numerical simulation of steam displacement field performance applications.pdf

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Numerical Simulation of Steam Displacement —

Field Performance Applications

C.

Chu,

SPE-AIME, Geuy Oil Co.

A. E. Trirnble, SPE-AIME, Getty Oil Co.

Introduction

The three-dimensional, three-phme numerical model for

steam displacement, described by Coats et al . , 2 was

previously tested with three sets of laboratory experimen-

tal data and was used in various applications, including a

representative field-scale problem and a steam-

stimulation example. This paper concerns the same

model and consists of two parts. The fust part is a further

validation of the numerical model by history matching

the performance of a single pattern in the Kern “A”

project in Kern River field, Calif. (Fig. 1). Related

studies are included on the simulation of steam stimula-

tion, the effects of grid orientation, upstream weighting

of viscosities, and the temperature dependence of relative

permeabilities. In the second part, the model is used to

optimize an objective function for the cases of constant

and varying steam rates. While the constant-steam-rate

cases maintained the same rates for the entire project life,

the varying-steam cases used decreasing sequences of

steam rates, with each maintained for a prespecified

length of time.

Performance of a Single Patkrn

Field Pattern Description

A

five-spot steam-displacement pattern was selected in

the Kern “A” project (Fig. 2) for histo~matching pur-

poses. The basis for this selection was primarily the

following criteria: (1) the field operation typifies Kern

River, (2) the availability of good reservoir data, (3) the

newly complete cycle life of the displacement zone, and

(4) near symmetry of the pattern. The Pattsm seleeted is

about 430 ft in the east-west direetion and 270 ft in the

north-south direction, and covers an area of 2,7 acres,

East-west and north-south cross-sections through this

pattern are gi (en in Fig. 3. The displacement sand shown

in the cross-seetions is locally referred to as the “RI”

interval. Note the existence of a tight streak in a part of

this interval. Core-hole data are available from Weli 503,

which was cored before displacement of the RI zone, and

Well C, H. 1, which was cored near the depletion of

displacement in the R, sand. The patterns including

Wells 503 and C. H. 1were not chosen because the wells

are not as symmetrically spaced as those in the pattern

around Well 68.

Data from Well 503, shown in Table 1, were used for

the vertical distribution of permeability and saturation.

The permeability of Layer 2 inTable 1was assumed to be

only 1 percent of that of Layer 1, which reflects the

existence of the tight streak shown in Fig. 3. The fluid

volumes, as indicated by core analysis, were assumed to

completely fill the available pore space, and normalized

saturations were calculated. This method tends to restore

the core data of the unconsolidated sand to reservoir

conditions, as suggested by Elk ins, 4 Calculations were

also made to correet core porosities to vahes closer to

reservoir conditions.

Additional Fluid and Reek Data

This type of reservoir simulation requires a wide range of

input data. Where possible, field data were used to sup-

plement data obtained from the literature. In Table 2, Set

?

A three-dimensional three-phase numerical steam-displacement model was used to history

match 5 years offield aktafiom a representativefive-spot pattern inKernRiver Calif. The

model was usedjimther to effect optimization of steam-injection ratesfor typicaljive-spot

patterns.

JUNE, 1975

765


2/12

.

w temperature viscosity data used in the history

match. These data were obtained from produced oil in

Kern WelI 68, the model pattern producer, Table 3 shows

the relative-permeability data used to obtain the final

history match. Note that temperature dependence is

shown in the table. Previous simulations of experimental

data led to the conclusion that the temperature depen-

dence of relative permeabilities has a considerable effect

on calculated results, Studies in this project showed that

an increase in oil relative permeabilityy and a decrease in

water relative permeability at an elevated temperature

resulted in better representation of the field performance

inthe Kern” A” Well 68 pattern. Subsequent laboratory

measurements of relative permeability for temperatures

up to 400”F have shown that these effects are indeed

representative of the behavior of Kern River core mate-

rial. Similar effects on consolidated rocks were also re-

ported by Weinbrandt and Ramey. 7The remaining basic

data used in the history match are shown in Table 4. The

injection wells are completed inthe lower one-third of the

in:srval, whereas the production well is completed over

the entire interval.

Required Numerieal Techniques

Before the history match, several problems were encour -

tered. One of these problems was the excessive pressure

buildup upon steam injection into the reservoir. There are

at least three methods that can be used to cope with this

problem. The first method involves the use of infinite

boundaries. An obvious disadvantage of this method is

that it requires a very large number of grid blocks and,

therefore, an excessive expenditure of computer time.

Reduction of :he number of grid blocks in the x and y

directions leads to enormous pressures, The second

method is to use fictitious wells on the boundary of a

confined region to remove the fluids pushed out by the

injected steam. The disadvantage of this methoc is that

the fluids being produced in the fictitious wells are perm-

anently lost. These two methods gave unsatisfactory

results in several trials.

The third method involves the concept of “spongy

rock. ” It is assumed that voids, or gas saturations, ex-

isted in the reservoir before the start of the steam dis-

placement process, either because the original formation

contained voids or because they were created during

primary operation. The present steam model cannot ac-

count for any gas other than steam. To simulate the

cushioning effect of the existing gas saturation, the com-

pressibility of the rock was assumed to be a composite of

the compressibilities of rock and gas. The following

equation was used and can be readily derived, assuming

ideal gas behavior:

(1–s s)cH + ; s@J

C* =

(1)

l–++sg+ ‘“”’”””””””””””’

where

C, = compressibility of the spongy rock

(composite of rock and nonsteam gas)

CR = cornpressibilit y of the rock

~ = pxosity

So = initial saturation of the nonsteam gas

P =

absolute pressure.

In the spongy-rock method, a confined region can be used

that allows the fluid to be released when the pressure is

lowered during the production stage. Many computer

runs using this method have given satisfactory results.

