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Improved models to predict gas–water relative permeability in fractures and porous media
Journal of Natural Gas Science and Engineering 19 (2014) 190e201
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Improved models to predict gasewater relative permeability in fractures and porous media Yuansheng Li a, *, Xiangfang Li a, Sainan Teng b, Darong Xu a a b
MOE Key Laboratory of Petroleum Engineering, China University of Petroleum (Beijing), ChangPing, Beijing 102249, China Shanghai Off Shore Petroleum Bureau Research Institute, Shanghai 200120, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 March 2014 Received in revised form 4 May 2014 Accepted 5 May 2014 Available online
Fractures and pores are major flow channels of fluid in low permeability and unconventional reservoirs. So, it is very important to accurately predict multiphase flow in fractures and porous media. In this paper, the empirical, semi-empirical, and theoretical relative permeability models in fractures and porous media are investigated. Among these models, four models have been usually suggested to represent two-phase flow behavior: X-model, Corey or BrookseCorey model, viscous-coupling model and Chima’s model. However, these models have not considered some factors that affect relative permeability significantly. For example, X-model doesn’t reflect nonlinear characteristic, Corey or Brooks eCorey model neglects gaseliquid interaction, and viscous-coupling model and Chima’s model don’t consider irreducible phase. According to Chima’s method, when the surface geometry of fracture is assumed to be two ideal parallel planes and that of fracture or pore is assumed to be a pipe, new analytical relative permeability models of fractures and porous media considering irreducible water have been proposed respectively, based on cubic law for flow in rectangular fractures, Poiseuille law for flow in pores; and the momentum balance and Newton’s law of viscosity both in fractures and pores. Furthermore, we also improved these relative permeability models by considering the influence of tortuosity based on BrookseCorey model. The results show that the relative permeability models are nonlinear functions of mobile water, viscosity, irreducible water and the fracture or pore-size distribution index. The gas and water relative permeabilities are different in fractures and pores, and the gas phase relative permeability is more sensitive to the surface geometry of flow channels. And compared with model of fractures, the gas phase relative permeability deviates by more than 50% while the water phase relative permeability deviates by less than 20% in pores. What’s more, the deviation will become smaller with the increase of irreducible water saturation. The proposed model with irreducible water saturation of fractures is validated with experimental data from literature and good agreements between experimental data and those evaluated by the proposed model. And the model of porous media is validated by BrookseCorey model and there is consistency in the forecasting result of two models when there is no irreducible water. The relative permeability models proposed in this work will be useful to professionals involved in modeling well performance, and gas production forecasting in low permeability and unconventional reservoirs. Ó 2014 Elsevier B.V. All rights reserved.
Keywords: Irreducible water Surface geometry Relative permeability Multi-phase Fractures Porous media Tortuosity Capillary pressure
1. Introduction Two-phase flows in fractures and porous media are of great importance in petroleum recovery. And the relative permeability functions are the most common and useful methods for understanding flow mechanisms and predicting petroleum recovery in
* Corresponding author. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.jngse.2014.05.006 1875-5100/Ó 2014 Elsevier B.V. All rights reserved.
fractures and porous media. The surface geometry of flow channel, the corresponding flow structure, residual saturation, and phase interference are major factors that control multiphase flow behavior (Pruess and Tsang, 1990; Nicholl et al., 2000). So, Studies of multiphase flow need not only to consider single-phase flow properties, but also to take account for complex interaction between phases and the surface geometry. Many empirical and theoretical models of gasewater relative permeability have been proposed to describe flow in fractures and
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
pores. The earliest and simplest model was linear model which could be called Romm’s model or X model (Romm, 1966) that is Krw ¼ Sw, Krg ¼ Sg and Krw þ Krg ¼ 1, which assumed no phase interference existence between two phases. But many numerical studies (Murphy and Thomson, 1993; Rossen and Kumar, 1992; Mendoza and Sudicky, 1991; Pyrak-Nolte et al., 1992; Diomampo et al., 2001) showed that the sum of gas and water relative permeability is less than 1, and X-model is not suitable to describe relative permeability as functions of saturation. Based on observation and from the finding of burdine concerning the nature of the tortuosity-saturation function, Corey (1954) proposed a semiempirical relative permeability model in classical porous media as follows:
Krw ¼
Krg ¼
S*w
1 S*w
2
1 S*w
2
(2)
Sw Swc 1 Swc Sgc
(3)
Brooks (1966) proposed a semi-empirical model which has been widely used for modeling two-phase relative permeability. The BrookseCorey relative permeability functions are given as follows:
Krw ¼
Krg ¼
S*w
ð2þ3lÞ=l
1 S*w
2
ð2þlÞ=l 1 S*w
(4)
(5)
where, l is pore-size distribution index. The media with a wide range of pore sizes distribution should have small values of l, while media with uniform pore sizes could have value of l close to infinity. The value of l equals 2 for typical porous media, which makes BrookseCorey model become Corey model (Eqs. (1) and (2)). Because the flow in rough-walled fracture may be approximated to that in porous media, the value of l in rough-walled fracture may approach to that in porous media, while the value of l should approach infinity in the case of fractured media. Therefore, Eqs. (4) and (5) are modified with l / N, which leads to the extreme behavior of BrookseCorey model for fractures:
Krw ¼
Krg ¼
S*w
3
1 S*w
(6) 3
(7)
Fourar and Lenormand (1998) proposed viscous-coupling model which derived from integrating Stokes’ equation for each stratum:
Krw ¼
Krw ¼ S2w
S2g
(1)
where, S*w is the normalized water saturation defined by:
S*w ¼
Comparison with BrookseCorey model, viscous-coupling model is an analytical model and represents the influence of viscosity. Then, Chima and Geiger (2012) also proposed an analytical model of relative permeability, based on shell momentum balance, Newton’s law of viscosity and cubic law for flow in fractures. But this model may have coefficients of Krw and Krg that are 1/6 and 1/ 12 which make Krw or Krg unequal to 1 or 0 when Sw ¼ 0, and we rewrite both these as 1/2 as follows:
Krg ¼
4
S2w ð3 Sw Þ 2
3 Krg ¼ ð1 Sw Þ þ mr Sw ð1 Sw Þð2 Sw Þ 2
(8)
(9)
where mr is viscosity ratio, mr ¼ mg/mw. Eqs. (9) and (11) show that relative permeability of nonwetting fluid depends on viscosity ratio mr. When mr 1, the term related to mr on the right-hand-side of Eqs. (9) and (11) plays less significant role on relative permeability.
191
2S2w þ 3Sg Sw
(10)
2 2S2g þ 3mr S2w þ 6mr Sg Sw
!
2
(11)
What’s more, in order to get Eqs. (10) and (11), the gas and water pressure drops are defined DPw ¼ DP/Sw and DPg ¼ DP/Sg which should be DP ¼ DPw ¼ DPg (Chen et al., 2004; Pieters and Graves, 1994; Fourar and Lenormand, 1998) when capillary pressure is neglected. Then, Chima’s model becomes viscous-coupling model after these are modified. The flow characteristics for relative permeability of fractures or porous media were different in experimental observation and models. Some fracture models were suggested to represent the flows in fractures as flows in two ideal parallel planes (Fourar et al., 1993; Chen et al., 2004; Chima and Geiger, 2012). Another model thought that flow in fractures was treated as flow in pipes based on experimental observation (Fourar and Bories, 1995), and flow in porous media also could be considered as flow in pipes (Yang and Wei, 2004). In this study, new analytical equations that consider the influence of irreducible water, the surface geometry and the corresponding flow structure are proposed to predict gasewater relative permeability curves in fractures and porous media respectively, in which fracture is assumed to be two ideal parallel planes and fracture or pore is assumed to be a pipe. Furthermore, these relative permeability models are improved by considering the influence of tortuosity based on BrookseCorey model. Then, validate the proposed models with experimental data. Finally, study the characteristics of the relative permeability of different surface geometry of flow channels and irreducible water. 2. Model Both analytical equations that predict gasewater relative permeability in fractures and porous media could be obtained by following five key steps (Chima et al., 2010): 1 Perform momentum balance within fractures and pores to obtain differential equations, respectively; 2 Define boundary conditions to solve the previous differential equations; 3 Use Newton’s law of viscosity to define average velocities; 4 Use cubic law in fractures and Poiseuille law in pores to obtain the final set of equations related with saturation, respectively; 5 Use Darcy law for two phase flow in fractures and pores to obtain the final relative permeability. The models must satisfy the following assumptions: (1) Gas and water are Newtonian fluid. (2) Gas is compressible and water is slightly compressible, and both have constant properties. (3) No phase transformation occurs between gas and water.
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(4) Fracture walls are planar and impermeable, and flow occurs in an open fracture, and flow in fracture is laminar and in steady state. (5) The pore throat is treated as a pipe and the flow in a pipe takes into account of the gas core fluid; (6) The irreducible water is absorbed on the walls of fractures and pores, and distributes uniformly on the walls; (7) No transformation and friction occur between irreducible water and mobile water; (8) The pipes and fractures are oriented at horizontal, fluid gravity and buoyancy effects are neglected. According to Chima’s method (Chima and Geiger, 2012), applying momentum balance, cubic law for flow in rectangular fractures and Newton’s law of viscosity to the fracture configuration and the flow configuration shown in Fig. 1, leads to the following analytical gasewater relative permeability of fractures that considers irreducible water saturation. Part of detailed mathematical derivation is given in Appendix A.
Krw ¼
Krg ¼
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg 2 ð1 Swc Þ3
(12)
Sg
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3
2mw
!
makes a between 1 and N. When tortuosity is neglected, a equals to zero, and Eqs. (14) and (15) can be simplified into Eqs. (12) and (13).When d / N, Eqs. (14) and (15) change into the model for ideal fractures:
Krw ¼
Krg ¼
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc 2 1 Swc ð1 Swc Þ3
(16)
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
Sg
ð1 Swc Þ3 Sw Swc 1 1 Swc
!
2mw
(17) Based on the observation, the flow structures observed in a fracture show more similar to the structures observed in a pipe than to those expected for a porous media (Fourar and Bories, 1995), so we also treat flow in fractures or porous media as flow in pipes. Applying momentum balance, Poiseuille law in a pipe and Newton’s law of viscosity to the pipe configuration and the flow configuration shown in Fig. 2, then the gasewater relative permeability of fractures or porous media that considers irreducible water saturation is given as follows. The detailed mathematical derivation is given in Appendix B:
(13) If Swc ¼ 0, Eqs. (12) and (13) are the same as viscous-coupling model (1998). Tortuosity could be approximated by the relationship of capillary pressure and saturation. Considering the influence of tortuosity, an improved gasewater relative permeability of fractured media is proposed as follows. The detailed mathematical derivation is given in Appendix C.
