Equivalent Transmissivities in Heterogeneous Porous Media under Radially Convergent Flow

Abstract

Most field methods used to estimate transmissivity values rely on the analysis of radial flow towards a single point (pump tests). Using a perturbation approach, we present an analytical solution to the problem of equivalent transmissivity (T _eq ) under radially convergent steady-state flow conditions produced by constant pumping from a well of radius r _w . In this context T _eq is defined as the value that best fits Thiem’s equation, and would be the transmissivity assigned to the well location in a pump test. Using a Green’s function technique, we derive an expression for T _eq , up to second-order in the expansion, which is given as a weighted average of the fluctuations in log-T throughout the domain. Previous work (Desbarats, 1992) hypothesized, based on empirical evidence, that the weighting function for log-T _eq could be written as a function of the log-T fluctuations normalized by the square of their distance from the well. We find that this is indeed the case, although our second-order terms differ from that of Desbarats. We conclude with a highly heterogeneous example that illustrates the relative ranges over which the empirical formula and our second order expansion are accurate.