Scientific Machine Learning
The integration of numerical PDEs with neural networks is reducing the computational cost for modeling physical phenomena. We proposed unsupervised learning methods to solve elliptic problems. The algorithm does not require any training data.
COMP-M has developed Numerics Informed Neural Networks (NINNs) to solve PDEs. NINNs are mesh-aware and produce physically constraint numerical solutions using deep learning and carefully designed loss functions. One approach is to learn the approximation of the finite difference solution to the PDEs.
Multiphase Flows in Porous Media
Multiphase flows in porous media occur in the production of hydrocarbons from reservoirs, or in the clean-up of contaminated groundwater sites. The mathematical models are coupled nonlinear systems of partial differential equations that are challenging to solve. Some numerical challenges are: heterogeneity of the porous media, degeneracy of the relative mobilities and mass transfer between phases. For instance, absolute permeability can vary over several orders of magnitude in a geological formation.
COMP-M has developed high order and locally mass conservative methods for incompressible two-phase, three-phase and black-oil models. Our publications show the advantages of using discontinuous polynomial approximations of high degree. The spatial discretization is based on interior penalty discontinuous Galerkin methods, or on hybridizable discontinuous Galerkin methods. For time stepping, we have compared sequential and fully implicit formulations. For two-phase flow, we have analyzed and implemented several schemes using any combination of wetting and non-wetting phase pressure and phase saturation. Our method for solving black-oil uses for primary unknowns, the liquid phase pressure, the aqueous phase saturation and the total mass fraction of gas component.
Convergence of FEM solution for degenerate two-phase flows
The numerical analysis of two-phase flows in porous media is challenging because of (i) the degeneracy of the relative permeabilities in the elliptic parts of the equations and (ii) the unboundedness of the closure models for the capillary pressure. Practitioners prefer to use numerical schemes that solve for physical quantities, such as phase pressure and phase saturation. Therefore the use of the artificial global pressure to remove the degeneracy in the schemes is not permitted. We formulate and prove convergence of a finite element method with mass lumping for solving the incompressible
two-phase flows. The proof is based on a compactness argument. To our knowledge, this is the first theoretical convergence result of a variational-based method for degenerate two-phase flows.
This project is funded by NSF: grant NSF-DMS 1913291.
V. Girault, B. Riviere and L. Cappanera. A Finite Element Method for Degenerate Two-Phase Flow in Porous Media. Part I: Well-Posedness, Journal of Numerical Mathematics, 29 (2), p.81–101, doi: 10.1515/jnma-2020-0004, 2021.
V. Girault, B. Riviere and L. Cappanera. A Finite Element Method for Degenerate Two-Phase Flow in Porous Media. Part II: Convergence, Journal of Numerical Mathematics, 29 (3) p.187–219, doi: 10.1515/jnma-2020-0005, 2021.
Elimination of overshoot/undershoot in saturation for two-phase flows
Overshoot and undershoot phenomena in discontinuous Galerkin methods are small oscillations localized in the neighborhood of a front in a convection-dominated problem. There are several ways to reduce the amount of overshoot/undershoot: implicit time-stepping, local grid refinement, magnitude of penalty parameter and limiters. These local oscillations remain small and bounded throughout the simulation in the best case scenarios. They are however never eliminated.
Our work on flux limiters addresses this challenge for DG methods applied to two-phase flows in heterogeneous media. In the case of incompressible two-phase flows, we prove that the numerical saturation is guaranteed to satisfy the maximum principle. The combination of flux limiters and slope limiters eliminate the overshoot and undershoot phenomena for realistic simulations.
This project is funded by NSF: grant NSF-DMS 1913291.
M.S. Joshaghani, B. Riviere and M. Sekachev. Maximum-principle-satisfying Discontinuous Galerkin Methods for Incompressible Two-Phase Immiscible Flow, Computer Methods in Applied Mechanics and Engineering, 391, p.114550, doi:10.1016/j.cma.2021.114550, 2022.
Multinumerics DG-FV method for two-phase flows
The DG-FV method is a multi numeric approach that combines the advantages of both discontinuous Galerkin method and cell-centered finite volume method in a non-overlapping domain partition. We propose special discretization of the coupling terms at the interface of the DG and FV subdomains, for the two-phase flow problem in heterogeneous porous media.
This project is funded by NSF: grant NSF-DMS 1913291.
B. Doyle, B. Riviere and M. Sekachev. A Multinumerics Scheme for Incompressible Two-Phase Flow, Computer Methods in Applied Mechanics and Engineering, 370, 113213, doi:10.1016/j.cma.2020.113213, 2020.
