Asymptotic Expansions for Moench’s Integral Transform of Hydrology

Theis’ theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform H [ g ] ( r , t ) of the pumping function g that depends on the time t and the relative position r to the pumping point as well as on other physical parameters. Several analytic approximations of H [ g ] ( r , t ) have been investigated in the literature that are valid and accurate in certain regions of r , t and the mentioned physical parameters. In this paper, the analysis of possible analytic approximations of H [ g ] ( r , t ) is completed by investigating asymptotic expansions of H [ g ] ( r , t ) in a region of the parameters that is of interest in practical situations, but that has not yet been investigated. Explicit and/or recursive algorithms for the computation of the coefficients of the expansions and estimates for the remainders are provided. Some numerical examples based on an actual physical experiment conducted by Layne-Western Company in 1953 illustrate the accuracy of the approximations.