On the new Approaches for Modeling Water Flow in Heterogeneous Media with Numerical Manifold Method

Material interfaces and typical boundary conditions are boundaries influencing the existence, uniqueness and stability of numerical solutions of water flow in heterogeneous media. The Numerical Manifold Method (NMM) with a two-cover-meshing system is an ideal method to handle boundaries, considering its flexibility with no need to adjust mathematical nodes onto boundaries, the meshing efficiency, and its integration precision. In this study, we derive and compare two new approaches for boundary constraints, i.e., a discontinuous approach with Lagrange multiplier and a continuous approach with jump function in the frame of NMM. In the discontinuous Lagrange multiplier method (LMM), Dirichlet boundaries or the material interfaces are regarded as discontinuities which are linked by Lagrange multipliers. In the continuous Jump function method (JFM), the material interfaces are regarded as inner interfaces, where the refraction law is realized by constructing jump terms with discontinuity of hydraulic gradient in its normal direction. Both the continuous and discontinuous approaches make full use of the aforementioned advantages of NMM and are established based on a clear physical meaning of water flow by our energy-work seepage model. The newly derived boundary-constraint approaches were coded into an NMM water flow model and tested for precision and efficiency on different simulation examples. We first tested both LMM and JFM for an idealized heterogeneous model, comparing the numerical results with analytical solutions. Then we tested both approaches with a heterogeneous model and compared the results of hydraulic head and specific discharge. We proved that both approaches are very suitable for material boundaries considering the accurate realization of the boundary constraints and the flexibility with no need for mesh-conforming.