Dake L.P. Fundamentals of reservoir engineering

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CONTENTS XXI

Fig. 7.22 Numerical simulation model showing the physical no-flow boundary drained by

well A and the superimposed square grid blocks used in the simulation 198

Fig. 7.23 Horner buildup plot, infinite reservoir case 201

Fig. 7.24 Position of the well with respect to its no-flow boundary; exercise 7.7 203

Fig. 7.25 Pressure buildup analysis to determine the average pressure within the no-flow

boundary, and the dynamic grid block pressure (Exercise 7.7) 204

Fig. 7.26 Influence of the shape of the drainage area and degree of well asymmetry on the

Horner buildup plot (Exercise 7.7) 205

Fig. 7.27 Multi-rate oilwell test (a) increasing rate sequence (b) wellbore pressure

response 210

Fig. 7.28 Illustrating the dependence of multi-rate analysis on the shape of the drainage

area and the degree of well asymmetry. (Exercise 7.8) 214

Fig. 7.30 Multi-rate test analysis in a partially depleted reservoir 219

Fig. 7.31 Examples of partial well completion showing; (a) well only partially penetrating

the formation; (b) well producing from only the central portion of the formation; (c)

well with 5 intervals open to production (After Brons and Marting

) 220

Fig. 7.32 Pseudo skin factor S

as a function of b and h/r

(After Brons and Marting

)

(Reproduced by courtesy of the SPE of the AIME) 220

Fig. 7.33 (a) Amerada pressure gauge; (b) Amerada chart for a typical pressure buildup

survey in a producing well 222

Fig. 7.34 Lowering the Amerada into the hole against the flowing well stream 223

Fig. 7.35 Correction of measured pressures to datum; (a) well position in the reservoir, (b)

well completion design 223

Fig. 7.36 Extreme fluid distributions in the well; (a) with water entry and no rise in the

tubing head pressure, (b) without water entry and with a rise in the THP 224

Fig. 7.37 Pressure buildup plot dominated by afterflow 225

Fig. 7.38 Russell plot for analysing the effects of afterflow 226

Fig. 7.39 (a) Pressure buildup plot on transparent paper for overlay on (b) McKinley type

curves, derived by computer solution of the complex afterflow problem 227

Fig. 7.40 McKinley type curves for 1 min <∆t < 1000 min. (After McKinley

) (Reproduced

by courtesy of the SPE of the AIME) 228

Fig. 7.41 Buildup plot superimposed on a particular McKinley type curve for T/F = 5000 230

Fig. 7.42 Deviation of observed buildup from a McKinley type curve, indicating the

presence of skin 230

CONTENTS XXII

Fig. 7.43 Russell afterflow analysis Exercise 7.9) 234

Fig. 7.44 Match between McKinley type curves and superimposed observed buildup

(Exercise 7.9). o − match for small ∆t (T/F = 2500)

• − match for large ∆t (T/F =

5000) 235

Fig. 8.1 Radial numerical simulation model for real gas inflow 241

Fig. 8.2 Iterative calculation of p

using the p

formulation of the radial, semi-steady

state inflow equation, (8.5) 244

Fig. 8.3 Real gas pseudo pressure, as a function of the actual pressure, as derived in

table 8.1; (Gas gravity, 0.85; Temperature 200°F) 248

Fig. 8.4 2p/µZ as a function of pressure 248

Fig. 8.5 Calculation of p

using the radial semi-steady state inflow equation expressed in

terms of real gas pseudo pressures, (equ. 8.15) 250

Fig. 8.6 p/µZ as a linear function of pressure 251

Fig. 8.7 Typical plot of p/µZ as a function of pressure 252

Fig. 8.8 Laboratory determined relationship between

and the absolute permeability A

relationship is usually derived of the form 256

Fig. 8.9 Gas well test analysis assuming semi-steady state conditions during each flow

period. Data; table 8.3 267

Fig. 8.10 Gas well test analysis assuming semi-steady state conditions and applying equ.