The latest version of the model (Coast3), with the added

--1---

I

~

ymg} COMPANY

- KERN ‘s’PROJECT

~ W“ ,. ,,0,,,,S

7

—.

28S. 2f3E

?L

2,

J

-ly

I

s

;Z

/

29 S.-2SE.

/

Fig.

l—Map of

Kern

River field.

~-~ 5-SPOTPATTEP.NFlNT’WST

~ ,WLLSCOREOHROUGHI SANO

AoA’ ,BsS GEOLOGICROSS-SECTIONSSEEFIG. 3)

Fi g. 2—lsopech of net product ive R, oil send — Kern “ A” pil ot

area

JOURNAL OF PETROLEUM TECHNOLOGY


3/12

—.

TABLE 1—VERTICAL DISTRIBUTION OFPERMEABILITIES ANO

SATURATIONS

Layer Thickness (ft) k (red)

Sw

1

12

5,500

0.350

2

0.350

3 2; 1,9% 0.630

4 21

1,359

0.650

5

9 1,408

0,670

capabllit y to handle gas saturation, obviates this proce-

dure. On the other hand, the rock compressibilities calcu-

lated based on the spongy-rock concept are within the

range af the ex~rimentally determined rock compres-

sibilities of unconsolidated sands reported by Sawabini,

et al . 6

A second problem was the effect of grid orhmtation on

calculation results of the steam displacement process in

five-spot patterns, as noted previously.z The parallel grid

that favors flow of steam in a direction along the

streamline joining the injector and the producer gave

more satisfactory representation of laboratory experi-

mental data than did the diagonal grid. This effect was

noted again in this study.

To obtain the correct direction of fluid flow, a quadrant

of the pattern with dimensions equal 10the averages ofthe

dimensions of the four quadrants was used. In this way,

the steam front moves directly from tne injector toward

the producer. This is shown in the isotherms in Fig. 4. An

advantage of using a qua&ant instead of the entire pattern

isthat the number of grid blocks necessary for the simula-

tion can be substantially reduced.

A “+”

c.7A”* mk ,

B

.

TABLE2-41-VISCOSITY DATA

& (Cp)

Temperature (“F)

Set 1

Set

2

75.0 3,000.0

5,780.0

100.0 740.0 1,380.0

150.0

107.0 187.0

200.0 24.0

47.0

250.0 9.0

17.4

303.0

2.6 8.5

350.0

1.7

400.0

1.0

::;

500.0

1.5

600.0

0.8

TABLE3-WATER-OIL AND GAS-OIL RELATIVE-PERMEABILITY DATA

Water-Oi l Data

s

k w km.

__S__

at 7 YF

0.270

0.000

1.00

0.420

0.002

0.99

0.510

0,004

0.80

0.560

0.006

0,60

0.620

0.008 040

0,650

0.010 0.30

0.680

0.012 0.20

0.720

0.014

0.10

C.800

0.021

0.00

0.940

0.045

0.00

0.970

0.100 0.00

1.000 1.000 0.00

at 400”F

0.500’” 0.000 1.00

0.620 0.004 0.99

0,690 0.010

0.80

0.720

0.012 0.60

0.740 0.014

0.40

0,760

0,016

0.30

0.780

0.018 0.20

0.810 0.022 0.10

0.850

0.028

0.00

0.940 o.f345 0.00

0.970

0.100

0.00

1.000 1.000 0.00

Gas-Oil Data

Sw so

0.30

0.59

0.61

0.63

0.65

0.68

0.71

0.74

0.78

0.83

0.89

1.00

0.63

0.64

0.65

0.67

0,69

0.71

0.74

0.77

0.80

0.84

0.90

1.00

k

r9

krcw

at 75°F

0.51

0.000

0.50

0.005

0.45

0.010

0.40

0.020

0.35

0.030

0.30

0.040

0.25

0.060

0.20 0.080

0.15

0.130

0.10 0.190

0.05

0.300

0.00

1.000

at 400°F

0.51 0.000

0.50

0.005

0.45 0010

0.40

0.020

0.35

0.030

0.30 0.040

0,25

C.070

0.20

0.090

0.15 0.130

0.10 0.190

0.05

0.300

0.00

1.000

LJW,,mol/res bbl

b.,,

STB/res bbl

(2W,vol/vol-psi

CO,vol/vol-psi

Cfi, vol/vol-psi

Cm, vol/vol-°F

Cm, vol/vol-°F

CPW,Btu/lb-°F

Cm, Btu/lb-°F

4

KR ,

Btu/ft-D-°F

K ,

Btu/ft-D-°F

(P), i Btu/cu ft-” F

(@)@, Btu /cu f t -°F

1,,

“ F

PI

psia

Sfl

pimlPsi

pm

psi

Steam qual i ty

o.. ib/c u ft

1

Z.*

It-

,.-

TABLE4-BASIC OATA

1.00

1.00

0.000003

0.000005

0.000735

(0.000586)

(for spcmgy-rock concept)

0.00049

0.00039

1.00

0.50

0.345

38.4

38.4

35.0

35.0

95.0

50.0

14.5

0.7

60.3

(0.3)

(80.0)

r“ .–

II’

PCIC9

0.0

\

>

---

-w ?M?u

Pcm

0 0

Pig.3-Cross-sec ti on s p assing throu gh Wel l 68. Up per p or ti on :

No te: Mo st o f th e da te l is te d h ere are co mm on to bo th run s for hi st ory m at chi ng

east-west

d iract ion. Lower pot t ion: nor th-south d irect ion.

a nd o pt im ize ti n. I n c aa e the d at a d iffe r, t ho se p ert ai ni ng to op ti mi zat io n

runs are placed within parentheses

JUNE, 1975

767


4/12

.-

Fi g. Wlculat ed i sot hf irms us ing a Farallel gri d.

