An New Analytical Equation to Predict Gas-Water Two-Phase Relative Permeability Curves in Fractures


 
Copyright © the author(s). This work is licensed under a Creative Commons Attribution 4.0 International License. 

DOI: 10.14800/IOGR.424  

Received March 15, 2017; revised April 7, 2017; accepted April 14, 2017. 

*Corresponding author: lg1987cup@126.com                                                                               1         

An Analytical Equation to Predict 

Oil-Gas-Water Three-Phase Relative 

Permeability Curves in Fractures 
 

Gang Lei* and Cai Wang, Peking University, Beijing, China; Yuan Tian, Tongji University, 

Shanghai, China; Limin Yang, China University of Petroleum, Beijing, China 

 

 

Abstract 

As fractures are the major flow channels for multiphase flow in naturally and hydraulically fractured 

reservoirs, the accurate prediction of multiphase flow in fractures is highly important. The oil-gas-water 

three-phase relative permeability relations in fractures define the hydrodynamics of multiphase fluids 

flow and are necessary for modeling of multi-phase flow in fractured reservoirs.  

  In this work, a novel flow model based on the concept of shell momentum balance, Newton's law of 

viscosity, and the cubic law, is derived to determine analytic functions for the three-phase relative 

permeability curves versus phase saturation and viscosity in a single fracture.  

  The results show that the equations describing three-phase relative permeability curves in a fracture are 

function of saturations and viscosities. Water phase relative permeability depends on water saturation, gas 

phase relative permeability depends on gas saturation when µg is much lower than µo and µw. However, 

oil phase relative permeability is function of all-phase saturations. The isoperms of water phase and gas 

phase are straight lines. However, oil phase isoperms are functions of all phase saturations and have 

significant curvature. The curvatures of oil phase isoperms increase with the increase of µo. Gas 

saturation decreases oil phase relative permeability with a given oil saturation, while the viscosity ratio 

increases it. 

Introduction 

Multiphase flow in naturally fractured reservoirs and hydraulically fractured reservoirs, which holds 

major part of the world's remaining hydrocarbon reservoirs, is strongly influenced by fractures in the 

geological formations (Lei et al. 2014). Fractures are the major flow channels for fluid flowing in 

fractured reservoirs. Thus, it is highly important to predict multiphase flow in fractures accurately.  

  The three-phase relative permeability relations for fractures defines hydrodynamics of fluid flow and 

are necessary for modeling of multiphase flow in reservoirs. 

  The study of three-phase relative permeability was reported as early as 1941 by Leverett and Lewis 

(1941). They conducted steady-state three-phase relative permeability measurements in a tightly packed 

sand core. Corey et al. (1956) reported results of three-phase relative permeability measurements in Berea 

sandstone and proposed a model for prediction of three-phase relative permeability with assuming that the 

oil relative permeability depends on two saturations due to the dependence of residual oil saturation on 

two saturations. They suggested that the water phase isoperms and gas phase isoperms were straight lines. 

Sarem (1966) modified three-phase relative permeability measuring techniques by using unsteady-state 

technique. Donaldson and Dean (1965) used Sarem’s technique to measure three-phase relative 

mailto:lg1987cup@126.com


2 

permeability in Berea sandstone. Saraf and Fatt (1967) developed a new technique using NMR for in-situ 

saturation measurements in three-phase flow system. Stone (1970) proposed the Stone I model for 

prediction of three-phase relative permeability. Stone (1973) proposed the Stone II model using four 

two-phase flow relative permeability curves (two oil-water and two oil-gas) for predicting three-phase 

relative permeability. 

  Dietrich and Bonder (1976) proposed a model accounted for reduction in oil relative permeability due 

to the presence of a third phase. Spronsen (1982) measured a three-phase system in Berea sandstone using 

the centrifuge method. Saraf et al. (1982), Grader and O’Meara (1988) and Maini et al. (1990) measured 

the three-phase relative permeability using steady-state and unsteady-state methods. In addition, there 

have been more models (Maini et al. 1989; Hustad and Hansen 1995; Oosrom and Lenhard 1998; 

Balbinski et al. 1999) proposed for prediction of three-phase relative permeability. Oak et al. (1990) and 

Oak (1990) conducted a three-phase relative permeability measurement on water-wet fired Berea 

sandstone core and presented about 1,800 data collected for three-phase measurements for different 

saturation paths to investigate the effect of saturation history on relative permeability curves. These 

studies have great significance for studying three-phase relative permeability. However, these studies are 

for multiphase flow in porous media but not for fracture systems.  

