Non-Darcian two-phase flow in a transparent replica of a rough-walled rock fracture

Authors


Abstract

[1] This article presents experimental results for single- and two-phase flows at high flow rates through a replica of an actual rough-walled rock fracture. The results of the single-phase flow are interpreted using non-Darcian laws: the weak inertia cubic law, Forchheimer's law, and the full cubic law. They allow the determination of the fracture's intrinsic properties (absolute permeability and inertial coefficient). These laws are then generalized to describe non-Darcian two-phase flows. The generalized cubic and Forchheimer's laws are shown to be inconsistent in terms of the relative permeabilities. The generalization of the full cubic law is used to describe the liquid and gas relative permeabilities in the non-Darcian regime with a good accuracy.

1. Introduction

[2] Two-phase flows through rough-walled rock fractures occur in several industrial processes, including the underground storage of nuclear wastes, oil and gas recovery, and the production of geothermal energy. In many cases, the fluids flow in a very few open fractures or porous layers with high permeabilities, leading to flow rates that are high enough to generate inertial effects. Such flows cannot be modeled by Darcy's law and are known as non-Darcian flows. High flow rates of gas and liquids in porous media and fractures can be found in petroleum engineering, chemical engineering [Lockhart and Martinelli, 1949], geological storage of carbon dioxide near injection wells, and in problems related to the cooling of debris after an accident in a nuclear plant. In the last years, single-phase flow at high flow rates in rough-walled rock fractures and porous media has been widely studied experimentally, numerically and theoretically [Barrère, 1990; Mei and Auriault, 1991; Firdaouss et al., 1997; Andrade et al., 1999; Skjetne et al., 1999; Fourar et al., 2004; Panfilov and Fourar, 2006].

[3] While the physics of flow in porous media and rough-walled fractures is the same, there is a significant difference in terms of tortuosity. For fractures, the tortuosity is lower than for porous media. As a result, the transition from Darcy's law to a fully inertial regime in a fracture requires a substantial increase of the flow rate while in porous media this transition is achieved quickly. In other words, in terms of Reynolds number, the transition regime (also known as the weak inertia regime) is wider for fractures than for porous media [Zimmerman et al., 2004; Lo Jacono et al., 2005]. Therefore, for two-phase flows at high flow rates, one would reasonably expect also a wider transition regime for fractures than for porous media.

[4] Although there have been successful studies of two-phase flow in the Darcian flow regime where the inertia effects are negligible [Corey, 1954; Merrill, 1975; Chen et al., 2004; Chen and Horne, 2006; M. Fourar and R. Lenormand, A viscous coupling model for relative permeabilities in fractures, paper SPE 49006 presented at 1998 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 27–30 September, 1998], the fundamentals of non-Darcian two-phase flow in fractures are still poorly understood. The direct measurement of the flow properties often is not possible mainly because of the small fracture volume but also because there are several possible flow structures [Fourar and Bories, 1995] that have not been completely explored nor understood. As a result, the calculation of the pressure drop and saturation must rely on empirical models [Lipinski, 1981; Buchlin and Stubos, 1987; Fourar et al., 1993; Fourar and Lenormand, 2000; B. D. Turland and K. A. Moore, One-dimensional models of boiling and dryout: Post accident debris cooling, paper presented at 5th Post Accident Heat Removal Information Exchange Meeting, Nuclear Research Center, Karlsruhe, Germany, 28–30 July 1982, 1983; T. Schulenberg and U. Muller, A refined model for the coolability of core debris with flow entry from bottom, paper presented at 6th Information Exchange Meeting on Debris Coolability, University of California, Los Angeles, California, November, 1984].

[5] The modeling of two-phase flows through porous media and fractures is based on the notion of relative permeability, which is a generalization of the single-phase Darcy's law [Bear, 1972]:

equation image
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where the subscripts l and g stand for the liquid and gas phases, respectively. ΔP/L is the pressure drop per unit of length, q is the volumetric flow rate, A is the cross-sectional area, μ is the dynamic viscosity, K is the intrinsic permeability of the media, and kr is the relative permeability.

