Comparison of the three-phase oil relative permeability models

Transport in Porous Media 4 (1989), 59-83. 59 �9 1989 by Kluwer Academic Publishers.

Comparison of the Three-Phase Permeability Models

Oil Relative

M O J D E H D E L S H A D and G A R Y A. P O P E Department of Petroleum Engineering, The University of Texas at Austin, CPE 2, 502, Austin, TX 78712, U.S.A.

(Received: 3 November 1987; revised: 16 May 1988)

Abstract. A comparative study of seven different methods for predicting three-phase oil relative permeabilities in the presence of gas and water phases is presented. Predicted oil relative permeabilities from these correlations have been compared with published three-phase experimental data obtained in Berea sandstone core samples. Some of the correlations under study have been recently developed and have never been tested against the laboratory data.

The comparison shows that the commonly used models such as Stones' often do not give accurate predictions of the experimental data. It is concluded that the recently developed models fit the experimental data as well as or better than the previously developed and widely used three-phase oil relative permeability models.

Key words. Three-phase flow, relative permeability models, enhanced oil recovery, ground water.

1. Nomenclature

a

e/ Cow ~ eog

G h

k~ow, k~og Sj S# SLrg

Sorw, Sorg

St

Subscripts g o

w

go o w

Free parameter Exponent of relative permeability curve of phase j Two-phase oil exponent when the other phase is water or gas Interpolating function Capillary head Relative permeability of phase j Endpoint relative permeability of phase j Two-phase oil relative permeability flowing with water or gas Saturation of phase j Residual saturation of phase j Total residual liquid satuation to gas during two-phase flow of gas-oil Residual oil saturation to flowing water or gas phases Total liquid saturation

G a s

Oil Water Gas-oil Oil-water

60

Superscripts gow Gas-oil-water ow Oil-water

MOJDEH DELSHAD AND GARY A. POPE

2. Introduction

Accurate predictions of three-phase oil relative permeabilities are required for a variety of petroleum processes and ground water pollution problems. Simul- taneous flow of water-gas-oil mixtures are encountered in petroleum reservoirs producing under primary, secondary, and tertiary processes. Enhanced oil recovery methods such as thermal recovery, carbon dioxide immiscible displacement, or any gas injection process generates simultaneous flow of three phases. Three- phase flow is also encountered where a relatively large amount of an oleic pollutant is transported through the unsaturated zone under the ground surface. This usually is assumed to occur either preceding or following rainfall infiltration events. The list of case studies of immiscible pollutants shows that the most commonly investigated incidents involve petroleum products such as crude and refined mineral oils [6, 14, 19, 27, 37]. Several reasons suggested for this are

(1) The EPA estimated that 75 000 to 100 000 of approximately 2 million underground gasoline storage tanks in the U.S. are leaking [17].

(2) Other sources of petroleum pollutants are pipelines and refineries. (3) Petroleum spills are the most common source of petroleum pollutants.

Despite their importance, only a few sets of experimental three-phase relative permeabilities for a specific type of porous medium is available in the literature. Due to the scarcity of reliable three-phase laboratory data, engineers have relied on theoretical models which make use of such information as capillary pressure or two-phase relative permeabilities to predict three-phase flow behavior.

Saraf and McCaffery [30] presented an extensive review of two- and three- phase relative permeabilities. The three-phase relative permeability models discussed were

�9 Corey [7, 8] and Brooks-Corey [3, 4] equations for drainage. �9 Naar-Wygal equations for imbibition [23-25]. �9 Land's equations for both drainage and imbibition [18]. �9 Stone's equations for both drainage and imbibition [33, 34].

Although there is no comparison of the above-mentioned models with experimental data in the paper, Saraf and McCaffery pointed out that because of the scarcity of three-phase experimental data, each has not been tested for more than one or two sets. Therefore, it is difficult to recommend any one of these models.

Fayers and Matthews [15] have examined normalized forms of Stone's two models for predicting three-phase oil relative permeabilities. The models were

THREE-PHASE OIL RELATIVE PERMEABILITY MODELS 61

tested against experimental data of Corey et al. [8], Dalton et al. [9], Saraf and Fatt [29], Holmgren and Morse [16], Schneider and Owens [32], and Saraf et al.

[28]. Fayers and Matthews [15] concluded that the normalized forms of Stone's Models I and II give very similar results when oil saturations are large, but differ considerably as residual oil saturation is approached. They also noticed that Model I is superior to Model II if residual oil saturation calculated from the equation proposed in [15] is used in Model I.

Manjnath and Honarpour [20] have also presented a review of experimental and mathematical methods to predict three-phase oil relative permeabilities. In their review paper, they have discussed three-phase relative permeability models proposed by Corey et al. [8], Naar and Wygal [24], Land [18], Stone [33, 34], normalized Stone's Model II by Dietrich and Bonder [12] and by Nolen [11]. However, they compared only Corey's equation against Donaldson and Dean's three-phase data [13]. The conclusions were that (1) Corey's equation gives higher oil relative permeabilities than those obtained by Donaldson and Dean, (2) computed oil isoperms are concave towards the 100% oil saturation and decreasing gas saturation similar to what was observed by Donaldson, and (3) the discrepancy between the computed and experimental data are greater at low oil saturations.

We present a brief description of the three-phase oil relative permeability models under study, which includes Stone's Models I and II. To test the accuracy of relative permeability predictions, the computed relative permeabilities using the models are compared with experimental data from the literature.

The majority of the proposed models use the two-phase oil-water and oil-gas data to predict the three-phase oil relative permeabilities. The three-phase water and gas relative permeabilities are assumed to be the same as those of two-phase flow. The assumptions made by these authors are that gas and water three-phase relative permeabilities are functions of their own saturations only, whereas those of the oil phase are functions of two saturations.

3. Description of Three-Phase Oil Relative Permeability Models

3.1. MODEL 1 (BAKER)

This model was proposed by Lee Baker [2] and uses an interpolation between the two-phase data as follows

k,o = ( S w - Swr)krow + ( S g - Sg~) k~og, (1) (Sw - Swr) -]- (Sg - Sgr)

where the two-phase relative permeabilities can be experimental data or they can be estimated using two phase models such as the following

So--Sorw )~ow k~o. = k r~ 1 - Swr- Sor w (2)

62 MOJDEH DELSHAD AND GARY A. POPE

(1--Sg--SLrg~ e~ (3) kr~ = kr~ \1 - S L r g - Sgr]

and Sg = 1 - So-min(Sw, Swr) and SLrg is the total residual liquid saturation to gas phase during two-phase flow of gas and oil (SLrg = Swr q-Sorg ).

3.2. MODEL 2 (POPE)

This is the model proposed by Pope and is independent of the two-phase data. The model is described as below:

kro = o -o~ k~ow[aSo(1 - ,~w) t3 +(1 a)g~(l - ~)~] ,

where

(4)

So = 1 - Sw- S g - So~ (5) 1 - Sw~- S~r- So~'

Sw = S w- Swr 1 - S ~ - S g r - Sor ' (6)

Sg - Sgr Sg = 1 - Sw~- Sgr - Sor" (7)

In the absence of experimental data, the residual oil saturation during three- phase flow can be estimated using the relationship proposed by Fayer et al. [15].

