Dake L.P. Fundamentals of reservoir engineering

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CONTENTS XI

2.6 COMPLETE PVT ANALYSIS 69

REFERENCES 70

CHAPTER 3 MATERIAL BALANCE APPLIED TO OIL RESERVOIRS 71

3.1 INTRODUCTION 71

3.2 GENERAL FORM OF THE MATERIAL BALANCE EQUATION FOR

A HYDROCARBON RESERVOIR 71

3.3 THE MATERIAL BALANCE EXPRESSED AS A LINEAR EQUATION 76

3.4 RESERVOIR DRIVE MECHANISMS 77

3.5 SOLUTION GAS DRIVE 78

3.6 GASCAP DRIVE 86

3.7 NATURAL WATER DRIVE 91

3.8 COMPACTION DRIVE AND RELATED PORE COMPRESSIBILITY

PHENOMENA 95

REFERENCES 98

CHAPTER 4 DARCY'S LAW AND APPLICATIONS 100

4.1 INTRODUCTION 100

4.2 DARCY'S LAW; FLUID POTENTIAL 100

4.3 SIGN CONVENTION 104

4.4 UNITS: UNITS CONVERSION 104

4.5 REAL GAS POTENTIAL 110

4.6 DATUM PRESSURES 111

4.7 RADIAL STEADY STATE FLOW; WELL STIMULATION 112

4.8 TWO-PHASE FLOW: EFFECTIVE AND RELATIVE

PERMEABILITIES 117

4.9 THE MECHANICS OF SUPPLEMENTARY RECOVERY 121

REFERENCES 125

CHAPTER 5 THE BASIC DIFFERENTIAL EQUATION FOR RADIAL FLOW IN A

POROUS MEDIUM 127

5.1 INTRODUCTION 127

5.2 DERIVATION OF THE BASIC RADIAL DIFFERENTIAL EQUATION 127

CONTENTS XII

5.3 CONDITIONS OF SOLUTION 129

5.4 THE LINEARIZATION OF EQUATION 5.1 FOR FLUIDS OF SMALL

AND CONSTANT COMPRESSIBILITY 133

REFERENCES 135

CHAPTER 6 WELL INFLOW EQUATIONS FOR STABILIZED FLOW

CONDITIONS 136

6.1 INTRODUCTION 136

6.2 SEMI-STEADY STATE SOLUTION 136

6.3 STEADY STATE SOLUTION 139

6.4 EXAMPLE OF THE APPLICATION OF THE STABILIZED INFLOW

EQUATIONS 140

6.5 GENERALIZED FORM OF INFLOW EQUATION UNDER SEMI-

STEADY STATE CONDITIONS 144

REFERENCES 146

CHAPTER 7 THE CONSTANT TERMINAL RATE SOLUTION OF THE RADIAL

DIFFUSIVITY EQUATION AND ITS APPLICATION TO OILWELL TESTING 148

7.1 INTRODUCTION 148

7.2 THE CONSTANT TERMINAL RATE SOLUTION 148

7.3 THE CONSTANT TERMINAL RATE SOLUTION FOR TRANSIENT

AND SEMI-STEADY STATE FLOW CONDITIONS 149

7.4 DIMENSIONLESS VARIABLES 161

7.5 SUPERPOSITION THEOREM: GENERAL THEORY OF WELL

TESTING 168

7.6 THE MATTHEWS, BRONS, HAZEBROEK PRESSURE BUILDUP

THEORY 173

7.7 PRESSURE BUILDUP ANALYSIS TECHNIQUES 189

7.8 MULTI-RATE DRAWDOWN TESTING 209

7.9 THE EFFECTS OF PARTIAL WELL COMPLETION 219

7.10 SOME PRACTICAL ASPECTS OF WELL SURVEYING 221

7.11 AFTERFLOW ANALYSIS 224

REFERENCES 236

CONTENTS XIII

CHAPTER 8 REAL GAS FLOW: GAS WELL TESTING 239

8.1 INTRODUCTION 239

8.2 LINEARIZATION AND SOLUTION OF THE BASIC DIFFERENTIAL

EQUATION FOR THE RADIAL FLOW OF A REAL GAS 239