[6] The relative permeabilities in a two-phase flow account for the interference of each phase with the flow of the other. It is generally assumed that they depend uniquely on the phase saturation; this assumption is valid if the inertial forces are negligible compared with the viscous forces. Several models for the relative permeabilities have been proposed in the literature: the X curve model [Merrill, 1975], the Corey model [Corey, 1954], the viscous coupling model (Fourar and Lenormand, presented paper, 1998) and the tortuous channel model [Chen et al., 2004; Chen and Horne, 2006]. These models are valid for the capillary and viscous regimes, and they depend on the saturation only. They are not suitable for non-Darcian flows, where the inertial forces are not negligible compared with the viscous forces. In those flows, the relative permeabilities have been shown to depend on the saturation and the Reynolds number [Buchlin and Stubos, 1987; Fourar et al., 1993; Fourar and Lenormand, 2000].

[7] The approach most commonly used to account for inertial effects in two-phase flows through porous media and fractures is a generalization of Forchheimer's law [Forchheimer, 1901]. This law accounts for inertial effects during single-phase flow, and is usually generalized to two-phase flows by introducing two functions for each fluid: the relative permeability in the viscous term, and the relative inertial coefficient in the inertial term [Buchlin and Stubos, 1987; Fourar and Lenormand, 2000].

equation image
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where ρ is the fluid density, β is the inertial coefficient and βr is the relative inertial coefficient. The weakness of this approach is that it introduces four unknown functions. Generally, kr (the Corey or viscous coupling model) is assumed from a correlation, and βr is determined experimentally. However, the quality of the results depends highly on the correlation chosen for kr [Lipinski, 1981; Turland and Moore, presented paper, 1983; Schulenberg and Muller, presented paper, 1984].

[8] Another alternative is to assume that kr and 1/βr have the same values for each phase [Lipinski, 1982; Saez and Carbonell, 1985; H. S. Lee and I. Catton, Two-phase flow in stratified porous media, paper presented at 6th Information Exchange Meeting on Debris Coolability, University of California, Los Angeles, California, November, 1984]. This assumption is equivalent to using the Lockhart-Martinelli model [Lockhart and Martinelli, 1949], initially developed for two-phase flow in pipes, and which leads to empirical predictive laws very useful in chemical engineering. The two-phase flow pressure gradient is always greater than the single-phase pressure gradient of each phase flowing at the same flow rate. The Lockhart-Martinelli model accounts for this property by introducing two factors also known as liquid and gas multipliers:

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where subscript s stands for single-phase flow. Φl and Φg express the ratio between the two-phase and the single-phase pressure gradients. Instead of saturation, a new variable X = Φgl, known as the Martinelli parameter, represents the relative importance of the flow of the liquid to the gas flow. By comparing equations (5) and (6) with equations (1) and (2), it is apparent that Φl and Φg are analogous to 1/krl and 1/krg. Therefore the Lockhart-Martinelli model can be seen as a generalization of Darcy's law for non-Darcian flows. However, as for kr in the inertial regime, the Lockhart-Martinelli parameter depends highly on the Reynolds number of the two fluids.

[9] To explicitly account for the Reynolds numbers, Fourar and Lenormand [2001] proposed a new model that is also based on a generalization of the single-phase Forchheimer's law. In their approach, only one function instead of two is introduced to account for the presence of the second fluid; it is simply assumed that the superficial velocity of each fluid is multiplied by a function F that depends on the fluid saturation only.

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[10] This model was tested by Fourar and Lenormand [2001] using experimental results performed with air and water at high flow rates (Forchheimer's regime) through a single artificially roughened fracture obtained by gluing a layer of glass beads to two glass plates. As the F functions are assumed to depend only on saturation, they were calculated analytically using a model derived and validated in the purely viscous regime in a single smooth fracture (Fourar and Lenormand, presented paper, 1998):

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where Sl is the saturation of the liquid phase in the fracture.

[11] However, since no accurate method of measuring the saturation was available, the model was tested only in terms of pressure drop.

[12] In this article we present experimental results of single- and two-phase flows through a replica of an actual rough-walled rock fracture. The fluids are flowing at various high rates, and we use different approaches for their interpretation. The article is organized as follows. First, the experimental setup and procedures are described. Next, the experimental results of the single- and two-phase flows are presented. The intrinsic properties of the fracture are determined from the single-phase flow by following these three approaches: the weak inertia cubic law, Forchheimer's law, and the full cubic law. Finally, a generalization of the full cubic law based on the F function approach is proposed. This model is used to describe the experimental results of two-phase flows at high flow rates.