Sor= bSo~w + (1 - b)Sorg, (8)

b = 1 Sg (9) 1 - Swr- Sorg"

This formulation requires that b = 1 when Sg = 0 and b = 0 if Sw = Swr and So = Sorg. This implies that Sot in three-phase flow behaves linearly between the two limiting residual oil saturations (Sor,, Sorg).

The parameters in Model 2 are calculated using the following two methods:

Method A: The parameters a , /3 , 7, 8, and a were defined such that the model at the two-phase limits takes the values of experimental two-phase data. In other words, a and 2/are set such that 7 = a +/3 = eow, a = 7 + 8 = eog, and a = �89 so there are no free parameters.

Method B: To take advantage of free parameters in this model, parameters a, and T were fixed by history matching the two-phase oil-gas and oil-water relative permeability data, while parameters a , /3 , and 8 (chosen as free parameters) were estimated by history matching the three-phase data.


3.3. MODEL 3 (LAKE)

Lake [5] proposed this model to predict the microemulsion relative permeability during three-phase oil-water-microemulsion flow. He introduces an interpolation function based on saturations of the wetting and nonwetting phases. The residual saturation, endpoint relative permeability, and exponent for the intermediate wetting phase can then be estimated using those of the other two phases. Here we apply the same concept to estimate the oil relative permeability during the flow of gas-water-oil. The model, with some modification [10], is given by

G = (Sg - Sgr)[1 - (Sw- Swr)] (10) (Sg -- Sgr) "}- (Sw -- Swr) '

Sor = Sgr q- O ( S w r - Sgr), (11)

eo = eg + G ( e w - eg), (12)

o kr~ = kr~ + O ( k ~ - krg), (13)

and

( So-So kro = kr~ 1 - S ~ - Sg~- S o r ] "

(14)

In the absence of the wetting (water) phase, oil takes the properties of the wetting phase and in the absence of the nonwetting (gas) phase, oil takes the properties of the nonwetting phase.

3.4. MODEL 4 (STONE I)

This is the model proposed by Stone [33] and was normalized by Aziz and Settari [1]. Stone's unnormalized model was valid only if kr~ (kro at residual water saturation) happened to be unity. Aziz and Settari adjusted the equation by normalizing it with kr~ since kro at Sw~ is frequently less than one.

The model is described as

kro --

where

So krow krog k~ - S~)(1 - Sg)'

So - Sot

(15)

(16) 1 - S w r - S o r - Sgr'

g w - Sw- Swr 1 -- Swr- Sor- Sgr' (17)

Sg- Sgr (18) ~qg = 1 - Swr- Sot- Sgr"


Stone's formula uses the two-phase relative permeabilities defined by oil-water in the absence of the gas phase and gas-oil displacements in the presence of residual water. Either experimental data can be used for the two-phase relative permeabilities or a model such as Equations (2) and (3) can be used.

The residual oil saturation (Sor) is computed using Equation (8).

3.5. MODEL 5 (STONE II)

This is Stone's Method II [34] normalized by Nolen [11] as

kr~ k~~ L\kro-~w k~w)~row + krg - ( k ~ w + k~g) , (19)

where one choice for the two-phase relative permeabilities is

S w - Sw r ~ ew

k ~ w = k ~ 1 - S w ~ - S o ~ / ' (20)

k~g=krg(1 Sg-Sg~ )eg (21) -- S L r g - S g r / "

The two-phase oil relative permeabilities (k~ow and krog) can be calculated using Equations (2) and (3).

Stone's model II in an unnormalized form does not go to the proper limits at residual water saturation and zero gas saturation unless k~~ = 1.0. Since kro at Sw~ is normally less than one, Stone's model II had to be normalized to produce realistic results.

3.6. MODEL 6 (COREY-TYPE)

In this model we assume that the dependence of oil relative permeability on two saturations is through the dependence of residual oil saturation on two saturations. The model is given as

( So-So~ )eo o (22) kro = k ro 1 - 7 '

where Sot = f(Sg, Sw). The first functional form used for estimating Sot is the linear form proposed by

Fayers et al. [t5, Eq. (8)]. The functional forms with higher order in Sg can also be used if the linear function is not producing satisfactory results.

In the absence of experimental data for three-phase oil exponent and endpoint relative permeability, the following can be used

o __ o kro - bk~ow + (1 - b) kr~ (23)

eo = beow+ (1 - b)eog, (24)

THREE-PHASE OIL RELATIVE PERMEABILITY MODELS

where

b = 1 Sg 1 --Swr--Sorg"

65

(25)

3.7. MODEL 7 (PARKER)

This is the model proposed by Parker et aL [26] to describe relative permeability- saturation-fluid pressure functional relationships in two- or three-phase flow subject to monotonic saturation paths. The model is developed under the assumption that fluid wettability follows the sequence w a t e r > o i l > g a s . The expression for oil relative permeability is derived from the scaled capillary head-saturation function using a flow channel distribution model to estimate effective mean fluid conducting pore dimensions [35].

The scaled capillary head-saturation relation is

how > O, gggOW = [1 + (aowhow)"]-"*, (26)

how < 0, Sgw ~ = 1, (27)

and for total liquid saturation

hgo > 0, ggow = [1 q-(Otgohgo)n] -m, (28)

hgo < O, ~gow = 1, (29)

where the effective saturations are defined as

Cgow _

~gwO w __ ~ w Sr (30) 1 - & '

~gow _ S , g ~ & t I- Sr ' (31)

and

S g~ ~weg~ _* ~oeg~ . (32)

The assumption employed is that the apparent 'irreducible' fluid saturation (&) is independent of fluid properties or saturation history. Implicit in the above equations is the assumption that no gas-water interfaces occur in the three-phase region until oil saturation diminishes to a level at which oil exists only as discontinuous blobs or pendular rings at particle contact.

Using Mualem's model [22] to evaluate effective hydraulic radii from the capillary pressure-saturation function yields the expression

kro -- ( K - ~w)1"-{[1 - g'w/m] m - [ l - g y m ] ~ } 2 (33)

indicating that, in general, oil relative permeability is a function of both water- and oil-phase saturation.

6 6 MOJDEH DELSHAD AND GARY A. POPE

4. Three-Phase Relative Permeability Data

A brief review of the experimental data on three-phase relative permeability in chronological order is presented here. The data reviewed was obtained using only Berea sandstone samples.

Corey, Rathjens, Henderson, and Wyllie, 1956 [8] reported results of three- phase oil, water, and gas relative permeability experiments on Berea sandstone using Hassler's capillary pressure method. Water was kept immobile by using semi-permeable barriers so that water relative permeability could not be measured. The relative permeabilities were measured on nine cores with water saturation ranging from 17 to 71%. The oil relative permeability data show a definite dependence on two saturations based on the curvature of oil isoperms. Gas isoperms are straight lines, implying that three-phase gas relative permeability is a function of gas saturation only. Because of the technique used by Corey et al., water relative permeability was not measured, but water isoperms were generated based on the assumption that water relative permeability is the same as the oil relative permeability in gas-oil two-phase flow in a water-wet system. Relative permeability contours generated for water are straight lines parallel to water isosaturation lines.

Donaldson and Dean, 1966 [13] measured the three-phase oil (soltrol), water (NaC1 brine), and gas (air) on Berea sandstone cores using an extension of Welge's two-phase unsteady-state method [38] to three-phase flow. The oil isoperms are concave toward the 100% oil saturation apex. It seems that the three-phase oil relative permeability is not a function of just its own saturation since (1) the oil relative permeabilities are much higher when water is the other phase (two-phase oil-water flow) than when gas is the second phase (two-phase oil-gas flow), and (2) the three-phase oil relative permeability contours show significant curvature. The relative permeability contours for three-phase water and gas phases are not straight lines, indicating that the relative permeabilities are affected by the saturation distribution of the other fluids in the medium.