In

the original version of the numerical model, up-

stream weighting of the viscosity of steam was used,

along with ~hhetic averagng of viscosities of oil and

water. [Suppose a fluid moves from Block i

to

Block i +

1 and its viscosities in these blocks are pi and p.~+1,

respectively. Upstream weighting means using w for

calculating intertbek transmissibility, whereas arithme-

tic weighting means using % (M i- W+,) instead. ] In the

simulation of the laboratory data, the calculated ~ival of

the steam fron? was later than the actual arrival in the

experiments. A similar discrepancy was observed when a

history match of the five-spot pattern was attempted in

this study. Using upstream weighting for all phases re-

sults in faster propagation of the steam front and in

improved matching of the field performarice.

Field Pattern Op’ation

In Kern River, when displacement is planned in patterns

of a projeet, the producing wells are heavily stimulated

before and during the early stages of steam displacement.

If production starts to drop after displacement has started,

the well is stimulated again. This procedure has proved

successful in improving pattern sweep efllciency in this

low-pressure reservoir. It was necessary to consider this

stimulation history of Well 68 in the history match.

Well 68 was steam stimulated and subsequently pro-

duced before inception of the Kern “A” displacement

project. In Feb. 1968, it was stimulated for 6 days. In

March 1968, continuous steam injection started in Well

504 (Fig. 2). Wells 505,507, and 508 started injection in

TABLE6-STEAM-STIMULATION DATAOFWELLS8

Total Steam

Year

Month Injected (STB) Remarks

——

1968 Feb.

6,990

6 days of

1968

July 7,957

stimulation

fol lowed by

3 days of

soaking

the

following month. In July 1968, when oil-production

rate declined inWell 68, it was steam stimulated a second

time. Soon thereafter, production response to displace-

ment was realized and further stimulation was not neces-

sary. The steam-injection rates were averaged during

periods when they stayed fairly constant (Table 5). The

steam-stimulation data of Well 68 are given in Table 6,

whereas its oil- and water-production rates and cumula-

tive productions throughout the period between March

1968 and Aug. 1973 are presented on a monthly basis in

Table 7.

Pattern History Match

History matching was performed using the actual

steam-injection rates, together with availabie experimen-

tal reservoir rock and fluid data. Two ways of ascertain-

ing the initial oil and water saturations were investigated.

One way assumed that the initial oil saturations reported

in core analysis were correct and that initial water satura-

tions were just the balances. Another way assumed that,

since the total fluid saturations reported in core analysis

did not add up to unity, the saturations should be nor-

malized. The latter method was chosen Oinsure beuer

simulation, Water-oil and gas-oil relative-permeability

data were found to ha~e a profound effect on calculated

production. Adjustments were made on the relative-

permeability data to achieve better agreement between

calculated and actual oil and water product ions, The

results of history matching are presented in Figs. 5

through 7.

Fig. 5 compares the cumulative oil and water produc-

tion of Well 68 with that predicted by the model. The

calculated cumulative water production deviatez only

slightly from the field data, from the b:ginning of the

project up to the end of available data. The comparison

between calculated and actual cumulative oil productions

reveals more appreciable deviations exist ing in several

segments; nevertheless, the calculated ultimate oil pro-

duction at the end of Aug. 1973 matches the field value.

The comparison of oil- and water-production rates are

shown in Fig. 6. Several variances can be noted in the

comparisons of the oil-rate curves. The comparisons are

not as good as we would have liked, but the matching of

field thermal-production data is difficult. Part of this is

From

—

Year

Month

1968 March

1968 April

196 Aug.

1969 Jan.

1969 July

1970 July

1971 Jan.

TABLE5-AVERAGED INJECTIONRATESOF WELLS

(Well s 504, 505,507, and 508)

To Steam Rate (STB/D)

Year

Month

Well 504 Well 505

Well 507 Well 5=

——

—.

127 0

1968

July

214 25

251

25?

1968 Dec.

245 295

245

246

1969 June

307 307

307

310

1970 June

290 260

246

269

1970 Dec.

216 216

210

249

1973 Aug.

272

272

218

258

768



5/12

because of erratic production, despite fairly constant

operating conditions.

The match between calculated and field water rates in

Fig. 6 is considered quite good. The calculated and field

oil rates show a reasonable match through 1970; how-

ever, a decline began in 1971 that could not be matched

).000,000

-4...——

- L--L— —.:.-— ..- :. .+. ——;

Ir

,— ;- ...+._ 4..

.+ —- .—+ ——.’:

I

10

~

IN’, ooo --–:g

F

s

:

/ : :

‘ :-—-”

E

.

. . . . . .

... . . . . . . . . . . . . .

~

2

: 10.00 ?

5

u

i

“

“

‘ “ :

““-j -”-;---

.. -,. . . . . .

1

t

—

1,W?

i968 ,

:969 ,

;a-o

: 1971

1972

1973

L

1

Fig. 5--A h is to ry match — cumulat ive o il an d water p ro du ct io ns .

— field data

---- calculated

f

r

v

v

10

l:.

. .... . . :..- -..

.. .:

,.. ,“.

. . . . . . .

:1

,., -: ...

,:

,.:

1

1968

1969 i

I

1970,

197] ;

I

1972

‘ 1973

I

Fi g. 6 -A hi st or y mat ch

— o il an d water p ro du ct io n rates .

