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Capillary end effects in coreflood calculations
Journal of Petroleum Science and Engineering 19 Ž1998. 103–117
Capillary end effects in coreflood calculations David D. Huang ) , Matt M. Honarpour Mobil Technology Company, 13777 Midway Road, Dallas, TX 75244, USA
Abstract Capillary end effects in coreflood experiments, in some cases, can significantly influence the computation of end-point relative permeabilities and final saturation levels. Because capillary end effects arise from capillarity, these corrections for relative permeability and saturation can be quantified if the capillary pressure curve is known a priori. Based on Darcy’s law and the relative permeability–capillary pressure relationships, we present in this paper how these corrections can be done in corefloods if the in situ saturation profile is unavailable. Under the same principle, if the in situ saturation profile of the capillary end is available, it is therefore possible to simultaneously predict both relative permeability and capillary pressure using pressure data. Such a method of prediction is presented in the paper and experimentally verified. In a reservoir-conditioned oilflood using a carbonate core sample, we obtained the capillary end information through in situ microwave monitoring, and the relative permeability and capillary pressure were predicted based on the theory. The predictions were compared with separate capillary pressure Žmercury injection, porous-plate, and centrifuge. and relative permeability Žsteady-state. measurements. The predictions and experiments agree well with one another. q 1998 Elsevier Science B.V. Keywords: capillary end effects; Corey–Burdine equations; relative permeability; capillary pressure
1. Introduction The capillary end effect is an important issue in coreflood experiments, because it can cause serious errors in the calculation of saturation and relative permeabilities from pressure drop and production information. Capillary end effects arise from the discontinuity of capillarity in the wetting phase at the outlet end of the core sample. Capillary end effects commonly appear in situations of oil displacing water in water-wet cores, and gas-displacing-oil cases. In other displacement processes, capillary end effects are less pronounced than these two situations. Experiments of oil displacing water in a water-wet core, or a drainage process, are important, because they establish the end-point oil relative permeability at the interstitial water saturation, k ro at Siw , which is the starting point of a waterflood Žor an imbibition. process. Experiments of the gas-displacing-oil cases are also important, because they represent internal or external gas displacement in oil reservoirs. This paper shows how to correct errors in saturation and relative permeabilities exploiting the capillary end effect. With this theory, we will show how to estimate relative permeability and capillary pressure simultaneously from in situ capillarity information. )
Corresponding author.
0920-4105r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 0 - 4 1 0 5 Ž 9 7 . 0 0 0 4 0 - 5
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
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Fig. 1. The capillary end effect in an oil–water co-injection experiment. The core, assuming water-wet, was initially saturated with water, with or without Sor .
1.1. The capillary end effect formulation For the purpose of illustration, we discuss an experiment of co-injecting oil and water into a water-presaturated, water-wet core sample. The saturation profiles during the experiments are schematically shown in Fig. 1. After oil breakthrough, the system approaches steady-state Ž t ™ `., and water capillarity tends to keep a higher water saturation toward the core end. This non-uniform saturation profile at the steady-state is called the capillary end effect, and can be formulated based on Darcy’s law for both the fluids: sy
kk rw E pw
sy
kk ro E po
qw qo A
Ž 1.
mw E x
A
Ž 2.
mo E x
Water–oil capillary pressure is Pc s po y pw
Ž 3.
Combining Eqs. Ž1. – Ž3. gives 1 kA
ž
qw m w
y
k rw
qo m o k ro
/
s
E Pc
Ž 4.
Ex
Since the capillary pressure is only a function of water saturation, the right-hand term of Eq. Ž4. can be written as E Pc d Pc E S w s Ž 5. Ex d Sw E x Eqs. Ž4. and Ž5. together give
E Sw Ex
1 s kA
ž
qw m w
y
k rw
qo m o k ro
/
d Pc
Ž 6.
d Sw
At steady-state Ž t ™ `., water saturation is not a function of time, thus d Sw dx
1 s kA
ž
qw m w k rw
y
qo m o k ro
/
d Pc d Sw
Ž steady state.
Ž 7.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
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This equation can be integrated if all quantities are known Žthat is k ro , k rw , and Pc are known functions of S w ; q w , qo , m w , mo , k, A are all known.. The integration results predict the steady-state saturation profile as a function of distance Lyxs
L
S w ,out
d Pc
w
d Sw
Hx d x s kAHS
r
ž
qw m w
y
k rw
qo m o k ro
/
d Sw
Ž 8.
In this equation, L y x represents the distance from the core outlet. The upper limit in the integration on the right-hand side is the water saturation at the core outlet end. In a water-wet system, a relatively large amount of water may be withheld toward the core outlet because of the capillarity. Therefore, without causing much error, S w,out can either be assumed as 100% in the absence of residual oil, or 1 y Sor in the presence of residual oil. Eq. Ž8. can be subsequently written as Lyxs
L
1yS or
d Pc
w
d Sw
Hx d x s kAHS
r
ž
qw m w k rw
y
qo m o k ro
/
d S w Ž steady-state, water-wet, oilrwater system.
Ž 9. If initially there is no residual oil, Sor is set to zero in the above equation. Since Eq. Ž9. governs the saturation profile, we will refer it to as the ‘profile equation.’ 1.2. PreÕious theoreticalr experimental work Richardson et al. Ž1952. conducted a series of gas-displacing-oil experiments using a Berea sandstone to study capillary end effects. The oil saturation in all experiments was initially 100%. Thus, for gasroil system, the profile equation is L y x s kA
1
d Pc
o
d So
HS
r
ž
qo m o k ro
y
qg mg k rg
/
d So Ž steady-state, gasroil with initial So s 100% .
Ž 10 .
Richardson et al. Ž1952. compared their experimental results with that predicted by Eq. Ž10., and demonstrated excellent agreements between measurements and predictions. Their results support the theoretical profile equation.
2. The capillary end effect by the Corey–Burdine equation We return to the oil-displacing-water situation. Brooks and Corey Ž1964. assumed that the capillary pressure can be represented by Pc s Pd
ž
S w y Siw 1 y Siw y Sor
y1 r l
/
Ž 11 .
where Pd is the displacement pressure Žsometimes called ‘threshold pressure’., and l is a material constant determined experimentally. Eq. Ž11. is schematically shown in Fig. 2. We define normalized water saturation based on the mobile saturation range, 1 y Siw y Sor , S w) s
S w y Siw 1 y Siw y Sor
Ž 12 .
and Eq. Ž11. becomes Pc s Pd Ž S w) .
y1 r l
Ž 13 .
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
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Fig. 2. In oil-displacing-water case, the measured capillary curve Ždashed. can be represented by an idealized curve Žsolid.. The intercept of the idealized curve at residual oil saturation is the displacement pressure, Pd .
Based on the work of Purcell Ž1949., Corey combined Burdine’s research ŽBurdine, 1953. on tortuosity and developed the Corey–Burdine Equations ŽCorey, 1954; Honarpour et al., 1986.. For oil–water systems, the Corey–Burdine equations predict the following interrelationships between relative permeabilities and capillary pressure k ro s k ro < S iw Ž 1 y S w) . k rw s k rw < S or Ž S w) .
2
2
1
d S w)
) w
Pc2
HS
) Sw
H0
d S w) Pc2
1
H0
r
1
H0
r
d S w)
Ž 14 .