2. Experimental Setup and Procedures

2.1. Experimental Setup

[13] The schematic of the experimental setup is shown in Figure 1. The apparatus was designed to measure the pressure drop of two immiscible fluids as they flow through a transparent replica of a natural fracture. The centerpiece of the apparatus is a transparent epoxy resin cast of both sides of a natural rough-walled rock fracture. The original fracture is a Vosges sandstone sample with dimensions approximately 26 cm long and 15 cm wide. There are four pressure ports drilled along the centerline of the lower plate to allow access to the pressure transducer. The air and water at the inlet were dispersed evenly through eight alternating channels; the outflow was drained through a single outlet. Water was injected by a volumetric pump (PCM, EcoMoineau M Series), and its flow rate was measured by a positive displacement flowmeter (Oval, Model LSM45). The airflow was regulated by a mass flow controller (Brooks Instrument, Model 5851S). One differential pressure transducer (Validyne, Model DP45) was used to measure the pressure drop through the fracture.

Figure 1.

Schematic of experimental apparatus.

2.2. Experimental Procedures

[14] To calculate the experimental krl and krg according to equations (1) and (2), it is necessary to know K, A, ΔPl, ΔPg, L, μl, and μg. The fracture length and viscosities are known, and the pressure drop is measured (in our experiments, ΔPl = ΔPg), but the intrinsic permeability K and the cross-sectional area A are not known for the fracture. To calculate K and A, a single-phase flow experiment was conducted at room temperature. In this experiment, the pressure drop across the fracture was measured while the fracture was fully saturated with water flowing at different rates. These values of the pressure drop were then used to calculate K and A, as discussed below in section 3.1.

[15] To determine the relative permeabilities, the fracture was initially saturated with water which was injected at a constant rate at the beginning of each experiment. Air injection was then added at a constant rate, which was increased stepwise. When a steady state was reached for each flow rate, the pressure drop across the fracture was measured. The instantaneous value of the pressure drop fluctuated rapidly, because of the successive arrival of different fluid phases at the pressure ports. Therefore only the time-averaged values were measured. Once the maximum airflow rate was reached, the fracture was resaturated with water and the experiment was repeated using a higher water flow rate.

[16] The relative permeabilities were determined for three different liquid flow rates. The corresponding liquid phase Reynolds numbers (as defined in section 4.1) were 0.07, 0.29, and 0.45.

3. Experimental Results

3.1. Single-Phase Flow

[17] Single-phase flow in porous media is generally modeled using Darcy's law, which is valid if the inertial forces are negligible compared with the viscous forces [Darcy, 1854]. This same approach also applies to flow in fractures. For a horizontal flow of an incompressible fluid with no gravity effects, Darcy's law is written in the form:

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[18] If the inertial forces cannot be neglected, the pressure drop per unit length is approximated by a cubic or quadratic function of the superficial velocity. At low Reynolds numbers (i.e., weak inertial regime), the nonlinear behavior can be described by the weak inertia cubic law [Barrère, 1990; Mei and Auriault, 1991; Firdaouss et al., 1997; Fourar et al., 2004]:

equation image

where γ is a dimensionless parameter for the nonlinear term.

[19] This equation was verified numerically for a two-dimensional periodic porous medium [Barrère, 1990; Firdaouss et al., 1997], and was also obtained theoretically using the homogenization approach for an isotropic homogeneous porous medium [Mei and Auriault, 1991].

[20] At higher Reynolds numbers (i.e., strong inertial regime), the empirical Forchheimer's law is used to account for the deviation from Darcy's law [Forchheimer, 1901]:

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[21] Also, in the strong inertial regime, a model of a high flow rate can be obtained by asymptotic expansions over the channel diameter, which is much smaller than the channel length [Panfilov and Fourar, 2006]. The resulting series for a corrugated channel provides a full cubic law:

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where the parameters β and γ may be negative or positive, depending on the channel geometry.

[22] In equations (12), (13), and (14), the intrinsic permeability K and the cross-sectional area A can be written as functions of the hydraulic aperture of the fracture [Konzuk and Kueper, 2004]:

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where h is the hydraulic aperture and w is the fracture width. It is important to note that for a rough-walled fracture, the hydraulic aperture is commonly defined for a given flow rate as the aperture of a smooth fracture of same dimensions generating the same pressure drop following Darcy's law. Therefore h is variable and decreases as the flow rate increases because of non-Darcian effects. In this article, we used the same approach as Fourar et al. [1993] in which experimental pressure drop measurements are modeled using non-Darcian models and thus, h is a unique value for all flow rates.

