Type-curve estimation of statistical heterogeneity

The analysis of pumping tests has traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains, consisting of one or at most a few units having uniform hydraulic properties. Recently, attention has been shifting toward methods and solutions that would allow one to characterize subsurface heterogeneities in greater detail. On one hand, geostatistical inverse methods are being used to assess the spatial variability of parameters, such as permeability and porosity, on the basis of multiple cross-hole pressure interference tests. On the other hand, analytical solutions are being developed to describe the mean and variance (first and second statistical moments) of flow to a well in a randomly heterogeneous medium. We explore numerically the feasibility of using a simple graphical approach (without numerical inversion) to estimate the geometric mean, integral scale, and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate. By local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test. Experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to exhibit an integral scale well below the maximum well spacing. This is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform (on the scale of each test), which tend to exhibit regional-scale spatial correlations. We show that whereas the mean and integral scale of local log transmissivity can be estimated reasonably well based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well, estimating the log transmissivity variance is more difficult. We obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown about its mean.