Calculation of the Production Rate of a Thermally Stimulated Well T. C. BOBERG R. B. LANTZ JUNIOR MEMBERS AIME
ABSTRACT This paper presents a method for calculating the producing rate of a well as a function of time following steam stimulation. The calculations have proved valuable in both selecting wells for stimulation alld ill determining optimum treatment sizes. The heat transfer model accounts for cooling of the oil sand by both vertical alld radial conduction. Heat losses for any number of productive sands separated by unproductive rock are calculated for the injection. shut-in and production phases of the cycle. The oil rate increase caused by viscosity reduction due to heating is calculated by steady-state radial flow equations. The response of Sllccessive cycles of steam injection call also he calculated with this method. Excellent agreement is shown betweell calculated and actual field results. Also included are the results of several reservoir and process variable studies. The method is best suited for wells producing from a multiplicity of thin sands where the bulk of the stimulated production comes from the unheated reservoir. The flow equations used neglect gravity drainage and sall/ration changes within the heated region.
INTRODUCTION This paper presents a calculation method which can be used to predict the field performance of the cyclic steam stimulation process. The calculation method enables the engineer to select reservoirs that have favorable characteristics for steam stimulation and permits him to determine how much steam must be injected to achieve favorable stimulation. While the calculation represents a considerable simplification of physical reality and the results are subject to numerous assumptions which must be made about the reservoir, it has been found that realistic calculations can be made of individual well performance following steam injection. The duration of the stimulation effect will depend primarily on the rate at which the heated oil sand cools which, in turn, is determined by the rate at which energy is removed from the formation with the produced fluids and conducted from the heated oil sand to unproductive rock. A complete mathematical solution to this problem is a formidable task, and finite difference techniques would undoubtedly have to be used. The calculation method preOriginal manuscript received in Society of Petroleum Engineers office July 8, 1966. Revised manuscript received Oct. 31, .1966. Pa~er (SPE 1578) was presented at SPE 41st Annual Fall Meetmg held m Dallas. Tex., Oct. 2-5, 1966. © Copyright 1966 American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. DECEMBER, 1966
ESSO PRODUCTION RESEARCH CO. HOUSTON, TEX.
sented here utilizes analytic solutions of simple related heat transfer and fluid flow problems. The method is sufficiently simplified that it can be used as a hand calculation, although the calculations are somewhat lengthy and laborious. For that reason, the analysis was programmed for an IBM 7044 digital computer. Well responses observed at the Quiriquire field in eastern Venezuela" have been matched using this program after making suitable approximations for reservoir and well bore conditions. One of the most valuable uses of this calculation method is to assess the effect of reservoir and proccess variables on the stimulation response. This paper contains results of several studies made of key reservoir and process parameters. Among the most important of these is the influence of prior well bore permeability damage. If a well is severely damaged prior to stimulation, a higher stimulation response will be observed than if it is undamaged. If a portion of this damage is removed. a permanent rate improvement will occur. THEORY DESCRIPTION OF CALCULATION METHOD
The process of cyclic steam stimulation is essentially one of reducing oil viscosity around the well bore by heating for a limited distance out into the formation through the injection of steam. Suitable modifications of the calculation technique presented here can be made so that stimulation of wells by hot gas injection or in situ combustion can also be calculated. A schematic drawing of the heat transfer and fluid flow considerations included in the calculation method is shown in Fig. 1. In brief, the calculation assumes that the oil sand is uniformly and radially invaded by injected steam. For wells producing from several sands, each sand is assumed to be invaded to the same distance radially. In calculating the radius heated rh , energy losses from the wellbore and conduction to impermeable rock adjacent to the producing sands are taken into account. After steam injection is stopped, heat conduction continues and oil sands with r < r" cool as previously unheated shale and oil sand at r> rh begin to warm. The effect of warming of oil sand out beyond 1'" has little effect on the oil production rate compared to the effect of cooling of the oil sand nearer the wellbore than rho Thus, in computing the oil production rate, an idealized step function temperature distribution in the reservoir is assumed where the original temperature exists for r > rh and where an average elevated temperature exists for r < rho The average temperature in the oil sand for the region r < 1'" is computed as a function of - - - - - - - -_._-
9References given at end of paper. 1613
time following termination of steam injection by an energy balance. From the average temperature, the oil viscosity in this region is determined. The oil production rate is calculated by a steady-state radial flow approximation which accounts for the reduced oil viscosity in this region.
casing radius; for insulated tubing. 1', may be roughly approximated as the inside tubing radius. The average down-hole steam quality X i for the entire steam injection period is then
X,
SIZE OF THE HEATED REGIO'.;
= X""'f -
_ 2r.DKr,
Q" -
.( T,-T +2 aD) I r
--------~--------~-
(1)
a
where I is read from Fig. 2 as a function of dimensionless time at, 11',". For uninsulated tubing or the case where steam is in direct contact with the casing wall, r, is the inside
-./VVVv
H EAT CONDUCTION OIL SAND SHALE OIL SAND
SHALE OIL SAND
(2)
Heated Radills
During steam injection the oil sand near the well bore is at condensing steam temperature L. the temperature of saturated steam at the sand face injection pressure. Pressure fall-off away from the well during injection is neglected in this analysis. and T, is assumed to exist out to a distance rio where the temperature falls sharply to Tr, the original reservoir temperature. In reality, the temperature falls more gradually to reservoir temperature because of the presence of the hot water bank ahead of the steam, but this is neglected to simplify the calculation. The heated zone radius is calculated by the equation of Marx and Langenheim: In the case of multisand reservoirs, if it is assumed that each sand is invaded uniformly as though all sands had the same thickness h and were invaded by equal amounts of steam:
r.' _ _ FLOW OF OIL, WATER & GAS
.
, If,!
Wellbore Heat Losses
To calculate the size of the region heated by steam, it is necessary to estimate the quantity of heat actually injected after well bore heat losses are taken into account. Various methods are available for estimating wellbore heat losses. ]0,11 A simple method which assumes a constant, average temperature of the injected steam and an average initial geothermal gradient computes the cumulative energy lost during injection Q" as:
MQ'J"
= hM.
