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Upscaling of Single- and Two-Phase Flow in Reservoir Engineering [Kõva köide]

(AAPG Student Chapter Delft, The Netherlands)
  • Formaat: Hardback, 222 pages, kõrgus x laius: 246x174 mm, kaal: 600 g, 6 Tables, black and white; 95 Line drawings, black and white; 95 Illustrations, black and white
  • Ilmumisaeg: 15-Nov-2021
  • Kirjastus: CRC Press
  • ISBN-10: 0367767430
  • ISBN-13: 9780367767433
Teised raamatud teemal:
Upscaling of Single- and Two-Phase Flow in Reservoir EngineeringSuurem pilt
  • Formaat: Hardback, 222 pages, kõrgus x laius: 246x174 mm, kaal: 600 g, 6 Tables, black and white; 95 Line drawings, black and white; 95 Illustrations, black and white
  • Ilmumisaeg: 15-Nov-2021
  • Kirjastus: CRC Press
  • ISBN-10: 0367767430
  • ISBN-13: 9780367767433
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780367767433.html
Märksõnad:

This book describes fundamental upscaling aspects of single phase / two-phase porous media flow for application in petroleum and environmental engineering. Many standard texts have been written about this subject. What distinguishes this work from other available books is that it covers fundamental issues that are frequently ignored but are relevant for developing new directions to extend the traditional approach, but with an eye on application.

Our dependence on fossil energy is 80-90 % and is only slowly decreasing. Of the estimated 37 (~ 40) Gton/year anthropogenic emissions about 13 Gton/year of carbon dioxide remains in the atmosphere. An Exergy Return on Exergy Invested analysis shows how to obtain an unbiased quantification of the exergy budget and the carbon footprint. The intended audience of the book learns thus to quantify his method of optimization of recovery efficiencies supported by spreadsheet calculations.

As to single phase-one component fluid transport, it is shown how to deal with inertia, anisotropy, heterogeneity and slip. Upscaling requires numerical methods. The main application of transient flow is to find the reasons for reservoir impairment. The analysis benefits from solving the porous media flow equations using (numerical) Laplace transforms. Multi-phase flow requires the definition of capillary pressure and relative permeabilities. When capillary forces dominate we have dispersed (Buckley-Leverett flow). When gravity forces dominate we obtain segregated flow (interface models). Miscible Flow is described by a convection-dispersion equation. We give a simple proof that the dispersion coefficient can be approximated by Gelhar's relation, i.e., the product of the interstitial velocity, the variance of the logarithm of the permeability field and a correlation length.

The book will appeal most to students and researchers of porous media flow in connection with environmental engineering and petroleum engineering.



This book describes fundamental upscaling aspects of single-phase and two-phase porous media flow for application in petroleum and environmental engineering.
Preface xi
About the author xv
Chapter 1 Dutch and Worldwide Energy Recovery; Exergy Return on Exergy Invested 1(14)
Introduction
1(1)
1.1 Fraction Fossil in Current Energy Mix
2(2)
1.2 Possible New Developments
4(3)
1.3 Exergy
7(1)
1.4 Exergy Return on Exergy Invested (ERoEI) Analysis
7(8)
1.4.1 Exercise ERoEI
12(1)
1.4.2 Anthropogenic Emissions versus Natural Sequestration
12(1)
1.4.3 Exercise: Trees to Compensate for Intercontinental Flights
13(2)
Chapter 2 One-Phase Flow 15(56)
Introduction
15(1)
2.1 Mass Conservation
16(1)
2.2 Darcy's Law of Flow in Porous Media
17(4)
2.2.1 Definitions Used in Hydrology and Petroleum Engineering
17(2)
2.2.2 Exercise, EXCEL Naming
19(1)
2.2.3 Empirical Relations for Permeability (Carman-Kozeny Equation)
19(2)
2.3 Examples that Have an Analytical Solution
21(6)
2.3.1 One Dimensional Flow in a Tube
21(2)
2.3.2 Exercise, Two Layer Sand Pack
23(1)
2.3.3 Exercise, Numerical Model
23(1)
2.3.4 Exercise, EXCEL Numerical 1-D Simulation
24(1)
2.3.5 Radial Inflow Equation
25(1)
2.3.6 Boundary Conditions for Radial Diffusivity Equation
26(1)
2.3.7 Exercise, Radial Diffusivity Equation
27(1)
2.4 Modifications of Darcy's Law
27(10)
2.4.1 Representative Elementary Volume
27(2)
2.4.2 Exercise, Slip Factor
29(1)
2.4.3 Space Dependent Density
29(3)
