Pressure Preconditioning Using Proper Orthogonal Decomposition

Abstract

We developed and implemented a new first-stage preconditioning method for large-scale reservoir simulation as an alternative to the popular Algebraic Multi Grid (AMG) method. We used Proper Orthogonal Decomposition (POD) to derive a reduced-order model for the linearized pressure equation in a proprietary reservoir simulator. A small set of pre-computed pressure solutions are used to transform the equation into a lower-order system that can be solved economically yet provides a relatively accurate estimation of the solution of the original high-order system. We present results for a two-phase (oil-water) reservoir model under water flooding conditions. Use of the POD-based preconditioner lead to a significantly faster convergence of the overall simulation compared to AMG, and reduced the linear solver times between approximately 70 % and 80%, i.e. we obtained accelerations with factors between three and five. The highest speed up was achieved for a case in which the flow rates in the eight injection wells were changed frequently during the entire production period, using ten POD basis functions obtained from short training simulations in which the injectors were operated only once each. These first results are very encouraging, especially because there is room to optimize the trade-off between the number of basis functions used in the POD method and the required number of linear iterations. However, further research is required to assess the applicability of the POD-based preconditioner to more complex cases including e.g. strongly compressible flow or compositional effects. The overhead required to pre-compute the POD solutions implies that the new method will be particularly attractive when many solutions of near-similar simulation models are required such as in computer-assisted history matching or flooding optimization.

Physics-based Pre-conditioners for Large-scale Subsurface Flow Simulation

“…Therefore is important to use only the correct number of deflation vectors. Techniques such as Proper Orthogonal Decomposition (POD) [9,10] could help us to chose the correct set of deflation vectors. 1 500* 500* 500* Figure 14.…”

On POD-based Deflation Vectors for DPCG applied to porous media problems

“…A potential ROM to reduce the computing time for large-scale problems is Proper Orthogonal Decomposition (POD). A method that has been investigated for flow problems in porous media, in [6][7][8][9][10][11][12][13][14][15], among others. The use of a POD-based preconditioner for acceleration of the solution is proposed by Astrid et al [11] to solve the pressure equation resulting from two-phase reservoir simulation, by Jiang et al [14] for a similar application and by Pasetto et al [15] for groundwater flow models.…”

“…The POD method requires the computation of a series of 'snapshots' which are solutions of the problem with slightly different parameters or well inputs. Astrid et al [11] use snapshots in the form of solutions of the pressure equation computed in a small number of short pre-simulations, prior to the actual simulation. The snapshots are obtained with diverse well configurations.…”

“…Algebraic Multigrid (AMG) [18], Multi-level and Domain Decomposition [19] preconditioners have been studied in combination deflation techniques to accelerate the convergence of iterative methods. In [8,11,15], after computing a basis from the previously obtained snapshots, the solution is computed in the subspace generated by this basis and then projected back to the original high-dimensional system. Carlberg et al [20] also use POD to obtain information from the system, in particular, the previous time step solutions.…”

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

“…Markovinović et al [20] employ the reduced-order model solution as initial guess for the iterative method obtaining an acceleration of the convergence. Astrid et al [21] show the benefits of replacing the algebraic multigrid method with a two-stage iterative method based on a Constrained Pressure Residual solver in combination with POD for the solution of a two-phase reservoir model. Jiang [22] introduces a reduced-order model preconditioner in the stationary Richardson iteration.…”