3 - Steady-state saturated flow

Abstract

This chapter introduces various moment equation methods for multidimensional, steady-state flow in random heterogeneous porous media. The methods discussed include perturbative expansion method, Green's function method, stationary and nonstationary spectral methods, space-state method, Adomian decomposition, and closure approximations. The Monte Carlo method is a complementary approach to moment equation methods that usually serves as an independent tool to validate or invalidate them. The Monte Carlo approach involves three steps. The first step is to generate multiple realizations of the geological formation of interest on the basis of given statistical moments and distributions of formation properties. A given realization is deterministic and provides a complete representation of the formation properties, but it is selected by a probabilistic procedure. The second step is to solve—for each realization—the deterministic governing equations by numerical methods such as finite differences and finite elements. The third step is to average over solutions of many realizations to obtain statistical moments or distributions of dependent variables. The chapter also discusses techniques, including renormalization, renormalization group, and Feynman diagrams.