Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation

We consider 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. More precisely, we consider 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.” We derive a compact integral expression 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. We show 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. We demonstrate numerically that in two‐dimensional mean radial flow it may increase from the harmonic mean of K(x) near interior and boundary sources to the geometric mean far from such sources. For cases where r_κ(x) can neither be expressed nor approximated by a local expression, we propose a weak (integral) approximation (closure) 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; our theory 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 second moments of head and flux prediction errors, we conclude by discussing practical methods 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 lognormal.