A semi-analytical solution of the Richards Equation posed on a two-layered one-dimensional soil supplied with various boundary conditions is derived under a constraint that the constitutive relations are exponentially dependent on the pressure head. It allows for a transformation of the Richards Equation into a linear parabolic partial differential equation that governs a spatial–temporal function that represents the hydraulic conductivity. The procedure is proceeded with expressing this function as a linear combination of a set of eigenfunctions associated with a novel Sturm—Liouville problem that reflects the layer system and an auxiliary function that depends only on the spatial variable and the pressure head at the interface at the time of interest. All the relevant coefficients in the representation satisfy a nonlinear differential–algebraic system gathered from imposing the continuity of the pressure head and its flux at the interface. Two different approximations of the derivatives yield algebraic systems to be solved by the Newton method of iteration. Several pertinent numerical experiments demonstrating the approach are discussed and compared with the standard finite volume method.

在本构关系与压力水头呈指数关系的约束条件下，导出了在具有各种边界条件的两层一维土壤上提出的理查兹方程的半解析解。它允许将理查兹方程转换为线性抛物线偏微分方程，该方程控制代表水力传导率的时空函数。该过程继续将该函数表示为与反映层系统的新 Sturm-Liouville 问题相关的一组特征函数的线性组合和仅取决于空间变量和界面处的压力头的辅助函数感兴趣的时间。表示中的所有相关系数都满足一个非线性微分代数系统，该系统是通过在界面处施加压力头及其通量的连续性而收集的。导数的两种不同近似产生代数系统，要通过牛顿迭代法求解。讨论了证明该方法的几个相关数值实验，并与标准有限体积法进行了比较。