate as a function of the dimensionless rate is

xD

CA

pD q

pD2πkh

qμ-------------Δp=

k h μ

Δp pi pwf–=

pi pwf

A

tDAkt

cφμA-------------=

φ c

qD pwf

qDμ

2πkhΔp--------------------q t( )=

q t( ) 2πkhΔpμ

-------------------- qD×=


L.2 Pseudosteady Equations 729

where is the constant draw down pressure (the initial reservoir pres-

sure ( ) minus the flowing pressure ( )).

The productivity index ( ) is defined as

(L-4)

where is the flow rate, is the average reservoir pressure, and is the flowing

pressure. The dimensionless productivity index, , is defined as

(L-5)

L.2 Pseudosteady EquationsRamey (1971) defined pseudosteady state as the condition in a finite closed reser-voir when producing at a constant rate that “every point within the reservoir willeventually experience a constant rate of pressure decline.” Ramey also states “Thatthis condition has been referred to as pseudosteady, quasi-steady, semi-steady, andeven steady state in the literature.” Like Ramey, we will use the term pseudosteadyor pseudosteady state to describe this behavior.

Dimensionless PressureRamey (1971), Earlougher and Ramey (1973), and Gringarten (1974) showed thatat sufficiently large producing times the system (unfractured or fractured) eventu-ally reaches pseudosteady state and the dimensionless pressure may be evaluatedfrom

(L-6)

where is the shape factor and is the effective wellbore radius as originallyproposed by Prats (1961). Ramey (1971) showed that for pseudosteady flow

Δp pi pwf–=

pi pwf

J

J qp pwf–---------------- 2πkh

μ-------------JD= =

q p pwf

JD

JDμ

2πkh------------- J× μ

2πkh------------- q

p pwf–----------------×= =

pD tDA( ) 2πkhμq

------------- pi pwf–( ) 2πtDA 1 2⁄ 4A

eγCArw′ 2

--------------------⎝ ⎠⎜ ⎟⎛ ⎞

ln+= =

CA rw′



and

The dimensionless pressure solution can be written in the form (see e.g.,Ramey(1971) and Valko (1998)):

(L-7)

where

or

(L-8)

Effective Wellbore RadiusRearranging Eq. (L-8) in terms of effective wellbore radius

(L-9)

where the reservoir area for a rectangular reservoir hasbeen incorporated.

2πtDA2πkh

μq------------- pi p–( )=

1 2⁄ 4AeγCArw

′------------------

⎝ ⎠⎜ ⎟⎛ ⎞

ln 2πkhμq

------------- p pwf–( )=

pD 2πtDA 1 JD⁄+=

1 JD⁄ 1 2⁄ 4A

eγCArw′ 2

--------------------⎝ ⎠⎜ ⎟⎛ ⎞

ln=

JD 1 2⁄ 4A

eγCArw′ 2

--------------------⎝ ⎠⎜ ⎟⎛ ⎞

ln1–

=

1JD------ 4π

eγCA λ( )--------------------

re

rw′

-----ln 4

λeγCA

-------------------xe

rw′

-----⎝ ⎠⎜ ⎟⎛ ⎞

ln= =

1JD------ 4π

eγCA λ( )--------------------

rexf----

xf

rw′

-----ln+ln 4

λeγCA

-------------------xexf----

xf

rw′

-----ln+ln= =

A πre2 4xeye 4xe

2 λ⁄= = =


L.2 Pseudosteady Equations 731

The effective wellbore radius, , concept was first introduced by Prats (1961,

1962) as a means to define an equivalent reservoir with wellbore radius, , thatwould have a production behavior similar to that of a fractured reservoir. Pratsshowed that for high conductivity fractures ( ) and low fracture penetration

ratios ( ), the effective wellbore radius was given by

(L-10)

The effective wellbore radius concept is discussed in greater detail below as it per-tains to finite and infinite conductivity vertical fractures in closed systems.

Pseudo-Skin Relationships

The dimensionless productivity index as formulated by Cinco-Ley is of the form

(L-11)

where is the pseudo-skin function with respect to the fracture half-length ( ).

The pseudo-skin function and effective wellbore radius, , are related by

Eq. (L-9) can also be written in terms of a fracture skin, ,

The relationships between the pseudo-skin function, , dimensionless reciprocaleffective wellbore radius, , and fracture skin, , are given below

rw′

rw′

CfD ∞→

xf xe⁄ 0→

rw′ xf 2⁄→

1JD------ 4π

eγCA

------------rexf---- f+ln=

f xf

rw′

fxf

rw′

-----ln=

Sf

1JD------ 4π

eγCA

------------rerw----- Sf+ln=

f

xf rw′⁄ Sf

fxf

rw′

-----ln Sfrwxf-----ln– Sf

xfrw-----ln+= = =



(L-12)

and

or

L.3 Pseudosteady Fractured System ModelThe pseudosteady model for a fractured domain is based on Darcy’s law

(L-13)

where is a reference dimension. The pseudosteady flow rate from Eq. (L-13) is

(L-14)

ResistivityEq. (L-14) can be rearranged and placed in terms of a resistivity, , such that

(L-15)

The resistivity from Eq. (L-14) and Eq. (L-15) is

(L-16)

Sfrwxf----- f+ln

rwxf-----

xf

rw′

-----ln+lnrw

rw′

-----ln= = =

xf

rw′

----- ef xfrw-----e

Sf= =

rw′ rwe

Sf–xfe

f–= =

v kμ---

ξddp–=

ξ

q vA A kμ---

ξddp–⎝ ⎠

⎛ ⎞= =

ρ

ρ ξ( ) 2πkhμq

-------------–ξd

dp≡

ρ ξ( ) ω ξ( )A

2πh----------k ξ( )

k----------

---------------------=


L.3 Pseudosteady Fractured System Model 733

where is the permeability as a function of position and is the reference or farfield reservoir permeability. The fracture flow rate as a function of position is givenby .

Reservoir ResistivityThe reservoir resistivity, , for radial flow from Eq. (L-16) is

(L-17)

where the flow rate for pseudosteady radial flow is a constant and the flow area given by at any position

has been substituted.

Fracture ResistivitySubstituting the fracture flow area (i.e., flow rate through area ,

where is the single wing fracture flow rate) into Eq. (L-15), we find

(L-18)

where is the permeability as a function of position in the fracture ( ) and isthe reference or far field reservoir permeability. The fracture flow rate as a functionof position is given by .

The dimensionless fracture flow rate as a function of position can be represented by

(L-19)

where for slot flow, , and for a uniform flux, . The average fracture

resistivity, , from the wellbore to any position for a constant width fracture isfound by integrating Eq. (L-19)

k ξ( ) k

ω ξ( ) q ξ( ) q 0( )⁄=

ρr

ρr ξ( ) 1 ξk ξ( )k

----------⎝ ⎠⎛ ⎞⁄=

ω ξ( ) q ξ( ) q 0( )⁄ 1= = A 2πhξ= ξ

A wfh= q A

q

ρf ξ( ) ω ξ( )wf2π------

kf ξ( )k

-------------------------------=

kf ξ( ) ξ k

ω ξ( ) q ξ( ) q 0( )⁄=

ω ξ( ) q ξ( ) q 0( )⁄= 1 ξ xf⁄–( )αq=

αq 0= αq 1=

ρf ξ

ρf ζ( ) ρf 0( )⁄ ω ξ( ) ξd0

ζ

∫ ζ⁄ 1 1 ζ–( )αq 1+

–1 αq+( )ζ

-------------------------------------= =



where . The average fracture resistivity over the entire fracture length is

(L-20)

As illustrated, the average resistivity for slot flow is twice that of the uniform fluxsolution.

The average apparent conductivity as a function of position is

(L-21)

where

and

The average apparent conductivity parameter for a uniform-flux fracture ( )is

Inverse Dimensionless Productivity IndexThe inverse dimensionless productivity index ( ) is found from Eq. (L-5) by inte-grating the resistivity over the flow domain as given by

(L-22)

where is the constant of integration.

No FractureThe formation resistivity for an unfractured homogeneous reservoir ( )from Eq. (L-16) is .

ζ x xf⁄=

ρf ρf 0( )⁄ 11 αq+---------------=

κ ζ( )kfwf ζ( )

kfwf------------------

1 αq+( )ζ

1 1 ζ–( )αq 1+

–-------------------------------------= =

κ 0( )kfwf 0( )

kfwf------------------ 1= = κ 1( )

kfwf 1( )kfwf

------------------ 1 αq+( )= =

αq 1=

κ ζ( ) 2ζ

1 1 ζ–( )2–----------------------------=

JD

1JD------ ρ

ξ0

ξ

∫ ξ( )dξ c+=

c

k ξ( ) k⁄ 1=ρr ξ( ) 1 ξ⁄=



The dimensionless productivity index ( ) for radial flow in a homogeneous reser-

voir with a wellbore radius, , and drainage radius, , from Eq. (L-22) is

(L-23)

The constant of integration is found from Eq. (L-8)

(L-24)

or

(L-25)

where .

Eq. (L-23) can also be written as

(L-26)

The equivalent drainage radius, , in terms of rectangular reservoir dimensions

for a given drainage area ( ) is

(L-27)

where the reservoir aspect ratio is given by .

Rearranging Eq. (L-8) or Eq. (L-26) in terms of the rectangular reservoir coordi-nate, , with , we have

(L-28)

JD

rw re

1JD------ 1

ξ---

rw

re

∫ dξ c+rerw-----⎝ ⎠

⎛ ⎞ln c+= =

1JD------ 1 2⁄ 4A

eγCArw2

------------------⎝ ⎠⎜ ⎟⎛ ⎞ re

rw-----⎝ ⎠

⎛ ⎞ln c+=ln=

c 4π

eγCA λ( )--------------------ln=

re A π⁄≡

1JD------ 1

ξ---

rw

re

∫ dξ c+ 4π

eγCA λ( )--------------------

rerw-----ln= =

re

xe ye,( ) A 4xeye= 4xe2λ πre

2= =

xere---- π

4λ------=

λ xe= ye⁄

xe A 4xe2 λ⁄=

1JD------ 1

ξ---

rw

re

∫ dξ c+ 4π

eγCA λ( )--------------------

rerw-----ln 4

λeγCA

-------------------xerw-----

⎝ ⎠⎜ ⎟⎛ ⎞

ln= = =



The above equation can be written as

(L-29)

where

;

and

.

The constant of integration in Eq. (L-28) can also be represented by

(L-30)

Circular ReservoirThe beta integration constants for a well located at the center of a circular drainagearea are

(L-31)

where .

The pseudosteady inverse productivity solution for a well located at the center of aclosed circular homogeneous reservoir from Eq. (L-29) and Eq. (L-31) is

(L-32)

where the constant of integration can is now represented by

1JD------ βre

λ( )rerw-----ln βxe

λ( )xerw-----ln= =

βreλ( ) 4π

eγCA λ( )--------------------= βxe

λ( ) 4

λeγCA

-------------------=

βxeλ( ) βre

λ( ) 4λπ------=

c 4π

eγCA λ( )--------------------

rerw-----ln 1

ξ---

rw

re

∫ dξ– 4

λeγCA

-------------------xerw-----

⎝ ⎠⎜ ⎟⎛ ⎞

ln 1ξ---

rw

re

∫ dξ–= =

λ 1=

βre0.47237097=

βxe0.53301379=

CA λ 1=( ) 31.62=

1JD------ 1

ξ---

rw

re

∫ dξ c+ 0.472rerw-----⎝ ⎠

⎛ ⎞ln 0.533xerw-----⎝ ⎠

⎛ ⎞ln= = =



(L-33)

The dimensionless productivity index for a well of radius located at the centerof a closed circular drainage area from Eq. (L-32) is

(L-34)

Square ReservoirThe beta integration constants for a well located at the center of a square are

(L-35)

where .

The pseudosteady inverse productivity solution for a well located at the center of aclosed square homogeneous reservoir from Eq. (L-29) and Eq. (L-31) is

(L-36)

where the constant of integration can is now represented by

(L-37)

The dimensionless productivity index for a well of radius in the center of ahomogeneous square formation from Eq. (L-32) is

(L-38)

c 0.472rerw-----⎝ ⎠

⎛ ⎞ 1ξ---

rw

re

∫ dξ–ln=

rw

JD1

0.472rerw-----⎝ ⎠

⎛ ⎞ln------------------------------ 1

0.533xerw-----⎝ ⎠

⎛ ⎞ln------------------------------= =

λ 1=

βre0.47797569=

βxe0.53933781=

CA λ 1=( ) 30.8828=

1JD------ 1

ξ---

rw

re

∫ dξ c+ 0.478rerw-----⎝ ⎠

⎛ ⎞ln 0.539xerw-----⎝ ⎠

⎛ ⎞ln= = =

c 0.478rerw-----⎝ ⎠

⎛ ⎞ 1ξ---

rw

re

∫ dξ–ln=

rw

JD1

0.478rerw-----⎝ ⎠

⎛ ⎞ln------------------------------ 1

0.539xerw-----⎝ ⎠

⎛ ⎞ln------------------------------= =



Finite Conductivity Vertical Fracture SystemThe fracture for a rectangular reservoir with aspect ratio, , is mapped in a

fracture domain with dimensions or .

The total system resistivity within the fracture domain or radial fracture

domain (where is some reference (still to be determined) apparent oreffective domain radius) for the formation and fracture acting in parallel is

where

and the net fracture resistivity is

The average net inverse fracture resistivity integrated from the wellbore to someposition is

The total inverse resistivity (reservoir and fracture) within the fracture domain zoneis then

(L-39)

To illustrate this methodology we will first develop the equations for a square reser-voir and then extend the analysis to a rectangular shaped reservoir.

λ xe ye⁄=

rw′ rw

′ λ⁄× rw′ rw

′ λ⁄,( )

rw′ rw

′ λ⁄×

ξ rw′≤ rw

′

1ρ--- 1

ρr----- 1

ρf----+=

ρr 1 ξk ξ( )k

----------⎝ ⎠⎛ ⎞⁄=

ρfω ξ( )

wf ξ( )2π

-------------kf ξ( )

k------------ k ξ( )

k----------–⎝ ⎠

⎛ ⎞-------------------------------------------------=

ξ

ρf ρf0

ξ

∫ dξ′⎝ ⎠⎛ ⎞ ξ⁄=

1ρ ξ rw

′≤( )----------------------- ξk ξ( )

k---------- 1

ρf----+=



Square ReservoirThe inverse dimensionless productivity for a two wing fracture in a square reservoirwith a fracture domain of from Eq. (L-22) is

(L-40)

Eq. (L-40) is similar to an equation originally proposed by Raymond and Binder(1962) but with a few significant differences. Raymond integrated the above equa-tion over the fracture half-length ( ) without transforming the fracture into a

domain coordinate system (i.e., some effective wellbore radius, domain). He

also assumed slot flow in the fracture (i.e., ). Both of theseconditions were detrimental to his results.

Eliminating the constant of integration, Eq. (L-40) can be written as

(L-41)

where the constant of integration has been implemented with the aid of Eq. (L-30).

For an infinite conductivity fracture with a uniform formation permeability Eq. (L-41) becomes

(L-42)

As illustrated, the apparent domain radius to satisfy Eq. (L-42) must be the effec-tive wellbore radius for an infinite conductivity fracture as given by .

To prevent confusion, the dimensionless fracture half-length ratio with respect tothe effective wellbore radius (i.e., reciprocal effective wellbore radius) for an infi-nite conductivity fracture will be defined as

(L-43)

where is a function of the penetration ratio ( ) as discussed below.

rw′ rw

′,( )

1JD------ 1

ξk ξ( )k

---------- 1ρf----+

-------------------------rw

rw′

∫ dξ 1

ξk ξ( )k

------------------------

rw′

re

∫ dξ c+ +=

xf

rw′

ω ξ( ) q ξ( ) q 0( )⁄ 1= =

1JD------ βre

rexf----

xf

rw′

-----⎝ ⎠⎜ ⎟⎛ ⎞

ln 1ξ--- k

k ξ( )---------- 1–⎝ ⎠

⎛ ⎞rw

′

re

∫ dξ+= 1

ξk ξ( )k

---------- 1ρf----+

-------------------------rw

rw′

∫ dξ+

1JD------ βre

rexf----

xf

rw′

-----ln+ln βre

rexf----

xf

rw′

-----lnCfD ∞→

+ln= =

rw′ rw

′CfD ∞→

=

ζ∞ xf rw′⁄

CfD ∞→

≡

ζ∞ Ix( ) Ix xf xe⁄=



Defining a dimensionless fracture domain position, , with respect to the effectivewellbore radius for an infinite conductivity fracture ( ), we have

(L-44)

where

and

.

Eq. (L-41) can now be written in terms of the dimensionless domain parameter as

(L-45)

where

The fracture dimension is related to by

(L-46)

The dimensionless position as a function of fracture position from Eq. (L-46)is

(L-47)

Eq. (L-46) can be approximated by

ζ

rw′

CfD ∞→

ζ ξ rw′⁄

CfD ∞→

=

ζw rw rw′

CfD ∞→

⁄ rw xf⁄( )ζ∞= =

ζe re rw′

CfD ∞→

⁄ re xf⁄( )ζ∞= =

ζ

1JD------ βre

rexf----ζ∞⎝ ⎠

⎛ ⎞ln 1χ---

ζw

1

∫ dξ 1ζ--- k

k ζ( )---------- 1–⎝ ⎠

⎛ ⎞1

ζe

∫ dζ+ +=

χ ζk ζ( )k

---------- 1ρf----

ζ∞

xf------+=

x ζ

x rw–xf rw–---------------

ζ ζw–1 ζw–---------------=

ζ x

ζ 1 ζw–( )x rw–xf rw–--------------- ζw+=

x xf⁄ ζ≅



for most cases if and .

Eq. (L-45) can be represented in terms of a pseudo-skin function

(L-48)

where the pseudo-skin function is given by

(L-49)

The delta pseudo-skin function, , represents the skin of a finite conductivityfracture (i.e., additional skin of a finite conductivity fracture as opposed to an infi-nite conductivity fracture). This finite conductivity skin is given by

(L-50)

For a piece wise continuous fracture conductivity, Eq. (L-50) can be represented by

(L-51)

where is the number of fracture intervals and the integrand is represented by

and

Rectangular ReservoirThe inverse dimensionless productivity for a two wing fracture in a rectangular res-ervoir with a fracture domain of from Eq. (L-22) is

ζw 1« x rw»

1JD------ βre

rexf----ln f 1

ζ--- k

k ζ( )---------- 1–⎝ ⎠

⎛ ⎞1

ζe

∫ dζ+ +=

f ζ∞( )ln ΔSf+=

ΔSf

ΔSf1χ---

ζw

1

∫ dξ=

ΔSf1χ---

ζw

1

∫ dξ 1χi----

ζw

ζi

∫ dξ 1χi----

ζw

ζi 1–

∫ dξ–i 1=

N

∑==

N

χi ζk ζ( )k

---------- Ci κ ζi( )×+=

Ciwfi2π------- 1

xf----

kfik----- k ζ( )

k----------–⎝ ⎠

⎛ ⎞ ζ∞=

rw′ rw

′ λ⁄,( )



(L-52)

Eq. (L-41) can now be written in terms of the dimensionless domain parameter as

(L-53)

where

(L-54)

To determine the formulation of the geometric parameter , Eq. (L-54) can besimplified with the aid of the mean value theorem for a homogeneous reservoir andconstant fracture conductivity as

where

and represents the ratio of the reservoir to net fracture resistivity.

The geometric shape parameter is obtained from the definition of the dimen-sionless domain conductivity where

1JD------ βre

rexf----

xf

rw′

-----⎝ ⎠⎜ ⎟⎛ ⎞

ln 1

ξk ξ( )k

---------- g λ( )ρf

-----------+--------------------------------

rw

rw′

∫ dξ 1ξ--- k

k ξ( )---------- 1–⎝ ⎠

⎛ ⎞rw

′

re

∫ dξ+ +=

ζ

1JD------ βre

rexf----ζ∞ln 1

χ---

ζw

1

∫ dξ 1ζ--- k

k ζ( )---------- 1–⎝ ⎠

⎛ ⎞1

ζe

∫ dζ+ +=

χ ζk ζ( )k

---------- g λ( )ρf

-----------ζ∞

xf------+=

g λ( )

χ ζ C κ ξ( )×+≅

κ ξ( )kfwf ξ( )

kfwf------------------

1 αq+( )ξ

1 1 ξ–( )αq 1+

–-------------------------------------= =

C CfDg λ( )ζ∞

2π------------------=

CfDwfxf-----

kfk---- 1–⎝ ⎠

⎛ ⎞=

C

g λ( )

C ζ( )

C ζ( ) λ C ζ( ) λ 1=

ρr λ 1=( )ρr λ( )

------------------------ C ζ( ) λ 1= g λ( )×= =



This geometric parameter for high and low conductivity fractures for aspect ratiosgreater than or equal to unity ( ) is given by

and

respectively. The geometric parameter for aspect ratios less than one is equal tounity (i.e., ).

There are actually two parts to the function. Part one is the transformation ofvariable in Eq. (L-52) (i.e., integration from to for high conductivity frac-

tures and straight mapping to for low conductivity fractures) and a reservoir

area resistivity factor .

This follows since for low conductivity fractures the fracture domain is not a func-tion of fracture length (i.e., the domain is not ).

As illustrated for a square reservoir is not a function of conductivity (i.e.,). For an infinite-acting reservoir is also equal to unity, ( ).

The general form of for all conductivities and penetrations is

where

To simplify the nomenclature may be written interchangeably as or

.

