POD-based deflation techniques for the solution of two-phase flow problems in large and highly heterogeneous porous media

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.