Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Chapter 1: Semipermeable Wall Model of a Multilayer Reservoir with Crossflow

• Abstract
• 1.1 Introduction
• 1.2 Semipermeable Wall Model and Fundamental Differential Equations—Oil Reservoir
• 1.3 Semipermeable Wall Model and Fundamental Differential Equations—Gas Reservoir
• 1.4 Crossflow Behavior for Incompressible Flow
• 1.5 Summary

Chapter 2: Single-Phase Fluid Flow in a Two-Layer Reservoir with Significant Crossflow

• Abstract
• 2.1 Introduction
• 2.2 Pressure Drawdown and Buildup Behavior When Both Layers Are Completed at a Single Well
• 2.3 Crossflow Behavior in a Partially Perforated Two-Layer Reservoir
• 2.4 Maximum Effective Hole Mathematical Model and Analytical Solution for a Two-Layer Reservoir

Chapter 3: Pressure Transients and Crossflow Caused by Diffusivities in Multilayer Reservoirs

• Abstract
• 3.1 Assumption and the Flow Problem
• 3.2 Approximate Analytic Solutions
• 3.3 Numerical Solutions for Cases with n=3 and n=5
• 3.4 Numerical Solutions and Comparison with Theory
• 3.5 Analytical Solution of the Three-Layer Reservoir
• 3.6 Summary

Chapter 4: Crossflow Behavior and the Determination Reservoir Parameters by Drawdown Tests in Multilayer Reservoirs

• Abstract
• 4.1 Assumption and Mathematical Expression of the Problem
• 4.2 The Unsteady Flow Behavior in Drawdown and Buildup Tests When Partial Layers Produce with Crossflow
• 4.3 New Drawdown Test
• 4.4 Summary

Chapter 5: Determination of Total Productivity by a Constant Wellbore Pressure Flow Test and the Crossflow Behavior in Multilayer Reservoirs

• Abstract
• 5.1 Assumption and Approximate Theoretical Solution of the Problem
• 5.2 Numerical Results and Comparison with the Approximation Theory
• 5.3 Exact Solution of a Two-Layer Reservoir with Crossflow Under a Constant Pressure Condition
• 5.4 Summary

Chapter 6: Determination of Individual Layer Properties by Layer-by-Layer Well Tests in Multilayer Reservoirs with Crossflow

• Abstract
• 6.1 Assumptions and Mathematical Model
• 6.2 Approximate Solution of Unsteady Flow in a Three-Layered Reservoir and Its Extension
• 6.3 Discussion of Simulation Results
• 6.4 Interpretation Theory
• 6.5 Summary

Chapter 7: Determination of Parameters for Individual Layers by Transient Well Tests in Multilayer Reservoirs

• Abstract
• 7.1 Introduction
• 7.2 Approximation Solution of Unsteady-Flow and Crossflow Behavior Caused by Skin Factors
• 7.3 Determination of Parameters for Individual Layers by a Series of Transient Tests
• 7.4 Determination of Semipermeabilities by Steady Rates
• 7.5 Summary

Chapter 8: Interpretation Theory for Vertical Interference Testing Across a Low-Permeability Zone

• Abstract
• 8.1 Introduction
• 8.2 Model Description
• 8.3 The Interpretation Theory When q(t) is Known
• 8.4 The Interpretation Theory When the Well Produces with a Constant Surface Rate
• 8.5 The Influence of Different Parameters on Drawdown Curves
• 8.6 Summary

Chapter 9: Unsteady Crossflow Model for Leaky Vertical Interference Testing Across a Low-Permeability Zone

• Abstract
• 9.1 Introduction
• 9.2 Model Description
• 9.3 Relationships of Different Models
• 9.4 Solution of the Problem
• 9.5 Interpretation Method of Parameters
• 9.6 Comparison of Different Models
• 9.7 Summary

Chapter 10: Material Balance Equation of Multilayer Gas Reservoir

• Abstract
• 10.1 Material Balance of Two-Layer Gas Reservoir
• 10.2 Reserve Estimation of a Two-Layer Commingled Gas Well
• 10.3 Material Balance Equation of Gas Reservoir with a Supplying Region
• 10.4 Summary

Chapter 11: Computer-Aided Well Test Design and Well Test Analysis of Layered Reservoirs

• Abstract
• 11.1 Introduction
• 11.2 Computer Program
• 11.3 Examples of Well Test Design
• 11.4 Examples of Well Test Analysis
• 11.5 Summary

Nomenclature

Appendix A: Analytical Solution for the Stationary Value of f (Chapter 2)

Appendix B: Approximate Solution of the Diffusivity Crossflow Problem in n-Layer Reservoirs (Chapter 3)

Appendix C: Solution of the Crossflow Caused by Different Boundary Pressures (Chapter 4)

Appendix D: Approximate Solution for a Well Producing From All Layers

• D.1 Approximate Solution for the Short Time Period
• D.2 Approximate Solution for the Long Time Period

Appendix E: Approximate Solution and Its Extension When Each Layer of a Two-Layer Reservoir Produces Independently

• E.1 Approximate Solution for the Short Time Period
• E.2 Approximate Solution for the Long Time Period
• E.3 Buildup Case
• E.4 Extension of the Solution for Two-Layer Reservoirs to n-Layer Reservoirs

Appendix F: Case for Variable Bottom-Hole Rate (Chapter 8)

• F.1 Case for Unknown Function
• F.2 Case for a Known Leakage Rate
• F.3 Case for

Appendix G: Case for Constant Wellhead Rate (Chapter 8)

• G.1 Case for Unknown Leakage Rate q_D2
• G.2 Case for Known Function f(t_D)
• G.3 Case for Leakage Function
• G.4 Case for
• G.5 Limiting Behavior for Function

Appendix H: Solution of the Problem (Chapter 9)

• H.1 When Leakage Function f Is Known
• H.2 When Leakage Function
• H.3 When Leakage Function f Is Unknown

Appendix I: Limiting Behavior for f=α_D(p_wD1-p_wD2) (Chapter 9)

• I.1 Early Time Period
• I.2 Behavior When t→0 (Short Time Period)
• I.3 Behavior When t→∞ (Long Time Period)

Appendix J: Commonly Used Units in Different Unit Systems