Sarem, 1966 [31] used the unsteady-state method proposed by Welge for two-phase flow to obtain three-phase oil (soltrol), water (NaC1 brine), and gas (air) relative permeabilities in Berea sandstone core. To apply the Welge's method to three-phase flow, he assumed that the fractional flow and relative permeability for each phase are functions only of its own saturation. The conclusion based on the experimental results is that oil relative permeability is a function of its own saturation since the oil relative permeabilities lie on a single curve, regardless of the saturation of the other two phases. Water and gas isoperms are also parallel to the isosaturation lines. These results, however, may be affected by the assumption made for the analysis that the three-phase relative permeabilities are functions of their own saturations only. Sarem did not present enough data to draw the three-phase oil isoperms.

V a n Spronsen, 1982 [36] measured the three-phase relative permeability for oil (unknown), water (glycol solution), and gas (air) phases in Berea sandstone using


the centrifuge method. The data indicated that oil and water isoperms are slightly concave toward their respective apexes. The shape of oil and water isoperms shows that the three-phase relative permeability of oil or water is relatively insensitive to the presence of other phases. No data is available for three-phase gas relative permeability. Van Spronsen gives no information on two-phase oil-water or oil-gas relative permeabilities.

Saraf, Batycky, Jackson, and Fisher, 1982 [28] measured three-phase oil (soltrol), water (distilled water), and gas (nitrogen) relative permeabilities on Berea sandstone using both steady-state and unsteady-state methods. The three- phase relative permeabilities were measured for several saturation directions (Sw+So~SgT, Sw-..-~So~Sg+, Sw~So~Sg~). It seems that oil relative permeabilities are independent of, or show a weak dependence on, two saturations since (1) one can draw a single curve through the three-phase data when presented as oil relative permeability versus oil saturation, (2) the oil isoperms show no significant curvature, and (3) Saraf et al. made the comment that in view of the scatter in the data, the justification for drawing curves instead of straight lines for oil isoperms was marginal. Gas and water isoperms were straight lines parallel to the isosaturation lines, indicating that three-phase water and gas relative permeabilities are functions of their respective saturations only.

In summary, it is very difficult to draw a definite conclusion from the available three-phase oil relative permeability data with regard to the dependence of oil relative permeability on either one- or two-phase saturations considering scatter in the data, error in the procedures used, and differences in core sample properties. However, based on the available data, it appears that there is a slightly greater tendency for the oil relative permeabilities to depend on two saturations than on one saturation. The majority of oil relative permeability contours measured are concave toward the oil apex.

5. Comparison of Predicted and Measured Oil Relative Permeability

Predicted oil relative permeability using seven different relative permeability models described earlier are compared against three sets of experimental results (Corey et al., Donaldson and Dean, and Saraf et al.). These authors have provided the essential two-phase data which are required in most of the relative permeability models.

5.1. THE DATA OF COREY, RATHJENS, HENDERSON, AND WYLLIE

To test Models 1 through 6 against the experiments of Corey et al., data for krow and k~w are needed. Because the two-phase oil-water experimental data were lacking, values of k,~ and krow were estimated based on the following assumptions:


(1) Water permeability during oil/water flow is the same as the oil permeability during oil/gas flow.

(2) Oil permeability during oil/water flow is the same as the gas permeability during oil/gas flow.

The parameters in Model 2 were found using both methods A and B. Table I lists the best estimates for these parameters based on the minimum SSL function defined in the following section.

The parameters (St , M ) in Model 7 were treated as free parameters and were found by history matching the three-phase data since the capillary pressure data were not available.

The two-phase relative permeability parameters such as residual saturations, endpoint relative permeabilities, and exponents are based on the two-phase data of Corey (Table I). The resulting predictions for the seven models are shown in Figures 1 through 8. With the exception of Model 3, the predicted oil isoperms show similar trends and curvatures as those measured by Corey et al. The predicted oil isoperms show concave curvature toward 100% oil apex similar to those of measured data with isoperms computed from Model 3 showing the most curvature and those from Model 6 the least curvature. The predicted oil isoperms using Model 2 are closer to the experimental data when the parameters were found by history matching the three-phase data (Figures 2 and 3).

-~ EXP. DATA ( COREY ) 6 m EXP. DATA ( COREY ) o ~} KR2 = O. 05 ] KR2 = O. 05

KR2 O. 10 e KR2 D. 10 ! KR2 O. 20 i A KR2 O. 20

+ KR2 O. 30 ~ I + KR2 = O. 30 ~" X KR2 O. 40 = c; e KR2 O. 50 6 ~' KR2 , 50

1.05

H ' t ~,.1

m ~ 1

~ I ~

d7 I

z • .... 05#

I+l~ ~ ; ' o { ~, /+/~:

~ 1o o'.29 0'.,2 o'.58 0:74 ~.~o o.1o 0.26 0.42 o'.~8 OIL SATURATION OIL SATURATION

Fig. 1, Comparison of predicted oil isoperms using Model 1 with the data of Corey et al.

0'. 74 0'. 90

Fig. 2. Comparison of predicted oil isoperms using Model 2 (method A) with the data of Corey et al.

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7 0 MOJDEH DELSHAD AND GARY A. POPE

o EXP. DATA (COREY) i I ~ KRo ~ O'O5

o K R O ~ D, I

t~ KRO ~ O. 2 + KRO = O. 3 X KRO = O. 4

KRO = O. 5

.05 z i

// o , o I .2

/ ~' I t' t .3

?: j• ,~ +~/o

o

o'. 10 26 42 ; - - v ~ - 74 O - - 90 OIL SATURATION

Fig. 3. Comparison of predicted oil isoperms using Model 2 (method B) with the data of Corey e t a l .

EXP. DATA [ COREY )

o I~ KR2 D. I O

KR2 O. 20 ~i + KR2 D. 30

I X KR2 D. 40 c; r KR2 0. 50

m m

m

�9 o

I

,@/ ~,, " / . + * - - - . . 2

/ , + j o ~ .4 .~ .

,2 %'.10 0'.~6 0'.42 o'.sB D'.74 0'.9o

0 ' T L S A T U R A T ' T ON

Fig. 4. Comparison of predicted oil isoperms using Model 3 with the data of Corey e t a l .

z•

I -

Oq~ V I o

.< (D

EXP. DATA ( COREY ) KR2 = O. OS

O KR2 = O. 10 & KR2 ~ O. 20 + KR2 ~ O. 30 X KR2 O, 40 @ KR2 = O. 50

/ .05 /

/ m

I P.7 ] • ,

/ , .2 / / I 8 ~,.3

/ o / / .~ ,

/ ' ,/ /i

o o

%'.10 0'.2. 0'.42 0'.58- d. 74 D'.gD OIL SATURATION

Fig. 5. Comparison of predicted oil isoperms using Model 4 with the data of Corey e t al .