(XiII", + h, .. - /z".) 'f, . 4Kr.(T, - T,.)t,N,
(3)
The function l. is plotted as a function of dimensionless time T = 4Kt,! h'(pC)] in Fig. 3. The use of·this equation for multisand reservoirs also assumes that injection times are sufficiently short and that interbedded shale is sufficiently thick that no heating occurs at the mid-plane of the shale during the injection period. Eq. 3 further assumes that the value of average density times heat capacity pC for the barren strata is the same as that for the oil sands [(pC),h" Ie = (pC),]. For multisand reservoirs, a heated radius for each sand r"i could be calculated assuming equal steam injection per foot of net sand. An average heated radius rh could then be computed from rio' = !h,r,,,'/!h,. Practically speaking. the h ' for reservoirs consisting of more than three sands of reasonably uniform thickness can be approximated by using Eq. 3.
r
TEMPERATURE HISTORY OF THE HEATED REGION (r", r r.)
< <
The average temperature Tav. of the heated region (or FIG. I-SCHEMATIC REPRESE]'o;TATlOl'i OF HEAT TRANSFER Al'iD FLUID FLOW CALCULATED BY MATHEMATICAL MODEL.
IOr-----------------------------------------~
t ~
'"~-10 F
-
====:::;...-:
0.01 ';;;-----~--_;:_:--------__:__=_-----~~--------~ 0.01 0.1 1.0 10 FIG. 1614
2-1
FACTOR FOR WELLBORE HEAT
Loss
DETERMINATION.
FIG. 3--DIMENSIONLESS PUSITlOl'i OF STEAMI:D·OUT REGION. JOtTR'iAL OF PETROLEUM TECHl"OLO(;Y
regions in the case of a multisand reservoir) after termination of steam injection is calculated from an approximate energy balance around the region r" < I' < r h :
=
T,,"
+ (T,
T,
- T..) [v,~, (1 - 8) - 0], OF
(4)
In Eq. 4, ~. and ~ are unit solutions of component conduction problems in the radial and vertical directions, respectively. 8 is a correction term which accounts for the energy removed from the oil sand by the produced oil, gas and water. If little energy is removed by the produced fluids (the case of a low rate well), 8 can be neglected and Eq. 4 reduces to a product solution for the average temperature of a series of right circular cylinders of radius r" conducting three-dimensionally to initially unheated rock. The development of Eq. 4 is given in the Appendix. Values for V,' and ;;-, for the case of production from a single sand may be read from Fig. 4. Relations from which values of y, and Y, may be calculated are developed in the Appendix (Eqs. A-8 and A-16). ENERGY REMo\ED \\ITH THE PRODUCED FLums
The quantity 8 in Eq. 4 accounts for the energy removed from the formation with the produced fluids and is defined by:
If I
8= 2
Hfdx .. ZTrr,,'(pC),(L _ T..)' dlll1enslOnless .
(5)
I,
Eq. 4 must be solved in a stepwise manner since H" the energy removal rate, is a function of T""g. Because approximations are used in deriving Eq. 4, as 8 approaches unity 1',,," calculated by Eq. 4 can hecome less than T, .. This is physically impossible. When it happens, T,," should be taken as equal to T,. H" the heat removal rate with the produced fluids, is given by:
Hf
=
q""(H..,,
+
H,,), Btu/D
(6)
where H"" = [5.61(pC)"
+
R"c"](T,," - T,.), Btu/STB oil
(7) H" = 5.61 p,,[R,,(/lr - h r,.)
+
R",. h f .,], Btu/STB oil (8)
R
wv
=
0000 • 1356
(P"")R ". bblliquid water at .60F , bbl stock-tank 011
£I ... - P ... ,·
(9) when P" > p ... ,. and R".1' < R ... , R ... ,. = R" when p"", > p,,; if R"., calculated by the above formula is greater than R,,, then R"., = R"." 1.0
;:::-
0,8
r--.
~III ~
.........,
0.6
"'-
rh
"-
"
0.2
Vr
I ~
IIIII
I I III
_ _ 41l(t-t j )
Vz :8=---2-
IIIII I ~111 r VZ -II~I~GL~ ~AI~~)
'"
0.1
Steady-State A pproxill1ation for the Productivity Index
For a given stage of reservoir depletion, the ratio of the oil productivity index (q,J 6..P) in the stimulated condition i" to that in the unstimulated condition J e can be estimated by an equation of the form: - _ i" _ 1 J-----J, {t""
p.,,,
DEeEMIlER, 1966
c,
, dimensionless
(10)
+ c,
The geometric factors c, and c, include the pattern geometry and the skin factor of the well. For preliminary engineering calculations if no change in permeability occurs, suitable values can be calculated from the formulas given in Table I, which assume that r" « r, and r,r « r,. The effect of prior well bore permeability damage is accounted for by using the effective well radius r" in the relations (Table 1). The effective well radius is related to the and skin factor S by the equation actual well radius
r"
(11) Implicit in the assumptions used to derive Eq. 10 is the assumption that heating and fluid injection have a negligible effect on the permeability to oil. To adequately predict changes in the permeability to oil during the injection and production phases, prediction of two-dimensional, three-phase behavior is required. The complexity of adding these equations would undoubtedly necessitate the use of finite difference techniques to arrive at a solution and are beyond the scope of the simplified model proposed in this paper. When swelling clays are present, steam might reduce the permeability, although cases of plugging attributable to steam injection in permeable unconsolidated sands are apparently rare. Influence and Significance of the Skin Factor
The skin factor S has a large effect on the stimulation response. The greater the skin factor, the greater c, will be relative to c,; hence, the greater will be the influence of the heated to cold viscosity ratio {tu'./P-., on the ratio of stimulated to unstimulated productivity indices I. This can be seen by referring to Eqs. 10 and 11 together with the defining relations for c, and c, given in Table 1.
