Automated Estimation of Aquifer Parameters from Arbitrary-Rate Pumping Tests in Python and MATLAB

Inspired by the analysis by Mishra et al. (2012) of variable pumping rate tests using piecewise-linear reconstructions of the pumping history, this article contains a derivation of the convolutional form of pumping tests in which the pumping history may take any possible form. The solution is very similar to the classical Theis (1935) equation but uses the Green's function for a pumped aquifer given by taking the time derivative of the well function $W\left(u\left(t\right)\right)$. This eliminates one integration inside another and renders the convolution including the pumping history about as computationally demanding as calculating the well function alone, so that the convolution can be completed using handy mathematical software. It also allows nonlinear well losses, and because an easily-computed deterministic model exists for all data points and pumping history, an objective function may include all data, so that errors are reduced in calculating any nonlinear-well losses. In addition, data from multiple observation wells may be used simultaneously in the inversion. We provide codes in MATLAB and Python to solve for drawdown resulting from an arbitrary pumping history and compute the optimal aquifer parameters to fit the data. We find that the subtleties in parameter dependencies and constructing an appropriate objective function have a substantial effect on the interpreted parameters. Furthermore, the optimization from step-drawdown tests is typically nonunique and strongly suggests that a Bayesian inversion should be used to fully estimate the joint probability density of the parameter vector.