2.4.4 Why is the Flow Resistance Proportional to the Shear Viscosity?
32(1)
2.4.5 Forchheimer Equation Must Be Used for High Values of the Reynolds Number
32(2)
2.4.6 Exercise, Inertia Factor
34(1)
2.4.7 Adaptation of Carman-Kozeny for Higher Flow Rates
34(1)
2.4.8 Exercise, Carman Kozeny
35(1)
2.4.9 Anisotropic Permeabilities
35(1)
2.4.10 Exercise, Matrix Multiplication
36(1)
2.4.11 Substitution of Darcy's Law in the Mass Balance Equation
36(1)
2.5 Statistical Methods to Generate Heterogeneous Porous Media
37(4)
2.5.1 The Importance of Heterogeneity
37(1)
2.5.2 Generation of Random Numbers Distributed According to a Given Distribution Function
37(1)
2.5.3 Log-Normal Distributions and the Dykstra-Parson's Coefficient
38(1)
2.5.4 Exercise, Lognormal Distribution Functions
39(1)
2.5.5 Generation of a Random Field
40(1)
2.5.6 Exercise, Log-Normal Permeability Field
41(1)
2.5.7 Exercise, Average Permeability Field
41(1)
2.6 Upscaling of Darcy's Law in Heterogeneous Media
41(7)
2.6.1 Arithmetic, Geometric and Harmonic Averages
41(4)
2.6.2 The Averaged Problem in Two Space Dimensions
45(1)
2.6.3 Effective Medium Approximation
46(1)
2.6.4 Pitfall: A Correctly Averaged Permeability Can Still Lead to Erroneous Production Forecasts
47(1)
2.7 Numerical Upscaling
48(8)
2.7.1 Finite Volume Method in 2-D; the Pressure Formulation
48(2)
2.7.2 The Finite Area Method; The Stream Function Formulation
50(3)
2.7.3 Finite Element Method (After F. Vermolen)
53(2)
2.7.4 Flow Calculation
55(1)
2.A Finite Volume Method in EXCEL
56(6)
2.A.1 The Data Sheet
56(2)
2.A.2 The Sheet for Calculation of the X-dip Averaged Permeability
58(1)
2.A.3 The Harmonically Averaged Grid Size Corrected Mobility in the x-Direction
58(1)
2.A.4 The Geometrically Averaged Grid Size Corrected Mobility in the y-Direction between the Central P and the Cell S
59(1)
2.A.5 The Sheet for the Well Flow Potential
59(1)
2.A.6 The Sheet for Productivity/Injectivity Indexes
59(1)
2.A.7 The Sheet for the Wells
60(1)
2.A.8 The Sheet for Flow Calculations
60(2)
2.B Finite Element Calculations
62(1)
2.C Sketch of Proof of the Effective Medium Approximation Formula
62(2)
2.D Homogenization
64(7)
Chapter 3 Time Dependent Problems in Porous Media Flow 71(32)
3.1 Transient Pressure Equation
71(15)
3.1.1 Boundary Conditions
75(1)
3.1.2 The Averaged Problem in Two Space Dimensions
76(1)
3.1.3 The Problem in Radial Symmetry
77(1)
3.1.4 Boundary Conditions for Radial Diffusivity Equation
78(1)
3.1.5 Dimensional Analysis for the Radial Pressure Equation; Adapted from Lecture Notes of Larry Lake
79(2)
3.1.6 Solution of the Radial Diffusivity Equation with the Help of Laplace Transformation
81(1)
3.1.7 Laplace Transformation
82(2)
3.1.8 Self Similar Solution
84(1)
3.1.9 The Dimensional Draw-Down Pressure
85(1)
3.2 Pressure Build Up
86(2)
3.2.1 Superposition
86(2)
3.2.2 Time Derivatives of Pressure Response
88(1)
3.2.3 Practical Limitations of Pressure Build Up Testing
88(1)
