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

Abstract

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 doing 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.Several examples are provided to show the efficiency of the proposed fast multipole fully-coupled poroelastic displacement discontinuity model (FMPDDM) in problems with large degrees of freedom (in time and space). These examples include hydraulic fracturing of a horizontal well and randomly distributed pressurized fractures at different orientations with respect to principal horizontal stresses. The results of FMPDDM are compared to PDDM in terms of computational time and accuracy, which demonstrates that FMPDDM decreases the computation time by up to a factor of 70 for a case with 20000 piece-wise constant elements and a single time step. The difference between the computational times of FMPDDM and PDDM can be higher for a larger number of elements and time steps. Moreover, the associated absolute value of relative error for a problem of this size is observed to be less than 4% for the hydraulic fracture example, and less than 0.5% for the example of randomly distributed pressurized fractures. Furthermore, the solution of tip displacements using both methods are used to compare the computation of stress intensity factors (SIF) in mode I and II, which are needed for fracture propagation. The calculated displacement discontinuities reveal that the SIF calculation using FMPDDM has a negligible error compared to PDDM. Therefore, FMPDDM will not affect the estimation of the fracture propagation direction. Accordingly, the proposed algorithm may be used for fracture propagation studies to substantially reduce the computational time.