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A Steam-Soak Model for

Depletion-Type Reservoirs

P. J. Closmann, SPE-AIME ,SheUDevelopmentCo.

N.

W. Ratliff, SPE-AIME,Shell DevelopmentCo.

N. E. Truitt, SPE-AIME,ShellDevelopmentCo.

Introduction

The widespread application of the steam-soak proc-

essl’2 has made more essential an understanding of

the basic mechanism of the process. As presently

applied, it consists of injecting an arbitraxy quantity

of steam into a formation, stopping injection and

~:e~~g ~ the weld for ~ome “soak” time, and then

producing oil from the injection well. Recent reports

on field applications have been given by Bowman and

Gilbert,s Adams and Khan,’ and de Haan and van

Lookeren.5

Theones to describe the steam-soak process have

been presented by Boberg and Lantz~ Davidson et

al.7, Martin,s Seba and Perry,g and Kuo et af.10None

of these theones has attempted to include the detailed

distribution of the steam or the oil viscosity distribu-

tion. The present method is applicable to depletion-

type reservoirs and includes the specific interval of

steam penetration as well as the viscosity distribution

resulting from heating. The method assumes that dur-

ing the injection phase oil is displaced from the steam

zone until some residual value of oil saturation is at-

tained. During the production phase oil is allowed to

flow back across the outer radius of the steam zone.

The time to resaturate this zone is calculated. Heating

of oil in adjoining strata results in a greatly increased

flow of oil through the heated layers into the well dur-

ing backflow. To estimate this effect, it is necessary to

use the viscosity-temperature curve for the particular

oil being considered. We hope that t.hk method will be

I

useful to operating personnel and that it will provide

insight into some of the essential factors of the steam-

soak process.

Two types of formation are treated in the present

method. For the first case, zero vertical permeabtity

is assumed, and oil flows only horizontally. This cal-

-: 1. A A +011,7

culation should be applicable to eases UIlloda.-~

stratified reservoirs. For the second case, isotropic

permeability is assumed, and crossflow into the de-

pleted steam zone is estimated by means of crossflow

factors developed for a range of formation tilck-

nesses, steam-zone radii, and viscosity distributions.

In many practical cases the vertieal permeability will

be significant but still less than the horizontal per-

meability. Results for this situation will then be inter-

mediate between the two extremes calculated by this

model. However, it should also be possible to com-

pute crossflow for these cases ,as well. The model

should apply to both light and heavy oil reservoirs.

The relevant data used, such as steam-zone thickness,

residual oil saturation in the steam zone, and oil vis-

cosity, should be chosen accordingly. The effect of

steam distillation is not taken into account explicitly

but could tiect some of the data chosen.

Theory

General

Description

Our general description of the steam-soak process in

a depletion-type reservoir is as follows:

I

A mathematical model for predicting first-cycle performance of steam stimulation in

depletion-type reservoirs agrees reasonably well with field observations. It can be applied

to both stratified and nonstratified reservoirs.

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When steam is injected into a formation, a steam

zone is formed, and heat flows, mostly by conduction,

:--- AL..

-.... . . .. . . A...”na,l;..- PfimAa”..+a

thwc nllt

Inlu UK SU1l

uul luu l~ l l lGUIUU1. V“uuuhl.w .w Al ” .. “ “ ...

into the formation. The steam zone is depleted of oil

to some residual saturation. If the injection well is

then put on production, the reservoir fluids expand

and flow into the well. The fluids also flow into the

steam zone, which is gradually resaturated with oil.

Most of the injected heat remains somewhere in the

vicinity of the injection well and the steam zone. It is

thk heat that causes a reduction in viscous resistance

to oil flow near the well and enables greater produc-

tion rates to be obtained. The distribution of heat

changes with time. Local values of oil viscosity are

then functions of time.

Basic Mathematical Model

An exact mathematical description of the above proc-

ess is quite complicated. What is desired is a theo-

retical description that accounts for the chief physical

factors and that at the same time is not too difficult or

time-consuming to use. A model for computing pro-

duction response can be setup to include the follow-

ing ass-umptions (see Figs. 1 and 2):

1. Steam is injected and flows through the forma-

tion in a zone of constant thickness and uniform steam

temperature.

2. Heat is lost from the steam zone by vertical

conduction only.

3. Gravity drainage within the reservoir can be

neglected.

4. The outer radius of the heated region (numeri-

cfiy equal to the steam-zone radius) remains constant

with time.

5. The temperature, and hence the oil viscosity, at

any given depth below (or above) the steam zone is

taken as constant between the wellbore and the outer

radius of the heated region (steam-zone radius). The

initial temperature distribution is approximated from

the heat loss (Appendix B). Later temperature dis-

tributions are obtained by temperature decay from the

jni~~a~nrnfik w~~h h~~~ flow by Conduction in the

. ~...-,

vertical direction only (Appendix C).

6. Temperature in the steam zone is uniform with

both radial and vertical distances but declines with

time (Appendix C). The soak time constitutes an addi-

tional time increment in the temperature decay.

7. The reservoir pressure at the start of produc-

tion is assumed uniform. The actual value used would

depend on previous reservoir history.

8. Effects of heat in produced fluids are neglected.

9. All effects of dip are neglected.

10. Oil flux for the crossflow case may be approxi-

mated by first calculating horizontal flow for a strati-

fied model (no crossflow) and then multiplying this

value by a crossflow correction factor. This factor is

obtained by averaging results for an initial and a later

viscosity distribution (Appendix A). Oil produced by

crossflow appears immediately at the production well

and does not contribute to resaturation of the steam

zone.

Horizontal oil fluxes obtained in Assumption 10

,.L,...,.

