2008 branimir cvetkovic

NTNUFaculty of Engineering And Technology

Well Production

Decline

Branimir Cvetkovic

December 2008

Supervisor:Assessor:

ii

Abstract

E¤ective rate-time analysis during a declining production in an oil or gas wells isan important tool for establishing a successful management. The reason behindthe production decline include reservoir, fracture and well conditions. A well�sdecline rate is transient, signifying that the pressure wave propagates freely fromthe wellbore, leading to depletion when the outer boundary for the well is reachedand to the wave propagation coming to a halt. This thesis studies the transientdecline, with emphasis on a horizontal well with fracture wellbore responses. Italso deals with depletion decline, investigating the wellbore pressure responsesfor a vertical well producing under variable rate conditions of Arps decline.The well decline model solutions are analytical, and the modelling itself is

carried out in two steps. The �rst step involves modelling the transient wellresponses of a multi-fractured horizontal well. These responses originate froman in�nitive reservoir and are considered as full-time rate-time responses. Multi-fractured horizontal well rate-time responses represent the solutions to a di¤u-sion equation with varying boundary conditions and di¤erent fracture options(i.e., with or without fracture, a variety of fracture orientations, various frac-ture lengths, etc.). The transient model calculates individual fracture rates,productivity indexes and an equivalent wellbore radius for the multi-fracturedwell. For the transient decline of a fractured-horizontal well model, well datais matched and the reservoir diagnosis and production prognosis are improvedthrough the individual fracture production, with a model screening ability, andnovel model features that can handle wellbore conditions changing from rate-to-pressure. Screening analyses can generate valuable information for fracturediagnosis in addition to a well and fracture production prognosis. Further modelruns are carried out to match the real well data. The model solution is comple-mentary to the reservoir simulation. More geology features should be consideredto fully take advantage of the modelling �ndings. The starting point of the sec-ond modelling step concerns late time vertical-well responses or decline curvesinvolving empirical solution of Arps type. This includes an investigation of wellpressure responses for a rate decline of an Arps-type variable-rate of a wellborefor selected exponents, b. The modelling explores pressure wellbore responsesduring a variable-rate production, and the approach introduces a no-�ow speci-�ed boundary that moves outwards from a wellbore axis with a prede�ned speed.For the speci�c speed of a no-�ow moving boundary, the model generates pres-sure pro�les causing the decline in production. In the depletion decline, pressurepro�les were generated for various decline exponents, b. Known b values were se-lected and each of them was empirically derived through the in�ow performancerelationships to a drive mechanism. This modelling approach with analyticallyderived pressure solutions can be extended to a horizontal well. Furthermore,the continuously measured well rates and pressure models can be calibrated andveri�ed.E¤ective rate-time analysis during a declining production in an oil or gas

iii

wells is an important tool for establishing a successful management. The reasonbehind the production decline include reservoir, fracture and well conditions. Awell�s decline rate is transient, signifying that the pressure wave propagates freelyfrom the wellbore, leading to depletion when the outer boundary for the well isreached and to the wave propagation coming to a halt. This thesis studies thetransient decline, with emphasis on a horizontal well with fracture wellbore re-sponses. It also deals with depletion decline, investigating the wellbore pressureresponses for a vertical well producing under variable rate conditions of Arpsdecline.The well decline model solutions are analytical, and the modelling itself is

carried out in two steps. The �rst step involves modelling the transient wellresponses of a multi-fractured horizontal well. These responses originate froman in�nitive reservoir and are considered as full-time rate-time responses. Multi-fractured horizontal well rate-time responses represent the solutions to a di¤u-sion equation with varying boundary conditions and di¤erent fracture options(i.e., with or without fracture, a variety of fracture orientations, various frac-ture lengths, etc.). The transient model calculates individual fracture rates,productivity indexes and an equivalent wellbore radius for the multi-fracturedwell. For the transient decline of a fractured-horizontal well model, well datais matched and the reservoir diagnosis and production prognosis are improvedthrough the individual fracture production, with a model screening ability, andnovel model features that can handle wellbore conditions changing from rate-to-pressure. Screening analyses can generate valuable information for fracturediagnosis in addition to a well and fracture production prognosis. Further modelruns are carried out to match the real well data. The model solution is comple-mentary to the reservoir simulation. More geology features should be consideredto fully take advantage of the modelling �ndings. The starting point of the sec-ond modelling step concerns late time vertical-well responses or decline curvesinvolving empirical solution of Arps type. This includes an investigation of wellpressure responses for a rate decline of an Arps-type variable-rate of a wellborefor selected exponents, b. The modelling explores pressure wellbore responsesduring a variable-rate production, and the approach introduces a no-�ow speci-�ed boundary that moves outwards from a wellbore axis with a prede�ned speed.For the speci�c speed of a no-�ow moving boundary, the model generates pres-sure pro�les causing the decline in production. In the depletion decline, pressurepro�les were generated for various decline exponents, b. Known b values were se-lected and each of them was empirically derived through the in�ow performancerelationships to a drive mechanism. This modelling approach with analyticallyderived pressure solutions can be extended to a horizontal well. Furthermore,the continuously measured well rates and pressure models can be calibrated andveri�ed.

Contents

Abstract iii

Contents iii

List of Tables vii

List of Figures ix

Acknowledgements xxiii

1 INTRODUCTION 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 TRANSIENT RATE DECLINE REVIEW 92.1 Oil Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Vertical Well . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Horizontal Well . . . . . . . . . . . . . . . . . . . . . . . . 282.1.3 Vertical-Fractured Well . . . . . . . . . . . . . . . . . . . . 302.1.4 Horizontal-Fractured Well . . . . . . . . . . . . . . . . . . 35

2.2 Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.1 Vertical Well . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.2 Vertical-Fractured Well . . . . . . . . . . . . . . . . . . . . 412.2.3 Horizontal-Well . . . . . . . . . . . . . . . . . . . . . . . . 422.2.4 Horizontal-Fractured Well . . . . . . . . . . . . . . . . . . 43

2.3 Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Flow Under Variable Rate and Pressure . . . . . . . . . . . . . . . 462.5 Other Transient Models . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.1 Multilateral Model . . . . . . . . . . . . . . . . . . . . . . 492.5.2 Multiple Wells Model . . . . . . . . . . . . . . . . . . . . . 51

iii

iv CONTENTS

3 DEPLETION RATE DECLINE REVIEW 533.1 Empirical Models (Arp�s) . . . . . . . . . . . . . . . . . . . . . . . 533.2 Analytical-Numerical Models . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Oil Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2 Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2.3 Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Type-Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Vertical Well . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.2 Vertical Fractured Well . . . . . . . . . . . . . . . . . . . . 813.3.3 Horizontal Well . . . . . . . . . . . . . . . . . . . . . . . . 843.3.4 Horizontal Fractured Well . . . . . . . . . . . . . . . . . . 863.3.5 Multilateral Well . . . . . . . . . . . . . . . . . . . . . . . 893.3.6 Multi Wells . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4 Decline Curve Analysis Physics . . . . . . . . . . . . . . . . . . . 963.4.1 Solution Gas Drive Decline . . . . . . . . . . . . . . . . . . 983.4.2 Solution Gas Drive and Gravity Drainage Decline . . . . . 107

3.5 Analysis of Well Production data . . . . . . . . . . . . . . . . . . 1103.5.1 Type Curves and Decline Curve Analysis . . . . . . . . . . 110

4 RATE DECLINE OF A FRACTURED WELL 1274.1 Transient Oil Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.1.1 Fractured-Vertical Well Model . . . . . . . . . . . . . . . . 1284.1.2 Horizontal Well with Transversal Fractures . . . . . . . . . 1314.1.3 Horizontal Well with Longitudinal Fractures . . . . . . . . 136

4.2 Depletion Oil Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.2.1 Fractured Vertical Well Model . . . . . . . . . . . . . . . . 1424.2.2 Horizontal Well with Transversal Fractures . . . . . . . . . 1444.2.3 Model Solutions Comparison . . . . . . . . . . . . . . . . . 147

4.3 Additional Well-Fracture Features . . . . . . . . . . . . . . . . . . 1514.3.1 Fracture Conductivity . . . . . . . . . . . . . . . . . . . . 1514.3.2 Well Conductivity . . . . . . . . . . . . . . . . . . . . . . . 1524.3.3 Fracture-Well Limited Communication (Choking E¤ect) . 1544.3.4 Restart Option . . . . . . . . . . . . . . . . . . . . . . . . 1544.3.5 Late Time Approximations . . . . . . . . . . . . . . . . . . 157

5 RATE DECLINE WITH A MOVING BOUNDARY 1655.1 Vertical Well �Oil Flow . . . . . . . . . . . . . . . . . . . . . . . 167

5.1.1 Introduction to Moving Boundary Problems . . . . . . . . 1675.1.2 Fixed Boundary . . . . . . . . . . . . . . . . . . . . . . . . 1685.1.3 Moving Boundary . . . . . . . . . . . . . . . . . . . . . . . 1695.1.4 Variable Rate Production of Arps Type . . . . . . . . . . . 178

5.2 Vertical Well - Gas Flow . . . . . . . . . . . . . . . . . . . . . . . 1845.2.1 Pseudo-Pressure Transformation (Intermediate Pressures) . 185

CONTENTS v

5.2.2 Pressure Squared Transformation . . . . . . . . . . . . . . 1895.2.3 No-Flow Moving Boundary - Gas Flow Solutions . . . . . 190

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6 RATE DECLINE CURVES 1936.1 Transient Decline of a Fractured-Horizontal Well . . . . . . . . . . 194

6.1.1 Transient-Rate Response of a Well . . . . . . . . . . . . . 1956.1.2 Well Transient-Pressure Responses . . . . . . . . . . . . . 2136.1.3 Well Pressure-to-Rate Responses . . . . . . . . . . . . . . 2156.1.4 A Well with Longitudinal Fractures . . . . . . . . . . . . . 218

6.2 Well Depletion Responses . . . . . . . . . . . . . . . . . . . . . . 2206.2.1 Closed (BOX) Model . . . . . . . . . . . . . . . . . . . . . 220

6.3 Rate Decline with a No-�ow Moving Boundary . . . . . . . . . . . 2226.3.1 Hyperbolic Decline (b = 1=3) . . . . . . . . . . . . . . . . 2226.3.2 Hyperbolic Decline (b = 0:5) . . . . . . . . . . . . . . . . . 2256.3.3 Harmonic Decline (b = 1) . . . . . . . . . . . . . . . . . . 2326.3.4 Decline Exponent (b = 2) . . . . . . . . . . . . . . . . . . 242

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7 CASE STUDIES 2557.1 Model Comparison and Validation . . . . . . . . . . . . . . . . . . 255

7.1.1 Well Models Comparisons . . . . . . . . . . . . . . . . . . 2557.1.2 Semi-Analytical versus Numerical Model Validation . . . . 256

7.2 Fractured-Horizontal Well - Oil Production . . . . . . . . . . . . . 2597.2.1 Fractured-Horizontal Well (Eko�sk Oil Field - North Sea) . 2597.2.2 Fractured-Horizontal Well (North Sea Oil Field) . . . . . . 2627.2.3 Syd-Arne Oil Field (North Sea) . . . . . . . . . . . . . . . 2667.2.4 Syd-Arne Oil Field (North Sea) . . . . . . . . . . . . . . . 2747.2.5 Valhall Oil Field (North Sea) . . . . . . . . . . . . . . . . 282

7.3 Fractured-Horizontal Well - Water Injection . . . . . . . . . . . . 2897.3.1 Syd-Arne (North Sea) . . . . . . . . . . . . . . . . . . . . 289

7.4 Fractured-Horizontal Well - Gas Production/Injection . . . . . . . 2947.4.1 Syd-Arne Synthetic Data . . . . . . . . . . . . . . . . . . . 294

7.5 Vertical Well �Exponential Decline with the Moving Boundary . 2967.5.1 Variable Rate IBCs of b Almost Zero . . . . . . . . . . . . 296

8 DISCUSSION, CONCLUSIONS, RECOMMENDATIONS 3118.1 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.1.1 Transient Rate Decline . . . . . . . . . . . . . . . . . . . . 3118.1.2 Depletion Rate Decline . . . . . . . . . . . . . . . . . . . . 314

8.2 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 3158.3 RECOMMENDATIONS . . . . . . . . . . . . . . . . . . . . . . . 316

vi CONTENTS

9 NOMENCLATURE 3199.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.2 SI Metric Conversion Factors . . . . . . . . . . . . . . . . . . . . 321

10 REFERENCES 323

A No-Flow Moving Boundary Model Solutions 345

B "LAPLACE" Inversion Transforms 347

C Regression Techniques 351C.1 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . 351C.2 Linear Multiple Regression . . . . . . . . . . . . . . . . . . . . . 352C.3 Weighted Residuals Regression . . . . . . . . . . . . . . . . . . . 354

D Relevant Reports and Papers 355D.1 SPE Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355D.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

D.2.1 Conferences/Forums . . . . . . . . . . . . . . . . . . . . . 355D.2.2 Schlumberger Internal EUREKA Presentations: . . . . . . 356

D.3 Industry Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . 356D.4 NTNU Faculty Reports . . . . . . . . . . . . . . . . . . . . . . . . 356D.5 Other Related Presentations and Reports . . . . . . . . . . . . . . 357

List of Tables

1-1 Pressure testing and rate testing comparisons of events, interpre-tation methods and historical emphasis (Following the 2006 reviewby Gringarten and Anderson et al.] . . . . . . . . . . . . . . . . . 7

2-1 Pressure gradients and dimensionless pressure functions for radialreservoir �ow at the well - After Valko and Economides (1995) . . 11

2-2 Variable rate publications the history [After Gringarten (2006)] . 47

3-1 The rate-time equation for a gas well in terms of the back pressureexponent, n, with constant "pwf" of 0 as de�ned by Fetkovich (1980) 71

3-2 The dimensionless ratio as a function of dimensionless pressure asde�ned by Anash et al. (2000) . . . . . . . . . . . . . . . . . . . . 78

3-3 The vertical well, the vertical and the horizontal fracture . . . . . 82

4-1 Solution for a fractured-vertical (longitudinal) and a fractured-horizontal (transversal) well . . . . . . . . . . . . . . . . . . . . . 148

4-2 Finite and in�nite conductivity fractures - parameters) . . . . . . 1534-3 Model solutions for a fractured-vertical well as compared to those

of a fractured-horizontal (single-transversal-fracture) well . . . . 163

5-1 The dimensionless pressure, "pD", calculated by the model at time"tD", for a variable-rate production of Arps type, de�ned by ex-ponent b (ranging from 0.3; 0.5; 1., and 2) . . . . . . . . . . . . . 183

6-1 Reservoir, Well and Fracture Data . . . . . . . . . . . . . . . . . . 1956-2 The sensitivity to the initial reservoir pressure, "Pi", the porosity

and the e¤ective reservoir thickness, h . . . . . . . . . . . . . . . 1966-3 Sensitivity to fracture half length sizes, Lf . . . . . . . . . . . . . 2086-4 Sensitivity to the distance bewtween two fractures due to well

length, L changes (L= 2520, 2100, and 1680 ft . . . . . . . . . . . 2086-5 Sensitivity to the distance bewtween two fractures due to number

of fractures changes sor the same well length, L = 2100 ft . . . . . 2086-6 Sensitivity to fracture conductivity "FC" (mDft) equal to 2500,

1000, qnd 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

vii

viii LIST OF TABLES

6-7 The sensitivity to the fracture conductivity ,"FC" (mDft), whenit is in�nite, 1000, and 50 mDft . . . . . . . . . . . . . . . . . . . 211

6-8 Wellbore inner boundary conditions, IBCs of variable pressure forthe in�nite conductivity and �nite conductivity for fractures with"FC" = 50 mDft . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

6-9 The constant A as a function of the coe¢ cient of the no-�owmoving boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6-10 A as a function of the coe¢ cient of the no-�ow moving boundary 2316-11 A as a function of the coe¢ cient of the no-�ow moving boundary 2396-12 The constant A as the function of the coe¢ cient of the no-�ow

moving boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 2466-13 The dimensionless pressure, "pD", calculated by the model for

"tD" going to 0,and the "tD" going to in�nity, for a variable rateproduction of Arps type. The decline exponent, b (b with valuesof 0.333; 0.5; 1., and 2) de�nes the Arps type production decline . 252

7-1 The input data for fractured-horizontal well (MF), fractured-verticalwell (VFW) and partially perforated horizontal well (PP) model . 257

7-2 Input data from the Eko�sk �eld - North Sea (for a horizontalwell with 8 transversal-fractures) . . . . . . . . . . . . . . . . . . 260

7-3 Some general parameters of the Syd Arne North Sea �eld (SPE103282) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

7-4 Wellbore IBCs of variable pressure for the in�nite conductivityand �nite conductivity, for "FC" = 50 (mDft) fractures . . . . . . 285

7-5 Input parameters (n and "rD") to the semi-exponential decline inthe moving boundary model . . . . . . . . . . . . . . . . . . . . . 297

B-1 The inverse "Laplace" transform methods [after Davies and Mar-tin (1979)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

B-2 The comparrison of the inverse "Laplace" transform methods . . . 350

List of Figures

2.1 The domain in which the pseudo-pressure, ; varies linearly withp and p2[After Bourdarot (1998)]. . . . . . . . . . . . . . . . . . . 38

3.1 Dimensionless Arps curves (Decline b = 0:0; 0:5, and 1:0) [AfterCvetkovic (1992)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Semi-analytical dimensionless rate-time type curves (for variousdimensionless radii, rD) [After Cvetkovic (1992)]. . . . . . . . . . 64

3.3 Combined transient-depletion dimensionless Fetkovich (1973) rate-time type curves [After Cvetkovic (1992)]. . . . . . . . . . . . . . 65

3.4 Transient dimensionless rate-time curves (for two values of rD)[After Cvetkovic (1992)]. . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Transformed depletion dimensionless rate-time curves (for two di-mensionless rD) [After Cvetkovic (1992)]. . . . . . . . . . . . . . . 66

3.6 Transient dimensionless rate-time curves (for various dimension-less rD values) [After Cvetkovic (1992)]. . . . . . . . . . . . . . . . 67

3.7 Arps dimensionless rate-time curves [After Cvetkovic (1992)]. . . . 673.8 A type-curve match for a constant- pressure drawdown test with

variable property solutions [After Samaniego and Cinco (1980)]. . 693.9 The dimensionless �ow rate compared to the Arps�decline rates

[After Samaniego and Cinco (1980)]. . . . . . . . . . . . . . . . . 703.10 Radial-linear gas reservoir type curves [After Carter (1985)]. . . . 733.11 The linear and the radial �ow geometry [After Chen and Teufel

(2000)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.12 The dimensionless rate, qD, and the cumulative production, QD,

versus the dimensionless time, tD [After Chen and Teufel (2000)]. 753.13 The composite type-curves: (A) The �ow rate vs. time; (B) The

cumulative production vs. time; (C) The �ow rate vs. the cumu-lative production [After Chen and Teufel (2000)]. . . . . . . . . . 76

3.14 The distribution of the viscosity-compressibility function [AfterAnsah et al. 2000]. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.15 The "�rst-order" polynomial solution for real-gas �ow under boundary-dominated �ow conditions. A viscosity-permeability, �ct, is linearwith dimensionless pressure, pD [After Ansah 2000]. . . . . . . . . 79

ix

x LIST OF FIGURES

3.16 The "exponential" solutions for real-gas �ow under boundary-dominated �ow conditions [After Ansah (2000)]. . . . . . . . . . 79

3.17 "General polynomial" solution for real-gas �ow under boundary-dominated boundary conditions [after Ansah 2000)]. . . . . . . . 80

3.18 The vertical well, the vertical and the horizontal fracture. . . . . . 833.19 The vertical fractured well in a rectangular drainage area [After

Chen et al. (1991)]. . . . . . . . . . . . . . . . . . . . . . . . . . . 833.20 Type of �ow for a vertcal fractured well . . . . . . . . . . . . . . . 843.21 The dimensionless rate, qD versus the dimensionless time, tDXf

for the horizontal well [After Cox et al. (1996)]. . . . . . . . . . . 853.22 Decline curve for a horizontal well ina bounded reservoir [After

Poon (1991)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.23 The e¤ect of the aspect ratio on horizontal well productivity (the

ratio of the length to the width of a rectangular well pattern)[After Poon (1991)]. . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.24 The fracture orientation along a horizontal well. . . . . . . . . . . 903.25 A well in a three layered reservoir with perforated segments re-

placed by uniform-�ux fractures. . . . . . . . . . . . . . . . . . . . 913.26 The multibranch and multiple-fracture con�gurations for horizon-

tal wells [After Economides at al. (2001)]. . . . . . . . . . . . . . 913.27 The multilateral well types [After Louis J. Durlofsky TAML, 1999

presentation]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.28 A vertical and horizontal well with laterals positioning within an

oil reservoir [After Cvetkovic et al., (2007)]. . . . . . . . . . . . . 933.29 The multiple vertical, horizontal and deviated completioned wells

in the layered reservoir [After Gilchrist et al. (2007)]. . . . . . . . 953.30 The dimensionless rate vs. the dimensionless time Fetkovich type

curves [After Fetkovich (1980)]. . . . . . . . . . . . . . . . . . . . 1023.31 The �rst decline on Fetkovich�s type curve, for b > 0 [After Padilla

and Camacho (2004)]. . . . . . . . . . . . . . . . . . . . . . . . . 1073.32 The �rst decline on Fetkovich�s type curve, for b = 0 [After Padilla

and Camacho (2004)]. . . . . . . . . . . . . . . . . . . . . . . . . 1083.33 The second decline on Fetkovich�s type curve, for b < 0. The

decline exponent is negative and constant [After Padilla and Ca-macho (2004)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.34 Production decline curves for a �nite-conductivity, vertcally frac-tured well positioned in a closed rectangular reservoir.[after Pos-ton and Poe (2008)]. . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.35 The production decline curves for a vertical well positioned in acylindrical reservoir with a step-rate �ux outer-boundary condi-tion .[After Poston and Poe (2008)]. . . . . . . . . . . . . . . . . . 121

LIST OF FIGURES xi

3.36 The production decline curves for a vertical well positioned in acylindrical reservoir with a ramp�rate �ux outer-boundary condi-tion.[After Poston and Poe (2008)]. . . . . . . . . . . . . . . . . . 122

3.37 Production decline curves for an in�nite conductivity fracturedwell, centrally located in a closed, cyllindrical reservoir [After Po-stone and Poe (2008)]. . . . . . . . . . . . . . . . . . . . . . . . . 123

3.38 Production decline curves for a �nite-conductivity vertically frac-tured well centrally located in a closed [after Poston and Poe(2008) ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.1 A fractured-horizontal well of length, L, with three transversalfractures of half-lengths, Lf . The reservoir is non-bounded orin�nite in the x and y directions (Top view). . . . . . . . . . . . . 132

4.2 A fractured-horizontal well of length L , with three transversalfractures of half-lengt Lf . The reservoir is non-bounded or in�nitein the Xe and the Ye directions (Cross section view). . . . . . . . 137

4.3 An in�nite conductivity vertical fracture fully penetrating the xdirection of a reservoir and the formation in the vertical z direc-tion. The no-�ow outer boundary condition de�nes the closedrectangular reservoir (Areal cross section). . . . . . . . . . . . . . 143

4.4 A fractured-horizontal well of length L with three transversal frac-tures of half-length Lf . The reservoir is bounded and no-�owboundaries are Xe and Ye (Areal cross section). . . . . . . . . . . 145

4.5 A vertical-fractured, and a horizontal- fractured well (with transver-sal and longitudinal single fractures). . . . . . . . . . . . . . . . . 149

4.6 The model for a vertical-fractured well rate, qD; versus s. Thefracture is longitudinal and of in�nite conductivity (�Laplace"space solutions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.7 The model for a fractured-horizontal (with a transversal fractureof uniform �ux) well rate, qD ; versus s (�Laplace" space solutions).150

4.8 The two model solutions for the rate, qD ; versus s (�Laplace"space solutions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.9 Models for a fractured-horizontal well, the e¤ective wellbore ra-dius of a vertical well, and the e¤ective half-length of a horizontalwell with a single transversal fracture. . . . . . . . . . . . . . . . 158

5.1 The dimensionless pressure, pD, as a function of the dimensionlesstime, tD and radial distance, rD ( with K = 1, and speed x = 10). 172

5.2 The dimensionless pressure, pD; as a function of the dimensionlesstime, tD; for the dimensionless distance, rD = 1 ( with K = 1,and x = 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

xii LIST OF FIGURES

5.3 The dimensionless pressure, pD; as a function of the dimensionlesstime, tD; for the dimensionless distance, rD = 10 ( with K = 1and x = 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.4 The dimensionless distance, rD; of a no-�ow moving boundary asa function of a constant, �, and the dimensionless time tD . . . . . 176

5.5 The dimensionless position, rD; versus the dimensionless time, tD;for a coe¢ cient of no-�ow moving boundary � = 1,3,5,7 and 9. . 177

5.6 The velocity of a no-�ow moving boundary, vD = drDd�D

; as a func-tion of the dimensionless time, tD; for various constant values of ,�: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.7 The dimensionless distance, rD; of a no-�ow moving boundary asa function of the dimensionless time, tD; for various constants ofthe no-�ow moving boundary, � = 1; 3; 5; 7;and 9. . . . . . . . . . 178

5.8 A 3D plot of Arps equation rate, q; versus time, t; and initialdecline, Di ( for a decline exponent b = 0:33, b = 0:5, b = 1, andb = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.9 A 2D plot of Arps equation rate, q, versus time, t; and initialdecline, Di (for a decline exponent b = 0:33, b = 0:5 ; b = 1, andb = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.10 The rate, q; versus time, t, for b = 0:33 plotted in circles, andvarious decline exponents (b = 0:5; 1; and 2) all plotted as solidline. The rate versus time is calculated for a speci�c initial decline,qi = 5000, and an initial decline, Di = 0:01). . . . . . . . . . . . . 180

5.11 The constant A as a function of the constant � of a no-�ow movingboundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.1 Individual fracture rates, qfri (i = 1; :::5) and the rate of a well(with fractures), q (bbl=d), versus time, t, in days. . . . . . . . . . 196

6.2 Individual cumulative fracture production, Qfri (i = 1; :::; 5), andthe cumulative production of a well with fractures, Q (bbl), versustime, t, in days. . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.3 Rate and cumulative production pro�les for various values of poros-ity, �, (i.e., 0.24, 0.34, 0.44). . . . . . . . . . . . . . . . . . . . . . 198

6.4 Rate and cumulative production pro�les for various e¤ective thick-nesses, h, (i.e., 95 , 75 and 55 ft). . . . . . . . . . . . . . . . . . . 198

6.5 The individual-fracture rate and individual-cumulative fractureproduction versus time for various values of initial pressure, Pi,(i.e., 6425, 5700, and 4975 psi). . . . . . . . . . . . . . . . . . . . 199

6.6 The rates and cumulative productions vs. time for varying valuesof the initial wellbore pressure, Pwf , (i.e., 2176, 2900 and 3626 psi).199

6.7 Individual fracture rates and cumulative productions (qfr2 andQfr2) vs. time for various values of wellbore pressure, Pwf , (i.e.,2176, 2900 and 3626 psi). . . . . . . . . . . . . . . . . . . . . . . 200

LIST OF FIGURES xiii



6.10 Individual fracture rates, q and cumulative productions, Q vs.time for various values of permeability, Kh= Kv (i.e., 40, 4, 0.4mD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.11 The individual fracture rate, q; and cumulative production, Q; vs.time for various vertical permeabilities, Kv (i.e., 0.4, 2, and 4 mD). 202

6.12 The individual fracture rate, q; and cumulative production, Q;vs. time for various total compressibility values, cT (i.e., 7.25e-05, 6.26 e-05, 1.86 e-06). . . . . . . . . . . . . . . . . . . . . . . 203

6.13 The individual fracture rate, q, and the cumulative production,Q, vs. time for various oil viscosities, � (i.e., 3, 4 and 5 cp). . . . 203

6.14 The individual fracture rate, q, and the cumulative production,Q, vs. time for various oil viscosities, Bo (i.e., 1.35, 1.55 and 1.75rb/stb). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

6.15 The individual fracture rate, q, and the cumulative production, Q,vs. time for various fractures (longitudinal fracture vs. transversalfractures). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

6.16 Productivity index, PI for a horizontal well with 5 transversalfractures and cumulative rate, Q with time for various fracturepartial penetrations with height, h (of 75, 55 and 35 ft). . . . . . 206

6.17 Productivity index, PI for a horizontal well with a longitudinalfracture positioned along the 2100 ft horizontal well, and cumu-lative rate, Q with time for various fracture partial penetrationswith height, h (of 75, 55 and 35 ft). . . . . . . . . . . . . . . . . 206

6.18 Productivity index, PI for a horizontal well with a longitudinalfracture and transversal fractures, and cumulative rate, Q withtime for the partial penetrated fracture the height, h = 55 ft. . . 207

6.19 Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the various half-length,Lf (of 170, 85 and 42.5 ft). . . . . . . . . . . . . . . . . . . . . . 207

6.20 Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the unequal fracture(case 2 and 3) compared to the equally sized fractures (case 1). . 209

6.21 Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the well length, L (of2520, 2100 and 1680 ft). . . . . . . . . . . . . . . . . . . . . . . . 209

xiv LIST OF FIGURES

6.22 Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the various numberof fractures, on a �xed well-length, L=2100 ft for various numberof fractures, n (of 7, 5 and 3). . . . . . . . . . . . . . . . . . . . . 210

6.23 The productivity index, PI; for a horizontal well with transversalfractures, and the cumulative production, Q; vs. time for the var-ious fracture characters (uniform �ux, in�nite conductivity, and�nite conductivity (with FC = 1000 mDft). . . . . . . . . . . . . . 211

6.24 The productivity index, PI; for a horizontal well with transversalfractures, and the cumulative rate, Q; vs. time for the in�niteconductivity fracture and �nite conductivity fractures (with FCvalues of 1000 and 50 mDft). . . . . . . . . . . . . . . . . . . . . . 212

6.25 The rate, q; for a horizontal well with transversal fractures, andthe cumulative rate, Q; vs. time for an in�nite conductivity frac-ture, and �nite conductivity fractures with FC values of 1000 and50 mDft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.26 The rate, q; for a horizontal well with transversal fractures, andindividual fracture rates qfri , where i = 1; :::; 5 versus the cu-mulative rate, Q, for an in�nite conductivity fracture and �niteconductivity fractures with FC values of 1000 and 50 mDft. . . . . 213

6.27 The rate, q, versus the cumulative production, Q, for a wellborewith 5 transversal fractures. Each fracture conductivity, FC ; is 50(mDft). The wellbore friction reduces both the well rate produc-tion and the cumulative production. . . . . . . . . . . . . . . . . 214

6.28 Responses of rate, q, versus time, t. The wellbore inner boundaryconditions, IBC, correspond to a variable pressure for the in�niteand �nite conductivity fractures of 50 mDft. . . . . . . . . . . . . 215

6.29 The rate, q, versus cumulative production ,Q, responses. The well-bore inner boundary conditions, IBC are variable pressure for thein�nite conductivity and �nite conductivity, 50 (mDft), fractures.Each individual fracture rate vs.cumulative rate is graphically pre-sented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.30 In�uence of fracture choking e¤ect (for variable rate IBC) on pres-sure di¤erence and well with fractures PI. Fractures are �nite con-ductivity (2500 mDft). . . . . . . . . . . . . . . . . . . . . . . . . 217

6.31 In�uence of fracture choking e¤ect (for variable rate IBC) on cu-mulative indifvidual fracture production Qfri (i=1,5 and 3) andwell with fractures PI. Fractures are �nite conductivity (2500mDft). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

LIST OF FIGURES xv

6.32 Wellbore contant rate to constant pressure IBCs. The value ofthe employed �owing pressure following the constant rate periodcorresponds to the pressure determined by the model at the endof this period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.33 Longitudinal versus transversal fracture rates and the cumulativeproduction (for a horizontal well with �ve equally spaced in�niteconductive, and equal half-length fractures). . . . . . . . . . . . . 219

6.34 Longitudinal vs.transversal fracture rates and the cumulative pro-duction (for selective fractures: 1, 5 and 3). . . . . . . . . . . . . 219

6.35 The productivity index, PI, versus time, t, for a horizontal wellwith longutudinal fractures of n = 7, 5, and 3. . . . . . . . . . . 220

6.36 The rate, q, versus time, t, for the BOX model (5000 ft by 5000ft). IBCs of variable pressure of 4900, 3900, and 2900 psi. In�niteconductive fractures with varying fracture-half lengths, Lf , of 120,85 and 50 ft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.37 The pressure di¤erence, P, versus time, t, for the BOX model(5000 ft by 5000 ft). IBCs of variable rate of 150, 110, and 70bbl/d. In�nite conductive fractures with varying fracture half-lengths, Lf , of 120, 85 and 50 ft. . . . . . . . . . . . . . . . . . . 221

6.38 The rate, q, versus time, t, for b=0.33 plotted in circles, and forother values of the decline exponents (b=0.5, 1, and 2) all plottedas solid line. The rate versus time is calculated for a speci�c initialdecline qi = 5000 and a initial decline, Di = 0:01 ). . . . . . . . . 223

6.39 The constant A as a function of the constant , �; of the no-�owmoving boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . 224

6.40 The dimensionless pressure, pD, versus the dimensionless time, tD,and the dimensionless distance, rD (for � = 1 and b = 1

3). . . . . . 225

6.41 The dimensionless pressure, pD; versus the dimensionless time, tD;and with a distance rD = 1 (for � = 1, and the decline exponentb = 1

3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

6.42 The dimensionless pressure, pD; versus the dimensionless time, tD;and with a distance rD = 4 (for � = 1, and the decline exponentb = 1

3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

6.43 The dimensionless pressure, pD, versus the dimensionless time, tD,and with a distance rD = 10 (for � = 1, and the decline exponent,b = 1

3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.44 The dimensionless pressure, pD, versus a distance, rD, at a di-mensionless time tD = 16 (for � = 1, and the decline exponentb = 1

3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.45 The dimensionless pressure, pD, versus the dimensionless time, tD,and versus the dimensionless distance, rD (for � = 3). . . . . . . . 228

xvi LIST OF FIGURES



6.48 The dimensionless pressure, pD; versus the dimensionless time, tD;and versus the dimensionless distance, rD (for � = 9). . . . . . . . 229

6.49 The rate, q, versus time, t, for b = 0:5 (plotted in circles). Theother parameters were considered constant, i.e., the initial declinedecline Di = 0:01, and the initial decline rate qi = 5000). . . . . . 230

6.50 The constant A(�) as a function of the constant , �; of no-�owmoving boundary: . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

6.51 The dimensionless pressure, pD; versus the dimensionless time,�D; calculated at a dimensionless radius rD = 1 (� = 1, and adecline exponent b = 0:5). . . . . . . . . . . . . . . . . . . . . . . 232

6.52 The pressure di¤erence, pD versus the dimensionless time, tD;calculated at a dimensionless distance, rD = 4 (coe¢ cient: � = 1,and decline exponent: b = 0.5). . . . . . . . . . . . . . . . . . . . 233

6.53 The pressure di¤erence, pD; versus the dimensionless time, �D,calculated at the dimensionless distance rD = 10 (for � = 1, anddecline exponent: b = 0:5). . . . . . . . . . . . . . . . . . . . . . . 233

6.54 The dimensionless pressure, pD; versus the distance, rD; at a di-mensionless time �D = 16 (for � = 3, and decline exponen: b = 0:5).234

6.55 The dimensionless pressure, pD, versus the dimensionless time,�D; and versus the dimensionless distance, rD (for � = 1, b = 0:5). 234

6.56 The dimensionless pressure, pD, versus the dimensionless time,�D, and versus the dimensionless distance, rD (for � = 3, b = 0:5). 235


6.58 The dimensionless pressure, pD, versus the dimensionless time, tD;and versus the dimensionless distance, rD (for � = 7, b = 0:5). . . 236


6.60 The rate, q, versus time, t, for b = 1:0 (plotted in circles). Witha speci�c initial decline qi = 5000 and an initial decline rate Di =0:01). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6.61 The constant A as a function of the coe¢ cient �. A(�) is cal-culated for an inner boundary condition of variable rate. Arpsdecline exponent b = 1. . . . . . . . . . . . . . . . . . . . . . . . . 238

6.62 The dimensionless pressure, pD, versus the dimensionless time, tD,calculated at a dimensionless radius rD = 1 (for: � = 1 and b = 1). 240

LIST OF FIGURES xvii

6.63 The pressure di¤erence, pD, versus the dimensionless time, tD,calculated at a dimensionless distance rD = 4 (for: � = 1, andb = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.64 The pressure di¤erence, pD, versus the dimensionless time, tD,calculated at a dimensionless distance rD = 10 (for: � = 1, andb = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

6.65 The dimensionless pressure, pD, versus the dimensionless distance,rD, at a dimensionless time tD = 16 (for: � = 1, and b = 1). . . . 241

6.66 The dimensionless pressure, pD, versus the dimensionless time, tD,and versus the dimensionless distance, rD (for � = 1, b = 1). . . . 242


6.68 The dimensionless pressure, pD versus the dimensionless time, tDand versus the dimensionless distance, rD (for � = 5, b = 1). . . . 243



6.71 The rate, q, versus time, t, for decline exponent b = 2:0 (plottedin circles). The initial decline qi = 5000, and the initial declinerate, Di = 0:01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

6.72 The constant, A (�), as a function of the coe¢ cient, �, of theno-�ow moving boundary, : . . . . . . . . . . . . . . . . . . . . . . 246

6.73 The dimensionless pressure, pD, versus the dimensionless time,tD,calculated at the dimensionless radius, rD = 1 (for, � = 1 andfor, b = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6.74 The pressure di¤erence, pD; versus the dimensionless time, tD,calculatedat a dimensionless distance, rD = 4 (for, � = 1, and for, b = 2). . 247

6.75 The pressure di¤erence, pD; versus the dimensionless time, tD,calculated at a dimensionless distance, rD = 10 (for � = 1, anddecline exponent, b = 2). . . . . . . . . . . . . . . . . . . . . . . . 248

6.76 The dimensionless pressure, pD, versus the distance, rD, at thedimensionless time, tD = 16 (for � = 1, and and decline exponent,b = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.77 The dimensionless pressure, pD, versus the dimensionless time, tD,and the dimensionless distance, rD (for � = 1, b = 2). . . . . . . . 249


6.79 The dimensionless pressure, pD, versus dimensionless time, tD,and versus the dimensionless distance, rD (for � = 5, b = 2). . . . 250

6.80 The dimensionless pressure, pD, versus the dimensionless time, tD,and versus dimensionless distance, rD (for � = 7, b = 2). . . . . . 251

xviii LIST OF FIGURES

6.81 The dimensionless pressure, pD, versus the dimensionless time, tD,and versus dimensionless distance, rD (for � = 9, b = 2). . . . . . 251

7.1 The pressure-di¤erence versus time for three models (multi-fracturedhorizontal well (MFW-open outer boundary-SLAB), vertical-fracturedwell (VFW-open outer boundary-SLAB) and partially perforatedhorizontal well (PPHOW-closed, BOX ) [After Cvetkovic et al.(2000)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

7.2 The semi-analytical model cumulative production compared tothe other models (2 numerical and a semi-analytical models). . . 258

7.3 Comparisons of model-calculated cumulative productions (two nu-merical models and a semi-analytical single-phase model). Theproduction history comprises 1200 days. . . . . . . . . . . . . . . 259

7.4 A comparison of observed (oil rate) data with that calculated bythe model at an IBC of variable pressure (from 7 selected timeintervals). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.5 A comparison of observed (oil rate) data with that calculatedby the model at IBCs of variable pressure. The daily measuredpressures at the wellbore are devided into 3 pressure intervals. . . 261

7.6 The matching of observed and calculated pressure di¤erences ver-sus time (for variable-rate IBCs and changes in fracture �niteconductivities FC). . . . . . . . . . . . . . . . . . . . . . . . . . . 262

7.7 The matching of observed and calculated pressure-di¤erences ver-sus time (for variable-rate IBCs and changes in the fracture half-length, Lf , from 50 ft and 25 ft, and assuming a maintained frac-ture conductivity, FC , of 20 mDft). . . . . . . . . . . . . . . . . . 263

7.8 The matching of observed and calculated pressure-di¤erences ver-sus time (for variable-rate IBCs and changes in the fracture half-length, Lf , from 50 ft and 25 ft, and assuming a maintained frac-ture conductivity, FC , of 20 mDft). The initial pressure, Pi, isreduced from 4400 psi to 3850 psi. . . . . . . . . . . . . . . . . . . 264

7.9 Observed and calculated rates as functions of time. The IBC ofthe model are of variable pressure, and the fracture conductivities,FC , change from 70, 40 down to 20 mDft. . . . . . . . . . . . . . . 265

7.10 Matching of well observed cumulative oil data with a model cal-culated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

7.11 Observed and model pressure di¤erences vs. time, in addition tocalculated and observed rates vs. time. The IBC of the model areof variable rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

7.12 The fractured horizontal well, SA-P1 penetrating 14 transversal-fractures in an oil reservoir (cross section). . . . . . . . . . . . . . 267

7.13 The measured wellbore pressure data, pwf (psi), versus time, t (d). 268

LIST OF FIGURES xix

7.14 The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe(bbl/d), and the gas rate, qg (Scf3/d), versus time, t (d), for ahorizontal well SA-P1 with 14 transversal-fractures. . . . . . . . . 268

7.15 The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe(bbl/d), and the gas rate, qg (Scf3/d), versus time, t (d), for ahorizontal well SA-P1 with 14 transversal-fractures on a log-linscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

7.16 The well and fracture input data. . . . . . . . . . . . . . . . . . . 2697.17 The reservoir input data. . . . . . . . . . . . . . . . . . . . . . . . 2707.18 A comparison of the calculated and measured well data (model

data obtained with an IBC of constant pressure). . . . . . . . . . 2707.19 The well cumulative production, Q (bbl), and the fracture pro-

duction, Qfri(i = 1; :::14) (bbl), versus time (d). The IBC of themodel is of constant pressure. . . . . . . . 271

7.20 The model well rate, q (bbl/d), and the fracture rates, qfri(i =1; :::14) (bbl/d), versus time (d). The IBC of the model is ofconstant pressure. . . . . . . . . . . . . . . 272

7.21 The pressure di¤erence, Pi�Pwf (psi), versus time (d) for an IBCof variable rate. (Well production rates for the �rst 800 days areconsidered as 1 rate-interval). . . . . . . . . . . . . . . . . . . . . 272

7.22 The pressure di¤erence, Pi�Pwf (psi), versus time (d) for an IBCof variable rate (The well production rates for the �rst 800 daysof are devided into 3 rate intervals). . . . . . . . . . . . . . . . . 273

7.23 The step function match obtained with an IBC of constant-rate toconstant-pressure processed in a single run. Both pressures andrates are matched within the single run. . . . . . . . . . . . . . . 273

7.24 A fractured horizontal well, SA� P2, penetrating 14 transversalfractures in an oil reservoir. Cross section view. . . . . . . . . . . 274

7.25 The measured wellbore pressure, pwf (psi), versus time, t (d). . . 2757.26 The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe

(bbl/d), and the gas rate, qg (Scf3/d), versus time, t (d), for ahorizontal well SA-P2 with 14 transversal fractures. . . . . . . . . 275

7.27 The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe(bbl/d), and the gas rate, qg (Scf3/d), versus time, t (d), for ahorizontal well SA-P2.with 14 transversal fractures on a log-linscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

7.28 The well and fracture input data. . . . . . . . . . . . . . . . . . . 2767.29 The reservoir input data. . . . . . . . . . . . . . . . . . . . . . . 2777.30 A comparison of the calculated and measured well data (model

data obtained with an IBC of constant pressure). . . . . . . . . . 278

xx LIST OF FIGURES

7.31 The measured versus calculated data for the cumulative rate, Q(bbl), versus time (d). The calculated data are de�ned with thehalf-length, Lf , (of 34 and 50 ft) and the fracture penetrationheight, hf , (of 40 and 50 ft). . . . . . . . . . . . . . . . . . . . . 278

7.32 The measured versus calculated data for the cumulative rate, Q(bbl) versus time (d). The calculated data are de�ned with thehalf-length, Lf (of 20, 34 and 50 ft) and the fracture perforation,hf(of 40 and 50 ft). . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.33 The well cumulative production, Q (bbl), and the individual frac-ture cumulative production, Qfri (i = 1; :::; 14), versus time, t(d). (Each fracture half-length Lf=34 ft and the fracture partialpenetration height, hf=40 ft). . . . . . . . . . . . . . . . . . . . . 280

7.34 The well rate production, q (bbl/d), and the individual fracturerate production, gfri (i = 1; :::; 14), versus time, t(d) (Each frac-ture half-length Lf = 34 ft, and the fracture partial penetrationheight hf = 40 ft). . . . . . . . . . . . . . . . . . . . . . . . . . . 280

7.35 Fracture rate for individual fractures (1, 6, 7, and 14) for varyingfracture half-lengths, Lf (34, 50 ft) and partial penetration heights(40, 50 ft). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.36 Valhall �eld with several multi-fractured horizontal wells [AfterNorris et al. (2001)]. . . . . . . . . . . . . . . . . . . . . . . . . . 282

7.37 Rate and PI data versus measured values of the cumulative wellproduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.38 The wellbore pressure and GOR data versus time. . . . . . . . . 2847.39 Productivity Index versus Time (IBC = Constant Rate). [Model

and Well Data: PI - MATCH]. . . . . . . . . . . . . . . . . . . . . 2867.40 Model and well data PI - match (for an IBC of costant rate - the

fracture permeability and fracture width are constant). . . . . . . 2867.41 Rate versus time (for an IBC of variable pressure). . . . . . . . . 2877.42 Calculated wellbore pressure matches model observed data for

variable rate IBCs. . . . . . . . . . . . . . . . . . . . . . . . . . . 2877.43 The step-function procedure calculates dimensionless pressure for

the IBC of constant-rate and rates for the IBC of constant-pressure.Within the same run the IBCs are changing from constant-rate toconstant-pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . 288

7.44 The match of the models (SLAB & BOX) with the well rates. . . 2887.45 The water injection rate and the cumulative injection rate versus

time for a horizontal well with 16 transversal fractures.Well: SA-WI1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.46 The wellbore pressure versus time for the SA-WI1 well. . . . . . . 2907.47 The well and fracture input data. . . . . . . . . . . . . . . . . . . 2917.48 The reservoir input data. . . . . . . . . . . . . . . . . . . . . . . 291

LIST OF FIGURES xxi

7.49 The model water injection rate, qi (bbl/d), and the water injectioncumulative production, Q (bbl), versus time, t (d), for the SA-WI1water injection well. . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.50 The fracture water injection rate, q (bbl/d), for a horizontal wellwith 16 fractures, and the individual fracture injection rates, gfri(i = 1; :::; 16). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.51 The cumulative fracture water injection, Q (bbl), for a horizontalwell with 16 fractures and the individual fracture water injection,Qfri (i = 1; :::; 16). . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7.52 The productivity index, PI (bbl/d psi), versus tme, t (d), for thewater injection horizontal well penetrating 16 fractures. . . . . . . 293

7.53 A horizontal well with 14 transversal-fractures producing from asynthetic gas reservoir. The cumulative oil production is con-verted into its cumulative gas equaivalent. The IBCs are eitherconstant or of variable pesudo-pressures. . . . . . . . . . . . . . . 295

7.54 The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponernt n = 1 and a dimensionless radiusrD = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7.55 The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and a dimensionless radius, rD= 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7.56 The dimensionless pressure, pD, versus the dimensionless time,� , for an inverse decline exponernt n = 1 and a dimensionlessradius, rD = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.57 The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and dimensionless radius, rD =50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.58 The dimensionless pressure, pD, versus dimensionless time, � ; foran inverse decline exponernt n = 1 and dimensionless radius, rD =100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

7.59 The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and dimensionless radius, rD =200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

7.60 The dimensionless pressure,pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and a dimensionless radius,rD = 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

7.61 The dimensionless pressure, pD, versus the dimensionless time,� , for an inverse decline exponent, n = 10 and a dimensionlessradius, rD = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

7.62 The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radiusrD = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

xxii LIST OF FIGURES



7.65 The dimensionless pressure, pD,versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radiusrD = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

7.66 The dimensionless pressure, pD; versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radiusrD = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

7.67 The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radiusrD = 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

7.68 The dimensionless pressure, pD, versus the dimensionless time,� , for an inverse decline exponent n = 100 and a dimensionlessradius rD = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

7.69 The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponent n = 100 and a dimensionless radiusrD = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

7.70 The dimensionless pressure, pD; versus the dimensionless time,� , for an inverse decline exponent n = 100 and a dimensionlessradius rD = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

7.71 The dimensionless pressure, pD; versus the dimensionless time,� , for an inverse decline exponent n = 100 and a dimensionlessradius rD = 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

7.72 The dimensionless pressure, pD, versus the dimensionless time,� , for an inverse decline exponent n = 100 and a dimensionlessradius rD = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.73 The dimensionless pressure, pD, versus the dimensionless time,� , for an inverse decline exponernt n = 100 and a dimensionlessradius rD = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

7.74 The dimensionless pressure, pD, versus the dimensionless time,� , for an inverse decline exponent n = 100 and a dimensionlessradius rD=500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

7.75 The dimensionless pressure, PD, versus the dimensionless time, �for an inverse decline exponent n = 1000 and a dimensionlessradius rD=10. The singularity case. . . . . . . . . . . . . . . . . . 310

Acknowledgements

I would like to take this opportunity to express my sincere gratitude to myadvisor Professor Jon Steinar Gudmundsson for his encouragement, guidance,and patience throughout this work.I owe a special thanks to Dr. Gotskalk Halvorsen for his mathematical sup-

port and enthusiasm in designing the model and for all our useful discussions.Thanks to Jan Sagen for his contribution in programming the model options.I was privileged to work and study at NTNU, the Department of Petroleum

Engineering and Applied Geophysics, where I got the opportunity to learn fun-damental and advanced issues of various disciplines in petroleum engineering,thus making years spent there enjoyable and unforgettable.I also extend my sincere thank to Professor Jon Kleppe and to other profes-

sors for the provided support throughout my studies. Appreciation is extendedto Mrs. Marit Valle Raaness for all administrative assistance continuous encour-agements.Financial aid during the course of study at NTNU was provided by the

Phillips Petroleum Company Norway, and is gratefully acknowledged. I wouldalso like to thank to IFE (Institute for Energy Technology at Kjeller) for sup-porting the development of the model and enabling me to �nalise the thesis. Inparticular, I would like to thank Arne Westeng and Jan Egil Arneberg in Bay-erngas Norge AS, for the provided support. I wish to express my gratitude toall individuals who aided in the completion of the thesis.

xxiii

xxiv 0. Acknowledgements

Preface

The work presented in this thesis has developed from the Dr.ing. study periodat NTNU (NTH) between 1991-1994 at the �Institute for Petroleum Engineeringand Applied Geophysics�where I was given the task to study the decline curveanalysis in order to derive how the drive mechanism is related to the declineexponent, b. Without monitoring the full production history data, but merelyinvestigating the connection of the curvature of decline curves, de�ned by thedecline exponent, b, it was di¢ cult to establish a relation between the decline ex-ponent, b, and the drive mechanism. The study ended with 3 reports, all relatedto decline curve analysis. Due to rate time relations being empirical and derivedfor a vertical well operating under conditions of constant pressure during the welldepletion time, it was extremely challenging to derive a single relation. This wasdue to the fact that there were numerous possible solutions (as a result of theproblem being inverse). Until now, decline curve analysis has been consideredas a convenient empirical procedure for analysing well performance. However,only limited signi�cance has been put in relation to the values of exponent b.Fetkovich (1980) related the empirical solutions of Arps (1945) to single-phase�ow solutions, thereby providing the theoretical framework for Arps solutions.Fetkovich also related exponent b to the exponent of the deliverability curve, n.The drive mechanism and its relation to the rate decline have yet to be the-

oretically determined. The work of providing explanations and unique solutionshas been challenging, particularly for a well operating in the North Sea. Suchwells generally operate under conditions of restricted well pressure, changingfrom transient to depletion mode. This is caused by the production plateau, andthe pipeline transportation constraints. In addition, the well is mostly deviatedor even horizontal with fractures.My interest in well-rate decline continued while working at IFE Kjeller, par-

ticularly when Wiggo Holm from Phillips Petroleum Company, Norway, askedif it was possible to provide a well response for a fractured-horizontal well pen-etrating up to 50 fractures. The theoretical model approach for coupling a frac-tured well to a reservoir was presented under a poster session of the conference�Mathematical Modelling of Flow through Porous Media�which was held in St-Etienne, France in (1995). The tentative model was presented to Martin Rylancefrom BP, who decided to �nance the project. The model development was fur-

xxv

xxvi 0. Acknowledgements

ther �nanced by industry (BP, CONOCO, and PHILLIPS), NFR, and IFE, andended in 2001. The �nal model was successfully presented at the Seminar withworkshop organised by BP-AMOCO, Norway, in 2001. Obtained project resultswere summarised in IFE internal publications, two published SPE papers (2000and 2001) and two posters (presented at the Schlumberger GeoQest FORUMin London, England, in 2001, and at the International Petroleum EngineeringConference in Zadar, Croatia, in 2001).The fractured-horizontal well data helped me to evaluate and validate the

overall model design and its solutions. Model features were further evaluatedwith fractured-horizontal well data from North Africa provided by ENI (2007).North Sea �eld test data were obtained from Vallhal, Eko�sk and Syd-Arne �elds(last obtained in 2008).I conducted the unconventional stimulation study and investigated the rate

decline (while working at Schlumberger). The physics of such decline is unknown,and was not considered by Fetkovich (1980). Unconventional stimulations causean increased production rate and there is no physical explanation why a stimu-lated well increases the oil production of nearby wells. The overall �eld projectwas proposed by Schlumberger and ENI, with the task to physically explainthe unconventional drive mechanism. The project idea was presented in KualaLumpur, Malaysia, in 2007, and further inspired my interest in developing themodel for examining the nature of pressure in the vicinity of a producing wellwith an Arps rate decline. By solving a di¤usion equation with speci�ed bound-ary conditions, it was possible to obtain analytical solutions presenting values ofpressure within the drainage area of a producing vertical well.The present thesis comprises analyses of transient and depletion well rate

decline. I have carried out their design and contributed in their developmentwith implementations of the fractured-horizontal well transient-rate-pressure so-lutions. I also created the model for a vertical well with a variable rate declineof Arps type. The following statements summarise the novelty and contributionin the thesis:1. The design of a fractured horizontal well model, involving the integra-

tion of numerical and program routines into a screening tool for quick transienttest analyses of a fractured horizontal well. The study on transient decline pro-vides semi-analytical derived solutions related to the fractured horizontal wellproduction as:a. Screening options that include the wellbore condition or inner boundary

conditions of constant and variable rate or pressure, with the speci�c constant-rate-to-constant-pressure feature.b. Late time approximations including equivalent wellbore radii, and equiv-

alent fracture half-lengths as measures of the e¢ ciency of a horizontal well withfracture production.c. Individual fracture production quantities with productivity indices.

0. Acknowledgements xxvii

d. A validation of mathematical and software model features, with fracturedwell case studies from the North Sea.e. The proposition of novel solutions for more sophisticated reservoirs with

extended heterogeneity options, new fracture features (e.g., fracture skin) andwell features (e.g., well skin) to be adapted in the future. Since the existingmodel is robust and stable, it can be extended to the multilateral options.2. The investigation of the nature of a vertical well with a rate decline

production of Arps type. The creation of a physical model and the provisionof analytical pressure responses to a variable rate wellbore condition of Arpstype. In order to solve di¤usion equations with variable rate wellbore conditions(approximating Arps rate decline for large times), the approach introduces a no-�ow speci�ed outer moving boundary. The inclusion of the velocity of the no-�owmoving boundary proportional to the square root of time renders it possible toanalytically derive wellbore pressure responses. Further modelling includes:a. That of pressure responses for declining rates, each de�ned with the se-

lected decline exponent, b, in turn represented by the drive mechanism followingthe concept introduced by Fetkovich (1980).b. The contribution of the overall work to a transient and depletion rate

decline that is relevant to wells producing oil and gas.c. A study on the depletion decline; investigating the nature of pressure

responses for variable-rate wellbore conditions of Arps decline. These pressureresponses are solutions to the di¤usion equation with inner boundary conditionsof variable-rate (i.e., a known decline exponent, b) and no-�ow speci�ed outerboundary conditions (moving outward from a vertical well axis).d. Solutions that can be extended to a horizontal well, and further calibrated

with measured pressure and rate data from a wellbore.I have contributed to the mathematical, numerical, and programming work,

and a particular contribution consists in the novel late-time approximations andstep-function features. I worked together with Gotskalk Halvorsen regarding themathematical and numerical modelling of the designed model features. Also,work was equally shared with Jan Sagen, Frederik Martin and Aage Stangelandconcerning the programming of the designed model options.

xxviii 0. Acknowledgements

Chapter 1

INTRODUCTION

1.1 Background

Well testing and rate testing have been the subjects of frequent in depth studiesover the last few decades (from the 1950�s). Unlike reservoir simulation thatdeals with multidimensional and multiphase �uid �ow in porous media with so-lutions obtained by �nite di¤erence methods, both well testing and rate testingtransient responses are mostly solutions based on di¤usion equation. These so-lutions are predominantly semi-analytical thus yielding results quickly, which iswhy single well studies are helpful in de�ning basic reservoir parameters. Due tothe methodology being thoroughly tested, well responses are able signi�cantlyto improve reservoir descriptions. Well testing or pressure-time analysis hasbeen used less frequently in the North Sea �elds during the last decades due toincreased operational costs during testing (as a consequence of the augmenteddaily drilling-rig costs). Pressure testing and rate testing analyses are basedon an identical modelling theory and respective solutions. Pressure testing re-quires experimental data (various test data), further data modelling and theinterpretation of the measured data. Rate testing data, on the other hand, arecontinuously monitored and further modelled and tested.The method for analysing "rate response with time" while keeping the well

in production has improved continuously since the 1980�s and made a huge leapforward during the last twenty years. Although rate-time analysis predates thepressure-time analysis of well testing the break-through came �rst in the 1970�swhen Fetkovich combined empirical and analytical equations and as a resultprovided type curves to be matched with real observation of well-productiondata. This historical milestone introduced type curves and interpretation tech-niques thus questioning the physical understanding of the in�ow performance of awell. A drive mechanism causing the rate curvature was introduced by Fetkovich(1980) who also made the �rst attempt to de�ne the initial rate decline, qi, thedecline exponent, b, and the initial decline, Di, by physical means.

1

2 1. INTRODUCTION

Well testing without closing of a well is now possible due to new measurementtechniques and novel modelling solutions. Made possible by new near wellboremeasuring instrumentation developed during the last decade, current research inwell testing is based on modelling the de-convolving variable pressure with timeto a rate condition.In parallel, rate-time analysis continues to be developed throght the combina-

tion of new modelling techniques and new type curves, in addition to improvingthe interpretation of rate-time responses. Anderson et al. (2006) de�ned pressuretransient data as "high-frequency high resolution data" and rate testing data as"low-frequency low-resolution data". Compared to well-testing data analysis,rate-testing analysis is characterised by poor quality data with reduced quan-tity. Methodology and data interpretation of rate transient data depends on thefrequency and accuracy of the recorded information. Also, pressure transientdata are acquired as part of a controlled �experiment�, performed as a speci�cevent (pressure build-up, BU or pressure draw-down, DD). The production datarepresent long term monitoring data, usually followed by considerable varianceoccurring during its acquisition. Poston and Poe (2008) have provided recentadvances in decline-curve analysis that have aided in improving well productionperformance analysis in addition explaining production-forecasting techniquescurrently in use in the industry.