.......
1.0
TABLE I-EXPRESSIONS FOR c, AND c, USED IN EQ. 10 FOR CASE OF NO ALTERATION OF PERMEABILITY BY STIMULATION
"-.....
;;r
c,
System
10
10C
Radial-Pe Constant
8 - DIMENSIONLESS TIME FIG. 4--S0LUTION FOR
OF THE OIL RATE
For conventional heavy oil reservoirs which have sufficient reservoir energy to produce oil under cold conditions, the use of steady-state radial flow equations appears to be adequate for predicting the oil production rate response to steam injection. These equations are inadequate for tar sands and pressure-depleted reservoirs where the bulk of the stimulated production must come from oil sand which is actually heated rather than from the portion of the reservoir that is still cold. The equations outlined in the following section for this latter case will predict an unrealistically low rate response until the heated region itself has become depleted or reduced in oil saturation.
",-
I
0.01
tj
Vr: 8=--2-
0.4
o
Il(!- l)
CALCULATlO~
AN[);;z (SINGLE SA;,\[).
I"(rhl rw J/ In(r,,/r Ie) In ( -
'I, )
r
Radial-pi' Declining
U'
'I,' 2r,--
--"
I"(rhl TtIJ )/ln(re/ru:)
(")
In -
Th
-
"," V, +-. 2 :J
re
In("I,"·)-'h 1615
Another way to understand the effect of S is to consider its relationship to k,j and rd, the permeability and radius of the region around the well bore where permeability differs from the formation permeability k:
r" k - J ) In-=-. S = ( -k d rtl If kd is small relative to k and r" large positive value.
TABLE 2-STEAM STIMULATION TEST DATA FOR WelL Q-594 Reservoir Characteristics
Net sand thickness, ft Number of sands Oil viscosity. cp
(12)
> r".,
4,050 470 183 16 133
Depth. It Section thickness, ft
Pre-stimulation
then S will have a
Alteration oj Skin Factor hy Heating
Oil rate, BID WOR, bbl/bbl GOR, scflbbl
135 0.83 985
Stimulation First Cycle
In some wells asphaltene precipitation may be a cause of high skin factors. If this and similar damage can be alleviated by heating, suitable modification of c, and c, can be made. For the constant p, case where the skin factor S is reduced to S, following stimulation and r" > rd: c,
+ In rh/r". = S, S + In relr",
1n r,/rh ; c:, = -----'---
S
+
In rJf",
Determination of the Oil RaIl'
To obtain the oil rate as a function of time. It IS necessary to know the unstimulated productivity index J, and the static reservoir pressure p, as a function of the cumulative fluid withdrawals. Then the stimulated oil rate q"h can be calculated:
J J,.
:'P. STB/D
(14)
where T is determined from Eq. 10; J, is obtained from an extrapolated plot of the productivity index history of the well prior to stimulation; and :'P the pressure drawdown is Pe - p", during the period of stimulated production. The static pressure p, can be estimated from an extrapolation of a plot of p, vs cumulative oil produced prior to stimulation for those cases where the pressure maintenance benefits of the injected fluids can be neglected. ESTIMATION OF PERFORMANCE FOR SUCCEEDING CYCLES
To calculate the performance of cycles following the first, account must be made for the residual heat left in the reservoir during preceding cycles. The energy remaining in the oil sand after preceding cycles can be approximated by: Heat remaining
= u,,'(pC)J1N, (T,,," -
T,.)
Injection time, including shut-in, days Durat,ion of cycle including injection, days
Stimulated producing time. days Actual oil production, bbl Calculated oil production, bbl Estimated cold production, bbl
(13)
A similar form can be written for the case where p, is declining.
qoh =
Wellhead iniection pressure, psig
(15)
An approximate method of taking this energy into account is to add it to that injected during the succeeding cycle while assuming that injection takes place into a reservoir that is at original temperature T,. This additional energy will result in new cycles having a larger heated radius for the same injection time and steam injection rate. The major assumption involved in this approximation is that interbedded shale and overburden and underburden are at initial reservoir temperature at the beginning of each cycle. This is a conservative assumption since calculated heat losses for cycles after the first will be higher than those actually observed. A more optimistic analysis would assume that all of the energy remaining, including that in the shale, is added to the energy injected on the succeeding cycle.
Second Cycle
19.2 800 55 354 288 47,813 53,700 30,400
18.1 770 46 487 378 80,803 79,140 47,600
Steam injected, MM Ib
RESULTS MATCH OF FIELD TEST PERFORMANCE WITH CALCULATIONS
A stimulation field test conducted in the Quiriquire field of eastern Venezuela provides an excellent check of the utility of the calculation method. Comparisons of actual to calculated results are presented for the first two cycles of steam stimulation of Well Q-594 which produces from 16 sands in a 470-ft interval (Table 2)." A comparison between calculated and actual oil production rates after stimulation for Well Q-594 is given in Fig. 5. It should be noted that the calculated curve is a match of observed data rather than a true prediction. The observed pressure drawdown. down time and water-oil ratio were used as input data for the calculations. The only skin effect used in the match of the production history of the well was an effective skin due to the perforated casing. The effective well radius used in the calculation was 0.00176 ft compared to the actual casing radius of 0.292 ft to account for a perforation density of four holes per foot. A plot of the effective radius for various perforation densities based on Muska!' is illustrated in Fig. 6. The external drainage radius r, was estimated as 570 ft. The estimated decline in static pressure p, and cold productivity index J, as a function of cumulative oil produced following steam injection is given in Table 3. The water cut for this well following stimulation is given in Ref. 9. The agreement of the calculated and observed rate history for the first cycle is excellent. Calculated cumulative oil produced at the end of the first cycle differs by less than -ACTUAL ---CALCULATED DATA: SEE TABLE 2
400,----------------------------------, >-
~ 300
........
....
III
~200
...... : 100....
