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.
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.
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
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—.
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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
JOURNAL OF PETROLEUM TECHNOLOGY