3.3 Formulation in a Bounded Reservoir
88(2)
3.4 Non-Darcy Flow
90(2)
3.A About Boundary Condition at r = reD
92(1)
3.A.1 Exercise, Stehfest Algorithm
92(1)
3.B Rock Compressibility
93(3)
3.B.1 Physical Model
93(1)
3.B.2 Mass Balance in Constant Control Volume
94(2)
3.C Equations Disregarding the Grain Velocity in Darcy's Law
96(1)
3.D Superposition Principle
96(1)
3.E Laplace Inversion with the Stehfest Algorithm
97(1)
3.F EXCEL Numerical Laplace Inversion Programme
98(5)
3.F.1 Alternative Inversion Techniques
99(4)
Chapter 4 Two-Phase Flow 103(70)
Introduction
103(4)
4.1 Capillary Pressure Function
107(11)
4.1.1 Interfacial Tension and Capillary Rise
108(1)
4.1.2 Exercise, Laplace Formula
109(1)
4.1.3 Exercise, Young's Law
110(1)
4.1.4 Application to Conical Tube; Relation between Capillary Pressure and Saturation
111(1)
4.1.5 Relation between the Pore Radius and the Square Root of the Permeability Divided by the Porosity
112(1)
4.1.6 Non-dimensionalizing the Capillary Pressure
113(1)
4.1.7 Exercise, Ratio Grain Diameter/Pore Throat Diameter
114(1)
4.1.8 Three-Phase Capillary Pressures
115(1)
4.1.9 Experimental Set Up and Measurements of Capillary Pressure
115(1)
4.1.10 Cross-Dip Capillary Equilibrium
116(2)
4.1.11 Exercise, Capillary Desaturation Curve
118(1)
4.2 Relative Permeabilities
118(6)
4.2.1 Exercise, Brooks-Corey Rel-perms
121(1)
4.2.2 LET Relative Permeability Model
121(2)
4.2.3 Estimate of the LET Parameters
123(1)
4.2.4 Exercise, Residual Oil and Rel-perm
123(1)
4.3 Theory of Buckley-Leverett
124(2)
4.3.1 Exercise, Vertical Upscaling Relative Permeability
125(1)
4.4 Material Balance
126(14)
4.4.1 Solutions of the Theory of Buckley-Leverett
129(1)
4.4.2 Equation of Motion (Darcy's Law) and the Fractional Flow Function
129(1)
4.4.3 Analytical Solution of the Equations
130(2)
4.4.4 Construction of the Analytical Solution; Requirement of the Entropy Condition
132(2)
4.4.5 Exercise, Buckley Leverett Profile with EXCEL
134(1)
4.4.6 Derivation of the Shock Condition
135(1)
4.4.7 Analytical Calculation of the Production Behavior
136(1)
4.4.8 Exercise, Buckley Leverett Production File
137(1)
4.4.9 Exercise, Analytical Buckley Leverett Production Curve
137(1)
4.4.10 Determination of Relative Permeabilities from Production Data and Pressure Measurements
138(1)
4.4.11 Determination of the Relative Permeabilities by Additional Measurement of the Pressure Drop
139(1)
4.5 Finite Volume Approach to Obtain the Finite Difference Equations for the Buckley Leverett Problem
140(3)
4.5.1 Exercise, Numerical Solution of Buckley Leverett Problem
143(1)
4.6 Vertical Equilibrium as a Basis for Upscaling of Relative Permeabilities and Fractional Flow Functions
143(6)
4.6.1 Dake's Upscaling Procedure for Relative Permeabilities
144(3)
4.6.2 Exercise, Sorting Factor Dependence
147(1)
4.6.3 Hopmans's Formulation
148(1)
4.7 Physical Theory of Interface Models
149(5)
4.7.1 Derivation of Interface Equation of Motion and Productions for Segregated Flow
