Application of the Fast Multipole Fully Coupled Poroelastic Displacement Discontinuity Method to Hydraulic Fracturing Problems
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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 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 becomes slow as the number of temporal, or spatial elements increases, or necessary details such as poroelasticity, that makes the solution history-dependent, are added to the model. FMM is a technique to expedite matrix-vector multiplications within a controllable error without forming the matrix explicitly. Fully-coupled poroelastic formulation of DDM involves the multiplication of a dense matrix with a vector in several places. A crucial modification to DDM is suggested in two places in the algorithm to leverage the speed efficiency of FMM for carrying out these multiplications. 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) to iteratively solve for the problem unknowns. Several examples are provided to show the efficiency of the proposed approach in problems with large degrees of freedom (in time and space). Examples include hydraulic fracturing of a horizontal well and randomly distributed pressurized fractures at different orientations with respect to horizontal stresses. The results are compared to the conventional DDM in terms of computational processing time and accuracy. Accordingly, the proposed algorithm may be used for fracture propagation studies while substantially reducing the processing time with a controllable error.
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