Modelling Energy Dissipation Due to Soil-Moisture Hysteresis

  • Martin Brokate
  • Stephen MacCarthy
  • Alexander PimenovEmail author
  • Alexei Pokrovskii
  • Dmitrii Rachinskii


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.


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



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.


  1. 1.
    Hillel, D. (1982). Introduction to Soil Physics. New York: Academic PressGoogle Scholar
  2. 2.
    Sander, G. (2008). Proceedings of Workshop on Multi-rate Processes and Hysteresis. Journal of Physics: Conference Series.Google Scholar
  3. 3.
    O’Kane, J. P. (2005). The FEST model-a test bed for hysteresis in hydrology and soil physics. Journal of Physics: Conference Series, 22, 148–163. Scholar
  4. 4.
    Krejci, P., O’Kane, J.P., Pokrovskii, A., & Rachinskii, D. (2010). Mathematical models of hydrological systems with Preisach hysteresis. 58.pdf. BCRI Preprint series, 57, 46, Dept. of Applied Mathematics, UCC, Ireland.
  5. 5.
    Appelbe, B., Flynn, D., McNamara, H., O’Kane, J.P., Pimenov, A., Pokrovskii, A., et al. (2009). Rate-independent hysteresis in terrestrial hydrology. Control Systems Magazine, 29(1), 44–69.CrossRefGoogle Scholar
  6. 6.
    Flynn, D., Zhezherun, A., Pokrovskii, A., & O’Kane, J. P. (2008). Modeling discontinuous flow through porous media using ODEs with Preisach operator. Physica B, 403, 440–442.CrossRefGoogle Scholar
  7. 7.
    Chen, H. H., Chang, L. C., Shan, H. Y., & Tsai, J. P. (2008). Joint impact of hysteresis and scaling rule on NAPL flow simulation. Environmental Modeling and Assessment, 14(6), 715–728.Google Scholar
  8. 8.
    Chang, L. C., Chen, H. H., Shan, H. Y., & Tsai, J. P. (2008). Effect of connectivity and wettability on the relative permeability of NAPLs. Environmental Geology, 56(7), 1437–1447.CrossRefGoogle Scholar
  9. 9.
    Parlange, J. (1980). Water transport in soils. Annual Review of Fluid Mechanics, 12, 77–102.CrossRefGoogle Scholar
  10. 10.
    Haverkamp, R., Reggiani, P., Ross, P. J., & Parlange, J. Y. (2002). Soil water hysteresis prediction model based on theory and geometric scaling. In P. A. C. Raats, D. Smiles, & A. W. Warrick (Eds.), Environmental mechanics: Water, mass and energy transfer in the biosphere (pp. 213-246). American Geophysical Union.Google Scholar
  11. 11.
    Chang, L. C., Tsai, J. P., Shan, H. Y., & Chen, H. H. (2009). Experimental study on imbibitions displacement mechanisms of two-phase fluid using micro model. Environmental Earth Science, 59, 4, 901–911.CrossRefGoogle Scholar
  12. 12.
    Migliori, L. (2004). The Hydraulics and Hydrology of a Pumped Polder-a Case Study in Hydroinformatics. Ph.D. dissertation, University College Cork.Google Scholar
  13. 13.
    Haines, W. B. (1930). Studies in the Physical Properties of Soil: V. The hysteresis effect in Capillary Properties, and the Modes of Moisture Distribution Associated Therewith. Journal of Agricultural Science, 20, 97–116.CrossRefGoogle Scholar
  14. 14.
    Campbell, G. (1985). Soil physics with BASIC-Transport models for soil-plant systems. Developments in Soil Science. New York: Elsevier.Google Scholar
  15. 15.
    de Gennes, P.-G., Brochard-Wyart, F., & Quere, D. (2003). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. New York: Springer.Google Scholar
  16. 16.
    Angulo-Jaramillo, R., Elrick, D., Parlange, J., Gerard-Marchant, P., & Haverkamp, R. (2003). Analysis of short-time single-ring infiltration under falling-head conditions with gravitational effects. AGU Hydrology Days Proceedings, 23, 16–23.Google Scholar
  17. 17.
    Hogarth, W., Hopmans, J., Parlange, J.-Y., & Haverkamp, R. (1988). Application of a simple soil-water hysteresis model. Journal of Hydrology, 9, 21–29.CrossRefGoogle Scholar
  18. 18.
    Parlange, J., Steenhuis, T., Haverkamp, R., Barry, D., Culligan, P., Hogarth, W., et al. (1999). Soil Properties and Water Movement, Vadose Zone Hydrology - Cutting Across Disciplines (pp. 99–129). New York: Oxford University Press.Google Scholar
  19. 19.