Compared with the studies on relative permeability in porous medium, relative permeability in fractures 

has received less attention. Many researchers have studied flow regimes in a fracture (Persoff et al. 1991; 

Persoff and Pruess 1995; Diomampo 2001; Fourar et al.1993; Fourar and Bories 1995; Pan 1999; Chen 

2005) and revealed that the flow patterns not only depend on fracture geometry but also on phase 

properties. In order to examine flow of multiphase flow in fractures, some researchers have done different 

experimental studies (Chen et al. 2004; Chen 2005; Habana 2002; Kneafsy and Pruess 1998; Nicholl and 

Glass 1994; Nicholl et al. 2000; Pan 1999). The first relative permeability models for fracture systems 

were established by Romm (1966) based on experimental results using kerosene and water. Romm 

suggested that two-phase flow in fractures can be modeled by straight-line. However, many scholars 

(McDonald et al. 1991; Pieters and Graves 1994; Fourar and Bories 1995; Diomampo 2001; Speyer et al. 

2007) had proved that relative permeability curves in fractures were not a simple linear function of 

saturation with experimental evidence. Many different theoretical studies have been conducted to 

examine multiphase flow in fractures (Bodin et al. 2003; Iwai 1976; Reis 1990; Shad and Gates 2010; 

Chima et al. 2010; Chima and Geiger 2012). Shad and Gate (2010) developed a new model for 

multiple-phase layer flow in a single fracture and concluded how relative permeability in fractures 

depended on flow structures as well as fluid properties. Chima et al. (2010) derived an analytical equation 

to calculate oil-water relative permeability curves in fracture systems. And they concluded that relative 

permeability curves in two-phase flow in fractures are not a linear function of saturation. Chima and 

Geiger (2012) presented a new model to predict gas-water relative permeability curves in a fracture. The 

model was validated with experimental data and showed much better agreement than original models. 

Although these experimental and theoretical studies relative permeability in a fracture and the influence 

of flow structures and fluid properties on relative permeability, they are not for oil-gas-water three-phase 

relative permeability. Based on Chima and Geiger’s (2012) two-phase models, a novel model that can 

predict three-phase relative permeability in fracture systems was proposed in this paper. Although our 

three-phase relative permeability model is for fracture systems and different from previous models, the 

results of our studies match the previous studies (Corey et al. 1956; Donaldson and Dean 1966; Saraf et al. 

1982; Oak 1990; and Maini et al. 1990).  

Mathematical Model 

The following assumptions are made to derive the proposed mathematical model of oil-gas-water relative 

permeability curves in the fracture: 



3 

1. Flow is laminar and in steady-state; 

2. Fluids are Newtonian. Gas is compressible, oil and water are slightly compressible, all the phases 

have constant properties; 

3. No phase transformation between the three phases; 

4. Fracture walls are planar and impermeable, e.g. no fluid is exchanged between matrix and 

fracture; 

5. The wettability of fracture walls sequences from water>oil>gas. Water phase flow occurs close to 

fracture surface, gas phase flows in the center of the fracture, oil phase flow occurs in-between 

water phase and gas phase; 

6. Flow occurs in an open fracture, e.g. the planar surfaces representing the fracture walls remain 

parallel and thus are not in contact at any point; 

7. The fracture is oriented horizontally and gravity is negligible. 

 

 

Figure 1—Proposed fracture model used in the mathematical model. For a perfectly smooth fracture placed 

horizontally with negligible gravity and buoyancy effects. 

 

  Applying the shell momentum balance to the fracture configuration shown in Figure 1 leads to the 

following equations that allow reservoir engineers to estimate oil-gas-water relative permeability curves 

in fractures.    

  The detailed mathematical derivation is given in Appendix A. 

