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

Abstract

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 CO 2 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 RVEs 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 1 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.

Seismic Signatures of Fractured Porous Rocks: The Partially Saturated Case

“…In order to obtain the seismic response, we locally solve Equations 1 to 4 for a representative elementary volume (REV) of the formation of interest by means of a finite element method (Favino et al., 2019). We employ a set of boundary conditions, which can be classified in the form of two compressional and one shear oscillatory relaxation tests (Figure 1).…”

“…In all tests, we apply periodic boundary conditions for the solid displacement and the fluid pressure along the remaining boundaries of the sample. Antiperiodic boundary conditions are considered for the traction and the fluid flux (Favino et al., 2019).…”

“…A detailed description of the numerical upscaling procedure and its boundary conditions is given by Favino et al. (2019).…”

An Equi-Dimensional Finite Element Approach for Flow Problems in Fractured Porous Media

“…We briefly describe the AMR strategy for the generation of adapted equi-dimensional meshes for arbitrary fracture networks. We follow the strategy presented in [4] even if we employ a different criterion to choose the elements to refine. Such an approach has been used for the simulation of hydro-mechanical coupling in fractured porous media in [9,18].…”

“…In this work, we present an alternative approach based on an equi-dimensional representation of the fractures and a continuous FE method on adapted meshes that allows to simulate fractured media of realistic complexity. Employing an idea similar to the one presented in [4], we use an adaptive mesh refinement (AMR) strategy to create a hierarchy of meshes that approximate the interfaces between the fractures and the embedding background with increasing accuracy without the need to explicitly resolve them. Starting from an initial regular mesh unrelated to the fracture distribution, at each step of the AMR strategy, we create a new mesh level by refining the elements which have non-empty overlap with the interfaces between fractures and background.…”