—.—

water rate, f iel d d ata

. . . . . . wat er r at e, c al cul at ed

o il r ate, f iel d d ata

–--– oil rate, calculated

TABLE7—PRODUCTION RATES AND CUMULATIVE PRODUCTION

OFWEU 68

Cumulat ive Cumulat ive

Oil Water

Oil

Water

Rate Rate

Product io rl Production

Date

Mai ch 1968

Apr i l 1968

May 1968

June 1968

July 1966

Aug. 1968

Sept. 1968

Ot t, 1968

NOV. 1968

Dec. 1968

Jan. 1969

Feb. 1969

Match 1969

Apr i l 1969

May 1969

June 1969

JUIY 1969

Aug. 1969

Sept. 1969

Oct. 1969

NOV. 1969

Dec. 1969

Jan. 1970

Feb. 1970

March 1970

Apr i l 1970

May 1970

June 1970

July 1970

Aug. 1970

Sept. 1970

Oct. 1970

Nov. 1970

Dec. 1970

Jan, 1971

Feb. 1971

March 1971

Apr i l 1971

May 1971

June 1971

July 1971

Aug. 1971

Sept. 1971

Oct. 1971

Nov. 1971

Dec. 1971

Jan. 1972

Feb. 1972

March 1972

April 1972

May 1972

June 1972

July 1972

Aug. 1972

Sept. 1972

Oct. 1972

NOV. 1972

NC. 1972

Jan. 1973

Feb. 1973

March 1973

Apr il 1973

May 1973

June 1973

July 1973

Aug. 1973

(ST8/D) (STB/D) (STB)

(STB)

3:

51

33

39

25

3

8

8

;:

114

161

193

166

163

118

90

110

112

104

185

122

68

111

93

106

72

40

62

129

123

88

84

78

40

42

33

34

;;

45

39

37

34

33

30

30

20

27

21

17

17

21

18

10

20

23

18

15

14

;:

16

12

16

17

18

132

194

279

100

167

2::

261

318

440

555

327

359

267

206

212

205

20 I

301

222

182

170

,

162

189

I 50

95

164

286

310

205

188

171

145

167

157

157

208

176

284

237

436

205

201

111

198

117

177

116

180

181

165

186

173

259

236

172

258

338

284

271

358

293

221

282

303

1,023

2,013

3,222

3,972

4,065

4,313

4,554

5,266

7,816

11,350

16,341

21,745

26,891

31,781

35,439

38,139

41,549

45,021

48,141

53,876

57,536

59,644

63,085

65,689

68,975

71,135

72,375

74,235

78,234

82,047

84,687

87,291

89,631

90,871

92,173

93,097

94,151

94,661

95,808

97,158

98,367

99,514

100,534

101,557

102,457

103,387

104,007

104,790

105,441

105,951

106,478

107,108

107,666

107,976

108,576

109,289

i09,829

110,294

110,728

111,260

111,942

112,422

112,794

113,274

113,801

114,359

1,581

5,541

11,555

19,925

23,025

28,202

30,212

36,815

44,645

54; 503

68,143

83,683

93,820

104,590

112,867

119,047

125,619

131,974

138,004

147,335

153,995

159,637

164,907

169,443

175,302

179,302

182,747

187,667

196,533

206,143

212,293

218,121

223,251

227,746

232,923

237,319

242,186

248,426

253,882

262,402

269,749

283,265

289,415

295,646

298,976

305,114

308,741

313,874

317,470

322,870

328,481

333,431

339,197

344,560

352,330

359,646

364,806

372,804

383,282

391,234

399,635

410;375

419,458

426,088

434,830

444,223

JUNE, 1975

769


6/12

with the model. By the end of Aug. 1973, the calculated

oil rate was about 60 percent higher than the field ti.a.

The history match was not perfect; however, given

more adequate input data to describe the reservoir charac-

teristics and more time to make various reasonable ad-

justmen~, a perfect match would be readily accessible.

The main objective of this effort was to validate the steam

model; the reasonable history match obtained proved

the viability of the model, making efforts to pdrsue a

perfect match unnecessary. Furthermore, the discrep-

ancies in oil-production rate toward the latter part of

a project become less significant when the oil is dis-

counted based on the present-worth concept. Such dis-

crepancies would therefore produce insignificant effect

on the results of optimization studies such as the study

on optimal steam rates presented below.

Fig. 7 shows calculat&l temperature distributions at

various times of the steam-displacement operation. It is

interesting to note that, in spite of the intervening shale

break, steam reaches the top of the formation and propa-

gates along the ceiling toward the reducer, The same

phenomenon can be noticed in Fig. 8, where oil-

saturation distributions are presented at various times.

The S-shaped curve of the Oto 0.10 saturation line in the

cross-section from injector to producer at 3 and 5 years is

evidently caused by the impedence to oil flow by the tight

streak. The islands of increased cil saturations in the

.,

1

yr

bottom plan views demonstrate the possibility of a small

oil-bank formation,

Optimization of Steam Injection Rates

Steam injection in a displacement project is a major

operating cost, Any attempt at optimizing steam-

injection rates has potential impact on a project’s profit-

abilityy. The numerical steam model, as discussed in the

first part of the paper, gives a reasonable history match

to field production data, In this section, we show how

the model can be used to evaluate the important variable

of steam rate.

Criterion for Optimization

In any optimization, the initial problem is to establish a

realistic objective function to maximize or to minimize.

In this study, an objective function was selected that

considers the present worth of produced oil at-d con-

sumed fuel. This function is referred to as the cumulative

discounted net oil (CDNO).

To define CDNO, cumulative net oil is fwst defined as

equal to the cumulative oil production minus the barrels

of fuel needed to produce that oil, This can be expressed

as

Cumulative net oil = Cumulative oil moduced

– Cumulative oil ~urned, . . . .(2)

3 yr

TOP

PIAN

VIEW

,. p—,

p-

CROSS

SECTION

FROM INJECTOR

TO PRODUCER

MIDWAY

CROSS SECTION

NORMAL TO LINE

JOINING INJECTOR

AND PRODUCER

770

5 yl’

L

95- 200°F

lzz

200- 300°F

m

AEOVE 300°F

Fig .7-Calculated temperature d ist ribu tions.