Krw ¼
Krg ¼
Krw ¼ (14)
Krg ¼
Or
a ¼
Krg ¼
Sg ð1 Swc Þ3
Sg
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
(19)
!"
ðSw Swc Þ4
(20)
ð1 Swc Þ4
2þd # Sw Swc d 1 1 Swc 2mw ! 2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ Sw Swc a 1 1 Swc 2mw
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3
(18)
ð1 Swc Þ2
The improved gasewater relative permeability considering the influence of tortuosity on the assumption of pipe bundle is given in Appendix B.
2þd ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc d Krw ¼ 2 1 Swc ð1 Swc Þ3 ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc a Or Krw ¼ 2 1 Swc ð1 Swc Þ3
Sg
ðSw Swc Þ2
2þd
d
where, the fracture-size distribution index, d, which is constant for a given fracture system, is measured directly from best-fit line drawn through the data points. Taking into consideration of the influence of tortuosity, the value of d ranges from 0 to N, which
Krg ¼
Sg ð1 Swc Þ2
(15)
" 2mg ðSw Swc Þ þ mw Sg
mw
1
ðSw Swc Þ2
#
ð1 Swc Þ2 (21)
The influence of tortuosity is different for various pore-size distributions. And considering the influence of tortuosity, an improved gasewater relative permeability of fractured or porous
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
p0
pL
Krg ¼
Q
Sg ð1 Swc Þ
Or Krg ¼
193
2mg ðSw Swc Þ þ mw Sg
2
Sg
mw
"
1
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
"
Sw Swc 1 Swc
2þl # l
# Sw Swc b 1 1 Swc
(23)
Fig. 1. Fracture to be used in the mathematical model proposed. For fracture is taken as two parallel planes placed in horizontal with negligible gravity and buoyancy effects, ideally the wetting phase contacts the top and bottom fracture surfaces, and part of the wetting phase is adsorbed at fracture surfaces that is irreducible phase, while nonwetting phase flows in between.
b¼
2þl
l
where, the pore-size distribution index, l, which is constant for a given porous media system, is also measured directly from best-fit line drawn through the data points. Taking into consideration of the influence of tortuosity, the value of l varies from 0 to N, which makes b between 1 and N. When tortuosity is neglected, b equals to zero, and Eqs. (22) and (23) can be simplified into Eqs. (18) and (19). When the porous media is treated as the pipe bundle, b ¼ 2 (Brooks, 1966).
3. Model validation Fig. 2. Fractures or porous media to be used in the mathematical model proposed. For Fracture or pore is taken as a pipe placed horizontal with negligible gravity and buoyancy effects, ideally the wetting phase contacts pipe surfaces, and part of the wetting phase is adsorbed at pipe surfaces that is irreducible phase, while nonwetting phase flows in between.
media is proposed as follows. The detailed mathematical derivation is also given in Appendix C.
Krw ¼ Or
2þl ðSw Swc Þ2 Sw Swc l 1 Swc ð1 Swc Þ2
Krw
(22)
ðSw Swc Þ2 Sw Swc b ¼ 1 Swc ð1 Swc Þ2
Several experimental studies on two-phase flow in fractures have been performed. Romm (1966) used kerosene and water two-phase flow through an artificial fracture by using parallel plates. Persoff et al. (1991) and Persoff and Pruess (1995) also used air and water two-phase flow through rough-walled fractured rocks. Fourar et al. (1993) and Fourar and Bories (1995) studied airewater two-phase flow in a fracture consisting of two parallel glass plates (1 m 0.5 m) with an opening equal to 1 mm. Diomampo et al. (2001) performed experiments of nitrogen and water flow through smooth-walled artificial fractures. Other experimental datasets (e.g., McDonald, et al., 1991; Pieters and Graves, 1994; Speyer et al., 2007) use oil-water systems. The databases of Fourar and Bories (1995) and Diomampo et al. (2001) can be used to validate our model because their experiments are gasewater systems conducted in smooth-walled fractures. In their
1000
diomampo fourar & Bories X-model Viscous coupling model Brooks-Corey model(λ=2,Swc=0) Proposed model(α=1,Swc=0.081)
100
Kg/Kw
10
1
0.1
0.01
0.001
0.0001
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 3. Comparison of Fourar and Bories (1995) and Diomampo et al. (2001) laboratory measurement data (yellow and blue points) with X-model (green line), VeC model (yellow line), Corey model (brown line), proposed model in fractures with a ¼ 1, Swc ¼ 0.081 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
194
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
1000
diomampo fourar & Bories Brooks-Corey model(λ=2,Swc=0.081) Brooks-Corey model(λ=infinite,Swc=0.081) Proposed model(α=1,Swc=0.081)
100
10
Kg/Kw
1
0.1
0.01
0.001
0.0001 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 4. Comparison of Fourar and Bories (1995) and Diomampo et al. (2001) laboratory measurement data (yellow and blue points) with Corey model (yellow line) and proposed model with a ¼ 1, Swc ¼ 0.081 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
experiments, the fractures were initially saturated with water which was injected at a constant flow rate. Air injection was then started at a constant flow rate and increased stepwise. Saturation was calculated as total pixels of the liquid group over the sum of the air and liquid groups through a digital video. When the steady state was reached for each flow rate, the relative permeability was measured. In the experiments, the measured irreducible water saturation was 0.081. And, the smooth-walled fractures could be considered as idea fractures, so d approaches to infinity and a equals to 1 in this experiment. Fig. 3 compares the experimentally measured relative permeability ratio curves for air and water (blue and yellow points) with relative permeability ratio curves estimated from X-model, VeC model, BrookseCorey model, and proposed model (a ¼ 1, Swc ¼ 0.081). It is obvious that proposed model agrees with experimental data better than other models.
1 Krw Krg Krw(model) Krg(model)
0.9 0.8 0.7
4. Discussion
Kr
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
Considering irreducible water, Fig. 4 compares the experimentally measured relative permeability ratio curves for air and water (blue and yellow points) with the relative permeability ratio curves estimated from BrookseCorey model (l ¼ 2, Swc ¼ 0.081), Brookse Corey model (l ¼ infinite, Swc ¼ 0.081)and proposed model ( a ¼ 1, Swc ¼ 0.081, red line), It is apparent that the proposed model matches the experimental data better than other models. What’s more, Fig. 5 compares Diomampo et al. (2001) laboratory measured relative permeability with proposed model (a ¼ 1, Swc ¼ 0.081). And it shows that the proposed model agrees with the experimental data very well. To validate Eqs. (20) and (21) in porous media, we compared the proposed model with Corey model without irreducible water (Eqs. (1) and (2)), the water phase relative permeabilities of the two models are the same, but the gas phase relative permeability of proposed model has an extra part that reflects the interaction of gas and liquid comparing with Eq. (2). Fig. 6 compares proposed model in porous media with BrookseCorey model (l ¼ 2, Swc ¼ 0) and BrookseCorey model (l ¼ infinite, Swc ¼ 0). And the curves of proposed model are just consistent with the two models. What’s more, the influence of tortuosity has significant effect on the relative permeability.
0.4
0.6
0.8
1
Sw Fig. 5. Comparison of Diomampo et al. (2001) laboratory measurement relative permeability (red and blue points) with proposed model with a ¼ 1, Swc ¼ 0.081. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In this paper, we assume that the surface geometry of flow channels is rectangle and pipe, and propose different relative permeability models as shown in Eqs. (12), (13), (18) and (19), respectively. Comparing with the curves of the two models without irreducible water shown in Fig. 7, we find the relative permeability curves are different between rectangle and pipe. And from pipe to rectangle, the gas phase relative permeability has larger deviation than that of water phase. As shown in Fig. 7, the deviation of the gas phase relative permeability between pipe and fracture is larger than 50%, but the deviation of the water phase relative permeability between pipe and fracture is smaller than 20%. That may mean that the surface geometry mainly impacts gas phase relative permeability. And the irreducible water saturation also influences the deviation of the relative permeability between the fracture and pipe shown in Fig. 8. The deviation of the relative
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
195
10000
Brooks-Corey model(λ=infinite,Swc=0) Brooks-Corey model(λ=2,Swc=0) Proposed model in porous media(β=2,Sw=0) Proposed model in porous media(β=1,Sw=0) Proposed model in porous media(β=0,Sw=0)
1000
Kg/Kw
100
10
1
0.1
0.01
0.001 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 6. Comparison of BrookseCorey model with and proposed model in porous media with b ¼ 0, 1, 2; Swc ¼ 0.
1 0.9
Pipe(Krg)
0.8
Pipe(Krw)
0.7
Fracture(Krg) Fracture(Krw)
Kr
0.6
50%
20%
0.5 0.4 0.3
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 7. Comparison relative permeability curves of fracture (red points) and pipe (blue points) without irreducible water. The dotted line of left is 50% deviation of the relative permeability of gas phase in fracture, and the dotted line of right is 20% deviation of the permeability of water phase in fractures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
1 0.9
Pipe(Krg),Swc=0.1
0.8
Pipe(Krw),Swc=0.1 Fracture(Krg),Swc=0.1
0.7
Fracture(Krw),Swc=0.1 Kr
0.6
Pipe(Krg),Swc=0.3
0.5
S0.1
0.4
S0.3
S0.5
Pipe(Krw),Swc=0.3 Fracture(Krg),Swc=0.3 Fracture(Krw),Swc=0.3
0.3
Pipe(Krg),Swc=0.5 0.2
Pipe(Krw),Swc=0.5
0.1
Fracture(Krg),Swc=0.5 Fracture(Krw),Swc=0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 8. The relative permeability curves of different irreducible water (Swc ¼ 0.1, 0.3, 0.5). And the proportion (Si) enclosed by water or gas relative permeability curves of fractures and pipes reflects deviation of relative permeability of different irreducible water and S0.1 > S0.3 > S0.5.
