Equivalent and effective conductivities of heterogeneous aquifers for steady source flow, with illustration for hydraulic tomography

Abstract

The effective conductivity of heterogeneous aquifers Kefu relates the mean flux to the mean head gradient in mean uniform (natural gradient) flow. While Kefu depends on the hydraulic conductivity K statistical moments (KG, the geometric mean and σY2, the logconductivity variance for a given structure model) it is not a local property for strongly nonuniform well flow. In hydraulic tomography, flow to a short well segment which is considered here, can be approximated by that toward a source in a three-dimensional heterogeneous medium. We investigate the equivalent conductivity Keq, which unlike Kefu, is a measurable property, directly related to the head H (or its mean) and the discharge. The dependence of Keq on K statistics and distance r from the source is determined for steady source flow in a formation of random stationary and statistically isotropic K. For the first time Keq is derived by numerical simulations for σY2≤ 4. The derivation of the transition of Keq from the local exact value for r/I << 1 (I is the integral scale) to Kefu for r/I⪆1 is one of the main results of the study. Application to a synthetic example of hydraulic tomography to derive KG, σY2and I, from H measurements, demonstrates the promise of the method and calls for its further development and validation.