2̀ EXP. DATA ( CDREY )

c ~ ~ [] KR2 = O, OS ! , O KR2 D, 10 ! & KR2 = O, 20

+ KR2 = O. 30 X KR2 O. 40

8 - ~ KR2 = O. 50

.05

~- 8 9.7 < / m

I d ~ u ) ~ /

/ / / , 3 <~ I / §

e

o

8 o l o o'.2o o'.42 0'.58 0'.74 ~.~o O T L S A T U R A T T O N

Fig. 6. Comparison of predicted oil isoperms using Model 5 with the data of Corey e t a l .

5.2. T H E D A T A O F D O N A L D S O N A N D D E A N

Donaldson and Dean provided necessary two-phase oil-water and oil-gas data. However, they presented the relative permeabilities in two- and three-phase flow as a function of terminal saturation (saturation at outflow end of the core) rather than average saturation. Table II shows the values for adjusting parameters in Models 2 and 7 and the two-phase parameters based on their two-phase measured data used in the rest of the relative permeability models. The predicted


6" 6"

Z •

.< (5

m

6"

EXP. DATA ( OOREY ) [] KR2 = O. 050 �9 KR2 = O. I 0

A KR2 = O. 20 + KR2 = O. 30 X KR2 = D. 40 '~ KR2 = 0, 50

-05 I

/ '[] [ ] , ~

I ' [.2

] 'r ] , + +

o o

Fig. 7.

D: 2~ ot ,~ o: s~ o: 7. o: oo OIL SATURATION

Comparison of predicted oil isoperms using Model 6 with the data of Corey et al.

H ~

% I--

(,'3~

< C~

6

o

~

EXP. DATA ( COREY ) KR2 = O. 050

e KR2 = O. 10 A KR2 = O, 20 + KR2 "= O. 30 X KR2 = O. 40

KR2 = D. 50

/.05 / []j

, . 7

/ ~ ~, '

I ] ,ix t,

I 1 �9 +~

I

I

/.2

.3 / ~/ • / .4

+ ~"://.5 m

o 26 ot ,~2 ot 58 o: 7~ ot 90 OIL SATUkATION

Fig. 8. Comparison of predicted oil isoperms using Model 7 with the data of Corey et al.

and experimental three-phase isoperms are shown in Figures 9 through 16. With the exception of Model 7 and Method B of Model 2, there was no adjustment of the input parameters in any of the models. The oil isoperms generated by the models with the exception of those by Model 3 show less curvature compared to those measured by Donaldson and Dean. The predicted oil isoperms have a concave curvature similar to those observed by Donaldson and Dean. The relative permeabilities computed using Model 2 (Method B) are closer to the experimental data at low values of gas saturation compared to those computed

EXP. DATA (DONALDSON) o ~l KR2 . 0 . 0 1

~ �9 K R 2 . 0 . 0 8 KR2 - O. 15

+ KR2 - 0 . 20 X KR2 ~ D. '30 �9 KR2 - 0 . 40

,5 . 07

I I I , . 0 5 I / '~ ' . 15 1

c

c~O. 10 0 : 2 2 0 . 3 4 0 ' . 4 6 ~ 0 ! 5 8 O I L S A T U R A T I O N

C•Z H v ~ ~6

0 3 ~

0 3 6 "

2

oi 70

i

EXP. DATA (DONALDSON) B KR2 = O. 01 O KR2 = O. 06

KR2 : O, 15 + ~R2 = 0 . 2 0 X KR2 = O. 30 r ~(R2 = D, 40

~.0/ \

I ~ ' :06

~ 'a .2 I e| ! ~ A

m 1 ~, i3+

[] l \ |

[] ~ o ', , , + ' \ ~ , \ 2x \ . \ ,,~

10 0122 m 0:34 o 0 t 4 6 " ' ^ o [ 5 8

O I L SATURATION 0'. 70

Fig. 9. Comparison of predicted oil isoperms using Model 1 with the data of Donaldson and Dean.

Fig. 10. Comparison of predicted oil isoperms using Model 2 (method A) with the experimental data of Donaldson and Dean.

tO

Tab

le 1

1.

Tw

o-ph

ase

data

use

d in

the

rel

ativ

e pe

rmea

bili

ty m

odel

s (B

ased

on

Don

alds

on a

nd D

ean

data

)

Res

. sa

tura

tio

n

En

dp

oin

t re

l. p

erm

. E

xp

on

en

t

Sw

r S

g r

Sorg

S

....

Sr

S

Lrg

kr

~ k

~

kr~

k

~

e w

eg

eo

g

eow

a

g 3'

~

a m

Mo

de

l 1

Mo

de

l 2

a

(me

tho

d

A)

Mo

de

l 3

Mo

de

l 4

Mo

de

l 5

Mo

de

l 6

Mo

de

l 7

Mo

de

l 2

(me

tho

d

B)

0.3

1

0.0

0.23

-

0.53

-

0.71

6 0.

75

-

0.31

0

,0

11.2

2 -

- -

11

,71

6

- -

- 1

.66

2

.94

-

- -

- 1

,66

1

.28

2

.94

-1

.28

(}

,50

0,3

1

0,0

-

- -

0,4

0

0,6

0

- 3

.33

1

,94

-

-

0.3

1

0,0

0

.22

0

.23

-

0,5

3

0.7

16

-

- -

1.6

6

2.9

4

0,3

1

0,0

-

0.2

3

- 0

,53

0

.40

0

.60

0

.71

6

- 3

.33

1

,94

1

,66

2

.94

0.3

1

0.0

0

.22

0

.23

-

- 0

,71

6

0.7

5

- -

1,6

6

2,9

4

- -

0,3

0

..

..

.

0.3

1

0.0

0

.22

0

.23

.

..

..

- 0

.70

1.6

0

.50

2

.8

-0.5

0

0.5

0

- Z:

ro,3

at~

+~

=

T

=

eow

,

3~

+ 6

=

a =

e

og

.

�9


,5-

00•

I-- < . e

e~

6

EXP. DATA (DONALDSON) m KR2 = O. Ol ~] ~ KR2 O. O8

KR2 0 . 1 5 + KR2 0 . 2 0 X KR2 0 . 3 0 ~ ]

KR2 0 . 4 0 ~ I

~o ot22 o':70

~.O1

[ ~ . 0 6 I []

i m ',,,~ .15

t , \Dr2 o , , i § 2 4 7

/ o , ~1 , , \'•

I ', . ~ x~\ I i o , " ~ i • I , , \ ml e , ~ \ + 1 x~

I , . +

-m I , 'e i .& i X - - - 0 . 3 4 O. 46 O. 58

OTL SATURATTON

EXP. DATA(DONALDSON) KR2 ~ O. 01

�9 KR2 = O. 06 KR2 = O. ! 5

+ KR2 - - O. 20 X KR2 . O. 30 4, KR2 O. 40

H m m

<~6" m

~ , ~ ~ - ~ ~ ~ . 0 1 6

. " ++ - 2~ r A x

J

/ " + / .. - f - - : . 4 m - , r i x ~ i

O 0 ' ~ 2 2 ~ 0134 0 . 4 6 0 . 5 8 0 . 7 0 OIL S A T U R A T Z ON

Fig. 11. Comparison of predicted oil isoperms using Model 2 (method B) with the experimental data of Donaldson and Dean.


o E~ZP. DATA (DONALDSON) 6" ] • KR2 : O. 01

O KR2 = O. 06 a KR2 O. 15 I i + KR2 = 0 . 2 0

~ X t KR2 = 0 . 3 0 6" t r KR2 = O. 4D

/

i 1.O7 I

~61 i

~ i ~ ' . t 5

I \ . 3

'~ o~ 5?; ? /

\ ooi +, o 6 - J - - ~ - - [ D4 t~ f - - e - " ; 3:

0 . 1 0 , 2 2 0 . 5 4 0146 ~ 0r, 58 O Z L S A T U R A T T O N

~TD

Fig. 13. Comparison Of predicted oil isoperms using Model 4 with the data of Donaldson and Dean.

o EXR. DATA (DONALDSON)

6 " m KR2 = O. Ol e KR2 = O. 06

KR2 = 0. 15 + KR2 = O, 20 X KR2 = O. 50

6 " ~ NR2 = 0. 40

\ .Ol \

m 1 , eo I e!