Pc2
d S w)
Ž 15 .
Pc2
Substituting Eq. Ž13. into Eqs. Ž14. and Ž15., Brooks and Corey Ž1964. obtained the following results: 2
k ro s k ro < S iw Ž 1 y S w) . 1 y Ž S w) . k rw s k rw < S or Ž S w) .
Ž2r l .q1
Ž 16 .
Ž2r l .q3
Ž 17 .
With these expressions, the profile equation, Eq. Ž9., becomes Lyx s L
kk ro < S iw APd Lqo mo
°1
1
¢
) w
~ H l S
¶
Ž . Ž . 2 Ž Sw) . 1rl q2 Ž 1 y Sw) . 1 y Ž Sw) . 2rl q1
Ž
Ž2r l .q3 S w) y
.
k ro < S iw q w m w k rw < S or qo mo
Ž
2 1 y S w)
. 1y Ž
•
d S w)
Ž2r l .q1 S w)
.
ß
Ž 18 .
which applies to steady-state oil-displacing-water situations in a water-wet core. Ideally, one would integrate Eq. Ž18. to obtain the steady-state saturation profile, S w as a function of distance, in an oil-displacing-water experiment. Realistically, this equation is difficult to use. In this equation, Pd and l can be determined by a capillary pressure experiment; the terminal water relative permeability, k rw < S or , can be determined by a water-displacing-oil Ži.e., a waterflood. experiment, assuming no capillary end effects if the core is rendered water-wet. The only term causing problem is the terminal oil relative permeability, k ro < S iw , which cannot be evaluated a priori in an experiment. The reason is that one gets the value of an aÕeraged oil relative permeability Ž k ro . at an aÕeraged terminal condition Ž Siw . based on the pressure drop over the whole core and the water production. In other words, k ro < S iw is a part of the solution, not a known parameter.
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Our goals are to compute the actual terminal oil relative permeability Ž k ro < S iw . and the actual water saturation Ž Siw . values from the averaged k ro < S iw and Siw . In the next section, we will show step by step how this can be accomplished. Using the word ‘actual’ does not mean ‘true’ values of relative permeability and saturations, which should be directly measured in the absence of capillary end effects. ‘Actual,’ in our phrase, means derived values, where the use of the Corey–Burdine equations and the idealized capillary pressure curve ŽEq. Ž11.. are implicit. 3. Error correction in relative permeability and saturation calculations We consider the oil-displacing-water case again. The calculations presented in this work can be as well extended to other coreflood situations such as the gas-displacing-oil cases, provided that the subscripts are appropriately changed Žw to o and o to g.. For example, change the kk ro < S iw APdrLqo m o term in our work to kk rg < S or APdrLqg mg for a gas-displacing-oil case, where Pd becomes the displacement pressure Ža positive value. for gas displacing oil. To further simplify our discussion, we consider the scenarios of single-phase oil injection. Setting q w s 0 Žno water co-injection. in Eq. Ž18. Lyx kk ro < S iw APd 1 1 Ž . Ž . 2 s Ž 1 y Sw) . 1 y Ž Sw) . 2r l q1 r Ž Sw) . 1rl q1 d Sw) L Lqo mo l Sw)
H
½
5
Ž oil displacing water, oil-injection alone, water-wet core.
Ž 19 .
Integrating term by term of Eq. Ž19. gives Lyx kk ro < S iw APd 2 l2 Ž l q 2 . Ž 6 l2 q l q 1 . 2 y1r l 1y1r l s y 2 q Ž Sw) . Ž Sw) . 2 L Lqo mo Ž l y 1 . Ž 4l y 1 . Ž 3 l q 1 . l y 1
½
1 y
2ly1
Ž Sw) .
2y 1r l
1 q
lq1
Ž Sw) .
1q1r l
2 y
2lq1
Ž Sw) .
2q1r l
1 q
3l q 1
Ž Sw) .
3q1r l
5
Ž oil displacing water, oil-injection alone, water-wet core. Ž 20 . Using this equation, Fig. 3 shows, as an example, the normalized water saturation profiles using l s 2.1 at different values of kk ro < S iw APdrLqo mo . In Fig. 3, the inlet-end water-saturation is governed by Eq. Ž20. where x is set to zero kk ro < S iw APd 2 l2 Ž l q 2 . Ž 6 l2 q l q 1 . 2 1 y1r l 1y1r l ) 1s S y q y Ž w ,inlet . Ž Sw),inlet . 2 2 Lqo mo 2ly1 Ž l y 1 . Ž 4l y 1 . Ž 3 l q 1 . l y 1
½
= Ž S w),inlet .
2y 1r l
1 q
lq1
Ž Sw),inlet .
1q1r l
2 y
2lq1
Ž Sw),inlet .
2q1r l
1 q
3l q 1
Ž S w),inlet .
3q1r l
5
Ž 21 .
) This is a nonlinear equation which can be solved numerically. The solution is shown in Fig. 4 where S w,inlet is < plotted as a function of kk ro S iw APdrLqo m o for l s 1.5 to 2.9.
3.1. AÕerage water saturation The aÕerage water saturation over the whole core Žpresumably close to interstitial level except in the vicinity of the outlet end. can be calculated as follows Žnote that the water saturations are all normalized.: ) Siw s
1
L )
H L 0
Sw d x s
1
1
H L S
) w ,inlet
S w)
d S w)
ž / dx
y1
d S w) s
kk ro < S iw APd Lqo mo
1
HS
= 1 y Ž S w) .
) w ,inlet
½Ž1yS
Ž2r l .q1
) 2 w
.
rl Ž S w) .
1r l
5dS
) w
Ž 22 .
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Fig. 3. Normalized water saturation profiles of single-phase oil injection displacing water in a water-wet core for l s 2.1 and different values of kk ro < S iw APd r Lqo mo .
This equation can be expanded and integrated term by term to give ) Siw s
2 l2 Ž l q 2 . Ž 18 l2 q l q 1 .
Žy1r l . X y XS w),inlet
Ž l y 1 . Ž 4 l 2 y 1 . Ž 9l 2 y 1 . Ž 4 l q 1 .
1r l q XS w),inlet
ž
2 S w),inlet
2lq1
y
3 2 S w),inlet
3l q 1
q
4 S w),inlet
4l q 1
/
; with X '
ž
S w),inlet
ly1
kk ro < S iw APd Lqo mo
y
2 2 S w),inlet
2ly1
q
3 S w),inlet
3l y 1
/ Ž 23 .
With the solutions of Eq. Ž21. presented in Fig. 4, Eq. Ž23. is used to compute the normalized interstitial water saturation for different values of l and kk ro < S iw as shown in Fig. 5.
Fig. 4. Normalized inlet-end water saturation as a function of kk ro < S iw APd r Lqo mo for single-phase oil displacing water Žwater-wet core; 1.5F l F 2.9..
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
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Fig. 5. Normalized average water saturation as a function of kk ro < S iw APd r Lqo m o for single-phase oil displacing water Žwater-wet core; 1.5F l F 2.9..
