Journal of Petroleum Science and Engineering 146 (2016) 505–514

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Journal of Petroleum Science and Engineering

http://d0920-41

n CorrE-m

PleasPetro

journal homepage: www.elsevier.com/locate/petrol

More general relationship between capillary pressure and resistivitydata in gas-water system

Changwei Liu a, Kewen Li a,b, Dong Ma c,n, Youguang Chen d

a School of Energy Resources, China University of Geosciences (Beijing), Beijing 100083, Chinab Energy Resources Dept., Stanford University, CA 94305, USAc Petroleum Engineering College, Yangtze University, Wuhan 430100, Chinad Department of Petroleum and Geosystems Engineering, University of Texas, Austin, TX 78712, USA

a r t i c l e i n f o

Article history:Received 8 December 2015Received in revised form25 June 2016Accepted 5 July 2016Available online 6 July 2016

Keywords:Resistivity indexCapillary pressureWell logging

x.doi.org/10.1016/j.petrol.2016.07.00805/& 2016 Elsevier B.V. All rights reserved.

esponding author.ail address: madong@yangtzeu.edu.cn (D. Ma)

e cite this article as: Liu, C., et al., Mol. Sci. Eng. (2016), http://dx.doi.org/1

a b s t r a c t

Capillary pressure data, with fundamental significance in reservoir engineering, can be determined inlaboratory through different methods. However, these methods are expensive, complex and time con-suming. Scarce literature has been published to describe the relationship between capillary pressure andresistivity data. In this study, a more general model inferring dimensionless capillary pressure directlyfrom resistivity index data was derived from correlating the modified Kr-RI (relative permeability andresistivity index) model with the widely-used Kr-Pc (relative permeability and capillary pressure) modelbased on the same function of Kr. This model demonstrated a linear relationship between capillarypressure and ( ⋅ ) *I SS / ww

1/2 (I is the resistivity index, Sw is wetting phase saturation, and *Sw is the nor-malized wetting phase saturation). The feasibility of this model was verified by experimental data fromdifferent literatures and ours. The results demonstrated that the model works satisfactorily in most casesexcept for cores with extra-low permeability (generally, permeability less than 10 mD). In addition, thisproposed model could also match the experimental data both at ambient and reservoir conditions, in-dicating that it reveals a more general relationship applicable for determining capillary pressure fromresistivity data both in laboratories and reservoirs.

& 2016 Elsevier B.V. All rights reserved.

1. Introduction

Capillary pressure is one of the most fundamental properties ofmultiphase flow transporting in porous media. It is widely used inpore size distribution analysis, irreducible water/residual oil sa-turation determination, numerical reservoir simulation, and manyother aspects of petroleum engineering. Capillary pressure curve isusually determined in the laboratory by mercury injection, porousplate, or centrifuge techniques. However, those methods havemany limitations. For example, too much time would be taken andthe capillary pressure is limited below 200 psi when porous platemethod is applied. For mercury intrusion method, the core can nolonger be reused and the mercury has environmental risks. Dataanalysis in centrifuge method is complicated and even likely toresult in errors. It would be useful if the desired capillary pressuredata could be obtained through some other methods.

It is known that resistivity measurement is relatively easierthan capillary pressure measurement in the laboratory. Besides,

.

re general relationship betw0.1016/j.petrol.2016.07.008i

large volumes of resistivity data are available from well-logging.Therefore, a new model inferring capillary pressure from re-sistivity data could be significant in many ways.

According to experimental results, Szabo found a linear re-lationship between capillary pressure and resistivity index (Szabo,1974). However, the proposed model has not been widely acceptedbecause single straight lines could not be obtained from the re-lationship between capillary pressure and resistivity index inmany cases. The model is expressed as:

= +( )

RR

a bP1c

t

o

where R0 is the resistivity of rock at a water saturation of 100%, Rtis the resistivity at a specific water saturation of Sw, Pc is the ca-pillary pressure, a and b are two constants.

Li formulated a new theoretical model to correlate capillarypressure and resistivity index based on the fractal scaling theory(Li and Williams, 2007). The proposed model indicates a powerlaw relationship between the two parameters and fits well withthe experimental results of fourteen core samples from two for-mations located in one oil reservoir. The model proposed by Li is

een capillary pressure and resistivity data in gas-water system. J.