(8.47). Data; table 8.4 268

Fig. 8.11 MBH chart for the indicated 4:1 rectangular geometry for t

< .01 (After

Earlougher, et.al

) 273

Fig. 8.12 Essis-Thomas analysis of a multi-rate gas well test under assumed transient flow

conditions. Data; table 8.6 274

Fig. 8.13 The effect of the length of the individual flow periods on the analysis of a multi-

rate gas well test; (a) 4×1 hr periods, (b) 4×2 hrs, (c) 4×4 hrs 277

Fig. 8.14 (a) Rate-time schedule, and (b) corresponding wellbore pressure response

during a pressure buildup test in a gas well 279

Fig. 8.15 Complete analysis of a pressure buildup test in a gas well: (a) buildup analysis

(table 8.12); (b) and (c) transient flow analyses of the first and second flow

periods, respectively (table 8.10) 284

Fig. 8.16 MBH plot for a well at the centre of a square, showing the deviation of m

D(MBH)

from p

D(MBH)

for large values of the dimensionless flowing time t

287

Fig. 8.17 Iterative determination ofp in a gas well test analysis (Kazemi

) 288

CONTENTS XXIII

Fig. 8.18 Schematic of a general analysis program applicable to pressure buildup tests for

any fluid system 294

Fig. 9.1 Radial aquifer geometry 300

Fig. 9.2 Linear aquifer geometry 300

Fig. 9.3 Dimensioniess water influx, constant terminal pressure case, radial flow. (After

Hurst and van Everdingen, ref. 1) 302

Fig. 9.4 Dimensionless water influx, constant terminal pressure case, radial flow (After

Hurst and van Everdingen, ref. 1) 303

Fig. 9.5 Dimensionless water influx, constant terminal pressure case, radial flow (After

Hurst and van Everdingen, ref. 1) 304

Fig. 9.6 Dimensionless water influx, constant terminal pressure case, radial and linear

flow (After Hurst and van Everdingen, ref.1) 305

Fig. 9.7 Dimensionless water influx, constant terminal pressure case, radial and linear

flow (After Hurst and van Everdingen, ref.1) 306

Fig. 9.8 Water influx from a segment of a radial aquifer 307

Fig. 9.9 Matching a continuous pressure decline at the reservoir-aquifer boundary by a

series of discrete pressure steps 309

Fig. 9.10 Aquifer-reservoir geometry, exercise 9.2 311

Fig. 9.11 Reservoir production and pressure history; exercise 9.2 311

Fig. 9.12 R

and B

as functions of pressure; exercise 9.2 312

Fig. 9.13 B

as a function of pressure; exercise 9.2 312

Fig. 9.14 Reservoir pressure decline approximated by a series of discrete pressure steps;

exercise 9.2 313

Fig. 9.15 Aquifer fitting using the interpretation technique of Havlena and Odeh 319

Fig. 9.16 Comparison between Hurst and van Everdingen and Fetkovitch for r

= 5 327

Fig. 9.17 Comparison between Hurst and van Everdingen and Fetkovitch for r

= 10 327

Fig. 9.18 Predicting the pressure decline in a water drive gas reservoir 329

Fig. 9.19 Prediction of gas reservoir pressures resulting from fluid withdrawal and water

influx (Hurst and van Everdingen) 330

Fig. 9.20 Prediction of gas reservoir pressures resulting from fluid withdrawal and water

influx (Fetkovitch) 332

Fig. 9.21 Conditions prior to production in a steam soak cycle 333

CONTENTS XXIV

Fig. 10.1 Hysteresis in contact angle in a water wet reservoir, (a) wetting phase increasing

(imbibition); (b) wetting phase decreasing (drainage) 338

Fig. 10.2 Water entrapment between two spherical sand grains in a water wet reservoir 339

Fig. 10.3 Drainage and imbibition capillary pressure functions 339

Fig. 10.4 Capillary tube experiment for an oil-water system 340

Fig. 10.5 Determination of water saturation as a function of reservoir thickness above the

maximum water saturation plane, S

= 1−S

, of an advancing waterflood 341

Fig. 10.6 Linear prototype reservoir model, (a) plan view; (b) cross section 344

Fig. 10.7 Approximation to the diffuse flow condition for H>> h 346

Fig. 10.8 (a) Capillary pressure function and; (b) water saturation distribution as a function

of distance in the displacement path 348

Fig. 10.9 Typical fractional flow curve as a function of water saturation, equ. (10.12) 349

Fig. 10.10 Mass flow rate of water through a linear volume element Adx

350

Fig. 10.11 (a) Saturation derivative of a typical fractional flow curve and (b) resulting water

saturation distribution in the displacement path 352

Fig. 10.12 Water saturation distribution as a function of distance, prior to breakthrough in

the producing well 353

Fig. 10.13 Tangent to the fractional flow curve from S

= S

354

Fig. 10.14 Water saturation distributions at breakthrough and subsequently in a linear

waterflood 355

Fig. 10.15 Application of the Welge graphical technique to determine the oil recovery after

water breakthrough 357

Fig. 10.16 Fractional flow plots for different oil-water viscosity ratios (table 10.2) 360

