Simulation of flow through highly heterogeneous porous media results in large ill-conditioned systems of equations. In particular, solving the linearized pressure system can be especially time-consuming. Therefore, extensive efforts to find ways to address this issue effectively are required. In this work, we introduce a POD-based deflation method that combines the advantages of two state of the art techniques: Proper Orthogonal Decomposition (POD) and the deflation method. The dominant features of the system are captured in a set of POD basis vectors, used later to accelerate the solution of linear systems with a deflation procedure. If all of the system information is contained in the POD basis, the deflation method converges in one iteration. This behavior was compared with the usual choices of deflation vectors, which require more than 18 iterations for the same number of deflation vectors. If only part of this information is obtained, the POD-based deflation method gives a good initial solution, after one iteration the error of the solution is of order 10^{-4}. The applicability of the POD-based deflation method does not depend on the test case. It is implemented for reservoir simulation problems, but it can be implemented for any time-varying problem. Furthermore, we study its applicability for various 2L-PCG methods, but it can also be implemented together with many other linear solvers, e.g., multigrid, multilevel, and domain decomposition techniques. The implementation can also be extended to include various preconditioners.@en

We study fast and robust iterative solvers for large systems of linear equations resulting from simulation of flow trough strongly heterogeneous porous media. We propose the use of preconditioning and deflation techniques, based on information obtained frfrom the system, to reduce the time spent in the solution of the linear system.An important question when using deflation techniques is how to find good deflation vectors, which lead to a decrease in the number of iterations and a small increase in the required computing time per iteration. In this paper, we propose the use of deflation vectors based on a POD-reduced set of snapshots. We investigate convergence and the properties of the resulting methods. Finally, we illustrate these theoretical results with numerical experiments. We consider compressible and incompressible single-phase flow in a layered model with variations in the permeability layers up to 10 3 and the SPE 10 benchmark model with a contrast in permeability coefficients of 10 7. Using deflation for the incompressible problem, we reduce the number of iterations to 1 or 2 iterations. With deflation, for the compressible problem, we reduce up to ∼ 80% the number of iterations when compared with the only-preconditioned solver.@en

We explore and develop POD-based deflation methods to accelerate the solution of large-scale linear systems resulting from two-phase flow simulation. The techniques here presented collect information from the system in a POD basis, which is later used in a deflation scheme. The snapshots required to obtain the POD basis are captured in two ways: a moving window approach, where the most recently computed solutions are used, and a training phase approach, where a full pre-simulation is run. We test this methodology in highly heterogeneous porous media: a full SPE 10 model containing O(10^6) cells, and in an academic layered problem presenting a contrast in permeability layers up to 10^6. Among the experiments, we study cases including gravity and capillary pressure terms. With the POD-based deflated procedure, we accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method, reducing the required number of iterations to around 10-30 %, i.e., we achieve speed-ups of factors three to ten. @en

Simulation of two-phase flow through highly heterogeneous porous media results in ill-conditioned large systems of linear equations for the pressure when using, e.g., sequential procedures. Solving the resulting linear system can be particularly time-consuming. Therefore, there have been extensive efforts to find ways to address this issue effectively. Iterative methods, together with preconditioning techniques [1, 2], are the most commonly chosen techniques to solve these problems. In the literature, we can also find Reduced Order Models (ROM) [3–5] and deflation methods [6, 7], where system information is reused to find a good approximation more quickly. For the deflation techniques, an optimal selection of deflation vectors is crucial for a good performance. The construction of deflation vectors based on information captured with ROM, in particular, Proper Orthogonal Decomposition (POD), was recently presented for a single-phase flow problem [8, 9]. The goal of this work is to further explore and develop the possibilities of combining POD and deflation techniques for a two-phase flow simulation. We propose selecting deflation vectors from a POD basis in two different ways. In the first one, we obtain the basis on-the-fly during the simulation, to accelerate the upcoming time steps. For the second one, the basis is obtained off-line in a training phase, and it is used later to solve slightly different problems. The convergence properties of the resulting POD-based deflation method is studied for an incompressible two-phase flow problem in heterogeneous porous media, presenting a contrast in permeability coefficients of O(107). We compare the number of iterations required to solve the above-mentioned problem using the Conjugate Gradient method preconditioned with Incomplete Cholesky (ICCG), against the deflated version of the same method (DICCG). The efficiency of the method is illustrated with the SPE 10 benchmark, our largest test case, containing O(106) cells. For this problem, the DICCG method requires only 20% of the number of ICCG iterations. @en

We study fast and robust iterative solvers for large systems of linear equations resulting from simulation of flow trough strongly heterogeneous porous media. We propose the use of preconditioning and deflation techniques, based on information obtained frfrom the system, to reduce the time spent in the solution of the linear system.An important question when using deflation techniques is how to find good deflation vectors, which lead to a decrease in the number of iterations and a small increase in the required computing time per iteration. In this paper, we propose the use of deflation vectors based on a POD-reduced set of snapshots. We investigate convergence and the properties of the resulting methods. Finally, we illustrate these theoretical results with numerical experiments. We consider compressible and incompressible single-phase flow in a layered model with variations in the permeability layers up to 10 3 and the SPE 10 benchmark model with a contrast in permeability coefficients of 10 7. Using deflation for the incompressible problem, we reduce the number of iterations to 1 or 2 iterations. With deflation, for the compressible problem, we reduce up to ∼ 80% the number of iterations when compared with the only-preconditioned solver.@en

@en

We consider deflation-based pre-conditioning of the pressure equation for large-scale reservoir models with strong spatial variations in the permeabilities. The use of deflation techniques involves the search for good deflation vectors, which usually are problem-dependent. We propose the use of proper orthogonal decomposition (POD) to generate physics-based problem-specific deflation vectors. The use of POD to construct pre-conditioners has been attempted before but in those applications, a snap-shot-based reducedorder basis was used as pre-conditioner directly whereas we propose the use of basis vectors as deflation vectors. We investigate the effectiveness of the method with numerical experiments using the conjugate gradient iterative method in combination with Incomplete Cholesky preconditioning (ICCG) and PODbased deflation (DICCG). We consider incompressible and compressible single-phase flow in a layered model with large variations in the permeability coefficients, and the SPE10 benchmark model. We obtain an important reduction for the number of iterations with our proposed DICCG method in comparison with the ICCG method. In some test problems, we achieve convergence within one DICCG iteration. However, our method requires a number of preparatory reservoir simulations proportional to the number of wells and the solution of an eigenvalue problem to compute the deflation vectors. This overhead will be justified in case of a large number of subsequent simulations with different control settings as typically required in numerical optimization or sensitivity studies. @en