Uncertainty quantification of unsteady source flows in heterogeneous porous media

Abstract

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$\unicode[STIX]{x1D70E}^{2}$of the zero meanrsf$\unicode[STIX]{x1D700}=1-K/\langle K\rangle$(hereafter$\langle \rangle$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$\unicode[STIX]{x1D70C}$for$\unicode[STIX]{x1D700}$. 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.