Effect of Capillary Number and Its Constituents on Two-Phase Relative Permeability Curves R.A. Fulcher Jr., SPE, ARCO Oil and Gas Co. Turgay Ertekin, SPE, Pennsylvania State U. C.D. Stahl, SPE, Pennsylvania State U.

Summary One primary goal of any enhanced recovery project is to maximize the ability of the fluids to flow through a porous medium (Le., the reservoir). This paper discusses the effect of capillary number, a dimensionless group de-scribing the ratio of viscous to capillary forces, on two-phase (oil-water) relative permeability curves. Specifical-ly, a series of steady-state relative permeability measure-ments were carried out to determine whether the capillary number causes changes in the two-phase permeabilities or whether one of its constituents, such as flow velocity, fluid viscosity, or interfacial tension (1FT), is the con-trolling variable.

For the core tests, run in fired Berea sandstone, a Soltrol 170 oil/calcium chloride (CaCI 2) brine/isopropyl alcohol (IPA)/glycerin system was used. Alcohol was the 1FT reducer and glycerin was the wetting-phase viscosifier.

The nonwetting-phase (oil) relative permeability showed little correlation with the capillary number. As 1FT decreased below 5.50 dyne/cm [5.50 N/m], the oil permeability increased dramatically. Conversely, as the water viscosity increased, the oil demonstrated less ability to flow. For the wetting-phase (water) relative perme-ability, the opposite capillary number effect was shown. For both the tension decrease and the viscosity increase (i.e., a capillary number increase) the water permeability increased. However, the water increase was not as great as the increase in the oil curves with an 1FT decrease. No velocity effects were noted within the range studied.

Other properties relating to relative permeabilities were also investigated. Both the residual oil saturation (ROS) and the imbibition-drainage hysteresis were found to decrease with an increase in the capillary number. The irreducible water saturation was a function of 1FT ten-sion only.

A relative permeability model was developed from the experimental data, based on fluid saturations, 1FT, fluid viscosities, and the residual saturations, by using regres-sion analysis. Both phases were modeled for both the im-bibition and the drainage processes. These models demonstrated similar or better fits with experimental data of other water- and oil-wet systems, when compared with existing relative permeability models. The applicability of these regression models was tested with the aid of a two-phase reservoir simulator.

Copyright 1985 SOCiety of Petroleum Engineers

FEBRUARY 1985

Introduction As world oil reserves dwindle, the need to develop EOR techniques to maximize recovery is of great importance. Methods such as chemical flooding, miscible flooding, and thermal recovery involve altering the mobility and/or the 1FT between the displacing and the displaced fluids.

Recovery efficiency was found to be dependent on the capillary number, defined as

p.v N c=- ................................ (1)

'Yep

The viscous forces were defined as the fluid viscosity, flow velocity, and the flow path length. Capillary forces vary with the fluid 1FT and the pore geometry of the medium.!

Taber defined the capillary number in terms of the pressure drop between two points, the flow length, and the IFT.2

IIp N c=-. ................................ (2)

L'Y

He concluded that as this ratio increased to a value of 5 psi/ft/dyne/cm [0.2 kPa/m/N/m] the ROS was reduced significantly. By decreasing the 1FT by using surface-active agents, or by decreasing the path length by alter-ing the field geometry, the capillary number could be in-creased.

Others have shown similar results. Melrose and Brand-ner,3 for example, indicated that as the capillary number rose to a value of 10 -4, the microscopic displacement efficiency, which accounts for the residual saturations to both oil and water, increased. The effects of the capillary number on the recovery of residual oil 'are given by Chat-zis and Morrow 4 and by other authors 5 (Fig. 1).

Few studies, however, have shown the effect of the capillary number on the two-phase flow between the residuals. The variables within this group have been researched, but their combined effect on relative permeabilities has been largely ignored.

Several authors have noted that the viscosity ratio of oil and water alters the oil relative permeability but has little effect on that of water. 6-8 Few or no changes by fluid flow velocity were observed, provided that no boundary effects were present during the core tests. 9-11

249

'i o

'"

Wagner & Leach 1.0 I-~~~S{- -'""-----.......

, 'C '% \.,-\&

\~. ~ 0.5 \9" \1:9

\~ '"

--- Nonwetting Residual - - - Wetting Residual

\:> f;:: , , ,

" O~~L-~~~---L~~--~~~--~~~ 10.8 100

Capillory Number, Nc

Fig. 1-Recovery of residual oil vs. capillary number.4

Studies on oil and gas permeabilities showed that as the 1FT decreased by increasing the temperature and the equilibrium pressure between two phases, the relative permeability curves increased and straightened out. 12 The results of other tests on reducing the tension between oil and water indicated that (1) few or no relative permeability alterations occurred for tension above 0.1 dyne/cm [0.1 N/m]; however, larger increases were observed below 0.1 dyne/cm [0.1 N/m] for both phases; (2) the curves tend-ed toward linearity; (3) the imbibition-drainage hysteresis lessened; and (4) the residual saturations to both oil and water decreased. 13-15