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Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
permeability becomes smaller with the increase of irreducible water saturation. The gas and water phase permeability is determined by relative permeability and effective permeability. For the same effective permeability, gas phase relative permeability may be more sensitive to the surface geometry than that of water phase as shown in Figs. 7 and 8. So, it’s very important to choose the appropriate relative permeability model of gas phase. 5. Conclusion The following key conclusions can be drawn from this study: 1. Gasewater relative permeability models are investigated in this study. Among these, four models have been suggested to approximate the two-phase flow behavior in the fracture and porous media: the X-model, Corey or BrookseCorey models, viscous-coupling model and Chima’s model. However, gase liquid interaction, the irreducible water (wetting phase), the surface geometry and the corresponding flow structure in porous media are not all considered in these models. So, considering irreducible water, the surface geometry and the corresponding flow structure, and gaseliquid interaction, according to Chima’s method, the new analytical relative permeability models are developed in fractures and porous media, respectively. What’s more, the proposed relative permeability model of fractures has been improved by considering the influence of tortuosity. The result shows that the relative permeability models are functions of mobile water, viscosity, irreducible water and the fracture or pore-size distribution index. Furthermore, we study the characteristic of relative permeability curves in different surface geometry and corresponding flow structure in fractures and porous media. 2. The proposed relative permeability model with irreducible water saturation of fractures is validated with experimental data and shows superior agreement compared with the X-model, Corey or BrookseCorey models, and viscous-coupling model. 3. The proposed relative permeability model considering the influence of tortuosity of fracture or porous media has also been studied. This model is validated with Corey or BrookseCorey models. The expressions of the two models without irreducible water are very similar and the curves of proposed model are in consistency with BrookseCorey models (l ¼ 2) and Brookse Corey models (l ¼ infinite). What’s more, the tortuosity has significant effects on the relative permeability in fractured media or porous media. 4. Due to the influence of surface geometry, the relative permeability models are very different between parallel planes and pipes, and the gas phase relative permeability is more sensitive to the surface geometry than that of water phase. So, the optimized model of gas phase relative permeability is more important than that of water phase for field development. Because of the influence of irreducible water, the deviation of the relative permeability between the fracture and pore becomes smaller with the increase of irreducible water saturation. Acknowledgments This research was supported by National Natural Science Foundation Project (U1262113) and Science Foundation of China University of Petroleum, Beijing (YJRC-2013-37). We also recognize the support of MOE Key Laboratory of Petroleum Engineering in China University of Petroleum (Beijing).
Nomenclature A Aw Ag AS C1
area of flow channel area of water bed in flow channel area of gas bed in flow channel area of moved fluid integration constant
CWM 1 C1g
integration constant 1 of mobile water phase integration constant 1 of gas phase
CWM 2
integration constant 2 of mobile water phase
C2g H D hg hw hwc KrW Krg KeW Keg Kabs L P0 PL pb pc Q Qw Qg Sw Swm Swc Sg Sgc
integration constant 2 of gas phase total thickness of the fracture diameter of the pore thickness of gas bed in flow channel thickness of water bed thickness of irreducible water bed relative permeability to water relative permeability to gas effective permeability to water effective permeability to gas absolute permeability fracture or pipe length pressure on flow cross section in z ¼ 0 pressure on flow cross section in z ¼ L bubbling pressure capillary pressure flow rate water flow rate in z direction flow rate in z direction water saturation mobile water saturation irreducible water saturation gas saturation residual gas saturation
VzW Vzg
velocity of water phase velocity of gas phase
VzWM
average velocity of mobile water phase
Vzg
average velocity of gas phase volume of mobile water volume of irreducible water volume of gas width of fracture difference operator viscosity of gas viscosity of water force in the z direction on a unit area perpendicular to the r direction in the mobile water; force in the z direction on a unit area perpendicular to the r direction in the gas; the proportion (Si) enclosed by water or gas relative permeability curves of fracture and pipe when irreducible water saturation Swc ¼ i.
Vw Vwc Vg w Dr
mg mw sWM rz
sgrz Si
Appendix A Here we briefly describe the derivation of the proposed gase water relative permeability model when the fracture is taken as two parallel planes. According to Chima’s equation, perform momentum balance within a fracture, and then use Newton’s law of viscosity to define average velocities, finally gas phase and water phase rate could be calculated as follows (Chima and Geiger, 2012):
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
Apply the momentum balance (Bird, 2002) in pipes, as shown in Fig. 2:
For gas phase:
Krg Keg AS DPg ¼ mg L ! 2mw h2g þ 3mg h2w þ 6mg hw hg P PL ¼ Ag 0 2L 12mg mw
Qzg
ðL$2pr$srz Þr ðL$2pr$srz ÞrþDr þ 2pr$Dr$ðp0 pL Þ ¼ 0 (A1)
For water phase:
Qzw ¼
! 2h2w þ 3hw hg Krw Kew AS DPw P PL ¼ Aw 0 mw L 2mw L 12
¼ Swc
Vw wLðhw þ hwc Þ hw þ hwc ¼ ¼ wLðhw þ hwc Þ þ wLhg Vw þ Vg hw þ hwc þ hg
sgrz ¼
hwc H
g C P0 PL $r þ 1 2L r
(B3)
mw
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg 2 ð1 Swc Þ3
vVzWM
ð1 Swc Þ
C WM P0 PL $r þ 1 2L r
¼
# C1WM P0 PL $r vr 2Lmw r mw
C WM P PL $r 2 1 ln r þ C2WM VzWM ¼ 0 mw 4mw L
g
(A3)
For gas phase:
! 2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ 3
Z "
Z
Vz ¼
Substituting DPg, DPw, AS, Ag, Aw, Kew, Keg, hg and hw into above equations, then simplify: For water phase:
Sg
vVzWM ¼ vr
(B4)
For gas phase:
! 2mw h2g þ 3mg h2w þ 6mg hw hg Ag m g L P0 PL ¼ 2L 12mg mw Keg AS DPg
Krg ¼
(B2)
After this, use Newton’s law of viscosity to define velocities: For mobile water phase
! 2h2w þ 3hw hg Aw mw L P0 PL ¼ Kew AS DPw 2mw L 12
Krw ¼
C WM P0 PL $r þ 1 2L r
For gas phase:
For gas phase:
Krg
(B1)
Apply Eq. (B1) to both gas (g) phase and water (W) phase. Integration of Eq. (B1) for these regions gives: For mobile water phase
hw þ hwc H
So: hw ¼ (Sw Swc)H, hg ¼ SgH If capillary pressure is ignored, then the pressure is DPw ¼ DPg ¼ P0 PL The effective permeability in the fracture is: Kew ¼ Keg ¼ (hg þ hw)2/12 Translating Eqs. (A1) and (A2) into expressions of relative permeability, then simplify: For water phase:
Krw
vðr srz Þ P PL ¼ 0 r vr L
sWM ¼ rz
Vwc wLhwc hwc ¼ ¼ ¼ Vw þ Vg wLðhw þ hwc Þ þ wLhg hw þ hwc þ hg ¼
ðr srz ÞrþDr ðr srz Þr ðP PL Þ lim r ¼ 0 Dr/0 Dr L Above equation simplifies:
(A2)
Then, build the relationship between saturation and thickness of gas bed or water bed in the fracture geometry. For the fracture geometry and flow configuration shown in Fig. 1, we obtain:
Sw ¼
197
2mw (A4)
(B5)
Then, define boundary conditions to solve the previous differential equations. g Boundary Condition 1: at r ¼ hg/2; sWM ¼ srz rz Boundary Condition 2: at r ¼ 0; VzWM ¼ finite Boundary Condition 3: at r ¼ hg/2; VzWM ¼ Vzg Boundary Condition 4: at r ¼ (hg þ hw)/2; VzWM ¼ 0 Substituting boundary condition 1 into Eqs. (B2) and (B3), boundary condition 2 into Eqs. (B4) and (B5), we find:
C1WM ¼ C1g ¼ C1 ¼ 0
(B6)
Substituting boundary condition 3, 4 into Eqs. (B4) and (B5), we find:
Appendix B The modeled fracture or porous media geometry and flow configuration are shown in Fig. 2. The flow in fracture or pore is taken as flow in a pipe. Within the pipe, the water and gas are flowing in a steady-state condition. A part of wetting phase (water) adheres to walls of the pipe and doesn’t flow. The nonwetting phase (gas) flow is surrounded by wetting phase (water). The flow is in z direction and the diameter of the pipe is D.
! Cg P0 PL g $r 2 1 ln r þ C2 mg 4mg L
g C2
¼
! ! mg h2w þ 2hw hg mg þ mw h2g P0 PL $ 4mw mg L 4
C2WM ¼
hw þ hg 2 P0 PL $ 4mw L 2
(B7)
(B8)
Substituting Eqs. (B6), (B7) and (B8) into Eqs. (B4) and (B5), velocities in r direction are:
198
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
For mobile water phase:
hw þ hg 2 P PL P0 PL $r 2 þ $ VzWM ¼ 0 4mw L 4mw L 2
(B9)
For gas phase:
! P0 PL $r 2 4mg L ! ! mg h2w þ 2hw hg mg þ mw h2g P0 PL $ þ 4mw mg L 4
g
Vz ¼
(B10)
The average velocity for each phase is calculated as follows:
2 2 pLðD2 pL h2g D2 h2g Vw Sw ¼ ¼ ¼ 2 Vw þ Vg D2 pL D2 2 pL h2g h2g Vg Sg ¼ ¼ ¼ 2 2 Vw þ Vg D pL D2 2 2 2 w pLðD2 pL hg þh D2 hg þ hw Vwc 2 Swc ¼ ¼ ¼ 2 Vw þ Vg D2 pL D2 2 2 2 h w pL hg þh pL 2g hg þ hw h2g Vwm 2 Swm ¼ ¼ ¼ 2 Vw þ Vg D2 pL D2 h2w þ 2hg hw D2
Swm ¼ Z ðhw þhg Þ=2 h g =2
VzWM ¼
2prVzWM vr
Z ðhw þhg Þ=2 h g =2
¼
P0 PL mw L
2pr dr
h i 4 2 hw þ hg 2 hw þ hg h2g þ h4g i h 2 32 hw þ hg h2g (B11)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi hg ¼ So: hw ¼ ð Sw Sg D; ffi Swc þ Sg Sg ÞD; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hg þ hw ¼ Sw Swc þ Sg D If capillary pressure is ignored, so the pressure is DPw ¼ DPg ¼ P0 PL. Using Poiseuille law, the effective permeability in the pipe is Kew ¼ Keg ¼ (hg þ hw)2/32 Translating Eqs. (A1) and (A2) into expression of relative permeability, then simplify: For water phase:
Z Vzg ¼
hg =2 0
Z 0
¼
Krw ¼ 2prVz dr
hg =2
g
!
2mg h2w þ 4hw hg mg þ mw h2g
!
32mw
(B12)
The flow for each phase satisfies Darcy law, so the flow rate is calculated as follows: For mobile water phase:
Krw Kew AS DPw mw L h i h þ h 4 2h þ h 2 h2 þ h4 w g w g g g P0 PL i h ¼ Aw 2 mw L 32 hw þ hg h2g
Qzw ¼ Aw VzWM ¼
For gas phase:
Krg Keg AS DPg mg L ! ! 2mg h2w þ 4hw hg mg þ mw h2g P0 PL mg L 32mw
Krg
Krw ¼
2mg h2w þ 4hw hg mg þ mw h2g
!
