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 σ_Y ², 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 σ_Y ²≤ 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⪆1 is one of the main results of the study. Application to a synthetic example of hydraulic tomography to derive K_G, σ_Y ²and I, from H measurements, demonstrates the promise of the method and calls for its further development and validation.