Transport in Porous Media

, Volume 119, Issue 3, pp 499–519 | Cite as

A Heuristic Insight on End-Point Calculation and a New Phase Interference Parameter in Two-Phase Relative Permeability Curves for Horizontal Fracture Flow

  • Mohammad Ranjbaran
  • Saeed Shad
  • Vahid Taghikhani
  • Shahab AyatollahiEmail author


Relative permeability curves of two-phase flow in a fracture have been a subject of study in recent years. The importance of these curves have been widely observed in multidisciplines, such as water subsurface resources, geothermal energy and underground hydrocarbon resources, especially fractured oil and gas reservoirs. Extensive experimental studies have been cited alongside the numerical studies in this area. However, simple analytical and practical solutions are still attractive. In the current study, wettability effects and phase interference explicitly were tried to be implemented in a simple analytical formula. The wettability effects are represented by residual saturations which resulted in direct calculation of relative permeability end points. In addition, the phase interference part affected the shape of the curves that allowed to quantify the degree of phase interference from no phase interference, assigned as zero, to ultimate phase interference, assigned as infinity. The results were compared and validated with the available experimental data in the literature. The proposed formulation is applicable for both smooth and rough fracture assemblies.


Analytical study Phase interference Smooth and rough fractures Two-phase flow 

1 Introduction

Ever since the study of relative permeability in fractures has commenced, numerous works that cover both theoretical and experimental studies have been published. The analytical solutions to the proposed model of relative permeability range from very simple and practical techniques to the complex methods, while experimental results rarely follow the analytical models.

Few of research studies can be found investigating the analytical and simple approaches to the relative permeability curves. Romm (1966) was the first to propose the practical X-curve system for a two-phase flow in a fracture. The proposed approach included only fluid–wall interaction with no phase interference. Fourar and Lenormand (1998) applied fluid–fluid interaction effect of viscosity coupling with the aid of changing the flow regime from channeling to sandwich flow as wetting phase spread on the wall and non-wet in the middle. This approach signified the importance of viscosity ratio for the relative permeability measurement of the non-wetting phase and resulted in a curvature for both relative permeabilities. Subsequently, Shad and Gates (2010) proposed a multilayer flow theory in a horizontal fracture. Their theory generalized the viscous coupling effects of Fourar and Lenormand (1998) to multiphase flow and represented flow pattern effect on fracture relative permeability curves. Recently, Saboorian-Jooybari (2016) studied wettability as well as gravity effects on relative permeability curves for a two-phase flow in an inclined fracture. They assumed that for strongly wet systems, the flow regime was sandwich flow which became stratified flow for neutral wet systems. In addition, Liu et al. (2013) developed a closed form for calculation of relative permeability curves of two-phase flow in a horizontal fracture using the idea of two-phase flow in a porous medium and parametric model of Burdine. Their model was able to match Chen and Horne (2006) data by fitting three adjusting parameters. Afterward, Ye et al. (2017) extended Liu et al. (2013) model for Mualem parametric model. In addition, they applied the power-law form of two-phase flow relative permeability as the macroscopic model. All of their models with four adjusting parameters (apart from mean aperture and aperture standard deviation) were able to match Chen and Horne (2006) rough fracture data as well as \(\nu \)-type curves developed by Watanabe et al. (2015) as well as Pruess and Tsang (1990). However, the macroscopic model was better than the others. Furthermore, their model was able to match strong phase interferences. However, no physical interpretation for the fitted tortuosity factors related to the fracture features were obtained.

Over the past two decades a comparative increase in the experimental studies investigating relative permeability curves for two-phase flow in a fracture has been observed. For alternating channel flow configuration, Rangel-German et al. (2006) realized that X-curves properly works when the slopes are changed to a value of 0.6, which is in agreement with Pan et al. (1998), who stated that the interference of the two phases led to the summation of relative permeability values being less than one. Moreover, for a water–gas system Chen et al. (2004) and Chen and Horne (2006) obtained relative permeability curves for a water-wet smooth/rough fracture. They also spotted the channel flow as the dominant regime for the two-phase flow in fractures. In addition, it was reported that the summation of two relative permeabilities was less than one and the interference was related to channeling of flow within each other. In addition, it is worth noting that in case of more phase interference, more residual water saturation was achieved. In a recent study, Alturki et al. (2014) examined a rough-wall oil-water fracture flow in a Plexiglas Hele-Shaw model. They analyzed both horizontal and inclined fractures to detect the fluid flow scenarios. In addition, they detected hysteresis when drainage was transformed to imbibition process while the channel flow regime was dominant. However, their injection system differed from the systems used by Pan (1999) and Pan et al. (1998) which was close enough to the condition happened for matrix–fracture interaction in fracture reservoirs. Indeed, it caused large values of residual saturations inside the fracture which is in disagreement with other works (e.g., Pan 1999; Chen et al. 2004; Chen and Horne 2006). Huo and Benson (2016) analyzed some of the literature data in the term of phase interference. Lian et al. (2012) reported fracture relative permeability data for a water–oil system in a naturally fractured carbonate rock. Their results showed the same degree of phase interference in relative permeability curves as in Huo and Benson (2016). In addition, large values in wetting phase and non-wetting phase were retained as residual saturations due to a permeable matrix. In an earlier work, Bertels et al. (2001) studied a natural fracture two-phase flow in an impermeable rock. Their relative permeability curves were S-shape and showed almost a specific value for relative permeability summation, but much less than one. In addition, Babadagli et al. (2015a, b) ran several observable tests in single-phase and two-phase rough fractures. They concluded that wettability effect and fracture roughness control residual saturation of displaced phase. Finally, they estimated the effect of roughness on relative permeability curvature within power-law approach. Moreover, Watanabe et al. (2015) conducted semi-dynamic and dynamic two-phase flow tests in a natural fracture with a variety of rock and fluid features for low capillary numbers. For the dynamic flow test, no influence of capillary pressure as well as no phase interference was observed. However, numerically they proposed a new \(\nu \)-type model for relative permeability curves under log-normal aperture distribution and capillary pressure dominance. Moreover, their new model was not experimentally verified. Their numerical study was in line with Pruess and Tsang (1990) who modeled a rough-walled fracture as a two-dimensional non-homogeneous porous medium. They generated a two-dimensional synthetic statistical aperture distribution in order to calculate each section’s permeability using local parallel-plate approach. Their study showed that interference between phases was generally strong. In addition, they mentioned that high values of non-wetting phase saturation might become immobile in the system, and thus \(\nu \)-type relative permeability curves could appear. They also emphasized that the relative permeability features were significantly related to the selected aperture distribution. Therefore, their numerical experiment could be viewed as a first demonstration of trends.

From the above mentioned review, most of the previous works both for theoretical and experimental studies on relative permeabilities in fractures just focused on the shape of the curves rather than the relative permeability end points. In addition, there has only been a few studies on quantifying degree of phase interference for flow in fractures. Therefore, in the current study the model of relative permeability curves for alternating channel flow configuration suggested by Romm (1966) and Pan (1999) is modified thoroughly to explore the analytical models that could be utilized in the oil industry. In the first stage, the effect of solid–fluid interaction (wettability effect) is explicitly inserted into the formulation to affect end points which was not considered by Saboorian-Jooybari (2016). Secondly, the significant concept of phase interference that was comprehensively discussed by Huo and Benson (2016) is quantified by a new parameter named here as the degree of phase interference. This new parameter is obtained after modification of Brinkman’s equation of single-phase flow in a porous medium for two-phase flow in a fracture. In addition, all of the developed models are validated by comparing with the available experimental data. Eventually, a sensitivity analysis is performed to evaluate the significance of all the defined key parameters for the proposed analytical model.