32mw
ðSw Swc Þ2
(B15)
ð1 Swc Þ2 Sg
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
(B16)
where, Eqs. (B15) and (B16) are gasewater relative permeabilities which do not consider the influence of tortuosity. Corey (1954) proposed the relative permeability of porous media, taking into account of the influence of tortuosity:
(B14)
Then, build the relationship between saturation and thickness of gas bed or water bed in fracture or pore geometry. For the pipe geometry, and flow configuration shown in Fig. 2, we obtain:
!
Substituting DPg, DPw, AS, Ag, Aw, Kew, Keg, hg and hw into above equations, then simplify:
Krw ¼
Qzg ¼ Ag Vzg ¼
P0 PL mg L
Ag mg L ¼ Keg AS DPg
Krg ¼
(B13)
¼ Ag
h i 4 2 hw þ hg 2 hw þ hg h2g þ h4g i h 2 32 hw þ hg h2g
For gas phase:
2pr dr
P0 PL mg L
Aw mw L P0 PL mw L Kew AS DPw
ðSw Swc Þ2 ð1 Swc Þ2
Krg ¼
1
Sw Swc 1 Swc
ZSw
! Z1 ! . . 2 2 dSw pc dSw pc
0
2
(B17)
0
Z1
! Z1 ! . . dSw p2c dSw p2c
Sw
(B18)
0
1/p2c
And, the relationship between w Sw can be got from derivations and experiments (Corey, 1954), and if the porous media is considered as capillary tubes, the influence of tortuosity is:
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
ZSw
. dSw p2c
!, Z1
0
Z1
. dSw p2c
! ¼
0
. dSw p2c
!, Z1
Sw
. dSw p2c
ðSw Swc Þ2
(B19)
2
ð1 Swc Þ
¼ 1
ðSw Swc Þ2 ð1 Swc Þ2
Krg ¼
Sg
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
¼
1 pc
(C2)
2
If Eq. (C1) is substituted into Eqs. (B17) and (B18), a change of variable is made for Sw to S*w, and the constant pb is eliminated in the substitution process, the result is: *
ðSw Swc Þ2 ð1 Swc Þ2
(B21)
4
l
p2b
Krw ¼
ðSw Swc Þ4
S*w
for pb pc
2
(B20)
Comparison Eqs. (B15) with (B17), the influence of tortuosity is exactly Eqs. (B19) and (B20), so the gasewater relative permeability considering the influence of tortuosity is
ð1 Swc Þ
pb pc
Or
!
0
Krw ¼
h il
S*w ¼
199
ZSw
S*w
2 l
dS*w
!, Z1
S*w
2 l
! dS*w
0
0
2þl ðSw Swc Þ Sw Swc l ¼ 1 Swc ð1 Swc Þ2 2
# " ðSw Swc Þ2 1 ð1 Swc Þ2
(C3)
(B22) Krg ¼ Appendix C
¼
Eqs. (A3), (A4), (B15) and (B16) are derived on ideal conditions when the influence of tortuosity has not been considered. Though Eqs. (21) and (22) have taken into account of the influence of tortuosity, but the assumption of tube bundle is too ideal. So, for general fractured and porous media, the exponential coefficient may be different in various reservoirs, and is not always 2.0, just as Eqs. (B19) and (B20). What’s more, the tortuosity may be different between fractured media and porous media, and tortuosity could be approximated by the relationship of capillary pressure and saturation (Brooks, 1966). Based on large number of desorption on consolidated porous rocks, Brooks (1966) proposed the relationship of effective saturation and capillary pressure which could be considered the influence tortuosity:
ln S*w ¼ l ln pc þ l ln pb
pb pc
. dSw p2c
!, Z1
0
Z1 Sw
. dSw p2c
! ¼
b¼
!, Z1 0
2 l
dSw
!, Z1
0
0
. dSw p2c
S*w
. dSw p2c
! ¼
Z1 Sw
S*w
2 l
dSw
!, Z1
log paper. The definitions of the constants Swc and pb are implicit in the method of their determination. According to Eq. (C1), theoretical relationships for permeability as a function of saturation and capillary pressure could be derived by taking the antilog of Eq. (C1):
l
dS*w
!, Z1
S S 2þl w wc l 1 1 Swc
S*w
2 l
! dS*w
0
l
!
2
S*w
0
Sw Swc 1 Swc
0
2
2þl
l
¼
dSw
0
S*w
1
2
S*w
where, the pore-size distribution index, l, is measured directly from best-fit line drawn through the data points proposed by Brooks (1966). By experimentally determining S*w versus pc, the pore-size distribution index, l, can be determined (provided Swc is known) for predicting relative permeability as a function of capillary pressure and saturation. The pore-size distribution index, l may be different in various porous media. Eqs. (C3) and (C4) is exactly the same as BrookseCorey model, the exponential term of the right of Eq. (C3) is 2, which could also be obtained experimentally by Burdine (1953) and analytically by Wyllie and Gardner (1958). But the integration term of the right of Eq. (C3) is different due to the pore-size distribution index, l.
(C1)
ZSw
*
ZSw
Sw Swc 2 1 1 Swc
(C4)
This equation was found to be generally valid for the core analyzed. To calculate S*w, the residual saturation Swc, must be known or assumed, in above analysis, Swc was chosen such that data fit as closely as possible to a straight line when plotted on loge
ZSw
2 l
! dSw
2þl Sw Swc l 1 Swc "
¼
Sw Swc 1 1 Swc
(C5)
2þl # l
(C6)
Comparison Eqs. (B15) with (C3), the integration term of the right of Eq. (C3) could be considered as the influence of tortuosity. Therefore, the relative permeability of porous media considering the influence of tortuosity could be got by multiplying Eqs. (C5) and (C6) into the analytical equation Eqs. (B15) and (B16).
200
Krw
Or
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
2þl ðSw Swc Þ2 Sw Swc l ¼ 1 Swc ð1 Swc Þ2 Krw
Krg ¼
Or
(C7)
ðSw Swc Þ2 Sw Swc b ¼ 1 Swc ð1 Swc Þ2
2þd # Sw Swc d 1 1 Swc 2mw ! 2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ Sw Swc a 1 1 Swc 2mw
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
Sg ð1 Swc Þ3
Krg ¼
Sg ð1 Swc Þ3
!"
2mg ðSw Swc Þ þ mw Sg
1
(C8)
b¼
2þl
l
The fractured media could be considered as a special porous media and the derivation method of relative permeability of fractured media is the same as that of porous media. So, for given fracture-size distribution, the relative permeability also has the same expression on influence of tortuosity. Therefore, the relative permeability could also be got through analytical equations Eqs. (A3) and (A4) multiplying by Eqs. (C5) and (C6):
Krw ¼
!, ZSw . ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg 2 p dS w c 2 ð1 Swc Þ3 . dSw p2c
(C9)
Krg ¼
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3 Z1 Sw
. dSw p2c
!
2mw !, Z1 dSw
.
2þd
d
where, the fracture-size distribution index, d, is also measured directly from best-fit line drawn through the data points. Chen (2005) pointed according to Brooks and Corey’s reasoning, if the porous media is the case of fractured media, the value of l should approach infinity, so the value of d may also approach infinity. Therefore, Eqs. (C11) and (C12) are modified into the model for fractured media with d / N:
Krw ¼
Krg ¼
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc 3 2 1 Swc ð1 Swc Þ Sg
(C13)
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3 Sw Swc 1 1 Swc
!
2mw
References
!
0
Sg
a ¼
(C12)
(C14)
0
Z1
(C11)
"
2þl # Sw Swc l mw 1 Swc ð1 Swc Þ2 " # 2mg ðSw Swc Þ þ mw Sg Sg Sw Swc b 1 Or Krg ¼ mw 1 Swc ð1 Swc Þ2 Sg
Krg ¼
2þd ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc d 2 1 Swc ð1 Swc Þ3 2 ðSw Swc Þ 2ðSw Swc Þ þ 3Sg Sw Swc a Or Krw ¼ 2 1 Swc ð1 Swc Þ3 Krw ¼
! p2c
0
(C10) As a special porous media, the fractured media may have the same function of capillary pressure and saturation like Eq. (C2). For distinguishing fractured media from porous media, the fracturesize distribution index, d, is used to differ from the porous-size distribution index, l. So, we could get the relative permeability of fractured media considering the influence of tortuosity by substituting Eqs. (C7) and (C8) into Eqs. (C9) and (C10).
Bird, R.B., 2002. Transport phenomena. Appl. Mech. Rev. 55 (1), R1eR4. Brooks, R.H., 1966. Properties of porous media affecting fluid flow. J. Irrig. Drain. Div. Proc. Am. Soc. Civ. Eng. 92, 61e88. Burdine, N., 1953. Relative permeability calculations from pore size distribution data. J. Pet. Technol. 5 (03), 71e78. Chen, Z., 2005. LiquideGas Relative Permeabilities in Fractures: Effects of Flow Structures, Phase Transformation and Surface Roughness. Stanford University. Chen, C.Y., Horne, R.N., Fourar, M., 2004. Experimental study of liquidegas flow structure effects on relative permeabilities in a fracture. Water Resour. Res. 40 (8). Chima, A., Geiger, S., 2012, January. An analytical equation to predict gas/water relative permeability curves in fractures. In: SPE Latin America and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers. Chima, A., Chavez Iriarte, E.A., Carrillo, C., Himelda, Z., 2010, January. An equation to predict two-phase relative permeability curves in fractures. In: SPE Latin American and Caribbean Petroleum Engineering Conference. Society of Petroleum Engineers. Corey, A.T., 1954. The interrelation between gas and oil relative permeabilities. Prod. Mon. 19 (1), 38e41. Diomampo, G., Chen, C.Y., Li, K., Horne, R.N., 2001, August. Relative permeability through fractures. In: Proceedings of the 27th Workshop on Geothermal Reservoir Engineering, pp. 28e30. Fourar, M., Bories, S., 1995. Experimental study of air-water two-phase flow through a fracture (narrow channel). Int. J. Multiph. Flow. 21 (4), 621e637. Fourar, M., Lenormand, R., 1998. A Viscous Coupling Model for Relative Permeabilities in Fractures. SPE, 49006, pp. 27e30.
