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Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems

Received: 20 February 2017     Accepted: 14 March 2017     Published: 28 October 2017
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Abstract

This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 5)
DOI 10.11648/j.ijtam.20170305.12
Page(s) 163-166
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Richards’ Equation, Partial Differential Equations, Irrigation, Hydrology, Soil Moisture

References
[1] Gardner, W. “Some Steady-State Solutions of the Unsaturated Moisture Low Equation with Application to Evaporation from a Water Table”, Soil Science, Vol. 85, 1958, pp. 228-232.
[2] Kuraz, M., Mayer, P. and Pechm P. “Solving the Non-linear Richards’ Equation Model with Adaptive Domain Decomposition”, Journal of Computational and Applied Mathematics, Vol. 270, 2014, pp 2-11.
[3] Bouchemella, S., Seridi, A. and Alimi-Ichola, I. “Numerical Solution of Water Flow in Unsaturated Soils: Comparative Study of Different Forms of Richards’ Equations” European Journal of Environmental and Civil Engineering, Vol. 9, 2014, pp 1.
[4] Bhosale P. A. and Dixit V. V. ” Water Saving-irrigation Automatic Agricultural Controller”, International Journal of Science and Technology, Volume 1, Number 2, 2012, pp. 34-36.
[5] Israelsen O. W. and Hansen V. E., Irrigation Principle and Practice, John Wiley and Son Inc., 3rd Edition, New Delhi, p 171, 2011.
[6] Hillel, D., Introduction to Environmental Soil Physics, Academic Press, Hardcover, Boston, 2003.
[7] Philip, J. R., “Hydrostatics and Hydrodynamics in Selling Soils”, Water Resources, Res. 5, 1969, pp 1070-1077.
[8] Celia, M. A., Bouloutas, E. T. and Zarba, R. L. “A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation”, Water Resources Research, Volume 26, Number 7, 1990, pp. 1483-1696.
[9] Majumdar D. K., Irrigation Water Management: Principles and Practice, Eastern Economy Edition, Prentice – Hall of India Private Limited. New Delhi, 2006.
[10] Tracy, F. “Clean Two- And Three-Dimensional Analytical Solutions of Richards' Equation for Testing Numerical Solvers”, Water Resources Research, Vol. 42, Number 8, 2006, pp. 3106-109.
[11] Tracy, F. “Three-Dimensional Analytical Solutions of Richards' Equation for a Box-Shaped Soil Sample with Piecewise-Constant Head Boundary Conditions on the Top”, Journal of Hydrology, Vol. 336, 2007, pp. 391-400.
[12] Warrick, A., Soil Water Dynamics, Oxford University Press, Paperback, New York, 2003.
[13] Genuchten, M. “A Closed-Form Equation for Producing the Hydraulic Conductivity of Unsaturated Soils”, Soil Science American Journal, Vol. 44, 1980, pp. 892-898.
Cite This Article
  • APA Style

    Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo. (2017). Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. International Journal of Theoretical and Applied Mathematics, 3(5), 163-166. https://doi.org/10.11648/j.ijtam.20170305.12

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    ACS Style

    Ise Ise Ekpoudom; Anamekere Ime Jacob; Umana Thompson Itaketo. Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. Int. J. Theor. Appl. Math. 2017, 3(5), 163-166. doi: 10.11648/j.ijtam.20170305.12

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    AMA Style

    Ise Ise Ekpoudom, Anamekere Ime Jacob, Umana Thompson Itaketo. Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems. Int J Theor Appl Math. 2017;3(5):163-166. doi: 10.11648/j.ijtam.20170305.12

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  • @article{10.11648/j.ijtam.20170305.12,
      author = {Ise Ise Ekpoudom and Anamekere Ime Jacob and Umana Thompson Itaketo},
      title = {Analytical Solution of Richards’ Equation for Application in Automatic Irrigation Systems},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {3},
      number = {5},
      pages = {163-166},
      doi = {10.11648/j.ijtam.20170305.12},
      url = {https://doi.org/10.11648/j.ijtam.20170305.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170305.12},
      abstract = {This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.},
     year = {2017}
    }
    

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    AB  - This paper presents the analytical solution to Richards’ equation of hydrology for unsaturated soils. In order to facilitate the design and analysis of a real-time automatic irrigation system, an accurate model must be developed for the system. Richard’s equation of water hydrology may be used to model part of the irrigation system. The major problem with the application of Richard’s equation is in the linearization of the non-linear partial differential equation (PDE). In this paper, an empirical relationship stated by Gardner [1] was used to linearize the nonlinear PDE. The solution for the PDE was obtained using separation of variables technique.
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Author Information
  • Department of Electrical, Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria

  • Department of Electrical, Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria

  • Department of Electrical, Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria

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