λ 1≥

g λ( ) CfD ∞→ λA λ 1=( )A λ( )

----------------------→ 2λ1 1 λ⁄+-------------------=

g λ( ) CfD 0→A λ 1=( )

A λ( )----------------------→ 2

1 1 λ⁄+-------------------=

g λ 1<( ) 1=

g λ( )

rw′ λ⁄ rw

′

rw′

A λ 1=( ) A λ( )⁄

rw′ rw

′,( ) rw′ rw

′ λ⁄,( )

g λ 1=( )

g λ 1=( ) 1= g g 1=

g λ CfD Ix, ,( )

g λ CfD Ix, ,( ) φg λ( ) CfD 0→ 1 φ–( )g λ( ) CfD ∞→+=

φ e2 CfDIx

2×–≅

g λ CfD Ix, ,( ) g

g λ( )



L.4 Pseudosteady Fracture SolutionsGeneral Solution for Homogeneous ReservoirsThe governing inverse productivity index equation for a piece-wide continuous andlinearly varying fracture conductivity in homogeneous reservoir ( ) fromEq. (L-53) is

(L-55)

or

where

(L-56)

The delta pseudo-skin function (i.e., additional skin of a finite-conductivity frac-ture) for a piece-wise continuous varying fracture conductivity in a homogeneousreservoir as derived from the mean value theorem is

(L-57)

where

and .

k ζ( ) k⁄ 1=

1JD------ βre

rexf----ln ζ∞ln ΔSf+ +=

1JD------ βre

rexf----ln f+=

f ζ∞ln ΔSf+=

ΔSf1χ---

ζw

1

∫ dζ ΔSfi

i 1=

N

∑= =

ΔSfi ΔSf ζw ζi,( ) ΔSf ζw ζi 1–,( )–=

ΔSf ζw ζ,( ) 1χ---

ζw

ζ

∫ dξ ζ κ ζ( )C+ζw κ ζ( )C+----------------------------⎝ ⎠

⎛ ⎞ln= =

κ ζ( )1 αq+( )ζ

1 1 ζ–( )αq 1+

–-------------------------------------=

κ 0( ) 1=


L.4 Pseudosteady Fracture Solutions 745

The conductivity variable is given by

and is a function of the aspect ratio, the dimensionless conductivity,and the penetration ratio.

Slot FlowA fundamental solution to Eq. (L-57) can be obtained if we assume slot flow in thefracture ( ). Although, slot flow is not a very realistic assumption for mostfractured systems (i.e., assuming all the flow enters at the tip of the fracture andflows with a constant rate in fracture) it will provide a limiting solution for the max-imum pressure in the fracture/reservoir domain. That is, slot flow results in themaximum value for the pseudo-skin function (minimum value for ).

Assuming slot flow ( , ) in the fracture with a constant conduc-tivity fracture, Eq. (L-57) is integrated to obtain the delta pseudo-skin as a functionof position

(L-58)

The inverse fracture productivity index for slot flow in a fracture with a constantconductivity from Eq. (L-58) is

(L-59)

Cwfxf-----

kfk---- 1–⎝ ⎠

⎛ ⎞ g λ CfD Ix, ,( )ζ∞

2π-------------------------------------=

CfDg λ CfD Ix, ,( )ζ∞

2π-------------------------------------=

g λ CfD Ix, ,( )

ω ζ( ) 1=

JD

ω ζ( ) 1= κ ζ( ) 1=

ΔSf ζ( ) 1χ---

ζw

ζ

∫ dζ ζ C+ζw C+----------------⎝ ⎠

⎛ ⎞ln==

1JD------ βre

rexf----ln ζ∞ln 1 C+

ζw C+----------------⎝ ⎠

⎛ ⎞ln+ +=

βre

rexf----ln

ζ∞

C------ ζ∞+

ζwC------ 1+

-------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

ln+=



Placing Eq. (L-59) in terms of the dimensionless fracture conductivity for a con-

stant conductivity fracture , the productivity index for slot flow can

be written as

(L-60)

Uniform FluxThe uniform flux assumption is a more realistic solution for high conductivity ver-tical fracture systems. Since the fracture flux is a function of the fracture conductiv-ity as illustrated by Cinco-Ley (1981), the flux (flow rate in the fracture) must beknown prior to solving Eq. (L-55).

To obtain a more realistic solution to this domain problem, we will assume a con-stant flux into the fracture (at least for high conductivity fractures). As illustratedabove, the resistivity for a uniform fracture flux is one-half that of slot flow (i.e.,

). This makes the effective conductivity look like it is two times higher

since the full flow rate is not seen over the entire length of the fracture. Conse-quently, one may without much more thought make the following incorrect analogy

Then for a uniform flux

The above analogy of course is false for all positions in the fracture. This is easilyillustrated by noting that at or near the wellbore, the fracture flow rate is at the max-imum value (like slot flow) from which ( ). However, at the fracturetip, the flow rate is less than the value at the well .

Thus, from Eq. (L-55) we would find (see below) that for a uniform flux

C CfDg λ( )ζ∞

2π------------------=

1JD------ βre

rexf----ln

2πCfDg λ( )-------------------- ζ∞+

2πCfDg λ( )--------------------

ζwζ∞------ 1+

-------------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

ln+=

ρf12---ρf 0( )=

ρfρf 0( )------------ 1

1 αq+---------------≅

CfDCfD--------- 1 αq+( )≅⇒

ρfρf 0( )------------ 1

2---≅

CfDCfD--------- 2≅⇒

C ζ 0→( ) C≅

C ζ 1→( ) C»



(L-61)

However, applying this analogy to the entire fracture domain is approximately true,as will be illustrated. Thus, integration over the entire fracture domain for a highconstant conductivity fracture, we find

(L-62)

and

(L-63)

The resulting pseudo-skin function from Eq. (L-63) is shown to be in excellentagreement with the numerical results of Cinco-Ley as discussed in the main body ofthe report.

SummaryThe inverse dimensionless productivity index and related equations for a piece wisecontinuous linearly varying fracture conductivity in a homogeneous reservoir is

(L-64)

where

(L-65)

and

(L-66)

The delta pseudo-skin function (i.e., skin of a finite conductivity fracture) for apiece wise continuous varying fracture conductivity in a homogeneous reservoir forslot and a uniform fracture flux are

ΔSf1χ---

ζw

ζ

∫ dζ ζ 2C+ζw 2C+-------------------⎝ ⎠

⎛ ⎞ln≠=

ΔSf1χ---

ζw

1

∫ dξ 1 2C+ζw 2C+-------------------⎝ ⎠

⎛ ⎞ln≅=

1JD------ βre

rexf----ln ζ∞

1 2C+ζw 2C+-------------------⎝ ⎠

⎛ ⎞ln+≅

1JD------ βre

rexf----ln f+=

f ζ∞( )ln ΔSf+=

ΔSf1χ---

ζw

1

∫ dζ 1χi----

ζi 1–

ζi

∫ dζ

i 1=

N

∑= =



General Flow SolutionA more general solution as derived from the mean value theorem that we will use is

where

(L-67)

and .

For a uniform flux fracture ( ) and we have

(L-68)

The conductivity variable is given by

(L-69)

and is a function of aspect ratio, penetration, and conductivity as givenbelow by the following limits

and

The general form of for all conductivities and penetrations is

ΔSf1χ---

ζw

ζ

∫ dξ ζ κ ζ( )C+ζw κ ζ( )C+----------------------------⎝ ⎠

⎛ ⎞ln= =

κ ζ( )1 αq+( )ζ

1 1 ζ–( )αq 1+

–-------------------------------------=

κ 0( ) 1=

αq 1=

κ ζ( ) 2ζ

1 1 ζ–( )2–----------------------------=

Cwfxf-----

kfk---- 1–⎝ ⎠

⎛ ⎞ g λ CfD,( )ζ∞

2π------------------------------=

CfDg λ CfD,( )ζ∞

2π------------------------------=

g λ CfD,( )

g λ CfD,( )CfD 1»

2λ1 1 λ⁄+-------------------=

g λ CfD,( )CfD 1«

21 1 λ⁄+-------------------=

g



(L-70)

where

To simplify the nomenclature, will be written interchangeably as or

. It is also noted, that for aspect ratios less than unity, we have.

Effective Wellbore RadiusThe inverse dimensionless effective wellbore radius for a constant (uniform) con-ductivity fracture in a rectangular reservoir from Eq. (L-57) is

(L-71)

or

(L-72)

Eq. (L-72) can be simplified if we assume a uniform fracture flux ( and

) and if as given by

(L-73)

Consequently, for large and small fracture conductivities with aspect ratios greaterthan or equal to unity ( ), we have

(L-74)

and

g λ CfD Ix, ,( ) φg λ CfD Ix, ,( )CfD 0→

1 φ–( )g λ CfD Ix, ,( )CfD ∞→

+=

φ e2 CfDIx

2×–≅

g λ CfD Ix, ,( ) g

g λ( )

g λ 1< CfD Ix, ,( ) 1=

xf rw′⁄

ζ∞ κ 1( )Cζ∞+ζw κ 1( )C+

-----------------------------------ζ∞ κ 1( )C⁄ ζ∞+ζw κ 1( )C⁄ 1+

---------------------------------------= =

xf rw′⁄ 2π κ 1( )⁄

CfDg λ( )---------------------- ζ∞+⎝ ⎠

⎛ ⎞ 2π κ 1( )⁄CfDg λ( )----------------------

ζwζ∞------ 1+⎝ ⎠

⎛ ⎞⁄=

αq 1=

κ 1( ) 2= πCfDg λ( )---------------------

ζwζ∞------ 1«

xf rw′⁄ π

CfDg λ( )-------------------- ζ∞+⎝ ⎠

⎛ ⎞=

λ 1≥

xf rw′⁄

CfD 1»

π

CfD2λ

1 1 λ⁄+-------------------⎝ ⎠

⎛ ⎞---------------------------------- ζ∞+=



(L-75)

where in Eq. (L-75) we have assumed that .

Eq. (L-75) illustrates that the effective wellbore radius at low fracture conductivi-ties is not a function of the fracture length

Then from the above equation for low conductivity vertical fractures in a squarereservoir ( ), we have

The remainder of this section is devoted to finding solutions to the above set ofequations for special cases of interest.

The following analysis will primarily be based on dimensionless productivity solu-tions for finite conductivity vertical fractures with a well at the center of a rectangu-lar reservoir.

L.5 Pseudosteady Cases

Constant Finite Conductivity FractureOne very important and fundamental solution for the dimensionless productivityindex, , is for the case of a finite conductivity fracture of constant width and per-meability (conductivity) in a homogeneous square reservoir. The boundary condi-tions for this case are:

xf rw′⁄

CfD 1«

π

CfD2

1 1 λ⁄+-------------------⎝ ⎠

⎛ ⎞---------------------------------- ζ∞+=

πCfDg λ( )---------------------

ζwζ∞------ 1«

rw′

CfD 1«

1π--- 2

1 1 λ⁄+-------------------⎝ ⎠

⎛ ⎞ wfkfkr

---------→

λ 1=

rw′

CfD 1«

1π---

wfkfkr

---------→

JD


L.5 Pseudosteady Cases 751

(L-76)

or

(L-77)

The fundamental productivity index solution for a uniform fracture conductivity(and uniform flux) is found by substituting the boundary condition of Eq. (L-77)into Eq. (L-64) and rearranging

(L-78)

where

and

Placing Eq. (L-78) in terms of the dimensionless fracture conductivity and assum-ing a uniform flux ( ), we have

(L-79)

Eq. (L-78) can be simplified if the fracture permeability is much greater than theformation permeability and the wellbore radius is much less than the width perme-ability ratio, as given by

k ζ( ) k ζw ζ ζe≤ ≤=

kf ζ( ) kf ζw ζ 1≤ ≤=

w ζ( ) wf ζw ζ 1≤ ≤=

CfD ζw ζ 1≤ ≤( ) CfD=

1JD------ βre

rexf----ζ∞ln ΔSf+=

ΔSf1χ---

ζw

1

∫ dζ 1 κ 1( )C+ζw κ 1( )C+----------------------------⎝ ⎠

⎛ ⎞ln= =

C CfDg λ( )ζ∞

2π------------------=

κ 1( ) 2=

ΔSf

πCfDg λ( )ζ∞--------------------------- 1+

πCfDg λ( )--------------------

ζwζ∞------ 1+

-------------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

ln=



The resulting simplification of Eq. (L-78) with the above assumptions can be writ-ten as

(L-80)

where the dimensionless fracture conductivity, , for is approximatedby

(L-81)

The pseudo-skin function, , from Eq. (L-65) and Eq. (L-80) is

(L-82)

The reciprocal effective wellbore radius, , from Eq. (L-12) and Eq. (L-80) for

a uniform finite conductivity fracture in a square homogeneous reservoir (and ) is

(L-83)

Figure L.1 shows the effective dimensionless wellbore radius (Eq. (L-83)) as afunction of fracture conductivity. A comparison with the work of Cinco-Ley andRamey illustrates the excellent agreement.

kf k⁄ 1»

ζwwfxf-----

kfk---- 1–⎝ ⎠

⎛ ⎞ g λ( )ζ∞

π------------------ or rw wf

kfk---- 1–⎝ ⎠

⎛ ⎞ g λ( )ζ∞

π------------------««

1JD------ βre

rexf----ln π

CfDg λ( )-------------------- ζ∞+⎝ ⎠

⎛ ⎞ln+=

CfD kf k⁄ 1»

CfDwfkfxfk---------=

f

f πCfDg λ( )-------------------- ζ∞+⎝ ⎠

⎛ ⎞ln=

xf rw′⁄

λ 1=g λ( ) 1=

xf rw′⁄ π

CfD--------- ζ∞+=



Figure L.1: Dimensionless Effective Wellbore radius for Finite Conductivity vertical fracture.

Figure L.2 shows the dimensionless pseudo-skin (Eq. (L-82)) as a function of frac-ture conductivity. A comparison with the work of Cinco-Ley and Ramey is alsoshown.

Figure L.2: Pseudo-skin function versus Dimensionless Conductivity. Comparison with the work of Cinco-Ley and Ramey.



Non-Uniform Fracture ConductivityConsider a fracture with uniform conductivity zones. The boundary conditionsfor this general case are:

(L-84)

These boundary conditions are typical for propped fractures with a pinch zone, atail-in, over flushing, choked flow, etc.

The incremental pseudo-skin from Eq. (L-78) for the above boundary conditions is

where

(L-85)

and from Eq. (L-67)

The conductivity constant from Eq. (L-69) is defined as

and .

N

k ζ( ) k ζw ζ ζe≤ ≤=

CfD ζ( ) CfD1 ζw ζ ζ1≤ ≤=

CfD ζ( ) CfD2 ζ1 ζ ζ2≤ ≤=

......CfD ζ( ) CfDN ζN 1– ζ 1≤ ≤=

ΔSf ΔSfi

i 1=

N

∑=

ΔSfiζi κ ζi( )Ci+ζw κ ζi( )Ci+-------------------------------⎝ ⎠

⎛ ⎞lnζi 1– κ ζi 1–( )Ci+ζw κ ζi 1–( )Ci+

-------------------------------------------⎝ ⎠⎛ ⎞ln–=

κ ζ( )1 αq+( )ζ

1 1 ζ–( )αq 1+

–-------------------------------------=

Ci CfD i

g λ( )ζ∞

2π------------------=

ζ0 ζw=



The general productivity solution for uniform fracture conductivity zones fromand Eq. (L-78) is

(L-86)

Then for three zones we have

and

where for a uniform flux distribution.

Pinched Fracture or BlockageA pinched fracture is defined as a propped fracture that has a restricted width orpinch zone somewhere within the fracture. A limiting restriction is when the frac-ture and formation permeabilities are equal (i.e., no propped fracture). The resultingboundary conditions with a uniform conductivity on either side of a pinch zone are:

(L-87)

The reciprocal productivity index from Eq. (L-86) with the boundary conditions ofEq. (L-87) is

N

1JD------ βre

rexf----ζ∞⎝ ⎠

⎛ ⎞ln ΔSfi

i 1=

N

∑+=

ΔSf ΔSfi

i 1=

3

∑=

ΔSf1ζ1 κ ζ1( )Ci+ζw κ ζ1( )Ci+--------------------------------⎝ ⎠

⎛ ⎞ln=

ΔSf2ζ2 κ ζ2( )C2+ζw κ ζ2( )C2+---------------------------------⎝ ⎠

⎛ ⎞lnζ1 κ ζ1( )C2+ζw κ ζ1( )C2+---------------------------------⎝ ⎠

⎛ ⎞ln–=

ΔSf31 κ 1( )C3+

ζw κ 1( )C3+-------------------------------⎝ ⎠

⎛ ⎞lnζ2 κ ζ2( )C3+ζw κ ζ2( )C3+---------------------------------⎝ ⎠

⎛ ⎞ln–=

κ 1( ) 2=

CfD1 CfD ζw ζ ζ1≤ ≤=

CfD2 0 ζ1 ζ ζ2≤ ≤=

CfD3 CfD ζ2 ζ 1≤ ≤=



and

The finite conductivity skin for , is

where for a uniform flux fracture.

Placing Eq. (L-86) in terms of a pinch point or zone skin, , for a uniform fluxfracture, we have

(L-88)

where

(L-89)

and

ΔSf ΔSfi

i 1=

3

∑=

ΔSf1ζ1 κ ζ1( )C+ζw κ ζ1( )C+-------------------------------⎝ ⎠

⎛ ⎞ln=

ΔSf2ζ2ζ1-----⎝ ⎠

⎛ ⎞ln=

ΔSf31 κ 1( )C+

ζw κ 1( )C+----------------------------⎝ ⎠

⎛ ⎞lnζ2 κ ζ2( )C+ζw κ ζ2( )C+-------------------------------⎝ ⎠

⎛ ⎞ln–=

ζw C«

ΔSfζ1 κ ζ1( )C+ζw κ ζ1( )C+-------------------------------⎝ ⎠

⎛ ⎞lnζ2ζ1-----⎝ ⎠

⎛ ⎞ln 1 κ 1( )C+ζw κ 1( )C+----------------------------⎝ ⎠

⎛ ⎞lnζ2 κ ζ2( )C+ζw κ ζ2( )C+-------------------------------⎝ ⎠

⎛ ⎞ln–+ +=

ζ1κ ζ1( )C------------------ 1+⎝ ⎠

⎛ ⎞lnζ2

κ ζ2( )C------------------ 1+⎝ ⎠

⎛ ⎞ln–ζ2ζ1-----⎝ ⎠

⎛ ⎞ln 1κ 1( )C--------------- 1+⎝ ⎠

⎛ ⎞ln+ +=

κ 1( ) 2=

Spp

1JD------ βre

rexf----⎝ ⎠

⎛ ⎞ln πCfDg λ( )-------------------- ζ∞+⎝ ⎠

⎛ ⎞ln + += Spp

ΔSppζ1

κ ζ1( )C------------------ 1+⎝ ⎠

⎛ ⎞lnζ2

κ ζ2( )C------------------ 1+⎝ ⎠

⎛ ⎞ln–ζ2ζ1-----⎝ ⎠

⎛ ⎞ln+=

C CfDg λ( )ζ∞

2π------------------=



Then for an infinite conductivity fracture ( ), the pinch point skin is

Tail-inThe boundary conditions for a two zone conductivity fracture is

The pseudo skin function from Eq. (L-85) is

and

The total skin is

Then for and , we have

If the far field conductivity is infinite ( ), we have

C ∞→

ΔSppζ2ζ1-----⎝ ⎠

⎛ ⎞ln=

CfD CfD1 ζw ζ ζ1≤ ≤=

CfD CfD2 ζ1 ζ 1≤ ≤=

ΔSfiζi κ ζi( )Ci+ζw κ ζi( )Ci+-------------------------------⎝ ⎠

⎛ ⎞lnζi 1– κ ζi 1–( )Ci+ζw κ ζi 1–( )Ci+

-------------------------------------------⎝ ⎠⎛ ⎞ln–=

ΔSf1ζ1 κ ζ1( )C1+ζw κ ζ1( )C1+---------------------------------⎝ ⎠

⎛ ⎞ln=

ΔSf21 κ 1( )C2+

ζw κ 1( )C2+-------------------------------⎝ ⎠

⎛ ⎞lnζ1 κ ζ1( )C2+ζw κ ζ1( )C2+---------------------------------⎝ ⎠

⎛ ⎞ln–=

ΔSfζ1 κ ζ1( )C1+ζw κ ζ1( )C1+---------------------------------⎝ ⎠

⎛ ⎞ln1 κ 1( )C2+

ζw κ 1( )C2+-------------------------------⎝ ⎠

⎛ ⎞lnζ1 κ ζ1( )C2+ζw κ ζ1( )C2+---------------------------------⎝ ⎠

⎛ ⎞ln–+=

ζw C2« ζw C1«

ΔSfζ1

κ ζ1( )C1-------------------- 1+⎝ ⎠

⎛ ⎞lnζ1

κ ζ1( )C2-------------------- 1+⎝ ⎠

⎛ ⎞ln– 1κ 1( )C2------------------ 1+⎝ ⎠

⎛ ⎞ln+=

C2 ∞→



(L-90)

Over FlushOver flushing or over displacing the proppant near the wellbore is a special case ofa pinched fracture where the pinch zone is at the wellbore. The boundary conditionsfor this case are:

The pseudo skin function is

Then for , the total pseudo skin is

The total skin for an infinite conductivity fracture in the main body is

where

and

ΔSfζ1

κ ζ1( )C1-------------------- 1+⎝ ⎠

⎛ ⎞ln=

CfD 0 ζw ζ ζ1≤ ≤=

CfD CfD ζ1 ζ 1≤ ≤=

ΔSf ΔSfi

i 1=

2

∑=

ΔSf1ζ1ζw------⎝ ⎠

⎛ ⎞ln=

ζw C«

ΔSfζ1ζw------⎝ ⎠

⎛ ⎞ln 1 κ 1( )C+ζw κ 1( )C+----------------------------⎝ ⎠

⎛ ⎞lnζ1 κ ζ1( )C+ζw κ ζ1( )C+-------------------------------⎝ ⎠

⎛ ⎞ln–+=

ζ1ζw------⎝ ⎠

⎛ ⎞ln 1 κ 1( )C+ζ1 κ ζ1( )C+------------------------------⎝ ⎠

⎛ ⎞ln κ 1( )κ ζ1( )--------------⎝ ⎠

⎛ ⎞ln–+=

ΔSfζ1ζw------⎝ ⎠

⎛ ⎞ln=

ζw rw rw′

CfD ∞→

⁄ rw xf⁄( )ζ∞= =



Choked FractureCinco-Ley defined a choked fracture as an infinite conductivity fracture with a flowrestriction near the wellbore as a result of reduced fracture permeability. A chokedfracture is therefore a special case of a tail-in with an otherwise infinite conductiv-ity fracture.

