Mechanics of Oil and Gas Flow in Porous Media

This book discusses various aspects of percolation mechanics. It starts with the driving forces and driving modes and then examines in detail the steady state percolation of single-phase incompressible fluids, percolation law of natural gas and percolation of non-Newtonian fluids. Progressing from simple to complex concepts, it also analyzes Darcy’s law, providing a basis for the study of reservoir engineering, oil recovery engineering and reservoir numerical simulation. It serves as a textbook for undergraduate students majoring in petroleum engineering, petroleum geology and groundwater engineering, and offers a valuable reference guide for graduate students, researchers and technical engineers engaged in oil and gas exploration and development. Preface 5 Contents 7 1 Basic Law of Percolation 12 1.1 Static State in Reservoirs 13 1.1.1 Distribution of Oil, Gas and Water in Reservoirs 13 1.1.2 Concept of Various Pressures 14 1.2 Driving Forces and Driving Modes in Reservoir 16 1.2.1 Hydrostatic Pressure 17 1.2.2 Elastic Energy of Formation and the Fluid in It 17 1.2.3 Elastic Energy of Dissolved Gas 20 1.2.4 Elastic Energy of Gas Cap 21 1.2.5 Action of Gravity 21 1.3 Basic Laws of Percolation-Darcy’s Law 22 1.3.1 Continuous Medium 22 1.3.2 Darcy’s Law 22 1.3.3 Percolation Velocity and Actual Velocity 25 1.3.4 Dimension of Permeability 26 1.4 The Limitations of Darcy’s Law and Non-Darcy Percolation 27 References 32 2 Steady-State Percolation in Porous Media of Single-Phase Incompressible Fluid 33 2.1 Three Basic Forms of Percolation in Porous Media 33 2.2 Planar One-Dimensional Steady Percolation of Single-Phase Incompressible Fluid 35 2.3 Planar Radial Steady Percolation of Single-Phase Incompressible Fluid 42 2.4 Imperfect Well 49 2.4.1 Classification of Imperfect Well 49 2.4.2 Research Method of Imperfect Well 50 2.5 Steady Well Testing 52 2.6 Differential Equation of Single-Phase Fluid Percolation in Porous Media 54 2.6.1 Continuing Equation of Percolation of Single-Phase Fluid 55 2.6.2 State Equation 58 2.6.3 Darcy Equation 59 2.6.4 Basic Differential Equation of Single-Phase Slightly Compressible Fluid Percolation 60 2.6.5 Boundary Condition and Initial Condition of Reservoir 63 2.6.6 Basic Differential Equation and Its Solutions of Single-Phase Incompressible Fluids Percolation 65 References 69 3 Interference Theory of Wells Under Rigid Water Drive 70 3.1 Phenomenon of Interference Between Wells 70 3.2 Superposition Principle of Potential 72 3.3 Method of Mirror 78 3.3.1 Interference of Uniform-Intensity One Source and One Sink—The Source-Sink Image Method 79 3.3.2 The Interference of Two Sinks Equal in Intensity—Sink Point Image Method 86 3.3.3 Popularization and Application of Mirror Image 89 3.3.4 Partial Penetration Well 93 3.3.5 Equivalent Flow Resistance Method 99 3.4 Application of Complex Function Theory in the Planar Percolation Field 104 3.4.1 Foundation of Complex Function 104 3.4.2 Complex Potential of Planar Percolation Field 106 3.4.3 Superposition Principle of Complex Potential 112 3.4.4 Application of Conformal Mapping in Planar Percolation Field 116 3.4.5 Production Formula of the Multi-fracture Well and Multi-branched Horizontal Well 128 References 133 4 Unsteady Percolation of Slightly Compressible Fluid 134 4.1 The Physical Process of Elastic Fluid Percolation Towards Wells 134 4.2 The Planar One-Dimensional Unsteady Percolation in Semi-infinite Formation 137 4.2.1 Unsteady Percolation with Constant Bottom Pressure 137 4.2.2 Unsteady Percolation When Production Is Constant 140 4.3 Pressure Transient Law of Elastic Fluid Unsteady Percolation Towards Well Bore 142 4.4 Approximate Solution of Pressure Change of Fluid Percolation Towards Well in Finite Closed Elastic Formation 149 4.5 Multiple-Well Interference of Elastic Unsteady Percolation 154 4.5.1 Superposition Principle of Pressure Drop 154 4.5.2 Image Method in the Formation with Linear Boundary 156 4.5.3 Duhamel Principle Solving the Problem of Changing Production 157 4.6 Unsteady Well-Test Analysis 160 4.6.1 The Pressure Fall-off Test 161 4.6.2 Pressure Build-up Test 165 References 171 5 Percolation Law of Natural Gas 172 5.1 Properties of Natural Gas and Its Basic Differential Equation