Mathematical forms and numerical schemes for the solution of unsaturated flow equations

Hamid Taheri Shahraiyni, Behzad Ataie-Ashtiani

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


Prediction of water infiltration into the soil, fluid movement in the unsaturated soils, and groundwater recharge are important problems in different fields of science and engineering. Moreover, the transfer of the different pollutants (e.g., pesticides) from ground surface to groundwater occurs through the unsaturated zone. In this paper, a comprehensive evaluation of different finite difference schemes (e.g., fully implicit, Crank-Nicolson, and Runge-Kutta) is presented for the solution of head-based and mixed forms of the Richard's equation. Two examples of water infiltration in very dry and relatively wet unsaturated soils are used for the evaluation of schemes. In addition, the effects of various approximations of moisture capacity function, convergence criteria, and time stepping methods on the performance of the schemes are investigated, and the results showed their significant influences on mass balance, number of iterations, and convergence condition of the numerical schemes. The results of numerical simulations showed that the generally mixed form has better performance than the head-based form. In addition, the Crank-Nicolson scheme showed better results than the Runge-Kutta scheme, but both have convergence problems. In general, the comparison of numerical methods showed that the fully implicit scheme has the best performance among various finite difference schemes and can be selected as a reliable scheme with acceptable solutions for different infiltration problems in unsaturated soils.

Original languageEnglish
Pages (from-to)63-72
Number of pages10
JournalJournal of Irrigation and Drainage Engineering
Issue number1
Publication statusPublished - 30 Apr 2011
Externally publishedYes


  • Finite difference method
  • Mathematical model
  • Numerical analysis
  • Unsaturated flow


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