7/12

where

Cumulative oil burned =

Cumulative heat

Cumulative heat

input to sand face –

from inlet water

Heating value X

Efficiency

of fu_el factor -

Values used in this study were 70”F feedwater, 6.2 MM

Btu/bbl fuel, 0.68 efficiency factor, and 200 psia steam at

70 percent quality.

Substituting these values in Eq. 2 gives

Cumulative net oil = Cumulative oil produced

_ Cumulative steam

13.36

— ..,., (3)

With cumulative net oil defined, CDNO is now de-

fined as

N

CDNO =

z

n=]

(Cumulative net oil).+, – (Cumulative net oX., . . . (4)

t

1.1 *

where CDNO isinstock tank barrels, n = time index, and

r~+ Y2 = time in days, corresponding to the midpoint

between the times denoted by

n

and n+ 1. An annual

discount rme of 10percent has been assumed,

In Eq. 4, N corresponds to the time to which the

CDNO refers. This study uses the CDNO atthe economic

limit as the objective function. This particular CDNO is

termed the final CDNO (FCDNO). It is assumed that the

total operating cost for steam injection is equal to twice

the fuel cost. Inother words, the economic limit for steam

injection is reached when the following occurs:

Instantaneous

= % X steam/fuel ratio . . . . (5)

steam/oil ratio

The number 13.36 in Eq.

3 is

the steam/fuel ratio; there-

fore, the steam-oil ratio used as the economic limit in this

study is 6.68 bbl/bbl,

In a related study, Ferguson5 performed a more com-

plete economic analysis in which he included operating

expenses other than fuel cost and capitalization of

steam-generation facilities, He found that CDNO k ade-

quate as a criterion for optimization if inflation premises

are used, whereas some refinement is necessary with the

use of the coilstant doHar basis,

Scope

of Optimization

All variables related to the steam-displacement process,

whether controllable or uncontrollable, can affect the

3

yr

yr

TOP

PLAN

VIEW

BOTTOM

PLAN

Vmw

CROSS SECTION

FROM INJECTOR

TO PRODUCER

MIDWAY

CROSS SECTION

NORMAL TO LINE

JO INItW3 INJECTOR

AND PRODUCER

n

0-0.10

Ezl

0.10-0.35

= ABOVE 0.35

Fig. &Calculated oil -saturation dist ribut ions.

JUNE, 1975


8/12

TABLE8-OATA FORRATEOPTIMIZATION

Pattern ccmfiguration

Five-spot

Relative permeability

Cu rves ap pl icab le to K ern “A ”

68 pat ter n (Tab le 3)

Oil viscosity

Cu rve ap pl icab le to K ern Ri ver

c ru de (Set 2 of Tabl e 2)

Permeabi l ity, md

3,000

Porosity, percent

30

Initial oil saturation, percent 50

St eam i nj ec ti on pr es su re, ps ia 200

Steam qual i ty , percent

70

Completion interval

Inj ec ti on wel l - l ower thi rd

Product ion wel l - en ti re interval

Production rate

Determined by del iverabi li ty

based on a rwoduction-well

p res su re 0f i4.7 p si a,

sub ject to a specif ied

max imum p ro du ct io n o f

total f luids

Other pertinent data Given in Table 4

optimal choice

of

st eam-injecti on rat es.

Although

the

ideal optimization is to vary all variables at the same

time, this approach is too unwieldy to yield any meaning-

ful results without an unjustifiably large amount of com-

putational effort and expense. To make the problem more

tractable, the scope of optimization was narrowed in this

study by fixing the quantities as shown in Table 8.

In addition, the pattern producer was steam stimulated

at the beSinning of steam displacement and at every

6-month interval when needed. Each stimulation lasted 6

days, followed by 3 days of soaking. Steam-stimulation

rates were 1,200, 800, and 400 B/D for respect ive thick-

nesses of 90, 60, and 30 ft. This thickness range covers

the bulk. of individual displacement zones in the 668

inverted five-spot patterns now in operation by Getty Oil

in Kern River. Average pattern size is about 2.5 acres,

which is considered near optimal for the area, Therefore,

this study was based prima)”ily on 2.5 acres. However,

because of future expansion into areas with potentially

larger spacing, some data on 5-acre spacing are included.

The five specific cases reported here are the following,

Pattern Size Thickness

Case

(acres) (ft)

——

2.5

90

; 2.5

60

3 2.5

30

4

5.0

90

5 5.0

30

Cases 1 through 3 were studied using both constant and

variable steam rates, while Cases 4 and 5 involved only

constant steam rates.

Constant Steam Rates

Most of Kern River production history is based on con-

stant steam-injection rates. Therefore, our initial s udy of

the field production was based on these constant injection

rates, providing a good basis of comparison for the wri-

able injection-rate cases discussed in the next section.

Computational results for constant steam-injection

rates are shown in Table 9. Based on these tabulated

results, Figs, 9 and 10were plotted, giving the variation

of FCDNO with steam rate for Cases 1 through 3

(2.5-acre spacing) and Cases 4 and 5 (5. O-acre spacing),

respectively. From these figures, the optimal choice of

steam rates can bemade for various pattern sizes and sand

thicknesses, as shown in Table 10.