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201 Fourar, M., Bories, S., Lenormand, R., Persoff, P., 1993. Two-phase flow in smooth and rough fractures: measurement and correlation by porous-medium and pipe flow models. Water Resour. Res. 29 (11), 3699e3708. McDonald, A.E., Beckner, B.L., Chan, H.M., Jones, T.A., Wooten, S.O., 1991, January. Some important considerations in the simulation of naturally fractured reservoirs. In: Low Permeability Reservoirs Symposium. Society of Petroleum Engineers. Mendoza, C.A., Sudicky, E.A., 1991, November. Hierarchical scaling of constitutive relationships controlling multi-phase flow in fractured geologic media. In: NIPER/DOE Third International Conference on Reservoir Characterization Requirements for Different Stages of Development. Dept. of Energy, Tulsa, Okla. Murphy, J.R., Thomson, N.R., 1993. Two-phase flow in a variable aperture fracture. Water Resour. Res. 29 (10), 3453e3476. Nicholl, M.J., Rajaram, H., Glass, R.J., 2000. Factors controlling satiated relative permeability in a partially-saturated horizontal fracture. Geophys. Res. Lett. 27 (3), 393e396. Pieters, D.A., Graves, R.M., 1994, January. Fracture relative permeability: linear or non-linear function of saturation. In: International Petroleum Conference and Exhibition of Mexico. Society of Petroleum Engineers. Pruess, K., Tsang, Y.W., 1990. On two-phase relative permeability and capillary pressure of rough-walled rock fractures. Water Resour. Res. 26 (9), 1915e1926.
201
Pyrak-Nolte, L.J., Helgeson, D.J., Haley, G.M., Morris, J.W., 1992, January. Immiscible fluid flow in a fracture. In: The 33rd US Symposium on rock Mechanics (USRMS). American Rock Mechanics Association. Romm, E.S., 1966. Fluid Flow in Fractured Rocks. Nedra Publishing House, Moscow (English translation, W.R. Balke, Bartlesville, OK, 1972). Rossen, W.R., Kumar, A.T., 1992, January. Single-and two-phase flow in natural fractures. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Yang Shenglai, Wei Junzhi, 2004. Reservoir physics. Petroleum Industry Press. Speyer, N., Li, K., Horne, R., 2007. Experimental measurement of two-phase relative permeability in vertical fractures. In: Proceedings, Thirty-Second Workshop on Geothermal Reservoir Engineering. Stanford University, Stanford, California. Persoff, P., Pruess, K., Myer, L., 1991. Two-Phase Flow Visualization and Relative Permeability Measurement in Transparent Replicas of Rough-Walled Rock Fractures. Lawrence Berkeley Lab. Persoff, P., Pruess, K., 1995. Two-phase flow visualization and relative permeability measurement in natural rough-walled rock fractures. Water Resour. Res. 31 (5), 1175e1186. Wyllie, M.R.J., Gardner, G.H.F., 1958. The generalized KozenyeCarman equation. World Oil 146 (4), 210e213.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Improved models to predict gasewater relative permeability in fractures and porous media Yuansheng Li a, *, Xiangfang Li a, Sainan Teng b, Darong Xu a a b
MOE Key Laboratory of Petroleum Engineering, China University of Petroleum (Beijing), ChangPing, Beijing 102249, China Shanghai Off Shore Petroleum Bureau Research Institute, Shanghai 200120, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 March 2014 Received in revised form 4 May 2014 Accepted 5 May 2014 Available online
Fractures and pores are major flow channels of fluid in low permeability and unconventional reservoirs. So, it is very important to accurately predict multiphase flow in fractures and porous media. In this paper, the empirical, semi-empirical, and theoretical relative permeability models in fractures and porous media are investigated. Among these models, four models have been usually suggested to represent two-phase flow behavior: X-model, Corey or BrookseCorey model, viscous-coupling model and Chima’s model. However, these models have not considered some factors that affect relative permeability significantly. For example, X-model doesn’t reflect nonlinear characteristic, Corey or Brooks eCorey model neglects gaseliquid interaction, and viscous-coupling model and Chima’s model don’t consider irreducible phase. According to Chima’s method, when the surface geometry of fracture is assumed to be two ideal parallel planes and that of fracture or pore is assumed to be a pipe, new analytical relative permeability models of fractures and porous media considering irreducible water have been proposed respectively, based on cubic law for flow in rectangular fractures, Poiseuille law for flow in pores; and the momentum balance and Newton’s law of viscosity both in fractures and pores. Furthermore, we also improved these relative permeability models by considering the influence of tortuosity based on BrookseCorey model. The results show that the relative permeability models are nonlinear functions of mobile water, viscosity, irreducible water and the fracture or pore-size distribution index. The gas and water relative permeabilities are different in fractures and pores, and the gas phase relative permeability is more sensitive to the surface geometry of flow channels. And compared with model of fractures, the gas phase relative permeability deviates by more than 50% while the water phase relative permeability deviates by less than 20% in pores. What’s more, the deviation will become smaller with the increase of irreducible water saturation. The proposed model with irreducible water saturation of fractures is validated with experimental data from literature and good agreements between experimental data and those evaluated by the proposed model. And the model of porous media is validated by BrookseCorey model and there is consistency in the forecasting result of two models when there is no irreducible water. The relative permeability models proposed in this work will be useful to professionals involved in modeling well performance, and gas production forecasting in low permeability and unconventional reservoirs. Ó 2014 Elsevier B.V. All rights reserved.
Keywords: Irreducible water Surface geometry Relative permeability Multi-phase Fractures Porous media Tortuosity Capillary pressure
1. Introduction Two-phase flows in fractures and porous media are of great importance in petroleum recovery. And the relative permeability functions are the most common and useful methods for understanding flow mechanisms and predicting petroleum recovery in
* Corresponding author. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.jngse.2014.05.006 1875-5100/Ó 2014 Elsevier B.V. All rights reserved.
fractures and porous media. The surface geometry of flow channel, the corresponding flow structure, residual saturation, and phase interference are major factors that control multiphase flow behavior (Pruess and Tsang, 1990; Nicholl et al., 2000). So, Studies of multiphase flow need not only to consider single-phase flow properties, but also to take account for complex interaction between phases and the surface geometry. Many empirical and theoretical models of gasewater relative permeability have been proposed to describe flow in fractures and
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
pores. The earliest and simplest model was linear model which could be called Romm’s model or X model (Romm, 1966) that is Krw ¼ Sw, Krg ¼ Sg and Krw þ Krg ¼ 1, which assumed no phase interference existence between two phases. But many numerical studies (Murphy and Thomson, 1993; Rossen and Kumar, 1992; Mendoza and Sudicky, 1991; Pyrak-Nolte et al., 1992; Diomampo et al., 2001) showed that the sum of gas and water relative permeability is less than 1, and X-model is not suitable to describe relative permeability as functions of saturation. Based on observation and from the finding of burdine concerning the nature of the tortuosity-saturation function, Corey (1954) proposed a semiempirical relative permeability model in classical porous media as follows:
Krw ¼
Krg ¼
S*w
1 S*w
2
1 S*w
2
(2)
Sw Swc 1 Swc Sgc
(3)
Brooks (1966) proposed a semi-empirical model which has been widely used for modeling two-phase relative permeability. The BrookseCorey relative permeability functions are given as follows:
Krw ¼
Krg ¼
S*w
ð2þ3lÞ=l
1 S*w
2
ð2þlÞ=l 1 S*w
(4)
(5)
where, l is pore-size distribution index. The media with a wide range of pore sizes distribution should have small values of l, while media with uniform pore sizes could have value of l close to infinity. The value of l equals 2 for typical porous media, which makes BrookseCorey model become Corey model (Eqs. (1) and (2)). Because the flow in rough-walled fracture may be approximated to that in porous media, the value of l in rough-walled fracture may approach to that in porous media, while the value of l should approach infinity in the case of fractured media. Therefore, Eqs. (4) and (5) are modified with l / N, which leads to the extreme behavior of BrookseCorey model for fractures:
Krw ¼
Krg ¼
S*w
3
1 S*w
(6) 3
(7)
Fourar and Lenormand (1998) proposed viscous-coupling model which derived from integrating Stokes’ equation for each stratum:
Krw ¼
Krw ¼ S2w
S2g
(1)
where, S*w is the normalized water saturation defined by:
S*w ¼
Comparison with BrookseCorey model, viscous-coupling model is an analytical model and represents the influence of viscosity. Then, Chima and Geiger (2012) also proposed an analytical model of relative permeability, based on shell momentum balance, Newton’s law of viscosity and cubic law for flow in fractures. But this model may have coefficients of Krw and Krg that are 1/6 and 1/ 12 which make Krw or Krg unequal to 1 or 0 when Sw ¼ 0, and we rewrite both these as 1/2 as follows:
Krg ¼
4
S2w ð3 Sw Þ 2
3 Krg ¼ ð1 Sw Þ þ mr Sw ð1 Sw Þð2 Sw Þ 2
(8)
(9)
where mr is viscosity ratio, mr ¼ mg/mw. Eqs. (9) and (11) show that relative permeability of nonwetting fluid depends on viscosity ratio mr. When mr 1, the term related to mr on the right-hand-side of Eqs. (9) and (11) plays less significant role on relative permeability.
191
2S2w þ 3Sg Sw
(10)
2 2S2g þ 3mr S2w þ 6mr Sg Sw
!
2
(11)
What’s more, in order to get Eqs. (10) and (11), the gas and water pressure drops are defined DPw ¼ DP/Sw and DPg ¼ DP/Sg which should be DP ¼ DPw ¼ DPg (Chen et al., 2004; Pieters and Graves, 1994; Fourar and Lenormand, 1998) when capillary pressure is neglected. Then, Chima’s model becomes viscous-coupling model after these are modified. The flow characteristics for relative permeability of fractures or porous media were different in experimental observation and models. Some fracture models were suggested to represent the flows in fractures as flows in two ideal parallel planes (Fourar et al., 1993; Chen et al., 2004; Chima and Geiger, 2012). Another model thought that flow in fractures was treated as flow in pipes based on experimental observation (Fourar and Bories, 1995), and flow in porous media also could be considered as flow in pipes (Yang and Wei, 2004). In this study, new analytical equations that consider the influence of irreducible water, the surface geometry and the corresponding flow structure are proposed to predict gasewater relative permeability curves in fractures and porous media respectively, in which fracture is assumed to be two ideal parallel planes and fracture or pore is assumed to be a pipe. Furthermore, these relative permeability models are improved by considering the influence of tortuosity based on BrookseCorey model. Then, validate the proposed models with experimental data. Finally, study the characteristics of the relative permeability of different surface geometry of flow channels and irreducible water. 2. Model Both analytical equations that predict gasewater relative permeability in fractures and porous media could be obtained by following five key steps (Chima et al., 2010): 1 Perform momentum balance within fractures and pores to obtain differential equations, respectively; 2 Define boundary conditions to solve the previous differential equations; 3 Use Newton’s law of viscosity to define average velocities; 4 Use cubic law in fractures and Poiseuille law in pores to obtain the final set of equations related with saturation, respectively; 5 Use Darcy law for two phase flow in fractures and pores to obtain the final relative permeability. The models must satisfy the following assumptions: (1) Gas and water are Newtonian fluid. (2) Gas is compressible and water is slightly compressible, and both have constant properties. (3) No phase transformation occurs between gas and water.