3.2. AÕerage relatiÕe permeability Now we continue to discuss the pressure drop in the core sample. Because of steady state and no water injection, the pressure drop in the core at an arbitrary distance x can be obtained by integrating the Darcy’s equation for oil: D P < x sp< x yp< 0 sy s
) cŽ w .
HPPŽ SS c
qo m o Ak
d Pc s Pd
) w ,inlet .
1
x
H0
k ro
Ž Sw) .
d xsy
y1 r l
qo m o Ak
y Ž S w),inlet .
x
H0
1
d x d S w)
k ro d S w)
y1r l
d Pc
d Pc s
qo m o Ak
) cŽ w .
HPPŽ SS c
) w ,inlet .
1 k ro
Akk ro
ž / qo m o
d Pc
Ž 24 .
) At given kk ro < S iw APdrLqo mo and l values, Fig. 5 shows S w) being a function of x, and the value of S w,inlet Ž . Ž . can be determined from Fig. 4. Thus, using Eq. 24 , the pressure gradient curve in dimensionless form can be plotted as shown in Fig. 6. Fig. 6 shows that a significant pressure drop exists across the last 20% of the core length for all situation considered. The right-hand side end point of each curves Ži.e., the outlet end. in Fig. 6 is related to the ratio of the actual terminal relative permeability to that of the averaged, as now discussed. Steady-state terminal oil relative permeability, in the laboratory, is calculated based on the pressure drop over the entire core. This calculation requires correction because of the capillary end effect. Denote the aÕerage relative permeability Žbased on the entire core pressure drop Ž D P < xs L s D PL ., k ro , from Darcy’s law and using Eq. Ž24.
k ro < S iw s
Lqo mo k Ž yD PL . A
s
Lqo mo kPd A
Ž Sw),inlet .
y1r l
y1
y1
Ž 25 .
or k ro < S iw k ro < S iw
sX
Ž Sw),inlet .
y1r l
y1 ; X'
kk ro < S iw APd Lqo mo
Ž 26 .
Eq. Ž26. represents the ratio of the actual terminal oil relative permeability to that of the averaged k ro . This ratio is graphically shown in Fig. 7. Fig. 7 shows that the actual oil relative permeability Žk r o at Si w . at terminal
110
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
Fig. 6. Pressure gradient, Žy D p < x . kk ro < S iw A r xqo mo , as a function of distance from the injection end, x r L. Condition considered: single-phase oil displacing water Žwater-wet core; l s 2.1..
condition is greater than the averaged k ro at Si w . Physically, this means that the higher pressure gradient near the outlet end caused by the capillary end effect results in a lower averaged relative permeability. 3.3. Corrections of terminal oil relatiÕe permeability computation Based on the previous calculations, the terminal oil relative permeability ratio, k ro r k ro, can be expressed in terms of measurable quantities.
Fig. 7. Ratio of actual terminal oil relative permeability to the averaged value as a function of kk ro < S iw APd r Lqo mo for 1.5F l F 2.9. The averaged terminal oil relative permeability is calculated based on the total pressure drop across a core sample.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
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Fig. 8. Ratio of the actual terminal oil relative permeability to the aÕeraged terminal oil relative permeability as a function of Pd r < D PL < for 1.5F l F 2.9.
The ratio of the two quantities shown as the x-coordinate in Figs. 8 and 9, is Pdr< D PL <, the ratio of the displacement pressure to the viscous pressure drop over the entire core:
ž
kk ro < S iw APd Lqo mo
k ro < S iw
/ ž / %
k ro < S iw
s
kk ro < S iw APd Lqo mo
s
Pd < D PL <
Ž 27 .
Fig. 8 shows k ro < S iw rk ro < S iw as a function of Pdr< D PL < for various l. Based on measured kk ro < S iw APdrLqo mo Ži.e., Pdr< D PL <. and l, we can find the value of k ro < S iw rk ro < S iw from Fig. 8. This value and the averaged k ro < S iw can be used to compute the actual terminal oil relative permeability k ro < S iw .
Fig. 9. Normalized average water saturation as a function of Pd r < D PL < for 1.5F l F 2.9.
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3.4. Corrections of interstitial water saturation computation ) Fig. 9 shows the normalized average interstitial water saturation, Siw , as a function of the measurable kk ro < S iw ) APdrLqo mo Ži.e., Pdr< D PL <. and l. Based on measured kk ro < S iw APdrLqo mo and l, we find the value of Siw using Fig. 9. From Eq. Ž12., we have
) Siw s
Siw y Siw
Ž 28 .
1 y Siw y Sor
Solving for Siw Siw s
) Siw y Siw Ž 1 y Sor .
Ž 29 .
) 1 y Siw
) Ž For known values of Siw from Fig. saturation at the terminal condition, Siw
9., Siw Žmeasured. and Sor Žmeasured., the actual interstitial water Žfree of end effect. can be calculated using Eq. Ž29..
3.5. Examples of relatiÕe permeability and saturation corrections The computations of corrected terminal oil relative permeability and interstitial water saturation are demonstrated in this section. Values obtained from Figs. 8 and 9 using l s 2.1, a typical value for consolidated sandstone, are tabulated in Table 1. Using Fig. 4, the inlet water saturation values are also tabulated. Table 1 shows that the higher the capillary-to-viscous pressure ratio Ž Pdr< D PL <., the more significant are corrections in Siw and k ro . In an example shown in the table Ž Pdr< D PL < s 0.4, the capillary pressure being quite significant., the averaged relative permeability is 81% lower than the actual value, and the averaged interstitial water saturation is 0.164 PV higher than the actual value Ži.e., Siw y Siw s 0.164..
4. Simultaneous estimation of relative permeability and capillary pressure In this section, we will show how to simultaneously estimate the relative permeabilities and capillary pressure of a core sample, if the averaged and actual values of interstitial water saturation and terminal oil relative permeability are given. To obtain the latter information, we conducted a coreflood experiment using a carbonate core sample, which possesses water-wet characteristics at reservoir temperature.
Table 1 Capillary end effects calculation for oil displacement of water in a water-wet core using l s 2.1 Capillary to viscous Viscous to capillary Normalized inlet Normalized avg. water For primary drainage, Relative permeability ) pressure ratio pressure ratio, water saturation, saturation, Siw at Siw s 20% and correction, k ro < S iw r k ro < S iw ) < D PL
100 50.0 25.0 12.5 10.0 5.00 3.33 2.50
4.2Ey5 2.5Ey4 1.1Ey3 4.2Ey3 6.7Ey3 2.3Ey2 4.6Ey2 7.2Ey2
4.5Ey3 8.8Ey3 1.8Ey2 3.6Ey2 4.5Ey2 8.8Ey2 1.3Ey1 1.7Ey1
19.64 19.29 18.53 17.01 16.23 12.28 8.05 3.61
1.02 1.03 1.07 1.13 1.17 1.36 1.58 1.81
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Fig. 10. Water saturation profiles before and after the coreflood experiments. Initial condition: top bullets with a mean S w s 0.749; final condition with the capillary end effect: open diamonds with an average Siw s 0.324; final condition without the capillary end effect: bottom bullets with a mean Siw s 0.292. Saturation profiles are obtained by microwave attenuation techniques.