Nomenclature

A Apparent across area of rockAa Across area of tortuous capillary tubeAw Across area of wetting phase effective flow pathI Resistivity indexKa Permeability at a water saturation of 100%Kw Permeability at a specific water saturation of Swkrnw Relative permeability of non-wetting phasekrw Relative permeability of wetting phaseL Apparent length of rockLa Length of tortuous capillary tubeLw Length of wetting phase effective flow pathn Saturation exponentPc Capillary pressurePcD Dimensionless capillary pressurePe Entry capillary pressureΔP Pressure gradientQ Volumetric rate of flowra Movable water radius of tortuous capillary tube at a

water saturation of 100%rw Movable water radius of tortuous capillary tube at

specific saturation of Swrai Radius containing irreducible water at a water sa-

turation of 100%rwi Radius containing irreducible water at specific sa-

turation of SwRo Resistivity of rock at a water saturation of 100%Rw Resistivity of rock at a specific water saturation of SwSw Wetting phase saturationSw

∗ Normalized wetting phase saturationSwi Irreducible water saturationμ Viscosity of waterλ Pore size distribution indexλrw Tortuosity ratio (τa/τw)ϕ Porosity of the rock.τa Tortuosity at a water saturation of 100%τw Tortuosity at a specific water saturation of Swβ Exponent in the Li model

Fig. 1. Fluid flow path at different water saturation in tortuous capillary tube.

Table 1Properties of core samples.

Core number ϕ (%) k (md) Swi (%)

1 25 280 0.372 26 250 0.393 19 70 0.354 20 15 0.485 21 10 0.466 27 100 0.57 17 40 0.288 26 75 0.4

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514506

written as follows:

= ( ) ( )βP I 2cD

where PcD is the dimensionless capillary pressure (Pc/Pe); I isthe resistivity index. β is the exponent.

According to Li's research, the value of parameter β, not aconstant, varies with core permeability. The value of parameter βcould be obtained from Li's model. In addition, based on Toledo et

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008

al.'s (1994) conclusion, Li model works better in a specific range oflow water saturations than in high water saturations. According toLi, the power law relationship between capillary pressure andresistivity index does not exist at high wetting phase saturations(Li and Williams, 2007).

Literature on the relationship between capillary and resistivityindex has been scarce. In this study, a more general model wasderived theoretically to infer dimensionless capillary pressurefrom resistivity index data. Compared with previous researches,the dimensionless capillary pressure from our model is the func-tion of irreducible water saturation (Swi) and resistivity index (I)without any indeterminate coefficients. In order to verify theproposed model, capillary pressure data calculated from this newapproach are compared with the experiment data of 24 coresamples from different literatures and our experiments.

2. Mathematics

After modification of the Kr-RI (relative permeability and re-sistivity index) model proposed by Ma et al. (2015), a more generalrelationship between capillary pressure and resistivity index data

een capillary pressure and resistivity data in gas-water system. J.i

Rela

tive p

erm

eabl

ity

Water saturation

Rel

ativ

e per

mea

blity

Water saturation

Rela

tive p

erm

eabl

ity

Water saturation

Rela

tive p

erm

eabl

ity

Water saturation

Rela

tive p

erm

eabl

ity

Water saturation

Rela

tive p

erm

eabl

ity

Water saturation

Rela

tive p

erm

eabl

ity

Water saturation

Rela

tive p

erm

eabl

ity

Water saturation

(a) (b)

(c) (d)

(e) (f)

(g) (h)Fig. 2. Calculated gas-water relative permeability and the comparison to the experimental data in eight different rocks.

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514 507

could be derived by correlating the Kr-RI modified model with thewidely-used Kr-Pc (relative permeability and capillary pressure)model based on the same function Kr.

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008i

2.1. Kr-RI modified model

Scarce literature on the relationship between relative

een capillary pressure and resistivity data in gas-water system. J.

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514508

permeability and resistivity index has been reported. Ma et al.(2015) derived a model of obtaining two-phase relative perme-ability from resistivity index data on the basis of Poiseuille's lawand Darcy's law.

Ma's model was modified by characterizing the irreduciblewater property which should be ignored in the Poiseuille's law,but essential in conductivity for resistivity calculation; Irreduciblewater can't move in the fluid flow, but it can conduct electricity. Inaddition, after modification, the Kr-RI modified model can becorrelated with Kr-Pc model for the derivation of Pc-RIrelationship.