2 Formulation

2.1 Single-Phase Equations

To consider fluid flow in a horizontal fracture, especially in 2D models, the fracture is modeled as two parallel plates with a narrow gap(e.g., Fourar and Lenormand 1998; Pan 1999; Shad and Gates 2010; Saboorian-Jooybari 2016), see Fig. 1. However, in this section of study the classic 2D model is modified innovatively for imperfection (Eqs. 410) as well as roughness (Eqs. 1117, 1921, and 2324).
Fig. 1

Schematic of a fracture as two parallel plates with a narrow gap

For a parallel plate flow, Hanks (1963) proposed a generalized stability parameter from pipe flow and calculated the critical Reynolds number (\(Re_{c}\)) of 2800 for laminar-turbulent transition. In their definition of Reynolds number, the characteristic length was exactly the gap between the two plates (h). Therefore, for \(\textit{Re} \le 2800\), the laminar flow is dominant and with assumption of fully developed flow, the governing equation of flow between two parallel plates for a Newtonian fluid is as follows (Bird et al. 2002, p. 85):
$$\begin{aligned} -\frac{\partial p}{\partial z}+\mu \frac{{{\partial }^{2}}{{v}_{z}}}{\partial {{y}^{2}}}=0, \end{aligned}$$
where p is pressure, \(\mu \) is fluid viscosity, and \(v_z\) is the axial fluid velocity. In addition, the accepted boundary condition is no-slip boundary condition (Bird et al. 2002, p. 42).
$$\begin{aligned} v_z=0 \ @\ y=-h/2, \end{aligned}$$
$$\begin{aligned} v_z=0 \ @\ y=+h/2. \end{aligned}$$
If the parallel plate setup is not perfect, it can have a dead volume or the wall will be slightly permeable to a wetting phase with an undetermined penetration length (\(\delta \)), see Fig. 2. Therefore, the no-slip boundary condition should be applied to a deeper length based on the following equations:
$$\begin{aligned} v_z=0 \ @\ y=-(h/2+\delta ), \end{aligned}$$
$$\begin{aligned} v_z=0 \ @\ y=+(h/2+\delta ). \end{aligned}$$
Fig. 2

Schematic of an imperfect fracture or a fracture with slightly permeable walls

Solving Eq. (1) with boundary conditions (Eqs. 4 and 5) leads to the following equation for axial velocity (\(v_z\)).
$$\begin{aligned} v_z=\frac{1}{2\mu }\frac{\mathrm{d}p}{\mathrm{d}z}\left[ y^2-(h/2+\delta )^2\right] . \end{aligned}$$
As the next step, \(v_z\) is averaged over the fracture open cross-sectional area and the fracture superficial velocity (\(\bar{v}_z\)) is calculated.
$$\begin{aligned} {\bar{v}}_z=\frac{1}{h}\int \limits _{-h/2}^{+h/2}v_z\mathrm{d}y=\frac{h^2}{12}\left[ 1+6\frac{\delta }{h}\left( 1+\frac{\delta }{h}\right) \right] \frac{1}{\mu }\left( -\frac{\mathrm{d}p}{\mathrm{d}z}\right) . \end{aligned}$$
The part of the fluid which is trapped in the dead or permeated zone can be interpreted as an irreducible wetting phase saturation (\(S_\mathrm{wtr}\)).
$$\begin{aligned} S_\mathrm{wtr}=\frac{2\delta }{h+2\delta }. \end{aligned}$$
Introducing Eq. (8) into Eq. (7) leads to Eq. (9).
$$\begin{aligned} {\bar{v}}_z=\frac{h^2}{12}\left[ 1+\frac{3}{2}\frac{S_\mathrm{wtr}(2-S_{wtr})}{(1-S_\mathrm{wtr})^2} \right] \frac{1}{\mu }\left( -\frac{\mathrm{d}p}{\mathrm{d}z}\right) . \end{aligned}$$
The absolute permeability of the fracture, K, is obtained by comparing Eq. (9) with Darcy’s law.
$$\begin{aligned} K=\frac{h^2}{12}\left[ 1+\frac{3}{2}\frac{S_\mathrm{wtr}(2-S_{wtr})}{(1-S_\mathrm{wtr})^2} \right] . \end{aligned}$$
When the fracture is rough, see Fig. 3, behavior of the flow stands between smooth fracture and porous medium. At this condition Brinkman’s heuristic equation can be applied (Durlofsky and Brady 1987).
$$\begin{aligned} -\frac{\partial p}{\partial z}+\mu \frac{{{\partial }^{2}}{{v}_{z}}}{\partial {{y}^{2}}}-\frac{\mu }{k}{{v}_{z}}=0\,, \end{aligned}$$
where k is characteristic permeability of the system. However, it is more like an intermediate term and it will be replaced by a more meaningful parameter in the next few lines.
Fig. 3

Schematic of an imperfect rough fracture or a rough fracture with slightly permeable walls

Solution of Eq. (11) with Eqs. (4) and (5) as boundary conditions, leads to Eq. (12) for \(v_z\).
$$\begin{aligned} {{v}_{z}}=-\frac{k}{\mu }\frac{\mathrm{d}p}{\mathrm{d}z}\left[ 1-\frac{\cosh \left( {y}/{\sqrt{k}}\; \right) }{\cosh \left( {\left( {h}/{2}\;+\delta \right) }/{\sqrt{k}}\; \right) } \right] . \end{aligned}$$
Similar to Eq. (6), Eq. (12) is averaged over the open cross-sectional area in order to obtain fracture superficial velocity (\(\bar{v}_z\)).
$$\begin{aligned} {{\bar{v}}_{z}}=\frac{1}{h}\int \limits _{-h/2}^{+h/2}{{{v}_{z}}dy}=k\left[ 1-\frac{\sinh \left( {{h}/{2}\;}/{\sqrt{k}}\; \right) }{\left( {{h}/{2}\;}/{\sqrt{k}}\; \right) \cosh \left( {\left( {h}/{2}\;+\delta \right) }/{\sqrt{k}}\; \right) } \right] \frac{1}{\mu }\left( -\frac{\mathrm{d}p}{\mathrm{d}z} \right) . \end{aligned}$$
Introducing Eq. (8) into Eq. (13) leads to Eq. (14).
$$\begin{aligned} {{\bar{v}}_{z}}=k\left[ 1-\frac{\sinh \left( {{h}/{2}\;}/{\sqrt{k}}\; \right) }{\left( {{h}/{2}\;}/{\sqrt{k}}\; \right) \cosh \left( h/2/\sqrt{k}\;/{\left( 1-S_\mathrm{wtr} \right) } \right) } \right] \frac{1}{\mu }\left( -\frac{\mathrm{d}p}{\mathrm{d}z} \right) . \end{aligned}$$
Finally, the absolute permeability of the fracture is obtained by comparing Eq. (13) with Darcy’s law.
$$\begin{aligned} K=\frac{{{h}^{2}}}{12}{{\left( {2\sqrt{k}}/{h}\; \right) }^{2}}\left[ 3-\frac{3\left( {2\sqrt{k}}/{h}\; \right) \sinh \left( {{h}/{2}\;}/{\sqrt{k}}\; \right) }{\cosh \left( {{{h}/{2}\;}/{\sqrt{k}}\;}/{\left( 1-{{S}_\mathrm{wtr}} \right) }\; \right) } \right] . \end{aligned}$$
In Eq. (15), \(h/2/\sqrt{k}\) is defined as \(\lambda \). For a better understanding of this parameter, the absolute permeability value was calculated while \(\lambda \) approaching its limits.
$$\begin{aligned}&\displaystyle \lim \limits _{\lambda \rightarrow 0}K=\frac{{{h}^{2}}}{12}\frac{1}{{{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}}}\left[ \frac{3}{2}-\frac{1}{2}{{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}} \right] ,\end{aligned}$$
$$\begin{aligned}&\displaystyle \lim \limits _{\lambda \rightarrow \infty }K=k. \end{aligned}$$
For calculation of Eq. (16), \(\lambda \) and h were considered as independent variables. However, for calculation of Eq. (17), \(\lambda \) and k were considered as independent variables. Equation (16) is exactly the same as Eq. (10) and it is equivalent to smooth condition, whereas Eq. (17) means that the fracture has turned into a porous medium. This comparison confirms that the proposed rough fracture model can change between the two limiting conditions of two parallel slits (The classic cubic law) and a porous medium (where permeability is independent of fracture aperture) only by changing \(\lambda \). In addition, these two limits are well known in the literature as regimes III and I, respectively (e.g., Pyrak-Nolte et al. 1988; Renshaw 1995; Sisavath et al. 2003; Yu 2015). Investigation on regime I and its features is beyond the scope of this paper. Therefore, for further analysis, h, \(S_\mathrm{wtr}\), as well as \(\lambda \) are considered as rough fracture parameters. In the next few lines we endeavor to find a physical meaning for term \(\lambda \).
It is useful to relate \(\lambda \) to a geometrical property. To do so, in the following lines, two models for absolute permeability in the rough fracture are reviewed and equalized with the proposed model. For small scale roughness, Lomize (1951) suggested the following correlation for absolute permeability:
$$\begin{aligned} {{K}_{L}}=\frac{{{h}^{2}}}{12}{{\left[ 1+17{{\left( {\varepsilon }/{h}\; \right) }^{1.5}} \right] }^{-1}}, \end{aligned}$$
where \(\varepsilon \) is half of the mean asperity height.
To use Eq. (18), at first, \(\lambda \) is introduced into Eq. (15) and \(S_\mathrm{wtr}\) is set to zero.
$$\begin{aligned} K=\frac{h^2}{12}\frac{3}{\lambda ^2}\left[ 1-\frac{\tanh \left( \lambda \right) }{\lambda }\right] . \end{aligned}$$
Secondly, the main hyperbolic function of \(\lambda \) is replaced with some terms of Taylor series expansion at \(\lambda =0\), because Eq. (18) is used for small scale roughness. Therefore,
$$\begin{aligned} K=\frac{{{h}^{2}}}{12}{{\left[ \frac{{{\lambda }^{3}}}{3\left( \lambda -\tanh \left( \lambda \right) \right) } \right] }^{-1}}\approx \frac{{{h}^{2}}}{12}{{\left( 1+\frac{2}{5}{{\lambda }^{2}}+O\left( {{\lambda }^{4}} \right) \right) }^{-1}}. \end{aligned}$$
Comparing Eqs. 18 and 20 results in Eq. (21).
$$\begin{aligned} \lambda =\sqrt{\frac{85}{2}}{{\left( {\varepsilon }/{h}\; \right) }^{{3}/{4}\;}}. \end{aligned}$$
Equation (21) does not provide a physical attribute for \(\lambda \). The reason might be the empirical nature of Eq. (18). Furthermore, for large-scale roughness, Renshaw (1995) suggested Eq. (22) for absolute permeability based on theoretical interpretation.
$$\begin{aligned} {{K}_{R}}=\frac{{{h}^{2}}}{12}{{\left[ 1+{{\left( {\sigma }/{h}\; \right) }^{2}} \right] }^{-1.5}}, \end{aligned}$$
where \(\sigma /h\) is the standard deviation per mean aperture of the rough fracture which is dimensionless fracture roughness (e.g., see Pan 1999; Zimmerman et al. 1991).
In order to compare Eq. (19) with Eq. (22), Eq. (19) is changed to a proper form, and then, the main hyperbolic function of \(\lambda \) is replaced with its Taylor series expansion at \(\lambda =0\).
$$\begin{aligned} K=\frac{{{h}^{2}}}{12}{{\left[ \frac{{{\lambda }^{2}}}{{{3}^{{2}/{3}\;}}{{\left( \lambda -\tanh \left( \lambda \right) \right) }^{{2}/{3}\;}}} \right] }^{-1.5}}\approx \frac{{{h}^{2}}}{12}{{\left( 1+\frac{4}{15}{{\lambda }^{2}}+O\left( {{\lambda }^{4}} \right) \right) }^{-1.5}}. \end{aligned}$$
Comparing Eqs. 22 and 23 results in Eq. (24).
$$\begin{aligned} \lambda =\frac{\sqrt{15}}{2}\left( {\sigma }/{h}\; \right) . \end{aligned}$$
Equation 24 does provide a physical attribute for the proposed parameter \(\lambda \) as a parameter proportional to measurable fracture roughness.
Finally, Fig. 4 compares Eqs. 19 and 22 with numerical results of Patir and Cheng (1978) and Brown (1987). The prediction for Eq. (19) is closer to numerical results than Renshaw’s equation, see Table 1, which confirms that the basic assumptions of this study for single-phase flow in a rough fracture are correct.
Fig. 4

