Some analytical solutions for sensitivity of well tests to variations in storativity and transmissivity

The theory of a pumping test or a slug test to measure aquifer transmissivity or storativity assumes that the aquifer properties are uniform around the well. The response of the drawdown to small spatial variations in aquifer properties in the volume of influence is determined by spatial weighting functions or Fréchet kernels, which in general are functions of space and time. The Fréchet kernels determine the effective “volume of influence” of the measurements at any time. Under the assumption that the well is a line sink we derive explicit analytical expressions for the Fréchet kernels for storativity and for transmissivity for both pumping and slug tests. We also derive the total sensitivity functions for uniform variations in storativity and transmissivity and show that they are the spatial integrals of the Fréchet kernels. We consider both the case of separate pumping and observation wells and also the radially symmetric case of observations made at the pumped or slugged well. The “volume of influence” is symmetric with respect to the pumping or slugged well and the observation well, and far from the well the contours of equal spatial sensitivity approach the shapes of ellipses with a well at each focus, rather than circles centered on the pumping well. We use the analytical solutions to investigate the nature of the singularities in the spatial sensitivity functions around the wells, which govern the importance of inhomogeneities close to the well or observation point.