Fig. 10.17 Dimensionless oil recovery (PV) as a function of dimensionless water injected

(PV), and time (exercise 10.2) 364

Fig. 10.18 Displacement of oil by water under segregated flow conditions 365

Fig. 10.19 Illustrating the difference between stable and unstable displacement, under

segregated flow conditions, in a dipping reservoir; (a) stable: G > M−1; M > 1;

. (b) stable: G > M−1; M < 1;

. (c) unstable: G < M−1. 366

Fig. 10.20 Segregated displacement of oil by water 369

Fig. 10.21 Linear, averaged relative permeability functions for describing segregated flow in

a homogeneous reservoir 370

Fig. 10.22 Typical fractional flow curve for oil displacement under segregated conditions 371

CONTENTS XXV

Fig. 10.23 Referring oil and water phase pressures at the interface to the centre line in the

reservoir. (Unstable segregated displacement in a horizontal, homogeneous

reservoir) 372

Fig. 10.24 Comparison of the oil recoveries obtained in exercises 10.2 and 10.3 for

assumed diffuse and segregated flow, respectively 376

Fig. 10.25 The stable, segregated displacement of oil by water at 90% of the critical rate

(exercise 10.3) 378

Fig. 10.26 Segregated downdip displacement of oil by gas at constant pressure; (a)

unstable, (b) stable 380

Fig. 10.27 (a) Imbibition capillary pressure curve, and (b) laboratory measured relative

permeabilities (rock curves,- table 10.1) 382

Fig. 10.28 (a) Water saturation, and (b) relative permeability distributions, with respect to

thickness when the saturation at the base of the reservoir is S

= 1 − S

=0)383

Fig. 10.29 Water saturation and relative permeability distributions, as functions of thickness,

as the maximum saturation, S

= 1 − S

, is allowed to rise in 10 foot increments

in the reservoir 385

Fig. 10.30 Averaged relative permeability curves for a homogeneous reservoir, for diffuse

and segregated flow; together with the intermediate case when the capillary

transition zone is comparable to the reservoir thickness 387

Fig. 10.31 Capillary and pseudo capillary pressure curves. 387

Fig. 10.32 Comparison of oil recoveries for different assumed water saturation distributions

during displacement. 387

Fig. 10.33 Variation in the pseudo capillary pressure between +2 and -2 psi as the

maximum water saturation S

= 1−S

rises from the base to the top of the

reservoir 388

Fig. 10.34 Example of a stratified, linear reservoir for which pressure communication

between the layers is assumed 390

Fig. 10.35 (a)-(c) Rock relative permeabilities, and (d) laboratory measured capillary

pressures for the three layered reservoir shown in fig. 10.34 391

Fig. 10.36 (a) Water saturation, and (b) relative permeability distributions, with respect to

thickness, when the saturation at the base of the layered reservoir (fig. 10.34) is

= 1−S

° = 2 psi) 392

Fig. 10.37 (a)-(h) Water saturation and relative permeability distributions,as functions of

thickness,for various selected values of

(three layered reservoir, fig 10.34) 394

Fig. 10.38 Averaged relative permeability functions for the three layered reservoir,

fig. 10.34: (a) high permeability layer at top, (b) at base of the reservoir 396

CONTENTS XXVI

Fig. 10.39 (a) Pseudo capillary pressure, and (b) fractional flow curves for the three layered

reservoir, fig. 10.34). (——high permeability at top; − − −at base of the reservoir).396

Fig. 10.40 Individual layer properties; exercise 10.4 397

Fig. 10.41 Averaged relative permeability curves; exercise 10.4 399

Fig. 10.42 (a) Pseudo capillary pressures, and (b) fractional flow curves, exercise 10.4 (——

High permeability layer at top; − − −at base of reservoir) 400

Fig. 10.43 Methods of generating averaged relative permeabilities, as functions of the

thickness averaged water saturation, dependent on the homogeneity of the

reservoir and the magnitude of capillary transition zone (H). The chart is only

applicable when the vertical equilibrium condition pertains or when there is a total

lack of vertical equilibrium 404

Fig. 10.44 Numerical simulation model for linear displacement in a homogeneous reservoir406

Fig. 10.45 Spatial linkage of the finite difference formulation of the left hand side of

equ (10.83). 408

Fig. 10.46 Example of water saturation instability (oscillation) resulting from the application

of the

IMPES solution technique (− − − correct, and  incorrect saturations) 411

Fig. 10.47 Determination of the average, absolute permeability between grid blocks of

unequal size 415

Fig. 10.48 Overshoot in relative permeability during piston-like displacement 416