(5
OLi__
o
~~_LilWLll
200
__
400
~~
__________
600
800
~
__
~
1000
TIME SINCE START OF FIRST STEAM INJECTION - DAYS FIG. 5-C0Il1PARISOl'\ OF CALCULATED AND ACTUAL OIL RATES AFTEft STEAII1 INJECTlOl'\ FOR QUlRIQUlRE WELL Q-594.
JOliR:\AL ,oFPETIIOLEUM TECHl\Ol.OGY
3 per cent from the cumulative oil actually produced from the well. Temperature profiles obtained some time after production had been initiated indicated reasonably uniform heating of all 16 sands. Temperatures of highest and lowest sands were cooler than the middle sands; however, the average temperature across the producing interval agreed closely with the value of T,,," calculated by this method. The difference between the observed and calculated oil rate for the second cycle of the Quiriquire well is greater. Observed cumulative oil at the end of the second cycle differs by about 10 per cent from that calculated. However, agreement between calculated and observed performance is still adequate for engineering estimates. PROCESS
YARIABLE
STUDIES
The calculation method described here has been used to study the effects of several reservoir and process variables on the steam stimulation process. Studies of this type provide preliminary estimates of the energy required to achieve desired stimulation levels and additional insight into which variables have the greatest influence on the production response of a steam-stimulated well (Figs .. 7 through 10). The incremental oil-steam ratio has been selected as the primary dependent variable for these studies since it can be directly related to the economics of the process. The incremental oil-steam ratio is defined as the ratio of the stimulated less primary oil production to the amount of steam injected expressed as barrels of condensate. The values of this ratio are maximum values which occur when the computed oil production rate has returned to the extrapolated value of the cold rate. The influences of _.
_.-
~-.
-
--
I
---+~~--+-
./
/
:
I
:
i
/
...
I-
/
I
...
/
~
I
:
VI
oct ar:: -'
'/ [7
~-+-
::::>
IO-J
-' w
~
/
w
> ~
/
u
...... w
/
1'1
w
I
Static
Index
(M bbl)
Pressure (psio)
(B/D/psi)
0 100 200 300 400
490 410 330 250 170
0.312 0.281 0.256 0.231 0.206
skin damage, sand-shale ratio, stabilized water-oil ratio, pre-stimulation oil production rate, gross sand thickness, steam injection rate, total heat input and back-pressuring of the well early in the production phase on the maximum incremental oil-steam ratio are discussed in the following sections. Effect of Skin Damage
The amount of skin damage present in a well prior to stimulation can have a tremendous effect on the production response of the well when it is steam stimulated. This is true even if no well bore cleanup (i.e., damage removal) is obtained although the stimulation benefits are greater when well bore cleanup is achieved. The effect of damage became evident when stimulated production histories for some California wells became available which showed better than 20-fold increases in productivity following steam stimulation."'" This compared with three- to four-fold calculated increases which would be predicted by steady-state flow formulas assuming no permeability damage. As an example, the effect of different assumed skin factors on the stimulation response of a hypothetical well (Well A) is shown in Fig. 7. Reservoir and fluid properties used in the example are shown in Table 4. For all wells in Table 4, a temporary increase in the water-oil ratio following steam injection was assumed. This increase is shown in Fig. 8 as a function of the cumulative water produced divided by the cumulative water injected. This plot represents the average of actual data from two Quiriquire wells. The calculated curves in Fig. 7 assume that the wells in question have no change in the skin factor following steaming; thus, production returns to the pre-stimulation rate. A permanent improvement in productivity would result if post-stimulation skin factors were reduced.
I
I
Effect of Oil Viscosity
!
For a given temperature rise, the viscosity reduction of a low viscosity oil is much less pronounced than for a high viscosity crude. Thus, peak stimulated oil rates following steam injection will be smaller, the lower the original oil
, ,
o lOOOr---,---,---,----,---,---,---,----,--,
T rw= 0.25 FT I
Cold Productivity
Cumulative Ojl Production
,
/
--
I
TABLE 3-ESTIMATED DECLINE IN STATIC PRESSURE AND COLD PRODUCTIVITY WITH CUMULATIVE OIL PRODUCED FOLLOWING STEAM INJECTION fOR WELL Q·594
~
I PERF. RADIUS = 0.25 IN.
o
~ 800r-----+---+---+----~- -+--+---.+--, ~
i
~ 600r----+--1-+-"<--+--~---+----L--~.. -· ,
z
-_.
._-_._,
, I
'--"'---
i
,
o
5
I
i
I
Ti
2
4
6
i I,
8
400f---+--++-~""'
DATA GIVEN IN TABLE 4
;:)
o
o
'" .... ~
i 10
200r----+-~+---~--~--~~~---
STEAM INJ
""'~;;;;;;;;;;;;;;;;:;:;:::::d i
(5
NO. OF HOlES/FT FIG. 6--EFFECTIVE \VELL RADIUS FOR PERFORATED CASED HOLE (AFTER MUSKAT'). DECEMIIEII.1966
FIG. 7-EFFECT OF SKIN DAMAGE ON STEAM STIMULATION RESPONSE. 1617
TABLE 4-STEAM STIMULATION TEST AND CALCULATION DATA
Reservoir Characteristics Depth, It Section thickness, ft Net sand thickness, ft
Well A (Fig. 7)
Well B (Fig. 9)
867 546
3,000 200 67
335 7
Number of sands
6
Well C (Fig. 10)
Effect of Sand-Shale Ratio
Well D (Fig. 11)
3,740 200 to 1 ,088 234 18
4,000 1,400 467 36
Reservoir temperature,
of
Oil viscosity, cp: at TI"
at 300F Skin factor Effective well radius,
ft'
97
100
120
125
900 13 20 to 60
4010 1,000 1 to 6
133 8
70 3
o
o
o
0.45
0.00176
0.00176
0.25
30
300
99
230
0.128 0.35 63
1.0 1.0 1,000
0.3 0.57 600
0.5 0-1.0
6.9
40
16.6
42 to 126
Pre-stimulation
Oil role, BID Oil productivity index, B/D/psi WOR, bbl/bbl GOR, scf/bbl
soo
Stimulation Steam injected, MM Ib Wellhead injection
surface conditions Pressure, psig 440 780 770 Temperature, of 450 520 518 Steam quality, dimensionless 1.0 0.95 0.95 Injection time, days 21 80 55 Shut-in time, days 13 4 5 *Includes effect of well completions, perforations, etc., jf any.