One study did show relative permeability increases with increases in the capillary number; however, these ex-periments were run using artificial cores of Teflon ,

alumina, and stainless steel rather than reservoir-type rock samples. 16

Also, with the increased use of mathematical reservoir simulators to predict recovery from different EOR proc-esses, the need to model the various flow properties, such as relative permeability, becomes important. Thus, ex-perimental results need to be applied to empirical or statistical models for use in numerical simulators. Materials and Experimental Procedure For this study, a series of steady-state relative permeability tests were carried out at 77F [25C] in 2-fi- [0.6-m- ] long, 2-in.- [5-cm- ] diameter Berea sandstone cores fired at 1,832F [1000C]. The cores were baked to prevent large decreases in their absolute permeability because of clay swelling during and between the experimental runs. Cores were reused for several tests before being discard-ed to minimize differences in porosity, permeability, and lithology. The properties of the cores for each test are given in Table 1. The fluids used were mixtures of Soltrol 170 oil, * 2% CaCl2 brine, IPA, and glycerin, as shown in Table 2. Later, core measurements were taken using Bradford crude oil, ** Kendex 0837 , t and 2 % sodium chloride (NaCl) brine.

1FT effects were studied with the Soltrol170, IPA, and CaCl2 brine system. Alcohol acted as the tension reducer. CaCl2 brine was used instead of NaCI brine because it does not form a precipitate with Soltrol and IPA as does NaCl. 17 Glycerin was used as the viscosi-fying agent to increase the aqueous-phase viscosity.:j: 'Phillips Petroleum Co., Bartlesville, OK.

"Pennzoil Co., Bradford, PA. tKendall Oil Co., Bradford, PA. :j:Crookston, R., Ehrlich, R., and Bae, J.: private communication, Gulf R&D Co., Pittsburgh (Feb. 13, 1981).

TABLE 1-CORE PROPERTIES

Permeability Permeability Length PV Porosity (Prerun) (Post-run) Change

Run Core ~ (cm) 3 (%) (md) (md) (%) 1 1* 24 276 22.46 241.3 115.1 -52.3 2 2* 12 148 23.96 198.1 136.0 -31.3 3 3** 24 240 19.42 240.6 115.1 -52.2 4 4 24 280 22.48 325.3 278.4 -14.4 5 4 24 280 22.48 278.4 247.8 -11.0 6 4 24 280 22.48 247.8 251.0 + 1.3 7 4 24 280 22.48 251.0 219.7 -12.5 8 4 24 280 22.48 219.7 219.7 +0.0 9 4 24 280 22.48 219.7 182.1 -17.1 10 4 24 280 22.48 123.2 128.3 +4.1 11 5 24 287 23.05 365.9 408.2 + 11.6 12 5 24 287 23.05 408.2 355.3 -13.0 13 5 24 287 23.05 355.3 353.4 -0.5 14 5 24 287 23.05 353.4 311.1 -12.0 15 5 24 287 23.05 311.1 259.3 -16.7 16 6 24 279 22.40 433.1 433.1 +0.0 17 6 24 279 22.40 433.1 416.4 -3.9 18 6 24 279 22.40 416.4 384.5 -7.7 19 6 24 279 22.40 384.5 20 7 12 146 23.48 531.6 21 8 24 330 26.50 388.1 398.5 +2.7 22 9 24 276 22.16 357.7 23 10* 24 249 20.15 177.3 24 11* 24 235 19.02 279.1 25 12* 24 272 22.01 160.2

Unfired core . Core fired at 250C.

250 JOURNAL OF PETROLEUM TECHNOLOGY

Runs Phase 1-7,9,

13,21,22 aqueous 8 aqueous

10,11 aqueous 12,16 aqueous

14 aqueous 15 aqueous 17 aqueous 18 aqueous 19 aqueous 20 aqueous

23,24 aqueous 25 aqueous

1-7,9, 12,21,22 oleic

8 oleic 10,11 oleic 12,16 oleic

14 oleic 15 oleic 17 oleic 18 oleic 19 oleic 20 oleic

23,24 oleic 25 oleic

2% NaCI brine. "Bradford crude oil. tKendex 0837 oil.

Fluid Reservoirs

FEBRUARY 1985

TABLE 2-PROPERTIES OF THE AQUEOUS AND OLEIC PHASES

2% CaCI 2 Brine (%)

100.0 60.0 10.9

3.9 40.0 17.5 0.0 0.0 0.0 0.0

100.0' 100.0'

0.0 0.2 0.9 1.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Pump

Ruska Pump

Soltrol170 Oil (%)

0.0 0.0

14.0 43.5

0.0 0.0 9.6 0.1

20.6 0.0 0.0 0.0

100.0 95.3 83.2 71.5

100.0 100.0 76.1 89.7 67.0

100.0 100.0" 100.0"

JPA (%)

0.0 40.0 75.1 52.6

0.0 0.0

63.5 38.3 62.8

0.0 0.0 0.0

0.0 4.5

15.9 27.4

0.0 0.0

22.4 10.0 30.2

0.0 0.0 0.0

Glycerin (%)

0.0 0.0 0.0 0.0

60.0 82.5 27.9 61.6 16.6

100.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 1.5 0.3 2.8 0.0 0.0 0.0

r-I

Manometer

Specific Viscosity Refractive Gravity at 25C Index at 25C (cp) at 25C

1.007 0.947 1.3385 0.917 2.761 1.3664 0.804 2.613 1.3879 0.785 2.475 1.3996 1.156 13.795 1.4231 1.212 128.58 1.4516 0.914 13.636 1.4115 1.069 126.62 1.4418 0.853 6.100 1.4050 1.258 954.00 1.4735 1.009 0.892 1.3356 1.009 0.892 1.3356