Fig. 10.49 Alternative linear cross sectional models required to confirm the existence of

vertical equilibrium 418

LIST OF TABLES

TABLE 1.1 Physical constants of the common constituents of hydrocarbon gases

and a typical gas composition 15

TABLE 2.1 Results of isothermal flash expansion at 200°F 57

TABLE 2.2 Results of isothermal differential liberation at 200º F 58

TABLE 2.3 Separator flash expansion experiments performed on the oil sample whose

properties are listed in tables 2.1 and 2.2 62

TABLE 2.4 Field PVT parameters adjusted for single stage, surface separation at 150

psia and 80°F;

c = .7993 (Data for pressures above 3330 psi are taken

from the flash experiment, table 2.1) 64

TABLE 2.5 Differential PVT parameters as conventionally presented by laboratories, in

which B

and R

are measured relative to the residual oil volume at 60°F 67

TABLE 3.1 88

TABLE 3.2 89

TABLE 3.3 89

TABLE 4.1 Absolute and hybrid systems of units used in Petroleum Engineering 106

TABLE 6.1 Radial inflow equations for stabilized flow conditions 139

TABLE 7.1 157

TABLE 7.2 167

TABLE 7.3 167

TABLE 7.4 187

TABLE 7.5 200

TABLE 7.6 201

TABLE 7.7 203

TABLE 7.8 207

TABLE 7.9 208

TABLE 7.10 211

TABLE 7.11 213

TABLE 7.12 213

TABLE 7.13 213

CONTENTS XXVIII

TABLE 7.14 214

TABLE 7.15 218

TABLE 7.16 219

TABLE 7.17 232

TABLE 7.18 233

TABLE 8.1 Generation of the real gas pseudo pressure, as a function of the actual

pressure; (Gas gravity, 0.85, temperature 200°F) 245

TABLE 8.2 265

TABLE 8.3 266

TABLE 8.4 269

TABLE 8.5 272

TABLE 8.6 274

TABLE 8.7 276

TABLE 8.8 276

TABLE 8.9 276

TABLE 8.10 282

TABLE 8.11 282

TABLE 8.12 283

TABLE 8.13 285

TABLE 8.14 292

TABLE 9.1 308

TABLE 9.2 314

TABLE 9.3 314

TABLE 9.4 316

TABLE 9.5 316

TABLE 9.6 317

TABLE 9.7 318

TABLE 9.8 322

TABLE 9.9 326

CONTENTS XXIX

TABLE 9.10 328

TABLE 10.1 358

TABLE 10.2 359

TABLE 10.3 359

TABLE 10.3(a) Values of the shock front and end point relative permeabilities calculated

using the data of exercise 10.1 361

TABLE 10.4 363

TABLE 10.5 363

TABLE 10.6 376

TABLE 10.7 377

TABLE 10.8 379

TABLE 10.9 Water saturation and point relative permeability distributions as functions of

the reservoir thickness; fig.10.28(a) and (b). 384

TABLE 10.10 Thickness averaged saturations, relative permeabilities and pseudo

capillary pressures corresponding to figs. 10.28 and 10.29 386

TABLE 10.11 Phase pressure difference, water saturation and relative permeability

distributions for

= 2 psi; fig. 10.36 392

TABLE 10.12 Pseudo capillary pressure and averaged relative permeabilities

corresponding to figs. 10.36 and 10.37 395

TABLE 10.13 398

TABLE 10.14 399

LIST OF EQUATIONS

()()

OIP V 1 S res.vol.

=− (1.1) 1

()

wc oi

STOIIP n v 1 S / B

== − (1.2) 2

OP FP GP=+ (1.3) 3

() ()

dFP dGP=− (1.4) 3

water

pD14.7(psia)

æö

×+

ç÷

èø

(1.5) 4

water

p D 14.7 C (psia)

æö

=×++

ç÷

èø

(1.6) 4

p0.45D15(psia)=+ (1.7) 5

()

p0.35D565 psia=+ (1.8) 6

()

p 0.08D 1969 psia=+ (1.9) 6

wc oi

Ultimate Recovery (UR) ( V (1 S ) / B ) RF

=− × (1.10) 9

∂

—

(1.11) 10

dV cV p=∆ (1.12) 10

pV nRT= (1.13) 12

()

(p ) V b RT

+−=

(1.14) 12

pV ZnRT= (1.15) 13

pc i ci

pnp=

(1.16) 14

pc i ci

TnT=

(1.17) 14

(1.18) 14

(1.19) 14

1.2(1 t)

0.06125p t e

−−

(1.20) 19