Coupled Geomechanical and Reservoir Modeling on Parallel Computers

Abstract

In coupled geomechanics and reservoir modeling, the finite element discretization of the force balance equation leads to very large linear systems, whose solution is both time and memory consuming. ICCG (Incomplete Cholesky Factorized Conjugate Gradient) is a popular technique for solving for displacements, but the technique is limited to about 60,000 nodal points on desktop machines. Most large 3D field scale problems will have to be run on parallel machines. In this paper, we present a reduced-communication, super coarsening multigrid method that can be combined with other domain decomposition-based preconditioner to achieve faster convergence with high parallel scalability. A preliminary test case of 1.5 million grid blocks with up to 59 processors shows a parallel efficiency of above 90%. Introduction The modeling of fluid-structure interactions is of growing importance for both energy and environmental applications. Due to complexities, nonlinearities, phase behavior, and the number of partial differential equations required to describe a coupled system of poro-elasticity and/or poro-plasticity with multiphase flow, extending a conventional reservoir model to a coupled fluid-flow and geomechanical model is not trivial, though considerable success has been achieved in recent years. Several authors1,2,3,4,5 have presented mathematical and numerical formulations for the coupled equations. Different coupling techniques have been investigated6,7 that are applicable to existing reservoir flow models and one's choice normally depends on speed, accuracy, or ease of implementation. Comparisons have been done between coupled and uncoupled simulations, and between different coupling techniques for accuracy and efficiency. Recently, stability issues have also been discussed concerning oscillations in low permeability zones. Individual models are becoming more sophisticated, and coupling methods are becoming more accurate and more stable. However, the computational bottleneck of solving the large linear system still remains. The finite element formulation of the geomechanics model has a 27-point stencil compared to the standard 7-point stencil in the finite difference formulation of multiphase flow equations. More than 70 percent of the total CPU time for a coupled simulation is spent in the geomechanics model solving the system of linear equations. So it is very important to develop efficient linear solvers to reduce the computational cost without losing scalability on parallel computers, so that coupled analyses can be economically and numerically feasible for practical field applications. Thomas et al.8 describe a coupled procedure running on multiple processors, but the parallel efficiency with 16 processors is less than 50 percent. In this work, we present a parallel poroelastic model combined with a multiphase flow model using the iteratively coupled technique. The linear solver we employ is PCG preconditioned by Incomplete Cholesky Factorization (IC) or block Jacobi. Faster convergence and higher scalability is achieved by calling a super coarsening multigrid routine to make further global corrections after each preconditioning step. The rest of the paper is organized as follows. First the governing flow and deformation equations are described briefly. Instability issues are also briefly discussed. Then the super coarsening multigird preconditioner and its parallel implementation are presented. Results of numerical experiments conducted on a 64-node Beowulf PC cluster are shown to demonstrate the efficiency and parallel scalability of the combined IC (or block Jacobi) and multigrid preconditioned BiCGSTAB (Biconjugate Gradient Stablized) method. Geomechanics and Multiphase Flow Equations Terzaghi first analyzed the fluid-flow-stress coupling equations in 1925 as a 1D consolidation problem. Later, Biot9 extended the theory to a more generalized three-dimensional case, based on a linear stress-strain relation and a single-phase fluid flow. Here we present an extension of Biot's equations for three-phase immiscible and isothermal flow.

Development of an Efficient Embedded Discrete Fracture Model for 3D Compositional Reservoir Simulation in Fractured Reservoirs

“…Parallel processing with high performance computing clusters can alleviate these challenges to a good extent. Wang et al (1997), Zhang et al (2001), Dogru et al (2002), Gai et al (2003), andDeBaun et al (2005) demonstrated the necessity, advantages, and applicability of using parallel processing for large-scale reservoir simulations.…”

Domain Decomposition Methods in Geomechanics

“…In both levels, i.e. reservoir and near borehole, the flow simulation needs to be coupled to geomechanics in order to predict these changes and their impact [40,69,74], which causes a significant increase in CPU time and memory requirements.…”

“…He also implemented a model that includes thermal effects and a plasticity model for the porous medium. Also at CSM, Gai [74] used continuous elements for displacements and a cell-centered finite difference method for pressure and implemented an iteratively coupled scheme to find the numerical solution. She also tackled the multiphase flow version of the poroelasticity equations.…”

Linear Thermo-Poroelasticity and Geomechanics

“…The mathematical formulation considers a bounded domain Ω ⊂ R n ,n ¼ 2, 3 and its boundary is Γ ¼ ∂Ω, and a time interval of interest 0, ℑ½.LetT h be a non-degenerate, quasi-uniform conforming partition of Ω composed of triangles or quadrilaterals for two-dimensional problems, and hexahedra or tetrahedra for three-dimensional problems. For instance, Gai [5] thesis showed that deformable porous media, i.e., the reservoir matrix, the single-phase flow model equation derives from the continuity equation, i.e., a mass balance statement, for slightly incompressible single-phase flow and Darcy's law which yields:…”