8.3 THE RUSSELL, GOODRICH, et. al. SOLUTION TECHNIQUE 240

8.4 THE AL-HUSSAINY, RAMEY, CRAWFORD SOLUTION

TECHNIQUE 243

8.5 COMPARISON OF THE PRESSURE SQUARED AND PSEUDO

PRESSURE SOLUTION TECHNIQUES 251

8.6 NON-DARCY FLOW 252

8.7 DETERMINATION OF THE NON-DARCY COEFFICIENT F 255

8.8 THE CONSTANT TERMINAL RATE SOLUTION FOR THE FLOW

OF A REAL GAS 257

8.9 GENERAL THEORY OF GAS WELL TESTING 260

8.10 MULTI-RATE TESTING OF GAS WELLS 262

8.11 PRESSURE BUILDUP TESTING OF GAS WELLS 278

8.12 PRESSURE BUILDUP ANALYSIS IN SOLUTION GAS DRIVE

RESERVOIRS 289

8.13 SUMMARY OF PRESSURE ANALYSIS TECHNIQUES 291

REFERENCES 295

CHAPTER 9 NATURAL WATER INFLUX 297

9.1 INTRODUCTION 297

9.2 THE UNSTEADY STATE WATER INFLUX THEORY OF HURST

AND VAN EVERDINGEN 298

9.3 APPLICATION OF THE HURST, VAN EVERDINGEN WATER

INFLUX THEORY IN HISTORY MATCHING 308

9.4 THE APPROXIMATE WATER INFLUX THEORY OF FETKOVITCH

FOR FINITE AQUIFERS 319

9.5 PREDICTING THE AMOUNT OF WATER INFLUX 328

9.6 APPLICATION OF INFLUX CALCULATION TECHNIQUES TO

STEAM SOAKING 333

REFERENCES 335

CONTENTS XIV

CHAPTER 10 IMMISCIBLE DISPLACEMENT 337

10.1 INTRODUCTION 337

10.2 PHYSICAL ASSUMPTIONS AND THEIR IMPLICATIONS 337

10.3 THE FRACTIONAL FLOW EQUATION 345

10.4 BUCKLEY-LEVERETT ONE DIMENSIONAL DISPLACEMENT 349

10.5 OIL RECOVERY CALCULATIONS 355

10.6 DISPLACEMENT UNDER SEGREGATED FLOW CONDITIONS 364

10.7 ALLOWANCE FOR THE EFFECT OF A FINITE CAPILLARY

TRANSITION ZONE IN DISPLACEMENT CALCULATIONS 381

10.8 DISPLACEMENT IN STRATIFIED RESERVOIRS 389

10.9 DISPLACEMENT WHEN THERE IS A TOTAL LACK OF VERTICAL

EQUILIBRIUM 402

10.10 THE NUMERICAL SIMULATION OF IMMISCIBLE,

INCOMPRESSIBLE DISPLACEMENT 405

REFERENCES 419

AUTHOR INDEX 422

SUBJECT INDEX 424

CONTENTS XV

EXERCISES

EXERCISE 1.1 GAS PRESSURE GRADIENT IN THE RESERVOIR 23

EXERCISE 1.2 GAS MATERIAL BALANCE 33

EXERCISE 2.1 UNDERGROUND WITHDRAWAL 50

EXERCISE 2.2 CONVERSION OF DIFFERENTIAL LIBERATION DATA TO GIVE THE

FIELD PVT PARAMETERS B

, R

AND B

EXERCISE 3.1 SOLUTION GAS DRIVE; UNDERSATURATED OIL RESERVOIR 79

EXERCISE 3.2 SOLUTION GAS DRIVE; BELOW BUBBLE POINT PRESSURE 80

EXERCISE 3.3 WATER INJECTION BELOW BUBBLE POINT PRESSURE 85

EXERCISE 3.4 GASCAP DRIVE 87

EXERCISE 4.1 UNITS CONVERSION 108

EXERCISE 6.1 WELLBORE DAMAGE 142

EXERCISE 7.1 ei-FUNCTION: LOGARITHMIC APPROXIMATION 154

EXERCISE 7.2 PRESSURE DRAWDOWN TESTING 157

EXERCISE 7.3 DIMENSIONLESS VARIABLES 162

EXERCISE 7.4 TRANSITION FROM TRANSIENT TO SEMI-STEADY STATE FLOW 166

