Mixed hybrid finite element method on CCAR grids in highly heterogeneous porous media

Abstract

Upscaling is critical in computational simulations of flow and transport in porous media, bridging fine-scale geological details with coarse-scale computational models, particularly in groundwater modeling and subsurface hydrology. Cartesian cell-based anisotropic refinement (CCAR) grids facilitate this process by adaptively refining grid cells in regions of significant heterogeneity or complex flow dynamics. This study proposes a novel mixed hybrid finite element (MHFE) method for solving flow equations on CCAR grids. The method employs hypothetical triangulation, subdividing each CCAR grid element based on the number of surrounding faces. This enables the elimination of internal face unknowns through flux conservation and pressure continuity rules, enhancing computational efficiency without increasing the degrees of freedom. The proposed method was validated using four models, including synthetic and highly heterogeneous domains with various boundary conditions. The accuracy of the proposed method is evaluated against a fine-grid reference solution and a standard finite volume (FV) method applied to uniformly coarsened grids. Across all test cases, the MHFE method demonstrates significantly improved velocity accuracy. Grid convergence analysis revealed consistent monotonic convergence with rates of α  ≈  0.38 for pressure error and α  ≈  0.32 for velocity error. Computational efficiency analysis demonstrated speedup factors of 30–40 x compared to fine-grid simulations while maintaining superior accuracy relative to conventional coarse-grid approaches. Results also demonstrated the ability of the proposed method to accurately compute velocity and pressure fields, as well as streamlines, without distortions or discontinuities, even in complex configurations. By assigning permeability at the hypothetical triangulation level, the method preserved fine-scale heterogeneity and produced a more accurate equivalent permeability field compared to conventional approaches. Additionally, the method imposes no restrictions on the number of neighbors per element, addressing challenges inherent to CCAR grids. These features establish the proposed MHFE method as a robust and efficient tool for advanced upscaling applications in porous media, offering accurate and reliable solutions for complex groundwater systems and geological formations.