�9 'r •

Q

~0 10 0122 0 . 3 4 0~,46 • 9 B

. 0 . 58 0 ~. 70 OZL S A T U R A T I O N


H w

~ - c ~ - <

c~61 ,<

using Method A. In general, the predicted oil relative permeabilities are far from the experimental data. This may be partly due to the assumption used in all the models that water and gas relative permeabilities are functions of their own saturation only despite the fact that Donaldson made the conclusion that gas and water relative permeabilities depend on the saturation distribution of other phases as well.


EXP. DATA (DONALDSON) `3] El KR2 = 0. 01

i O KR2 O. 06 ,i ~, KR2 = O, 15

+ KR2 = O, 20 X KR2 = O. 30

`3~ r KR2 = O, 40 ! , ~07

[]

R i \ ~.06 t ' . 7 5

z i i [] , ~

~ j ,,, ', ,, q',,t§

,3 a

I ~ \ ~ ',,, + \ '~, , \ r \ r o ~ , + t O. 10 O. 22 O. 34 (3 I. 46 O. 58 [3. 70

O I L S A T U R A T I O N


g 5-

EXP. DATA (DONALDSON) KR2 = O. 01 KR2 = O. 06 KR2 = O. 15 KR2 = O, 20 KR2 = O. 30 KR2 = O. 40

~)tD o p01

] r

~* I 6 u,,,, ~l d , " .0 0 ~ 6 I | ,

< ~ ' ~ .15 o �9 , A y~

,.,, I ~ .+ ..x.2

I g �9 / x . % . 3

~ / A / x ~ . / �9 g / ,/"+ . . / " %L.~D o'.= [] o : 3 ~ - o ' , E '~ D:58 otto



5.3. THE DATA OF SARAF, BATYCKY, JACKSON, AND FISHER

Saraf et al. provided three-phase oil relative permeability data using the unsteady-state technique for increasing gas saturation (Sw+ So I' Sg 1') and decreasing gas saturation (SwTSo~Sg~, S w ~ SoT Sg$). However, they reported oil relative permeability contours only for the increasing gas saturation case. The oil relative permeabilities measured using the steady-state method were not presented in a ternary plot and the individual data points were not reported in detail to construct the oil isoperms. Therefore, only the unsteady-state relative permeability data in the direction of increasing gas saturation are used in this paper. Hysteresis effects are taken into consideration by employing the appropriate two-phase data. The two-phase data used are the oil-gas relative permeabilities at Sw, = 0.23, SoL and SgT and oil-water relative permeabilities at Sg, = 0, Sw~, and So1'. The values of k,~ k,~ So~, So,g, Sw,, and Sg, were not specifically reported and the two-phase values plotted separately in their paper are not consistent with the three-phase ternary diagram. Our estimates of the two-phase parameters based on both the two- and three-phase data presented in Saraf's report are given in Table III. The predicted three-phase isoperms using the models are compared with the measured values of Saraf in Figures 17 through 24. No adjustment in the input parameters was made except for Models 2 (Method B) and 7, and the predicted values are based on the measured two-phase data. With the exception of oil isoperms predicted by Models 2 (Method B) and 3, the computed oil isoperms show the same trends and curvatures as those measured by Saraf et al. The oil isoperms calculated by Models 1, 2 (Method A), and 4 through 7 show concave curvature similar to those measured by Saraf et al.

Tab

le

IIl.

T

wo

-ph

ase

dat

a u

sed

in

th

e re

lati

ve

per

mea

bil

ity

m

od

els

(Bas

ed

on

S

araf

et

al.

dat

a)

m

> �9

>

Res

. sa

tura

tio

n

En

dp

oin

t re

l. p

erm

. E

xp

on

ent

Sw

r S

gr

Sor

g S

o~,,

S

r SL

rg

k~v

k~

kr~

k~

ew

eg

eog

eow

~

/3

3'

8 a

m

Mo

del

1

0.3

1

0.0

-

0.3

3

0.5

3

- -

l.O

0

.60

-

- 2

,92

2

,58

-

-

< ~0

Mo

del

2

a

(met

ho

d

A)

Mo

del

3

Mo

del

4

Mo

del

5

Mo

del

6

Mo

del

7

Mo

del

2

(met

ho

d

B)

0.3

l 0

.0

0.3

0

0.3

3

..

..

1.

0 .

..

..

2

.92

0

.34

2

.58

0

.34

0

.50

-

0.3

1

0.0

.

..

.

0.0

8

0.7

0

- -

1.7

4

2.9

7

..

..

..

.

0.3

1

0,0

0

.30

0

.33

-

0.5

3

- -

1.0

0.6

0

- -

2.9

2

2.5

8

..

..

.

0.3

1

0.0

-

0.3

3

- 0

.53

0

.08

0

.70

1.

0 "

0.6

0

1.7

4

2.9

7

2.9

2

2.5

8

..

..

0.3

1

0.0

0

.30

0

.33

.

..

.

1.0

0.6

0

- -

2.9

2

2.5

8

..

..

.

- -

- 0

.39

.

..

..

..

..

..

.

1.0

0.3

1

0~0

0.3

0

0.3

3

..

..

1.

0 .

..

.

2.3

0

-0.8

0

3.0

-0

.50

0

.40

-

> =.

0

aa+

/3=

T

=

eow

, 3

/+

t$ =

a

=eo

g.

,.,q


Z O � 9

~ "

el

d"

{XP . DATA ($ARAF) B KR2 - O . 01 r KR2 - O. 10 a KR2 " D. 20 + KR2 " O. 3D X KR2 ~ O, 4D �9 KR2 ~ D. 50

H ~

<~,s

D

t.Ar

<~ o

.01 .

a m �9

m .1 m �9 a

~ ,~ " .2 ++ ' a + x

I ~ | , ~ l * , x 4 * ~ : ' ' + I x . . ~ - * 5

a .~, x e .