0.781 2.363 1.4339 0.771 2.246 1.4348 0.774 2.309 1.4277 0.775 2.352 1.4196 0.781 2.363 1.4339 0.781 2.363 1.4339 0.785 2.149 1.4207 0.781 2.029 1.4264 0.791 2.173 1.4168 0.781 2.363 1.4335 0.814 5.195 1.4504 0.845 11.297 1.4711

vocuumt:J I I 1 I

Core __ J

Sample Collector

Fluid Reservoir

Fig. 2-Relative permeability experimental apparatus.

1FT with Oleic/Aqueous

at 25 C (dyne/cm)

37.9 5.50 0.335 0.0389

30.3 29.7

0.454 2.91 0.118

25.3 24.5 10.8

37.9 5.50 0.335 0.0389

30.3 29.7

0.454 2.91 0.118

25.3 24.5 10.8

251

tPA tPA

2% CaCI 2 Brine Soh,ol 170 OH Glyce,in Soh,ol 170 Oil

Fig. 3-Equilibrium phase diagrams for Soltrol 170/lPN20f0 CaCI 2 and Soltrol 170llPNglycerin systems.

TABLE 3-SUMMARY OF CORE TESTS

Flow Wetting Phase Microscopic Rate 1FT Viscosity Capillary So, Sw;, Displacement

Run System (cm 3 ) (dyne/cm) (cp) Number (Ofo) (Ofo) Efficiency (Ofo) 1 SC 160 3.79 x 101 0.947 2.47x10- 6 40.4 38.2 34.7 2 SC 160 3.79 x 101 0.947 2.34x10- 6 43.5 34.3 33.8 3 SC 200 3.79 x 101 0.947 3.61 x 10-6 48.9 25.5 34.4 4 SC 200 3.79 x 101 0.947 3.09x10- 6 36.3 22.1 53.4 5 SC 80 3.79x 101 0.947 1.24x10-6 35.1 31.7 48.6 6 SC 120 3.79 x 101 0.947 1.86x10-6 42.3 29.9 39.7 7 SC 160 3.79 x 101 0.947 2.48 x 10- 6 44.1 26.6 39.9 8 SCA 200 5.50x 10 2.761 6.22x 10-5 33.1 39.4 45.4 9 SC 400 3.79x 101 0.947 6.18x10- 6 42.8 34.8 34.4 10 SCA 200 3.35 x 10-1 2.613 9.66x10- 4 0.0 56.3 100.0 11 SCA 200 3.35 x 10-1 2.613 9.43x10- 4 8.9 41.5 84.8 12 SCA 200 3.89x10- 2 2.475 7.65x10- 3 0.1 33.6 99.0 13 SC 200 3.79x101 0.947 3.02 x 10- 6 36.8 33.0 45.1 14 SCG 200 3.03x101 13.795 5.50 x 10-5 38.6 37.0 38.7 15 SCG 200 2.97 x 101 128.58 5.23 x 10-4 17.1 32.9 74.5 16 SCA 200 3.89 x 10-2 2.475 7.92 x 10- 3 0.0 32.0 100.0 17 SGA 200 4.54x 10-1 13.636 3.74x10- 3 2.2 40.2 96.3 18 SGA 200 2.91 x 10 126.62 5.41x10- 3 30.6 33.9 53.7 19 SGA 200 1.18x 10-1 6.100 6.43x10- 3 3.9 30.1 94.4 20 SC 200 2.59 x 101 954.00 4.37x10- 3 10.3 36.2 83.8 21 SC 200 3.79 x 101 22 SC 200 3.79 x 101 23 BN 200 2.45 x 101 24 BN* 200 2.45 x 101 25 KN 200 1.08x 101

OilWet System.

System Notes: S Soltrol 170 oil. B - Bradford crude oil. K - Kendex 0837 oil. C - 2% (by weight) CaCI 2 brine. N - 2% (by weight) NaCI brine. A - Isopropyl alcohol. G - Glycerin.

Glycerin mixed completely with the brine and was im-miscible with the oil in all proportions. Combined effects of interfacial tension and viscosity were studied by using a Soltrol/lPA/glycerin fluid system.

The experimental apparatus is shown in Fig. 2. The cores were prepared using the Lapp Pulsafeeder pump. Relative permeability tests required two Ruska constant-displacement pumps, one for each fluid phase. All fluids were flowed through a O.4-micron [O.4-/-tm] 252

0.947 2.62x 10-6 30.1 40.1 49.8 0.947 3.14x10- 6 20.9 38.5 66.0 0.892 4.41x10- 6 37.9 30.8 46.4 5.195 3.13x10- 5 39.4 19.7 51.0 0.892 1.06x10-5 33.1 33.0 50.6

filter before entering the core. Different procedures were used to prepare the fluids

for injection depending on the specific fluid system. For viscosity alterations, the glycerin/CaCl2 brine solutions were mixed separately from the oil. For the 1FT and the combined tension-viscosity systems, a tie line, on the ternary diagram, yielding the desired properties was chosen (Fig. 3). A mixture, lying approximately in the center of the tie line, was selected, and the appropriate

