Uncertainty Quantification of Unsteady Flows Generated by Line-Sources Through Heterogeneous Geological Formations

We consider an unsteady flow generated by a line-source within a geological formation where, owing to the erratic spatial variations of the conductivity K, the log-transform Y equiv ln K is modeled as a random space function. As a consequence, flow variables become random fields, and we aim at quantifying their uncertainty. Under ergodic conditions, simple (closed form) expressions for the second order statistical moments of the flow variables are derived by regarding the variance of Y as a small (perturbation) parameter. Then, we focus the discussion by dealing with a formation of stratified heterogeneity structure. In particular, we investigate (in view of their use in practical applications) moments' behavior in the large time regime.