Statistical moment equations for fluid flow and deformation in poroelastic randomly heterogeneous materials

Simulation of complex interactions between fluid flow and mechanical deformations in heterogeneous porous media involves several parameters whose spatial distribution cannot be measured directly. Quantification of this parametric uncertainty via Monte Carlo simulations (MCS) is computationally expensive. We present an efficient alternative that solves stochastic/statistical moment equations (SME) for state variables describing flow in, and deformations of, poroelastic media uncertain/random material properties (permeability and Lamé’s first coefficient). This entails developing a closure for SME, which has the form of evolution equations for the spatiotemporal covariance functions of the state variables. Our methodology is demonstrated on the Mandel and Barry–Mercer problems. These numerical experiments demonstrate that, within a fully implicit framework and for arbitrary time-step sizes, the SME approach is more than ten orders of magnitude faster than MCS. For an input log-permeability variance of 𝜎 2 𝑌 = 1 . 0 , the maximum relative difference is around 2% for the mean of the QoI and 7% for its variance.