Large-scale properties for two-phase flow in random porous media

Abstract

Hydrocarbon contaminants in the subsurface are important sources of pollution in aquifers. Hence, mathematical models of these flows have become key tools in environmental studies. In this paper we are interested in the flow in the saturated zone. Simulation of two-phase flow in large, complex heterogeneous domains often requires an unacceptably large number of computational grid blocks. Despite recent progress in computational methods and tools, we must call upon special techniques in order to use larger grid blocks while compensating for intracell variations in rock properties and fluid saturations. The use of pseudo-functions is one way of increasing grid dimensions to a more tractable level with a minimal loss of simulation representativeness. This change of scale problem has also been treated theoretically by different scaling-up techniques, such as large-scale averaging. This method calculates the transport equations and the effective properties at a given scale by an averaging process over the equations corresponding to a lower scale. This procedure leads to a closure problem which is very complex in the general case. Previously a first solution of the large-scale averaging problem was proposed in the quasi-static case corresponding to local capillary equilibrium. In the general case of a heterogeneous medium with a complex geometry, this boundary value problem (closure problem) can be solved by numerical methods. For this purpose, after having chosen a grid-block description of our system in accordance with the description used in reservoir simulators, we have implemented a three-dimensional (3-D) numerical resolution of the closure problem. The most important variations in the rock properties are associated with the permeability. We have therefore generated porous media with a random permeability distribution using different methods. Other multiphase properties are chosen to depend on the permeability. The properties of the closure problem in the case of randomly generated porous media are investigated. In particular, we give the conditions under which the general form of the local capillary pressure and relative permeability curves are recovered at the large scale. Particular properties related to each generation method are stated. The equivalent properties are calculated using averages over the results of many realizations of a given medium. The influence of the size of the averaging surface for a given correlation length as a function of the variance of the permeability is studied. We have therefore established some general rules for the calculation of the large-scale properties of random porous media.