The core dimensions and the experimental conditions are: L s 10 cm, D s 1.83 cm, k s 5 md, f s 22.5%, T s 1508F, mo s 6 cp, qo s 1.389 = 10y4 cm3rs. The experiment was monitored by microwave attenuation techniques along every centimeter of the core for in situ saturation profile information Žsee Fig. 10.. The experiment protocol and the results are the following: Step 1. Oilflood Ž qo s 0.5 cm3rh. a core that initially has a mean residual oil saturation, Sor s 0.251. At the end of oilflood, in the presence of capillary end effect, the averaged interstitial water saturation Ž Siw . is 0.324, and the averaged terminal oil relative permeability, k ro < S iw s 0.51. Step 2. Use a water-wet porous plate at the core outlet end. Transform the constant-rate injection mode into the constant-pressure mode of oil injection. In this way, the nonuniform water saturation profile becomes uniform Žsee Fig. 10.. At this point, we found the true interstitial water saturation, Siw s 0.292. Step 3. Remove the porous plate, and again conduct a constant rate oil injection Ž qo s 0.5 cm3rh. to determine the true terminal oil relative permeability Ži.e., without the capillary end effect.. We found k ro < S iw s 0.65. The experimental results give us k ro < S iw k ro < S iw ) Siw s
0.65 s 0.51
s 1.275
Siw y Siw
Ž 30 . 0.324 y 0.292
s
1 y Siw y Sor
1 y 0.292 y 0.251
s 0.07
Ž 31 .
Interpolating the value of l from Figs. 8 and 9 simultaneously using Eqs. Ž30. and Ž31., we found
l s 2.3, and
kk ro < S iw APd Lqo m o
s 0.18
Ž 32 .
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which further gives the value of the threshold pressure, Pd : Pd s
0.18 Lqo m o
s
kk ro < S iw A
psi Ž 0.18 . Ž 10 . Ž 0.5r3600 . Ž 6 = 0.01 . = s 3.28 psi y12 68947.57 dynercm2 Ž 5 = 9.87 = 10 . Ž 0.51 . Ž 2.63 .
Ž 33 .
Eqs. Ž11., Ž16. and Ž17., that predict relative permeabilities and capillary pressure, become
ž ž
Pc Ž psi . s 3.28 =
k ro s 0.65 = 1 y
S w y 0.292 0.457 S w y 0.292 0.457
k rw s Ž k rw < S or s 0.06 . =
ž
y1 r2.3
/ /
Ž 34 . 2
= 1y
S w y 0.292 0.457
ž
S w y 0.292 0.457
1.87
/
Ž 35 .
3.87
/
Ž 36 .
In order to check the validity of these equations, we conducted separate measurements for capillary pressure and steady-state waterroil relative permeabilities. The comparison between values from predictions and measurements are shown in Figs. 11 and 12. Fig. 11 shows the comparison of capillary pressure prediction ŽEq. Ž34.. with data from three different measurements: Ža. reservoir-condition centrifuge experiment, Žb. reservoir-condition porous-plate experiment through the use of microwave attenuation for water saturation determination, and Žc. ambient-condition mercury-injection experiments rescaled to the reservoir condition. Our prediction using Eq. Ž34. provides an overall agreement with the three different sets of data. Fig. 12 shows the comparison between the predicted relative permeabilities ŽEqs. Ž35. and Ž36.. with that measured in a reservoir-condition steady-state experiment using microwave attenuation for in situ water saturation determination. Again, the predicted curves agree well with the measurement.
Fig. 11. Comparison between the theoretically predicted capillary pressure with the measured ones from centrifuge, porous-plate, and mercury-injection experiments.
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Fig. 12. Comparison between the theoretically predicted relative permeabilities with the experimental data. The experiment was conducted at reservoir condition in steady-state mode.
To double check the calculations, we substitute the values of Pd and l from Eqs. Ž32. and Ž33. for Eq. Ž19., the profile equation, to determine the saturation profile. The predicted profile can be compared with the measured saturation profile Žshown earlier in Fig. 10. as a final check. Fig. 13 shows this comparison. Again, the theory agrees well with the experiment.
Fig. 13. Comparison between the predicted capillary-end-effect saturation profile with that measured using microwave attenuation techniques. Solid curve: predicted, bullets: measured values.
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5. Summary and conclusions In this work, we have addressed how capillary end effects can influence the computation of relative permeability and saturation in coreflood experiments. To illustrate, we studied the situations of oil displacing water in a water-wet core at the terminal condition as an example. Ø We showed how to convert the averaged terminal oil relative permeability to the actual oil relative permeability. This correction, in some cases we studied, can be very significant Žin the order of 100%. when the capillary force is of the same order of magnitude as the viscous force. Ø We showed how to compute the actual interstitial water saturation from an averaged value. The correction to the computation of Siw is found even more significant than that needed for the terminal relative permeability. Ø We also demonstrated how to simultaneously estimate relative permeabilities and capillary pressure in a drainage process, based on terminal-condition measurements with and without the capillary end effect. The estimated relative permeabilities and capillary pressure were in good agreement with experimental data from different sources. When applying this simultaneous estimation technique, the capillary-induced nonuniform saturation profile information should be obtained from in situ scanning methods.
6. Nomenclature A D E k kr L Pc Pd p q S Siw Sor T t X x
core cross-sectional area Žcm2 . core diameter Žcm. exponentiation based on 10 absolute permeability Žmd. relative permeability Ždimensionless. core total length Žcm. capillary pressure Ždynercm2 . capillary displacement pressure Ždynercm2 . phase pressure Ždynercm2 . phase volumetric flow rate Žcm3rs. saturation Ždimensionless. interstitial water saturation Ždimensionless. residual oil saturation Ždimensionless. temperature Ž8F. time Žs. a dimensionless group defined in Eq. Ž23. distance from core inlet Žcm.
Greek letters DP f l m
pressure different in Eq. Ž24. porosity Ždimensionless. a core-dependent constant for Pc in Eq. Ž11. viscosity Žcp.
Subscripts and superscripts L inlet
across the whole core evaluated at the core inlet
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
out o, w, g x 4
117
evaluated at the core outlet oil, water, and gas respectively evaluated at distance x from the core inlet normalized saturation
Bars <
averaged over the whole core evaluated at
Acknowledgements We thank Mobil management for permission to publish this paper. We thank the technical discussions with Rafi Al-Hussainy and George Hirasaki.
References Brooks, R.H., Corey, A.T., 1964. Hydraulic properties of porous media. Hydrology Papers of Colorado State University, No. 3. Burdine, N.T., 1953. Relative permeability calculations from pore size distribution data. Trans. AIME ŽAm. Inst. Min. Metall. Eng.. 198, 71–78. Corey, A.T., 1954. The interrelation between gas and oil permeabilities. Producers Monthly, November, 38–41. Honarpour, M.M., Koederitz, L., Harvey, A.H., 1986. Relative permeability of petroleum reservoirs. CRC Press. Purcell, W.R., 1949. Capillary pressures — their measurement using mercury and the calculation of permeability therefrom. Trans. AIME ŽAm. Inst. Min. Metall. Eng.. February, 39–48. Richardson, J.G., Kerver, J.K., Hafford, J.A., Osoba, J.S., 1952. Laboratory determination of relative permeability. Trans. AIME ŽAm. Inst. Min. Metall. Eng.. 195, 187–196.
