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Article Title: Viscous coupling Model and experimental data
Date:2005/09/21
Body:
The relative permeability can be calculated either from the Generalized Darcy’s law, which is a simple relationships based on the dimensionless two-phase pressure gradient, or from a viscous coupling model between two fluids flowing simultaneously in a single fracture (the latest approach). Four main flow patterns were studied, namely alternating channel flow, stratified flow, sandwich flow I (water wetting the walls), and sandwich flow II (oil wetting the walls). The viscous coupling model leads to simple analytical relationships between the relative permeabilities and either saturation or fluid velocities. This model also explains the dependence of the non-wetting fluid relative permeabilities on the viscosity ratio. Several series of data available in the literature depicts that the relative permeability to oil is a function not only of saturation but also of viscosity ratio, capillary number, and flow pattern. On the other hand, the relative permeability to water is a strong function of saturation, not affected significantly by the other parameters identified above. Results obtained from the extension of Darcy’s law were compared to the viscous coupling model. The available saturation data are found to correlate with the Lockhart-Martinelli, X, very well for both models. For use in numerical simulations, the saturation can be derived from the flow rates using the same model. The viscous coupling model can improve the relative permeability curves that are used in the numerical simulations.
The concept of relative permeabilities is very important to two-phase theory and practice. For two-phase flow in porous media, it is presumed that each fluid flows in its own separate network of interconnected pathways, and any body of disconnected oil is stranded behind pore throat. Based on the presumption, the relative permeabilities can be obtained from the extended Darcy's law. The conventional relative permeabilities, as defined in the extended Darcy's law, should be interpreted as a representation of the drag due to the flow of each fluid through a solid boundary. In fact, it is assumed that the solid surface and each one of the fluids form a new porous matrix through which the other fluid flows. This implies that the numerous interfaces between the two fluids remain immobile and, therefore, do not influence the two-phase flow.
With the accumulation of physical evidence that fluid interactions during two phase flow in porous media cannot be neglected, theories have also been advanced to take into account the interactions in terms of quantifying the drag due to momentum transfer across the fluid-fluid interfaces. It has been shown theoretically that additional interaction coefficients (cross terms) appear in the generalized Darcy's law for two-phase flow, which account for the viscous drag across the interfaces (Rose, 1988; de la Cruz and Spanos, 1983; Whittaker, 1986; Kalaydjian, 1987).
The generalized Darcy’s law, used to describe multiphase flow in porous media, neglects the interaction between the fluids. The conventional relative permeability to one fluid is a reflection of the drag due to the flow of that fluid as if it were flowing over a solid surface, i.e., the pore surface and the other fluids form a new porous matrix through which the fluid in question flows. This implies that all fluid-fluid interfaces remain static in steady-state flow as adopted to be the presumption of the conventional multiphase flow theory. However, this is not true, especially for flow in a fracture.
The neglected fact by conventional relative permeability concept is the viscous coupling due to momentum transfer across interfaces between two immiscible fluids. . The most common one is the viscous coupling theory which states, according to Rose (1988) “By coupling is meant a situation where the motion of elements of pore fluid reciprocally will be subject to viscous to viscous drag extending across the fluid-fluid interfaces that separate them from other contiguous element of immiscible fluid(s)…”. The governing equations for coupled immiscible two-phase flow in homogenous and isotropic porous media have been written in the form as (Rose,1988; de la Cruz and Spanos, 1983; Whitaker, 1986; Kalaydjian, 1987)
Visual observations of two-phase flow in fractures have shown a mixing at small scale and therefore a strong interaction between the fluids. For instance, Kouame and Fourar and Bories have identified several fluid configurations for air-water flows in an artificial fracture: bubbles, unstable bubbles, film flow, etc.; in the same way X. Pan, R.C.K. Wong, and B.B. Maini have identified several fluid configurations for oil-water flows in an artificial fracture. As a consequence of these observations, any physical model should account for this coupling between two-fluids flowing simultaneously. The real mechanisms are very difficult to model since the real geometry of the interface between the fluids is unknown and that one of the phases is generally discontinuous. Recently, Fourar and Lenormand assumed that the complexity of the real flow could be modelled, in a first approximation, by viscous coupling between the fluids. This mechanism has already been studied for two-phase flow in porous media (De Gennes; Kalaydjian and Legait; Zarcone and Lenormand). They have shown that viscous coupling can also explain the shape of the relative permeability curves for the parallel planes with small aperture fracture. The two fluids are flowing simultaneously and the interface is assumed to be a plane. Fluid L is considered as the wetting fluid and therefore is in contact with the walls, and fluid G (non-wetting) flows in between. Writing Stokes’s equation in each fluid with a no-slip condition in the interface derives the viscous coupling between the fluids.
Results of experimental and numerical studies of single-phase flow (based on the literature review) through a fracture indicate that the cubic law governs single-phase flow through the fracture if the walls of the fracture are either smooth or have good correlation between them. In this case, the fracture can be idealized as between a pair of parallel plates. However, if the fracture has numerous contacts and unmatched surfaces, deviation from the idealized parallel-plate fracture is expected. The approach of representing a rough-walled fracture by a set of parallel-plate cells with apertures generated by geostatistical methods, however, proves to be a reasonable physical simplification of the complex system.
SEYED EMAD HOSSEINI UNIVERSITY OF CALGARY
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