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Modeling variably saturated flow in the vadose zone is of vital importance to many scientific fields and engineering applications. Richardson-Richards equation (RRE, which is conventionally known as Richards' equation) is often chosen to physically represent the fluxes in the vadose zone when the accurate characterization of the soil water dynamics is required. Being a highly nonlinear partial differential equation, RRE is often solved numerically. Although there are mature software and codes available for simulating variably saturated flow by solving RRE, the numerical solution of RRE is nevertheless computationally expensive.Moreover, sometimes the robustness and the efficiency of RRE-based models can deteriorate rapidly when certain unfavorable conditions are met. These demerits of RRE hinder its application on large-scale vadose zone hydrology problems and uncertainty quantification, both of which requires many runs of the prediction model. To address these challenges, the accuracy, convergence, and efficiency of the numerical schemes of RRE should be further improved by testing a wide variety of cases covering different initial conditions, boundary conditions, and soil types. We reviewed and highlighted several critical issues related to the numerical modeling of RRE, including spatial and temporal discretization, the different forms of RREs, iterative and noniterative schemes, benchmark solutions, and available software and codes. Based on the review, we summarize the challenges and future work for solving RRE numerically.

The consistencies of the ensemble Kalman ilter (EnKF) and iterative EnKF algorithms for two-dimensional nonlinear unsaturated low in randomly heterogeneous soil are discussed. A modiied Restart EnKF with low computational effort and sound consistency is proposed. The effects of various factors, e.g., observation type and damping factor, are investigated.

Changes in soil hydraulic proper es with depth are common in fi eld soils and need to be described in numerical simula ons. Water content is undesirable to calculate Darcian fl ux since it is not con nuous across boundaries between diff erent soil layers. A general form of the one-dimensional water content-based Richards equa on, which was fi rst derived by Hills et al., is adopted here to simulate one-dimensional unsaturated fl ow in grada onal soils by adding a correc on term. For cell-centered grid, several algorithms are presented to fi nd the correc on term for the two nodes in the vicinity of the layer interface. For vertex-centered grid, the dyadic values of water content at the interface are represented by one sole water content value through the introduc on of composite soil water curve. The proposed algorithms are implemented in an itera ve model, and then the model is tested with several synthe c cases in grada onal soil and layered soil. The water content-based Richards equa on leads to superior numerical performance in terms of mass conserva on, accuracy, and effi ciency over the mixed form Richards equa on, especially for the fl ow in rela vely dry soil and with coarse grid. In layered soil, it is found that the algorithm based on vertex-centered grid is the most robust among all the methods discussed here.The movement of soil water in vadose zone can be represented by Richards' equation (RE) (Richards, 1931). Available numerical models to solve RE are distinguished by their RE forms, grid systems, and numerical methods, etc. Th ese diff erences can significantly infl uence computational effi ciency, accuracy and numerical behavior (Crevoisier et al., 2009). According to the types of unknown variables, the original RE can be written in three diff erent forms: head (h)-based form, water content (θ)-based form, and mixed form. Th e h-based RE may lead to serious mass balance problem unless very small time steps are used. Mixed form RE with primary variable of matric head (Celia et al., 1990) allows the reduction of mass balance errors. Th is form has been employed in the popular soft ware HYDRUS (Simunek et al., 2005). However, it is well-known that the mixed form RE performs relatively poorly if the head is used as primary variable, especially for problems involving infi ltration into initially very dry material (Forsyth et al., 1995). Furthermore, without adaptive spatial and temporal discretization, the numerical solution of sharp wetting front can be very expensive (Tocci et al., 1997;Miller et al., 2006). To solve these problems, many authors considered the primary variable switching technique (Forsyth et al., 1995;Diersch and Perrochet, 1999; Krabbenhoft , 2007). With this technique, the water content or the head is used as primary variable when solving the governing equations, depending on the degree of saturation at each node.Instead of using h-based or mixed form RE, the one-dimensional θ-based RE has been widely adopted as soil hydrology modules in land surface parameterization models, for e...