JOURNAL OF PETROLEUM TECHNOLOGY

1.0

0.9

0.8

0.7 kro 0.6

0.5

0.4

0.3

0.2

0.1

0.00

Y = 37.9 diem v = 40 ft,day P-w = 0.9468 ep Ne = 3.09 x 10-6

10 20 30 40 50

Run 4 --Drainage -- --Imbibition

60 70 80 90 Sw (%)

1.0

0.9

0.8

0.7

0.6 krw

0.5

0.4

0.3

0.2

0.1

108.0

1.0

0.9

0.8

0.7 kro

0.6

0.5

0.4

0.3

0.2

0.1 0.0

0

Y = 37.9 diem v = 40 ft/day P-w = 0.9468 ep Ne = 3.02 x 10-6

10 20 30 40 50

Run 13 Drainage

- - - - Imbibition

60 70 80 90 Sw (%)

1.0

0.9

0.8

0.7

0.6 krw

0.5

0.4

0.3

0.2

0.1

Fig. 4-Oil-water relative permeabilities for the base system and reproducibility of measurements.

amounts of each component were mixed. The resulting solution was shaken, then allowed to sit quietly until com-plete phase separation occurred.

The Penn State steady-state relative permeability method 19 then was run using the desired fluid system. The experimental technique involved flowing both phases simultaneously through the core and calculating the ef-fective permeabilities with Darcy's law applied to each phase. Fluid saturations were determined by material balance. Drainage curves were found by moving the wetting-phase saturation from 100% to its irreducible value. Imbibition curves then were determined by revers-ing the saturation direction from irreducible water to the ROS.

To avoid capillary end effects, the criterion given by

1.0

0.9

0.8

0.7 k,e

0.6

0.5

0.4

0.3

0.2

0.1

y = 5.50 diem y = 40 It/dey "w = 2.7606 ep He = 6.22 x 10-5

Run 8 --Drainage - - - -Imbibition

0.0 0 10 20 30 40 50 60 70 80 Sw (%)

1.0

0.9

0.8

1.0

0.9

0.8

0.7 k,e

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0

Y = 0.3346 diem y = 40 It/dey "w = 2.6126ep He = 9.43 x 10-4

10 20 30 40 50

Kyte and Rapoport,20 a 50-psi [345-kPa] pressure dif-ferential across the core, was required. Thus, an SO-mLlhr [SO-cm 3 /h] minimum flow rate at a velocity of 16 fi/D [4.9 mId] was needed for fluids of high 1FT (greater than 1.0 dyne/cm [1.0 N/m]), and 200 mLlhr [200 cm 3 Ih] at 40 fi/D [12.2 mId] was needed for low-tension (less than 1.0 dyne/cm [1.0 N/m]) systems. 21 Discussion of Results A summary of the experimental core tests is shown in Table 3. Two control tests were run (Fig. 4) to measure the reproducibility of the procedure, test the viability of using fired Berea cores, and establish a comparative base for the altered fluid runs. Soltrol 170 and 2 % CaCl2 brine were flowed at a total fluid flow rate of 200 mLlhr

Run II --Drainage - - - -Imbibition

60 70 80 90

0.8

1.0

0.9

0.8

0.7 0.7 krw kro

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1

0.0

Y = 0.03885 diem Run 16 ., :;: 40 h/doy --Drainage ""w :;: 2.4752 cp -- - -Imbibition He = 7.92 x 10-3

0

0.7 k,w

0.6

0.5

0.4

0.3

0.2

Sw (%)

Fig. 5-Behavior of oil/water relative permeabilities at low 1FT's.

FEBRUARY 1985 253

1. 0 rr---r--r.,-,-rncn--.----,-,--TTlTTr-,-TOOTnnr-----r-,---T"T1

0.9

0.8

0.7

0.6

kro 0.5

0.4

0.3

0.2

0.1

0.010

.2

1.0

0.9

0.8

0.7

0.6 kro 0.5

0.4

0.3

0.2

0.0 10.2

10.1 Interfacial Tension (diem)

Interfacial Tension (d 'em)

--Droinoge - - -Imbibition

Q = 200 cc/hr-"'w '" 2 cp

--Drainage - - -Imbibition

Q = 200 cc/hr '" w",2 cp

"0-_---0

Fig. 6-Variation of oil and water relative permeabilities against interfacial tension for different wetting-phase saturations.

[200 cm 3 /h] in both imbibition and drainage. These results compared favorably with one another and with those for unfired Berea cores.

