Simulating flow in porous media

The problem of two-phase fluid flow in statistically homogeneous but random porous media is addressed. Particular emphasis is placed on the role of the stochastic nature of the porous medium in the development of unstable interfaces at large mobility ratios. A formalism is developed in which the random nature of a real porous medium can be incorporated quantitatively into a numerical simulation scheme based on a discretization of the continuum fluid-mechanical equations. In particular, it is not necessary to set the discretization length close to the pore scale and to perform a detailed structural analysis of the microgeometry in order to characterize the random nature of the porous matrix. The formalism is based on the concepts of tubes and chambers which give rise, on the discretization length scale, to random hydrodynamic conductivity and fluid capacity, respectively, and on larger scales to the macroscopically determined permeability and porosity, respectively. In the absence of detailed information about the statistical properties of the random medium at the discretization length scale, a maximum entropy criterion is used to deduce the most random distribution of properties of tubes and chambers consistent with macroscopically observable quantities. This criterion reveals the fundamental significance of an exponentially distributed fluid capacity. A number of numerical experiments are reported. The relation of the present simulation algorithm to diffusion-limited aggregation (DLA) and related algorithms for the simulation of two-phase flows is revealed, and exponentially distributed fluid capacity is again found to be of fundamental significance. In particular, it is found that a number of further assumptions which are physically plausible but difficult to justify rigorously are required to relate the DLA and similar ‘‘random-walk’’ models to a fluid-mechanical model based on the equations of continuum mechanics applied to a stochastic medium.