1 - Introduction

Abstract

Formation material properties are observed at only a few locations, and they exhibit a high degree of spatial variability at all length scales. This combination of significant spatial heterogeneity with a relatively small number of observations leads to uncertainty about the values of formation properties. The theory of stochastic processes provides a natural method to evaluate uncertainties. In the stochastic formalism, uncertainty is represented by probability or by related quantities such as statistical moments. This chapter discusses saturated and unsaturated flows in one-space dimension. Flow in one-dimension is often unrealistic and of minor importance in applications. However, it serves the purpose of introducing stochastic differential equations and some methods for solving them. Stochastic differential equations are solved by using either probability density function (PDF) methods or statistical moment methods. The complete solution for solving stochastic differential equations are probability density functions, but it is usually difficult to construct PDF equations and to solve them. Statistical moments of dependent variables are expressed explicitly, governed by differential equations, or given in terms of integral equations that depend on the nature of the problem. This chapter also discusses the application of moment equation methods to various flows in porous media, and it introduces techniques to derive and solve moment equations.