The effect of the ratio of sand to shale thickness on the incremental oil-steam ratio for a single cycle is depicted in Fig. 10 for the conditions assumed for Well C in Table 4. The net sand thickness and number of sands were held constant in these calculations. and the gross section thickness was varied to vary the sand-shale ratio. The decline in the incremental oil-steam ratio as the sand-shale ratio decreases (increasing gross section thickness) is the result of increased heat losses to the interbedded shales. These greater heat losses result in an accelerated temperature decline; hence, an increase in the rate of decline of productivity occurs. The above conclusions about sand-shale ratio hold only as long as the average individual sand thick· ness is roughly constant. Thicker sands will cool more slowly and the stimulation response will last longer for the same sand-shale ratio than will be the case for thin sands. Effect of Post-Stimulation Water-Oil Ratio And Gross Sand Thickness
1,500 600 0.95 18 to 54
3
viscosity. However, it is not obvious how much less incremental oil will be produced over the entire cycle length since the lower heat removal rate with the produced fluids will cause stimulated production to last longer for the lighter oil. If there were no heat losses from the heated portions of the oil sands, the increased cycle length for the lower viscosity oil should give the same incremental oil recovery as for the heavier oil although at a reduced stimulated rate. Consequently, heat losses to unproductive rock are important in evaluating the effect of oil viscosity on incremental oil recovery. The effect of initial oil viscosity is shown in Fig. 9 for Well B of Table 4. These results indicate more than a 50 per cent increase in incremental oil recovered for a 1,000cp oil over a 40-cp oil.
The effects of post-stimulation water-oil ratio and level of steam input per foot of gross thickness are shown in Fig. 11 for a well having the conditions indicated for Well D in Table 4. Fig. 11 indicates that the incremental oil-steam ratio depends markedly on the quantity of steam injected per foot
iii 1.8 c:a
'..... ~
1.6
g
1.4
« CI:
:E 1.2
«
t;;
~
'..... 1.0
~
o 4.
0.8
~
...:E ...
~
CI:
~<: Ow CI:::l
u..:c(
CI:>
Oz 0.8 ~O z~
-c( W-A
C)::l
z~
I
1.6 1.2
0.4
I
\ \'
:I:ti CI: 0..
~
0.0
I
I i i 1
!
I
1.
1
!
i
i
i ,
1000
I
~
I
I
I
-T--t--'-
~ 0.5 ..."'al
1
!
I II I
J.
I
I
I
.1
OIL VISCOSITY - CP
i
I
.1
100
FIG. 9-EFFECT OF VISCOSITY 0:\ bCREMENTAL OIL/STEA~1 RATIO.
I
1
Ii
i
0.0
i
~ r-- 1
I
-0.4
I
!
1
I
0.4 40
I
I
I
0.6
CI: U
I
c(-
uJJ
BASIC DATA GIVEN TABLE 4
Z
2.0
~
~
-'
~~0.4 Oal
!
~ ~ 0.3
I
-+-- L
__~,_;~
-"11F---+--t----+-~____t__-~-~
"'0
~ i= 02 ~ « . ... co: co: u 0.1 f--i-----j
I I
DATA:!m
~ABl~ 4
l
z
0.2
0.4
0.6
0.8
1.0
1.2
CUM. WATER PROD.jCUM. WATER INJ. FIG. 8--EFFLCT OF STEAM STIMULATION ON WATER-OIL RATIO. 16111
FIG. 100CALCULATEU EFFECT OF SANU/SHALE RATIO 0:\ INCREMENTAL OIL/STEAM RATIO.
JOllIC\AL OF I'ETItOLEl·!\t TECHNOLOG\'
of gross section and on the stabilized water-oil ratio. The detrimental effect of a high pre-stimulation water-oil ratio is a consequence of the high heat capacity of the produced water. Since the heat capacity of water is approximately twice that of the crude oil. a high water-oil ratio results in a high rate of energy removal as fluids are produced from the formation. High producing gas-oil ratios will also have a detrimental effect on stimulation benefit. Both of these effects can be seen from Eqs. 8 and 9. Under high temperature conditions a high gas-oil ratio can remove a large amount of water in the vapor phase accompanied by a high latent heat of vaporization. Consequently, both high water-oil ratios and gas-oil ratios cause a more rapid decline in reservoir temperature. Thicker sections will require a proportionately greater input of steam to achieve a given stimulation response, all other variables being equal. The curves plotted in Fig. 11 apply specifically only for the conditions of Table 4 (i.e., an undamaged well having the indicated cold productivity with negligible cold rate decline expected over the cycle life, and a very thick section where heat loss rates for the many individual sands can be assumed essentially equal). Process COlltrol Variables
Three variables relating to process control are the rate of steam injection, the total heat input and the degree of back-pressuring of a well early in the production phase of a stimulation cycle. Within the limits of injection pressure considerations, it appears to be beneficial to inject the steam at as high a rate as possible. The high injection rates provide two benefits: (1) well bore heat losses as a percentage of total heat injected are reduced and (2) a given amount of energy can be injected in a shorter period of time, thus reducing the lost production while the well is being steamed. It should be noted in Fig. 11 that as the cumulative steam input is increased, the incremental oil-steam ratio curves pass through a maximum and begin to decline at higher steam input. Thus, there appears to be an optimum level of steam input for a given set of operating conditions, and it should be possible to optimize the length of steam injection to maximize the incremental oil-steam ratio. The -' 2.4 CD CD
ms/hn = STEAM RATE (LB/HR) /NET FT COLD OIL RATE 0.5 (STB/D)/FT
.......
-' CD CD
2.0
-
.
