Numerical Solution of the Pressure Equation for Fluid Flow in a Stochastic Medium

Abstract

When modelling pressure in oil reservoirs one can interpret the permeability in the medium as a random field. Such a model has been suggested by Holden et al. in [HLØUZ]. They consider the pressure equation

$$ {\nabla_x}\left( {k\left( {x,\omega } \right)\diamondsuit {\nabla_x}p\left( {x,\omega } \right)} \right) = - f\left( {f,\omega } \right)\;x \in D\;p\left( {x,\omega } \right) = g\left( {x,\omega } \right)\;x \in \partial D $$

(1.1)

where k(x, ω) is the permeability and D denotes the reservoir with x ∈ D. ω ∈ Ω is a probability space and the ◊ stands for a renormalization product of functions on this probability space called the Wick product. For the permeability, they use a lognormal random field, for which they are able to construct an explicit solution. Unfortunately, the solution for the stochastic pressure equation is singular and has to be understood in a distributional sense. This implies that it is difficult to study its stochastic properties.