𝑘𝑟𝑤 =
𝑆𝑤

3

2
(3 − 𝑆𝑤),…………….………………………………..……………………………..……....…(1) 

𝑘𝑟𝑔 = 𝑆𝑔
2 (𝑆𝑔

2 +
𝜇𝑔

𝜇𝑤
(3𝑆𝑤 −

3

2
𝑆𝑤

2 ) +
𝜇𝑔

𝜇𝑜
(3𝑆𝑔𝑆𝑜 +

3

2
𝑆𝑜

2)),..…….………..…………………..………(2) 

𝑘𝑟𝑜 = 𝑆𝑜
2 (

𝜇𝑜

𝜇𝑤
(3𝑆𝑤 −

3

2
𝑆𝑤

2 ) +
3

2
𝑆𝑔𝑆𝑜 + 𝑆𝑜

2),…………….…………..…………………………..………...(3) 

  Eqs. 1 through 3 are applied to water phase, gas phase, and oil phase, respectively. Eq. 1 implies that 

water relative permeability is only function of its saturation. Eq. 2 illustrates that gas relative permeability 

depends on all other phases’ saturation. Eq. 3 reveals that oil relative permeability is not only function of 

saturation but also depends on water and gas saturations and oil-water two-phase viscosities. However, if 

µg is much lower than µw and µo, Eq. 2 can be simplified as, 

𝑘𝑟𝑔 = 𝑆𝑔
4,……………………………………………………..……………….………….……………..(4) 

Eq. 4 illustrates that gas relative permeability is only function of gas saturation when gas viscosity is much lower 

than all other phases’ viscosity. 



4 

Model Validation and Model Analysis 

With the basic parameters (Table 1) applied in the novel model, the oil-gas-water three-phase relative 

permeability curves in the fracture systems was estimated by the Eq. 1 to 3 and oil-gas-water three-phase 

isoperms of the study are given in Figure 2.  

  The results of this study confirm the dependency of water and gas relative permeabilities on their own 

saturations, and oil phase relative permeability to all the phases. The study also shows that the isoperms 

of water and gas phases are function of their own saturations. However, oil isoperms are not only function 

of oil saturation but also had significant curvature (concave towards the 100% oil saturation) which has 

the same conclusions with the previous studies (Corey et al. 1956; Donaldson and Dean 1966; Saraf et al. 

1982; Oak 1990; and Maini et al. 1990).  

 

Table 1-Basic parameters applied in the model. 

Parameters Value 

Fracture length [m] 1.15 

Fracture width [m] 2.25 

Fracture thickness [mm] 0.75 

Gas bed thickness [mm] 0.30 

Water bed thickness [mm] 0.25 

Oil bed thickness [mm] 0.20 

Inlet pressure [MPa] 1.50 

Gas density [kg/m3] 0.82 

Water density [kg/m3] 1000 

Oil density [kg/m3] 810 

Gas viscosity [cp] 0.017 

Water viscosity [cp] 1.0 

Oil viscosity [cp] 4.5 

Outlet pressure [MPa] 1.45 

  



5 

     

                      (a)                                       (b)  

 

(c) 

Figure 2—The isoperms of different phase for fracture systems calculated for the Example. (a) Gas phase. (b) 

Water phase. (c) Oil phase. 

 

  With the proposed model, series of three-phase relative permeability calculations at various oil 

viscosities were performed. The results show that the trend and curvature of oil isoperms varied with oil 

viscosity. The larger the viscosity of oil phase is, the greater the curvatures of oil isoperms are. Oil 

isoperms (kro=0.08) with different oil viscosities of the data sets are shown in Figure 3. 

 

 

Figure 3—Oil isoperms for fracture systems with different viscosities of oil phase (kro=0.08). 

0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00
0.00

0.25

0.50

0.75

1.00

S
w

 krg=0.01

Sg

(a)

 krg=0.05

S
o

 krg=0.1

 krg=0.2

 krg=0.4

0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

Sw

 krw=0.2

S
o

0.00

0.25

0.50

0.75

1.00

(b)

S
g

 krw=0.1

 krw=0.05

 krw=0.01

 krw=0.4

S
g

 kro=0.001

S
w

0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

 So

 kro=0.01

0.00

0.25

0.50

0.75

1.00

 kro=0.05

 kro=0.08

 kro=0.1

(c)

 kro=0.4

 kro=0.2

 kro=0.6
 kro=0.65

0.00

0.25

0.50

0.75

1.00

S
g

 
w
=1cp,

o
=1.45cp

S
w

So
0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

 
w
=1cp,

o
=1cp

 
w
=1cp,

o
=0.59cp

 
w
=1cp,

o
=0.71cp



6 

  Figure 4 shows the predicted oil phase relative permeability results at various water saturations. And 

