Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Abstract

The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils

“…The assumption, that the gravity and viscosity of the air phase are negligible and thus eliminating the influence of the air dynamic system (Pinder & Celia, ), has led to a highly simplified but physically based equation, which is conventionally known as the Richards' equation (Richards, ). However, it has been recently discovered (e.g., Farthing & Ogden, ; Raats & Knight, ) that Lewis Frey Richardson () was the first one who attempted to solve the land surface water flux using “Richards' equation” in his book “Weather Prediction by Numerical Process” published in 1922. Thus, it will be more proper to refer this equation as RRE, which is adopted in this manuscript.…”

Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties

“…This solver used the method of lines (MOL), which allowed variable-order time integration and variable time step size while controlling the temporal error [37]. Note that, the MOL method works well with the nonlinear Richards Equation (RE) [38] and provides a mass conservative solution for its h form [39]…”

Iterative Linearisation Schemes for Doubly Degenerate Parabolic Equations

“…If the new variable is properly chosen, the Richards equation receives a that is more suited for the Newton method, in the sense that all nonlinearities are Lipschitz continuous. We refer to [10] for a review detailing on such aspects.…”