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(4) Fracture walls are planar and impermeable, and flow occurs in an open fracture, and flow in fracture is laminar and in steady state. (5) The pore throat is treated as a pipe and the flow in a pipe takes into account of the gas core fluid; (6) The irreducible water is absorbed on the walls of fractures and pores, and distributes uniformly on the walls; (7) No transformation and friction occur between irreducible water and mobile water; (8) The pipes and fractures are oriented at horizontal, fluid gravity and buoyancy effects are neglected. According to Chima’s method (Chima and Geiger, 2012), applying momentum balance, cubic law for flow in rectangular fractures and Newton’s law of viscosity to the fracture configuration and the flow configuration shown in Fig. 1, leads to the following analytical gasewater relative permeability of fractures that considers irreducible water saturation. Part of detailed mathematical derivation is given in Appendix A.
Krw ¼
Krg ¼
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg 2 ð1 Swc Þ3
(12)
Sg
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3
2mw
!
makes a between 1 and N. When tortuosity is neglected, a equals to zero, and Eqs. (14) and (15) can be simplified into Eqs. (12) and (13).When d / N, Eqs. (14) and (15) change into the model for ideal fractures:
Krw ¼
Krg ¼
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc 2 1 Swc ð1 Swc Þ3
(16)
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
Sg
ð1 Swc Þ3 Sw Swc 1 1 Swc
!
2mw
(17) Based on the observation, the flow structures observed in a fracture show more similar to the structures observed in a pipe than to those expected for a porous media (Fourar and Bories, 1995), so we also treat flow in fractures or porous media as flow in pipes. Applying momentum balance, Poiseuille law in a pipe and Newton’s law of viscosity to the pipe configuration and the flow configuration shown in Fig. 2, then the gasewater relative permeability of fractures or porous media that considers irreducible water saturation is given as follows. The detailed mathematical derivation is given in Appendix B:
(13) If Swc ¼ 0, Eqs. (12) and (13) are the same as viscous-coupling model (1998). Tortuosity could be approximated by the relationship of capillary pressure and saturation. Considering the influence of tortuosity, an improved gasewater relative permeability of fractured media is proposed as follows. The detailed mathematical derivation is given in Appendix C.
Krw ¼
Krg ¼
Krw ¼ (14)
Krg ¼
Or
a ¼
Krg ¼
Sg ð1 Swc Þ3
Sg
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
(19)
!"
ðSw Swc Þ4
(20)
ð1 Swc Þ4
2þd # Sw Swc d 1 1 Swc 2mw ! 2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ Sw Swc a 1 1 Swc 2mw
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3
(18)
ð1 Swc Þ2
The improved gasewater relative permeability considering the influence of tortuosity on the assumption of pipe bundle is given in Appendix B.
2þd ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc d Krw ¼ 2 1 Swc ð1 Swc Þ3 ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc a Or Krw ¼ 2 1 Swc ð1 Swc Þ3
Sg
ðSw Swc Þ2
2þd
d
where, the fracture-size distribution index, d, which is constant for a given fracture system, is measured directly from best-fit line drawn through the data points. Taking into consideration of the influence of tortuosity, the value of d ranges from 0 to N, which
Krg ¼
Sg ð1 Swc Þ2
(15)
" 2mg ðSw Swc Þ þ mw Sg
mw
1
ðSw Swc Þ2
#
ð1 Swc Þ2 (21)
The influence of tortuosity is different for various pore-size distributions. And considering the influence of tortuosity, an improved gasewater relative permeability of fractured or porous
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
p0
pL
Krg ¼
Q
Sg ð1 Swc Þ
Or Krg ¼
193
2mg ðSw Swc Þ þ mw Sg
2
Sg
mw
"
1
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
"
Sw Swc 1 Swc
2þl # l
# Sw Swc b 1 1 Swc
(23)
Fig. 1. Fracture to be used in the mathematical model proposed. For fracture is taken as two parallel planes placed in horizontal with negligible gravity and buoyancy effects, ideally the wetting phase contacts the top and bottom fracture surfaces, and part of the wetting phase is adsorbed at fracture surfaces that is irreducible phase, while nonwetting phase flows in between.
b¼
2þl
l
where, the pore-size distribution index, l, which is constant for a given porous media system, is also measured directly from best-fit line drawn through the data points. Taking into consideration of the influence of tortuosity, the value of l varies from 0 to N, which makes b between 1 and N. When tortuosity is neglected, b equals to zero, and Eqs. (22) and (23) can be simplified into Eqs. (18) and (19). When the porous media is treated as the pipe bundle, b ¼ 2 (Brooks, 1966).
3. Model validation Fig. 2. Fractures or porous media to be used in the mathematical model proposed. For Fracture or pore is taken as a pipe placed horizontal with negligible gravity and buoyancy effects, ideally the wetting phase contacts pipe surfaces, and part of the wetting phase is adsorbed at pipe surfaces that is irreducible phase, while nonwetting phase flows in between.
media is proposed as follows. The detailed mathematical derivation is also given in Appendix C.
Krw ¼ Or
2þl ðSw Swc Þ2 Sw Swc l 1 Swc ð1 Swc Þ2
Krw
(22)
ðSw Swc Þ2 Sw Swc b ¼ 1 Swc ð1 Swc Þ2
Several experimental studies on two-phase flow in fractures have been performed. Romm (1966) used kerosene and water two-phase flow through an artificial fracture by using parallel plates. Persoff et al. (1991) and Persoff and Pruess (1995) also used air and water two-phase flow through rough-walled fractured rocks. Fourar et al. (1993) and Fourar and Bories (1995) studied airewater two-phase flow in a fracture consisting of two parallel glass plates (1 m 0.5 m) with an opening equal to 1 mm. Diomampo et al. (2001) performed experiments of nitrogen and water flow through smooth-walled artificial fractures. Other experimental datasets (e.g., McDonald, et al., 1991; Pieters and Graves, 1994; Speyer et al., 2007) use oil-water systems. The databases of Fourar and Bories (1995) and Diomampo et al. (2001) can be used to validate our model because their experiments are gasewater systems conducted in smooth-walled fractures. In their
1000
diomampo fourar & Bories X-model Viscous coupling model Brooks-Corey model(λ=2,Swc=0) Proposed model(α=1,Swc=0.081)
100
Kg/Kw
10
1
0.1
0.01
0.001
0.0001
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 3. Comparison of Fourar and Bories (1995) and Diomampo et al. (2001) laboratory measurement data (yellow and blue points) with X-model (green line), VeC model (yellow line), Corey model (brown line), proposed model in fractures with a ¼ 1, Swc ¼ 0.081 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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1000
diomampo fourar & Bories Brooks-Corey model(λ=2,Swc=0.081) Brooks-Corey model(λ=infinite,Swc=0.081) Proposed model(α=1,Swc=0.081)
100
10
Kg/Kw
1
0.1
0.01
0.001
0.0001 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 4. Comparison of Fourar and Bories (1995) and Diomampo et al. (2001) laboratory measurement data (yellow and blue points) with Corey model (yellow line) and proposed model with a ¼ 1, Swc ¼ 0.081 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
experiments, the fractures were initially saturated with water which was injected at a constant flow rate. Air injection was then started at a constant flow rate and increased stepwise. Saturation was calculated as total pixels of the liquid group over the sum of the air and liquid groups through a digital video. When the steady state was reached for each flow rate, the relative permeability was measured. In the experiments, the measured irreducible water saturation was 0.081. And, the smooth-walled fractures could be considered as idea fractures, so d approaches to infinity and a equals to 1 in this experiment. Fig. 3 compares the experimentally measured relative permeability ratio curves for air and water (blue and yellow points) with relative permeability ratio curves estimated from X-model, VeC model, BrookseCorey model, and proposed model (a ¼ 1, Swc ¼ 0.081). It is obvious that proposed model agrees with experimental data better than other models.
1 Krw Krg Krw(model) Krg(model)
0.9 0.8 0.7
4. Discussion
Kr
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
Considering irreducible water, Fig. 4 compares the experimentally measured relative permeability ratio curves for air and water (blue and yellow points) with the relative permeability ratio curves estimated from BrookseCorey model (l ¼ 2, Swc ¼ 0.081), Brookse Corey model (l ¼ infinite, Swc ¼ 0.081)and proposed model ( a ¼ 1, Swc ¼ 0.081, red line), It is apparent that the proposed model matches the experimental data better than other models. What’s more, Fig. 5 compares Diomampo et al. (2001) laboratory measured relative permeability with proposed model (a ¼ 1, Swc ¼ 0.081). And it shows that the proposed model agrees with the experimental data very well. To validate Eqs. (20) and (21) in porous media, we compared the proposed model with Corey model without irreducible water (Eqs. (1) and (2)), the water phase relative permeabilities of the two models are the same, but the gas phase relative permeability of proposed model has an extra part that reflects the interaction of gas and liquid comparing with Eq. (2). Fig. 6 compares proposed model in porous media with BrookseCorey model (l ¼ 2, Swc ¼ 0) and BrookseCorey model (l ¼ infinite, Swc ¼ 0). And the curves of proposed model are just consistent with the two models. What’s more, the influence of tortuosity has significant effect on the relative permeability.
0.4
0.6
0.8
1
Sw Fig. 5. Comparison of Diomampo et al. (2001) laboratory measurement relative permeability (red and blue points) with proposed model with a ¼ 1, Swc ¼ 0.081. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In this paper, we assume that the surface geometry of flow channels is rectangle and pipe, and propose different relative permeability models as shown in Eqs. (12), (13), (18) and (19), respectively. Comparing with the curves of the two models without irreducible water shown in Fig. 7, we find the relative permeability curves are different between rectangle and pipe. And from pipe to rectangle, the gas phase relative permeability has larger deviation than that of water phase. As shown in Fig. 7, the deviation of the gas phase relative permeability between pipe and fracture is larger than 50%, but the deviation of the water phase relative permeability between pipe and fracture is smaller than 20%. That may mean that the surface geometry mainly impacts gas phase relative permeability. And the irreducible water saturation also influences the deviation of the relative permeability between the fracture and pipe shown in Fig. 8. The deviation of the relative
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
195
10000
Brooks-Corey model(λ=infinite,Swc=0) Brooks-Corey model(λ=2,Swc=0) Proposed model in porous media(β=2,Sw=0) Proposed model in porous media(β=1,Sw=0) Proposed model in porous media(β=0,Sw=0)
1000
Kg/Kw
100
10
1
0.1
0.01
0.001 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 6. Comparison of BrookseCorey model with and proposed model in porous media with b ¼ 0, 1, 2; Swc ¼ 0.
