Development of a Numerical Model for Furrow Irrigation by Coupling 1D Saint-Venant and 3D Richards’ Equations

Document Type : Research Paper


Department of Water Science and Engineering, Faculty of Agriculture, Ferdowsi University of Mashhad (FUM), Mashhad, Iran


Development of numerical models for management and assessment of irrigation systems is an important step for establishing farm decision support systems. In this study, a coupled model has been developed for simulation of furrow irrigation using 1D fully hydrodynamic form of Saint-Venant equations and 3D fully-form of Richards’ equation. The Saint-Venant equations have been discretized by an explicit scheme while the Richards’ equation has been solved by an implicit scheme. Furthermore, coordinate transformation technique was employed to handle non-orthogonal grids of 3D Richards’ equation. The model was subsequently validated using experimental and numerical data and in all cases acceptable accuracy was observed. Root mean square error and mean absolute error for the advance phase were 0.63 and 2.63 sec, respectively. Furthermore, the maximum root mean square error and the mean absolute error for pressure head distribution were obtained 0.24 and 0.45 m, respectively.  Finally, the proposed model was employed to simulate furrow irrigation for five irrigation events and the results were analyzed. The results showed that the proposed model is able to simulate advance phase of furrow irrigation.


Main Subjects

Abbasi, F., (2013). Principles of Flow in Surface Irrigation. Iranian National Committee on Irrigation and Drainage, 232 pp.
Abbasi, F., Šimůnek, J., van Genuchten, M. T., Feyen, J., Adamsen, F. J., Hunsaker, D. J., Strelkoff, T. S., Shouse, P. (2003). Overland water flow and solute transport: Model development and field-data analysis. Journal of Irrigation and Drainage Engineering, 129(2), 71-81.
Abdul, A. S., and Gillham, R. W. (1984). Laboratory studies of the effects of the capillary fringe on streamflow generation. Water Resources Research, 20(6), 691-698.
An, H, and Yu, S. (2014) Finite volume integrated surface‐subsurface flow modeling on nonorthogonal grids. Water Resources Research, 50.3: 2312-2328.
An, H., Ichikawa, Y., Tachikawa, Y., and Shiiba, M. (2010). Three‐dimensional finite difference saturated‐unsaturated flow modeling with nonorthogonal grids using a coordinate transformation method. Water Resources Research, 46(11).
An, H., Ichikawa, Y., Tachikawa, Y., and Shiiba, M. (2012). Comparison between iteration schemes for three-   dimensional coordinate-transformed saturated–unsaturated flow model. Journal of Hydrology, 470, 212-226.
Brunetti, G., Šimůnek, J., and Bautista, E. (2018). A hybrid finite volume-finite element model for the numerical analysis of furrow irrigation and fertigation. Computers and Electronics in Agriculture, 150, 312-327.
Celia, M. A., Bouloutas, E. T., and Zarba, R. L. (1990). A general mass‐conservative numerical solution for the unsaturated flow equation. Water Resources Research, 26(7), 1483-1496.
Chaudhry, M. H. (2007). Open-Channel Flow: Springer Science and Business Media
Clemmens, A. J. (1979). Verification of the zero-inertia model for border irrigation. Transactions of the ASAE, 22(6), 1306-1309.
Clemmens, A., and Dedrick, A. (1994). Irrigation techniques and evaluations. In Management of Water Use in Agriculture (pp. 64-103): Springer
Ebrahimian, H., A Liaghat, B Ghanbarian-Alavijeh, F Abbasi. (2010). Evaluation of various quick methods for estimating furrow and border infiltration parameters. Irrigation Science 28 (6), 479-488
Ebrahimian, H., Liaghat, A., Parsinejad, M., Playán, E., Abbasi, F., and Navabian, M. (2013). Simulation of 1D surface and 2D subsurface water flow and nitrate transport in alternate and conventional furrow fertigation. Irrigation Science, 31(3), 301-316.
Ebrahimzadeh, A,. Ziaei, A. N., Jafarzadeh, M. R., Beheshti, A. A., Sheikh Rezazadeh Nikou, N. (2017). Numerical Modeling of One-Dimensional Flow in Furrow Irrigation by Solving the Full Hydrodynamics Equations using Roe Approach. 28(3), 41-51.
He, Z., Wu, W., and Wang, S. S. (2008). Coupled finite-volume model for 2D surface and 3D subsurface flows. Journal of Hydrologic Engineering, 13(9), 835-845.
Kollet, S. J., and Maxwell, R. M. (2006).  Integrated surface–groundwater flow modeling: A free-surface overland flow boundary condition in a parallel groundwater flow model. Advances in Water Resources, 29(7), 945-958.
Kosugi, K. (2008). Comparison of three methods for discretizing the storage term of the Richards equation. Vadose Zone Journal, 7(3), 957-965.
Liu, K., Huang, G., Xu, X., Xiong, Y., Huang, Q., and Šimůnek, J. (2019). A coupled model for simulating water flow and solute transport in furrow irrigation. Agricultural Water Management, 213, 792-802.
