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Neural Solution to Elliptic PDE with Discontinuous Coefficients for Flow in Porous Media
- Publisher: European Association of Geoscientists & Engineers
- Source: Conference Proceedings, ECMOR 2022, Sep 2022, Volume 2022, p.1 - 17
Abstract
Locally conservative finite-volume schemes have been developed for solving the general tensor pressure equation of petroleum reservoir-simulation on structured and unstructured grids. The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O (1) errors introduced by standard reservoir simulation schemes when applied to full tensor flow approximation.
Two-point flux schemes (TPFA) are not applicable to full-tensor permeability and multi-Point flux approximation schemes (MPFA) has a major drawback that when it is applied to strongly anisotropic heterogeneous media as it fails to satisfy a maximum principle and result in loss of solution monotonicity for high anisotropy ratios causing spurious oscillations in the numerical pressure solution. Although variations of TPFA and MPFA have been proposed, challenges related to application on highly heterogeneous and anisotropic media still exists.
In this paper a Neural solution method to general tensor elliptic PDE with discontinuous coefficients is presented. The Neural solution to elliptic PDE is based on utilization of a deep learning multi-layer neural network, which could serve as a more effective alternative to TPFA or MPFA type schemes for fast and accurate results. Series of 2D test cases are presented, where the results of Neural solutions are compared with numerical solution using TPFA and MPFA schemes with range of heterogeneities demonstrating general applicability and accuracy of the Neural solution method. Order of accuracy of the method is compared with the numerical solution using a measure of error like the L2 norm, which shows convergence with refinement study. Neural solution for specific cases is also tested on a range of general grid types.