The schematic model was shown in the Fig. 1.A is the apparent across area of rock and L is the apparent

length of rock. In the Fig. 1a, Aa and La are across area and length ofeffective water phase flow path at a water saturation of 100% forthe tortuous capillary tube, respectively. For a rock saturated withwater partially, the effective water phase flow path at a specificwater saturation of Sw was depicted in Fig. 1b. At a specific watersaturation of Sw , the effective water phase flow path maintains theshape of tortuous capillary tube, but the across area and the lengthof effective water phase flow path are changed to Aw and Lw,respectively.

In addition, we redefine ra and rai as the movable water radiusof tortuous capillary tube and the radius containing irreduciblewater at a water saturation of 100%, respectively. rw and rwi are themovable water radius of tortuous capillary tube and the radiuscontaining irreducible water at specific saturation of Sw,respectively.

The derivation started with the combining of the Poiseuille'slaw and Darcy's law. The Poiseuille's law for a cylinder is:

πμ

=∆

( )Q

r pL8 3

a

a

where Q is the volumetric rate of flow, μ is the water viscosity, ΔPis the pressure gradient. Darcy's law is:

Table 2Fitting coefficients R2 of different models.

Corenumber

Fitting coefficients R2 of water Fitting coefficients R2 of gas

Li model Ma model Modifiedmodel

Li model Ma model Modifiedmodel

1 0.983 0.967 0.959 0.919 0.925 0.9292 0.932 0.891 0.748 0.820 0.867 0.9063 0.655 0.816 0.860 0.833 0.877 0.9074 0.954 0.958 0.866 0.775 0.810 0.8525 0.871 0.944 0.957 0.825 0.853 0.8866 0.873 0.933 0.900 0.896 0.915 0.9347 0.771 0.902 0.931 0.505 0.582 0.6278 0.861 0.774 0.647 0.905 0.911 0.913

Table 3Properties of core samples.

High permeability core group

Core number ϕ (%) k (mD) Swi (%) R2

1 27.2 941 11.2 0.95263 28.1 1192 11.6 0.90756 19.1 999 13.4 0.97178 22.7 3680 6.7 0.993410 32.1 437 16.7 0.996616 26.2 1916 07.8 0.9886

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008

μ=

∆( )

Qk A p

L 4a

where Ka is the water permeability at a water saturation of 100%,as well as the absolute permeability.

The volumetric rate of flow obtained from the two equationsare equivalent. Combining Eqs. (3) and (4), ka can be obtained:

π= ⋅ ⋅

( )k

LA

rL

r8 5a

a

a

a2 2

Using the same procedure, one can obtain:

π= ⋅ ⋅

( )k

LA

rL

r8 6w

w

w

w2 2

where kw is the water permeability at a specific water saturation ofSw. Combining Eqs. (5) and (6), the relative permeability of waterphase can be obtained:

ππ

= ⋅ ⋅( )

kLL

r

r

r

r 7rw

a

w

w

a

w

a

2

2

2

2

It is known that the resistivity index is the function of the ef-fective length and across area of electrical flow path. Irreduciblewater cannot flow in the medium but it contributes to electricconduction. Considering these problems, the resistivity indexshould be modified as follow:

ππ

= ⋅( )

ILL

r

r 8w

a

ai

wi

2

2

It is assumed that irreducible water distributes in the outer wallof capillary; it just affects the radius of the capillary and does notchange the lengths, so the Eq. (9) can be expressed as follows:

( )

ππ

ππ

ππ

= ⋅ ⋅ = ⋅ = ⋅ =

⋅( − )

= ⋅ ⋅ = ⋅ ⋅*

( )−

IL

L

r

r

LL

L

L

VV

L

L

VV

L

L

V S

V

L

L

r

r

LL

ILL

r

r

SS

/ 1

/ 9

w

a

ai

wi

a

w

w

a

ai

wi

w

a

ai

wi

w

a

a wi

wS S

S

w

a

ai

wi

a

w

w

a

a

w

w

w

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2w wi

w

The movable water volume in the tortuous capillary tube at awater saturation of 100% can be expressed as:

φ π( − ) = ( ) ( )AL S r L1 10wi a a2

where ϕ is the porosity of the rock. Using the same procedure, onecan obtain:

φ π( − ) = ( ) ( )AL S S r L 11w wi w w2

Combining Eqs. (10) and (11) yields:

=−

−⋅ ⋅

( )r

r

S SS

LL

LL1 12

w

a

w wi

wi w

a2

2

Low permeability core group

Core number ϕ (%) k (mD) Swi (%) R2

152 11.4 1.49 51.9 0.8629153 7.7 0.028 79.6 0.9745204 17.9 0.56 61.7 0.6331299 18.5 4.63 44.6 0.9915334 23.4 38.70 22.2 0.9817336 16.3 35.3 38.8 0.9840418 21.1 74.0 45.4 0.9442479 21.0 28.3 56.0 0.9784

een capillary pressure and resistivity data in gas-water system. J.i

(a) core 1 (b) core 3

(c) core 6 (d) core 8

0

10

20

30

40

50

60

70

80

0 30 60 90 120 150 180 210 240 270 300

Pc,psi

(I·Sw)1/2/Sw*

0

10

20

30

40

50

60

70

80

0 500 1000 1500 2000 2500 3000

Pc,psi

(I·Sw)1/2/Sw*

0

10

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30

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60

70

80

0 50 100 150 200 250

Pc,psi

(I·Sw)1/2/Sw*

0

10

20

30

40

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60

70

80

0 40 80 120 160 200

Pc,psi

(I·Sw)1/2/Sw*

0

10

20

30

40

50

60

70

80

0 50 100 150 200

Pc,psi

(I·Sw)1/2/Sw*

0

10

20

30

40

50

60

70

80

0 50 100 150 200 250 300 350

Pc,psi

(I·Sw)1/2/Sw*

(e) core 10 (f) core16Fig. 3. Relationship between capillary pressure and (I � Sw)1/2/Sw* of high permeability cores.

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514 509

Tortuosity, is the ratio of effective path length to the apparentrock length. By introducing tortuosity, Eq. (12) can be written as:

ττ

= ⋅*

⋅−

−⋅

( )k

ISS

S SS

11 13rw

w

w

w wi

wi

a

w

Tortuosity, is the ratio of effective path length to the apparentrock length. Burdine (1953) developed an empirical expression ofthe tortuosity ratio as a function of wetting phase, which is writtenas:

λ =ττ

= ⋅ =−

− ( )L

LLL

S S1 S 14rw

a

w w

a w wi

wi

where τa is the tortuosity at a water saturation of 100% (τa¼ L L/a )and τw is the tortuosity at a specific water saturation of Sw

(τw¼ L L/w ). λrw is the tortuosity ratio, which is defined as the ratioof the tortuosity at 100% water phase saturation to the tortuosityat a water phase saturation of Sw.

Substituting Eq. (14) into Eq. (13), one can obtain:

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008i

= ⋅*

( )k

ISS

115rw

w

w

3

The experimental data of resistivity and gas-water relativepermeability reported by Pirson and Boatman (1964) were used toverify the modified model. Table 1 presents the properties of coresamples used in the experiments. Fig. 2 shows the calculated gas-water relative permeability using the modified model can matchthe experimental data, and the fitting results of non-wetting phaseare better than other models. The comparison among Li andWilliams (2007) model, Ma et al.'s (2015) model and the modifiedmodel of fitting results with other models were shown in Table 2.The results demonstrated that the modified model could also beused to infer relative permeability from resistivity data.

2.2. Kr-Pc model

Many models, such as Purcell (1949) model, Burdine (1953)model, Corey (1954) model, have shown that relative permeability

een capillary pressure and resistivity data in gas-water system. J.

(a) core 152 (b) core 153

(c) core 204 (d) core 299

(e) core 334 (f) core 336

(g) core 418 (h) core 479

0

5

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20

25

30

35

0 20 40 60 80

Pc,psi

(I·Sw)1/2/Sw*

0

5

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15

20

25

30

35

0.0 2.0 4.0 6.0 8.0 10.0

Pc,psi

(I·Sw)1/2/Sw*

0

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25

30

35

0 2 4 6 8 10 12Pc,psi

(I·Sw)1/2/Sw*

0

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30

35

0 10 20 30 40 50

Pc,psi

(I·Sw)1/2/Sw*

0

5

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30

35

0 5 10 15 20 25 30

Pc,ps i

(I·Sw)1/2/Sw*

0

5

10

15

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25

30

35

0 5 10 15 20 25

Pc, psi

(I·Sw)1/2/Sw*

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

Pc,psi

(I·Sw)1/2/Sw*

0

5

10

15

20

25

30

35

0.0 0.5 1.0 1.5 2.0 2.5

Pc,psi

(I·Sw)1/2/Sw*

Fig. 4. Relationship between capillary pressure and (I � Sw)1/2/Sw* in low permeability cores.