..nl,-,.1,-,t~,-l,w. +:rn~.~.,~.~rr~v~l,,~~

mf thr=

auu v c ~i~ bcubumbtiu LU. LUIm-LL. WA a~u . w-w. “. . ..W

oil viscosity. These values are determined from the

relation

+M+w

2

/

(t, –t,) +... , . (1)

where the various values pn correspond to tempera-

tures at the various times

t.

after the start of produc-

tion. The temperatures are calculated for time-average

steam-zone heat capacities (Append~ C). With this

procedure the results will be tiected by the number

and size of time steps used. However, for initial time

steps of 20 days the effect is not significant. At longer

00

—— ———

VN

———

——

~

BASE ROCK

(

ARROWS INDICATE DIRECTION OF’

OIL FLOW DURING BACKFLOW

)

Fig. l—Diagram of stratified model.

758

STEAM

ZONE

T

nw

.

BASE ROCK

. -------- . . . . . .... “,nc . *, m., fit\

mLL’&’”HJ’&A: H&ix “r)

Fig.

2—Diagram of general model with crossflow.

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times, the size of time steps may be increased, since

production rates and temperature dktribution are not

changing so rapidly.

According to Assumption 2, radial heat conduc-

tion is neglected. Thk could be important for large-

interval injection. However, the effect of heat flow

into the cooler part of the reservoir would be to

decrease flow resistance in this zone and to increase

it in the hot zone — two effects that tend to offset

each other.

As formulated here, this model applies strictly to

the first steam-soak cycle only. For application to

repeated steam-soak operations, the calculations de-

scribed here should be modified to include the appro-

priate temperature dktribution, formation pressure,

and compressibility. The latter two properties can be

estimated from knowledge of the prior reservoir his-

tory, including preceding steam-soak cycles. Knowl-

edge of the total amount of heat injected and that

produced by the fluids should make it possible at

least to approximate the temperature dktribution.

With this knowledge the steam zone developed in later

injection cycles can be estimated and the oil produc-

tion calculation can be extended to this case.

Stratified Model

In the stratified model the oil-bearing regions adjacent

to the steam zone are divided into a number of hori-

zontal layers of uniform thickness (Fig. 1), The

horizontal fluxes through the layers are determined

and added. Cumulative oil flux through each layer is

determined from a QT,. function for a composite

mediumll analogous to that for a uniform medium.’~

(Relevant mathematical formulas are presented in

Ref. 11.) If N;(t) is the cumulative production per

unit thickness from the ith layer, then the total cumu-

lative production from the oil-bearing layers is

Ivo(t)=Azs Ni(t), . . . . . , . (2)

;

where Az k the chosen interval in Z. It has been found

that sufficient accuracy is obtained by limiting the

thickness of the layers to values no greater than 13 ft.

The time required to resaturate the steam zone with

oil, tfj1I, is calculated by equating the oil volume for

resaturation to the cumulative oil production from a

uniform layer of thickness equal to the steam-zone

thickness; i.e.,

z(r82 – r,,’) h,+tio

= 2~h,+c,r,2 (p~ – PJ

Q~(ltill) , . (3)

where Q~(t~i1,) is the dimensionless Q~ function, as

defined by van Everdingen and Hurst” for uniform

reservoirs, evaluated at A~1I. The outer radius r. of

the steam zone (Appendix B) is used as the well radius

in this calculation. After the steam zone has been

resaturated, it is treated as a single horizontal layer,

and the composite Q~Cfunction calculation applies.

Then the total cumulative production becomes

N(t) = NO(2)+ N*(t) , . . . . . . (4)

where

IV, t)

s the cumulative production from the

steam-zone layer. The model has been set up to in-

clude the case of an infinitely thin steam zone, as when

steam is injected into a fracture.