The total the pseudo-skin , (which Cinco-Ley called a choked skin ( ) and

we will later identify as ) from Eq. (L-85) with ,

and is

Then for , we have

(L-91)

where

The inverse dimensionless productivity index is

(L-92)

where is really a choked skin .

Eq. (L-92) simplifies to Eq. (L-80) as and illustrates the correct asymptotic

behavior for a finite conductivity fracture. That is checking the limits of as ,we find

ζ1 1 ζw–( )x1 rw–xf rw–---------------- ζw+=

ΔSf Sfs( )ch

Sch CfD ∞→

ζ1 1 ζw–( )x1 xf⁄ ζw+= x1 xf⁄ ζw»

ΔSfζ1 κ ζ1( )C1+ζw κ ζ1( )C1+---------------------------------⎝ ⎠

⎛ ⎞ln=

ζw C1«

ΔSfζ1

κ ζ1( )C1-------------------- 1+⎝ ⎠

⎛ ⎞ln=

C1wfxf-----

kf1k

------ 1–⎝ ⎠⎛ ⎞ g λ( )ζ∞

2π------------------=

1JD------ βre

rexf----⎝ ⎠

⎛ ⎞ln ζ∞( )ln Sch+ +=

ΔSf Sch

ζ1 1→

x1 xf→



(L-93)

where

Placing in terms of a choked skin, , (skin compared to an infinite conductivityfracture), we have

where we have assumed .

The choked fracture skin as defined by Cinco-Ley (SPE 10043) is

(L-94)

Placing our choked fracture skin in terms of , we find

(L-95)

Now, if the choked skin is small , Eq. (L-95) becomes

(L-96)

where for a uniform flux fracture is

1JD------ = βre

rexf----⎝ ⎠

⎛ ⎞ πCfD--------- ζ∞+⎝ ⎠

⎛ ⎞ln+ln

CfDwfkf1xfk

------------=

Sch

Schx1k

wfkf1------------ 2π

g λ( )κ ζ1( )ζ∞-------------------------------- 1+

⎝ ⎠⎜ ⎟⎛ ⎞

ln=

kf1 k 1»⁄

Sfs( )ch

Sfs( )chπx1kwfkf1------------=

Sch Sfs( )ch

Sch Sfs( )ch2

g λ( )κ ζ1( )ζ∞--------------------------------⎝ ⎠

⎛ ⎞ 1+⎝ ⎠⎛ ⎞ln=

Sfs( )ch 1«

Sch Sfs( )ch2

g λ( )κ ζ1( )ζ∞--------------------------------⎝ ⎠

⎛ ⎞=

κ ζ1( )

κ ζ1( )2ζ1

1 1 ζ1–( )2–------------------------------=


L.6 Infinite Fracture Conductivity 761

Eq. (L-96) simplifies to the Cinco-Ley choked skin for small choked distances (i.e.,, ) and low penetration values ( and ) in a square

reservoir with as shown below

(L-97)

Slot Solution

If one considers that for an infinite conductivity fracture with only a small nearwellbore choked region, most of the fluid enters the tip of the choked region andflows through the choked fracture region with little flowing through the formation

for small values of (i.e., and ).

Cinco-Ley assumed that all of the fluid entered the choked fracture near the tip ofthe choked region with the full flow rate going through the choked part of the frac-ture. This pressure loss in the choked fracture region (for all flow in the choked partof fracture) is

Thus, the choked skin for all flow through the choked fracture in the near wellregion can be calculated from

This result can also be obtained form Eq. (L-25) and Eq. (L-18)

L.6 Infinite Fracture ConductivitySolutions for infinite conductivity fractures in infinite and closed rectangular reser-voirs are presented based on the work of Gringarten.

x1 xf⁄ 1« κ ζ1( ) 1→ Ix 0→ ζ∞ 2→

g λ( ) 1=

Sch Ix 0→Sfs( )ch→

x1 x1 xf⁄ 1«wfkf1xfk

------------ 1»

ΔpchqA--- μ

kf1------ x1 rw–( ) q

2hwf------------ μ

kf1------ x1 rw–( )= =

Sch2πkh

qμ-------------Δpch

πkwfkf1------------ x1 rw–( )= =

Sch1

wfπ-----

kf1k

------⎝ ⎠⎛ ⎞

------------------rw

x1

∫ dξ Schπx1kwfkf1------------ 1 rw x1⁄–( )= = =



Vertical Fracture in an Infinite SystemThis analysis is for a vertical fracture in an infinite-acting system. The dimension-less pressure for a constant production rate is defined as

(L-98)

Gringarten (1974), as report by Earlougher (1977), presented two dimensionlesspressure solutions for a infinite conductivity vertical fracture in an infinite-actingsystem.

Uniform Flux Vertical Fracture This solution assumes that the fluid flux into the fracture is a uniform rate per unitarea of the fracture face with a pressure drop in the fracture (within the fractureplane). The dimensionless pressure in the fracture is calculated from

where the dimensionless fracture position is given by and the dimen-sionless time based on the fracture half-length is defined as

(L-99)

and

(L-100)

The dimensionless pressure at the well for the uniform-flux vertical frac-ture in an infinite-acting system is

pD2πkh

qμ-------------Δp=

pD tDxf( ) 12--- πtDxf erf

1 xD–

2 tDxf

----------------⎝ ⎠⎜ ⎟⎛ ⎞

erf1 xD+

2 tDxf

----------------⎝ ⎠⎜ ⎟⎛ ⎞

+⎩ ⎭⎨ ⎬⎧ ⎫

=

1 xD–( )4

--------------------Ei

1 xD–( )2–4tDxf

-------------------------⎝ ⎠⎜ ⎟⎛ ⎞

–1 xD+( )

4--------------------E

i

1 xD+( )2–4tDxf

--------------------------⎝ ⎠⎜ ⎟⎛ ⎞

–

xD x xf⁄=

tDxfλtL2-----=

λ kcφμ---------=

xD 0=



(L-101)

The dimensionless pressure solution at early times (i.e., , linear or 1Dflow) for a static (non-propagating) uniform flux fracture in an infinite system isgiven by

(L-102)

When , Eq. (L-101) becomes

(L-103)


Infinite-Conductivity Vertical FractureThe infinite conductivity solution assumes that the fracture has an infinite perme-ability and that the pressure is uniform throughout the fracture (no pressure drop inthe fracture). The numerical results of Gringarten determined that the long time

pressure at the wellbore was

(L-104)

Gringarten observed that the same result could be obtained in the uniform-flux frac-ture case by measuring the pressure drop at in the fracture. Therefore,Gringarten concluded that “This suggests that the pressure drop in the fracture forthe infinite-conductivity fracture can be obtained from that for the uniform fluxfracture, Eq. (L-101), with .” The resulting approximate solution for aninfinite-conductivity fracture as given by Gringarten (1974) is

(L-105)

Gringarten noted that Eq. (L-105) “yields the correct value of the wellbore pressurefor a well with an infinite-conductivity vertical fracture, at early and long times. It

pD tDxf( ) πtDxf erf 12 tDxf

----------------⎝ ⎠⎛ ⎞ 1

2---Ei

1–4tDxf------------⎝ ⎠

⎛ ⎞–=

tDxf 0.1<

pD πtDxf→

tDxf 10>

pD12--- tDxf 2.80907+ln( )=

tDxf 10>

pD12--- tDxf 2.2000+ln( )=

xD 0.732=

xD 0.732=

pD tDxf( ) 12--- πtDxf erf 0.134

tDxf

-------------⎝ ⎠⎛ ⎞ erf 0.866

tDxf

-------------⎝ ⎠⎛ ⎞+

⎩ ⎭⎨ ⎬⎧ ⎫

=

0.067Ei0.018–tDxf

----------------⎝ ⎠⎛ ⎞– 0.433Ei

0.750–tDxf

----------------⎝ ⎠⎛ ⎞–



can be assumed that it also yields the correct pressure values during the transitionperiod.” Gringarten also points out that “a similar result ( ) has beenobtained by Muskat for a well with partial penetration at steady state.”

The dimensionless pressure solution at early times (i.e., ) for an infiniteconductivity fracture in an infinite system is given by

(L-106)

and when , Eq. (L-105) becomes

(L-107)


Vertical Fracture in a Rectangular Closed ReservoirGringarten presented solutions for both the uniform flux and infinite conductivityvertical fracture cases in a closed rectangular reservoir. Gringarten noted that “as inthe infinite-reservoir case, it is only necessary to derive the dimensionless pressuredrop for the uniform flux fracture”. The Gringarten solution for a well at the centerof a rectangular reservoir is

or

(L-108)

xD 0.75=

tDxf 0.01<

pD πtDxf→

tDxf 10>

pD12--- tDxf 2.2000+ln( )=

pD xD tDA,( ) 2π 1 2 e n2π2λτ– cos2 nπ2

------⎝ ⎠⎛ ⎞

n 1=

∞

∑+0

tDA

∫=

1 2 e n2π2 λ⁄ τ–

nπ2

------Ix⎝ ⎠⎛ ⎞sin

nπ2

------Ix

------------------------ nπ2

------⎝ ⎠⎛ ⎞cos nπ

2------ 1 xDIx+( )⎝ ⎠

⎛ ⎞cos

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

n 1=

∞

∑+

pD xD tDA,( ) 2π 1 2 e 4n2π2λτ–

n 1=

∞

∑+0

tDA

∫=

1 2 e 4n2π2 λ⁄ τ– nπIx( )sinnπIx

------------------------ nπIxxD( )cosn 1=

∞

∑+



Gringarten stated that “The pressure drops on the fracture for a uniform flux frac-ture and for an infinite-conductivity fracture are obtained by evaluating Eq. (L-108)at and , respectively. The choice of the same point as the infi-nite case leads to reasonable results and can be justified a posteriori by the methodof desuperposition...” Gringarten also states that “it was found that desuperpositionof the analytical solution for a closed square yields a very good approximation ofthe analytical solution for an infinite reservoir, for values of 2, 5, and 10/3.

This justifies a posteriori the choice of for representing the wellborepressure for an infinite-conductivity fracture in the finite reservoir case.”.

The dimensionless pseudosteady solution can be calculated from Eq. (L-7)

where is calculated from Eq. (L-108) once pseudosteady behavior has beenreached.

Gringarten in 1978 presented a closed form analytical expression for the pseudo-steady state form of Eq. (L-108). The inverse dimensionless productivity index fora well located at the center of a rectangular reservoir based on Gringarten’s pseudo-steady solution is

(L-109)

where Gringarten obtained the solutions for a uniform flux and an infinite-conduc-tivity fracture by evaluating Eq. (L-109) at and , respectively.

From Eq. (L-109) for a fully penetrating fracture , the dimensionless produc-

tivity index is .

The problem with assuming a constant value for at all values of and aspect

ratios is that will not be a monotonically increasing function of . In fact

Gringarten’s solution predicts a greater value for at values of then at

. At all aspect ratios Eq. (L-109) exhibits similarities and anomalies as illus-trated in Figure L.3.

xD 0= xD 0.732=

xe xf⁄

xD 0.732=

1 JD⁄ 2πtDA pD–=

pD

1 JD⁄ π6---λ π

4---λIx 1 xD

2+( )– π4---λIx

2 13--- xD

2+⎝ ⎠⎛ ⎞ π

6--- 1

λIx-------+ +=

12π------ 1

λIx------- e

nπλIx 1 xD–( )–

n2 1 e 2nπλ––[ ]------------------------------------ 1 e

2nπλ 1 Ix–( )––[ ] 1 e

2nπλIxxD––[ ]

n 1=

∞

∑–

xD 0= xD 0.732=

Ix 1=

JD Ix 1=

6π---λ=

xD Ix

λ JD Ix

JD Ix 1<

Ix 1=



Figure L.3: Gringarten JD Solution for an Infinite Conductivity Fracture Versus Penetration Distance and Aspect Ratios.

Since must be a monotonically increasing function of , we have

(L-110)

or

Differentiating Eq. (L-109) with respect to for large aspect ratios , we find

JD Ix

dJDdIx--------- 0>

JD2

d 1 JD⁄( )dIx

--------------------- 0<

Ix λ 1»



(L-111)

Therefore for large aspect ratios when the fracture penetration approaches theextent of the reservoir , the value for must be less than 0.732, i.e.,

. As the aspect ratio approaches zero , the maximum

value for is about 2/3 for . Figure L.4 shows the maximum value

for at such that as a function of aspect ratio.

Figure L.4: Maximum values to prevent Similarities and Required

values at such that ( ) Versus Aspect Ratio.

This illustrates that assuming a constant value of for the infinite con-ductivity case to be used in the uniform flux solution is not correct.

Eq. (L-111) also illustrates that the singularity at large aspect ratios occurswhen

d 1 JD⁄( )dIx

---------------------λ 1»

π4---λ 1 xD

2+( )– π2---λIx

13--- xD

2+⎝ ⎠⎛ ⎞ π

6--- 1

λIx2

--------–+ 0<→

Ix 1→ xD

xD 1 3⁄ 0.577350269≅< λ 0→

xD dJD dIx⁄ 0=

xD Ix 1→ dJD dIx⁄ 0=

xD xD

Ix 1= dJD dIx⁄ 0>

xD 0.732=

λ 1»

π4---λ 1 xD

2+( )– π2---λIx

13--- xD

2+⎝ ⎠⎛ ⎞+ 0=



or at

Thus using , we find the singularity for large aspect ratios at .

At an aspect ratio of ten ( ), the singularity position increases to .

A low penetrations, , a value of will result in the effective

wellbore radius being equal to one-half the fracture length ( ) for aninfinite conductivity fracture.

Assuming is of the form

(L-112)

we now only have to find a value of to satisfy the condition

(L-113)

Differentiating Eq. (L-113) with respect to we find

(L-114)

Now differentiating Eq. (L-112) and substituting the results into Eq. (L-114), wefind

or

Ix1 xD

2+

2 13--- xD

2+⎝ ⎠⎛ ⎞

------------------------=

xD 0.732= Ix 0.8835≅

λ 10= Ix 0.93≅

Ix 0→ xD 0.740108≅

rw′ xf⁄ 1 2⁄→

xD

xD λ Ix,( ) xD Ix 0→xD Ix 0→

xD λ 1,( )–[ ] Ix( )α–=

α

dxDIxdIx

-------------- 0>

Ix

dxDdIx---------

xD–Ix

--------->

α xD Ix 0→xD λ 1,( )–[ ] Ix( )α 1– xD λ Ix,( )

Ix---------------------–>–



Then as we have

Noting that for a given aspect ratio near full penetration,

numerically we find the approximate solution for to be

where

The values for as a function of aspect ratio are illustrated in Figure L.4.

Figure L.5 shows the modified dimensionless productivity index solution for aninfinite conductivity fracture as a function of penetration and aspect ratio. As illus-trated increases with penetration distance for all aspect ratios.

αxD λ Ix,( )

Ix( )α xD Ix 0→xD λ 1,( )–[ ]

----------------------------------------------------------------<

Ix 1→

αxD λ 1,( )

xD Ix 0→xD λ 1,( )–

----------------------------------------------<

dJD dxD⁄Ix 1→

cons ttan≅

xD

xD λ Ix,( ) xD Ix 0→xD Ix 0→

xD λ 1,( )–[ ] Ix( )α–=

αxD λ 1,( )

xD Ix 0→xD λ 1,( )–

----------------------------------------------=

xD λ 1,( )

JD

JD



Figure L.5: Modified Gringarten JD Solution for an Infinite Conductivity Fracture Versus Penetration Distance and Aspect Ratios.

Figure L.6 illustrates the performance of an infinite conductivity fracture for rect-angular shaped reservoirs versus a square reservoir. As illustrated, these resultsmatch those presented by Valko and Economides.

Figure L.6: Performance of an infinite conductivity fracture for rectangular shaped reservoirs versus a square reservoir



Figure L.7 shows the dimensionless pressure versus dimensionless time ( ) and

penetration ( ) for a well located at the center of a square reservoir with aninfinity conductivity fracture. Figure L.8 shows the dimensionless pressure versusdimensionless time ( ) and penetration ( ) for a well located at the center ofa square reservoir with an infinity conductivity fracture. Figure L.9 shows thedimensionless pressure versus dimensionless time ( ) and penetration ( )for a well located at the center of a square reservoir for a uniform-flux fracture.

Figure L.7: Dimensionless pressure for a vertically fractured well in the center of a closed square system, infinite conductivity.

tDA

xe xf⁄

tDxf xe xf⁄

tDxf xe xf⁄



Figure L.8: Dimensionless pressure for a vertically fractured well in the center of a closed square system, infinite conductivity.

Figure L.9: Dimensionless pressure for a vertically fractured well in the center of a closed square system, uniform-flux fracture

Effective Wellbore Radius - Infinite Conductivity

The inverse dimensionless effective wellbore radius, , for an infinite conduc-tivity vertical fracture can be calculated from Eq. (L-9)

xf rw′⁄



(L-115)

where

In Eq. (L-115) the dimensionless productivity index is given by Eq. (L-109) and

by Eq. (L-112).

To prevent confusion, the dimensionless fracture half-length ratio with respect tothe effective wellbore radius (i.e., reciprocal effective wellbore radius) for an infi-nite conductivity fracture will be defined as

where is a function of the penetration ratio ( ) as illustrated in Eq.(L-115).

Figure L.10 shows the inverse dimensionless effective wellbore radius, , foran infinite conductivity vertical fracture for various aspect ratios as a function ofpenetration.

xf

rw′

-----CfD ∞→

e1 JD⁄

Ix βxeλ( )⁄=

βxeλ( ) 16

λeγCA λ( )------------------------=

JD

xD

ζ∞ xf rw′⁄

CfD ∞→

≡

ζ∞ Ix( ) Ix xf xe⁄=

xf rw′⁄



Figure L.10: Inverse Dimensionless effective wellbore radius for an infinite conductivity vertical fracture for various aspect ratios.

L.7 Shape FactorThe reservoir shape factor can be found numerically by using the method ofimages for a well in a closed reservoir. The number of image wells to approximatethe infinite series simulation is based on the analytical and numerical studies ofLarson (1985). Although Larson’s method works very well it is not an explicit solu-tion.

Observing that the inverse dimensionless effective wellbore radius, , for aninfinite conductivity vertical fracture is approximately constant for aspect ratiosless than unity, we can formulate an implicit analytical solution for .

From Eq. (L-115) the shape factor can be written as

Then for a fully penetrating fracture we have

CA

xf rw′⁄

CA

CA λ( )ζ∞ λ Ix,( )

Ix----------------------

2e

2– JD⁄ 16λeγ--------=

Ix 1=


L.7 Shape Factor 775

(L-116)

where from Eq. (L-109) has been substituted.

The shape factor ratio of a rectangular reservoir to a square reservoir from Eq. (L-116) is

Eq. (L-116) can be simplified for aspect ratios less than unity since. Using this result the shape factor can be approximated by

(L-117)

where is a slight correction for .

From symmetry for an unfractured reservoir with the well in the center of a rectan-gular reservoir we can Eq. (L-117) as

The slight correction based on numerical simulations is of the form

where the coefficients are related through symmetry

CA λ( ) ζ∞ λ( )2Ix 1= e

π3λ------– 16

λeγ--------=

JD λ Ix 1=,( ) 6π---λ=

CA λ( )CA λ 1=( )-------------------------

ζ∞ λ( )2Ix 1=

ζ∞ λ 1=( )2Ix 1=

-----------------------------------------⎝ ⎠⎜ ⎟⎛ ⎞ e

π3λ------–

eπ3---–

----------

16λeγ--------

16eγ--------------

ζ∞ λ( )2Ix 1=

ζ∞ λ 1=( )2Ix 1=

-----------------------------------------⎝ ⎠⎜ ⎟⎛ ⎞

eπ3---– 1 λ⁄ 1–( )1

λ---

⎝ ⎠⎜ ⎟⎛ ⎞

= =

ζ∞ λ 1<( ) ζ∞ λ 1=( )≅

CA λ 1≤( )CA λ 1=( )------------------------- e

π3---– 1 λ⁄ 1–( )1

λ---

⎝ ⎠⎜ ⎟⎛ ⎞

1 fc λ( )+[ ]≅

fc λ( ) ζ∞ λ 1<( ) ζ∞ λ 1=( )≠

CA λ 1≥( )CA λ 1=( )------------------------- e

π3---– λ 1–( )

λ⎝ ⎠⎜ ⎟⎛ ⎞

1 fc λ( )+[ ]≅

f λ( )

fc λ( ) a 1 e b– λ 1–( )–( )=

dCAdλ

----------λ 1=

0=



Thus from symmetry, we have

where from numerical simulations using Larson’s method of images the coeffi-

cients are found to be and .

The complete analytical implicit solution for the shape factor from the above equa-tions is

(L-118)

where

To calculate the shape factor for aspect ratios less then unity we can use the sym-metry identity

Figure L.11 and Figure L.12 show comparisons of the shape factor as a function ofaspect ratio for the numerical simulation results based on the method of Larson andthe analytical solution of Eq. (L-118). Comparison with the published values ofshape factors from Earlougher are also presented.

a π 3⁄ 1–b

-------------------=

b 2π≅ a π 3⁄ 1–2π

------------------- 0.00751≅=

CA λ 1≥( )CA λ 1=( )------------------------- e

π3---– λ 1–( )

λ⎝ ⎠⎜ ⎟⎛ ⎞

1 fc λ( )+[ ]≅

fc λ( ) π 3⁄ 1–2π

------------------- 1 e 2π– λ 1–( )–( )=

CA λ( ) CA 1 λ⁄( )=


L.8 Fracture Skin 777

Figure L.11: Shape Factor versus Aspect Ratio (1-6) - Numerical and Analytical Comparisons.