1.2 Scope of the Work

Decline curves and particularly decline curvature de�ning a decline exponent, b,have been discussed in literature; by Fetkovich (1973) and Raghavan (1993). Thequestion is how to relate drive mechanism (generally multiphase �ow) analyticalsolutions to the empirical models de�ned by Arps (1945) and later improvedby Fetkovich (1973). One possible approach involves combining multiphase �owsolutions with the empirical model solutions as done and discussed by Raghavan(1993). He evaluated the solution of a di¤usion equation solved for multiphase�ow and postulated conditions for which decline curvature empirical values couldbe matched. This Thesis describes an attempt of modelling the empiricallyderived depletion rate-time decline stems. A vertical well was produced from acircular reservoir with various decline stems. Rate-time stem curvature is createdby the drive mechanism and the reservoir heterogeneity. Developed di¤usionmodels use variable wellbore rates and no-�ow outer boundaries moving outwardsfrom a wellbore axis as inner and outer boundary conditions. The speed of anoutwards no-�owmoving boundary is prede�ned. The model solutions are uniquein the sense that, for a prede�ned rate and other model input parameters, it ispossible to model the system pressure responses at various points. The presentThesis provides a novel pressure solution to the empirical decline curvatures.Moreover it relates rate-time stems to the pressure responses within various

1. INTRODUCTION 3

points in the circular system.The aim of this work was to develop a novel approach in creating model

transient rate-time responses of a well with fractures to be matched to realobserved data, and thus characterising well performance and improving wellintervention in time. An analytical model was developed in order to provide arapid assessment of the productivity of various fracture con�gurations along ahorizontal well. New modelling techniques for a well with fractures were appliedand model solutions were veri�ed in several case studies. New e¤ective valueswere tabulated and listed for a well with fracture models. This work has beenconcentrated on a fractured horizontal well, its modelling and interpretation.Within the same tool, various IBC of pressure and rate are combined. The step-function features enable a change in wellbore conditions from a constant pressureto a constant rate within the same run. The transient rate decline features arecovered by a model introducing new late time solutions for a well with fracturescoupled to a reservoir.For selected rate-time stems each de�ned with the decline exponent, b;by

solving di¤usion equation it is possible to obtain pressure solution in time invarious points within a drainage area of a vertical well. So, at the same pointwithin drainage area it is possible to generate various pressure-time pro�les forselected Arp�s stems de�ned with the decline exponent, b. The wellbore variable-rate conditions (of Arp�s type) are combined with an outer-boundary that movesoutwards from a well with a certain speed. As drive mechanism and exponent,b, known relations are empirically derived this modelling approach provide basisfor analysis of decline exponent, b, analytically. This can be achieved once bothrate and pressure data are continuously monitored. The speed of the no-�owboundary should be related to decline stems by using this analytical approachas its basis. This forms the tentative proposal for future studies. The extensiveresearch on multiphase �ow modelling continues as the physical understandingof the curvature decline de�ned with the decline exponent, b, is still gainingattention in petroleum literature.The main objective of chapters 2 and 3 is to review well rate-time pro�les

based on transient-analytical and depletion-empirical models. These discussedmodels include the selection of the following parameters:- Model geology (homogeneous or heterogeneous such as varying permeability,

layering, composite, naturally fractured),- Types of �uid in a reservoir (incompressible, slightly-compressible and com-

pressible �uids),- Flow regime within a reservoir (steady state, unsteady state, pseudo-steady-

state),- Reservoir geometry (radial, linear, spherical and hemispherical),- Number of �owing �uids (single or two-phase),- Various well positioning (vertical or horizontal wells),

4 1. INTRODUCTION

- Well-fracture coupling to a reservoir (vertical fractured well, horizontal frac-tured well),- Well operating conditions (constant or variable pressure),- Special features, e.g., a well with laterals, and- Multiple well realizations.This thesis mainly consider solutions of the three fundamental combined

equations, i.e., the Darcy equation, the continuity equation and the equation ofstate given as:

�!u = �k�(rP � Z): (1.1)

r (��!u ) + A+@ (��)

@t= 0 (1.2)

� = �0e[c(p�p0] (1.3)

All three equations, solved with initial and boundary conditions describe �uid�ow through porous media. The initial condition is most often one of pressureboundary conditions are given in terms of pressure, P; and velocity �eld term,�!u .Further simpli�cation predominately related to the �uid and the rock proper-

ties include: a constant porosity, �, a constant compressibility, c, a and constantpermeability, k; that leads to solutions for pressure or rate distribution with time.For the slightly compressible �uid, van Everdinger and Hurst (1949) derived adi¤usion equation that is similar to that concerning the conduction of heat �owas presented by Carslaw and Jaeger (1947). Physical di¤usion signi�es that therate of pressure change at a given point is a function of the number of parametersdescribing porous media and the curvature of the pressure around the selectedpoint. In a case of linear �ow, the rate of pressure change, @P

@t; is proportional to

the permeability, k; as well as to a local curvature of the pressure pro�le, @2P@x2.

It is also inversely proportional to the �uid viscosity, to the reservoir porosity,and to the total compressibility.More complex semi-analytical or approximative solutions of di¤usion equa-

tion are derived by introducing:- Anisotropy and permeability, k variations with radius, r;- Porosity as a function of time,- Density and viscosity as functions of pressure,- Gas �ow,- Multiphase �ow,- Inclusion of fractures coupled to a well within a reservoir, and- Multiple wells.

1. INTRODUCTION 5

This thesis comprises a review of various formulations and presents a numberof analytical and semi-analytical solutions for a heterogeneous reservoir produc-ing at a constant pressure from a single fully-perforated well.One should be aware of potential limitations in using the semi-analytical ap-

proach. Particularly when simulating nature and describing a multidimensional�ow with a simpli�ed single phase �ow, ideal preconditions for the modelling set-up should be presumed. As a result the matching procedure, although fast, isnot accurate. Nevertheless these decline analyses, or DCA testing techniques are,due to their limited time of interpretation, still considered useful methodologiesthat are powerful for screening single-well response analyses.A single well that penetrates a reservoir with de�ned parameters and that

is �lled in with a liquid is considered to be at equilibrium. We would like todetermine the rate response of the well in time by imposing a perturbation, e.g.,starting the production of �uid through a well under speci�c conditions. Thesolution of such a physical problem determining the rate-time well response isunique and this is mathematically referred to a direct or forward problem.It is also possible to determine reservoir parameters if one knows rate-time

response to a given well side perturbation. This is perceived as an inverse prob-lem that is usually not unique. There are several realisations that can provideidentical responses to a given perturbation. Both pressure testing and rate test-ing disciplines have several common and complementary features that have beenhistorically considered as presented in the Table (1.3). Here, we refer to twocomprehensive review papers by Gringarten (2006), and Anderson et al. (2006).

1.3 Organisation

Chapter 2 reviews the transient rate decline caused by �uid expansion with acontinuously increasing drainage area, it considering a model with a well pro-ducing under constant BHP. A well positioned in a circular drainage area, forwhich there is no-�ow at the drainage boundaries, is the basic model for gener-ating transient rate-time pro�les. Various models provide the latest rate testing"state of the art technology" review of theory with selected solutions, all givenin transient mode.Chapter 3 provides a review of the depletion decline. This decline begins after

the drainage radius reaches the outer boundaries that de�ne the drainage area.We introduce decline curve analyses of the production data from a depletionperiod only. An extensive type curve summary comprises the latest theoreticalsolutions to �ow equations, all based on Fetkovich�s (1980) type curve approach.We mainly consider a special case of the transient solution i.e., the depletionsolution. This chapter also provides a "state of the art technology" review ofproduction data analysis. It gives a comprehensive review of reservoir mod-elling tools that are helpful in diagnosing a reservoir model and characterising a

6 1. INTRODUCTION

reservoir.

The �rst two chapters consider a number of analytical and semi-analyticalanswers to the forward and inverse solutions of a homogeneous or heterogeneousreservoir producing at constant pressure from a single well. Both areally andradially heterogeneous reservoirs have been taken into account. For each con-sidered case, the adequate system of units is stated. The available literature onrate testing is far too extensive to be summarised within the scope of these chap-ters. We thus refer only to selected available publications and present modellingmethodologies of rate testing analyses for certain complex well geometries.

Chapter 4 focuses on new solutions for a well with fracture full-time responsesand also present a late-time approximation of equivalent wellbore parameters(radius and fracture half-length). It summarises multi-fractured horizontal wellmodel features (both transient and depletion rate time solutions). As the modelis designed for wellbore inner-boundary-conditions of both, pressure and rate,it can be used in order to handle rate-to-pressure changes with regards to well-bore conditions. This unique feature is presented through step modelling. Allsolutions are solved in dimensionless form and converted into a rate-pressure-time for the prede�ned units. Individual fracture rate and cumulative rate arenovel model features. These fracture features are available in well-producing andwell-injection mode.

Chapter 5 comprises transient pressure-time solutions of a vertical well po-sitioned in an in�nite reservoir. New model features involve the moving no-�owouter boundary conditions with a selected speed. As the boundary moves out-wards, the drainage volume changes, giving rice to a transient model. At thesame time the rate decline is of Arp�s type. The inner-boundary conditions ofthe model are of variable-rate, and with such a wellbore condition it is possibleto generate pressure pro�les over time within a selected spatial distance froma wellbore axis. This model provides pressure pro�les for the wellbore variablerate conditions of Arp�s type. The model drainage volume changes with theoutward-moving boundary, and thus the model investigates the transient �owbehaviour for both oil and gas.

Chapter 6 describes the veri�cation of modelling features and the generationof various rate and pressure versus time curves (in a prede�ned unit system).Most model features presented graphically are evaluated with an input from acase study. A basic model with the no-�ow moving boundary generates pressure-time pro�les and provides type curves related to the inner boundary conditionof constant rate. The newly developed transient model is valid for the wellborevariable rate conditions. Even in a transient model, a rate-time steam declineis presumed to occur at the wellbore. The purpose is to derive the pressurepro�le that matches the decline stems. With our use of no-�ow and movingouter-boundaries, we further contribute the Raghavan (1993) observation to theinclusion of transient data while matching the decline in depletion. Several model

1. INTRODUCTION 7

Table 1-1: Pressure testing and rate testing comparisons of events, interpretationmethods and historical emphasis (Following the 2006 review by Gringarten andAnderson et al.]

8 1. INTRODUCTION

solutions are provided in a dimensionless forms of pressure versus time. Oncepseudo-steady-state time is reached it would be possible to derive solutions tothe case when a no-�ow boundary moves inwards. The transient-decline �owanalysis should also include the pressure normalization procedure and shouldrelate the speed of a moving boundary to the physics, and potentially to a drivemechanism that causes the curvature of the decline stems.Chapter 7 extends rate-time solutions for a horizontal well with transversal

and longitudinal fractures. New model solutions cover both the transient anddepletion rate-time solutions. At intermediate time, the model also includesthe pressure-time solutions. The new step function feature combines wellboreconditions of contant-pressure to those of constant-rate, thus allowing a quickand easy screening and matching of well data. Late time approximations arerecent solutions related to the e¤ective wellbore parameters (radius and fracturehalf-length). Several case studies have provided details to overall fractured-horizontal well model features. Investigations in an oil reservoir present thescreening potential of the available tool features. As the model is de�ned foroil and water �ow, it can also be used for water injection studies. Since it ismonophase it only provides individual fracture rates, and such individual waterinjection quantities should be further considered by reservoir simulation. Themodel is furthermore de�ned as a basic gas screening tool under the assump-tion that the pseudo-pressure, m(p), is used instead of the reservoir pressure, P .Overall, common features of the model is in creating individual fracture ratesand cumulative rates thus leading to a well with fracture productivity indices.These fracture injection-production features are complementary to the commer-cial numerical simulation. Further in Chapter 7, model solutions of a verticalwell with a no-�ow moving boundary are represented by the dimensionless pres-sure in surroundings of a wellbore that changes with time. Since the pressure iscalculated by the rate decline of an exponent b almost equal zero, it should beveri�ed with the measured pressure data at the site. The dimensionless pressureversus the dimensionless time is derived with no-�ow moving boundary solutionsin Appendix A.

Chapter 2

TRANSIENT RATE DECLINEREVIEW

This chapter review the transient rate-time performance of a well positioned inan oil and gas reservoir. Rate-time analysis or rate-testing investigates reservoirresponses measured as a rates in the producing well. During transient rate-time,a pressure wave created by the well has not reached the boundary of the reser-voir. A produced rate is a response to a speci�c pressure history at a wellbore.Generally it is possible to use rate-time transient analysis for the purpose of de-scribing reservoirs. We here distinguish rate-time analysis from its pressure-timecounter part. Pressure transient tests are early reservoir responses to wellboreconditions representing a constant �ow rate. Usually the constant �ow rate iseasier to control in a short transient test interval of a few hours or days. Pressuretransient tests (as drawdown, DD; buildup, BU; and interference test, IT) diag-nose near-wellbore conditions such as the reservoir conductivity de�ned by thepermeability-thickness product, kh, the wellbore storage and skin, s. There is nofundamental di¤erence between pressure-transient and rate-transient analyses.They describe the same process and are governed by identical reservoir charac-teristics. In both analyses, as discussed by Horne (1995), pressure transmissionis an inherently di¤usive process and is largely governed by average conditionsrather than by local heterogeneities. The di¤usion equation solutions can beinterpreted to estimate bulk reservoir properties due to them being insensitiveto most local scale heterogeneities.In general the equation of a well performance relates well rates and pressures

to the properties of reservoir formation and within a �uid. Such well performancerelations are solutions to the di¤usion equation for selected initial and boundaryconditions. This review comprises a number of selected published solutions forinner boundary conditions at a well with either constant or varying pressure.The outer boundary condition, denoted in�nite acting, de�nes unsteady-state�ow within the reservoir. In such unsteady-state �ow, �uid �ow is due entirelyto rock or �uid expansion. We refer to the type of reservoir heterogeneity that

9

10 2. TRANSIENT RATE DECLINE REVIEW

is penetrated by various well types.Several methods can be used for obtaining the solutions to the di¤usion equa-

tion and these methods may be grouped into either analytical, semi-analyticalor numerical methods (with �nite-di¤erence grids or �exible grids).The solution to the di¤usion equation was �rst derived for a heat conduction

problem (Kelvin line source solution), and applied by Theis (1935) to groundwater hydrology problems. The fundamental theoretical work of Carslaw andJaeger (1947) is used as a basis for engineering studies. A further developmentfor the petroleum industry followed by Muskat (1937), and van Everdingen andHurst (1949).The following three principals de�ne the �ow equation for unsteady-state

�ow: The Law of conservation of mass; Darcy�s Law, and Equation of state. Amathematical expression of the Law of conservation of mass is the continuityequation. For a liquid this continuity equation combined with Darcy�s Law andthe equation of state derives a radial di¤usion equation for a �uid of constantcompressibility. It applies to �ows of oil or water.In the Darcy equation the permeability, k, relates the driving force, macro-

scopic phase �uid velocity, u, to the phase pressure gradient, rp.The velocity �eld as de�ned by Darcy�s law is expressed as:

�!u = �k�(rP � Z): (2.1)

Liquid �ow is described with permeability of medium, k, phase viscosity, �,phase pressure, P , pressure gradient, , depth, Z. For the compressible gas �owand turbulence non-Darcy �ow the velocity �eld can be modi�ed.For a small compressibility, ct, and small pressure gradients equation is lin-

earised, to the known di¤usion equation:

1

r

@

@r(r@p

@r) =

��ctk

@p

@t(2.2)

where, hydraulic di¤usion, �, is constant and equal to k��ct

. Hydraulic di¤usionis a measure of the expendability of the system. The following:assumption aremade: the �ow take place along a radial path towards a wellbore; the physicaldimensions of the rock are not time dependent; the porosity and permeabilityare constant in space and time; the �uid saturations are constants and the �uid�ow is single phase; the �uid viscosity is constant, and both the compressibilityof the �uid and the pressure gradients are small.

It is possible to solve the above equation provided that the following condi-tions are met:- the outer boundary conditions, OBCs, are in�nite, no �ow and constant

pressure, and- the inner boundary conditions, IBCs are of constant or variable rate, q,

and constant or variable pressure, pwf . The boundary conditions are usually

2. TRANSIENT RATE DECLINE REVIEW 11

Table 2-1: Pressure gradients and dimensionless pressure functions for radialreservoir �ow at the well - After Valko and Economides (1995)

Radial �owLiquid �ow

Pressure gradient, �pGas �ow Pressuregradient, �m(p)

Dimensionlesspressure, pD

Transient(OBC - In�nite acting)

pi � pwf m(pi)�m(pwf ) pD = �12Ei(� 1

4tD)

Semilogarithmic approximationat tD>100 where tD= k

��ctr2wt

Steady state(OBC-Constant pressure)

pe � pwf m(pe)�m(pwf ) pD = lnrerw

Pseudosteady state(OBC-No �ow)

pi � pwf m(pi)�m(pwf ) pD =ln 0:472re

rw

expressed in terms of a Darcy velocity �eld, �!u , or pressure, p. The pressuresolution obtained at IBCs of a constant rate, q, can be converted to a rate,q, value according to by Valko and Economides (1995). The rates, q, for theconstant rate solutions are:

q =2�kh�p

�pD(2.3)

In order to reduce the number of unknowns and obtain solutions that are in-dependent of any unit system the dimensionless pressure, pD and dimensionlessrate, qD, are introduced. Earlougher (1977) demonstrated that the dimenionlesspressure, pD, for the constant rate production is almost equal to the 1

qDfor a

well producing at constant pressure. Thus the transient rate, q, obtained atIBCs with constant pressure, pwf ; IBC has the following form:

q =2�kh(pi � pwf )

� 1qD

(2.4)

The �ow pressure gradients and dimensionless pressure functions for a radialreservoir according to Economides and Ehlig-Economides (1994) and Valko andEconomides (1995) are presented in Table 2-1.Table 2-1 can also be used for the compressible gas. Instead of �p, Dake

(1978) and Economides et al. (1994) suggested an expression with �m(p). Thein�nite acting reservoir conditions are met under conditions with a signi�cantpressure drop at any outer boundary. This may be de�ned with a small dimen-sionless time at the outer boundary:

tDe =k

��cr2et � 0:1 (2.5)

A gas compressible �ow to be reviewed later in a the chapter is based on a real


gas pseudopressure, m(p) provided by Al-Hussainy and Ramey (1966) in thefollowing form:

m(p) =

pZp0

2p

�Zdp (2.6)

This yields to a di¤usion equation

1

r

@

@r(r@m(p)

@r) =

��ctk

@m(p)

@t

The present chapter reviews selected solutions de�ned with constant or vari-able IBCs and in�nite acting OBCs for various fractured wells positioned withinan oil and gas reservoir. Dake formulated a general solution of the radial di¤usionequation describing transient �ow in a reservoir as:

1

r

@

@r(r@�

@r) =

��ctk

@�

@t

where parameter � is de�ned for: undersaturated oil as � = �; real gas as

� = m(p); gas-oil (two-phase) � = m(p)0, where m(p)0 =PRP 0

kro(S0)�0B0

dp is pseudo-

pressure as stated by Raghavan (1993).Ikoku (1984) published general solutions by solving the dimensionless form

of the di¤usion equation for the inner boundary of constant rate.

1

rD

@

@rD(rD

@�D@rD

) =��ctk

@�D@t

The general solution for the constant rate inner boundary conditions is�a

q

�f(p) = �D(tD) + S

For undersaturated oil, real gas and two-phase �ow, the de�nition of the abovesolution terms includes: a

qbeing the function of kh

�; f(p) being the pressure

di¤erence; and S being the skin. For real gas, skin S is the function of S =S(S;Dq) where D stands for a non-Darcy �ow.

2.1 Oil Flow

To perform a conventional well test analysis on a well, one common procedure isto �ow the well at a constant rate for several days before carrying out the test.This procedure is not always e¤ective, and often the delay can be avoided byperforming transient rate tests instead. The most important test is the analysis


of the rate response to a step change in the producing pressure. This test allowsa type-curve analysis of the transient rate response without the complication ofwellbore storage e¤ects.The �ow of a single oil phase through such porous media is generally accepted

to be described by a linear di¤usion equation. This is strictly valid only forslightly compressible �uids, such as undersaturated oil or water for which theproperties are little a¤ected by changes in pressure. This transient rate analysisreview is focused on in�nite and circular reservoirs with concentric wells, i.e.,wells that are arbitrarily located in regularly or irregularly shaped reservoirs.

2.1.1 Vertical Well

Most reservoirs can be produced by the release of pressure and the consequentexpansion of underground �uid. During part of the production history of areservoir, �uid compressibility can be considered as small and constant. Solu-tion assumptions include a constant �owing pressure at the wellbore, which fullypenetrates the reservoir. The reservoir contains a slightly compressible �uid ofsingle phase and constant viscosity, and the �uid �ow is horizontal in a homoge-neous and isotropic porous medium of uniform thickness with constant perme-ability and porosity. Single-phase liquid solutions based on these assumptionsare widely used in hydrology and petroleum engineering.

Homogeneous Reservoir

When considering homogeneous porous media with a constant porosity, the per-meability is also isotropic provided that kv = kh. No saturation gradients occurswithin such a system. For an oil �ow, the oil saturation is equal wherever it isconstant i.e., So = (1� Siw). The pressure is considered to be above the bubblepoint of the �uid. The e¤ect of gravity is ignored, leading to �uid properties be-ing uniform over the constant thickness of a non-dipping formation. The in�owto a well is horizontal-radial since a well is perforated over the entire reservoirthickness. The pressure in time and space in the porous media for a single phase�uid is described with the general di¤usion equation as

1

r

@

@r

�k

��r@p

@r

�= ��ct

@p

@t(2.7)

This di¤usion equation is non-linear because of the rock and �uid properties(ct; �; k; �; �) being pressure dependent. The above equation is linearised byassuming @p

@tas being small to a most known form of

1

r

@

@r

�r@p

@r

�=1

�

@p

@t(2.8)


Here, the hydraulic di¤usion, � = k��ct

, for small changes in pressure is assumed tobe constant. By imposing small changes in pressure, or with a pressure increase,ct decreases and the parameters k, �, and � increase. For oil �ow, the totalcompressibility (�uid and pore volume), ct, is assumed to be ct = c0S0+cwSw+c�.The above linearisation is however valid only for

ctp� 1 (2.9)

as shown by Dranchuk and Quon (1967). The limitation of the linearisationapproach presented above is that, close to a wellbore, we may expect errorsin the local estimation of pressure, p(r; t), while the second derivative of thepressure on radius,

�@p@r

�2, is neglected.

The linearised di¤usion equation can be solved analytically. The above equa-tion is similar to the thermal di¤usion equation

1

r

@

@r

�r@T

@r

�=1

K

@T

@t(2.10)

where the thermal di¤usion, K(m2=s), corresponds to its hydraulic counter part,�, and the temperature, T (K), to the pressure, p. Solutions of the thermaldi¤usion equation for various initial and boundary conditions were publishedby Carslaw and Jaeger (1947) and can be applied in rate-testing analysis as anadvantage of similarity of the two di¤usion equations.The transient rate decline can be employed as a tool for identifying the char-

acteristics and predicting the behaviour of reservoir systems. The reservoir per-meability, porosity, and wellbore skin factor can be determined by matchingtype-curves. Conditions under which a constant-pressure �ow is maintained ata well include production into a constant-pressure separator or pipeline, or open�ow to the atmosphere. The dimensionless form of the di¤usion equation of awell producing at the constant pressure IBCs, or Dirichlet conditions and in�niteOBCs is:

1

rD

@

@rD

�rD@pD@rD

�=@pD@tD

(2.11)

Here the dimensionless radius, rD = rrw, the dimensionless time, tD =

�r2wt, and

the dimensionless pressure, pD(rD; tD) = 2�khq�(pi � p(r;t)). For an ideal well we

de�ne the in�nite acting or transient rate decline with the initial and boundaryconditions. The initial condition are: p = pi at t = 0 for all r. The outerboundary conditions are in�nite acting or: p = pi for all t, at r = 1. Theinner boundary conditions include: a constant rate where q is constant for allt and a constant pressure where p is constant for all t. For a constant rate,q = 2�khrw

�

�@p@r

�r+. By assuming that rw is negligibly small, lim

r!0(r @p@r) = qh

2�kh=

constant.


The pressure-time solution of the radial di¤usion equation is the line sourcesolution for the constant rate production at a wellbore. These solutions have beendeveloped by well testing. Moreover, the rate-time solution based on constantpressure inner boundary conditions has been the subject of rate-decline analysis.

Analytical Methods (Integral TransformMethods) The problem of con-stant pressure production from a well located at the centre of a homogeneousand isotropic cylindrical reservoir was �rst studied by Moore et al. (1933) and byHurst (1934). The results for an in�nite, unbounded reservoir were presented ingraphical form in terms of a dimensionless �ow rate decline with dimensionlesstime. Results were put forward for in�nite slightly compressible, single phaseradial systems. Furthermore, Jaeger and Clarke (1942) studied the heat con-duction problem. An integral solution for the temperature drop during a �owperiod in an in�nite cylinder was derived and numerical values of the integralwere tabulated. van Everdingen and Hurst (1949) presented a series of solutionsfor the �ow rate decline with time. Their work included the study of an in�nitereservoir. Jacob and Lohman (1952) described the dimensionless �ow rate be-haviour for a well producing at constant pressure. Both the dimensionless �owrate and the dimensionless time applied only to the in�nite reservoir system.The relation for dimensionless �ow rate versus the dimensionless time was givenas:

qD =2

ln tD + 0:80907(2.12)

with the qD and the tD de�ned as:

qD(rD; tD) =B�

2�kh(pi � pwf )q (2.13)

tD =�

r2wt (2.14)

Ferris et. al. (1962) tabulated rate-time values for the in�nite reservoir case.Furthermore, Earlougher (1977) provided constant pressure equations to obtainthe permeability and skin factor from rate-time production data.

Semi Analytical (Inverse "Laplace" Transform) Techniques The use ofthe integral "Laplace" transform technique to obtain a solution for the pressurebehaviour was attempted by Clegg (1967). In order to obtain the "Laplace"inverse function, Clegg (1967) employed an approximation de�ned by Schapery(1962). Ehlig-Economides (1979) and Ehlig-Economides and Ramey (1981) pre-sented the model set-up according to the given assumptions for a saturated liquid�owing as a single phase, in an isothermal reservoir of the constant �uid viscos-ity, �, and for a constant and small total compressibility of the �uid and porous


medium, ct. In the radial geometry, the radial �ow is described with negligiblegravity e¤ects according to the di¤usion equation. The �ow through a porousmedium exists from �nite wellbore radius, rw, to the in�nite or �nite externalreservoir radius, re. Here, the porous medium is homogeneous and isotropicwith a constant di¤usion (permeability, k, porosity, �, and thickness, h). Thus,the idealised �ow through porous media can be described by the fundamentalpartial di¤erential equation, also known as the di¤usion equation. The partialdi¤erential equation is reduced to the ordinary di¤erential equation with thespacial variable, rD, rD =

r

rwand the "Laplace" variable, s, as an unknown. It

is then solved analytically for the "Laplace" transform of the pressure pD by the"Laplace" integral transformation.A common method for solving the radial-di¤usion dimensionless Equation

(2.11) under de�ned inner and outer boundary conditions is to use the "Laplace"transformation. The advantages of this method consist in transforming partialdi¤erential equations into ordinary di¤erential equations that can be solved an-alytically for the "Laplace" variable, s, and the space variable, rD, as describedby van Everdingen and Hurst (1949). The "Laplace" transformation is de�nedby:

F (s) =

1Z0

e�stDF (tD)dtD (2.15)

Moreover, the "Laplace" transform applied to the dimensionless partial di¤eren-tial Equation (2.11) and can be expressed as:

d2�pDdr2D

+1

rD

d�pDdrD

= s�pD (2.16)

The "Laplace" space solution for a production from the centre of a circularreservoir under constant pressure inner boundary conditions and in�nite outerboundary conditions is given as:

�qD(s) =K1(s)p

s(K0

ps+ S

psK1

ps)

(2.17)

For the cumulative rate, QD we obtain

QD(s) =K1(s)

sps(ps(K0

ps+ S

psK1

ps))

(2.18)

Since the cumulative rate, QD, is de�ned as QD =

tZ0

q(t)dt, the dimensionless

�ow rate, qD, in �eld units becomes:


qD =141:2�B

kh(Pi � Pwf )q(t) (2.19)

and the dimensionless time, tD in �eld units is:

tD =0:0063kh

��ctr2wt (2.20)

These equations are exact integral transform solutions in "Laplace" space as pub-lished by Ehlig-Economides (1979). The inverse integral Laplace transformationcan only be obtained through the Mellin inversion integral transformation. TheStehfest (1970), Crump (1976) or other approximate numerical inversion proce-dure can be applied to the given solutions. van Everdingen and Hurst (1949)presented the relationship between the Laplace transformed solutions for theconstant pressure and the constant rate in the form of:

qDpD =1

s2(2.21)

where qD; is de�ned under constant pressure inner boundary conditions and pD;is de�ned under constant rate conditions. Expression (2.21) reveals that anysolution for pD; or constant rate production has an analog solution, qD;for pro-duction under constant pressure. Both variables are "Laplace" space variables.Equation (2.21) can be derived from the principle of superposition. A numericalinversion technique, consists in inverting "Laplace" space variables to a dimen-sionless rate, qD, and dimensionless time, tD; now in the space-time domain.

Numerical Methods (Finite Di¤erence) Juan-Camus (1977) presented analternative numerical method for obtaining solutions for a constant pressure �ow.He derived the constant pressure solutions from the constant rate solutions byway of superposition. The vertical direction of the reservoir was also consideredwhen solving the di¤usion equation. The derivation did not require the integraltransformation of "Laplace".Uraiet (1979) and Uraiet and Raghavan (1980) presented a study of the

transient rate behaviour through numerical methods. They considered the classicproblem of the �ow of a slightly compressible �uid in a cylindrical, homogeneous,isotropic reservoir of constant thickness. The well was located at the centreof the cylinder, and �uid was produced at a constant pressure. Initially, thepressure was uniform throughout the reservoir. The skin region in this modelwas assumed to be an annular region that was concentric with the wellboredisplaying a permeability di¤erent from the formation permeability. Wellborestorage e¤ects were not considered since the well �owed at a constant pressure.The authors determined that the semi-log analysis method for constant ratetesting could be applied to the transient rate if the reciprocal rate was used.


Kleppe and Cekirge (1980) presented approximate solutions to the di¤usionequation for an in�nitive radial reservoir for the constant well pressure underinner boundary conditions. They used a numerical simulation of well tests toverify the obtained expressions and to present examples of their applications.Raghavan (1993) observed that the late time solution for the �nite wellboreradius case was identical to the late-time solution for a line source well. Thisresult justi�es the use of the line source solution for practical problems.The "Laplace" time domain expansions published by van Everdingen and

Hurst (1949) applied to homogeneous reservoir only. Technique comprises inte-gral computation along contour with a search for poles of a function over thecomplex plane.

Heterogeneous Reservoir

Speci�c types of heterogeneous reservoir systems have received much attentionin the oil and gas industry in recent years. Several analytical and numericalmodels exist in order to consider the permeability, k, changes with distance,r, naturally fractured, layered and composite reservoir systems. Heterogeneousreservoirs have been well documented in research papers. In reality, reservoirsystems usually combine all the e¤ects of these types of heterogeneities.In a radially heterogeneous reservoir, the �ow is supposed to be purely radial

with a permeability as a function of only the radial distance from the well. Theporosity is here constant. The di¤usion equation is solved for systems wherethe permeability varies continuously from one well to another. Loucks (1961)presented transient pressure solutions for a case with a radial heterogeneousreservoir. The reservoir permeability varies with the power of the radial distance

from the wellbore, i.e., as k(r) = k(rn), for values of n = 0� 13� 1; and� 2.

Oliver (1990) presented an approximative solution to the problem of a wellproducing at inner boundary conditions of constant rate from an areally hetero-geneous reservoir. Here, the permeability is considered as an arbitrary functionof position, i.e., k = k (r;�). By using perturbation theory and the "Laplace"transform he derived approximate solutions for the transient wellbore pressure.A two-dimensional permeability distribution, k = k(r;�), is identical to the solu-tions for an equivalent radial permeability, k = k(r). The permeability k(r)weretaken as the harmonic average of k = k(r;�) over 2. This was valid for onlysmall variation of permeability in direction of �. A study by Feitosa (1993)considered a well producing a heterogeneous reservoir characterised by a contin-uously variable radial permeability. It was also possible to consider the e¤ect ofporosity variations in an areally heterogeneous reservoir.

Areally and Radially Heterogeneous Reservoir A reservoir with proper-ties, as permeability, porosity and thickness being arbitrary function of position


de�ned by radius, r, and azimuth, �; is called the areally heterogenous reservoir.The radially heterogeneous reservoir refers to a reservoir in which one or moreof the basic parameters (permeability, porosity, and thickness) varies only withthe radial distance, r; in the (r;�) coordinate system.The Oliver (1990) solution is based on a perturbation theory and the "Laplace"

transforms. By introducing a small variation in permeability about a meanvalue the dimensionless permeability, kD; as a function of a small number " andf(rD;�) that is on order of 1 becomes

kD(rD;�) =1

1� "f(rD;�)(2.22)

A perturbation series approach combined with "Laplace" transforms leads to theapproximate pressure solution in "Laplace" space

pD(rD = 1;�; s) = pD0(rD = 1; s) + "pD1(rD = 1;�; s) (2.23)

Here, pD0 refers to a constant-permeability solution given as

pD0(rD = 1; s) =K0(

ps)

spsK1(

ps)

(2.24)

and pD1 refers to the �rst order perturbation of constant permeability with atime-correspondent "Laplace" variable, s

"pD1(rD = 1;�; s) = �1

sK21(ps)

1Z1

�

8<:K21(�ps)

24 12�

+�Z��

�1� 1

kD(r;�)

�d�

359=; :d�

(2.25)These equations were converted to dimensionless rate-dimensionless time solu-tions through

qD =1

s2 [pD0(rD = 1; s) + "pD1(rD = 1;�; s)](2.26)

Both dimensionless-pressure and dimensionless-rate solutions can be numericallyinverted by Stehfest to obtain the real time solutions of pressure p(rD = 1; t)= pD0 + �pD1 and accordingly of rate q (rD = 1; t). Oliver (1990) derived alate time approximation to the dimensionless pressure solutions by letting the"Laplace" variable, s, approaching zero:

pD0(rD = 1; s) =K0(

ps)

s(2.27)

"pD1(rD = 1;�; s) = �1Z1

�

8<:K21(�ps)

24 12�

+�Z��

�1� 1

kD(r;�)

�d�

359=; d� (2.28)


An analytical inversion of the "Laplace" space solutions comprises a part con-sisting of a homogeneous solution (or constant-permeability solution)

pD0(rD = 1; tD) = �1

2Ei

�� 1

4tD

�for tD > 25, and when using a logarithmic approximation and Euler�s constant = 0:57721566

pD0(rD = 1; tD) =1

2ln

�4tDe

�The �rst order perturbation solution to the costant-permeability solution is

"pD1(rD = 1;�; tD) = �1Z1

8<:G(�; tD)24 12�

+�Z��

�1� 1

kD(�;�)

�d�

359=; d�

where G is the kernel function in terms of Whittaker�s function.Feitosa (1993) developed a numerical solution to the above stated areally

heterogeneous reservoir with the �nite-di¤erence approximations. For an arbi-trary heterogeneous reservoir it is possible to determine the pressure response ina reservoir with known radial permeability distribution, k = k(r). Feitosa (1993)calculated the pressure response with time in an areally homogeneous reservoirproducing a slightly compressible �uid from a single well. The physical modelthat was considered comprised: an inner boundary condition of constant rate,an outer boundary condition of closed upper and lower boundaries with a later-ally in�nite reservoir, a constant porosity, thickness and rock compressibility, asingle phase �uid with constant viscosity and compressibility; negligible gravityand capillary e¤ects; fully penetrating well, rock and �uid properties indepen-dent of pressure, a uniform initial pressure throughout the reservoir; as well asnegligible wellbore storage and skin e¤ects.In recent research Berard (2007) considered the use of a generalised Weber

expansion in the radius domain in well-testing analysis. A generalized Weberfunction is also used in the rate-time analysis in Appendix A.

Layered Reservoirs

Reservoir rocks are usually not uniform, whether in horizontal or vertical direc-tions. The reservoir is thus not homogeneous or isotropic. Heterogeneities canexist in rock and �uid properties from deposition, folding and faulting, post-depositional changes in reservoir lithology and changes in �uid-type properties.The small scale heterogeneities may result in carbonate reservoir rocks matrixand fractures, vugs or solution cavities. On the large scale, heterogeneities resultin physical barriers. Layering is the most common form of heterogeneity.


Layered reservoirs can be divided into reservoirs with cross�ow, commin-gled reservoirs and composite reservoirs. Layered reservoirs with cross�ow arehydrodynamically communicating at the contact planes, whereas for commin-gled systems, or layered reservoirs without cross�ow, the layers communicatethrough the wellbore. The �ow between adjacent and connected layers resultingfrom capillary, gravitational, or viscous forces, has been studied for many years.A composite reservoir is made up of two or more regions, and each region hasits own rock and �uid properties. A composite system can either be createdarti�cially or be naturally occurring.

Natural formations are vertically heterogeneous because of strati�cation andthe various depositions. The vertical sequence of deposits alternates between lay-ers with good and poor permeability. The di¤usivities, de�ned by the physicalproperties of strati�ed deposits, di¤er from layer to layer, and the layers re-spond to production at various rates. This causes di¤erential pressure depletionbetween the layers, thereby altering the radial �ow pattern of a well.

Pressure transient analysis of layered systems has been described by numer-ous papers. However, only few studies have addressed rate transient analysis.Most of the reviews discuss testing methods, well testing, and rate testing in-terpretation techniques. A comparison of the layered model solution methodsand a precise study of the available numerical techniques applied in the in-verse "Laplace" transform solution procedure has been treated. Although, the"Laplace" transform has been more widely used in pressure and rate transientmodelling, other integral transforms can be very useful in the solution of theboundary value problems of layered systems as presented by Cvetkovic (1992).

Layered reservoir models are developed assuming that the reservoir pressurecan be directly measured. The pressure recorder is located in the wellbore, andthe wellbore constitutes a link between the reservoir and the recorders. Therecorded pressures are representative of the pressure in the reservoir and can bea¤ected by the number of wellbore-related phenomena. These wellbore phenom-ena must be recognised in order for the diagnosis of the reservoir characteristicto be achieved. The wellbore dynamics, such as the e¤ects of temperature ona wellbore �uid, gas-oil solution-liberation or retrograde condensation, liquidin�ux-e ux, phase redistribution, wellbore and near-wellbore clean-up, di¤er-ences between drawdown and build-up, recorder and others, not precisely de-�ned, e¤ects all transient analysis.

Ehlig-Economides and Joseph (1987) reviewed various models of layeredreservoirs, of which the layered reservoir model treats the heterogeneous reser-voir system as consisting of separate homogeneous strata. These layers maycommunicate in the reservoir through cross-�ow. Sabet (1991) presented di¤er-ent physical and mathematical models according to the various solution methods,number of layers, interlayer �ow, well speci�cation, inner and outer boundaryconditions. The report published by Cvetkovic (1992) is an attempt to incorpo-


rate model understanding and numerical model analysis of the published layeredreservoir solution. Ehlig-Economides (1993) presented an investigation on modeldiagnosis for layered reservoirs. The question is when interpretation models forlayered reservoirs can be used e¤ectively, and how to identify a model for eachreservoir layer.Tariq and Ramey (1978) introduced the solutions of a bounded (circular)

multilayer reservoir system producing at constant rate in order for skin e¤ectsto be included in each layer. The total wellbore storage model was also includedinto solutions. The dimensionless "Laplace" space solutions were transformedto the real domain with the numerical inversion of Stehfest (1970). Spath etal. (1994) proposed a stable and robust algorithm to compute pressure andrate responses from a well producing the commingled reservoir. This approachdetermined the well responses for constant or variable-rate productions. Thefollowing dimensionless pressure for a commingled reservoir in the "Laplace"domain is based on the constant-rate dimensionless pressure solution for eachlayer:

pwDj(s) =1

nPj=1

kjhj

kh1

pwDj (s)

Further, Spath et al. (1990) formulated the rate of each layer:

qsDj(s) =kjhj

kh

pwsD(s)

spwDj(s)

Blasingame et al. (1991) computed the e¤ect of the wellbore storage andthe wellbore phase redistribution. The obtained solutions were accurate whencompared to the results obtained from the numerical inversion of the "Laplace"domain solutions.Lolon et al., (2008) formulated the new solution for the multilayer reservoir

and developed approximate semi-analytical solutions for the wellbore pressureand fractional �owrate responses for commingled layered reservoirs for which across�ow was permitted in the wellbore and not in the reservoir. He referredto the study of Lefkovits et al., (1961) concerning the pressure behaviour forlayered reservoir systems assumed to be homogeneous, isotropic and saturatedwith a �uid of small and constant compressibility. The reservoir comprised nlayers and each layer was de�ned by: permeability, k, thickness, h, porosity, �,viscosity , �, �uid compressibility, ct, wellbore and outer radii, rw and re, andskin, s. The obtained solutions provided accurate approximations only for latetimes.Lolon et al., (2008) validated the developed explicit and approximate time

solutions in the real domain for the modelling of the performance of a multilayer


reservoir system. This was done after considering certain approximate behav-iours of the individual layer solutions for the purpose of creating algebraicallyconvenient results in the "Laplace" domain for the total system behaviour. Thechosen basic functions were able to yield forms that were inverted analytically tothe real domain. The process provided solutions for multilayer reservoir system.Solutions for a single well in a multilayer reservoir system can be used to modelwell test or production data.

Cross�ow Models A cross�ow reservoir is considered to consist of continu-ous layers of permeabilities ki (for i = 1; 2; :::). For the 2 layers system, whenpermeability k1 is greater than permeability k2; the pressure gradient in the �rstupper layer is of substantial size. The vertical pressure gradient is created inthe lower layer of permeabilities k2 according to the pressure drop in the layer ofpermeability k1. The direction of �ow is determined by the di¤usivity contrastand is upwardly oriented towards the high permeability layer.Russell and Prats (1962) reviewed the practical features of the interlayer

cross�ow systems. They concluded that a well producing from a layered reservoirwith cross�ow behaves similarly to one in a homogeneous, single-layer reservoirwith the same pore volume, and a �ow capacity equal to the total �ow capacityof the strati�ed system.The e¤ect of contrasting permeabilities for two perforated layers with the

same storage and thickness was considered by Prijambodo at el. (1985). Onthe basis of several numerical simulation runs, it was noticed that whenever thevertical-to-horizontal permeability ratio of the �rst layer was greater than orequal to 0:01 and the permeability ratio between two layers was bigger thanor equal to 5, the interlayer cross�ow was a signi�cant factor in establishing thedrawdown behaviour. In the case of the lager contrasting permeabilities betweentwo layers, the cross�ow equalises the pressure so rapidly that the well behavesas if it were producing from a single layer.These results are similar to those derived by Javandel and Witherspoon

(1969) on the basis of a �nite element method. Moreover, Javandel and Wither-spoon (1969) concluded that the permeability contrast had a considerable e¤ecton the transitional pressure behaviour. The smaller the contrast in permeability,the more rapid is the convergence to a single-layer behaviour. In all cases, atlarge values of time, the direction of �ow becomes almost radial regardless of thepermeability contrast.

Commingled Reservoirs If the layers are separated by a completely imper-meable boundary, interlayer cross�ows will be absent within the reservoir. Cross-�ow can take place only at the wellbore itself and only if the well is completedin more than one layer. The pressure pattern for this case di¤ers signi�cantlyfrom that for interlayer cross�ow.


During production, cross�ow can occur at the wellbore provided that thelayers have an unequal initial pressure. Whenever the �owing pressure is higherthan that of the layer with the lower pressure, only a portion of the total �owfrom the layer with the higher pressure is produced at the surface. The remainderof the production enters into the lower pressure layer through cross�ow at thewellbore. When the �owing well pressure is lower than that of the lower pressurelayer, both layers contribute to the surface production.The solution of the fully penetrated well in a circular, bounded and commin-

gled reservoir with homogeneous and isotropic layers �lled with �uid of smalland constant compressibility and constant viscosity was given by Letkovits et al.(1961). Under their assumption, if the reservoir is initially at a uniform pressurepi at all times ti; and the production rate q, measured at initial reservoir condi-tions, is held constant, the pressure must satisfy at any point the known partialdi¤erential equation for �ow of a �uid. The solution to this equation for theproper boundary conditions is obtained with the "Laplace" transform. Letkovitset al. (1961) presented rigorous equations describing the pressure behaviour ata constant terminal rate well producing from a bounded, noncommunicating,layered system with contrasting properties. Letkovits et al. (1961) also providedthe report describing the pressure behaviour of a well producing at a constantterminal rate from an in�nitely large, multi-layer non communicating reservoir.

Composite Reservoir Often, the region surrounding the wellbore is eithermore or less permeable than the reservoir because of the various drilling and com-pletion practices. The drilling-�uid invasion reduces the permeability whereasthe operation of fracturing or acidising increases it. Composite reservoir mod-els consider reservoir systems made up of two concentric zones of varying rockand �uid properties separated by a discontinuity. Examples include reservoirsdamaged by drilling or completion �uid invasion, acid-stimulated wells. Suchcomposite systems were studied in connection with heat �ow by Jaeger (1941).He presented a solution for temperature distribution in a radial system with anin�nitely large outer radius. Penner and Sherman (1947) studied similar heat�ow problems. Closman and Ratli¤ (1967) presented a solution for a well pro-ducing at a constant pressure from a closed, radial composite reservoir. Turki(1986), and Olarewaju and Lee (1987) presented solutions in "Laplace" spacefor a well producing at a constant pressure from a radial, in�nite compositereservoir. Turki et al. (1989) studied a constant-pressure well in the centre of atwo-composite reservoir with wellbore skin. For in�nite composite reservoirs, thee¤ect of the mobility ratio, storativity ratio, wellbore skin and the discontinuitydistance on rate decline are described.The two-regions assumptions include: a single phase �ow with only one �uid,

a formation that is horizontal and of uniform thickness, a sandphase wellborepressure that is maintained constant, as well as a constant distance to the radial


discontinuity. Turki et al. (1989) described each region with the following partialdi¤erential di¤usion equation

@2pD1@r2D

+1

rD

@pD1@rD

=@pD1@rD

for 1 < rD < RD (2.29)

@2pD2@r2D

+1

rD

@pD2@rD

=@pD2@rD

for RD < rD <1 (2.30)

With initial conditions and boundary conditions de�ned for each composite re-gion, and after applying the "Laplace" transformation to the equations above,Turki and al.(1989) published the "Laplace" space well rates qD1 and qD2 (where� is the region di¤usion constant):

qD1= �rD

ps�C11I1(rD

ps)� C12K1(rD

ps)�; for1 � rD � RD (2.31)

qD2= �rD

p�s

�[C21I1(rD

p�s)� C22K1(rD

p�s)] ; forRD � rD <1 (2.32)

The well rate, qD, in "Laplace" space is

qD = �ps�C11I1(

ps)� C12K1(

ps)�

(2.33)

and the cumulative well production in "Laplace" space, QD is

QD = �1ps

�C11I1(

ps)� C12K1(

ps)�

(2.34)

To solve the transient rate decline in real space of composite reservoir inversetransform Stehfest algorithm can be applied. Constants C11, C12, C21 and C22 forthe OBC in�nite active are provided by Turki et al. (1989). Furthermore, Issakaand Ambastha (1998) investigated in�nite reservoir responses from a composite

reservoir with regards to a two-region mobility(k=�)1(k=�)2

and storativity (�c1)1(�c2)2

.

Naturally Fractured Reservoir

Laminar Flow The naturally fractured reservoir models, also referred to asdual-porosity reservoir models, consider a heterogeneous system made up of twodistinct porous media. The primary porosity medium contains a majority of�uid stored in the reservoir but possessing a low conductivity. The secondaryporosity medium acts as the conductive medium but has a low storativity. Thestudy of transient rate decline in a double-porosity system was started by Mavorand Cinco-Ley (1979) and continued by Da Prat et al. (1981), Raghavan and


Ohaeri (1981), and Ozkan et al. (1987). Sageev et al. (1985) published a typecurve method for analysing rate-time data in a double-porosity system with awell skin. Moreover, Grasman and Grader (1990) have presented an analyticalmethod to determine double-porosity reservoir properties.The model of Warren and Root (1963) was extended by Mavor and Cinco-Ley

(1979). Da Prat (1981) formulated the statement of the problem for naturallyfractured reservoirs with the following partial di¤erential equation

@2PfD@r2D

+1

rD

@PfD@rD

= (1� !)@pwD@tD

+ !@PfD@rD

(2.35)

(1� !)@pmD@tD

= �(PfD � PmD) (2.36)

where parameters ! and � are associated with reservoir and �uid properties.The storage of secondary porosity to a total storage (both matrix and fracture),

! is de�ned as ! =(�Vc)f

(�Vc)f+(�Vc)m. An interporosity �ow controls �, and the

interporosity �ow shape factor, �, is equal to � = �Km

Kfr2w. Further, the dimen-

sionless fracture pressure, PfD, and the dimensionless time, tD, are expressed asPfD =

Kfh(Pi�P )141:2q�B

and tD = 2:637x10�7Kt

[(�C)f+(�C)m]�r2w, respectively.

Appropriate initial and boundary conditions include: PfD(rD; 0) = 0 andIBCs of a constant producing pressure PfD � S(

@PfD@tD

) = 1, where S is theskin factor. The in�nite OBC gives lim

rD!1PfD(rD; tD) = 0. The dimensionless

�ow rate, qD, into the wellbore is qD(tD) = ��@PD@tD

�rD=1

, and the cumulative

production, QD, related to the �ow rate is QD =

tDZ0

qDdtD.

The "Laplace" transformations convert partial di¤erential equations into asystem of ordinary di¤erential equations that can be solved analytically. Solu-tions in "Laplace" space are functions of the complex variable, s, and the spacevariable, rD. Mavor and Cinco-Ley (1979) published in�nite or unbounded reser-voir transient rate solutions with approximations for early and late time. Thedimensionless �ow rate in "Laplace" space given by Da Prat (1981) is:

qD(s) =

psf(s)K1

hpsf(s)

isnKo

hpsf(s)

i+ spsf(s)

1

hpsf(s)

io f(s) =!(1� !)s+ �

(1� !)s+ �

(2.37)Moreover, with an Inverse "Laplace" transform it is possible to invert the abovesolutions to a real time and space. For early times, the dimensionless �ow rateis qD =

p��( tD!)�

12 and the cumulative dimensionless rate is QD =

2p��(!tD)

12 .


A late time approximation of equation (2.37) becomes equal to a homogeneousreservoir solution, as the one provided by Jacob and Lohman (1952). Da Pratet al. (1980) derived the late time approximation for a dimensionless rate qD =

2ln tD+0:80907

.

Non-Laminar Flow Rodriguez-Roman and Camacho-Velazquez (2002) pre-sented analytical expressions for non-Darcy liquid �ow in dual porosity systems.The di¤erential equation describing �ow of a matrix-fracture systems is given as:

1

rD

@

@rD

0BB@rD 2

1 +

r1 + 4�D

h@pfD@rD

i1CCA = (1� !)

@pmD@tD

+ !@pfD@tD

(2.38)

(1� !)@pmD@tD

= �(pfD � pmD) (2.39)

where, !, �, �D, rD, pfD, tD are dimensionless variables. In other words, ! =(�Vc)f

(�Vc)f+(�Vc)m, � = �Km

Kfr2w, �D =

k�q2�rW h�

, PfD = Pi�PPi�Pwf , rD =

rrw, with a unit

conversion constant �l, and the dimensionless time tD = �l k�ct�r2w

t . With theForchheimer equation, it is possible to include non-Darcy �ow:

@pf@r

=�

k�f + ��2f (2.40)

there, the coe¢ cient of inertial �ow resistance, �, is equal to � = 48511�5:5f k0:5f

. The

approximative transient rate solution for early times thus becomes:

qD =

r!

�tD��k2f�

�(pi � pw)

rw�

�!

�p�tD

p2 etD (2.41)

valid for a range of etD, 2 � 1pftD � 0:5. It was di¢ cult to assign a value toetD when comparing this analytical solutions to simulation results. The lam-inar �ow solution was obtained when neglecting non-Darcy �ow e¤ects. Theapproximative transient rate solution for late times is equal to:

qD = �

2421Z0

�4e2

�tD

�(x+ 1)dx

35x 241� 4��k2f�

�(pi � pwf )

rw�

� 1Z0

�4e2 t�xx

�(x+ 1)dx

35 (2.42)


Combined Reservoir Olarewaju and Lee (1991) presented a model that pre-dicts production performance from a naturally fractured reservoir with a ra-dial discontinuity around the wellbore. This type of reservoir (composite dual-porosity reservoir) con�guration exists when a well is damaged, acidised orgravel-packed. Zhang et al. (1993) presented solution techniques for rate declinebehaviours in a complex system comprising a well that is arbitrarily located ina regularly or an irregularly shaped reservoir or in a composite reservoir. Forthe composite model, the analytical "Laplace" solution is based upon placinga constant pressure well with an arbitrary location in a two-composite radiallyconcentric domain. The model can then provide the characteristic responsesof such composite systems by varying the properties and the geometries of thedomains.

2.1.2 Horizontal Well

Horizontal wells have received considerable attention in the literature. The hor-izontal well technology enables the exploitation of numerous reserves that maynot have been economically viable with conventional drilling methods. Naturallyfractured reservoirs, discontinuous reservoirs, water and gas conning reservoirs,tight reservoirs, heavy oil reservoirs and EOR applications are major applica-tions for horizontal wells. Theses wells have the potential of producing at higherproduction rates than their vertical counterparts but also have other advantagessuch as enhancing reservoir management and accessing reserves that cannot beexploited by other means. Advantages of horizontal wells consists is in an in-creased well productivity, enhanced reservoir management, and the access toincremental reserves. However, the parameters a¤ecting horizontal well perfor-mance involve a higher level of uncertainty as compared to vertical wells.Nevertheless, horizontal wells have a potential advantage over vertical or

deviated wells based on the following main reasons: an increased exposure tothe reservoir giving higher productivities (PIs); the ability to connect laterallydiscontinuous features; e.g. fractures, fault blocks; and the ability to change thegeometry of drainage, e.g., reduce drawdown in oil rims. A horizontal well hasa higher productivity in laterally extensive reservoirs, as the productivity indexis a function of the length of the reservoir drained by a well. The productivityimprovement factor (PIF) compares the productivity of a horizontal well that ofa vertical in a same reservoir. It is estimated to

PIF =

LpkhLpkv

=L

h

pkvpkh

where L is the length of the horizontal well section, h the height of the reser-voir, kh and kv horizontal and vertical permeabilities, respectively. The vertical


permeability, kv reduces the production of a horizontal well, rendering the pro-duction rate lower than in a vertical well.

Predicting the performance of a horizontal well for a wide range of reservoirapplications has constituted a continuous research topic as of 1977, when the �rsthorizontal well was drilled in the o¤shore Raspo Mare Field in the Adriatic area.Special opportunities for a horizontal well technology appeared with the decreaseof drilling costs to about 1.3 times those for vertical wells. Joshi (1987) and Norriset al. (1991) have reviewed the horizontal well technology. A large amount ofresearch has been focused on the topics of reservoir engineering. Many analyticalsolutions for productivity of transient pressure responses for various boundaryconditions have been derived. Besides boundary conditions, near wellbore e¤ects,such as formation damage, non-Darcy �ow and arbitrary completions have beenstudied. Several investigations have predicted in�ow performance relationships(IPRs) for horizontal wells. Transient pressure analysis studies further extendto various boundary conditions and derive pressure and rate responses. Brekke(1996) studied how wells are a¤ected by geological variations. A long horizontalwell increases the potential both for success and failure. A better understandingof the total reservoir and wellbore interaction and �ow behaviour raises thepotential to success before the well is completed. Moreover, horizontal wellsimprove the recovery factor in the oil and gas �elds.

Solutions for a horizontal well in a form of a vertical-stripe being a part ofa vertical-fractured well, are presented by Joshi (1991). Vicente et al. (2000)presented the fully implicit, three-dimensional simulator with local re�nementaround the wellbore, developed to simultaneously solve reservoir and horizontalwell �ow equations, for single-phase liquid as well as gas cases. The modelinvolves the conservation of mass and Darcy�s law in the reservoir, in additionto mass and momentum conservation in the wellbore for isothermal conditions.The coupling requirements are satis�ed by preserving the continuity of pressureand mass balance at the sandface. The proposed simulator is tested against andveri�ed with the results obtained from a commercial "Black Oil" code, availablepublic domain simulators and semi-analytical models. The model can be usedto simulate the transient pressure and �ow rate behaviour of both the reservoirand the horizontal wellbore. The e¤ects of permeability, formation thickness,well length, and �uid compressibility were also studied.

Cheng (2003) investigated the productivity evaluation and well test analysisof horizontal wells. The major components of this work consist of a 3D coupledreservoir /wellbore model , a productivity evaluation, a deconvolution technique,and a nonlinear regression method improving horizontal well test interpretation.The 3D-coupled reservoir and wellbore model was developed using the bound-ary element method for realistic description of the performance behaviour ofhorizontal wells. The model is able to �exibly handle multiple types of innerand outer boundary conditions, and can accurately simulate transient tests and


long-term production of horizontal wells. The work comprises a comprehensiveliterature review of the modelling of a horizontal well.Thompson et al. (1991) used a line source solution, while Besson (1990)

and Economides et al. (1996) employed a point source solution to develop theirsemi-analytical models for performance evaluation. Spivey et al. (1992) system-atically presented an e¤ective method for obtaining the new solution for pressuretransient responses of a horizontal well at an arbitrary azimuth in an anisotropicreservoir. This method transforms the relevant parameters (permeability, reser-voir thickness, wellbore length and radius, and vertical position of wellbore) to anequivalent isotropic system. Ding (1999) used the boundary integral equationmethod to obtain a coupled reservoir/wellbore model. Ding�s model not onlyconsidered wellbore hydraulics, but also considered the wellbore as a cylindricalsurface source instead of a line source.As mentioned earlier, the semi-analytical models described above are applica-

ble only to reservoirs with the same geometrical shape and boundary conditionsas those of the source functions used to develop the model. In reality, the bound-ary situation is problem-dependent; however, these models cannot �exibly dealwith changing conditions.A signi�cant breakthrough in Green�s function method involves the use of the

Green function in the free space and boundary element method, BEM in orderto develop the solutions. Based on the BEM, Koh and Tiab (1993) developeda reservoir model that can prescribe arbitrary boundary shapes and conditions.They discretised the reservoir boundaries and wellbore surface with triangularelements. Their solution was developed in the "Laplace" domain and the so-lution in the time�domain was numerically inverted using Stehfest�s algorithm.However, the frictional pressure loss in a horizontal wellbore was not consideredin their model.

2.1.3 Vertical-Fractured Well

Hydraulic fracturing is widely used to increase the productivity of damaged wellsor wells producing from low to moderate permeability formations. A fracture hasa much greater permeability than the formation it penetrates; hence, it in�uencesthe pressure or rate response of a well. This fracture can be either vertical orhorizontal and the relation between the overburden or vertical stress and thehorizontal stress de�nes the fracture type. If the overburden stress is larger thanthe horizontal stresses, the fracture is vertical whereas for the horizontal stressesgreater than the overburden stress, the fracture is horizontal.Hydraulically induced fractures are vertical for reservoir depths greater than

1000 m. Bellow such depths, in shallow formations, hydraulic fractures tendto be horizontal. The hydraulic fracture length and width vary according tothe formation permeability. In moderate or high-permeability formations, thehydraulic fracture should be short and wide, as opposed to long and narrow in


low permeability formations.Thus, much research has been carried out determine the e¤ect of hydraulic

fractures on pressure-rate transient behaviour and well performance. Besidesanalytical modelling for a single-phase �ow, numerical modelling of �ow hasbeen investigated. Based on fracture �ow, analytical models fall under threecategories: �nite conductivity, in�nite conductivity and uniform �ux. Thesethree concepts have been used to describe the �uid �ow behaviour within thefracture. In the �nite conductivity fracture, pressure drops due to �uid �owwithin the fracture has been found to be signi�cant. In in�nite conductivityfractures on the other hand pressure drops are negligible and close to zero. Thefracture conductivity is very large in comparison to the formation permeability.This ideal condition renders the production near fracture tips higher than that atthe well centre. A uniform �ux fracture corresponds to a fracture for which the�ux per unit length entering the fracture is constant along the fracture length.Moreover, the production along the fracture length is constant, causing the well�owing pressure at the fracture centre to be smaller than that at the fracturetips.A vertical fracture is generally described as fully penetrating the formation

symmetrically across the well and as having a uniform width, w. The fracturehalf-length, Lf , is de�ned as the distance from the axis of the well to the fracturetip. The fracture conductivity, CfD, is the product of dimensionless fractureconductivity, kfD, and the dimensionless fracture width, wfD. The dimensionlessterms, kfD and wfD, are equal to kf=k and w=Lf , respectively. Here kf is thefracture permeability and k is the formation permeability.The well productivity is known to increase with hydraulic fracturing and a

consequence of such fracturing is a single crack or a fracture. Fractures arehydraulically induced and do not resemble the natural fractures known as �s-sures. In general, fractures can be horizontal for depths less than 1000 m, butare mostly vertical. Rock mechanics describe stress forces and physical fracturemodelling processes that cause fracture propagation. The physics of appearanceof fractures will however not be discussed in the review. Rather, we presentproperties related to each fracture to be further considered in fracture �ow mod-elling, e.g., the fracture half-length, Lf ; the dimensionless radius, reD = re

xf,

the fracture height, hf , usually assumed to be equal to a formation thickness;the fracture permeability, kf ; the fracture width, wf , the fracture conductivity,FC = kfwf , the dimensionless conductivity (conductivity group) FCD =

kfk

wfxf,

the dimensionless radius (fracture group) reD = reLf.

The identi�cation of well-reservoir variables that impact future well perfor-mances have been carried out both through well testing and rate testing. Con-ductive fracture properties Lf , wf and hf ; are mostly unknown and as well arefracture geometric features. Since the fracture permeability is much greater thanthe formation permeability in�uence rate response signi�cantly. When analysing


rate tests data from a vertical-fractured well it is possible to consider three rate-transient models. Each model de�nes adequate fracture character.

