The application of the first-order second-moment method to analyze poroelastic problems in heterogeneous porous media

In this study, a stochastic model is proposed to solve poroelastic problems in heterogeneous porous media. The model is constructed using the first-order second-moment method to investigate the dynamic behaviors of statistical mean and covariance of the change in pore water pressure and displacement. Although several variables can be simultaneously treated as random in the model, the Darcy conductivity is selected as the only random variable for this preliminary investigation due to its large variation compared to other mechanical and hydrogeological properties in natural environments. The constructed model is general in multiple dimensions; however, the one-dimensional case is taken as an example to demonstrate the use of the stochastic model. This model is validated using analytical and numerical solutions from the literature. Numerical experiments are then performed to investigate the boundary effects on the coupled fluid pressure and mechanical deformation in elastic porous media. The results show that the dynamic behavior of a coupled flow–stress system is more complex than a system that does not consider the deformation of porous media. Loading effects on deformation and pore pressure are instantaneous while the effect of discharge takes time to propagate from the boundary through the whole domain. Both loading and discharge boundary conditions can significantly affect the uncertainty of the system response. In the scenario combining loading at the top boundary and discharge at the bottom boundary, the mean total settlement and the average flux satisfy the relationship of superposition to be the sum of the separated effects of loading at top boundary and discharge at bottom boundary, but the variances do not. The proposed stochastic poroelastic method can be applied to hydrological issues that concern the interaction of flow and geomechanics.