A Fast Adaptive Integral Method for Estimating Stress Distribution and Failure in Large Natural Fracture Networks

Abstract

Abstract The displacement discontinuity method (DDM) is a powerful method for hydraulic fracturing simulations. However, the computational time for this method increases exponentially as the number of fracture elements increases. This occurs primarily because the method involves the multiplication of a large dense matrix with a vector. In this study, a fast adaptive integral method (AIM) is used to reduce the computational time significantly when solving for the displacement field in a large complex fracture network. The key to the fast Fourier transform (FFT)-based adaptive integral method is the fast matrix-vector multiplication algorithm. The large dense matrix is decomposed into far-field and near-field components. The far-field component is computed by using the uniformly spaced Cartesian grids, and this component provides the foundation to perform discrete fast Fourier transform. The sparse near-field component is calculated by using the grids for fracture elements. Based on the split of dense matrix into far-field and near-field components, FFT is applied to accelerate the multiplication of matrix and vector since no dense matrices are used. A large fracture network is used to compare AIM with the standard DDM method. It is shown that the fast matrix-vector multiplication algorithm provides a very good approximation of the dense matrix. The accuracy of the fracture displacements is computed for different fracture orientation patterns. It is shown that the displacement calculated by AIM matches the displacement calculated by DDM very well. A comparison of the computational time for both the extended FFT-based AIM and DDM indicates that for small-scale problems, DDM performs as well or even better than AIM because extra computational time is needed for the correction between near neighbors and propagation on a regular grid. However, for large fracture networks with a large number of elements, the computational time needed by AIM is orders of magnitude less than that needed by DDM. The advantages of fast Fourier transform are then fully utilized to compute matrix and vector multiplication. The new algorithm (extending FFT-based AIM) provides a novel method for solving stresses and displacements in large-scale fracture networks. It is capable of drastically reducing the computational time for such problems with very little loss in accuracy.