Crossflow Model

To determine results with crossflow of oil into the

steam zone (Fig. 2), a simplified model has been set

up. In this model the crossflow is computed for a

time interval during which the viscosity distribution

in the heated oil-bearing zone is assumed to remain

~~~~fant, The model further assumes that all oil pro-

duced by crossflow appears immediately at the pro-

duction well and does not contribute to resaturation

.c .5. .*. - vfime lf the laffe~ effect were permitted,

 IL u G WGaus -“IA.. . . . . . -- .

the initial calculated production values would be

lower and the time for resaturation of the steam zone

would be shortened. The method is described in

Appendix A. The ratio of production with crossilow

to production without crossflow has been determined

over a wide range of variations in formation thick-

ness, steam-zone radius, and viscosity distribution.

These ratios, or “crossflow factors”, are then used to

multiply the cumulative obtained from the stratified

model calculation. Calculations for crossflow correc-

tion factors have been made for temperature distribu-

tions immediately after steam injection and 409 days

after steam injection. The actual correction for most

cases to be considered will be intermediate between

these two sets. Most of the production times exceeded

1*D= 5,000, after which the crossflow factors did not

change as widely as at shorter times. Hence, one aver-

age value was used for a given production curve.

Furthermore, all of the crossflow factors determined

for a given range of values of oil-zone thickness and

steam-zone radius apply to a single total quantity of

heat injected. In the cases presented, approximately

the same total quantity of heat was used. To make

estimates of the crossflow factors in any application,

it is therefore necessary to make calculations as out-

lined in Appendix A for a range of the variables

(steam-zone radius, oil-zone thickness, and time).

This technique has been chosen as one that should

be suitable for use on a production basis.

For this method, if

F, = crossflow factor for cumulative produc-

tion,

then

cumulative production from oil-bearing layers

= F, X cumulative production from

bearing layers in stratified model;

and

total production

= production from oil-bearing layers

+ production from steam zone.

Results

Comparison with Field Data

Presented here is the application of this theory to

oil-

(5)

(6)

two

wells, Wells A and B; ‘which produce by depletion.

The reservoir properties (Table 1) chosen for these

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cases are thought to be representative of the field. A

separate oil-viscosity vs temperature curve is used for

each well. The compressibility chosen corresponds to

that of a reservoir having a small gas saturation, with

the pressure indicated. Results are not so sensitive to

the value of compressibility as they are to that of

permeability.

Results for Well A, including the effect of cross-

flow, are shown in Fig. 3 for cumulative oil produc-

tion. Calculations were made for equal amounts of

steam injected into zone thicknesses of 10, 50, and

100 ft located at the top of the interval and also over

the full interval of 173 ft. Spinner surveys have shown

that steam enters at or near the top of the perforated

interval in most cases. The crossflow factors used for

these cases are shown below for Well A.

Steam Zone

Average

T%icicness

Crossflop Factor,

(ft)

10

5.;

50

3.1

100

2.0

Calculations of cumulative production for the 50-ft

steam zone agree well with the fieid data as shown

in Fig. 3. These same cases are shown in Fig. 4 for

no crossflow. The field production is much higher

than that calculated for no crossflow.

Reference to Fig. 3 shows that if cmssfimw

U12GIUS,

----

steam injection into thin zones yields higher long-

time cumulative production than equal steam injec-

tion into thick zones. This suggests the deslrab@ of

injecting into thin intervals wherever crossflow is

expected. However, in those cases with good cross-

flow (vertical and horizontal permeabilities approxi-

mately equal) the actual confinement of steam to the

interval chosen may be difficult to achieve, since

steam tends to rise to the top of the interval. If cross-

tlow is not expected, thick-zone injection is preferable,

as can be seen from Fig. 4.

It should be noted that a considerable length of

time is required in this model to resaturate the steam

TABLE 1—PROPERTIES USED IN CALCULATIONS

k. = 3 darcies

c = 0.0016 psi-’

T, = 109.4” F

pi =

285 psig

r~=

0.1875 ft

r. =

379 ft

p, C, = 36.8 Btu/(cu ft “F)

~, C, = 33.0 Btu/(cu ft ‘F)

K = 25.1 Btu/(ft D “F)

@J= 0.36

As. = 0.44

OIL VISCOSITY

Well A A = 0.037003 Well B: A = 0.036302

B = 2,409.8 B = 2,398.0

c = 100.195

C = 103.878

Oil Viscosity at 109.4”F

Oil Viscosity at 109.4”F

= 3,643 Cp = 2,773 Cp

zone with oil

— for example, 73.7 days for the case

of a 50-ft-thick steam zone in Well A. In our model

a thin steam zone produces less oil on backflow at

any given time, as its own contribution, but the adjoin-

ing oil-bearing layers produce more as a result of

greater crossflow. The crossflow factor increases with

the steam-zone radius for a given oil-zone thickness.

And also as the oil-zone thickness adjoining the steam

zone increases, the crossflow factor increases. It does

so at a constant steam-zone radius, up to a certain

value, and then decreases for larger values of oil-zone