EXERCISE 7.5 GENERATION OF DIMENSIONLESS PRESSURE FUNCTIONS 184

EXERCISE 7.6 HORNER PRESSURE BUILDUP ANALYSIS, INFINITE RESERVOIR

CASE 200

EXERCISE 7.7 PRESSURE BUILDUP TEST ANALYSIS: BOUNDED DRAINAGE

VOLUME 202

EXERCISE 7.8 MULTI-RATE FLOW TEST ANALYSIS 211

EXERCISE 7.9 AFTERFLOW ANALYSIS TECHNIQUES 231

EXERCISE 8.1 MULTI-RATE GAS WELL TEST ANALYSED ASSUMING STABILIZED

FLOW CONDITIONS 265

EXERCISE 8.2 MULTI-RATE GAS WELL TEST ANALYSED ASSUMING UNSTABILIZED

FLOW CONDITIONS 272

EXERCISE 8.3 PRESSURE BUILDUP ANALYSIS 282

EXERCISE 9.1 APPLICATION OF THE CONSTANT TERMINAL PRESSURE SOLUTION307

EXERCISE 9.2 AQUIFER FITTING USING THE UNSTEADY STATE THEORY OF HURST

AND VAN EVERDINGEN 310

CONTENTS XVI

EXERCISE 9.3 WATER INFLUX CALCULATIONS USING THE METHOD OF

FETKOVITCH 324

EXERCISE 10.1 FRACTIONAL FLOW 358

EXERCISE 10.2 OIL RECOVERY PREDICTION FOR A WATERFLOOD 361

EXERCISE 10.3 DISPLACEMENT UNDER SEGREGATED FLOW CONDITIONS 375

EXERCISE 10.4 GENERATION OF AVERAGED RELATIVE PERMEABILITY CURVES

FOR A LAYERED RESERVOIR (SEGREGATED FLOW) 397

LIST OF FIGURES

Fig. 1.1 (a) Structural contour map of the top of the reservoir, and (b) cross section

through the reservoir, along the line X−Y2

Fig. 1.2 Overburden and hydrostatic pressure regimes (FP = fluid pressure; GP = grain

pressure) 3

Fig. 1.3 Pressure regimes in the oil and gas for a typical hydrocarbon accumulation 6

Fig. 1.4 Illustrating the uncertainty in estimating the possible extent of an oil column,

resulting from well testing in the gas cap 8

Fig. 1.5 Primary oil recovery resulting from oil, water and gas expansion 11

Fig. 1.6 The Z−factor correlation chart of Standing and Katz

(Reproduced by courtesy of

the SPE of the AIME) 17

Fig. 1.7 Pseudo critical properties of miscellaneous natural gases and condensate well

fluids

Fig. 1.8 Isothermal Z−factor as a function of pressure (gas gravity = 0.85;

temperature = 200° F) 21

Fig. 1.9 Isothermal gas compressibility as a function of pressure (gas gravity = 0.85;

temperature = 200° F) 24

Fig. 1.10 Graphical representations of the material balance for a volumetric depletion gas

reservoir; equ. (1.35) 28

Fig. 1.11 Graphical representation of the material balance equation for a water drive gas

reservoir, for various aquifer strengths; equ. (1.41) 30

Fig. 1.12 Determination of the GIIP in a water drive gas reservoir. The curved, dashed

lines result from the choice of an incorrect, time dependent aquifer model; (refer