Comparison of normalized absolute permeability of a rough fracture calculated from our model (Eq. 19) with Renshaw model (Eq. 22), and numerical data of Patir and Cheng (1978) and Brown (1987)

2.2 Two-Phase Equations

In this study, for two-phase flow in a smooth fracture, the idealized flow configuration of alternating channel flow is modified, see Fig. 5.
Table 1

The calculated root mean square deviation (RMSD) and average absolute relative deviation (AARD) values for Both Renshaw’s (RM) and the proposed model (PM)



Reference of data


AARD\(^\mathrm{b} [\%]\)


AARD [%]

Brown (1987)





Patir and Cheng (1978)





\(^\mathrm{a}\) RMSD = \(\sqrt{\frac{1}{n}\sum \limits _{i=1}^{n}{{{\left( {{y}_{\mathrm{numeric.},i}}-{{y}_{\mathrm{anal.},i}} \right) }^{2}}}}\)

\(^\mathrm{b}\)AARD = \(\frac{100}{n}\sum \limits _{i=1}^{n}{\frac{\left| {{y}_{\mathrm{numeric.},i}}-{{y}_{\mathrm{anal.},i}} \right| }{{{y}_{\mathrm{numeric.},i}}}}\)

Fig. 5

Alternating channel flow configuration in an imperfect fracture or a fracture with permeable wall to wetting phase

This model interpolates between alternating channel flow (Pan 1999) and sandwich flow (Fourar and Lenormand 1998).