,
=
-~-.--------.--
----~
IX
---
-------
m s/h n =1500
0
i=
-
1.6
~
....
on
....... -' (5 -'
1.2
m s/h n =1500 ms/h n = 500
.S
....~
CONCLUSIONS I. The simplified calculation method presented herein can match history for conventional heavy oil wells and can be used with confidence to make preliminary field selection and process variable studies for these cases. Steadystate radial flow equations, while adequate for predicting steam stimulation response for most conventional heavy oil wells, should not be used for tar sands and depleted reservoirs. For these cases the bulk of the oil production comes from the region actually heated rather than from the unheated region. 2. Process variables studies using the calculation show that (a) wells having a high skin factor prior to stimulation will respond most favorably to steam stimulation; a permanent rate improvement results if heating removes a portion of the skin; (b) low produced gas-oil and water-oil ratios, high steam injection rate, high sand-shale ratio, thick sands and high original oil viscosity benefit the stimulation effect; (c) thick sections require a proportionately greater energy input to achieve a given incremental oil-steam ratio; (d) flashing of produced water which causes rapid cooling and deterioration of the stimulation response can be avoided by back-pressuring the well early in the producing cycle. This will permit more incremental oil to be produced than would be attained if drawdowns were maximized throughout the entire cycle. '::This is true while the heated radius Th is still much smaller than the drainage radius reo As rh becomes nearly equal to rtJ, productivity increases rapidly with increasing Tit again.
~
Z ....
maximum in the incremental oil-steam ratio is the result primarily of four factors: (I) increased heat losses associated with the larger heated radius and longer cycle times which resulted from higher energy inputs. (2) increased lost production as the length of the injection period is increased, (3) a lower rate of increase of the heated radius with energy injected as the heated radius becomes large. and (4) a diminishing incremental benefit to the productivity index by further increasing the heated radius. * The back-pressuring of a well early in the production phase of a stimulation cycle can result in substantial increases in the calculated cumulative oil produced at cycle end. Back-pressuring prevents or minimizes the flashing of produced water to steam which removes large quantities of energy which would otherwise be available to prolong the stimulation effect. A pumping well can be back-pressured by one of two methods. First. the annulus pressure can be controlled manually while the well is pumped off. Second, the well can be back-pressured more or less automatically by the column of liquid that will exist above the pump when pump capacities are rate limiting. Table 5 illustrates the effect of back-pressuring by the pump limiting method on the computed results for Well Q-594. Probably the optimum program of back-pressuring a well would be one in which the flowing bottom-hole pressure is maintained slightly above the saturation pressure for steam at the existing bottom-hole temperature. This would provide the maximum drawdown possible without flashing a large fraction of the produced water to steam.
t
IX
U
~
I
0 0
I
200 150 250 100 50 mst;/h - M LB STEAM/FT OF GROSS INTERVAL
FIG. II-THEORETICAL PREDICTlO'i OF Il'iCRDIE'iTAI. OlJjSTEA~I RATIO \"5 STEA!\OI }l'iJECTED. UE(;E'IIIIER. 1966
TABLE 5-EFFECT OF BACK· PRESSURE DURING THE EARLY PART OF PRODUCT ION PHASE
300
Calculated Cumulative Inc. Oil ot Cycle End. STB
Case
Method of Back·Pressuring
(Well Q·594)
2
Pump limited No bock-pressure
35,000 17,000 1619
NOMENCLATURE
r,
=
well radius used in Eq. 1 for well bore heat loss calculations, ft
rd
=
r,
= = =
radius of the region of damaged permeability k d , ft drainage radius of well, ft
a = geothermal gradient, °F/ft b
=
square root of dimensionless time for radial temperature decay (Eq. A-5, dimensionless) B, = constant appearing in Eq. A-14, ft c"
C,
C"
= =
constants appearing in Eq. 10, dimensionless (Table I) average specific heat of gas, over temperature interval T,. to T",", Btul scf, F 0
c,,,
C.'
=
average specific heats of oil and water over the temperature interval T" to T .. ,.,., Btu/lb of " ,
D
=
h
=
depth of the producing formation, ft average thickness of the individual sand mem_ bers, ft
hi
= enthalpy
of liquid water at T,,," above 32F.
r. r,
=
specific enthalpy of vaporization of water at T Btu/lb
rw
=
actual well bore radius, ft
R"
=
total produced gas-oil ratio. scflbbl at stocktank conditions
Rw
=
total produced water-oil ratio, bbll bbl at stocktank conditions
.lR w = Rw - Roo, bbllbbl at stock-tank conditions R"o
=
R"" =
DV" ,
hiT = specific enthalpy of liquid water at T" Btu/lb hIs = specific enthalpy of liquid water at T" Btu/lb h, = individual sand thicknesses, ft
h, = artificially increased sand thickness used in Eq. A-14, ft HI = rate at which energy is removed from the formation with the produced fluids at time I, Btu/D
Ho. = [5.61(pC)" + R,Ic',l(T,,," - T,.), Btu/STB oil Hon = 5.61 p,,[R ... (h I - hI) + R ... ,/zIgl, Btu/STB oil I = dimensionless factor read from Fig. 2 as a function of aI, I r,,' J = ratio of stimulated to unstimulated productivity indexes. dimensionless J., J" = stimulated (hot) and unstimulated (cold) productivity indexes. respectively, STBI D/psi J" = zero order Bessel function of the first kind J, = first order Bessel function of the first kind k = formation permeability, darcy kd = damaged permeability, darcy k" = oil permeability, darcy K = formation thermal conductivity. Btu/ft/D/oF I, = thickness of an individual interbedded shale. ft IJ = artificially reduced shale thickness used in Eq. A-14. ft M. = total mass of steam plus condensate injected in the current cycle, Ib N, = number of individual sand members P. = static formation pressure existing at a distance r, from the wellbore. psia P .. = producing bottom-hole pressure, psi a Pm, = saturated vapor pressure of water at T",", psia q. = oil production rate, STB/D qQ' = oil production rate during stimulated production, STB/D Q", = cumulative energy lost from wellbore during steam injection, Btu r = radial distance from the well bore, ft 1620
inside tubing radius, ft
r w = effective well bore radius, ft
Btu/lb
hi.