0'. 30 O. 40 O. 50 O. 60 O:IL SATURATION

E~P. DATA (SARAF) il9 KR2 - 0.01 | KR2 O. 10 a KR2 - 0 , 20 + KR2 - D. 30 x KR2 - D. 40 �9 KR2 - 0 .50

o o g

%: ~o d 70 %: ~D o: 3o o" 70

Fig. 17. Comparison of predicted oil isoperms using Model 1 with the data of Saraf et al.

m

p |

I g .

i �9 . 2+ x

"l b O. 40 D. 50 O. 60

OIL SATURATION

Fig. 18. Comparison of predicted oil isoperms using Model 2 (method A) with the data of Sara[ et al.

o

EXP. DATA" (SARAF) o " �9 KR2 = D. 01

{9 KR2 = D. 1D ~, KR2 = O. 20 + KR2 = O. 30

"~ x KR2 O. 40 c; ~ KR2 = D. 50

8o

m ~

(/3o

6 '

g

~o D'. 3o

,01 \

'/ m o

t ~ ,~ I o , 9 &

I e o \'~ .3 * * J , ! : . 4 ~

A I 0 / :? ~ [ O, ~ ie t �9

l m , + t :~ e , m r i ' z . i ~

D. 40 0 .50 O. 60 O. 7 O.TL S A T U R A T 2 0 N

d

Z

p- <l.,e

,<

e~

o"

g

%t 20

Fig. 19. Comparison of predicted oil isoperms Fig. 20, using Model 2 (method B) with the data of Saraf et al,

EXP. DATA (SA~AF) B. KR2 " O. 01 �9 KR2 = O. 10 ~, KR2 - D, 20 + KR2 = O. 30 X KR2 - O, 40 e, KR2 - 0 , 5 0

/ /

/ /

~ - - - . ~ . 0 1 | 7 ~ �9

m

&

_ - + . I + • ia . e l A " - + x

| ", ,+ .2~ ." f �9 _ _ + . _ x _ o . ~

�9 i "<. . . ~ . - ~ . s O[ 30 O, 40 0 . 5 0 O. S0 D. 70

O]IL SATURATION

Comparison of predicted oil isoperms using Model 3 with the data of Saraf et al.

The predicted oil isoperms of Model 2 employing Method B are closer to the experimental data than those computed using Method A even though the curvature is convex toward the 100% oil apex.


EXP. DATA (SARAF) O- [] KR2 - 0 . 0 1

�9 KR2 0 . 1 0 KR2 = 0 , 2 0

+ KR2 0 . 3 0 m X KR2 0 , 4 0 ~" �9 KR2 - 0 . 5 0

Z o | .01

%: ~o ot ~o

Fig. 21.

m .q | , |

, :g .2 �9 l , ' . 3 +

' , +,4 x " ~ I +1 x

~ ' 'C§ ~.5 r A ~ "+ x �9

o'. '~ - o'.Eo ' o'.~o" o7o O I L S A T U R A T I O N


H ~

0 3 ~

g

2

EXP. DATA (SARAF) KR2 = 0 , 0 1

e KR2 = 0 . 1 0 KR2 = 0 , 2 0

+ KR2 = 0.36 X KR2 = 0 4 0 r KR2 - 0 . 5 0

o o

~o " r ~ ~ . 20 . 30 0 ~. 70

.01 ~ - I m ./

/ "2 ,2 / []

I [ ] , o ~ i "~ +* , ~ ' + x

, o ~ ~

0. 40 0. 50 0. 60 O ] L S A T U R A T I O N

Fig. 22. Comparison of predicted oil isoperms using Model 5 with the data of Saraf et al.

z o m H l ' l

S d

EXP. DATA (SARAF) r m KR2 - o. oi le KR2 0 . 1 0

KR2 - 0 . 2 0 KR2 - 0 , 3 0

X KR2 - 0 . 4 0 �9 KR2 - 0 .50

EXP. DATA (SARAF)

.01 ~ | | Ig

�9 ' + +

'~ ~ . 2 . . 3 x t

p ~ Ix " ,,.b

l,~ | Xx i

0 . 4 0 0 , 5 0 0 , 8 0 O I L S A T U R A T I O N

g

%'. 2o o'. ~o o'. 70 o'. ?o

6 " B KR2 = 0 , 0 1 r KR2 0. 10 A KR2 O. 20 + MR2 O. 30 x k r 2 O. 40 �9 KR2 0. 50

z o r

0~ 30

Fig. 24. Fig. 23. Comparison of predicted oil isoperms using Model 6 with the data of Saraf et al.

|

e .o7 , :

' &

? , : 0 : 3 : ~ e �9 ', ~,+ x �9

! . o ; o , ., .. 0 , 4 0 0 . 5 0 0 . 6 0



6. Comparison of Three-Phase Oil Relat ive Permeabifity Predict ions

To evaluate how the relative permeability models behave in comparison to each other, two functions employed here are

S S E -- Z [(kro)e - (kro)c] 2 (34) t ip

and

S S L = ~ [log(kro)e - log(kro)c] 2 (35) n p


where the subscript e denotes the experimental value taken as the true value of kro function, and subscript c denotes the computed kro from the models. The number of experimentaI data points for any specific oil isoperm is np. The more commonly used function, SSE, gives a smaller value (better fit) for models that predict relative permeabilities that are too small than those that overpredict the relative permeabilities, even though the overpredicted values are closer to the experimental data. This is especially true for relative permeabilities close to zero. To avoid this, the logarithmic form of relative permeabilities has been used (SSL). The above functions were then evaluated using the data of Corey et al., Donaldson and Dean, and Saraf et al. Tables IV through VI summarize the results for each of the relative permeability models. The standard error of estimate (SEE) is also calculated and is presented in Tables IV through VI.

SEE = , / S S E (36) Y N

where N is the total number of data points used in the analysis. Models 1, 2 (Methods A and B), and 7 give smaller values of SSE and SSL

than the rest of the models, indicating a better fit of the experimental data of Corey et al. for all the oil isoperms. Based on the SSE and SSL values, it seems

Table IVa. Comparison of predicted and experimental data of Corey et al. using SSE function

Oil 0.001 0.01 0.05 0 .10 0 .20 0 .30 0 .40 S E E

Isoperm np = 6 np = 5 np = 4 np = 4 n o = 4 np = 3 np = 3 N = 29

Model1 8 . 6 E - 6 9 . 1 E - 4 2 . 1 E - 3 3 . 6 1 E - 3 9 . 5 E - 3 5 . 2 E - 3 2 . 2 E - 3 0 .0285

Model2A 6 . 2 E - 6 1 . 1 E - 3 1 . 5 E - 3 3 . 6 2 E - 3 2 . 0 E - 2 2 . 6 E - 2 2 . 1 E - 2 0 .0510

Model2B 3 . 1 E - 6 2 . 5 E - 4 3 . 7 E - 4 f . 2 E - 3 8 . 2 E - 3 1 . 6 E - 2 1 . 1 E - 2 0 .0362

Model3 6 . 0 E - 6 3 . 6 E - 4 7 . 7 E - 3 2 . 0 E - 2 4 . 0 E - 2 5 . 2 E - 2 2 . 3 E - 2 0 .0706

Model4 1 . 5 E - 4 1 . 0 E - 2 1 . 0 E - 2 1 . 8 E - 2 1 . 6 E - 1 4 . 0 E - 2 3 . 2 E - 2 0 .0968

Model5 1 . 3 E - 2 7 . 9 E - 3 1 . 0 E - 2 2 . 0 E - 2 4 . 8 E - 2 4 . 1 E - 2 3 . 3 E - 2 0 .0775

Model6 1 . 1 E - 5 1 . 3 E - 3 6 . 9 E - 3 1 . 4 E - 2 4 . I E - 2 3 . 9 E - 2 3 . 4 E - 2 0 .0689

Model7 6 . 0 E - 6 5 . 2 E - 5 6 . 5 E - 4 4 . 2 E - 3 1 . 5 E - 2 3 . 1 E - 2 6 . 1 E - 2 0 .0624

Table IVb. Comparion of predicted and experimental data of Corey et al. using SSL function

