a new model to calculate oil-water relative permeability...

Research ArticleA New Model to Calculate Oil-Water Relative Permeability ofShaly Sandstone

Huiyuan Bian ,1,2 Kewen Li,3,4 Binchi Hou,5 and Xiaorong Luo6

1College of Geology & Environment, Xi’an University of Science and Technology, Xi’an, Shaanxi 710054, China2Key Laboratory of Coal Resources Exploration and Comprehensive Utilization, MLR, Xi’an, Shaanxi 710021, China3School of Energy Resources, China University of Geosciences, Beijing, China4Stanford University, USA5Research Institute of Shaanxi Yanchang Petroleum (Group) Co., LTD, China6Tuha Branch of China National Petroleum Corporation Logging, Tuha, Xinjiang 839009, China

Correspondence should be addressed to Huiyuan Bian; [email protected]

Received 29 May 2020; Revised 5 August 2020; Accepted 7 September 2020; Published 24 September 2020

Academic Editor: Wen-Dong Wang

Copyright © 2020 Huiyuan Bian et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Oil-water relative permeability curves are the basis of oil field development. In recent years, the calculation of oil-water relativepermeability in sandstone reservoirs by resistivity logging data has received much attention from researchers. This article firstanalyzed the existing mathematical models of the relationship between relative permeability and resistivity and found that mostof them are based on Archie formula, which assumes the reservoir is clean sandstone. However, in view of the fact thatsandstone reservoir is commonly mixed with shale contents, this research, based on the dual water conductivity model,Poiseuille’s equation, Darcy’s law, and capillary bundle model, derived a mathematical model (DW relative permeability model)for shaly sandstone reservoir, which calculates the oil-water relative permeability with resistivity. To test and verify the DWrelative permeability model, we designed and assembled a multifunctional core displacement apparatus. The experiment of coreoil-water relative permeability and resistivity was designed to prove the effectiveness of the DW relative permeability model inshaly sandstone reservoirs. The results show that the modified Li model can well express the transformational relation betweenresistivity and relative permeability in sandstone reservoir with low clay content. Compared with the modified Li model and thePairoys model, the DW relative permeability model is more helpful to collect better results of relative permeability in shaly sand.These findings will play a significant role in the calculation of oil-water relative permeability in reservoirs based on resistivitylogging data and will provide important data and theory support to the shaly sandstone reservoir characterized oil fielddevelopment.

1. Introduction

The evaluation of tight reservoir has always been an impor-tant part of petroleum geology research, while oil-water rela-tive permeability, which is vital to the evaluation of fluid flowin porous media, is used in all aspects of the reservoir engi-neering [1, 2]. Traditionally, relative permeability is obtainedin laboratory. However, in many cases, especially in low per-meability reservoirs, or when phase transformation or masstransfer happens with the change of pressure, oil-water rela-tive permeability experiments are difficult, expensive, andtime consuming simultaneously [3, 4]. Alternately, it is diffi-

cult to maintain the samples the same as in the reservoirs;moreover, relative permeability is almost impossible toobtain in real time. Despite all these difficulties, experimentserves as the main method to calculate relative permeabilitycurves for oil fields.

Conventional resistivity logging data, which is the basicinformation of oil and gas well standard logging, is in largeamount and available. In recent years, more and morescholars indicated that there is a relationship between relativepermeability and resistivity [5–9]. Cai et al. [10] presented areview of the electrical conductivity models using fractal, per-colation, and effective medium theories. In another article,

HindawiGeofluidsVolume 2020, Article ID 8842276, 11 pageshttps://doi.org/10.1155/2020/8842276

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https://creativecommons.org/licenses/by/4.0/