Uniform-�ux fractures The �ow rate from a formation to a fracture is uni-form along the entire fracture length. Due to a variable pressure along the frac-ture, the transient pressure behaviour includes two �ow periods, a linear �ow,and an in�nnite-acting pseudo-radial �ow. Acid treatment is the most commonand creates uniform fractures.

In�nite-conductivity fractures The �ow into the wellbore occurs only throughthe fracture. The fracture is highly conductive and considered as in�nite. Thereis no pressure drop from the tip of the fracture to the wellbore and, accord-ingly no pressure is lost in the fracture. Since the �ow in the wellbore occursonly through the fracture, the transient pressure behaviour includes three �owperiods: a fracture linear �ow, a formation linear �ow, and an in�nite-actingpseudo-radial �ow.

Finite-conductivity fractures The �ow and pressure is characterised bymeasurable pressure drops in the fracture. The transient pressure behaviourincludes four �ow periods: linear �ow within the fracture, bilinear �ow, linear�ow in the formation, and in�nite-acting pseudo-radial �ow. The conventionalhydraulic fracturing with a large quantity of propping agent maintain the fractureopen. The fracture permeability, kf , is lower than that of in�nite-conductivityfractures.The literature on vertical fractures goes back to Muskat (1937), and maybe

even further. Prats (1961) presented a solution to a cylindrical, homogeneous,isotropic reservoir with a vertical well intercepted by a vertical fracture with a�nite fracture conductivity. His work was based on the assumption of an incom-pressible �uid. This study introduced for the �rst time the idea of modelling afractured well with an unfractured well having a larger equivalent wellbore ra-dius. The e¤ective wellbore radius has been presented as a function of fracturelength and relative fracture capacity.Prats et al. (1962) extended Prats work to a compressible �uid depleted

through a constant-pressure or constant-rate production. This work assumed anin�nite conductivity vertical fracture that fully penetrated the formation in thevertical direction. It was found here that both the terminal rate and terminalpressure cases could be modelled by an elliptical reservoir with a larger �e¤ective�wellbore radius. Prats et al. (1962) solved the problem of an in�nite conductivityfracture producing from a reservoir that had an elliptical outer boundary. The"Laplace" space solution was expressed as a series of Mathieu functions. Thesefunctions always arise when solving unsteady problems in elliptical geometries.


Scott (1963) studied the transient behaviour of a single vertical fracture inter-secting a vertical well by means of a heat �ow analogue. The results lie betweenthe cylindrical well and a line source for a cylindrical well with a wellbore radiusof one forth of the total fracture length, thus con�rming previous �ndings. Morseand von Gonten (1972) investigated the behaviour of in�nite conductivity frac-tures prior to pseudo steady state. Their work involved the productivity indexratio between fractured cases as well as unfractured pseudo steady state cases.

Gringarten and Ramey (1973) presented the use of Green and source func-tions to solve variety of fractured well problems, and their conclusions wererepresented in a library of functions that facilitate the use of the technique. The�rst work on pressure transients in fractured wells using the Green�s functiontechnique is that of Gringarten et al. (1974). These authors posed the problemof an in�nite conductivity fracture producing at a constant rate as an integralequation. In this integral equation, the �ux distribution was the unknown andthe free space Green�s function was the kernel. The solution procedure consistedin discretising the equation in time and space, thus numerically obtaining the�ux distribution. Such a procedure has been used in numerous �elds and has be-come known as the boundary integral equation method. Gringarten et al. (1974)presented the �ux distribution for various times, but apparently did not use itto calculate pressures. Instead, they investigated the discretised equations anddemonstrated that a uniform �ux fracture evaluated at a certain point along thefracture (they gave x = 0:732 Lf) equals in�nite conductivity fracture whetherat early or late times. Kuchuk et al. (1991) indicated that this �pressure pointmethod�will not capture the character of the in�nite conductivity fracture atintermediate times.

Cinco-Ley et al. (1978) provided a semi-analytical solution to a homoge-neous, isotropic, slab reservoir with a vertical well crossed by a vertical, �niteconductivity fracture. The results showed that, for times of interest, the wellborepressure solutions can be correlated to a single parameter (i.e., the dimensionlessfracture conductivity). The �ux within the fracture was found to depend on thefracture conductivity. For large fracture conductivities, approximately 67 % ofthe total �ow originates from the far end of the fracture while for low fractureconductivities, about 70 % of the total �ow comes from the half of the frac-ture nearest to the wellbore. The pressure distributions within the fracture werefound to depend on the fracture conductivity; i.e., the larger the conductivity,the smaller are pressure drops. Also for the fracture permeabilities close to thereservoir permeability, the fracture pressure drop corresponds to that of radial�ow. Furthermore, it was reported that the uniform �ux solution behave likean in�nite conductivity fracture at early times, then like a variable conductivityfracture at medium times and �nally approached a �nite, constant dimension-less fracture conductivity at late times. Cinco-Ley et al. (1978) used a versionof the boundary integral method to evaluate the pressure response of a �nite


conductivity fracture of rectangular cross-section. The �nite conductivity modelaccounts for pressure drops along the fracture. The procedure in question em-ployed the integral of Gringarten et al. (1974) to represent reservoir pressure anda second integral to account for fracture pressure. Equating these two integralsat the fracture face resulted in an integral equation that could be discretised andsolved numerically. The use of this technique provides more accurate pressureresults than the �nite di¤erence method used by Agarwal et al. (1979) or the�nite element treatment of Barker and Ramey (1978).Kucuk and Brigham (1979) used the approach given by Tranter (1951) to ex-

press the solution of an elliptical wellbore producing at a constant rate from anin�nite system. They investigated the in�nite conductivity fracture responses forthe fracture producing at the constant pressure and the constant rate. Agarwalet al. (1979) addressed the problem of a �nite conductivity fractures intersect-ing a vertical well being produced at constant rate or constant pressure. Theywere able to solve the di¤usion equation numerically. The authors used a two-dimensional, quarter of a square model. The reservoir simulation model wasre�ned at the wellbore, fracture tip and parallel to the fracture face.Cinco-Ley and Samaniego (1981) have reviewed the concepts of bilinear, lin-

ear and pseudo-radial �ow �nite conductivity, fractured wells. Their work alsoaddressed the e¤ects of wellbore storage. Di¤erent type curves are presentedto facilitate the analysis of fractured wells and several scenarios are given forlimited available well test data. The estimation of parameters, the uniquenessof the solution and the limitations encountered in each of these scenarios werediscussed. This work re-evaluated the e¤ective wellbore radius as a function ofthe dimensionless fracture conductivity.Uniform �ux �ow model is similar to the in�nite-conductivity �ow model.

The di¤erence only occurs at the boundary of the fracture. Among the variousmodels, that of Sheng-Tai and Brockenbrough (1986) provided an approximateanalytical solution for �nite-conductivity vertical fractures. The solution for thismodel account for the e¤ects of skin, wellbore storage and fracture storage inearly time.The use of the "Laplace" transformation eliminates the time integral, thereby

leaving only an integral in space. The "Laplace" transform has been used invertical fracture problems from 1987; in fact, the �rst evaluation of the uniform�ux solution in "Laplace" space was given by Kuchuk (1987).Papatzacos (1987) presented the solution for reservoir pressure for an in�nite

conductivity fracture producing at constant rate. He approached the problemthrough the integral formulation of Gringarten et al. (1974), and the exactsolution was derived by means of a Mathieu function expansion of the kernel ofthe integral. Papatzacos (1987) stated that the di¤erences between the exactsolution and the uniform �ux approximation were as large as four percent.Cinco-Ley et al. (1987) focused on analyses of wells with low fracture con-


ductivities (i.e., FCD < 0:1). Their work presented three important concepts: (1)the equivalent wellbore radius rw for FCD < 0:1 is not a function of the fracturelength, (2) given a value of kf wk , there is no increase in well productivity, and(3) for a low fracture conductivity, there are only three �ow regimes: bilinear,transition and pseudoradial. The paper presents a type curve and equations forthe estimation of fracture conductivity, the formation �ow capacity (kh), theequivalent wellbore radius and the skin factor. Finally, the paper put forwardthe idea, that in very low dimensionless fracture conductivities, up to 99 % ofthe wellbore �ow comes from the 33 % fracture length closest to the wellbore.The boundary integral method solution of the �nite conductivity problem

in "Laplace" space, for the case of double-porosity reservoirs, was obtained byvan Kruijsdijk (1988) and Cinco-Ley and Meng (1988). The most accurate ap-proximate model to date is that of Wilkinson (1980). This model neglects all ofthe �ow in the reservoir that is not adjacent to the fracture; the reservoir thusbecomes an in�nitely long, strip of �nite width for which a two-dimensional �owis allowed but the sides are closed to �ow. The solution was presented in termsof a "Fourier" cosine series. Wilkinson then combined this solution with thein�nite conductivity solution to obtain an approximate well solution. This wasfound to work well for high conductivity fractures, whereas a correction termwas required for low conductivity fractures.Riley et al. (1991) investigated the pressure solutions for a �nite conductivity

fracture with an elliptical cross-section. The main conclusion of this work wasthat the behaviour of an elliptical fracture was essentially the same as that of arectangular one.It is not our intent to give an exhaustive account of the literature, but rather

to highlight the studies that consider fracture �ow modelling. Cvetkovic (1992)reviewed in�nite conductivity and uniform �ux solutions and a review of greatdetails is given by Villegas (1997).

2.1.4 Horizontal-Fractured Well

Hydraulically fractured horizontal wells represents a proven technology for pro-ducing oil and gas from tight formations. Thus undeveloped low permeabilityreservoirs can be produced. Induced hydraulic fractures reduce well drawdown,and increase the productivity of horizontal wells by increasing the surface-areain contact with a formation making fracture as a high conductivity path to aformation. Hydraulic fractures, depending on in-situ stress orientations, can beeither parallel or perpendicular to the horizontal, longitudinal or transversal wellaxis. The question is how to drain a reservoir, what is, the number of hydraulicfractures and how to design an e¢ cient well spacing.A literature survey shows numerous analytical solutions for multi-fractured

horizontal well systems considering single-phase �ow. Although exact, solutionsobtained with analytical tools are limited and restricted to the scope of assump-


tions and simpli�cations imposed to describe the system. Few attempts havebeen made to employ numerical solutions. The di¢ culties are related to nu-merical instabilities, especially during the transient period, due to the use ofexcessively �ne grid blocks when representing the fractures. Several authorshave contributed with solutions of coupling a well with fractures to a reservoir.For instance Soliman et al. (1990) investigated pressure-transient analysis of awell with fractures.

In 1991, Mukherjee and Economides presented a parametric comparison be-tween fractured vertical wells and horizontal fractured and unfractured verticalwells. This work demonstrated how to calculate a minimum number of transversefractures in a horizontal well, and work uses simple relationships to treat trans-verse fractures as vertical fractures in vertical wells with an additional pseudo-skin. However, the study does not account for the interference between transversefractures and is therefore only valid for very short times. In 1991, Economideset. al., presented results from a numerical simulation study re-evaluating theproductivity expression for an unfractured horizontal well and extending it toanisotropic formations. Longitudinally fractured horizontal wells are simulatedshowing productivity indices for isotropic formations and for several values offracture conductivity. Transverse fractures are simulated and the numerical re-sults con�rm the analytical expression previously presented.

Hareland and Rampersad (1995) have put forward the fractured well perfor-mance model, being steady state and in which the variations in time are handledby using the production, e.g., a single time interval (as one day) to re-estimate anew �initial�pressure. This was followed by calculating the production of the nexttime period, etc. The include it being model limitations a 2-dimensional model,made quasi-3D (by using a technique originally justi�ed through an appeal tolaboratory experimental results in electrostatics). Interferences between frac-tures were absent. On the contrary, the given solutions are based on (arti�cial)prolongations of the fractures into no-�ow �walls� connecting to the outer no-�ow boundary. Only a very specialised geometry (reservoir/fractures/wellbore)is treated, and moreover, �uid �ow to the wellbore, both from the formationand from a fracture is forced to be linear/radial/uniform at �xed distances. Itse¢ cient numerical realisation has been acknowledged through the innovative useof Gauss-Jacobi quadrature methods.

Hegre and Larsen (1995) have reported on results with a theoretical basis,mainly presented in other papers. Their investigations concerned pressure tran-sient analysis of multifractured horizontal wells. The approach is a classicalsemianalytic one, where "Laplace" transformed pressure equations are solvedanalytically and then numerically inverted to real time by use of the Stehfestalgorithm. Although this was a very solid piece of work, its limitations, throughthe simplifying assumptions made in order to obtain a manageable system, needto be pointed out here. We would speci�cally like to comment on the discreti-


sation of �nite conductivity fractures used, in the manner of solving the inte-gral equations by the Cinco-Ley and Samaniego (1981) method. Here, no-�owedges (tips) of fractures were assumed which has become common in this kindof modelling. Speci�c assumptions are also made of simpli�ed �ow regimes inthe fractures. Flow directly to the wellbore was generally not treated.Raghavan, et al. (1994) presented a study explicitly treating a simple in�-

nite slab reservoir with longitudinal and fully penetrating fractures, for whichalso multiple perforations were allowed along the well. With proper alterations,transverse fractures may be treated as well. All fractures are produced at acommon wellbore pressure. Referring to one of the authors, the solution methodhas been found to be too slow for practical purposes. Thus, either a change ofmethod or a change of model, or both, would seem necessary. The basic methodrequiring excessive computer time in this model is used for treating discretised�nite conductivity fractures.A further contribution is made by Horne and Temeng (1995). In 1996, Valkó

and Economides presented the �rst semi-analytical solution for longitudinallyfractured wells in isotropic formations. This work shows that the productivityof a fractured horizontal well can be three to �ve-fold that of a fractured verticalwell. An economic analysis reveals that longitudinally fractured horizontal wellsare competitive in isotropic formations. Moreover, the publications by Raghavanet al. (1997), Cvetkovic et al. (1999), Wan and Aziz (1999), Cvetkovic et al.(2000, 2001), Al-Kobaisi et al. (2006), and Medeiros et al. (2007) contributeswith new solutions to a coupling of a horizontal well with fractures to a reservoir.

2.2 Gas Flow

2.2.1 Vertical Well

The very �rst solutions of a gas di¤usion equation were derived assuming smallpressure gradients and constant gas properties. The variation of gas and rockproperties with pressure was ignored due to analytical di¢ culties. The gas �owbehaviour is most accurately described using the real gas pseudo-pressure (orreal gas potential), since it takes into account the variation of gas viscosity andgas deviation factor as a function of the pressure. If one assumes the p

�Zproduct

to be constant, one obtains a solution in terms of pressure. This solution isappropriate only at higher pressures i.e., over a limited pressure range, of 3000psi (or 210 bar). At high pressure, gases presents a behaviour that is similar tothat of oil. The pseudo-pressure varies linearly with pressure, and the pressure-squared solution assumes the �Z product to be constant. It is only appropriateto use at lower pressures, such low pressure of less than 2000 psi (140 bar).To overcome the limitations of these solutions, Al-Hussainy et al. (1965)

proposed the use of the real gas pseudo-pressure, a term they de�ned as:


Figure 2.1: The domain in which the pseudo-pressure, ; varies linearly with pand p2[After Bourdarot (1998)].

(p) = 2

pZpb

p

�(p)Z(p)dp (2.43)

It has been demonstrated that the gas �ow behaviour can be most accurately

described using the pseudo-pressure function, (p), which takes into account thevariability of gas viscosity and the gas deviation factor as a function of pressure.The application of the real gas potential reduced a rigorous partial di¤eren-

tial equation for the transient �ow of real gases to a quasi-linear �ow equationwithout assumptions of small pressure gradients or a slow variation of the gasviscosity and gas deviation factor with pressure. Although the use of real gaspotential did not fully linearise the di¤usion equation, and the fact that thehydraulic di¤usion had to be assumed constant in order to arrive at solutions,the authors justi�ed its use for small �ow rates and small production times withcomparisons to numerically simulated data. According to Chien (1993), afterthe pressure had declined more than 10 percent from its initial value, however,the solutions started to deviate signi�cantly. In order to adjust the solutionsafter a reduction of the pressure by more than 10 percent, Kacir (1990) appliedan approach to reset bounding rates and times periodically for changes in theterm of the gas viscosity multiplied by the compressibility term. The results forcalculations of the �owing bottomhole pressure were satisfactory for a pressuredecline down to 70 percent of the initial reservoir pressure. Such adjustments orresets were however not based on any mathematical formulation. Furthermore,the pressure solution started to fall o¤ at lower pressure values. In 1990, Pratsintroduced a new form of the real gas potential to reduce the nonlinear gas �owequation to a quasi-linear di¤usion equation. Considering the hydraulic di¤u-


sivity, as a function of pressure the author successfully transformed the real gasdi¤usivity equation into a solvable form. Another formulation of the di¤usionequation for general �uid �ow was proposed in terms of a porosity-density prod-uct with the intent to account for pressure-dependent �uid and rock propertiesby Fair (1992).Due to the highly nonlinear variation of gas density and viscosity with re-

spect to pressure, no analytical solution to the real gas di¤usion equation hasever been presented in the literature. Analytical solutions used in gas well test-ing and pressure analysis are based on idealised assumptions, such as small andconstant gas compressibilities and constant hydraulic di¤usion. These solutionsthough widely used and easily applied, are inaccurate. They are neither ap-plicable to a broad range of pressure changes nor to di¤erent �ow periods. Asdiscussed in the literature, by Chien (1993) these solutions start to deviate sig-ni�cantly after the pressure has declined more than 10 percent. Moreover, avariety of approximate analytical solutions are used for various �ow periods, dueto a wide range of time and boundary conditions. All of these limitations arecaused by the inability to analytically solve the more general nonlinear gas �owequation. In 1993, Chien presented a new real gas potential that was rigorouslyimplemented. Moreover, a general solution with pressure-dependent �uid androck properties were analytically derived from the nonlinear gas �ow equation.The obtained solution was more accurate than those available in the literatureand also applicable to a wide pressure range.

Gas Di¤usion Equation The principle of conservation of mass for isothermal�ow of a single �uid through a porous medium, assuming negligible gravity e¤ectsis expressed by the continuity equation as:

r(�!u) = �@(��)@t

(2.44)

The �uid �ux is given by Darcy�s law for laminar �ow as:

!u = �k(p)

�(p)rp (2.45)

Consider �(p), �(p), k(p) and �(p) to be functions solely of pressure, and sub-stitute Equation (2.45) in Equation (2.44). This yields:

r(�(p)k(p)�(p)

rp) = �@(�(p)�(p))@t

(2.46)

Equation (2.46) is the most general form of the di¤usion equation describing anunsteady state �ow of �uid in porous media. A classical solution to the di¤usionequation is for single-phase, one-dimensional radial �ow to a well in an in�nite


reservoir. The one-dimensional radial form of Equation (2.46) in a cylindricalcoordinate system is:

1

r

@

@r

�r�(p)k(p)

�(p)

@p

@r

�=@(�(p)�(p))

@t(2.47)

The initial conditions set an initial reservoir pressure everywhere in the system:

IC p(r; t = 0) = pi

while the �rst boundary condition makes sure that the system remains in anunsteady-state �ow condition:

B:C:1 p(r !1; t) = pi

The second boundary condition states that the �owmust approach a steady-statecondition as the �uid approaches the in�nitely small wellbore:

B:C:2

�r@p

@r

�r=rw

= � qw�(pw)

2�hk(pw)

In order to transform Equation (2.47) into a di¤usion equation and to preserve�(p), �(p), k(p) and �(p) as functions of pressure during the entire derivations,Prats (1990) de�ned a new real gas potential as:

m(p) =�(pb)

�(pb)k(pb)

pZpb

�(p)k(p)

�(p)dp

where pb is a low base pressure (Chien (1993) used pb = 14:7 psia).To formulate a mathematical model for real gas �ow, the following assump-

tions were made: isothermal �ow; negligible gravity e¤ects; laminar Darcy �owthrough porous media; �owing gas of constant composition; single phase �ow;homogeneous porous media; constant reservoir thickness, isotropic porous me-dia. After taking derivatives on pressure and time and introducing the isothermalcompressibility of gas cg(p) = 1

�(p)�(p)d(�(p)�(p))

dp, the di¤usion equation in terms of

m(p) is equal to:

1

r

@

@r

�r@m(p)

@r

�=�(p)�(p)cg(p)

k(p)

@m(p)

@t(2.48)

The initial and boundary conditions are:

IC m(p)(r; t = 0) = m(pi)

B:C:1 m (p)(r !1; t) = m(pi)


B:C:2

�r@m(p)

@r

�r=rw

= �qw�w2�h

�(pb)

�(pb)k(pb)

It is possible to extend the above solutions for an anisotropic porous media withthe variable transformation developed by Caudle (1967). An analytical solutionof Equation (2.48) with a special case of constant ��cg

k, has been presented by

Al-Hussainy and Ramey (1966). Equation (2.48) cannot be solved directly sincem(p) is a function of pressure, which in turn is a function of both time andposition. In 1993, Chien transformed the partial di¤erential Equation (2.48)into an ordinary di¤erential equation by substituting variables according to aprocedure known as the Boltzmann transformation.

2.2.2 Vertical-Fractured Well

Finite-di¤erence reservoir simulation models can be used to history-match theproduction performance of fractured wells. These models are often are cum-bersome, requiring enormous amounts of data preparation and analysis time toobtain reasonable matches of the production performance of a fractured well.So, and alternative lies in modelling a fractured well with the semi-analyticalmethods.Prats, in 1961, presented a solution to a cylindrical, homogeneous, isotropic

reservoir with a vertical well intercepted by a vertical fracture with �nite frac-ture conductivity. This work was based on the assumption of an incompressible�uid, and showed the pressure distribution inside the fracture and in the reser-voir; thereby providing tools to calculate the productivity improvement due toa fracturing job and to analyse the e¤ects of fracture face damage. The e¤ec-tive wellbore radius was presented as a function of fracture length and relativefracture capacity; a parameter that is proportional to the inverse of the dimen-sionless fracture conductivity (i.e., a = �=2=FCD). It is demonstrated thatthe e¤ective wellbore radius decreases along with the fracture conductivity. Thee¤ective wellbore radius was shown to vary from a maximum of 0.5Lf for an in-�nite conductivity fracture to a minimum of rw for a fracture conductivity equalto w=Lf .In 1962, Prats et al., extended the earlier work of Prats to a compressible

�uid depleted through constant pressure or constant rate production. This workassumed an in�nite conductivity vertical fracture that fully penetrated the for-mation in the vertical direction. Moreover, it considers a maximum fracturepenetration or partial length of 50 % of the radius of investigation. It was foundhere that both the terminal rate and terminal pressure cases can be modelled byan elliptical reservoir with a larger �e¤ective�wellbore radius.In 1964, Russell and Truitt obtained a numerical solution to the case of a

fractured vertical well with an in�nite conductivity fracture in a square reservoir.


The reservoir was closed and was depleted at constant production rate. Theirwork shows that a transient �ow regime is characterised by a region near thefracture where the �ow is linear, and a region away from the fracture where the�ow is pseudo-radial (elliptical). The wellbore pressure behaviour during thepseudo-radial regime was shown to depend greatly on the partial fracture lengthor penetration. The larger the fracture penetration, the closer the performanceapproaches that for a pure linear �ow. The e¤ect of the in�nite conductivity frac-ture during pseudo-steady state can be represented by an equivalent reservoirthat has a wellbore radius of x=2. Additionally, this work numerically corrobo-rates previous observations on analogues and analytical solutions for linear �ow.Moreover, it demonstrates that in�nite conductivity fractured wells with a smallfracture penetration (i.e.< 0:1) can be analysed with radial unfractured modelswithin a 10 % error.In 1969, Wattenbarger and Ramey extended the theory of fractured wells to

fractured gas well testing including wellbore storage and turbulence. The workuses numerical methods similarly to those of Russell and Truitt, and conformalmapping similarly to that of Prats. Here, in�nite conductivity fractures in anin�nite reservoir were considered. The study gives a good explanation on treatingturbulence, fracture face damage and fracture length as pseudo-skins.In 1972, Morse and von Gonten investigated the behaviour of in�nite con-

ductivity fractures prior to pseudosteady-state. Their investigation revisits thework by Russell and Truitt and presents it in terms of productivity indices.The productivity index ratio between fractured cases and cases of unfracturedpseudosteady-state cases a decrease with time until stabilisation. The productiv-ity index ratio increases very rapidly as the partial fracture length, Lf

Leincreases.

A two-dimensional numerical simulation is run to constant pressure depletion,and the results reveal again that, the larger the L

Le, the larger the increase in the

productivity index ratio.

2.2.3 Horizontal-Well

The development of low permeability gas reservoirs, conventional or unconven-tional, is one of the solutions to the energy supply and demanding problems oftoday. In low-permeability gas reservoirs, the creation of a �ow path is critical,and horizontal wells have been extensively used to increase the reservoir contactarea. Hydraulic fracturing can further expand the contact between wellbores andformations. For horizontal wells, both with and without hydraulic fracturing,the well performance becomes very sensitive to permeability and the anisotropicratio when the reservoir permeability is low. If the vertical permeability in theformation is extremely low (high anisotropic ratio), then the bene�t of horizontalwells starts to diminishing. In such a case, hydraulic fracturing provides anotheroption to increase well productivity. When hydraulically fracturing a horizontalwell, created fractures can be single longitudinal, multiple longitudinal, single


transverse, or multiple transverse. The orientation and placement of fracturesalong a horizontal well greatly a¤ect its performance. Depending on the for-mation condition and fracturing design, a fracture may in some cases result inun�avoured productivity, which has been evidenced in the �eld. Predicting wellperformance for fractured and non-fractured horizontal wells can help to obtainthe best estimates of production from low permeability gas formations.In the early 1980�s, major production successes through horizontal wells were

reported at the Prudhoe Bay �eld and the Rospo Mare �eld, o¤shore Italy. Thereported increase in production was on the order of at least two to three timesthe equivalent production of vertical wells. The Rospo Mare �eld happens tobe the ideal application of horizontal wells because of its producing formationtype. Giger et al. reported that the Rospo Mare pay consists of karsts madeup of very-low-permeability, compact carbonates. The oil resides mainly in thefractures and vugs of the karstic matrix system. A horizontal well is more aptto intersect many of these discrete natural fractures or vugular systems in suchformations.Recently, with the improvement in horizontal well drilling and completion

technology, the feasibility of horizontal wells is seriously considered for suchdi¤erent reservoirs as the naturally fractured Austin chalk formations, the low-permeability Spraberry formations in west Texas, the Hugoton formations inthe Kansas/Oklahoma region, and the naturally fractured Bakken formation inthe Williston basin. Improvements in technology and operating procedures havealso resulted in a substantial cost reduction. Wilkinson et al. (1980) reporteda reduction in cost per foot of horizontal wells on the order of 40% based onthe average cost of the original three horizontal wells drilled at the Prudhoe Bay�eld. Drilling costs, however, are still reported to be 1.3 to 2 times higher thanfor comparable vertical wells. Abdat (2000) reviewed selected papers of transientbehaviour of a horizontal well.

2.2.4 Horizontal-Fractured Well

The evaluation of multifractured horizontal well performance, or the selection ofan optimum perforation/stimulation design for such wells, may be approachedthrough �ne grid reservoir simulations. However, while reservoir simulation isthe most advanced method of predicting well performance, it is often too timeconsuming to be used for a parametric screening studies. The data requiredis often unavailable and the e¤ort may be unwarranted. As an alternative tosimulation, the application of semi-analytical models can readily yield wellboreresponses to various boundary conditions. Frequently, this is su¢ cient to providean understanding of the factors with the most in�uence on well performance. Ifsimulation work is warranted, it can then proceed with the insight obtained fromthe analytical models.Tight gas reservoirs are becoming increasingly popular candidates for mul-


tifractured horizontal wells. An accelerated production accompanied by moremoderate recovery increases, pays for the initial capital expense, especially ashorizontal drilling and stimulation costs continue to decline. The multi-fracturedhorizontal well technology has recently been applied to enhance productivityfrom a tight gas �eld located o¤shore from the Netherlands. A horizontal wellwith two hydraulic fractures was completed in the tight (permeability = 0.2-1.0 md) Ameland East reservoir, which constitutes a classic example of how apoor candidate for horizontal wells can yield a substantially improved produc-tion when produced from a multi-fractured horizontal wellbore. The reservoirexhibits a �low ratio�of vertical to horizontal permeability rendering the non-stimulated horizontal well uneconomic. Simulations of vertical in�ll wells andvarious combinations of multi-fractured horizontal wells, combined with eco-nomic evaluations, have demonstrated that the case of a horizontal well withtwo hydraulic fractures provided the best economic return. The actual produc-tivity improvement of this well, over the horizontal well with no fractures, isestimated to be a factor of four.For wells with hydraulic fractures, Prats (1961) started working on analytic

in�ow performance correlations for a single fracture. Van Kruijsdijk (1988) useda combination of "Laplace" transformation and a Boundary Element formula-tion to model the transient response in fractured reservoirs. This model was laterextended to include tight gas reservoirs, where non-Darcy �ow in the fracturemust be taken into account. Kuppe and Settari (1996) have performed a num-ber of reservoir simulations to cover multi-fractured reservoirs, and to provideengineering correlations for variety of scenarios.

Methods for Predicting Productivity

With the increasing use of fractured and multi-fractured horizontal wells, itseems appropriate to expect a more accurate method for determining the po-tential productivity index enhancement of these wells. The manner in which aproductivity index is calculated, with consideration taken to a range of variables(i.e., fracture height and length, well length, reservoir and fracture permeability,etc.), could in�uence the size of hydraulic fractures, the number of fractures inthe horizontal wellbore or whether or not a horizontal well should be drilled. Asmentioned previously, the use of the available analytical solutions, for fracturedor unfractured horizontal wells, could lead to errors if the limiting assumptionsare not taken into consideration or overlooked. There is a growing trend tomarry numerical simulation technology with analytical solutions. Improvementscan be made to existing analytical solutions by comparing their predictions tonumerical simulation results. Economides et al. (1991): used a simulator witha ��exible grid scheme� (i.e., not following standard Cartesian orthogonality)to modify Joshi�s solution in anisotropic permeability conditions. The originalform of one version of Joshi�s equation was:


qH =2�khh�P

�B

�ln

�a+pa2�(L=2)2

L=2

��+ �h

Lln��h2rw

�where: L = well length, a = large half-axis of elliptical drainage area, � =pkh=kv.

Tight Gas Reservoirs

In 1998 El-Banbi presented a collection of models and solutions useful for analysingpressure and production data of tight gas reservoirs. An investigation was carriedout of the linear �ow since many tight gas wells produce predominantly underlinear �ow conditions for long times. The causes behind linear �ow in tightgas reservoirs are numerous. Among these can be mentioned: linear reservoirs;high permeability streaks; wells between two no-�ow boundaries; transient dualporosity behaviour for radial reservoirs; wells intercepted by vertical, horizontal,or diagonal fractures; horizontal wells; and horizontal wells with fractures. Lin-ear reservoirs are those that show predominantly linear �ow because of the shapeof the reservoir. The reservoir would impose one-dimensional linear �ow. Thissituation may occur in vertically fractured vertical wells whose fractures extendlaterally to the reservoir boundaries. It may also occur in horizontal naturalfractures and high permeability streaks. In this case, the linear �ow developsin the vertical direction. Such reservoir con�gurations may give rice to a linear�ow from the start of production. Linear �ow may also persist for long timesbefore any boundary e¤ects are felt.

2.3 Multiphase Flow

It is of common interest to describe multiphase �ow in a reservoir. It is how-ever di¢ cult to obtain a simple solutions of �ow within a reservoir as equationsdescribing multiphase �ow are highly nonlinear. Muskat and Meres (1936) for-mulated the fundamental equation that governs the multiphase �ow in porousmedia. The major contribution is the extension of Darcy�s law from single phaseto multiphase �ow problems. This was possible due to the key concept of e¤ec-tive (or relative) permeability. Kato and Serra (1991) commented on di¢ cultiesin characterising the reservoir parameters under multiphase �ow, according towhich there do not exist very many publications on multiphase welltest analysis.Perrine (1955) was able to modify modify single phase solution with a pressureapproach. Based on empirical observations, the single phase properties (mobility,compressibility) could thus be replaced by total system properties. Furthermore,it was possible to estimate e¤ective phase permeabilities (not absolute perme-ability) and wellbore skin. Martin (1959) showed that Perrine�s approach was


based on the pressure di¤usion equation derived when assuming negligible pres-sure and saturation gradients. Perrine�s approach was further investigated byWeller (1966), Chu et al. (1986) and Ayan and Lee (1988) and it has remainedthe most commonly applied approach. Fetkovich (1973) intuitively contributedin understanding the multiphase �ow, and Raghavan (1976) suggested pseudo-pressure for solution gas drive reservoirs. This approach is analogous to the oneproposed by Al-Hussainy and Ramey (1966). The pseudo-pressure approach wasfurther elaborated by Aanonsen (1985), who studied non-linear e¤ects of solutiongas-drive reservoirs and noticed that small inaccuracies in relative permeabilitydata greatly in�uence the exactness of the pseudo-pressure approach. Bøe etal. (1989) used a similarity transform (the Boltzman transform) to solve radialproblems in well test analysis under multiphase �ow. Al-Kalifah et al. (1987)derived a di¤usion equation for multiphase �ow with pressure squared, p2, as thedependent variable.

2.4 Flow Under Variable Rate and Pressure

Ilk et al. (2006) sorted the references involving methods for variable-rate reser-voir performance into the following categories:� Superposition and Convolution� Rate Normalization and Material Balance Deconvolution� DeconvolutionThe transient �ow rates are generated by a constant �owing pressure of the

well bottom hole and analysed by decline curve analysis. In reality, due tochanges in operating procedures, the downhole �owing pressure seldom remainsat a constant level over a long period of time. Recently, the deconvolutiontechnique, has become employed for well testing converts measured transientpressure due to variable sand-face rate into the transient pressure response as aresult of equivalent constant �owing rate. This technique can also be applied totransient �owing rate analysis. Since the wellbore pressure usually varies whendeconvolved, it appears to be equivalent to a constant pressure. Therefore,deconvolved constant pressure at a wellbore is able to generate rate-time wellresponses.The basic assumption of all deconvolution techniques resides in the consis-

tency of the measured pressure and rate data with the linear Duhamel model,which is based on the principle of superposition. The approach is limited due tolinearity of the system in which only one well disturbance appears. Moreover,the technique cannot be applied if nearby wells cause interference in pressurein a system. Further deconvolution cannot apply if well pressure behaviour isin�uenced by aquifer or gas cap. Additional requirements for linearity of thesystem include the single-phase �ow, which signi�es that deconvolution appliesto pressures above bubble point pressure in oil reservoirs. Finally the initial


Table 2-2: Variable rate publications the history [After Gringarten (2006)]


uniformity of pressure, within the whole investigated part of the reservoir andwell rate from the entire production by this well, must be satis�ed all the wayfrom the initial equilibrium state.The existing deconvolution methods can be classi�ed into spectral and time

domain techniques. The spectral methods are based on the convolution theo-rem of spectral analysis, and the convolution product is obtained by applying aspectral transform such as the "Laplace" or "Fourier" transform. Kuchuk andAyestaran (1985), Roumboutsos and Stewart (1988), Cheng et al. (2003), andIlk et al. (2006) have all applied the spectral method. The time domain meth-ods discretise the convolution integral using an interpolation scheme and thenproceed to solve the linear system. A set of smoothing constraints were imposedon the solution when reducing the solution oscillation. The time domain methoddeconvolution algorithm was recently published by von Schroeter et al. (2004).His method works in a time-domain when a reasonable level of noise is presentin both pressure and rate data. In 2005 Levitan improved the algorithm.Zheng and Fei (2008) created a deconvolution algorithm and code, which were

only tested with single phase oil data. They considered both pressure-rate andrate-pressure deconvolution. Generally deconvolution methods applied to thereservoir system are given by Duhamel�s integral or principle of superposition,as a function of time. The pressure drop across the reservoir corresponds to theconvolution product of rate and reservoir response as given below:

�p(t) = pi � p(t) =

tZ0

q(�)g(t� �)d� (2.49)

there, q(t) is the measured �ow rate, p(t) the pressure at the wellbore, and pithe initial pressure. Equation (2.49) is referred to as the impulse response ofthe reservoir system. With the deconvolution, or an inversion of the convolutionintegral, it is possible to estimate the reservoir system response. Two types ofdeconvolution are related to pressure-time and rate-time analyses.Well testing deals with pressure-rate deconvolution. The reservoir system

responds with transient pressure due to wellbore constant-rate conditions asdescribed with the following convolution integral:

�p(t) = pi � p(t) =

tZ0

q(�)dpur(t� �)

dtd� (2.50)

where q(�) is the measured �ow rate, p(t) is the measured bottomhole pressure,pi is the initial reservoir pressure and pur is the unit-rate pressure response. Weare interested in rate testing that deals with rate-pressure deconvolution in whichthe unit constant pressure transient rate response of the reservoir system is givenby the following convolution integral:


q(t) =

tZ0

qup(t� �)d�p(�)

d�d� (2.51)

there qup is the transient rate response of the reservoir system obtained whena well produces under unit constant pressure conditions. Multi-phase �ow andmulti-well interferences need further investigations and new algorithms.

2.5 Other Transient Models

2.5.1 Multilateral Model

When obtaining solution for both vertical and horizontal wells the line sourcemethod has been commonly used. The study of well behaviour by solving di¤u-sion equations is di¢ cult due to a di¤erence in scale between the wellbore diam-eter and the size of a reservoir size (scale of 105). The well behaviour was solvedwith the a semi-analytical source function method, �rst presented by Carlslawand Jaeger (1959) and Gringarten and Ramey (1973). The following authorshave since then studied pressure transient response for a horizontal well with thesource function method: Clonts and Ramey (1986), Goode and Thambynayagam(1987), Daviau et al. (1988), Ozkan et al. (1998), Rosa and de Carvalho (1989),and Odeha and Babu (1990). A further extension of the method to advancedwell studies has been performed by Besson (1990), Economides et al. (1994),Azar-Nejad et al. (1996), Jasti et al. (1997). In 1998 Ouayang et al. appliedthe line source approach to model �nite conductivity wells.Multi-lateral wells are increasingly used in reservoir engineering to improve

the oil recovery. To optimise the e¢ ciency of these wells, it has been necessary todevelop semi-analytical methods which are simple, fast and accurate for study-ing the transient pressure behaviour. The pressure drop in the wellbore has astrong impact on the pressure transient behaviour. Ding (1999) presented theboundary integral equation, based on a single layer heat potential, to describethe transient phenomena This approach included the pressure drop along thewell length. A mathematical method of Galerkin-type was employed to solvethe boundary integral equation leading to a quick and accurate evaluation ofthe pressure drop along the length. The method was used to study the pressuresolution for multilateral wells and also for selectively perforated wells. Boundaryintegral methods (BIEs) represent an alternative for determining the advancedwell behaviour. Scale problems between the wellbore diameter and the reservoirsize could be solved by representing the equation at the wellbore boundary andthe reservoir boundary. Among the authors that have used BIEs to study the�ow behaviour by applying the integral equation to the reservoir boundary canbe mentioned Kikani and Horne (1993), Sato and Horne (1993), Pechera and


Stanislav (1997), Oguztorelli and Wong (1998) and Jongkittinarukorn and Tiab(1998). In 1998 and 1999 publications Ding applied the BIE to the wellboreboundary. This approach permitted an accurate modelling of pressure and �owin the near well region. Furthermore, in 1999, Ding published more accuratenumerical techniques in order to solve the BIE for advanced well modelling. Thehigh-order Galerkin approach was used for space discretisation, and, comparedto the line source method, the linear Galerkin approach. The study includedcalculations of the pressure drop along the wellbore. For each well in a clusteredsystem, both inner boundary conditions of pressure and rate were imposed. Itwas possible to obey the coupled modelling of the reservoir and wellbore �owfor the single phase �owing in a homogeneous anisotropic media. The transientpressure behaviour of an advanced multilateral well included calculations of thepressure drop along the well length. The transient pressure behaviour and in�owdistribution can be calculated with any well con�gurations by for each well im-posing both the rate and bottom hole pressure. The advantage of the modellingapproach was to validate the well modelling features in a reservoir simulator.Calculations of the numerical PI and transmissibility in the vicinity of a wellwere performed to improve the well modelling in a reservoir simulator. Furthermodel extensions were considered to a multi-layer reservoir with an arbitraryreservoir geometry by applying the boundary integral equation to the reservoirboundary and to the interfaces between layers.

Ozkan et al. (1998) investigated the transient pressure behaviour of dual-lateral wells. The in�uences of the length, phase angle and vertical and horizontalseparations of the laterals were discussed, and the in�uence of anisotropy in thehorizontal plane on dual lateral well responses were determined. The e¤ect of thehorizontal anisotropy may reduce the e¤ective total length, and results indicatedthat the best dual-lateral con�guration was obtained when opposing laterals aredrilled along the minimum permeability direction.

Umnauayponwiwat et al. (2000) studied the transient pressure responsesand in�ow performance of a multiple well. The analytical model developed byUmnauayponwiwat et al. (2000) evaluated the in�ow performance of multiplehorizontal wells in a closed system. Moreover, the transient model consideredthe �ow of a single phase liquid in a simple homogeneous and isotropic porousmedium. It was possible to simulate a mixture of vertical, fractured and hori-zontal wells, arbitrarily located in a reservoir, with varying production and shutin sequences. The model corresponds to a "Laplace" transformation domain andthe results are inverted into a time domain by the Stehfest numerical inversionalgorithm. The approach investigated the e¤ects of transient �ow periods onthe estimation of in�ow performances and the analysis of build-up responses ofhorizontal wells. A simple homogeneous and isotropic porous medium can beextended to a naturally fractured reservoir and anisotropy can be incorporatedinto the model.


2.5.2 Multiple Wells Model

Numerous solutions of the di¤usion equation were reviewed under various bound-ary conditions, providing rate responses of a producing single well. The con-ventional theories of transient pressure or rate analysis and in�ow performanceconsider a single well with �xed drainage boundaries, that can be constitutedof either the physical boundaries of the reservoir or the �xed boundaries result-ing from a stabilised �ow conditions (pseudo-steady state or steady �ow). In amulti-well production system, wells interfere with each other due to the tran-sient �ow conditions resulting from changing wellbore conditions. A multi-welllproduction system may be a mixture of vertical, horizontal, and fractured wellscreating a system with complex interactions among the wells.In a closed system with multiple wells, Onur et al. (1991) and Valko et

al. (2000) studied the most relevant items related to the pressure transientbehaviour and well in�ow performances. In 2000, Umnuayponwiwat et al. in-vestigated transient pressure behaviour of multiple wells in closed rectangularsystems. The main objective was to provide an understanding of the complexinteraction among wells in a multi-well system. The study involved the e¤ectof pressure transients due to changes in the production rates and estimations ofthe wells drainage areas.Fokker et al. (2005) presented a new semi-analytical method for calculating

the productivity of vertical, horizontal or multilateral wells draining either gasor oil reservoirs. They considered well interference e¤ects and the presence ofnatural or induced fractures. By introducing moving pressure boundaries theycalculated the pressure �eld in a three-dimensional reservoir containing multiplewells and �nite-conductivity fractures. Anyhow the moving-boundary approxi-mation of the transient pressure response had limitations in accuracy for smalltimes at large distances from the well. Thus, this approach was considered valu-able for screening purposes and for quick analysis. In their approach the solutionof a fully penetrating vertical well was further used for the complex well geome-try or fracture geometry. The moving pressure boundary was developed for thevertical well. Hence, for short times there is no in�uence of the boundaries ofthe reservoir, thus the reservoir can be treated as if it were in�nite. In 1978Dake provided the solution of the pressure behaviour within a reservoir with aproducing fully penetrating vertical well as

p(r; t) = pi �q

4��h

1Zx= r2

4Dt

e�sds

s(2.52)

further, simplifying they got

p(r; t) � pi �q

4��hln(e

r2

4Dt) = pi �

q

4��hln(

p4Dt=e

r) (2.53)


with � = k�, and D = k

��cand Euler�s constant = 0:5772. This pressure

approximative solution is based on the simpli�cation valid for small value of x,i.e., x < 0:01, where the integral in Equation (2.52) is equal to

1Zx= r2

4Dt

e�sds

s� � � ln(x) (2.54)

They calculated the time-dependent external boundary

re(t) =

r4Dt

e

by comparing the simpli�ed Equation (2.53) to the steady-state solution ofa vertical well with a drainage radius, re. Thus, by using a moving externalboundary it was possible to approximate the transient solution as to solution tothe steady-state problem.Busswell et al. (2006) presented novel analytical solutions to a single layer

model for a cuboid shaped reservoir by employing a method of integral trans-forms. The model solutions of a di¤usion equation apply to variety of boundaryand initial conditions. The well model comprises partially penetrating vertical,horizontal and fractured wells and provides solutions in multi-well and multi-rate scenarios. All three fracture conditions are implemented (uniform �ux,�nite and in�nite conductivity). The gas model also comprises non-Darcy �ow,wellbore storage, and a naturally fractured reservoir. Wellbore conditions in-clude both pressure and rate. Model solutions are constituted of "Laplace"space solutions and are inverted to real space. This method may solve problemswhere any permutation of the Neuman (�ux) and Dirichlet (pressure) and Rubin(�ux+pressure) are speci�ed over the multi-well closed box model boundaries.The key reference on integral transform methods is that of Thambynayagem(2006), unfortunately unavailable. Several other authors have recently presenteda multi-well concept that appears to be a research topic of great interest. De-spite the existence of many advanced reservoir simulation technology models,there is a need for an alternative analytical tool that is quick and at the sametime honours the physics of �uid �ow providing a broad understanding of thereservoir dynamics.Jordan et al. (2008) contributed by creating a simple method for predict-

ing the performance of multiple gas wells in complex reservoir shapes. Usingan approximation of the traditional "image well" method, pressure and produc-tion pro�les could be generated for arbitrarily shaped reservoirs. The inclusionof pseudo-time, which handles variable viscosity and compressibility (�ct), im-proved the quality of late-time forecasting of gas productions.

Chapter 3

DEPLETION RATE DECLINEREVIEW

Depletion rate decline also known as decline curve analysis, is a method formatching the observed production rates of an individual well, group wells orreservoirs by a mathematical function for reserve estimation and production fore-cast. In the 1950�s, Arps presented traditional decline curve analysis includingexponential, hyperbolic and harmonic methods. In the 1980�s, Fetkovich cre-ated type curves. He used constant-pressure analytical mono-phase solutions fortransient production analysis with empirical multiphase depletion decline curvestaken from Arps. As a result, the interpretation of wellbore rate responses wasmore rigorous due to the possibility of dividing rates into transient and the de-pletion parts. It is generally di¢ cult to precisely measure transient rates. Also,the depletion rates can be determined with a certain accuracy, assuming thatthe down-hole �owing pressure is constant. However, in reality, due to conditionconstraints or changes in operating procedures, the down-hole �owing pressureis seldom kept constant over long periods of time. In other words, the methodcannot be directly applied. Further improvements in rate time interpretationswere made in the 1990�s with methods that include various types of superposi-tion and normalisation among which the most important are the investigationsconducted by Palacio and Blasingame (1993) and Agarwal et al. (1999). Kuchuket al. (2005) proposed a deconvolution-based method for diagnostics in decline-curve analysis.

3.1 Empirical Models (Arp�s)

The earliest reported e¤ort to study the production drop over time was per-formed by Arnold and Anderson in 1908. They proposed that the productionrates during equal time intervals formed a geometric series and stated that theproduction drop expressed as a fraction was approximately constant. Arnold and

53

54 3. DEPLETION RATE DECLINE REVIEW

Anderson called this fraction �the decline�. These authors were also the �rst tonotice that the rate-versus-time curve exhibited a straight line on semilog paper.Between 1908 and 1944, extensive research was carried out in this area and

although the studies are too numerous for all of them to be cited, the workperformed by Cutler in 1924 warrants highlighting. Cutler noticed that thegeometric or exponential type of decline curve gives conservative estimates ofvolume as well as a conservative production forecast. He also stated that ahyperbolic relationship on log-log paper would better describe the productiondecline.Attempts to theoretically model production rate decline and cumulative pro-

duction curves of gas and oil wells, date as far back as the early part of the 20th

century. In 1921, a detailed summary of the most important �ndings of the earlyresearch activities in this area was documented in the Manual for the Oil andGas Industry. This treatise, which is mainly a compilation of the research workof the U.S. Bureau of Mines personnel, �rst noted the exponential decline modelfor oil wells, as well as the use of graphical techniques in the form of percentagedecline curves (i.e., �hyperbolic� declines) for the analysis of production ratedata from gas wells.In 1927, Johnson and Bollens introduced the so-called �loss ratio method�,

which was de�ned as the ratio between production rates and production dropsat equal time intervals. This ratio was found to remain approximately constant,thus providing an easy method for extrapolation.

Arps Model

In 1945, Arps published a comprehensive review of the previous e¤orts regard-ing decline curve analysis. Based on these results, he was able to empiricallyverify the equations for the exponential and hyperbolic decline behaviour. Thecontinuous use of these equations, even presently, is basically due to the easeof application and their acceptance in industry. Arps also showed how to ex-trapolate rate-time data following an exponential or hyperbolic decline. Whilethe exponential decline represents the simplest model to use, it also yields con-servative estimates and remains the most popular method within the petroleumindustry.Several e¤orts were made during the years that followed, and the most signif-

icant contribution towards the development of the modern decline curve analysisconcept is probably the classic paper by Arps presented in 1945. In this paper,Arps described a set of exponential and hyperbolic equations for production rateanalysis. Although the basis of Arps�development was purely statistical, andtherefore empirical in nature, these historic results have found widespread ap-peal in the oil and gas industry. The continuous use of these so-called �Arps�equations�to date is basically due to the explicit nature of the relations and theease of application of these equations to �eld data.

3. DEPLETION RATE DECLINE REVIEW 55

Arps also introduced the concepts of a decline exponent, b, and a declinerate constant, Di, both of which have become the cornerstones of many sub-sequent research e¤orts. By estimating these important parameters throughhistory matching, Arps demonstrated the technique of extrapolating rate-timedata following exponential and hyperbolic declines using a semilog plot. Al-though the exponential decline is the simplest model to use, especially in declinecurve analysis of oil wells, this model also yields the most conservative estimatesof in-place �uids and production rates. On the other hand, the harmonic declinemodel, provides the most optimistic estimates when used for predicting futureproduction rates.Research following Arps� publication concentrated on improving the fore-

casting of production data based on a general hyperbolic model. In 1968, Sliderpresented a new curve matching technique for obtaining a more accurate ex-trapolation of production rate data following the hyperbolic decline model. Thisapproach was also based on semilog analysis. In addition, the author demon-strated a practical curve-�tting method using preconstructed theoretical declinecurves based on Arps�equations. The technique was presented as simple andmore e¤ective than other decline curve analysis methods although a signi�cantamount of work was required in data preparation.Arps (1945) also carried out a fundamental study of the mathematical basis

of decline curves. He showed that equations for the semilog and log-log graphscould be derived from the basic di¤erential equations. Moreover, he introducedexponential, hyperbolic and harmonic decline curves, which were determined bythe decline exponent, b. The curves were de�ned according to the drop in pro-duction rate per unit time, represented by the fraction of production rate directlyproportional to the production rate and fractional power of the production rate.The fractional power is de�ned by the decline exponent, b, which has a valuebetween zero and one. Expressed in the form of a di¤erential equation, the abovestatement becomes:

D(t) = Kq(t)b = �dq(t)dt

q(t)(3.1)

This equation can be solved and presented in the form of a general empiricalhyperbolic equation:

q(t) = qi(1 + bDit)� 1b (3.2)

With the integration of the rate time relationship, the cumulative productioncan be expressed as:

Np =

Zq(t)dt =

qib

(1� b)Di

(q(1�b)i � q(t)(1�b)) (3.3)


The Arps equation (3.2) is the most commonly used. It is empirical and selectedfor depletion behaviour under producing conditions where the compressibilityis modi�ed. An example of compressibility changes involves solution gas drivemechanisms. It was noticed that the decline exponent, b, can be in�uenced by thereservoir �ow conditions and that value of b determines the degree of curvaturefrom the straight line or exponential semi-log decline (b= 0.0) to the harmonicdecline (b = 1.0). Exponential and harmonic decline equation are particularsolutions to the general hyperbolic solution.An exponential decline can also be referred to as a constant percentage decline

since the decline rate,D, remains constant with production. A hyperbolic declineequation contains two unknowns: the hyperbolic exponent, b, and the initialdecline rate, Di. Once these two unknowns are established the determination ofremaining reserves and future production can be done.The extrapolation of production decline curves provides us with the remain-

ing quantity of oil and gas reserves as well as the time of abandonment of a wellor lease. Arps assumed that the extrapolation procedure was strictly empiricaland that everything causing the decline curve trend in the past uniformly con-tinues to maintain it. The decline exponent, b, must be between zero and one.Empirical extrapolation signi�es a wide range of interpretations. Experience,integrity and objectiveness of the evaluator are related to the interpretation.Available data also can be controlled by the interpretation. Production datacan be controlled by the nature of reservoir rocks, �uid characteristics and drivemechanisms or by the production strategy, works on wells, producing equipmentlimitations or personnel policies.

3.2 Analytical-Numerical Models

3.2.1 Oil Flow

The purpose of this subsection is to discuss, based on examples, the generalmodel that generates the rate responses of a vertical perforated well in a strati�edhomogeneous reservoir. A single-phase oil production takes place only throughvertical well perforations. Boundaries are circular and either no-�ow or bounded.Generally, the model can incorporate inner boundary conditions of variable rateor variable pressure at the wellbore. This study considers constant pressureIBCs and does not include a wellbore skin analysis. Outer Boundary Conditions,OBCs, are no-�ow and �xed, or located from a wellbore axis at a distance re.The model yields a reservoir rate that varies with radial distance and time.

Volumetric �Bounded Reservoir In 1981, Ehlig-Economides and Rameyestablished an overview of the transient rate decline analysis for well produc-tion at constant pressure. From 1981 until today, the area of decline rate solu-


tions were partly presented in various publications and literature. Transient rateanalysis is currently an alternative to the well test analysis.The conditions considered di¤ered from the previous model in outer boundary

condition. The closed or bounded reservoir model was characterised by a no-�uid�ow across an outer boundary q(t) = 0 and in dimensionless form, expressed forrD = rDe as:

@pD@rD

= 0.For a well producing at constant pressure from the limited drainage volume,

the resultant behaviour is an exponential decline in the rate. This case wasdenoted exponential depletion. A well producing at a constant rate from thelimited drainage volume represented a pseudosteady state behaviour. The expo-nential depletion for the closed boundary system derived by Ehlig-Economidesand Ramey (1981) was given as:

qD(tD) =1

ln 4A CAr2w

�4�tDAln 4A

CAr2w

!(3.4)

This was an exact equation for the exponential depletion for tDA > (tpss)D,where tDA was the dimensionless time based on the drainage area, A, and (tpss)was the dimensionless time at the beginning of the pseudosteady state �ow, orin other words the time required for the development of the true pseudosteadystate at the constant rate inner boundary condition. It was dependent on thereservoir shape as published by Earlougher and Ramey (1968). The e¤ect of skinwas included by using an apparent, e¤ective wellbore radius rwa, instead of theradius rw in (3.4). This �ctitious radius, rwa, was de�ned as rwa = rwe

(�s).Tsarevich and Kuranov (1956) were the �rst to publish that the exponential

decline was the �nal form of �ow rate decline for constant pressure inner bound-ary production from a circular reservoir. They provided a theoretical basis fordecline curve analysis and presented tabulated solutions for the cumulative pro-duction for the closed boundary reservoir.A special case derived for a well located in the centre of a circular reservoir

was proposed by Fetkovich (1980). An analytical solution applied to a circularreservoir case con�rmed that the exponential rate time decline was a late timesolution of the volumetric reservoir under constant pressure inner boundary con-ditions:

qD(tD) =1

ln(0:472rDe)e� 2tDr2De

ln(0:472rDe) (3.5)

The unsteady �ow rate declined to the point where the cumulative productionwas constant depending on the reservoir size, rDe, as published by Uraiet andRaghavan (1980):

QpD =1

2(r2De � 1) (3.6)


Uraiet and Raghavan solved partial di¤erential equations with de�ned bound-ary conditions by the �nite di¤erences method. They analysed build-up be-haviour of a well producing at a constant wellbore pressure. Pressure buildupequations can be obtained by the principle of superposition, and a �nite di¤er-ences model was developed due to the di¢ culty in obtaining a simple analyticalexpression that can describe the bottomhole pressure buildup behaviour.Ehlig-Economides and Ramey presented solutions for three existing outer

boundaries by implementing the numerical "Laplace" transform inversion algo-rithm according to Stehfest.A method for converting constant rate solutions to constant pressure so-

lutions developed by Cox (1979) was also applicable to bounded reservoirs.Bounded reservoirs producing under pseudosteady state �ow conditions againstwell drawdown displayed exponential declines in the production rate. The pres-sure response for a well producing under pseudosteady state �ow conditions wasexpressed by Ramey and Cobb (1971):

pD = 2�TDA +1

2ln(

A

rw2) +

1

2ln(2:2458

CA) + s (3.7)

Equation (3.7) was transformed by Cox (1979) in the form of an exponentialdecline for the production rate, qD:

qD(tDA) =1

PDOe(�2�tDAPDO) (3.8)

where PDO is the intercept in equation (3.8).The constant pressure outer boundary associated with a gas cap or bottom

water did not change the pressure distribution with time. A steady state condi-tion was described with r = re and the pressure p = pi . The outer boundarycondition in dimensionless form involved, for rD = rDe dimensionless pressurepD = 0. The exact solution, including the skin factor, was derived by Ehlig-Economides and Ramey (1981) was:

qD =1

ln rDe + S(3.9)

The solution was valid for a dimensionless time, tDA of:

tDA =1

2:2458�(3.10)

3.2.2 Gas Flow

The decline analysis of gas wells has been reported by Stewart (1970) and Gurley(1963). The gas well rate equation can be expressed as:

qg(t) = Cg(p2R � P 2wf )

m (3.11)


If pwf = 0, and assuming that the gas compressibility Z = 1, we get:

PR = ��PR

GPi

�Gp + PRi (3.12)

Alternatively, the cumulative gas production as a function of the initial gas canbe expressed as:

Gp = GPi

"1�

�PR

PRi

�k#(3.13)

For Gp = Gp( PR,t), Equation (3.13) can be written as:

dGPdt

= �kGPiPR

k�1

PRk

dPRdt

(3.14)

Moreover, the rate equation for gas wells was de�ned as:

qg(t) = Cg(Pr2 � P 2wf )

m (3.15)

In comparison to the rate oil equation, this results in:

Cg�J�0

PR

PRi(3.16)

For the known initial rate and pressure, the gas well backpressure curve coe¢ -cient, Cg, can be calculated by:

Cg =qgi�

PRi2 � P 2wf

�m (3.17)

which yields:

qgqgi=

PR

2 � P 2wf

PRi2 � P 2wf

!m(3.18)

By assuming that Pwf is very small, this equation can be simpli�ed to:

qgqgi=

PR

2

PRi2

!2m(3.19)

A combination of Equations (3.14)) and (3.19) results in:

qg =dGPdt

= qgi

�PR

PRi

�2m= kGPi

PRk�1

PRk

i

dPRdt

(3.20)

By separating the variables and integrating, the equation can take the form:


PRZPRi

PR(�2m+k�1)

dPR =�qgiPRi

(�2m+k)

kGPi

tZ0

dt (3.21)

qg(t) = qgi1�

2m�kk

qgiGPi

t� 1� 2m2m�k

(3.22)

The rate time equation for a gas well in the case where (-2m+k)6=0 and k6=2mcan be expressed as:

PRZPRi

1

PRdPR = �

qgikGPi

tZ0

dt (3.23)

The general rate time equation for gas wells:

qg(t) = qgi1

e

�2mk

qgiGPi

t� = 1

e

�� qgiGPi

t� (3.24)

The unit solution of Equation (3.24) was plotted as a log-log type curve. Forthe lower limit of the backpressure curve slope i.e., m = 0:5 decline exponentwas b = 0:0, which was recognised as an exponential decline. The upper limit ofm = 1:0 resulted in the decline exponent b = 0:5 The e¤ect of the backpressureon the gas well was thus found to alter the type of decline. This situation di¤eredfrom the liquid case solution. The backpressure was expressed as a pf=pi ratioand for the pwf ! pi (i.e., �p! 0), the type curve approached the exponentialdecline with b = 0. Cumulative rate time log-log type curves could be preparedby integration of the rate time Equation (3.24).

