Fast multipole displacement discontinuity method (FM-DDM) for geomechanics reservoir simulations

Abstract

Summary The displacement discontinuity method (DDM) is frequently used in geothermal and petroleum applications for modeling the behavior of fractures in linear‐elastic rocks. The DDM requires O(N2) memory and O(N3) floating point operations (where N is the number of unknowns) to construct the coefficient matrix and solve the linear system of equations by direct methods. Therefore, the conventional implementation of the DDM is not computationally efficient for very large systems of cracks, often limiting its application to small‐scale problems. This work presents an approach for solving large‐scale fracture problems using the fast multipole method (FMM). The approach uses both the DDM and a kernel‐independent version of the FMM along with a preconditioned generalized minimal residual algorithm to accelerate the solution of linear systems of equations using desktop computers. Using the fundamental solutions for constant displacement discontinuity in a two‐dimensional elastic medium, several numerical examples involving fracture networks representing fractured reservoirs are treated. Numerical results show good agreement with analytical solutions and demonstrate the efficiency of the FMM implementation of the DDM for large‐scale simulations. Copyright © 2015 John Wiley & Sons, Ltd.

Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation

“…Scientific applications of FMM for boundary integrals include acoustics [95,57], biomolecular electrostatics [103], electromagnetics [33,41], fluid dynamics for Euler [94] and Stokes [86] flows, geomechanics [90], and seismology [21,93]. Application areas of FMM for discrete volume integrals are astrophysics [14], Brownian dynamics [73], classical molecular dynamics [82], density functional theory [88], vortex dynamics [104], and force directed graph layout [105].…”

The Shear Mechanisms of Natural Fractures during the Hydraulic Stimulation of Shale Gas Reservoirs

“…The propagation of complex fracture networks with hundreds of fractures can be easily simulated by DDM based models [ 13 , 28 , 29 , 30 , 31 , 32 , 33 ]. Moreover, based on the traditional DDM, Verde and Ghassemi [ 34 , 35 ] proposed a fast multipole DDM by considering the difference when calculating the induced stress of far field and near field elements. Using this technique, a fracture system with up to 100,000 boundary elements can be simulated on current mainstream computers.…”

Large-scale poroelastic fractured reservoirs modeling using the fast multipole displacement discontinuity method

“…Previous works combining DDM and FMM for large‐scale problems account only for elastic rocks, neglecting the fluid leak‐off from the fractures into matrix and its influence over the stresses and pore pressure. This could be mainly because of the difficulty of understanding FMM and lack of mathematical developments for the fundamental solutions of interest as well as the associated high programming complexity.…”

“…In although the efficiency is sometimes guaranteed under certain element arrangements by adopting the conventional FMM formulation for two‐dimensional potential problems, the accuracy is compromised by the selected kernel approximations. More than a decade later, Verde and Ghassemi proposed the so called Fast Multipole‐Displacement Discontinuity Method (FM‐DDM) using a kernel‐independent version of the FMM which does not require approximations of kernel to compute the geomechanical interactions in naturally fractured reservoirs containing up to 100 000 fractures. In the accuracy and efficiency are satisfied, but the approach is still focused to solve elastic fracture problems of limited applicability for the petroleum and geothermal industries by neglecting fluid flow through the fracture network.…”

“…More than a decade later, Verde and Ghassemi proposed the so called Fast Multipole‐Displacement Discontinuity Method (FM‐DDM) using a kernel‐independent version of the FMM which does not require approximations of kernel to compute the geomechanical interactions in naturally fractured reservoirs containing up to 100 000 fractures. In the accuracy and efficiency are satisfied, but the approach is still focused to solve elastic fracture problems of limited applicability for the petroleum and geothermal industries by neglecting fluid flow through the fracture network. More recently, Verde and Ghassemi presented a novel approach for the efficient dynamic solution of the coupled fluid flow‐geomechanical problem in large‐scale naturally fractured reservoirs solving up to 200 000 unknowns.…”