The flow of each phase is governed by Stokes equation (Bird et al. 2002, p. 85).
$$\begin{aligned}&\displaystyle -\frac{\partial {{p}_{i}}}{\partial x}=0,\end{aligned}$$
$$\begin{aligned}&\displaystyle -\frac{\partial {{p}_{i}}}{\partial y}+{{\rho }_{i}}g=0,\end{aligned}$$
$$\begin{aligned}&\displaystyle -\frac{\partial {{p}_{i}}}{\partial z}+{{\mu }_{i}}\left( \frac{{{\partial }^{2}}{{v}_{z,i}}}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}{{v}_{z,i}}}{\partial {{y}^{2}}} \right) =0, \end{aligned}$$
where g is the earth gravitational field intensity and \(\rho _i\) is the \(i^{th}\) fluid density. At the interfaces, continuity of fluid velocity and coupling condition are applied (Gross and Reusken 2011, p.9 and 19).
$$ \begin{aligned}&\displaystyle v_\mathrm{nwt}=v_\mathrm{wt}\ @\ x=l-a\ \& \ 0\le y\le h/2, \end{aligned}$$
$$ \begin{aligned}&\displaystyle p_\mathrm{nwt}-p_\mathrm{wt}=\sigma _\mathrm{{wt,nwt}}\kappa \ @\ x=l-a\ \& \ 0\le y\le h/2, \end{aligned}$$
$$ \begin{aligned}&\displaystyle {{\mu }_\mathrm{{nwt}}}\frac{\partial {{v}_\mathrm{{z,nwt}}}}{\partial x}-{{\mu }_\mathrm{{wt}}}\frac{\partial {{v}_\mathrm{{z,wt}}}}{\partial x}=0\ @\ x=l-a\ \& \ 0\le \ y\le h/2, \end{aligned}$$
$$ \begin{aligned}&\displaystyle v_\mathrm{{nwt}}=v_\mathrm{{wt}}\ @\ 0\le x\le l-a\ \& \ y=h/2, \end{aligned}$$
$$ \begin{aligned}&\displaystyle {{p}_\mathrm{{nwt}}}-{{p}_\mathrm{{wt}}}={{\sigma }_\mathrm{{wt,nwt}}}\kappa \ @\ 0\le x\le l-a\ \& \ y=h/2, \end{aligned}$$
$$ \begin{aligned}&\displaystyle {{\mu }_\mathrm{nwt}}\frac{\partial {{v}_\mathrm{{z,nwt}}}}{\partial y}-{{\mu }_\mathrm{{wt}}}\frac{\partial {{v}_\mathrm{{z,wt}}}}{\partial y}=0\ @\ 0\le x\le l-a\ \& \ y=h/2, \end{aligned}$$
where \(\sigma _\mathrm{{wt,nwt}}\) is the interfacial tension coefficient and \(\kappa \) is the interface curvature.
In this study Eqs. 2533 are solved for a limiting condition. When \(h\ll a\) and \(h\ll l\), all the partial derivatives in x-coordinate are negligible. For this situation, the main features of the model is still preserved, but the solution is much simplified. For pressure the generalized form is:
$$\begin{aligned} {{p}_{i}}={{p}_{i,0}}+\rho _{i}gy+c_{i}z, \end{aligned}$$
where \(p_{i,0}\) and \(c_i\) are constants. Introducing Eq. (34) into Eqs. 29 and 32 leads to:
$$\begin{aligned} p_\mathrm{{nwt,0}}-p_\mathrm{{wt,0}}+\left( {{\rho }_\mathrm{{nwt}}}-{{\rho }_\mathrm{{wt}}} \right) gy+\left( {{c}_\mathrm{{nwt}}}-{{c}_\mathrm{{wt}}} \right) z= & {} \sigma _\mathrm{{wt,nwt}}\kappa _{x}, \end{aligned}$$
$$\begin{aligned} {{p}_\mathrm{{nwt,0}}}-{{p}_{wt,0}}+\left( {{\rho }_\mathrm{{nwt}}}-{{\rho }_\mathrm{{wt}}} \right) g{h}/{2} + \left( {{c}_\mathrm{{nwt}}}-{{c}_\mathrm{{wt}}} \right) z= & {} {{\sigma }_\mathrm{{wt,nwt}}}{{\kappa }_{y}}. \end{aligned}$$
Both Eqs. 35 and 36 result into \(c_\mathrm{{wt}}=c_\mathrm{{nwt}}\) when it is assumed that the cross-sectional configuration of flow does not change in z-direction. This equality is very important because it means that although pressure in each phase is different than the other, their derivative with respect to z, which is the main driving force, is equal in both phases and it is named \({\partial p}/{\partial z}\). No more analysis on pressure is required.
For velocities, the simplified solutions are:
$$\begin{aligned}&\displaystyle {{v}_\mathrm{{z,nwt}}}=\frac{1}{2}\frac{1}{{{\mu }_\mathrm{{nwt}}}}\frac{\partial p}{\partial z}\left[ {{y}^{2}}+\left( \frac{{{\mu }_\mathrm{{nwt}}}}{{{\mu }_\mathrm{wt}}}-1 \right) {{\left( {h}/{2} \right) }^{2}}-\frac{{{\mu }_\mathrm{{nwt}}}}{{{\mu }_\mathrm{{wt}}}}{{\left( {h}/{2}+\delta \right) }^{2}} \right] , \end{aligned}$$
$$\begin{aligned}&\displaystyle {{v}_\mathrm{{z,wt}}}=\frac{1}{2{{\mu }_\mathrm{{wt}}}}\frac{\partial p}{\partial z}\left[ {{y}^{2}}-{{\left( {h}/{2}\;+\delta \right) }^{2}} \right] . \end{aligned}$$
As the next step, \(v_{z}s\) are averaged over the fracture open cross-sectional area and the superficial velocities of fracture (\({{\bar{v}}_{z,i}}\)) are calculated.
$$\begin{aligned} {{\bar{v}}_\mathrm{{z,nwt}}}= & {} \frac{\int \limits _{0}^{l-a} {\int \limits _{-{h}/{2}}^{{h}/{2}}{{{v}_\mathrm{{z,nwt}}}dydx}}}{hl}= \frac{{h}^{2}}{12}\frac{1}{\mu _{nwt}}\frac{\partial p}{\partial z}\frac{l-a}{l} \nonumber \\&\times \left[ \frac{1}{2}+\frac{3}{2}\left( \frac{{{\mu }_\mathrm{{nwt}}}}{{{\mu }_\mathrm{wt}}}-1 \right) -\frac{3}{2}\frac{{{\mu }_\mathrm{{nwt}}}}{{{\mu }_\mathrm{wt}}}{{\left( 1+\frac{2\delta }{h} \right) }^{2}} \right] , \end{aligned}$$
$$\begin{aligned} {{\bar{v}}_\mathrm{{z,wt}}}= & {} \frac{\int \limits _{l-a}^{l}{\int \limits _{-{h}/{2}\;}^{{h}/{2}\;}{{{v}_\mathrm{{z,wt}}}\mathrm{d}y\mathrm{d}x}}}{hl}=\frac{{{h}^{2}}}{12}\frac{1}{2{{\mu }_\mathrm{wt}}}\frac{\partial p}{\partial z}\left[ 1-3{{\left( 1+\frac{2\delta }{h} \right) }^{2}} \right] \frac{a}{l}. \end{aligned}$$
According to Fig. 5, saturations for the wetting phase and the non-wetting phase are defined as follows:
$$\begin{aligned} {{S}_\mathrm{wt}}= & {} \frac{2\delta l+ah}{\left( h+2\delta \right) l}, \end{aligned}$$
$$\begin{aligned} {{S}_\mathrm{{nwt}}}= & {} \frac{h\left( l-a \right) }{\left( h+2\delta \right) l}. \end{aligned}$$
Introducing Eqs. 41 and 42 into Eqs. 39 and 40 results in:
$$\begin{aligned} {{\bar{v}}_\mathrm{{z,nwt}}}= & {} \frac{{{h}^{2}}}{12}\frac{1}{{{\mu }_\mathrm{{nwt}}}}\frac{\partial p}{\partial z}\frac{{{S}_\mathrm{{nwt}}}}{1-{{S}_\mathrm{wtr}}}\nonumber \\&\times \left[ \frac{1}{2}+\frac{3}{2}\left( \frac{{{\mu }_\mathrm{{nwt}}}}{{{\mu }_\mathrm{wt}}}-1 \right) -\frac{3}{2}\frac{{{\mu }_\mathrm{{nwt}}}}{{{\mu }_\mathrm{wt}}}\frac{1}{{{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}}} \right] , \end{aligned}$$
$$\begin{aligned} {{\bar{v}}_\mathrm{{z,wt}}}= & {} \frac{{{h}^{2}}}{12}\frac{1}{{{\mu }_\mathrm{wt}}}\frac{\partial p}{\partial z}\frac{{{S}_\mathrm{wt}}-{{S}_\mathrm{wtr}}}{1-{{S}_\mathrm{wtr}}}\left[ \frac{1}{2}-\frac{3}{2}\frac{1}{{{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}}} \right] , \end{aligned}$$
and finally by comparing Eqs. 16, 43, and 44 with extended form of Darcy’s law for two-phase flow, relative permeability equations are derived.
$$\begin{aligned} {{k}_\mathrm{{rnwt}}}= & {} \frac{3\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}}\; \right) +\left[ 2-3\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}}\; \right) \right] {{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}}}{3-{{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}}}\frac{{{S}_{nwt}}}{1-{{S}_\mathrm{wtr}}}, \end{aligned}$$
$$\begin{aligned} {{k}_\mathrm{rwt}}= & {} \frac{{{S}_\mathrm{wt}}-{{S}_\mathrm{wtr}}}{1-{{S}_\mathrm{wtr}}}. \end{aligned}$$
For two-phase flow in a rough fracture, the idealized flow configuration of alternating channel flow is modified, see Fig. 6.
Fig. 6

Alternating channel flow configuration for two-phase flow in an imperfect or permeable wall to wetting phase rough fracture

