Theory And Methods For Finite Analytic Numerical Simulation In The Unsaturated Zone

The vadose zone plays an important role in groundwater recharge, discharge, particularly for the ecological environment of arid and semiarid zone. The governing equations of the mathematic models for unsaturated flow are nonlinear since their unsaturated hydraulic parameters of the equations depend on their solutions. Therefore, it is hard to solve the type of equation and obtain the analytical solution. It can be obtain analytical solutions to simulate water flow in vadose zone under the conditions of relatively easy boundary conditions. However, numerical methods are the most efficient way to simulating moisture movement in the unsaturated zone under the complicated conditions. Richard’s equation is a second order elliptic differential equation in space, however, it also has the character of parabolic equation in time. Thus, the traditional numerical methods, such finite different method and finite element method, could lead to the numerical oscillation, dispersion, and the divergence of the solution if improper time and space steps were selected. In the 80’s, Chen developed a new method —finite analytic method(FAM). The basic idea of FAM is incorporation of the analytical solution in a small local element to formulate the algebraic representation of the partial differential equation of the unsaturated flow. Therefore, finite analytic method can effectively control the numerical oscillation and dispersion. To overcome these problems, such as numerical oscillation, dispersion, FAM is applied to solve Richards’ equation.This paper derived four different finite analytic calculation formats for homogeneous soil and one format for layered soils. Through the qualitative and quantitative investigating, it can obtain the following results:1. The convergence and stability of finite analytic numerical scheme are proven by a rigorous mathematical analysis. It demonstrated that FAM can obtain stability and convergence solutions.2. It derived four types of finite analytic calculation formats of Richards’ equations, it can be concluded that all formats can obtain stability, convergence, high accuracy numerical solutions and effectively control mass balance error, except for the format based on pressure head form Richards’ equation. Comparison with four formats and analytical solutions, the finite analytic calculation format of mixed form Richards’ equation can obtain the highest accuracy numerical solutions and most efficiently control the global mass error. The formats of moisture content form and Kirchhoff transfer are the second. The type of pressure head form is worst. Therefore, the format of pressure head form is not recommended.3. The finite analytic calculation formats of mixed form RE, moisture content form RE and Kirchhoff transfer RE can obtain relatively high accuracy solutions under the condition of coarse grid sizes.4. FAM, based on Kirchhoff transform, was first introduced to solve one-dimensional heterogeneous soil. The proposed algorithm can deal with the discontinuous of the variable of Kirchhoff transform, and obtain high accuracy and stability solution under condition of coarse spatial steps. But the solutions of finite difference method deviate significantly from the analytical solutions near the layer interface and the observed local inaccuracy even extend far from this internal boundary. Beside that finite difference method is sensitive to the space steps.5.Application of FAM simulating a laboratory experiment, it can be seen that the results of FAM are in excellent with the values of observation. It illustrates that FAM reproduces results of laboratory experiments, and it reveals FAM can be used to solve the complex practice issues.