Nonlinear flow and well test analysis in porous media, livre ebook

Based on the analysis of factors influencing fluid infiltration in porous media, this book systematically summarizes the characteristics and expressions of low-velocity nonlinear flow and high-velocity nonlinear flow infiltration in porous media and provides a set of evaluation methods. Using the exponential formula, the starting pressure gradient formula, and the binomial equation of motion, the authors present a detailed comparison and analysis of the production, pressure, and dimensionless background pressure of the nonlinear flow and Darcy linear flow for steady and unsteady flow. In addition, based on the equation of motion of the starting pressure gradient, a mathematical model of the one-, two-, and three-medium nonlinear seepage flow is established, and approximate analytical solutions are given while the graph of the corresponding well test curve is drawn. Finally, based on the mathematical model of the well test established from the exponential equation of the high-velocity nonlinear flow motion, the atypical well test curve and the relational surface of the time- and space-varying infiltration index are obtained. The authors also discuss the relationship between reservoir and fluid properties and the nonlinear flow test curve. This book is intended to serve as a reference for technical personnel, researchers, teachers, and students involved in oil and gas development. Its research contents provide a theoretical basis for the identification of water flow dominant channels in the long-term water injection development of high-water-cut oilfields, profile control and water shut-off, productivity evaluation of carbonate reservoirs and formation parameters.

