Stochastic analysis of spatial variability in subsurface flows: 2. Evaluation and application

Abstract

The stochastic differential equation describing one‐dimensional flow in a statistically homogeneous porous medium is solved exactly, and the results are compared with an approximate solution considering small perturbations in the logarithm of the hydraulic conductivity. The results show that the logarithmic approximation is valid when the standard deviation of the natural logarithm of the hydraulic conductivity σ f is less than 1; the errors increase rapidly for σ f > 1. The effective hydraulic conductivity of statistically homogeneous media with one‐, two‐, and three‐dimensional perturbations is determined to the first order in σ f 2 . The effective conductivity is found to be the harmonic mean for one‐dimensional flow, the geometric mean for two‐dimensional flow, and (1 + σ f 2 /6) times the geometric mean for three‐dimensional flow. The application of stochastic analysis is illustrated through two elementary network design problems that demonstrate the importance of the correlation length of the hydraulic conductivity and the role of measurement error.