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514510

and capillary pressure are coupled. Li and Horne (2006) conducteda model fitting the best with experimental data. The result in-dicated that the Purcell relative permeability model was the bestfit for the wetting-phase relative permeability. According to Li andHorne (2006), the wetting-phase relative permeability can be

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008

calculated using the Purcell (1949) model:

= ( *) ( )λλ+

k S 16rw w2

where λ is the pore size distribution index, defined by Brooks and

een capillary pressure and resistivity data in gas-water system. J.i

Table 4Properties of core samples.

Corenumber

ϕ (%) k (mD) At ambient condition At reservoir condition

Swi (%) R2 Swi (%) R2

78 10.39 6.01 14.2 0.8786 13.9 0.876795 13.29 60.11 10.9 0.9742 13.3 0.9910143 10.27 13.95 10.5 0.9969 15.8 0.9898240 12.94 167.96 5 0.9980 9.5 0.9717

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514 511

Corey (1966) and can be inferred by the following equation:

= ( *) ( )λ−P P S 17c e w1

Combining Eqs. (16) and (17), one can obtain:

=*

( )k

S

P 18rw

w

cD2

Eq. (12) is the relationship between relative permeability anddimensionless capillary pressure for the Purcell model, which isalso reported by Li (2011).

2.3. New Pc-RI model

Combining Eqs. (15) and (18), one can obtain:

= * ( )P

IS

S 19cDw

w

Eq. (19) is the new relationship between dimensionless capil-lary pressure and resistivity index. At 100% water saturated con-dition, it is known that I¼1 and Sw*¼1, thus the value of di-mensionless capillary pressure calculated by Eq. (19) would beequal to 1. At irreducible water saturation, it is known that Sw*¼0,and the dimensionless capillary pressure approaches infinity ac-cording to Eq. (19), which is consistent with the definition of PcD.

It is obvious that the dimensionless capillary pressure can becalculated directly by Eq. (19) once the resistivity index and irre-ducible water saturation are available.

3. Model evaluation

The model was verified by three groups of resistivity and ca-pillary pressure data: Li and Williams (2007), Aboujafar and Amara(2013) and experiments done by ourselves. The results are pre-sented and discussed in this section.

3.1. Case 1

The experimental data reported by Li and Williams (2007) areused to test our model. The experiments were conducted at aroom temperature to measure the gas-water capillary pressureand resistivity simultaneously by porous plate method. Fourteensandstone cores were divided into two groups based on perme-ability and other properties of the core samples, which are listed inTable 3. In addition, all the cores were cleaned by toluene andmethanol before capillary pressure experiments.

The relationship between capillary pressure and (I � Sw)1/2/Sw* ofhigh permeability cores and low permeability cores are shown inFigs. 3 and 4, respectively. It can be seen that a straight line can befitted by plotting capillary pressure versus (I � Sw)1/2/Sw* both inFigs. 3 and 4, as the proposed model predicted.

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008i

3.2. Case 2

Another experiment, reported by Aboujafar and Amara (2013),was used to further test the proposed model. The rock samplesrestored from a Libyan reservoir were measured both at ambientand reservoir conditions in this case. All samples were cleaned bytoluene and methanol before capillary pressure experiments.Confining pressure and temperature were set to 1500 psi and75 °C respectively, similar to the reservoir conditions. The per-meability of four rock samples ranges from 6.01 to 167.96 mD.Table 4 presents the properties of these rock samples both atambient and reservoir conditions.

The experimental data of capillary pressure and (I � Sw)1/2/Sw*

are plotted in Fig. 5. As one can see, the proposed model matchesthe data well both at ambient and reservoir conditions.

3.3. Case 3

In order to further verify the proposed relationship, we con-ducted experiments to determine the capillary pressure and re-sistivity. All of the six samples are sandstone cores with differentpermeability, and the properties of those samples are listed inTable 5. All samples were cleaned by toluene and methanol beforeexperiments.