Preface..................................................... III

CHAPTER 1

Mechanism of Nonlinear Flow andAnalysis of Characteristics............ 1

1.1 Analysis of Factors AffectingFluid Flow in Porous Media .......... 1

1.2 Mechanism of Nonlinear Flowin Low Permeability Reservoirs....... 3

1.2.1 Characteristics ofNonlinear Flow in Low Permeability

Reservoirs........................................ 4

1.2.2 Model of Nonlinear Flow inLow Permeability Reservoirs ..... 6

1.3 Mechanisms of StressSensitivity in Low-Permeability Reservoirs..... 9

1.3.1 Stress SensitivityCharacteristics of Low-Permeability

Reservoirs........................................ 9

1.3.2 Stress Sensitivity Modelfor Low Permeability Reservoirs ..... 10

1.4 Characteristics ofHigh-Velocity Nonlinear Flow ................. 13

1.4.1 Introduction....................................... 13

1.4.2 Characteristics ofHigh-Velocity Nonlinear Flow ............ 15

1.4.3 Judgment Method forHigh-Velocity Nonlinear Flow ........ 17

1.4.4 Equation for Description ofHigh-Velocity Nonlinear Flow .... 19

CHAPTER 2

Model of Low-Velocity NonlinearFlow in Single Media ................. 25

2.1 Threshold Pressure GradientModel of Low-Velocity Nonlinear Flow . . 25

2.1.1 Stable Flow Model.................................. 26

2.1.2 Unstable Flow Model................................ 26

2.1.3 Dimensionless Well TestModel Under Unstable Flow ........ 28

2.2 Exponential Model ofLow-Velocity Nonlinear Flow .............. 34

2.2.1 Stable Flow Model.................................. 35

2.2.2 Unstable Flow Model................................ 37

2.2.3 Dimensionless Well TestModel of Unstable Flow ........... 41

CHAPTER 3

Theory of Well Test forLow-Velocity Nonlinear Flow in Multiple Media .... 47

3.1 Mathematical Model ofLow-Velocity Nonlinear Flow in Double

Media................................................. 47

3.1.1 Motion Equation................................... 48

3.1.2 Channeling Equation................................ 48

3.1.3 State Equation..................................... 49

3.1.4 Continuity Equation................................ 50

3.1.5 Simplified Models of MatrixPermeability and Fracture

Porosity.......................................... 50

3.1.6 Simplified Model of MatrixPermeability ................. 52

3.2 Theory of Well Test ofLow-Velocity Nonlinear Flow in Double Media 53

3.2.1 Mathematical Model of WellTest in Dual-Porosity Reservoir

and Its Solution.................................... 53

3.2.2 Mathematical ModelConsidering Wellbore Storage Effect

and Skin Factor.................................... 56

3.2.3 Characteristics of WellTest Curve of Low-Velocity Nonlinear

Flow in Double Media............................... 56

3.2.4 Analysis of FactorsAffecting the Well Test Curves

of Low-Velocity Nonlinear Flow inDouble Media Reservoirs . . 59

3.3 Mathematical Model ofLow-Velocity Nonlinear Flow in Triple

Media................................................. 67

3.3.1 Ideal Model of Triple Media........................... 68

3.3.2 Mathematical Model and ItsSolution for Triple Media

Reservoirs........................................ 69

3.3.3 Well Test CurveCharacteristics of Low-Velocity Nonlinear

Flow in Triple Media Reservoirs........................ 73

3.3.4 Analysis of FactorsAffecting the Well Test Curves

of Low-Velocity Nonlinear Flow inTriple Media ............ 76

CHAPTER 4

Well Test of Nonlinear FlowConsidering Stress Sensitivity in Low

Permeability Reservoirs........................................ 89

4.1 Factors Affecting StressSensitivity and Mathematical Models ....... 89

4.1.1 Change Characteristics ofStress Sensitivity ............... 89

4.1.2 Factors Affecting StressSensitivity ...................... 92

4.1.3 Mathematical Model ofStress Sensitivity ................. 94

4.2 Nonlinear Stable andPseudo-Stable Flow Model Considering Stress

Sensitivity.............................................. 97

4.2.1 Productivity and PressureModels of Stable Flow Considering

Stress Sensitivity................................... 97

4.2.2 Productivity and PressureModels of Pseudo-Stable Flow

Considering Stress Sensitivity.......................... 97

4.3 Nonlinear and Unstable WellTest Considering Stress Sensitivity ..... 99

4.3.1 Assumptions and the Model........................... 99

4.3.2 Solution and Charts forWell Test Model Considering Stress

Sensitivity in Low PermeabilityReservoirs ................ 101

4.4 Well Test in Fracture-PoreDouble Media Considering Stress

Sensitivity.............................................. 108

4.4.1 Sensitivity of FracturePermeability ..................... 108

4.4.2 Well-Test Model of DoubleMedia Considering Stress

Sensitivity and Typical Curves......................... 111

CHAPTER 5

High-Velocity Nonlinear StableFlow Model ......................... 119

5.1 Darcy Linear Radial Flow.................................. 119

5.2 Binomial Nonlinear RadialFlow ............................. 120

5.3 Exponential Nonlinear RadialFlow ........................... 122

5.4 Comparison of the CalculationResults for Darcy’s Linear Flow

and Nonlinear Flow...................................... 123

5.4.1 Producing in a ConstantProduction .................... 124

5.4.2 Producing at a ConstantBottom-Hole Pressure ............ 129

CHAPTER 6

Transient Well Test UnderHigh-Velocity Nonlinear Flow ............... 139

6.1 Exponential High-VelocityNonlinear Flow Model ................ 139

6.1.1 Exponential High-VelocityNonlinear Flow Model ........... 139

6.1.2 Analytical Solution to theExponential High-Velocity

Nonlinear Flow Model............................... 141

6.1.3 Numerical Solution for theExponential High-Velocity

Nonlinear Flow Model............................... 144

6.1.4 Results and Discussion............................... 148

6.2 Binomial High-VelocityNonlinear Flow Model .................. 152

6.2.1 Establishment of Flow Model.......................... 152

6.2.2 Solution of theModel................................ 153

6.2.3 Results and Discussion............................... 155

CHAPTER 7

Transient Well Test of MovingBoundary in High-Velocity Nonlinear Flow . . 159

7.1 Assumptions............................................ 159

7.2 Establishment and Solution ofthe Mathematical Model ........... 160

7.3 Results and Discussion.................................... 161

Nomenclatures............................................... 167

Appendix: Virtual ArgumentInteger Bessel Function .................. 173

References.................................................. 177