https://doi.org/10.1155/2020/8842276

Cai et al. [11] proposed a combined model including pore-throat ratio, tortuosity, and connectivity, exactly estimatingthe influence of complex pore structure on the transportbehavior associated with electrical parameters. Li [12] basedon Archie formula put forward a mathematical model thatuses resistivity to calculate the relative permeability of gas-water, and Li verified the model with experimental data. Liet al. [13, 14] conducted a lot of research on this field. Litogether with Horne and Williams worked out methods tocalculate two-phase relative permeability with resistivity log-ging data in uniform medium. Mohammed and Birol [15]modified Li model by taking the fluid viscosity and the aver-age water saturation at the time of water breakthrough intoconsideration. Alex et al. [16] proposed a method using resis-tivity to calculate relative permeability in dual porositymodel, but the model has not been verified experimentally.Pairoys et al. [17] verified the Li model and Brooks-Coreymodel [6] with gas-water relative permeability experimentsand found that the Li model works better than Brooks-Corey model in that situation. Then, the Li model is modifiedby replacing the pore size distribution index λ to index satu-ration exponent n Li model [18]. Pairoys [19] analyzed thechange of resistivity under different frequency in the processof unsteady two-phase flow displacement, based on whichthe Li model was verified again with gas-water relative per-meability experimental data and oil flooding data by Bianand Li [20]. The above-modified models based on the Limodel were established under the condition of homogeneousclean sandstone reservoir. However, most real sandstone res-ervoirs contain shale contents, which influence rock resistiv-ity and relative permeability significantly.

This study, based on the dual water conductivity model,Poiseuille’s equation, Darcy’s law, and capillary bundlemodel, proposed and verified a mathematical model (DWrelative permeability model) to calculate relative permeabilityusing resistivity. To improve the Li model, a new modelnamed “dual water relative permeability model (DWmodel)”was proposed in the consideration of better expressing thetransformational relations between resistivity and relativepermeability in the shaly sand reservoir. According to exper-iments, the DW model achieved the goal of reflecting therelation between resistivity and relative permeability in a bet-ter way than the modified Li model and Pairoys model, whichis helpful in both the calculation of oil-water relatively per-meability in shaly sand reservoir based on resistivity andthe oil field development.

2. Mathematical Background

2.1. Relationship between Water Saturation and RelativePermeability. There exist many relationship models betweenwater saturation and relative permeability, among which themost common one is as shown below [20]:

krw = krwmax Sw′′� �nw , ð1Þ

krnw = krnw max 1 − Sw′′� �nnw , ð2Þ

Sw′′ =Sw − Swr

1 − Swr − Snwr, ð3Þ

where krw and krnw are relative permeabilities of the wettingand nonwetting phase, Sw and Swr are the saturation andthe irreducible saturation of wetting phase, krwmax is the max-imum krw when Sw = 1 − Snwr, krnwmax is the maximum krnwwhen Sw = Swr, Snwr is the residual saturation of nonwettingphase, and Sw′ is the normalized saturation of wetting phase.

2.2. Li Model for the Relationship between RelativePermeability and Resistivity. Fluid flow in porous media issimilar to current flow in conductive media [13]. Accordingto the Li model, gas/water relative permeability is calculatedusing resistivity.

S∗w = Sw − Swr1 − Swr

, ð4Þ

k∗rw = S∗w1I, ð5Þ

k∗rw = S∗wð Þ 2+λð Þ/λ, ð6Þ

k∗rnw = 1 − S∗wð Þ2 1 − S∗wð Þ 2+λð Þ/λh i

, ð7Þ

where I is the resistivity index, and k∗rw is the wetting-phasenormalized relative permeability.

2.3. Pairoys Model.Many modified models grew out of the Limodel that is suitable for gas-water two-phase flow and oilflooding, but not for water flooding. Pairoys worked out thefollowing model after analyzing water flooding situation [19].

Sw′ =Sw − Swc

1 − Swc − Sor, ð8Þ

k∗rw = Sw′I, ð9Þ

k∗rnw = 1 − Sw′� � I

Imax, ð10Þ

where Sw is the saturation of the wetting phase, Swc is the irre-ducible saturation of the wetting phase, Sor is the residual sat-uration of the nonwetting phase, Sw′ is the normalizedsaturation of the wetting phase, I is the resistivity index,when Sw = 1 − Sor, and k∗rw and k∗rnw are normalized relativepermeabilities of the wetting and nonwetting phases Imax isthe resistivity index.

2.4. Modified Li Model. Based on the Li model, Bian and Liproposed a model for the relationship between resistivityand oil-water relative permeability of water wet sandstonereservoirs with low shaly contents [20].

Sw′ =Sw − Swc

1 − Swc − Sor, ð11Þ

2 Geofluids

k∗rw = Sw′RorRt

, ð12Þ

k∗ro = 1 − Sw′� �2

1 − krwð Þ, ð13Þ

where Ror is the formation resistivity when Sw = 1 − Sor andk∗rw and k∗ro are normalized relative permeabilities of the waterand oil phases.