thickness. For a given quantity of steam, the use of

a smaller

steam-zone

thickness results in both a larger

steam-zone radius and a greater oil-zone thickness.

At very small oil-zone thicknesses (about 20 ft) the

effect of crossflow tends to be reduced. This is a result

of a smaIler viscosity contrast across the oil-bearing

interval. The effect of crossflow is reduced also at

small values of steam-zone radius.

Additional results are shown on Figs, 5 and 6 for

Well B. The general agreement between calculated

and observed results for Well B is about the same as

that for Well A. Again, the 5U-ft steam zone gives a

good fit. In this example a reasonable fit to the data

was obtained by adjusting values of the steam-zone

thickness,

h,.

If ‘tie v-alueof

h,

prevailing during steam

injection is known, this value should be used in the

computations. There is usually enough uncertainty in

values of permeability and compressibility to permit

some adjustment of these within reasonable limits for

a good fit of theory to the observations.

These calculations emphasize the need for reliable

information on permeabfiity, compressibility, and oil

viscosity at reservoir conditions. For example, use of

a lower oil viscosity at reservoir conditions in the

calculations would yield a higher production curve.

Effect of Location of Steam Zone

Plots of temperature profles for steam-zone thick-

nesses of 50 ft and zero ft are shown on Fig. 7. Fig. 8

shows, for a 100-ft steam zone, a plot of cumulative

oil production vs time for the individual layers away

from the steam zone, Well A, for a set of constants

slightly dtierent from those in Figs. 3 and 4. In this

Fig. 3-Cumulative oil produtilon (for crossflow model)

as a function of time, Well A.

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140,

0

0

0

0 //

:1 n

o FIELO OATA

— CALCULATE

o

~=3 darcies

— ..,,.737

c =0.0016pSi-l

‘2:k

100

TIME IN DAYS

I

I

I

  300

400

500

Fig. Cumulative oil production (for stratified model) as a function of time, Well A.

example a change in well pressure after 63 days of

production was chosen to simulate a lower fluid level

in the well. The same change was made in the calcu-

lations for Figs. 9 through 11. Fig. 8 shows that the

oil-bearing layer closest to the steam zone, being

hottest, makes the most sif@cant contribution to the

production. ‘Ilk suggests the desirability of locating

the steam zone somewhere near the middle of the

formation so that oil can be heated by conduction

both above and below the steam zone. Results illus-

trating the effect of the location of a 50-ft steam zone

are shown on Fig. 9 (with crossflow) and on Fig. 10

(without crossflow). The top of the steam zone is con-

sidered to be (1) at the top of the formation, (2) 25 ft

below the top of the formation, and (3) 61.5 ft below

the top (steam zone located in the middle of the in-

terval). The presence of only 25 ft of oil-bearing layer

above the steam zone increases the production 43 per-

cent for the stratified case and 39 percent for the non-

stratified case at 400 days.

Fig. 5-Cumulative oil production (for crossflow model)

as a function of time, Well B.

As mentioned previously, however, for those cases

where crossflow is important, the steam zone will tend

to rise to the top of the formation. In such cases the

injection interval could be located near the bottom

of the formation. In many cases of interest the vertical

permeability is less than the horizontal permeability.

The steam then does not rise so rapidly as expected.

The effect of heating both above and below the steam

zone will still be present. For those cases in which

steam rises rapidly to the top, because of any of

various reasons, the advantage of selective injection

is, of course, much less.

For those cases in which crossflow is not signi ican~

Figs. 4 and 6 show that for zones located at the top

of the interval, the thicker the steam zone the greater

the production once the steam zone has been resatu-

rated. Since location of a steam zone near the middle

of the formation yields greater production response,

it isof interest to compare production curves for steam

zones of various thicknesses located at the middle of

-.

2,c-

0

0

“

:-4

Coooes-(

o   , ,, DA,.

— C.

LCULA7,

,.:, dorcl. s

Fig. ~umulative oil production (for stratified model)

as a function of time, Well B.

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500

INJECTION STEAM

F

EMPER

20

80

400

L

400

k 300

-

z

u

m

2

k

<

u

w

 

z

u

‘_ 200 -

I00 —

RE =464° F

—PROFILE AFTER

STEAM INJECTION

NOTE:

SOAK TIME = 9 DAYS

NuMBERS ON CURVES

~EFER To -,. .r ,., ~:y~

llm L H

AFTER SOAK PERl OO

ORIGINAL FORMATION

\

EMPERATURE=I09.4°F

20

80

i’

 ’

,

~PROFILE AFTER STEAM

INJECTION

(0)STEAM ZONE THICKNESS =50 FEET

o

I

I

I

I

I

o 20

40

60

80

100

50C

40C

INJECTION STEAM

‘TEMPERATURE = 464°F

—PROFILE AFTER

STEAM INJECTION

I

NOTE

SOAK TIME =9 OAYS

$ 300

-k

NUMBERS ON CURVES REFER

z

20

TO TIME IN DAYS AFTER SOAK

PERIOD

a

>

N

ORIGINAL FORMATION

m

* 200

TEMPER ATu RE=I094°F

1

100

t

\

PROFILE AFTER STEAM

INJECTION

I

 

I

(b)l NFINITELY THIN STEAM ZONE

01

I

I

I

o 20

40

60 80

100

olsTANCE FROM MI OPOINT OF STEAM ZONE IN FEET

OISTANCE FROM MI OPOINT OF STEAM ZONE IN FEET

Fig. 7—Formation temperature distribution for Well A.

8000

700C

I00(

STEAM ZONE THICKNESS =100 FEET

DISTANCE (IN FEET) OFT