Capillary end effects in coreflood calculations David D. Huang ) , Matt M. Honarpour Mobil Technology Company, 13777 Midway Road, Dallas, TX 75244, USA
Abstract Capillary end effects in coreflood experiments, in some cases, can significantly influence the computation of end-point relative permeabilities and final saturation levels. Because capillary end effects arise from capillarity, these corrections for relative permeability and saturation can be quantified if the capillary pressure curve is known a priori. Based on Darcy’s law and the relative permeability–capillary pressure relationships, we present in this paper how these corrections can be done in corefloods if the in situ saturation profile is unavailable. Under the same principle, if the in situ saturation profile of the capillary end is available, it is therefore possible to simultaneously predict both relative permeability and capillary pressure using pressure data. Such a method of prediction is presented in the paper and experimentally verified. In a reservoir-conditioned oilflood using a carbonate core sample, we obtained the capillary end information through in situ microwave monitoring, and the relative permeability and capillary pressure were predicted based on the theory. The predictions were compared with separate capillary pressure Žmercury injection, porous-plate, and centrifuge. and relative permeability Žsteady-state. measurements. The predictions and experiments agree well with one another. q 1998 Elsevier Science B.V. Keywords: capillary end effects; Corey–Burdine equations; relative permeability; capillary pressure
1. Introduction The capillary end effect is an important issue in coreflood experiments, because it can cause serious errors in the calculation of saturation and relative permeabilities from pressure drop and production information. Capillary end effects arise from the discontinuity of capillarity in the wetting phase at the outlet end of the core sample. Capillary end effects commonly appear in situations of oil displacing water in water-wet cores, and gas-displacing-oil cases. In other displacement processes, capillary end effects are less pronounced than these two situations. Experiments of oil displacing water in a water-wet core, or a drainage process, are important, because they establish the end-point oil relative permeability at the interstitial water saturation, k ro at Siw , which is the starting point of a waterflood Žor an imbibition. process. Experiments of the gas-displacing-oil cases are also important, because they represent internal or external gas displacement in oil reservoirs. This paper shows how to correct errors in saturation and relative permeabilities exploiting the capillary end effect. With this theory, we will show how to estimate relative permeability and capillary pressure simultaneously from in situ capillarity information. )
Corresponding author.
0920-4105r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 0 - 4 1 0 5 Ž 9 7 . 0 0 0 4 0 - 5
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
104
Fig. 1. The capillary end effect in an oil–water co-injection experiment. The core, assuming water-wet, was initially saturated with water, with or without Sor .
1.1. The capillary end effect formulation For the purpose of illustration, we discuss an experiment of co-injecting oil and water into a water-presaturated, water-wet core sample. The saturation profiles during the experiments are schematically shown in Fig. 1. After oil breakthrough, the system approaches steady-state Ž t ™ `., and water capillarity tends to keep a higher water saturation toward the core end. This non-uniform saturation profile at the steady-state is called the capillary end effect, and can be formulated based on Darcy’s law for both the fluids: sy
kk rw E pw
sy
kk ro E po
qw qo A
Ž 1.
mw E x
A
Ž 2.
mo E x
Water–oil capillary pressure is Pc s po y pw
Ž 3.
Combining Eqs. Ž1. – Ž3. gives 1 kA
ž
qw m w
y
k rw
qo m o k ro
/
s
E Pc
Ž 4.
Ex
Since the capillary pressure is only a function of water saturation, the right-hand term of Eq. Ž4. can be written as E Pc d Pc E S w s Ž 5. Ex d Sw E x Eqs. Ž4. and Ž5. together give
E Sw Ex
1 s kA
ž
qw m w
y
k rw
qo m o k ro
/
d Pc
Ž 6.
d Sw
At steady-state Ž t ™ `., water saturation is not a function of time, thus d Sw dx
1 s kA
ž
qw m w k rw
y
qo m o k ro
/
d Pc d Sw
Ž steady state.
Ž 7.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
105
This equation can be integrated if all quantities are known Žthat is k ro , k rw , and Pc are known functions of S w ; q w , qo , m w , mo , k, A are all known.. The integration results predict the steady-state saturation profile as a function of distance Lyxs
L
S w ,out
d Pc
w
d Sw
Hx d x s kAHS
r
ž
qw m w
y
k rw
qo m o k ro
/
d Sw
Ž 8.
In this equation, L y x represents the distance from the core outlet. The upper limit in the integration on the right-hand side is the water saturation at the core outlet end. In a water-wet system, a relatively large amount of water may be withheld toward the core outlet because of the capillarity. Therefore, without causing much error, S w,out can either be assumed as 100% in the absence of residual oil, or 1 y Sor in the presence of residual oil. Eq. Ž8. can be subsequently written as Lyxs
L
1yS or
d Pc
w
d Sw
Hx d x s kAHS
r
ž
qw m w k rw
y
qo m o k ro
/
d S w Ž steady-state, water-wet, oilrwater system.
Ž 9. If initially there is no residual oil, Sor is set to zero in the above equation. Since Eq. Ž9. governs the saturation profile, we will refer it to as the ‘profile equation.’ 1.2. PreÕious theoreticalr experimental work Richardson et al. Ž1952. conducted a series of gas-displacing-oil experiments using a Berea sandstone to study capillary end effects. The oil saturation in all experiments was initially 100%. Thus, for gasroil system, the profile equation is L y x s kA
1
d Pc
o
d So
HS
r
ž
qo m o k ro
y
qg mg k rg
/
d So Ž steady-state, gasroil with initial So s 100% .
Ž 10 .
Richardson et al. Ž1952. compared their experimental results with that predicted by Eq. Ž10., and demonstrated excellent agreements between measurements and predictions. Their results support the theoretical profile equation.
2. The capillary end effect by the Corey–Burdine equation We return to the oil-displacing-water situation. Brooks and Corey Ž1964. assumed that the capillary pressure can be represented by Pc s Pd
ž
S w y Siw 1 y Siw y Sor
y1 r l
/
Ž 11 .
where Pd is the displacement pressure Žsometimes called ‘threshold pressure’., and l is a material constant determined experimentally. Eq. Ž11. is schematically shown in Fig. 2. We define normalized water saturation based on the mobile saturation range, 1 y Siw y Sor , S w) s
S w y Siw 1 y Siw y Sor
Ž 12 .
and Eq. Ž11. becomes Pc s Pd Ž S w) .
y1 r l
Ž 13 .
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
106
Fig. 2. In oil-displacing-water case, the measured capillary curve Ždashed. can be represented by an idealized curve Žsolid.. The intercept of the idealized curve at residual oil saturation is the displacement pressure, Pd .
Based on the work of Purcell Ž1949., Corey combined Burdine’s research ŽBurdine, 1953. on tortuosity and developed the Corey–Burdine Equations ŽCorey, 1954; Honarpour et al., 1986.. For oil–water systems, the Corey–Burdine equations predict the following interrelationships between relative permeabilities and capillary pressure k ro s k ro < S iw Ž 1 y S w) . k rw s k rw < S or Ž S w) .
2
2
1
d S w)
) w
Pc2
HS
) Sw
H0
d S w) Pc2
1
H0
r
1
H0
r
d S w)
Ž 14 .
Pc2
d S w)
Ž 15 .
