Fully-automated adaptive mesh refinement for media embedding complex heterogeneities: application to poroelastic fluid pressure diffusion

Relating the attenuation and velocity dispersion of seismic waves to fluid pressure diffusion (FDP) by means of numerical simulations is essential for constraining the mechanical and hydraulic properties of heterogeneous porous rocks. This, in turn, is of significant importance for a wide range of prominent applications throughout the Earth, environmental, and engineering sciences, such as, for example, geothermal energy production, hydrocarbon exploration, nuclear waste disposal, and CO₂ storage. In order to assess the effects of wave-induced FDP in heterogeneous porous rocks, we simulate time-harmonic oscillatory tests based on a finite element (FE) discretization of Biot’s equations in the time-frequency domain for representative elementary volumes (REVs) of the considered rock masses. The major challenge for these types of simulations is the creation of adequate computational meshes, which resolve the numerous and complex interfaces between the heterogeneities and the embedding background. To this end, we have developed a novel method based on adaptive mesh refinement (AMR), which allows for the fully automatic creation of meshes for strongly heterogenous media. The key concept of the proposed method is to start from an initially uniform coarse mesh and then to gradually refine elements which have non-empty overlaps with the embedded heterogeneities. This results in a hierarchy of non-uniform meshes with a large number of elements close to the interfaces, which do, however, not need to be explicitly resolved. This dramatically simplifies and accelerates the laborious and time-consuming process of meshing strongly heterogeneous poroelastic media, thus enabling the efficient simulation of REVs containing heterogeneities of quasi-arbitrary complexity. After a detailed description of the methodological foundations, we proceed to demonstrate that the FE discretization with low-order FE has a unique solution and hence does not present spurious modes. We assess the practical effectiveness and accuracy of the proposed method by means of four case studies of increasing complexity.