of Percolation 172 5.1.1 Equation of State of Gas 173 5.1.2 The Compressibility of Natural Gas 174 5.1.3 The Standard Condition of Natural Gas 175 5.1.4 The Basic Differential Equation of Gas Percolation 175 5.2 Steady Percolation of Gas 180 5.2.1 The Planar One-Dimensional Steady Percolation 180 5.2.2 The Planar Radial Fluid Steady Flow 182 5.2.3 The Percolation of Gas Obeying the Binomial Theorem 183 5.3 Unsteady Percolation of Gas 186 6 Foundation of Water/Oil Displacement Theory 192 6.1 Piston Like Displacement of Oil by Water 193 6.1.1 Planar One-Dimensional Piston-Like Displacement of Oil by Water 194 6.1.2 Planar Radial Piston Like Displacement of Oil by Water 196 6.1.3 Shrinkage of Oil Edge 198 6.2 Bottom Water Coning 201 6.2.1 The Equilibrium Condition of Water Coning 202 6.2.2 The Pressure Distribution on the Surface of Water Coning 204 6.2.3 Approximate Solving Method of Biggest Water-Free Production 204 6.2.4 Problem of Bottom Water Coning of Horizontal Well 214 6.2.5 The Rising Time of Water Coning of Vertical Well 219 6.2.6 The Behavior After Water Breakthrough of Vertical Well 220 6.2.7 The Stability of Oil-Water Interface of Vertical Well 223 6.3 Theoretical Foundation of Non-piston Like Displacement of Oil by Water 226 6.3.1 The Buckley-Leverett Solution 227 6.3.2 The Pressure and Production on Oil-Water Two-Phase Percolation 238 6.3.3 Planar Radial Non-piston Displacement of Oil by Water 240 References 246 7 Percolation Theory of Oil-Gas Two Phases (Dissolved Gas Drive) 247 7.1 Basic Differential Equation of Oil-Gas Two-Phases Percolation 247 7.2 Oil-Gas Two-Phase Steady Percolation 251 7.3 Unsteady Percolation of Gassy Fluid 259 Reference 267 8 Percolation in Dual-Medium 268 8.1 Percolation Characteristics in Dual-Medium 268 8.2 Basic Percolation Equation of Single-Phase Slightly Compressible Fluid in Dual-Medium 270 8.3 Pressure Distribution in Infinite Dual-Medium 274 8.4 Oil-Water Two-Phase Percolation in the Dual-Medium 277 8.5 Percolation of Fluid in Pure Fracture Formation 284 8.5.1 The Steady State Flow Towards the Well of Fluid 285 8.5.2 The Unsteady State Flow Towards the Well of Fluid 287 References 288 9 Percolation of Non-Newtonian Liquid 289 9.1 Mechanical Behavior and Type of Non-Newtonian Liquid 289 9.1.1 The Viscosity 290 9.1.2 The Viscoelasticity 294 9.2 Percolation of Non-Newtonian Power Law Liquid 295 9.2.1 The Method of van Poollen and Jargon 295 9.2.2 The Method of Ikoku and Ramey 297 9.2.3 The Method of Odeh and Yang 302 References 305 10 Theory of Similarity 306 10.1 Concept of Similarity 306 10.2 Kinematical and Dynamic Similarity 307 10.3 Standard Configuration Model and Law of Similarity 309 10.4 Dimensional Analysis Examples for the Known Equations of Physical System 311 10.5 Examples of Analogy Model 316 10.6 Partial Similarity in Standard Configuration Model 319 10.7 Special Aspects of Standard Configurations of Porous Medium Model 322 Reference 323 Appendix A Percolation Equation in Cylindrical Coordinate 324 Appendix B To Obtain the Solution of One-Dimensional Unsteady Percolation by the Laplace Transformation 329 Appendix C Calculation Procedure of Error Function erf(x) 331 Appendix D Calculation Procedure of Exponential Integral Function—Ei(−x) 337 Appendix E Calculation Procedure of Function Γ(X) 339 Appendix F Conversion Relationship Between the Commonly Used Units in Reservoir Engineering 341 Front Matter ....Pages i-xi Basic Law of Percolation (Dang Li, Junbin Chen)....Pages 1-21 Steady-State Percolation in Porous Media of Single-Phase Incompressible Fluid (Dang Li, Junbin Chen)....Pages 23-59 Interference Theory of Wells Under Rigid Water Drive (Dang Li, Junbin Chen)....Pages 61-124 Unsteady Percolation of Slightly Compressible Fluid (Dang Li, Junbin Chen)....Pages 125-162 Percolation Law of Natural Gas (Dang Li, Junbin Chen)....Pages 163-182 Foundation of Water/Oil Displacement Theory (Dang Li, Junbin Chen)....Pages 183-237 Percolation Theory of Oil-Gas Two Phases (Dissolved Gas Drive) (Dang Li, Junbin Chen)....Pages 239-259 Percolation in Dual-Medium (Dang Li, Junbin Chen)....Pages 261-281 Percolation of Non-Newtonian Liquid (Dang Li, Junbin Chen)....Pages 283-299 Theory of Similarity (Dang Li, Junbin Chen)....Pages 301-318 Back Matter ....Pages 319-339