A three-dimensional Generalized Finite Element Method for the simulation of wave propagation in fluid-filled fractures

This paper proposes a novel methodology for 3-D simulations of pressure waves propagating in fluid-filled fractures in an elastic body — the so-called Krauklis waves. The problem is governed by an approximation to the compressible Navier–Stokes equations for the viscous fluid in the fracture cavity coupled to the elastic-wave equation in the surrounding solid. The solid is assumed to be isotropic linear elastic and the fluid to be Newtonian. The elastic-wave equation is discretized in space with a quadratic Generalized Finite Element Method (GFEM) and in time by Newmark's method while the Krauklis waves equations are discretized in space by a quadratic standard FEM and in time by the α-method. A monolithic solver is adopted for the coupled problem and is numerically shown to be stable for the adopted approximation spaces. The GFEM is particularly appealing for the discretization of 3-D fractures since it does not require meshes fitting their geometry. Furthermore, analytical asymptotic solutions are used to enrich the GFEM approximation spaces, increasing the accuracy of the method. Mesh adaptivity around the fracture fronts is employed to further decrease the discretization error while controlling the computational time. The methodology is verified with analytical solutions and compared with experimental data. Different fracture geometries are investigated to demonstrate the complex 3-D effects of the physical phenomenon and the robustness of the proposed GFEM methodology.