Determining Flow Equations from Stochastic Properties of a Permeability Field: the MHD Model

Numerical reservoir simulations can be improved by using "homogenized" transport equations which account for real flow mechanisms: instability, fingering, channeling, etc. In this paper, the homogenized transport equations and relative permeabilities are derived by using a standard stochastic representation of the permeability field and an original stochastic calculation for the flow equations which uses the concept of streamtubes. An approximate equation, called "MHD", is proposed for displacements in reservoirs for various types of permeability fields. The equation contains a dispersive term (dispersion coefficient D) and a nonlinear convective term with the product H (heterogeneity factor) multiplied by viscosity ratio M. The main advantage of the method is that the D and H parameters that characterize the permeability field are determined from a one-phase numerical simulation, a simulation that is not time consuming.