oil viscosity is 1.45 cp. The results of this study show that oil phase relative permeability increased with 

the decrease of gas saturation (e.g. oil phase relative permeability increased with the increase of water 

saturation) with the same oil saturation. The study illustrates that gas saturation decreased oil relative 

permeability for the same oil saturation. The reason for this is that water phase flow occurs close to 

fracture surface, gas phase flows in the center of the fracture and oil phase flow occurs in-between water 

phase and gas phase. Under the same oil saturation, gas saturation decreases with the increase of water 

saturation. For the same thickness of oil bed, the larger water saturation is (e.g. the larger the thickness of 

water bed in the fracture is), the lower the thickness of gas bed is, the closer to the center of the fracture 

oil phase flow occurs and the faster oil phase flows in the fracture. 

   

 

Figure 4—Oil relative permeabilities for fracture systems. The curves represent three-phase data at various 

water saturations when oil viscosity is 1.45 cp. 

 

Figure 5 shows oil phase relative permeability results at various water saturations and viscosities of oil 

phase (as µw=1cp, µo represents viscosity ratio). The results show that oil phase relative permeability 

increases with the increase of viscosity ratio. The result can be explained as: wetting phase (water phase) 

flow occurs close to fracture surface, the flow of the adjacent high-viscosity non-wetting phase (oil phase) 

passing by it, to some extent, can be considered as a sliding motion in which the wetting phase (water 

phase) provides lubrication. Figure 5 also shows that water saturation intensifies the influence of viscosity 

ratio to oil relative permeability. The effect increases with the increase of water saturation. 

 

 

               (a)                             (b)                             (c)  

Figure 5—Oil relative permeabilities for fracture systems. The curves represent three-phase data at various 

oil viscosities with different water saturations. (a) The value of water saturation is 0.1. (b) The value of water 

saturation is 0.2. (c) The value of water saturation is 0.3. 

0.0 0.2 0.4 0.6 0.8 1.0
0.0

0.2

0.4

0.6

0.8

1.0

 

 

O
il
 r

e
la

ti
v

e
 p

e
rm

e
a

b
il
it

y

Oil saturation

 Sw=0

 Sw=0.1

 Sw=0.2

 Sw=0.3

 Sw=0.4

0.0 0.2 0.4 0.6 0.8
0.0

0.2

0.4

0.6

0.8

1.0

 

 

O
il
 r

e
la

ti
v

e
 p

e
rm

e
a

b
il
it

y

Oil saturation

S
w
=0.1

 
o
=1.45cp

 
o
=1cp

 
o
=0.85cp

 
o
=0.75cp

 
o
=0.65cp

 
o
=0.55cp

 
o
=0.45cp

(a)

0.0 0.2 0.4 0.6 0.8
0.0

0.2

0.4

0.6

0.8

1.0

(b)

 

 

O
il
 r

e
la

ti
v

e
 p

e
rm

e
a

b
il
it

y

Oil saturation

S
w
=0.2

 
o
=1.45cp

 
o
=1cp

 
o
=0.85cp

 
o
=0.75cp

 
o
=0.65cp

 
o
=0.55cp

 
o
=0.45cp

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0

0.2

0.4

0.6

0.8

1.0

(c)

 

 

O
il
 r

e
la

ti
v

e
 p

e
rm

e
a

b
il
it

y

Oil saturation

S
w
=0.3

 
o
=1.45cp

 
o
=1cp

 
o
=0.85cp

 
o
=0.75cp

 
o
=0.65cp

 
o
=0.55cp

 
o
=0.45cp



7 

Conclusions 

The following main conclusions can be drawn from this study: 

1. A novel analytical model for oil-gas-water three-phase relative permeability in fracture systems 

has been proposed in this study. The equations describing oil-gas-water three-phase relative 

permeability curves in a fracture are function of saturations and viscosities.  

2. The novel model was validated with the previous studies. The isoperms of water and gas phases 

are function of their own saturations, however, oil isoperms are not only function of oil saturation 

but had significant curvature. 

3. The study illustrated that the trend and curvature of oil isoperms varied with oil viscosity. The 

curvatures of oil isoperms increase with the increase of the viscosity of oil phase. Gas saturation 

decreases oil relative permeability, however, the viscosity ratio increases it. 

Conflicts of Interest 

The author(s) declare that they have no conflicting interests. 

