Applications of the fast multipole fully coupled poroelastic displacement discontinuity method to hydraulic fracturing problems

摘要

In this study, a fast multipole method (FMM) is used to decrease the computational time of a fully-coupled poroelastic hydraulic fracture model with a controllable effect on its accuracy. The hydraulic fracture model is based on the fully-coupled poroelastic formulation of the displacement discontinuity method (DDM) which is a special formulation of the boundary element method (BEM). DDM is a powerful and efficient method for problems involving fractures. However, this method involves the multiplication of a dense matrix with a vector in several places. Thus, the DMM algorithm slows down drastically as the number of elements increases, and even more so with the inclusion of necessary details such as poroelasticity, which makes the solution history-dependent. Rather, FMM is a technique to expedite matrix-vector multiplications, within a controllable error range, by approximating the far-range interactions without forming the matrix explicitly. In doi ng so, FMM offers a leverage in the computational efficiency by modifying the algorithm of the fully-coupled poroelastic displacement discontinuity method (PDMM) in two places. The first modification is in the time-marching scheme, which accounts for the solution of previous time steps to compute the current time step. The second modification is in the generalized minimal residual method (GMRES), where the unknowns are solved for iteratively.