Figure L.12: Shape Factor versus Aspect Ratio (6-20) - Numerical and Analytical Comparisons.

L.8 Fracture SkinThe pressure drop, , as a result of a skin effect is defined asΔps



(L-119)

where is the fracture skin.

External Skin

The velocity through the filter cake as a function of the cake permeability ( ),

cake thickness ( ), and pressure drop across the cake ( ) from Darcy’s law

can be written as

(L-120)

The total flow rate over the fracture face is

(L-121)

The pressure drop across the cake in terms of the flow rate is

(L-122)

The fracture skin for an external filter cake from Eq. (L-119) is

Internal SkinThe pressure drop of the fluid in the formation for linear flow adjacent to the frac-ture as result of mobility effects (see Eq. (L-122)) can be written as

Δpsqμ

2πkh-------------⎝ ⎠

⎛ ⎞ S=

s

kc

δc Δps

vkcμc-----

Δpsδc

---------=

q 4hL( )kcμc-----

Δpsδc

---------=

Δpsqμc4h---------

δcL----- 1

kc----=

qμ2πkh------------- π

2---

δcL----- k μ⁄

kc μc⁄--------------⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

sf external

π2---

δcL----- k μ⁄

kc μc⁄--------------⎝ ⎠

⎛ ⎞=


L.9 Radial Flow Differential Equations 779

(L-123)

where the subscript refers to the leakoff fluid properties, is the extent of the

damage region into the formation normal to the fracture face, and is the

leakoff fluid mobility.

The mobility ratio, , is given as

(L-124)

The internal fracture skin from Eq. (L-119) is

(L-125)

L.9 Radial Flow Differential EquationsThe basic equation for the radial flow of a single phase fluid in a homogeneousporous medium is a combination of mass conservation and Darcy’s law (e.,g., seeEarlougher (19977, pg4), Drake (1978, pg 131,295) etc.). The well known govern-ing general partial differential equation in radial form is

(L-126)

This equation is non-linear and in order to obtain an analytical solution it is firstnecessary to linearized (e.g., see Drake). This section will present a general linear-ization methodology using pseudopressure and pseudo time concepts. However,prior to linearizing the above equation we will present the derivation of the radialpartial differential equation. This is necessary for a fundamental understanding ofthe basic underlying assumptions in the above diffusivity equation.

Δpsq

4h------

δsL---- 1

kc---- μ

k---

s

μk---–⎝ ⎠

⎛ ⎞=

qμ2πkh------------- π

2---

δsL---- k μ⁄

ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

s δs

ks μs⁄

M

M k μ⁄ks μs⁄--------------=

sf internal

π2---

δsL---- k μ⁄

ks μs⁄-------------- 1–⎝ ⎠

⎛ ⎞⎩ ⎭⎨ ⎬⎧ ⎫

=

1r---

r∂∂ kρ

μ------r r∂

∂p⎝ ⎠⎛ ⎞ φcρ t∂

∂p=



Derivation of Radial Diffusivity EquationThe general derivation of the diffusivity equation will be in radial form based onmass conservation and Darcy’s law. This derivation is not dependent on the fluidunder consideration. The following simplifying assumptions will be made in thederivation of the partial differential equation in radial flow (see Drake):

1. The reservoir is homogeneous and isotropic in permeability and porosity.

2. The flow is fully developed radial flow, The well is therefore produced overthe entire pay thickness.

The principle of mass conservation applied over a stationary volume elementthrough which fluid flows is

(L-127)

For simplicity, we will use a control volume for radial flow. The fluid velocity inthe r-direction is designated by , the density is indicated by . The rate of mass inthrough the face at is and the rate out through the face at is

. The rate of mass accumulation within the volume element is

.

The continuity equation for radial flow through an imaginary volume fixed in posi-tion or control volume is

Upon simplification for a constant formation height, we have

(L-128)

This is the continuity equation which describes the rate of change of density withrespect to time as a function of the changes in the mass velocity vector at a fixedpoint.

rate ofmass

in⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫ rate of

massout⎩ ⎭

⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

–rate ofmass

accumulation⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫

=

υ ρ

r ρυr( ) r 2πh( ) r Δr+

ρυr( ) r Δr+ 2πh

t∂∂ ρ2πrhΔr( ) 2πrhΔrφ t∂

∂ρ=

ρυr( ) r 2πh( ) ρυr( ) r Δr+ 2πh– 2πrhΔrφ t∂∂ρ=

1r---

r∂∂ ρυr( )– φ t∂

∂ρ=



The radial flux is now placed in terms of the pressure gradient by using Darcy’s lawfor radial flow

(L-129)

Substituting Eq. (L-129) into Eq. (L-128), we find

(L-130)

The density in terms of the pressure from the definition of isothermal compressibil-ity is

or (L-131)

Substituting Eq. (L-131) into Eq. (L-130) and rearranging we find

(L-132)

This is general diffusivity equation for radial flow.

Linearization of Radial Diffusivity Equation

Pseudo-Pressure and Pseudo-Time FunctionsDrake states that “Prior to obtaining useful solutions of this equation it must first belinearized (or partially linearized) and the method by which this can be achieveddepends on the nature of the fluid under consideration”.

Al-Hussainy (1966 see lee ref 7 pg 167) et. al. linearized the above equation inspace by replacing pressure with a pseudopressure to account for the change in vis-cosity and density with pressure. Al-Hussainy defined real gas pseudopressure as

Following is a summary of Drake’s methodology for linearization:

ν kμ---

r∂∂p–=

1r---

r∂∂ kρ

μ------r r∂

∂p⎝ ⎠⎛ ⎞ φ t∂

∂ρ=

c p( ) 1ρ---

p∂∂ρ

T= cρ p∂

∂ρ=

1r---

r∂∂ kρ

μ------r r∂

∂p⎝ ⎠⎛ ⎞ φcρ t∂

∂p=

m p( ) 2 pμ p( )z p( )---------------------- pd

pb

p

∫=



Undersaturated Oil

Drake (pg 138, 295) shows that for liquid flow, the above equation can be partiallylinearized if 1) the viscosity is independent of pressure , 2) the fluid is

slightly compressible, and 3) the term .

Real Gas

Partial linearization using the integral transformation

Gas-Oil

Partial linearization using the integral transformation

or as Drake points out the correct form of this transformation should be

Drake illustrates that the substitution of these transformations leads to the followingre-formulation of Eq. (L-132)

(L-133)

where for an undersaturated oil, , for a real gas, , and for a gas-oilor two-phase flow, .

Total linearization of Eq. (L-132) is only achieved if coefficient is a con-stant (i.e., under saturated oil with the above assumptions). Since for both a real-gasand two-phase flow the product is a function of pressure Eq. (L-132) is still notlinear.

μ constant≅

cp 1« r∂∂p

⎝ ⎠⎛ ⎞

20≅

m p( ) 2 pμ p( )z p( )---------------------- pd

pb

p

∫=

m′ p( )kro So( )

μo p( )Bo p( )---------------------------- pd

pb

p

∫=

m′ p( )kro So( )ρo p( )

μo p( )-------------------------------- pd

pb

p

∫=

1r---

r∂∂ r r∂

∂β⎝ ⎠⎛ ⎞ φμc

k---------

t∂∂β=

β p= β m p( )=β m′ p( )=

φμc k⁄

μc



Agarwal(1979, SPE 8279) developed a new time function which considered varia-tions in gas viscosity and compressibility as a function of pressure to linearize theabove equation. Agarwal’s pseudo-time was defined as

The adjusted time proposed by Lee (pg 115) was

The linearization methodologies are of the general form

Carter (SPE 12917) developed a set of pseudo-steady decline curves which used alamda parameter to represent variations in the decline curve from real gas proper-ties. Carter’s correction to the dimensionless time was

where

Gardner (2000) proposed a dimensionless time of the form

based on the mass balance

ta1

μ p( )c p( )---------------------- td

0

t

∫=

ta μc 1μ p( )c p( )---------------------- td

0

t

∫=

dtaμc

μ p( )c p( )----------------------dt=

tD′ tDλ=

λμc( )i2

------------m pi( ) m pwf( )–[ ]

pz---⎝ ⎠

⎛ ⎞i

pz---⎝ ⎠

⎛ ⎞wf

–------------------------------------------=

tD′ kt

φμcrw2

----------------=

q t′( )μ p( )c p( )---------------------- t′d

0

t

∫Q t( )μc t( )-------------=



Gardner found the average viscosity compressibility product to be

(L-134)

where

This equation can be simplified based on the following simple mass balance rela-tionship

(L-135)

where is the original gas in place.

Substituting Eq. (L-135) into Eq. (L-134) and rearranging we find

Thus lamda factor based on Gardner’s formulation is then

(L-136)

The dimensionless form of Eq. (L-133) can be expressed as

where and

μc

μc2piQ t( )

ziGiΔm p( )--------------------------=

Δm p( ) 2 pμ p( )z p( )---------------------- pd

p

pi

∫=

Q t( ) Gi⁄zipi----

pizi---- p

z p( )----------–⎝ ⎠

⎛ ⎞=

Gi

μc 2pizi---- p

z p( )----------–⎝ ⎠

⎛ ⎞ Δm p( )⁄=

λμc( )i2

------------m pi( ) m p( )–[ ]

pz---⎝ ⎠

⎛ ⎞i

pz---⎝ ⎠

⎛ ⎞p

–-------------------------------------=

1rD-----

rD∂∂ rD rD∂

∂β⎝ ⎠⎛ ⎞

tD∂∂β=

rD r rw⁄=



The lamda factor given by Eq. (L-136) will be shown as the correct formulationwhich satisfies the mass balance equations.

The general solution for a constant production rate at the wellbore ( ) asgiven by Drake is

The inverse dimensionless productivity is given as

or

For the above equation, the various parameters for the specific type of flows aregiven in Table L.1.

Table L.1: Parameter Definitions for Various Types of Flow

UndersaturatedOil

Real Gas Two PhaseGas-Oil

field units field units field units field units

tDkt

φμcrw2

---------------- ktφ μc( )irw

2----------------------λ= =

rD 1=

aq---⎝ ⎠

⎛ ⎞ f p( ) βD tD( ) S+=

aq t( )---------⎝ ⎠

⎛ ⎞ f p( ) 1JD------=

JD1a---q t( )

f p( )---------=

aq--- kh

141.205qoμoBo-------------------------------------- kh

1424qscT----------------------- kh

141.205qo-------------------------

aq--- 2πkh

qoμoBo------------------ 2πkh

qμ-------------μZ

2p-------

πTscpscT----------- kh

qsc-------=

2πkhqo

-------------

a2πkhμoBo------------- πTsc

pscT-----------kh 2πkh

f p( ) Δp pi pwf–= Δm m pi( ) m pwf( )–= Δm′ m′ pi( ) m′ pwf( )–=



Improved Pseudo-Pressure and Pseudo-Time FunctionsA more general methodology of linearizing the governing partial differential equa-tion for radial flow will be presented in this section. The general diffusivity equa-tion for radial flow from Eq. (L-132) is

(L-137)

From Eq. (L-137) the pseudopressure integral transformation function, , that willlinearize the above equation for all fluids is

(L-138)

where the fluid density and viscosity are assumed to be functions of only pressure.Differentiating Eq. (L-138), we find a relationship between the differentialpseudopressure function and pressure

(L-139)

Substituting Eq. (L-139) into Eq. (L-137), the diffusivity equation in terms of thepseudopressure is

(L-140)

A partial dimensionless form of Eq. (L-140) can be expressed as

Table L.1: Parameter Definitions for Various Types of Flow

UndersaturatedOil

Real Gas Two PhaseGas-Oil

f p( ) Δp p pwf–= Δm p( ) m p( ) m pwf( )–= Δm′ p( ) m′ p( ) m′ pwf( )–=

βD tD( ) pD tD( ) mD tD( ) pD tD( )= m′D tD( ) pD tD( )=

1r---

r∂∂ kρ

μ------r r∂

∂p⎝ ⎠⎛ ⎞ φcρ t∂

∂p=

Ψ

Ψ p( ) ρμ--- p′d

pb

p

∫=

ζ∂∂Ψ ρ

μ---

ζ∂∂p=

1r---

r∂∂ kr r∂

∂Ψ⎝ ⎠⎛ ⎞ φμc t∂

∂Ψ=



(L-141)

where and the permeability is assumed constant. Modification of toaccount for relative permeability effects can also be handled as described above byDrake.

To fully linearize Eq. (L-141) we will now introduce a dimensionless pseudotimefunction

(L-142)

where

The lamda factor considers variations in gas viscosity and compressibility as afunction of time. As illustrated, this is essentially the dimensionless form of Agar-wal’s pseudotime transformation in terms of a lamda factor. Evaluation of lamdawill be discussed later on in this report.

The final dimensionless form of the diffusivity equation, obtained by substitutingthe pseudotime transformation (i.e., Eq. (L-142) into Eq. (L-141)), is

(L-143)

Dimensionless ParametersA new set of dimensionless parameters will now be defined that are independent ofthe fluid. The dimensionless pressure function for slightly compressible fluids wasdefined as

1rD-----

rD∂∂ rD rD∂

∂Ψ⎝ ⎠⎛ ⎞ φμcrw

2

k----------------

t∂∂Ψ=

rD r rw⁄= Ψ

τDk

φμcrw2

---------------- t′d0

t

∫=

ktφ μc( )irw

2----------------------

μc( )iμc

------------ t′d0

t

∫ t⁄⎝ ⎠⎛ ⎞=

tDλ t( )=

λ t( )μc( )iμc

------------ t′d0

t

∫ t⁄μ pi( )c pi( )

μc-------------------------= =

1rD-----

rD∂∂ rD rD∂

∂Ψ⎝ ⎠⎛ ⎞

τD∂∂Ψ=



From the definition of the pseudopressure function, we have

or

Therefore, the dimensionless pseudopressure ( ) for a constant production rate

( ) is

(L-144)

where is the formation permeability, is the formation height, is the reservoirfluid density, and is the differential pseudopressure.

The dimensionless pseudotime based on the drainage area, , is defined as

(L-145)

where is the formation porosity and is the initial formation compressibil-ity-viscosity product.

The dimensionless flow rate ( ) for a constant flowing pressure ( ) is definedas

(L-146)

where and the mass flow rate (i.e., mass flowrate per unit time). Therefore, Eq. (L-146) represents the dimensionless mass flowrate

The mass flow rate as a function of the dimensionless mass rate is

pD tDA( ) 2πkhμq

------------- pi pwf–( )=

dΨ ρμ---dp= Δp μ

ρ---ΔΨ=

ΨD

q

ΨD2πkh

ρq-------------ΔΨ=

k h ρ

ΔΨ Ψ pi( ) Ψ pwf( )–=

A

τDAkt

φ μc( )iA--------------------λ tDAλ= =

φ μc( )i

qD pwf

qDρq t( )

2πkhΔΨ---------------------- m· t( )

2πkhΔΨ----------------------= =

ΔΨ Ψ pi( ) Ψ pwf( )–= m· t( ) ρq t( )=

m· D qDm· t( )

2πkhΔΨ----------------------= =



(L-147)

The flow rate at standard conditions is

where is the constant draw down pseudopressure (i.e.,

also constant drawdown pressure since the initial reservoir pressure ( ) minus the

flowing pressure ( ) is a constant).

The dimensionless productivity index, , is defined as

(L-148)

where .

The productivity index ( ) is defined as

(L-149)

where is the mass flow rate, is the flow rate at standard conditions, is the

average reservoir pressure, and is the flowing pressure.

Pseudopressure Relationships and LimitsThis section presents various pseudopressure relationships and limiting solutions of

for undersaturated oil, gas and gas-oil mixtures. The general pseudopressurefunction is

m· t( ) 2πkhΔΨ pi( ) m· D×=

qsc t( ) 2πkhΔΨ pi( )m· Dρsc-------×=

ΔΨ pi( ) Ψ pi( ) Ψ pwf( )–=

pi

pwf

JD

JD1

2πkh------------- m·

Ψ p( ) Ψ pwf( )–------------------------------------×=

12πkh------------- m·

ΔΨ p( )-----------------× 1

2πkh-------------

ρscqscΔΨ p( )-----------------×==

ΔΨ p( ) Ψ p( ) Ψ pwf( )–=

J

J 2πkhμi

-------------JDm· μi⁄

Ψ p( ) Ψ pwf( )–------------------------------------= =

m· μi⁄ΔΨ p( )-----------------

ρscqsc( ) μi⁄ΔΨ p( )

----------------------------==

m· qsc p

pwf

Ψ



Undersaturated Oil

If we assume the viscosity for undersaturated oil is independent of pressure, and that the fluid is slightly compressible, Eq. (L-138) can be

represented by

The dimensionless pseudopressure for a constant flow rate from Eq. (L-144) is

(L-150)

The ratio of the stock tank rate (volume) to the reservoir rate (volume) as definedby the formation volume factor, is

or

where is stock tank flow rate.

Placing Eq. (L-150) in terms of the stock tank flow rate, we have

(L-151)

Real Gas

The constitutive relationship for the density of a real gas is

Ψ p( ) ρμ--- p′d

pb

p

∫=

μ constant≅ cp 1«

Ψ p( ) ρμ--- p′d

pb

p

∫ρoμo----- p pb–( )= =

ΨD2πkhρoq

-------------ΔΨ 2πkhρoq

-------------ρoμo----- pi pwf–( )= =

2πkhμoq

------------- pi pwf–( )=

Bo

Boqq0-----= q qoBo=

qo

ΨD2πkh

μoqoBo------------------ pi pwf–( )=

ρρscTscpscT

--------------- pZ---=



Substituting the above constitutive relationship into Eq. (L-138) we have

or

(L-152)

where the real gas pseudopressure integral transformation by Drake for a real gas is

The dimensionless pseudopressure for a constant flow rate (mass rate) from Eq. (L-144) is

or in terms of Drakes pseudopressure

The above result is identical to Drake’s dimensionless pressure function as given inTable L.1.

Gas-Oil

For a gas oil mixture the density is replaced by the relative permeability densityproduct (i.e., ) and the pseudopressure function is

Ψ p( ) ρμ--- p′d

pb

p

∫ρscTscpscT

--------------- p′μ p′( )Z p′( )--------------------------- p′d

pb

p

∫= =

Ψ p( )ρscTscpscT

--------------- p′μZ------- p′d

pb

p

∫12---

ρscTscpscT

--------------- 2 p′μZ------- p′d

pb

p

∫⎩ ⎭⎨ ⎬⎧ ⎫

= =

12---

ρscTscpscT

---------------m p( )=

m p( ) 2 p′μ p′( )Z p′( )--------------------------- p′d

pb

p

∫=

ΨD2πkh

ρq-------------ΔΨ 2πkh

ρsqs------------- Ψ pi( ) Ψ pwf( )–( )= =

ΨD2πkhρsqs-------------ΔΨ 2πkh

ρsqs-------------1

2---

ρscTscpscT

--------------- m pi( ) m pwf( )–( )= =

πkhqs

---------Tsc

pscT---------- m pi( ) m pwf( )–( )=

ρ kroρo→



(L-153)

or

where the pseudo pressure transformation given by Drake for an oil-gas mixture is

The above equations are applicable above and below the bubble point (i.e., produc-tion above and below the bubble point pressure).

Above the bubble point (undersaturated oil), the compressibility is

The relationship between density and formation factor for undersaturated oil is

or (L-154)

or

where is the initial reservoir pressure and is any reference pressure above thesaturation pressure (bubble point).

The pseudopressure for undersaturated oil in terms of the formation factor is foundby substituting Eq. (L-154) into Eq. (L-153)

(L-155)

Ψ p( )kroρo

μo------------- p′d

pb

p

∫=

Ψ p( ) m′ p( )=

m′ p( )kro So( )ρo p′( )

μo p′( )---------------------------------- pd

pb

p

∫=

c 1ρo-----

dρodp--------- 1

Bo------

dBodp

---------–= =

dρoρo

---------dBoBo

---------–=ρo p( )ρo pi( )---------------

Bo pi( )Bo p( )---------------=

ρo p( )Bo pi( )ρo pi( )

Bo p( )-------------------------------

Bo pr( )ρo pr( )Bo p( )

--------------------------------= =

pi pr

Ψ p( )kroρo

μo------------- p′d

pb

p

∫ ρo pi( )Bo pi( )kro p′( )

μo p′( )Bo p′( )-------------------------------- p′d

pb

p

∫= =


L.10 Dimensionless Rate & Pressure Solutions 793

The dimensionless pseudopressure for a constant flow rate (mass rate) from Eq. (L-144) in terms of Drake’s pseudopressure is

where the stock tank flow rate ( ) to reservoir rate ( ) is .Drake’s pseudopressure in terms of formation factor is given by the integral trans-formation

L.10 Dimensionless Rate & Pressure SolutionsThis section presents a summary of dimensional pressure and rate solutions forradial or infinite acting systems and pseudosteady behavior in closed systems. Thestarting point for the development of the dimensionless rate solutions will be basedon the dimensionless pressure solutions and mass conservation in closed systems.

Solutions are presented for undersaturated oil (liquid), real gas, and two phase gas-oil flow in a homogeneous formation. The solutions are applicable for fractured andunfractured reservoirs.

The general dimensionless pressure and rate solutions are applicable forflow of oil, gas and two phase gas-oil flows as illustrated in Table L.1. The dimen-sionless times will be defined in terms of a lamda parameter to account for the timedependent behavior of viscosity and compressibility effects as illustrated below

(L-156)

and

m′ p( )

ΨD2πkh

ρq-------------ΔΨ 2πkh

ρo pi( )q pi( )----------------------------ρo pi( )Bo pi( )Δm′= =

2πkhqo

------------- m′ pi( ) m′ pwf( )–( )=

qo q pi( ) qo q pi( ) Bo pi( )⁄=

m′ p( )kro So( )

μo p( )Bo p( )---------------------------- pd

pb

p

∫=

pD qD

τDk

φμcrw2

---------------- t′d0

t

∫kt

φ μc( )irw2

----------------------λ tDλ= = =

τDxfk

φμcxf2

--------------- t′d0

t

∫kt

φ μc( )ixf2

----------------------λ tDxfλ= = =



The time average lamda factor for a real gas in a closed system will be shown to be

This lamda formulation for closed systems with pseudosteady state behavior isbased on mass conservation (i.e., evaluated at the average reservoir pressure).

