Effects of First‐Order Approximations on Head and Specific Discharge Covariances in High‐Contrast Log Conductivity

Abstract

The hydraulic head and the specific discharge fluctuations depend nonlinearly on the hydraulic conductivity. However, the methods most commonly used in the stochastic analysis of groundwater flow are based upon the linearization of these relations. In this paper we apply a numerical spectral approach to investigate the range of validity of the small perturbation approximation for head and specific discharge moments in two‐dimensional finite domains. We find that the small perturbation approximation tends to underestimate the variance of large‐scale head and specific discharge fluctuations and error increases with increasing log‐conductivity variance and increasing domain size. The head fluctuations do not appear ergodic even when the small perturbation approximation predicts they will be ergodic. The specific discharge fluctuations, on the other hand, do appear ergodic. The small perturbation approximation performs well in estimating specific discharge variance in the longitudinal direction but significantly underestimates transverse specific discharge variance.