Hybrid-Dimensional Biot Problem with an Optimization Based Domain Decomposition Approach

The present work proposes a numerical approach for solving coupled flow and mechanics problems in fractured porous media, represented as mixed-dimensional domains.

In this formulation, the elements of the 3D mesh are allowed to arbitrarily intersect the fractures.

Displacements are discontinuous across fractures through the use of the eXtended Finite Element Method (XFEM) on the 3D mesh.

The mechanical problem is formulated as a saddle-point problem, in which Lagrange multipliers are used to enforce displacement continuity across the fractures.

The resulting Lagrange multipliers represent the stress field acting on the fracture surfaces.

Likewise, the pressure field is allowed to be discontinuous across fractures through the XFEM formulation on the non-conforming mesh and is computed using an optimization-based domain decomposition strategy specifically designed for mixed-dimensional problems.

The fixed-stress splitting scheme is employed to decouple the flow and mechanics subproblems, while the mixed-dimensional pressure problem is solved at each fixed-stress iteration using the Conjugate Gradient (CG) method.

The combination of the fixed-stress scheme and the CG solver proves to be highly effective for this class of problems.