The lamda factor for an undersaturated slightly compressible oil is unity, .

The lamda factor for an infinite acting system should be evaluated at some reason-able average pressure in the flow domain. A good starting point for flow in infiniteor infinite acting systems (unbounded radial flow) is

Infinite or Infinitely Acting SystemThe dimensionless time for an infinite acting system can be represented by

where is evaluated at some average domain pressure as given approxi-mately by

Constant RateThe general dimensionless pressure solution for pseudo-radial flow in an infiniteacting system producing at a constant rate is of the form

τDAk

φμcA------------- t′d

0

t

∫kt

φ μc( )iA--------------------λ tDAλ= = =

λ p( )μc( )i2

------------m pi( ) m p( )–[ ]

pz---⎝ ⎠

⎛ ⎞i

pz---⎝ ⎠

⎛ ⎞p

–-------------------------------------=

λ 1=

λ p( ) λpi pwf+

2------------------⎝ ⎠

⎛ ⎞=

τD tDλ=

λ λ p( )=

λ p( ) λpi pwf+

2------------------⎝ ⎠

⎛ ⎞≅



where is a dimensionless time and is a constant.

No FractureThe dimensionless pressure solution for an infinite-acting system as given by Drake(pg 157) is

(L-157)

where the exponential integral is given by

The exponential integral of the first kind can be approximated by (see mathematicalhand book pg 229)

( )

where is Euler’s (i.e., Euler-Mascheroni) constant.

A simplification to Eq. (L-157) with less than 1% error

when is

(L-158)

where

ΨD12--- τD C+ln( )=

τD C

ΨD12---– Ei

rD2

4τD---------–⎝ ⎠

⎛ ⎞ E1rD

2

4τD---------⎝ ⎠

⎛ ⎞= =

Ei x( )

Ei x–( ) e s–

s------- sd

x

∞

∫–=

E1 z( ) γ– zln– 1–( )nzn

nn!------------------

n 1=

∞

∑–≅ zarg π<

γ 0.5772156649…=

τD 10>

ΨD12--- 4

eγ----τDln⎝ ⎠

⎛ ⎞ 12--- τD 0.80907+ln( )≅ ≅

τD tDλ=



The constant in Eq. (L-158) is actually . However, this infi-

nite series is usually reported as . The constant is also used in theclosed system formulation rather than the equivalent numeric value (i.e.,

).

The exponential integral solution is only applicable if , or when

. The solution at very early times ( ) as given by the Everdingen-Hurst solution is

Vertical Fracture

The dimensionless pressure solution for a vertical fracture from Eq. (L-158) and theeffective wellbore radius concept in an infinite-acting system when is

where

or

4 eγ⁄ 0.809078697…=ln

0.80907 4 eγ⁄ln

12--- 4 eγ⁄ln 0.40453934…≅

rD 20> tD rD2⁄ 0.5>

rD2 25> tD 0.01<

pD2π--- πtD 2 tD π⁄= =

τDxf 10>

ΨD12--- 4

eγ----τDln Sf+=

12--- 4

eγ----τDxfln f+=

Sfrwxf----- f+ln

rwxf-----

xf

rw′

-----ln+lnrw

rw′

-----ln= = =



where

and is a pseudo-skin parameter that is a function of fracture conductivity as origi-nally presented by Cinco-Ley (SPE 10179) (see also spe 23630). The pseudo-skinparameter for a uniform flux fracture is , and for an infinite conductivity frac-ture (i.e., ). Gringarten gave the constant as

from which the dimensionless reciprocal effective wellboreradius is found to be . Numerically Gringarten’s solution with

results in a value of .

The pseudo-skin parameter for a constant fracture conductivity in an infinite actingsystem as illustrated below (i.e., see Eq. (L-83)) is

The apparent fracture conductivity for a uniform flux fracture in pseudosteady or

radial flow with is .

Constant Flowing PressureThe approximate dimensionless rate solution for pseudo-radial flow in an infiniteacting system producing at a constant flowing bottomhole pressure as given by Ear-lougher (pg. 40) is

ΨD12--- 4

eγ----τD⎝ ⎠

⎛ ⎞ rw

rw′

-----⎝ ⎠⎛ ⎞

2ln=

12--- 4

eγ----τDxf⎝ ⎠

⎛ ⎞ xf

rw′

-----⎝ ⎠⎛ ⎞ln+ln=

12--- 4

eγ----τDxf f+ln=

12--- τDxf 0.80907+ln( ) f+=

τDxf tDxfλ=

f

f 1=

f xf rw′⁄ln 0.69314718= = xf rw

′⁄ 2=

2f 0.80907+ 2.2000=

xf rw′⁄ 2.00464≅

xD 0.732= xf rw′⁄ 2.015≅

f xf rw′⁄( )ln π

CfD--------- 2+⎝ ⎠

⎛ ⎞ln= =

f 1= CfD uniform flux

πe 2–----------- 4.37≅=



(L-159)

where is a dimensionless time and is a constant. The corresponding equation

for a slightly compressible fluid is given by . The constant for an unfrac-

tured well (with no skin) is .

Lee (pg. 105) defines a dimensionless cumulative production, (for an unfrac-tured oil well with no skin) as

and for , can be approximated by

The dimensionless rate is

This solution is with 6% of Eq. (L-159) for at . The differ-

ence in these two solutions decreases to zero as .

The dimensionless rate solution at very early times ( ) as given by the Ever-

dingen-Hurst or Laplace solution ( ) is

The general dimensionless mass rate solution for pseudo-radial flow in an infiniteacting system producing at a constant flowing bottomhole pressure is

m· D1

ΨD--------≅ 2

τD C+ln----------------------=

τD C

qD 1 pD⁄≅

C 0.80907=

QpD

QpD qD tDd0tD∫=

tD 200> QpD

QpD4.29881– 2.02566tD+

tDln-------------------------------------------------------≅

qDdQpDdtD

-------------Q– pD tD⁄ 2.02566+

tDln------------------------------------------------= =

C 0.80907= tD 200=

tD ∞→

tD 0.01<

qD2π--- 1

pD------=

qD πtD=

m· D1

ΨD tD( ) S+----------------------------=



The general mass and flow rate equations in terms of the dimensionless mass ratefor an infinite acting systems are

and

The resulting mass and flow rates are summarized below for undersaturated oil,real gas and two phase fluids.

Undersaturated Oil

If we assume that for undersaturated oil that the viscosity is independent of pressure, and that the fluid is slightly compressible, Eq. (L-138) can be

represented by

The dimensionless pseudopressure for a constant flow rate from Eq. (L-144) is

(L-160)

The ratio of the stock tank rate (volume) to the reservoir rate (volume) as definedby the formation volume factor, is

or

where is stock tank flow rate.

Placing Eq. (L-160) in terms of the stock tank flow rate we have

m· t( ) 2πkhΔΨ m· D×=

q 2πkhΔΨm· Dρ

-------×=

μ constant≅ cp 1«

Ψ p( ) ρμ--- p′d

pb

p

∫ρoμo----- p pb–( )= =

q 2πkhΔΨm· Dρo------- 2πkh

ρoμo----- pi pwf–( )

⎩ ⎭⎨ ⎬⎧ ⎫m· D

ρo-------= =

2πkhμo

------------- pi pwf–( )m· D=

Bo

Boqq0-----= q qoBo=

qo



(L-161)

where and .

Real Gas


Substituting the above constitutive relationship into Eq. (L-138), we have

(L-162)

The mass and flow rates in terms of the dimensionless flow rate is

and

or in terms of Drake’s dimensionless pseudopressure

or

qo2πkhμoBo------------- pi pwf–( )m· D=

2πkhΔpμoBo

--------------------qD=

qD m· D≡ Δp pi pwf–=

ρρscTscpscT

--------------- pZ---=

Ψ p( ) 12---

ρscTscpscT

---------------m p( )=

m· D

m t( )· 2πkh( )ΔΨm· D=

qsc t( ) 2πkh( )ΔΨm· Dρsc-------=

m· 2πkh( )ΔΨm· D 2πkh( ) 12---

ρscTscpscT

--------------- m pi( ) m pwf( )–( )⎩ ⎭⎨ ⎬⎧ ⎫

m· D= =

πkhρscTscpscT

--------------- m pi( ) m pwf( )–( )m· D=



where and

The above result is identical to Drake’s dimensionless pressure and rate function asgiven in Table L.1.

Gas-Oil

The mass flow rate for gas-oil two phase flow is

(L-163)

where for a gas oil mixture, the density the pseudopressure function is

(L-164)

and is Drake’s pseudopressure as a function of oil density.

The reservoir flow rate from Eq. (L-163) is

The stock tank rate is

where .

The above equations are applicable above and below the bubble point (i.e., produc-tion above and below the bubble point pressure).

Above the bubble point (undersaturated oil), the pseudopressure in terms of the for-mation factor from Eq. (L-164) is

qsc t( )πTscpscT-----------khΔm pwf( )qD=

Δm pwf( ) m pi( ) m pwf( )–= qD m· D≡

m· t( ) 2πkhΔΨ m· D× 2πkhΔm′ m· D×= =

Ψ p( ) m′ p( )kroρo

μo------------- p′d

pb

p

∫= =

m′ p( )

q pi( ) 2πkhΔΨm· D

ρo pi( )---------------×=

qo 2πkhΔΨm· D

Bo pi( )ρo pi( )-------------------------------×=

q pi( ) qoB pi( )=



(L-165)

where the Drake pseudotime function in terms of the formation volume factor is

The mass flow rate for a constant producing pressure in terms of the Drakepseudopressure is

The stock tank rate is

or

where the stock tank flow rate ( ) to reservoir rate ( ) is .

Closed SystemThe analysis for closed systems producing at a constant flowing pressure must alsoaccount for mass conservation. That is the initial mass of the hydrocarbon systemmust be equal to the mass of system at any time plus the mass of hydrocarbons pro-duced. As we shall see this the underlying basic formulation for a pseudo-timefunction in variable compressible systems.

Constant Mass RateThe general dimensionless pressure solution for pseudosteady flow in a closed sys-tem producing at a constant rate (mass rate) is

Ψ p( )kroρo

μo------------- p′d

pb

p

∫ ρo pi( )Bo pi( )kro p′( )

μo p′( )Bo p′( )-------------------------------- p′d

pb

p

∫= =

ρo pi( )Bo pi( )m′ Bo p,( )=

m′ Bo p,( )kro p′( )

μo p′( )Bo p′( )-------------------------------- p′d

pb

p

∫=

m′ Bo p,( )

m· 2πkh( )ΔΨmD· 2πkh( )ρo pi( )Bo pi( )Δm′ Bo pwf,( )mD

·= =

qo 2πkhΔΨm· D

Bo pi( )ρo pi( )-------------------------------×=

qo 2πkh( )Δm′ Bo pwf,( )mD·=

qo q pi( ) qo q pi( ) Bo pi( )⁄=



(L-166)

where the dimensionless time is

The productivity index as formulated by Cinco-Ley is of the form

Lamda should be evaluated at the average reservoir pressure as given by

The lamda factor for a gas formation will be shown to be of the form

Lamda for a slightly compressible fluid is , as will be shown below.

The formulation of the lamda equation is based on mass conservation principlesand is presented in the constant pressure boundary condition for closed systems.

The above equations are applicable for undersaturated oil, gas, and two phase flow.in fractured and unfractured wells.

Constant Flowing PressureThe dimensionless rate solution for production at a constant flowing bottomholepressure can be obtained from the constant flow rate (mass flow rate) solution (i.e.,

) in laplace space as proposed by van Everdingen and Hurst(see spe 26424 pg5). Their result is given as

ΨD 2πτDA 1 JD⁄+=

τDA tDAλ=

1JD------ 4π

eγCA

------------rexf---- f+ln=

λ p( ) μc( )iΨ pi( ) Ψ p( )–ρ pi( ) ρ p( )–---------------------------------⎝ ⎠

⎛ ⎞=

λ p( )μc( )i2

------------m pi( ) m p( )–[ ]

pz---⎝ ⎠

⎛ ⎞i

pz---⎝ ⎠

⎛ ⎞p

–-------------------------------------=

λ 1=

ΨD 2πτDA 1 JD⁄+=



(L-167)

The dimensionless pressure solution Eq. (L-166) in Laplace space is

(L-168)

The dimensionless mass flow rate solution in Laplace space from Eq. (L-167) andEq. (L-168) is

The inverse transformation of the dimensionless rate solution is

(L-169)

The mass and reservoir flow rate as a function of dimensionless rate are given by

(L-170)

or

and

or

where .

m· D s( ) 1s2ΨD s( )--------------------=

ΨD s( ) 2π

s2------ s JD⁄+=

m· D s( ) 1s2ΨD s( )--------------------

JD2πJD s+---------------------= =

m· D τDA( ) JDe2πJDτDA–

=

m· t( ) 2πkhΔΨ pi( ) m· D×=

m· t( ) 2πkhΔΨ pi( ) JDe2πJDτDA–

×=

q t( ) 2πkhΔΨ pi( ) m· D ρ⁄( )=

q t( )2πkhΔΨ pi( )

ρ-------------------------------JDe

2πJDτDA–=

ΔΨ Ψ pi( ) Ψ pwf( )–=



Pseudopressure as a Function of TimeThe dimensionless pseudopressure as a function of time during draw down can befound from Eq. (L-169), Eq. (L-170), and the definition of the inverse productivityindex Eq. (L-148)

(L-171)

where

The mass flow in terms of the inverse productivity from rate from Eq. (L-171) is

(L-172)

Equating Eq. (L-172) and Eq. (L-170) we have

The dimensionless pressure ratio as a function of dimensionless time from Eq. (L-169) is

(L-173)

or

(L-174)

The above equation relates the change in dimensionless pressure as a function ofpseudotime. The only remaining parameter to be formalized in the above set ofequations is the dimensionless pseudotime lamda factor.

Alternate Method

The mass flow in terms of the inverse productivity from rate from Eq. (L-171) is

JD1

2πkh------------- m·

Ψ p( ) Ψ pwf( )–------------------------------------× 1

2πkh------------- m·

ΔΨ p( )-----------------×= =

ΔΨ p( ) Ψ p( ) Ψ pwf( )–=

m· 2πkh ΔΨ p( )JD×=

m· 2πkh ΔΨ p( )JD× 2πkhΔΨ pi( ) m· D×= =

ΔΨ p( )ΔΨ pi( )------------------

m· DJD------- e

2πJDτDA–= =

ΔΨ p( )ΔΨ pi( )------------------

Ψ p( ) Ψ pwf( )–Ψ pi( ) Ψ pwf( )–-------------------------------------- e

2πJDτDA–= =

m· 2πkh ΔΨ p( )JD×=



The total mass of fluid produced must be equal to the decrease of the fluidmass in the formation. That is

. (L-175)

The mass flow rate is found by differentiating Eq. (L-175) with respect to time

(L-176)

Now from the compressibility and pseudopressure equations, we find

or (L-177)

and

or (L-178)

Then from Eq. (L-176), Eq. (L-177), and Eq. (L-178), we find

(L-179)

Equating Eq. (L-172) with Eq. (L-179), we have

(L-180)

or

(L-181)

ΔM

ΔM ΔM– system φhA ρ pi( ) ρ p( )–[ ]= =

m· t( )td

d ΔM– system( ) φhAdρ p( )dt

--------------–= =

c p( ) 1ρ p( )-----------dρ p( )

dp--------------= dρ p( )

dp-------------- c p( )ρ p( )=

Ψ p( ) ρμ--- pd

pb

p

∫= dΨ p( )dp

---------------- ρ p( )μ p( )-----------=

m· t( ) φhAdρ p( )dt

--------------– φhAdΨ p( )dt

----------------dρ p( )dp

-------------- dpdΨ p( )----------------–= =

φhAdΨ p( )dt

----------------c p( )ρ p( )μ p( )ρ p( )-----------–=

φhAc p( )μ p( )dΨ p( )dt

----------------–=

φhAc p( )μ p( )dΨ p( )dt

----------------– 2πkh ΔΨ p( )JD×=

1ΔΨ p( )-----------------dΨ p( )

pi

p

∫–2πkJD

φμ p( )c p( )A------------------------------dt

0

t

∫=



Integrating Eq. (L-181), we find an expression for pseudo pressure as a function ofthe dimensionless pseudotime

or

where

and

It is worth noting that and are time averaged values. A consequent ofthis is that the total mass production as a function of time can be determined, butthe mass rate can not be directly found from

(L-182)

since is defined with an average viscosity compressibility product over thetotal time. That is, we need an instantaneous value for lamda.

ΔΨ p( )ΔΨ pi( )------------------ln

2πkJDφμ p( )c p( )A------------------------------dt

0

t

∫– 2πJDτDA–= =

ΔΨ p( )ΔΨ pi( )------------------ e

2πJDτDA–=

τDAk

φμ p( )c p( )A------------------------------dt

0

t

∫kt

φμc t( )A-------------------- μc t( )

μ p( )c p( )----------------------dt

0

t

∫= =

ktφμc t( )A--------------------=

ktφ μc( )iA--------------------

μc( )i

μc t( )------------- kt

φ μc( )iA--------------------λ t( )==

1μc t( )------------- 1

μ p( )c p( )----------------------dt

0

t

∫⎝ ⎠⎛ ⎞ t⁄=

λ t( )μc( )i

μc t( )-------------=

μc t( ) λ t( )


×=

τDA



Mass Conservation and LamdaThe total mass of fluid produced from mass conservation is

(L-183)

Recalling that the dimensionless pseudotime from Eq. (L-142) based on drainagearea is

or

where is a constant over the time interval and will be represented as to pre-vent confusion.

Eq. (L-183) can now be written as

(L-184)

ΔM m· t′d0

t

∫=

τDAk

φμcA------------- t′d

0

t

∫=

ktφ cμ( )iA--------------------

μc( )iμ p( )c p( )---------------------- t′d

0

t

∫ t⁄⎝ ⎠⎛ ⎞=

ktφ cμ( )iA--------------------λ t( )=

dτDAkλ

φ cμ( )iA--------------------dt′=

λ λ p( )

ΔM m· t′ kφ cμ( )iA--------------------λ p( )⎝ ⎠

⎛ ⎞ φ cμ( )iAkλ p( )

--------------------⎝ ⎠⎛ ⎞×d

0

t

∫=

m· τ′φ cμ( )iAkλ p( )

--------------------⎝ ⎠⎛ ⎞d

0

τDA

∫=

2πkhΔΨ pi( ) m· D× τ′φ cμ( )iAkλ p( )

--------------------⎝ ⎠⎛ ⎞d

0

τDA

∫=

ΔΨ pi( )λ p( )

------------------ φ cμ( )ihA( ) 2π m· D× τ′d0

τDA

∫=




(L-185)

Integrating Eq. (L-185), we find

Substituting the average pseudopressure ratio from Eq. (L-174), we have

(L-186)

The total mass of fluid produced must be equal to the decrease of the fluidmass in the formation. That is,

. (L-187)

Then substituting Eq. (L-187) into Eq. (L-186) and rearranging, we find that thevalue of lamda to satisfy mass conservation is

or

(L-188)

Simplifying Eq. (L-188) for a slightly compressible fluids and a real gas is pre-sented below.

ΔMΔΨ pi( )

λ p( )------------------ φ cμ( )ihA( ) 2πJDe

2πJDτ′–τ′d

0

τDA

∫=

ΔMΔΨ pi( )

λ p( )------------------ φ cμ( )ihA[ ] 1 e–

2πJDtDA–[ ]=

ΔMΔΨ pi( )

λ p( )------------------ φ μc( )ihA[ ] 1 ΔΨ p( )

ΔΨ pi( )------------------–⎝ ⎠

⎛ ⎞=

ΔΨ pi( )λ p( )

------------------ φ μc( )ihA[ ]ΔΨ pi( ) ΔΨ p( )–

ΔΨ pi( )-----------------------------------------⎝ ⎠

⎛ ⎞=

1λ p( )----------- φ μc( )ihA[ ] Ψ pi( ) Ψ p( )–[ ]=

ΔM

ΔM ΔM– system φhA ρ pi( ) ρ p( )–[ ]= =

λ p( )φ μc( )ihA[ ] Ψ pi( ) Ψ p( )–[ ]

φhA ρ pi( ) ρ p( )–[ ]-------------------------------------------------------------------=


⎛ ⎞=



Undersaturated Oil or Slightly Compressible Fluid

The change in fluid density as a function of pressure can be determined from thedefinition of fluid compressibility

or

(L-189)

Integrating Eq. (L-189) from the initial pressure to the average reservoir pressurefor a constant compressibility ( ), we find

or

(L-190)

Now if the fluid is slightly compressible ( ), we can rewrite Eq. (L-190)as

or

.

The pseudopressure function for a constant viscosity and slightly compressiblefluid is

c 1ρ---dρ

dp------=

dρρ

------ cdp=

c ci=

dρρ

------ρi

ρ

∫ cdppi

p

∫=

ρρi----ln ci p pi–( )=

ci p pi–( ) 1«

ρρi---- e

ci p pi–( )1 ci p pi–( )+≅=

ρi ρ– ρici pi p–( )≅

Ψ p( ) ρμ--- p′d

pb

p

∫=ρiμi---- p pb–( )≅



Consequently, lamda for a slightly compressible fluid with a constant viscosity is

or

.

The reservoir flow rate as a function of the dimensionless rate is

where is the constant draw down pressure (the initial reservoir pres-

sure ( ) minus the flowing pressure ( )). The stock tank flow rate is related to

the reservoir rate by the formation volume factor (i.e., ).