Pc2
Substituting Eq. Ž13. into Eqs. Ž14. and Ž15., Brooks and Corey Ž1964. obtained the following results: 2
k ro s k ro < S iw Ž 1 y S w) . 1 y Ž S w) . k rw s k rw < S or Ž S w) .
Ž2r l .q1
Ž 16 .
Ž2r l .q3
Ž 17 .
With these expressions, the profile equation, Eq. Ž9., becomes Lyx s L
kk ro < S iw APd Lqo mo
°1
1
¢
) w
~ H l S
¶
Ž . Ž . 2 Ž Sw) . 1rl q2 Ž 1 y Sw) . 1 y Ž Sw) . 2rl q1
Ž
Ž2r l .q3 S w) y
.
k ro < S iw q w m w k rw < S or qo mo
Ž
2 1 y S w)
. 1y Ž
•
d S w)
Ž2r l .q1 S w)
.
ß
Ž 18 .
which applies to steady-state oil-displacing-water situations in a water-wet core. Ideally, one would integrate Eq. Ž18. to obtain the steady-state saturation profile, S w as a function of distance, in an oil-displacing-water experiment. Realistically, this equation is difficult to use. In this equation, Pd and l can be determined by a capillary pressure experiment; the terminal water relative permeability, k rw < S or , can be determined by a water-displacing-oil Ži.e., a waterflood. experiment, assuming no capillary end effects if the core is rendered water-wet. The only term causing problem is the terminal oil relative permeability, k ro < S iw , which cannot be evaluated a priori in an experiment. The reason is that one gets the value of an aÕeraged oil relative permeability Ž k ro . at an aÕeraged terminal condition Ž Siw . based on the pressure drop over the whole core and the water production. In other words, k ro < S iw is a part of the solution, not a known parameter.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
107
Our goals are to compute the actual terminal oil relative permeability Ž k ro < S iw . and the actual water saturation Ž Siw . values from the averaged k ro < S iw and Siw . In the next section, we will show step by step how this can be accomplished. Using the word ‘actual’ does not mean ‘true’ values of relative permeability and saturations, which should be directly measured in the absence of capillary end effects. ‘Actual,’ in our phrase, means derived values, where the use of the Corey–Burdine equations and the idealized capillary pressure curve ŽEq. Ž11.. are implicit. 3. Error correction in relative permeability and saturation calculations We consider the oil-displacing-water case again. The calculations presented in this work can be as well extended to other coreflood situations such as the gas-displacing-oil cases, provided that the subscripts are appropriately changed Žw to o and o to g.. For example, change the kk ro < S iw APdrLqo m o term in our work to kk rg < S or APdrLqg mg for a gas-displacing-oil case, where Pd becomes the displacement pressure Ža positive value. for gas displacing oil. To further simplify our discussion, we consider the scenarios of single-phase oil injection. Setting q w s 0 Žno water co-injection. in Eq. Ž18. Lyx kk ro < S iw APd 1 1 Ž . Ž . 2 s Ž 1 y Sw) . 1 y Ž Sw) . 2r l q1 r Ž Sw) . 1rl q1 d Sw) L Lqo mo l Sw)
H
½
5
Ž oil displacing water, oil-injection alone, water-wet core.
Ž 19 .
Integrating term by term of Eq. Ž19. gives Lyx kk ro < S iw APd 2 l2 Ž l q 2 . Ž 6 l2 q l q 1 . 2 y1r l 1y1r l s y 2 q Ž Sw) . Ž Sw) . 2 L Lqo mo Ž l y 1 . Ž 4l y 1 . Ž 3 l q 1 . l y 1
½
1 y
2ly1
Ž Sw) .
2y 1r l
1 q
lq1
Ž Sw) .
1q1r l
2 y
2lq1
Ž Sw) .
2q1r l
1 q
3l q 1
Ž Sw) .
3q1r l
5
Ž oil displacing water, oil-injection alone, water-wet core. Ž 20 . Using this equation, Fig. 3 shows, as an example, the normalized water saturation profiles using l s 2.1 at different values of kk ro < S iw APdrLqo mo . In Fig. 3, the inlet-end water-saturation is governed by Eq. Ž20. where x is set to zero kk ro < S iw APd 2 l2 Ž l q 2 . Ž 6 l2 q l q 1 . 2 1 y1r l 1y1r l ) 1s S y q y Ž w ,inlet . Ž Sw),inlet . 2 2 Lqo mo 2ly1 Ž l y 1 . Ž 4l y 1 . Ž 3 l q 1 . l y 1
½
= Ž S w),inlet .
2y 1r l
1 q
lq1
Ž Sw),inlet .
1q1r l
2 y
2lq1
Ž Sw),inlet .
2q1r l
1 q
3l q 1
Ž S w),inlet .
3q1r l
5
Ž 21 .
) This is a nonlinear equation which can be solved numerically. The solution is shown in Fig. 4 where S w,inlet is < plotted as a function of kk ro S iw APdrLqo m o for l s 1.5 to 2.9.
3.1. AÕerage water saturation The aÕerage water saturation over the whole core Žpresumably close to interstitial level except in the vicinity of the outlet end. can be calculated as follows Žnote that the water saturations are all normalized.: ) Siw s
1
L )
H L 0
Sw d x s
1
1
H L S
) w ,inlet
S w)
d S w)
ž / dx
y1
d S w) s
kk ro < S iw APd Lqo mo
1
HS
= 1 y Ž S w) .
) w ,inlet
½Ž1yS
Ž2r l .q1
) 2 w
.
rl Ž S w) .
1r l
5dS
) w
Ž 22 .
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
108
Fig. 3. Normalized water saturation profiles of single-phase oil injection displacing water in a water-wet core for l s 2.1 and different values of kk ro < S iw APd r Lqo mo .
This equation can be expanded and integrated term by term to give ) Siw s
2 l2 Ž l q 2 . Ž 18 l2 q l q 1 .
Žy1r l . X y XS w),inlet
Ž l y 1 . Ž 4 l 2 y 1 . Ž 9l 2 y 1 . Ž 4 l q 1 .
1r l q XS w),inlet
ž
2 S w),inlet
2lq1
y
3 2 S w),inlet
3l q 1
q
4 S w),inlet
4l q 1
/
; with X '
ž
S w),inlet
ly1
kk ro < S iw APd Lqo mo
y
2 2 S w),inlet
2ly1
q
3 S w),inlet
3l y 1
/ Ž 23 .
With the solutions of Eq. Ž21. presented in Fig. 4, Eq. Ž23. is used to compute the normalized interstitial water saturation for different values of l and kk ro < S iw as shown in Fig. 5.
Fig. 4. Normalized inlet-end water saturation as a function of kk ro < S iw APd r Lqo mo for single-phase oil displacing water Žwater-wet core; 1.5F l F 2.9..
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
109
Fig. 5. Normalized average water saturation as a function of kk ro < S iw APd r Lqo m o for single-phase oil displacing water Žwater-wet core; 1.5F l F 2.9..
3.2. AÕerage relatiÕe permeability Now we continue to discuss the pressure drop in the core sample. Because of steady state and no water injection, the pressure drop in the core at an arbitrary distance x can be obtained by integrating the Darcy’s equation for oil: D P < x sp< x yp< 0 sy s
) cŽ w .