3. Relationship between Resistivity and RelativePermeability of Shaly Sand Reservoir

The relationship models of resistivity and relative permeabil-ity mentioned above all assume that the reservoirs are homo-geneous and pure sandstone. However, in reality, mostsandstone reservoirs contain shale contents. Therefore, inorder to calculate the relative permeability in shaly sand res-ervoirs accurately, a new model suitable for shaly sandstoneshould be established.

The cross-sectional area, length, and volume of the waterwet shaly sandstone areA, L, andV , respectively (as shown inFigure 1(a)). The effective pore space of rock is considered tobe composed of n large bore capillary columns with equalcross-sectional area andm small bore capillary columns withequal cross-sectional area. The large columns are filled withmovable water and oil, while the small columns are filled withimmovable water (irreducible water) and residual oil. Thecross-sectional area, cross-section radius, length, and volumeof the large bore capillary columns are Aa, ra, La, and Va i,while those of the small columns are Ab, rab, Lb, and Vb i,respectively (as shown in Figure 1(b)).

When the water saturation of the rock is Sw, in the ith

(i = 1, 2,⋯, n) large capillary columns, the oil cross-sectional area and oil cross-section radius are Aof and rof ,while the cross-sectional area, length, and volume of themovable water are Awf , Lwf , and Vwf i, respectively. In thejth (j = 1, 2,⋯,m) small capillary columns, the oil cross-sectional area and oil cross-section radius are Aor and ror,while the cross-sectional area, length, and volume of theimmovable water are Awc, Lwc, and Vwc j, respectively. Dueto the existence of shale contents, it is assumed that theimmobile water in the small capillary columns contains claywater. The cross-sectional area, length, and volume of theclay water are Awb, Lwb, and Vwb j, respectively (as shownin Figure 1(c)).

When the core sample is saturated with water, accordingto Poiseuille flow formula, the liquid flow in the ith

(i = 1, 2,⋯, n) large capillary column qa i is calculated asfollows.

qa i =πr4aΔp8μLa

, ð14Þ

where Δp is the pressure difference, and μ is the fluidviscosity.

The total flow of water in the rock q is

q = 〠n

i=1

πr4aΔp8μLa

+ 〠m

j=1

πr4bΔp8μLb

: ð15Þ

According to Darcy’s formula,

Q = kAΔpμL

: ð16Þ

Suppose that the length of the large capillary column isequal to that of the small capillary column. The permeabilityk is obtained as follows.

k = 18 φa

r2aτ2a

+ φbr2bτ2b

� �: ð17Þ

Similarly, when the water saturation is SwðSwc ≤ Sw ≤ 1− SorÞ, the small capillary columns are filled with boundwater and residual oil, while the large capillary columns arefilled with movable water and movable oil.

According to the Poiseuille flow formula, the total flow ofmovable water in the rock qwf is

qwf = 〠n

i=1

A2wfΔp

8πμLwf: ð18Þ

According to Darcy’s formula,

Qwf = kwAΔpμL

: ð19Þ

So, the permeability kw is

kw = 18φSwf

r2a‐r2of� �τ2wf

: ð20Þ

And the water relative permeability krw is obtained asfollows.

krw = φSwf r2a‐r2of� �

τ2wf φa r2a/τ2að Þ + φb r2b/τ2b� �� : ð21Þ

The electrical conductivity of the model is analyzedbelow. The resistivity of free water is Rw, while the resis-tivity of clay water is Rwb. In the jth (j = 1, 2,⋯,m) smallcapillary columns, the cross-sectional area, length, and vol-ume of the bound free water are Awz, Lwz, and Vwz j,respectively. When the core sample is saturated with

3Geofluids

water, the resistivity is R0. When the saturation is Sw andthe resistivity is Rt, there is the following equation.

1Rt L/Að Þ = 〠

n

i=1

1Rw Lwf /Awfð Þ + 〠

m

j=1

1Rw Lwz/ Ab − Awbð Þð Þ

+ 〠m

j=1

1Rwb Lwb/Awbð Þ :

ð22Þ

Assuming that all the capillary columns have the samelength,

La = Lb = LA, ð23Þ

Lwf = Lwc = Lwz = Lwb = Lw: ð24ÞThe resistivity index I is

I = RtR0

: ð25Þ

So, the water relative permeability krw can be obtained.

krw = Sw − Swcð Þ21 − Swc − Sorð Þ2 + Swc + Sorð Þ2

τ2Aτ2w

: ð26Þ

Define the resistivity Rb.

