Numerical methods for modelling infiltration processes in heterogeneous soils and their interactions with surface flows

拟议的研究计划的目的是开发新的数值模拟方法在非均质土壤中的渗透过程及其与地表流的相互作用。我们将提出稳健的计算技术，用于模拟非均质土壤中的不稳定入渗，开发高效的耦合地表-地下流数值模型，提出误差估计器，并开发新的时空技术，以提高所提出的数值模型的精度和效率。我们将使用新的连续模型，我们最近开发的准确地描述了在土壤渗透过程中形成的重力指。将为称为相场模型的新一代多孔介质多相流模型开发稳健和有效的二维和三维数值近似方法。相场模型是基于这样一个事实，即系统的非均匀性和宏观界面应考虑计算系统的能量。在系统中分离不同相的尖锐界面应通过引入在薄厚度区域上平滑变化的扩散序参数或相场参数而被扩散界面取代。在这些地区的时空演变可以描述一组耦合偏微分方程，并使用适当的离散技术数值求解。从变分原理出发，通过最小化系统的自由能表达式，导出了相场参数随时间的演化关系。高阶精确的离散化将被开发和一个合适的时间积分将被用来处理的问题的不稳定性与刚度的数值解的来源。地表-地下流耦合系统的控制方程的通量项可分为双曲线部分和椭圆部分。非振荡高阶连续/不连续重建的解决方案将开发，以设计稳定和准确的数值方案。一个合适的时间格式需要处理的问题，与刚度的来源，在系统的数值解的不稳定性。该系统的半离散形式可分为线性刚性项和剩余非线性非刚性项，刚性项采用隐式时间积分方法，非刚性项采用显式时间积分方法。在我们的方法中，我们将使用误差估计器，网格自适应，并提出新的技术，以准确地捕捉湿润锋的传播。所提出的数值模型的效率，收敛性和精度将通过几个数值试验进行检查。所提出的计划的空间收敛速度将被确定数值和这些方法的时间顺序的半隐式方法的影响将被研究。

The aim of the proposed research program is to develop new numerical methods for modelling infiltration processes in heterogeneous soils and their interactions with surface flows. We will propose robust computational techniques for the simulation of unstable infiltration in heterogeneous soils, develop efficient numerical models for coupled surface-subsurface flows, propose error estimators and develop new space-time techniques to improve the accuracy and efficiency of the proposed numerical models. We will use the new continuum model that we have recently developed which accurately describes the formation of gravity fingers during infiltration in soils. Robust and efficient two-dimensional and three-dimensional numerical approximation methods will be developed for a new generation of multiphase-flow models in porous media known as phase-field models. Phase-field models are based on the fact that the non-homogeneity of the system and the macroscopic interfaces should be considered for computing the energy of the system. The sharp interfaces separating different phases in the system should be replaced by diffusive interfaces by the introduction of a diffusive order parameter or phase-field parameter that varies smoothly over thin thickness regions. The spatiotemporal evolution in these regions can be described by a set of coupled partial differential equations and solved numerically using suitable discretization techniques. The time-evolution of the phase-field parameter is derived from the variational principle by minimizing the free energy expression of the system. High-order accurate discretizations will be developed and a suitable temporal integration will be used to deal with the issues of the instability associated with the sources of stiffness in the numerical solutions. The flux terms of the governing equations of the coupled system of surface-subsurface flows can be split into a hyperbolic part and an elliptic part. Non-oscillatory high-order continuous/discontinuous reconstructions of the solutions will be developed in order to design stable and accurate numerical schemes. A suitable temporal scheme is required to deal with the issues of the instability associated with the sources of stiffness in the numerical solutions of the system. The semi-discrete form of the system can be split into linear stiff terms and residual nonlinear non-stiff terms, and implicit time integration methods will be used for stiff terms and the explicit temporal schemes can be used to integrate non-stiff terms. In our approach, we will use errors estimators, mesh adaptation, and propose new technique to accurately capture the propagation of the wetting fronts. The efficiency, convergence and accuracy of the proposed numerical models will be examined by several numerical tests. The spatial convergence rate of the proposed schemes will be determined numerically and the impact of the semi-implicit approach on the temporal order of these methods will be studied.