Temporal moments in geoelectrical monitoring of salt tracer experiments

Abstract

Monitoring of salt tracer experiments by electrical resistivity tomography (ERT) has been shown to be a valuable tool for characterizing the hydraulics of an aquifer, but efficient approaches of determining the spatial hydraulic conductivity distribution from ERT data are still missing. Standard inversion of ERT data obtained during salt tracer tests may even lead to estimates of the concentration distribution that are in contradiction to flow and transport of conservative compounds in porous media. In order to avoid nonphysical behavior, we consider the governing equations of groundwater flow, solute transport, and geoelectrics as a coupled system. While the tracer passes through the part of the domain that is sensitive for a particular electrode configuration, the measured electrical potential differences are perturbed. We characterize these perturbations by their temporal moments and relate them to the temporal moments of concentration, which themselves depend on hydraulic conductivity. We present temporal moment‐generating equations leading from the hydraulic conductivity field via heads and velocities to the temporal moments of concentration and electrical potential perturbations. The approach makes use of a linearized version of the Poisson equation. On the basis of this system of coupled steady state equations, we compute the sensitivity of electrical potential perturbations with respect to the log hydraulic conductivity distribution by the continuous adjoint state method for coupled systems. For demonstration, we simulate salt tracer experiments in a virtual quasi‐two‐dimensional sandbox, monitored by ERT. We show that the ratio of the first over the zeroth temporal moment of potential perturbation is less affected by the linearization of the Poisson equation than the zeroth and first moments themselves. Thus, it appears recommendable to use the ratio of first to zeroth moments also as data in inversion. We compare sensitivity patterns resulting from different electrode configurations. The methods of forward simulations and sensitivity calculations presented in this paper can be combined with any inverse kernel to develop a complete inverse model. Altogether, using temporal moments of potential perturbation appears promising for fully coupled hydrogeophysical inversion of ERT surveys during salt tracer tests.