Velocity Effects. The initial variable altered was the fluid flow velocity. The rates ranged from 80 mLlhr [80 cm3 /h] (16 ftlD [4.9 mid)), the minimum rate to avoid capillary end effects, to 400 mLlhr [400 cm3 /h] (80 ftlD

y 0 lO.l diem Run 14 y 0 29.7 diem v 0 40 Itlday --D,oinag~ v = 40 It/day "'w 0 13.795 ep -- - -Imbibition "'w = 128.58 ep Ne 0 5.50 x 10.5 Ne 0 5.2l x 10'4

1.0 1.0 1.0

0.9 0.9 0.9

0.8 0.8 0.8

0.7 0.7 0.7 kro krw k,o

0.6 0.6 0.6

0.5 0.5 0.5

0.4 0.4 0.4

O.l O.l O.l

0.2 0.2 0.2

0.1 0.1 0.1

0.0 0.0 0 10 20 lO 40 50 60 0 10 20 lO

Sw (%)

[24.4 m/d)) , the maximum rate attainable with the Ruska pumps. Little or no significant change occurred in the relative permeability curves within this range, which agreed with the work of previous authors. 9- 11 Thus, fur-ther studies are required to test fluid velocity to attain at least a two-fold order of magnitude increase in the capillary number, although such rates are not seen ex-cept around a wellbore.

Oleic/Aqueous IFf Effects. The range of oleic/aqueous 1FT's for Soltrol and brine (Fig. 4) was from 37.9 dyne/cm [37.9 N/m] to 0.0389 dyne/cm [0.0389 N/m). The relative permeability curves for low-tension systems are shown in Fig. 5. The total flow velocity was main-tained at 40 ft/D [12 mid] and the fluid viscosities were kept at approximately 2.0 cp [0.002 Pa' s]. At a tension of 5.50 dyne/cm [5.50 N/m], a slight increase was ob-served in the water permeabilities but none in the oil values. The former result may have been attributed to the slight increase in the brine viscosity from 0.947 to 2.761 cp [9.47 X 10-4 to 27.61x1O-4 Pa's], as is discussed in the next section. At 0.335 dyne/cm [0.335 N/m] , both sets of curves showed large increases, indicating less resistance to flow for each phase. At the lowest attainable tension for the oillbrine/IPA system (0.0389 dyne/cm [0.0389 N/m)), further increases in the permeabilities were noted. Also, the curves started to approach lineari-ty, as was reported for fluids of zero 1FT. 22 To further illustrate the tension effects, the relative permeabilities of both the oleic and the aqueous phases at various aqueous saturation values are given in Fig. 6. For both drainage and imbibition, larger increase occurred below 5.50 dyne/cm [5.50 N/m] , indicating a critical point when fluid began to move more easily.

Wetting-Phase Viscosity Effects. The wetting-phase viscosity range was from that of CaCl2 brine (0.947 cp [9.47 X 10 -4 Pa' s)) to that of glycerin (954.0 cp [0.954 Pas)). The fluid velocity was 40 ftlD [12 mid] and the 1FT was maintained at approximately 30 dyne/cm [30 N/m]. The relative permeability curves for four viscosity values are shown in Fig. 4 (for brine) and in Fig. 7. As the wetting phase (aqueous) viscosity increased, its

Run 15 Y = 25.9 diem Run 20 --Drainage v 0 40 Itlday __ Drainage - - - -Imbibition "'w 0 954.0ep -_ - - Imbibition

Ne 0 4.37 x 10l

1.0 1.0

0.9 0.9

0.8 0.8

0.7 0.7 0.7 k,w kro krw

0.6 0.6 0.6

0.5 0.5 0.5

0.4 0.4 0.4

O.l 0.3 O.l

0.2 0.2 0.2

0.1 0.1 0.1

0.0 0.0 0 10 20 lO 100

Sw (%) Sw (%)

Fig. 7-Behavior of oil/water relative permeabilities at different aqueous-phase viscosities.

254 JOURNAL OF PETROLEUM TECHNOLOGY

permeability curves also increased and tended toward linearity. However, the oleic (Soltrol 170) curves de-creased in approximately the same order of magnitude as the water curves increased. These effects are seen in Fig. 8, where the relative permeabilities for different satura-tions are plotted against aqueous viscosity values. Note that caution must be exercised when applying these results since the wettability of the glycerin/brine systems must be verified as strongly water-wet.

Oil-Water Relative Permeabilities at Different Capillary Numbers. To demonstrate the capillary number effect, tests were carried out to measure the results of combining 1FT reductions and aqueous viscosity increases (Fig. 9). For a low-tension system of 0.118 dyne/cm [0.118 N/m] and a wetting-phase viscosity of 6.10 cp [61.0 x 10 -4 Pa' s], the curves showed near linearity, but the oleic values were slightly less than those for the 0.335-dyne/cm [0.335-N/m] run for tension effects alone. This observation indicated an interaction between the aqueous phase viscosity and the 1FT. For a system of higher tension (0.454 dyne/cm [0.454 N/mD and higher viscosity (13.64 cp [136.4 x 10 -4 Pa' s]), the aqueous curves remained relatively unchanged but the oleic curves showed further decreases. However, the oil permeabilities still were larger than those for the 13. 80-cp [138.0 x 10 -4 -Pa' s] run at approximately 30 dyne/cm [30 N/m] (Fig. 7). At combinations of even larger variable values, the 1FT effect was almost negligible.

