Title: Application 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 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.