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A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0)

Ireson, Andrew M.; Spiteri, Raymond J.; Clark, Martyn P.; Mathias, Simon A.

A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0) Thumbnail


Andrew M. Ireson

Raymond J. Spiteri

Martyn P. Clark


A simple numerical solution procedure – namely the method of lines combined with an off-the-shelf ordinary differential equation (ODE) solver – was shown in previous work to provide efficient, mass-conservative solutions to the pressure-head form of Richards' equation. We implement such a solution in our model openRE. We developed a novel method to quantify the boundary fluxes that reduce water balance errors without negative impacts on model runtimes – the solver flux output method (SFOM). We compare this solution with alternatives, including the classic modified Picard iteration method and the Hydrus 1D model. We reproduce a set of benchmark solutions with all models. We find that Celia's solution has the best water balance, but it can incur significant truncation errors in the simulated boundary fluxes, depending on the time steps used. Our solution has comparable runtimes to Hydrus and better water balance performance (though both models have excellent water balance closure for all the problems we considered). Our solution can be implemented in an interpreted language, such as MATLAB or Python, making use of off-the-shelf ODE solvers. We evaluated alternative SciPy ODE solvers that are available in Python and make practical recommendations about the best way to implement them for Richards' equation. There are two advantages of our approach: (i) the code is concise, making it ideal for teaching purposes; and (ii) the method can be easily extended to represent alternative properties (e.g., novel ways to parameterize the K(ψ) relationship) and processes (e.g., it is straightforward to couple heat or solute transport), making it ideal for testing alternative hypotheses.


Ireson, A. M., Spiteri, R. J., Clark, M. P., & Mathias, S. A. (2023). A simple, efficient, mass-conservative approach to solving Richards' equation (openRE, v1.0). Geoscientific Model Development, 16(2), 659-677.

Journal Article Type Article
Acceptance Date Jan 4, 2023
Online Publication Date Jan 27, 2023
Publication Date 2023
Deposit Date Jan 31, 2023
Publicly Available Date Jan 31, 2023
Journal Geoscientific Model Development
Print ISSN 1991-959X
Publisher European Geosciences Union
Peer Reviewed Peer Reviewed
Volume 16
Issue 2
Pages 659-677


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