Fig. 11 shows the variation of optimal steam rate with

thickness for both 2.5- and 5 .O-acre spacings, For

2.5-acre spacing, it is seen that, as thickness decreases

from 90 to 60 ft, the optimal rate decreases. However, the

TABLEUOMPUTATiONAt RESULTS— CONSTANT STEAM RATES

Steam Rate Stimulation cutoff Cumulative

Run

(BID)

Time (years) Time (years)

Oil (STB)

Case

1 — 2.5ac re, 90f t

Clol

150 0,0.5, 1,0, 1.5

?102 200 0,0.5, 1.0

C104 250 0,0.5

C105 300 0,0.4

C106 400 0,0.4

CI07 400 0

C108

500 0

Case 2 — 2.5 acre, 60 ft

C201 150 0,0.5, 1.0

C202 175 0,0.5, 1.0

C203

200 0,0.5

C204

250 0,0.5

Case3— 2.5acre, 30ft

C301

100

0,0.5, 1.0, 1.5

C302

150

0,0.5

C303

200 0,0.5

C304

250 0,0.5

C305 350 0,0.5

Case 4— 5.Oacre, 90 ft

C401 300 0,0.5, 1.0

C402 ‘ 400

0,0,5, 1.0

C403

500

0,0.5, 1.0

Case 5— 5.0 acre, 30 ft

C501

300

0,0.5

C502 400

0,0.5

C503 500

0,0.5

11.4

9.3

:::

3.9

4. I

2.7

8.5

7.5

6.6

5.3

6.o

3.6

2,9

2.3

1.7

13.2

10.4

8.7

4.7

3.6

2.9

155,000

153,500

149,000

138,900

110,100

112,500

90,503

104,000

103,600

99,900

95,400

41,100

41,100

40,800

41,000

37,600

310,500

299,600

285,500

86,000

85,800

79,800

FCDNO

(STB)

62,700

67,500

67,400

65,400

55,600

55,600

47,100

46,300

4“/,200

46,700

46,000

17,500

20,500

21,300

21,400

19,100

111,460

115,800

113,600

35,100

36,400

32,600

772



9/12

decrease is not in proportion to the decrease in thickness.

As thickness decreases further from 60 to 30 ft, the

optimal steam rate increases again so that the optimal

rates for 90 and 30 ft are the same. Assuming that the

optimal rate should be proportional to the oil content of

the reservoir, and therefore, proportional to sand thick-

ness (assuming constant oil saturation), i t would be ex-

pected that since the optimal rate for 90-ft thickness is

225 B/D, the optimal constant rates for 60 and 30 ft

should be 150 and 75 B/D, respectively. The calculated

optimal constant rates for 60 and 30 ft are higher than

these values because heat losses to the overburden and

underburden become more and more important as thick-

ness decreases. An increase in steam rate should be

provided to compensate for the heat lcsses.

Fig. 11 shows that, for 5.O-acre spacing, the optimal

rate decreases slightly when thickness decreases from 90

to 30 ft. Furthermore, although the curve joining the three

points for the 2.5-acre-spacing case shows a dip at 60 ft,

the variation in optimal rates for the entire range of 39 to

90 ft is not large. It may, therefore, be deduced that the

optimal rate for 2,5-acre spacing is in the neighborhood

of 200 B/D. By the same token, the optimal rate for

5 .O-acre spacing is about 400 B/D. This indicates the

optimal constant steam rate is relatively independent of

sand thickness, but is proportional to pattern size. This

conclusion was reached based on thicknesses in the range

of 30 to 90 ft and applies to that range only, It is conceiv-

able that the optimal constant steam rate could be propor-

tional to both thickness and pattern size (that is, the

volume) for reservoirs much thicker than 90 ft, and that

the cptimal rate could increase as thickness decreases for

resewoirs less than 30 ft thick.

Fig. 12 shows the FCDNO at the optimal steam rate

I

I

I

I

I

.—. . :. —. —.. .— —..

---- t- -

.. ---- ..-. ..+ .. .,-

1

I

.—__ ,.

. . . .

-----

I

1

30 fr

–“ 7= =

1 I

::

~—––

“/”-–;-

11

I

I

1 I

00 100

200 300

400

CtINSTANT STE.M RATE, BPD

Fi g. k Fi nal cu mu lat iv e di sc ou nt ed net o il as a f unc ti on o f

con stan t s team rate — 2.5 ac res .

JUNE, 1975

plotted against thickness. If the optimal steam rates are

used, the FCDNO is proportional to sand thickness.

Fig. 13 gives the FCDNO at the optimal steam rate

plotted against pattern size. Since the points for 5.O-acre

spacing lie beneath the dotted lines joining the corre-

sponding 2.5-acre points and the origin, 5-acre spacing is

less favorable than 2.5-acre spacing.

121

10

8’

61

4’

2

,

I

L–

I

II

1

I

I

“1--Lu_

00

300 400

500

I

CONSTANT STEAM

RATE - BPD

-1

600

Fi g. 10-Fi nai c umui a i ve di sc ount ed n et oi l as a f unc ti on of

con stan t s team rate — 5.0 ac res .

c1

Ck

m

500

I

400

300

200

100

0

TT

0 ACRES

t

I

o

30 60

90

THICKNESS , FEET

Fig. 1 l—Var iat io n o f o pt imal s team rate w ith thi ckness.

773


10/12

120

100

80

60

40

70

c1

c

30

60

90

TIIIcK?:CSS, t’ECT

h. 12—Final cumulat ive d isco un ted n et o il at o pt imal s team rate

as

a funct ion of thickness.

120

100

80

60

40

20

0

f

—.—-——.-. .

1

I

,

.

——

Q

. ..~.-- —— - -- — . -.

c

2,5

5.0

PA TTE RN S 1 ZE , A CW 3

Fi g. 13-Fi nal c umu lat iv e d is co un ted net Oi l at o pt imal s team rate

as a fun ct io n o f p at tern s ize.

774

Variable Steam Rates

Initially, constant steam rates were used at Kern River,

but production history and closer analysis indicated a

ste~m-rate reduction was desirable at some time after the

start of injection, The basis fort his operations change is

given by Bursell and Pittman, 1.Todetermine the effect of

rate cutback of injection on production, this study was

initiated.