How the capillary number affected the oil-water relative permeabilities is seen in Fig. 10 for the imbibition process at a 50% wetting-phase saturation. The specific variable within the capillary number determined the direction of the oil curves. As the aqueous-phase viscosity increased, the oleic permeability values decreased. As the tension decreased, the curves shifted upward. Conversely, as the capillary number increased, so did the ability of the wet-ting phase to flow. The correlation was not a perfect fit because changes from the viscosity were larger than those from the 1FT. Thus, the capillary number may not be a true predictor of the wetting-phase permeability. Similar effects were observed in both the oleic and the aqueous drainage curves. 21

kro

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00

Y = 0.1180 diem v = 40 Itlday U w = 6.100 ep Ne = 6.43 x 10,3

10 20 30 40 50 Sw (%)

Run ]9 --Drainage - - --Imbibition

60 70 80 90

1.0

0.9

0.8

1.0

0.9

0.8

0.7 0.7 krw kro

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 0.0 0.0

100 0

y = 0.4539 diem ... = 40 h/doy U w = 13.6361 ep Ne = 3.74 x 10,3

10 20 30 40 50

0.6

O.S

0.4

kro 0.3

0.2

0.1

0.0

0.5

0 .

0.3

0.2

0.1

100

Q = 200 cc/hr y .. 30 d/cm

'VisCOlil, (cp)

-- Drainage Imbibition - - - Imbibition

Viscosit, (cp)

--Drainage - - -Imbibition

s,.= Sw-

------1w=6""

Fig. 8-Variation of oil and water relative permeabilities against aqueous-phase viscosity for different wetting-phase saturations.

Run 17 --Drainage - - - - Imbibition

60 70 80 90

1.0

0.9

0.8

0.7 kro

0.6

0.5

0.4

0.3

0.2

0.1

Y = 2.91 diem v = 40 Itlday U w = 126.62 ep Ne = 5.41 x 10,3

Run 18 --Drainage - - - - Imbibition

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 10 20 30 40 50 60 70 80 90 lr:O

krw

Sw (%) Sw (%)

Fig. 9-Behavior of oil/water relative permeabilities at different capillary numbers,

FEBRUARY 1985 255

1.0

0.9

.~ O.B

1 0.7 E ~ 0.6

~ 0.5 ~ 0.4 6 0.3

0.2

0.1

0.5

i 0.4 1

~ 03 D-.

~ 02 '" . .

.; 0.1

o-Ok,.IQ) [',-k,.IY) o--ok ro (j4.) +-t kro(comb.)

... J 0 00 0 ~

"------- + o

'I o-okrw(Q) 6-!;,k

rw(Y)

o-Okrw (j4.)

+-t krw(comb.)

Capillary Number

"I 'I

+

0.0 1 10-6 10-5 10-4 10-3

Copi Ilory Number

'I

10-2

Fig. 10-lmbibition oil and water relative permeabilities as func-tions of capillary number at 50% water saturation.

40

g 30 c .

J 20

,

0, o 0 ~_--

__ 0----

o 0

f-------- ----

e---4Sor (bose; v) ~ - -oSwir (bose; 'I) ----"5.,IY) ..... --"5w;, IY)

10 ----Sor ("d ~--DSwir (JA,)

Capillary Number

o - -_ + tJ,.

-~ '+

Fig. 11-Experimental residual saturations as functions of the capillary number.

Residual Saturation Changes. The effects on the residual saturations of both phases also were noted. The ROS showed a large reduction from 40 % to 0 % for both decreases in the 1FT and increases in the wetting-phase viscosity, corresponding to a capillary number increase from 10-6 to 10-2 , respectively. As the tension de-creased to 5.50 dyne/cm [5.50 N/m], the irreducible water saturation showed very little change, but at the lowest-tension value of 0.0389 dyne/cm [0.0389 N/m] it de-creased back to 32 %. The residual saturations as func-tions of the capillary number are given in Fig. 11, with the irreducible water saturation showing changes because of tension changes only. As a check on our experimental

256

u

6 4i

..0

10-2

10-3

5 10-4 z >-o 0.. C

U

10-5

10~

-- Melrose ond Brandner Data - - - Experimental Data

o

o o 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Microscopic Displacement Efficiency (Em)

Fig. 12-Microscopic displacement efficiency as a function of capillary number.

results, the microscopic displacement efficiency, defined as

....................... (3)

was plotted and compared with the results of Melrose and Brandner3 (Fig. 12). Although Em increased at approx-imately one order of magnitude lower for the experimental capillary numbers than those reported, the data fell within the same general range.

As the capillary number increased, no matter what variable was being altered, the imbibition-drainage hysteresis decreased but never disappeared entirely. This result held true for both the aqueous and oleic phases. 21

Relative Permeability Model. To develop a relative permeability model based on the experimental results, the Minitab II@23 statistical computation system was used. This system uses regression analysis to determine the best coefficients for an equation and the statistical parameters to evaluate the function.

The following functional forms were found for both im-bibition and drainage. For the oil (nonwetting-phase) relative permeabilities,

kro(dr) =AS* (B+CID ")')(/-tw ) D .............. (4) /-to

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TABLE 4-SUMMARY OF RELATIVE PERMEABILITY MODEL COEFFICIENTS

Model S* A B C 0 ,2(%)* MSER" Ft ; plot; kro(dr) So 0.72899 1.2861 0.08043

kro(im) So - Sor 1.56878 1.33874 0.09187 1-Sor

k rw1 (dr) Sw -Swir

0.70216 1.25579 0.0 1-Swir

k rw1 (im) Sw -Sw;r

0.61135 1.25875 0.0 1-Swir

krw2(dr) Sw-Swif 0.70340 0.66596 0.0 1-Sw;r

K rw2(im) Sw -SWir

0.61135 0.69580 0.0 1-Swir

Sor +5.846x10-' +2.96x 10-' +4.62x 10- 2

SWif +4.0214 x 10- 1 + 3.976 x 10- 3 - 7.065 X 10-3

Correlation coefficient for linear regression. uMean square error ratio. tFdistribution. *' -= random scatter of ; values.