3.2.3 Multiphase Flow

A multiphase �ow approach based on the assumptions employed for Arps�equa-tions was done by Camacho and Raghavan (1989). They examined a well per-formance in solution-gas-drive reservoirs with a closed boundary �ow. The Arpsdecline exponent, b, and the initial decline rate,Di, expressed in terms of physicalproperties. The conditions for decline analysis can be described by a homoge-neous closed cylindrical model with a fully penetrating well located in its centre.The inner boundary condition was de�ned for a well producing at constant well-bore pressure. The e¤ect of a skin region was included through an annular re-gion with a permeability di¤ering from that of the formation. Gravity, capillarypressure and non-Darcy �ow e¤ects were not considered. Cammancho (1987)developed the dimensionless pseudopressure:


ppD(r; t) =

kh

141:2q0(t)

8><>:r(t)Zr

�� (p; S0)

@p

@r

�t

dr +

tZ0

�� (p; S0)

@p

@t0

�r

dt0

9>=>; (3.25)

where � is a function of pressure and saturation, �(p; S0) = kro(S0)= [(�0(p)B0(p)],and r is radius corresponding to the position in the reservoir at which pressure,p(r), is equal to average pressure, p. During the boundary-dominated �ow pe-riod, r � 0:54928 re. The dimensionless pseudopressure was calculated for inter-val close to a wellbore, where 1 6 rD 6 rsD, where rD = r=rw is dimensionlessradius, , and rsD is the dimensionless radius of the skin zone. For an outerinterval:

ppD(r; t) = ppD(t) +

�lnreDrD

� 34

�

+

�k

ks� 1�"

1

4

r4sD

r4eD�

r2sD

r2eD+1

2

(r2sD � 1)r2sD

#+1

2

(r2sD � 1)r2sD

(3.26)

for rsD � rD � reD: The volumetric average of the pseudopressure is calculatedby using Muskat (1945) material balance equation:

ppD(t) =

kh

141:2q0(t)

piZp(t)

�(p0; r)dp0 = 2�ggtAD (3.27)

This dimensionless pseudopressure may also be considered a generalization of thematerial-balance equation for production at a variable rate in solution-gas-drivereservoirs. Here, ggtAD = eetD r2W

A(3.28)

and the dimensionless time, eetD de�ned aseetD = 0:006328k

�r2q0(t)

tZ0

q0(t0)�t(t

0)

ct(t0)dt0 (3.29)

The system compressibility, ct, and mobility, �t,corresponding to p (and S0) are

ct =S0

B0(p)

�dB0dp

�p

� Sg

Bg(p)

�dBgdp

�p

+ S0Bg(p)

B0(p)

�dRsdp

�p

(3.30)

and


�t =

�kro�0+krg�g

�(p;S0)

(3.31)

For the constant oil-rate dimensionless pressure in term of time, tD = 0:006328k�r2

tR0

�t(t0)

ct(t0)dt0

is:

ppD(t) = 2�tD

r2wA

(3.32)

Equation (3.32) is an extension of the material-balance equation for single-phaseliquid �ow.

3.3 Type-Curves

3.3.1 Vertical Well

Authors such as Tsarevich and Kuranov (1956), Ehlig-Economides and Ramey(1981), Uraiet and Raghavan (1980) have considered production rate declineanalysis given a constant wellbore pressure. They assumed a constant di¤usiv-ity, � for various dimensionless radii rD. The analytical solutions presented bythese authors form the transient portion of Fetkovich�s (1980) type-curves whereconstant pressure in�nite (early transient period) solutions were combined withthe empirical decline curve equation developed by Arps (1945).

Fetkovich Type-Curve

Decline curve analysis is founded on the same basic �uid �ow principles that areused in pressure transient analysis. Rate time curve analysis is based on constantwellbore pressure solutions for various physical models. This concept includedthe depletion period and pressure transient period of time. Constant wellborepressure solutions and their corresponding log-log type curve plots representedthe inverse of the constant rate solution.Single dimensionless unit type curves were composed of an analytical constant

wellbore pressure solution and the Arps exponential, hyperbolic and harmonicdecline curve solution. Depletion steam values range between an exponential,b = 0:0, and a harmonic, b = 1:0, decline accepted as the lower and upper limits.The exponential depletion was taken as common to the Arps equation depletionpart and to the transient part of the analytic solution on the dimensionless plot.Dimensionless qDd and tDd values were de�ned by Fetkovich (1980). He basedthis de�nition on the Arps exponential equation:

qDd(tD) =q(t)

qi=

1

eDit=

1

etDd(3.33)


From the hyperbolic equation, the dimensionless variables qDd and tDd can beformulated as:

qDd(tDd) =q(t)

qi=

1

(1 + bDit)1b

=1

(1 + btDd)1b

(3.34)

Moreover, for a dimensionless harmonic decline, b = 1, was de�ned as:

qDd(tDd) =1

(1 +Dit)1b

=1

(1 + tDd)(3.35)

The unit solutions, for an initial decline exponent Di = 1, were plotted as aset of log-log type curves. For each decline curve the decline exponent, b, wasbetween 0 and 1, increasing with increments value of 0:1. On the dimensionlessgraph, data previous tdD = 0:3 will be on the exponential decline regardless oftrue value of b. All curves were de�ned in the depletion area on the plot, andthe dimensionless time between 0.2 and 0.3 separated the depletion from thetransient period.The dimensionless time and rate of the decline curve were de�ned in terms

of reservoir variables for the transient period with the following expressions,

tDd =tD

12[ln( re

rw)2 � 1][ln( re

rw)� 1]

(3.36)

tD =0:00634k

��ctr2wat

qD = qD[ln(rerw� 12)] and qD =

141:3�B

kh(pi � pwf )q(t):

All analytically derived depletion stems become exponential solutions and col-lapse into a single curve with the above de�nition of qDd and tDd.The published values of tD and qD for the in�nite solution data were ob-

tained from Ferris et al.(1962). For the �nite constant pressure solutions, datawere obtained from Tsarevich and Kuranov (1966). The tD and qD values weretransformed into a de�ned decline dimensionless time, tDd, and rate, qDd, forvarious values of rDe = re

rw.

Late-time production analysis is based on the Arps decline curves (by match-ing real data to empirical Arps expressions). Fetkovich (1973) combined Arps�decline curves with semi-analytical rate-time well responses, which were thentransformed and plotted in dimensionless form of associated rate and time. Semi-analytical expressions are solutions of a di¤usion equation solved for inner bound-ary conditions of constant pressure and outer boundary conditions of no-�ow ata �xed distance re. Again, the distance re from a well situated in a middle of acylinder was kept constant and the drainage area does remain unchanged withtime. Under single-phase �ow conditions in a homogeneous reservoir of heighth, the well rate response with time declines exponentially after a time tPSS.


Figure 3.1: Dimensionless Arps curves (Decline b = 0:0; 0:5, and 1:0) [AfterCvetkovic (1992)].

Figure 3.2: Semi-analytical dimensionless rate-time type curves (for various di-mensionless radii, rD) [After Cvetkovic (1992)].


Figure 3.3: Combined transient-depletion dimensionless Fetkovich (1973) rate-time type curves [After Cvetkovic (1992)].

Pseudo-Steady-State Time Single-phase �ow solutions for various drainageareas are plotted in a transient part of the type curves transformed by Fetkovich(1973). Each drainage radius is represented with a transient decline curve asdemonstrated in Figure (3.2). After a time tPSS, this single-phase response isexponential, i.e., all drainage areas end with the same exponential decline asin Figure (3.5). Contrary to transient decline, depletion or Arps decline mayalso be relevant to a multiphase �ow. The well produces the same drainagearea. After the time tPSS (the time in which the reservoir boundary is reached),the semi-analytical expression for the rate becomes exponential, overlaying theArps exponential decline curve. Other Arps curves have rate-time curvaturesexpressed with a decline exponent, b, ranging from 0:1 to 1:0. In other words,the rate decline changes with time from exponential to hyperbolic and �nally toharmonic. Arps expressions for rate-time decline are empirical, based on analyseof data collected through numerous years of well production involving certainde�ned drainage areas. The approach of Fetkovich (1973, 1980) introduced morephysics into the decline parameters (initial rate, qi, initial decline, Di, and declineexponent, b). The time tPSS, on a unique combined curve, actually divides rate-time well responses into the transient decline and the depletion decline. Aftertime tPSS, the well is producing a constant drainage volume and the radius ofdrainage reaches the outer boundaries of no-�ow. In the depletion decline ofa well, several drive mechanisms may be considered (solution gas drive, gravitydrainage or partial water drive). The well may also be producing a single layered,multilayered or heterogeneous reservoir.Transient decline curves have been derived for a vertical well with one �ow

regime during transient �ow. Golan and Whitson (1986) derived rD expressionsfor early times for a conventional vertical well situated in the centre of a radialreservoir. The extension of Fetkovich type curves to a nonconventional well as


Figure 3.4: Transient dimensionless rate-time curves (for two values of rD) [AfterCvetkovic (1992)].

Figure 3.5: Transformed depletion dimensionless rate-time curves (for two di-mensionless rD) [After Cvetkovic (1992)].


Figure 3.6: Transient dimensionless rate-time curves (for various dimensionlessrD values) [After Cvetkovic (1992)].

Figure 3.7: Arps dimensionless rate-time curves [After Cvetkovic (1992)].


a well with fractures should include several �ow regimes in the transient declinepart.Blasingame et al. (1991) implemented method for analysing rate-time data

when the bottom hole pressure is variable. He introduced a material balance timefunction, tcp, (that was calculated by dividing the cumulative oil by the oil ratefor each time period) and applying it to convert the constant pressure solutionsfor liquid and gas to an equivalent constant rate liquid solution for a single layersystem. During transient �ow conditions the constant pressure and constant ratemethods are the same, while they are quite di¤erent during depletion, when theconstant rate system solutions are harmonic. However, the constant pressuresolutions are exponential. The method required that the drawdown normalisedrate be plotted vs. the material balance time, tcp. The method smooth the dataand may thus improve the type-curve matching. Nevertheless, it was noticedthat, with this method, the depletion data was "forced" to match the harmonicdepletion stem and not the value of exponent b < 1. So, it is evident that byapplying the method someone loose information on drive mechanism, recoverye¢ ciency, and layered no cross-�ow behaviour.Callard et al. (1995) presented type curves in plots of the pressure-normalised-

rate versus the pressure-normalised cumulative production. Both curves for con-stant rate and constant pressure overlaid the same rate-cumulative curve duringtransient and pseudosteady-state �ow conditions. The equation for the dimen-sionless cumulative production, QD, was given by

QD =0:8936Q(t)B0�hr2wa(pi � pwf )

(3.37)

Pressure-Dependent Fluid and Rock Properties Samaniego and Cinco-Ley (1980) performed a numerical investigation of the in�uence of pressure-dependent �uid and rock properties, during a single-phase �ow, on well produc-tion decline caused by constant wellbore pressure conditions. The variable rockproperties included the permeability, porosity, pore compressibility and forma-tion thickness and the variable �uid properties included the density, compress-ibility and viscosity. A �ow equation considering the pressure dependence of rockand �uid properties when expressed as a function of a pseudo-pressure, m(p),resembled the di¤usion equation. Samaniego et al. (1976 and 1977) evaluatedthis variable property problem for various �ow conditions.The mathematical model was based on: horizontal �ow, with no gravity, a

fully penetrating well, an isothermal single phase �uid obeying Darcy�s law, andan isotropic homogeneous formation. Samaniego at al., (1976 and 1977) andSamaniago (1974) noted that the assumption of horizontal �ow was not quitevalid.With �uid and rock properties held constant, the computer model obtained

data that was veri�ed against published solutions of van Everdingen and Hurst


Figure 3.8: A type-curve match for a constant- pressure drawdown test withvariable property solutions [After Samaniego and Cinco (1980)].

(1949) and Fetkovich (1973). A good match with dimensionless solutions ofvan Everdingen and Hurst was obtained. Numerical results obtained from thecomputer model and the analytical technique by Fetkovich (1973) were createdfor 3 reservoirs of varying reD and a good agreement was found. It was noticedthat, for all ratios of pi=pwf during transient �ow conditions, the productionrate decline expressed in terms of a dimensionless rate, qD, was the same asthe production rate decline for constant property liquid �ow as given in thetransient part of Figure (3.8). Nevertheless, solutions for a bounded reservoirdeviated from the classic qD solutions once the �ow was a¤ected by the outerboundaries, as in Figure (3.9). It was concluded that the production rate inpressure sensitive-systems declined faster than in constant-property systems.

Strati�ed No-Cross�ow Reservoirs Most reservoir are heterogeneous andconsist of several layers without cross�ow, each with its reservoir properties.In a reservoir with cross�ow the adjacent layers can be combined into a singleequivalent layer that can be described as homogeneous through an averaging ofthe reservoir properties of the cross�owing layers. The decline curve exponent,b, for a single homogeneous layer ranges from 0, to 0:5, whereas for layered no-cross�ow systems, values of b range from 0:5 to 1. Thus, those b values greaterthe 0:5 can identify the reservoir strati�cation.Fetkovich et al. (1996) suggested that the decline exponent, b, and the decline

rate, Di, can be expressed in terms of the back-pressure curve exponent, n. Bothexpressions were derived from the back-pressure equation,


Figure 3.9: The dimensionless �ow rate compared to the Arps� decline rates[After Samaniego and Cinco (1980)].

qg = C(pr � pwf )n (3.38)

where n is the back-pressure curve exponent; C is the performance coe¢ cient;andpr is the reservoir pressure. The Arps decline exponent, b, and the decline rate,Di, (with an initial in place gas in place, G) were respectively de�ned as:

b =1

2n

"(2n� 1)�

�pwfpi

�2#(3.39)

Di = 2n(qiG) (3.40)

Equation (3.39) shows that, as the pi approaches pwf , the hyperbolic declineshifts to the exponential decline, thus changing the b exponent from a valuenot equal to zero, to zero. Further, for those wells producing at a very lowbottom-hole �owing pressure with pwf = 0, the decline exponent is reduced to

b = 1� 1

2n

Fetkovich (1980) derived the expressions in Table (3-1), by combining Arps�hyperbolic equation with the material balance equation that relates p=Z withGp, and the back-pressure equation. He expressed the rate-time equation for agas well in terms of the back-pressure exponent, n, with a constant pwf of 0 thatalso implies that qi = qimax, as given by


Table 3-1: The rate-time equation for a gas well in terms of the back pressureexponent, n, with constant "pwf" of 0 as de�ned by Fetkovich (1980)

n qt Gp(t)0:5 < n < 1:00 < b < 0:5 qi

[1+(2n�1)( qiG )t]2n

2n�1Gn1�

�1 + (2n� 1)

�qiG

�t� 11�2n

on = 0:5b = 0 qie

�( qiG )t Gh1� e�(

qiG )ti

n = 1b = 0:5 qi

[1+( qiG )t]2 G

�1� 1

1+( qitG )

�

qi = qimax =khp2i

1422T (�gZ)avg

hln( re

rw)� 3

4+ si (3.41)

Here, qimax (in Mscf=d) is a stabilized absolute open-�ow potential, i.e., atpwf = 0; G (in Mscf) is the initial gas in place; qi (inMscf=d) is gas �ow rate attime t; and Gp(t) (in Mscf) is the cumulative gas production at time t. Ahmed(2006) presented a case of a commingled well producing from two layers at aconstant pwf . The total �ow rate, (qt)total is the sum of the �ow rates of thetwo layers according to (qt)total = (qt)1 + (qt)2. For a hyperbolic decline with anexponent, b = 0:5, using the expression from Table (3-1), we get

(qmax)total�1 +

�qmaxG

�total

�2 = (qmax)1�1 +

�qmaxG

�1

�2 + (qmax)2�1 +

�qmaxG

�2

�2 (3.42)

Evidently, the composite rate-time value of b = 0:5 can be achieved only if�qmaxG

�1=�qmaxG

�2, assuming that decline exponent, b, of each layer is equal to

0:5.

Carter Type Curves

Fetkovich (1980) type-curves were developed for a well producing under constantpressure in an oil and gas reservoir. For a pressure drawdown that is moderate tolarge, Fetkovich (1980) liquid �ow curves are however not recommended for gasproduction type-curve analysis. Carter (1985) developed type-curves for a gaswell production from a boundary reservoir. These type curves are theoretical,and provide understanding and implicit guidelines to �eld data analysis. Carteralso noticed that the changes in �uid properties with pressure a¤ect the reser-voir performance predictions, especially the gas viscosity-compressibility prod-uct, �gcg. In order to represent changes in �gcg during depletion and to measure


the magnitude of pressure drawdown on �gcg, he introduced the "dimensionlessdrawdown", �, according to

� =

��gcg

�i�

�gcg�avg

(3.43)

� =

��gcg

�i

2

"m(pi)�m(pwf )

piZi� pwf

Zwf

#(3.44)

By introducing the magnitude of the pressure drawdown in gas wells, �, Carter(1985) presented gas type-curves. These curves are based on dimensionless para-meters: the dimensionless time, tD; the dimensionless rate, qD; the dimensionlessgeometry parameter, �; the dimensionless radius, reD, and the �ow geometry;dimensionless drawdown correlating parameter, �. The-type curves were gener-ated with the radial gas simulation model. For the exponential decline, b = 0,corresponding to � = 1:0 and indicating a negligible drawdown e¤ect, also thegas decline was de�ned as exponential. An increasing magnitude of pressuredrawdown was de�ned with � = 0:75 and � = 0:55. Gas reserves are betterestimated with Carter type curves as those presented in Figure (3.10). This setof curves is similar that of Fetkovich in the aspect of plotting scales, but are notas straightforward and general as those of Fetkovich.

Chen and Teufel Type Curves

In 2000, Chen and Teufel extended the "Fetkovich" type-curves to linear/near-linear-�ow features that are important in tight-gas production data analysis.They used Fetkovich�s transient decline and Arps�depletion decline by simul-taneously considering Carter�s linear and radial single phase �ow. The derivedsolutions in "Laplace space" for a vertical well producing at constant pressure ina closed drainage area were provided for linear �ow and derived from the tem-perature solution published by Carslaw and Jaeger (1959, p. 309) in the formof

qD =1pstanh(

ps) (3.45)

The radial �ow in "Laplace" form was presented by van Everdingen and Hurst(1949, Eq. VII-4) as

qD =1ps

K1(ps)� I1(

ps)K1(reD

ps)

I1(reDps)

K0(ps) + I0(

ps)K1(reD

ps)

I1(reDps)

(3.46)

Linear and radial �ow are presented in Figures (3.11). Dimensionless log-logtype curves with linear and radial �ow of a reservoir with reD < 10 are plotted


Figure 3.10: Radial-linear gas reservoir type curves [After Carter (1985)].


Figure 3.11: The linear and the radial �ow geometry [After Chen and Teufel(2000)].

in Figure (3.13) with qD and QD there functions of �: (� is also de�ned asthe fraction of 2� radians that is open to �ow). Chen and Teufel referred themain di¢ culty when constructing "Fetkovich" curves to de�ning dimensionlessplotting variables. Finding a proper set of dimensionless variables should giverise to a unique curve during theoretical boundary-dominated �ow period forboth linear and radial �ow, and thus also for all types of �ow in a closed system.The simpli�ed dimensionless set of Fetkovich curves (1980 and 1996) was foundto be inadequate for cases of small values of reD. Carter�s dimensionless setwas used for a smooth transition from linear to radial �ow and for a decentconvergence of boundary dominated �ow. Further, Chen and Teufel de�ned thedimensionless set, the �ow rate, qdD, the cumulative rate, QdD, and the time,tdD, according to Figure (3.12). Moreover the authors provided detailed analysesof dimensionless parameters with an explanation of the coupling procedure.

Palacio and Blasingame Type Curves

For a varying bottom�owing pressure, Blasingame et al. (1991) created an equiv-alent rate liquid solution from liquid and gas constant pressure solutions. Thesemethods smooth data, thus improving the type-curve match. The depletion datawas forced to match the harmonic depletion stem instead to the Arps declineexponent b < 1. This approach exclude the Fetkovich (1980) concept of the drive


Figure 3.12: The dimensionless rate, qD, and the cumulative production, QD,versus the dimensionless time, tD [After Chen and Teufel (2000)].

mechanism, recovery e¢ ciency, and layered no-cross �ow behaviour. For eachtime period, the cumulative oil can be divided by an oil rate. Consequently,in a single-layer system, it is possible to �nd a material balance time function,tcp. Both constant pressure and constant rate solutions during transient �ow areequivalent. Contrarily during depletion decline, constant rate solutions follow aharmonic decline and constant pressure solutions follow an exponential decline.This method requires that the drawdown-normalised rate be plotted versus thematerial balance time function, tcp. As constant pressure data are replotted withthe time, tcp, and compared to the constant rate solution, they become equiva-lent and overlay the harmonic stem. Palacio and Blasingame (1993) representedCarter�s curves in terms of Fetkovich plotting variables, including the issues ofchanging �uid properties and operating conditions.Aqarwal et al. (1999) commented that a constant rate solution takes ad-

vantage of pressure transient analysis techniques for plotting decline curve data.Moreover type curves utilise plots of pressure, rate, cumulative rates, time andderivatives for result veri�cation.

Mattar and Anderson Type Curves

Mattar and Anderson (2003) applied the concept of a normalised rate and amaterial balance pseudo-time to create a simple linear plot, which can be ex-trapolated to the �uid in place. The method is similar to that of Palacio and


Figure 3.13: The composite type-curves: (A) The �ow rate vs. time; (B) Thecumulative production vs. time; (C) The �ow rate vs. the cumulative production[After Chen and Teufel (2000)].


Blasingame regarding the use of available production data. The �ow system of adepletion drive gas reservoir under peeudosteady-state conditions was describedby

q

m(pi)�m(pwf)=

q

�m(p)= (� 1

Gb0pss)QN +

1

bpss0(3.47)

where, QN , is the normalised cumulative production

QN =2qipita

(ct�iZi)�m(p)

and ta is the Blasingame normalised material balance pseudo-time

ta =

��qcq

�i

qi

ZiG

2pi[m(pi)�m(p)]

Furthermore, bpss0 was de�ned as the inverse productivity index (psi2=cp �MMscf) according to

bpss0 =1:417106T

kh

�ln(

rerwa)� 3

4

�(3.48)

Ansah-Knowles-Blasingame Type Curves

Ansah et al. (2000) noticed that, during depletion, a signi�cant change in gasproperties a¤ect the reservoir characteristics. This change is due to a variationin the gas viscosity-compressibility product, �gcg, with pressure.The gas property changes were described by material balance equation in

dimensionless form

pD = (1�GpD) (3.49)

which was derived from

p

Z=p

Z(1� Gp

G)

where, pD =p=Zpi=Zi

and GpD = Gp=GThe authors proposed that �gct be expressed in a the form of dimensionless

ratio(�gct)i(�gct)

: Here, cti is the total system compressibility at pi (psi�1), and �iis the gas viscosity (cp) at pi. Further, they stated the dimensionless ratio as afunction of dimensionless pressure according to Table (3-2).The type-curves by Ansah et al., are given as a set of dimensionless variables,

qDd, tDd, reD, and the correlation parameter is a function of the dimensionlesspressure, �, as presented in Figures (3.15, 3.16, and 3.17).


Figure 3.14: The distribution of the viscosity-compressibility function [AfterAnsah et al. 2000].

Table 3-2: The dimensionless ratio as a function of dimensionless pressure asde�ned by Anash et al. (2000)

dimensionless rationvs. dimensionless pressure

Pressure ranges

First order polynomial �icti�ct

= pD pi < 5000

Exponential model �icti�ct

= �0e(�1pD) pi > 8000

General polynomial model�icti�ct

= a0 + a1pD+a2p

2D + a3p

3D + a4p

4D

any


Figure 3.15: The "�rst-order" polynomial solution for real-gas �ow underboundary-dominated �ow conditions. A viscosity-permeability, �ct, is linearwith dimensionless pressure, pD [After Ansah 2000].

Figure 3.16: The "exponential" solutions for real-gas �ow under boundary-dominated �ow conditions [After Ansah (2000)].


Figure 3.17: "General polynomial" solution for real-gas �ow under boundary-dominated boundary conditions [after Ansah 2000)].

Analysis of Production Data The paper by Mattar and Anderson (2003)provides a comprehensive presentation of the methods developed by Arps, Fetkovich,Blasingame, and Agarwal-Gardner, as well as a new method named the Flow-ing Material Balance. Some methods yield recoverable reserves, while othersgive hydrocarbons in place. Traditional (Arps) decline analysis (exponential orhyperbolic) can underestimate or overestimate the reserves as it ignores the �ow-ing pressure data. Nonetheless, it gives reasonable answers in many situations.With new developed electronic data measurements both �owing pressure and�ow rate are readily available and thus more sophisticated techniques of datainterpretation are needed in data interpretation. Each method has its strengthsand limitations.Arps methodology is simple and does not require knowledge of reservoir or

well parameters. It uses an empirical curve match to predict the future per-formance of the well. It applies to production through any type of reservoir orreservoir drive mechanism. A practical guidelines has been assembled throughextensive �eld analysis which suggests what curves belong to which type curvesthrough Fetkovich (1980). Arps decline analysis is not able to include changes inoperating constraints as it inherently assume that historical operating conditionsremain constant in the future. It is not applicable to transient �ow conditions.Thus, predicting ultimate recovery with Arps curves has very limited application.Fetkovich (1980) was the �rst to use type curves to analysis of production

data. Type curves combine depletion stems describing boundary dominated �owwith constant pressure type curves that originated by Van Everdingen and Hurstfor transient type of �ow. This type curves valuable feature lies in diagnosticand less in analysis of production data. Matching production data with type


curves it is possible to de�ne transient and depletion decline. Method calculatesexpected ultimate recovery and is constrained to existing operating conditions.The transient part of type curves assumes constant bottomhole �owing pressurewhat is a limitation. Usually when well is rate restricted approach does not apply.This technique can quantify hydrocarbon-in-place by using recovery factor only.Blasingame et al. (1989) and Agarwal et al. (1999) methods are similar to

Fetkovich, as being used for production data analysis. The main di¤erence isthat the these new methods incorporate the �owing pressure data along withproduction rates. In addition these methods use analytical solutions to calculatehydrocarbon-in-place meaning that expected recoverable reserves can be quan-ti�ed independently of production constraints The main two features are innormalizing of rates using �owing pressure drop and handling changing com-pressibility of gas with pressure Mattar and Anderson (2003) explained howpseudo time works.Moreover, instead of type curves there are modern analytical methods as

Flowing Material Balance. For a reservoir under volumetric depletion by usingproduction rate and �owing pressure data it is possible to calculate hydrocarbon-in-place. As these methods are analytical it comprises simpli�cations about thereservoir and production data. Mostly method assume single phase �ow andaccount for interference e¤ects and a volumetric reservoir. The non volumetrice¤ects such as water-drive and interference among multiple wells can be handlede¤ectively using in�uence functions. As an example, Blasingame type curveshave a multiple-well feature that can accommodate and account for interfer-ence e¤ects. Single phase �ow is considered valid especially for gas wells. Formultiphase �ow pressure loss from surface to bottomhole conditions should beconsidered.

3.3.2 Vertical Fractured Well

Special type-curves designed for hydraulically fractured wells have also beenproposed for various degree of complexity, e.g., planar/elliptical fracture within�nite/�nite conductivity partially/fully penetrated in an in�nite/closed reser-voir. Thus many type curves have been developed. In 1975, Locke and Sawyergenerated a constant pressure type curve numerically that included boundarye¤ects and an in�nite conductivity fracture. The constant pressure solution fora �nite reservoir exhibits an exponential decline for large value of time. To rep-resent the change in �owrate they used a stepwise method. Further, an integralwas developed for converting the constant rate solution to constant pressure.In 1979, Agarwal et al., determined dimensionless rates for constant terminal

pressure from wells with �nite-conductivity fractures. Type-curves were intro-duced at early times by To simulate the e¤ect of a �nite conductivity verticalfracture on �ow behaviour they developed a �nite-di¤erence model. Verticalfractures were assumed to have high storage capacity. In 1985, Fetkovich et al.


Table 3-3: The vertical well, the vertical and the horizontal fractureVertical Fractured Well

Vertical HorizontalPlane is perpendicularto earth�s surface

due to overburden stressbeing too great to overcome.

Plane is parallel to theearth surface, and is usuallyassociated with shallow wells

of less than 3000 ft (914 m) depth.Fracture gradient < 0:8 psi=ft Fracture gradient > 1:0 psi=ft

demonstrated that low permeability hydraulically fractured wells can be analysedusing the transient radial �ow solution with very little di¤erence in results fromthe same data matched to an in�nite conductivity fracture type curve. In 1986Fraim developed semi-analytical type curves for history matching hydraulicallyfractured wells. In 1996 Cox et al. presented new hydraulically fractured typecurves that comprised the e¤ect of reservoir geometry, skin e¤ect, and varyingcompressibility.

Agarwal et al. (1999) used the constant rate solution in combination withtransformation Blasingame applied before to create rate-time, and cumulative-time plots along with their derivative plots. Type curves were generated forradial systems and vertically fractured �nite and in�nite conductivity system.Fetkovich et al. (2006) found that in same case the derivative may be noisy.Production type curves that assume a constant bottom �owing pressure havebeen developed and used to predict production.

Vertically fractured well can exhibit �ve distinct �ow regimes: fracture linear,bilinear, formation linear, pseudo-radial and boundary dominated �ow. Flowregimes are separated by the transition periods. During linear �ow �uid in thefracture expands and enters the wellbore with the following characteristics: itends quickly; not usually seen in production data; fracture �uid cleanup occur-ring; an remark is that it is not useful for analysis. During the bilinear �ow thefollowing occurs: �uid �ows linearly from the formation to fracture and fromfracture to wellbore; the lower CfD the longer is bilinear �ow; the formation �owis compressible and fracture �ow is incompressible;and the fracture tip e¤ectscould not yet be seen at the well. During formation linear �ow the pressure dropin fracture is negligible compared to other factors driving the system by meansthat the higher is CfD the longer is formation linear �ow. Further, during thepseudo radial �ow, after bilinear or formation linear �ow, it is independent ofCfD;the higher the CfD the larger tD before the pseudo radial �ow is reached; itcan be approximated by the well large rwa; this �ow may not be exhibited dueto the boundary e¤ects.


Figure 3.18: The vertical well, the vertical and the horizontal fracture.

Figure 3.19: The vertical fractured well in a rectangular drainage area [AfterChen et al. (1991)].


Figure 3.20: Type of �ow for a vertcal fractured well

3.3.3 Horizontal Well

Several theoretical studies of in�ow performance of a horizontal well are madelong time ago by Slichter (1897) in 2 D space, and further Kozeny (1933) andMuskat (1937) studies in 3 D space.Decline curves for horizontal wells under various boundary conditions have

been developed using the Green�s and source functions. These decline curvescan be used to estimate the production forecast and the ultimate recovery ofa horizontal well. Poon (1991) studied in�uence of wellbore stimulation, wellspacing and length on e¤ectiveness of a horizontal well. Most of equations forpredicting the production rates of a horizontal well are based on the assumptionof stady-state �ow conditions. These equations have been developed using theGreen�s and source functions.The most widely used methods are the steady-state-eqations published by

Merkulov (1958), Borisov (1964), Giger (1983), Karcher et al. (1986), Renardand Dupuy (1990) and Joshi (1986 and 1991). These equations require an es-timation of the horizontal well drainage radius which is not known until thewell has been on production and well tests were conducted. Also, steady stateproduction rate cannot be used to estimate the ultimate recovery of the well.Recently Michelevichius and Zolotukhin (2002) provided an alternative approachto the productivity evaluation of a horizontal well. The approach was based onthe idea of representing a well as a chain of spheres and on averaging techniquederived by Muskat.However, several analytical horizontal well analytical model have been devel-


Figure 3.21: The dimensionless rate, qD versus the dimensionless time, tDXf forthe horizontal well [After Cox et al. (1996)].


oped which do not require the assumption of the drainage radius meaning thatthere are not based on the steady-state Darcy equation. Babu and Odeh (1988)presented an equation for calculating the productivity of a horizontal well in abounded reservoir during pseudosteady-state �ow. The �rst is the geometric fac-tor which accounts for the e¤ect of anisortopic permeability, the well location andthe drainage area. The second is the skin factor which accounted for the e¤ect ofwell length. Goode and Kuchuk (1991) developed an equation for predicting thein�ow performance of a horizontal well in a bounded rectangular reservoir beingunder constant pressure outer boundary. Duda and Aminian (1989) and Changet al. (1989) developed type curves for predicting the cumulative production ofa horizontal well with numerical simulation. Poon (1991) presented the declinecurve development. He used the analytical model solutions for a single well de-veloped by Clonts and Ramey.(1986) and implemented the Duhamel�s theoremintroduced by van Everdingen and Hurst (1949). For a single horizontal wellin a bounded rectangular reservoir the Green�s function can be integrated withrespect to time and "Laplace" transform was taken directly, thus providing anexact solution. However, approximate solution method should be consideredto avoid di¢ culties in applying "Laplace" transform. Thus, de Carvalho andRosa (1988) suggested the use of the short and long time approximations to theGreen�s function to evaluate the "Laplace" transform of the pressure function.Further, Cox (1979) introduced another approximate solution method that �ta simple correlation equation through the transient pressure data. He provideddimensionless production rate versus time for a wellbore producing under theconstant pressure conditions. Poon plotted the dimensionless rate, qD, versusdimensionless cumulative production, NpD, for a single horizontal well locatedin a bounded reservoirLocke and Sawyer (1975) developed a constant pressure type curve for a in-

�nite conductivity fracture with a numerical simulator that included boundarye¤ect. Agarwal et al. (1979) considered well with �nite-conductivity verticalfracture. Further, Fetkovich et al. (1985) used the transient radial �ow solutionto analyse rates from low permeability hydraulically fractured well. Fraim etal. (1986) developed semi-analytical type curves to history match hydraulicallyfractured wells. Cox et al. (1996) presented new hydraulically fractured typecurves that included the e¤ect of reservoir geometry, skin e¤ect and varying com-pressibility. Chen and Teufel (2000) extended Fetkovich type curves by includingthe linear �ow regime.

3.3.4 Horizontal Fractured Well

It is known that horizontal wells have been successful in naturally fracturedreservoirs and in reservoirs with gas and water coning problems. However, thereare situations where a fractured horizontal well is preferable. Thus, the fractur-ing of a horizontal well must be considered before the well is completed. Many


Figure 3.22: Decline curve for a horizontal well ina bounded reservoir [AfterPoon (1991)].

Figure 3.23: The e¤ect of the aspect ratio on horizontal well productivity (theratio of the length to the width of a rectangular well pattern) [After Poon (1991)].


authors studied the productivity aspects of fractured horizontal wells. One im-portant aspect of a positioning of a horizontal well is the determination of thestress �eld about the proposed well. Once, principal stress is known it is possibleto create transversal fractures or longitudinal fractures as presented in Figure(3.24). In 2006 Demarchos et al., discussed the operational challenges of a frac-turing project and provided recommendations for the successful treatment of atransversally fractured horizontal well. Wei and Economides (2005) had studiedthe performance of horizontal wells with multiple transversal fractures. Further,it was found that at depth where exists producing formation, the stress �eldsleads to a hydraulic fracture that is vertical and normal to the minimum horizon-tal stress. Thus, the fracture direction and azimuth a¤ect the well orientation.Study by Villegas et al. (1996) concluded that with equal fracture length andconductivity, the performances of a fractured vertical well and a longitudinallyfractured horizontal well are almost identical. It was found that almost all of thereported applications of fracturing of horizontal wells are transversal fractures.The fracturing of horizontal wells has been mostly considered in the UnitedStates and the North Sea. Fractured horizontal gas wells were also consideredin Germany and Australia.

The increased productivity of multiple transversal fractured horizontal wellhas been studied by several authors. as Giger (1985), Karcher et al. (1986),Mukherje and Economides (1988). Soliman (1990) was among �rst to studythe behaviour of a horizontal well with a single fracture. By disregarding thepresence of the well they created the simple model that was a solution for a �niteconnectivity disk in a one dimensional in�nite slab. In 1989 van Kruijsdijk andDullart provided a boundary element solutions of the transient pressure responsesof multiply fractured horizontal well. Larsen and Hegre (1991) examined wellperformance during the linear-radial �ow regime. Further, Roberts et al., (1991)investigated a well with fractures pressure responses and major �ow regimes. In1993 Guo and Evans provided a two-dimensional solution for a horizontal wellwith multiple fractures. Raghavan et al. (1994) provided model solutions witha comprehensive report of understanding of the performance of multi-fracturedhorizontal wells. Horizontal wells with multiple fractures were further studiedby Guo and Evans (1994), Kuchuk and Kadar (1994).

An extensive analytical work has been done to investigate pressure-transientanalysis and short and long term productivity of horizontal well with singleor multiple hydraulic fractures by Soliman et al. (1990), Larsen and Hegre(1994), and Kuchuk and Hubusky (1994). Moreover, Horne and Temang (1995),Raghavan et al. (1997), Soliman (1998), further investigated the e¤ect of numberof transversal fractures on a well productivity.

In 1997 Guo and Schechter presented a simple mathematical model for esti-mating productivity of a vertical and horizontal wells intersecting long fractures.They observed that the rapid decline in wellbore productivity was mainly at-


tributed to stress-sensitive fracture conductivity. Their study provided overviewon several analytical solutions for transient �ow in fractured reservoirs and se-lected numerical models developed for simulating �uid �ow in fractured reser-voirs. The review comprised Economodes et al. (1991) general simulation schemethat can handle horizontal wells. Hydraulic fractures that penetrate the well inboth the transversal and longitudinal directions were e¤ectively simulated, andpredicted performance was in excellent agreement with analytical solutions.In 1999 Wan and Aziz derived analytical solution for the well pressure that

can be combined with a numerically computed gridblock pressure to obtain thewell pressure(WI). They presented an overview of existing methods for hydraulicfractures modelling. They used in the study the modifying the e¤ective wellboreradius method. The re�ning the fracture grid method; and modifying the frac-ture transmissibilities method were only reviewed.Al-Kobaisi et al. (2006) provided a hybrid, numerical-analytical model for

the pressure-transient response of a �nite-conductivity fracture intercepted by ahorizontal well.In addition Mederios (2006 and 2007) presented semi-analytical solutions of

fractured horizontal wells with transverse and longitudinal fractures in hetero-geneous, tight gas formations.

3.3.5 Multilateral Well

Ozkan et al.(1998) presented computational methods with solutions for duallateral wells in homogeneous formations. In 1998 Larsen derived solutions forpressure-transient behaviour of multibranched wells in layered reservoirs. Thecomputational methods was based on Laplace transforms and numerical inver-sion to generate type curves for use in direct analyses of pressure-transient data.The approach can handle any number of branches with arbitrary direction anddeviation. Earlier in 1997 Larsen provided solutions for deviated wells in layeredreservoirs. The results applied for any deviation, and hence, also for horizon-tal segments within di¤erent layers. The approach was restricted, however, tocover at most one segment within each layer with no overlap vertically. Theapproach cannot handle the boundary condition at the wellbore for nonverticalsegments, thus, each perforated layer segment has to be replaced by a uniform-�ux fracture. This approach is illustrated in Figure (3.25) for a deviated well ina three-layered reservoir. Further, in 1998 Larsen investigated productivity offractured and non-fractured deviated well in commingled layered reservoir Directanalytical methods are introduced to determine productivity indices of fracturedand nonfractured deviated wells in commingled layered reservoirs, including caseswith horizontal wells. For non-fractured wells, the total productivity index (PI)can be obtained by adding the individual layer PI�s. For a fractured wells, amore direct approach covering all well and fracture elements was needed if atleast one of the fractures penetrates more than one additional layer above or


Figure 3.24: The fracture orientation along a horizontal well.


Figure 3.25: A well in a three layered reservoir with perforated segments replacedby uniform-�ux fractures.

Figure 3.26: The multibranch and multiple-fracture con�gurations for horizontalwells [After Economides at al. (2001)].

below the wellbore layer.In 2001 Economides et al., presented advances of the complex well-fracture

con�gurations. A rather sophisticated conceptual con�guration involved thecombination of multiple-fractured vertical branches from a horizontal "mother"well drilled above the producing formation in a non-producing interval. Thesimplify perforation strategy of a vertical section over a highly deviated or hori-zontal section makes multibranch well with vertical branches more advantageouscompared to a horizontal well with multiple transverse fractures as presented inFigure (3.26). However, a complex well design procedures need advance mod-elling tool for better understanding of a complex fractured well performances.Cvetkovic et al. (2007) numerically investigated a complex well with a com-

plex lateral geometry. Numerical simulation were performed with synthetic reser-voir and �uid data. A producing well was positioned vertically and horizontally


Figure 3.27: The multilateral well types [After Louis J. Durlofsky TAML, 1999presentation].


Figure 3.28: A vertical and horizontal well with laterals positioning within anoil reservoir [After Cvetkovic et al., (2007)].

with over 100 laterals. Liu (2009) presented an overview of multilateral wellsthat become a standard IOR practice to enhance hydrocarbon recovery in bothoil and gas reservoirs. Accurately forecasting the performance of multilateralwells is challenging particularly in a complex reservoir such as highly faulted ornaturally fractured reservoirs. Reservoir simulation is considered as a reliableand economic method to asses the bene�t of multilateral wells in terms of in-creased oil production and improved sweep e¢ ciency. Technological advances inmeasurement and geological modelling provide a detailed description of the reser-voir, especially in the vicinity of the wells, thus accurate well models are essentialfor reservoir and production engineering applications. Recently Karimi-Fard etal. (2009) presented an overview of di¤erent numerical techniques developed tostudy the well productivity in complex situations including fractured reservoir.

3.3.6 Multi Wells

Rodriquez and Cinco-Ley (1993) developed a model for production decline in abounded multi-well system. The primary assumptions in their model are thatthe pseudosteady-state �ow conditions exists at all points in the reservoir, andthat all wells produce at a constant bottomhole pressure. They concluded thatthe production performance of the reservoir was shown to be exponential in allcases , as long as the bottomhole pressures in individual wells are maintained


constant. Later in 1996, Camacho and Galindo-Nava, improved the Rodriquezand Cinco-Ley model by allowing individual wells to produce at di¤erent times.Moreover, Camacho et al., also assumed the existence of the pseudosteady-statecondition and that all wells produce at constant bottomhole pressures.Valko at al. (2000) presented a concept for an arbitrary number of wells in a

bounded reservoir system named as "muti-well productivity index". These au-thors also assumed the existence of pseudosteady-state �ow, but proved that theconcept was valid for constant rate, constant pressure, or variable rate/variablepressure production. Marhaendrajana and Blasingame (2001) developed a gen-eral multi well solution that was accurate and provided mechanism for the analy-sis of production data from a single well in a muti-well reservoir system. Themethodology was applicable for both oil and gas reservoirs. Their approach usedthe single well decline type curve (i.e., Fetkovich-McCray type curve) coupledwith the appropriate data transforms for the multi-well reservoir system. Fur-ther, method includes a "total material balance time" plotting function thatcomprised the performance from all of the producing wells in the multi wellreservoir system. Method was applicable to estimate �ow capacity (i.e., per-meability), the original �uid in place. Method applied for homogeneous andheterogeneous reservoir systems.In 2007 Gilchrist et al., presented novel semi-analytical solutions to the lay-

ered reservoir produced with multiple wells. Applying a method of integraltransform they derived an analytical solution within each layer. Solutions wereapplicable to partially penetrating vertical, horizontal, deviated and fracturedwell taking into account superposition e¤ects in multi-well and multi-rate sce-narios. Further, they derived solutions for in�nite conductivity fracture anda �nite conductivity fracture with non-Darcy �ow. Inner boundary conditionswere both, constant pressure and rate for the overall multi-well scenario. Allpublished solutions were related to the interpretation of generalized multiplelayer, multiple well problems in single phase hydrocarbon reservoirs as presentedin Figure (3.29). Selected derived expressions were taken from the Thamby-nayagam�s work that is internal and unfortunately not available. According toauthors, Thambynayagam provided practical and elegant solutions to a varietyof con�gurations by the use of successive integral transforms. In an earlier paperBusswell et al., (2006) presented the "Laplace space" solutions derived from ageneralised single layer analytical model that handle multi vertical and horizon-tal wells. In order to avoid problems in converting "Laplace space solutions" to areal domain, mainly caused by the Stehfest inversion algorithm and its handlinga discontinuous nature of rate history, they chose the Chen and Raghavan (1996)approach. They implemented Chen and Raghavan approach in order to handlediscontinuities of the Stehfest algorithm.


Figure 3.29: The multiple vertical, horizontal and deviated completioned wellsin the layered reservoir [After Gilchrist et al. (2007)].


3.4 Decline Curve Analysis Physics

The following constitutes a review of Arps�gas decline analysis. Arps�equationcontinues to apply as an empirical relation in both history match and productionforecast. The initial rate, qi, initial decline rate, Di, and decline exponent, b, areconstants that de�ne the rate-vesus-time relation. An exponential decline givesrise to a decline exponent b = 0 , and a decline rate D that is constant withtime, or D = 1

q(dqdt). The plot of such a rate logarithm versus cumulative gas

production corresponds to a straight line. A hyperbolic decline is obtained withb between 0 and 1, and a decline rate D that decreases constantly with time.A plot of the rate logarithm versus the cumulative gas production (and time)appear upward concave. Moreover, the reserves calculated for the same initialdecline rate vary. Consequently, the exponentially calculated reserves are moreconservative as compared to those calculated hyperbolically.The choice of the decline exponent, b, in�uences the estimates of reserves,

and economic evaluations of well production (duration of a well production andwell rate). We presume that the empirical hyperbolic equation (3.50) is validonly for the boundary-dominated �ow and when the well�s �owing pressure isconstant.

q =qi

(1 + bDit)1b

(3.50)

Fetkovich type-curves combine transient mono-phase solutions with the em-pirical boundary-dominated stems of the Arps equation. The Fetkovich type-curves allow the entire transient-decline data set analysis with limitations in-volving the fact that a well produces under a constant �owing pressure. Recentwork introduced the variable �owing pressure analysis. The theoretical exponen-tial (mono-phase) depletion decline and the empirical depletion (multi-phase)decline with an exponent b = 0 can be superimposed. Fetkovich et al. (1996)investigated the exponential decline b and the drive mechanism relation, andfound the volumetric depletion driving force to be in�uenced by the total systemcompressibility. In a gas system (contrarily to a single-phase liquid system), thecompressibility varies approximately as the inverse of the average reservoir pres-sure and is not constant. As a result, the b value of the gas system is larger than0. Among factors that a¤ect gas rate-decline can be mentioned: wellbore �ow-ing pressure, turbulence and multiple no cross�ow layers. Okuszko et al. (2008)investigated the e¤ect of various parameters on gas decline by means of reservoirsimulation. It was found that the b value depended on the magnitude of the�owing pressure. For a low drawdown (pi�pwf ) or higher backpressure, pwf , theexponent b approached exponential decline. As the backpressure, pwf ,decreasedthe decline exponent, b, increased from the exponential to the hyperbolic type.In a near-wellbore region the wellbore turbulence, n, was found to also a¤ectthe decline exponent, b. In the backpressure equation (3.51), the turbulence


factor, n, relates to the degree of turbulence. As a consequence, the laminar�ow represents a value of n = 1, and the turbulence �ow represents a value ofn = 0:5.

q = C(p2R � p2wf )n (3.51)

The second relationship is between the decline exponent, b, and the tur-bulence factor, n, was made by Fetkovich et. al. (1996). They coupled thebackpressure equation (3.51) with the material balance equation to obtain theexpression (3.52), valuable for conditions of a very low �owing pressure. Withan increase in turbulence, or for a decreasing n, b approaches zero.

b =2n� 12n

(3.52)

In a single layer gas reservoir the decline exponent, b, is between 0 and 0:5. If�uid properties are constant during depletion (a liquid-like �uid behaviour), b isequal to zero. If �uid properties changes signi�cantly during depletion, declineexponent, b, approaches a value of 0:5. The change in �uid properties is moresigni�cant at lower reservoir pressures. Moreover, that exists a proportionalitybetween decline exponent b and reservoir depletion. Both drawdown and tur-bulence a¤ect the pressure depletion and the decline exponent, b. The reservoirpressure is reduced with a high draw-down, which is followed by higher �uidproperty changes and a higher value of the decline exponent, b.Fetkovich et. al. (1996) declared that the decline exponent can be as high

as one provided that the gas reservoir is layered. In a tight gas reservoir, itis possible to obtain exponent b higher than one. This seems to signify thatthe well produces partly in transient mode instead of exclusively in a boundarydominated mode.The decline exponent, b, may change at the end of production. Okuszko et

al. (2008) found that deviation from a constant decline exponent, b,occurs afterproduction of 90% of the expected reserves. For a constant value of the declineexponent, b, they noticed only a marginal overestimation of reserves of ca. 5%.As the decline exponent decreases, the late-time e¤ect on gas reserve estimatesis below 5%. For this reason, a constant decline exponent, b, represents a goodapproximation.

Drive Mechanisms Decline curve analysis is used for interpreting rate-timedata and to predict the future performance of a well or reservoir. The productiontrend in a well or �eld is a re�ection of the characteristics of the formation, the�uid in the formation, the well, the mechanism by which the �uid is driveninto the well and the mechanism by which it is lifted to the surface. If thesecharacteristics remain unchanged, the past trend will continue into the future.


The following presents certain comments on reservoirs with various drivemechanisms. The oil in an oil reservoir is not ordinarily produced by only oneor two of the principal oil-recovery mechanisms. Gas-cap and edge-water drivesmay function simultaneously. This can take place in the early oil-productivereservoir. Gravity drainage might predominate as the oil-recovery mechanismin the later oil-productive life of the reservoir. Lefkovits and Matthews (1958)studied gravity-drainage reservoirs and found a decline exponent b of 0:5. Ahigh-pressure oil-reservoir can, in its early life, have unlimited edge-water drive,assuming that it is combined with gravity drainage as the �eld is developed,especially if the oil productive strata are thick and dip steeply.Gravity drainage is an ine¤ective oil-recovery mechanism in a thin and �at-

lying reservoir. If such reservoir-rock is �lled with a high-pressure gas-saturatedoil as well as immobile interstitial water, and if no natural gas-cap energy ornatural water-drive energy is available, the solution-gas-drive mechanism wouldbe the primary agent for recovering the oil. Some of the gas that is originallyin solution in the oil is released when the �uid pressure is reduced below thatof the gas-saturated oil. This gas expands, thus displacing liquid oil into thewell. As the process continues, the �uid pressure becomes reduced at increasingdistances from the well. Moreover, the fraction of pore space of the oil reservoirrock occupied by gas increases and the oil fraction decreases. The volume ratioof gas to oil experiences a rise.The solution gas-drive has received the most attention in theoretical studies.

An exponential decline of b = 0 is obtained for n = 0:5. The harmonic decline isnot possible. The highest value of n given by Fetkovich was n = 1:0 which leadsto b = 1=3 or 0:333. According to Fetkovich, pwf = 0 is a realistic assumption fora well on wide-open decline. Fetkovich�s in�ow-performance curve equation orIPR curve equation for n = 1:24 are approximately identical to Vogel�s referencecurve, and provide b = 0:43. This �ts Arps� �nding that, for the majorityof decline curves, the range of b is 0 < b < 0:4. However, the result is indisagreement with that of Ramsey and Guerrero (1969).

3.4.1 Solution Gas Drive Decline

Raghavan (1993) provided the rate, q0(t), as

q0(t) =dNppp

= �ct��t

�Ah

5:614(3.53)

The decline rate, Di,was expressed as

Di =d ln q0dt

t2�0:006328k

12

�ln 4A

e CAr2w+ 2s

��A

�tct

(3.54)

Moreover, the decline exponent, b, can be written


b = � d

dt

1

d ln q0dt

!t

12

�ln 4A

e CAr2w+ 2s

��A

2�0:006328k

d

dt

�ct

�t

�(3.55)

which states that as long as the ratio of the average total compressibility andmobility varies linearly with time, the decline exponent, b; is constant Boththe initial decline rate, Di, and the decline exponent, b, depend on the relativepermeability and �uid properties, and thus material-balance can be used tostudy the variation of the mobility and compressibility. It is also indicated thatpredictions of the future performance are strong functions of the �uid properties,the well conditions and well spacing. Fetkovich (1973, 1987) suggested that bshould be in the range 0:333 � b � 0:667, while Camacho and Raghavan (1989)published that b was in the range 0:4 � b � 0:8. They noticed that responsescut across several curves as a result of the total compressibility-mobility rationot being a linear function of time. They also introduced the idea of a variableskin factor causing non-Darcy �ow to yield a constant value of b.

Decline Exponent b Raghavan (1993) studied the character of exponent b,and clearly stated conditions under which the decline exponent can be constant.By assuming a �xed drainage area, he stated that exponent b should vary withtime, i.e., it can not be constant for a well producing under IBCs of constantpressure. He also mentioned that it would be possible for a well to produce withthe �xed decline exponent, by assuming the existence of well skin incorporatedinto solution. Moreover, he pointed out that this assumption was relevant tothe well producing from a �xed drainage area where the distance to the no-�owboundaries remained unchanged with time.The following summarises the discussion of decline exponent b, based on a

modelling approach and analytically obtained expressions. The decline exponent,b, is the measure of the change in loss ratio

b = � d

dt

qdqdt

!(3.56)

According to the above expression, the �rst di¤erence of the loss ratio is constant,signifying that the loss ratio is a linear function of time. The decline exponent,Di, is, in turn, related to the loss ratio by

1

Di=

qdqdt

!i

(3.57)

By integrating and combining these two equations, it is possible to obtain thedimensionless rate, qDd, known as the hyperbolic decline, which describes theimplicit boundary dominated �ow


qdD =q

qi=

1

(1 + bDit)1b

(3.58)

For an exponential decline

qdD =q

qi= e�Dit (3.59)

Decline curve analysis was considered as a convenient empirical procedureuntil Fetkovich (1973) tried to attribute signi�cance to b and Di. Fetkovich typecurves combined empirical Arps (1945) solutions to the analytical single phase�ow solutions. Following the Raghavan (1993) nomenclature, the dimensionless�ow rate qD is

qD(tAD) =1

ae�

2�tADa

a =1

2ln

�4A

e CAr2w

�+ S

Introducing the time, tAD;i, and the corresponding rate, qDi, that relates to qD,we obtain

qdD =qDqDi

=q

qi= e�

2�(tAD�tAD;i)a (3.60)

Di =2��2k

a�ct�A(3.61)

tdD = Dit =2��2k

a�ct�At (3.62)

With an initial decline qi

qi =2�kh

�1B�

(pi � pwf )h12ln�

4Ae CAr2w

�+ S

i (3.63)

qdD(tdD) =q(t)

qi=

�1B�

2�kh(pi � pwf )

�1

2ln

�4A

e CAr2w

�+ S

�q(t) (3.64)

For two �owing phases, Raghavan (1993) provided an expression for the porevolume, Vp,

Vp =�1�2B�

(pi � pwf )�ct

�t

tdD

�M

�q

qdD

�M�

qdDq

�M

=�1B�a

2�kh(pi � pwf )and

�t

tdD

�M

=a�ct�A

2��2k


Fetkovich (1980) related decline exponent, b, to the production mechanismand to the exponent of the deliverability curve. The deliverability equation formultiphase �ow as derived by Fetkovich (1973), with n ranging from 0:5 � n � 1can be written

q0 = J 00(p2 � p2wf )

n (3.65)

The starting production at initial conditions, i, and producing cumulative oil,Np, that can be expressed with the material balance for the single phase �ow(under boundary dominated conditions) as

p� pi = �piNpi

Np

Further, assuming pwf � p and J 00 _ �, it is possible to obtain the followingexpression where exponent n varies between 1

2� n � 1

q0 = J 00ip

pip2n =

qoih1 + 2n

�qoiNpi

�ti 2n+12n�2

(3.66)

The decline exponent b = 12for n = 1

2, and b = 2

3for n = 1. For the multiphase

�ow condition Fetkovich (1980) derived the following expression

p2 � p2i = �p2iNpi

Np (3.67)

and the derived rate can be expressed according to

q0 =qoih

1 + 2n�12

�qoiNpi

�ti 2n+12n�2

Here the decline exponent b = 13for n = 1 and b = 0 for n = 1

2. Presuming

that reservoir boundaries in�uence the rate response Fetkovich (1980) related bto the drive mechanism as:

b = 0 for undersaturated oil or gravity drainage without a free surfaceb = 1

2for gravity drainage with a free surface

b = 23for a solution gas drive

During the transient rate, the decline exponent b > 1. The dimensionlessrate and time according to Fetkovich (1973) can be expressed as

qdD(tdD) =�1B�

2�kh(pi � pwf )

�ln

�rerw

�� 12

�q(t)

tdD =2��2k

�ct�Ahln�rerw

�� 1

2

it


Figure 3.30: The dimensionless rate vs. the dimensionless time Fetkovich typecurves [After Fetkovich (1980)].

Fetkovich et al. (1986) explained that the termhln�rerw

�� 1

2

i, as opposed toh

ln�rerw

�� 3

4

i, better matched the depletion exponential decline stem of b = 0 on

the right-hand side of the combined type curves. The term also better correlatedthe transient re

rwstems on the left side of the type-curves in Figure (3.30). So,

the term 12better match analytical transient re

rwstems and at the same time

provided a better �t to the exponential empirical decline stem of b = 0. Raghavancommented in 1993 that it is important to realise the di¤erence and consider theerror of this matching approach by using type-curves.According to Fetkovich (1973), a unique match of transient stems re

rwis only

possible if the same set of data matches one of the depletion stems de�ned byb. If a match is obtained only for the transient re

rwstems, the decline exponent b

should be greater than 1.As presented earlier, Camacho and Raghavan (1989b) provided a semi-analytical

explanation for matching b values in transient rate decline. Material balanceequations derived by Muskat (1945) with terms �t and ct are

dS0dp

=S0 (p)

B0(p)

dB0dp

+�o

�tct (3.68)

�t = �o + �g =

�kro�o

�+

�krg�g

�(3.69)


ct = �S0

B0(p)

dB0dp

� SgBg(p)

dBgdp

+S0Bg(p)

B0(p)

dRsdp

(3.70)

where�kri�i

�is the volume average of

�kri�i

�with i = o, g. Combining the above

equations with

d�

dt= ��3q0(t)

�Ah

Raghavan (1993) expressed

�ct�t

�kr0�0B0

dp

dt= ��3q0(t)

�Ah(3.71)

and de�ned pseudo-pressure related to average pressure as an integral with areference pressure independent of time pr as

m(p) =

p(t)Zpr

kr0�0B0

dp (3.72)

The above equation then becomes

�ct�t

�dm(p)

dt= ��3q0(t)

�Ah(3.73)

The Constant Pressure Production Conditions After integration and

assuming a rate-normalised pseudopressure, mD(wtAD), corresponding to the av-

erage pressure m(p), the following expression is obtained

mD(wtAD) = 2�

wtAD

mD(wtAD) =

2�kh

�1q0(t)[mp(pi)�mp(p)] =

2�kh

�1q0(t)

piZp(t)

�kr0�0B0

�dp


wtAD =

�2k

�Aq0(t)

tZo

q0(t0)�t(t

0)

ct(t0)dt0

The Constant Rate Production Conditions

mD(=tAD) = 2�

=tAD

mD(=tAD) =

2�kh

�1q0(t)[mp(pi)�mp(p)] =

2�kh

�1q0(t)

piZp(t)

�kr0�0B0

�dp

=tAD =

�2k

�A

tZo

�tctdt

Both inner boundary obtained expressions can be used under conditions changingfrom a constant rate to a constant production. The gas drive is given by theexpression:

mD(wtAD) = 2�

wtAD: (3.74)

During late-time and boundary-dominated �ow, the obtained results di¤er forliquid �ow. For a well producing under constant pressure, a gas drive value for

mD(wtAD) is obtained from

mD(wtAD) = �a

�1� e

2�=

tADa

�(3.75)

a =1

2ln

�4A

e CAr2w

�+ S

A single liquid dimensionless pressure drop pD = 2�kh�1q(t)B�

(pi � p) satis�es theabove expression.

2�mD(wtAD) = a

�e2�

=tADa � 1

�Expanding the exponential function e

2�=

tADa leads to inequality

wtAD >

=tAD (3.76)

The derivative of the cumulative oil, Np, over the average pressure, p, is obtainedby di¤erentiating above equations with time


dNp

dp= ��3

ct�0

�t�Ah

This expression corresponds to the prediction performance as a function of rela-tive permeabilities, �uid properties and pore volume. Further, Raghavan (1993)derived the following relation:

d ln q0dt

� �2��2k�Aa

�tct

(3.77)

Here, if �tctis constant then log q0 versus time t can be presented as a straight

line. This suggests that exponential decline b = 0 solutions can be used for the

performance prediction if=tD is employed. Raghavan (1993) introduced decline

curve parameters as a function of reservoir properties. Based on the transient-�ow analytically derived equations, he obtained the following expressions for Di

and b:

Di =

�2��2k

�Aa

�tct

�

b = � d

dt

1

d ln q0dt

!=

��A

2��2k

d

dt

�ct

�t

�With theses two expressions, derived from Arps�(1945) exponential decline re-lation, it is possible to comment on both rate data analyses and future rateperformances.

