Analytical Solutions for Water Infiltration into Unsaturated–Semi-Saturated Soils Under Different Water Content Distributions on the Top Boundary

  • Hamed Reza Zarif SanayeiEmail author
  • Nasser Talebbeydokhti
  • G. Reza Rakhshandehroo
Research Paper


Modeling and simulation of water infiltration from the ground surface into unsaturated–saturated soil is very important in the study of the hydrological cycle process. Transient fluid flow through unsaturated–saturated soil is generally described by Richards’ equation. The equation is a nonlinear parabolic partial differential equation for which many numerical and limited analytical solutions exist. In this paper, exponential models for water content and hydraulic conductivity are selected so the nonlinear equation becomes linear and then separation of variables and Fourier series expansion techniques are used to derive analytical solutions to 2D water infiltration into unsaturated–semi-saturated soils with different distributions of water content on a part and/or parts of the top boundary. 2D water infiltrations are investigated in a rectangular soil domain, where high water content zone is considered as semi-saturated zone and residual water content zone is considered as unsaturated zone at initial time. A total of three cases are investigated in this paper. In case 1, a uniform distribution of water content on a part at the top boundary and no-flow boundaries at side edges are considered. Also, horizontal water table in unsaturated–semi-saturated domain is assumed as initial condition. In case 2, an inclined water table initial condition and a uniform distribution of water content on a part at the top boundary are investigated. Finally, in case 3, an analytical solution for ramp distributions of water content on two parts at the top boundary and an inclined water table initial condition is obtained. To illustrate the use of the derived equations, water content values are obtained from a numerical solution and compared to those from analytical solutions for two cases, showing less than 2% errors. These analytical solutions may be used as a benchmark for verification and efficiency assessment of numerical approaches.


Analytical solution Water infiltration Unsaturated–semi-saturated soil Two dimension Richards’ equation Inclined water table 


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Copyright information

© Shiraz University 2019

Authors and Affiliations

  • Hamed Reza Zarif Sanayei
    • 1
    Email author
  • Nasser Talebbeydokhti
    • 2
    • 3
  • G. Reza Rakhshandehroo
    • 2
  1. 1.Faculty of EngineeringShahrekord UniversityShahrekordIran
  2. 2.Department of Civil and Environmental EngineeringShiraz UniversityShirazIran
  3. 3.Environmental Research and Sustainable Development CenterShiraz UniversityShirazIran

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