Bradford Crude-NaCI brine system (water-wet)

kro

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Il

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00

Bradford Crude-NaCI brine system (oil-wet) jJ.o = 5.195 ep 0--0 Experimental Data jJ. = 0.8918 ep - - - Model Data

w ___ Naar-Hendersan Y = 24.5 diem Data (2

problem was set up by the implicit pressure, explicit saturation (1M PES) method and was solved using the general band algorithm.

The runs were carried out in a 40-acre [162 000-m2 ] five-spot pattern with 3-darcy permeability, 35 % porosi-ty, and 58 % initial oil saturation. A horizontal 6 X 6 grid system was established with block lengths and widths of 110 ft [34 m] and a 30-ft [9-m] thickness. Initial reser-voir temperature and pressure were 80F [27C] and 2,000 psia [13 789 kPa], respectively.

Fig. 15 illustrates the effect on the total oil recovery by the 1FT expressed as a percentage of the original oil in place (OOIP) for approximately 16 years. For higher-tension floods, production leveled at 1,000 days. For 0.335 dyne/cm [0.335 N/m], recovery slowed at 3,500 days while production at the minimum tension was still increasing at 6,000 days. Total recovery improved from approximately 30% to 89% OOIP. The aforementioned results of the simulation runs clearly indicate the impor-tant role of a low-tension flood in achieving a high mobilization efficiency, and more importantly, how in-fluential the relative permeability characteristics are in the recovery performance.

Conclusions

The following are the conclusions based on the experimen-tal and computational observations.

1. The nonwetting (oil) relative permeabilities were found to be functions of the 1FT and the viscosity variables individually rather than a function of the capillary number.

2. The wetting (brine) relative permeabilities behaved as functions of the capillary number but were better modeled using the individual variables.

3. Insignificant 1FT effects were observed on both k ro and k rw until a value of 2.0 dyne/ cm [2.0 N / m] was ob-tained. Below this value, the relative permeabilities in-creased with decreasing tension.

4. Increases in aqueous- (wetting-) phase viscosity yield-ed reductions in k ro and increases in k rw' provided that the glycerin systems remained strongly water-wet.

5. At very low 1FT values, the relative permeability curves straightened out and approached the theoretical X-shape present at zero tension. For large aqueous viscosities, only the water curves behaved in a similar manner.

6. Low tensions 5.50 dyne/cm [< 5.50 N/m]) and viscosity values in the range between 2.0 and 13.6 cp [0.002 and 0.0136 Pa' s] interacted to yield the "X" ap-pearance for the relative permeability curves. However, above 5.50 dyne/cm [5.50 N/m] , tension seemed to have little effect on k ro or k rw .

7. No rate effects were observed for the limited range within this study (16 to 80 ftlD [4.9 to 24 mid]).

8. As the capillary number increased to 0.01, the ROS decreased from approximately 40% to zero.

9. The irreducible water saturation showed no consis-tent change above a tension of 5.50 dyne/cm [5.50 N/m], but decreased to 32% at 0.0389 dyne/cm [0.0389 N/m].

10. For increases in the capillary number, the imbibition-drainage hysteresis was reduced for both k ro and krw.

11. Mathematical relative permeability models were developed from the experimental data. These models

FEBRUARY 1985

yielded similar results with experimental data for different fluid systems.

12. Proposed relative permeability models were tested with the aid of a two-phase reservoir simulator. Results of the simulation studies showed that production increased from 30% to 89% OOIP as 1FT decreased from 37.9 dyne/cm to 0.0389 dyne/cm [37.9 N/m to 0.0389 N/m].

Nomenclature A, B, C,

D, S * = coefficients of variables in linear regression model

Em = microscopic displacement efficiency, fraction

kro(dr) = drainage relative permeability to oil, fraction

k ro(im) = imbibition relative permeability to oil, fraction

krw(dr) = drainage relative permeability to water, fraction

k rw(im) = imbibition relative permeability to water, fraction

k rwl = variable modeled water relative permeability, fraction

krw2 = capillary number modeled water relative permeability, fraction

L = flow length [m] N c = capillary number, fraction

p = pressure, psia [kPa] Sor = residual oil saturation, fraction

Swir = irreducible water saturation, fraction v = flow velocity, in.lsec [cm/s] 'Y = interfacial tension, dyne/cm [N/m]

IL 0 = oil viscosity, cp [Pa' s] ILw = water viscosity, cp [Pa' s]

cf> = porosity

References 1. Moore, T.F. and Slobod, R.C.: "The Effect of Viscosity and

Capillarity on the Displacement of Oil by Water," Producers Monthly (Aug. 1956) 20-30.

2. Taber, J.J.: "Dynamic and Static Forces Required to Remove a Discontinuous Oil Phase from Porous Media Containing Both Oil and Water," Soc. Pet. Eng. J. (March 1969) 3-12.

