Algebraic Multigrid Methods (AMG) for the Efficient Solution of Fully Implicit Formulations in Reservoir Simulation (original) (raw)
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This paper demonstrates an application of element-based Algebraic Multigrid (AMGe) technique developed at LLNL (19) to the numerical upscaling and preconditioning of subsurface porous media flow problems. The upscaling results presented here are further extension of our recent work in 3. The AMGe approach is well suited for the solution of large problems coming from finite element discretizations of systems of partial differential equations. The AMGe technique from 10,9 allows for the construction of operator-dependent coarse (upscaled) models and guarantees approximation properties of the coarse velocity spaces by introducing additional degrees of freedom associated with non-planar interfaces between agglomerates. This leads to coarse spaces which maintain the specific desirable properties of the original pair of Raviart-Thomas and piecewise discontinuous polynomial spaces. These coarse spaces can be used both as an upscaling tool and as a robust and scalable solver. The methods employed in the present paper have provable O(N) scaling and are particularly well suited for modern multicore architectures, because the construction of the coarse spaces by solving many small local problems offers a high level of concurrency in the computations. Numerical experiments demonstrate the accuracy of using AMGe as an upscaling tool and comparisons are made to more traditional flow-based upscaling techniques. The efficient solution of both the original and upscaled problem is also addressed, and a specialized AMGe preconditioner for saddle point problems is compared to state-of-the-art algebraic multigrid block preconditioners. In particular, we show that for the algebraically upscaled systems, our AMGe preconditioner outperforms traditional solvers. Lastly, parallel strong scaling of a distributed memory implementation of the reservoir simulator is demonstrated.
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.
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.
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.
IMA Journal of Numerical Analysis, 1999
We analyze the use of fast solvers as preconditioners for the discretized pressure equation arising in reservoir simulation. Under proper conditions on the permeability functions and the source term, we show that the number of iterations for the conjugate gradient method is bounded independently of both the lower bound δ of the permeability and the discretization parameter h. Such results are obtained for a special class of self-adjoint second-order elliptic problems with discontinuous coef"cients. We also discuss how fast solvers can be utilized in the presence of non-rectangular domains by applying a domain imbedding procedure. The theoretical results are illustrated and supplemented by a series of numerical experiments.
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...
Simultaneous Solution of Multiphase Reservoir Flow Equations
Society of Petroleum Engineers Journal, 1970
A strongly implicit iterative procedure has been developed to solve systems of equations arising in multiphase, two-dimensional reservoir flow problems. The two-dimensional, two-phase and problems. The two-dimensional, two-phase and two-dimensional, three-phase algorithms have been evaluated by several test problems and compared with the corresponding alternating direction iterative routines. The strongly implicit procedure (SIP) has been found to have several advantages in the solution of reservoir problems. It is fast, and in problems with extreme anisotropy in the transmissibilities and/or highly irregular geometries it can obtain a solution where the alternating direction procedure many times cannot. For the problems tested, it bas been found that a reliable set of iteration parameters is easily calculated from the coefficient matrix. Finally, SIP appears to be relatively insensitive to the rounding errors inherent in machine computations. Introduction The efficient solution of ...
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.
A Preconditioned Conjugate Gradient Based Algorithm for Coupling Geomechanical-Reservoir Simulations
Oil & Gas Science and Technology, 2002
Un algorithme de gradient conjugué préconditionné pour le couplage de codes géomécanique et réservoir -Cet article présente un nouvel algorithme de couplage entre l'écoulement des fluides en milieux poreux du simulateur de réservoir et le code de géomécanique modélisant la compaction du milieu poreux. Le couplage est réalisé entre des périodes de simulations de réservoir et les calculs géomécaniques à la fin de chaque période de temps. L'approche proposée repose sur un algorithme de type gradient conjugué préconditionné appliqué au champ de déplacement mécanique. Cet algorithme est comparé, sur un exemple monodimensionnel, à l'algorithme de couplage décalé relaxé par un paramètre de compressibilité de roche. Nous concluons que l'algorithme de gradient conjugué est bien plus robuste et converge plus rapidement que l'algorithme décalé, avec un coût supplémentaire par itération qui reste négligeable en pratique.
Deflation AMG Solvers for Highly Ill-Conditioned Reservoir Simulation Problems
Proceedings of SPE Reservoir Simulation Symposium, 2007
In recent years, deflation methods have received increasingly particular attention as a means to improving the convergence of linear iterative solvers. This is due to the fact that deflation operators provide a way to remove the negative effect that extreme (usually small) eigenvalues have on the convergence of Krylov iterative methods for solving general symmetric and non-symmetric systems. Hence, the present approach offers the possibility of applying AMG to more general large-scale reservoir settings without further modifications to the AMG implementation or algebraic manipulation of the linear system (as suggested by two-stage preconditioning methods). Promising results are supported by a suite of numerical experiments with extreme permeability contrasts.