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

The effective conductivity of heterogeneous aquifers K-efu relates the mean flux to the mean head gradient in mean uniform (natural gradient) flow. While K-efu depends on the hydraulic conductivity K statistical moments (K-G, the geometric mean and sigma(2)(Y), 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 K-eq, which unlike K-efu, is a measurable property, directly related to the head H (or its mean) and the discharge. The dependence of K-eq 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 K-eq is derived by numerical simulations for sigma(2)(Y) <= 4. The derivation of the transition of K-eq from the local exact value for r/I << 1 (I is the integral scale) to K-efu for r/I greater than or similar to 1 is one of the main results of the study. Application to a synthetic example of hydraulic tomography to derive K-G, sigma(2)(Y) and I, from H measurements, demonstrates the promise of the method and calls for its further development and validation.