Uncertainty Quantification in Scale-Dependent Models of Flow in Porous Media

Equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale dependent. In the moment equation (ME) method, exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations. Computable approximations of the MEs can be derived via perturbation expansion in orders of the standard deviation of the random model parameters. As such, their convergence is guaranteed only for standard deviation smaller than one. Here, we consider steady-state saturated flow in a porous medium with random second-order stationary conductivity field. We show it is possible to identify a support scale, $\eta*$, where the typically employed approximate formulations of MEs yield accurate (statistical) moments of a target state variable. Therefore, at support scale $\eta*$ and larger, MEs present an attractive alternative to slowly convergent Monte Carlo (MC) methods whenever lead-order statistical moments of a target state variable are needed. We also demonstrate that a surrogate model for statistical moments can be constructed from MC simulations at larger support scales and be used to accurately estimate moments at smaller scales, where MC simulations are expensive and the ME method is not applicable.