In this study, similar to the single-phase model, the rough fracture is assumed as a medium between a channel and a porous medium. Therefore, a formulation like Brinkman’s equation is used here, which is modified for two-phase flow.
$$\begin{aligned} -\frac{\partial p}{\partial z}+{{\mu }_{i}}\frac{{{\partial }^{2}}{{v}_{z,i}}}{\partial {{y}^{2}}}-\frac{{{\mu }_{i}}}{{{k}_{eff,i}}}{{v}_{z,i}}=0. \end{aligned}$$
All the x-direction dependencies are ignored because \(h\ll a\) and \(h\ll l\).
In Eq. (47), \(k_{e\!f\!f,i}\) is an effective permeability that phase i will have, if the fracture converges to a porous medium. Equation (47) is solved using Eqs. (4) and (5) as boundary conditions, and Eqs. (31) and (33) as interface conditions.
$$\begin{aligned} v_{z,nwt}= & {} -\left[ \begin{array}{l} 1-\frac{\cosh \frac{\delta }{\sqrt{k_{eff,w\!t}}}+\left( 1-\cosh \frac{\delta }{\sqrt{k_{eff,w\!t}}}\right) \frac{\mu _{n\!w\!t}}{\mu _{w\!t}}\frac{k_{eff,w\!t}}{k_{eff,n\!w\!t}}}{\left[ \begin{array}{c} \cosh \frac{h/2}{\sqrt{k_{eff,n\!w\!t}}}\cosh \frac{\delta }{\sqrt{k_{eff,w\!t}}} +\frac{\mu _{n\!w\!t}}{\mu _{w\!t}}\sqrt{\frac{k_{eff,w\!t}}{k_{eff,n\!w\!t}}}\sinh \frac{h/2}{\sqrt{k_{eff,n\!w\!t}}}\sinh \frac{\delta }{\sqrt{k_{eff,w\!t}}}\end{array}\right] }\\ \times \cosh \frac{y}{\sqrt{k_{eff,n\!w\!t}}}\end{array} \right] \nonumber \\&\times \frac{k_{eff,n\!w\!t}}{\mu _{n\!w\!t}}\frac{\partial p}{\partial z} , \end{aligned}$$
$$\begin{aligned} {{v}_\mathrm{{z,wt}}}= & {} -\left[ 1-\frac{\cosh \left( {y}/{\sqrt{{{k}_{eff,w\!t}}}}\; \right) }{\cosh \left( {\left( {h}/{2}\;+\delta \right) }/{\sqrt{{{k}_{eff,w\!t}}}}\; \right) } \right] \frac{{{k}_{eff,w\!t}}}{{{\mu }_{w\!t}}}\frac{\partial p}{\partial z}. \end{aligned}$$
Similar to Eq. (12), Eqs. (48) and (49) are averaged over the open cross-sectional area in order to obtain fracture superficial velocities (\({{\bar{v}}_{z,i}}\)).
$$\begin{aligned} {{{\bar{v}}}_\mathrm{{z,nwt}}}= & {} \frac{\int \limits _{0}^{l-a}{\int \limits _{-{h}/{2}}^{{h}/{2}}{{{v}_\mathrm{{z,nwt}}}\mathrm{d}y\mathrm{d}x}}}{hl}=-\frac{{{k}_{eff,n\!w\!t}}}{{{\mu }_{n\!w\!t}}}\frac{\partial p}{\partial z}\frac{l-a}{l} \nonumber \\&\times \left[ \begin{array}{l} 1-\frac{\cosh \frac{\delta }{\sqrt{k_{eff,w\!t}}}+\left( 1-\cosh \frac{\delta }{\sqrt{k_{eff,w\!t}}}\right) \frac{\mu _{n\!w\!t}}{\mu _{w\!t}}\frac{k_{eff,w\!t}}{k_{eff,n\!w\!t}}}{\left[ \begin{array}{c} \cosh \frac{h/2}{\sqrt{k_{eff,n\!w\!t}}}\cosh \frac{\delta }{\sqrt{k_{eff,w\!t}}}\\ +\frac{\mu _{n\!w\!t}}{\mu _{w\!t}}\sqrt{\frac{k_{eff,w\!t}}{k_{eff,n\!w\!t}}}\sinh \frac{h/2}{\sqrt{k_{eff,n\!w\!t}}}\sinh \frac{\delta }{\sqrt{k_{eff,w\!t}}}\end{array}\right] }\\ \times \frac{2\sqrt{k_{eff,n\!w\!t}}}{h}\sinh \frac{h/2}{\sqrt{k_{e\!f\!\!f,n\!w\!t}}}\end{array} \right] , \end{aligned}$$
$$\begin{aligned} {{\bar{v}}_{z,wt}}= & {} \frac{\int \limits _{l-a}^{l}{\int \limits _{-{h}/{2}}^{{h}/{2}}{{{v}_\mathrm{{z,wt}}}\mathrm{d}y\mathrm{d}x}}}{hl}=-\frac{{{k}_{e\!f\!\!f,w\!t}}}{{{\mu }_{w\!t}}}\frac{\partial p}{\partial z}\frac{a}{l}\left[ 1-\frac{\sinh \frac{{h}/{2} }{\sqrt{{{k}_{e\!f\!\!f,w\!t}}}} }{\frac{h/2}{\sqrt{{{k}_{e\!f\!\!f,w\!t}}}}\cosh \frac{\ {h}/{2}+\delta }{\sqrt{{{k}_{e\!f\!\!f,w\!t}}}}} \right] .\qquad \end{aligned}$$
Similar to the single-phase flow, in the two-phase flow \({h/2}/{\sqrt{k_{e\!f\!\!f,i}}}\) is defined as \(\lambda _i\). Then, Eqs. 8, 41, and 42 are introduced into Eqs. 50 and 51. Therefore,
$$\begin{aligned} \begin{array}{rl} {{{\bar{v}}}_\mathrm{{z,nwt}}}=&{}-\frac{h^2}{12}\frac{1}{\mu _{nwt}}\frac{\partial p}{\partial z}\frac{S_{nwt}}{1-S_\mathrm{wtr}} \\ {} &{} \times \frac{3}{\lambda _{nwt}^2} \left[ \begin{array}{l} 1-\frac{\cosh \frac{S_\mathrm{wtr}\lambda _\mathrm{wt}}{1-S_\mathrm{wtr}}+\left( 1-\cosh \frac{S_\mathrm{wtr}\lambda _{wt}}{1-S_\mathrm{wtr}} \right) \frac{\mu _{n\!w\!t}}{\mu _{w\!t}}\left( \frac{\lambda _\mathrm{nwt}}{\lambda _\mathrm{wt}}\right) ^2}{\left[ \begin{array}{l} \cosh \lambda _\mathrm{nwt}\cosh \frac{S_\mathrm{wtr}\lambda _\mathrm{wt}}{1-S_\mathrm{wtr}}\\ +\frac{\mu _{n\!w\!t}}{\mu _{w\!t}}\frac{\lambda _\mathrm{nwt}}{\lambda _\mathrm{wt}}\sinh \lambda _\mathrm{nwt}\sinh \frac{S_\mathrm{wtr}\lambda _{wt}}{1-S_\mathrm{wtr}}\end{array}\right] }\\ \times \frac{\sinh \lambda _\mathrm{nwt}}{\lambda _\mathrm{nwt}}\end{array} \right] \end{array}, \end{aligned}$$
$$\begin{aligned} {{\bar{v}}_\mathrm{{z,wt}}}=-\frac{{{h}^{2}}}{12}\frac{1}{{{\mu }_\mathrm{wt}}}\frac{\partial p}{\partial z}\frac{{{S}_\mathrm{wt}}-{{S}_\mathrm{wtr}}}{1-{{S}_\mathrm{wtr}}}\frac{3}{\lambda _{wt}^{2}}\left[ 1-\frac{1}{{{\lambda }_\mathrm{wt}}}\frac{\sinh {{\lambda }_{wt}} }{\cosh \left( {{{\lambda }_\mathrm{wt}}}/{\left( 1-{{S}_\mathrm{wtr}} \right) }\; \right) } \right] . \end{aligned}$$
Comparing Eqs. 52 and 53 with extended Darcy’s law for two-phase flow and introducing Eq. (15) for absolute permeability in a rough fracture and \(\lambda \), lead to:
$$\begin{aligned} k_{rnwt}= & {} \frac{\lambda ^2}{\lambda _{nwt}^2}\frac{1-\frac{1+\left[ \frac{1}{\cosh \frac{S_\mathrm{wtr}\lambda _\mathrm{wt}}{1-S_\mathrm{wtr}}}-1\right] \frac{\mu _{nwt}}{\mu \mathrm{wt}}\left( \frac{\lambda _{nwt}}{\lambda _\mathrm{wt}}\right) ^2}{1+\frac{\mu _\mathrm{nwt}}{\mu \mathrm{wt}}\frac{\lambda _\mathrm{nwt}}{\lambda _\mathrm{wt}}\tanh \lambda _\mathrm{nwt}\tanh \frac{S_\mathrm{wtr}\lambda _{wt}}{1-S_\mathrm{wtr}}}\frac{\tanh \lambda _\mathrm{nwt}}{\lambda _\mathrm{nwt}} }{1-\frac{\cosh \lambda }{\cosh \left( \lambda /\left( 1-S_\mathrm{wtr}\right) \right) }\frac{\tanh \lambda }{\lambda } }\frac{S_{nwt}}{1-S_\mathrm{wtr}}, \end{aligned}$$
$$\begin{aligned} k_{rwt}= & {} \frac{\lambda ^2}{\lambda _{wt}^2}\frac{1-\frac{\cosh \lambda _{wt}}{\cosh \left( \lambda _{wt}/\left( 1-S_\mathrm{wtr}\right) \right) }\frac{\tanh \lambda _{wt}}{\lambda _{wt}} }{1-\frac{\cosh \lambda }{\cosh \left( \lambda /\left( 1-S_\mathrm{wtr}\right) \right) }\frac{\tanh \lambda }{\lambda } }\frac{S_{wt}-S_\mathrm{wtr}}{1-S_{wtr}}, \end{aligned}$$
where \(\lambda ^2/\lambda _i^2\) is equal to \(k_{e\!f\!\!f,i}/k\). It means the relative permeability partition of porous medium in relative permeability of the rough fracture. To calculate an expression for \(\lambda ^2/\lambda _i^2\), first it is necessary to obtain values of \(k_{ri}\) at limits of \(\lambda \) with the constraint of finite value for \(\lambda ^2/\lambda _i^2\).
$$\begin{aligned} \lim \limits _{\lambda \rightarrow 0}k_\mathrm{rnwt}= & {} \frac{\frac{2}{3}+2\frac{\mu _\mathrm{nwt}}{\mu _\mathrm{wt}}\frac{S_\mathrm{wtr}}{1-S_\mathrm{wtr}}+\frac{\mu _\mathrm{nwt}}{\mu _\mathrm{wt}}\left( \frac{S_\mathrm{wtr}}{1-S_\mathrm{wtr}}\right) ^2 }{\frac{1}{\left( 1-S_\mathrm{wtr}\right) ^2}-\frac{1}{3}}\frac{S_{nwt}}{1-S_\mathrm{wtr}}, \end{aligned}$$
$$\begin{aligned} \lim \limits _{\lambda \rightarrow 0}k_\mathrm{rwt}= & {} \frac{S_\mathrm{wt}-S_\mathrm{wtr}}{1-S_\mathrm{wtr}}, \end{aligned}$$
$$\begin{aligned} \lim \limits _{\lambda \rightarrow \infty }k_\mathrm{rnwt}= & {} \frac{\lambda ^2}{\lambda _\mathrm{nwt}^2}\frac{S_\mathrm{nwt}}{1-S_\mathrm{wtr}}, \end{aligned}$$
$$\begin{aligned} \lim \limits _{\lambda \rightarrow \infty }k_\mathrm{rnwt}= & {} \frac{\lambda ^2}{\lambda _\mathrm{wt}^2}\frac{S_\mathrm{wt}-S_\mathrm{wtr}}{1-S_{wtr}}. \end{aligned}$$
According to our earlier hypothesis that at \(\lambda \rightarrow +\infty \) a rough fracture converges to a porous medium, Eqs. 58 and 59 must be equal to relative permeabilities of a porous medium. Therefore, the Brook–Corey equation is applied to this limit. In addition, in order to have the same relative permeability end points in both limits (a smooth fracture and a porous medium), the end-point values at \(\lambda \rightarrow 0\) are also implemented to the other limit. Therefore, \(\lambda ^2/\lambda _i^2\) is calculated as follows:
$$\begin{aligned}&\displaystyle \frac{{{\lambda }^{2}}}{\lambda _\mathrm{nwt}^{2}}=\frac{3\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}}\; \right) +\left[ 2-3\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}}\; \right) \right] {{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}}}{ 3-{{\left( 1-{{S}_\mathrm{wtr}} \right) }^{2}} }{{\left( \frac{{{S}_{nwt}}}{1-{{S}_\mathrm{wtr}}} \right) }^{2}}, \end{aligned}$$
$$\begin{aligned}&\displaystyle \frac{{{\lambda }^{2}}}{\lambda _\mathrm{wt}^{2}}={{\left( \frac{{{S}_{wt}}-{{S}_\mathrm{wtr}}}{1-{{S}_\mathrm{wtr}}} \right) }^{2}}. \end{aligned}$$