STEAM ZONE AT TOP OF FORMATION

CENTER OF OIL-eEARING’

hO : 2 do fcies

LAYER BELOW BOTTOM

OF STEAM ZONE

h

c= OOO15 psi-’

i

AS. =044

/ 3.65

/

I00 200

300

400 500

TIME IN DAYS

Fig. 8-Cumulative oilproduction (for stratified model) from individual layers, Well A.

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.

the formation (Fig. 11) for the stratified case. The

same total amount of steam is injected in all cases.

For the cases shown, the maximum advantage occurs

for the 100-ft steam zone after about 300 days. It has

produced about 20 percent more oil after 400 days

than the steam zone occupying the full formation

thickness. The 10-ft zone produces more oil initially,

because it heats the oil at greater radial distances in

the reservoir. At longer times, however, the tempera-

ture of the 10-ft zone falls faster than the others.

Hence, this thickness tends to become less favorable

at longer times. These results as well as those in Fig.

10 show that the greatest improvement results when

steam zones of moderate thickness are selectively

placed near the middle of the formation rather than

at the top for stratified formations.

Effect of Sfze of Interval Open to Production

The foregoing results were obtained with the assump-

tion that oil is allowed to flow into the well throughout

. .

the helgnt of the oil-bearing ti.-.,.. .,.-

m+~m,~lmn=n t-h.~~gh

steam is injected over a limited interval. In the field,

ditliculties might arise in completing a well so as to

allow injection to be selective but production to be

from the entire interval. It is therefore desirable to

compare the foregoing results with some in which

horizontal flow is prevented from the oil layer directly

into the well. This last situation would correspond to

L.hatof ~ we whose casing was perforated only over

the interval for steam injec~ion. Modifying the bound-

ary conditions as outlined in Appendix A so that

ap,D/& = O over the height of the oil layer, we ob-

tained results for dimensionless cumulative produc-

180,

tion (computed as volumetric average pressure drop,

Appendix A) for two cases shown in Table 2.

From Table 2, the greatest change from the cases

of a well perforated over the entire oil zone to one

perforated over only the steam zone is a decrease of

3.8 percent. It appears, therefore, that the special

completion techniques mentioned above may not be

necessary for the steam-zone radii and oil-zone thick-

nesses likely to be encountered. It should generally

suiiice, if vertical permeabiMy is high, to perforate

only over the zone where steam is to be injected. If,

in an actual case, production appears limited, it may

then be desirable to perforate over a wider interval.

Effect of Produced Heat

According to the mathematical model used to describe

the steam-soak process, the effect of heat. lost in the

produced fluids is not included in the calculation. This

heat loss would gradually reduce the production rate

below that calculated. In order to estimate the maxi-

mum effect of this heat loss, we have recalculated the

production curve for one case in which the injected

heat of the stratified model is reduced by the amount

of heat that would be produced in our model. This

calculation would then represent too great a correc-

tion for heat loss in produced fluids, since our produc-

tion rates should be high. The curves showing the

production of Well A with and without this reduction

of heat loss are shown in Fig. 12. The difference

between the two curves after 400 diq~s i s a%ut 4.??

percent. In view of the uncertainties in such quantities

as permeabdity, compressibility, steam-zone thick-

ness, and in-situ oil viscosity, this difference due to

I  

&..

STEAM ZONE THICKNESS * 50 FEET

~ **

, 25 FEET FROM TOP

160

t

FORMATION THICKNESS * 173 FEET

..@v

 1 7

CHANGE IN PRODUCING WELL PRESSURE AT 63 OAYS

I

I

I

‘o I00

200

300 :00

~

TIME IN OAYS

Fig.Cumulative oilproduction (for crossflow model) as a function of time for various positions of steam zone.

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TABLE 2-COMPARISON OF CALCULATED RESULTS WITH AND WITHOUT WELL OPEN OVER ENTIRE INTERVAL

DimensionlessCumulativeProduction

Timeof

Dimensionless

ViscosityProfile

Production

AfterSteam

Time,

Inimtinn

(Dam) tm

... . . . .. . . , --, .,

\

104

0

105

\

104

409

105

r.frW= 441.1

r~lh = 0.001875

St;i:d

om358

0.0143

0.0141

0.0480

;OvJ::

ilrossfhw into

SteemZone

0.01s4

0.0583

0.0434

0.1145

crossflow

:--- N----

lnlUiLWI1l

n if f . ..” .-a

“,,,

e,.m,w

ZoneOnly

(percent)

0.0177

0.0658

0.0430

0.1141

\

STEAM ZONE THICKNESS x 50 FEET

I 20 -

FORMATION THICKNESS x 173 FEET

z

n

m

~ 100 –

z

z

g

~ 80 –

POSITION OF STEAM ZONE

~ ‘0 - F~’’’ss”RERE L:?L’ROM

u

>

~ 40

-1

_\/~l

~ ,ill = 1 86.5 OAYS

3

x

>

>

0

20

0

0

100

200

300

400

TIME IN OAYS

3.8

3.7

0.9

0.3

10

Fig. 10-Cumulative oil production (for stratified model) as a function of time for various positions of steam zone.

I 40

120

=

n

.

Q 100

z

FoRMATlo N THIcKNEss = 175 FEET

STEAM ZONE THICKNESS IN

/ 100

:

eo

- 50

z

g

a

: ‘0 FSHEP’:SS”RE -o(F”L’

~ 40

2

>

s

>

u

20

y y ,

0

I

o

I 00

200

300

400

TIME IN OAYS

FEET

THICK NES

o

Fig. n-Cumulative oil production (for stratified model) as a function of time for various thicknesses of steam

zone located in the middle of the formation.

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heat loss does not appear excessive. In cases of high