Chapter 9) 31

Fig. 1.13 Gas field development rate−time schedule (Exercise 1.2) 35

Fig. 1.14 Phase diagrams for (a) pure ethane; (b) pure heptane and (c) for a 50−50 mixture

of the two 37

Fig. 1.15 Schematic, multi-component, hydrocarbon phase diagrams; (a) for a natural gas;

(b) for oil 38

Fig. 2.1 Production of reservoir hydrocarbons (a) above bubble point pressure, (b) below

bubble point pressure 44

Fig. 2.2 Application of PVT parameters to relate surface to reservoir hydrocarbon

volumes; above bubble point pressure. 45

CONTENTS XVIII

Fig. 2.3 Application of PVT parameters to relate surface to reservoir hydrocarbon

volumes; below bubble point pressure 47

Fig. 2.4 Producing gas oil ratio as a function of the average reservoir pressure for a

typical solution gas drive reservoir 47

Fig. 2.5 PVT parameters (B

, R

and B

), as functions of pressure, for the analysis

presented in table 2.4; (p

= 3330 psia). 49

Fig. 2.6 Subsurface collection of PVT sample 52

Fig. 2.7 Collection of a PVT sample by surface recombination 54

Fig. 2.8 Schematic of PV cell and associated equipment 56

Fig. 2.9 Illustrating the difference between (a) flash expansion, and (b) differential

liberation 56

Fig. 3.1 Volume changes in the reservoir associated with a finite pressure drop ∆p; (a)

volumes at initial pressure, (b) at the reduced pressure 72

Fig. 3.2 Solution gas drive reservoir; (a) above the bubble point pressure; liquid oil, (b)

below bubble point; oil plus liberated solution gas 78

Fig. 3.3 Oil recovery, at 900 psia abandonment pressure (% STOIIP), as a function of the

cumulative GOR, R

(Exercise 3.2) 82

Fig. 3.4 Schematic of the production history of a solution gas drive reservoir 84

Fig. 3.5 Illustrating two ways in which the primary recovery can be enhanced; by downdip

water injection and updip injection of the separated solution gas 85

Fig. 3.6 Typical gas drive reservoir 87

Fig. 3.7 (a) Graphical method of interpretation of the material balance equation to

determine the size of the gascap (Havlena and Odeh) 88

Fig. 3.7 (b) and (c); alternative graphical methods for determining m and N (according to

the technique of Havlena and Odeh) 90

Fig. 3.8 Schematic of the production history of a typical gascap drive reservoir 91

Fig. 3.9 Trial and error method of determining the correct aquifer model (Havlena and

Odeh) 93

Fig. 3.10 Schematic of the production history of an undersaturated oil reservoir under

strong natural water drive 95

Fig. 3.11 (a) Triaxial compaction cell (Teeuw); (b) typical compaction curve 95

Fig. 3.12 Compaction curve illustrating the effect of the geological history of the reservoir

on the value of the in-situ compressibility (after Merle) 97

Fig. 4.1 Schematic of Darcy's experimental equipment 101

CONTENTS XIX

Fig. 4.2 Orientation of Darcy's apparatus with respect to the Earth's gravitational field 102

Fig. 4.3 Referring reservoir pressures to a datum level in the reservoir, as datum

pressures (absolute units) 112

Fig. 4.4 The radial flow of oil into a well under steady state flow conditions 113

Fig. 4.5 Radial pressure profile for a damaged well 114

Fig. 4.6 (a) Typical oil and water viscosities as functions of temperature, and (b) pressure

profile within the drainage radius of a steam soaked well 116

Fig. 4.7 Oil production rate as a function of time during a multi-cycle steam soak 116