3 Results and Discussion

The new developed formulations in this study (Eqs. 45, 46, 54, and 55) are validated, by comparing the results with the available experimental datasets in the literature. Four datasets were extracted from the literature and are placed in the two following subsections. Afterward, sensitivity analysis is applied to the formulation to investigate the effect of each parameter on non-wet relative permeability end point and the crossover point.

3.1 Water–Nitrogen System

For the smooth water-wet fracture data of Chen et al. (2004), a comparison is made with the results of Eqs. 4546 as well as Eqs. 5455.
Fig. 7

Comparison of Chen et al. (2004) data for smooth fracture with our model of alternating channel with a \(S_\mathrm{wtr}=0.15\) and \(\lambda = 0\) and b \(S_\mathrm{wtr}=0.15\) and \(\lambda = 0.46\) \(\left( R^2=0.906\right) \)

Generally, for a smooth fracture the common assumption is no residual saturation for fluid inside the fracture. However, Fig. 7a clearly shows that the tuned value of 0.15 for the irreducible water saturation is necessary due to the fact that this value meets gas-phase end point much better. Figure 7a also shows that the linear model for relative permeability values might work, especially for the wetting phase within the range of uncertainty of data. However, fitting the value of \(\lambda \) to the experimental data, see Fig. 7b, shows that liquid–gas phase interference plays a major role in both wet and non-wet relative permeabilities, especially after crossover point.

Chen and Horne (2006) investigated two water-wet rough fracture cases. It is interesting to note that the residual water saturation, \(S_\mathrm{wtr}\), the mean aperture, h, and the standard deviation, \(\sigma \), for their fractures were given. This helps to examine the predictive capability of the proposed model. For the first case of homogeneous rough fracture (HR), the parameters are \(S_\mathrm{wtr} = 0.25\), \(\sigma = 0.03\) mm, and \(h = 0.145\) mm. Therefore, from Eq. (24), degree of phase interference, \(\lambda \), can be calculated as \(\lambda _{H\!R} = 0.40\). For the second case or random rough fracture (RR), \(S_\mathrm{wtr} = 0.39\), \(\sigma = 0.05\) mm, and \(h = 0.24\) mm. So, \(\lambda _{RR} = 0.40\). In addition to the prediction for each rough case, \(\lambda \) was also used as a fitting parameter. The difference between \(\lambda \) values for the tuned and predictive cases is interpreted as the effect of fluid–fluid phase interference.
Fig. 8

Calculated relative permeability curves for the homogeneous rough fracture model of Chen and Horne (2006): a Predictive model \(\left( R^2=0.937\right) \) with parameters \(S_\mathrm{wtr} = 0.25\) and \(\lambda _{H\!R} = 0.40\); b fitted model \(\left( R^2=0.976\right) \) with parameters \(S_\mathrm{wtr} = 0.25\) and \(\lambda _{H\!R} = 0.69 \)

Figure 8 shows the calculated relative permeability curves for the homogeneous rough fracture model of Chen and Horne (2006). It also shows that the predictive model (Fig. 8a) could calculate \(k_{rw}\) data points as well as \(k_{rg}\) boundary points very well. Besides, the tuned model (Fig. 8b) follows both relative permeability curves adequately. This comparison also demonstrates that the value “0.29” added to the predicted value of \(\lambda _{H\!R}\) can be regarded as the result of fluid–fluid phase interference which has a larger influence on the middle saturations. In brief, in the total phase interference 42% is for fluid–fluid interaction and 58% for fluid–solid interaction.
Fig. 9

Calculated relative permeability curves for the random rough fracture model of Chen and Horne (2006): a Predictive model \(\left( R^2=0.906\right) \) with parameters \(S_\mathrm{wtr} = 0.39\) and \(\lambda _{RR} = 0.40\); b fitted model \(\left( R^2=0.910\right) \) with parameters \(S_\mathrm{wtr} = 0.39\) and \(\lambda _{RR} = 0.46\)

Figure 9 demonstrates the calculated relative permeability curves for the random rough fracture model of Chen and Horne (2006). The results indicate that the predictive model as well as the tuned model only agrees with the water phase data points for the entire saturation domain. Considering the gas phase, the proposed model as shown in Fig. 9 could only meet the experimental data range. The reason might be hidden in the abnormal S-shape form of relative permeability data. Bertels et al. (2001), for their S-shape relative permeability observations, mentioned that the capillary pressure monotonically increases with the increase of wetting phase saturation. It suggests that for a particular value of \(S_w\) (here \(~S_w=0.52\)), a change in capillary vs. viscous forces dominance occurs. Therefore, it is reasonable to expect a change in the degree of phase interference. However, in this study \(\lambda \) is a set as a constant. Accordingly, it cannot fully meet the S-shape behavior.