149(1)
4.7.2 Stationary Interface (Mobility Number < Gravity Number +1)
150(3)
4.7.3 Exercise, Interface Angle Calculations
153(1)
4.7.4 Production Behavior for Stationary Solution, i.e., M < G + 1
154(1)
4.8 Non-stationary Interface
154(12)
4.8.1 The Volume Balance in the Form of an Interface Equation
155(2)
4.8.2 Dietz-Dupuit-Approximation
157(1)
4.8.3 Approximate Equilibrium Equation
157(1)
4.8.4 Derivation of Flow Rate Qwx from Darcy's Law
158(1)
4.8.5 Quasi Stationary Solution of the Dietz-Dupuit Equation for M < G+1
159(1)
4.8.6 Exercise, Shock Solution versus Interface Angle Solution
160(1)
4.8.7 Analytical Solutions
160(2)
4.8.8 Analytical Expressions for the Interface as a Function of Position in the Reservoir
162(1)
4.8.9 Analytical Expressions for the Production Behavior
162(3)
4.8.10 Summary of Analytical Procedure for Interface Models
165(1)
4.8.11 Exercise, Advantage of M < or = to G+ 1
166(1)
4.A Numerical Approach for Interface Models
166(2)
4.A.1 Exercise. Behavior for M > G+ 1
167(1)
4.B Numerical Approaches for Buckley Leverett and Interface Models Implemented with EXCEL
168(1)
4.B.1 Simple Sheet for Buckley-Leverett Model
168(1)
4.C Numerical Diffusion for First Order Upstream Weighting Scheme
169(4)
Chapter 5 Dispersion in Porous Media 173(28)
5.1 Introduction
173(5)
5.2 Molecular Diffusion Only
178(3)
5.3 Solutions of the Convection-Diffusion Equation
181(3)
5.3.1 Injection in a Linear Core
181(3)
5.3.2 Taylor's Problem in a Cylindrical Tube
184(1)
5.4 Derivation of the Dispersion Equation
184(2)
5.5 Statistics and Dispersion
186(2)
5.5.1 Random Walk Models
186(2)
5.6 Variance of Concentration Profile and Dispersion
188(1)
5.7 Dispersivity and the Velocity Autocorrelation Function
189(2)
5.8 Exercise, Numerical/Analytical 1D Dispersion
191(2)
5.9 Exercise, Gelhar Relation
193(2)
5.10 Numerical Aspects
195(1)
5.A Higher-Order Flux Functions for Higher-Order Schemes
196(2)
5.B Numerical Model with the Finite Volume Method
198(3)
Glossary 201(2)
List of Symbols 203(4)
References 207(12)
Index 219
Hans Bruining is a professor emeritus in geoenvironmental engineering of the Technical University of Technology of Delft, which is ranked 10 as one of the top technical institutes worldwide in Engineering. He holds a PhD degree from the University of Amsterdam. He is the founder of the Dietz-De Josselin de Jong laboratory. His special interests are the environmental aspects of fossil fuel recovery, enhanced oil recovery and theory and experiments of complex flow processes in porous media. He is review chairman of the Society of Petroleum Engineering Journal (SPEJ). He is the recipient of the SPE Distinguished Achievement Award for Petroleum Engineering Faculty (2012). The international award recognizes superiority in classroom teaching, excellence in research, significant contributions to the petroleum engineering profession and/or special effectiveness in advising and guiding students.