Total volume of oil produced, , as a function of producing time assumingpseudo-steady state behavior is

(L-191)

where the maximum produced reservoir volume for an undersaturated oil (liquid)formation is given by

(L-192)

The average reservoir pressure for a slightly compressible fluid producing at a con-stant bottomhole flowing pressure is

λ p( ) μc( )iΨ pi( ) Ψ p( )–

ρi ρ–---------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞

μc( )i

ρiμi---- pi p–( )

ρici pi p–( )---------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

≅=

λ 1≅

q t( ) 2πkhΔpμ

-------------------- qD× 2πkhΔpμ

-------------------- JDe2πJDtDA–

×= =

Δp pi pwf–=

pi pwf

βo qstb q Bo⁄=

Q t( ) t

Q t( ) q τ( ) τd0

t

∫2πkhΔp

μ-------------------- JDe

2πJDkτ φμcA( )⁄–× τd

0

t

∫= =

2πkhΔpμ

--------------------φμcA2πk

------------- 1 e2πJDtDA–

–( )=

Qmax 1 e2πJDtDA–

–( )=

Qmax φhAcΔp=



or

(L-193)

Now from Eq. (L-5) with the definition of the dimensionless productivity index

we find as a check on our solution that indeed

The above analysis follows from the assumptions that the formation fluid is slightlycompressible and has a constant compressibility.

Real Gas


The lamda factor is

Substituting the above constitutive relationship into Eq. (L-138), we have

(L-194)

pi p–pi pwf–------------------ Q t( )

Qmax------------ 1 e

2πJDtDA––( )= =

p pwf–pi pwf–------------------ e

2πJDtDA–=

JDμ

2πkh------------- q t( )

p pwf–----------------×=

q t( ) 2πkhΔpμ

-------------------- JDe2πJDtDA–

×=

ρρscTscpscT

---------------pz---=


⎛ ⎞=

Ψ p( ) 12---

ρscTscpscT

---------------m p( )=



The lamda factor for a real gas in terms of is

Ideal Gas

The constitutive relationship for the density of an ideal gas is

(L-195)

The compressibility for an ideal gas is found from

(L-196)

The pseudopressure for an ideal gas with a constant viscosity is

(L-197)

The lamda factor for an ideal gas is found by substituting Eq. (L-195), Eq. (L-196),and Eq. (L-197) into Eq. (L-188)

or

(L-198)

The lamda factor for an ideal gas (with a constant viscosity) is shown to range from1 to 1/2.

m p( )

λ p( )μc( )i2

------------m pi( ) m p( )–

piz pi( )------------ p

z p( )----------–

--------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

ρρscTscpscT

---------------p=

c 1ρ---dρ

dp------ 1

ρ---

ρscTscpscT

--------------- 1p---= = =

Ψ p( ) ρμ--- p′d

pb

p

∫=ρi

piμi--------- p2 pb

2–( )≅

λ p( ) cμ( )i

ρipiμi--------- pi

2 p2–( )

ρi 1 ppi----–⎝ ⎠

⎛ ⎞--------------------------------

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

cipi

2 p2–( )pi p–( )

----------------------⎝ ⎠⎛ ⎞= =

λ p( ) 12--- 1 p pi⁄+( )=



Numerical Solution -Total Mass and Mass RateConstant Lamda SolutionA very simple methodology for finding the mass rate and total mass produced is touse a constant value for . From mass conservation, the correct average value for

is

(L-199)

The mass rate is then calculated from

where

The total mass produced as a function of time is then calculated from

where

The above mass balance equation can also be written as

(L-200)

where

λ

λ

λ pwf( ) μc( )iΨ pi( ) Ψ pwf( )–ρ pi( ) ρ pwf( )–--------------------------------------⎝ ⎠

⎛ ⎞=


×=

τDA tDAλ pwf( )=

ΔM t( ) m· t( ) td0

t

∫=

2πkhΔΨ pi( ) JDe2πJDτ–

× τd0

τDA

∫dτDA

dt------------⎝ ⎠

⎛ ⎞⁄=

dτDAdt

------------kλ pwf( )φ μc( )iA--------------------=

ΔM t( ) ΔM pwf( ) 2πJDe2πJDτ′–

τd0

τDA

∫=

ΔM pwf( ) 1 e2πJDτDA–

–( )=



It is easy to illustrate that Eq. (L-199) is the correct choice for lamda. From Eqn wehave

Then as time goes to infinity , we find

or

Rearranging, we find

which is identical to Eq. (L-199).

Eq. (L-200) can also be written in terms of the total mass in place

where the total mass in place is

ΔM pwf( ) φhA ρ pi( ) ρ pwf( )–( )=

ΔM t( ) 2πkhΔΨ pi( ) JDe2πJDτ–

× τd0

τDA

∫kλ pwf( )φ μc( )iA--------------------⎝ ⎠

⎛ ⎞⁄=

φhA( )μc( )iΔΨ pi( )

λ pwf( )------------------------------- 2πJDe

2πJDτ–τd

0

τDA

∫=

φhA( )μc( )iΔΨ pi( )

λ pwf( )------------------------------- 1 e

2πJDτDA––( )=

τDA ∞→

ΔM t( ) ΔM pwf( )→

φhA[ ]μc( )iΔΨ pi( )

λ pwf( )------------------------------- φhA[ ] ρ pi( ) ρ pwf( )–[ ]→

λ pwf( ) μc( )iΔΨ pi( )

ρ pi( ) ρ pwf( )–-----------------------------------⎝ ⎠

⎛ ⎞→

ΔM t( ) ΔM total

ρ pi( ) ρ pwf( )–ρ pi( )

-----------------------------------⎝ ⎠⎛ ⎞ 1 e

2πJDτDA––( )=

ΔM total φhAρ pi( )=



The above equations can be solved using a non-iterative numerical procedure for aconstant lamda value and a given average reservoir pressure:

Step 1

Given:

Step 2

Find: , , , , , and

where

Step 3

Find: , , ,

The total mass of hydrocarbons produced is

The dimensionless pseudotime is then calculated from

or

p

ΔM max ΔΨ pi( ) ΔΨ p( ) ρ p( ) λ p( ) t

Ψ p( ) ρμ--- p′d

pb

p

∫=

dρρ

------ρi

ρ

∫ cdppi

p

∫=

λ pwf( ) μc( )iΨ pi( ) Ψ pwf( )–ρ pi( ) ρ pwf( )–--------------------------------------⎝ ⎠

⎛ ⎞=

ΔM max φhA ρ pi( ) ρ pwf( )–( )=

ΔM p( ) φhA ρ pi( ) ρ p( )–( )=

ΔM p( ) τDA tDA t

M p( ) ΔM– system φhA ρ pi( ) ρ p( )–[ ]= =

τDA

ΔM p( ) ΔM pwf( ) 1 e2πJDτDA–

–( )=



The dimensionless time and time are then found from

and

Step 4

Find:

Time dependent Lamda Solution

As stated above, since and are time average values, the total mass pro-duction as a function of time can be determined, but the mass rate can not bedirectly found from

(L-201)

since was defined with an average viscosity compressibility product over thetotal time. That is, we need an instantaneous value for lamda to evaluate the aboveequation. However, all is not lost.

The numerical procedure to find the total mass and mass rate as a function of timefor a non-constant lamda is outlined below:

Step 1

Given:

Step 2

Find: , , , , and

where

τDA1

2πJD------------- 1 ΔM p( )

ΔM pwf( )----------------------–⎝ ⎠

⎛ ⎞ln=

tDA τDA λ pwf( )⁄=

t τDAφ μc( )iA

k--------------------⎝ ⎠

⎛ ⎞=

m· t( )


×=

μc t( ) λ t( )


×=

τDA

λ t( )

p

ΔΨ pi( ) ΔΨ p( ) ρ p( ) λ p( ) t



Step 3

Find: , , ,

The total mass of hydrocarbons produced is


The dimensionless time and time are then found from

and

Step 4

Find:

Ψ p( ) ρμ--- p′d

pb

p

∫=

dρρ

------ρi

ρ

∫ cdppi

p

∫=


⎛ ⎞=

ΔM p( ) τDA tDA t

ΔM p( ) ΔM– system φhA ρ pi( ) ρ p( )–[ ]= =

τDAΔΨ p( )ΔΨ pi( )------------------ e

2πJDτDA–=

τDA1

2πJD-------------

ΔΨ pi( )ΔΨ p( )------------------⎝ ⎠

⎛ ⎞ln=

tDA τDA λ p( )⁄=

t τDAφ μc( )iA( )

k-------------------------⎝ ⎠

⎛ ⎞=

m· t( )

m· t( )ΔM tn( ) ΔM tn 1–( )–

tn tn 1––-------------------------------------------------=



where is the current time step.

General Dimensionless Numerical Solution

The dimensionless flow rate or mass rate as a function of dimensionless time isgiven by

(L-202)

where is the average reservoir pressure at dimensionless time . This equationcan also be written as

(L-203)

where

The dimensionless flow (or mass) rate as a function of average reservoir pressurefrom Eq. (L-173) is

(L-204)

where .

The numerical procedure to find the total mass and mass rate as a function ofdimensionless time for a non-constant lamda is outlined below:

Step 1

Given: , , , , , , and

Step 2

Find: , , , and .

n

QD p( ) qD tDAd0

tDA

∫1

2πλ p( )------------------

m pi( ) m p( )–m pi( ) m pwf( )–-------------------------------------= =

p tDA

QD p( ) QD pwf( )pi zi⁄ p z⁄–

pi zi⁄ pwf zwf⁄–-------------------------------------×=

QD pwf( ) 12πλ pwf( )-----------------------=

qD p( ) m· D p( ) JDΔΨ p( )ΔΨ pi( )------------------× JD

Δm p( )Δm pi( )------------------×= = =

Δm p( ) m p( ) m pwf( )–=

λ t( )

pn pn 1– qD pn 1–( ) QD pn 1–( ) p z⁄( )n 1– Δm pn 1–( ) tDA pn 1–( )

qD pn( ) QD pn( ) p z⁄( )n Δm pn( )



Step 3

Find: ,

The average dimension flow rate from to is given by

The change in the formation dimensionless mass from to isgiven by

Step 3

Find:


The dimensionless time and real time are then calculated from mass bal-ance

and

qD avgΔQD

tDA pn 1–( ) tDA pn( )

qD avg

qD pn 1–( ) qD pn 1–( )–qD pn 1–( ) qD pn 1–( )⁄( )ln

--------------------------------------------------------------=

tDA pn 1–( ) tDA pn( )

ΔQD QD pn( ) QD pn 1–( )–=

tDA pn( )

τDAΔΨ p( )ΔΨ pi( )------------------ e

2πJDτDA–=

τDA1

2πJD-------------

ΔΨ pi( )ΔΨ p( )------------------⎝ ⎠

⎛ ⎞ln=

tDA pn( )

tDA pn( ) tDA pn 1–( ) ΔQD qD avg⁄+=

tn pn( ) tDA pn( )φ μc( )iA( )

k-------------------------⎝ ⎠

⎛ ⎞=


L.11 Real Gas Potential and Related Equations 821

L.11 Real Gas Potential and Related EquationsThe above equations were transformed into pseudo-pressures for gas systems sincethe gas viscosity and density vary significantly with pressure.

Agarwal PseudopressureTo handle this non-linear behavior problem, a real gas potential (or real gaspseudopressure or pseudopressure) was defined by Agarwal (1978), Al-Hussainy(1996) (see also Earlougher (1977, pp 17) and Economides and Nolte (1987, 1-5))

(L-205)

where is an obituary base pressure. Then,

(L-206)

and

(L-207)

Earlougher [1977] reports that as a rule of thumb, at low pressures ( ) is essentially constant and at high pressures ( ) is essentially

constant.

Another useful relationship from Eq. (L-213) is

The dimensionless real gas potential is obtained from the identity . Nowfrom Eq. (L-1), we have

and

m p( ) 2 pμ p( )z p( )---------------------- pd

pb

p

∫=

pb

m p( )∇p∂

∂ m p( ) p∇ 2pμZ------- p∇= =

t∂∂ m p( ) 2p

μZ-------

t∂∂p=

p 2000 psi<

μZ p 3000 psi< p μZ⁄

dm p( )dp

--------------- 2pμZ-------=

mD pD=

dpDd Δp( )--------------- 2πkh

qμ-------------=



(L-208)

The gas density from the real gas law is

(L-209)

where is the number of moles, is the molecular weight, is the gas volumeand is the non-ideal gas or Z-factor. Eq. (L-209) can be arranged as

(L-210)

or

(L-211)


or

(L-212)

General PseudopressureThe general pseudopressure to linearize the radial diffusivity equation is

(L-213)

where is an obituary base pressure. Then,

dmDd Δm( )----------------

dmDd Δp( )--------------- d Δp( )

d Δm( )----------------

dpDd Δp( )--------------- d Δp( )

d Δm( )----------------×= =

2πkhqμ

------------- μZ2p-------× πkh

q---------= Z

p---×=

ρ nMV

-------- nMZnRT p⁄--------------------- Mp

ZRT-----------= = =

n M VZ

M ρZRTp

--------------ρscRTsc

psc-------------------= =

Zp---

ρscρpsc----------

TscT

-------=

dmDd Δm( )---------------- πkh

q---------

ρscρpsc----------

TscT

-------×πTscpsc

----------- khqscT----------×= =

mDπTscpsc

----------- khqscT---------- Δm×=

Ψ p( ) ρ p( )μ p( )----------- pd

pb

p

∫=

pb


L.12 Forchheimer Equation 823

(L-214)

and

(L-215)

Another useful relationship from Eq. (L-216) is

The dimensionless real gas potential is obtained from the identity . Nowfrom Eq. (L-1), we have

and

(L-216)

Rearranging Eq. (L-216), for a constant mass flux ( ), we find

(L-217)

where

L.12 Forchheimer EquationThe pressure loss in a fracture for Darcy and non-Darcy flow as given by the Forch-heimer equation is

Ψ p( )∇ p∂∂Ψ p∇ ρ

μ--- p∇= =

t∂∂Ψ ρ

μ---

t∂∂p=

dΨ p( )dp

---------------- ρμ---=

ΨD pD=

dpDd Δp( )--------------- 2πkh

qμ-------------=

dΨDd ΔΨ( )-----------------

dΨDd Δp( )--------------- d Δp( )

d ΔΨ( )-----------------=

dpDd Δp( )--------------- d Δp( )

d ΔΨ( )-----------------× 2πkh

qμ------------- μ

ρ---×==

2πkhρq

-------------=

ρq

ΨD2πkh

ρq------------- ΔΨ×=

ΔΨ Ψ pi( ) Ψ p( )–=



(L-218)

where the non-Darcy Reynolds number is defined as

where is the fracture permeability in the fracture, is the fluid density, is thefluid velocity (i.e., is the superficial or average velocity), and is the total flowrate (2-wings) at the wellbore.

Since gas is a compressible fluid, the gas density and velocity are functions of thepressure in the fracture. From mass conservation, the mass flux per unit area in thefracture is related to the gas density and velocity at standard conditions by

(L-219)

Multiplying Eq. L-218 by the gas density and placing in terms at standard condi-tions, we have

(L-220)

or

(L-221)

where the non-Darcy gas Reynolds number which may be a function of position isdefined as

(L-222)

xddp– μ

kf----υ β ρυ2( )+ μ

kf----υ 1

βkfρυμ

---------------+⎝ ⎠⎛ ⎞= =

μkf----υ 1 Reβ x( )+( )=

Reβ x( )βkfρυ x( )

μ-----------------------

βkfρqfμAf

----------------βkfρqt2μAf

----------------= = =

k ρ υ

υ qt

ρυ( )p T, ρυ( )psc Tsc,=

ρ dpdx------–⎝ ⎠

⎛ ⎞ μkf---- ρscυsc( ) β ρscυsc( )2+=

ρμ--- dp

dx------–⎝ ⎠

⎛ ⎞ ρscυsc( ) 1kf---- 1

βkfρscυscμ

-----------------------+⎩ ⎭⎨ ⎬⎧ ⎫

=

ρscυsc( ) 1kf---- 1 Reβ x( )+{ }=

Reβ x( )βkfρυ

μ---------------

βkfρscυscμ

-----------------------βkfρscqsc

2μAf-----------------------= = =



Placing Eq. (L-220) in terms of the dimensionless pseudopressure, , we have

or

The governing equation based on Argwal’s pseudopressure function is

(L-223)

In the above equations, the mass flux ( ) is a function of positionand therefore, the non-Darcy Reynolds number is also a function of position (i.e.,

). Assuming a mass flux distribution of the form

where the alpha power coefficient is a function of fracture conductivity. The flux isapproximately uniform for large conductivity fractures ( ) and for low con-

ductivity fractures the flux power coefficient is much greater than unity ( ).

The total pseudopressure or pressure loss in the fracture (one wing) is

Ψ

dΨdx--------– ρscυsc( ) 1

kf----

β ρscυsc( )2

μ--------------------------+=

dΨdx--------– ρscυsc( ) 1

kf----

β ρscυsc( )2

μ--------------------------+=

ρscυsc( ) 1kf---- 1

βkf ρscυsc( )μ

----------------------------+⎩ ⎭⎨ ⎬⎧ ⎫

=

ρscυsc( ) 1kf---- 1 Reβ x( )+{ }=

dmdx-------–

ρscυsc( )ρ

--------------------dmdp------- μ

kf---- 1

βkf ρscυsc( )μ

----------------------------+⎩ ⎭⎨ ⎬⎧ ⎫

=

2pscTTsc

--------------υsckf

------- 1βkf ρscυsc( )

μ----------------------------+

⎩ ⎭⎨ ⎬⎧ ⎫

=

m· ″ ρv ρv( )sc= =

Reβ x( )

m″· xD( ) m· ″ 0( ) 1 xD–( )αq=

αq 1→

αq 1»



or

(L-224)

The beta Reynolds number is defined as

(L-225)

and

(L-226)

where the gamma coefficient for slot flow ( ) is unity ( ), for large

conductivity fractures or a uniform flux fracture ( ) the gamma coefficient is

about 2/3 ( ) and for low conductivities fractures ( ) the gamma coef-

ficient is about 1/2 ( ).

The single wing mass flux at the wellbore is . The total mass flux at the well-

bore (two-wings) is . It is important to note that the Reynolds numberas define above is based on single wing fracture mass rate. Some authors use a Rey-nolds number based on the total 2 wing mass rate.

ΔΨxfm· ″ 0( )-------------------

1 xD–( )αq

kf------------------------

βm· ″ 0( ) 1 xD–( )2αq

μ----------------------------------------------+ xDd

0

1

∫=

11 αq+---------------⎝ ⎠

⎛ ⎞ 1kf---- 1

1 2αq+-------------------⎝ ⎠

⎛ ⎞ βm· ″ 0( )μ

-------------------+=

11 αq+---------------⎝ ⎠

⎛ ⎞ 1kf---- 1

1 αq+1 2αq+-------------------⎝ ⎠

⎛ ⎞ βkfm· ″ 0( )μ

-----------------------+⎝ ⎠⎜ ⎟⎛ ⎞

=

ΔΨxfm· ″ 0( )------------------- 1

1 αq+---------------⎝ ⎠

⎛ ⎞ 1kf---- 1

1 αq+1 2αq+-------------------⎝ ⎠

⎛ ⎞ Reβ+⎝ ⎠⎛ ⎞=

11 αq+---------------⎝ ⎠

⎛ ⎞ 1kf---- 1 γqReβ+( )=

Reββkfm· ″ 0( )

μ-----------------------

βkfmt· ″

2μ----------------= =

γq1 αq+

1 2αq+-------------------=

γq αq 0= γq 1=

αq 1≅

γq 0.67≅ αq 1»

γq 0.5≅

m· ″ 0( )

mt· ″ 2m· ″ 0( )=



Effective ConductivityThe effective permeability and dimensionless conductivity for non-Darcy flowfrom Eq. (L-224) are given by

(L-227)

and

(L-228)

Rearranging Eq. (L-228), and placing the effective conductivity in terms of a Darcyand non-Darcy dimensionless conductivities, we have

(L-229)

where the non-darcy dimensionless conductivity is

where . The dimensionless beta conductivity can be written as

Eq. (L-229) also illustrates that for an infinite conductivity fracture ( ), the

effective conductivity will be equal to non-darcy value .

kf ekf

-------- 11 γqReβ+------------------------=

CfD eCfD

------------- 11 γqReβ+------------------------=

CfD e

11

CfD--------- 1

CfD β

-------------+-----------------------------=

CfD β

CfDγqReβ-------------- μhw2

γqβρqkxf----------------------- 2μ

γqβρqt----------------- hw2

kxf---------×= = =

ρυ( )p T, ρυ( )psc Tsc,=

CfD β

2μβρqt----------- hw2

kxf---------× 2μ

βρsc qsc( )t------------------------- hw2

kxf---------×= =

CfD ∞→

CfD eCfD β

→



L.13 References1. Gringarten, A.C., Ramey, H.J., and Raghavan, R.: “Unsteady-State Pressure

Distributions Created by a Well with a Single Infinite-Conductivity Fracture,”SPEJ, August 1974, 347-360.

2. Gringarten, A.C.: “Reservoir Limit Testing for Fractured Wells,” SPE 7452,October, 1978.


4. Lee, S.T., and Brockenbrough, J.R.: “A New Analytical Solution for FiniteConductivity Vertical Fractures with Real Time and Laplace Space ParameterEstimation,” SPE 12013, 1983.

5. Ramey, H.J. and Cobb, W.M: “A General Pressure Buildup Theory for a Wellin a Close Drainage Area,” JPT December, 1971, 1493-1505

6. Earlougher, R.C., and Ramey, H.J.: “Interference Analysis in Bounded Sys-tems,” JCPT December 1973, 33-45.

7. Valko, P.P. and Economides, M.J.: “Heavy Crude Production from ShallowFormations: Long Horizontal Wells Versus Horizontal Fractures,” SPE 50421,November, 1998.

