PERCOLATION MODELS FOR TRANSPORT IN POROUS MEDIA

Theory and Applications of Transport in Porous Media

Series Editor: Jacob Bear, Technion- Israel Institute of Technology, Haifa, Israel

Volume9

The titles published in this series are listed at the end of this volume.

Percolation Models for Transport in Porous Media With Applications to Reservoir Engineering

by

V.I. Selyakov Laboratory of Heterogeneous Media, Department of Theoretical Problems, Russian Academy of Science, Moscow, Russia

and

V. V. Kadet Department of Oil & Gas Hydromechanics, State Gubldn Oil & Gas Academy, Moscow, Russia

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-4771-7 ISBN 978-94-015-8626-9 (eBook) DOI 10.1007/978-94-015-8626-9

Printed on acid-free paper

All Rights Reserved © 1996 Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 Softcover reprint of the hardcover 1st edition 1996

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents

Foreword ix

Abstract xi

Introduction 1

I Fluid- and Electric Conductivity in Porous Media. Theoretical Analysis 5

1 Percolation Model of Micro Heterogeneous Media 7 1.1 Percolation Theory. Basic Concepts . . . . . . . . . . . . . . . . . 7 1.2 Conductivity of a Network with the Random Distribution of Elements 12 1.3 Effect of Electric Current on Conductivity of Heterogeneous Media 18

2 Percolation Models of One-Phase Flow in Rocks of Different Types 23 2.1 Conductivity of Grained Media . . . . 23 2.2 "Permeability- Porosity" Correlation 26 2.3 Conductivity of Cavernous Media . . . 28 2.4 Conductivity of Fractured Media . . . 30 2.5 Conductivity As a Function of the Strained State . 34

3 Percolation Model of Fluid Flow in Heterogeneous Media 41 3.1 Flow at the Micro Level . . . . . . . . . . . . . . . . . . . . . 41 3.2 Effect of Pore Space Structure on Laws for Macroscopic Flow 47 3.3 Results of Numerical Calculations and Comparison with Experiment 53

4 Percolation Model of Steady State Multiphase Flow in Porous Media 57 4.1 Steady State Flow of Immiscible Newtonian Fluids . . . . . . 57 4.2 Effect of Plastic Properties of Fluids on Phase Permeabilities 64

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4.3 Phase Permeabilities of a Medium with Mixed Wettability . . . . . 69 4.4 Three-Phase Steady State Flow of Immiscible Newtonian Fluids . 79 4.5 Stability of Percolation Methods for Calculation of Phase Perme-

abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Percolation Model of Non-Steady State Two-Phase Flow in Porous Media 89 5.1 Immiscible Displacement of a Viscous Fluid by a Non-Viscous One 89 5.2 Effects of Viscosities and Interfacial Tension . . . . . . . . . . . 98

6 Determination of Pore Size Distribution in Grained and Cavernous Rocks 105 6.1 Percolation Model for the Mercury Injection Test . . . . . . . . 106 6.2 Percolation Model for the Electric Porometry Method . . . . . 112 6.3 Percolation Model for the Combined Mercury and Electric Poro-

metry Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4 Numerical Calculations and Core Data Processing with the Electric

Porometry Method . . . . . . . . . . . . . . . . . . . . . . . . 122

7 Methods for Determining Parameters of Fractured Rocks 129 7.1 Concentration and Average Length of Fractures Determined from

the Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.2 Determination of Fracture Length Distribution from Fracture Traces

on the Core . . . . . . . . . . . . . . . . . . . . . . . . 132 7.3 Determination of Fracture Parameters from the Core . . . . . . . . 136

II Effects of Physical Fields on Recovery of Mineral Resources 139

8 Conductivity of a Heterogeneous Medium under Impulse and Alternating Current 143 8.1 Threshold Values for Electric Treatment with Impulse Current 143 8.2 Permeability and Electric Conductivity under Impulse Current 147 8.3 Determination of Threshold Values for Electric Treatment . . . 151 8.4 Permeability and Electric Conductivity after Electric Treatment 152

9 Changing Conductivity and Pore Space Structure with Electric Current. Experiments 157 9.1 Conductivity of Sandy-Argillaceous Medium . . . . . . . . . . . . . 157 9.2 Pore Space Structure of a Sandy-Argillaceous Medium after Electric

Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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9.3 Research on Irreversible Change of Conductivities for Sandy-Argillaceous Media after Electric 'freatment . . . . . . . . . . . . . . . . . . . . 164

