INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Beil & Howell Information Company 300 North Zeeb Road, Arm Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 NOTE TO USERS This reproduction is the best copy available UMI THE UNP/ERSITY OF OKLAHOMA GRADUATE COLLEGE A DUAL-POROSITY MODEL FOR TWO-PHASE FLOW IN DEFORMING POROUS MEDIA A Dissertation SUBmTTED TO THE GRADUATE FACULTY in partial fuifillnient of the requirements for the degree of Doctor of Philosophy By ZHENGYING SHU Norman, Oklahoma 1999 ÜMI Number: 9930844 UMI Microform 9930844 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 ©Copyright by Zhengying Shu 1999 All Rights Reserved A DUAL-POROSITY MODEL FOR TWO-PHASE FLOW IN DEFORMING POROUS MEDIA A Dissertation APPROVED FOR THE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING BY Dedication To my beautiful wife, Hui, and lovely son, David Jiaqi. IV Acknowledgments I would like to express my most siacere gratitude to my advisor Dr. Jeaa- Claude Roegiers for his invaluable advice, encouragement and guidance throughout this work. His leadership and broad knowledge have been of benefit to me and led to excellence of the Rock Mechanics Institute at the University of Oklahoma. I also highly appreciated Dr. Mao Bai for his suggestions and help in conducting my research and for drawing me into the field of poroelasticity. I would hke to extend my appreciation to Dr. Roy M. Knapp, Dr. Michael L. Wiggins, Dr. Anuj Gupta and Dr. James M. Forgotson Jr., for serving on my dissertation committee and for their helpful guidance. A special note of gratitude to all my research group-mates, especially Dr. M. Zaman and Dr. Fanghong Meng, for innumerable discussions from which I learned a lot. My appreciation goes to my parents, Gengwu Shu and Jinming Zheng, my parents-in-law, Qifan Zhang and Meifang Liu, and my sister-in-law, Jie Zhang, for taking care of my son in both China and the US. I want also to thank my sister, Lianfang Shu, and my brother-in-law, Gaosong Liu, for their encouragement, help and taking part of my responsibility in taking care of our parents. Contents Acknowledgments v List of Figures ix List of Tables xii Abstract xiv 1 Latroduction 1 1.1 Objectives of the Study ................................................................................. 3 1.2 Organization of the Dissertation ................................................................. 3 2 Critical Literature Review 6 2.1 Dual-Porosity M odel .................................................................................... 7 2.1.1 Uncoupled Dual-Porosity M odel .................................................... 7 2.1.2 Coupled Single Porosity M odels ........................................................12 2.1.3 Coupled Dual-Porosity Models ........................................................... 16 2.2 Equivalent Porous M edia .................................... ,. ......................................... 19 2.3 Single Fracture Models .....................................................................................20 2.4 Fracture Network Models ..................................................................................21 2.5 Models C om parisons ........................................................................................ 22 3 Governing Equations for Two-phase Fluid Flow in a Deformable vi Fractured Porous Medium 24 3.1 Equilibrium Equations for Solid Mixtures (Displacement Equations) . 24 3.2 Continuity equations for fluid phases .......................................................... 28 3.2.1 Oil-water s y s te m ..................................................................................28 3.2.2 Darcy’s l a w ............................................................................................34 3.2.3 Oil-gas system ........................................................................................ 34 4 Numerical Implementation 45 4.1 Finite Diflerence Formulae for Derivatives .................................................47 4.2 Finite Difference Approximation of Equilibrium E quations ................... 48 4.3 Finite Difference Approximation of Continuity Equations .......................50 4.3.1 Water-Oil S y stem ..................................................................................50 4.3.2 Gas-Oü S y s te m.....................................................................................55 4.4 Solution procedure ...........................................................................................61 5 Model Validation 63 5.1 Comparison with Analytical Solution ...........................................................63 5.1.1 Analytical S o lu tio n ...............................................................................63 5.1.2 Numerical S o lu tio n .................................................................... 65 5.2 Comparison with a Finite Element Model .................................................... 72 6 Model Applications 75 6.1 Two-Phase Flow Coupled with Fractured Rock Consohdation .... 75 6.2 0Ü Production from a Deformable Fractured Reservoir with Water In jectio n ...............................................................................................................81 6.2.1 Description of the case studied............................................................82 6.2.2 Approximation of Boundary Conditions for Boundaries and Comer Points ........................................................................................ 83 vu 6.2.3 Analysis of Simiilaticn R esults .......................................................... 87 7 Sensitivity Analyses 98 7.1 Description of the Model U sed .......................................................................99 7.2 Approximations of Boundary Conditions for Bormdaries and Comer P o in ts ............................................................................................................... 102 7.3 Analysis of Rock Deformations .....................................................................106 7.4 Effects of Injection Rate on Pore Pressure .............................................. 110 7.5 Effects of Ks (Solid Grain Bulk Modulus) on Displacements and Pore P re ssu re ............................................................................................................I l l 7.6 Effects of Yoimg’s Modulus on Displacements and Pore Pressiue . .112 8 Conclusions and Recommendations 119 8.1 Contributions and Conclusions ..................................................................... 119 8.2 Recommendations for future w o rk ...............................................................122 Nomenclature 123 References 126 A Fundamental Concepts 140 A.l Porosity ............................................................................................................140 A.2 Saturation ........................................................................................................ 141 A.3 Perm eability ..................................................................................................... 142 A.4 Capillary Pressure ........................................................................................ 144 A.5 Formation Volume F actor ............................................................................145 A.6 Solution Gas-Oü R atio .................................................................................. 148 A.7 Fractured Reservoirs ..................................................................................... 148 A.7.1 Definit i o n ........................................................................................... 148 vm A.7.2 Origin ...................................................................................................149 A.8 Stress and Strain ...........................................................................................151 A.8.I Stress-Strain Relationships ...............................................................153 A-9 P oroelasticity ................................................................................................