Abstract
One major challenge in modeling Darcy flow in heterogeneous porous media is simulating the material interfaces accurately. To overcome this defect, the refraction law is fully introduced into the numerical manifold method (NMM) as an a posteriori condition. To achieve a better accuracy of the Darcy velocity and continuous nodal velocity, a high-order weight function with a continuous nodal gradient is adopted. NMM is an advanced method with two independent cover systems, which can easily solve both continuous and discontinuous problems in a unified form. Moreover, a regular mathematical mesh, independent of the physical domain, is used in the NMM model. Compared to the conforming mesh of other numerical methods, it is more efficient and flexible. A number of numerical examples were simulated by the new NMM model, comparing the results with the original NMM model and the analytical solutions. Thereby, it is proven that the proposed method is accurate, efficient, and robust for modeling Darcy flow in heterogeneous porous media, while the refraction law is satisfied rigorously.
Highlights
Heterogeneity is the natural property of the geologic structure in groundwater systems
We developed a high-order numerical manifold method (NMM) model for Darcy flow in heterogeneous porous media with a nodal-continuous weight function, as well as a local cover function derived by first order approximation of Taylor expansion
Due to the lack precision of measuring the velocity solution by ordinary methods, a new weight function with a continuous nodal gradient was adopted into the original NMM model
Summary
Heterogeneity is the natural property of the geologic structure in groundwater systems. The two major limitations of modeling Darcy flow in heterogeneous porous media by conventional numerical methods are as follows. The first way is using some post-processing techniques to improve the velocity solution after the hydraulic head is solved [8,9,10] These methods are efficient, as they can be modified from the original FEM approach. Xie et al [15] developed a domain decomposed finite element method for solving the Darcy velocity in heterogeneous porous media without iterations. Aiming to simulate Darcy flow in heterogeneous media, Hu et al [33,34,35] developed a continuous approach with jump functions and a discontinuous approach with Lagrange multipliers, by enforcing the continuity of normal flux across material interfaces, while the refraction law for tangential flux is ignored.
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