SwcRb

= Swc − SwbRw

+ SwbRwb

: ð27Þ

Define SR can be calculated as follows.

SR = RbRw

1 − Sor − SwcSwc

: ð28Þ

Therefore, the normalized water relative permeabilityin the dual water relative permeability model can beexpressed as follows:

k∗rw = krwkrw Sw = 1 − Sorð Þ , ð29Þ

k∗rw = Sw′� �2 Ror/R0ð Þ SR + 1ð Þ − 1

I SRSw′ + 1� �

− 1: ð30Þ

When Sw = Swc, k∗rw = 0. When Sw = 1 − Sor, Sw′ = 1 and

k∗rw = 1, which satisfies the boundary condition.When the clay water content is 0 (Swb = 0), the model is

simplified to a clean sandstone model, and the normalizedwater relative permeability is as follows.

k∗rw = Sw′� �2 Ror 1 − Sorð Þ − R0Swc

RtSw − R0Swc: ð31Þ

3.1. Determination of the Parameters in the DW RelativePermeability Model

3.1.1. Calculation of the Resistivity of Clay Water Rwb. Diffu-sion factor of Na+ ion diffusion layer α is calculated asfollows [21]:

L

A

(a) Shaly sand core

L

AAwf

LwfLa LwcLb

Aof

AwcAor

(b) Capillary columns pack bundle (water wetting)

Radius ResistivityCross-sectional area

Awfra

rofAof Ro

Rwb

Rw

Ro

Rw

ror

rb

rwzAwz

Awb

Aor

(c) The cross section of capillary columns pack bundle (water wetting)

Figure 1: Capillary bundle model.

4 Geofluids

α =1, when Pw > Pwo,ffiffiffiffiffiffiffi

PwoPw

s, when Pw ≤ Pwo,

8>><>>: ð32Þ

where Pw is the salinity of formation water, Pwo is thesalinity of formation water when xd = xH, and xd is thethickness of Na+ ion diffusion layer (10-8 cm).

The pore volume occupied by clay water VQ when Qv= 1mmol/cm3 is calculated as follows:

VQ = 12:853 + 0:019T °Cð Þ : ð33Þ

The equivalent conductivity β of compensation Na+ ionin clay water (S/m) (mmol/L) is calculated as follows:

β = 0:0857T °Cð Þ − 0:143: ð34Þ

The clay water resistivity Rwb is calculated as follows:

Rwb =αVQβ

, ð35Þ

Rwb = α0:0857T °Cð Þ − 0:1432:853 + 0:019T °Cð Þ : ð36Þ

It can be seen from the above formula that the resistivityof formation water is affected by both α and temperature T .When the water salinity is high, α = 1. Therefore, the resistiv-ity of clay water Rwb is independent from the equilibriumcation concentration and clay types.

3.1.2. Calculation of the Clay Water Saturation Swb.

Swb = αVQQv: ð37Þ

Substitute equation (33) into equation (37),

Swb =αQv

2:853 + 0:019T °Cð Þ : ð38Þ

From the above formula, it can be seen that the clay watersaturation Swb increases with the increase of α. As thetemperature T increases, Swb decreases. With the increaseof Qv, Swb increases.

3.1.3. Calculation of the Resistivity Rb.

SwcRb

= Swc − SwbRw

+ SwbRwb

: ð39Þ

3.1.4. Calculation of the Parameter SR.

SR = RbRw


: ð40Þ

3.1.5. Calculation of the Normalized Water SaturationSw′ .

Sw′ =Sw − Swc

1 − Swc − Sor: ð41Þ

3.1.6. Calculation of the Normalized Water RelativePermeability k∗rw.

k∗rw = Sw′� �2 Ror/R0ð Þ SR + 1ð Þ − 1

I SRSw′ + 1� �

− 1: ð42Þ

In conclusion, the DW relative permeability model canbe expressed as follows.

