On Solving Groundwater Flow and Transport Models with Algebraic Multigrid Preconditioning (original) (raw)
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Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media
SIAM Journal on Scientific Computing, 2017
Multiphase flow is a critical process in a wide range of applications, including carbon sequestration, contaminant remediation, and groundwater management. Typically, this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase (e.g., oil, water), along with constitutive laws for the relationship of phase velocity to phase pressure. In this study, we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time-implicit treatment of models of multiphase flow. We explore the performance of several preconditioners based on algebraic multigrid (AMG) for solving the linearized problem, including "black-box" AMG applied directly to the system, a new version of constrained pressure residual multigrid (CPR-AMG) preconditioning, and a new preconditioner derived using an approximate Schur complement arising from the block factorization of the Jacobian. We show that the new methods are the most robust with respect to problem character as determined by varying effects of capillary pressures, and we show that the block factorization preconditioner is both efficient and scales optimally with problem size.
2007
Application of algebraic multigrid method and approximate sparse inverses are applied as preconditioners for large algebraic systems arising in approximation of diffusion-reaction problems in 3-dimensional complex domains. Here we report the results of numerical experiments when using highly graded and locally refined meshes for problems with non-homogeneous and anisotropic coefficients that have small features and almost singular solutions. For the discretization of the domain and the finite element approximation we have used the system AGGIEFEM, a universal computational tool for PDEs developed in the VIGRE seminar in Introduction to Scientific Computing at TAMU. For solving the algebraic system we have used ParaSails and BoomerAMG preconditioners that are part of the HYPRE (High Performance Preconditioners) library developed in CASC at Lawrence Livermore National Laboratory. *
Journal of Computational Physics, 1998
Parallel preconditioners are considered for improving the convergence rate of the conjugate gradient method for solving sparse symmetric positive de nite systems generated by nite element models of subsurface ow. The di culties of adapting e ective sequential preconditioners to the parallel environment are illustrated by our treatment of incomplete Cholesky preconditioning. These di culties are avoided with multigrid preconditioning, which can be extended naturally to many processors so that the preconditioner remains global and e ective. The coarse grid correction which de nes the multigrid preconditioner is outlined and its parallel implementation with the distributed nite element data structure is presented, along with some examples of its use as a parallel preconditioner.
SPE Reservoir Simulation Symposium, 2007
A primary challenge for a new generation of reservoir simulators is the accurate description of multiphase flow in highly heterogeneous media and very complex geometries. However, many initiatives in this direction have encountered difficulties in that current solver technology is still insufficient to account for the increasing complexity of coupled linear systems arising in fully implicit formulations. In this respect, a few works have made particular progress in partially exploiting the physics of the problem in the form of two-stage preconditioners. Two-stage preconditioners are based on the idea that coupled system solutions are mainly determined by the solution of their elliptic components (i.e., pressure). Thus, the procedure consists of extracting and accurately solving pressure subsystems. Residuals associated with this solution are corrected with an additional preconditioning step that recovers part of the global information contained in the original system. Optimized and highly complex hierarchical methods such as algebraic multigrid (AMG) offer an efficient alternative for solving linear systems that show a "discretely elliptic" nature. When applicable, the major advantage of AMG is its numerical scalability; that is, the numerical work required to solve a given type of matrix problem grows only linearly with the number of variables. Consequently, interest in incorporating AMG methods as basic linear solvers in industrial oil reservoir simulation codes has been steadily increasing for the solution of pressure blocks. Generally, however, the preconditioner influences the properties of the pressure block to some extent by performing certain algebraic manipulations. Often, the modified pressure blocks are "less favorable" for an efficient treatment by AMG. In this work, we discuss strategies for solving the fully implicit systems that preserve (or generate) the desired ellipticity property required by AMG methods. Additionally, we introduce an iterative coupling scheme as an alternative to fully implicit formulations that is faster and also amenable for AMG implementations. Hence, we demonstrate that our AMG implementation can be applied to efficiently deal with the mixed elliptic-hyperbolic character of these problems. Numerical experiments reveal that the proposed methodology is promising for solving large-scale, complex reservoir problems.
International Journal for Numerical Methods in Engineering, 2010
A new library called FLEXMG has been developed for a spectral/finite-element incompressible flow solver called SFELES. FLEXMG allows to use various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, of smooth aggregation-type and non nested finite-element-type. Unlike gridless multigrid, both of these families use the information contained in the initial fine mesh. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function while the non nested finite-element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled to a GMRES method. After analyzing the accuracy of our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption are compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study.
