Uncertainty quantification of unsteady source flows in heterogeneous porous media

Unsteady flow generated by a point-like source takes place into a d -dimensional porous formation where the spatial variability of the hydraulic conductivity K is modelled within a stochastic framework that regards K as a stationary, normally distributed random space function (rsf). As a consequence, the hydraulic head H 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 [STIX]x1D70E² of the zero meanrsf [STIX]x1D700=1-K/ K (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 [STIX]x1D70C for [STIX]x1D700 . This enables one to obtain simple results providing straightforward physical insight into the spatial distribution of H as a consequence of the heterogeneity of K . 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.