WOR’s, however, the heat carried off by the water

could be significant.a’8

Genera cQIlclusions

A model has been developed that gives a good de-

scription of the steam-soak process in depletion-type

fields. It takes into account in an approximate manner

the effect of crossflow in reservoirs with isotropic per-

meability. The model may also be applied to stratified

reservoirs. For purposes of calculation it is essential

to know the thickness and vertical position of the

steam zone, as well as the reservoir permeabdity,

compressibility, and oil viscosity.

Important conclusions demonstrated by the model

are as follows:

1. For a steam zone of given size, location of the

zone somewhere near the middle of the formation

..

a I. fin,, ~ the oil is heated

enhances production of ou Ue.-& .- ---

both above and below the steam zone. Thk objective

should be more easily attainable in case of limited

vertical permeability.

2. When crossflow is important, confining steam

injection to a thin interval rather than allowing it to

enter the entire interval will give a larger production

response for a given quantity of steam injected, if the

steam remains in a thin zone.

3. When crossflow is not important, thick steam

zones are preferable to thin ones for instances of top-

located steam zones.

STRATIFIED MOfJEL

WELL A

Nomenclature

~: ~

h—

  =

h, =

H, =

k=

ko =

K=

N(t) =

Ni(t) =

No(t) =

N,(t) =

P=

pD =

jD =

p, =

pm =

compressibility, psi- 1

crossflow factor for cumulative produc-

tion

height of producing interval, exclusive of

steam zone, ft

steam zone thickness, ft

enthalpy of steam relative to the forma-

tion, Btu/lb

permeability, darcies

oil permeability, darcies

thermal conductivity of cap and base

rock, Btu/(ft D ‘F)

total cumulative oil production at time

t, bbl

cumulative production per unit thickness

from the ith layer at time t, bbl

cumulative oil production at time

t

from

“ -

ownrc b~]

oii-bearm~ LJ -..,

cumulative production from steam-zone

layer at time t,bbl

pressure, psig

dimensionless pressure drop,

P – P

Pf – Pw

volumetric average dimensionless pres-

sure drop

original formation pressure, psig

producing well pressure, psig

TIME, OAYS

., -mm,. ,

Fig. 12—Effect of produced heat on cumulative oil produ&ion.

765

8/16/2019 SPE-2516-PA

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p,.o = dimensionless pressure drop in Region 1

p,~ z dimensionless pressure drop in Region 2

Ap = PI – PW,psi

Q = total heat loss, Btu

Q,., = oil produced by expansion of the cold

j~~~i, Vd ft

Q~il, = oilc:~me to resaturate the steam zone,

Q. = dimensionless cumulative production

from uniform reservoir

n

= dimensionless cumulative production

TC

from two-region composite reservoir

r = radial variable, ft

r, = outer reservoir boundary radius, ft

rn = r/rW

r8 = steam-zone radius, ft

rm =

well radius, ft

reo = dimensionless outer reservoir boundary

radius = r,/r~

rSLI= dimensionless steam-zone radius = r8/r10

tie = change in oil saturation of steam zone

t = time, measured from start of production,

days

td = delay or soak time, days

t,~ = dimensionless time based on Region 2

constants (6.328 kt/@Kicr,c:)

f. = nth value of time after start of produc-

tion at which production is calculated,

n=0,1,2, . . .. days

t~ = total time, t + td, days

tfiII = time for oil to resaturate steam zone, days

— .taom {qi..finn time fl~vs

finj — 3LGU11L 11 J-W.. V.. . . ...-7 - , -

Ir = equivalent heating time for layers adja-

cent to steam zone, days

T, = temperature in cooling steam zone, ‘F

T, =

temperature in heated regions adjacent to

steam zone, ‘F

Ti = original formation temperature, ‘F

T,, = injection steam temperature, OF

~ = vertical distance measured from center

of steam zone, ft

~D = h r~

x~D= h re/rW

xsII = h r,V/rtO

z = vertical distance measured from base of

steam zone, ft ( = lxI – hs/2)

ZD= z/h

a = thermal diffusivity of cap and base rock,

sq ft/D

y = 4.

KP,C,/h,.’(P,C,)’,

days-’/ [(h,’(P,C,)21

p = od viscosity, cp

Pi = oil viscosity at original formation tem-

perature, cp

p. = oil viscosity corresponding to t., cp

~,Cl = volumetric heat capacity of steam zone,

Btu/cu ft ‘F

p,C, = volumetric heat capacity of cap and base

rock, Btu/cu ft ‘F

p,i, = mass steam injection rate, tons/D

T = ~ tinj

+ = porosity

Subscripts

1 = medium 1, hot

2 = medium 2, cold

References

1 Qh.=a ~. B,: “SeGQndary

Recovery of Oil by Steam

In-

. “.1.,..,

jection in the United States”, Proc., Third Symposium

on the Development of Petroleum Resources of Asia

and the Far East, New York (1967 ) IL

2. Bums, James:

‘{A Review of $te~~l Soak @r~&~~~~

in California”, J.

Pet. Tech.

(Jan., 1969) 25-34.

3. Bowman, C. H. and Gilbert, S.: “A Successful

CYCliC

ct-nm Tni.ction Project in the Santa Barbara Field,

U.W................

Eastern Venezuel~;, J. Pet. Tech. (Dec., 1969) 1531-

1s39.

4. ‘Adams, R. H. and Khm,, A. M.: “Cyclic Steam Injec-

tion Performance Analysls and Some Results of a Con-

tinuous Steam Displacement Pilot”, J. Pet. Tech. (Jan.,

1969) 95-100.

5. de Haan, H. J. and van Lookeren, J.: “Early Results of

the First Large-Scale Steam Soak Project in the Tia

Juana Field, Western Venezuela”, J. Pet. Tech. (Jan.,

1969) 101-110.

6. Boberg, T. C. and Lantz, R. B.: “Calculation of the

Production Rate of a Thermally Stimulated Well”, J.

Pet. Tech.

(Dec.,

1966) 1613-1623.

7. Davidson, L. B., Miller, F. G. and Mueller, T. D.: “A

