Stabilized Finite Element Methods for Coupled Geomechanics - Reservoir Flow Simulations

Abstract

Abstract Coupled geomechanical-fluid flow models are needed to account for rock deformation resulting from flow-induced pressure changes in stress sensitive reservoirs. There are, however, issues of numerical stability that must be addressed before these coupled models can be used reliably. Specifically, it is known that standard procedures can lead to pressure oscillations as a result of the violation of the Babuska-Brezzi (B-B) condition, which requires unequal order interpolation of the displacement and pore-pressure variables. In this paper, a number of different types of coupled models are considered. A novel finite element method is developed to circumvent the B-B condition. The method applies a stabilized finite element technique to solve the force balance and pressure equations along with a finite volume method to solve the remaining component mass balance equations. All of the equations are solved in a fully coupled fashion. This method is compared with fully coupled and iteratively coupled models developed using non-stabilized finite elements for the force balance with finite volume methods for all of the component mass balance equations. These comparisons demonstrate that all methods perform reliably on homogeneous reservoirs over long time scales. The stabilized method is shown to provide improved stability at early times and for reservoirs with very low permeability barriers. Introduction Geomechanics can play an important role in oil recovery and oil field operations. In particular, geomechanics is crucial in such problems as production-induced compaction and subsidence, borehole stability, and hydraulic fracturing. In these areas, a key mechanism is the rock deformation due to changing in situ stress. Several investigations1,2,3 have shown that models that do not couple flow and geomechanics may give inaccurate predictions. For this reason, recent attention has been focused on modeling of the coupled system1,2,3,4,5. Here we present an accurate and stable numerical technique for coupled problems. In order to build robust and accurate numerical models, it is important to first consider the accuracy and stability properties of existing methods. Vermeer and Verruijt6 presented a stability condition for a simple consolidation problem. This condition required that time steps be larger than a certain value, otherwise spatial oscillations would occur. Such a requirement may seem counterintuitive, as we generally expect to observe improved accuracy as the time step is decreased. Within the context of coupled soil mechanics - fluid flow problems, Zienkiewicz et al.7 showed that the problem could be understood in terms of the Babuska-Brezzi stability condition (B-B condition)8,9, which requires nonuniform (i.e., different order) interpolation of the displacement and pore-pressure variables to ensure stability. Murad and Loula10,11 showed that an incompressible Stokes flow system at early times led to spatial instability if the B-B condition was not satisfied. Hughes et al.12 developed stabilized methods to circumvent the B-B restrictions for incompressible Stokes flow problems. These methods are within the category of weighted residual methods. Weighted residual methods modify the bilinear form associated with the problem so that improved numerical stability is achieved without compromising consistency. Zienkiewicz andWu13 showed similar stabilization by modifying the form of the time stepping in transient problems. In this work, we adopt the general approach of stabilized methods and develop stabilized finite element schemes to control the numerical instability arising in consolidation problems without losing consistency. Using this approach, any combination for interpolations of pressure and displacement can be used. We demonstrate both the instability that can be observed using standard methods as well as the improved performance of our stabilized method.