Maxwell, R. M., Putti, M., Meyerhoff, S., Delfs, J. O., Ferguson, I. M., Ivanov, V., Kim, J., Kolditz, O., Kollet, S. J., Kumar, M., Lopez1, S., Niu, J., Paniconi, C., Park, Y., Phanikumar, M. S., Shen, C., Sudicky, E. A., and Sulis, M. (2014). Surface‐subsurface model intercomparison: A first set of benchmark results to diagnose integrated hydrology and feedbacks. Water Resources Research, 50(2), 1531-1549.
Naghedifar, S. M., Ziaei, A. N., and Ansari, H. (2018). Simulation of irrigation return flow from a Triticale farm under sprinkler and furrow irrigation systems using experimental data: A case study in arid region. Agricultural Water Management, 210, 185-197.
Naghedifar, S. M., Ziaei, A. N., Playán, E., Zapata, N., Ansari, H., and Hasheminia, S. M. (2019). A 2D curvilinear coupled surface–subsurface flow model for simulation of basin/border irrigation: theory, validation and application. Irrigation Science, 1-18.
Narasimhan, T. N. (2005). Buckingham, 1907. Vadose Zone Journal, 4(2), 434-441.
Playán, E., Faci, J. M., and Serreta, A. (1996). Modeling microtopography in basin irrigation. Journal of Irrigation and Drainage Engineering, 122(6), 339-347.
Playán, E., Walker, W. R., and Merkley, G. P. (1994). Two-dimensional simulation of basin irrigation. I: Theory. Journal of Irrigation and Drainage Engineering, 120(5), 837-856.
Richards, L. A. (1931). Capillary conduction of liquids through porous mediums. Physics, 1(5), 318-333.
Roe, P. L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43(2), 357-372
Saad, Y. (2003). Iterative Methods for Sparse Linear Systems (Vol. 82): SIAM
Sanders, B. (2001). High-resolution and non-oscillatory solution of the St. Venant equations in non-rectangular and non-prismatic channels. Journal of Hydraulic Research, 39(3), 321-330.
Šimůnek, J. (2006). Models of water flow and solute transport in the unsaturated zone. Encyclopedia of Hydrological Sciences.
Singh, V., and Bhallamudi, S. M. (1997). Hydrodynamic modeling of basin irrigation. Journal of Irrigation and Drainage Engineering, 123(6), 407-414.
Strelkoff, T., and Souza, F. (1984). Modeling effect of depth on furrow infiltration. Journal of Irrigation and Erainage Engineering, 110(4), 375-387.
Tabuada, M., Rego, Z., Vachaud, G., and Pereira, L. (1995). Modelling of furrow irrigation. Advance with two-dimensional infiltration. Agricultural Water Management, 28(3), 201-221.
Van Genuchten, M. T. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5), 892-898.
VanderKwaak, J. E. (1999). Numerical simulation of flow and chemical transport in integrated surface-subsurface hydrologic systems.
Versteeg, H. K. and Malalasekera, W. (2007). An Introduction to Computational Fluid Dynamics: The Finite Volume Method: Pearson education
Walker, W. R., and Humpherys, A. S. (1983). Kinematic-wave furrow irrigation model. Journal of Irrigation and Drainage Engineering, 109(4), 377-392.
Weill, S., Mouche, E., and Patin, J. (2009). A generalized Richards equation for surface/subsurface flow modelling. Journal of Hydrology, 366(1-4), 9-20.
Wöhling, T., and J. C. Mailhol. (2007), Physically based coupled model for simulating 1D surface–2D subsurface flow and plant water uptake in irrigation furrows. II: Model test and evaluation. Journal of Irrigation and Drainage Engineering 133(6), 548-558.
Wöhling, T., and Schmitz, G. H. (2007). Physically based coupled model for simulating 1D surface–2D subsurface flow and plant water uptake in irrigation furrows. I: Model development. Journal of Irrigation and Drainage Engineering, 133(6), 538-547.
Wöhling, T., Singh, R., and Schmitz, G. (2004). Physically based modeling of interacting surface–subsurface flow during furrow irrigation advance. Journal of Irrigation and Drainage Engineering, 130(5), 349-356.
Xu, D., Zhang, S., Bai, M., Li, Y., and Xia, Q. (2013). Two-dimensional coupled model of surface water flow and solute transport for basin fertigation. Journal of Irrigation and Drainage Engineering, 139(12), 972-985.
Zha, Y., Yang, J., Yin, L., Zhang, Y., Zeng, W., and Shi, L. (2017). A modified Picard iteration scheme for overcoming numerical difficulties of simulating infiltration into dry soil. Journal of Hydrology, 551, 56-69.
Zhang, S., Xu, D., Bai, M., and Li, Y. (2014a). Two-dimensional surface water flow simulation of basin irrigation with anisotropic roughness. Irrigation Science, 32(1), 41-52
Zhang, S., Xu, D., Bai, M., Li, Y., and Liu, Q. (2016). Fully coupled simulation for surface water flow and solute transport in basin fertigation. Journal of Irrigation and Drainage Engineering, 142(12), 04016062.
Zhang, S., Xu, D., Bai, M., Li, Y., and Xia, Q. (2014b). Two-dimensional zero-inertia model of surface water flow for basin irrigation based on the standard scalar parabolic type. Irrigation Science, 32(4), 267-281.