Multigrid reduction preconditioning framework for coupled processes in porous and fractured media
Computer Methods in Applied Mechanics and Engineering, 2021
Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long timescales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Because of the wide range of processes and couplings that may be involved-e.g. formation and propagation of fractures, deformation of the solid porous medium, viscous flow of one or more fluids in the pores and fractures, complicated well sources and sinks, etc.-it is difficult to develop general-purpose but scalable linear solver frameworks. This challenge is further aggravated by the range of different discretization schemes that may be adopted, which have a direct impact on the linear system structure. To address this obstacle, we describe a flexible strategy based on multigrid reduction (MGR) that can produce purely algebraic preconditioners for a wide spectrum of relevant physics and discretizations. We demonstrate that MGR, guided by physics and theory in block preconditioning, can tackle several distinct and challenging problems, notably: a hybrid discretization of single-phase flow, compositional multiphase flow with complex wells, and hydraulic fracturing simulations. Extension to other systems can be handled quite naturally. We demonstrate the efficiency and scalability of the resulting solvers through numerical examples of difficult, field-scale problems.
A Block-Diagonal Algebraic Multigrid Preconditioner for the Brinkman Problem
The Brinkman model is a unified law governing the flow of a viscous fluid in cavity (Stokes equations) and in porous media (Darcy equations). In this work, we explore a novel mixed formulation of the Brinkman problem by introducing the flow's vorticity as an additional unknown. This formulation allows for a uniformly stable and conforming discretization by standard finite element (Nédélec, Raviart-Thomas, discontinuous piecewise polynomials). Based on the stability analysis of the problem in the H(curl) − H(div) − L 2 norms ([24]), we study a scalable block diagonal preconditioner which is provably optimal in the constant coefficient case. Such preconditioner takes advantage of the parallel auxiliary space AMG solvers for H(curl) and H(div) problems available in hypre ([11]). The theoretical results are illustrated by numerical experiments.
Application of an energy-minimizing algebraic multigrid method for subsurface water simulations
International Journal of Numerical Analysis and Modeling
Efficient methods for solving linear algebraic equations are crucial to creating fast and accurate numerical simulations in many applications. In this paper, an algebraic multigrid (AMG) method, which combines the classical coarsening scheme by [J. W. Ruge and K. Stüben, “Algebraic multigrid”, Multigrid methods 3, 73–130 (1987)] with an energy-minimizing interpolation algorithm by [J. Xu and L. Zikatanov, Comput. Vis. Sci. 7, No. 3-4, 121–127 (2004; Zbl 1077.65130)], is employed and tested for subsurface water simulations. Based on numerical tests using real field data, our results suggest that the energy-minimizing algebraic multigrid method is efficient and, more importantly, very robust.
Combined Preconditioning with Applications in Reservoir Simulation
Multiscale Modeling & Simulation, 2013
We develop a simple algorithmic framework to solve large-scale symmetric positive definite linear systems. At its core, the framework relies on two components: (1) a norm-convergent iterative method (i.e. smoother) and (2) a preconditioner. The resulting preconditioner, which we refer to as a combined preconditioner, is much more robust and efficient than the iterative method and preconditioner when used in Krylov subspace methods. We prove that the combined preconditioner is positive definite and show estimates on the condition number of the preconditioned system. We combine an algebraic multigrid method and an incomplete factorization preconditioner to test the proposed framework on problems in petroleum reservoir simulation. Our numerical experiments demonstrate noticeable speed-up when we compare our combined method with the standalone algebraic multigrid method or the incomplete factorization preconditioner.
An efficient algebraic multigrid preconditioned conjugate gradient solver
Computer Methods in Applied Mechanics and Engineering, 2003
In this paper, we present a robust and efficient algebraic multigrid preconditioned conjugate gradient solver for systems of linear equations arising from the finite element discretization of a scalar elliptic partial differential equation of second order on unstructured meshes. The algebraic multigrid (AMG) method is one of most promising methods for solving large systems of linear equations arising from unstructured meshes. The conventional AMG method usually requires an expensive setup time, particularly for three dimensional problems so that generally it is not used for small and medium size systems or low-accuracy approximations. Our solver has a quick setup phase for the AMG method and a fast iteration cycle. These allow us to apply this solver for not only large systems but also small to medium systems of linear equations and also for systems requiring low-accuracy approximations.