Stochastic approach to steady state flow in nonuniform geologic media

Abstract

This dissertation considers the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict steady state flow within a bounded domain, driven by random source and boundary functions. That is, the work concerns the prediction of local hydraulic head h(x) and Darcy flux q(x) by means of their unbiased ensemble moments (h(x))(κ) and (q(x))(κ) conditioned on measurements of K(x). These predictors satisfy a deterministic flow equation in which (q(x))(κ) = -(κ)(x)∇(h(x))(κ) + r(κ)(x) where κ(x) is a relatively smooth unbiased estimate of K(x) and r(κ)(x) is a "residual flux." A compact integral expression is derived for r(κ)(x) which is rigorously valid for a broad class of K(x) fields, including fractals. It demonstrates that (q(x))(κ) is nonlocal and non-Darcian so that an effective hydraulic conductivity does not generally exist. It is shown analytically that under uniform mean flow the effective conductivity may be a scalar, a symmetric or a nonsymmetric tensor, or a set of directional scalars which do not form a tensor. For cases where r(κ)(x) can neither be expressed nor approximated by a local expression, a weak (integral) approximation (closure) is proposed, which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of K(x) measurements increase. The nonlocal deterministic flow equation can be solved numerically by standard methods; the theory here shows clearly how the scale of grid discretization should relate to the scale, quantity and quality of available data. After providing explicit approximations for the prediction error moments of head and flux, some practical methods are discussed to compute κ(x) from noisy measurements of K(x) and to calculate required second moments of the associated estimation errors when K(x) is log normal. Nonuniform mean flow is studied by conducting high resolution Monte Carlo simulations of two dimensional seepage to a point sink in statistically homogeneous and isotropic log normal K(x) fields. These reveal the existence of radial effective hydraulic conductivity which increases from the harmonic mean of K(x) near interior and boundary sources to geometric mean far from such sources for σ^2/Υ (the variance of ln K) at least as large as 4. They suggest the possibility of replacing r(κ)(x) by a local expression at distances of few conditional integral scales from the interior and boundary sources. Special attention is paid to the "art" of random field generation, and comparisons are made between four alternative methods with five different random number generators.