High‐order models for hydro‐mechanical coupling problems in multiscale porous media

Abstract

An accurate prediction of nonlinear hydro‐mechanical (HM) coupling in subsurface structures with pronounced heterogeneity at multiple spatial scales is still an open topic and crucial for numerous engineering applications, for example, hydraulic fracturing and enhanced geothermal systems. In this study, novel high‐order multi‐scale asymptotic solutions are developed to accurately capture the locally oscillating characteristics of gas flow and solid displacement fields at multiple scales. First, the formal macro‐meso two‐scale asymptotic expansion is performed to establish the homogenized solutions and macro‐meso high‐order models which can predict the HM coupling problems at the macroscale and mesoscale. By directly expanding the cell functions defined at the mesoscale to the microscopic levels, the high‐order two‐scale expansion for the mesoscopic cell functions is built, and the upscaled relations for the flow parameters and constitutive coefficients from the microscale to the mesoscale and macroscale are developed correspondingly. The multi‐scale low/high‐order models are established by combining the high‐order expansion of all cell functions at the mesoscale and the developed macro‐meso two‐scale models. The main contributions are that the present approaches follow the reverse thought processes of reiteration homogenization methods, and offer a very innovative and efficient way to establish high‐accuracy high‐order models for the nonlinear HM coupling problems in the heterogeneous porous medium with any number of spatial scales. The effectiveness and accuracy of the present solution are validated by several representative cases with different constitutive coefficient contrasts. The results demonstrate that the high‐order solutions provide an accurate prediction for gas transport and solid deformation of heterogeneous porous media with multi‐scale configurations.