Nomenclature 

As    = area of the fracture 

Aw   = area of water bed in the fracture 

Ag   = area of gas bed in the fracture 

Ao    = area of oil bed in the fracture 

C1 = integration constant 

C11 = integration constant 

C12 = integration constant 

C13 = integration constant 

C14 = integration constant 

C15 = integration constant 

𝐶1
𝑤𝑏 = integration constant of water phase at the bottom 

𝐶1
𝑔

 = integration constant of gas phase 

𝐶1
𝑤𝑡 = integration constant of water phase at the top 

𝐶1
𝑜𝑏 = integration constant of oil phase at the bottom 

𝐶1
𝑜𝑏 = integration constant of oil phase at the top 

hg  = thickness of gas bed in the fracture 

hw1   = thickness of water bed in the fracture 

ho1   = thickness of oil bed in the fracture 

krw   = relative permeability to water phase 

krg   = relative permeability to gas phase 

kro   = relative permeability to oil phase 

ke   = absolute permeability 

L  = fracture length 

P1   = pressure in z=0 

P2  = pressure in z=L 

Sw   = water saturation 

Sg   = gas saturation 



8 

So   = oil saturation 

𝑣𝑧
𝑤𝑏 = velocity of water phase at the bottom 

𝑣𝑧
𝑤𝑡 = velocity of water phase at the top 

𝑣𝑧
𝑜𝑏 = velocity of oil phase at the bottom 

𝑣𝑧
𝑜𝑡 = velocity of oil phase at the top 

𝑣𝑧
𝑔

 = velocity of gas phase 

𝑉𝑧
𝑤 = average velocity of water phase 

𝑉𝑧
𝑜 = average velocity of water phase 

𝑉𝑧
𝑔

 = average velocity of gas phase 

W   = width of fracture 

Δx    = difference operator 

µg   = viscosity of gas 

µo    = viscosity of oil 

µw    = viscosity of water 

𝜏𝑥𝑧
𝑤𝑏 = shear stress for the water phase at the bottom 

𝜏𝑥𝑧
𝑤𝑡 = shear stress for the water phase at the top 

𝜏𝑥𝑧
𝑜𝑏 = shear stress for the oil phase at the bottom  

𝜏𝑥𝑧
𝑜𝑡 = shear stress for the oil phase at the top 

𝜏𝑥𝑧
𝑔

 = shear stress for the gas phase 

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Paper presented at International Petroleum Conference and Exhibition of Mexico, Veracruz, Mexico, 10-13 

October. SPE-28701-MS. 

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SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, 22-25 April. SPE-20204-MS. 

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Water-oil-gas Mixtures Through Water-wet Sandstones. Paper presented at SPE California Regional Meeting, 

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Petroleum Technology 12(4): 53-61. 

Appendix A 

The fracture geometry and flow configuration for the proposed model is shown in Figure 1. For the 

proposed model, the modeled fracture comprises two smooth and planar fracture walls. Within the 

fracture, oil-gas-water three fluids are flowing in a steady-state condition. The wettability of fracture 

walls sequences from water>oil>gas. Gas phase flows in the center of the fracture. Water phase flow 

occurs close to fracture surface. Oil phase flow occurs in-between water phase and gas phase. 

  Applying the shell momentum balance (Bird et al. 2002; Chima et al. 2010; Chima and Geiger 2012) in 

the fracture, the following equation can be written as, 

∂𝜏𝑥𝑧

∂𝑥
=

𝑃1−𝑃2

𝐿
,………………..………………………………………………………………………..(A-1) 



11 

  With the integration of the Eq. A-1 for the five regions, the shear stress for water phase at the bottom, 

oil phase at the bottom, gas phase, oil phase at the top and water phase at the top can be written as, 

𝜏𝑥𝑧
𝑤𝑏 =

𝑃1−𝑃2

𝐿
𝑥 + 𝐶1

𝑤𝑏,…………………………………………………..………..…..……...….……(A-2) 

𝜏𝑥𝑧
𝑜𝑏 =

𝑃1−𝑃2

𝐿
𝑥 + 𝐶1

𝑜𝑏,…………………….…………………………….………..….…………….….(A-3) 

𝜏𝑥𝑧
𝑔

=
𝑃1−𝑃2

𝐿
𝑥 + 𝐶1

𝑔
,…………..………………………………………...…………..………....……..(A-4) 