10 Changing Well Production with Electric Treatment 167 10.1 Calculation of Change in Well Production after Electric 'freatment 167 10.2 Reversible Change of Permeability. Determination of Optimal Regime

for Electric 'freatment . 171 10.3 Results of Field Studies . . . . . . . . . . . . . . . . . . . . . . . . 176

11 Gas Colmatation Effect during Electric Action on Saturated Porous Media 181 11.1 Temperature Effects in Capillaries Caused by Electric Current 182 11.2 Movement and Growth of Bubbles in Capillaries 184 11.3 Permeability Change under Electric Field . . . . . . . . . . . . 192

12 Effects of Acoustic Waves on Irreversible Change of Permeability of a Saturated Porous Medium 199 12.1 Dissipation of Energy in Viscous Poiseuille Flow . . . . . . . . . . 200 12.2 Destruction of Surface Layer in Pore Channels under Tangential

Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 12.3 Cavitation in Pore Channels under Acoustic Action . . . . . 203 12.4 Dissipation of Acoustic Energy Due to Thermal Slide . . . . 208 12.5 Gas Colmatation During Acoustic Action on Porous Media 210

13 Effect of Acoustic Action on Well ,Production 215 13.1 Laboratory Research on Permeability of Media after Acoustic 'freat-

ment ................................... 216 13.2 Determination of Size for the Acoustic Action Zone. . . . . . . . . 220 13.3 Calculation of Well Rate after Acoustic Action and Results of Field

Experiments . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Optimization of Acoustic 'freatment of Porous Media .

Bibliography

Index

223 225

229

239

Foreword

It is an honour and pleasure to write a foreword to this useful and interesting book. Authors are very well known researchers who pioneered percolation modelling of transport in porous media in Russia from the early 80-th till nowadays.

The main scope of the work presented in the book was developed when bright papers by A. Aharony, H.T. Davis, F.A.L. Dullien, A.A. Heiba, R.G. Larson, R. Lenormand, M.Sahimi, L.E. Scriven, D. Stauffer, M. Yanuka, Y.C.Yortsos were not available at the "other" side of the Iron Curtain.

Nowadays hundreds of works and papers with the "percolation" keywords ap­pear in petroleum and related applied research areas. The book will take a re­markable place in the "petroleum percolation" bibliography.

There are two important features of novelty in the monograph presented. First of all the authors developed a generalization of percolation clusters theory

for grids with varying conductivity. Technique of representation of an infinite cluster as an hierarchial set of trees (so called r-chain model) allows to present conductivity of a stochastic grid in a closed form of explicit formulae.

This method differs from those known in the West, such as effective media theory, solutions for the Bethe-lattice, etc. It has his own area of successful appli­cations.

This technique was applied for modelling of transport of multiphase systems in complex porous media. The majority of cases discussed have been investigated by other authors in the West in more detail using other methods for conductivity calculation (two- and three-phase flows, fractured media); some effects are pretty new (flow of non-Newtonian fluids, deformable porous media).

Quasi static percolation models have been further developed into dynamic ones (so-called model of the forest growth). Theory of non-equilibrium displacement on the pore level in micro heterogeneous media is developed (Chapter 5). This approach could give a hint for solution of a very desirable problem of dynamic transport in fractals.

The second important novelty of the book is percolation modelling of electric current in porous media. The theory provides basics for fast-developing methods

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of well stimulation and improved oil recovery by application of different electro­magnetic fields to the well drainage areas.

I hope the technique of r-chains and dynamical "forest growth" will be suc­cessfully applied and further developed in the area of upscaling of multiphase multicomponent flows in reservoirs with complex heterogeneity.

The book is strong both in theoretical fundamentals and engineering. Wide audience of petroleum engineers, researchers and graduate students will find it useful and informative. Wish them pleasant reading and further inspiration!

Professor Pavel Bedrikovetsky, MSc, PhD, DSc Moscow State Oil and Gas Academy, Russia Presently with PETROBRAS, CENPES CIDADE UNIVERSI­TARIA Q.7 ILHA DO FUNDAO 21949-900- RIO DE JANEIRO- RJ- BRAZIL 30th October 1994

Abstract

Results of theoretical analysis and experimental investigations for transport in porous media are presented. A new approach to modelling of transport in porous media is developed and a number of new percolation models is considered. The models allow to obtain analytical correlations for relative phase permeabilities for different porous media. Different methods of intensification of economic minerals based on new physical effects of reconstruction of the rock's pore space structure, are analysed.

The monograph is of interest for reservoir and chemical engineers, for specialists in reservoir characterization and simulation, for core analysts and researchers, and for post-graduate students in the above-mentioned areas.

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