1ð Þ α =1, when Pw > Pwo,ffiffiffiffiffiffiffiPwoPw

s, when Pw ≤ Pwo,

8>><>>:

2ð ÞRwb = α0:0857T °Cð Þ − 0:1432:853 + 0:019T °Cð Þ ,

3ð Þ Swb =αQv

2:853 + 0:019T °Cð Þ ,

4ð Þ SwcRb

= Swc − SwbRw

+ SwbRwb

,

5ð Þ SR = RbRw


,

6ð Þ Sw′ =Sw − Swc

1 − Swc − Sor,

7ð Þ k∗rw = Sw′� �2 Ror/R0ð Þ SR + 1ð Þ − 1

I SRSw′ + 1� �

− 1,

8ð Þ k∗ro = 1 − Sw′� �2

1 − krwð Þ:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð43Þ

3.2. Sensitivity Analysis of Parameters of DW Model ofShaly Sandstone

3.2.1. The Effect of Irreducible Water Saturation Swc on k∗rw.Suppose that Sor = 0:2, Swb = 0:05, R0 = 30Ω · m, Ror = 35Ω· m, Rw = 0:5Ω · m, and Rwb = 0:1Ω · m, oil-water relativepermeability curves under different irradiated water satura-tion Swc are shown in Figure 2. It indicates that the normal-ized water relative permeability k∗rw decreases while Swcincreases. The reason is that the movable water saturationSwf decreases under the same normalized water saturationSw′ as Swc increases, thus the normalized relative permeabilityof water phase k∗rw decreases.

3.2.2. The Effect of Residual Oil Saturation Sor on k∗rw. Suppose

that Swc = 0:2, Swb = 0:05, R0 = 30Ω · m, Ror = 35Ω · m, Rw= 0:5Ω · m, and Rwb = 0:1Ω · m, the oil-water relative per-meability curves under different residual oil saturation Sorare shown in Figure 3. It illustrates that the normalized waterrelative permeability k∗rw decreases with the increase of Sor

5Geofluids

because the movable water saturation Swf decreases under thesame normalized water saturation Sw′ when Sor increases,which leads to the decrease of normalized relative permeabil-ity of water phase k∗rw.

3.2.3. The Effect of Cation Exchange Capacity Qv on k∗rw. Sup-pose that there are a set of cores with the same parameters asfollows. T = 20°C, Pw = 8000 ppm, Swc = 0:5, Sor = 0:2, Rw =5Ω · m, R0 = 33Ω · m, and Ror = 35Ω · m. Normalized oil-water relative permeability curves under different Qv areshown in Figure 4. It illustrates that the normalized waterrelative permeability k∗rw decreases with the increase of Qv.

Studies [22–24] show that clay mineral content is one ofthe main factors affecting the shape of oil-water relative per-meability curve of rock. When water is injected into the coresample, it first enters into larger pores, where the relative per-meability of the water phase increases rapidly. Soon after the

injection, water gradually enters into small pores, where theflow resistance increases. At the same time, the oil in largepore paths is separated into small oil droplets by the water.If the oil droplets migrate to the vicinity of the pore throat,the so-called “liquid resistance effect” will emerge when thediameter of the oil droplets is similar to that of the porethroat. In this case, the capillary force of the orifice throatmust be overcome if the oil droplets want to move [25, 26].The hydrophilic particles in the pores will move to the porethroat and cause blockage. With the increase of water satura-tion Sw, the amount of plugging particles will increase, andthe relative permeability of water phase will decreaseaccordingly.

3.2.4. The Effect of Total Salinity Pw on k∗rw. Suppose that T= 20°C, Qv = 0:25mmol/L, Swc = 0:5, and Sor = 0:3. Figure 5

krw∗ :DW(Swc=0.01)

kro∗ :DW(Swc=0.01)







0

0.2

0.4

Nor

mal

ized

rela

tive p

erm

eabi

lity

(frac

tion)

Normalized water saturation (fraction)

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Figure 2: Normalized oil-water relative permeability curves underdifferent Swc.

krw∗ :DW(Sor=0.01)

kro∗ :DW(Sor=0.01)







0

0.2

0.4

Nor

mal

ized

rela

tive p

erm

eabi

lity

(frac

tion)


0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Figure 3: Normalized oil-water relative permeability curves underdifferent Sor.

