# Modelling Energy Dissipation Due to Soil-Moisture Hysteresis

• Martin Brokate
• Stephen MacCarthy
• Alexander Pimenov
• Alexei Pokrovskii
• Dmitrii Rachinskii
Article

## Abstract

Cyclic wetting and drying of soils due to changing weather conditions is characterised by a well-documented hysteresis in the relationship between the pressure or matric potential and the water content in the soil. Hysteresis manifests itself through a difference in drying and wetting curves. Its presence suggests that there is dissipation of energy associated with intermittent wetting and drying of the soil, which can be released in the form of heat. In this paper, we discuss a model for evaluation of the energy dissipation rate due to soil-moisture hysteresis. The model has three main ingredients: (a) a hysteresis constitutive relationship between soil-water potential and moisture content in the soil, which is modelled by the Preisach operator; (b) a lumped form of Darcy’s law assuming that the water flux is proportional to the difference of potentials in the soil and on its surface; (c) a rainfall term governing the evolution of the outer potential, which is modelled by a periodic forcing or a Poisson process. We propose a combination of analytic and numerical methods to evaluate the energy dissipation rate and how it is affected by the variation of rain parameters, such as the frequency of arrivals of rain cells and their shape, intensity and duration.

## Keywords

Hysteresis Hydrology Energy dissipation Mathematical modelling Differential-operator equation Preisach operator

## Notes

### Acknowledgements

This publication has emanated from a research conducted with the financial support of Science Foundation Ireland. M. Brokate and D. Rachinskii acknowledge the support of Alexander von Humboldt Foundation and Royal Irish Academy. A. Pokrovskii and D. Rachinskii were partially supported by the Russian Foundation for Basic Research, grants 06-01-72552, 06-01-00256 and 10-01-93112, and Federal Programme ‘Scientists of Innovative Russia’, grant 2009-1.5-507-007.

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## Authors and Affiliations

• Martin Brokate
• 1
• Stephen MacCarthy
• 2
• Alexander Pimenov
• 2
Email author
• Alexei Pokrovskii
• 2
• Dmitrii Rachinskii
• 2
1. 1.Zentrum MathematikTechnische Universität MünchenGarchingGermany
2. 2.Department of Applied MathematicsUniversity College CorkCorkIreland