Computational results for variable steam-injection

rates are given in Table 11. A graphical representation of

the variation of steam rate with time for various computer

runs is given in Fig. 14.

To compare sorre of the results from these variable

steam-rate cases with those of the constant-rote cases,

Table 12 was constructed. (This is merely a “con-

venience” table constructed from Table 9 and 11 for

discussion.)

Note that in comparing Runs V101 and Cl 04 (Table

12), the cumulative oil is about the same, bat the FCDNO

is larger by 4,700 STB for the variable-rate case. The

dominant factor in this increase appears to be the shorter

production life of the variable-rate case, which would

increase present worth.

In Group 2, Runs VI02, VIO1, and CI04 are com-

pared. In Run V102, Run V1OI is modified by adding a

third rate reduction to 100 B/D. This rate reduction in-

creases the life of the displacement zone and increases

both cumulative production and FCDNO. Comparing all

three runs suggests a field procedure of high initial steam

rates, a middle steam-rate reduction, and a final steam-

rate reduction as most advantageous. Limitations on

these multiple rate changes would be those imposed by

field operation,

To obtain a better look at the effect of increasing the

initial steam rate, Run V103 was made. Note that the

initial rate increase from 500 to 750 B/D resulted in about

the same cumulative oil, but that the FCDNO increased

by 2,400 STB. This increase was apparently caused by

the faster production rate.

This comparative study shows that carefully selected

rate changes over a displacement-zone life can improve

protlabilit y either by increased oil recovery or by present

worth. The actual steam-rate reduction in any given proj-

ect will need to be chosen so that an injection-rate reduc-

tion does not make a significant change in production

trend.

Fig. 15shows the type of study required for analysis of

the effect of steam-rate change. Note that in this run (Run

V103), the reduction in steam rate does not cause a

proportional reduction inoil rate and, therefore, results in

an immediate drop in steam-oil ratio,

The variation of steam rates with time in Run V103

may be visualized as a discrete approximation to a hyper-

bola, described by the equation

TABLE 1O-OPTIMAL CHOICEOFCONSTANT STEAM RATES

FCDNO

at

Pattern Thi ck nes s Opt imal St eam Optimal Rate

Size (acres)

(ft)

Rate (B/D)

(STB)

——

2.5 90 225

67,800

2.5

175

47,200

2.5 %

225

22,400

5.0 90 400

115,800

5.0 30 375

36,600



11/12

Stearnrate XTime Q= Constant, . . . . . . . . . ...(6)

where a is an index. In a similar vein, Run V104 shows a

straight-line variation. Run V105 is similar toRun V 103,

except that the initial rate Mlowered and another step is

added. The FCDNO of these two runs was not as good as

that in Run V103. Run VI I6 used a geometric sequence,

800,400, 200, and 100 B/D, as the rates for the succes-

sive stages. At each stage, the rate was maintained until

the residual oil saturation reached or approached 6.68

bbl/bbl. At the end of the last stage (100 B/D), steam

injection was discontinued. The FCDNO for this run was

78,600 B/D, slightly higher than that of Run VI03.

Although a more definitive study is needed to deter-

mine the optimal variation of steam rate, Runs V101

through V106 tend to support the contention that a hyper-

bolic variation is more favorable than a linear variation.

For the case of 60-ft thickness, Run V201 represents a

linear variation, whereas Run V202 approximates a

hyperbolic variation. Here again, the latter is more favor-

able than the former. Run V202 gives an FCDNO of

51,400 STB, 9 percent higher than the value of 47,200

STB shown in Table 10 for the optimal constant rate of

175 BID.

Results in the case of the 30-ft thickness indicate that

the FCDNO’S of Runs V301 through V304 are not sig-

nificantly better tlm the value of the optimum constant-

injection rate. Apparently, heat loss is an overriding

factor with thin sands,

Summary

Performance of a Single Pattern

1. A reasonable history match was made with the oil

and water production of Well 68, Kern “A” project,

Run

Cas e 1

Viol

V102

V103

V104

V105

V106

throughout the 5-year, 6-month period. This match, al-

though short of being perfect, served to validate the

model.

2. With the present model, which does not have the

capability to handle hydrocarbon gas, using the concept

of spongy rock alleviates the excessive pressure created

during the stimulation stage.

3. Using upstream weighting of viscosities of oil and

water and including the temperature dependence of rela-

tive permeabilities tends to improve the simulation.

Optimization of Steam Iqjection Rates

1. The final cumulative discounted net oil at the

economic limit (FCDNO) is an adequate criterion for

comparison and optimization. This criterion takes into

consideration the present worth of the produced oil and

consumed fuel and yet avoids the use of monetary values,

which are affected by price and ccst changes.

2. The optimal choice of constant steam rate is rela-

tively independent of sand thickness but is proportional to

pattern size. As sand thickness decreases, the total oil

content in the reservoir decreases, and this calls for a

lower steam rate. At the same time, a higher steam rate is

needed to compensate for the increased prcentage heat

loss with a decrease in thickness. These two counteract-

ing factors nmdt in only a small variatkn in the optimal

steam rate as thickness changes from 90 to 30 ft.

3. With the same thickness, the FCDNO at the opti-

mal constant steam rate for 5.O-acre spacing is less favor-

able than 2.5-acre spacing for the situation studied.