0

0.2

0.4

Nor

mal

ized

rela

tive p

erm

eabi

lity

(frac

tion)


0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

krw∗ :DW(Qv=0.0 mmol/L)

kro∗ :DW(Qv=0.0 mmol/L)







Figure 4: Normalized oil-water relative permeability curves underdifferent Qv .

krw∗ :DW(Pw=500 ppm)

kro∗ :DW(Pw=500 ppm)







0.0

0.2

0.4

Nor

mal

ized

rela

tive p

erm

eabi

lity

(frac

tion)


0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 5: Normalized oil-water relative permeability curves underdifferent Pw .

6 Geofluids

shows the normalized oil-water relative permeability curvesunder different Pw. It illustrates that the normalized waterrelative permeability k∗rw increases with increased Pw.

In shaly sand reservoir, with the decrease of salt contentin free water, the salinity in free water decreases, clay mineralcrystal layer expands, and the formation permeability con-tinues to decline, which is called the salt-sensitive phenome-non (Meng, 2012). There are a large number of clay mineralsin shaly sand reservoir. Therefore, with the decrease of freewater salinity, the salt content in free water decreases, the

salt-sensitive phenomenon gets worse, and the relativepermeability of water phase goes down.

4. Experimental Verification of DW RelativePermeability Model

4.1. Experiments. In order to verify the relationship betweenresistivity and relative permeability in DW Model, a multi-functional core displacement experiment device was designed

K6 K5K3

K11

K4

K7

K8

K1K2

K10

K9

Pump

LCR meter Pump Balance Flow meter Hand pump Vacuum pump Computer

Air receiver Core Pressure gauge Valve Pressure transducer Burette

K66 K5K3K3

K11

K4

K7

K8

K11K2K2

K10000

K9

(a) Schematic of the apparatus used for the simultaneous measurements (b) Experimental apparatus

Figure 6: Multifunctional core displacement experimental apparatus.

Table 1: The basic physical parameters of core samples.

No. Samples L (cm) D (cm) φ (%) kw ( × 10−3 μm2) Qv (mmol/mL)

Group I

A-7 5.158 2.552 26.61 548.22 0.068

A-8 7.572 2.555 23.53 51.80 0.097

A-9 6.724 2.549 24.83 349.98 0.062

B-1 5.150 2.554 17.62 14.06 0.100

B-9 7.363 2.554 27.33 647.10 0.103

B-10 6.697 2.580 30.70 250.67 0.092

Group II

A-1 7.224 2.556 27.66 123.36 0.155

A-3 6.325 2.535 27.42 119.43 0.168

A-5 7.424 2.551 24.64 58.09 0.514

A-10 7.871 2.549 26.86 1200.05 0.857

B-7 7.063 2.566 24.87 33.02 0.142

B-8 6.737 2.548 24.31 42.38 0.240

7Geofluids

(Figure 6). The resistivity and relative permeability of coresamples with different water saturations were measured.

Core samples from Wells A and B are tested to explorethe relationship between resistivity and relative permeabil-

ity. Core samples are divided into two groups. Group Icontains less clay and smaller Qv than group II. The basicphysical parameters of cores are shown in Table 1. Thesalinity of formation water in wells A and B are

0

0.2

0.4

Nor

mal

ized

rela

tive

perm

eabi

lity

(frac

tion)

Nor

mal

ized

rela

tive

perm

eabi

lity

(frac

tion)

Nor

mal

ized

rela

tive

perm

eabi

lity

(frac

tion)

Nor

mal

ized

rela

tive

perm

eabi

lity

(frac

tion)





0.6

0.8

1

0 0.2

Core A-7

Core A-8

Core A-9

Core B-1

0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Nor

mal

ized

rela

tive

perm

eabi

lity

(frac

tion)

Nor

mal

ized

rela

tive

perm

eabi

lity

(frac

tion)



Core B-9

Core B-10

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

krw∗ :experiment

kro∗ :experiment

krw∗ :Li model

kro∗ :Li model

krw∗ :pairoys

kro∗ :pairoys

krw∗ :modified Li

kro∗ :modified Li

Rela

tive p

erm

eabi

lity

(frac

tion)

Water saturation (fraction)




Rela

tive p

erm

eabi

lity

(frac

tion)

Rela

tive p

erm

eabi

lity

(frac

tion)

Rela

tive p

erm

eabi

lity

(frac

tion)

Core A-7

Core A-8

Core A-9

Core B-1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1



Rela

tive p

erm

eabi

lity

(frac

tion)

Rela

tive p

erm

eabi

lity

(frac

tion)

Core B-9

Core B-10

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

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Figure 7: Comparison of relative permeability curves in sandstone reservoir.