The Variation of ct�tand Decline Exponent b For a solution gas drive,

Fetkovich (1980) de�ned b as 13� b � 2

3whereas Raghavan (1993) de�ned it as

25� b � 4

5. Since, b = constant for ct

�tcorresponds to a linear function of time, the

decline exponent b is constant when �uid properties, relative permeabilities andthe in place volume all combine into a linear function of time. Since, b 6= constantfor ct

�tis not a linear function of time, �0(p) ! �0(pwf ) as time increases and

calculated rates should cross over several stems on type-curves or over severalvalues of the decline exponent b. This was also stated by Carter (1985) andFraim and Wattenbarger (1987).Camacho and Raghavan (1989) considered the following: the possibility to

derive analytically the constant decline exponent, b; the wellbore pressure be-haviour supporting the constant decline exponent, b.They published that, for the single layer system; the decline exponent, b, was

not constant for a constant pressure production case. Thus, b can be obtainedunder the following conditions: when a well produces under a variable pressure-variable rate; if the wellbore pressure increases with time (while skin and drainage


area remain constant); if the expression aAd�ct�t

�dt

is constant during the constantpressure production; and when the variable skin factor S exists in term a: Fora layered reservoir (both commingled or with interlayer communication), it ispossible to achieve the conditions yielding a constant decline exponent b.

The Use of Pseudo Time The following expression relates q0(t) to the

pseudotime=tD and is derived from Equation (3.77)

q0(t) = q0ie

�2�

=t ADa

!

The expression is based on the following simpli�cation

d ln q0

d=tAD

� �2�a

Plots of q0(t) versus pseudotime=t gives rise to a straight line. We now need

to compute the pseudotime=t . All above calculations are based on a simple

material balance equation of Muskat (1945). Raghavan (1993) suggested the use

of other simple material balance equations for studying the in�uence of�ct�t

�on

parameters Di and b.

Decline Curves and Transient �ow If only transient rates are available andno depletion rates are measured, the value of decline exponent, b, is greater thenunity. Camacho and Raghavan (1991) derived the following expression duringradial �ow for skin S

S =1

2

1� 1

d ln q0d ln t

� ln 4tDe

!and then combined it with decline expression for b thus giving

b = 2S + ln

�4tDe

�(3.78)

Raghavan (1993) stated that in Equation (3.78) during transient �ow the declineparameter b is a function of time t. For a decline exponent b greater then unity,the following inequality is valid for large and small reservoirs as well as for largeand small values of the dimensionless time tD

S >1

2

�1� ln 4tD

e

�Thus, parameter b is in most cases greater than 1 provided that transient re-sponses are used to predict the performance.


Figure 3.31: The �rst decline on Fetkovich�s type curve, for b > 0 [After Padillaand Camacho (2004)].

3.4.2 Solution Gas Drive and Gravity Drainage Decline

Padilla and Camacho (2004) examined the well and reservoir performance underthe combined e¤ects of solution gas-drive and gravity drainage in homogeneoussystem. The in�uence of various parameters like oil rate, position of the pro-ducing interval, wellbore pressure level, skin factor, and vertical permeabilitywere investigated by numerical simulation. Moreover, they found that duringthe boundary dominated �ow period, when gravitational forces are important,the production decline presents two decline periods with a stabilisation periodin between. This stabilisation period depends on wellbore pressure, geometrical,petrophysical and �uid properties. Camacho and Raghavan (1989) and Cama-cho (1987) showed that the production decline during the boundary dominated�ow under solution-gas-drive does not follow a �xed decline curve Figure (3.31)shows a match of several rate responsescorresponding to di¤erent values of wellbore pressure and three skin values

of Fetkovich�s type curve. It is evident that the data points, that also includegravity segregation forces, do not follow a �xed decline curve. As a consequence


Figure 3.32: The �rst decline on Fetkovich�s type curve, for b = 0 [After Padillaand Camacho (2004)].

of variation of �tctwith time production decline under gravity segregation does

not follow the type curves of Fetkovich. It was also observed that the datapoints do not follow the b = 0:5 curve as pointed out by Mathews and Lefkovits(1956). During the �rst decline period, when gravitational forces are importantthe �t

ctfunction is approximately constant with time and the production decline

is exponential as presented in Figure (3.32). However, when a second declineperiod is present the production decline is does not follow an exponential formas given in Figure (3.33). This was in agreement with results of Gentry andMcCray (1978).


Figure 3.33: The second decline on Fetkovich�s type curve, for b < 0. The declineexponent is negative and constant [After Padilla and Camacho (2004)].


3.5 Analysis of Well Production data

3.5.1 Type Curves and Decline Curve Analysis

A Transient Radial Flow Regime Transient �ow conditions start as a wellbecomes to �ow. Transient rate and pressure data are used to calculate thepermeability, thickness and skin. Once production from a well a¤ects the entiredrainage area the well �ows under pseudo-steady-state conditions also knownas boundary-dominated �ow conditions. Pseudo-steady state data are used tocalculate the decline exponent b and to determine the corresponding originaloil in place. Transient and pseudo-steady state or boundary-dominated declinebehaviour are di¤erent. It is found that a value of the decline exponent greaterthan one matches the transient rate-time data. Fetkovich et al. (2006) statedthat a b value greater than one is physically impossible.It is possible to determine unknown parameters from de�ned reservoir e¤ects

and �uid characteristics. The future production can thus be calculated with-out a prior production history. Such a method was implemented in a reservoirwith known physical quantities and composition. A numerical simulation of thede�ned geology and reservoir showed that a �uid system can change its initialdecline rate Di. A deviation in the relative permeability has a greater e¤ecton decline exponent b than changing �uid properties The initial production rateqi(t) depends on the permeability of the formation and initial water saturationand its magnitude depends on the �uid characteristics. Reservoir heterogeneitiesthus have a predictable e¤ect on the production history.Production history data can be plotted on a logarithmic scale versus time on

a linear scale and further extrapolated as a straight line into the future. Thisextrapolation is denoted as a constant percentage decline or exponential decline(b = 0) under the estimated production. A hyperbolic decline with a betterreliability is used to describe future production trends. On the same plots ofsemilog scale rate versus time, hyperbolic declines have been found to exhibitconcave upward behaviour. A technique used to �t the production data to ahyperbolic curve involves repetitive plotting of data points by trial and errorto obtain a straight line. Decline curve determination of the future productioncan be used by a trial and error procedure, graphical methods, and de�nedmathematical expressions.Data displaying a concave upwards trend indicate transient �ow, while data

presenting a concave downward bend indicate a pesudosteady-state �ow. From-early time data, it is possible to determine a dimensionless external radius andto calculate permeability and skin,S. The two dimensionless plots of qDd and tDdand the real data plot of q versus t are during matching shifted by the coe¢ cientsof rate, q in qDd, and time, t in tDd. Once matching qDd, tDd, reD and b has beendone it becomes possible to calculate the following reservoir variables: kh, re=rw,re and skin, S. Rate-time data is history matched on an appropriate log-log type


curve and then extrapolated to make a forecast.OOIP was calculated from type curves by:

OOIP =

��0B0

�icti (pi � pwf )

��t

tDd

q(t)

qDd

�match

The initial declining rate period can be considered as an extended drawdowntest. Early-time data on a rate-time type curve can be matched to obtain thepermeability, k. Depletion data exhibit a value of the decline exponent, b > 0,indicating changing values of

�kro�0B0

�and (�)p (ct)p, thus representing a re�ec-

tion of an increasing total compressibility with an increasing gas saturation, asstated by Fetkovich et al. (2006). The permeability-thickness, kh, is calcu-lated with rate-time transient data using the dimensionless rate, qD, versus. thedimensionless time, tD, as in Figure (3.21), also known as Cox type curves:

kh =

�141:2�0B0(pi � pwf )

��q(t)

qD

�match

The skin, S, is calculated from S = � ln rware.

A Transient Linear Flow Regime Due to a low mobility (heavy oil), theorientation of a horizontal well and the well length, the �ow regime can last fora long time until either a transient pseudo-radial or pseudo-steady state �owbegins. Pseudo-radial �ow is not typically observed in low mobility horizontalwells. Their �ow usually undergoes a transition from being linear transient tobecoming boundary-dominated. The estimated time needed for a horizontal wellpositioned in a centre of a square to reach pseudo-steady-state can be determinedwith the time, tpss, according to

tpss =379��0ctA

k

Joshi compared a horizontal well to a controlled in�nite conductivity verticalfracture of limited height. Well productivity of a horizontal well is a strongfunction of the reservoir thickness, h, and the permeability ratio, kv

kh:

Pseudo-steady-state productivity indices, PI, were estimated by Babu andOdeh (1989). They assumed a uniform�ux along a horizontal wellbore, describedby a set of simpli�ed equations. Goode and Kuchuk (1991) took for granted auniform pressure that was represented by a more complicated in�nite series.

Rate Testing and Well Testing

Transient and Boundary-Dominated Flow Stems The coupling of thetransient and boundary-dominated �ow stems may be accomplished in an em-pirical manner, such as that used by Fetkovich et al. (1980, 1987), or with a


theoretical basis as that used by Doublet and Blasingame (l995a, 1995b) andShih and Blasingame (1995). The Fetkovich empirical approach was previouslyaddressed in this text. The present section provides, a theoretical basis for cou-pling the transient and boundary-dominated �ow production decline behavioursinto a composite production decline curve set. The production decline behaviourof a well is governed by a number of variables, among which can be mentionedthe time level of interest and the speci�c properties of the reservoir and wellcompletion.The transient behaviour of the well is governed by the intrinsic properties of

the reservoir and the well completion e¢ ciency. The e¤ects of a �nite drainageareal extent of the reservoir do not a¤ect the transient production decline be-haviour of the well. Examples of the properties of the reservoir which govern theearly transient behaviour of the well include its e¤ective permeability,.porosity,�uid saturations, and �uid and rock properties. The e¤ects of a dual perme-ability or dual-porosity system are also factors in the early transient behaviourof the well. Some of the well-completion e¢ ciency properties that a¤ect theearly transient behaviour of the well are the system�s characteristic length (L),the fraction of the productive formation height that is open to �ow to the well,near-well stimulation or damage, and the speci�c completion design e¢ ciencies,such as those caused by perforations and gravel-pack completions. Other factorsmay also a¤ect the early transient behaviour of a well, such as inertial and/ormultiphase �ow in porous media that are �ow-rate- or time-dependent.The late-time boundary-dominated �ow behaviour of a well is also gov-

erned by the previously addressed reservoir and completion properties controllingthe transient behaviour. However, the boundary-dominated �ow behaviour ofthe well is more predominantly governed by the extent and shape of reservoirdrainage area, the location of the well within that drainage area, and the typesof boundaries that exist along the perimeter of the drainage area of the well,just as suggested by the name of the �ow regime. During the fully developedboundary-dominated �ow regime, the e¤ects of all boundaries of the reservoirare exhibited in the production decline behaviour of the well. A unique pro-duction decline-curve analysis of the historical production performance of a wellcan actually only be obtained when at least some of the production performancehistory spans at least a portion of both the transient and boundary-dominated�ow regimes.

Theoretical Basis for Coupling Stems The development of a set of compos-ite production decline curves for evaluating the reservoir and completion prop-erties using solutions of the rate-transient behaviour of a well in a �nite closedreservoir is actually quite simple and straightforward. Doublet and Blasingame(1995) presented both pressure and rate-transient approaches for establishing thenecessary ordinate and abscissa scaling parameters required to obtain conjuga-


tion of the transient and boundary-dominated �ow regime production decline be-haviour stems. The rate-transient approach is more applicable and directly pro-vides the required scale shift parameter values. The dimensionless rate-transientbehaviour of any well type (e.g., unfractured vertical, vertically fractured, orhorizontal) located in a closed �nite reservoir, during the late-time fully devel-oped boundary-dominated �ow regime, can be generalised in the form given byEquation (3.79).

qwD(tD) =1

�exp(��2�tDA

�) (3.79)

The dimensionless time referenced to drainage area, tDA, is de�ned in terms ofthe dimensionless time, tD, and the dimensionless drainage area, AD. The di-mensionless superposition time is determined in a manner similar to that usedfor the dimensionless material balance time, except that it relates the superposi-tion time function of a variable �ow rate history to its equivalent for a constantdrawdown pressure (i.e., inner-boundary condition) history.

tD(t) =k

��ctL2ete(t) (3.80)

The dimensionless time referenced to the drainage area, tDA, is thus de�ned as

tDA =tD(t)

ADThe dimensionless drainage area is simply the drainage area of the reservoirdivided by the square of the system characteristic length, Lc, or

AD =A

L2c

The system imaging function, �, that appears in Equation (3.79), is speci�cfor a given set of well completion and reservoir properties at a certain well lo-cation. The mathematical de�nitions of the system imaging function for themore commonly considered well completion and reservoir types were presentedby Postone and Poe (2008) in Chapter 8. They observed from such a construc-tion of a dimensionless decline analysis reference decline rate variable, and adimensionless decline time function results in a collapse of the whole family ofboundary-dominated �ow production decline stems to a single decline stem forthe late-time �ow regime. The dimensionless decline analysis reference declinerate variable is the product of the dimensionless well �ow rate and the sys-tem imaging function, �. The dimensionless decline time function is shown toconveniently incorporate the elements of the argument of the exponential func-tion. The resulting dimensionless decline �ow rate relationship obtained withthis variable substitution, applicable to the boundary-dominated �ow, is givenby


qDd(tDd) = exp(�tDd) (3.81)

Here, the dimensionless decline �ow rate, qDd, and the the dimensionless declinetime, tDd, reference functions are respectively given by:

qDd(tDd) = �qwD (3.82)

tDd =2�

�tDA (3.83)

The dimensionless cumulative production during the boundary-dominated �owregime can also be generalised for the rate-transient production decline behaviourof a well. The dimensionless cumulative production of a well (e.g., unfracturedvertical, vertically fractured, or horizontal) in a �nite closed reservoir is describedfor the boundary-dominated �ow by the expression:

QpD(tD) =AD2�

�1� exp

��2��tDA

��(3.84)

The de�nition of the dimensionless decline cumulative�production function, QpDd,is therefore an integration of the dimensionless decline �ow rate with respect tothe dimensionless decline time (including all values of tDd):

QpDd(tDd) =

tDdZ0

qDd(�)d� =2�QpDAD

(3.85)

For the boundary-dominated �ow we have:

QpDd(tDd) = 1� exp(�tDd) = 1� qDd(tDd)

Imaging Function The non dimensional image function, �, applicable for anunfractured vertical well that is centrally located in a closed, circular reservoir,is given by:

� = ln(reD)�3

4

there, the dimensionless drainage radius, reD, is de�ned in a conventional mannerby:

reD =reLc=rerw:

Similarly, the image function appropriate for a fully penetrating, unfracturedvertical well, located at a reservoir spatial position given by the coordinates(XwD; YwD) in a closed, rectangular reservoir of dimensions (XeD; YeD) ; whose


reference origin according to Poe (2003) is located at the lower left corner of therectangle, is given by:

� = 2�YeDXeD

�1

3� YDYeD

+Y 2D + Y 2

wD

2Y 2eD

�+ 2

1Xm=1

1

mcos(m�

XwD

XeD

) cos(m�XD

XeD)

:

:cosh

hm� Y eD�jYD�YwDj

XeD

i+ cosh

hm� Y eD�jYD+YwDj

XeD

isinh(m� YeD

XeD)

: (3.86)

Here, the dimensionless spatial parameters (XD; YD; XwD; YwD; XeD and YeD)are de�ned as the ratio of the corresponding dimensional spatial dimensionsX; Y;Xw; Y w;Xe;and Y e) to the characteristic system length (Lc = rw).During the early transient behaviour of a vertical well, the wellbore solution is

commonly evaluated using a line source well solution at a dimensionless reservoirspatial position away from the centre of the well, equal to the dimensionlesswellbore radius, rD = 1. However, under boundary-dominated �ow conditions,it is generally su¢ cient to simply evaluate the solution at the reservoir spatialposition, equal to the midpoint of the wellbore (XD = XwD; YD = YwD).It follows that the above solution can be simpli�ed into a more readily com-

putable form given as:

� = 2�YeDXeD

�1

3� YwDYeD

+Y 2wD

Y 2eD

�+ 2

1Xm=1

1

mcos(m�

XwD

XeD

) cos(m�XD

XeD)

:

�1 + exp(�2�m YeD

XeD

) + exp(�2�mYwDXeD

) + exp(�2�mYeD � YwDXeD

)

�

:

"1 +

1Xn=1

exp(�2mn� YeDXeD

#(3.87)

The dimensionless fracture conductivity, FCD, is de�ned as a relative mea-sure of the ratio of the fracture conductivity (kfbf ) to the formation e¤ectivepermeability, k, and the characteristic system length (Lc = Xf ) ; or convention-ally de�ned as:

FCD =kfbfkLc

=kfbfkXf

Poe (2005) evaluated the equivalent dimensionless fracture-spatial position X�D

as:


X�D = 0:7355�1:5609(

1

FCD)+1:5313(

1

FCD)2�179:4346( 1

FCD)3+3928:97:(

1

FCD)4

� 40211:24( 1

FCD)5 + 183267:48(

1

FCD)6 � 305367:26( 1

FCD)7 (3.88)

Moreover, X�D was used to accurately reproduce the wellbore rate or pressure-

transient behaviour of a �nite-conductivity fracture (for FCD � 4:1635) duringthe pseudoradial and boundary-dominated �ow regime. The equivalent fracturespatial position away from the well location at which to evaluate the uniform�ux fracture solution to obtain the equivalent wellbore response as that of anin�nite-conductivity vertical fracture, XD, obtained in above equations is equalto 0.7355. This value is only slightly larger than that commonly reported in theliterature (i.e., 0.732) for evaluating an in�nite-conductivity fracture responsefrom the uniform �ux solution, originally presented by Gringarten et al. (1974).The �nite conductivity fracture responses and the in�nite-conductivity fracturedwell responses are evaluated with the uniform �ux solution by the X�

D correla-tion. This X�

D correlation was used by Ozkan (1988) for de�ning the pseudoskinfunction caused by the bounded nature of the reservoir (�), determined as:

�(X�D; reD) =

�(x�D + 1)

3 � (x�D � 1)3�

12r2eD

The image function that applies for a �nite-conductivity vertically fracturedwell located in a closed, rectangular reservoir of dimensions XeD by YeD with themidpoint of the fracture located at (XwD; YwD) has been given by Poe (2002) inthe form of:

� = 2�YeDXeD

�1

3� YDYeD

+Y 2D + Y 2

wD

2Y 2eD

�

+2XeD

�

1Xm=1

1

m2sin(m�

1

XeD

) cos(m�XwD

XeD

) cos(m�XD

XeD)

:cosh

hm� Y eD�jYD�YwDj

XeD

i+ cosh

hm� Y eD�jYD+YwDj

XeD

isinh(m� YeD

XeD)

(3.89)

The solution for the image function of a vertically fractured well in a closed rec-tangle can also be simpli�ed into a more readily computable form for a wellborespatial position of (XwD; YwD) with the �nite-conductivity fracture evaluationspatial location of (XD = XwD +X�

D; YD = YwD), as given by:


� = 2�YeDXeD

�1

3� YDYeD

+Y 2D + Y 2

wD

2Y 2eD

�

+2XeD

�

1Xm=1

1

m2sin(m�

1

XeD

) cos(m�XwD

XeD

) cos(m�XD

XeD)

:

�1 + exp(�2�m YeD

XeD

) + exp(�2�mYwDXeD

) + exp(�2�mYeD � YwDXeD

)

�

:

"1 +

1Xn=1

exp(�2mn� YeDXeD

#(3.90)

Similar expressions of the appropriate image functions can also be derivedfor a horizontal well located in a closed circular or rectangular reservoirs. For auniform �ux horizontal well with the midpoint of the e¤ective wellbore length ex-posed to the reservoir centered in a closed, cylindrical reservoir, the appropriateimage function is presented in a readily computable form by Ozkan (1988):

� = ln(reD)+1

4+ �(XD; 0)+ �(XD; 0; reD)+

1

�LD

1Xn=1j

cos(n�ZD) + cos(n�ZwD)

n

8<:n�LD(1+XD)Z0

K0(u)du+

n�LD(1�XD)Z0

K0(u)du+K1(n�LDreD)

I1(n�LDreD)

24n�LD(1+XD)Z0

I0(u)du +

n�LD(1�XD)Z0

I0(u)du

359=;The above solution is expressed in terms of modi�ed Bessel functions of the �rstkind of orders zero and one (Io and I1), modi�ed Bessel functions of the secondkind, of orders zero and one (Ko and K1), as well as integrals of the modi�edBessel functions of the �rst and second kind, of order zero.For a horizontal well, centrally located in a cylindrical, bounded reservoir,

the dimensionless drainage radius is de�ned as the ratio of the e¤ective drainageradius of the circular reservoir divided by the characteristic system length, whichin this case is equal to half the e¤ective horizontal wellbore length in the pay

reD =reLc=2reLh

The imaging function for a uniform-�ux, horizontal well, located in a closed,rectangular reservoir, can also be derived using the late-time solutions reportedby Ozkan (1988). The image function for an in�nite-conductivity, horizontal


well located in a closed rectangular reservoir can be evaluated as the sum of thecorresponding image function of an in�nite-conductivity, vertical fracture in aclosed, rectangular reservoir, �f , given by Equation (3.90).Composite production decline curves for in�nite-conductivity horizontal wells

and its uniform �ux solution are evaluated at Xo = XwD+0:732 after Gringartenet a1. (1974) in closed, rectangular reservoirs have also been reported by Shih andBlasingame (1955). The appropriate wellbore solution is evaluated at the reser-voir spatial position (XD = XwD + X�

D; YD = YwD;and ZD = ZwD + rwD);withX�D = 0:732. The late-time imaging function, �, for a horizontal well (with all

its components) can be written as:

� = �f + PDb + PDb1 + PDb2 + PDb3

Values of PDb, PDb1, PDb2 and PDb3 when expressed in a readily computableform is quite lengthy and has not been here included. The interested reader can�nd it in the book of Postone and Poe (2008) on page 131-132.The use of the dimensionless decline variables given by Equations (??, 3.83

and 3.84) and the corresponding scaling associated with each variable for boththe transient and boundary-dominated �ow behaviours of the production declineof a well results in a composite reference decline curve set with only a single late-time stem for the sake of ease with regards to graphical matching purposes.

Integral Decline Curve Analysis Functions The dimensionless decline�ow rate-integral and rate-integral�derivative functions are introduced to im-prove the uniqueness of the graphical matching procedure. The dimensionlessdecline �ow rate-integral function is equivalent to the dimensionless decline cu-mulative�production function (3.85), normalised by the dimensionless declinetime (3.83). Moreover, the dimensionless �ow rate integral�derivative is equal tothe derivative of the dimensionless �ow rate integral with respect to the naturallogarithm of the dimensionless decline time function. These graphical analysisrelationships tend to display the same general trend as the dimensionless decline�ow rate function with respect to the dimensionless decline time. Moreover, theycan provide a clearer demarcation of the �ow regimes exhibited in the declinebehaviour of production performance of a well.The dimensionless �ow rate integral function introduced as an aid in graphical

production decline-curve analysis matching procedures is equal to the dimension-less decline time normalised dimensionless decline cumulative-production func-tion. The function, qDdi, is applicable for all values of dimensionless time, tDd,and is expressed as:

qqDdi(tDd) =1

tDd

tDdZo

qDd(�)d� =QpDd(tDd)

tDd


The function, qDdi, only applicable for boundary-dominated �ow, is expressedwith

qqDdi(tDd) =1

tDd[1� qqD(tDd)]

This function is signi�cantly smoother than the dimensionless decline �ow ratefunction (3.81), yet, despite this, it does not su¤er any appreciable loss in char-acter as a result of the production decline trend for a particular �ow regime.The derivative of the dimensionless �ow rate integral function with respect

to the natural logarithm of the dimensionless decline time function has alsobeen used to provide a more distinctive character of the transient productiondecline behaviour than either the dimensionless decline �ow rate or the �owrate integral functions. The enhanced signature that is characteristic of thedimensionless decline �ow rate integral-derivative function renders it extremelyuseful for identifying the start and end of a particular �ow regime, as well asfor improving the uniqueness of the decline-curve analysis graphical matchingprocedure. The derivative function applicable for all values of tDd is expressedas:

qDid(tDd) = �dqDdi(tDd)

d ln(tDd)= �tDd

dqDdi(tDd)

tDd= qDdi(tDd)� qDd(tDd) (3.91)

For the boundary-dominated �ow, qDid, is represented by

qDid(tDd) =1

tDd[1� qDd(tDd) [1 + tDd]] (3.92)

A graphical representation of the reference production decline curves for anunfractured vertical well centrally located in a closed, cylindrical reservoir, pre-sented in the more conventional manner (qDd versus tDd) is given in Figure (3.34).The dimensionless decline �ow rate response is displayed as black curves,

the dimensionless �ow rate integral behaviour is given as red curves, and thederivative response is presented in blue.Two types of single-phase �ow boundary �ux models were proposed by Dou-

blet and Blasingame (1995) for an unfractured vertical well centered in a cylin-drical bounded reservoir. These are the step-rate and ramp-rate boundary �uxmodels. The in�ux at the outer boundary is initially equal to zero (no-�owouter-boundary condition), which permits the use of the closed-boundary rate-transient decline curve analysis development procedure previously discussed, atleast for the limiting case of a no-in�ux outer-boundary condition.The no-in�ux case results in the exponential production decline given by

equation comprising terms of the modi�ed Bessel function of the �rst and sec-ond kinds, of orders of zero and one. The Laplace space transform parameter


Figure 3.34: Production decline curves for a �nite-conductivity, vertcally frac-tured well positioned in a closed rectangular reservoir.[after Poston and Poe(2008)].

(s) corresponds to the dimensionless time values at which solution should beevaluated.

vpwD(s) =

K0(ps)I1(

psreD) + I0(

ps)K1(

psreD)

Sps�K1(

ps)I1(

psreD)� I1(

ps)K1(

psreD)

�+

vqDext(s) [K0(

ps)I1(

ps) + I0(

ps)K1(

ps)]

psreD

�K1(

ps)I1(

psreD)� I1(

ps)K1(

psreD)

� (3.93)

a speci�ed �ux condition, qDext to several in�ux models is given as:

vqDext =

8<:0 no� flow

�1sqDext exp(�tDstarts) step� rate

�qDexts(1+tDstarts)

ramp� rate

9=;At a speci�c point in time, tDstart, in the production history, the outer-boundarycondition is switched from the initial no-�ow condition to a speci�ed �ux condi-tion, qDext.The real space rate-transient solution of the boundary �ux model can be

readily obtained by evaluation of the Laplace space dimensionless wellbore �owrate solution given by application of Duhamel�s theorem followed by a numericalinversion of the result into the real space domain by Stehfest (1970).The ramp-rate boundary �ux model assumes that the outer-boundary condi-

tion (initially at zero in�ux) smoothly and slowly increases from time-zero to a


Figure 3.35: The production decline curves for a vertical well positioned ina cylindrical reservoir with a step-rate �ux outer-boundary condition .[AfterPoston and Poe (2008)].

�xed larger value at a later time. This decline analysis model was developed toapproximate the production decline behaviours of reservoirs with natural waterin�ux or slowly responding water�ood systems. The production decline behav-iour obtained with the ramp-rate boundary �ux model is often found to be virtu-ally indistinguishable from the production decline behaviour of a dual-porosityreservoir.The step-rate boundary �ux model assumes that the outer-boundary condi-

tion is abruptly switched to a speci�ed �ux condition at a prescribed point intime, after which the in�ux rate is held constant. This decline analysis modelwas developed to address issues related to water�ood operations. The produc-tion decline behaviour obtained with the model exhibits production "humps"similar to those commonly observed in producing wells in water�ooded �elds.Multiphase numerical reservoir simulation runs were carried out in the inves-

tigation by Doublet and Blasingame (1995) to validate the use of the single-phaseanalysis step-rate and ramp-rate boundary �ux models for the production declineanalysis of two-phase systems, in which water displaced oil. It was found thatthe single-phase model developed in Doublet and Blasingame (1995) should beapplicable for oil/water two-phase systems with mobility ratios close to unity,potentially relevant to an injection-production system analogous to that of a �veor nine-spot pattern.The application of the recent developments in decline-curve model construc-

tion was also employed for developing production/injection decline curves for


Figure 3.36: The production decline curves for a vertical well positioned ina cylindrical reservoir with a ramp�rate �ux outer-boundary condition.[AfterPoston and Poe (2008)].

in�nite-conductivity vertically fractured wells located in closed, cylindrical reser-voirs (Doublet and Blasingame 1995). The uniform �ux fracture solution ofOzkan (1988) was in that study used to rapidly (and with reasonable accu-racy) estimate the in�nite-conductivity fractured well response at a dimension-less fracture spatial position from the wellbore equal to 0.732 after Gringartenet al. (1974). The generated set of reference decline-curves based on this solu-tion is presented in Figure (3.37) for a set range of dimensionless drainage radii,reD =

reXf, ranging from 1 to 1000.

A generally more appropriate production reference decline-curve set can beconstructed for an in�nite conductivity vertical fracture in a closed, rectangu-lar reservoir using the uniform �ux solution developed by Ozkan (1988). It hasbeen reported in the literature by a number of investigators that the e¤ectivelydraining the reservoir area (over which the pressure distribution was in�uenced)by a vertically fractured well in a low permeability reservoir is a direct functionof the e¤ective fracture half-length. This results in a drained area in the reser-voir that is typically more elliptical in shape during the transient �ow regimes.An elongated rectangular drainage area is therefore generally considered to bemore appropriate than a circular one during modelling of transient behaviourof vertically fractured wells in �nite reservoirs. Since the in�nite-conductivityfracture is simply a special case of the �nite-conductivity (FCD > 500) fracturedwell in a closed, rectangular reservoir, this limiting case can be readily includedin the reference decline-curve sets for a �nite conductivity vertical fracture in a


Figure 3.37: Production decline curves for an in�nite conductivity fracturedwell, centrally located in a closed, cyllindrical reservoir [After Postone and Poe(2008)].

rectangular reservoir.

Vertical Wells Intersected by Finite�Conductivity Fractures An ex-ample of a dimensionless production decline-curve set for a �nite-conductivityfracture is given by Figure (3.38). This decline-curve set was developed for a frac-tured well, centrally located in a rectangular drainage area. A complete familyof reference production decline curves for a �nite-conductivity fractured well in-cludes other decline-curve sets for a range of dimensionless fracture conductivityand drainage area and/or aspect ratio.

ln�nite-Conductivity Horizontal Wellbore Dimensionless production declinecurves for an in�nite-conductivity horizontal wellbore also have three or moreindependent parameters that govern their transient dimensionless productiondecline behaviour. For an in�nite conductivity horizontal wellbore centrally lo-cated in a closed, circular reservoir, the required image function for coupling thetransient and boundary-dominated �ow production decline behaviours is givenby Postone and Poe (2008). The independent variables determining the tran-sient behaviour of the well in this case include the dimensionless wellbore length,LD =

Lh2h, the dimensionless drainage radius, reD = re

Lc= 2re

Lh, the dimensionless

wellbore radius, rwD = rwh, the dimensionless wellbore vertical spatial position

(or stando¤ from the bottom) in the reservoir, and the dimensionless declinetime.


Figure 3.38: Production decline curves for a �nite-conductivity vertically frac-tured well centrally located in a closed [after Poston and Poe (2008) ].

The production decline behaviour of an in�nite-conductivity horizontal welllocated in a closed, rectangular reservoir is even more complex. The transientproduction decline behaviour of an in�nite-conductivity horizontal well locatedin a closed rectangle is a function of the dimensionless wellbore length, the di-mensionless wellbore radius, the stando¤ from the bottom of the reservoir, thereservoir drainage area, the wellbore midpoint location in the drainage area,and the dimensionless decline time. This list of parameters represents the mini-mum that must be considered. Additionally, the e¤ects of reservoir permeabilityanisotropy and dual-porosity behaviour may also be included in the family ofreference production decline curves, thus expanding the required number of ref-erence curve sets even further.

Calculation Procedure for Plotting Functions The analysis of the histor-ical production decline data of a well using the dimensionless production declinesolutions that are expressed in terms of the dimensionless decline variables of �owrate, �ow rate integral, and �ow rate integral-derivative requires that the appro-priate dimensional production decline functions be directly computed from thehistorical production data. The dimensional graphical analysis variables derivedfrom the historical production data used are the pressure drawdown normalised�ow rate, the pressure drawdown normalised �ow rate integral, and the pressuredrawdown normalised �ow rate integral-derivative functions.

Production Decline-Curve Analysis With Partial or Absent PressureRecord A problem commonly encountered in the evaluation of the production


performance of a well not all o¤ the required production data being available foranalysis. This problem arises quite frequently and may result from a numberof reasons. Whether performing a production decline-curve analysis using thematerial balance time approach, or a history match of the well performance databased on a numerical model employing superposition of the varying �ow rate and�owing pressure history, the �ow rates of each of the �uid phases as well as thebottomhole-�owing pressure are required at each time level in the productionhistory to correctly perform the analysis.

Assumptions and Limitations The e¤ective convolution analysis techniquepreviously described in this chapter is subject to limitations and assumptions.One such limitation, related to the use of the Horner approximation of thepseudoproducing time (e.g., material balance time), concerns the assumptionthat the �ow rate is only permitted to be smoothly varying. Therefore, an er-ratic well-�ow history shortly before a production data time level of interest canresult in a signi�cant error in the estimation of the equivalent superposition timefunction value for that time level. This limitation is also true for other produc-tion analysis methods that use the material balance time function as a substitutefor the superposition time function.

Chapter 4

RATE DECLINE OF AFRACTURED WELL

Remaining world-wide recoverable hydrocarbon resources exist in reservoirs pos-sessing poor permeability. At present, low production rates accompanying suchpoor permeability imply that some form of permeability enhancement or stimu-lation must be carried out within these reservoirs in order for hydrocarbons tobe economically exploited. Even where initial permeabilities are relatively high,stimulation may still be required to overcome problems associated with localisedpermeability damage due to, for example, drilling mud invasion.Hydraulic fracturing is widely used to increase the productivity of damaged

wells or wells producing from low permeability formations, and consists in in-creasing well e¤ective areal contact within a reservoir. Fractures can be posi-tioned either parallel or transversally to well length. Moreover the stimulationof a horizontal well in a low-permeability reservoir may further increase its pro-ductivity. Unlike a vertical well, a horizontal well may be fractured at morethan one point along the well length. A fracture has a much greater permeabil-ity than the formation it penetrates; hence, it in�uences the pressure and rateresponse of a well. Thus, much research has been carried out to determine thee¤ect of hydraulic fractures on pressure-transient and rate-transient behavioursin addition to well performance.This chapter discusses and develops the fractured well model responses. The

fractured well is positioned in a non-bounded or in�nite reservoir, and in abounded or closed reservoir. Solution for a vertical-fractured well solutions areextended from pressure-time to rate-time solutions. Furthermore solutions fora horizontal fractured well are new in development A vertical well fracture islongitudinal or parallel to the wellbore axis, whereas horizontal well fracturesare transversal and longitudinal. Here, we present model solutions that couplea well with fractures to an oil reservoir. Within the model, both fracture �owand wellbore �ow may be simulated, and for each model, initial and boundaryconditions for a di¤usion equation are set up. The di¤usion equation is then

127

128 4. RATE DECLINE OF A FRACTURED WELL

solved for the inner boundary condition of constant pressure, giving rise to well-rate responses in time. The inner boundary condition of the constant pressurecan be extended to a variable pressure condition. The new inner boundarycondition feature is the restart option. This feature combines the constant rateand constant pressure inner boundary condition within the same time interval,where the IBCs consist in a constant rate for a selected time. At the end of thattime, the model �nds the wellbore pressure and changing IBC to the constantwellbore pressure that is maintained unaltered until the end of the time interval.This restart feature is developed, implemented and tested, and the solution isdescribed in the third section.In the section fourth, we present the late time approximation in an in�nite

reservoir for a horizontal well that is transversally fractured. The developedapproximation relates rates of a multi-fractured horizontal well to those of avertical well.The approximations are useful and may convert complicated rate-time solu-

tions to simple expressions to be used in a screening analysis of a single wellproduction. The rate-time approximation developed for a horizontal-fracturedwell can be compared to a vertical well, a vertical-fractured well or a horizon-tal well. E¤ective wellbore radii and half-length expressions are developed tomeasure the e¤ectiveness of a multi-fractured horizontal well.The models presented in this chapter do not consider formation damage.

This parameter, a common problem associated with �eld operations which isof great importance for the e¢ cient exploration and production of hydrocarbonresources, should be considered in future studies, in which fracture skin and non-Darcy �ow may also be incorporated. The model is applicable to compressiblegas �ow, i.e., pressure, p, should be replaced by the pseudo-pressure, m(p), witha remark that inclusion of non-Darcy �ow should be further implemented.

4.1 Transient Oil Flow

The choice was made to develop rate-time solutions for two models; a vertical-fractured well model, and a horizontal fractured well. The two model solutionswere then summarised and well rates were plotted versus time solutions. Sincethe reservoir was an oil non-bounded, the �ow regime was transient or in�niteacting.

4.1.1 Fractured-Vertical Well Model

Hydraulic fracturing is a widely used method for increasing well productivity.Such productivity increases occur due to fracturing e¤ectively increases the wellsurface area, thus rendering �ow to the well much more e¢ cient. The increasein a surface area is achieved by injecting �uids into the formation at pressures

4. RATE DECLINE OF A FRACTURED WELL 129

above the formation parting pressure. Injection of a �uid at high pressure ini-tiates the fracture and causes it to propagate. The subsequent injection of aproppant allows the fracture to remain open. The �ow of �uids to a proppedfracture is much more e¢ cient than �ow to a wellbore. This is due to the wellborehaving a small surface area whereas that of a fracture may be very large. Frac-turing thus drastically increases well production, often rendering unpro�tablewells highly pro�table. Although hydraulic fracturing is usually very e¤ective,it is also very costly. It is thus essential that methods be available to evaluatethe e¤ectiveness of the fracturing process. The most widely used evaluation toolfor hydraulic fractures is pressure transient testing or recently, equivalently, ratedecline analysis.

Rate-time transient analysis consists of two phases, of which the �rst concernsposing and solving the equations that govern �ow in an idealised model. Thismodel is assumed to re�ect the mechanisms at work in the reservoir. The secondphase consists in matching the rate and pressure measurements, taken in the�eld, to those from the assumed model. The present work mainly investigatesthe �rst aspect of pressure transient testing of fractured wells posing and solvingthe equations that govern �ow in the model. Nevertheless the prediction ofthe pressure response of fractured wells is not a new topic. Numerous modelshave been investigated which consider various aspects of the problem. However,these models either consider only part of the problem, or only allow approximatesolutions of the governing equations.

The most comprehensive model to have been investigated is the �nite con-ductivity fracture model developed by Cinco and co-authors and presented inseveral papers. The two most important are Cinco-Ley et al. (1978), wherethe model is proposed and the governing equations solved, and Cinco-Ley andSamaniego (1981), where the behaviour of the solution is investigated.

The solution procedure used in the �rst of these papers is a numerical solutionof an integral equation. This technique has become known as the boundaryintegral equation method. The computation method is intensive, and since it isnumerical, yields only approximate results. The pressures computed using thismethod appear to be accurate, although it is di¢ cult to state just how accuratethey are.

In Cinco-Ley and Samaniego (1981), the behaviour of the well pressure isinvestigated. The intent of the present work is to put forward a model thatdepicts the �ow of �uids into and through a �nite conductivity vertical fracture.We seek an exact solution that can be used to determine pressures anywhere inthe fracture and reservoir system. The reason for pursuing an analytic solution isthat it is expected to be highly accurate and but time consuming. The ultimategoal is to provide a solution to which the computation is su¢ ciently rapid to beof use in computer-aided rate-time interpretations.

Here, we �nd a rate-time solution from the pressure solution of Cinco-Ley�s


model. The solution, was determined by altering the inner boundary conditionof constant rate production to a constant pressure production. The followingexpressions were implemented into the existing code and the output was testedversus cases found in the literature. We further compare the vertical fracturedwell model, the model by Cinco-Ley at al. (1978) to that of a horizontal wellwith one transversal fracture within an oil in�nite reservoir.By presenting rate solutions for a vertical well with a �nite conductivity

fracture and a horizontal well with a transversal �nite conductivity fracture,we �rst compared solutions for rate versus time. Moreover model di¤erencesand model similarities were discussed. The model for a horizontal well with atransversal fracture has more options that may be included into the vertical wellmodel. The model comparison helps to understand the di¤erences between ahorizontal fractured versus a vertical fractured model.The work of Cinco-Ley and Samaniego (1981) describes modelling features

of a vertical fractured well in an oil reservoir. Based on this, a pressure solutionwas extended to a rate solution for a vertical fractured well.The di¤usivity equation describes unsteady-state �ow in the system (of �nite

conductivity fractured vertical well in an in�nite reservoir of SLAB geometry).The properties of both the reservoir and the fracture are independent of pressureand the �ow in the entire system obeys Darcy�s law. The pressure gradients aresmall, gravity e¤ects are negligible, and the �ow into the wellbore is believed tocome through the fracture.The dimensionless wellbore pressure (drop) is given in "Laplace" space as:

pwD =C

s (s+ dps)

1

2

(4.1)

where C and d are constants, given by

C =� �

12fD

(kf � bj)D; d =

2�jD(kf bj)

(4.2)

where again

�fD =kf � ctk �f cft

(4.3)

(kj bf )D =kj bfk � xf

(4.4)

with bf , xf representing the fracture width and half-length.The corresponding dimensionless time is given by

tDxf =�kt

��Ctx2f(4.5)


where parameter � = 3:6 � 10�9Also,

pwD =kh (pi � pwf )

�o g Bo�(4.6)

with �o = 1:842 (�eld unit solutions).The model by Cinco-Ley (1982) de�nes the dimensionless pressure, pwD, and

we calculate the dimensionless rate, qD; with the known transformations in Equa-tion (4.7). Subsequently, all transformations and solutions are carried out in�Laplace�space.Hence,

q � pwD =1

s2(4.7)

which gives

q =1

s2

�1

pwD

�(4.8)

q =1

s2s (s+ d

ps)

12

C(4.9)

q =1

s

(s+ dps)

12

C(4.10)

there, q; is the rate ("Laplace" space) solution for a vertical fractured well basedon the work of Cinco-Ley and Samaniego (1981). A real rate solution can beobtained by the known Stehfest (1970) inverse "Laplace" technique (presentedin Appendix B).

4.1.2 Horizontal Well with Transversal Fractures

The general assumptions and limitations for a model of a multi-fractured hori-zontal well include: an in�nite (or geometry SLAB) reservoir that is isotropic oranisotropic, with no-�ow boundaries above and below. The well passes throughthe centre of transversal, rectangular, and fully penetrating fractures. Flow toa well occurs only through fractures, that are either of uniform �ux or �niteconductivity type, the latter being introduced through the "equivalent pres-sure point" method from references Gringarten and Ramey (1972), Chen et al.(1991), of Raghavan and Joshi (1993), Blasingame and Poe (1993). For the gen-eral case, fractures are uniformly spaced and sized. However, a limited numberof variations of sizing and spacing may be treated additionally. Inner boundaryconditions include: a constant rate or constant pressure, nevertheless, variable


Figure 4.1: A fractured-horizontal well of length, L, with three transversal frac-tures of half-lengths, Lf . The reservoir is non-bounded or in�nite in the x andy directions (Top view).


rate inner boundary condition may be employed. A top view of the model isgiven in Figure 4.1.The work presented in the next two sections are selected topics related to the

fractured horizontal well published by Cvetkovic et al. (1999). The transversaland longitudinal model solutions are utilised in section 3 (for the derivation of therestart option) and section 4 (for the late-time approximations, the equivalentwellbore radius and the equivalent half-length).The basic pressure (di¤usion) equation (with sources) can be expressed as:

r ��k

�rp�� c

@p

@t= q (4.11)

Here, the ordinary simplifying assumptions are made i.e., one-phase Darcy �ow,Newtonian �uids, isothermal conditions, gravity negligible, small pressure gra-dients, constant compressibility, viscosity, porosity and permeability.�, �uid viscosityk, permeability tensor�, porosity of the medium, constantq, source volumetric production rate per unit volume.For the principal axes of permeability (anisotropy) coinciding with the coor-

dinate the axes, we may write

r2p� 1�

@p

@t� �

kq = 0 (4.12)

where � = k�c�is the di¤usivity coe¢ cient of the porous medium. We have written

k � 3pkxkykz, to obtain an invariance of volume elements and the �uxes, while

the new coordinate variables are x0 =q

kkx� x etc., reducing to isotropy. By

introducing the usual dimensionless variables xD = x0=l, yD = y0=l, zD = z0=l,tD = �t=l2, qD = l2

�q, we �nally obtain the expression in dimensionless form:

r2Dp�

@pD@tD

� qD�c= 0 (4.13)

The application of the Laplace transform, de�ned by

f(s) = L ff(tD)g =1Z0

e�stDf(tD)dtD (4.14)

gives the simpler equation

r2Dp� sp =

~qD�c� pi (4.15)

where pi � p(tD = 0). Leaving out the details, we can assume a uniform initialpressure and write


�p =pis� p (4.16)

Further, assuming outer boundary conditions of a normally occurring type,Green0s identities can be used to obtain an integral representation of the fol-lowing kind:

�p =�

kl

ZS

q � dS 0 (4.17)

there, q is the (Laplace-transformed) volumetric rate of extraction through thesource S, while is a fundamental solution or Green0s function, depending onthe geometry and boundary conditions.Assuming now an in�nite reservoir geometry with impermeable boundaries

at z = 0 and z = h, one form of is:

=1

4�

+1X�1

exph�ps �p(x� x0)2 + (y � y0)2 + (z � z0 � 2nh)2

ip(x� x0)2 + (y � y0)2 + (z � z0 � 2nh)2

+exp

h�ps �p(x� x0)2 + (y � y0)2 + (z + z0 � 2nh)2

ip(x� x0)2 + (y � y0)2 + (z + z0 � 2nh)2

(4.18)

This is the direct result obtained by using the method of images on the funda-mental (Lord Kelvin0s) point source solution

o =exp(�R

ps)

4�R(4.19)

The integration in (4.18) is performed over the system of fractures in our case.Having assumed uniform �ux fractures, we �rst obtained the simpli�cation

�p =�

klq

ZS

dS 0 (4.20)

For several fractures, we get the linear superposition

�p =�

kl

nX1

qi

ZSi

idS0 (4.21)

To make things somewhat less general, the fractures are presumed to be rectan-gular, fully penetrating and transversal.Another basic assumption is that there is no direct �ow to the wellbore, which

is designated as being of in�nite conductivity and passing centrally through thefractures. Finally, we assume an equal spacing and size, choosing our referencelength,l, as equal to the fracture half-length, Lf .


Making use of Poisson0s summation formula enables us to express our solutionin a (formally) simple manner via modi�ed Bessel functions K0(R

ps):

2�kh

�q�pi � piD =

rkzkxk2

nXj=1

qjD

qkkxZ

0

K0

�ps �qu2 + (i� j)2�2

�du (4.22)

Here,

� =

skxky� L=Lfn� 1 (4.23)

where L is the distance between the outermost fractures. piD is then the nor-malised pressure (in "Laplace" space ) evaluated at the centre of the fractures.When demanding that these pressures be equal, a system of equations is

obtained:

q1I0 + q2I1 + ::::+ qnIn�1 = pq1I1 + q2I0 + ::::+ qnIn�2 = p

:::

q1In�1 + q2In�2 + :::+ qnI0 = pq1 + q2 + :::+ qn = q

9>>>>>>>>=>>>>>>>>;(4.24)

Here we have again employed a simpli�cation of k = kx = ky = kz, and can thuswrite:

Ij � Ij(s) �1Z0

K0

�ps �qx2 + j2�2

�dx: (4.25)

Where, Ij(s) is the coe¢ cient in the set of Equations (4.24) , and K0; is themodi�ed Bessel function of the second kind of zero order. From this set ofequations (4.24) it is desirable to eliminate q1, q2,......, qn. This gives us a relationof the following kind:

p = Bn(s) � q (4.26)

For one fracture it can be fairly easy determinated that n = 1 and

B1(s) = I0(s) (4.27)

Further, if we increase the number of fractures n = 2; :::; 4 we obtain


B2(s) =1

2I0(s) +

1

2I1(s) (4.28)

B3(s) =I20 (s) + I0(s)I2(s)� 2I21 (s)3I0(s)� 4I1(s) + I2(s)

(4.29)

B4(s) =I20 (s) + I0(s)I1(s)� I21 (s) + I0(s)I3(s)� 2I1(s)I2(s) + I1(s)I3(s)� I22 (s)

4I0(s)� 2I1(s)� 4I2(s) + 2I3(s)(4.30)

The expressions rapidly become bigger, with B7(s) being the last of theseexpressions to �ll less than one page. B10(s); instance needs more than 8 pages!The parameter B(n) relates the dimensionless pressure, p to the dimensionlessrate,q both in "Laplace" space. Late time solution for the equally sized andspaced fractures details are reported by Cvetkovic et al.(1996).

4.1.3 Horizontal Well with Longitudinal Fractures

Fractures of a horizontal well are of uniform �ux type, fully penetrating, evenlyspaced and have the same rectangular shape - with half-length, Lf . The wellboreruns through the middle of the reservoir (and fractures); there is no direct �ow tothe wellbore, and hence its length, L, is considered to extend from the beginningof the �rst fracture to the end of the last one. The top view of a model for ahorizontal well with three fractures is given in Figure (4.2).The dimensionless pressure calculated in formula (4.22) is further used in

[4.26]. For the normalised dimensionless pressure, an evaluation was carriedout at some point along the wellbore at fracture number i with (dimensionless)coordinate y:

�piD =1

2�2

NXj=1

j�+2Zj�

K0

�ps� jy � y0j

�dy0 (4.31)

y = 1 + i� + �

�1 � � � 1

Here,K0 is the standard modi�ed zero-order Bessel function andN is the numberof fractures. In this case we obtain:

� = 6

rkvkh

; � =

LLf� 2

N � 1


Figure 4.2: A fractured-horizontal well of length L , with three transversal frac-tures of half-lengt Lf . The reservoir is non-bounded or in�nite in the Xe andthe Ye directions (Cross section view).

valid for N � 2; and evidently when for N = 1; � = 0. Further for L � 2NLfimplies � � 2(N � 2): Moreover, changing variable from y0 = 1 + j� + � 0 to � 0

gives dimensionless pressure

�piD =1

2�2

NXj=1

�qjD

1Z�1

K0

�ps� j(i� j)� + � � � 0j

�d� 0 (4.32)

�piD =1

2�2

NXj=1

�qjDJji�jj(�)

Choosing i = 1; 2; :::; N; provides us with the same form of equations as in (4.22),(4.24), connecting fracture rates, �qjD; to the fracture (i.e., wellbore) pressure �piD.In this case, however, the pressure varies along each fracture, and it is crucial tomake a choice with regard to presenting the wellbore pressure. One possibilityis to choose the midpoint (� = 0) pressure (the natural choice for transver-sal fractures), but we will instead use the integral average, as suggested byRaghavan and Ozkan (1995) (of page 72) to obtain a good approximation foran in�nite-conductivity wellbore, although this renders the computations morecumbersome. In other words in (4.33), each Jk(�) is replaced by its integral


average:

1

2

1Z�1

Jk(�)d� � ~Jk

Changing the variables once again, this time by a factor �ps , we �nd:

~Jk =1

4s

2�psZ

0

2�psZ

0

K0

��k��ps+ y � x�� dxdy (4.33)

Dealing �rst with the special case where coe¢ cient, k = 0 (coinciding with thecase � = 0; i.e., N = 1) we �nd after some calculations:

~J0 =1

4s

2644�ps 2�psZ

0

K0(u)du� 22�psZ

0

uK0(u)du

375Simpli�ed, with properties of Bessel functions, we obtain:

~J0 =�ps

264 2�psZ

0

K0(u)du+K1(2�ps)

375� 1

2s(4.34)

Moreover, when k > 0 further calculations give

~Jk =1

4s

�f((k� � 2)�

ps)� 2f(k��

ps) + f((k� + 2)�

ps)�

(4.35)

where we have

f(x) � x

xZ0

K0(u)du+K1(x) (4.36)

For a numerical evaluation ~Jk given in (4.34), (4.35) and (4.36), we use Chebyshev-polynomial expressions to compute In for large s-values. However, di¤erent ver-sions are required here; furthermore, we will also cover �small� and moderates-values by the same kind of expansions. (small s corresponds to the large timeand large s to the early time).In addition to formula (4.36) we need also the modi�ed version for large

x-values

f(x) = x

24�2�

1Zx

K0(u)du+K1(x)

35 (4.37)


From Luke (1969) (of pages 341-343) we may directly use tabulated values forcoe¢ cients en and dn in

K1(x) =

r�

2xe�x

1Xn=0

enT�n

�5

x

�(x � 5) (4.38)

1Zx

K0(u)du =

r�

2xe�x

1Xn=0

dnT�n

�5

x

�(x � 5) (4.39)

Once again,

T �n(n) (4.40)

correspond to shifted Chebyshev polynomials, given by

T �n(x) = Tn(2x� 1) (4.41)

Tn(x) = cos(n cos�1 x) (4.42)

with explicit formulas for polynomial expansions given by Oberhettinger andBadii (1973) (page 15). However, there are smarter ways of carrying out thesecomputations, e.g., via recursion formulas given by Wimp (1962). Prior to this,in addition to (4.38) and (4.39) we need similar expansions valid for

jxj � 5

From Luke, Y. (1969), ( page 452-453) we have

K1(ax) =� + log

ax

2

�I1(ax) +

1

ax+

1Xn=0

HnT2n+1(x) (4.43)

I1(ax) =1Xn=0

DnT2n+1(x); (0<x � 1) (4.44)

with coe¢ cients Hn, Dn tabulated for a = 5 in Wimp (1962), (page 454). Similar

coe¢ cients for expansions ofxR0

K0(u)du have to be developed specially, which is

done below. We shall need the following coe¢ cients an

axZ0

I0(u)du =1Xn=0

anT2n+1(x); jxj � 1 (4.45)

In order to determine an a di¤erentiation is carried out giving:


aI0(ax) =1Xn=0

anT02n+1(x) (4.46)

Furthermore, we have:

I0(ax) =1Xn=0

CnT2n(x) (4.47)

with Cn calculated for a = 5 in Wimp (1962) (of page 454). Generally,

T2n(x) =1

2

"T 02n+1(x)

2n+ 1�T 0j2n�1j(x)

2n� 1

#(4.48)

see e.g., Luke (1969) (page 300). Hence, we obtain directly:

an =a

2(2n+ 1)(Cn � Cn+1) (4.49)

(n � 1)

a0 = aC0 (4.50)

Similarly, starting with the assumption

axZ0

K0(u)du = �� + log

ax

2

� axZ0

I0(u)du+1Xn=0

bnT2n+1(x) (4.51)

and then di¤erentiating, gives us:

aK0(ax) = �( + logax

2)aI0(ax)� x�1

axZ0

I0(u)du+

1Xn=0

bnT02n+1(x) (4.52)

This is to be compared to a formula from Wimp (1962) (of page 453):

K0(ax) = �( + logax

2)I0(ax) +

1Xn=0

GnT2n(x) (4.53)

Again, Gn is tabulated for a = 5 in Wimp (1962) (page 454).We now have a connection between bn, Gn and an; but in order to solve for

bn; according to Luke (1969), (page 298), we need to observe that:

xT 02n+1(x) = (2n+ 1)

"T2n+1(x) + 2

n�1X0

T2k+1(x)

#(4.54)


Deleting some steps of computation, this can be �inverted�to yield:

x�1T2n+1(x) =1

2n+ 1T 02n+1(x) + 2

n�1Xk=0

(�1)n�k2k + 1

T 02k+1(x) (4.55)

After some additional computations, we can conclude that:

bn =n

a=22n+1

(Gn �Gn+1)o+

1

2n+ 1

an + 2

1Xm=n+1

(�1)m+nam

!(4.56)

(n � 1)

Finally, we obtain the following very e¢ cient nesting (recursion) procedure an-nounced above to evaluate the sums occurring in (4.38), (4.39), (4.43), (4.44),(4.45) and (4.46), of the forms

f(x) =NX0

AnT�n(x) (4.57)

f(x) =NX0

AnT�n(x) (4.58)

g(x) =NX0

BnT2n+1(x) (4.59)

f(x) = a0 + a1(1� 2x) an = (4x� 2)an+1 � an+2 + An (4.60)

g(x) = (b0 � b1)x bn = (4x2 � 2)bn+1 � bn+2 +Bn (4.61)

Starting with aN+1 = aN+2 = bN+1 = bN+2 = 0 and using the backwardsrecursion in (4.60), we end up with our evaluations of (4.57). The procedure isbased on the work of Clenshaw (1955).

4.2 Depletion Oil Flow

This section deals with two semi-analytical model solutions, one for a verti-cal fractured well and another for a horizontal-fractured well. For each casewe develop and discuss the rate-time solutions by considering a close reservoirgeometry. Two fracture conductivities are considered: the in�nite conductivitiesand a uniform �ux. For each model, we set up several assumptions to facilitatea mathematical formulation that is made both accurate and simple. In the twomodel approaches, the following is assumed:


� The porous medium is isotropic, homogeneous, and bounded by upper andlower impermeable strata.� The reservoir is bounded, corresponding to a BOX geometry, of constant thick-ness, h. Other reservoir parameters such as porosity, �, and permeability, k, arealso assumed constant.� The reservoir contains a slightly compressible single-phase �uid of constantcompressibility, c, and constant viscosity, �. The �ow in the formation is as-sumed to be of Darcy type.� The well is produced at constant pressure and an incompressible non-Darcy�ow is assumed to occur within the fracture.

4.2.1 Fractured Vertical Well Model

Suppose that a vertical well is located in a closed or bounded rectangular reser-voir (a reservoir geometry also known as the BOX geometry). The reservoir ispositioned horizontally and extends in directions x, y and z. A fracture fullypenetrates the reservoir in its z and x directions. The above assumption for afracture geometry makes this model one dimensional with the variable, y, as aspace variable.The inner boundary condition is of a constant well �owing pressure, pwf .

Carslaw and Jaeger (1959) presented the dimensionless rate, qD; versus the di-mensionless time, tD; as a late time behaviour of the linear closed system. Vil-legaz (1997) solved the di¤usion equation in one dimension for the inner bound-ary condition of constant pressure.The system in Figure 4.3 simulates the behaviour of a vertical fractured well.

The �uid and the formation compressibility relate to pressure according to:

cf =1

�

d�

dp; c =

1

�

d�

dp(4.62)

The one-dimensional di¤usion equation is:

@2p

@y2=

��c

0:00633k

@p

@t(4.63)

and the initial and boundary (inner and outer) conditions are:

p(y; 0) = pi; p(0; t) = pwf ;@p

@y= 0 (4.64)

The solution is given for the pressure, p(y; t): The rate is calculated by

q(y; t) =@p(y; t)

@t y=0: (4.65)

Moreover, the variables for dimensionless time, tDLf


Figure 4.3: An in�nite conductivity vertical fracture fully penetrating the xdirection of a reservoir and the formation in the vertical z direction. The no-�owouter boundary condition de�nes the closed rectangular reservoir (Areal crosssection).

tDLf =0:00633k

��cLft (4.66)

and the dimensionless rate, qD; is

qD =141:2B�

(pi � pwf )q(t) (4.67)

This �nally de�nes qD(tDLf ) as,

qD�tDLf

�=4

�

�Lfye

� 1Xn=0

e

8<:�(2n+1)2 �2

!20@Lfye

1AtDLf

9=;(4.68)

The above solution can be inverted into the �Laplace�space. For this, Equation(4.68) is �rst subject to the following transformation:

qD =

�4

�

��Lfye

�r�

�t

1

2�3

�1

2j ��2t

�(4.69)


with

� =

�yeLf

�2: (4.70)

Now, by taking the "Laplace" transformation of qD; we have

L (qD) =2

�p�

1Xn=�1

(�1)nr�

se�2

p�sjnj

=2

�ps

1 + 2

1Xn=1

(�1)n e�2p�sn

!2

�ps

�1� 2 e�2

p�s

1 + e�2p�s

�

=2

�ps

1� e�2p�s

1 + e�2p�s=

2

�pstanh

p�s (4.71)

and �nally, after substituting Villegaz (1997) obtained for the "Laplace" trans-form of qD :

qD =2

�

tanh

�yeLf

�ps

: (4.72)

By using the inverse "Laplace" transformation of Stehfest (1970), it is possibleto invert the rate de�ned in the "Laplace" space, qD; to the real time solution,qD :

qD = L�1 (qD) (4.73)

The �nal comment is that �ow to a wellbore from the reservoir is facilitated bya vertical fracture of the in�nite fracture conductivity. The fracture is of length2Lf , width bf , and fully penetrates the wellbore. The fracture permeability,kf , is assumed constant. This simpli�ed modelling approach is one-dimensional.The vertical fracture fully penetrates the formation and completely extends inthe x;direction of the reservoir.