4. The FCDNO for the constant steam rate can be

improved by increasing the steam rate in the initial stages

and decreasing the steam rate with time. Although a more

definitive study is needed to determine the optimal varia-

TABLE1140MPUTATIONAL RESULTS — VARIABLESTEAM RATES

Stimulation cutoff Cumulative FCDNO

Steam Rate (Du rat io n) B /D (y r)

Time (years)

Time (years)

Oil (STB)

(STB)

– 2.5 acre, 90 ft

500(1) — 250

0

6.9 148,800

72,100

500(1) — 250(4) — 100 0

9.7 157,400

75,900

750(1)— 250(3) — 100 0

9.3

157,500 78,300

400(1) — 300(2) — 200(2) — 100

0,0.4

9.2

161,500 76,200

600(1) — 400(1) — 200(2) — 100

9.8 156,800

76,600

800(1,3) —400(1.1)—200(4 .1)—100 :

7.2

163,400

78,600

Case

2— 2.5

acre,

60 f t

V201 250 2 — 200 2 — 150 2 — 100

0,0.5 7.7

104,200 49,800

V202 450(1) — 250(1) — 150(2) — 100 0,0.5

8.0

104,000

51,400

Case 3— 2,5acre, 30 ft

V301 250(1)—200(1)— 150

0,0.5

2.7

40,800

21,800

V302 300(1) — 250(1) — 200 0,0.5

40,600

21,200

V303 350(1)— 100

0,0.5

:

41,100

21,500

V304 350(1)— 200(1)–. 100

0,0.5

2.4 41,000

21,700

TADLE12-COMPARISON OFSELECTEDCONSTANTAND VARtABt.ESTEAM-RATE CASES

Number

Comparison Cases

1 Viol

C104

2 V102

Viol

C104

3 V103

V102

C104

steamRdteS

(BPD)

500/250

250

500/250/100

500/250

250

750/250/100

500/250/100

250

Cuto f f Time

6.9

7.8

9.7

6.9

7.8

9.3

9.7

7.8

Cumulat ive Oi l

(STB)

148,800

149,000

157,400

148,800

149,000

157,500

157,400

149,000

FCDNO (STB)

72,100

67,400

75,900

72,100

67,400

78,300

75,900

67,400

JUNE,

1975

775


12/12

CASE

1

2.5 AcRE 90 FT.

v

101

Iw

WI

b

00246

v 102

420

:W

k

002468

CASE 2

2.5 ACttE: 60 FT.

v 201

v 202

4W

too

L

0ot48

Sco

L

v 103

w

SW

00 4,8

em

L

v

105

400

200

@0246a

8(W

W

L

V 106

w

200

0

0248

CASE 3

2.5 AcM: 30 F1’.

v 301

2C4

b

0

z

too

c1

0 2

Zw

L

00 2

200

b

01

v 302

v 303

V304

TIME,

Y RS

Fig, 14-Var iat io n o f s team rate w ith t ime for var io us compu ter

runs.

tions of steam rates, evidence so far obtained tends to

support the contention that a hyperbolic variation is

superior to a linear variation.

5, Improvement in FCDNO by reducing steam rates

with time increases with sand thickness. For a 2.5-acre

pattern, a greater improvement in the FCDNO was

realized for a 90-ft sand than for a 60-ft sand, No signifi-

cant improvement was noticed for a 30-ft sand,

Nomenclature

b =

formation volume factor, STB/res bbl

for oil, molkes bbl for water and steam

CDNO = cumulative discounted net oil, STB

C = compressibility, volhol-psi ..

CP = specific heat, Btu/lb-°F

CT= thermal expansion coefficient,

vol/vol –“F

0 :,, .;, ; ;,

1

,,

.m m.?.,

f ig . 16-Var iat io n o f cumulat ive d isco un ted n et o il an d s team-o il

rat io with t ime — 2.5 acres, 90 f t.

FCDNO = final cumulative discounted net oil, STB

k

permeability, md

k relative peirneability

K thermal conductivity, Btu/ft-12-°F

p = pressure, psia

P

capillary pressure, psi

S = saturation

t= time, days

a = index

~ = viscosity, cp

p = density, lb/cu ft

4 = porosity, fraction

Subscripts

g = gas

i =

initial condition

n = time index

o = oil

ob = overburden

R rock

w = water

References

1.

2.

3.

4.

5.

6.

7.

Bursell , C. G. and Pittmtm. i;. M.: “Performance of Steam Dis-

placement -

Kern River FWI.i,” paper SPE 5017 presenled at the

SPE-AIME 49th Annual Fa t Meeting. Houston, Oc[. 6-9, 1974.

Coats, K. H,, George, .

D., Chu, C., and Marcum, B. E.:

“Three-Dimensional Sim&’ion of Steamflooding.” Sot.

Pet.

Etrg. J.

(Dec. 1974) 573-59.’:

‘ram.

AIME., 2S7.

Coats, K. H.: “Simulation ~’Steamflooding With Distillation and

Solut ion Gas,” paper SPE ‘J’5 presented at the SPE-AIME 49th

Annual Fall Meeting, Houst, ~.Oct. 6-9, 1974.

Elkins, L. F.: “Uncertain\ of Oil-in-Place in Unconsolidated

Sand Reservoirs - A Case }:r;tory ,” J.

Per. Tech. Nov.

1972)

1315-1319.

Ferguson, N. B.: private comr]lunication.

Sawabini, C.,T., Chilingar, G. V., and Allen, D. R.: “Compressi-

bility of Unconsolidated, Arkosia Oil Sands,”’ Sot,

Per.Eng. J.

(April 1974) 132-138.

Weinbrandt, R. M. and Ramey, H. J., Jr., “The Effect ofTempera-

ture on Relative Permeability-of Consolidated Rocks,” paper-SPE

4142 presented at the SPE-AIME 47th Annual Fall Meeting. San

Antonio, Tex., Oc . 8-11, 1972.

mT

Original manuscript received m SocLefy of Petroleum Engmaers offnce Aug. 1.1974.

Revised manuscnpt recewed March 17, 1975. Paper [SPE 5016 was f l rat presented at

the SPE-AIME 49th Annual Fal l Meeting,

held m Houston, Oct.

6-9, 1974. @COpyright

] 9?5 American Instdute of Mining, Metal lmglcal, and Petroleum Enginee6. inc.

776


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