1 0.9
Pipe(Krg)
0.8
Pipe(Krw)
0.7
Fracture(Krg) Fracture(Krw)
Kr
0.6
50%
20%
0.5 0.4 0.3
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 7. Comparison relative permeability curves of fracture (red points) and pipe (blue points) without irreducible water. The dotted line of left is 50% deviation of the relative permeability of gas phase in fracture, and the dotted line of right is 20% deviation of the permeability of water phase in fractures. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
1 0.9
Pipe(Krg),Swc=0.1
0.8
Pipe(Krw),Swc=0.1 Fracture(Krg),Swc=0.1
0.7
Fracture(Krw),Swc=0.1 Kr
0.6
Pipe(Krg),Swc=0.3
0.5
S0.1
0.4
S0.3
S0.5
Pipe(Krw),Swc=0.3 Fracture(Krg),Swc=0.3 Fracture(Krw),Swc=0.3
0.3
Pipe(Krg),Swc=0.5 0.2
Pipe(Krw),Swc=0.5
0.1
Fracture(Krg),Swc=0.5 Fracture(Krw),Swc=0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sw Fig. 8. The relative permeability curves of different irreducible water (Swc ¼ 0.1, 0.3, 0.5). And the proportion (Si) enclosed by water or gas relative permeability curves of fractures and pipes reflects deviation of relative permeability of different irreducible water and S0.1 > S0.3 > S0.5.
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permeability becomes smaller with the increase of irreducible water saturation. The gas and water phase permeability is determined by relative permeability and effective permeability. For the same effective permeability, gas phase relative permeability may be more sensitive to the surface geometry than that of water phase as shown in Figs. 7 and 8. So, it’s very important to choose the appropriate relative permeability model of gas phase. 5. Conclusion The following key conclusions can be drawn from this study: 1. Gasewater relative permeability models are investigated in this study. Among these, four models have been suggested to approximate the two-phase flow behavior in the fracture and porous media: the X-model, Corey or BrookseCorey models, viscous-coupling model and Chima’s model. However, gase liquid interaction, the irreducible water (wetting phase), the surface geometry and the corresponding flow structure in porous media are not all considered in these models. So, considering irreducible water, the surface geometry and the corresponding flow structure, and gaseliquid interaction, according to Chima’s method, the new analytical relative permeability models are developed in fractures and porous media, respectively. What’s more, the proposed relative permeability model of fractures has been improved by considering the influence of tortuosity. The result shows that the relative permeability models are functions of mobile water, viscosity, irreducible water and the fracture or pore-size distribution index. Furthermore, we study the characteristic of relative permeability curves in different surface geometry and corresponding flow structure in fractures and porous media. 2. The proposed relative permeability model with irreducible water saturation of fractures is validated with experimental data and shows superior agreement compared with the X-model, Corey or BrookseCorey models, and viscous-coupling model. 3. The proposed relative permeability model considering the influence of tortuosity of fracture or porous media has also been studied. This model is validated with Corey or BrookseCorey models. The expressions of the two models without irreducible water are very similar and the curves of proposed model are in consistency with BrookseCorey models (l ¼ 2) and Brookse Corey models (l ¼ infinite). What’s more, the tortuosity has significant effects on the relative permeability in fractured media or porous media. 4. Due to the influence of surface geometry, the relative permeability models are very different between parallel planes and pipes, and the gas phase relative permeability is more sensitive to the surface geometry than that of water phase. So, the optimized model of gas phase relative permeability is more important than that of water phase for field development. Because of the influence of irreducible water, the deviation of the relative permeability between the fracture and pore becomes smaller with the increase of irreducible water saturation. Acknowledgments This research was supported by National Natural Science Foundation Project (U1262113) and Science Foundation of China University of Petroleum, Beijing (YJRC-2013-37). We also recognize the support of MOE Key Laboratory of Petroleum Engineering in China University of Petroleum (Beijing).
Nomenclature A Aw Ag AS C1
area of flow channel area of water bed in flow channel area of gas bed in flow channel area of moved fluid integration constant
CWM 1 C1g
integration constant 1 of mobile water phase integration constant 1 of gas phase
CWM 2
integration constant 2 of mobile water phase
C2g H D hg hw hwc KrW Krg KeW Keg Kabs L P0 PL pb pc Q Qw Qg Sw Swm Swc Sg Sgc
integration constant 2 of gas phase total thickness of the fracture diameter of the pore thickness of gas bed in flow channel thickness of water bed thickness of irreducible water bed relative permeability to water relative permeability to gas effective permeability to water effective permeability to gas absolute permeability fracture or pipe length pressure on flow cross section in z ¼ 0 pressure on flow cross section in z ¼ L bubbling pressure capillary pressure flow rate water flow rate in z direction flow rate in z direction water saturation mobile water saturation irreducible water saturation gas saturation residual gas saturation
VzW Vzg
velocity of water phase velocity of gas phase
VzWM
average velocity of mobile water phase
Vzg
average velocity of gas phase volume of mobile water volume of irreducible water volume of gas width of fracture difference operator viscosity of gas viscosity of water force in the z direction on a unit area perpendicular to the r direction in the mobile water; force in the z direction on a unit area perpendicular to the r direction in the gas; the proportion (Si) enclosed by water or gas relative permeability curves of fracture and pipe when irreducible water saturation Swc ¼ i.
Vw Vwc Vg w Dr
mg mw sWM rz
sgrz Si
Appendix A Here we briefly describe the derivation of the proposed gase water relative permeability model when the fracture is taken as two parallel planes. According to Chima’s equation, perform momentum balance within a fracture, and then use Newton’s law of viscosity to define average velocities, finally gas phase and water phase rate could be calculated as follows (Chima and Geiger, 2012):
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
Apply the momentum balance (Bird, 2002) in pipes, as shown in Fig. 2:
For gas phase:
Krg Keg AS DPg ¼ mg L ! 2mw h2g þ 3mg h2w þ 6mg hw hg P PL ¼ Ag 0 2L 12mg mw
Qzg
ðL$2pr$srz Þr ðL$2pr$srz ÞrþDr þ 2pr$Dr$ðp0 pL Þ ¼ 0 (A1)
For water phase:
Qzw ¼
! 2h2w þ 3hw hg Krw Kew AS DPw P PL ¼ Aw 0 mw L 2mw L 12
¼ Swc
Vw wLðhw þ hwc Þ hw þ hwc ¼ ¼ wLðhw þ hwc Þ þ wLhg Vw þ Vg hw þ hwc þ hg
sgrz ¼
hwc H
g C P0 PL $r þ 1 2L r
(B3)
mw
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg 2 ð1 Swc Þ3
vVzWM
ð1 Swc Þ
C WM P0 PL $r þ 1 2L r
¼
# C1WM P0 PL $r vr 2Lmw r mw
C WM P PL $r 2 1 ln r þ C2WM VzWM ¼ 0 mw 4mw L
g
(A3)
For gas phase:
! 2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ 3
Z "
Z
Vz ¼
Substituting DPg, DPw, AS, Ag, Aw, Kew, Keg, hg and hw into above equations, then simplify: For water phase:
Sg
vVzWM ¼ vr
(B4)
For gas phase:
! 2mw h2g þ 3mg h2w þ 6mg hw hg Ag m g L P0 PL ¼ 2L 12mg mw Keg AS DPg
Krg ¼
(B2)
After this, use Newton’s law of viscosity to define velocities: For mobile water phase
! 2h2w þ 3hw hg Aw mw L P0 PL ¼ Kew AS DPw 2mw L 12
Krw ¼
C WM P0 PL $r þ 1 2L r
For gas phase:
For gas phase:
Krg
(B1)
Apply Eq. (B1) to both gas (g) phase and water (W) phase. Integration of Eq. (B1) for these regions gives: For mobile water phase
hw þ hwc H
So: hw ¼ (Sw Swc)H, hg ¼ SgH If capillary pressure is ignored, then the pressure is DPw ¼ DPg ¼ P0 PL The effective permeability in the fracture is: Kew ¼ Keg ¼ (hg þ hw)2/12 Translating Eqs. (A1) and (A2) into expressions of relative permeability, then simplify: For water phase:
Krw
vðr srz Þ P PL ¼ 0 r vr L
sWM ¼ rz
Vwc wLhwc hwc ¼ ¼ ¼ Vw þ Vg wLðhw þ hwc Þ þ wLhg hw þ hwc þ hg ¼
ðr srz ÞrþDr ðr srz Þr ðP PL Þ lim r ¼ 0 Dr/0 Dr L Above equation simplifies:
(A2)
Then, build the relationship between saturation and thickness of gas bed or water bed in the fracture geometry. For the fracture geometry and flow configuration shown in Fig. 1, we obtain:
Sw ¼
197
2mw (A4)
(B5)
Then, define boundary conditions to solve the previous differential equations. g Boundary Condition 1: at r ¼ hg/2; sWM ¼ srz rz Boundary Condition 2: at r ¼ 0; VzWM ¼ finite Boundary Condition 3: at r ¼ hg/2; VzWM ¼ Vzg Boundary Condition 4: at r ¼ (hg þ hw)/2; VzWM ¼ 0 Substituting boundary condition 1 into Eqs. (B2) and (B3), boundary condition 2 into Eqs. (B4) and (B5), we find:
C1WM ¼ C1g ¼ C1 ¼ 0
(B6)
Substituting boundary condition 3, 4 into Eqs. (B4) and (B5), we find:
Appendix B The modeled fracture or porous media geometry and flow configuration are shown in Fig. 2. The flow in fracture or pore is taken as flow in a pipe. Within the pipe, the water and gas are flowing in a steady-state condition. A part of wetting phase (water) adheres to walls of the pipe and doesn’t flow. The nonwetting phase (gas) flow is surrounded by wetting phase (water). The flow is in z direction and the diameter of the pipe is D.