4.2.2 Horizontal Well with Transversal Fractures

Suppose that a model with a horizontal well is located in the middle of a closedand bounded (BOX type geometry reservoir). All BOX sides are no-�ow bound-aries. A well can be fractured with transversal fractures, and the fundamentalassumptions of a multi-fractured horizontal well model include:� Fractures that are vertically fully penetrating, of equal size and spacing, foreach of which is assumed a uniform �ux �ow.


Figure 4.4: A fractured-horizontal well of length L with three transversal frac-tures of half-length Lf . The reservoir is bounded and no-�ow boundaries are Xeand Ye (Areal cross section).

� No direct �ow to the wellbore, which passes perpendicularly through the mid-points of the fractures.� Additionally, there is no pressure drop along the wellbore.Omitting many details of the solution to the di¤usion equation with the inner

boundary condition of pressure or rate and the no-�ow outer boundary condition,we obtain an expression for the normalised dimensionless pressure (drop), pD,at the midpoint of a fracture number, i, as a linear combination of individualfracture rates �all in �Laplace" space.

piD =

NXj=1

Iij qjD (i = 1; 2; :::; N) (4.74)

For the coe¢ cients, Iij; by solving the di¤usion equation with sources throughGreen�s function method �omitting details we �nd:

Iij =��Lfxe

ch�ps+ ch�

psp

s sh�ps

+xe�Lf

1Xk=1

sin2k�Lfxe

k

ch�ps+ k2� + ch�

ps+ k2�p

s+ k2� sh�ps+ k2�

(4.75)


with coe¢ cient � equal to a

a =�yeLf

; � =4�2L2f�2x2e

� � �ij =�yeLf

� ji� jjd

� = �ij =�yeLf

� (i+ j � 2)d

d =�

Lf

L

N � 1 (N > 1); :::::::d = 0 (for N = 1)

� = 6

rkvkh

New variables, speci�c to the model geometry, are xe and ye giving the horizontalextension (side lengths) of the �BOX�model.Lf is the fracture half-length (inthe x-direction), and L is the well length (in the y-direction). The midpoint ofthe well coincides with the midpoint of the reservoir.The expression for Iij is numerically unsuitable, and thus it needs to be

transformed, which is re-entered possible through the connection with elliptictheta-functions. We �nd (omitting details) that:

Iij =Lf2ye

x0Z0

L�3

�x

2�

�� t�2�� 4

��

2a

�� ta2�

+ �3

�x

2�

�� t�2�� 4

��

2a

�� ta2�g dx (4.76)

where L stands for the "Laplace" transform with respect to t (dimensionlesstime), where x0 =

2� Lfxe

;and where the theta-functions �3 , �4 are given by thedual formulas:

�3

�x

2�

�� t�2�=

+1X�1

e��n2t cos nx (4.77)

�4

��

2a

�� ta2�=

+1X�1

(�1)n e��2

a2n2 t

cos� �

an (4.78)

�3

�x

2�

�� t�2�

=

r�

� t

+1X�1

e�(x+2� n)4� t

2 (4.79)

�4

��

2a

�� ta2�

=ap�t

+1X�1

e�[� + (2n+1)a]2

4t (4.80)


The two sets of formulas for �3, �4 have extremely good convergence properties,each one perfectly complementing the other. Using formulas (4.77, 4.78, 4.79,and 4.80) in the expression for Iij, we �nally arrive at the following:

Iij =� �

2 a

�aps� ch �

ps+ ch �

ps

sh aps

�1X0

"n (�1)n �cos

� �

an + cos

� �

an

s +�2

a2n2

�sh

"��p��

�rs +

�2

a2n2

#

sh

"�

�

rs +

�2

a2n2

# (4.81)

where

�n =2

1

n = 0

n > 2(4.82)

This �nal formula is very suitable for numerical calculations on the basic geom-etry BOX model.

4.2.3 Model Solutions Comparison

Model solutions of a vertical-fractured well and a horizontal-fractured well thatis coupled with a single transversal (longitudinal) fracture to an oil reservoirare considered, and rate-time obtained results are compared. There exist modelrate-time solutions for a vertical and a horizontal single-fracture well, as sketchedin Figure (4.5), positioned in a �nite reservoir geometry. Table (4-1) sumarises"Laplace" space rate-time solutions for the vertical fractured well and the hori-zontal fractured well (with a transversal fracture). We summarise the two modelsolutions: the vertical well-fractured model and the horizontal well-fracturedmodel. A vertical well fracture is vertical, longitudinal and has in�nite con-ductivity. A horizontal well is fractured with one transversal and uniform �uxfracture. The two model solutions are given in �Laplace" space. From the"Laplace" space solutions, the real time solutions may be obtained by the In-verse "Laplace" transform with the procedure of Stehfest (1973). The solutionfor both a vertical-fractured well, and a horizontal-fractured well (transversaland uniform �ux fracture) are developed for a closed bounded reservoir withan inner boundary condition of constant pressure. Model solutions are given inTable ??.From Table (4-1), we plot the two model rates, qD, versus, s, both in the

�Laplace" space. Figure (4.6) represents a fractured vertical-well rate qD versuss. Figure (4.7) presents a fractured horizontal-well rate, qD; versus, s. Forcomparison both solutions are plotted in Figure (4.8). Here we can commentthat a small s means a large time in a real space, whereas large s correspondsto early time solutions.


Table 4-1: Solution for a fractured-vertical (longitudinal) and a fractured-horizontal (transversal) well


Figure 4.5: A vertical-fractured, and a horizontal- fractured well (with transver-sal and longitudinal single fractures).

Table ( 4-1) model the vertical-fractured well (with a vertical in�nite con-ductivity fracture) and the horizontal-fractured (with a transversal uniform �uxfracture) well solutions. Both solutions are plotted in Figure (4.8).The real space solution should be obtained by the Inverse Laplace transform

method. Several Inverse Laplace transform methods are described in AppendixB.


Figure 4.6: The model for a vertical-fractured well rate, qD; versus s. Thefracture is longitudinal and of in�nite conductivity (�Laplace" space solutions).

Figure 4.7: The model for a fractured-horizontal (with a transversal fracture ofuniform �ux) well rate, qD ; versus s (�Laplace" space solutions).


Figure 4.8: The two model solutions for the rate, qD ; versus s (�Laplace" spacesolutions).

4.3 Additional Well-Fracture Features

4.3.1 Fracture Conductivity

If we further consider, the same reservoir pressure equation as before, with asolution for the dimensionless pressure (drop) in "Laplace" space we get:

�prD =NXi=1

1Z�1

�qji(x0)K0

hpspjx� x0j+ jy � yij

idx0 (4.83)

On the other hand, now also we have an equation for the (one-dimensional) �owin each fracture, yielding a pressure solution of the form (i=1,2,...,N)

�pfD = �qwif(x; s)�1Z

�1

�qji(x0)g(x� x0; s)dx0 (4.84)

Identifying the pressure expressions �prD and �pfD along each fracture gives asystem of integral equations for the fracture rates. By dividing each fractureinto a number of 2M gridblocks and assuming a uniform �ux over each block, wecan discretise the system of integral equations into a set of M.N linear algebraicequations for the discretised �uxes,

_qmn: The centre of each gridblock is chosen

for identi�cation of pressures.Further, as reported by Cvetkovic, Halvorsen and Sagen (1999), the total

�uxes,_qn,correspond to the term, �qwi; in Equation (4.84). For the total rate

_q

we have the following relation:


NXi=1

_qn =

_q (4.85)

This gives us a total of M:N +N + 1 equations for as many unknown qualities_qmn(m = 1; 2; :::;M:and n = 1; 2; :::; N),

_qn(n = 1; 2; :::; N), and either rate,

_q;

or pressure,_p: With a simpli�ed notation, we obtain the following system of

equations:

NXn=1

MXm=1

cm0n0(m;n)_qm0n0 = d(m)

_qn �

MXm0=1

dm(m)_qmn (4.86)

MXi=1

cm_qmn = c0

_qn �

_p (4.87)

NXi=1

_qn =

_q (4.88)

More details, including formulas are given by Cvetkovic et al. (1999).

4.3.2 Well Conductivity

To calculate the well pressure drop resulting from the �nite conductivity, weassume that the well�ow is stationary and laminar (viscous or streamline �ow).We use the so-called Hagen-Poiseuille law Equation (4.89) which states that thepressure drop is proportional to the well �ow and the distance travelled. Thewell �ow or the volumetric �ow rate, q , is a Newtonian �uid in steady laminar�ow through a pipe.

dp

dx=8�

�r4wq (4.89)

The Haugen-Poiseulle law, Equation (4.89), results from an averaging over thewellbore cross section of a parabolic velocity pro�le. This, in turn, is obtainedfrom integrating an assumed radially linear shear stress relation for a Newtonian�uid. With the usual notation for dimensionless quantities we get from Equation(4.89):

�pwD =16khLfr4w

qwD�xD (4.90)

Using this relation along the wellbore, summing the contributions from fractures1, 2, ... up to n, we obtain:


Table4-2:Finiteandin�niteconductivityfractures-parameters)

FiniteConductivity

In�niteConductivity

c 02� pscoth� p s

�2� s=

2�Lf

ws�

1

c m4� s

cosh

� M�m+1 2

M

� p s sinh

ps

2M

sinhps

4� s

12M=

2�Lf

Mws�

1

d(m)

2� ps

cosh

M�m+1 2ps

M

sinhps

2� s=

2�Lf

ws�

1(=limc 0)

dm0(m)

2� s

shps

2M

shps

h chM�jm�m0 j

ps +ch

M�m�m0 +1

M

ps

i2� s

12M2=

2�Lf

Mws�

1(=limc m)

dm

4� s

shps

4M

shps

h chps

4M

i chM�2m+1

M

ps +ch

4M�1

4M

ps

4� s

24M=

2�Lf

Mws�

1(=limc m

0 )

c m0n0

m�m0 +

1 2 Rm�m0 �

1 2M

Ko

� p sq x

2+(n�n0 )2�2

� dx+m+m0 +

1 2 Rm+m0 �

1 2M

Ko

� p sq x

2+(n�n0 )2�2

� dx

c m0n0as�niteconductivity

�=

L=Lf

N�1(N

>1)&�=0(N=1);�=

�Lf

b

q k k f; =q k f k

kf!1)f�!0& !1g


_pwD =

16khLwr4w(N � 1)

n�1Xi=1

(n� i)_q iD; n = 1; :::; N (4.91)

In Equation (4.91), fractures are numbered beginning at n = 1 at the toe of thewell and ending at n = N at the well�s heel. Incorporating this extra pressuredrop into the equations valid for an in�nitely conductive wellbore, gives us thefollowing modi�ed set of equations, to be substituted for Equation (4.87), thusleading to:

MXi=1

cm_qmn = c0

_qn�

_p+

16khLw(N � 1)r4w

"N�1Xn

(N � i)_q i + (N � n)

n�1X1

_q i

#; n = 1; 2; :::; N

(4.92)We note here that for n = N; the last bracketed expression equals 0, correspond-ing to the choice of a new wellbore pressure (drop), p;to be measured at thewell�s heel.

4.3.3 Fracture-Well Limited Communication (Choking Ef-fect)

Following the work of Schulte (1986), which disccused the in�uence of a limitedcommunication interval on the transient pressure behaviour and the long-termproductivity of a fractured well, Equation ( 4.94) presents the dimensionlesspressure (drop) or the "skin factor", Slc. This option was only implementedwithin the model for the IBC of constant rate. Derivation details was publishedby Cvetkovic, Halvorsen and Sagen (1999) (pages 8-11). From the formula forthe pressure (drop)

pD = �p2�kh

Nq�=

�k

Nkfbf�z (4.93)

and after averaging the total in�ow over N fractures the expression for the skinfactor, Slc or the dimensionless pressure (drop)

Slc = pD =hk

Nkfbflog sin

�erwh

(4.94)

Here, lf = erw corresponds to an "equivalent fracture half-length". This chokinge¤ect option is presented as a case study in Chapter 6.

4.3.4 Restart Option

Here we describe changing the Inner Boundary Condition (IBC) of constant-rateto one of constant-pressure within time for a multi-fractured horizontal well in


an in�nite reservoir SLAB model and closed reservoir BOX model. We generallyhave a linear relationship in "Laplace" space as follows:

p = B � q (4.95)

For the basic restart option treated here, the input parameters include a periodof constant rate, followed by the pressure maintained constant:

q = q0 for 0 < t < t0

p = p0 for t > t0 (4.96)

We assume that

p (0) = 0

and further that

p0 = limt!t0

p (t) (4.97)

The value of p (t) for t < t0 is easily calculated in the usual way.We need a new means of computing q (t) for t > t0, and the easiest way isdetailed bellow:From (4.95) we have:

q = B�1 � p (4.98)

which we rewrite

q = B�1 s�1 : sp (4.99)

This gives the convolution integral

q (t) =

t0Z0

L�1�B�1 s�1

�(t� �) � L�1 (sp) (�) d� (4.100)

generally now we have

L�1 (sp) = p0(t) (4.101)

provided that

p (0) = 0

as was also assumed. For


t > t0 ; p0(t) � 0

we hence obtain

q (t) =

t0Z0

F (t� �) � p0 (�) d� (t < t0) (4.102)

In Equation (4.102) F (t) represents the rate response to a unit pressure (drop)corresponding to

p = s�1 (4.103)

while p(t) is the pressure response of the system to the constant rate, q0; fort < t0. We may factor out the constant q0, relating also p(t) to a unit rate

(q � 1)

�nally obtaining

q (t) = q0

t0Z0

F (t� �) � p0 (�) d �

(where p(t) is now being a unit rate response.)A simple numerical approximation can be written:

q (t) � q0

N�1Xi=0

F (t� � i) [p (� i+1) � p (� i)] (4.104)

with � i = iNt0 and a su¢ ciently large N:Equation (4.104) is valid for t > t0

and. F (t) and p(t) are responses to the unit pressure and unit rate, respectively.Formula (4.95) results from the elimination of (n) individual fracture rates

from a linear system of (n+1) equations. These results are obtained from solvingthe basic pressure equation for the system of n fractures connected by a wellbore.In the simplest case concerning transversal, equally spaced and fully penetrating,uniform �ux fractures in a SLAB reservoir, our basic "transfer" function B(s)from formula (4.95) can be rationally expressed by integrals of the following type:

Ij � Ij (s) = �

�Z0

K0

�ps� �q

x2 + j2�2�dx (4.105)

with


� = 6

rkvkh

(4.106)

� =�L

(n� 1)Lf(4.107)

Where, L; length of wellbore, Lf ; half length of fracture. For 4 fractures, n = 4;we �nd function B(s)

B(s) =I20 (s) + I0 (s)I1(s) � I21 (s) + I0(s) I3(s)

4I0(s)� 2I1(s)� 4I2(s) + 2I3(s)

+I1(s) I3(s)� 2I1(s) I2(s) � I22 (s)

4I0(s)� 2I1(s)� 4I2(s) + 2I3(s)

With an increasing number of fractures, the expression for B(s) becomes verylong. We can nevertheless, easily handle 50 or more fractures, as we have to ourdisposal fast and robust numerical methods that approximate the Bessel functionintegrals by Chebychev polynomials that is combined with a �exible numerical"Laplace" inversion method.

4.3.5 Late Time Approximations

In this section, we develop approximate late time expressions for a fractured-horizontal well. These expressions are related to a vertical well production, and aproduction of a horizontal well (without fractures) production and to a horizontalwith a single transversal fracture. In order to measure the production e¢ ciencyof a fractured-horizontal well, we introduce the e¤ective wellbore radius and thee¤ective single-fracture half-length. The e¤ective wellbore radius is de�ned for avertical well without fractures, and for a horizontal well, also without fractures.Furthermore the e¤ective half-length is de�ned for a horizontal well with a singletransversal fracture. It is assumed that a transversal fracture �ow signi�es auniform �ux. Late time approximations are developed for a fractured-horizontalwell positioned in the middle of an in�nite reservoir. The developed expressionsfrom Section (4.1) for a multi-fractured horizontal well are utilised for de�ningan e¤ective wellbore radius and an e¤ective fracture half-length. The productionof a horizontal-fractured well may be related to the vertical well production inaddition to the single-fracture horizontal well production. Figure (4.9) illustratesthe models mentioned above.

Vertical-Well E¤ective Wellbore Radius

The following relation is developed for a vertical-well e¤ective wellbore radius,rwv:


Figure 4.9: Models for a fractured-horizontal well, the e¤ective wellbore radiusof a vertical well, and the e¤ective half-length of a horizontal well with a singletransversal fracture.

rwv = 2Lfe�( +AN ) (4.108)

Here, Lf ; as before, refers to the half-length of the fractures, is Euler�s constant( � :577), N is the number of fractures, and AN is given as:

AN = A0 +K00; (4.109)

for which the generalised expression K00 is de�ned in Equation (4.112) and A0corresponds to:

A0 = log 2� + 1; (4.110)

yielding

rwv =Lfe� e�K00 (4.111)

As an example, from Equation (4.111) we calculate the vertical-well e¤ectivewellbore radius for a fractured-horizontal well with number of fractures N= 1,2, and 3.For N = 1, we obtain K00 = 0, which gives the same value for the e¤ective

wellbore radius, rwv; as noted by Hegre and Larsen (1994). We can obtain a


general, explicit value for K00, and hence also for rwv, in the case of a generalN . The generalised expression for K00 is:

K00 =�eNF

�10 eTN

��1(4.112)

where the matrix elements of F0 are:

(F0)m;n = �r cot�1 r �1

2log(1 + r2) (4.113)

and where r = jm� nj � L=LfN�1 (N � 2). Explicitly, we also �nd, for N = 2 and

r = LLf

K00 = �1

2r cot�1 r � 1

4log(1 + r2); (4.114)

for N = 3:

K00 =2a2

4a� b; (4.115)

where

a = r cot�1 r+1

2log(1+r2); b = 2r cot�1(2r)+

1

2log(1+4r2); r =

L

2Lf(4.116)

Hence, for various N;we obtain expressions for rwv including dimensionless ra-dius r as r =

�LLf

�:

N = 1 : rwv =Lfe

(4.117)

N = 2 : rwv =Lfe� 4p1 + r2 � e

1

2r cot�1 rLf

e� e1=2r1=2 (r !1); (4.118)

N = 3 : rwv =Lfe� exp

8>>><>>>:2

�r cot�1 r +

1

2log(1 + r2)

�22r cot�1

2r3

1 + 3r2+1

2log

(1 + r2)4

1 + 4r2

9>>>=>>>;

� Lfe� 22=9 � e4=3 � r2=3(r !1); for r =

�L

2Lf

�(4.119)

corresponding to an identi�cation of the late time behaviour of a cylindrical,fully penetrating well and our fractured horizontal well. When we actually com-pare "Laplace" transforms for large time, t , (or small s-values), starting with a


classical expression for the vertical well dimensionless pressure transform, pD isgiven by Raghavan (1993), as

pD = C � K0 (rwDps)

rwDpsK1 (rwD

ps)� s�1 (4.120)

where�C = �qB�

2�kh

�. For late time behaviour (or small s-values), we get the di-

mensionless pressure, pD;

pD �1

2Cs�1 log

�e2 sr2wD4

�(4.121)

where,

rwD =rwLf

(4.122)

For the fractured horizontal well:

pD = �C

2s�1 (log s� 2AN +O(s log s))� C

2s�1 log

�se�2AN

�(4.123)

By identifying the two expressions for pD, and dropping terms corresponding toterms on the order of t�1(t!1) in the real time variable, we �nd the result fora vertical well e¤ective wellbore radius, rwv reported at the start of this sectionin Equation (4.111).

Horizontal-Well E¤ective Wellbore Radius

An e¤ective wellbore radius converts the productivity of a horizontal well intoan equivalent vertical well productivity.

r0w = rwe�s

Joshi (1991) presented a steady-state solution for an e¤ective radius. For ahorizontal and a vertical well, the drainage volumes were set to be equal, reh =rev, as were the productivity indices:�

q

�p

�v

=

�q

�p

�h

(4.124)

By substituting the steady-state rate solutions for a horizontal well rate, qh anda vertical well rate, qv, we can de�ne the expression for an e¤ective wellboreradius of a horizontal well, rwh:

�q

�p

�v

=

2�khh

�oBo

lnrerw

�q

�p

�h

=


=

0BBBBBBBBBBBBBBB@

2�khh

�oBo

ln

266664a+

sa2 �

�L

2

�2L

2

377775+�h

L

�ln

�h

2rw

�

1CCCCCCCCCCCCCCCAh

(4.125)

where

a = (L=2)

r0:5 +

q0:25 + (2rehL)

4 (4.126)

The equal productivity indices from above provide an e¤ective horizontal well-bore radius in an isotropic reservoir:

rwh =rehL

2a

�1 +

q1� [L= (2a)]2

�[h=(2rw)]

(h=L)

(4.127)

Now, by including

Lfe= rwv =

Lf (N)

ee�Koo(N) (4.128)

we are able to express the horizontal wellbore radius, rw; to a single fracture halflength, Lf .For an anisotropic reservoir the e¤ective wellbore radius becomes:

rwh =rehL

2a

�1 +

q1� [L= (2a)]2

�[�h=(2rw)]

(�h=L)

(4.129)

The above work can be extended to several e¤ects such as shape factor, wellposition and well skin.

A Multifractured-Horizontal Well E¤ective Wellbore Radius

The e¤ective wellbore radius for a horizontal-well with N transversal fracturesbeing associated to a vertical well corresponds to:

rwv(N) =rweN=Lf (N)

ee�Koo(N); (4.130)


For one fracture, N = 1 and Koo = 0, so

rwv(1) =Lf (1)

e(4.131)

or by equalizing rwv(1) = rwv(N), we get,

Lf (1)

e=

Lf (N)

ee�Koo(N) (4.132)

and

Lf (1) = Lf (N) e�Koo(N) (4.133)

or

Lfeff = e � rwv (N) (4.134)

Further, the comparison of solutions a horizontal-fractured well (with a singletransversal fracture) and those from literature concerning a vertical-fracturedwell in an in�nite reservoir are given in Table (4-3).


Table 4-3: Model solutions for a fractured-vertical well as compared to those ofa fractured-horizontal (single-transversal-fracture) well

Chapter 5

RATE DECLINE WITH AMOVING BOUNDARY

Arps empirical depletion-rate solutions from the 1950�s involving conventionalinterpretation procedures were in the 1980�s enlarged into combined transient-depletion rate type curves. Together with these curves, a decline exponent,b; and an initial decline, Di; were for the �rst time introduced by means ofreservoir-production parameters through the extensive study of Fetkovich (1980).Raghavan (1993) extended the analysis of the multiphase �ow in a reservoir andpresented the theoretical view on the decline exponent, b. He summarised themost critical items required for matching the decline exponent, b. He furtherstated limitations to the empirical and universal decline curve theory experiencedin the 1980�s. Raghavan�s (1993) study evaluates in a more sophisticated mannerthe possibility of matching the decline curvature de�ned by the decline exponent,b.Among others, the following issues were discussed by Raghavan (1993) (pages

527-528) concerning the production from a single-layer reservoir:1. The possibility to obtain a constant value of exponent b for a wellbore

constant-pressure production.2. The nature of the wellbore pressure response when the production is forced

to follow a speci�c value of parameter b in the Arps equations.3. The inclusion of the variable skin factor and a constant value of exponent

b.Camacho and Raghavan (1989) reported that exponent b in Arps equation

is not constant for an inner boundary condition of constant pressure. Moreoverthey stated a wellbore condition of variable pressure for which the rate may beforced to follow a speci�c stem of exponent b. They come to the conclusion that,in some cases, even the wellbore pressure must increase with time leading to thewellbore response following the speci�c value of the exponent b. This was basedon the premise that a single-layer system was produced, and was relevant to awell without skin zone properties and with the constant drainage area. Carter

165

166 5. RATE DECLINE WITH A MOVING BOUNDARY

(1985) with Fraim and Wattenbarger (1987) supported the statement that thetheoretical solutions can not match the exponent b, and moreover that theyshould cut accross several values of the decline exponent, b. Another option wasto include the wellbore skin. Then, by varying skin factor for a well producingat a constant pressure condition, a production characterised by the constantdecline exponent, b is obtained. If the reservoir is layered (with a commingledproduction or a system with interlayer communication), a case can be establishedfor a constant value of exponent b. Both Fetkovich and Raghavan noticed thattransient rate data matched depletion rate data, usually with an exponent bthat is grater than 1. Several case studies have demonstrated such a match ascommented by Raghavan (1993).

Further, this chapter mainly investigates the nature of the wellbore pressureresponse when production is forced to follow a speci�c value of exponent b. Wehere present a physical model and solve the di¤usion equation with variable-rateinner-boundary-conditions for several selected Arps exponents, b, and for no-�ow moving outer-boundary conditions. All obtained variable pressure functionswith time were analytically derived. This way, we demonstrate the nature of thewellbore pressure response (and also the pressure response within a drainagearea of a vertical well) when production is forced to follow a speci�c parameter,b, in the Arps equations.

The present chapter deals with pressure solutions of a well positioned in anoil and gas reservoir. In order to investigate the pressure behaviour with time,we de�ne the physical model with the wellbore variable-rate condition of Arp�stype, and no-�ow speci�ed moving outer boundary. The model input parametersare rates de�ned by Arps and moving outer boundary, whereas output data arepressure solutions that are analytically derived. The no-�ow outer boundarymoves with one speed of outwards moving. How this speed may be relatedto the reservoir condition and drive mechanism should be further investigated.Since no-�ow boundary moves with time, its speed of a no-�ow boundary alsochanges with time. Ideally, the speed of a no-�ow boundary will stop when adrainage area is reached. For such moving boundary production forced to followa speci�c value of the decline exponent, b; it would be possible to solve thedi¤usion equation for wellbore pressure responses. The model for the change indrainage volume is transient, but at the same time the rate-time is depletionkind of variable rate being of Arps type. Also a no-�ow boundary moves anddoes not have �xed position at the drainage radius. Ideally, the moving no-�owboundary should reach the drainage radius at the end of the Arps decline.

Solution to the di¤usion equation for initial and �xed boundary conditionshave been well explained in the petroleum literature. We solve the di¤usionequation for a single well situated in a circular homogeneous reservoir. To be ableto match Arps exponent, it is necessary to assume a moving no-�ow boundary.We estimate the position of a no-�ow boundary, that moves outwards from a

5. RATE DECLINE WITH A MOVING BOUNDARY 167

wellbore axis with the speed proportional to square a root of time. By assumingsuch speed, it is possible to solve the di¤usion equation for the variable-rate IBCof Arps type. The constant of proportionality, �, may have a physical meaning.It is supposed to be related to a driving force moving a no-�ow boundary. Ateach point in the drainage area, the solution consists in a variable-pressure withtime. Pressure response solutions of the di¤usion equation are presented in thefollowing sections, for both for oil and gas �ow.

5.1 Vertical Well �Oil Flow

This section considers the following topics related mainly to transient rate decline:� Solutions to the di¤usion equation for an oil well in which the wellbore

production varies with time.� A well variable rate production of Arps type, i.e., where the decline expo-

nent, b is known.� A no-�ow boundary moving (not �xed) outwards from a wellbore axis.� A solution corresponding to the pressure response to a variable rate decline,

de�ned by the exponent b and the speed of the moving boundary.� The basis for further studying the physical meaning of a coe¢ cient, � (�

relates the dimensionless position, rD; and speed, drDdtD

; of the no-�ow movingboundary with the dimensionless time, tD).In this chapter we consider the moving boundary solutions for the wellbore

conditions of variable rates (empirical rates de�ned by Arps). The function is introduced instead of rate q such function correspond to specially ratesof power with a negative exponent .The solutions are thus limited to speci�cchoices of rates, . In order to solve di¤usion equation with Arps rate decline,it is necessary to assume a moving no-�ow boundary. With time the no-�owboundary moves outwards from a wellbore axis. In addition, we have in thisstudy assumed that the zone skin is zero. To our knowledge, for the di¤usionequation analytical solutions for the inner boundary condition of an Arps ratedecline have as of yet not been presented. The production in a such case is forcedto follow a speci�c value of exponent b in the Arps equation. As the drainagevolume changes due to a moving no-�ow boundary, we consider the rate to betransient, i.e., combined with a decline de�ned by the Arps exponent, b.

5.1.1 Introduction to Moving Boundary Problems

Tarzia (2000) provided a comprehensive bibliography of moving and free bound-ary problems for the heat-di¤usion equation. This review contained almost6000 references in various kinds of publications; mostly western and from amathematical-physical-engineering literature. A very �rst publication on a mov-ing boundary is that by Lame and Clapeyron (1981). Italian literature (Fasano-


Primicerio�s group) has classi�ed problems for either the heat or di¤usion equa-tion, due to boundary problems being divided into: �xed, moving and free.

Cryer (1978) discussed the relationship between moving boundary problems(parabolic and time-dependent) and free boundary problems (elliptic and steadystate). Free boundary details are well presented by Tarzia (2000) in the followingcitation: "The free boundary problems for the heat equation are those in whichthe spatial domain of the unknown function varies with time because of a lawof movement not known a priori. The fact of not knowing boundary or part ofit, determines, of course, the mathematical need to impose new condition on theunknown function, which will depend on the physical model studied. In general,the new condition to be imposed on the unknown function is deduced from theprinciple of conservation of energy across the boundary. Thus it follows thatthis boundary is the complementary unknown of the problem, and is called freeboundary of the problem under analysis." The free moving boundary will not bea subject of study in this thesis.

5.1.2 Fixed Boundary

Inner Boundary Conditions of (Constant) Pressure

Boundary-dominated or depletion rate decline models for a vertical well are em-pirical, analytical and numerical. Ehlig-Economides and Ramey (1981) solvedthe di¤usion equation analytically for an inner boundary condition of constantpressure (de�ned at a wellbore), and an outer boundary condition of no-�ow(de�ned at the distance of drainage radius, re). They provided an expression forthe rate-time decline of various dimensionless radii, rD. For a reservoir closedor no-�ow boundary the rate declines exponentially with time. Thus, an ana-lytically derived exponential decline overlays the empirical Arps decline de�nedby the decline exponent, b = 0. This analytically derived model solution is idealsince it is limited to a single-phase �ow within a homogeneous oil reservoir.

In the empirical modelling approach by Arps (1945), the rate declines bothexponentially and hyperbolically with time. An extensive summary of solutionsare given in Chapter 2. Cvetkovic (1992) reviewed transient and depletion lay-ered reservoir solutions, and Cvetkovic and Gudmundsson (1993) have writtena bibliography of analytical models for constant wellbore pressure IBC. Thetransient-depletion rate production was de�ned by OBCs that were being bothin�nite acting and no-�ow but �xed. It would be challenging to consider a tran-sient rate decline with IBCs of constant and variable pressure and a no-�owboundary moving outwards. We also consider rate depletion (de�ned as variablerate at a wellbore) controlled by a no-�ow boundary moving inwards, from thetime when the pseudo-steady-state conditions are reached.


Inner Boundary Conditions of (Variable) Pressure

Deconvolution is a diagnostic tool that provides the equivalent rate or constantpressure response of a reservoir system a¤ected by variable-rate or pressure pro-duction. The method applies to both production and pressure test data analysis.Due to the fact that production data are generally of poor quality, their analysismay not be realistic. Deconvolution may provide signi�cant diagnostics valuesfor production data, as found in recently published references: Agarwal et al.(1999), Araya and Ozkan (2002), von Scroeter et al. (2002), Levitan (2005),Levitan et al. (2006), Ilk and Valko (2005), Ilk and Valko (2006), and Kuchuket al. (2005).

5.1.3 Moving Boundary

We consider here an isotropic, homogeneous, strati�ed or horizontal that is in-�nite in two horizontal directions of a moving no-�ow boundary. Within thereservoir, the thickness, permeability and porosity are uniform The formationproperties are independent of pressure. An upper and a lower impermeable layerbind the slab reservoir, which contains a single-phase, slightly compressible �uid.The compressibility and viscosity are assumed constant.Fluid is produced through a fully perforated vertical well, i.e., the perforations

extend over the total vertical height of the reservoir. Originally, the well isassumed as a line source. At outer boundary conditions, of the reservoir weincorporate no-�ow boundaries, nevertheless moving outwards from the wellboreaxis. The �uid �ows towards a wellbore of a single-phase slightly, compressible�uid in a porous medium assuming constant reservoir parameters governed bythe linear partial di¤erential equation in one independent variable pressure, p.This parabolic pressure equation is given in a known radial from by:

1

r

@

@r

�r@p

@r

�=1

�

@p

@t(5.1)

The IBC of the wellbore consists in a constant rate condition and a variablerate condition. The pressure, p; is a function of the radial distance, r; andthe time, t. The coe¢ cient, � = �

k�c; is known as the di¤usion coe¢ cient. By

introducing the dimensionless time, tD; as tD =�r2wt, the dimensionless distance,

rD =rrw; and dimensionless pressure, pD =

pi�p(r;t)pi�pwf ; we obtain a dimensionless

form of equation (5.1):

1

rD

@

@rD

�rD

@pD@rD

�=@pD@tD

(5.2)

We now introduce the time variable, � , such that � = � t. The di¤usion equation(5.1) thus becomes:


1

r

@

@r

�r@p

@r

�=

@p

@(�t)=@p

@�(5.3)

Alternatively, in dimensionless form with �D = � tD; the Equation (5.3) can bewritten:

1

rD

@

@rD

�rD

@pD@rD

�=@pD@�D

(5.4)

corresponding to the di¤usion equation in the dimensionless form of rD, �D,and the dimensionless pressure, pD. The outer boundary is no-�ow and movesoutwards from the wellbore axis according to the following expression: rD =�0ptD, in which the distance, rD; is proportional to the square root of time, tD:

This expression is now changed for time � to �rD = ��0ptD, or �rD = �

12�0p�tD,

and further rD = ��12�0p�tD = �

p�D, where � = �0p

�; is the no-�ow moving

boundary constant. The no-�ow boundary moves with dimensionless velocity,drDdtD

; equal to:

drDd�D

=�

2

1p�D

(5.5)

Inner Boundary Conditions of Constant Rate

Di¤usion equation solutions for pD as a function of the radius rD and time �Dare given below:

pD = '�rD�

�12D

�= '(x) (5.6)

We �rst calculate p�D on the left-hand side and prD on the right-hand side ofEguation (5.4)

p�D =@pD@ �D

=@'

@ �D

�rD �

� 12

D

�= �1

2rD�

� 32

D '0�rD �

� 12

D

�(5.7)

prD =@pD@rD

= �� 12

D '0�rD�

� 12

D

�(5.8)

Further derivatives of Equation (5.4) include:

@

@rD(rDprD ) =

@

@rD

�rD@ pD@ rD

�=

@

@rD

hrD �

� 12

D '0�rD �D

� 12

�i

= �� 12

D '0�rD �

� 12

D

�+ rD �

�1D '

00�rD �

� 12

�(5.9)

Equation (5.4) now becomes:


1

rD

h�� 12

D '0�rD �

� 12

D

�+ rD t

�1D '

00�rD �

� 12

D

�i= �1

2rD �

� 32

D '0�rD �

� 12

D

�(5.10)

or

h�� 12

D '0�rD �

� 12

D

�+ rD �

�1D '

00�rD �

� 12

D

�i= �1

2r2D �

� 32

D '0�rD �

� 12

D

�(5.11)

and

�rD �

�1D '

00�rD �

� 12

D

�+ �

� 12

D '0�rD �

� 12

D

�+1

2r2D �

� 32

D '0�rD �

� 12

D

��= 0

By multiplying with �12D we have:�

rD �� 12

D '00�rD �

� 12

D

�+ (1 +

1

2r2D �

�1D )'

0�rD �

� 12

D

��= 0

for,

rD t� 12

D = x (5.12)

x'00(x) +

�1 +

1

2x2�'0(x) = 0 (5.13)

'00(x) +

�1

x+1

2x

�'0(x) = 0 (5.14)

'00(x)

' 0 (x)+

�1

x+1

2x

�= 0 (5.15)

Hence,

log '0(x) + log x +

1

4x2 = C (5.16)

'0(x) = K x�1 e�

14x2 (5.17)

where K is the integration constant that can be adapted for any additionalconditions. Further integration gives, for ' (x) ;

' (x) = K

x0Z0

x�1 e�x2

4 d x (5.18)


Figure 5.1: The dimensionless pressure, pD, as a function of the dimensionlesstime, tD and radial distance, rD ( with K = 1, and speed x = 10).

rD@pD@rD

= rD t� 12

D '0�rD t

� 12

D

�= K '

0�rD t

� 12

D

�(5.19)

rD@pD@rD

= K e�14x2 ; x = rD t

� 12

D (5.20)

The obtained solutions can be considered as basic due to the imposed premises.Figure (5.1) presents the dimensionless pressure, plotted versus the dimensionlesstime, tD, and radial distance, rD. The dimensionless pressure can be calculatedfor a constant rate production given by the integral constant K = 1, and speedof a moving boundary proportional to the constant x = 10).A 2D plot of the dimensionless pressure versus the dimensionless time, given

in Figure (5.2) is calculated for a rate production de�ned by the constant K = 1,

a no-�ow moving boundary constant, x = 10; and a constant x = rD t� 12

D .For an IBC of constant production rate, the decline in dimensionless pressure

is more evident at times. In both Figure (5.2) and Figure (5.3), the constantrate parameter, K; and the moving boundary parameter, x; are constant.

Variable Rate Production

A di¤usion equation with an inner boundary condition of variable rate and anouter boundary condition of no-�ow, and moving outwards from the wellboreaxis, can be reduced to a system of ordinary di¤erential equations by the tech-nique of separation of variables. We assume a solution that is dependent on thedimensionless radial space, rD; and the dimensionless time, tD. As a mathe-matical entity, the di¤usion equation has two very important features: linearity


Figure 5.2: The dimensionless pressure, pD; as a function of the dimensionlesstime, tD; for the dimensionless distance, rD = 1 ( with K = 1, and x = 10).

Figure 5.3: The dimensionless pressure, pD; as a function of the dimensionlesstime, tD; for the dimensionless distance, rD = 10 ( with K = 1 and x = 10).


and separability. The fact that the equation is separable means that we candecompose the original equation into a set of uncoupled expressions for the eval-uation of pD in a space dimension R and a time dimension T . This signi�es thatwhatever solution one �nds for pD, it can be represented as the multiplicationof separate solutions. Thus pD can be written as a product of terms that eachdepends on a single coordinate:

pD = R (rD)T (tD) (5.21)

In order to solve the problem, this yields to:

rDRT0= T

�R

0+ rD R

00�

(5.22)

After we substitute the separable form of pD and divide both sides by rDRT; wehave:

T0

T= r�1D

�R

0

R+ rD

R00

R

�= �k (5.23)

Dividing both sides by rDRT results in the left-hand side having no space de-

pendence, and the right-hand side having no time dependence. The only waythat an equation strictly in time (the left-hand side) can be equal to an equationstrictly in space (right-hand side), is if they are both equal to a constant. This isdue to the time and space variables being arbitrarily varied. By convention, theconstant was taken to be �k; and this peculiar constant will make more sensewhen compared to the actual solution.The left-hand side of the equation can be integrated to �nd T (tD)

T (tD) = Ae�ktD (5.24)

The right-hand side, with rD; is a Bessel equation:

rD R00+ R

0+ k rD R = 0 (5.25)

Its solution is given by Bessel functions

R (rD) = Z0

�rDpk�

(5.26)

where a general zero-order Bessel function of �rst kind, Z0 is a linear combinationof Bessel functions J0 and Y0: A and B are constants that should be chosen.

Z0 = AJ0 + BY0 (5.27)

Being a linear equation means the equation itself has no powers or complicatedfunctions of the function pD This renders it possible to think of the equationas a so-called �linear di¤erential operator�, L; acting on the function pD, sothat L [pD] = 0. According to mathematical physics, the di¤erential equation is


linear, and has more than one solution, a new solution can be obtained by addingtogether other solutions. This means that if L [pDk ] = 0 for some solution pDk ,

we have L�P

k

akpDk

�= 0 for any arbitrary constants ak. From this we may

form linear combinations:

pD =Xk

C (k) e�k� Z0

�rDpk�

(5.28)

or

pD =

bZa

C (k) e�k� Z0

�rDpk�dk (5.29)

The last expression is very useful, especially when choosing pD as:

pD =

1Z0

C (k) e�k� Z0

�rDpk�d k (5.30)

We thus get:

rD

@p

@rD= r

D

1Z0

C(k)e�k�pkZ1(rD

pk)dk

Further the derivatives of the Bessel functions, Z0, Y0 and J0; are:

Z 00 = �Z1;Y 00 = �Y1; J 00 = �J1

Z1 = AJ1 + B Y1 (5.31)

Now, as an arbitrary chosen z ! 0 (which can also be equal to rDpk), we have

for the Bessel functions, J1(z) and Y1(z):

J1 (z) v1

2z:and:Y1 (z) � �

2

�z�1

Hence,

limrD!0

rD@ pD@ rD

= � 2�� B �

1Z0

C (k) e�k� d k (5.32)

which is a "Laplace" integral. Putting B = ��2, the constant B is not dependent

on time. Meanwhile, A remains an indeterminate. Further writing:


Figure 5.4: The dimensionless distance, rD; of a no-�ow moving boundary as afunction of a constant, �, and the dimensionless time tD .

limrD! 0

rD@ pD@ rD

= (�) (5.33)

where the rate in�ow from the left-hand side is expressed as any function of time,we get:

(�) = L [C (k)] , or C (k) = L�1 ( ) (5.34)

Here, L�1 ( ) is an inverse "Laplace" transform of the function.Analytical models, when applicable, have been of great importance because

of their power. The cost for this power is however a limited applicability. Inthat sense, we may now in our analytical model demand that the boundary beno-�ow, moving outwards according to (5.35),

@pD@rD

= 0 for rD = �ptD (5.35)

where � is a constant that should be further investigated and that is possiblyrelated to the driving force that moves the no-�ow outer boundary with thespeed de�ned by Equation (5.5).Figure (5.4) represents the perspective of a position, rD of, the no-�ow bound-

ary as a function of the dimensionless time, tD; and a constant, �. The 2D plotof the dimensionless position, rD; versus the dimensionless time, tD; for a coe¢ -cient of no-�ow moving boundary, � = 1,3,5,7 and 9 is given in Figure (5.5).Theconstant � can be related to the driving force or physics of the �ow, i.e., to adriving force that causes the rate to decline by the decline exponent b. Thederivative of the distance of a no-�ow boundary with time gives the speed of amoving no-�ow boundary and is presented by Figure (5.6).


Figure 5.5: The dimensionless position, rD; versus the dimensionless time, tD;for a coe¢ cient of no-�ow moving boundary � = 1,3,5,7 and 9.

Figure 5.6: The velocity of a no-�ow moving boundary, vD = drDd�D

; as a functionof the dimensionless time, tD; for various constant values of , �:


Figure 5.7: The dimensionless distance, rD; of a no-�ow moving boundary asa function of the dimensionless time, tD; for various constants of the no-�owmoving boundary, � = 1; 3; 5; 7;and 9.

According to the expression for rD@pD@rD

in Equation (5.34), we now get:

1Z0

C (k) ek�pk

264AJ1 ��pk��

2Y1

��pk��375 dk = 0 (5.36)

For Equation (5.36) we need to determine A when C(k) = L�1 ( ) is given.The constant A values for various exponents b are provided in Chapter 6.

Overall pressure responses, pD; within a drainage area are derived for Arpsvariable-rate wellbore conditions for selected decline exponents, b. Chapter 7describes the derivation of such responses for a selected decline exponent, b al-most equal to zero.

5.1.4 Variable Rate Production of Arps Type

Further, we present di¤usion equation solutions for a variable rate production ofArps type. The decline exponent, b; in Arps equation varies, and in each case,we de�ne an inner boundary condition of variable-rate and an outer boundarycondition of no-�ow and moving boundaries. The inner boundary condition is ofArps type, with a decline exponent, b = 0:33; 0:5; 1 and 2. The rate, q; versustime, t; and initial decline, Di, for four decline exponents, b (with b raging from0.33 for the lowest rate to 2 for the highest rate) are presented in 3D based onthe Arps equation in Figure (5.8), and in 2D in Figure (5.9).Figure (5.9) represents the 2D plot of the rate, q; versus time, t;based on Arps

equation. All rates are calculated with the same initial decline rate, qi = 5000,


Figure 5.8: A 3D plot of Arps equation rate, q; versus time, t; and initial decline,Di ( for a decline exponent b = 0:33, b = 0:5, b = 1, and b = 2).

Figure 5.9: A 2D plot of Arps equation rate, q, versus time, t; and initial decline,Di (for a decline exponent b = 0:33, b = 0:5 ; b = 1, and b = 2).


Figure 5.10: The rate, q; versus time, t, for b = 0:33 plotted in circles, andvarious decline exponents (b = 0:5; 1; and 2) all plotted as solid line. The rateversus time is calculated for a speci�c initial decline, qi = 5000, and an initialdecline, Di = 0:01).

and the same initial decline, Di = 0:001, ranging the decline exponent, b;from0:33 to 2:

Hyperbolic Decline (b=1/3)

In the case of hyperbolic decline the inner boundary condition of variable ratefor the model, was set by taking an Arps decline exponent, b; equal to 1=3 (Thisis known as the hyperbolic decline). Figure (5.10) displays a plot of Arps-derivedcurves of rate, q; versus time, t. The decline curve of b = 1=3 is plotted withcircles, for a de�ned initial decline ,Di; and an initial decline rate, qi. Here, werefer again to work of Fetkovich (1973) which relates the decline exponent, b; of1=3 to 2=3, for the solution gas drive mechanism. Such rates de�ned with thedecline exponent b = 1=3 correspond to the IBC of the variable-rate model.In Figure (5.10), all rates, q, are related to time, t, through Arps hyperbolic

relation:

q(t) = qi (1 + bDit)� 1b (5.37)

By simplifying (5.37) to

(t+ t0)� 1b (bDi)

� 1b qi = (t+ t0)

� 1b � Const: (5.38)

and by a further simpli�cation (by taking out t0) of equation (5.38) we obtainthe expression:


q(t) � t�1b � Const: (5.39)

Instead of using an Arps type expression (5.37) for rate q(t), we introduce anexpression (tD) similar to (5.39). Correspondence of (tD) to Arps decline isvalid for late values of time, tD ! 1. At early times (tD) ! 1. We cannow solve the di¤usion equation by de�ning the variable rate b = 1=3: The ratefunction (tD) thus becomes:

(�D) = ��3D :

The expression is simpli�ed and valid for late times in order to obtain inverse"Laplace" transform of the rate function (tD). This inverse "Laplace" trans-form of the rate function, (tD), is:

L�1 ( ) =k2

2� C(k):

Now the dimensionless pressure, pD, is:

pD =

1Z0

1

2k2hAJ0(rD

pk)� �

2Y0(rD

pk)ie�k�Ddk

=2

� 3D

�e� r2D4�D

�1� 1

2

r2D�D

+1

32

r4D� 2D

��A� 1

2Ei

�r2D4�D

��+1

16

r2

�D� 34

�(5.40)

By requiring that a no-�ow boundary moves with a distance r from a wellbore,we get:

@pD@rD

= 0; for rD = �p�D (5.41)

The distance r, where the no-�ow boundary is spaced in time, � , is related by aconstant �. Here, A and � are related according to:

A =1

2Ei

��2

4

�� 2

�21� 5

8�2 + 1

32�4

3 � 34�2 + 1

32�4

e�2

4 (5.42)

We can now plot the constant A as a function of the constant, �, as shown inFigure (5.11).

By imposing variable rate IBCs of various Arps selected decline exponent,b model generates production pro�les derived from the analytically solved dif-fusion equation. Table (5-1), summarises all developed model solutions. In thetable solutions for pD and A(�) are given for each variable-rate inner boundary


Figure 5.11: The constant A as a function of the constant � of a no-�ow movingboundary.

condition (the Arps rate decline exponents are: b = 0:33, b = 0:5, b = 1, andb = 2). Further, Table (5-1) presents the solution for the dimensionless pressure,pD; calculated for the early and the late dimensionless times, tD.


Table 5-1: The dimensionless pressure, "pD", calculated by the model at time"tD", for a variable-rate production of Arps type, de�ned by exponent b (rangingfrom 0.3; 0.5; 1., and 2)


5.2 Vertical Well - Gas Flow

This section presents the gas �ow associated with a moving boundary model. Alinear parabolic di¤usion equation is derived in Equation (5.1) for an oil �ow.Under the assumption that oil has a small and constant compressibility, thedi¤usion equation was derived by combining the continuity equation in radialcoordinates with Darcy�s law and an appropriate equation of state relating thedensity to the pressure. The assumption of a small and constant compressibilityis not valid for gas whose compressibility is on the order of the reciprocal of thepressure. In addition, the assumption further states that the gas-compressibilityand pressure product, cgp; is certainly not much less than unity, for gas �ow.Other di¤erences between oil and gas include that gas viscosities are a hundredtimes smaller than their lowest oil counter parts (gas viscosities are on the orderof 0:002 cp). The gas volumetric �ow-ratio and hence also gas velocities, arethus much higher. The Reynolds number near the wellbore region is such thatnon-Darcy �ow is occurring. Usually, this non-Darcy �ow e¤ect can be lumpedinto the skin e¤ect.The basic continuity equation is also valid for a real gas �ow:

@

@r(�u) +

�u

r= ��(@p

@t) (5.43)

and the Darcy velocity, u is:

u = �k(p)�(p)

@p

@r(5.44)

In Equation (5.44) we may neglect the dependence of permeability on the pres-sure, k(p), or the Klinkenberg e¤ect which can be expressed as:

k(p) = keff (1 +slp) (5.45)

there, keff is the e¤ective permeability to liquid, and sl is slippage in porousmedia (related to the mean free path of the gas molecules, controlled by tem-perature, pressure, and the nature of the gas). Klinkenberg and others haveobserved the permeability for a gas as a straight-line function of the reciprocalof the mean pressure of the measurement, as given by Equation (5.45) relatingpermeability to gas, k(p), to the mean pressure, p:This e¤ect is important only atvery low pressures, and for most practical purposes, the gas permeability can beassumed constant. Further, from the equation of state for real gases, we expressdensity as:

� =M

RT

�p

Z(p)

�(5.46)


Subsequently, the fundamental non-linear partial di¤erential equation describingisothermal �ow of real gases becomes:

1

r

@

@r

�p

�(p)Z(p)r@p

@r

�=�

k

@

@t

�p

Z(p)

�(5.47)

The consequence is that the di¤usion equation becomes non-linear. A specialtechnique is necessary to linearise the basic di¤erential equation for the radial�ow of a real gas.

5.2.1 Pseudo-Pressure Transformation (Intermediate Pres-sures)

Al-Hussainy et al. (1965) have introduced transformation by using the real gaspseudo-pressure or a real gas potential, m(p):

m(p) = 2

pZp0

p

�(p)Z(p)dp (5.48)

The limits of integration range from an arbitrary base pressure, p0, and thepressure of interest, p. In order to calculate m(p), we must know the viscosity, �,(de�ned by a correlation method) and compressibility factor Z (de�ned by theequation of state). Moreover the following assumptions need to be established.We assume that both the permeability, k; and the porosity, �, are constant.From the equation of state follows that the speci�c gravity, �; is

� =M

RT

p

Z(p)(5.49)

By neglecting the water compressibility, cw, and the formation compressibility,cf , the total compressibility ct can be expressed as:

ct = (1� swc) cg (5.50)

and the gas compressibility, cg, as

cg =1

p� 1

Z

@Z

@p(5.51)

Equation (5.48) then becomes

m(p) = 2

pZp0

pdp

�Z=2RT

M

pZpo

�

�dp (5.52)

The real gas-pseudo pressure derivatives in space:


@m(p)

@r=dm(p)

dp

@p

@r=2p

�Z

@p

@r(5.53)

and time:

@m(p)

@t=dm(p)

dp

@p

@t=2p

�Z

@p

@t(5.54)

Further derivatives give

2@

@t(p

Z(p)) = ��c

@m

@t=2RT

Mc�@p

@t(5.55)

This corresponds to

@�

@t= c�

@p

@t(5.56)

and

� = �0ec(p�p0) (5.57)

The di¤usion equation with a pseudo pressure as a dependent variable now be-comes:

1

r

@

@r

�r@m (p)

@r

�=� (�ct)i

k

@m (p)

@t(5.58)

The partial di¤erential equation describing an unsteady state radial gas �owis linearised by integral transformation. In drawdown, the product of viscos-ity and compressibility (�ct) is assumed constant at (�ct)i the initial value ofpressure, pi. The viscosity, �, is proportional to the pressure and compressibil-ity, ct, is inversely proportional to the pressure. In other words, the product(�ct) reduces the pressure dependence. The dimension of the real gas pseudo-pressure, m(p); is equal to the pressure squared over the viscosity or (psia2=cp)in the �eld system of units. The inner boundary condition of constant rate withpseudo-pressure, m(p) for a �nite wellbore radius are:

qsf2�hrw

=k

�

@p

@rjrw (5.59)

where the sandface �ow rate, qsf ;under standard conditions is,

qsf = ZQpscp

T

Tsc(5.60)

Further derivatives give:

@p

@r=

@p

@m(p)

@m(p)

@r=�Z

2p

@m(p)

@p(5.61)


We than obtain the pseudo-pressure gradient at the wellbore as a constant equalto:

@m(p)

@rjrw =

qpscT

�khrwTsc(5.62)

Here, all pressure-dependent variables disappear (�Z; p). The inner boundarycondition for a line source well is:

limr!0

�r@m (p)

@r

�=

qpscT

�khTsc(5.63)

The outer boundary conditions concern a no-�ow, but moving boundary. Thewell is assumed to be located in the centre of a cylindrical reservoir of radiusr = re. The no �ow boundary condition implies the local pressure gradient tobe zero, i.e., for t > 0;

@p

@rjr=rei = 0 (5.64)

Here, rei; stands for a no-�ow boundary that is moving with a certain velocityas previously described in this chapter.It is convenient to de�ne a dimensionless pseudo-pressure, m(pD), as:

mD(pD) =m (pi)�m (p)

qpscT

�khTsc

(5.65)

with the dimensionless time, tD, as:

tD =kt

� (�ct)i r2w

(5.66)

and the dimensionless radius, rD, (related to a no �ow and moving boundary)as:

rD =reirw

(5.67)

The above dimensionless equations (5.65), (5.66) and (5.67) are substitutedinto the linearised gas �ow equation (5.58) thus giving the dimensionless pressuregas �ow equation that is valid for 1 � rD � rDei:

1

rD

@

@rD

�rD@mD(pD)

@rD

�=@mD(pD)

@tD(5.68)

The initial condition speci�es a uniform initial pressure m(pi): It is further spec-i�ed that at time tD = 0 and for all rD:

mD(pD) = 0 (5.69)


The inner boundary condition (�nite wellbore radius) at rD = 1 and for all tDis:

@mD(pD)

@rD= �1 (5.70)

Alternatively, the line source form can be written as:

limrD!0

�rD@mD(pD)

@rD

�= �1 (5.71)

For all tD > 0 and moving rD = rDei, with no �ow across the external boundarythe outer boundary conditions are:

@mD(pD)

@rD= 0 (5.72)

Thus, the linearised di¤usion partial di¤erential equation and boundary condi-tions describing the unsteady state �ow of gas have been put in exactly the samegeneralised form as the dimensionless equations derived for the liquid case. Thisis achieved with pseudo-pressure rather than actual pressure, along with ap-propriate de�nitions of the dimensionless variables. All the analytical solutionsthat have been found useful in oil �ow are also applicable to gas �ow. Here, wecomment that pressure analysis must be carried out in terms of pseudo-pressure,m(p). This means that for a given recorded pressure against time we need toconvert pressure into a pseudo-pressure versus time by using the transform.It is possible to instead use the normalised pseudo-pressure, (p), which can

be obtained by dividing m(p) withpi(�Z) i

. (�Z)i is the product evaluated at

pressure pi and reservoir temperature T. The normalised pseudo-pressure (p)has all the linearising properties of m(p) as well as the units of pressure. Itsmagnitude is similar to the real pressure. When normalised pressure (p) isused rather than m(p) gas well plots look very similar to those obtained in oilwell analysis. This is typical for high pressure p > 3000 psi (210 bar). To be ableto use the normalised pressure, (pD), we changed the dimensionless pressurem(pD) to:

(pD) =m (pi)�m (p)

QpscT

�khTsc

=(pi)�(p)QpscT

�khTsc

(�Zi)

pi

(5.73)

It is important to mention that the pseudo-pressure concept does not result ina complete linearisation of the radial �ow equation.With the normalised dimensionless pressure, (pD), Equation (5.58) be-

comes:


1

rD

@

@rD

�rD@(pD)

@rD

�=@(pD)

@tD(5.74)

The inclusion of the normalised pseudo-pressure, (pD); into the dimensionlessquantity pD from equation (5.73) means that the analytical solutions termeddimensionless functions for oil, pD, and gas, (pD) �ow, are identical. In highrate oil wells with limited entry, there is some evidence of rate dependent skinbeing present. This skin should also be included in gas wells. Agarwal (1978)introduced the real gas pseudo-time pseudo-time, ta; by the following expression,comparable to the pseudo-pressure:

ta = (�ct)i

tZt0

dt

�ct(5.75)

When implemented into the di¤usion, we get:

1

r

@

@r

�r@m (p)

@r

�= �

(�ct)ik

@m (p)

@ta(5.76)

The same pseudo-time, ta, can be put into the dimensionless form, i.e., in tD:

tD =kta

� (�ct)i r2w

(5.77)

The conditions under which this pseudo-time transformation linearises the radial�ow equation have been studied by Lee and Holdich (1980). It is important tomention that the pseudo-pressure concept does not result in a complete lineari-sation of the radial �ow equation.

5.2.2 Pressure Squared Transformation

The present section concerns the special case of the pseudo-pressure transfor-mations valid for low pressures. This low pressure is below p < 2000 psi (140bar), where the squared pressure p2, is considered instead of the pseudo-pressurefunction m(p). Under the following assumptions, pressure-squared solutions canbe obtained:1. The gas behaviour is ideal (Z = 1) and the gas viscosity, �g, is assumed

to be independent of pressure. Equation (5.58) becomes:

1

r

@

@r

�r@p2

@r

�=��

kp

@p2

@t(5.78)

2. When the product of viscosity, �; and the gas deviation factor, Z, areconstant, the equation becomes:


1

r

@

@r

�r@p2

@r

�=��c

k

@p2

@t(5.79)

3. For pressure gradients assumed to be small, i.e.,�@p

@r

�2! 0 (5.80)

The dimensionless form of the di¤usivity equation is:

1

rD

@

@rD

�rD@pD(p

2)

@rD

�=@pD(p

2)

@tD(5.81)

Here, the only new de�ned dimensionless variable is now pD(p2), equal to:

pD(p2) =

p2i � p2

qpscT

�khTsc

(5.82)

5.2.3 No-Flow Moving Boundary - Gas Flow Solutions

All dimensionless pressure solutions developed for an oil �ow are now also validfor a gas �ow. The only di¤erence is that the dimensionless form of the pressurevariable for oil �ow has to be replaced by an adequate dimensionless variable forgas. Gas dimensionless pressure variables have been previously de�ned based onpseudo-pressure, normalised pseudo-pressure and squared pressure.

No-Flow Moving Boundary - Variable Rate of Arps Decline (b=0.5)

Here we present gas �ow solutions with variable rate of Arps decline for a hy-perbolic decline of b = 0:5 and a dimensionless pseudo-pressure, pD, is equalto:

pD = �1

2t2D

8><>:2641 + �1� r2D

4tD

�e�r2D4tDEi

�r2D4tD

�375� 2A�1� r2D4tD

�e�r2D4tD

9>=>;(5.83)

The dimensionless pseudo-pressure, pD, is:

pD =pi � pqpscT

�khTsc

(5.84)

the dimensionless time, tD, is:


tD =kt

� (�ct)i r2w

and the dimensionless radius, rD, is:

rD =reirw

Also here is the change of a no-�ow moving boundary with time de�ned as:

rD = �ptD

The constant A as a function of the coe¢ cient of no-�ow moving boundary, �,is given as:

A (�) =1

2Ei

��2

4

�� 2

�2

1� �2

4

2� �2

4

e

�2

4 (5.85)

Here, are presented only the pseudo-pressure gas �ow solutions. For a hyper-bolic decline of b = 0:5 dimensionless pseudo-pressure m(pD) is equal to:

mD(pD) = �1

2t2D

8><>:2641 + �1� r2D

4tD

�e�r2D4tDEi

�r2D4tD

�375� 2A�1� r2D4tD

�e�r2D4tD

9>=>;(5.86)

where dimensionless pseudo-pressure, m(pD) is:

mD(pD) =m (pi)�m (p)

qpscT

�khTsc

(5.87)

We can further normalised the dimensionless pseudo-pressure, m(pD), to thenormalised pseudo-pressure, (pD), according to:.

(pD) = �1

2� 2D

8><>:2641 + �1� r2D

4�D

�e�r2D4�DEi

�r2D4t�D

�375� 2A�1� r2D4�D

�e�r2D4�D

9>=>;(5.88)

where, D(pD) is:

(pD) = (pi)�(p)qpscT

�khTsc

(�Zi)

pi

(5.89)


Pressure squared solutions, pD(p2), are equal to:

pD(p2) = � 1

2� 2D

8><>:2641 + �1� r2D

4�D

�e�r2D4�DEi

�r2D4�D

�375� 2A�1� r2D4�D

�e�r2D4�D

9>=>;(5.90)

We can further write all solutions obtained for oil in an adequate form for gas�ow.

