Uncertainty quantification of unsteady source flows in heterogeneous porous media

Unsteady flow generated by a point-like source takes place into a -dimensional porous formation where the spatial variability of the hydraulic conductivity is modelled within a stochastic framework that regards as a stationary, normally distributed random space function (rsf). As a consequence, the hydraulic head becomes also stochastic, and we aim at quantifying its uncertainty. Towards this aim, we have derived the head covariance by means of a perturbation expansion which regards the variance of the zero mean rsf (hereafter being the ensemble average operator) as a small parameter. The analytical results are expressed in terms of multiple quadratures which are markedly reduced after adopting specific autocorrelation for . This enables one to obtain simple results providing straightforward physical insight into the spatial distribution of as a consequence of the heterogeneity of . In view of those applications (pumping tests) aiming at the identification of the hydraulic properties of geological formations, we have focused on a flow generated by a source of instantaneous and constant strength. The attainment of the large time (steady-state) regime is studied in detail.