Integration of moment equations in a reduced-order modeling strategy for Monte Carlo simulations of groundwater flow

We illustrate and test an approach grounded on embedding moment equations (MEs) of groundwater flow within a Monte Carlo based modeling strategy to yield a Reduced-Order Model (ROM) that enables the efficient and accurate evaluation of probability distributions of hydraulic heads in randomly heterogeneous transmissivity fields. The projection space determining the accuracy of the ROM solution is typically computed through the principal component analysis of a selected number of full system model solutions (the so-called snapshots). Computationally expensive sensitivity analyses are then required to assess the independence of the ROM from the snapshots. Here, we propose to compute the projection vectors upon relying on the hydraulic head covariance evaluated from the solution of corresponding MEs of groundwater flow. Our workflow to compute hydraulic head distributions is organized according to the following steps: (i) approximation of mean hydraulic head and head covariance matrix through (second-order accurate) solutions of MEs; (ii) computation of the leading eigenvectors of the head covariance matrix to form the basis set for the ROM projection space; and (iii) construction of the ROM. Sample probability density functions of hydraulic heads are then efficiently obtained via Monte Carlo simulations relying on the developed ROM. The proposed methodology is compared against snapshot-based ROMs and the full system model in a two- and a three-dimensional steady-state groundwater flow setting where pumping from a point source is superimposed to a mean uniform flow. Our results show that the projection space computed by relying on MEs provides a more accurate ROM solution than the one resulting from reliance on snapshots.