Oil 0.001 0.01 0.05 0 .10 0 .20 0 .30 0.40

Isoperm

Model 1 2 .5E - 0 6 .3E - 1 9 . 4 E - 2 4 . 2 E - 2 3 .3E - 2 9 .4E - 3 2 .4E - 3

Model 2 A 9 . 4 E - 0 7 .0E - 1 7 .4E - 2 4 . 8 E - 2 6 . 6 E - 2 3 .6E - 2 1 .9E - 2

Model 2B 1 .0E + 1 2 . 2 E - l 3 .4E - 2 3 .0E - 2 3 . 7 E - 2 2 .7E - 2 1.0E - 2

Model 3 5 .4E + 1 1.2E + 1 3 .6E + 1 2 . 3 E 0 4 . 9 E - 1 1 .4E - 1 2 .5E - 2

Model 4 1 .2E + 1 2 . 4E0 3 .2E - 1 1 .9E - 2 1 .2E - 1 5 .4E - 2 2 .8E - 2

Model 5 5 .4E + 1 1 .7E0 3 .1E - 1 2 . 0E0 1 .3E - 1 5 .5E - 2 2 .9E - 2

Model 6 9 . 9 E 0 8 . 4 E - 1 2 . 4 E - 1 1 . 3 E - 1 1 . 2 E - 1 5 . 6 E - 2 3 . 1 E - 2

Model 7 5 . 4 E + 1 5 . 9 E - 2 7 . 0 E - 2 1 . 2 E - 1 1 . 1 E - 1 1 . 0 E - 1 I . I E - 1

T H R E E - P H A S E OIL R E L A T I V E P E R M E A B I L I T Y M O D E L S

Table Va. Compar i son of predicted and experimental data of Donaldson and Dean using SSE funct ion

7 9

Oil 0.01 0.06 0.15 0.20 0.30 0.40 0.55 SEE Isoperm np = 6 np = 9 np= 5 n p = 7 np = 3 tip = 4 np = 3 N = 37

M o d e l l 3 . 0 E - 3 1 . 4 E - 2 9 . 0 E - 3 2 . 0 E - 2 3 . 1 E - 3 1 . 0 E - 2 5 . 7 E - 3 0.042 M o d e t 2 A 1 . 7 E - 3 1 . 3 E - 2 1 . 2 E - 2 2 . 3 E - 2 6 . 9 E - 3 1 . 0 E - 2 8 . 4 E - 3 0.045 M o d e l 2 B 1 . 1 E - 3 6 . 8 E - 3 5 . 7 E - 3 1 . 0 E - 2 2 . 7 E - 3 3 . 6 E - 3 4 . 3 E - 3 0.030 M o d e l 3 1 . 4 E - 3 1 . 8 E - 2 4 . 1 E - 2 8 . 5 E - 2 6 . 2 E - 2 1 . 2 E - I 9 . 9 E - 2 0.108 M o d e l 4 4 . 0 E - 3 1 . 3 E - 2 8 . 0 E - 3 1 . 7 E - 2 3 . 3 E - 3 8 . 7 E - 3 7 . 4 E - 3 0.041 M o d e l 5 1 . 1 E - 3 1 . 6 E - 2 7 . 5 E - 3 1 . 5 E - 2 2 . 5 E - 3 7 . 7 E - 3 6 . 8 E - 3 0.039 M o d e l 6 8 . 2 E - 4 1 . 0 E - 2 2 . 1 E - 2 3 . 4 E - 2 2 . 2 E - 2 2 . 6 E - 2 1 . 9 E - 2 0.060 M o d e l 7 4 . 0 E - 4 3 . 8 E - 3 9 . 5 E - 3 2 . 8 E - 2 1 . 2 E - 2 2 . 2 E - 2 1 . 3 E - 2 0.049

Table Vb. Compar i son of predicted and experimental data of Donaldson and Dean using SSL funct ion

Oil Isoperm 0.0l 0.06 0.15 0.20 0.30 0.40 0.55

Model 1 1.1E0 5 . 9 E - i 9 . 0 E - 2 8 . 6 E - 2 7 . 9 E - 3 1 . 2 E - 2 3 . 9 E - 3 Model 2A 1.2E0 1.1E0 1.6E - 1 1.3E - 1 1.6E - 2 1.4E - 2 5.9E - 3 Model 2B 6.0E - 1 4.4E - 1 6.3E - 2 5.4E - 2 6.5E - 3 4.6E - 3 2.9E - 3 Model 3 1.0E + 1 4.8E0 1.1E0 1.0E0 2 . 8 E - 1 3 . 1 E - 1 9.8E - 2 Model 4 1.3E0 5 . 1 E - 1 8 . 0 E - 2 7 . 5 E - 2 8 . 6 E - 3 1 . 1 E - 2 5 . 1 E - 3 Model 5 2 . 0 E + 1 2 .1E0 7 . 5 E - 2 6 . 6 E - 2 6 . 2 E - 3 9 . 8 E - 3 4 . 8 E - 3 Model 6 1.4E0 1.4E0 3.2E - 1 2.6E - t 6.6E - 2 4.0E - 2 1.4E - 2 Model 7 1.6E + 1 2.4E - 1 9.9E - 2 1.2E - 1 3.4E - 2 3 . 4 E - 1 9.4E - 3

t h a t M o d e l 2 u s i n g M e t h o d B is s u p e r i o r t o M o d e l 1 w h e n o i l r e l a t i v e p e r -

m e a b i l i t i e s a r e s m a l l ( k r o < 0 . 1 0 ) , w h i l e t h e S E E is s m a l l e r f o r M o d e l 1. F o r

a l m o s t a l l t h e o i l i s o p e r m s , M e t h o d B o f M o d e l 2 p e r f o r m s b e t t e r w h e n c o m p a r e d

w i t h M e t h o d A .

F o r t h e d a t a o f D o n a l d s o n a n d D e a n , M o d e l s 1, 2 ( M e t h o d B ) , 4 , a n d 5

p e r f o r m b e t t e r t h a n t h e r e s t o f t h e m o d e l s b a s e d o n t h e S S E a n d S S L v a l u e s .

Table VI. Compar i son of predicted and experimental data of Saraf et al. using SSE function

Oil 0.001 0.01 0.05 0.10 0.20 0.30 0.40 SEE Isoperm n p = 8 np = 8 n p = 8 np = 6 np = 5 np = 6 n v = 4 N = 45

Model 1 6.4E - 4 1.5E - 3 1.4E - 3 9.7E - 3 2.2E - 2 1.5E - 2 8.4E - 3 0.0365 M o d e l 2 A 5 . 4 E - 5 2 . 0 E - 4 4 . 9 E - 3 t . 6 E - 2 2 . 5 E - 2 5 . 2 E - 3 2 . 5 E - 3 0.0350 M o d e l 2 B 4 . 4 E - 4 2 . 5 E - 3 1 . 7 E - 3 4 . 9 E - 2 5 . 3 E - 3 4 , 2 E - 3 1.9E 2 0.0290 M o d e l 3 1 . 6 E - 2 3 . I E - 2 2 . 1 E - 2 1 . 5 E - 2 3 . 7 E - 2 1 . 2 E - 1 1 . 4 E - 1 0.0930 M o d e l 4 2 . 7 E - 3 9 . 4 E - 3 2 . 1 E - 3 9 . 5 E - 4 8 . 4 E - 4 1 . 2 E - 2 2 . 7 E - 2 0.0351 M o d e l 5 1 . 1 E - 3 9 . 5 E - 3 3 . 3 E - 3 1 . 5 E - 3 1 . 9 E - 3 1 . 9 E - 2 3 . 3 E - 2 0.0395 M o d e l 6 4 . 9 E - 5 1 . 8 E - 4 5 . 8 E - 3 1 . 9 E - 2 2 . 9 E - 2 7 . 8 E - 3 2 . 0 E - 3 0.0380 M o d e l 7 7 . 5 E - 6 1 . 1 E - 2 2 . 0 E - 2 1 . 3 E - 2 2 . 5 E - 2 6 . 0 E - 2 3 . 5 E - 2 0.0610