HPPŽ SS c
qo m o Ak
d Pc s Pd
) w ,inlet .
1
x
H0
k ro
Ž Sw) .
d xsy
y1 r l
qo m o Ak
y Ž S w),inlet .
x
H0
1
d x d S w)
k ro d S w)
y1r l
d Pc
d Pc s
qo m o Ak
) cŽ w .
HPPŽ SS c
) w ,inlet .
1 k ro
Akk ro
ž / qo m o
d Pc
Ž 24 .
) At given kk ro < S iw APdrLqo mo and l values, Fig. 5 shows S w) being a function of x, and the value of S w,inlet Ž . Ž . can be determined from Fig. 4. Thus, using Eq. 24 , the pressure gradient curve in dimensionless form can be plotted as shown in Fig. 6. Fig. 6 shows that a significant pressure drop exists across the last 20% of the core length for all situation considered. The right-hand side end point of each curves Ži.e., the outlet end. in Fig. 6 is related to the ratio of the actual terminal relative permeability to that of the averaged, as now discussed. Steady-state terminal oil relative permeability, in the laboratory, is calculated based on the pressure drop over the entire core. This calculation requires correction because of the capillary end effect. Denote the aÕerage relative permeability Žbased on the entire core pressure drop Ž D P < xs L s D PL ., k ro , from Darcy’s law and using Eq. Ž24.
k ro < S iw s
Lqo mo k Ž yD PL . A
s
Lqo mo kPd A
Ž Sw),inlet .
y1r l
y1
y1
Ž 25 .
or k ro < S iw k ro < S iw
sX
Ž Sw),inlet .
y1r l
y1 ; X'
kk ro < S iw APd Lqo mo
Ž 26 .
Eq. Ž26. represents the ratio of the actual terminal oil relative permeability to that of the averaged k ro . This ratio is graphically shown in Fig. 7. Fig. 7 shows that the actual oil relative permeability Žk r o at Si w . at terminal
110
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
Fig. 6. Pressure gradient, Žy D p < x . kk ro < S iw A r xqo mo , as a function of distance from the injection end, x r L. Condition considered: single-phase oil displacing water Žwater-wet core; l s 2.1..
condition is greater than the averaged k ro at Si w . Physically, this means that the higher pressure gradient near the outlet end caused by the capillary end effect results in a lower averaged relative permeability. 3.3. Corrections of terminal oil relatiÕe permeability computation Based on the previous calculations, the terminal oil relative permeability ratio, k ro r k ro, can be expressed in terms of measurable quantities.
Fig. 7. Ratio of actual terminal oil relative permeability to the averaged value as a function of kk ro < S iw APd r Lqo mo for 1.5F l F 2.9. The averaged terminal oil relative permeability is calculated based on the total pressure drop across a core sample.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
111
Fig. 8. Ratio of the actual terminal oil relative permeability to the aÕeraged terminal oil relative permeability as a function of Pd r < D PL < for 1.5F l F 2.9.
The ratio of the two quantities shown as the x-coordinate in Figs. 8 and 9, is Pdr< D PL <, the ratio of the displacement pressure to the viscous pressure drop over the entire core:
ž
kk ro < S iw APd Lqo mo
k ro < S iw
/ ž / %
k ro < S iw
s
kk ro < S iw APd Lqo mo
s
Pd < D PL <
Ž 27 .
Fig. 8 shows k ro < S iw rk ro < S iw as a function of Pdr< D PL < for various l. Based on measured kk ro < S iw APdrLqo mo Ži.e., Pdr< D PL <. and l, we can find the value of k ro < S iw rk ro < S iw from Fig. 8. This value and the averaged k ro < S iw can be used to compute the actual terminal oil relative permeability k ro < S iw .
Fig. 9. Normalized average water saturation as a function of Pd r < D PL < for 1.5F l F 2.9.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
112
3.4. Corrections of interstitial water saturation computation ) Fig. 9 shows the normalized average interstitial water saturation, Siw , as a function of the measurable kk ro < S iw ) APdrLqo mo Ži.e., Pdr< D PL <. and l. Based on measured kk ro < S iw APdrLqo mo and l, we find the value of Siw using Fig. 9. From Eq. Ž12., we have
) Siw s
Siw y Siw
Ž 28 .
1 y Siw y Sor
Solving for Siw Siw s
) Siw y Siw Ž 1 y Sor .
Ž 29 .
) 1 y Siw
) Ž For known values of Siw from Fig. saturation at the terminal condition, Siw
9., Siw Žmeasured. and Sor Žmeasured., the actual interstitial water Žfree of end effect. can be calculated using Eq. Ž29..
3.5. Examples of relatiÕe permeability and saturation corrections The computations of corrected terminal oil relative permeability and interstitial water saturation are demonstrated in this section. Values obtained from Figs. 8 and 9 using l s 2.1, a typical value for consolidated sandstone, are tabulated in Table 1. Using Fig. 4, the inlet water saturation values are also tabulated. Table 1 shows that the higher the capillary-to-viscous pressure ratio Ž Pdr< D PL <., the more significant are corrections in Siw and k ro . In an example shown in the table Ž Pdr< D PL < s 0.4, the capillary pressure being quite significant., the averaged relative permeability is 81% lower than the actual value, and the averaged interstitial water saturation is 0.164 PV higher than the actual value Ži.e., Siw y Siw s 0.164..
4. Simultaneous estimation of relative permeability and capillary pressure In this section, we will show how to simultaneously estimate the relative permeabilities and capillary pressure of a core sample, if the averaged and actual values of interstitial water saturation and terminal oil relative permeability are given. To obtain the latter information, we conducted a coreflood experiment using a carbonate core sample, which possesses water-wet characteristics at reservoir temperature.
Table 1 Capillary end effects calculation for oil displacement of water in a water-wet core using l s 2.1 Capillary to viscous Viscous to capillary Normalized inlet Normalized avg. water For primary drainage, Relative permeability ) pressure ratio pressure ratio, water saturation, saturation, Siw at Siw s 20% and correction, k ro < S iw r k ro < S iw ) < D PL
100 50.0 25.0 12.5 10.0 5.00 3.33 2.50
4.2Ey5 2.5Ey4 1.1Ey3 4.2Ey3 6.7Ey3 2.3Ey2 4.6Ey2 7.2Ey2
4.5Ey3 8.8Ey3 1.8Ey2 3.6Ey2 4.5Ey2 8.8Ey2 1.3Ey1 1.7Ey1
19.64 19.29 18.53 17.01 16.23 12.28 8.05 3.61
1.02 1.03 1.07 1.13 1.17 1.36 1.58 1.81
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113
Fig. 10. Water saturation profiles before and after the coreflood experiments. Initial condition: top bullets with a mean S w s 0.749; final condition with the capillary end effect: open diamonds with an average Siw s 0.324; final condition without the capillary end effect: bottom bullets with a mean Siw s 0.292. Saturation profiles are obtained by microwave attenuation techniques.