Fig. 4.8 (a) Effective and (b) corresponding relative permeabilities, as functions of the

water saturation. The curves are appropriate for the description of the

simultaneous flow of oil and water through a porous medium 118

Fig. 4.9 Alternative manner of normalising the effective permeabilities to give relative

permeability curves 119

Fig. 4.10 Water saturation distribution as a function of distance between injection and

production wells for (a) ideal or piston-like displacement and (b) non-ideal

displacement 121

Fig. 4.11 Illustrating two methods of mobilising the residual oil remaining after a

conventional waterflood 124

Fig. 5.1 Radial flow of a single phase fluid in the vicinity of a producing well. 128

Fig. 5.2 Radial flow under semi-steady state conditions 130

Fig. 5.3 Reservoir depletion under semi-steady state conditions. 132

Fig. 5.4 Radial flow under steady state conditions 132

Fig. 6.1 Pressure distribution and geometry appropriate for the solution of the radial

diffusivity equation under semi-state conditions 136

Fig. 6.2 Pressure profile during the steam soak production phase 140

Fig. 6.3 Pressure profiles and geometry (Exercise 6.1) 142

Fig. 6.4 Dietz shape factors for various geometries

(Reproduced by courtesy of the SPE

of the AIME). 146

Fig. 7.1 Constant terminal rate solution; (a) constant production rate (b) resulting decline

in the bottom hole flowing pressure 148

Fig. 7.2 The exponential integral function ei(x) 153

Fig. 7.3 Graph of the ei-function for 0.001 ≤ × ≤ 5.0 154

CONTENTS XX

Fig. 7.4 Single rate drawdown test; (a) wellbore flowing pressure decline during the early

transient flow period, (b) during the subsequent semi-steady state decline

(Exercise 7.2) 159

Fig. 7.5 Dimensionless pressure as a function of dimensionless flowing time for a well

situated at the centre of a square (Exercise 7.4) 168

Fig. 7.6 Production history of a well showing both rate and bottom hole flowing pressure

as functions of time 169

Fig. 7.7 Pressure buildup test; (a) rate, (b) wellbore pressure response 171

Fig. 7.8 Multi-rate flow test analysis 172

Fig. 7.9 Horner pressure buildup plot for a well draining a bounded reservoir, or part of a

reservoir surrounded by a no-flow boundary 175

Fig. 7.10 Part of the infinite network of image wells required to simulate the no-flow

condition across the boundary of a 2 : 1 rectangular part of a reservoir in which

the real well is centrally located 176

Fig. 7.11 MBH plots for a well at the centre of a regular shaped drainage area

(Reproduced by courtesy of the SPE of the AIME) 178

Fig. 7.12 MBH plots for a well situated within; a) a square, and b) a 2:1 rectangle

(Reproduced by courtesy of the SPE of the AIME) 179

Fig. 7.13 MBH plots for a well situated within; a) a 4:1 rectangle, b) various rectangular

geometries

(Reproduced by courtesy of the SPE of the AIME) 180

Fig. 7.14 MBH plots for a well in a square and in rectangular 2:1 geometries

181

Fig. 7.15 MBH plots for a well in a 2:1 rectangle and in an equilateral triangle

(Reproduced by courtesy of the SPE of the AIME). 181

Fig. 7.16 Geometrical configurations with Dietz shape factors in the range, 4.5-5.5 184

Fig. 7.17 Plots of ∆p

(calculated minus observed) wellbore flowing pressure as a function

of the flowing time, for various geometrical configurations (Exercise 7.5) 186

Fig. 7.18 Typical Horner pressure buildup plot 190

Fig. 7.19 Illustrating the dependence of the shape of the buildup on the value of the total

production time prior to the survey 192

Fig. 7.20 Analysis of a single set of buildup data using three different values of the flowing

time to draw the Horner plot. A - actual flowing time; B - effective flowing time; C -

time required to reach semi-steady state conditions 194

Fig. 7.21 The Dietz method applied to determine both the average pressure

p and the

dynamic grid block pressure

p 197