radius of region originally heated, ft
Sk
=
S = S, = I
=
t, =
1',,,"
=
T, =
T, =
v
=
v
= =
v" v,
v" V, = V =
normal (unstimulated> water-oil ratio, bbllbbl at stock-tank conditions water produced in the vapor state per stocktank bbl oil produced, bbl water vapor (as condensed liquid at 60F)/STB kth term in the series solution for v" dimensionless skin factor of well, dimensionless skin factor remaining after a stimulation treatment, dimensionless time elapsed since start of injection for the current cycle, days time of injection (current cycle), days average temperature of the originally heated oil sand at any time t, OF original reservoir temperature, OF condensing steam temperature at sand-face injection pressure. OF temperature difference at r, Z and I above initial reservoir temperature, OF T,,,," - T", of unit solution for the component conduction problems in the rand z direction, respectively. dimensionless integrated average of v" v, for 0 < r < rio and all h" respectively, dimensionless T, - T,., OF
w = constant appearing in Eq. A-16 equal to 4a (t - t,). sq ft
W, = constant appearing in Eq. A-16, ft
Xi = X,,,,(
=
y
=
Yo =
= z=
Y,
average downhole steam quality during mJeclion phase, Ib vapor/lb liquid plus vapor wellhead steam quality, Ib vapor/lb liquid plus vapor hypothetical thickness used in Eq. A-14, ft zero order Bessel function of the second kind first order Bessel function of the second kind vertical distance from bottom of lowest sand in interval, ft N.~
Z
=
~ ;=1
_
h,
a = overburden thermal diffusivity. sq ft/D
/30 = 8
=
oil formation volume factor, STB oill res bbI quantity defined in Eq 5, dimensionless JOUR:-'AL OF ,PETROLEUM TECJ\:\,OLOGY
7i =
dimensionle~s time _ _ 2 _ $, = eTerfc hiT) +--;=\I T -
The average unit solution forv, is obtained by solving the one-dimensional heat conduction problem in the radial direction:
v ....
~~',--( r~) r?r (Jr
= volumetric
heat capacity of reservoir rock including interstitial fluids, Btu/ cu ft, OF p", pw = stock-tank fluid density of oil and water, Ib/ cu ft T = 4Kt,/ h'(pC)" dimensionless (pC)1
ACKNOWLEDGMENTS The authors wish to express their appreciation to Esso Production Research Co, for permission to publish this paper. Also, the comments and help of A. R. Hagedorn and A. G. Spillette are gratefully acknowledged.
~, ill
(A-3)
with boundary and initial conditions: v" = 1
t = t,
0< r< r
v, = 0
t = t,
r> r
v,. = 0
t? t,
"
(A-4)
r __ " Ct::
If the thermal properties do not vary with r, the temperature solution for r < r,. is' if'
4
f
v, = .... '
REFERENCES
=
e-b'!fi,(y)J,,(ry/r,,)dy y' [J, (y) Y n (y) - in (y) Y, (y)]'
(A-5)
o
1. Armstrong, Ted A.: "Steam Injection Has Quick Payout in Wilmington Field", Oil & Gas Jour. (March 21, 1966) 78. 2. Carslaw. H. S. and Jaeger, J. c.: Condurtion of Heat in Sol· ids, 2nd Ed., Oxford U. Prf'ss, London, England (1959) 346. 3. Carslaw, H. S. and Jaeger, J. c.: ibid., 5.1. 4. Long, Robf'rt .T.: "Case History of Steam Soaking in Kern River Field. California". JOUf. Pet. Tech. (Sept .. 19(5) 91\9· 99.3. 5. Luk ... Y. L.: Intewals of Ressel Functions. McGraw Hill Book Co.. N. Y. (1962) 314. 6. Marx, J. W. and Langenheim. R. H.: "Reservoir Heating by Hot Fluid Injection", Tum"., AIME 11959) 2]6, 312·314. i. Muskat, M.: Physical Principles of Oil Produrtion. lst Ed .. McGraw Hill Book Co., N. Y. (1949) 194. 215. R Owens. \Y. D. and Suter. V. E.: "Steam Stimulation-Nf'wf'st Form of Secondary Pf'troleulll Rf'co\'f'ry", Oil & Gas JOUf. (April 26, 1965) 82. 9. Payne, R. \V. and Zambrano, G.: "Cyt:iie Steam Injedion Helps Raise Venezuf'la Production". Oil tIC Gas Jllur. (May 24. 19(5) 78. 10. Ramey. H. J .• Jr.: "Wellborf' Heat Transmission", Jour. Pel. Tech. (April, 19(2) 427-435. II. Squier, D. P., Smith. D. D. and Dougherty, E. L.: "Calclllatf'd Telllperatnre Behavior of Hot· \Vatpr InjPetion Wells", Jour. Pet. Tech. (April, 1962) 436-440.