The core dimensions and the experimental conditions are: L s 10 cm, D s 1.83 cm, k s 5 md, f s 22.5%, T s 1508F, mo s 6 cp, qo s 1.389 = 10y4 cm3rs. The experiment was monitored by microwave attenuation techniques along every centimeter of the core for in situ saturation profile information Žsee Fig. 10.. The experiment protocol and the results are the following: Step 1. Oilflood Ž qo s 0.5 cm3rh. a core that initially has a mean residual oil saturation, Sor s 0.251. At the end of oilflood, in the presence of capillary end effect, the averaged interstitial water saturation Ž Siw . is 0.324, and the averaged terminal oil relative permeability, k ro < S iw s 0.51. Step 2. Use a water-wet porous plate at the core outlet end. Transform the constant-rate injection mode into the constant-pressure mode of oil injection. In this way, the nonuniform water saturation profile becomes uniform Žsee Fig. 10.. At this point, we found the true interstitial water saturation, Siw s 0.292. Step 3. Remove the porous plate, and again conduct a constant rate oil injection Ž qo s 0.5 cm3rh. to determine the true terminal oil relative permeability Ži.e., without the capillary end effect.. We found k ro < S iw s 0.65. The experimental results give us k ro < S iw k ro < S iw ) Siw s
0.65 s 0.51
s 1.275
Siw y Siw
Ž 30 . 0.324 y 0.292
s
1 y Siw y Sor
1 y 0.292 y 0.251
s 0.07
Ž 31 .
Interpolating the value of l from Figs. 8 and 9 simultaneously using Eqs. Ž30. and Ž31., we found
l s 2.3, and
kk ro < S iw APd Lqo m o
s 0.18
Ž 32 .
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
114
which further gives the value of the threshold pressure, Pd : Pd s
0.18 Lqo m o
s
kk ro < S iw A
psi Ž 0.18 . Ž 10 . Ž 0.5r3600 . Ž 6 = 0.01 . = s 3.28 psi y12 68947.57 dynercm2 Ž 5 = 9.87 = 10 . Ž 0.51 . Ž 2.63 .
Ž 33 .
Eqs. Ž11., Ž16. and Ž17., that predict relative permeabilities and capillary pressure, become
ž ž
Pc Ž psi . s 3.28 =
k ro s 0.65 = 1 y
S w y 0.292 0.457 S w y 0.292 0.457
k rw s Ž k rw < S or s 0.06 . =
ž
y1 r2.3
/ /
Ž 34 . 2
= 1y
S w y 0.292 0.457
ž
S w y 0.292 0.457
1.87
/
Ž 35 .
3.87
/
Ž 36 .
In order to check the validity of these equations, we conducted separate measurements for capillary pressure and steady-state waterroil relative permeabilities. The comparison between values from predictions and measurements are shown in Figs. 11 and 12. Fig. 11 shows the comparison of capillary pressure prediction ŽEq. Ž34.. with data from three different measurements: Ža. reservoir-condition centrifuge experiment, Žb. reservoir-condition porous-plate experiment through the use of microwave attenuation for water saturation determination, and Žc. ambient-condition mercury-injection experiments rescaled to the reservoir condition. Our prediction using Eq. Ž34. provides an overall agreement with the three different sets of data. Fig. 12 shows the comparison between the predicted relative permeabilities ŽEqs. Ž35. and Ž36.. with that measured in a reservoir-condition steady-state experiment using microwave attenuation for in situ water saturation determination. Again, the predicted curves agree well with the measurement.
Fig. 11. Comparison between the theoretically predicted capillary pressure with the measured ones from centrifuge, porous-plate, and mercury-injection experiments.
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
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Fig. 12. Comparison between the theoretically predicted relative permeabilities with the experimental data. The experiment was conducted at reservoir condition in steady-state mode.
To double check the calculations, we substitute the values of Pd and l from Eqs. Ž32. and Ž33. for Eq. Ž19., the profile equation, to determine the saturation profile. The predicted profile can be compared with the measured saturation profile Žshown earlier in Fig. 10. as a final check. Fig. 13 shows this comparison. Again, the theory agrees well with the experiment.
Fig. 13. Comparison between the predicted capillary-end-effect saturation profile with that measured using microwave attenuation techniques. Solid curve: predicted, bullets: measured values.
116
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
5. Summary and conclusions In this work, we have addressed how capillary end effects can influence the computation of relative permeability and saturation in coreflood experiments. To illustrate, we studied the situations of oil displacing water in a water-wet core at the terminal condition as an example. Ø We showed how to convert the averaged terminal oil relative permeability to the actual oil relative permeability. This correction, in some cases we studied, can be very significant Žin the order of 100%. when the capillary force is of the same order of magnitude as the viscous force. Ø We showed how to compute the actual interstitial water saturation from an averaged value. The correction to the computation of Siw is found even more significant than that needed for the terminal relative permeability. Ø We also demonstrated how to simultaneously estimate relative permeabilities and capillary pressure in a drainage process, based on terminal-condition measurements with and without the capillary end effect. The estimated relative permeabilities and capillary pressure were in good agreement with experimental data from different sources. When applying this simultaneous estimation technique, the capillary-induced nonuniform saturation profile information should be obtained from in situ scanning methods.
6. Nomenclature A D E k kr L Pc Pd p q S Siw Sor T t X x
core cross-sectional area Žcm2 . core diameter Žcm. exponentiation based on 10 absolute permeability Žmd. relative permeability Ždimensionless. core total length Žcm. capillary pressure Ždynercm2 . capillary displacement pressure Ždynercm2 . phase pressure Ždynercm2 . phase volumetric flow rate Žcm3rs. saturation Ždimensionless. interstitial water saturation Ždimensionless. residual oil saturation Ždimensionless. temperature Ž8F. time Žs. a dimensionless group defined in Eq. Ž23. distance from core inlet Žcm.
Greek letters DP f l m
pressure different in Eq. Ž24. porosity Ždimensionless. a core-dependent constant for Pc in Eq. Ž11. viscosity Žcp.
Subscripts and superscripts L inlet
across the whole core evaluated at the core inlet
D.D. Huang, M.M. Honarpourr Journal of Petroleum Science and Engineering 19 (1998) 103–117
out o, w, g x 4
117
evaluated at the core outlet oil, water, and gas respectively evaluated at distance x from the core inlet normalized saturation
Bars <
averaged over the whole core evaluated at
Acknowledgements We thank Mobil management for permission to publish this paper. We thank the technical discussions with Rafi Al-Hussainy and George Hirasaki.
References Brooks, R.H., Corey, A.T., 1964. Hydraulic properties of porous media. Hydrology Papers of Colorado State University, No. 3. Burdine, N.T., 1953. Relative permeability calculations from pore size distribution data. Trans. AIME ŽAm. Inst. Min. Metall. Eng.. 198, 71–78. Corey, A.T., 1954. The interrelation between gas and oil permeabilities. Producers Monthly, November, 38–41. Honarpour, M.M., Koederitz, L., Harvey, A.H., 1986. Relative permeability of petroleum reservoirs. CRC Press. Purcell, W.R., 1949. Capillary pressures — their measurement using mercury and the calculation of permeability therefrom. Trans. AIME ŽAm. Inst. Min. Metall. Eng.. February, 39–48. Richardson, J.G., Kerver, J.K., Hafford, J.A., Osoba, J.S., 1952. Laboratory determination of relative permeability. Trans. AIME ŽAm. Inst. Min. Metall. Eng.. 195, 187–196.