8. Prats, M.: “Effect of Vertical Fractures on Reservoir Behavior - Incompress-ible Fluid case,” SPEJ June, 1961,105-118.

9. Prats, M Hazebroek, P., and Strickler, W.R.: “Effect of Vertical Fractures onReservoir Behavior - Compressible Fluid case,” SPEJ June, 1961,105-118.

10. Riley, M.F., Brigham, W.E., and Horne, R.N.: “Analytical Solutions for Ellip-tical Finite-Conductivity Fractures,” October 1991, SPE 22656.

11. Cinco-Ley, H. and Samaniego-V: “Transient pressure Analysis: Finite Con-ductivity Fracture case versus Damaged Fracture Case,” October 1981, SPE10179.

12. Cinco-Ley: “Evaluation of hydraulic fracturing by Transient Pressure AnalysisMethods,” March 1982, SPE 10043.

13. Cinco-Ley, H., Samaniego-V and Dominguez, A.N.: “Transient PressureBehavior for a Well with a Finite-Conductivity Vertical Fracture,” SPEJ Aug.1978.


L.13 References 829

14. McGuire, W.J. and Sikora, V.J.: “The Effect of Vertical Fractures on Well Pro-ductivity,” SPEJ Vol. 219 401-403, 1960.

15. Fetkovich, M.J. et.al: “Decline-Curve Analysis Using Type Curves-Case His-tories,” SPE Formation Evaluation Journal, 637-656, Dec. 1987.

16. Lee, J., and Wattenbarger, R.A.: Gas Reservoir Engineering, SPE TextbookSeries Vol. 5, SPE, 1996.

17. Drake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Science Publi-cations, Amsterdam, 1990.

18. Economides, M., Oligney, R., and Valko, P.: Unified Fracture Design, OrsaPress, Alvin, Texas, 2002.

19. Fraim, M.L. and Wattenbarger, R.A.: Gas Reservoir Decline Analysis usingType Curves with Real gas Pseudo-Pressure and Pseudo-Time,” SPE 14238,September, 1985.

20. Gardner, D.C., Hager, C.J., and Agarwal, R.G.: “Incorporating Rate-TimeSuperposition into Decline type Curve Analysis,” SPE 62475, March 2000.

21. Meyer, B.R. and Jacot, R.H.: “Pseudosteady-State Analysis of Finite Conduc-tivity Vertical Fracture,” SPE 95941, October, 2005.


Subject Index

A

Acidconvection 190diffusion 190diffusivity 148, 228molecular weight 226specific gravity 226treatment schedule 146

Acid concentration 147equilibrium 148inlet 147

Acid data 190closure stress 192conductivity damage factor 191in-situ temperature 193minimum conductivity 192non-reactive fines factor 194rock embedment strength 192rock specific gravity 193

Acid database 224activation energy 228, 229diffusivity 228dissolving power 226heat of reaction 226molecular weight 226reaction order 227reaction rate 227reference code 225reference temperature 228specific gravity 226

Acid treatment schedule 146, 185Acquire real-time data 257Acquired data file 246Acquisition setup 258

add computer’s time 258communication type 263direct connection 260ignore first line of data 260modem 263

Acquisition tool bar 255data window 256

Activation energy 228Active zone 118After closure analysis xxxix, 699

examples 385graphical method 721horner time 719impulse injection 718summary and implementation 718

Agarwalpseudopressure 821

Alpha coefficient 363Alpha parameters 81Analysis wizard 333

analysis type 335regression options 335report options 336select analyses 334wizard window 337

Analytical production simulator 397Angle build rate 106


832 Index

Annulus 105API gravity 418Apparent closure time 712Apparent conductivity 734Apparent conductivity parameter 734Apparent domain radius 739Apparent number of perfs 361Apparent viscosity 90Area 235, 413

drainage 735Arrhenius equation 227, 228ASCII files 246Aspect ratio 735, 738

ellipsoidal 482reservoir 414thermal and water 523

Associate a column of data with a pa-rameter 247Auto design 76, 78, 94, 130, 539

fracture length 132, 136proppant concentration 136proppant mass 132, 136slurry volume 132, 136

Auto scale plots 67Auto select zones 173Automatically find points 371Automatically find points menu 344Average fracture conductivity 421Average reservoir pressure 321Axes 374

delta pressure 375derivative/rate 376pressure 375time 374

B

Ballooning 97Base calculations 389Baud rate 260, 263Beep after each time step 67Best fit 325Beta correlation 232Beta factor 232, 236Beta integration constants 736

square reservoir 737BHTP

maximum 139maximum allowable 106reference depths (3) 115

Biot’s constant 161, 318, 481, 498, 500Biot’s energy balance equation 601Bottom of fracture 119Bottomhole data 273Bottomhole flowing pressure 406, 434, 435Bottomhole pressure 290

maximum 139Breakdown 295Bridge-out 144

criteria 144Bridging 144Bridging criteria 187Bubble point 801Bubble point pressure 418Build a plot 276Building plots 275

name 276Building plots in MView 275Bulk modulus 179, 321, 506Button conventions 11


Index 833

C

Cakebuild coefficient 519erosion coefficient 520erosion rate power coefficient 518erosion to build rate 517erosion to build rate ratio 521external 516permeability 518permeability damage factor 515permeability power coefficient 515porosity 518properties

external 511internal 512

resistance 517Cake thickness

maximum 519minimum 519

Calculate fracture perm damage factor 235Carbonate specific gravity 193Carter’s solution 668Casing 105, 110

database 111overlap 111

Casing database 111, 229Characteristic fracture dimension 162, 316, 479Check list 10Choked fracture 759

skin 760Circular reservoir 736Closed reservior 764Closed square system

dimensionless pressure solution 771

Closed system 802Closure

determining 291finding using derivative method 293lower bound 363upper bound 348

Closure data 323Closure equations 611Closure pressure 192, 220, 288, 324, 391Closure pressure on proppant 187Closure stress 192Closure time 324, 371, 391

put TC at intersection 373CO2 flow meter 155Coefficient 592

filtercake 182, 509thermal expansion 498wall building 182, 509

Coefficient of thermal expansion 499Coiled tubing 112Com port 260, 263Communication

input or output 260link 255setup 259type 260

Composite plots 201Compressibility 179, 506, 728

constant 810fluid 810gas 415total reservoir 415

Concentration 235Concentration per unit area 137Conclusions 589, 621Conduction 189Conductivity 421, 427


834 Index

Conductivity damage factor 191Configure real-time view 270, 271Configuring

real-time data window 269Conservation of momentum 602Consistency index 319, 482Constant critical stress 582Constant finite conductivity fracture 750Constant flowing pressure 797, 803Constant fluid loss 176Constant injection rate 711Constant mass rate 802Constant rate

dimensionless pressure solution 794Constitutive relationship

ideal gas 813Containment 159Continuity 553Continuity equation 780Convection 189, 190Coordinates 735Cortona VRML plug-in 204Cost

fixed equipment 456fluid 454hydraulic power 455miscellaneous 454, 456proppant 455share 458, 460

Cost share 458Coupled fracture and proppant solutions 96Crack initiation 161Critical stress 92, 162, 163Critical stress intensity 161, 316, 479Cross-flow 87, 88Crossover port 105

Crossover pressure loss coefficient 105Cumulative production plots 441, 464Currency

escalation rate 457name 466symbol 466

Currency escalation rate 458Customer information 3Customer support 10

D

Darcy friction factor 472Darcy's law 180, 416, 507Darcy’s law 781Data acquisition system 245Data bits 261Data channels

maximum number 239Data editing 251Data preferences 253

filtering 254identical lines 254merging 254

Data range in MinFrac 332Data sets 244

importing an ASCII file 246importing replay data file 245maximum number of lines 239, 248, 328preview plot 248reference name 244sample frequency 248, 328setup 246start and end rows 248units 248

Data setup


Index 835

dialog box 250Data sources 166Data viewing 267

digital 268translated 268

Databaseaccess 146acid 224, 230casing 229coiled-tubing 229installation 5rock 164tubing 229

Default growth 88Degree of interaction 126, 127

flow rate 126fluid loss 126stiffness 126

Delimiters 246Delta pressure 360, 375Delta pseudo-skin function 741, 744Delta time 297, 375Dendritic 127, 577Deposited concentration ratio 515Depth

bottom of zone 177, 179, 503, 505MD 158scale 200TVD 158, 177, 503

Derivative method 293, 297Derivative min-max range 307Derivative options 307, 376Design mode 75, 538, 549Desuperposition 635Determining closure 291Deviation 106Deviation angle 106

Diagnostic plots 198, 296Diagnostic plots and derivatives 299, 721Differential equations

radial flow 779Diffusion 190Diffusivity 148, 416, 439Diffusivity equation 781

general solution 785linearization 781

Diffusivity model 780Diffusivity multiplier 148Digital data view 268Dilatancy 93, 312, 475, 551Dimensionless conductivity

non-darcy 827Dimensionless domain conductivity 742Dimensionless domain parameter 740, 742Dimensionless effective wellbore radius 749, 774Dimensionless flow rate 800, 819Dimensionless fracture conductivity 124, 746Dimensionless fracture domain position 740Dimensionless fracture flow capacity 124Dimensionless fracture flow rate 733Dimensionless fracture half-length ratio 739Dimensionless fracture position 727Dimensionless inverse fracture diffusiv-ity 439Dimensionless net pressure slope 614Dimensionless parameters 630, 728

dimensionless rate 728


836 Index

pressure 728productivity index 729radial flow 787time 728

Dimensionless positionfracture 740

Dimensionless pressure 729, 762Dimensionless pressure ratio 805Dimensionless pressure solution 669, 730

constant rate 794early times 763infinite acting vertical fracture 796infinite conductivity 764infinite-acting system 795late time 763uniform flux 762

Dimensionless productivity index xxxix, 727, 729, 731, 734

circular reservoir 737definition 789modified 769square reservoir 737summary of equations 747

Dimensionless productivity solution 727Dimensionless pseudopressure

Drake 800time dependent 805

Dimensionless pseudopressure solutionconstant flow rate 799

Dimensionless pseudo-skinversus conductivity 753

Dimensionless pseudosteady solution 765Dimensionless rate 728, 798, 811Dimensionless rate & pressure solutions 793

closed system 802infinite or infinitely acting system 794

Dimensionless rate solution 671, 777, 797, 803Dimensionless reservoir aspect ratio 414Dimensionless time 728, 762, 794Dimensionless well location 414Dimensionless wellbore storage factor 437Dimensions

reservoir 735Direct connection 255, 260Discharge coefficient 95, 121, 122

initial and final 123typical values 122

Discount well revenue 646Discounted return on investment 647Discretization methodology 80, 541Discussion 580Dissolving power 226Domain

apparent radius 739conductivity 742dimensionless parameter 740dimensionless position 740fracture 738, 739radius 739resistivity xxxix, 727

Draglines 373points 373

Drag reduction 79, 540Drainage

area 735radius 735

Drainage area 413, 414, 502


Index 837

Duhamel’s theorem 700

E

Economic data 453Edit selections 374Editing imported data 328Effective conductivity 827Effective dimensionless wellbore radius

versus conductivity 752Effective wellbore radius 729, 730, 731, 749, 772

infinite conductivity fracture 740uniform flux 749

Efficiency 289Ellipsoidal aspect ratio 83, 119, 319, 482Empirical option 597Equivalent drainage radius 735Equivalent reservoir

permeability 416porosity 322viscosity 322

Equivalent reservoir permeability 416Equivalent reservoir porosity 417Equivalent reservoir viscosity 417Erosion to build rate 517Error checking 20, 67, 440

disable 67entering data 20file version 22min./max. 67run-time 21

Escalation rate 458Euler’s constant 795Excess pressure 92, 311Exodus reservoir simulator 421

export data to 73

Export to Exodus 73Exporting data 252Extend range to end of data in MinFrac 333External cake filtration equations 690External cake properties 516External skin 676, 778

F

Far field 578File

copying 15extensions 12name 15new 14opening 13save 14save as 14version 22

File management 12File name extensions 7

application summary 8general 7

Filter cakebuild rate 694coefficient 509, 691erosion rate 695resistance 519, 693thickness 691

maximum 519minimum 519

Filter cake coefficient 182, 323Filtering real-time data 254Filtrate viscosity 181, 508Filtration coefficient 514Filtration law


838 Index

conventional 492Finite conductivity skin 741Finite conductivity vertical fracture 727, 738Fixed depth 107Fixed equipment cost 456Flow area 733Flow behavior index 319, 482Flow control 261Flow rate 811

dimensionless 733Flow resistance at fracture tip 92Flowback 87, 153, 292, 293Flowback rate 292, 320Flowback time 320Fluid and proppant unit cost table 463Fluid code 213Fluid database 211Fluid filtrate viscosity 322Fluid gradient 88Fluid loss xxxviii, 75

conventional 489data 175, 502

layers 175, 502ellipsoidal 75, 492, 493fluid dependent 77, 529fluid type dependent 184, 529history 77interaction 127linear 75models 77, 534multiplier 510pressure dependent 183, 511time dependent 182, 510volume 177, 504

Fluid loss during pumping 606Fluid loss interaction 127

Fluid loss models 176, 494, 502constant 493dynamic 493harmonic 493

Fluid name 213Fluid temperature 82, 494, 542Fluid type 135, 144, 405, 453

gas 405oil & water 405

Fluid type dependent fluid loss 77, 184, 492Fluid type for MNpv 449Fluid unit cost 454Flush fluid type 141, 497Foam 78, 539Foam quality 154Foam treatment schedule 153Forchheimer 232Forchheimer equation 823Formation

compressibility 728porosity 728resistivity 734shape factor 728volume factor 799

Formation data 412, 418Formation volume factor 436Frac fluid leakoff viscosity 322Frac fluid unit cost 454Frac-pack 97Frac-pack screen 105Fraction of PAD 367Fraction of well filled 140Fractional deposition 518Fracture azimuth 414Fracture characteristic plots 464Fracture closure 295


Index 839

Fracture closure pressure 288Fracture conductivity 421, 427

infinite 761piece-wise continuous 741zero conductivity 125

Fracture data source 410Fracture design optimization 403Fracture diffusivity 439Fracture dimension 740Fracture domain 738, 739

rectangular reservoir 741Fracture efficiency 289, 296, 367Fracture flow rate 733Fracture fluid gradient 88, 164Fracture friction model 89, 90, 308, 472Fracture geometry models 83

2-D 2903-dimensional 86, 119GDK 84, 119horizontal ellipsoidal 83, 84, 119PKN 85, 119vertical ellipsoidal 83, 119

Fracture height 319, 482Fracture initiation interval 89

min. stress interval 89perforated interval 89

Fracture interaction 127Fracture interaction factors 126Fracture length

input 130propped 420

Fracture length (input) 132Fracture net pressure 367Fracture permeability 235Fracture propagation criteria 554Fracture propagation solution 604Fracture proppant effects 101

Fracture resistivity 733average 733

Fracture skin 632, 676, 731, 777external 778input 509internal 778

Fracture skin factor 438Fracture solution

homogeneous reservoirs 744slot flow 745

Fracture toughness 92, 161, 162, 311, 316, 474, 479

typical values 317, 480Fracture width 235Fractured system model 732

resistivity 732Fractured well/closed system 413Fractures

dendritic 127multiple parallel 127

Friction factor a and b coefficients 309Friction factor model 597Friction factor multiplier 90, 309, 473

empirical correlation 474Friction factors

Darcy 89, 90, 308, 472Fanning 79, 540

Friction loss multiplier 112, 145Friction tables 217Frictional dissipation 90Fronts

thermal 500thermal and water (plot) 524water 500

Future worth 457


840 Index

G

Gasconstitutive relationship 812

Gas compressibility 415Gas pseudopressure 781Gas PVT data 429Gas PVT table 406, 429Gas specific gravity 417Gas viscosity 429Gas-oil 801Gauge pressure 290General dimensionless rate solution 674General equations 645General flow solution 748General productivity solution

N uniform fracture conductivity zones 755

General solutiondiffusivity equation 785

Geometric parameter 742high conductivity fracture 743low conductivity fracture 743

Geometric shape parameter 742Getting started 1Governing equations 551, 569, 578, 602, 630Governing fluid loss equations 667Graphical edit

shift menu 283Graphical edit menu 280

add shift 281linear interpolation 281range 280remove points 282set to average 281set to value 280

show derivative 283show statistics 283smooth 282standard deviation 282

Graphical edit screen 278Graphical point menu 283

remove point 283set values 283

Graphical technique 304Graphical technique options 305Graphical treatment schedule 149

restart time 150Graphically editing data 277

active curve 278drag single point 279drag y-point 279menu 280select range 278single point 279zoom 279

Graphically editing zones 174Gringarten’s solution 761, 797

anomalies 765infinite conductivity 727infinite conductivity-long time 763pseudosteady state 765rectangular reservoir 764singularities 765

Growthdefault 88

H

Hardware security key 9Harmonic fluid loss 290Heat of reaction 226Heat transfer


Index 841

bottomhole 189coefficients 190conduction 189convection 189fracture 190inlet temperature 189internal database 190mean formation temperature 190model 189option 82surface 189wellbore 190

Heightfracture 482gross fracture 119leakoff 318, 481pay zone 414, 419wellbore 485

Help 19accessing 19menu 19

History match 325, 380, 401estimate parameters 402

History match calculations 391History match data 325Horizontal wells 105Horner analysis 295, 363

Horner plot 366select points 365select ranges 364

Horner plot 295Horner time 363Hydraulic diameter 216

annulus 216circular pipe 216

Hydraulic fracturingoptimization 447

theory xxxviiiHydraulic fracturing theory 551Hydraulic perforation diameter 123Hydraulic power unit cost 455Hydrocarbon saturation 508Hyphenating 246

I

Ideal gas 813Ignore first line of data 260Import data 326Import data from MFrac output file 426, 429Import FOD button 131Import log 165

data sources 166import properties 168parameters 165zones 169, 172

Import MFrac NPV data to MProd 428Import MFrac proppant data to MProd 426Import properties 168Import RT button 128, 548Import stress log 165Imported stress log

import 174Importing a replay data file 245Importing an ASCII data file 246Importing an MFrac file 249Importing an MView ADT file 246Importing data into MinFrac 304Importing real-time data 245Impulse injection 701Infinite conductivity 772Infinite conductivity fracture solution


842 Index

square vs. rectangular reservoirs 770Infinite conductivity solution 727, 763Infinite fracture conductivity 761

effective wellbore radius 772vertical fracture in a rectangular closed reservoir 764vertical fracture in an infinite system 762

Infinite systeminfinite fracture conductivity 762

Infinite vertical conductivity fractures 727Infinite-acting time period 714Infinite-conductivity fracture solution 763Initial reservoir pressure 415Initial stress 499Initiation interval 89Injected fluid 501Injection down 104

annulus 105both 105casing 105tubing 105

Injection pressure 290Injection rate 324, 482Inlet temperature 189Inperpolate generated data 172Input

fracture length 471, 483volume 471, 483

Input parameter menu 132Input treatment schedule 141Insert from database 157, 164, 533In-situ acid temperature 193In-situ fluid 190, 501In-situ stresses 159

Installation 2directories 5

Installing the Meyer Software 2Instantaneous shut-in pressure (ISIP) 288, 391Interaction factors 126, 578

flow rate 578fluid loss 579momentum conservation 579stiffness 579width-opening pressure 580

Interest rate 458Internal and external filter cakes 681Internal cake build and erosion 689Internal cake permeability 686Internal cake properties 512Internal cake skin 688Internal filtration equations 682Internal PVT table 405, 406, 429Internal skin 678, 778Interpolate stress gradient 163Inverse diffusivity 440Inverse dimensionless effective well-bore radius 749, 772Inverse dimensionless productivity

solution 785Inverse dimensionless productivity in-dex

summary 747Inverse fracture diffusivity 439Inverse fracture productivity index

slot flow 745Inverse resistivity 738Irreducible water saturation 503, 508Iterations 80, 541


Index 843

K

Koning 75

L

Lamda factor 784ideal gas 813infinite acting system, 794slightly compressible fluid 811time averaged 794

Lamda parameter 793Laminar flow 89, 585Laplacian operator 632Larson’s method of images 776LAS data 157Layer temperature 498, 500Leakoff 591

area 177, 182, 320, 392, 504coefficient

total 289coefficients 176, 502height 318, 481pressure 179, 180, 506, 507velocity 177, 504viscosity 322

Leakoff area 483Leakoff coefficient

constant 176, 502, 534total 177, 482, 504

Leakoff models 77, 175, 492, 494, 502, 533

constant 176, 502, 534dynamic 178, 504, 534harmonic 178, 534

Leakoff viscosity 322Least principal stress 289Lee and Brockenbrough 629

Legend 55font 56show 55tilde 55turn off 55

Linestyle 52, 55turn off 52, 55

Linear Elastic Fracture Mechanics (LEFM) 161Linear fluid loss 668Linear segments 108Linear solution 671, 705

constant pressure boundary condi-tion 706constant velocity boundary condi-tion 705summary 713time dependent velocity boundary condition 708

Linear-elastic deformation 159, 315Lithology symbols 156, 157Load parameters 243Log file importing 165Long file names 8

M

MAcq 246Magnetic windows 66Manage points 393Marker

style 52, 55turn off 52, 55

Mass accumulation 569Mass conservation 552, 569, 572, 605, 780


844 Index

after pumping 552after shut-in 608during pumping 552

Mass flow rategas-oil two phase flow 801

Mass transfer coefficient 228Maximize well profitability 447Maximum power 428Maximum proppant concentration 483Maximum time step 81McGuire and Sikora curves 404MD at bottom of zone 158Mean formation temperature 190Measured data 401Measured depth (MD) 111, 158Merged data sets

default data range 254Merging data sets 252

range 253time step 253time step value 253

Method of images 636, 776fracture 638no fracture 638

Methodology 650design criteria 651procedures 651

Meyer CD 3Meyer software xxxiiMFast

basic steps 469, 470calculations 484data 477data menu 470generating reports 486introduction 469menu bar 470

options screen 471overview 469plot 485simulation results 484

MFast description xxxivMFrac

basic steps 71, 165data input 102databases 211description xxxiifracture options 82general options 74menu bar 72options 73plot categories 197plots 196proppant options 93references 236reports 207run/performing calculations 194stop menu 195