Furthermore, the tuned results through modifying \(\lambda \) show that only 13% of the phase interference is caused by fluid–fluid interaction, which means that the predicted fluid–solid interaction is the dominant interference in this fracture.

In addition, to evaluate the effectiveness of the proposed model, in Table 2 RMSD values of each relative permeability curve are compared with the tortuosity modeling results of Chen and Horne (2006) as well as Liu et al. (2013) parametric model. The results are comparable. However, for the gas-phase relative permeability, the proposed model is able to meet the laboratory data slightly better than Chen and Horne (2006) tortuosity model. Therefore, it confirms that for water/gas system this model is physically correct.
Table 2

The calculated root mean square deviation (RMSD) values for Both \(k_\mathrm{rw}\) and \(k_\mathrm{rg}\) using the proposed model (PM), Chen and Horne (2006) tortuosity model (TM), and Liu et al. (2013) parametric model (LM)

Reference of data











Chen et al. (2004), smooth





\(\textendash \)

\(\textendash \)

Chen and Horne (2006), HR







Chen and Horne (2006), RR







\(^{a}\) RMSD = \(\sqrt{\frac{1}{n}\sum \limits _{i=1}^{n}{{{\left( {{y}_{exp.,i}}-{{y}_{\mathrm{anal.},i}}\right) }^{2}}}}\)

3.2 Water–Oil System

In another experimental study, Raza et al. (2016) tested a smooth wall oil-wet horizontal fracture and produced the relative permeability curves. The reported data file is compared here with the proposed model for two different scenarios, with and without phase interference.
Fig. 10

A comparison of Raza et al. (2016) data for water–oil system \(\left( \mu _o/\mu _w=18.4\right) \) in a smooth oil-wet fracture with the proposed model for \(S_\mathrm{wtr}=0.14\) and \(\lambda = 0\)

Figure 10 shows that the end-point calculation method proposed in this study causes the water relative permeability to approach the measured data. However, it still overestimates the laboratory data which indicates that the imperfection hypothesis cannot completely cover all the physical mechanisms for water/oil system. Furthermore, careful assessment of error bars of Raza et al. (2016) data for oil phase beyond the crossover point shows that there exists a region with curvature. According to Raza et al. (2016) this curvature is due to phase interference. Therefore, in Fig. 11 the required phase interference is added. In addition, viscous coupling model (Eqs. 62 and 63) is also applied for better comparison.
Fig. 11

A comparison of Raza et al. (2016) data for water–oil system \(\left( \mu _o/\mu _w=18.4\right) \) in a smooth oil-wet fracture with the proposed model for \(S_\mathrm{wtr}=0.14\) and \(\lambda = 0.46\) \(\left( R^2=0.91 \right) \) as well as viscous coupling model \(\left( R^2=0.90\right) \)

In Fig. 11 the proposed model for water relative permeability curve is superior to viscous coupling model before the crossover point. The viscous coupling model better predicts relative permeability values after the crossover point. However, the curvatures are much better met by the proposed model which demonstrates that the real flow regime for the whole domain is a regime between sandwich flow and ideal channel flow as assumed in this study. Since this fracture is smooth, no interference for fluid–solid interaction is considered and the \(\lambda = 0.46\) is totally related to fluid–fluid interaction. This value is the same as the value fitted for both smooth and randomly rough fracture data reported by Chen and Horne (2006), see Figs. 7b and 9b. This equality might qualitatively show that the defined \(\lambda \) parameter could be interpreted as a universal parameter. However, further study is needed to prove this claim. In addition, both the proposed model and the viscous coupling model are unable to predict the water relative permeability end point, which demonstrate that none of them have completely met the physical mechanisms active there.

3.3 Sensitivity Analysis

3.3.1 Phase Interference Definition

Saboorian-Jooybari (2016) stated that when \(k_\mathrm{rwt}+k_\mathrm{rnwt}=1\), negligible phase interference existed. In addition, Raza et al. (2016) quoted that at a fixed saturation, the lower the relative permeability value of a given phase, the higher the effect of other phase interference on the mentioned phase. It was also stated that fluid–fluid interference and fluid–solid interference behave the same. Sufficient physical reasons for phase interference were presented by Honarpour et al. (1986). However, as it was mentioned earlier, more quantitative studies for this effect are required.

In addition, in the authors’ point of view, the viscous coupling effect must be separated from phase interference effect, where it is not considered separately in the literature (e.g., see Huo and Benson 2016). In the following reasoning, the need for portioning of these effects is explained through a simple example. Fourar and Lenormand (1998) derived Eqs. 62 and 63 for two-phase sandwich flow (non-wet in the middle) in a fracture.
$$\begin{aligned} {{k}_\mathrm{rwt}}= & {} \frac{S_{wt}^{2}}{2}\left( 3-{{S}_\mathrm{wt}} \right) , \end{aligned}$$
$$\begin{aligned} {{k}_\mathrm{rnwt}}= & {} {{\left( 1-{{S}_\mathrm{wt}} \right) }^{3}}+\frac{3}{2}\frac{{{\mu }_\mathrm{nwt}}}{{{\mu }_\mathrm{wt}}}{{S}_\mathrm{wt}}\left( 1-{{S}_\mathrm{wt}} \right) \left( 2-{{S}_\mathrm{wt}} \right) . \end{aligned}$$
The summation of these two relative permeability values leads to:
$$\begin{aligned} {{k}_{r,tot}}={{k}_\mathrm{rwt}}+{{k}_\mathrm{rnwt}}=1+\frac{3}{2}\left( \frac{{{\mu }_\mathrm{nwt}}}{{{\mu }_\mathrm{wt}}}-1 \right) {{S}_\mathrm{wt}}\left( 1-{{S}_\mathrm{wt}} \right) \left( 2-{{S}_\mathrm{wt}} \right) . \end{aligned}$$
The summation of relative permeabilities could be one, less than one or even more than one, which depends on viscosity ratio of the phases. However, the phase interference is known for lowering the summation of relative permeabilities to less than one. In addition, phase interference has a symmetrical nature which means it affects both phases in the same way, but viscous coupling effect is different.

In brief, three mechanisms are involved in relative permeability behavior of two-phase flow in a fracture: (1) imperfection or permeated wall hypothesis which controls the end points, (2) viscous coupling effect which breaks the balance in flow of the two phases, and (3) phase interference which is a binary interaction property that affects both phases’ relative permeability values in a symmetric manner with respect to saturation. In addition, phase interference mostly affects the curvature of the relative permeability curves. These three mechanisms have a primary effect, but they intervene in each other’s role as well. Therefore, to be able to analyze the effect of each, one needs to calculate their linear and quadratic effects based on distinct dimensionless parameters which represent each mechanism.

Each mechanism is known through its distinct parameter, where \(S_\mathrm{wtr}\) is for the wall effect, \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\) for the viscous coupling, and \(\lambda \) for the phase interference, respectively.

The following equation is used for sensitivity analysis (Saltelli et al. 2000, p. 82):
$$\begin{aligned} \delta f=\sum \limits _{i=1}^{n}{\frac{\partial f}{\partial {{x}_{i}}}\delta {{x}_{i}}}+\frac{1}{2}\sum \limits _{i=1}^{n}{\sum \limits _{j=1}^{n}{\frac{{{\partial }^{2}}f}{\partial {{x}_{i}}\partial {{x}_{j}}}\delta {{x}_{i}}\delta {{x}_{j}}}}+o\left( \delta {{x}^{2}} \right) , \end{aligned}$$
where \(x_i\) is \(i\mathrm{th}\) decision variable and \(\delta x_i\) is its variation. In this study \(S_\mathrm{wtr}\), \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\), and \(\lambda \) are the chosen decision variables.