where b' = aU - t,)/r,,'. Since J 1 (y) Y" (y) - J" (y) Y 1 (y)
=2... Eq. A-5 reduces to: ....y
(f)
v,,
f
e-I>'II"J,(y)J,,(ry/r,.)dy
(A-6)
o The average of I', from 0 to r" is given by:
OVERBURDEN
,
---;---
----f------1---
APPENDIX DERIVATION OF EQUATIONS NEGLIGIBLE ENERGY REMOVAL WITH PRODUCED FLUIDS
,
The heat transfer model for conduction cooling of the heated oil sands following termination of steam Injection consists of approximating the sand-shale sequence by a stack of variable thickness. equal radii sand cylinders. These cylinders initially at constant temperature T, COII_ duct heat as a function of time to variable thickness interbedded shales and to semi-infinite overburden and underburden having thermal properties identical to the oil sands. The geometric approximation of the sand-shale sequence is illustrated in Fig. 12. When conduction radially and vertically are the only mechanisms of heat transfer. the temperature of any point within the originally heated cylinder can be expressed as the product: v
=
V v,
v~
DECEMBER, 1966
--t I
---t---I
I I
I
(A-l)
where v, and v~ are unit solutions of component conduction problems in the rand z directions, respectively. Similarly, an integrated average temperature for the heated regions may be computed:
v= V v, ~~
12
(A-2)
UNDERBURDEN FIG. 12-GEOMETRIf. ApPROXIMATION OF MULTIPLE SAND,SHALE SEQUENCE. 1621
h, = thickness of sand j I, = thickness of shale j e- h'''''},(y)dy Vr =
Luke' gives the following solution of the integral A-7: 1
00
-b" "-
--
Vr
y = [M. (X, h'q + hI, - hl,V(-:rr,,'(pC), (L - T,) NJ)- h (y is a hypothetical thickness which. when added to the individual sand thicknesses, accounts for all the energy injected including that lost to the shale during the injection phase.)
(A-7)
y
III
Eq.
The average temperature V, can then be found by integrating over all th~ in.dividual sands: Bl+hl F,
where s, is defined as:
S,
(-ITl),
=
f
(A-8)
Sf.
/:=0
1'(1.5
yr. k!
+
1'(2
k)T'( 1
+
+
k)
k) 1'(3
+
N., ~ ;=1
=
F,dz /
k)
F,
IN., ~
=
~;' _
4
(A-lO)
and it follows that:
[( ITI)"
l
W, + -- W, erl U.' , - W, ert-;=-
-
V
(2
+
k) (3
+
] k)
(A-II)
.1',.
where
U', = B",
The average unit solution for is obtained by solving the one-dimensional heat conduction problem in the vertical direction:
W;
=
B"
If
=
4a( I -
(t
il'v,
ill',
ilz' =
Tt
with boundary conditions \', itial condition is
( )_,0
all ~ I all
v. t"z .
'j
v,
(A-12l =
0, t
>
t, as
z ~ :x. The in-
-
z outside the regions of thickness h z within the regions of thickness II,.
j'
The solution of the conduction equation subject to the initial and boundary conditions is' C/J
I
('
J
2\/-:rl'it
(A-l3)
Carrying out the integration with the initial condition gives: I [ z . h, - z z - B, = 2 erl-=+ erf-=+ erl------===-+ erl
2yat
+ ert z
2\/at
+ h" -
2y'at
- _B,
+ el-I
B.,
2v'at
+ ii., -
z
+ . . . . .]•
2v'al
2v'at
= B,_, + h + i = 0 h =h +y i, =-~ I, - y (if i, < 0, j _,
j _,
CASES \\HERE ENERGY REMOYAI. WITH PRODUCED FLUIDS MUST HE TAKEN INTO ACCOUNT
If significant energy is removed with the produced fluids, some account should be made for this energy removal. Consider an energy balance taken on the sum of the thicknesses of the originally heated regions Z and radius rio: t
Zr.r,,'(pC)l~
=
Z-:rr,,' (pC), V -
1622
f
H,dx -
H.., (A-17)
ti
where H, = the energy conducted to the shale and oil-sand outside the originally heated region, Btu. Eq. A-17 states that the energy contained within the originally heated region at any time is equal to that contained immediately prior to production less that produced with the fluids removed from the formation and less that conducted to shale and oil sand outside of the originally heated region. A rigorous evaluation of H, has not been possible. However, a satisfactory approximation for H, is as follows. (A-IS) where t
j
the two sands of thickness Iz j and h )+, plus the shale of thickness I, are treated as a single sand. The y is then recalculated to account for one less sand).
(A-16)
For a single sand reservoir. \', is plotted vs 4(Y(t - l')lh,' in Fig. 4.
j
j
1]
Bm - ji",
(A-14) where B, B
W·.,-Iw) +
"
t,) .
l',(t"z') exp [ - (z - z')'/4l'it)dz'
-Cf)
Z
~w --=- [exp( -
\I
+ iim - B" + h" - Bm
W, = B" I·V, = B,., - B"
+ ii, -
yw
VW
exp( - W,'lw) -exp( - W,'lw) -exp(- W/lw) (k+1.5)
v,
B,
~
TI'I! m=ln=l
Hence, Eq. A-8. together with Eqs. A-lO and A-II. form the solution for V,. A plot of \'s b' is given in Fig. 4.
VZ
[W W. W, ert, ' + w, erl ,-
N.
-----:--c-.--
yw
=
(A-l5)
.
m= 1
1
vz
II,
Substituting Eq. A-l4 into Eq. A-15 and integrating gives:
2 S,,=
=
_
~ ;=1
B,
(A-9)
From Eq. A-9:
S'tl
N,
1,12
V= V -
J
Hldx
t,
Z-:rr,,' (pC),
(A-19)
Where conduction is the sole mechanism of temperature JOt:R:'I'AL OF PETROLEIJM TECH:,\,OLOGY
Ii
decay, and the energy removed with the produced fluids is negligible:
= V(I -
Combining Eq. A-20 with Eqs. A-I7 and A-IS. and dividing by Z.".r,,'(pCl, gives = V - 2 V8 - V(l - 8) (I which simplifies to E4. 4. Of several approximations for H, which were tried, the one used here gave best agreement with field data. Note that for the limiting case where conduction IS negligible -7 1), this approximation gives:
v,
v,vol
and Eqs. A-I7, A-IS and A-20 reduce to Eq. A-2.17 may be considered an effective driving force for conduction which takes into account the effect of temperature reduction due to the removal of energy by the produced fluids. The definition of 8 from Eq. 5 combined with Eqs. A17 through A-19 gives Eq. 4. By Eqs. A-19 and 5:
DECEMBER, 1 .. 66
(A-20)
8)
(v,v,
T,,," = T,.
+
(T, - T,)[l - 28]
(A-22)
From the definition of 8 this may be seen as giving the correct value for Ttlvg in this limiting case.
***
)623