Mathematical Model of Reservoir Response During the

Cyclic Injection of Steam”, SOC. Pet. Eng. J. (June,

1967) 174-188.

8. Martin, John C.:

“A Theoretical Analysis of Steam

Stimulation”,

J. Per. Tech.

(March, 1967) 411-418.

9. Seba, R. D., Jr., and Perry, G. E.: “A Mathematical

Model of Repeated Steam Soaks of Thick Gravity

Drainage Reservoirs”, J.

Pet, Tech.

(Jan., 1969) 87-94.

10. Kuo, C. H., Shain, S. A. and Phocas, D. M.: “A Gravity

Drainage Model for the Steam-Soak Process”, Sot. Pet.

Eng. J.

(June, 1970).

11. Closmann, P. J. and Ratliff, N. W.: “Calculation of

Transient Oil Production in a Radial Composite Re-

servoir”, ~OC.Per.Errg:J: (Dec.; 1967) 355-358.

12. van Everdingen, A. F. and Hurst, W.: “The Application

of the Laplace Transformation to Flow Problems in

Reservoirs”, Tram., AIME ( 1949) 186, 305-324.

13. Marx, J. W. and Langenheim, R. H.: “Reservoir Heat-

i -b;5Hot Fluid Injection”,

Trans.,

AIME ( 1959) 216,

14. Closmann, P. J.:

“Steam Zone Growth During Multiple-

Layer Steam Injection”,

Sot. Pet. Eng. J. (March,

1967) 1-10.

15. Carslaw, H. S. and Jaeger, J. C.: “Conduction of Heat

in Solids, 2nd cd., Oxford at the Clarendon Press

(1959) 60.

APPENDIX A

Estimation of Crossflow Effects

An estimate of the effects of crossflow can be obtained

by comparing the numerical results of two models

shown in Figs. 13 and 14. In each of these models,

the viscosity of oil in the inner region (radius less

than or equal to steam-zone radius) is assumed to be

a function of vertical distance from the steam zone,

and the viscosity of oil in the outer region is assumed

to be the oil viscosity at reservoir temperature. In the

crossflow model shown in Fig. 13, flow in the vertical

direction is allowed. The boundaries of the steam zone

(z = O, r,. < r < r,) and of the wellbore (r = rW)are

maintained at a constant unit pressure drop. Only

radial flow is permitted in the model without cross-

flow, with a constant unit pressure drop on the weU-

bore. A dimensionless cumulative production for each

of these models can be defined as the volumetric aver-

-..

10T?RN AT OF Pl=TRO1

FIJJ&f TFf7HNOI.OC,Y

8/16/2019 SPE-2516-PA

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.

age dimensionless pressure drop. The cumulative

~ro@ow factors can then be obtained as the ratio

of the average pressure drop in the crossflow case to

that without crossflow at any given time.

We shall discuss in some detail the formulation and

numerical solution to the crossflow model, since the

model represented by Fig. 14 may be considered as

a special case of the crossflow model.

Model with Crossflow

Consider the system in Fig. 13 to be composed of two

regions, as follows:

Region 1—

the heated region, r,. < r < ’83 where

P

= p(z).

Region 2 —

the cold region, r, < r < r., where

v

= pi.

In Region 1, we must solve

. . . . . . . . .,

(A-1)

where

plD

may be regarded as the flow potential (or

pressure if gravity is neglected). It is convenient to

make the following substitutions:

Eq. A-1 now becomes

,-,r.a,,,

I

P

, ~ : ,p,j,

1

a’pl. ~ r,O- _ ___

~-2xD

ax,,’

h’ ‘&,,’ y

ai h

azD

— I

1

Pi

J

(A-2)

In Region 2, Eq. A-2 reduces to the normal two-

dimensional diffusion equation

~_2rD

?’pm ~

r,.’ a2p2D _ ?P21)

ax~’

—.—. . . (A-3)

F t?z,;’ at, , ,

Thus, we must solve Eqs. A-2 and A-3 subject to

the boundary conditions

STEAM ZONE BOUNOARY

c-

I

onxD=(),oszDsl,t2D> 0>- 0 -

(A-4)

p,~ =

1

onzD=l,o0> . .

(A-5)

aPID _  J

——

azo

onzD=(),()sxDgx8D>r2D >05 ‘ “

(A-6)

ah _

o

——

az.

onzD=oj x DO>. -

(A-7)

@2D _ o

——

azD

onzD =

1,x*D

‘~ . .

(A-8)

onxD=xeD, o

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2

JJ

X.D

:D=~

PD XDY zD e2’Ddx~zD “

—–lo o

rWz

(A-13)

. . . . ..”” ‘

The dimensiodess production rate may be obtained

by numerical dtierentiation of the cumulative-vs-time

results.

Model Without Crossflow

For the case of the model without flow in the vertical

direction, the partial diflerentid equation for Region

1 is

~zp:= _ u aD,n [A-1A)

e-2~D — —

L-. . . . .

ax~’

\ n . r

pi

at2D

and the equation for Region 2 is

e.,zD 32P2D _

@D

—.. . . . .

(A-15)

8x*’

 

The boundary conditions applicable to this case are

pA~= 1

onxD=o,o ‘Y “ “ ‘A-17)

8P2D _ o

——

ihD

)2

t ‘, ‘,

‘i

~,

08L LAYER ,.,c..~=N

‘~ET

—————

20

11–

— ,00

. . . . . . . . . .- 200

‘4 ‘\’”\

,./,.

02 I 3

\

\

,s/, ” . 441 I

\

\

\

,\

), \

\ ,,, \

\

\

‘, \

\

‘ \

\

‘,

\

\t’\

‘i

,..

\

‘

1. ~

I

----

-_tyJJ __ -

t

onxD=xeD, o Q o 0

(A-18)

PID

= P2D

onxD = X8D, < ZD < l,t2D>0t “ “ ‘A-lg

PID = P2D = O

On~lxD, zD; tD=O” “ “ “

(A-20)

q~mrficmlv one space dimension appears in Eqs.---- -—,

A-14 and A-15, the Crank-Nlcolson implicit pro-

~a~eo~n~e~=

cedure W= used h the soiuUOIi Of thk

tially the same computational scheme at each time

step was employed as for the crossflow model. Eq.

A-14 was first solved for a given horizonti plane

with the first and second boundary conditions. Then

Eq. A-15 was solved for each plane with the third

-- —-.

and fourth boundary conditions: Finally, Eq. A-13

was solved by numerical integration. The Crossflow

factors calculated in this manner are plotted on Figs.

15 through 17.

APPENDIX B

Initial Tempemtire Distribution

After termination of steam injection, the initial tem-

perature distribution in

the medhm adjoining the

steam zone depends upon radkl position. In order

to simplify calculations, we used a dktribution func-