3. Melrose, J.C. and Brandner, C.F.: "Role of Capillary Forces in Determining Microscopic Displacement Efficiency for Oil Recovery by Waterflooding," J. Cdn. Pet. Tech. (Oct.-Dec. 1974) 54-62.

4. Chatzis, I. and Morrow, N.R.: "Correlation of Capillary Number Relationships for Sandstones," Soc. Pet. Eng. J. (Oct. 1984) 555-62.

5. Stegemeier, W.: Oil Recovery by Surfactant and Polymer Flooding. Academic Press Inc., Washington, DC (1977).

6. Downie, J. and Crane, F.E.: "Effect of Viscosity on Relative Permeability," Soc. Pet. Eng. J. (June 1961) 59-60.

7. Odeh, A.S.: "Effect of Viscosity Ratio on Relative Permeability," J. Pet. Tech. (Dec. 1959) 346-52; Trans. AIME, 216.

8. Schneider, F.N. and Owens, W.W.: "Steady-State Measurements of Relative Permeability for Polymer-Oil Systems," paper SPE 9408 presented at the 1980 SPE Annual Technical Conference and Ex-hibition, Dallas, Sept. 21-24.

9.0soba, J.S. et al.: "Laboratory Measurements of Relative Permeability," Trans., AIME (1951) 192,47-55.

10. Richardson, J.G.: "Calculation of Waterflood Recovery from Steady-State Relative Permeability Data," J. Pet. Tech. (May 1957) 64-66; Trans., AIME, 210.

259

11. Sandberg, C.R., Gournay, L.S., and Sippel, R.F.: "The Effect of Fluid-Flow Rate and Viscosity on Laboratory Detenninations of Oil-Water Relative Permeabilities," J. Pet. Tech. (Feb. 1958) 36-43; Trans., AIME, 213.

12. Bardon, C. and Longeron, D.G.: "Influence of Very Low Inter-facial Tensions on Relative Permeability," Soc. Pet. Eng. J. (Oct. 1980) 391-401.

13. Amaefule, J.O. and Handy, L.L.: "The Effect of Interfacial Ten-sions on Relative Oil/Water Permeabilities on Consolidated Porous Media," Soc. Pet. Eng. J. (June 1982) 371-81.

14. Batycky, J.P. et al.: "Interpreting Relative Permeability and Wet-tability for Unsteady-State Displacement Measurements," Soc. Pet. Eng. J. (June 1981) 296-308.

15. Lo, H.P.: "The Effect ofInterfacial Tension on Oil-Water Relative Permeabilities," Research Report RR-32, Petroleum Recovery Inst., Calgary, Alta., Canada (Nov. 1976) 5-9.

16. Lefebvre du Prey, E.J.: "Factors Affecting Liquid-Liquid Relative Permeabilities of a Consolidated Porous Medium," Soc. Pet. Eng. J. (Feb. 1973) 39-47.

17. Taber, J.J., Kamath, LS.K., and Reed, R.L.: "Mechanism of Alcohol Displacement of Oil from Porous Media," Soc. Pet. Eng. J. (Sept. 1961) 195-212; Trans., AIME, 222.

18. Morse, R.A., Terwilliger, P.L., and Yuster, S.T.: "Relative Permeability Measurements on Small Core Samples," Producers Monthly (Aug. 1947)19-25.

19. Kyte, J .R. and Rapoport, L.A.: "Linear Waterflood Behavior and End Effects in Water-Wet Porous Media," J. Pet. Tech. (Oct. 1958) 47-50; Trans., AIME 213.

20. Fulcher, R.A.: "The Effect of the Capillary Number and Its Con-stituents on Two-Phase Relative Permeabilities," PhD dissertation, Pennsylvania State U., University Park (1982)

260

21. Batycky, J.P. and McCaffery, F.G.: "Low Interfacial Tension Displacement Studies," paper 78-29-26 presented at the 1978 Petroleum Soc. of C.I.M. Annual Technical Meeting, Calgary, Alta., Canada, June 13-16.

22. Ryan, T.A. Jr.: "Minitab Release 81.1," Computation Center writeup, Dept. of Statistics, Pennsylvania State U., University Park (1981).

23. Naar, J. and Henderson, J.H.: "An Imbibition Model - Its Ap-plication to Flow Behavior and the Prediction of Oil Recovery," Soc. Pet. Eng. J. (June 1961) 61-70; Trans., AIME, 222.

24. Ertekin, T.: "Numerical Simulation of the Compaction-Subsidence Phenomena in a Reservoir for Two-Phase Nonisothermal Flow," PhD dissertation, Pennsylvania State U., University Park (1978).

SI Metric Conversion Factors ep X 1.0*

dyne x 1.0* ft X 3.048*

OF (OF-32)/1.8 in. X 2.54*

mL X 1.0*

Conversion factor is exact.

E-03 E-02 E-Ol

E+OO E+OO

Pa's mN m C em em 3

JPT Original manuscript received in the Society of Petroleum Engineers office Oct. 5, 1983. Paper accepted for publication June 6, 1984. Revised manuscript received Oct. 31, 1984. Paper (SPE 12170) first presented at the 1983 SPE Annual Technical Conference and Exhibition held in San Francisco Oct. 5-8.

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