Augmented multiscale linear solver for mechanics in fractured formations

Abstract

High-resolution geo-models with millions of grid cells are becoming a standard for modern reservoir simulators in oil and gas industry. In many applications such as uncertainty qualification, well location optimization, hydraulic fracturing studies, history matching, etc dynamic modeling of the global domain requires repetitive computation of the discretized linear system, which oftentimes ends up being the bottleneck for fast simulation (taking over 50% of the total simulation time). Mechanical deformation of fractured formations significantly affect the flow pattern of a reservoir through fracture closure/opening and slip reactivation due to stress state and fluid pressure changes. Multilevel iterative approaches use restriction and prolongation operators to effectively transfer information across the scales, and coupled with a local smoother lead to a superior convergence rate. However, fractures cause additional challenges for iterative linear solvers since they introduce discontinuity, irregular grids and ill-conditioned matrices, slowing down the solver. An Augmented Two-Step Algebraic Multiscale (TAMS) preconditioner is formulated for mechanical deformation of reservoirs with explicitly defined fractures to aid an iterative solver to a faster convergence to the fine solution. In the core of the proposed Augmented TAMS preconditioner as a global stage preconditioner is an adaptive enrichment of the standard multiscale basis function space with the special, numerically computed, discontinuous shape functions honoring the physics of the fractures. These functions propagate additional fracture-related information to the coarse scale and thus able to better approximate the local variation of the solution on the fine scale. The method is incorporated into a 3D general purpose research simulator AD-GPRS on an iterative linear solver level. Demonstration of the performance of the method is done by a numerical error analysis on a series of test cases for fractured formations with an arbitrary fracture geometry and distribution. A significant speed-up of about 30-50% in the number of linear iterations in comparison with the standard TAMS methods is achieved. The approach is also tested against the state-of-the art SAMG solver and shows relative reduction in the number of iterations in the presence of fractures.