MFrac NPV file 410MFrac plots 196

acid transport 199diagnostic 198fracture characteristics 197heat transfer 199leakoff/rheology 197multilayer 200net present value 199proppant transport 199treatment 199treatment schedule 199viewing 196wellbore hydraulics 197

MFrac reportsdiscussion 209


Index 845

full report 208selected sections 209summary report 208

MFrac-Lite 527data input 531description xxxivfeature comparisons 527fluid loss data 533fracture options 529general options 528perforation erosion 532proppant options 530rock properties 533zones 531

Micro-frac test 291Min/Max range bar 343, 372MinFrac

analysis 329analysis menu 329analysis wizard 333axes 374

delta pressure 375derivative/rate 376pressure 375time 375

base data 314basic concepts 288, 294basic steps 286closure data 323data input 313description xxxiiiediting imported data 328fracture options 308general options 304generating reports 392graphical technique 304, 329Horner analysis 363

import data 326introduction 285leakoff data 320methodology 287options 302output 388overview 286regression analysis 367select ranges 330simulated calculations 389step down analysis 355step rate analysis 348user specified closure 304wizard window 337

MinFrac time functions 297, 375data time 297, 375delta time 297, 375Nolte G time 297, 375Nolte time 297, 375root delta time 297, 375root Nolte time 297, 375root theta’ time 297, 375time 297, 375

Minifrac methodology xxxviiiMinimum concentration per unit area 187Minimum conductivity for etched width 192Minimum critical stress 162Minimum horizontal stress 159, 289Minimum stress interval 89Miscellaneous cost 454, 456MKey 11MNpv

basic steps 448data Input 452description xxxiv


846 Index

economic data 453fluid type 449introduction 447MProd output file selection 451options 448overview 447plots 464report 465run 463share % table 460unit revenue 458units 466variable unit costs 462

Mobility 181, 416, 508Mobility ratio 779Modem

auto answer 263dial 255, 265initialization 266re-dial 263send all data 263start 255stop 255

Modem connection 261, 265Modem problems 266Modify 4Modifying stages graphically 152Molecular weight 822Momentum conservation 553, 570Momentum equations 571MProd

basic steps 398data description 412data input 411database 443description xxxiiiformation data 412, 418

fracture options 407gas PVT data 429general options 401introduction 397model 404NPV 410NPV fracture data source 425NPV option 410options 399overview 397, 398pay zone height 414, 419plot categories 441plots 440production data 433references 445reports 442reservoir models 404run/performing calculations 440user specified NPV fracture data 428viewing pots 441well data 435

MProd output file selection for NPV 451MPwri

basic steps 490data input 496description xxxivfracture options 495general options 492introduction 489menu 490options 491run/performing calculations 521treatment schedule 497, 521

Multi-Axes plots 202Multi-Case 410Multilayer fracturing xxxviii, 87, 117Multilayer plots 200


Index 847

depth scale 200legends 200

Multilayer zones 117Multi-phase system 417Multiple fractures xxxviii, 125

degree of interaction 127fluid loss interaction 127interaction 126, 127number 127stiffness interaction 127

Multiple parallel fractures 127MView

basic steps 241building plots 275concentration 76, 538data 244data preferences 253data sets 244graphical edit screen 278graphically editing data 277introduction 239menu bar 241overview 239parameters 241plot name 276plots 274preferences 253real-time data collection 254test mode 255viewing plots 276

MView concentration to MFrac 148MView description xxxiiiMWell

basic steps 535data input 545description xxxvgeneral options 538

menu bar 536options 537proppant options 542zones 545

N

Near wellbore 584dissipation function 587governing equations 585losses 355momentum conservation 585pressure loss 92, 127, 362, 586pressure loss table 128, 547pressure table 128, 547width-opening pressure 586

Net cumulative production plots 441, 464Net flow rate 462Net fracture resistivity 738Net fracturing pressure 324Net pay zone thickness 318, 414, 419, 481Net present value (NPV)

MFrac option 76MProd data source 410MProd option 410

Net present value solution 466Net present value theory xxxviiiNet pressure 289, 296, 581

excess 92GDK 84PKN 85

Net pressure ratio 582constant critical stress 583toughness dominated 583viscous dominated 583


848 Index

Network administrator 9No fracture system 413Nolte after closure time 720Nolte G time 297, 375Nolte time 297, 375Nomenclature 563, 574, 623, 639, 696Non-Darcy 824Non-Darcy database 222, 443Non-Darcy effects 407Non-linear elastic effects 312Nonlinear regression analysis 371Non-uniform fracture conductivity 754NPV fracture characteristic dialog box 426, 429NPV fracture data source 425Numerical simulation 654Numerical solution 571, 619, 814

general 819Nusselt number 228

O

Oil API gravity 418Oil displacement factor 503, 508Optimization 403Optimum

fracture performance 404Over flush 758Over-pressure 311, 474Over-pressure factor 92, 311, 475

P

PAD fraction 296PAD volume 296Parameter list 241, 242

load units 243name 243

output unit 243save units 243specifying 242templates 243unit type 243

Parameters 165, 241load units 243maximum number 239save units 243template 243

Parameters Save 243Parametric relationships 555Parity 261, 263Partner share option 450, 458Partner share treatment cost & NPV plots 464Partnerships 460Paused mode 256Pay zone 123

depth 125interval 123permeability 124

Pay zone height 414, 419Pay zone thickness 318, 481Penetration ratio 739Percent propped 485Perforated interval 89, 117, 119, 531, 545, 546Perforation erosion 95, 119, 128, 547

critical proppant mass 122discharge coefficient 122hydraulic diameter 123pressure loss ratio 123rate 122, 123

Perforations 121, 547apparent number 361diameter 121, 122, 547


Index 849

erosion 120, 121number 121, 547tab 120, 547

Permeability 180, 507equivalent 416reservoir 416

Permeability and reservoir pressure 298, 721Phone book 264Pinched fracture 755Pipe roughness 598Plot configuration 49

area colors 58aspect ratio 53axes 54chart type 59color filled 57contours 57curve attributes 41, 51font size 53fonts 42, 52general colors 51grid 57labels 50layout 43, 44left & right axes 45legend 55legend font 56line style 52line width 52marker style 52markers 50plain contour 57scale 53shaded 57use defaults 51

Plot menu 37

add text block 47colors 40configuration 47curve attributes 41default attributes 39font size 42fonts 42general colors 40layout 43left & right axes 45mouse coordinates 44organize templates 62recall templates 62save templates 60templates 60zoom 100% 35, 49zoom out 35, 49

Plot templates 67Plots 32

3D plug-in 203arranging 33auto scale (run time) 67close all 34closing 33composite 201configuring 49copying to clipboard 38diagnostic 198exporting 38footer 37menu 37moving 33multi-axes 202multilayer 196, 200printer setup 36printing 36save as a metafile 39


850 Index

save file as type 39three-dimensional 203virtual reality 205VRML 204windows menu 32zooming 35

Plug-in 204Poisson’s ratio 159, 161, 317, 318, 480, 481Pore pressure 179, 506Poro-elastic input parameters 499Poroelastic stresses 498, 666Poroelasticity 92Porosity 181, 503, 504, 508, 728

equivalent 417mobile 508reservoir 417total 508

Power law model 319, 482Prandtl’s universal law 114, 309, 472, 598Prandtl-Karman law 90, 598Present worth 457Pressure

bubble point 418closure 288excess 92, 311gauge 290injection 290net 84, 92, 289pore 179, 506propagation 348reservoir 415

Pressure decline 297, 617Pressure dependent fluid loss 183, 511Pressure during injection 290Preview plot 248

Printerselecting 15setup 16, 36

Produced water reinjection theory xxx-viiiProduction boundary condition 406Production data 433Production data dialog table 433Production model theory xxxviiiProduction rate 406, 433, 435Production specified option 434, 435Productivity increase 634

constant flow rate 634constant pressure 635

Productivity index 729definition 789

Productivity solutioninfinite conductivity fracture 763

Program basics 11Program check list 10Program descriptions xxxiiProgram maintenance 4

modify 4remove 4repair 4

Propagation parameters 88Propagation pressure 348Property generation 169Proppant

bridging criteria 187closure pressure 220concentration/area 235conductivity damage 191damage factor 235, 431mass 428permeability 220, 221, 430porosity 220, 221, 233, 409, 431


Index 851

profile power law coefficient 138Proppant bridging 186Proppant calculator 231, 445

average diameter 233porosity 233specific gravity 234

Proppant code 219Proppant concentration 145

final 137incremental 136initial 136input 148maximum 137maximum inlet 97MView to MFrac 148

Proppant criteria 185closure pressure on proppant 187minimum concentration per area 187proppant layers to prevent bridging 186

Proppant damage factor 145Proppant database 144, 187, 218

average diameter 221code 219, 223, 444description 219, 223, 444permeability 221, 430porosity 221, 409, 431specific gravity 220, 431

Proppant effectsfracture 101wellbore 100, 543

Proppant flowback 95Proppant mass 132, 136

input 130Proppant mass (input) 132Proppant options 93, 530Proppant ramp 136, 142

linear 139non-linear 139

Proppant ramp option 95, 543Proppant settling options 97

cluster settling 99convective transport 98empirical 98user specified 100, 145

Proppant settling rate 142, 145Proppant solution (on/off) 94Proppant transport methodology 78, 95, 130, 539

conventional 96conventional (link) 96, 97frac-pack 97tip screen-out (TSO) 96

Proppant type 135, 144, 476Proppant unit cost 455

table 463Propped fracture

fraction 312minimum concentration per area 187percent 485

Propped fracture width 235Propped length 420, 427Pseudopressure 632, 821

gas 790general relationships 822ideal gas 813relationships 789undersaturated oil 790

Pseudopressure functionconstant viscosity & slightly com-pressible 810gas-oil 791rule of thumb 821

Pseudo-radial flow 295


852 Index

Pseudo-Radial flow solution 634Pseudo-skin 731Pseudo-skin function 731, 741, 747

delta 741, 744Pseudo-skin parameter 797

constant fracture conductivity 797Pseudo-skin relationships 731Pseudosteady behavior xxxix, 727Pseudosteady cases 750

choked fracture 759constant finite conductivity fracture 750non-uniform fracture conductivity 754over flush 758pinched fracture 755tail-in 757

Pseudosteady equations 729dimensionless pressure 729effective wellbore radius 730pseudo-skin relationships 731

Pseudosteady flow rate 732Pseudosteady fracture solutions 744

homogeneous reservoirs 744slot flow 745

Pseudosteady fractured system model 732

finite conductivity vertical fracture 738fracture resistivity 733inverse dimensionless productivity index 734reservoir resistivity 733resistivity 732

Pseudosteady inverse productivity solu-tion

circular reservoir 736

Pseudosteady model 732Pseudosteady productivity solution

square reservoir 737Pseudosteady solution xxxix, 727

constant lamda 814Pseudosteady state analysis 727Pseudosteady state analysis of finite conductivity vertical fractures xxxixPseudo-time 783Pump rate 136Pump time 324Pump-in/decline test 291Pump-in/flowback test 292PVT table 406

Q

Qualitative parametric effects 558Quality

both external 154external phase 154internal 154internal phase 154Mitchell 154

R

Radial diffusivity equationderivation 780linearization 781

Radial flowdifferential equations 779dimensionless parameters 787pseudo-pressure functions 786pseudopressure relationships and limits 789pseudo-time functions 786

Radial solution 714


Index 853

Horner time 715Nolte’s after-closure function 717

Ramplinear 139non-linear 139

Rangeedit selections 332extend to end of data 332

Raw data 267Raw data viewing 267Reaction order 227Reaction rate 227Readme file 3Real gas

constitutive relationship 812pseudopressure 790

Real gas equation 790Real gas potential and related equations 821Real gas pseudopressure 781Real-time 76, 244, 538

template file 264test mode 264

Real-time datarecover 271

Real-time data window 266configuring 269show 266

Real-time/replay treatment schedule 148Recirculation volume 141, 497Recover real-time data 271, 272Rectangular reservoir

dimensionless productivity index 741

Rectangular reservoir coordinates 735Reel friction loss 112Reference code 223, 444

Reference depthBHTP 115volume 106

Reference temperature 228References 565, 575, 600, 626, 641, 660, 697, 724, 828Refresh button 250Regression analysis 296, 367

deterministic information 296examples 377history match 380main menu bar 370, 371purpose 367, 382select points 369, 370, 371select ranges 368

Regression analysis window 325Regression technique 296Relative pipe roughness 80, 112, 540Remote data acquisition 255Remove 4Repair 4Replay 75, 244, 538, 549

concentration 76, 538Replay/real-time data from MView 305Reports 68

adding a bitmap 69configuring 68exporting 69HTML file 207saving as a text file 69saving as an HTML file 69saving as an RTF file 69viewing 68

Reservoiraspect ratio 414, 735circular 736compressibility 179, 321, 415, 506


854 Index

coupling 492dimensions 735drainage area 413half-length 501layers 87mobile porosity 508permeability 180, 392, 416, 507pore pressure 179, 506porosity 181, 322, 417, 503, 504, 508pressure 179, 321, 415, 811resistivity 733square 737temperature 406, 418viscosity 181, 322, 417, 508

Reservoir Coupling 75Reservoir coupling

ellipsoidal 75linear 75

Reservoir models 404fractured well/closed system 405fractured well/infinite reservoir 405no fracture 404

Reservoir pressure 506Residual hydrocarbon saturation 508Residual oil saturation 503Resistance 517Resistivity 732

average fracture 733formation 734fracture 733net fracture 738reservoir 733, 734square reservoir 739system 738total 738

Restart time 81, 150, 542

Restore default layout 29Results and conclusions 659Revenue as a function of time 459Revenue escalation rate 450Revenue share 458Revenue/unit volume 450, 457, 459, 462Reynolds number 89, 308, 472Rheology data 214Rheology model 319, 482Rock database 164, 230Rock embedment strength 192Rock layers 87Rock properties 156Rock specific gravity 193Rock symbols 157Rock/acid system 147Root delta time 297, 375Root Nolte time 297, 375Root theta' time 297, 375Row selection 248Run options 66

S

Saturation 508irreducible 508residual 508

Saturations 417Save data as a text file 252Screen-out 96, 144

criteria 144Section length 111Security key 9

MKey 11updating 11

Select pointsautomatically find points menu 344


Index 855

Select points in MinFrac 343Select points menu 340Select ranges 330Select ranges in MinFrac 330Semi-log asymptotic 629Sending data to MFrac & MinFrac from MView 272Sending MView concentration to MFrac 148Sending surface or bottomhole data 273Serial cable 255, 260Serial connection 245Serial link 260Setup 4Setup templates 251Shape factor 728, 729, 774

analytical solution 776Share

% table 460net flow rate table 461of cost 458, 460of revenue 458

Sherwood number 228Shift data 248Show simulation data windows 66Show units 55Shut-in time 371Simulate closure 195Simulate to closure 87Simulation data windows 65

show 66Simulation setup 272Skin

calculate 505, 509choked fracture 759external 510finite conductivity 741

input 505internal 509

Skin factorfracture 438wellbore 437, 438

Slightly compressible fluid 810Slip 84Slot flow 745

solution 761Slurry rate 136, 142Slurry volume 132, 136, 428, 483

input 130Slurry volume (input) 132Software installation 2

check list 10Solution

general 748time dependent lamda 817

Solution methodology 554Specific gravity

fluid 213gas 417

Sphericity 233, 234Spreadsheets 23

action keys 23column configuration 30freezing panes 25keyboard commands 23movable columns 30options 26

active cell border style 27alternate background color 27display ellipsis for truncated strings 27frozen rows background color 27move selection after enter 27

shift data 28


856 Index

speed buttons 27, 492y=mx+b 28

Spurt loss 177, 182, 319, 483, 594coefficient 177, 182, 319, 483volume 182, 319, 483

Spurt loss coefficient 182, 320, 483Spurt loss volume 483Square reservoir 737Stage

friction multiplier 145liquid volume 142modifying 152modifying graphically 152slurry volume 142time 143type 143

Stage slurry volume 142Staging profile 138Step down analysis 295, 355

diagnostic plot 361power coefficient 362pressure table 359select points 357select points plot 358select ranges 356

Step down test 295Step rate analysis 295, 348

diagnostic plot 354pressure table 352select points 350select range 349

Step rate test 291, 295Stiffness interaction 127Stock tank flow rate 799Stokes law 98Stop bits 261Stop menu 195

Storage 437Storage factor 437Stress 159

bottom of layer 163initial 499layer 163minimum horizontal 161, 179, 318, 481, 506tectonic 161, 318, 481thermal-elastic 500top of layer 163vertical 161, 318, 481

Stress gradient 156, 159, 163Stress intensity 161, 316, 479Stress intensity factor 161, 316, 479Stress log 165, 175Superposition 434, 700Surface data 273Surface line volume 106Symbols and conventions xlSynchronization 149System & hardware requirements 1System crash

recover real-time data 272System database 211, 224System units 17

T

Tail-in 757Target conductivity 138Target dimensionless conductivity 137Taylor series 572Technical support 10Technical support documentation xxxviiTectonic stresses 161, 318, 481Temperature


Index 857

fluid 494layer 498, 500mean formation 190reservoir 418

Templates 60axis title 63load 63organize 62recall 62save 60, 63

Tensile strength 161, 316, 479Test mode 255, 264The infinite-conductivity solution 670The uniform fracture flux solution 670Thermal & water fronts aspect ratio plot 523Thermal and Water Front Equations 661Thermal and water front equations 661Thermal front 499, 500Thermal stress

input parameters 499option 493

Thermoelastic and poroelastic stresses 664Thermoelastic stresses 664Time dependent fluid loss 182, 183, 510, 511Time dependent revenue 459Time scales 297, 375Time step 80, 81, 434, 542

maximum 81, 542synchronization 149

Tip effects 91, 310, 474Tip over-pressure 92, 93, 311, 474Tip screen-out (TSO) 96Tools menu 231Top of fracture 119

Tortuosity 90, 128, 472, 547, 577Total fracture height 319, 482Total leakoff coefficient 177, 289, 482, 504, 592Total leakoff height 318, 481Total pay zone height 414, 419Total proppant mass 428Total reservoir compressibility 321, 415Total suspended solids 514Total system resistivity 738Toughness 316Toughness dominated 582Transition volume loss per unit area 685Translated data viewing 268Transmit data 255Treatment cost 466Treatment cost & NPV plots 464Treatment design options 78, 539Treatment schedule 129

acid 146auto design 130database access 146fluid type 144foam 153, 155graphical 149input 141liquid volume 142proppant concentration 145proppant damage factor 145, 431proppant database 144proppant settling rate 145proppant settling rate (input) 142proppant type 144real-time 148slurry rate 142slurry volume 142stage time 143


858 Index

stage type 143variable column 142variable column list box 498

Treatment type 78, 539Trilinear solution 633

constant flow rate 633constant pressure 633

True vertical depth (TVD) 158, 320TSO & frac-pack methodology xxxviiiTSO option 96Tubing 105Tubing database 112, 229Turbulent flow 89, 586TVD at bottom of zone 158TVD versus MD plot 110

U

Undersaturated oil 799, 810pseudopressure 790

Uniform flux 746fracture 748fracture case 763solution 727vertical fracture 762

Unit revenue 457, 458as a function of time 459escalation rate 458

Units 17input 19load 17output 19save 17

User database 211, 224User specified closure 304, 305User specified pumping data 306User’s guide

how to use xxxixoverview xxxiprogram descriptions xxxiiwhat’s in xxxv

V

Variable column list box 142Variable injection rate 709Variable percentage vs. rate 460Vertical fracture

finite conductivity 738rectangular reservoir 764

Vertical wells 118, 546Viewing plots 276Virk's maximum drag reduction asymp-tote 598Viscosity 181, 508

equivalent 417reservoir 417

Viscous dominated 581Volume

oil produced 811Volume factor 799Volume injected 483VRML 204

plug-in 204

W

Wall building coefficient 182, 323, 509Wall roughness 90, 309, 472, 473Water front 500Water/thermal front plots 522Waterflood theory xxxviiiWeb browser 204Well data 435Well location 414


Index 859

WellboreBHTP reference depth 115deviated 104, 545deviation data 106fixed depth 107fluid friction multiplier 141fluid type 141fraction filled 140linear segment 107proppant effects 543tapered 104, 545volume 141volume reference depth 106

Wellbore deviation 106Wellbore friction factor xxxviiiWellbore heat transfer 190Wellbore hydraulics

general tab 104linear segments 108

Wellbore hydraulics model 78, 103, 539, 545

casing data 110empirical 79, 539general data 104none 79, 539profile 116restrictions data 114tubing data 110user database 80, 541

Wellbore hydraulics screen 104Wellbore positioning within the drain-age area 414Wellbore power coefficient 362Wellbore proppant effects 100Wellbore radius 436Wellbore skin factor 437, 438

base 437

prefrac 438reference 438stimulated 437

Wellbore solutionsynchronize 76, 81, 538, 542

Wellbore specific gravity 320Wellbore storage 437Wellbore storage factor 437Wellbore volume 106, 141, 497Width

average at well 485average in fracture 485maximum at well 485

Width-opening pressure 603Width-opening pressure elasticity condi-tion 553Windows fundamentals 11

Y

Young’s modulus 159, 315, 478typical values 315, 478

Z

Z-factor 429, 822Zone data 119, 546Zones 117, 172, 531, 545

active 118dialog screen 118, 532, 546name 118, 546

Zooming 35zoom 100% 35zoom out 35


860 Index


meyer user's guide 3

Documents

meyer software programs

warranty meyer

trademarks of meyer

meyer grants

licensed software

associated software

nontransferable license

end user