[23] Finally, equations (12), (13), and (14) can be rewritten in the following forms:

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[24] Several studies have been devoted to the prediction of the values of h, β, and γ. In most of these studies [Cornell and Katz, 1953; Geertsma, 1974; Noman and Archer, 1987; J. W. Neasham, The morphology of dispersed clay in sandstone reservoirs and its effects on sandstone shaliness, pore space and fluid flow properties, paper SPE 6858 presented at 52nd Annual Fall Meeting of the SPE, AIME, Denver, Colorado, 9–12 October, 1977], the inertial factors were essentially related to the porous medium (porosity, permeability, and roughness). However, Tiss and Evans [1989] showed that β is also a function of the fluid properties. These various results show that the parameters related to the viscous (K) and inertial (β, γ) effects cannot be predicted analytically and must be determined experimentally.

[25] The same approaches can be used for flow in a fracture. The viscous and inertial parameters are determined by fitting equations (17), (18), and (19) to the experimental pressure drop data points. However, as explained above, the cubic and Forchheimer's laws are valid in different flow regimes. Therefore they cannot be used to provide separate fits to the complete set of experimental data points.

[26] To distinguish between the different flow regimes, a more suitable representation of the experimental data can be obtained by dividing the pressure drop per unit length by the fluid flow rate. Thus equations (17), (18), and (19) are rewritten as follows:

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[27] Figure 2 clearly shows the flow regimes associated with the cubic and Forchheimer's laws. A constraint was placed on equations (20) and (21) so that the transition between the two flow regimes is continuous in its value and slope. The transition flow rate (2.10−5 m3/s) was chosen to minimize the mean absolute relative error between the experimental and model pressure drop per unit length for the complete set of data points.

Figure 2.

Pressure drop per unit length divided by the liquid flow rate versus liquid flow rate (single-phase flow experiment).

[28] As the full cubic law contains both quadratic and cubic terms, equation (22) was used to fit all of the experimental data points for the pressure drop per unit length (Figure 2).

[29] The viscous and inertial parameters estimated by all three of these models are listed in Table 1.

Table 1. Model Parameters for Equations (12), (13), and (14)
Modelh (μm)K (m2)β (m−1)γA (m2)
Weak inertia cubic law4091.4 × 10−81.37 × 10−46.1 × 10−5
Forchheimer's law4411.6 × 10−8103.76.5 × 10−5
Full cubic law4291.5 × 10−882.21.3 × 10−56.3 × 10−5

3.2. Two-Phase Flow

[30] As stated in section 2.2, all two-phase flow experiments were initiated as a liquid single-phase flow at a constant flow rate. Air was then injected into the apparatus at a fixed flow rate; this flow rate was increased in a stepwise fashion. For each fixed airflow rate, the pressure drop along the fracture was measured when steady state was reached. Figure 3 shows the pressure drop per unit length as a function of airflow rate for three different liquid Reynolds numbers. As the liquid Reynolds number increases, the pressure drop increases in a highly nonlinear way, which suggests that inertial effects become dominant. The following shows an analysis of these results using the generalization of Darcy's law.

Figure 3.

Pressure drop per unit length versus gas flow rate (two-phase flow experiments).

[31] As explained before, the modeling of two-phase flows through porous media and fractures is based on the notion of relative permeability, which is a generalization of the single-phase Darcy's law (equations (1) and (2)).

[32] The relative permeability expresses the degree to which each fluid phase impedes the flow of the other fluid in a two-phase flow. Since our experiments take place in non-Darcian regimes, the capillary forces are negligible, and the flow is controlled by viscous and inertial forces (i.e., Pc = PgPl = 0). Therefore the pressure drops per unit length in the two fluids are equal. Consequently, ΔPl = ΔPg, and dividing equation (1) by equation (2) leads to:

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[33] Measuring fluid saturation with a good accuracy was not possible because of the small fracture volume (33 ml). An attempt was made using a water collector and injecting water through a closed circuit. However, the free surface fluctuations in the collector prevented any measurements of water volume fluctuations in the fracture. Since no accurate experimental measurement of saturation is available, the measured and calculated relative permeabilities are plotted vs. parameter X, which is widely used for two-phase flow in pipes [Lockhart and Martinelli, 1949].

[34] The experimental values of krl and krg are calculated by substituting equations (15) and (16) into equations (1) and (2):

equation image
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where ΔP/L is the measured pressure drop per unit length under two-phase flow conditions, and h is the fracture's hydraulic aperture obtained from single-phase flow experiments.