The relationship between capillary pressure and (I � Sw)1/2/Sw* isshown in Fig. 6. Similar to case 1 and case 2, a straight line can befitted by plotting capillary pressure versus (I � Sw)1/2/Sw*. The valuesof fitting coefficient (R2) of most cores are greater than 0.90, whichindicates that the proposed model could work satisfactorily.

4. Discussions

The values of fitting coefficient, R2, are listed in Tables. It can beseen that the values of fitting coefficient (R2) in most core samplesare greater than 0.90, which indicates good correlations in mostcases.

However, the proposed model does not work very well in thecore sample No. 152 (k¼0.028 mD), No.204 (k¼0.56 mD) and No.78 (k¼6.01 mD). We divided all the core samples into threegroups based on permeability: extra-low permeability (o10 mD),low permeability (10–100 mD), medium and high permeability(4100 mD). Statistical results presented in Table 6 suggested thatthe cores whose fitting coefficients are less than 0.9 all came fromthe extra-low permeability group. Therefore, one of the possiblereasons that the modified model does not performs very well inthose three cores may be the extra-low permeability.

The range of capillary pressure during porous plate test waslimited below 200 psi, as reported by Li (Li and Williams, 2007).However, those three extra-low permeability cores require highdisplacement pressure at low water saturation. Therefore, the ca-pillary pressure may not be measured accurately, which may beanother reason of the poor fitting results for those three extra-lowpermeability cores.

Another possible reason is that Poiseuille's law is based on theassumptions of laminar flow. At extra-low permeability, the highdisplacement pressure leads to relatively high flow speed, whichmay cause turbulent flow near the pore-throat, so the proposedmodel may not work very well.

Based on the discussions, we found that the model could worksatisfactorily in most cases, except for cores with extra-low per-meability (generally, permeability less than 10 mD).

In addition, case 2 was shown that the irreducible water sa-turation values obtained at ambient and reservoir condition aremuch different in the same sample. Even so, the linear relationshipbetween dimensionless capillary pressure and (I � Sw)1/2/Sw* is

een capillary pressure and resistivity data in gas-water system. J.

(a) core 78 at ambient condition (b) core 78 at reservoir condition

(c) core 95 at ambient condition (d) core 95 at reservoir condition

(e) core143 at ambient condition (f) core 143 at reservoir condition

0

5

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20

25

30

35

40

0 10 20 30 40

Pc,psi

(I·Sw)1/2/Sw*

0

5

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20

25

30

35

40

0 10 20 30 40 50

Pc,psi

(I·Sw)1/2/Sw*

0

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100

120

140

0 10 20 30 40 50 60

Pc,psi

(I·Sw)1/2/Sw*

0

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40

0 20 40 60 80

Pc,psi

(I·Sw)1/2/Sw*

0

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120

140

0 20 40 60 80 100 120 140

P c,ps i

(I·Sw)1/2/Sw*

0

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30

35

40

0 100 200 300 400 500 600 700

Pc,ps i

(I·Sw)1/2/Sw*

0

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30

40

50

60

70

0 10 20 30 40 50

Pc,psi

(I·Sw)1/2/Sw*

0

10

20

30

40

50

60

70

0 20 40 60 80 100 120 140

Pc,psi

(I·Sw)1/2/Sw*

(g) core 240 at ambient condition (h) core 240 at reservoir conditionFig. 5. Relationship between capillary pressure and (I � Sw)1/2/Sw* at ambient and reservoir conditions.

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514512

Please cite this article as: Liu, C., et al., More general relationship between capillary pressure and resistivity data in gas-water system. J.Petrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008i

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514 513

observed in both conditions. The reason may be that capillarypressure and resistivity vary simultaneously when the experimentcondition is changed. In other words, the capillary pressure couldbe characterized by resistivity to some extent. Accordingly, theresistivity and irreducible water saturation should be obtained

Table 5Properties of core samples.

