Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media (original) (raw)
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On Solving Groundwater Flow and Transport Models with Algebraic Multigrid Preconditioning
Groundwater
Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow solve and a pure advective transport solve with different right-hand sides. The new solver was compared with traditional preconditioned iterative methods and direct sparse solvers on several two-and three-dimensional benchmark problems spanning homogeneous and heterogeneous formations. For the groundwater flow problems, using the algebraic multigrid preconditioning speeds up the numerical solution by one to two orders of magnitude. Contrarily, a sparse direct solver was the most efficient for the pure advective transport processes such as the forward travel time simulations. Hence, the best sparse solver for the more general advection-dispersion transport equation is likely 2 to be Péclet number dependent. When equipped with the best solvers, processing multimillion grid blocks by the dual delineation approach is a matter of seconds. This paves the way for routine time-consuming tasks such as sensitivity analysis. The paper gives practical hints on the strategies and conditions under which algebraic multigrid preconditioning for the class of nonlinear and/or transient problems would remain competitive.
All Days, 2009
The solution of the linear system of equations for a large scale reservoir simulation has several challenges. Preconditioners are used to speed up the convergence rate of the solution of such systems. In theory, a preconditioner defines a matrix M that can be inexpensively inverted and represents a good approximation of a given matrix A. In this work, two-stage preconditioners consisting of the approximated inverses M1 and M2 are investigated for multiphase flow in porous media. The first-stage preconditioner, M1, is approximated from Ausing four different solution methods: (1) constrained pressure residuals (CPR), (2) lower block Gauss-Seidel, (3) upper block Gauss-Seidel, and (4) one iteration of block Gauss-Seidel. The pressure block solution in each of these different schemes is calculated using the Algebraic Multi Grid (AMG) method. The inverse of the saturation (or more generally, the nonpressure) blocks are approximated using Line Successive Over Relaxation (LSOR). The second...
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.
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 and Analytical Methods in Geomechanics, 2010
Large-scale simulations of flow in deformable porous media require efficient iterative methods for solving the involved systems of linear algebraic equations. Construction of efficient iterative methods is particularly challenging in problems with large jumps in material properties, which is often the case in geological applications, such as basin evolution at regional scales. The success of iterative methods for this type of problems depends strongly on finding effective preconditioners. This paper investigates how the block-structured matrix system arising from single-phase flow in elastic porous media should be preconditioned, in particular for highly discontinuous permeability and significant jumps in elastic properties. The most promising preconditioner combines algebraic multigrid with a Schur complement-based exact block decomposition. The paper compares numerous block preconditioners with the aim of providing guidelines on how to formulate efficient preconditioners.
Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media
Computational Geosciences
This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approachand in particular, FSMSN-results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance-i.e., measured by wall-clock time-will be studied in a subsequent publication.
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. *
Physics-based preconditioners for porous media flow applications
… Techniques For Large Sparse Matrix Problems …
Eigenvalues of smallest magnitude have known to be a major bottleneck for iterative solvers. Such eigenvalues become a more dramatic bottleneck when the underlying physical properties have severe contrasts. These contrasts are commonly found subsurface geological properties such as permeability and porosity. We intend to construct a method as algebraic as possible. In particular, we propose an algebraic way of using the underlying permeability field to mark certain degrees of freedom as high permeable when they exceed a certain threshold. This marking process will define a permutation matrix which allows us to collect the degrees of freedom that causes the smallest eigenvalues in a subblock. We claim the responsibility of ill-conditioning to this subblock of the system matrix. The remaining of the matrix will then be well-conditioned if certain heuristics about the permeability field are satisfied. In our two-stage preconditioning approach, the first stage comprises the process of collecting small eigenvalues and solving them separately; the second stage deals with the remaining of the matrix possibly with a deflation strategy if needed. Numerical examples are shown for one-and two-phase flow scenarios in reservoir simulation applications. We demonstrate that our two-stage preconditioners are more effective and robust compared to deflation methods. Due to their algebraic nature, they support flexible and realistic reservoir topology.
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.
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.