[35] Three different models were used to fit the single-phase flow experiment; these resulted in three different values of h (Table 1). One of these values must be chosen to calculate krl and krg. Although the cubic and Forchheimer's laws correctly fit the single-phase experiment (within their corresponding flow regimes), these laws lead to different values of h and therefore to different experimental values of krl and krg. Figure 4 shows an example of the experimental relative permeabilities determined using equations (24) and (25), using values of h associated with the cubic and Forchheimer's laws. Because these values of h differ, no transition was observed between the resulting relative permeabilities. Moreover, there is no simple criterion available to guide the selection of h.

Figure 4.

Relative permeabilities versus parameter X = μlql/μgqg (Rel = 0.07).

[36] The full cubic law appears to be an appropriate alternative, since it accounts for both quadratic and cubic terms, and results in a unique value for the fracture aperture h.

[37] Figure 5 shows the experimental relative permeabilities obtained with h = 429 μm (from the full cubic law) for the three different liquid flow rates. It is generally assumed that the relative permeabilities depend uniquely on the phase saturation (or liquid hold up) in the viscous flow regime [Corey, 1954; Lipinski, 1982; Saez and Carbonell, 1985; Fourar and Bories, 1995; Turland and Moore, presented paper, 1983; Schulenberg and Muller, presented paper, 1984; Fourar and Lenormand, presented paper, 1998]. In Figure 5, we also plotted the analytical viscous coupling model of Fourar and Lenormand (presented paper, 1998), which corresponds to these same assumptions.

Figure 5.

Relative permeabilities versus parameter X = μlql/μgqg (generalized Darcy's law, h = 429 μm).

[38] As expected, Figure 5 clearly shows that krl and krg depend on the liquid flow rate. Since our experiments are in the non-Darcian regime, they therefore depend on the flow regime. Another confirmation is that the relative permeabilities do not superimpose on a single curve as function of the parameter X. Therefore it appears that the relative permeabilities are not only functions of the saturation, but also of the flow regime.

4. Non-Darcian Two-Phase Flow Model

[39] The model is based on the generalization of the single-phase full cubic law (equation (14)), that accounts for non-Darcian effects by using the F function approach [Fourar and Lenormand, 2000]. In this approach, the presence of a second fluid is taken into account through a multiplier function introduced into the superficial velocity of each fluid. These functions (Fl and Fg) are assumed to depend on the fluid saturation only.

4.1. Calculation of the Pressure Drop

[40] With the introduction of the F functions for the liquid and the gas phases, the standard full cubic law (equation (14)) applied to each fluid is written as:

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[41] The fluids Reynolds numbers, Rel and Reg, are defined respectively as [Geertsma, 1974; Whitaker, 1988]:

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equations (26) and (27) become:

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[42] There are two main advantages to this approach. First, it shows explicitly that the pressure drop per unit length is dependent on the flow regime, through the Reynolds numbers and fracture intrinsic properties. Second, since Fl and Fg are assumed to depend on the saturation only, these functions can be determined at a low flow rate in the viscous regime, either experimentally by measuring the saturation, or analytically with a model if accurate saturation is unavailable.

[43] To summarize, the proposed model requires the following data to calculate the pressure drop for given liquid and gas flow rates:

[44] 1. the intrinsic permeability (K) and the non-Darcian coefficients (β, γ), derived from a standard single-phase flow experiment (equation (22) and Table 1),

[45] 2. the functions Fl(Sl) and Fg(Sl), which are obtained in two different ways:

[46] 1. from a series of experiments at different liquid or gas flow rates when saturation can be measured, using equations (30) and (31), or

[47] 2. using a theoretical model or empirical relationships.

4.2. Calculation of Relative Permeabilities

[48] The coupling of the basic equations of the model (equations (30) and (31)) with the definition of the relative permeabilities (Darcy's law, equations (1) and (2)) leads to:

equation image
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[49] Using these equations, the relative permeabilities can be calculated at any flow rate provided that their values are known in the viscous regime (i.e., when FRe ≪ 1):

equation image
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where subscript v stands for the viscous regime. Consequently, the relative permeabilities for flows including both viscous and inertial effects can be easily derived from the kr,v measured in the viscous regime.

[50] Since F(Sl) is assumed to depend only on the fluid saturation and not on the flow regime, we can use a model derived and validated in a purely viscous regime. The relative permeabilities in the viscous regime for a single fracture were modeled using classic Corey's saturation power laws. Thus equations (34) and (35) become:

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where exponents a and b are chosen to minimize the mean absolute relative error between the measured and calculated relative permeabilities. Therefore the relative permeabilities become functions of Sl, a, b and the fluids Reynolds numbers.