Core number Φ (cm) ℓ (cm) ϕ (%) k (mD) Swi (%) R2

L1 2.555 5.151 27.3 160.36 22.0 0.9169L5 2.535 6.439 20.7 113.26 22.0 0.9804L6 2.547 6.890 26.8 654.00 21.3 0.9573L7 2.566 7.063 26.61 620.32 23.4 0.9698L8 2.548 6.737 23.53 93.42 27.5 0.9114L9 2.540 6.888 29.26 1023.55 16.0 0.9849

0

10

20

30

40

50

60

0 20 40 60 80 100 120

P c,ps i

(I·Sw)1/2/Sw*

0

5

10

15

20

25

30

0 5 10 15 20 25

Pc,psi

(I·Sw)1/2/Sw*

(c) core L6

(e) core L8

0

5

10

15

20

25

0 2 4 6 8 10 12

P c,psi

(I·Sw)1/2/Sw*

(a) core L1

Fig. 6. Relationship between capillary pres

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008i

under the same condition for calculation of the dimensionlesscapillary pressure accurately by our model.

As mentioned above, the proposed model could match theexperimental data in both low and high water saturation, but thepower law relationship between capillary pressure and resistivity

(d) core L7

(f) core L9

0

10

20

30

40

50

60

0 10 20 30 40 50

Pc,psi

(I·Sw)1/2/Sw*

0

5

10

15

20

25

30

0 5 10 15 20

Pc,psi

(I·Sw)1/2/Sw*

0

5

10

15

20

25

0 5 10 15

Pc,psi

(I·Sw)1/2/Sw*

(b) core L5

60

sure and (I � Sw)1/2/Sw* in experiments.

Table 6Statistical result of the core samples based on fitting coefficient and permeability.

Category Extra-lowpermeability

Low permeability Medium and highpermeability

o10 mD 10–100 mD 4100 mD

R240.90 2 7 12R2o0.90 3 0 0Total 5 7 12

een capillary pressure and resistivity data in gas-water system. J.

Table 7Comparisons between Li's model and proposed model.

Core no. Porosity Permeability(mD)

Irreducible wa-ter saturation

Fitting coefficient R2

Li model Liu model

Group 11 0.272 941 0.112 0.8460 0.95263 0.281 1192 0.116 0.7652 0.90756 0.191 999 0.134 0.8165 0.97198 0.227 3680 0.067 0.9724 0.993410 0.321 437 0.167 0.8134 0.996616 0.262 1916 0.078 0.8372 0.9886Group 2152 0.114 1.47 0.519 0.9246 0.8629153 0.077 0.028 0.796 0.9811 0.9745204 0.179 0.56 0.617 0.6699 0.6331299 0.185 4.63 0.446 0.9793 0.9915334 0.234 38.7 0.222 0.9776 0.9817336 0.163 35.3 0.388 0.9505 0.9840418 0.211 74 0.454 0.9855 0.9442479 0.21 28.3 0.56 0.9633 0.9784

C. Liu et al. / Journal of Petroleum Science and Engineering 146 (2016) 505–514514

based on Li Pc-RI model only exists in a specific range of low watersaturations and permeability. Comparisons between Li's modeland proposed model were made on the basis of data from Li'sliterature (Li and Williams, 2007), shown in Table 7.

The formulation indicates that the dimensionless capillarypressure is the function of irreducible water saturation (Swi) andresistivity index (I) without any indeterminate coefficients. Ob-viously, it is more general and easier to use both in laboratoriesand reservoirs.

5. Conclusions

Based on the present study, the following conclusions may bedrawn:

1. A model between relative permeability and resistivity index hasbeen modified, which fits the assumptions of Poiseuille's lawbetter and performs better for non-wetting phase. The results

Please cite this article as: Liu, C., et al., More general relationship betwPetrol. Sci. Eng. (2016), http://dx.doi.org/10.1016/j.petrol.2016.07.008

indicated that it could also be used to infer relative permeabilityfrom resistivity data, and provide a foundation for the deriva-tion of Pc-RI model.

2. A more general relationship has been derived theoretically be-tween dimensionless capillary pressures and resistivity data.This model predicts a linear relationship between capillarypressure and (I � Sw)1/2/Sw*.

3. The proposed model has been verified by the experimental datafrom different literatures and ourselves’. The results demon-strated that the model works satisfactorily, except for thosecores with extra-low permeability (generally, permeability lessthan 10 mD). In addition, this model can be applied to bothambient and reservoir conditions.

Acknowledgments

This research was supported by Natural Science Foundation ofHubei Province [Grant numbers 2015CFB635].

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een capillary pressure and resistivity data in gas-water system. J.i