𝜏𝑥𝑧
𝑜𝑡 =

𝑃1−𝑃2

𝐿
𝑥 + 𝐶1

𝑜𝑡,.……………………………………………………………….…….……..……(A-5) 

𝜏𝑥𝑧
𝑤𝑡 =

𝑃1−𝑃2

𝐿
𝑥 + 𝐶1

𝑤𝑡,...……….………………………………………………….…………………..(A-6) 

  Based on the boundary conditions (Chima et al. 2010; Chima and Geiger 2012; Lei et al. 2014) in the 

fracture, the following equations can be written as, 

Condition 1: at 𝑥 = 0, 𝜏𝑥𝑧
𝑤𝑏 = 𝜏𝑥𝑧

𝑜𝑏; 𝑣𝑧
𝑤𝑏 = 𝑣𝑧

𝑜𝑏 

Condition 2: at 𝑥 = ℎ𝑜1, 𝜏𝑥𝑧
𝑔

= 𝜏𝑥𝑧
𝑜𝑏; 𝑣𝑧

𝑔
= 𝑣𝑧

𝑜𝑏 

Condition 3: at 𝑥 = ℎ𝑜1 + ℎ𝑔, 𝜏𝑥𝑧
𝑔

= 𝜏𝑥𝑧
𝑜𝑡; 𝑣𝑧

𝑔
= 𝑣𝑧

𝑜𝑡 

Condition 4: at 𝑥 = 2ℎ𝑜1 + ℎ𝑔, 𝜏𝑥𝑧
𝑤𝑡 = 𝜏𝑥𝑧

𝑜𝑡 , 𝑣𝑧
𝑤𝑡 = 𝑣𝑧

𝑜𝑡 

Condition 5: at 𝑥 = −ℎ𝑤1, 𝑣𝑧
𝑤𝑏 = 0 

Condition 6: at 𝑥 = ℎ𝑤1 + 2ℎ𝑜1 + ℎ𝑔, 𝑣𝑧
𝑤𝑡 = 0 

  With the boundary conditions 1-4, we can find that 𝐶1
𝑔

= 𝐶1
𝑤𝑏 = 𝐶1

𝑜𝑏 = 𝐶1
𝑜𝑡 = 𝐶1

𝑤𝑡 = 𝐶1. When 

Newton's law of viscosity is substituted into Eqs. A-2 through A-6, the velocity equations for water phase 

at the bottom, oil phase at the bottom, gas phase, oil phase at the top and water phase at the top can be 

written as, 

𝑣𝑧
𝑤𝑏 = − (

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑤
+

𝐶1

𝜇𝑤
𝑥) + 𝐶11,………………………………………….………........................(A-7) 

𝑣𝑧
𝑜𝑏 = − (

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑜
+

𝐶1

𝜇𝑜
𝑥) + 𝐶12,………………………………………….………..…….....……...(A-8) 

𝑣𝑧
𝑔

= − (
𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑔
+

𝐶1

𝜇𝑔
𝑥) + 𝐶13,……………..………………………………..…….……….....….(A-9) 

𝑣𝑧
𝑜𝑡 = − (

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑜
+

𝐶1

𝜇𝑜
𝑥) + 𝐶14,……………………………………………..…….….............….(A-10) 

𝑣𝑧
𝑤𝑡 = − (

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑤
+

𝐶1

𝜇𝑤
𝑥) + 𝐶15,……………………………………………….…..………..…...(A-11) 

  With the boundary conditions 1-5, the velocity profiles for the oil-gas-water three-phase can be written 

as, 

𝑣𝑧
𝑤𝑏 = −

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑤
+

𝑃1−𝑃2

𝐿

ℎ𝑔+2ℎ𝑜1

2

𝑥

𝜇𝑤
+

𝑃1−𝑃2

𝐿

ℎ𝑤1
2 +ℎ𝑤1ℎ𝑔+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
,………………............................(A-12) 

𝑣𝑧
𝑜𝑏 = −

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑜
+

𝑃1−𝑃2

𝐿

ℎ𝑔+2ℎ𝑜1

2

𝑥

𝜇𝑜
+

𝑃1−𝑃2

𝐿

ℎ𝑤1
2 +ℎ𝑤1ℎ𝑔+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
,……………..……......…………..(A-13) 