    Haverkamp, R., Bouraoui, F., Zammit, C., & Angulo-Jaramillo, R. (2007). Soil properties and moisture movement in the unsaturated zone. In J. Delleur (Ed.), Handbook of groundwater engineering (pp. 5–13).Google Scholar
  20. 20.
    Bertotti, G., & Mayergoyz, I. D. (eds.) (2006). The Science of Hysteresis, 3-volume set. New York: Academic Press.Google Scholar
  21. 21.
    Brokate, M., & Sprekels, J. (1996). Hysteresis and phase transitions, ser. Applied Mathematical Sciences. New York: Springer.Google Scholar
  22. 22.
    Visintin, A. (1994). Differential Models of Hysteresis. Berlin: Springer.Google Scholar
  23. 23.
    Mayergoyz, I. (1991). Mathematical Models of Hysteresis. Berlin: Springer.CrossRefGoogle Scholar
  24. 24.
    Krasnosel’skii, M., & Pokrovskii, A. (1989). Systems with Hysteresis. New York: Springer (translated from the Russian by Marek Niezg’odka).Google Scholar
  25. 25.
    Flynn, D., McNamara, H., O’Kane, J. P., & Pokrovskii, A. (2005). Application of the Preisach model to soil-moisture hysteresis. In G. Bertotti and I. Mayergoyz (Eds.), Science of Hysteresis (vol. 3, pp. 689–744). no. ISBN: 0-12-480874-3. New York: Academic Press.Google Scholar
  26. 26.
    Flynn, D. (2005). Soil moisture hysteresis. Available at: appliedmath/soilhyst/index.html
  27. 27.
    O’Kane, J. P. (2006). The hysteretic linear reservoir-a new Preisach model. Physica B, 372, 1–2.Google Scholar
  28. 28.
    Krejci, P., O’Kane, J. P., Pokrovskii, A., & Rachinskii, D. (2008). Stability results for a soil model with singular hysteretic hydrology. WIAS preprint 1341, 1–26. Journal of Physics: Conference Series 268, 012016.
  29. 29.
    Pimenov et al. (2009). Numerical solution of equations with hysteresis. Available at:
  30. 30.
    Crossm, R., McNamara, H., & Pokrovskii, A. (2008). Modelling macroeconomic flows related to large ensembles of elementary exchange operations. Physica B, 403, 451–455.CrossRefGoogle Scholar
  31. 31.
    Rodriguez-Iturbe, I., Cox, D., & Isham, V. (1987). Some models for rainfall based on stochastic point processes. Proceedings of the Royal Society of London. Series A, 410, 269–288.CrossRefGoogle Scholar
  32. 32.
    Barndorff-Nielsen, O. E., et al. (1998). Stochastic methods in hydrology: Rain, landforms, and floods (pp. 8–10). Singapore: World Scientific Publishing Co. Pte. Ltd.Google Scholar
  33. 33.
    Cowpertwait, P., Isham, V., & Onof, C. (2007). Point process models of rainfall: Developments for fine-scale structure. Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences, 463(2086), 2569–2587.CrossRefGoogle Scholar
  34. 34.
    Mondonedo, C. A., Tachikawa, Y., & Takara, K. (2006). Quantiles of the Neyman-Scott rectangular pulse rainfall model for hydrologic design. Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 49 B.Google Scholar
  35. 35.
    Amann, A., Brokate, M., Rachinskii, D., & Temnov, G. (2011). Distribution of return point memory states for systems with stochastic inputs. Journal of Physics: Conference Series, 268.Google Scholar
  36. 36.
    Van Wijk, W. R., & De Vries, D. A. (1963). Periodic temperature variations in a homogeneous soil. In W. R. Van Wijk (Ed.), Physics of plant environment (pp. 103–143). Amsterdam: North-Holland.Google Scholar
  37. 37.
    Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids. London: Oxford University Press.Google Scholar
  38. 38.
    Gao, Z., Horton, R., Wang, L., Liu, H., & Wen, J. (2008). An improved force-restore method for soil temperature prediction. European Journal of Soil Science, 59, 972–981.CrossRefGoogle Scholar
  39. 39.
    Chacko P. T., & Renuka, G. (2002). Temperature mapping, thermal diffusivity and subsoil heat flux at Kariavattom of Kerala. Journal of Earth System Science, 111(1), 79–85.CrossRefGoogle Scholar
  40. 40.
    Basics of a soil layer model (2003). Available at:
  41. 41.
    Thermal Conductivity Measurement (2008). Available at:

Copyright information

© Springer Science+Business Media B.V. 2011

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

Personalised recommendations