## 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]:

where the subscripts *l* and *g* stand for the liquid and gas phases, respectively. Δ*P*/*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 *k*_{r} 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].

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

[8] Another alternative is to assume that *k*_{r} and 1/*β*_{r} 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:

where subscript *s* stands for single-phase flow. Φ_{l} and Φ_{g} express the ratio between the two-phase and the single-phase pressure gradients. Instead of saturation, a new variable *X* = Φ_{g}/Φ_{l}, 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 Φ_{l} and Φ_{g} are analogous to 1/*k*_{rl} and 1/*k*_{rg}. Therefore the Lockhart-Martinelli model can be seen as a generalization of Darcy's law for non-Darcian flows. However, as for *k*_{r} 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.

[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):

where *S*_{l} 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.