! Cg P0 PL g $r 2 1 ln r þ C2 mg 4mg L
g C2
¼
! ! mg h2w þ 2hw hg mg þ mw h2g P0 PL $ 4mw mg L 4
C2WM ¼
hw þ hg 2 P0 PL $ 4mw L 2
(B7)
(B8)
Substituting Eqs. (B6), (B7) and (B8) into Eqs. (B4) and (B5), velocities in r direction are:
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Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
For mobile water phase:
hw þ hg 2 P PL P0 PL $r 2 þ $ VzWM ¼ 0 4mw L 4mw L 2
(B9)
For gas phase:
! P0 PL $r 2 4mg L ! ! mg h2w þ 2hw hg mg þ mw h2g P0 PL $ þ 4mw mg L 4
g
Vz ¼
(B10)
The average velocity for each phase is calculated as follows:
2 2 pLðD2 pL h2g D2 h2g Vw Sw ¼ ¼ ¼ 2 Vw þ Vg D2 pL D2 2 pL h2g h2g Vg Sg ¼ ¼ ¼ 2 2 Vw þ Vg D pL D2 2 2 2 w pLðD2 pL hg þh D2 hg þ hw Vwc 2 Swc ¼ ¼ ¼ 2 Vw þ Vg D2 pL D2 2 2 2 h w pL hg þh pL 2g hg þ hw h2g Vwm 2 Swm ¼ ¼ ¼ 2 Vw þ Vg D2 pL D2 h2w þ 2hg hw D2
Swm ¼ Z ðhw þhg Þ=2 h g =2
VzWM ¼
2prVzWM vr
Z ðhw þhg Þ=2 h g =2
¼
P0 PL mw L
2pr dr
h i 4 2 hw þ hg 2 hw þ hg h2g þ h4g i h 2 32 hw þ hg h2g (B11)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi hg ¼ So: hw ¼ ð Sw Sg D; ffi Swc þ Sg Sg ÞD; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hg þ hw ¼ Sw Swc þ Sg D If capillary pressure is ignored, so the pressure is DPw ¼ DPg ¼ P0 PL. Using Poiseuille law, the effective permeability in the pipe is Kew ¼ Keg ¼ (hg þ hw)2/32 Translating Eqs. (A1) and (A2) into expression of relative permeability, then simplify: For water phase:
Z Vzg ¼
hg =2 0
Z 0
¼
Krw ¼ 2prVz dr
hg =2
g
!
2mg h2w þ 4hw hg mg þ mw h2g
!
32mw
(B12)
The flow for each phase satisfies Darcy law, so the flow rate is calculated as follows: For mobile water phase:
Krw Kew AS DPw mw L h i h þ h 4 2h þ h 2 h2 þ h4 w g w g g g P0 PL i h ¼ Aw 2 mw L 32 hw þ hg h2g
Qzw ¼ Aw VzWM ¼
For gas phase:
Krg Keg AS DPg mg L ! ! 2mg h2w þ 4hw hg mg þ mw h2g P0 PL mg L 32mw
Krg
Krw ¼
2mg h2w þ 4hw hg mg þ mw h2g
!
32mw
ðSw Swc Þ2
(B15)
ð1 Swc Þ2 Sg
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
(B16)
where, Eqs. (B15) and (B16) are gasewater relative permeabilities which do not consider the influence of tortuosity. Corey (1954) proposed the relative permeability of porous media, taking into account of the influence of tortuosity:
(B14)
Then, build the relationship between saturation and thickness of gas bed or water bed in fracture or pore geometry. For the pipe geometry, and flow configuration shown in Fig. 2, we obtain:
!
Substituting DPg, DPw, AS, Ag, Aw, Kew, Keg, hg and hw into above equations, then simplify:
Krw ¼
Qzg ¼ Ag Vzg ¼
P0 PL mg L
Ag mg L ¼ Keg AS DPg
Krg ¼
(B13)
¼ Ag
h i 4 2 hw þ hg 2 hw þ hg h2g þ h4g i h 2 32 hw þ hg h2g
For gas phase:
2pr dr
P0 PL mg L
Aw mw L P0 PL mw L Kew AS DPw
ðSw Swc Þ2 ð1 Swc Þ2
Krg ¼
1
Sw Swc 1 Swc
ZSw
! Z1 ! . . 2 2 dSw pc dSw pc
0
2
(B17)
0
Z1
! Z1 ! . . dSw p2c dSw p2c
Sw
(B18)
0
1/p2c
And, the relationship between w Sw can be got from derivations and experiments (Corey, 1954), and if the porous media is considered as capillary tubes, the influence of tortuosity is:
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
ZSw
. dSw p2c
!, Z1
0
Z1
. dSw p2c
! ¼
0
. dSw p2c
!, Z1
Sw
. dSw p2c
ðSw Swc Þ2
(B19)
2
ð1 Swc Þ
¼ 1
ðSw Swc Þ2 ð1 Swc Þ2
Krg ¼
Sg
2mg ðSw Swc Þ þ mw Sg
ð1 Swc Þ2
mw
¼
1 pc
(C2)
2
If Eq. (C1) is substituted into Eqs. (B17) and (B18), a change of variable is made for Sw to S*w, and the constant pb is eliminated in the substitution process, the result is: *
ðSw Swc Þ2 ð1 Swc Þ2
(B21)
4
l
p2b
Krw ¼
ðSw Swc Þ4
S*w
for pb pc
2
(B20)
Comparison Eqs. (B15) with (B17), the influence of tortuosity is exactly Eqs. (B19) and (B20), so the gasewater relative permeability considering the influence of tortuosity is
ð1 Swc Þ
pb pc
Or
!
0
Krw ¼
h il
S*w ¼
199
ZSw
S*w
2 l
dS*w
!, Z1
S*w
2 l
! dS*w
0
0
2þl ðSw Swc Þ Sw Swc l ¼ 1 Swc ð1 Swc Þ2 2
# " ðSw Swc Þ2 1 ð1 Swc Þ2
(C3)
(B22) Krg ¼ Appendix C
¼
Eqs. (A3), (A4), (B15) and (B16) are derived on ideal conditions when the influence of tortuosity has not been considered. Though Eqs. (21) and (22) have taken into account of the influence of tortuosity, but the assumption of tube bundle is too ideal. So, for general fractured and porous media, the exponential coefficient may be different in various reservoirs, and is not always 2.0, just as Eqs. (B19) and (B20). What’s more, the tortuosity may be different between fractured media and porous media, and tortuosity could be approximated by the relationship of capillary pressure and saturation (Brooks, 1966). Based on large number of desorption on consolidated porous rocks, Brooks (1966) proposed the relationship of effective saturation and capillary pressure which could be considered the influence tortuosity:
ln S*w ¼ l ln pc þ l ln pb
pb pc
. dSw p2c
!, Z1
0
Z1 Sw
. dSw p2c
! ¼
b¼
!, Z1 0
2 l
dSw
!, Z1
0
0
. dSw p2c
S*w
. dSw p2c
! ¼
Z1 Sw
S*w
2 l
dSw
!, Z1
log paper. The definitions of the constants Swc and pb are implicit in the method of their determination. According to Eq. (C1), theoretical relationships for permeability as a function of saturation and capillary pressure could be derived by taking the antilog of Eq. (C1):
l
dS*w
!, Z1
S S 2þl w wc l 1 1 Swc
S*w
2 l
! dS*w
0
l
!
2
S*w
0
Sw Swc 1 Swc
0
2
2þl
l
¼
dSw
0
S*w
1
2
S*w
where, the pore-size distribution index, l, is measured directly from best-fit line drawn through the data points proposed by Brooks (1966). By experimentally determining S*w versus pc, the pore-size distribution index, l, can be determined (provided Swc is known) for predicting relative permeability as a function of capillary pressure and saturation. The pore-size distribution index, l may be different in various porous media. Eqs. (C3) and (C4) is exactly the same as BrookseCorey model, the exponential term of the right of Eq. (C3) is 2, which could also be obtained experimentally by Burdine (1953) and analytically by Wyllie and Gardner (1958). But the integration term of the right of Eq. (C3) is different due to the pore-size distribution index, l.
(C1)
ZSw
*
ZSw
Sw Swc 2 1 1 Swc
(C4)
This equation was found to be generally valid for the core analyzed. To calculate S*w, the residual saturation Swc, must be known or assumed, in above analysis, Swc was chosen such that data fit as closely as possible to a straight line when plotted on loge
ZSw
2 l
! dSw
2þl Sw Swc l 1 Swc "
¼
Sw Swc 1 1 Swc
(C5)
2þl # l
(C6)
Comparison Eqs. (B15) with (C3), the integration term of the right of Eq. (C3) could be considered as the influence of tortuosity. Therefore, the relative permeability of porous media considering the influence of tortuosity could be got by multiplying Eqs. (C5) and (C6) into the analytical equation Eqs. (B15) and (B16).
200
Krw
Or
Y. Li et al. / Journal of Natural Gas Science and Engineering 19 (2014) 190e201
2þl ðSw Swc Þ2 Sw Swc l ¼ 1 Swc ð1 Swc Þ2 Krw
Krg ¼
Or
(C7)
ðSw Swc Þ2 Sw Swc b ¼ 1 Swc ð1 Swc Þ2
2þd # Sw Swc d 1 1 Swc 2mw ! 2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ Sw Swc a 1 1 Swc 2mw
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
Sg ð1 Swc Þ3
Krg ¼
Sg ð1 Swc Þ3
!"
2mg ðSw Swc Þ þ mw Sg
1
(C8)
b¼
2þl
l
The fractured media could be considered as a special porous media and the derivation method of relative permeability of fractured media is the same as that of porous media. So, for given fracture-size distribution, the relative permeability also has the same expression on influence of tortuosity. Therefore, the relative permeability could also be got through analytical equations Eqs. (A3) and (A4) multiplying by Eqs. (C5) and (C6):
Krw ¼
!, ZSw . ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg 2 p dS w c 2 ð1 Swc Þ3 . dSw p2c
(C9)
Krg ¼
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3 Z1 Sw
. dSw p2c
!
2mw !, Z1 dSw
.
2þd
d
where, the fracture-size distribution index, d, is also measured directly from best-fit line drawn through the data points. Chen (2005) pointed according to Brooks and Corey’s reasoning, if the porous media is the case of fractured media, the value of l should approach infinity, so the value of d may also approach infinity. Therefore, Eqs. (C11) and (C12) are modified into the model for fractured media with d / N:
Krw ¼
Krg ¼
ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc 3 2 1 Swc ð1 Swc Þ Sg
(C13)
2mw S2g þ 3mg ðSw Swc Þ2 þ 6mg Sg ðSw Swc Þ
ð1 Swc Þ3 Sw Swc 1 1 Swc
!
2mw
References
!
0
Sg
a ¼
(C12)
(C14)
0
Z1
(C11)
"
2þl # Sw Swc l mw 1 Swc ð1 Swc Þ2 " # 2mg ðSw Swc Þ þ mw Sg Sg Sw Swc b 1 Or Krg ¼ mw 1 Swc ð1 Swc Þ2 Sg
Krg ¼
2þd ðSw Swc Þ2 2ðSw Swc Þ þ 3Sg Sw Swc d 2 1 Swc ð1 Swc Þ3 2 ðSw Swc Þ 2ðSw Swc Þ þ 3Sg Sw Swc a Or Krw ¼ 2 1 Swc ð1 Swc Þ3 Krw ¼
! p2c
0
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