tion that represents the same total amount of heat

loss from the steam zone and yet is independent of

radkd position. Such a function may be obtained in

the following mannec

If tilection heat equals 2,000 pSi8HStinj, then totid

heat lo~s~Uds

Q = 2000f%i8H,ti.j – iTr,2&&(~8 – ~i)

(B-1)

. . . . . ..” ““

The steam zone radius r, is obtained, for a given

steam-zone t.lickness h. > 0, from Marx and Lang-

enheim’s formula13 for steam-zone growth:

r, = 12.616

where

[

1

8i,H,h,plClv(7) %

(B-2)

(T. – 7’i)

Kp2c2 ‘ “

s

k

o,,. L,, E. T“(c NNESS

IN FEET

:

—  00

:5

.---- .------200

g

:

z+

W,w . 202 I 3

<

%

*, /, W . 2060

.

:3

.

s

r,

1.0

--------- ------- -

:2

-----

g

1-

------- ----

.

~

:,

‘-’l

,

1,1 11

,

1

1

111111

,

,

I II IILI

O,oz

I

,

lo~

IO*

w’

D,. C.5,01+ LESS TIME 19,, )

I

I

I

1

1 11 1

I

1

I

I

, I

, II

1

,

I

 

,

,o~

0,02

,0,

,04

.,”, ”S,ON LESS TIME I,,. )

Fig. N5-Curnulative crossflow factors for Wells A and B.

Fig. 16-Cumulative crossflow factors for Wells A and B.

8/16/2019 SPE-2516-PA

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—

d

(T)= e’efic V~- 1+2 ~ , . (B-3)

and

(B4)

7

=y~inj, . . . ...” “

with

4K&,

(B-5)

. . . . “

y = ~,z (Plc,)z “ “

(ti.j = steam injection time). In.this calculation i is

assumed that the thermal properties of the od-beafmg

Iavers adjacent to the steam zone and of the cap and

b~seock are equal.

For the case of h. =

O, then r, can be obtained

from”

[

p,i,H,

1

?4

= 18.952

tinj% .

r~

(T, –

Ti) ~=

(B-6)

. . . . “

. . . . ““

Let Q be distributed such tha:

Q = 2p2C,~r,2 ~ [T,(z) – Td dz, . (B-7)

where

z=lx\—h*/2 , . . . . .

and

T,(z) = Ti + T. – Ti) erfc 3

_–——— —

‘4

rr—r----”

(B-8)

B-9)

;

~,

:

“

.

2

,,, ,;

o

4

,0’

 0’

m~o, McN~, oNLE,, ,,ME , 20’)0

where the temperature was assumed to have the one-

dimensional error function distribution.”

From these equations it is found that

i ’ r c ’

(B-1O)

“ – 4cr2

——, . . . . . .’”

where

(B-n)

. . . .

. . . . .“

The initial dimensionless temperature distribution can

then be represented by

1X1 – hs/2

(B-12)

= erfc

c&””’”’

APPENDIX C

General Temperature Distr~bution

Subsequent To Steam Injection

After steam injection ceases, the temperature distribu-

tion may be represented, for purposes of computation,

by the result of Appendix B. Then> during ‘he ‘~~1

period and later,

the temperature dlstrlbutlon

change with time as heat leaks away from the system.

To represent this changing temperature distribu-

tion, the following assumptions are made:

1. Temperature within the resaturating steam zone

remains uniform across the height of the zone but

declines with time.

2. Heat loss occurs in the vertical direction only.

These conditions may be restated as follows:

Let

h, = s team-zone thickness ,

T, = steam-zone temperature.

If

Iz’=t+ td, . . . . . . . .

(c-1)

32T,

1.?L ()O

  c-3

. . . . . “

. . . . ““

 

,?JT, =_K ~

. .

 c- 4

—

plcl -=j-~

&y ICI=+ “

The solutions for T, and T, were obtained by means

of Laplace transforms.

The temperature

T, t)

in the

steam zone is as follows:

8/16/2019 SPE-2516-PA

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‘;(t)_–T~{ @(f+ ‘d) erfc ~y(t +

td)

8

1

, ; ::,~,(z,+ )e-+(t+td)

 

z’

 

erfc

1

Vy(t

+

(i) dz’ ,

2~a, (t + td)

. . . . . . . . . . .

(c-5)

where

4Kp2c,

(C-5a)

y = h,z(plcl)’ ‘ o “ “ . .

and f(z + h,/2) is the initial dktribution obtained

from Appendix B.

Temperature T,(z, t) in the adjacent oil-bearing

layers can be obtained from

T,(z, t) – Ti =

T, – T,

2v.a;t+ ,J’(z’+ +)

o

[

(2’-2)2

(2’+2)2

  -

4a*(t+f,j)

e

—e

1

4a2(t+$.i) dz, + e-+l’(~+td)

[

z

erfc

2~a* t + td

+ ~y t + td

1

m

2(2’+.2) pzcl

——

+

‘p’c’ j f(z’ + ~) e h xc, ~y(t+td)

pLhs

o

[

2’+2

1

~y t

d) dz’ ,

2~~,(t + td)

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

(C-6)

where z = l-xi—

h2/z.

For the case h, = O,Eq. C-6 becomes

T,(z, t) – Ti =

1

T, – T,

2~7raz t + td

co

H

(z_z, )2

(7+*,)2

_—

1

 

j z’

e-

4a2(’+d’+ e 4a2(’+d)

dz’

 

As with the calculations in Appendix B, it is as-

sumed that the thermal properties of the oil-bearing

layers adjacent to the steam zone and of the cap and

base rock are equal.

Since the value of the heat capacity

P,CI

of the

depleted (but resaturating) zone is changing because

of resaturation with oil, a time average value of plC1

has been used in Eqs. C-5, C-5a, and C-6 above. This

value was computed for each time

t

as follows.

Before fillup of steam zone,

tfill — t

 plcl av = * P2C2 + —

tfill

pl c,“. .

(C-8)

After fillup of steam zone,

(plcl)w = ;

[

1

P~C~(t– tfi]] + tfill plcl .

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

 c- 9

If expansion of the cold layer adjoining the steam zone

is insutiicient to resaturate the steam zone, then

. . . . . . . . .

(c-lo)

where

Q~i,l = oil volume to resatumte the steam zone

= ~(rs’ – rW2) h,AS~ , . . . (C-11)

Q,,, = oil produced by expansion of the cold

layer

Original manuscript received in Society of Petroleum Engineers

office June 30, 1969. Revieed manuscript received Feb. 9, 1970.

Paper SPE 2516) was presented at SPE 44th Annual Fall Meeting,

held in Denver, Colo., Sept. 28-Ott. L 1969. 0 copyright 1970

American Intiitute of Mining, Metallurgical, and petroleum En-

gineers, Inc.

 c- 7

This paper will be printed in Transactions volume 249, which

.:. . . . . . . . . . .

will cover 1970.