5.3 Summary

Chapter 5 describes an investigation of the nature of the wellbore pressure re-sponse when production is forced to follow the Arps decline de�ned by the spe-ci�c value of a decline exponent, b. The physical model is de�ned with the innerboundary condition of variable rate of Arps type and a no-�ow boundary movingoutwards from a wellbore axis. The velocity of the no-�ow moving boundary isproportional to a coe¢ cient, �, and a square root of time. In order to solve adi¤usion equation for the variable-rate IBC of Arps type and a no-�ow movingboundary, the rates are modi�ed in such a way that Arps decline exponent, b,is matched with the late time rates of the model. In addition, the velocity ismade proportional to the square root of time. The no-�ow boundary moves andfor late time it reaches a condition where the pressure di¤erence is zero. Thesolutions obtained for the di¤usion equation correspond to the pressure chang-ing with time. Each variable pressure pro�le can be de�ned within the drainagearea of a vertical well. The obtained pressure solutions con�rm Raghavan�s ob-servation that matching rates de�ned by exponent b can only be achieved witha pressure that changes in time. Variable pressure solutions are developed foran oil �ow but are also applicable for gas �ows. Such solutions are available forboth low-pressure and high-pressure conditions.

Chapter 6

RATE DECLINE CURVES

This chapter comprises two parts, the �rst deals with transient and depletionresponses for oil fractured-horizontal wells oil and the second investigates the na-ture of the pressure responses when the rate response of a vertical well is of Arpstype for selected values of the decline exponent, b. The value of horizontal mul-tiple fractured wells can be maximised when the parameters that most impactthe productivity of such wells are understood. As a step towards achieving max-imum values from horizontal multi-fractured wells, a computer model, presentedin Chapter 4, has been developed, allowing the user to quickly assess the in�u-ence of various well or reservoir properties on the well performance. The modelis three-dimensional and considers full-time variability. It assumes a single-phaseoil �ow and is based on semi-analytical techniques. Modelling of transversal orlongitudinal fractures of varying types (uniform �ux, in�nite conductivity, �niteconductivity) is possible. The �exibility to use various constraints and the op-tion of a multi-run sensitivity generation can be used for obtaining prognosis ofa multi-fractured horizontal well productivity its diagnosis.B. Cvetkovic contributed to the teamwork in designing the software tool, to

the physical and mathematical modelling with software development and test-ing, based on the implemented model. Its main features comprehend the abilityto provide a fast screening analysis (for various inner boundary conditions andspecial features of step-function and late time approximations). Most of thesefeatures are presented in this chapter. The model solution that is developedcan be employed for a general screening and optimisation of a multi-fracturedhorizontal well thanks to the implementation of fast numerical algorithms aspublished by Cvetkovic et al. (1999). For any given set of reservoir, fracture andwell parameters, one can determine the oil rate cumulative production, well pres-sure or productivity index. The evaluation of a multi-fractured horizontal wellperformance, or the selection of an optimum perforation with stimulation designfor such wells, may be approached through �ne grid reservoir simulation. How-ever, while reservoir simulation is the most advanced method for predicting wellperformance, it is often too time consuming for conducting parametric screening

193

194 6. RATE DECLINE CURVES

studies. Often the data required are not available and the e¤ort may not bewarranted. As an alternative to simulation, the application of semi-analyticalmodels can readily yield wellbore and fracture responses to various boundaryconditions. This is often su¢ cient to provide an understanding of factors withthe most in�uence on well performance. If simulation work is warranted, it canthen proceed with the insight obtained from the analytical models.A variable-rate and speci�ed boundary model generates pressure pro�les for

the decline exponent, b, (b = 0:33; 0:5; 1 and 2). Obtained solutions havecontributed to Raghavan�s (1993) discussion on wellbore pressure conditions re-quired for the well to produce Arps-type rates with a speci�ed decline exponent,b. Although the producing well is vertical, the physical model describing itssolutions can be extended to a horizontal producing well.

6.1 Transient Decline of a Fractured-HorizontalWell

We chose to use several options from the fractured-horizontal well model (forwhich the general model solutions are presented in Chapter 4) to demonstratescreening analysis capabilities with data from a North Sea well. In cases wherewell data were lacking, properties that were believed to be appropriate for thesake of this review were assumed. The model is an extremely fast mini-simulator,modelling one-phase (slightly compressible) liquid �ow into a multifractured hor-izontal well in an open (slab) or closed (box) reservoir. The fractures are rec-tangular and vertical, and either transversal or longitudinal relative to the welldirection. They are also alternatively of �nite conductivity, in�nite conductivityor uniform �ux type. Further, the fractures are fully or partially penetrating,of equal or unequal length and spacing. There is no limitation to the numberof fractures. The well is either open or perforated only at the fractures; how-ever, one special option is a partially perforated well with no fractures. Thesimulation setup is made easy for the user through a screen input interface de-sign. Special features include multirun (several parameter values given and usedconsecutively), linear or logarithmic scales on axis, and a choice of units (�eld,metric or dimensionless). To be given as input are various relevant geometric andphysical parameters for the reservoir-well-fracture system, including the alterna-tive of well rate or pressure, each with the option of being constant or variableor varying in time in a freely stepwise manner.The output given is correspondingly the well pressure or rate as a function

of time, in addition to individual fracture rates. Further output parameterscomprise pressure derivatives (logarithmic and �square root�) and all cumulativerates. Also, productivity indices and e¤ective wellbore radii are given. Output(and input) data are immediately available, both in tabular form and through a

6. RATE DECLINE CURVES 195

Table 6-1: Reservoir, Well and Fracture Data

display of curves by a specially developed application of the Excel software. Welldata required for analysis using the fractured-horizontal well model are presentedin Table (6-1).

6.1.1 Transient-Rate Response of a Well

Sensitivity analysis to reservoir, �uid and well data

To start with, the ability to run sensitivities to various parameters is demon-strated by varying the number of reservoir, �uid and well data for a �xed numberof fractures and a �xed well length. Reservoir parameters include: the initialpressure, Pi, the e¤ective thickness, h, the horizontal and vertical permeabilitykh, kv, the porosity, �, and the total compressibility, cT . Fluid parameters are:the viscosity, �, and the formation volume factor, Bo. Further, well data includethe well length and the wellbore IBC of constant pressure. The initial reservoirpressure, Pi; was kept constant at 5700 psi. Values of the sensitivity to initialpressure, Pi, porosity, �, and e¤ective thickness, h, are given in Table (6-2).The case concerns production from a horizontal well in an oil reservoir with

5 transversal fractures. The fracture �ow is of uniform �ux. Figures 6.1 and6.2 represent plots of rate and cumulative rate for each fracture and for a well.


Table 6-2: The sensitivity to the initial reservoir pressure, "Pi", the porosity andthe e¤ective reservoir thickness, h

Figure 6.1: Individual fracture rates, qfri (i = 1; :::5) and the rate of a well (withfractures), q (bbl=d), versus time, t, in days.

Figure 6.1 shows the production pro�le of fracture 1 overlaying that of fracture5. One can also see the overlay of the fracture 2 production pro�le with that offracture 4. As expected, the pro�le of fracture 3 was the lowest, due to the sizeof the drainage area being small as compared to those of the other producingfractures. Production pro�les of cumulative rates with time are presented inFigure (6.2). Again, fracture 1 and 5 overlay each other. The same is truefor fracture 2 and 4. The cumulative rate of fracture 3 is low as compared tothe other plotted fracture cumulative rates, again in�uenced by the size of thedrainage area.The sensitivity of the rate and cumulative rate to the variation of initial

pressure, Pi, is presented in Figure (6.5). Plots for three initial pressures (Pi= 6425, 5700 and 4970 psi) demonstrate how the input pressure changes a (5-


Figure 6.2: Individual cumulative fracture production, Qfri (i = 1; :::; 5), andthe cumulative production of a well with fractures, Q (bbl), versus time, t, indays.

fractured) well rate and cumulative rate pro�le. The sensitivity to changes inporosity is presented in Figure (6.3) and to the e¤ective thickness, h (ft), inFigure (6.4).Figure 6.2 gives the variation in the fracture production pro�les and the frac-

ture pro�les for a well with 5 producing fractures. The �gure shows changes inthe individual fracture rate, qfr, and the individual cumulative fracture produc-tion rate, Qfr, with time for each of 5 transversal fractures positioned along thehorizontal well conditioned to a well �owing pressure of 2900 psi.For a selected initial pressure, Pi, equal to 5700, psi we considered three

wellbore pressures (with values of 2176, 2900 and 3626 psi). Further, in Figure(6.6), we plot the rate and cumulative rate with time at varying values of wellborepressure (pwf = 2176, 2900 and 3626 psi). The pressure, pwf , represents theaverage pressure in the wellbore.The contribution from fractures 1 and 5 to the well production is presented

in Figure (6.7). For each choice of the porosity production rate and cumulativeproduction pro�les, the data from fractures 1 and 5 overlay each other.The contribution from fractures 2 and 4 to the well production is presented

in Figure (6.8). Both the production rate and cumulative production with timecorrespond to overlays, and di¤er for each of three selected wellbore pressure.The production pro�les of fracture 2 and 4 are lower as compared to thoseof fracture 1 and 5.The contribution to the well production from fracture 3 is


Figure 6.3: Rate and cumulative production pro�les for various values of porosity,�, (i.e., 0.24, 0.34, 0.44).

Figure 6.4: Rate and cumulative production pro�les for various e¤ective thick-nesses, h, (i.e., 95 , 75 and 55 ft).


Figure 6.5: The individual-fracture rate and individual-cumulative fracture pro-duction versus time for various values of initial pressure, Pi, (i.e., 6425, 5700,and 4975 psi).

Figure 6.6: The rates and cumulative productions vs. time for varying values ofthe initial wellbore pressure, Pwf , (i.e., 2176, 2900 and 3626 psi).


Figure 6.7: Individual fracture rates and cumulative productions (qfr2 and Qfr2)vs. time for various values of wellbore pressure, Pwf , (i.e., 2176, 2900 and 3626psi).




presented in Figure (6.9). The time evolution of both the production rate andcumulative production di¤ers for each of the three selected wellbore pressures.The production pro�le of fracture 3 is the lowest compared to other fractureproduction pro�les. This is due to the small size of the drainage area.

For the case of an isotropic permeability (kh = kv), Figure (6.10) showsthe rate and cumulative production plotted against time while changing thepermeability, kh, from 40 mD, 4 mD and 0.4 mD. Increasing the permeabilitytenfold signi�cantly changed production pro�les. On the other hand, a tenfolddecrease in permeability reduced the rate and cumulative rate pro�les.

For a homogeneous reservoir with a permeability of 4 mD, the vertical per-meability, kv; was chosen as either 4, 2 or 0.4 mD. Figure (6.11) shows thechange in production pro�le due to a variation of the vertical permeability. Anincreased heterogeneity, caused by the kv value being only 10 % of a horizontalpermeability causes a doubling of the cumulative production pro�le.

In Figure (6.12), the plots of the rate and cumulative production vary due tothe changes in total compressibility, cT .

Furthermore, changes in �uid properties with viscosity, �, in Figure (6.13),and the formation volume factor, Bo, in Figure (6.14), illustrate how the pro-duction pro�les vary with time.


Figure 6.10: Individual fracture rates, q and cumulative productions, Q vs. timefor various values of permeability, Kh= Kv (i.e., 40, 4, 0.4 mD).

Figure 6.11: The individual fracture rate, q; and cumulative production, Q; vs.time for various vertical permeabilities, Kv (i.e., 0.4, 2, and 4 mD).


Figure 6.12: The individual fracture rate, q; and cumulative production, Q; vs.time for various total compressibility values, cT (i.e., 7.25 e-05, 6.26 e-05, 1.86e-06).

Figure 6.13: The individual fracture rate, q, and the cumulative production, Q,vs. time for various oil viscosities, � (i.e., 3, 4 and 5 cp).


Figure 6.14: The individual fracture rate, q, and the cumulative production, Q,vs. time for various oil viscosities, Bo (i.e., 1.35, 1.55 and 1.75 rb/stb).

Fracture position

Ideally, fractures can be positioned perpendicular to a well axis, known astransversal fractures, and also along a wellbore axis, known as longitudinal frac-tures. Figure (6.15) displays production pro�les for longitudinal and transversalfractures. A longitudinal fracture production rate pro�le oscillates. Model ratesare usually smooth functions, but for certain choices of the input parameters,the model may occasionally generate oscillating pro�les. This is due to the in-stability cased by the implemented Stehfest inversion techniques. One way toavoid oscillation in production pro�le is to tune the Stehfest inversion procedure.Another is to consider new inversion techniques (several of which are presentedin Appendix B).Up to now, all fractures were found to fully penetrate the reservoir. If

transversal fractures partially penetrate the reservoir, the productivity indexPI and cumulative rate production pro�les show small variations, as plotted inFigure (6.16). For the longitudinal transversal fracture positioned along the hor-izontal wellbore axis which partially penetrates the reservoir, the productivityindex PI and cumulative production pro�les are almost identical to those plottedin Figure (6.17). The very strong modelling feature represents fracture interfer-ences. Opposite to the fully perforated fractures, where each fracture producesits drainage area, in the partially penetrating fracture drainage area is distrib-uted by another fracture production. Due to fracture interferences, the resultant


Figure 6.15: The individual fracture rate, q, and the cumulative production, Q,vs. time for various fractures (longitudinal fracture vs. transversal fractures).

pro�les are varying slightly.Figure (6.18) plots productivity indices and cumulative rates versus time for

a horizontal well with a longitudinal fracture and 5 transversal fractures (eachwith a length of 85 ft). Two of the longitudinal fracture have lengths of: 2100 ftand 425 ft. All fractures are partially penetrating (with height of 55 ft). For afracture area of (425 ft x 85 ft), a single longitudinal fracture displays a higherproduction as compared to a production with 5 transversal fractures. Also, fortwo longitudinal fractures producing oil, a single longitudinal fracture that istwice as large will have twice the production.Production pro�les change rapidly with unequal fracture half-lengths, as can

be seen in Figure (6.19). The production increases for larger half-lengths aspresented in Figure (6.19). Table (6-3) shows the variation of the fracture half-length for 5 transversal fractures equally positioned along a horizontal well. Theproduction pro�les and PI�s change for a choice of fracture half-lengths. Forequally sized fractures (in Case 1), the production pro�le is higher as comparedto the unequally sized fractures (in Case 2 and Case 3). Again the interferencebetween producing fractures (in Case 2 and Case 3) reduces the productionpro�le. This is demonstrated in Figure (6.20).In a case where the fractures are unequally sized, the fracture interferes giv-

ing low productivity indexes in comparison to the equally sized fractures aspresented in Figure (6.20). The same �gure comprises plots of two unequallysized fractures. For each case, fracture half-length sizes are provided in Table


Figure 6.16: Productivity index, PI for a horizontal well with 5 transversal frac-tures and cumulative rate, Q with time for various fracture partial penetrationswith height, h (of 75, 55 and 35 ft).

Figure 6.17: Productivity index, PI for a horizontal well with a longitudinalfracture positioned along the 2100 ft horizontal well, and cumulative rate, Qwith time for various fracture partial penetrations with height, h (of 75, 55 and35 ft).


Figure 6.18: Productivity index, PI for a horizontal well with a longitudinalfracture and transversal fractures, and cumulative rate, Q with time for thepartial penetrated fracture the height, h = 55 ft.

Figure 6.19: Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the various half-length, Lf (of 170,85 and 42.5 ft).


Table 6-3: Sensitivity to fracture half length sizes, Lf

Table 6-4: Sensitivity to the distance bewtween two fractures due to well length,L changes (L= 2520, 2100, and 1680 ft

(6-3). The PI�s and cumulative rates are signi�cantly lower as compared to thecase with the equally sized fractures. This is due to fracture interferences causedby the unequal fracture size. Also, between unequally sized fractures (Case 2 andCase 3), there exists a di¤erence between two production pro�les. The higherproduction pro�le is de�ned by the fractures positioned at the beginning andthe end of a horizontal well. The size of a fracture half-length (Fracture 1 and 5)de�nes the production pro�le. This screening may help in understanding to whatdegree a production pro�le can be in�uenced by an unequal fracture half-length.Up to now, the fracture space between two fractures, has been equal, so for a

horizontal well of 2100 ft with 5 fractures, the space between two neighbouringfractures is 520 ft. This size can vary by increasing the well length, L, as given inTable (6-4). Productivity indexes and production pro�les are provided in Figure(6.21).The distance between two fractures changes with the number of fractures for

a well length, L (L = 2100 ft), as demonstrated in Table (6-5). Accordingly,production indexes and production pro�les change according to Figure (6.22).

Table 6-5: Sensitivity to the distance bewtween two fractures due to number offractures changes sor the same well length, L = 2100 ft


Figure 6.20: Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the unequal fracture (case 2 and 3)compared to the equally sized fractures (case 1).

Figure 6.21: Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the well length, L (of 2520, 2100 and1680 ft).


Figure 6.22: Productivity index, PI for a horizontal well with transversal frac-tures, and cumulative rate, Q with time for the various number of fractures, ona �xed well-length, L=2100 ft for various number of fractures, n (of 7, 5 and 3).

In all �gures, the distance between two fractures was kept equal. In onecase, where the fractures were unequally distributed along the well length, L,the production pro�les for each fracture varied with time, as presented in Figure(6.22).The fracture character changes from a uniform �ux to an in�nite and �nite

conductivity. Up to now, fractures have been considered to have a uniform �ux.In reality, a fracture is either in�nite conductive or �nite conductive. Figure(6.23) presents how the fracture character in�uences the production. The PI�sand the cumulative rate pro�les are similar for the uniform �ux and the in�niteconductivity fracture, whereas for the �nite conductivity fracture, the pro�lesare lower due the fracture being less conductive.Productivity indexes and the sensitivity to cumulative production pro�les to

selected values of �nite conductivity, FC , as provided in Table (6-6), are presentedin Figure (6.24). The rate and cumulative production versus time are plottedin Figure (6.25). Moreover, the fracture conductivity values are given in Table(6-7), and Figure (6.26) shows plots of both the rate and individual fracture rateversus the cumulative rate.Figure (6.27) presents 5-transversal �nite conductivity fractures, for which

each fracture conductivity, FC , is 50 mDft, and compares them to the wellborepro�les with calculated wellbore frictions. The wellbore friction signi�cantlyreduces the cumulative production, according to the friction model feature pre-


Figure 6.23: The productivity index, PI; for a horizontal well with transversalfractures, and the cumulative production, Q; vs. time for the various fracturecharacters (uniform�ux, in�nite conductivity, and �nite conductivity (with FC =1000 mDft).

Table 6-6: Sensitivity to fracture conductivity "FC" (mDft) equal to 2500, 1000,qnd 100

Table 6-7: The sensitivity to the fracture conductivity ,"FC" (mDft), when it isin�nite, 1000, and 50 mDft


Figure 6.24: The productivity index, PI; for a horizontal well with transversalfractures, and the cumulative rate, Q; vs. time for the in�nite conductivityfracture and �nite conductivity fractures (with FC values of 1000 and 50 mDft).

Figure 6.25: The rate, q; for a horizontal well with transversal fractures, andthe cumulative rate, Q; vs. time for an in�nite conductivity fracture, and �niteconductivity fractures with FC values of 1000 and 50 mDft.


Figure 6.26: The rate, q; for a horizontal well with transversal fractures, andindividual fracture rates qfri , where i = 1; :::; 5 versus the cumulative rate, Q,for an in�nite conductivity fracture and �nite conductivity fractures with FCvalues of 1000 and 50 mDft.

sented in Chapter 4.3.2.

Variable Wellbore Pressure Conditions

All plots generated up until now have been for inner boundary conditions, IBCs,of constant pressure. For variable IBCs, as provided in Table (6-8), the rateversus time responses for a well with 5 transversal fractures are de�ned as beingof in�nite and �nite conductivity. These are plotted in Figure (6.28). Figure(6.29) presents rates versus cumulative rates for a well with in�nite conductivityand �nite conductivity fractures. Individual fracture rates are plotted for eachof the �ve transversal fractures versus the cumulative fracture rate.

6.1.2 Well Transient-Pressure Responses

As for the constant pressure IBC, it is also possible to repeat all cases by choosingIBCs of constant rate instead of constant pressure. All features presented beforeare available with IBC of both constant and variable rate. Also, regarding thefracture character, it is possible to consider fracture �ow as either of uniform �ux,in�nite conductivity or �nite conductivity. The limited communication betweentransverse fractures and the wellbore creates a choking e¤ect near the well and


Figure 6.27: The rate, q, versus the cumulative production, Q, for a wellborewith 5 transversal fractures. Each fracture conductivity, FC ; is 50 (mDft). Thewellbore friction reduces both the well rate production and the cumulative pro-duction.

Table 6-8: Wellbore inner boundary conditions, IBCs of variable pressure for thein�nite conductivity and �nite conductivity for fractures with "FC" = 50 mDft


Figure 6.28: Responses of rate, q, versus time, t. The wellbore inner boundaryconditions, IBC, correspond to a variable pressure for the in�nite and �niteconductivity fractures of 50 mDft.

may cause an apparent reduction in fracture conductivity.The limited communication of a fractured-well (choking e¤ect) is included

in the model, as discussed in Chapter 4.3.3, and causes a decrease in the rateproduction and PI, as demonstrated in Figure (6.31). The fractures are of �niteconductivity (each of 2500 mDft) and the IBCs are of variable rate. For suchvariable rate IBCs, the PI and individual fracture cumulative production pro�lesare given in Figure (6.31).

6.1.3 Well Pressure-to-Rate Responses

Inner boundary conditions of constant and variable rates are usually applied inpressure and rate testing analyses. Operating conditions generally change fromrate to pressure and vice versa. Most of the wells in the North Sea, for example,operate under plateau production or constant rate wellbore conditions for a yearor longer. In Figure (6.32), we plot solutions to varying wellbore conditionsfrom constant rate to constant pressure. The well starts producing with therate of 4000 bbl/d and after 700 days it switches to the calculated constantwellbore pressure conditions as presented in Figure (6.32). For the basic restartoption treated here, we have as input a period of constant rate followed by aconstraint involving a bottomhole �owing pressure. The value of the employed


Figure 6.29: The rate, q, versus cumulative production ,Q, responses. The well-bore inner boundary conditions, IBC are variable pressure for the in�nite con-ductivity and �nite conductivity, 50 (mDft), fractures. Each individual fracturerate vs.cumulative rate is graphically presented.


Figure 6.30: In�uence of fracture choking e¤ect (for variable rate IBC) on pres-sure di¤erence and well with fractures PI. Fractures are �nite conductivity (2500mDft).

Figure 6.31: In�uence of fracture choking e¤ect (for variable rate IBC) on cumu-lative indifvidual fracture production Qfri (i=1,5 and 3) and well with fracturesPI. Fractures are �nite conductivity (2500 mDft).


Figure 6.32: Wellbore contant rate to constant pressure IBCs. The value of theemployed �owing pressure following the constant rate period corresponds to thepressure determined by the model at the end of this period.

�owing pressure following the constant rate period corresponds to the pressuredetermined by the model at the end of this period.

6.1.4 A Well with Longitudinal Fractures

Up until now, all fractures have been transversal. The following �gures compriserate changes for the longitudinal fracture positioned along a horizontal well.


Figure 6.33: Longitudinal versus transversal fracture rates and the cumulativeproduction (for a horizontal well with �ve equally spaced in�nite conductive,and equal half-length fractures).

Figure 6.34: Longitudinal vs.transversal fracture rates and the cumulative pro-duction (for selective fractures: 1, 5 and 3).


Figure 6.35: The productivity index, PI, versus time, t, for a horizontal wellwith longutudinal fractures of n = 7, 5, and 3.

6.2 Well Depletion Responses

6.2.1 Closed (BOX) Model

Rate Responses

A closed (BOX) model is very basic and thus needs further improvements. Calcu-lated production rates are low as compared to expected values, and consequentlysome modi�cations must be applied before use. Implemented model features in-clude constant and variable IBCs, along with a variation with regard to thefracture half-length.

Pressure Responses

By varying the IBC of rate, we generate pressure di¤erence pro�les with time fora horizontal well with 5 transversal fractures. As the IBC of rate is changed thewell starts producing at the rate 150 bbl/day. After 700 days, this rate reducesto 110 bbl/day and after 1400 days it goes down to 90 bbl/day. As mentionedbefore, closed or BOX model rates and pressures need to be normalised. Figures(6.36, 6.37) plot selected BOX model features of changing fracture half-lengthsand simultaneously with changing IBCs of pressure and rate.


Figure 6.36: The rate, q, versus time, t, for the BOX model (5000 ft by 5000ft). IBCs of variable pressure of 4900, 3900, and 2900 psi. In�nite conductivefractures with varying fracture-half lengths, Lf , of 120, 85 and 50 ft.

Figure 6.37: The pressure di¤erence, P, versus time, t, for the BOX model (5000ft by 5000 ft). IBCs of variable rate of 150, 110, and 70 bbl/d. In�nite conductivefractures with varying fracture half-lengths, Lf , of 120, 85 and 50 ft.


6.3 Rate Decline with a No-�owMoving Bound-ary

The following sections further investigate the nature of the wellbore pressureneeded for the Arps rate decline production de�ned with a �xed exponent, b.Chapter 5 presented the physical model together with its analytical solutions.This section aims at demonstrating the nature of pressure pro�les for severalb values. Each b value is selected, and can as such be related to the drivemechanism (i.e., b = 1/3, for the solution-gas-drive, b = 0.5, for the gravity-drainage, and b = 1, for the multilayered no-cross �ow production).

6.3.1 Hyperbolic Decline (b = 1=3)

In this case, we set the inner boundary condition of variable rate for the modelby taking the Arps decline exponent, b, equal to 1/3 (a known as hyperbolicdecline). In Figure (6.38), we plot curves of rate, q, vs. time, t. A decline curveof b = 1=3 is plotted with circles, for a de�ned initial decline, Di; and an initialdecline rate, qi. Fetkovich (1980) related values of the decline exponent, b; of1=3 to 2=3, for the solution gas drive mechanism.In Figure (6.38) all rates, q; are related to time, t, through Arps hyperbolic

relation

q(t) = qi (1 + bDit)� 1b (6.1)

By simplifying Equation (6.1) to

(t+ t0)� 1b (bDi)

� 1b qi = (t+ t0)

� 1b � Const: (6.2)

and a further simpli�cation of Equation (6.2), we obtain the expression:

(t+ t0)� 1b � Const: � t�

1b � Const: (6.3)

Instead of using the Arps-type expression (6.2) for the rate q(t), we introducea an expression (tD) that is similar to (6.3). For large values of time thecorrespondence of (tD) to Arps decline is tD ! 1. At early times (tD) !1.Now, we solve the di¤usion equation by de�ning the variable rate b = 1=3,i.e., the rate function (tD) then becomes:

(tD) = t�3D : (6.4)

Moreover, an inverse "Laplace" transform of the rate function is:

L�1 ( ) =�2

2� C(�): (6.5)


Figure 6.38: The rate, q, versus time, t, for b=0.33 plotted in circles, and forother values of the decline exponents (b=0.5, 1, and 2) all plotted as solid line.The rate versus time is calculated for a speci�c initial decline qi = 5000 and ainitial decline, Di = 0:01 ).

The dimensionless pressure, pD; is now:

pD =

1Z0

1

2�2hAV0(rD

p�)� �

2Y0(rD

p�)ie��tDd� (6.6)

pD =2

t3D

�e� r2D4tD

�1� 1

2

r2DtD+1

32

r4Dt2D

��A� 1

2Ei

�r2D4tD

��+1

16

r2

tD� 34

�(6.7)

From this, and provided that the no-�ow boundary moves with a distance r froma wellbore:

@pD@rD

= 0 ; for rD = �ptD (6.8)

The distance r, where the no-�ow boundary is spaced in time � ; is related by theconstant � (i.e., the coe¢ cient of the no-�ow moving boundary). The relationbetween A and � is given by the following equation:

A =1

2Ei

��2

4

�� 2

�21� 5

8�2 + 1

32�4

3 � 34�2 + 1

32�4

e�2

4 (6.9)

We can now plot the constant A as a function of the constant, �, as shown inFigure (6.39).The constant � relates the dimensionless distance of a no-�ow moving bound-

ary, rD; to a dimensionless time, tD.


Figure 6.39: The constant A as a function of the constant , �; of the no-�owmoving boundary.

Table 6-9: The constant A as a function of the coe¢ cient of the no-�ow movingboundary


Figure 6.40: The dimensionless pressure, pD, versus the dimensionless time, tD,and the dimensionless distance, rD (for � = 1 and b = 1

3).

Further, for a coe¢ cient � = 1, we plot the dimensionless pressure, pD; asa function of the dimensionless time, tD; and the dimensionless pressure, pD;as a function of the dimensionless radius, rD. Subsequently, we also plot thedimensionless pressure, pD; as a function of time, tD; and distance, rD, for variouscoe¢ cients, i.e., for � = 3; 5; 7 and 9.The following plots present the variation of the dimensionless pressure with

time and distance, for inner boundary conditions of variable rate (an Arps declineof b = 1

3) and the coe¢ cient � = 1.

In Figure (6.41) and Figure (6.43), all times between 0 and 16, correspondto times where the dimensionless pressure, pD; is outside the no-�ow boundary.Consequently the calculated dimensionless pressure, pD; is inside the no-�owboundary between the dimensionless time of 16 and 20. At a dimensionlesstime, tD = 16; and a dimensionless pressure, pD = 0:00218, the no-�ow boundaryreaches a dimensionless radius, rD = 4.In Figure (6.44), we plot the dimensionless pressure, pD; versus the dimen-

sionless radius, rD; for the particular time tD = 16. The no-�ow boundary movesfrom the wellbore axis with a speed that is inversely proportional to the squareroot of time. Moreover we have assumed a constant of proportionality � = 1,and the no-�ow boundary reaches the position of rD = 4.For distances, of rD greater that 4, the calculated pressures, pD; are outside

the no-�ow boundary.

6.3.2 Hyperbolic Decline (b = 0:5)

In this case, the inner boundary conditions are again of variable rate, and ofArps decline with exponent b = 0:5: This corresponds to the hyperbolic rate


Figure 6.41: The dimensionless pressure, pD; versus the dimensionless time, tD;and with a distance rD = 1 (for � = 1, and the decline exponent b = 1

3).

Figure 6.42: The dimensionless pressure, pD; versus the dimensionless time, tD;and with a distance rD = 4 (for � = 1, and the decline exponent b = 1

3).


Figure 6.43: The dimensionless pressure, pD, versus the dimensionless time, tD,and with a distance rD = 10 (for � = 1, and the decline exponent, b = 1

3).

Figure 6.44: The dimensionless pressure, pD, versus a distance, rD, at a dimen-sionless time tD = 16 (for � = 1, and the decline exponent b = 1

3).


Figure 6.45: The dimensionless pressure, pD, versus the dimensionless time, tD,and versus the dimensionless distance, rD (for � = 3).




Figure 6.48: The dimensionless pressure, pD; versus the dimensionless time, tD;and versus the dimensionless distance, rD (for � = 9).


Figure 6.49: The rate, q, versus time, t, for b = 0:5 (plotted in circles). The otherparameters were considered constant, i.e., the initial decline decline Di = 0:01,and the initial decline rate qi = 5000).

decline, as presented in Figure (6.50).The rate function (tD) is:

(tD) = t�2D (6.10)

and the inverse "Laplace" transform of the rate function is:

L�1 ( ) = k: (6.11)

Now, the dimensionless pressure pD becomes:

pD = � 1

2t2D

��1 +

�1� r2D

4tD

�e� r2D4tD Ei

�r2D4tD

�� 2A

�1� r2D

4tD

�e� r2D4tD

�(6.12)

and, provided that,

@pD@rD

= 0 ; for rD = �ptD (6.13)

we get

A (�) =1

2Ei

��2

4

�� 2

�21� �2

4

2 � �2

4

e�2

4 (6.14)

We can now plot the constant A as a function of the coe¢ cient of the no-�owmoving boundary, �; as shown in Figure (6.50).As previously mentioned the coe¢ cient � from Table (6-10), relates the di-

mensionless position, rD; of a no-�ow moving boundary to the dimensionless


Figure 6.50: The constant A(�) as a function of the constant , �; of no-�owmoving boundary:

Table 6-10: A as a function of the coe¢ cient of the no-�ow moving boundary


Figure 6.51: The dimensionless pressure, pD; versus the dimensionless time, �D;calculated at a dimensionless radius rD = 1 (� = 1, and a decline exponentb = 0:5).

time, tD. By calculating the coe¢ cient A(�) from Equation (6.14), we can, foreach choice of a coe¢ cient, �; compute a dimensionless pressure, pD: These pres-sure conditions are those for which we are able to get the wellbore rate declinewith a decline exponent b = 1

2. Further, for the coe¢ cient � = 1; we plot the

dimensionless pressure, pD; as a function of the dimensionless time, tD; and asa function of the dimensionless radius, rD, for various coe¢ cients, � = 1; 3; 5; 7;and 9.From the above �gures it is possible to determine how the dimensionless

pressure, pD, varies with the dimensionless time, tD. For example, in Figure(6.52 ), we know that at the time tD = 16, the no-�ow boundary reaches adimensionless radius rD = 4. All times tD < 16 are times where the dimensionlesspressure, pD; is outside of the no-�ow boundary. Consistently, the calculateddimensionless pressure, pD; is inside the no-�ow boundary for dimensionless timetD > 16.In Figure (6.55), it is clear that, at the time tD = 16, the no-�ow boundary

reaches a distance rD = 4. In order words, the pressure pD, calculated betweenrD = 0 and rD = 4; is considered to be within the no-�ow moving boundary.All values of pD for rD > 4 are considered to be outside the no-�ow movingboundary.

6.3.3 Harmonic Decline (b = 1)

The third case is equivalent to the rate change at IBC of Arps, with the declineexponent rate (the so called harmonic decline).Here, the rate function (tD) is:


Figure 6.52: The pressure di¤erence, pD versus the dimensionless time, tD; cal-culated at a dimensionless distance, rD = 4 (coe¢ cient: � = 1, and declineexponent: b = 0.5).

Figure 6.53: The pressure di¤erence, pD; versus the dimensionless time, �D, cal-culated at the dimensionless distance rD = 10 (for � = 1, and decline exponent:b = 0:5).


Figure 6.54: The dimensionless pressure, pD; versus the distance, rD; at a di-mensionless time �D = 16 (for � = 3, and decline exponen: b = 0:5).

Figure 6.55: The dimensionless pressure, pD, versus the dimensionless time, �D;and versus the dimensionless distance, rD (for � = 1, b = 0:5).


Figure 6.56: The dimensionless pressure, pD, versus the dimensionless time, �D,and versus the dimensionless distance, rD (for � = 3, b = 0:5).



Figure 6.58: The dimensionless pressure, pD, versus the dimensionless time, tD;and versus the dimensionless distance, rD (for � = 7, b = 0:5).



Figure 6.60: The rate, q, versus time, t, for b = 1:0 (plotted in circles). With aspeci�c initial decline qi = 5000 and an initial decline rate Di = 0:01).

(tD) = t�1D (6.15)

or

L�1 ( ) = 1 � C (�) (6.16)

For part I, we have:

�Z0

e��tDp� J1

��p�tD�d � =

�ptD2

t�2D e��2

4 (6.17)

and for part II:

�Z0

e��p�Y1

��p�tD�d � =

�

2�t� 32

D Ei

��2

4

�e�

��24 � 2

��t� 32

D (6.18)

where

Ei (z) � + log z +1X1

zn

n � n! ; with = 0:577 (6.19)

and

1X1

zn

n � n! �zZ0

eu � 1u

du (6.20)


Figure 6.61: The constant A as a function of the coe¢ cient �. A(�) is calculatedfor an inner boundary condition of variable rate. Arps decline exponent b = 1.

Hence, we get for A,

A = � 2�2e�2

4 +1

2Ei

��2

4

�(6.21)

We plot A from Equation (6.21), as a function of �. From Figure (6.61), wechoose several values for coe¢ cient � and calculate constant A. By substitutingA from Equation (6.22), into Equation (6.23), we �nd the dimensionless pressure,pD as a function of the radius, rD and the time, tD: We plot pD = pD(rD; tD) inFigure (6.62), to Figure (6.65).For pD:

pD =

1Z0

e�kthAJ0

�rpk��

2Y0

�rpk�i

dk (6.22)

and

pD = t�1D e� r2D4tD

�A � 1

2Ei

�r2D4tD

��(6.23)

The formulas developed above may be found either as "Laplace" transforms, oras "Hankel" transforms, e.g., from speci�c tables.As already mentioned, the coe¢ cient, �; from Table (6-11), relates the dimen-

sionless position, rD; of a no-�ow moving boundary to the dimensionless time,tD. By calculating coe¢ cient A from Equation (6.21), we can, calculate for eachchoice of the coe¢ cient, � compute the dimensionless pressure, pD: These pres-sure conditions are those for which we are able to obtain the the wellbore ratedecline with a decline exponent b = 1. Further, for the coe¢ cients, � = 1 we


Table 6-11: A as a function of the coe¢ cient of the no-�ow moving boundary

can plot the dimensionless pressure, pD; as a function of the dimensionless time,tD; as well as a function of dimensionless radius, rD.

From the above �gures, it is visible how the dimensionless pressure, pD,varies with the dimensionless time, tD: We can also see whether it is positionedinside or outside the no-�ow moving boundary. For all times tD <

�r�

�2; the

calculated pressure, pD; lies outside the no-�ow moving boundary. In Figure(6.61), for example, the calculated pressures, pD; for all times tD > 1, resideoutside the no-�ow moving boundary, due to rD = 1 and � = 1. In Figure(6.62), where, rD = 4, we see that for tD = 16, the no-�ow boundary reaches thedimensionless radius, rD = 4. In Figure (6.62), all times tD < 16 are such thatthe dimensionless pressure, pD; is outside the no-�ow boundary. In the same�gure, the calculated dimensionless pressure, pD; is inside the no-�ow boundaryfor dimensionless times, tD > 16. In Figure (6.63) all calculated pressure values,pD; lie outside the no-�ow moving boundary. In Figure (6.54), it is clear that,tD = 16, the no-�ow boundary reaches a distance rD = 4. In other words,the pressure pD, calculated between rD = 0 and rD = 4, is considered to bewithin the no-�ow moving boundary. All values of pressure, pD; for rD > 4 aretherefore considered to be outside the no-�ow moving boundary. In Figure (6.55)to Figure (6.59), we plot the dimensionless pressure, pD; calculated according toEquation (6.23), versus the dimensionless radius, rD; and versus dimensionlesstime, tD. For each coe¢ cient of the no-�ow moving boundary, �; we can forvalues of rD and times tD, plot the pressure solution pD and perform a similar


Figure 6.62: The dimensionless pressure, pD, versus the dimensionless time, tD,calculated at a dimensionless radius rD = 1 (for: � = 1 and b = 1).

Figure 6.63: The pressure di¤erence, pD, versus the dimensionless time, tD,calculated at a dimensionless distance rD = 4 (for: � = 1, and b = 1).


Figure 6.64: The pressure di¤erence, pD, versus the dimensionless time, tD,calculated at a dimensionless distance rD = 10 (for: � = 1, and b = 1).

Figure 6.65: The dimensionless pressure, pD, versus the dimensionless distance,rD, at a dimensionless time tD = 16 (for: � = 1, and b = 1).


Figure 6.66: The dimensionless pressure, pD, versus the dimensionless time, tD,and versus the dimensionless distance, rD (for � = 1, b = 1).

analysis evaluating the position in time where the calculated pressure lies.The above calculations describe the dimensionless pressure, pD; as a function

of time, tD; and distance, rD, for various coe¢ cients �= 1, 3, 5, 7, and 9. Thesepressure values are calculated for an inner boundary condition of variable rate.This corresponds to a harmonic rate decline with an Arps decline exponent,b = 1. From Table (6-11), we can take any A(�) and implement it in Equation(6.23) in order to generate more pressure plots.In the following subsection, we consider an inner boundary condition of vari-

able rate. We use the Arps hyperbolic expression with a decline exponent b > 1.Referring to Fetkovich (1980), such solutions are considered to be of transienttype depletion. Nevertheless, it is rather than interesting to include the solutionfor b > 1 in this study.

6.3.4 Decline Exponent (b = 2)

Figure (6.71) presents a plot of the rate, q, vs. time, t, for a decline exponentb = 2 (thick line). This decline is usually not considered as a depletion decline.The b exponent indicates that the transient region is used instead of its depletioncounterpart from combined curves of Fetkovich (1980).For the rate change (t) we have:

(tD) =

r�

tD(6.24)

and with an inverse "Laplace" transform:

L�1 [ (tD)] = k�12 ; (6.25)



Figure 6.68: The dimensionless pressure, pD versus the dimensionless time, tDand versus the dimensionless distance, rD (for � = 5, b = 1).


Figure 6.71: The rate, q, versus time, t, for decline exponent b = 2:0 (plotted incircles). The initial decline qi = 5000, and the initial decline rate, Di = 0:01.

we get the pressure, pD; according to:

pD =

r�

tDe�

r2

8�

�I0

�r2D8 tD

�A +

1

2K0

�r2D8 tD

��(6.26)

which gives:

A = �12

K0

��2

8

�+K1

��2

8

�I0��2

8

�� I1

��2

8

� (6.27)

With the values of � from Table (6-12), relating the dimensionless position,rD; of a no-�ow moving boundary to the dimensionless time, tD, we can calculatethe dimensionless pressure, pD;for which we are able to get a wellbore rate declinewith the decline exponent, b = 2. Further plots have been made for, �= 1From the above �gures, we can see how the dimensionless pressure, pD, varies

with the dimensionless time, tD; and where it is positioned, i.e., inside or outsidethe no-�ow moving boundary. For all times tD <

�r�

�2; the calculated pressure,

pD; lies outside the no-�ow moving boundary. This is exempli�ed in Figure(6.73), where pD is outside the no-�ow moving boundary, for tD > 1; rD = 1 and� = 1. Figure (6.75), shows that at the time, tD = 16, the no-�ow boundaryreaches dimensionless radius, rD = 4. Further Figure (6.74), correspond tothose all times tD < 16, where the dimensionless pressure, pD is outside theno-�ow boundary. In the same �gure, the calculated dimensionless pressure, pD;is inside the no-�ow boundary for dimensionless times tD > 16: In Figure (6.75)all calculated pressure, pD; values lie outside the no-�ow moving boundary.In Figure (6.76), at the time tD = 16, the no-�ow boundary reaches a dis-

tance rD = 4. In order words, the pressure pD, calculated between rD = 0 and


Figure 6.72: The constant, A (�), as a function of the coe¢ cient, �, of the no-�owmoving boundary, :

Table 6-12: The constant A as the function of the coe¢ cient of the no-�owmoving boundary


Figure 6.73: The dimensionless pressure, pD, versus the dimensionless time,tD,calculated at the dimensionless radius, rD = 1 (for, � = 1 and for, b = 2).

Figure 6.74: The pressure di¤erence, pD; versus the dimensionless time,tD,calculated at a dimensionless distance, rD = 4 (for, � = 1, and for, b = 2).


Figure 6.75: The pressure di¤erence, pD; versus the dimensionless time, tD,calculated at a dimensionless distance, rD = 10 (for � = 1, and decline exponent,b = 2).

Figure 6.76: The dimensionless pressure, pD, versus the distance, rD, at thedimensionless time, tD = 16 (for � = 1, and and decline exponent, b = 2).


Figure 6.77: The dimensionless pressure, pD, versus the dimensionless time, tD,and the dimensionless distance, rD (for � = 1, b = 2).

rD = 4 is considered to be within the no-�ow moving boundary. Consistently,all values of pressure, pD; for rD > 4 are considered to be outside the no-�owmoving boundary.

In Figure (6.77) to Figure (6.81), we plot the dimensionless pressure, pD;according to Equation (6.26), versus the dimensionless radius, rD; and versusthe dimensionless time, tD.Table (6-13) summarises all developed model solutions. In the table solutions

for pD and A(�); rate inner boundary conditions are given for each variable (withArps rate decline exponents of: b = 0:33, b = 0:5, b = 1, and b = 2). Further,Table (6-13) presents dimensionless pressure, solutions, pD, calculated for earlyand late dimensionless times, tD.The computations con�rm that the production mode must be of variable-

pressure if the rate is to follow a speci�c value of b, which is in agreementwith the work of Raghavan (1993). The model solutions are analytical and areavailable within a certain drainage area and at the wellbore for selected b values.Furthermore, the model assumes that a vertical well produces under variable-rateconditions and with a no-�ow outward-moving boundary



Figure 6.79: The dimensionless pressure, pD, versus dimensionless time, tD, andversus the dimensionless distance, rD (for � = 5, b = 2).


Figure 6.80: The dimensionless pressure, pD, versus the dimensionless time, tD,and versus dimensionless distance, rD (for � = 7, b = 2).

Figure 6.81: The dimensionless pressure, pD, versus the dimensionless time, tD,and versus dimensionless distance, rD (for � = 9, b = 2).


Table 6-13: The dimensionless pressure, "pD", calculated by the model for "tD"going to 0,and the "tD" going to in�nity, for a variable rate production of Arpstype. The decline exponent, b (b with values of 0.333; 0.5; 1., and 2) de�nes theArps type production decline


6.4 Summary

The aim of the multiple-fractured horizontal well model is to provide a tool foran improved modelling of oil production from horizontal or near-horizontal wellswith induced fractures. This is achieved by implementing original analytical so-lutions. The state-of-the art model is useful diagnostic and prognostic screeningtool for the oil industry, and its robustness and speed render the programmefor well production optimisation of a horizontal well with induced fractures veryuser-friendly. Another bene�t is the bringing-together of rate-time and pressure-time analyses, thereby a total package to better characterise the behaviour of ahorizontal well with induced fractures. The oil solutions from the model are alsoapplicable for gas �ows. If pseudopressure, m(p)D is employed instead of pres-sure, pD; the corresponding solution will not take into account non-Darcy e¤ectsand can be considered as an approximative gas solution. To include non-Darcy�ow, we should derive additional equations for gas �ow for the system when awell with fractures is coupled to a gas reservoir.Model solutions for a no-�ow moving boundary further investigate the nature

of the wellbore pressure required for the Arps rate decline production, de�nedwith the �xed exponent b. Chapter 5 presented the physical model together withits analytical solutions. The nature of the pressure pro�les for several b valueswere demonstrated. Since each b value was selected, they may as such be relatedto the drive mechanism (i.e., b = 1/3, for the solution-gas-drive, b = 0,5, forthe gravity-drainage, and b = 1, for the multilayered no-cross �ow production).The introduction of the dimensionless pesudo-pressure, m(p)D; in the place ofthe dimensionless pressure, pD;renders possible an extension from oil �ow to gas�ow.

Chapter 7

CASE STUDIES

The application of horizontal wells and, more speci�cally, that of multi-fracturedhorizontal wells for exploiting oil and gas reservoirs, is �rmly established withinthe industry. Various authors have made signi�cant contributions to improvingthe understanding of the �ow behaviour of such wells.The evaluation of multi-fractured horizontal well performances, or the se-

lection of optimum perforation/stimulation designed for such wells, may be ap-proached through �ne grid reservoir simulation. However, while reservoir simu-lation is the most advanced method for predicting well performance, it is oftentoo time-consuming when conducting parametric screening studies. Often, therequired data is unavailable and the e¤ort may not be warranted.

7.1 Model Comparison and Validation

7.1.1 Well Models Comparisons

The multiple-fractured horizontal well model is a tool for an improved modellingof oil production from horizontal or near-horizontal wells with induced fractures.The modelling is achieved by implementing original analytical solutions.The state-of the-art technology model represents a useful diagnostic and prog-

nostic screening tool for the oil industry, and its robustness and speed make thisprogramme for well production optimisation of a horizontal well with inducedfractures easy to use. Another bene�t is the bringing-together of rate-time andpressure-time analyses to provide a total package for an improved characterisa-tion of the behaviour of a horizontal well with induced fractures.For a sake of comparison, a fractured-horizontal well code was coupled to

a code of a fractured-vertical well and a partially-perforated-horizontal well.Thus, one comparison feature consists in available output variables for the threemodels. A fractured-horizontal well model is internally developed (as describedin chapter 4) and two other model codes was provided by BP for comparison

255

256 7. CASE STUDIES

Figure 7.1: The pressure-di¤erence versus time for three models (multi-fracturedhorizontal well (MFW-open outer boundary-SLAB), vertical-fractured well(VFW-open outer boundary-SLAB) and partially perforated horizontal well(PPHOW-closed, BOX ) [After Cvetkovic et al. (2000)].

purposes only. These two model solutions are partly reviewed in chapters 2, 3and 4.Table (7-1) presents input data for the fractured-horizontal well, the fractured-

vertical well, and the partially-perfortated-horizontal well models. The corre-sponding well response plots are presented in Figure (7.1).

7.1.2 Semi-Analytical versus Numerical Model Valida-tion

As an alternative to simulation, the application of semi-analytical models canreadily yield wellbore responses to various boundary conditions. This is oftensu¢ cient to provide an understanding of which factors that have the most in�u-ence on well performance. If simulation work is warranted, it can then proceedwith the insight obtained from the analytical models. The model is validated tothe numerical simulators (GeoQuest Schlumberger ECLIPSE). Recently, modelsolutions were compared to other commercial solutions and the obtained resultswere satisfactory. In�nite-acting outer boundary and in�nite conductivity frac-ture solutions were similar, whereas the �nite conductivity varied due to thefracture �ow modelling not being equal between the two models.

7. CASE STUDIES 257

Table 7-1: The input data for fractured-horizontal well (MF), fractured-verticalwell (VFW) and partially perforated horizontal well (PP) model

258 7. CASE STUDIES

Figure 7.2: The semi-analytical model cumulative production compared to theother models (2 numerical and a semi-analytical models).

A semi-analytical model was also compared to several commercial simulators.For the in�nite conductivity fractures (SLAB model), solutions from the semi-analytical and numerical model were equal, provided that the numerical modelboundary in the x and y directions were large enough (since the outer boundaryof the numerical model was not in�nite acting). Finite conductivity fracturesolutions varied due to the manner in which the fracture �ow was modelled.A semi-analytical solution was compared to several commercial numerical andsemi-analytical simulators.Similar results were obtained for the simulator with improved gridding (as

local grid re�nement and PEBI gridding), and have been presented by Cvetkovic(2000). The comparison of the recent study with the commercial simulatordemonstrates an almost equal in�nite conductivity fracture production for ahorizontal well with 5 fractures, whereas the �nite conductivity results vary.Obtained plots cannot be presented due to con�dentiality reasons.The validation of the recent model is based on the data provided by the

Amoco Exploration&Production Technology Group (1997). The semi-analyticalmodel is compared to 3 other model solutions (2 numerical and 1 semi-analytical)Figure (7.2) demonstrates how the semi-analytical model solution matches othermodels (for a fracture conductivity, FC , ranging from 50 to 100 mDft). Thedi¤erence between model solutions is mainly caused by the modelling of a �-nite fracture conductivity, FC . Another di¤erence is in the availability of themodelling solutions when handling the early-time production rate with fracture

7. CASE STUDIES 259

Figure 7.3: Comparisons of model-calculated cumulative productions (two nu-merical models and a semi-analytical single-phase model). The production his-tory comprises 1200 days.

responses. Despite these early-time di¤erences in the production pro�les, it ispossible to match other model pro�les for the larger time interval. Figure (7.3)shows that, within 1200 days of production, the semi-analytical model matchesother single-phase model cumulative productions. The cumulative productionsfor two numerical models and the single-phase model vary with �ow within afracture that is not equally modelled. The semi-analytical model is veri�ed bycomparing it to other model production pro�les as presented in Figure (7.3).

7.2 Fractured-Horizontal Well - Oil Production

7.2.1 Fractured-Horizontal Well (Eko�sk Oil Field - NorthSea)

Case history data have been obtained from the Eko�sk �eld in the North Sea.Data for the reservoir, a horizontal well and fractures are provided in Table (7-2). We assume that 8 transversal fractures are equally spaced along a horizontalwell with a length of 2090 ft. Further, each fracture half-length and fractureheight are the same. After performing the sensitivity analysis, we select inputparameters to be �ne-tuned. Further, observed well data are compared to datacalculated by the model, as obtained when varying the IBC of pressure and/orrate.

260 7. CASE STUDIES

Table 7-2: Input data from the Eko�sk �eld - North Sea (for a horizontal wellwith 8 transversal-fractures)

Variable-Pressure IBC

Figures (7.4 and 7.5) compare observed data to data obtained from the modelat a variable-rate IBC. Daily measured wellbore pressures, used as input data inthe model, are divided into 7 pressure intervals, and each interval is averaged.Matching of the observed rate data with that calculated by the model was

obtained with an IBC of variable-pressure, as given in Figure (7.5). The over-all time scope of 200 days was divided into 3 time intervals and each intervalcomprised the averaged pressure data calculated within the speci�c increment oftime.

Variable-Rate IBC

Consistently, matching of the observed pressure-di¤erence data with that cal-culated by the model was obtained for variable-rate IBC. For identical fracturehalf-lengths and fracture heights, the three fracture conductivities were consid-ered (for a �nite conductivity FC = 20, 15 and 5 mDft). The model �t ispresented in Figure (7.6).Thus, selecting a fracture conductivity FC = 20 mDft and choosing two

fracture half-lengths, Lf of 50 ft and 25 ft; we plot the pressure-di¤erence versustime as in Figure (7.7). The match is not yet achieved, so further tuning isneeded. Due to the initial pressure, Pi, not being precisely de�ned, it can be

7. CASE STUDIES 261

Figure 7.4: A comparison of observed (oil rate) data with that calculated by themodel at an IBC of variable pressure (from 7 selected time intervals).

Figure 7.5: A comparison of observed (oil rate) data with that calculated bythe model at IBCs of variable pressure. The daily measured pressures at thewellbore are devided into 3 pressure intervals.

262 7. CASE STUDIES

Figure 7.6: The matching of observed and calculated pressure di¤erences versustime (for variable-rate IBCs and changes in fracture �nite conductivities FC).

varied.Then, by selecting a lower value of Pi (i.e., 3580 psi instead of 4400 psi) and

further maintaining the fracture conductivity and fracture sizes as previouslyde�ned, it become possible to better match the observed data. Observed andcalculated pressure-di¤erence well data are presented in Figure (7.8).In a time-interval of only 200 days, calculated data for IBCs of both variable-

pressure and variable-rate were found to be within the range of observed datafor a horizontal well with 8 fractures. These results demonstrate that IBCs ofvariable-pressure and variable-rate were integrated in one tool and can be usedfor the screening analyses of a producing well with fractures.

7.2.2 Fractured-Horizontal Well (North Sea Oil Field)

Variable-Pressure IBC

The following study compares the model production data to observed well datafor 160 days of production. The inner boundary condition of the model is ofvariable pressure, and the fracture conductivity varies as FC = 70, 40 and 20mDft. Figure (7.9) compares the results from the model to observed productionwell rates. Observed well data corresponded to those created by the model forvarying fracture conductivities.The cumulative production, according to model calculation, for a horizontal

7. CASE STUDIES 263

Figure 7.7: The matching of observed and calculated pressure-di¤erences versustime (for variable-rate IBCs and changes in the fracture half-length, Lf , from 50ft and 25 ft, and assuming a maintained fracture conductivity, FC , of 20 mDft).

well with 6 fractures producing oil for 700 days is presented in Figure (7.10).Each fracture conductivity is FCi = 70 mDft, and the wellbore IBC is of variablepressure. With an increase a pressure intervals it would be possible to achieve abetter match.

Variable-Rate IBC

Figure (7.11) presents calculated pressure di¤erences for several fracture conduc-tivities, FC , with values of 75mDft, 100 mDft up to 150 mDft. For the 50 days ofwell production, the observed pressure di¤erence corresponded to that calculatedby the model (for a fracture conductivity of 150 mDft). Between 50 and 140 daysof production the fracture conductivity was reduced to 100 mDft, and after 140days the conductivity was further reduced to below 50 mDft. This screening wasachieved with variable-rate IBCs and varying fracture conductivities.

By using the fractured-horizontal well features with IBCs of variable-pressureand variable-rate, it is possible to match the observed wellbore data. The IBCsof variable-pressure and variable-rate were integrated into one tool that was usedin the screening analyses.

264 7. CASE STUDIES

Figure 7.8: The matching of observed and calculated pressure-di¤erences versustime (for variable-rate IBCs and changes in the fracture half-length, Lf , from 50ft and 25 ft, and assuming a maintained fracture conductivity, FC , of 20 mDft).The initial pressure, Pi, is reduced from 4400 psi to 3850 psi.

7. CASE STUDIES 265

Figure 7.9: Observed and calculated rates as functions of time. The IBC of themodel are of variable pressure, and the fracture conductivities, FC , change from70, 40 down to 20 mDft.

Figure 7.10: Matching of well observed cumulative oil data with a model calcu-lated.

266 7. CASE STUDIES

Figure 7.11: Observed and model pressure di¤erences vs. time, in addition tocalculated and observed rates vs. time. The IBC of the model are of variablerate.

7.2.3 Syd-Arne Oil Field (North Sea)

The Syd-Arne �eld, operated by Amerada Hess Corporation, is located in theDanish part of the North Sea. The �eld was originally discovered in 1969 andits initial reserves estimated by the Danish Energy Agency are 185MMstb of oiland 434 Bcf of gas. The �eld came into production in 1999 and is currentlyproducing with a total of 19 development wells (from which 12 are fractured-horizontal oil producers and 7 are fractured-horizontal water injectors). The �eldis an elongated chalk anticline, 12 km by 3 km, with a depth between 2700-2940m subsea. Over the crest of the �eld, the oil column is restricted to the thicknessof the reservoir. A more detailed description of the Syd-Arne �eld is providedby Christensen et al. (2006). The reservoir consists of the Maastrichtian bestreservoir layer (Upper Cretaceus) to the Danian (Paleocene) chalk of the Torand Eko�sk Formation (Fm) with reservoir parameters presented in Table (7-2).The reservoir, fracture and well data were provided by Amerada Hess for thepurpose of this thesis in 2008.

A Horizontal Well with 14 Transversal Fractures

A fractured-horizontal well penetrates 14 transversal fractures, as presented inFigure (7.12). Such a well produces for almost 10 years. Observed well data

7. CASE STUDIES 267

Table 7-3: Some general parameters of the Syd Arne North Sea �eld (SPE103282)

Figure 7.12: The fractured horizontal well, SA-P1 penetrating 14 transversal-fractures in an oil reservoir (cross section).

are given in Figures (7.13, 7.14 and 7.15), and model input variables includereservoir, fracture and well data, as presented in Figures (7.16 and 7.17).

Measured Well Data Measured well data are presented in Figures 7.13 to7.15.

Model Input Data. Model input data for the IBC of constant pressure areshown in Figures 7.16 and 7.17.

Obtained Match Figure 7.18 shows the obtained match between the calcu-lated and measured well data.

Additional Model Information A model with inner boundary conditionsof the constant pressure also matches the cumulative production. Individualfracture rates, fracture cumulative production and productivity index, PI, as afunction of time are additional information available by the model.For a well that has been producing for almost 10 years, it is with the provided

input data and performed sensitivity analysis, possible to match the observedwell rate production, as given in Figure (7.18). The wellbore IBCs were of

268 7. CASE STUDIES

Figure 7.13: The measured wellbore pressure data, pwf (psi), versus time, t (d).

Figure 7.14: The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe (bbl/d),and the gas rate, qg (Scf3/d), versus time, t (d), for a horizontal well SA-P1 with14 transversal-fractures.

7. CASE STUDIES 269

Figure 7.15: The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe (bbl/d),and the gas rate, qg (Scf3/d), versus time, t (d), for a horizontal well SA-P1 with14 transversal-fractures on a log-lin scale.

Figure 7.16: The well and fracture input data.

270 7. CASE STUDIES

Figure 7.17: The reservoir input data.

Figure 7.18: A comparison of the calculated and measured well data (model dataobtained with an IBC of constant pressure).

7. CASE STUDIES 271

Figure 7.19: The well cumulative production, Q (bbl), and the fracture produc-tion, Qfri(i = 1; :::14) (bbl), versus time (d). The IBC of the model is of constantpressure.

constant pressure. Additional model features to the screening analysis consist incalculating individual fracture rates and cumulative production, as presented inFigures (7.19, 7.20).

Model Input (IBC of variable rate) An IBC of variable rate matches thepressure-di¤erence during the �rst interval of the variable-rate IBC given inFigure (7.21). The �rst interval lasted ca. 800 days. More time intervals willpermit an improved match of the observed data, as presented in Figure (7.22).The programme set- up is limited to only limited interval changes, and theavailability of more intervals would allow a better match for the entire pressure-di¤erence history of over 3000 days.

