On the Ergodicity Hypothesis in Heterogeneous Formations

Abstract

Application of the ergodicity hypothesis to the stochastic subsurface hydrology has been checked by investigating the hydraulic conductivity field. The relative variance of the spatial average of conductivity, which is denoted as R, and the error index E.I. = \(\sqrt R \), were employed to justify the uncertainties and errors of using the ergodicity hypothesis. Six factors influence R: autocorrelation function; variance of logconductivity; spatial correlation scales of logconductivity; domain sizes; anisotropy; and dimensionality of the problem. Closed-form analytical solutions of R for the linear autocorrelation function were derived and the numerical integration of R for the exponential autocorrelation function for one-, two-, and three-dimensional problems calculated. It is easy to fulfill the ergodicity hypothesis under these conditions: weak heterogeneity and large ratio of domain size vs. correlation length (L/I). The uncertainties and errors of using this hypothesis increase rapidly when the variances of logconductivity increase and/or the ratios of L/I decrease. The ergodicity hypothesis has less error when applied to a problem with higher dimensionality.