In addition, Eq. (65) is a multivariable Taylor series expansion of the arbitrary function, f. It contains first- and second-order derivatives of the function, which represents each variables individual effect and binary interaction with the others, respectively. It is a well-known format used in commercial design of experiment software. Furthermore, all the derivatives are normalized by the use of the domain span of each decision variable involved. This normalization makes calculated values easy to be compared. For \(S_\mathrm{wtr}\) the domain span is assigned 0.3, suggested by Persoff and Pruess (1995). For \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\) the domain span is assigned 2 and the mean value 1, while according to Fourar and Lenormand (1998) the value 1 is the neutral value. In addition, value 0 is a typical value for gas–water system and value 2 is reasonable for a water-wet light oil–water system. Finally, for \(\lambda \) the domain span of 2 with a mean value of 1 is assigned according to data gathered by Renshaw (1995) for \(\sigma /h\), where for most of the data \(0<\sigma /h<1\). Therefore, from Eq. (24), \(0<\lambda <2\).

3.3.2 Non-wetting Phase Relative Permeability End Point

The non-wetting phase relative permeability end point, \(k_\mathrm{{rnwt,0}}\), is chosen to be studied through Eq. (65). Therefore \(f=k_\mathrm{{rnwt,0}}\).

All the needed partial derivatives were calculated numerically from Eq. 54 for \(S_\mathrm{nwt} = 1\). The results of the partial derivatives at different points are listed in Table 3.
Table 3

Estimated coefficients of non-wetting phase relative permeability end point,\({f}={k}_\mathrm{{rnwt,0}}\), where significant values are given in bold \(\left( \Delta S_\mathrm{wtr} = 0.3, \varDelta \left( \mu _{nwt}/\mu _\mathrm{wt}\right) = 2, \mathrm{and}\ \varDelta \lambda = 2 \right) \)

\(\left[ \begin{array}{ccc} S_\mathrm{wtr}&\mu _{nwt}/\mu _{wt}&\lambda \end{array}\right] \)





[0.15 1 0]

[0.15 1 1]

[0.15 1 2]

\({{\left( \frac{\partial {k}_{rnwt,0}}{\partial {{S}_\mathrm{wtr}}} \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\)




\({{\left[ \frac{\partial {k}_\mathrm{{rnwt,0}}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }\; \right] }_{0}}\varDelta \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_{wt}}} \right) \)




\({{\left( \frac{\partial {k}_\mathrm{{rnwt,0}}}{\partial \lambda }\; \right) }_{0}}\varDelta \lambda \)




\( { \left[ { \frac{\partial ^{2}{k}_\mathrm{{rnwt,0}}}{\partial {{S}_\mathrm{wtr}}\partial \left( \mu _{nwt}/\mu _\mathrm{wt} \right) } }\right] _{0}} \varDelta {{S}_\mathrm{wtr}}\varDelta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt} \right) \)




\({{\left( \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial {{S}_\mathrm{wtr}}\partial \lambda } \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\varDelta \lambda \)




\({{\left[ \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \partial \lambda } \right] }_{0}}\varDelta \left( {{{\mu }_{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \varDelta \lambda \)




\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial S_\mathrm{wtr}^{2}} \right) }_{0}}{{\left( \varDelta {{S}_\mathrm{wtr}} \right) }^{2}} \)




\(\frac{1}{2}{{\left[ \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial {{\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }^{2}}} \right] }_{0}} \left[ \varDelta {\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_{wt}}} \right) }\right] ^{2} \)




\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial {{\lambda }^{2}}} \right) }_{0}} \left( \varDelta {\lambda }\right) ^{2}\)




In Table 3 the linear and quadratic terms for the non-wetting phase relative permeability end point are calculated at three different points. The first column represents a two-phase system with no phase interference, the second column represents the same system with moderate phase interference, and the third column represents the system with high phase interference.

Moreover, Table 3 shows that \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\) is a key parameter which has linear effect and interacts with \(S_\mathrm{wtr}\) and both of these effects are crucial to this study. The interaction between \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\) and \(S_\mathrm{wtr}\) demonstrates that for a water-wet system, when the non-wetting phase is gas-like, i.e., \(\delta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt}\right) \sim -1\), \(k_\mathrm{{rnwt,0}}\) reduces critically as \(S_\mathrm{wtr}\) increases. On the contrary, when the non-wetting phase is oil-like, i.e., \(\delta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt}\right) \sim +1\), \(k_\mathrm{{rnwt,0}}\) increases with the increase of \(S_\mathrm{wtr}\). The reduction in non-wet relative permeability end point is in line with the reports in the literature (e.g., see Chen et al. 2004; Raza et al. 2016). Although in Table 3 an interactive effect for \(\lambda \) and \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\) is revealed at moderate phase interferences, it is small compared with the interaction of \(S_\mathrm{wtr}\) and \(\mu _\mathrm{nwt}/\mu _\mathrm{wt}\). Therefore, it is neglected.

3.3.3 Relative Permeability Crossover Point

The normalized form of relative permeability crossover point is considered for sensitivity analysis. Its definition is given in Eq. (66).
$$\begin{aligned} {f}=\frac{{{k}_{r,c.o.}}}{1+{{k}_\mathrm{{rnwt,0}}}}. \end{aligned}$$
This form of normalization is implemented in order to diminish the effect of non-wet relative permeability end point on the crossover point. The results for the partial derivatives at different points are listed in Table 4.
Table 4

Estimated coefficients of normalized relative permeability crossover point, \({f}={{{k}_{r,c.o.}}}/\left( {1+{{k}_\mathrm{{rnwt,0}}}}\right) \), where significant values are given in bold \(\left( \varDelta S_\mathrm{wtr} = 0.3, \varDelta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt}\right) = 2, and\ \varDelta \lambda = 2 \right) \)

\(\left[ \begin{array}{ccc} S_\mathrm{wtr}&\mu _{nwt}/\mu _{wt}&\lambda \end{array}\right] \)





[0.15 1 0]

[0.15 1 1]

[0.15 1 2]

\({{\left( \frac{\partial {f}}{\partial {{S}_\mathrm{wtr}}} \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\)




\({{\left[ \frac{\partial {f}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }\; \right] }_{0}}\varDelta \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \)




\({{\left( \frac{\partial {f}}{\partial \lambda }\; \right) }_{0}}\varDelta \lambda \)




\( { \left[ { \frac{\partial ^{2}f}{\partial {{S}_\mathrm{wtr}}\partial \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt} \right) } }\right] _{0}} \varDelta {{S}_\mathrm{wtr}}\varDelta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt} \right) \)




\({{\left( \frac{{{\partial }^{2}}{f}}{\partial {{S}_\mathrm{wtr}}\partial \lambda } \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\varDelta \lambda \)




\({{\left[ \frac{{{\partial }^{2}}{f}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \partial \lambda } \right] }_{0}}\varDelta \left( {{{\mu }_{nwt}}}/{{{\mu }_{wt}}} \right) \varDelta \lambda \)




\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{f}}{\partial S_\mathrm{wtr}^{2}} \right) }_{0}}{{\left( \varDelta {{S}_\mathrm{wtr}} \right) }^{2}} \)




\(\frac{1}{2}{{\left[ \frac{{{\partial }^{2}}{f}}{\partial {{\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }^{2}}} \right] }_{0}} \left[ \varDelta {\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_{wt}}} \right) }\right] ^{2} \)




\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{f}}{\partial {{\lambda }^{2}}} \right) }_{0}} \left( \varDelta {\lambda }\right) ^{2}\)




The results presented in Table 4 show the right definition of \(\lambda \) as the degree of phase interference. The first column of the results in Table 4 shows that f is highly sensitive to \(\lambda ^2\). In a moderate phase interference condition, or the second column, it is less sensitive, but still important and it distributes between linear and quadratic terms of \(\lambda \). At the third column or the high phase interference, all the sensitivities are negligible. This analysis demonstrates that f and \(\lambda \) represent the same concept.

4 Conclusion

An analytical study was performed on single- and two-phase flows inside 2D smooth and rough fractures. Application of the novel technique of fracture imperfection or permeated wall hypothesis gave a simple method to calculate non-wetting phase relative permeability end point. In addition, the application of Brinkman’s equation to single- and two-phase flows in a rough fracture led to a practical definition for the degree of phase interference. Moreover, this approach suggested that three mechanisms were active in two-phase fracture flow and defined three easily measurable parameters as their representatives.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran
  2. 2.Department of Chemical and Biomolecular EngineeringRice UniversityHoustonUSA
  3. 3.Sharif Upstream Petroleum Research Institute, Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran

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