[51] It is important to note that in the present work, equations (36) and (37) were preferred to equations (9) and (10) used by Fourar and Lenormand [2001]. The reason for that choice is that our fracture is significantly different from their artificially roughened fracture. We used a replica of a real rough Vosges sandstone fracture with a real 3D topography and therefore, the viscous coupling model is not suitable.

[52] Except for the fluids Reynolds numbers, other parameters must be calculated to obtain the relative permeabilities. The parameter X, which is measured, can be used to calculate these parameters. X can be rewritten in the following form:

equation image

[53] Using the given values of the exponents a and b, and using all of the experimental data points, equation (38) was solved for Sl using a nonlinear least squares method. Subsequently, we used equations (36) and (37) to calculate Fl and Fg, and finally, equations (32) and (33) to calculate the relative permeabilities.

[54] The values of a and b were set to vary in the range of 0.5 to 5, and for each pair of these exponents, krl and krg were calculated as described above. The corresponding mean absolute relative errors between the model and the experimental values of the relative permeabilities were calculated. The optimal values of a and b are those that result in the minimum mean absolute relative error. For the experiments presented here, the minimum error was 11.5%, for a = 1.15 and b = 3.05. Figure 6 shows good agreement between the experimental and semiempirical values of relative permeabilities. The model considers the dependency of the relative permeabilities on the Reynolds number.

Figure 6.

Experimental and model relative permeabilities versus parameter X = μlql/μgqg (h = 429 μm, β = 82.2 m−1, γ = 1.3 × 10−5).

5. Conclusions

[55] Experimental results for single- and two-phase flow in the non-Darcian regime through a rough-walled fracture are presented. The intrinsic properties of the fracture were determined from single-phase laws described in the literature. Two-phase flow results were interpreted by generalizing the full cubic single-phase law. The basis of this method is to introduce a function F as a multiplying factor of the superficial velocity during two-phase flow. This function is assumed to depend only upon the saturation, and therefore does not depend on the flow regime (or Reynolds number). The F function can be derived from experiments performed at a given flow rate, or from a theoretical model of the relative permeability kr in the viscous flow regime.

[56] Using classical Corey-like power laws of the saturation for the F functions, this model can be used to calculate the relative permeabilities in two-phase flows occurring in rough-walled fractures. The relative permeabilities obtained by the model explicitly show their dependence on the Reynolds number. The generalized full cubic law allows us to describe the relative permeabilities for air-water non-Darcian two-phase flows in a rough-walled fracture with good accuracy.

[57] Despite the achievement of the descriptions of relative permeability in rough-walled fractures using the proposed generalized full cubic law model, there are some limitations to this work. At this moment, the proposed model was only validated for one single real rough fracture (a replica of a Vosges sandstone real fracture). To generalize the proposed model, it will be applied to experimental results obtained using another fracture replica from a granite rock sample having a different surface topography. Also, as explained before, since no accurate experimental measurement of saturation was available, the parameter X was used to calculate the relative permeabilities. In the future, this experimental study may be extended to measure the fluid saturation with a good accuracy to test this model also in terms of saturation.

Notation
a

Corey saturation power law exponent.

b

Corey saturation power law exponent.

A

Cross-sectional area of the fracture or porous media, m2.

F

F function.

h

Hydraulic aperture of the fracture, m.

K

Intrinsic permeability, m2.

kr

Relative permeability.

q

Volumetric flow rate, m3/s.

Re

Reynolds number.

S

Fluid saturation.

w

Width of the fracture, m.

X

Martinelli parameter.

ΔP/L

Pressure drop per unit length, Pa/m.

β

Inertia coefficient, m−1.

βr

Relative inertia coefficient.

Φ

Lockhart and Matinelli fluid multiplier.

γ

Parameter of the cubic nonlinear term.

μ

Dynamic viscosity, Pa.s

ρ

Density, kg/m3.

Subscripts

g

l

Liquid.

s

Single-phase flow.

v

Viscous regime.

Acknowledgments

[58] Authors acknowledge financial support for this work funded by the French National Research Agency (ANR) within the frame of the Geocarbone Integrity project and by the French National Research Center (CNRS) within the frame of the APIT-Geothermie project.

Ancillary