8 0 M O J D E H D E L S H A D A N D G A R Y A. P O P E

T a b l e V I b . C o m p a r i s o n of p r e d i c t e d a nd e x p e r i m e n t a l da t a of Saraf et al. using SSL func t ion

Oil 0 .001 0.01 0.05 0 .10 0 ,20 0 .30 0 .40

I s o p e r m

Mode l 1 7 .9E - 0 1 .0E0 1 .5E - 1 3 .0E - 1 1 .5E - 1 4 . 0 E - 2 1 .1E - 2

Mode l 2 A 1 .8E0 2 .0E - 1 7 . 8 E - 1 6 . 4 E - 1 1 .8E - 1 1 .2E - 2 2 .7E - 3

Mode l 2B 5 .3E 0 1 .3E0 1 .3E - 1 1 .3E - 1 3 .0E - 2 7 . 9 E - 3 1 .9E - 2

Mode l 3 1 .6E + 1 4 . 2 E 0 1 .0E0 7 . 5 E - 1 3 .7E - 1 7 . 1 E - 1 4 .4E - 1

Mode l 4 1 . 3 E + 1 3 . 2 E 0 1 . I E - 1 1 . 8 E - 2 4 . 3 E - 3 2 . 1 E - 2 2 . 6 E - 2

Mode l 5 3 .3E + 1 3 . 2E0 1 .7E - 1 2 .5E - 2 8 .3E - 3 3 .3E - 2 3 .2E - 2

Mode l 6 1 .5E0 1 .8E - 1 1 .0E0 7 . 9 E - 1 2 .2E - 1 1 .9E - 2 2 .2E - 3

Mode l 7 7 . 2 E + 1 1 . 8 E + 1 8 . 7 E - 1 5 . 7 E - 1 2 . 1 E 0 8 .7 E0 1 . 8 E + 1

With the exception of oil isoperms of 0.06 and 0.70, Method B of Model 2 performs better than the rest of the models and gives the smallest SEE. Stone's Method II (Model 5) is superior to Stone's Method 1 (Model 4).

For data of Saraf et al., Model 6 gives a better fit for low- and high-oil isoperm values, while Models 1, 4, and 5 perform better for midrange values of oil relative permeability (kro = 0.05, 0.10, 0.20). Both Methods A and B of Model 2 perform reasonably well for all the oil isoperms considered in the analysis. Method A is ranked higher for oil isoperms of 0.001, 0.01, and 0.4, while Method B gives smaller SSE and SSL values for oil isoperms of 0.05, 0.10, 0.20, and 0.30 with the smallest SEE value.

It should be emphasized that there was no adjustment of the input parameters in the models with the exception of Models 2 (Method B) and 7 for the data presented in this paper. The parameters Sr and M of Model 7 are found by history matching the data since the pressure-saturation data were not available for the data considered. The advantage that Models 2, 4, and 6 have over other models is that they allow for the adjustment of parameters which can be used to decrease the differences between the computed and experimental relative permeabilities (as was noticed in Model 2 by employing Method B). The three-phase relative permeability computation is also influenced by the choice of the two- phase data extracted from the literature.

7. Summary and Conclusions

Estimates of three-phase oil relative permeabilities and residual oil saturations are required for a variety of petroleum and ground water processes. Because of the enormous experimental difficulties and expense in measuring three-phase relative permeabilities, engineers have relied on theoretical models. In order to test the accuracy of relative permeability models, seven different oil-water-gas relative permeability correlations, proposed by several authors to predict three- phase oil relative permeabilities, have been compared. The proposed models assume that three-phase oil relative permeabilities depend on two saturations.


However, this assumption is only slightly supported by the experimental data studied here.

The comparative study shows that the agreement between the computed oil isoperms from Models 1 (Baker), 2 (Pope), and 7 (Parker) and the experimental data of Corey et al. is reasonably good. Models 1 (Baker), 2 (Pope) (Method B), 4 (Stone I), and 5 (Stone II) give closer predictions than other methods for the data of Donaldson and Dean. Using the data of Saraf et al., Model 6 (Corey- type) gives good predictions for high- and low-oil isoperms, whereas Models 4 (Stone I) and 5 (Stone II) perform better for midrange relative permeability values. Method B of Model 2 (Pope) performs reasonably well for all the oil isoperms. The overall performance of Models 1 (Baker) and 2 (Pope) (Method B) is superior to that of the other models for at least three sets of experimental data studied here. More experimental data based on an appropriate relative permeability measurement procedure is required to better test the accuracy of predictions from these models as well as to recommend one of them.

The advantage that Model 2 (Pope) has is that it is more flexible in terms of the matching parameters as opposed to Models 1 (Baker), 3 (Lake), 5 (Stone II), and ' 7 (Parker) which do not allow any adjustment, and thus it can be applied in such a way to take advantage of the available three-phase data. We recommend the use of Model 2 (Pope) in those cases where it can be used to produce a better fit of the available data.

Models 2 (Pope), 4 (Stone I), and 6 (Corey-type) require the knowledge of residual oil saturation in the presence of gas and water phases. The residual saturation in the three-phase flow region needs to be determined experimentally since this type of data is almost nonexistent in the literature. A direct fit to the Sot function may improve the predictions from the models incorporating the residual oil saturation. Due to limited available residual oil saturation measurements [16], a linear function based on the limiting two-phase values (Sorw, Sorg) is presently adopted in the models. If residual oil data in the three-phase region are available, they can be used to test their influence on the accuracy of the oil relative permeability computations.

This study has clearly shown the need for better three-phase relative permeability models and the importance of carefully testing available models by comparison with experimental data. We have proposed a new model (Model 2) which is flexible and simple, yet agrees with several sets of data.

Acknowledgement We appreciate the comments and criticisms made by Dr Lee Baker from Amoco after reviewing this paper and also for allowing us to use his proposed model (Model 1).

We acknowledge support by the U.S. Department of Energy grant # D E - AC 19-85BC 10846 and the companies funding the Center for enhanced Oil and


Gas Recovery Research at the University of Texas at Austin: Amoco, Arco, ADREF, Chevron, Cities Service, Conoco Core Data, Cray Research, Digital Equipment, Elf Aquitaine Production, Enterprise Oil, Institute for Energy Technology, INTEVEP, Japan National Oil Corporation, JAPEX, Mobil, J. S. Nolen and associates, Norsk Hydro, Phillips, Rogaland Research Institute, Shell, Standard Oil, Sun, Tenneco, Texaco U.S.A., and Unocal. Computing resources for this work were provided by the University of Texas System Center for High Performance Computing.

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Comparison of the three-phase oil relative permeability models

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