12 

𝑣𝑧
𝑔

= −
𝑥2

2𝜇𝑔

𝑃1−𝑃2

𝐿
+

ℎ𝑔+2ℎ𝑜1

2

𝑃1−𝑃2

𝐿

𝑥

𝜇𝑔
+

𝑃1−𝑃2

𝐿
 (

ℎ𝑤1
2 +ℎ𝑔ℎ𝑤1+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
+

ℎ𝑔ℎ𝑜1+ℎ𝑜1
2

2𝜇𝑜
−

ℎ𝑔ℎ𝑜1+ℎ𝑜1
2

2𝜇𝑔
),…………..……(A-14) 

𝑣𝑧
𝑜𝑡 = −

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑜
+

𝑃1−𝑃2

𝐿

ℎ𝑔+2ℎ𝑜1

2

𝑥

𝜇𝑜
+

𝑃1−𝑃2

𝐿

ℎ𝑤1
2 +ℎ𝑔ℎ𝑤1+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
,…………………………………..(A-15) 

𝑣𝑧
𝑤𝑡 = −

𝑃1−𝑃2

𝐿

𝑥2

2𝜇𝑤
+

𝑃1−𝑃2

𝐿

ℎ𝑔+2ℎ𝑜1

2

𝑥

𝜇𝑤
+

𝑃1−𝑃2

𝐿

ℎ𝑤1
2 +ℎ𝑔ℎ𝑤1+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
,………………….…………......(A-16) 

  The average velocities for each phase are calculated as (Chima et al. 2010; Chima and Geiger 2012; 

Lei et al. 2014), 

𝑉𝑧
𝑤 = 𝑉𝑧

𝑤𝑏 = 𝑉𝑧
𝑤𝑡 =

1

ℎ𝑤1
∫ 𝑣𝑧

𝑤𝑏𝜕𝑥
0

−ℎ𝑤1
=

𝑃1−𝑃2

𝐿

4ℎ𝑤1
2 +3ℎ𝑤1ℎ𝑔+6ℎ𝑤1ℎ𝑜1

12𝜇𝑤
,………..……..….………....(A-17) 

𝑉𝑧
𝑜 = 𝑉𝑧

𝑜𝑏 = 𝑉𝑧
𝑜𝑡 =

1

ℎ𝑜1
∫ 𝑣𝑧

𝑜𝑏 ∂𝑥
ℎ𝑜1

0
=

𝑃1−𝑃2

𝐿
(

ℎ𝑤1
2 +ℎ𝑤1ℎ𝑔+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
+

3ℎ𝑔ℎ𝑜1+4ℎ𝑜1
2

12𝜇𝑜
),…….………….(A-18) 

𝑉𝑧
𝑔

=
1

ℎ𝑔
∫ 𝑣𝑧

𝑔
∂𝑥

ℎ𝑜1+ℎ𝑔

ℎ𝑜1
=

𝑃1−𝑃2

𝐿
(

ℎ𝑔
2

12𝜇𝑔
+

ℎ𝑤1
2 +ℎ𝑔ℎ𝑤1+2ℎ𝑤1ℎ𝑜1

2𝜇𝑤
+

ℎ𝑔ℎ𝑜1+ℎ𝑜1
2

2𝜇𝑜
),……….……..………..(A-19) 

  With the Darcy law, the following equations for oil-gas-water three-phase can be written as, 

𝑘𝑟𝑜𝑘𝑒𝐴𝑠

𝜇𝑜

𝛥𝑃𝑜

𝐿
= 𝐴𝑜𝑉𝑧

𝑜,………………………………..……………………..……………..…........….(A-20) 

𝑘𝑟𝑔𝑘𝑒𝐴𝑠

𝜇𝑔

𝛥𝑃𝑔

𝐿
= 𝐴𝑔𝑉𝑧

𝑔
,………………………………..……………………….…………..…...…..…(A-21) 

𝑘𝑟𝑤𝑘𝑒𝐴𝑠

𝜇𝑤

𝛥𝑃𝑤

𝐿
= 𝐴𝑤𝑉𝑧

𝑤,………………………………………………………………………......…...(A-22) 

