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

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## Abstract

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.

## Keywords

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

*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):

*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).

*K*, is obtained by comparing Eq. (9) with Darcy’s law.

*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.

*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 \(\lambda \).

### 2.2 Two-Phase Equations

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

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

*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).

*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.

## 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

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.

*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.

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.

Reference of data | PM | TM | LM | |||
---|---|---|---|---|---|---|

\(k_\mathrm{rw}\) | \(k_\mathrm{rg}\) | \(k_\mathrm{rw}\) | \(k_\mathrm{rg}\) | \(k_\mathrm{rw}\) | \(k_\mathrm{rg}\) | |

Chen et al. (2004), smooth | 0.0779 | 0.0737 | 0.0578 | 0.0647 | \(\textendash \) | \(\textendash \) |

Chen and Horne (2006), HR | 0.0467 | 0.0370 | 0.0372 | 0.0597 | 0.0682 | 0.0445 |

Chen and Horne (2006), RR | 0.0594 | 0.0481 | 0.0385 | 0.0456 | 0.0450 | 0.0287 |

### 3.2 Water–Oil System

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 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.

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 \(f=k_\mathrm{{rnwt,0}}\).

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

\(\left[ \begin{array}{ccc} S_\mathrm{wtr}&\mu _{nwt}/\mu _{wt}&\lambda \end{array}\right] \) | Point\(_1\) | Point\(_2\) | Point\(_3\) |
---|---|---|---|

[0.15 1 0] | [0.15 1 1] | [0.15 1 2] | |

\({{\left( \frac{\partial {k}_{rnwt,0}}{\partial {{S}_\mathrm{wtr}}} \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\) | 0.000 | 0.018 | 0.041 |

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

\({{\left( \frac{\partial {k}_\mathrm{{rnwt,0}}}{\partial \lambda }\; \right) }_{0}}\varDelta \lambda \) | 0.000 | 0.008 | 0.008 |

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

\({{\left( \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial {{S}_\mathrm{wtr}}\partial \lambda } \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\varDelta \lambda \) | 0.000 | 0.050 | 0.044 |

\({{\left[ \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \partial \lambda } \right] }_{0}}\varDelta \left( {{{\mu }_{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \varDelta \lambda \) | 0.000 | | 0.042 |

\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial S_\mathrm{wtr}^{2}} \right) }_{0}}{{\left( \varDelta {{S}_\mathrm{wtr}} \right) }^{2}} \) | 0.000 | 0.037 | 0.077 |

\(\frac{1}{2}{{\left[ \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial {{\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }^{2}}} \right] }_{0}} \left[ \varDelta {\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_{wt}}} \right) }\right] ^{2} \) | 0.000 | −0.012 | −0.022 |

\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{k}_\mathrm{{rnwt,0}}}{\partial {{\lambda }^{2}}} \right) }_{0}} \left( \varDelta {\lambda }\right) ^{2}\) | 0.016 | 0.000 | 0.000 |

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

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 \(\left( \varDelta S_\mathrm{wtr} = 0.3, \varDelta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt}\right) = 2, and\ \varDelta \lambda = 2 \right) \)

\(\left[ \begin{array}{ccc} S_\mathrm{wtr}&\mu _{nwt}/\mu _{wt}&\lambda \end{array}\right] \) | Point\(_1\) | Point\(_2\) | Point\(_3\) |
---|---|---|---|

[0.15 1 0] | [0.15 1 1] | [0.15 1 2] | |

\({{\left( \frac{\partial {f}}{\partial {{S}_\mathrm{wtr}}} \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\) | 0.000 | −0.031 | −0.018 |

\({{\left[ \frac{\partial {f}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }\; \right] }_{0}}\varDelta \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \) | 0.000 | 0.002 | 0.000 |

\({{\left( \frac{\partial {f}}{\partial \lambda }\; \right) }_{0}}\varDelta \lambda \) | −0.004 | − | −0.034 |

\( { \left[ { \frac{\partial ^{2}f}{\partial {{S}_\mathrm{wtr}}\partial \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt} \right) } }\right] _{0}} \varDelta {{S}_\mathrm{wtr}}\varDelta \left( \mu _\mathrm{nwt}/\mu _\mathrm{wt} \right) \) | 0.000 | 0.000 | −0.002 |

\({{\left( \frac{{{\partial }^{2}}{f}}{\partial {{S}_\mathrm{wtr}}\partial \lambda } \right) }_{0}}\varDelta {{S}_\mathrm{wtr}}\varDelta \lambda \) | −0.003 | 0.038 | 0.015 |

\({{\left[ \frac{{{\partial }^{2}}{f}}{\partial \left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) \partial \lambda } \right] }_{0}}\varDelta \left( {{{\mu }_{nwt}}}/{{{\mu }_{wt}}} \right) \varDelta \lambda \) | 0.000 | −0.006 | −0.002 |

\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{f}}{\partial S_\mathrm{wtr}^{2}} \right) }_{0}}{{\left( \varDelta {{S}_\mathrm{wtr}} \right) }^{2}} \) | 0.000 | −0.005 | −0.004 |

\(\frac{1}{2}{{\left[ \frac{{{\partial }^{2}}{f}}{\partial {{\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_\mathrm{wt}}} \right) }^{2}}} \right] }_{0}} \left[ \varDelta {\left( {{{\mu }_\mathrm{nwt}}}/{{{\mu }_{wt}}} \right) }\right] ^{2} \) | −0.033 | −0.016 | −0.009 |

\(\frac{1}{2}{{\left( \frac{{{\partial }^{2}}{f}}{\partial {{\lambda }^{2}}} \right) }_{0}} \left( \varDelta {\lambda }\right) ^{2}\) | − | | 0.042 |

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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