An inverse problem to determine the piecewise homogeneous hydraulic conductivity within rocks

Abstract

Abstract The experimental measurement of the piecewise homogeneous hydraulic conductivity of a rock sample in which the location of the discontinuities is unknown appears difficult, even in a one-dimensional situation. However, the values of these quantities can be obtained numerically using additional boundary and/or interior measurements of the pressure from transient hydraulic experiments. In the direct problem, the hydraulic conductivity is known, and only a solution for the pressure field is sought. However, in the inverse formulation of the problem, both the hydraulic conductivity and the pressure are unknown and have to be determined using additional pressure measurements. The numerical method employed for solving the diffusion equation is based on the boundary element method as a direct solution procedure, combined with an ordinary least-squares technique. The sensitivity coefficients are calculated for the unspecified boundary conditions and for interior pressures, and clearly show the need for interior measurement information to be imposed on the solution of the inverse ill-posed problem. The uniqueness of the solution of the inverse problem is thoroughly numerically investigated using additional pressure measurements at several prescribed times from various numbers of wells and different well locations. For a material presenting a single discontinuity in the hydraulic conductivity, and subject to certain experimental conditions, it was found that when the number of time measurements is limited then two interior well pressure measurements are necessary and sufficient in order to render a good estimate of the exact solution. Otherwise, it is necessary to increase the number of time measurements at a single interior well location to accomplish the same result.