Mixed-dimensional poromechanical models of fractured porous media

Abstract

We combine classical continuum mechanics with the recently developed calculus for mixed-dimensional problems to obtain governing equations for flow in, and deformation of, fractured materials. We present models in both the context of finite and infinitesimal strain, and discuss nonlinear (and non-differentiable) constitutive laws such as friction models and contact mechanics in the fracture. Using the theory of well-posedness for evolutionary equations with maximal monotone operators, we show well-posedness of the model in the case of infinitesimal strain and under certain assumptions on the model parameters.

Discrete fracture‐matrix model of poroelasticity

“…In contrast, our operators ∇ and div, which naturally appear as the result of semi-discretization, do not satisfy the integration by parts, therefore the additional hypothesis (2.4) of permeable fracture is required to obtain a mathematically well-posed problem. Another point of difference between [31] and the present paper stems from the use of different measures of volumetric change (cf. the quantity { { div𝐔} } in our paper with 𝔗 ′ (𝔢) and Example 3.6 in [31]).…”

“…Another point of difference between [31] and the present paper stems from the use of different measures of volumetric change (cf. the quantity { { div𝐔} } in our paper with 𝔗 ′ (𝔢) and Example 3.6 in [31]).…”

“…Remark 4.1. In [31], another DFM model of linearized poroelasticity was obtained using a different approach (see problem (4.11) therein). That model was derived from a mixed-dimensional version of continuum-mechanics laws, postulated directly in a domain containing the reduced geometry of fractures.…”

Fluid flow and mass transport in fractured media with curved fractures and varying apertures: A 3D modeling approach

“…Combining with Delaunay algorithm and hybrid-dimensional modeling approach, the mesh-dependent method had been extensively studied and applied in coupling processes. 31,32 In contrast, the grids in the mesh-independent method does not need to be confined along the fractures. A popular one is the embedded discrete fractured model (EDFM).…”

“…Delaunay algorithm is an efficient method to generate unstructured grids, where the rock matrix and fractures are discretized by tetrahedrons and triangles, respectively. Combining with Delaunay algorithm and hybrid‐dimensional modeling approach, the mesh‐dependent method had been extensively studied and applied in coupling processes 31,32 . In contrast, the grids in the mesh‐independent method does not need to be confined along the fractures.…”