Two-Phase Flow Through Porous Media

We want to develop the equations for the two-phase flow of fluids in porous media and discuss numerical methods for their solution. These problems are important in describing the flow of water and oil in petroleum reservoirs. The process of waterflooding is where water is injected into a reservoir in order to recover the oil that is residual in the void spaces of rocks such as sandstone and limestone. In order to describe this process, a mathematical model must be developed for the two-phase flow of water and oil through the porous rock material. 

Syj = water saturation (cm water/cm water plus oil) Vy, = water phase velocity (cm/sec) 

The momentum balance is given by Darcy s law which states that (Green-kom, 1983) 

The velocity can be expressed in terms of the fractional flow of water, fwf 

Now we want to discuss the numerical solution of equation (8.9.9). The boundary condition for waterflooding, i.e. the injection of water, is 

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Macroscopic experiments such as core flooding have been used to obtain relative permeabilities, dispersion coefficients, and other variables relevant to reservoir flow. However, they cannot reveal details of how immiscible phases interact on the pore level. Instead visual experiments have been used to elucidate microscopic flow mechanisms. The latter approach is taken here with experiments using a novel flow cell and state-of-the-art video equipment. The pore level phenomena observed provide a basis for the proper modeling of two-phase flow through porous media at high capillary numbers. 

Tung, V.X. Dhir, V.K. A hydrodynamic model for two-phase flow through porous media. Int. J. Multiphase Flow 1988, 14 (1), 47-65. 

Zaitoun, A., Kohler, N., 1988. Two phase flow through porous media Effect of adsorbed polymer layer. Paper SPE 18085 presented at the SPE 63" Annual Technical Conference, Houston, 2-5 October. 

V. X. Tung, and V. K. Dhir, A Hydrodynamic Model for Two-Phase Flow Through Porous Media, Int. J. Multiphase Flow, (14) 47-64,1988. 

Charpentier, J.-C., N. Favier. Hydrodynamics of Two-Phase Flow Through Porous Media. Chemical Engineering of Gas-Liquid-Solid Catalyst Reactions, G. P. L Homme Editor. Precedings of an International Symposium Held at the University of Liege, March 1978, 78-108. 

Levine, S. and Cuthiell, D. L. (1986) Relative permeabilities in two-phase flow through porous media an application of effective medium theory. J. Can. Pet. Tech. 25, 74-84. 

Many water-management advances have been made (Figure 2). However, fundamental properties associated with two-phase flow in porous media, including capillary pressure, liquid water content and, intrinsic and relative permeabilities, are still lacking and are needed for predictive models. This situation is due to the absence of measurement methods for thin materials. Contact porosimetry [57] does not presently allow measurement of the permeabilities or during wetting (to evaluate the presence of hysteresis). Fluorescence microscopy [58] can provide additional information about liquid water flow through porous media. 

Multiphase flow in porous media is a very important topic for low-temperature PEFCs because water produced at the cathode flows through the porous catalyst layers and porous gas diffusion media. Any local blockage of normally open pores restricts reactant flow to the reaction sites, a phenomenon known as flooding. The water balance and flooding in a PEFC is described in detail in Chapter 6. Here, the basic fundamentals that describe two-phase flow in porous media are described to guide and understanding. 

There is not enough space to describe the properties of this equation. Suffice it to say that the Buckley-Leverett equation has shock-like solutions, where the saturation front is a wave propagating through the reservoir. This combination of an elliptic equation for the total pressure and a parabohc, but nearly hyperbolic equation for the saturation, gives rise to great mathematical interest in two-phase flow though porous media. 

The mechanisms of steady state, cocurrent, two-phase flow through a model porous medium have been established for the complete range of capillary numbers of interest in petroleum recovery. A fundamental understanding of the mobile ganglia behavior observed requires a knowledge of how phases break up during flow through porous media. Several mechanisms have been reported in the literature and two have been observed in this flow cell. 

The two-phase flow option in FLAG allows numerical modelling of the flow of two immiscible fluids through porous media. A description of the concepts involved in the mathematical description of multi-phase flow may be found in reference books such as Fundamentals of Numerical Reservoir Simulation (Peaceman, D. W., 1977). Some of these concepts are addressed below. 

Liquid slip has implications to various macroscopic applications, i.e., flow through porous media, particle aggregation, liquid coating, and lubrication, etc., in addition to small scale, i.e., MEMS and bio-MEMS, applications. The movement of a three-phase contact line between two immiscible fluids and solid on a substrate during the advancing or receding film motion indicates... 

The interaction of the two phase flows in diffusion media is described through the relative permeabilities, kj. and fef, defined as the ratio of the intrinsic permeability of liquid and gas phases, respectively, to the total intrinsic permeability of a porous medium. Physically, these parameters describe the extent to which one fluid is hindered by others in pore spaces, and hence can be formulated as a function of liquid saturation. Most ofprevious work adopted the following correlations [11, 29] ... 

Newton s law is applied in order to derive the linear momentum balance equation. Newton s law states that the sum of all forces equals the rate of change of linear momentum. The derivation/application of this law ultimately provides information critical to any meaningful analysis of most chemical reactors. Included in this information are such widely divergent topics as flow mechanisms, the Reynolds number, velocity profiles, two-phase flow, prime movers such as fans, pumps and compressors, pressure drop, flow measurement, valves and fittings, particle dynamics, flow through porous media and packed beds, fluidization— particularly as it applies to fluid-bed and fixed bed reactors, etc. Although much of this subject matter is beyond the scope of this text, all of these topics are treated in extensive detail by Abulencia and Theodore. ... 

Formulations The equations of flow through porous media are generally too complicated to have exact solutions in analjdical form and numerical methods must be used. There are special cases, however, of great theoretical interest that are valuable for benchmarking numerical methods and validating computer implementations. As part of the process of deriving a numerical method it is sometimes useful to reformulate the differential equations to clarify the mathematical structure. It is also useful to relate the theory to diffusion, convection or wave propagation, for which there are models with canonical interpretations. To illustrate this idea, and to derive results needed in a later section, the reformulation of incompressible, two-phase flow will be developed. For simplicity, the case without sources is studied. 

A. Bourgeat and M. Panfilov (1998) Effective two-phase flow through highly heterogeneous porous media capillary nonequilibrium effects. Computational... 

Brownell, L.E., Katz, D.L., 1947. Plow of fluids through porous media, part II, simultaneous flow of two homogeneous phases. Chemical Engineering Process 43, 601-612. 

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

The major design concept of polymer monoliths for separation media is the realization of the hierarchical porous structure of mesopores (2-50 nm in diameter) and macropores (larger than 50 nm in diameter). The mesopores provide retentive sites and macropores flow-through channels for effective mobile-phase transport and solute transfer between the mobile phase and the stationary phase. Preparation methods of such monolithic polymers with bimodal pore sizes were disclosed in a US patent (Frechet and Svec, 1994). The two modes of pore-size distribution were characterized with the smaller sized pores ranging less than 200 nm and the larger sized pores greater than 600 nm. In the case of silica monoliths, the concept of hierarchy of pore structures is more clearly realized in the preparation by sol-gel processes followed by mesopore formation (Minakuchi et al., 1996). 

In this paper the problem of stationary flow of two-ionic species electrolyte through random piezoelectric porous media is studied, thus extending our earlier paper [14], where periodicity was assumed. To derive the macroscopic equations we use the method od stochastic two-scale convergence in the mean developed by [4], Solid phase was assumed to be piezoelectric since according to [9] wet bone reveals piezoelectric properties, cf. also [15], We recall that a strong conviction prevails that for electric effects in bone only streaming potentials are responsible. 

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