Abstract. 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.

Abstract. 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.

The phenomenon of freezing point depression in frozen soils results in the co-existence of ice and liquid water in soil pores at temperatures below 273.15 K (0°C), and is thought to have two causes: (a) capillary and adsorption effects, where the phase transition relationship is modified due to soil-air-water-ice interactions, and (b) solute effects, where the presence of salts lowers the freezing temperature. The soil freezing characteristic curve (SFC) characterizes the relationship between liquid water content and temperature in frozen soils. Most hydrological models represent the SFC using only capillary and adsorption effects with a relationship known as the Generalized Clapeyron Equation (GCE). In this study, we develop and test a salt exclusion model for characterizing the SFC, comparing this with the GCE-based model and a combined salt-GCE effect model. We test these models against measured SFCs in laboratory and field experiments with diverse soil textures and salinities. We consistently found that the GCE-based models under-predicted freezing-point depression. We were able to match the observations with the salt exclusion model and the combined model, suggesting that salinity is a dominant control on the SFC in real soils that always contain solutes. In modeling applications where the salinity is unknown, the soil bulk solute concentration can be treated as a single fitting parameter. Improved characterization of the SFC may result in improvements in coupled mass-heat transport models for simulating hydrological processes in cold regions, particularly the hydraulic properties of frozen soils and the hydraulic head in frozen soils that drives cryosuction.

The phenomenon of freezing point depression in frozen soils results in the co-existence of ice and liquid water in soil pores at temperatures below 273.15 K (0°C), and is thought to have two causes: (a) capillary and adsorption effects, where the phase transition relationship is modified due to soil-air-water-ice interactions, and (b) solute effects, where the presence of salts lowers the freezing temperature. The soil freezing characteristic curve (SFC) characterizes the relationship between liquid water content and temperature in frozen soils. Most hydrological models represent the SFC using only capillary and adsorption effects with a relationship known as the Generalized Clapeyron Equation (GCE). In this study, we develop and test a salt exclusion model for characterizing the SFC, comparing this with the GCE-based model and a combined salt-GCE effect model. We test these models against measured SFCs in laboratory and field experiments with diverse soil textures and salinities. We consistently found that the GCE-based models under-predicted freezing-point depression. We were able to match the observations with the salt exclusion model and the combined model, suggesting that salinity is a dominant control on the SFC in real soils that always contain solutes. In modeling applications where the salinity is unknown, the soil bulk solute concentration can be treated as a single fitting parameter. Improved characterization of the SFC may result in improvements in coupled mass-heat transport models for simulating hydrological processes in cold regions, particularly the hydraulic properties of frozen soils and the hydraulic head in frozen soils that drives cryosuction.