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7500 ppm and 8000ppm, respectively. The density of brineused is 1.02 g/cm3. The oil viscosity in both wells is8.8mPa s at 20°C, and its density is 0.845 g/cm3.

4.2. Verification of Relationship Models between Resistivityand Relative Permeability of Sandstone. The Li model, modi-fied Li model, and Pairoys model are verified with the

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Figure 8: Comparison of relative permeability curves in shaly sand reservoir.

9Geofluids

experimental data of core samples in group I. Figure 7 showsthe oil-water relative permeability curves of group I cores. InFigure 7(a), the solid blue triangular and pink dots are thenormalized water and oil relative permeability obtained fromthe water displacing oil experiment, respectively; the brownand sky blue chain dotted lines are the normalized waterand oil relative permeability calculated with resistivity basedon the Li model; the green and purple broken line are thenormalized water and oil relative permeability calculatedwith resistivity based on Pairoys model; and the blue andred solid line are the ones calculated with resistivity basedon the modified Li model. Figure 7(a) indicates that the Limodel does not work well in the data process of water flood-ing. The problems of Pairoys model is that the normalizationoil relative permeability it calculates is nonnegligibly largerthan the experimental value. However, the normalizationwater and oil relative permeabilities calculated by the modi-fied Li model are in good agreement with experimental data.

Figure 7(b) shows the oil-water relative permeabilitycurve of cores in group I. The water relative permeabilitiescalculated by the Li model and Pairoys model are smallerthan the experimental data, while the oil relative permeabil-ities are larger. The oil and water relative permeabilitiescalculated by the modified Li model fit well with the experi-mental data in sandstone reservoir with less shale contents.

4.3. Verification of Relationship Models between Resistivityand Relative Permeability of Shaly Sandstone. Experiment isdesigned to measure the resistivity and relative permeabilityof shaly sandstone samples in group II. Figure 8 shows thecomparison of the experimental results and the modelcalculated results.

Figure 8(a) shows the normalized relative permeabilitycurves, and Figure 8(b) shows the relative permeabilitycurves. The filled dots are the unsteady oil-water relative per-meability experiment data, the broken lines are the relativepermeability curves calculated with resistivity by the modi-fied Li model, and the solid lines are calculated with resistiv-ity by the DWmodel. As shown in Figure 8, the Li model, thewater relative permeability calculated by the modified Limodel fits well with the experimental data, but the calculatedoil relative permeability curve is smaller than the experimen-tal data. Meanwhile, the relative permeability curves of oiland water calculated by the DW relative permeability modelbetter fit the experimental data.

5. Conclusions

This study established the relationship model between theresistivity and oil-water relative permeability of the shalysandstone reservoir based on the rock physics experimentand the logging response of the shaly sandstone reservoir.According to existing research results, the following mainconclusions can be drawn.

(1) In view of the influence of shale, the DW relativepermeability model, suitable for shaly sandstonereservoir, was derived to calculate oil-water relativepermeability using resistivity based on the dual water

conductivity model, Poiseuille’s equation, andDarcy’s law.

(2) According to the sensitivity analysis, with other con-ditions being the same, the relative permeability ofwater phase will decrease as the irreducible water sat-uration increases, residual oil saturation increases,cation exchange capacity of rock increases, or freewater salinity decreases.

(3) With the core water flooding experimental device,the resistivity and oil-water relative permeability oftwo groups of sandstone samples with different shalecontents were tested. The experimental results showthat the modified Li model is suitable for clean sand-stone reservoirs, and the DW relative permeabilitymodel is suitable for shaly sandstone reservoirs.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declared that they have no conflicts of interest tothis work.

Acknowledgments

The study has been supported by the National Major Scienceand Technology Projects of China (No. 2017ZX05030-002),by the Natural Science Basic Research Plan in Shaanxi Prov-ince of China (Grant no. 2020JQ-747), and by the ScientificResearch Plan Projects of Shaanxi Education Department(Grant no. 18JK0517).

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a new model to calculate oil-water relative permeability...

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