  Where (see Appendix B for details), 

𝛥𝑃𝑜 =
𝑃1−𝑃2

𝑆𝑜
; 𝛥𝑃𝑔 =

𝑃1−𝑃2

𝑆𝑔
; 𝛥𝑃𝑤 =

𝑃1−𝑃2

𝑆𝑤
 

𝐴𝑜 = 2𝑊ℎ𝑜1;ℎ𝑜1 = 𝑆𝑜𝐻 = (1 − 𝑆𝑔 − 𝑆𝑤)𝐻   

𝐴𝑔 = 𝑊ℎ𝑔; ℎ𝑔 = 𝑆𝑔𝐻; 𝐻 = ℎ𝑔 + 2ℎ𝑤1 + 2ℎ𝑜1 

𝐴𝑠 = 𝑊𝐻 ;𝐴𝑤 = 2𝑊ℎ𝑤1 ;ℎ𝑤1 =
𝑆𝑤

2
𝐻 ;𝑘𝑒 =

𝐻2

12
 (Snow 1965; Witherspoon et al. 1980; Golf-Racht 

1982). 

  Rearranging Eqs. A20 to A22 give the relative permeability for oil-gas-water three-phase, 

𝑘𝑟𝑜 = 𝑆𝑜
2 (

𝜇𝑜

𝜇𝑤
(3𝑆𝑤 −

3

2
𝑆𝑤

2 ) +
3

2
𝑆𝑔𝑆𝑜 + 𝑆𝑜

2),…………………………………………………...….(A-23) 

𝑘𝑟𝑔 = 𝑆𝑔
2 (𝑆𝑔

2 +
𝜇𝑔

𝜇𝑤
(3𝑆𝑤 −

3

2
𝑆𝑤

2 ) +
𝜇𝑔

𝜇𝑜
(3𝑆𝑔𝑆𝑜 +

3

2
𝑆𝑜

2)),..……………………......…………....…(A-24) 

𝑘𝑟𝑤 =
𝑆𝑤

3

2
(3 − 𝑆𝑤),………..…….…………………………………….…………..........……….....(A-25) 

  Eqs. A-23 through A-25 are the proposed equations to estimate oil-gas-water relative permeability 

curves in fractures. 



13 

Appendix B 

For the fracture geometry and flow configuration shown in Figure 1, the following equations can be 

obtained as, 

ℎ𝑜1 = 𝑆𝑜
𝐻

2
,………………………………………………..………………..…..……..……....…...…(B-1) 

ℎ𝑔 = 𝑆𝑔𝐻,…………………………………………………………………….………..………….…(B-2) 

𝑆𝑤 =
𝑊𝐿(2ℎ𝑤1)

𝑊𝐿𝐻
=

2ℎ𝑤1

𝐻
,……………………………………………………………………………….(B-3) 

  The equalities 𝛥𝑃𝑜, 𝛥𝑃𝑔 and 𝛥𝑃𝑤 are given below (Chima et al. 2010; Chima and Geiger 2012; Lei 

et al. 2014), 

𝛥𝑃𝑜 =
𝑃1−𝑃2

𝑆𝑜
=

𝑃1−𝑃2

1−𝑆𝑤−𝑆𝑔
,……………………………………………………………..……………….(B-4) 

𝛥𝑃𝑔 =
𝑃1−𝑃2

𝑆𝑔
,…………………………………………………..…………...……….……….………..(B-5) 

𝛥𝑃𝑤 = 𝑃𝑤1 − 𝑃𝑤2 =
𝑃1−𝑃2

𝑆𝑤
,………………………….………………………..……………….….….(B-6) 

 

Gang Lei is a Postdoctoral fellow at the Peking University. His research interests include theory and 

laboratory studies of the fundamental properties and behavior of fractured reservoirs, numerical modeling 

of fractured horizontal wells, and enhanced oil recovery of tight sandstone reservoirs. Lei holds a PhD 

degree from the China University of Petroleum, Beijing. He is a member of SPE. 

 

Cai Wang is now a postdoctoral fellow in Peking University, Beijing, China. His research interests mainly 

contain contamination of shale gas development on underground water, productivity analysis and 

numerical modeling of shale gas and tight gas reservoirs. Wang hold a doctoral degree and a bachelor 

degree in China University of Geosciences (Beijing). 

 

Yuan Tian is now an undergraduate student at the School of Civil Engineering of Tongji University. Her 

research interests mainly contain mathematical modeling and fluid flow law in porous media. 

 

Limin Yang is associate professor of the Department of Mathematics at the China University of 

Petroleum, Beijing. Her research interests include multi-phase percolation theory in unconventional 

reservoirs.