IBC Step-Function A step-function or IBC of constant-rate to constant-pressure in the same run matches the production history.The choice of variety with regards to the IBCs makes it possible to fully eval-

uate the model features with the real daily measured well pressures, rates andGOR values. Such daily measured data were available within the entire produc-tion history of a well with 14 fractures. Although the details of the productionhistory were not given, it was possible to achieve a reasonably good match ofthe given measured well data. The obtained rate match was generated with the

272 7. CASE STUDIES

Figure 7.20: The model well rate, q (bbl/d), and the fracture rates, qfri(i =1; :::14) (bbl/d), versus time (d). The IBC of the model is of constant pres-sure.

Figure 7.21: The pressure di¤erence, Pi � Pwf (psi), versus time (d) for an IBCof variable rate. (Well production rates for the �rst 800 days are considered as1 rate-interval).

7. CASE STUDIES 273

Figure 7.22: The pressure di¤erence, Pi � Pwf (psi), versus time (d) for an IBCof variable rate (The well production rates for the �rst 800 days of are devidedinto 3 rate intervals).

Figure 7.23: The step function match obtained with an IBC of constant-rateto constant-pressure processed in a single run. Both pressures and rates arematched within the single run.

274 7. CASE STUDIES

Figure 7.24: A fractured horizontal well, SA � P2, penetrating 14 transversalfractures in an oil reservoir. Cross section view.

step-function, with which the overall time interval was divided into two parts.In the �rst interval the IBCs are of constant rate and in the second, they areof constant pressure. After the �rst interval, the model calculates the wellborepressure, maintaining it constant until the end. Thus, within a single run, theIBC changes from constant-rate to constant-pressure. The step function featureis applicable to the plateau production (as the production used in North Sea�elds). In the Figure (7.23) match of the pressure-di¤erence and the rate isachieved within the single run.Moreover, for the given rates, the pressure-di¤erence data were matched with

the IBC of variable-rate. The model is capable of handling IBCs of variablepressure and rate. Thus, by varying the IBC in this manner the model calculateswell production responses that match the measured well data. A combinationof the model rates and pressures may help to analysis well data. The processof matching well data is fast, so, overall, the developed tool can be used in thewell production screening analyses as demonstrated. An extension of the modelto naturally fractured reservoir features may be important for the advancedfractured reservoir study of the Syd-Arne wells.

7.2.4 Syd-Arne Oil Field (North Sea)


Measured Well Data Measured well data are presented in Figures 7.25 to7.27.

Model Input (IBC of constant pressure) Model input data for the IBCof constant pressure are shown in Figures 7.28 and 7.29.

7. CASE STUDIES 275

Figure 7.25: The measured wellbore pressure, pwf (psi), versus time, t (d).

Figure 7.26: The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe (bbl/d),and the gas rate, qg (Scf3/d), versus time, t (d), for a horizontal well SA-P2 with14 transversal fractures.

276 7. CASE STUDIES

Figure 7.27: The measured oil rate, qo (bbl/d), the equivalent oil rate, qoe (bbl/d),and the gas rate, qg (Scf3/d), versus time, t (d), for a horizontal well SA-P2.with14 transversal fractures on a log-lin scale.


7. CASE STUDIES 277


Obtained Match It is possible to select model-generated well production pro-�les and match them with the well production history as shown in Figures (??,7.31, 7.32). By varying the fracture half-length and partial fracture penetration,we are able to match the well production history. Cumulative production pro�leswere generated from the model by assuming that both the fracture half-length,Lf , and the fracture-height, hf , (for each of the 14 fractures) were equal. Themodel production pro�le (with Lf = 50 and hf = 50) matched the well dataduring the �rst 400 days of production. However, due to changes in the fractureshape, the last 700 days of production can only be matched by reducing boththe fracture height and the fracture half-length in the model to hf = 40 ft. andLf = 34 ft With more well and fracture data, it would be possible to achieve abetter match of the rates.Nevertheless with an IBC of constant pressure, the model matches the cu-

mulative production data, as shown in Figure (7.31). The two production cu-mulative rates from the model are de�ned with the fracture half-length, Lf , andthe fracture penetration height, hf . The measured production cumulative rateis matched for the �rst 500 days of production and from day 2750 until the endof the production history. In between the fracture character changes. In orderto investigate such change in fracture character, an additional production pro-�le is created, with an average fracture half-length, Lf , of 34 ft and an averagefracture penetration-height of 40 ft. In Figure (7.32), the cumulative productionobtained with the model matches the measured cumulative rate between 550and 1050 days of a well production history. It is evident from the �gure that

278 7. CASE STUDIES

Figure 7.30: A comparison of the calculated and measured well data (model dataobtained with an IBC of constant pressure).

Figure 7.31: The measured versus calculated data for the cumulative rate, Q(bbl), versus time (d). The calculated data are de�ned with the half-length, Lf ,(of 34 and 50 ft) and the fracture penetration height, hf , (of 40 and 50 ft).

7. CASE STUDIES 279

Figure 7.32: The measured versus calculated data for the cumulative rate, Q(bbl) versus time (d). The calculated data are de�ned with the half-length, Lf(of 20, 34 and 50 ft) and the fracture perforation,hf (of 40 and 50 ft).

the fracture character changes. Due to this variation, three models were usedto quantify the fracture closure e¤ects. These models generated the individualfracture cumulative production and were thus of help in the fracture closurediagnosis.

Additional Model Information The model provides 14 individual fracturecumulative production pro�les in addition to a production pro�le for a well withfractures as displayed in Figure (7.33). It is possible to investigate which fracturerate production causes the reduction in well production.The individual fracture rates versus time are given in Figure (7.35). Each

fracture half-length and fracture partial-penetration is the same.It is also possible to include fracture conductivity changes that will reduce

the production pro�le. In this study, the fractures were considered to be in�niteconductive. Further, it is possible to vary the IBC, and the well productionhistory can be matched accordingly (as presented in the previous well SA-P1case). With more information on the producing well, it is possible to improvethe well production match and thus obtain a better fracture production prognosisand diagnosis of the well in question.

E¤ective Wellbore Parameters and a Fractured Horizontal Well Pro-ductivity In 2009 Cvetkovic investigated the productivity of a well with frac-

280 7. CASE STUDIES

Figure 7.33: The well cumulative production, Q (bbl), and the individual fracturecumulative production, Qfri (i = 1; :::; 14), versus time, t (d). (Each fracturehalf-length Lf=34 ft and the fracture partial penetration height, hf=40 ft).

Figure 7.34: The well rate production, q (bbl/d), and the individual fracturerate production, gfri (i = 1; :::; 14), versus time, t(d) (Each fracture half-lengthLf = 34 ft, and the fracture partial penetration height hf = 40 ft).

7. CASE STUDIES 281

Figure 7.35: Fracture rate for individual fractures (1, 6, 7, and 14) for varyingfracture half-lengths, Lf (34, 50 ft) and partial penetration heights (40, 50 ft).

tures by means of semi-analytical late-time approximations. These expressionsare related to a vertical and a horizontal well with a single-fracture production.The �ow in a fracture that is coupled to a horizontal well is of uniform �ux (orof in�nite conductivity �ow) and the fractures can be positioned transversally orlongitudinally. To successfully measure the production e¢ ciency of a fractured-horizontal well, the e¤ective wellbore radius and the equivalent single-fracturehalf-length were introduced. The e¤ective wellbore radius was de�ned for a ver-tical well and could be extended for its horizontal counterpart. The equivalentfracture half-length was de�ned for a horizontal well with a single-transversal anda single-longitudinal fracture. The late-time approximations were developed fora fractured horizontal well positioned in the middle of an in�nite oil reservoir.The developed expressions were employed for a multi-fractured horizontal well inorder to de�ne an e¤ective wellbore radius and an equivalent fracture half-length(for both the transversal and the longitudinal fractures). A series of solutionscorresponding to various conditions were given in a multiple-fractured-horizontalwell model. It was possible to verify derived e¤ective parameters of a fracturedhorizontal well by using a fast, robust and facile software program, a screening-tool product, that has been of considerable bene�t to companies in the petroleumindustry.

Risk Assessments of a Fractured Horizontal Well Cvetkovic (2009) pre-sented a screening approach with risk analysis considering the fractured well

282 7. CASE STUDIES

Figure 7.36: Valhall �eld with several multi-fractured horizontal wells [AfterNorris et al. (2001)].

data from Syd-Arne. The analysis of the data obtained for a real reservoir-fracture-well was carried out by means of parameter changes, selecting objectiveparameters as rates and cumulative rates, analysing generated pro�les and �nallycomparing these pro�les to real data. Furthermore, the veri�ed input informa-tion was considered as the base data. Such input data were further used inrisk analysis thus generating a wide range of production pro�les. Such a studyprovides a work �ow that should be further tested for the optimal fracture posi-tioning along a horizontal well. A screening procedure is complementary to anycomplex simulation since semi-analytical tools are fast and require only limitedreservoir-fracture-well data.

7.2.5 Valhall Oil Field (North Sea)


In this case study, we were able to simulate production well responses with timefor an IBC of constant and variable pressure or rate, as well as constant-rateto constant-pressure changes; a step-function option. Valhall �eld with severalfractured horizontal well is given in Figure (7.36).Daily production rates and PIs of a horizontal well with 5 fractures and well

7. CASE STUDIES 283

pressures are given in Figures (7.37). The changing GOR with time in Figure(7.38) con�rms that the well was producing below the bubble point. Various�uid, reservoir, fracture and well properties are given in Table (7-5). Unfortu-nately, we do not have much information concerning the �uid characterisation,the geological model or any obtained simulation results. The choice was madeto match well data (oil production rate, cumulative rate, wellbore pressure andproductivity index, PI) with the data generated by the model.

Measured Well Data For an IBC of constant rate, it is possible to match thewell�s PI (with the �nite conductivity option in the model), i.e., for early times,the fracture conductivity is 50 D and for late time it is 2.5 D. Both have fracturesof 0.008 ft. The rate in Figure (7.39) is chosen to be constant at q = 7170 bbl.The next match is carried out for an IBC of variable pressure. The well

pressure as a function of time is split into three intervals, with s speci�c averagewellbore pressure. This pressure step function is for speci�c the model inputand the model output is rate. Without any �ne-tuning, we are able to obtain agood match of the rate data during a certain time period, as presented in Figure(7.41).For matching wellbore pressure, we use an IBC of variable rate. The time

interval is divided into several smaller intervals, each with a de�ned averagerate. This rate step function represents input for the model IBC as steps ofrate. Ideally, we should have applied variable rate changes within the entiretime-scale in order to obtain an adequate wellbore pressure response. Even withan imprecise input of variable rate, as in Figure (7.42), we are able to �t thewellbore pressure data quite well for almost 200 days.For the basic restart option treated here, we have as input a period of constant

rate followed by a pressure drop maintained constant. Both the SLAB and BOXmodels are used in the data analysis. The inner boundary condition changesfrom constant rate to constant pressure: the constant rate is kept for the �rst137 days, for which the model calculates the wellbore pressure. For the next479 days, the calculated wellbore pressure is maintained constant and the modelcalculates the rate decline, as shown in Figures (7.43, 7.44).

284 7. CASE STUDIES

Figure 7.37: Rate and PI data versus measured values of the cumulative wellproduction.

Figure 7.38: The wellbore pressure and GOR data versus time.

7. CASE STUDIES 285

Table 7-4: Wellbore IBCs of variable pressure for the in�nite conductivity and�nite conductivity, for "FC" = 50 (mDft) fractures

286 7. CASE STUDIES

Figure 7.39: Productivity Index versus Time (IBC = Constant Rate). [Modeland Well Data: PI - MATCH].

Figure 7.40: Model and well data PI - match (for an IBC of costant rate - thefracture permeability and fracture width are constant).

7. CASE STUDIES 287

Figure 7.41: Rate versus time (for an IBC of variable pressure).

Figure 7.42: Calculated wellbore pressure matches model observed data for vari-able rate IBCs.

288 7. CASE STUDIES

Figure 7.43: The step-function procedure calculates dimensionless pressure forthe IBC of constant-rate and rates for the IBC of constant-pressure. Within thesame run the IBCs are changing from constant-rate to constant-pressure.

Figure 7.44: The match of the models (SLAB & BOX) with the well rates.

7. CASE STUDIES 289

7.3 Fractured-Horizontal Well - Water Injec-tion

7.3.1 Syd-Arne (North Sea)

A Water Injection Horizontal Well with 16 Transversal Fractures

Figure (7.45) presents the water injection rate and cumulative injection. Thewellbore pressure as a function of time is given a horizontal well with 16 fracturesin Figure (7.46). A well injection lasts ca. 2500 days. The reservoir, fractureand well data are provided in Figures (7.47 and 7.48).

Measured Data Measured well data are presented in Figures 7.45 to 7.46.

Input Data Input data are shown in Figures 7.47 and 7,48.

Obtained Match Figure (7.49) show an almost perfect match for the waterinjection rates with the water cumulative injection rates.

Additional Information The individual fracture injection rates are presentedin Figure (7.50), and the individual cumulative fracture injection production isgiven in Figure (7.50). The water injectivity, and the PI, for a horizontal wellwith 16 fractures is provided in Figure (7.52). These data represent valuableadditional information and can be complementary to any simulation studies.

290 7. CASE STUDIES

Figure 7.45: The water injection rate and the cumulative injection rate versustime for a horizontal well with 16 transversal fractures.Well: SA-WI1.

Figure 7.46: The wellbore pressure versus time for the SA-WI1 well.

7. CASE STUDIES 291



292 7. CASE STUDIES

Figure 7.49: The model water injection rate, qi (bbl/d), and the water injec-tion cumulative production, Q (bbl), versus time, t (d), for the SA-WI1 waterinjection well.

Figure 7.50: The fracture water injection rate, q (bbl/d), for a horizontal wellwith 16 fractures, and the individual fracture injection rates, gfri (i = 1; :::; 16).

7. CASE STUDIES 293

Figure 7.51: The cumulative fracture water injection, Q (bbl), for a horizontalwell with 16 fractures and the individual fracture water injection, Qfri (i =1; :::; 16).

Figure 7.52: The productivity index, PI (bbl/d psi), versus tme, t (d), for thewater injection horizontal well penetrating 16 fractures.

294 7. CASE STUDIES

7.4 Fractured-Horizontal Well - Gas Produc-tion/Injection

7.4.1 Syd-Arne Synthetic Data

If the pressure, P; is substitute with the pseudo-pressure, m(p); we should beable to use model solutions for a horizontal well with fractures producing froma gas reservoir. Here, we �rst assume that solutions are only approximations,and that no non-Darcy �ows exist. Consequently the oil well production dataare converted to an equivalent gas data. Figure (7.53) presents a match of theoverall equivalent gas production history with the model data.The model pseudo pressure, m(p), is set up as an IBC of constant peudo-

pressure. The cumulative production obtained from the model matches thereservoir synthetic equivalent gas data. Moreover oil well rates are convertedinto equivalent values for gas, thus demonstrating that the model can be ex-tended also for gas production. In order to include the non-Darcy �ow themodel solutions should be developed further.

7. CASE STUDIES 295

Figure 7.53: A horizontal well with 14 transversal-fractures producing from asynthetic gas reservoir. The cumulative oil production is converted into its cu-mulative gas equaivalent. The IBCs are either constant or of variable pesudo-pressures.

296 7. CASE STUDIES

7.5 Vertical Well � Exponential Decline withthe Moving Boundary

This case further veri�es the rate-time analysis concept of a model with a no-�ow moving boundary. It provides new solutions to a di¤usion problem whenwellbore conditions are of variable rate decline with an exponent b close to zero,and the no-�ow boundary moving outwards from the wellbore axis. Generatedpressure pro�les comprise both the transient information of a changing drainagevolume and the prede�ned depletion rate decline corresponding to the rate pro�leas derived by the Arps exponential decline. The solutions are unique and openlead to a new approach in solving variable-rate inner boundary conditions witha no-�ow outwards-moving boundary. The variable rate is almost exponentialand the derived expression can thus be compared to the analytical solutions ofdepletion decline. One way of determining the velocity of the moving no-�owboundary is to employ the numerical streamline solutions.

7.5.1 Variable Rate IBCs of b Almost Zero

Variable rate IBCs should be considered for exponent b; almost zero, i.e., thevalue n = 1

bshould �rst be selected as for instance 100 or 1000. The velocity of a

no-�ow outwards-moving boundary, {, should be carefully chosen in order for sosingularity not to appear in the pressure di¤erence. Thus, for selected values of{ and n, it would be possible to receive positive and realistic pressure di¤erencedata. In order words, only certain solutions correspond to physical behaviourfor the selected values of { and n.The pressure di¤erence, PD, for the almost exponential decline b = 0 is

de�ned by the choice of the n value, which is the inverse b exponent, and thespeed of the no-�ow moving boundary, {. The expression is given in Equation(7.1)

pD sr2

�n�

14 ��nD (

r2D�D)�

14 e�

r2D

8�D

�A cos({ � r

p�D)� �

2sin({ � r

p�D)

� r

p�D

rn� 1

2� �

4

!(7.1)

there, constant A is de�ned asymptotically (for n !1) as:

A s ��2cot�{pn� �

4

�(7.2)

We now, choose a large value of n (i.e., a b value approaching zero). The valueof { is, accordingly, related to n by the relation:

7. CASE STUDIES 297

Table 7-5: Input parameters (n and "rD") to the semi-exponential decline inthe moving boundary model

{ =Npn� (7.3)

where N is an integer. Further, assuming that both n and � are approachingin�nity then, Equation (7.1) becomes:

pD sr2

�n�

14 ��nD (

r2D�D)�

14 e�

r2D

8�D (7.4)

We note that n�should be very small for calculating pD with time �D at a distance

rD. We plot pD versus �D for various rD values (1, 10, 100). Further we choosen and {: For rD = 1; and n = 100, the dimensionless pressure di¤erence is

pD =q

2�100�

14 ��100( 1

�)�

14 e�

18� .

Other possible plots should be made by evaluating solutions, signifying thatwe should select variables in order to avoid singularities that are present in themodel solutions.For rD = 1;and n = 100, pD =

q2�10�

14 ��10(100

�)�

14 e�

1008�

For rD = 100;and n = 100, p =q

2�100�

14 ��100(10000

�)�

14 e�

100008�

For rD = 100;and n = 1000, p =q

2�1000�

14 ��1000(10000

�)�

14 e�

100008�

In Table (7-5), b values are selected as b = 1 (when n = 1), b = 0:1 (whenn = 10), b = 0:01 (when n = 100).The following Figures (7.54, 7.55, 7.56, 7.57, 7.58, 7.59, and 7.60 ) are plots

of the inverse decline exponent, n = 1, and various dimensionless radii, rD, of 1,10, 20, 50, 100, 200, and 500. For increasing value of the dimensionless radius,rD, the dimensionless pressure, PD, is shifted from low to larger values as thedimensionless time, � , is increased.

298 7. CASE STUDIES

Figure 7.54: The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponernt n = 1 and a dimensionless radius rD = 1.

Figure 7.55: The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and a dimensionless radius, rD = 10.

7. CASE STUDIES 299

Figure 7.56: The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponernt n = 1 and a dimensionless radius, rD = 20.

Figure 7.57: The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and dimensionless radius, rD = 50.

300 7. CASE STUDIES

Figure 7.58: The dimensionless pressure, pD, versus dimensionless time, � ; foran inverse decline exponernt n = 1 and dimensionless radius, rD = 100).

Figure 7.59: The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and dimensionless radius, rD = 200.

7. CASE STUDIES 301

Figure 7.60: The dimensionless pressure,pD, versus dimensionless time, � , foran inverse decline exponernt n = 1 and a dimensionless radius, rD = 500.

302 7. CASE STUDIES

Figure 7.61: The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponent, n = 10 and a dimensionless radius, rD = 1.

Figures (7.61, 7.62, 7.63, 7.64, 7.65, 7.66, and 7.67) represent plots for adecline exponent n = 10, which almost corresponds to b = 0, and various dimen-sionless radii, rD, of 1, 10, 20, 50, 100, 200, and 500 and dimensionless pressure,PD, is shifted from low to high values for an increased dimensionless time, � .Figures (7.68, 7.69, 7.70, 7.71, 7.72, 7.73 and 7.74) are the corresponding

plots for a decline exponent n = 100, which is even closer to the exponentialdecline b = 0. The dimensionless pressure displays the same trend as in the twopreceding series of plots.Figure (7.75 ) plots the dimensionless pressure as a function of dimensionless

time, for a decline exponent n = 1000, which represents an exponential declineof approximated b = 0, and a dimensionless radius rD = 10. PD values for otherdimensionless radii, rD, are not available due to the solution singularity caused bythe models limitation to handle only selected values of input model parametersn, � and rD:The model is unstable for values of the decline exponent that areclose to zero. Further, the model allows the multiplication of the dimensionlespressure, PD, so tuning and calibration is needed before data can be validatedwith the measured pressure at various distances from the wellbore axis. Thedetails of the mathematical model are provided in Appendix A.With reference to other b values, the speed of a no-�ow boundary moving out-

wards from a wellbore axis may be further calculated by the streamline model.Further continuous pressure measurements at various dimensionless radii, rD,when available, may con�rm the exact model value and justify the modellingapproach. An additional e¤ort is required in order to model no-�ow boundarymoving inwards, i.e., from the drainage radius towards the wellbore axis. Thismodelling approach has great potential and provides a basis for further investi-

7. CASE STUDIES 303

Figure 7.62: The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radius rD = 10.


304 7. CASE STUDIES


Figure 7.65: The dimensionless pressure, pD,versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radius rD = 100.

7. CASE STUDIES 305

Figure 7.66: The dimensionless pressure, pD; versus the dimensionless time, � ,for an inverse decline exponent n = 10 and a dimensionless radius rD = 200.


306 7. CASE STUDIES


Figure 7.69: The dimensionless pressure, pD, versus dimensionless time, � , foran inverse decline exponent n = 100 and a dimensionless radius rD = 10.

7. CASE STUDIES 307



308 7. CASE STUDIES


Figure 7.73: The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponernt n = 100 and a dimensionless radius rD = 200.

7. CASE STUDIES 309

Figure 7.74: The dimensionless pressure, pD, versus the dimensionless time, � ,for an inverse decline exponent n = 100 and a dimensionless radius rD=500.

gations that can also include analyses and possible new methods for creating amore realistic physical model.

310 7. CASE STUDIES

Figure 7.75: The dimensionless pressure, PD, versus the dimensionless time, �for an inverse decline exponent n = 1000 and a dimensionless radius rD=10.The singularity case.

Chapter 8

DISCUSSION, CONCLUSIONS,RECOMMENDATIONS

The purpose of this thesis has been to take into account the production ratehistory of wells, which is usually of transient or depletion type, and to improvethe well decline analysis with new modelling and interpretation solutions. Whenthe recorded rate response is a¤ected by the no-�ow outer boundary or when thereservoir is closed, rates are in the depletion decline. If, on the other hand, thereservoir is in�nite, well rates are in the transient decline. This thesis providesfractured-horizontal-well solutions to a transient rate-decline and analyticallyderived pressure-time expressions to vertical wellbore conditions of a variable-rate decline of Arps type.

8.1 DISCUSSION

8.1.1 Transient Rate Decline

In the transient-decline scope of this work, emphasis has been placed on well rate-time responses. A well with fractures is coupled to an oil reservoir. Solutionsare full-time solutions and inner boundary conditions are of constant and vari-able rate and/or pressure. For long times it is possible to create well-equivalentradii, and fracture equivalent half-length can be obtained for a horizontal wellwith fractures. The bringing-together of rate-time and pressure-time analysesprovides a total package to better characterise the behaviour of a horizontal wellwith created fractures. The overall modelling work has been intended to con-tribute to the transient rate decline by including various features of a fracturedhorizontal well.By introducing both pressure and rate full time responses, it was possible

to better characterise wells with fractures as presented in numerous case studiesfrom North Sea wells. Several case studies with fractured horizontal wells have

311

312 8. DISCUSSION, CONCLUSIONS, RECOMMENDATIONS

illustrated and validated the procedure and applicability of a fractured-horizontalwell model in an oil reservoir. In the history match scenario, due to a rangeof input rate data being available, the model calculates well responses that aresimilar to observed well data. Even with a limited amount of data being available,the model predicts the cumulative rate target quite satisfactorily. The model isalso validated with data from a water-injection fractured-horizontal well. In thecase of a fractured-horizontal well producing only gas, the model is only partlyvalidated. The reason for this is that the model was primarily developed forthe only a liquid �ow. Since the obtained matching results were satisfactory,model features should be bene�cial in evaluating the fracture behaviour duringthe well�s production life.

This modelling work illustrates the use of a screening tool in optimising ahorizontal well with fractures located in an oil reservoir. This was achieved byimplementing new analytical solutions into the SLAB and BOX models, with theaim of providing a useful diagnostic and prognostic tool for the oil industry. Inaddition, although semi-analytical modelling requires some trial and error to �ne-tune the matching procedure, it o¤ers a unique �exibility in the diagnosis of wellswith fractures. With the special numerical features the implemented model couldbe considered e¢ cient in both accuracy and speed. The program was �exible andhad many options to accommodate the various wellbore operational propertiesand fracture properties. It presents improvements over several published semi-analytical tools that have been completed and tested as such.

The semi-analytical and mono-phase model o¤ers the following advantages:both the pressure and rate model IBC are constant and variable; by keepingthe wellbore production rate constant for a certain production time, the modelcalculates constant pressure wellbore conditions at the end of this time, and con-tinue to produce under these conditions until the end; implemented IBC featuresare ensured on the well plateau production data; the model includes fracturefeatures, such as fracture size (half-length, height) and fracture �ow character(uniform �ux, in�nite and �nite conductivity fractures); each fracture rate andcumulative rate is calculated in both transient and depletion mode; the fracturecharacter is represented by a uniform �ux, �nite and in�nite conductivity.

Although a mono-phase model is more simple that any commercial reservoirsimulation model, it has been used to attain complementary and useful results.These include individual-fracture cumulative-production and rates as well aswell-productivity-indexes.

Moreover, the use of an analytical model implies that there is no need forstatic-geology model data. The reservoir properties are restricted to only onezone, and thus input values such as zone height, porosity and permeability aremean values or homogenised reservoir model values. As the fracture character ismore or less unknown screening analysis is important for a better understand-ing of a complex reservoir-fracture-well system. The screening features render

8. DISCUSSION, CONCLUSIONS, RECOMMENDATIONS 313

this model superior to many others, especially commercial simulation models.It should also be noted that since the aim of the screening analysis has beento match target characteristics and provide a quick diagnosis to the fracturebehaviour, the program was self-validated.

A series of solutions corresponding to various conditions is given in a generalmodel that also includes various wellbore conditions. We have demonstratedthat it is possible to match well data with the options implemented in a module.The implemented method was very fast and provided an excellent history matchon the basis of a well. The individual fracture rate and cumulative rate responsesrepresents additional valuable information that is available. Fine-tuning of thevariable rate and/or variable pressure would have improved the match of the welldata. Additionally, we showed that the method was ideally suited for solvinga multiple fractured horizontal well in an oil reservoir when available data islimited and analysis cannot conducted with conventional reservoir simulationtools. The most used inversion procedure known as "Laplace", de�ned by theStehfest algorithm, was found to not always be stable. New inversion techniques(as listed in Appendix B) should be thus considered. The selected "Laplace"inversion techniques should be tested and implemented in the future modellingwork. In all case studies the simulations were very fast. The model has severaladvantageous features related to pressure-rate with time modelling. It includeswellbore friction, chocking e¤ects and the BOX model solutions that need to befurther improved. The gas model should include the non-Darcy �ow terms andshould be tested with more well data. As a horizontal well penetrates only onezone and accordingly becomes fractured, it should be extended to several zoneswhere each may have a selected number of fractures. The conductivity, angle andposition of each fracture additionally could be included. Nevertheless, from theresults of this study, a number of issues have been identi�ed. First, the programrequires trials and errors to �ne-tune a well with fracture responses. Second,although variable IBC are implemented, only a limited number of time steps areavailable in the resent releases. Any increase in the number of time-steps willimprove the matching data and only slightly reduce the processing time. Thetime-step function should be extended to more wellbore options, and the step-function could be improved in order for the constant rate to change in to constantor variable pressure. The implemented IBC changes from a constant rate toconstant pressure only. The model should also include more zones and each zonewill be penetrated by a certain number of producing fractures. Fracture mayalso penetrate several zones. Since the gas model is basic it should comprisenon-Darcy �ow and should be fully developed. Moreover, Stehfest algorithmshould be extended to other robust inversion techniques (several of which listedin Appendix B).

Finally, it could be concluded that the semi-analytical model has been ableto satisfactorily simulate a reservoir-fracture-well system for an oil-producing


and a water-injection well. With the current implemented and tested features,advantages of the physical, numerical and programming model were able toovercame its limited disadvantages. This was especially true when the individualfracture features of the production such as rate and cumulative rate may be usedin analysing the well production decline with time. In addition, due to the modelversatility, the existing model features could be improved with more heterogenousfeatures (as layering and a fractured reservoir).

8.1.2 Depletion Rate Decline

The depletion decline presented in Fetkovich type curves is analytically solvedwith the speci�ed moving boundary. The model can be extended to also accountfor the variation of the speed of the moving no-�ow boundary. This approachhas potential in improving knowledge of well behaviour, assuming that pressureresponses can also be continuously measured. A speed should be related to adriving force that causes a rate-time Arps stem.Raghavan (1993) discussed conditions under which the decline curvature can

be derived empirically. The inclusion of variable skin was suggested as a means ofanalytically deriving the rate-time curvature that de�nes the decline exponent,b.Raghavan also mentioned that, as long as the drainage radius and skin remainsunchanged, it is not possible to analytically derive solutions for various rate-timestems de�ned by the b exponent. The way to provide the analytical solutionsfor rate-time curves de�ned by the decline exponent b is to introduce a varyingdrainage volume that produces the Arps rate at the wellbore. In order to solvethe di¤usion equation, we specify the moving boundary as being no-�ow andmoving outwards from a wellbore axis with the certain speed. Thus, solvingthe di¤usion equation with the speci�ed moving boundary and at the same timeimposing wellbore conditions of variable rate of Arps type is a novel approachin determining pressure-time solutions. The model input values correspond torate-time data de�ned at the wellbore and speci�ed moving boundary, whereasthe model output values are pressure-time data calculated at any point withinthe vertical well drainage area. As soon as these data can be continuously mea-sured and thus become available this physical model can be tested and validatedaccordingly.A model extension is possible for a speci�ed moving boundary with an in-

wards motion i.e., moving towards the wellbore axis from a vertical well drainageradius. Further, in order to reduce existing model limitations, new methodsshould be considered.This work should be regarded as complementary to the decline curve analy-

sis. For selected stems, or Arps rates, each de�ned with a chosen declineexponent,b;the model generates pressure-time responses within a producing ver-tical well drainage area. These pressure-time pro�les represent the solution tothe di¤usion equation with the speci�ed moving boundary. Since it is a basic


model, it is limited to the choice of input parameters related to the no-�ow mov-ing boundary, the no-�ow boundary speed, � and the coe¢ cient n (n = 1

b). For

the right choice of � and n; it is possible to generate the positive pressure di¤er-ence, �P: For certain values of � and n; singularity is avoided, we are thus ableto calculate the �P expressions at any point within the vertical well drainageradius. The fact that the model is de�ned with only particularly chosen para-meters gives it limitations. The obtained pressures �P could be normalised.Further, the physical means of the speed of the no-�ow moving boundary, �;could be further investigated. Other model solutions could also be considered.The depletion rate-time analysis introduces new analytical pressure-time so-

lutions to the wellbore variable-rate conditions of Arps type. As depletion Arpsrate-time curves are empirical, the analytically obtained solved pressure-timesolutions are innovative. In order to solve the di¤usion equation, the speci�edmoving boundary conditions are introduced. These speci�ed moving boundarycondition are no-�ow with an outward motion from the wellbore axis with a pre-de�ned speed. As the drainage volume changes with no-�ow moving boundarythe model is transient. Nevertheless, at the same time the pressure is solved withthe wellbore conditions of Arps rate decline and considered as depletion. De-spite that the physical model is limited to the speci�ed boundary conditions andmodel parameters, this work has great potential and provides a basis for furtherinvestigations that may also include analyses and the possible understanding ofthe driving forces that de�ne the curvature of the rate-time production pro�les.

8.2 CONCLUSIONS

The overall study considers the transient rate decline and the depletion ratedecline. The transient study brings new solutions and interpretation techniquesthat can help in the analysis of a fractured-horizontal well. The depletion studyprovides pressure-time solutions for a vertical well producing under variable-rateconditions (of selected Arps stems).A general horizontal-fractured well model comprises a series of solutions cor-

responding to various wellbores with fractures conditions. It could be demon-strated that it was possible to match well data with the options implemented ina module. The implemented method was very fast and provided a good historymatch on the basis of a well. The individual fracture rate, the cumulative rateand the productivity index responses represented additional valuable informationthat was available. A �ne-tuning of the variable rate and/or variable pressurewould have improved the �tting of the well data. Additionally, the method wasshown to be ideally suited for solving multiple fractured horizontal wells in an oilreservoir when available data was limited and analysis could not be carried outwith conventional reservoir simulation tools. This thesis illustrates the use ofa screening tool for an improved modelling of oil production from horizontal or


near-horizontal wells with induced fractures. This was achieved by implement-ing new analytical solutions into the SLAB and BOX models, thus providing auseful diagnostic and prognostic tool for the oil industry.The following is a summary of a fractured-horizontal well that produces from

a SLAB/BOX model:1. A fast and robust algorithm is developed for calculating the transient

(SLAB model) and the basic depletion (BOX model) responses of a multiplefractured horizontal well in an anistropic homogeneous reservoir.2. The bringing-together of rate-time and pressure-time analyses in order

to obtain a total package for an improved characterisation of the behaviour ofhorizontal oil wells with induced fractures.3. The semi-analytical tool developed aided in optimising the well production

of a horizontal oil well with induced fractures.The following is the summary of a vertical well that produces under wellbore

variable-rate conditions of Arps type:1. The model provided new solutions to a di¤usion problem when wellbore

conditions were of variable rate decline and a speci�ed moving boundary (no-�owouter boundary that moves outwards from the wellbore axis).2. The model generated pressure pro�les and comprised the transient infor-

mation of a changing drainage volume, and simultaneously included the depletionrate decline as derived by Arps.3. The physical meaning of the speed of a no-�ow boundary moving outwards

from a wellbore axis should be further studied by means of a driving force thatcreates a rate decline shape.This work has also further improved the concept of depletion rate-time decline

curve analysis. The model solutions are unique and open for a new approach insolving di¤usion equations with variable-rate inner boundary conditions and aspeci�ed-moving outer boundary.

8.3 RECOMMENDATIONS

Both models have potential of being further developed, and a horizontal-fracturedwell model can possibly be coupled to other software.For a horizontal well with fractures the model recommendations are:1. To be extended to handle any fracture orientation, determined by the

stress orientation of the reservoir, a fracture wall damage or fracture skin andwellbore skin.2. To include heterogeneity as layering and naturally fractured reservoir

complexity.3. To be able to handle gas �ow (including the non-Darcy �ow)4. To employ continuous monitored pressure and rate data of a fractured-

horizontal well with the purpose of carrying out fracture closing diagnosis.


5. To integrate the semi-analytical model into a numerical simulator for animproved �t of calculated and observed data (commercial software).6. To couple a fractured-horizontal well model to the rate-testing and pressure-

time semi-analytical software (commercial software)7. To couple screening features of a fractured-horizontal well model to state-

of-technology risking software (commercial software)8. To improve fracture diagnosis by coupling microseismic modelling devices

to the fractured-horizontal well screening features.9. To combine a fractured reservoir and a horizontal-well with a prognosis of

the fracture production (commercial software).10. To verify the numerical model solutions with the semi-analytical solutions

(commercial software).For a model with the de�ned variable rate wellbore conditions (of Arps rate

decline) and a speci�ed moving boundary, recommendations are:1. A continuous monitoring of pressure and rate data for a vertical producing-

well as well as calibration and validation of the existing model.2. A possible reservoir production diagnosis based on pressure data calculated

by using values observed at any point within a drainage volume.3. The physical meaning of the speed of a moving boundary and its relation

to the drive mechanism.4. The derivation of new modelling solutions with the speci�ed moving

boundary.

Chapter 9

NOMENCLATURE

A = constantA = drainage area of a well, ft2 (m2 )AD = dimensionless drainage area, . AD = A=L2cb = Arps decline exponentbf = fracture width, ft (m)B = constantB = liquid formation volume factor, rb/sb (rcm/scm)B0 = oil formation volume factor, rb/sb (rcm/scm)Boi = initial oil formation volume factor, rb/sb (rcm/scm)Bw = water formation volume factor, rb/sb (rcm/scm)Bg = gas formation volume factor, rb/scf (rcm/scm)cf = formation pore compressibility, 1/psia (1/Pa)CfD = dimensionless fracture conductivitycg = gas compressibility, 1/psia (l/Pa)c0 = oil compressibility, 1/psia (l/Pa)ct = total compressibility, 1/psia (l/Pa)ct = average total compressibility, 1/psia (1/Pa)cw = water compressibility, 1/psia (1/Pa)Di = initial decline rate, 1/unit of timeQg = cumulative gas produced, MMscf (scm)h = reservoir net pay thickness, ft (m)I0 = modi�ed Bessel function of �rst kind of order zeroI1 = modi�ed Bessel function of �rst kind of order oneKo = modi�ed Bessel function of second kind of order zeroK1 = modi�ed Bessel function of second kind of order onek = average e¤ective permeability, mD (m2)ka = absolute permeability. mD (m2)kf = fracture permeability, mD (m2)kg = e¤ective permeability to gas, mD (m2)ko = e¤ective permeability to oil, mD (m2)

319

320 9. NOMENCLATURE

krg = relative permeability to gaskro = relative permeability to oilkw = e¤ective permeability to water, mD (m2)LC = system characteristic length, ft (m)LD = dimensionless e¤ective wellbore length, LD = Lh=2hLh = e¤ective horizontal well length in pay zone, ft (m)Qo = cumulative oil produced, sb (scm)Qoi. = original oil in place. sbbl (scm)po = base pressure, lower limit of integration, psia (Pa)p = pressure. psia (Pa)pi = initial reservoir pore pressure, psia (Pa)

m(p) = real gas pseudopressure, psia2/cp (Pa/s) m(p)=2pRp0

p0dp��gz

pwD = "Laplace" space dimensionless wellbore pressure solutionpwD =dimensionless wellbore pressurepwf = sandface �owing pressure, psia (Pa)qo = oil �ow rate. sb or Mscf/unit of time (scm/s)qi = initial �ow rate. sb or Mscf/unit of time (scm/s)qav = average well �ow rate, STB or Mscf/unit of time (semis)qDd = dimensionless decline �ow rate functionqDdi = dimensionless decline �ow rate integral functionqDdid = dimensionless decline �ow rate integral derivative functionqg = gas �ow rate. sb or Mscf/unit of time (scm/s)qt = total well �ow rate (all �uids) sb or Mscf/unit of time (scm/s)qw = water �ow rate. sb or Mscf/unit of time (scm/s)qdD = dimensionless wellbore �ow rateqdD = Laplace space dimensionless wellbore �ow rateQDd = dimensionless cumulative productionQpDd = dimensionless decline cumulative production functionrD = dimensionless radius, rD = r=Lc = r=rwre = e¤ective reservoir drainage radius, ft (m)reD = dimensionless e¤ective drainage radius .rw =.wellbore radius, ft (m)rwa =apparent or e¤ective wellbore radius, ft (m)rwD = dimensionless wellbore radius, rwD = rw=hs = "Laplace" space parameterSg = reservoir gas saturation, fraction of pore volumeSo = reservoir oil saturation, fraction of pore volumeSor = residual oil saturation, % of pore volumeSw = water saturation, % of pore volumeSwi = initial water saturation, fraction or % of pore volumeSwir = irreducible water saturation, % of pore volume

9. NOMENCLATURE 321

t = time, hours, days, months, years (s)t0 = time value parameter of integrationtD = dimensionless timetpss = time to reach pesudo-steady-state or boundary-dominated �ow,

units of time (s)�p =pressure di¤erential, psia (Pa) �p = pi � pwf� = reservoir average e¤ective porosity, fraction of bulk volume�t = average total system mobility function, 1=cp (1=Pa s)� =reservoir �uid viscosity, cp (Pa s)�o =oil viscosity, cp (Pa s)�g =gas viscosity, cp (Pa s)�w = water viscosity, cp (Pa s)

9.1 Functions

cos = cosine functioncosh = hyperbolic cosinee = exponential functionexp = exponential functionIn = natural logarithmsin = sine functionsinh = hyperbolic sine

9.2 SI Metric Conversion Factors

mD x 9.869 233 E - 04 = m2D x 9.869 233 E - 07 = m2psi x 6.894 757 E + 00 = kPapsi x 6.894 759 E - 02 = barin x 2.54* E - 02 = min2 x 6.4516* E - 04 = m2ft x 3.048* E - 01 = mft2 x 9.290 304* E - 02 = m2ft3 x 2.831 685* E - 02 = m3bbl x 1.589 873 E - 01 = m3cp x 1.0* E - 03 = Pa s*Conversion factors are exact

322 9. NOMENCLATURE

Chapter 10

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draulically Fractured Wells. Low Permeability Reservoirs Symposium. Denver,Colorado.Wimp, J .1980. Computation with recurrence relations. Boston, Pitman.Wimp, J. (1984). Computation with recurrence relations, Applicable Math-

ematics Series, Pitman (Advanced Publishing Program), Boston, MA.

Appendix A

No-Flow Moving BoundaryModel Solutions

In Chapter 7 we presented dimensionless pressure pD and constant A. In thisAppendix we provide solutions to the di¤usion equations with variable rate ofArp�s type. These solutions are of variable rate IBC close to an exponentialArp�s decline. To solve di¤usion equation we de�ne variable rate almost equal toexponential decline b s 0: Now we introduce n as an inverse of decline exponent,b so, we get

(tD) = t(�n)D (A.1)

We should consider multiplying right hand side with the constant C in order tonormalize the parameters n and {, so rate becomes now

(tD) = Ct(�n)D (A.2)

Further, the "Laplace" inverse transform is

L�1( ) =kn�1

�(n)(A.3)

and dimensionless pressure, pD is now

pD =t(�n)D

�(n)

1Z0

un�1e�u�AJ0(

rptD

pu)� �

2Y0(

rptD

pu)

�du (A.4)

from which by requiring that no-�ow boundary moves with distance r from awellbore as

@pD@rD

= 0; for rD = uptD (A.5)

and de�ning constant A as

345

346 A. No-Flow Moving Boundary Model Solutions

A =�

2

1Z0

un�12 e�uY1({

pu)du (A.6)

Further, referring to Obertrettinger and Badii (1973), page 155, we get the fol-lowing expression

L(tn�1D

�J0(a

ptD) + iY0(a

ptD)�) = (A.7)

2

�a[�(n)]2 p(

12�u)e(�

a2

8p)W 1

2�n;0

�a2

4pe�i�

�(A.8)

where W 12�n;0

�a2

4pe�i�

�is the Wittaker function, and L is the "Laplace" trans-

form

L(f(t)) =

1Z0

e�ptf(t)dt (A.9)

For the inverse decline exponent, n , referring to Buchholz.(1969), page.99,we have the following asymptotic formulas when n ! 1

W 12�n;0

�a2

4e�i�

�s 2�

12

�a2

4(n� 12)

� 14

(e

n� 12

)n�12 ei(a

pn� 1

2��4) (A.10)

In addition we need the Stirling�s formula:

�(n) =p2�(

n

e)nn�

12 (A.11)

Now, asymptotically when, n!1 we get

pD sr2

�n�

14 ��nD (

r2D�D)�

14 e�

r2D

8�D

�A cos({ � r

p�D)� �

2sin({ � r

p�D)

� r

p�D

rn� 1

2� �

4

!(A.12)

Where constant, A, is now de�ned asymptotically (for n!1) and considering(a = {) as:

A s ��2cot�{pn� �

4

�(A.13)

Appendix B

"LAPLACE" InversionTransforms

There are many types of partial di¤erential equations describing �uid �ow phe-nomena in porous medium whose solution may be found in terms of a "Laplace"transform. Layered reservoirs equations are too complicated for inversion us-ing the techniques of complex analysis. Numerous methods have been devisedfor the numerical evaluation of the "Laplace" inversion integral as presented byPiessens (1975).The main di¢ culty in applying "Laplace" transform technique is the deter-

mination of the original function f(t) from its transform:

F (s) =

1Z0

e�stf (t) dt (B.1)

Inverse "Laplace" transform methods are given in Table B-1.Three basic types of "Laplace" transform functions F(s), which should be

evaluated in the comparison analysis and they are not particularly the "Laplace"transform functions solution of the one phase �ow model:

-Functions which are continuous and for which F (s)! s� as s!1;-Functions which are continuous and for which there is not value � for

which F (s)! s� as s!1; and-Functions which have discontinuities.

A case of a F (s) about which little is known and for which an all-purposemethod is initially most convenient to apply is di¢ cult to evaluate. An F (s) forwhich the form of the solution f(t) is known are presented in Table (B-2).Numerical methods used in the numerical "Laplace" transform comparison

are classi�ed into methods which compute a sample, methods which expandfunction f(t) in exponential functions, methods based on Gaussian quadrature,methods based on a bylinear transformation and methods based on Fourier series.Extensive results comparison are presented by Davies and Martin (1979).

347

348 B. "LAPLACE" Inversion Transforms

Table B-1: The inverse "Laplace" transform methods [after Davies and Martin(1979)]

B. "LAPLACE" Inversion Transforms 349

A number of di¤erent numerical inversion techniques presented in Table (B-2)were evaluated by Davies and Martin (1979), according to the following criteriawhich are not fully independent: applicability to a variety of common types of in-version problems, numerical accuracy, relative computation times, programmingand implementation di¢ culties.Examples include problems with numerical data at arbitrary points, problems

with transforms in the form of rational fractions, problems with noisy data, orproblems for which the solution is known to be of particular form should betreated by a special available method. For many problems an all purpose methodfor which numerically inverting the "Laplace" transform may be inappropriate.It is usually worth using more than one method as recommended by Bellman

et al. (1966) on any unknown function as a check against peculiar behaviour ofthe function or of the numerical method, and against programming and imple-mentation errors. The di¤erent methods produce di¤erent results in the trou-blesome case. By trying the methods on a function whose analytical "Laplace"transform is known and is similar in form to the troublesome function, we canevaluate the bene�t of the used method. In the Table (B-2), there are testfunctions which are covering a wide range of functional forms which are coveringwell known and understandable functional behaviours, and because of the simpleanalytical solutions they have.Numerical accuracy can be presented for each function and each method by

two measures. By the used measures, the presentation of the comparison dataresults for each t value would not take up a large amount of space. Measure Lgives the root mean-square deviation between the analytical f(t) and numericalfa(t) solutions for various t values. L gives a fair indication of the success of amethod for large t and is given by the following expression:

Le =

30Xi=1

(f (i=2)� fa (i=2))2 =30

!1=2(B.2)

Measure Le is a similar measure weighted by the factor e�t and is used for rela-tively small t. Le measure is given as:

Le =

X(f (i=2)� fa (i=2))

2 e�ij2=

30Xi=1

e�i=2

!!1=2(B.3)

By the expressed measures we can evaluate the accuracy for each methodand for each function. It is also important to evaluate computation time require-ment of di¤erent functions and di¤erent numerical algorighms implemented. Aprogramming e¤ort should also be evaluated. Errors in programming can bedetected by comparing results on test problems with results in the papers orig-inally describing the method. The determination of the function F (s) which is

350 B. "LAPLACE" Inversion Transforms

Table B-2: The comparrison of the inverse "Laplace" transform methods

dedicated to the particular �uid �ow phenomena is easy for real values of s. Forcomplex values may be a source of error.A large number of numerical methods for inverting the "Laplace" transform

are variants of other methods. If the original method is successful, it is reasonableto test variants. Methods that show great promise are Dubner and Abate (1968),Silverberg (1970) and Crump (1976). On the opposite rather poor methods arethose of Bellman et al. (1966) and Piessens (1969).

Appendix C

Regression Techniques

The approaches that have been presented in the literature in decline curveanalysis are statistical method, least-square methods, log-log type-curve over-lays methods and the computer-automated curve �tting methods. These meth-ods are based on the general hyperbolic (exponential or harmonic is a specialcase) decline curve equation developed by Arps (1945). The hyperbolic decline-curve has three unknowns initial decline rate Di, initial production rate qi anddecline exponent b which have to be determined from the unknown measuredrate-time production data.

The least square method assumes the type of decline to be as exponential,hyperbolic or harmonic before curve �tting can be performed. The type curveanalysis also has disadvantages due to the non-uniqueness problem in type-curvematching. The computer-automated curve �tting is performed for decline curveanalysis by the linear multiple regression of the selected variables. The approachis based on the Arps (1945) general form equation and it is applied to the rate-time data.

C.1 Linear Regression

The various graphical and type curves approaches are less accurate and lessreliable than the statistical approach. Towler and Bansal (1993) proposed lin-ear regression method to determine the decline parameters Di, qi and b frommeasured rate-time data. A maximum in the regression coe¢ cient was used toindicate the straight-line �t. The two iterative methods to �nd the three un-known parameters in hyperbolic decline-curves are presented by Cvetkovic andGudmundsson (1993) in Table 6-1 (page 104). The regression coe¢ cient plottedversus chosen parameters shows when optimum linearity is achieved. The squareof regression coe¢ cient is de�ned as:

351

352 C. Regression Techniques

r2 = 1�

nPi=l

(yi �A�Bxi)2

nPi=1

(yi �A�Bxi)2 +BnPi=1

(xi � x) (yi � y)

where (xi; yi) is for the �rst method and second method de�ned as:

(xi; yi) = (log(1 + bDit); logqt)

(xi; yi) = (q1�bt ; Np)

Field examples were presented and discussed regarding decline exponent big-ger than 1 related to the fractured reservoir and the negative decline exponentb is believed to be caused by mechanical problems in the well. The analysis isbased on hyperbolic decline and by making comparison of both methods, we canguess harmonic b equal one decline.

C.2 Linear Multiple Regression

A linear multiple regression can be performed for the rate-time variables q, Np,and qt for optimum variables of qi, Di and b in empirical Arps (1945) equations.The time-cumulative relationship is derived from the hyperbolic solution of q(t)and can be expressed as:

Np =

tZ0

q (t) dt =[q (t) (1 + bDit)� qi]

[(b� 1)Di]

It can be rearranged to give general form, q(t) for the exponential (b = 0),hyperbolic (0 < b < 1) and harmonic solutions (b = 1) of Arps�equation in theform expressed as:

q (t) = qi + (b� 1)DiNp � bDiq (t) t (qi) + ANp +Bq (t) t

Duong (1989) suggested a procedure for the calculation of the unknown coe¢ -cients qi,A and B by using the linear multiple regression for variables q(t), Npand q(t)t. Decline rate Di = �(A + B) and decline exponent b = �B=Di.The cumulative production data can be calculated from rate-time data by theequations:

Np = 0:5nXj=1

�q (t)j � q (t)j�1

� �tj � tj�1

�The production rate can be calculated by the expression:

C. Regression Techniques 353

q (t) =[qi + ANp]

[1�Bt]

and the future production can be expressed as:

q (t) =[qi + A (Np) j � 1]

[1� A (t� tj�1)�Bt]

The future production rate can be calculated from the initial production rate, qi,previous cumulative production, (Np)j�1, and constants A and B and expressedas:

q (t) = qi + A (Np)j�1 + Aq (t) [t� tj�1] +Bqjt =

hqi + A (Np)j�1

i[1� A (t� tj�1)�Bt]

Chen (1991) introduced the relationship between cumulative production, Np rate,q and product of rate and time, qt in the form:

Np = A+Bq + Cqt

Where, A;B;C are de�ned as:

A =qiDi

1

1� b

B =1

Di

1

b� 1

C =1

b� 1and decline exponent b, initial decline Di and initial rate qi are related to thecoe¢ cients A, B and C through the following expressions:

b =C

C � 1

D1 =1

B(C � 1)

qi = �A

B

Again, Np, q and qt are not fully independent.

354 C. Regression Techniques

These approaches are related only to data that have been a¤ected by theboundary conditions. The linear multiple regression method is based on theassumption of full independency of the variables rate q, cumulative productionNp, and time t. The variable de�ned as a rate-time, qt can cause some di¢ cultiesfor certain sets of production data. Coe¢ cients in multiple linear regression arealso sensitive to the cumulative production term Np. If the values Np are notmeasured, it is suggested to integrate rate versus time with the trapezoid method.This term can change the coe¢ cient values so it is important to �nd the mostappropriate integration method for the given values of rate versus time.The Chen (1991) and Duong (1989) expressions were tested for various pro-

duction decline data. The interactive system for symbolic computation, MAPLE,was used for the linear multiple regression. The program can easily be extendedto the linear multiple regression calculation.

C.3 Weighted Residuals Regression

The extrapolation of the decline curves is a method of predicting the perfor-mance of the production wells. Ramsey and Guerrero (1969) used the leastsquare methods and could not reduce the e¤ect of large terms in the calculation.Least squares �tting expressed by Ramsey and Guerrero (1969) results in theshortcomings which can be eliminated by the use of weighted residuals. The oilproduction q(t) can be de�ned as:

q (t) = qi

�1 +

bt

a

�� 12

where decline exponent, b controls the degree of curvature of the decline curvesand has no units, qo is the initial oil production at time zero and parameter a isthe reciprocal initial of the initial decline rate Di with the same units as time t.

Appendix D

Relevant Reports and Papers

D.1 SPE Papers

Cvetkovic, B. (2009). E¤ective Wellbore Parameters and a Multi-fractured Hor-izontal Well Productivity. Paper SPE 120826. SPE Production and OperationsSymposium, Oklahoma City, Oklahoma, USA, 4�8 April.Cvetkovic, B. (2009). Semi-Analytical Modelling of a Horizontal Well with

Fractures in an Oil Reservoir - Screening Approach with Risk Analysis. SPEApplied Technology Workshop. Discussion Leader and Section Manager of theReservoir modelling Section, March 8-11, Penang, Malaysia.Cvetkovic,B., Halvorsen, H., Sagen, J. and E.N. Rigatos (2001). Modelling

the Productivity of a Multifractured-Horizontal Well. Paper SPE 71076, SPERocky Mountain Petroleum Technology Conference, Keystone, Colorado, May21�23.Cvetkovic,B., Halvorsen, G., and J.Sagen.(2000). Multiple-Fractured Hori-

zontal Well Case Study. SPE-65503 presented on the "4th International Con-ference and Exhibition on Horizontal Well Technology, November 4-8, Calgary,Canada, (http://www.petsoc.org/hw2000.html).

D.2 Presentations

D.2.1 Conferences/Forums

Cvetkovic, B., and G. Halvorsen, G. (2001). Multi-fractured horizontal wellstudy with HOWIF from a Vallhal Field. A Selected Software Day Seminar inBP-AMOCO, July 26, 2001, Stavanger.Cvetkovic, B., Halvorsen, G., Sagen,J., and R. Banerjee. (2001). E¤ective

Wellbore Radius For a Multi-fractured Horizontal Well Simulation. GeoQuestSchlumberger, FORUM 2001, London, UK, March 6-9.

355

356 D. Relevant Reports and Papers

Cvetkovic, B., Halvorsen, G. and J. Sagen. (1999). A Horizontal Well withInduced Fractures - Real Case Study. Presentation, The Heriot-Watt & StanfordUniversity �Reservoir Description and Modelling Forum, Crie¤ (Skotland).Cvetkovic�, B., Halvorsen, G., and E. Løw (1955). Modelling of a Horizontal

and a Vertical-Fractured Well�, Mathematical Modelling of Fluid Flow ThroughPorous Media Conference, Poster Session, St. Etienne, France, May 22-26

D.2.2 Schlumberger Internal EUREKA Presentations:

Cvetkovic, B. (2007). A Vertical/Horizontal Well with Laterals Simulation.Reservoir Symposium 2007 Schlumberger (EUREKA), Moscow, June.Cvetkovic, B. (2007). Unconventional Stimulation Studies. Reservoir Sym-

posium Schlumberger (EUREKA), Kula Lumpur 2007Cvetkovic, B. (2006). Fractured Carbonate-dolomite Gas-condensate Reser-

voir Simulation Studies. Reservoir Symposium 2006, Schlumberger (EUREKA),Bucharest, September.Cvetkovic, B (2006) Unconventional Stimulation Challenges, Schlumberger

Internal Seminar on Unconventional Reservoir Studies, Engineering and Man-ufacturing Russia and Schlumberger Moscow Research, SMR Novosibirsk, 6 thNovember.

D.3 Industry Reports

Cvetkovic, B., Halvorsen, G. and J.Sagen (1999). HOWIF-OIL BOX Model.(Final and Half-Year Report for PHILLIPS- Con�dential).Cvetkovic, B., Halvorsen, G. and J. Sagen (1998) HOWIF-OIL SLAB Ad-

vanced Model. (Final and Half-Year Report for BP, CONOCO, PHILLIPS -Con�dential);Cvetkovic, B., Halvorsen, G.and J. Sagen (1997). HOWIF-OIL SLAB Im-

proved Model. (Final and Half-Year Report for BP, CONOCO, PHILLIPS -Con�dential);Cvetkovic, B., Halvorsen, G.and J. Sagen. (1996). HOWIF-OIL SLAB

Model (Final Report and Half-Year Report for BP, CONOCO, PHILLIPS -Con�dential);Cvetkovic, B. and G. Halvorsen (1977) Horizontal Well Induced Fractures

Production Optimization. Pre-study work for Norwegian Research Council, NFR(December 1995-February 1996) IFE-Kjeller, March.

D.4 NTNU Faculty Reports

Cvetkovic, B. and J.S. Gudmundsson (1993). Analytical Models for Rate Declinein Oil and Gas Reservoirs. University of Trondheim, December.

D. Relevant Reports and Papers 357

Cvetkovic, B. (1992). Modelling and Solution Methods for Layered Reser-voirs. University of Trondheim, June.Cvetkovic, B. (1992). Fundamental Equations and Techniques Used in Decline

Curve Analysis. University of Trondheim, January.

D.5 Other Related Presentations and Reports

Cvetkovic, B. (2008). Risk Analysis of a Tri-Lateral Well Producing the GasCondensate Trym Reservoir with MEPO-ECLIPSE. Bayerngas internal presen-tation.Cvetkovic, B. (2003). A Horizontal Well Production Optimization�, simula-

tion study for Petrobaltic, 2003.Cvetkovic, B. (2001). A Multiple-Fractured Horizontal Well Case Study",

Invited presentation to Croatian SPE Section- January 30, INA-Naftaplin Oil &Gas Co., Zagreb, Croatia.Le Turdu,C., Cvetkovic,B., and S. Gacesa.(2001). Geological Modelling (Zu-

tica Field, Croatia) by PETREL & HOWIF. Poster presentation, InternationalPetroleum Engineering Conference, October 2001, Zadar, Croatia.Cvetkovic, B., Halvorsen, G., and J. Sagen, (1999). A Horizontal Well with

Multiple Fractures: Real Case Study. Poster presentation, Petroleum Associa-tion Meeting, Stavanger.Cvetkovic, B. (1999). Horizontal well with Induced Fractures Production

Optimization in Oil Reservoir-HOWIF Programme. SPE, Croatian Section, Za-greb, May.Cvetkovic�B., Halvorsen, G., Sagen, J., and Y. Bhushan (1999). Wellbore Re-

sponses of a Horizontal Well with Created Fractures Coupled to an Oil Reservoir.Petroleum Engineering Summer School, Workshop 4, Dubrovnik, 1999.Cvetkovic, B., Halvorsen, G., Sagen, J., and Y. Bhushan (1998). Coupling

of a Reservoir to a Wellbore with Created Fractures. Presentation of Semi-analytical software tool and reservoir simulation results, Reservoir SimulationUsers Meeting, Stavanger.Cvetkovic , B et al.(1999).Optimal Massive Gas Injection Conditions for Oil

Recovery Enhancement by Di¤usion in Fractured and Heterogeneous Reservoirs,MAGIC OR. Final Report 1996-1999, IFE-Kjeller, June (EU Research ProjectJOULE Final Report)Bekken C., Løw E., Cvetkovic�,B. and G. Halvorsen.(1996). Thermal In-

duced Fracturing model ALFAFRAC Output Upgrade. IFE Internal Report,IFE/KR/F-95/230, IFE-Kjeller, March.Cvetkovic, B., SPE, Halvorsen, H. and J. Sagen (2001). History Matching a

Multiple-Fractured-Horizontal-Well Responses with Step Function of Rates andPressures. Canadian International Petroleum Conference, June 12-14, Calgary,Alberta, Canada (Abstract accepted not realized).

358 D. Relevant Reports and Papers

Cvetkovic, B. and G. Halvorsen.(1998) Horizontal Well with Created Frac-tures Production Optimization�, Lectures with the HOWIF Programme Demon-stration, SPE Annual Technical Conference and Exhibition, Horizontal WellSeminar, New Orleans, LA. (Abstract accepted not realized).

2008 branimir cvetkovic

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