Coupled Surface–Subsurface Solute Transport Model for Irrigation Borders and Basins. I. Model Development D. Zerihun1; A. Furman2; A. W. Warrick3; and C. A. Sanchez4 Abstract: Surface fertigation is widely practiced in irrigated crop production systems. Lack of design and management tools limits the effectiveness of surface fertigation practices. The availability of a process-based coupled surface–subsurface hydraulic and solute transport model can lead to improved surface fertigation management. This paper presents the development of a coupled surface–subsurface solute transport model. A hydraulic model described in a previous paper by the writers provided the hydrodynamic basis for the solute transport model presented here. A numerical solution of the area averaged advection–dispersion equation, based on the split-operator approach, forms the surface solute transport component of the coupled model. The subsurface transport process is simulated using HYDRUS-1D, which also solves the one-dimensional advection–dispersion equation. A driver program is used for the internal coupling of the surface and subsurface transport models. Solute fluxes calculated using the surface transport model are used as upper boundary conditions for the subsurface model. Evaluation of the model is presented in a companion paper. DOI: 10.1061/͑ASCE͒0733-9437͑2005͒131:5͑396͒ CE Database subject headings: Surface irrigation; Coupling; Models; Unsaturated flow; Solutes; Salinity. Introduction of recent origin and are limited. Boldt et al. ͑1994͒ and Playan and Faci ͑1997͒ solved a spatially and temporally lumped form of Surface N-fertigation is widely practiced in the southwestern the advection equation for simulating nitrate transport in a surface ͑ ͒ United States. Flexibility, cost effectiveness, and the potential for irrigation stream. Garcia-Navarro et al. 2000 presented a nu- improved seasonal fertilizer application efficiency are advantages merical solution of the area averaged advection–dispersion equa- of fertigation over traditional fertilizer application methods. How- tion for surface fertigation applications. They used the split- ever, in the absence of appropriate management procedures, sur- operator method, in which the mechanism of advective transport ͑ ͒ face N-fertigation performance can be low compared to conven- was decoupled in the mathematical sense from hydrodynamic tional fertilizer application methods ͑Gardner and Roth 1984; dispersion and the resulting pair of equations were solved in two Jaynes et al. 1992͒. Thus, there is a potential to develop improved separate and consecutive steps, using numerical methods appro- management practices for surface N-fertigation using coupled priate to each subproblem. First, the advection equation was solved using a semi-Lagrangian integration scheme, then the dif- water flow and solute transport models. fusion equation was discretized using the central difference ap- The transport of N-fertilizer in an irrigation stream can be proximation. Solution algorithms based on the split-operator ap- modeled using the area averaged advection–dispersion equation. proach are designed primarily to minimize the numerical Studies related to the application of the one-dimensional ͑1D͒ problems ͑such as excessive numerical diffusion and nonphysical advection–dispersion equation in surface-fertigation modeling are oscillations near the vicinity of large concentration gradients͒ ac- cruing from the numerical treatment of the advective term in the 1 Assistant Research Scientist, Dept. of Soil, Water, and Environmental advection–dispersion equation ͑Holly 1975; Sauvaget 1985; Sciences, 429 Shantz Building #38, Univ. of Arizona, 1200 E. Campus Staniforth and Cote 1991; Islam and Chaudhry 1997; Karpick and Drive, Tucson, AZ 85721. Crockett 1997; Komatsu et al. 1997; Manson et al. 2001͒. Abassi 2Institute of Soil, Water, and Environmental Sciences, ARO-Volcani et al. ͑2003͒ proposed a solution for the advection–dispersion Center, P.O. Box 6, Bet Dagan 50250, Israel; formerly, Research Associate, Dept. of Hydrology and Water Resources, Harshbarger equation using an implicit finite difference scheme. They solved Building #11, Univ. of Arizona, Tucson, AZ, 85721. the complete equation in a single step and reported to have over- 3Professor, Dept. of Soil, Water, and Environmental Sciences, 429 come the numerical problems that plagued the discretization of Shantz Building #38, Univ. of Arizona, 1200 E. Campus Drive, Tucson, the advection term through proper selection of grid Péclet number AZ 85721 ͑corresponding author͒. E-mail: [email protected] and Courant number. 4Professor, Dept. of Soil, Water, and Environmental Sciences and The models described above deal exclusively with the trans- Director, Yuma Agricultural Center, Univ. of Arizona, W. 8th St., 6425 port of solutes in overland flow. Although these models calculate Yuma, AZ 85364. solute flux into the crop root zone as part of the solution, they Note. Discussion open until March 1, 2006. Separate discussions must lack the capability to simulate the movement, and spatial distri- be submitted for individual papers. To extend the closing date by one bution, of the fertilizer material in the soil profile. Such a capa- month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible bility requires the availability of a physically based coupled publication on July 26, 2004; approved on December 30, 2004. This surface–subsurface water flow and solute transport model. Cur- paper is part of the Journal of Irrigation and Drainage Engineering, rently, there exists no coupled model that is designed to simulate Vol. 131, No. 5, October 1, 2005. ©ASCE, ISSN 0733-9437/2005/5-396– water flow and solute transport, both on the surface and through 406/$25.00. the soil profile, in a surface-fertigation setting. Watershed scale 396 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 coupled hydrologic and solute transport models, like the SHE monly referred to as dispersion ͑Holly 1975; Rutherford 1994͒. model ͑Abbott et al. 1982 and 1986; Bathurst et al. 1995͒, are The 1D advection–dispersion equation that describes the transport primarily designed for simulating basin scale hydrologic pro- of a neutrally buoyant conservative solute in a surface irrigation cesses involving water flow and solute transport both on the sur- stream ͑in a unit width border or basin͒ can be derived using face and through the subsurface medium. However, such models either a rigorous approach that involves progressive averaging, are expensive and require maintenance. In addition, they lack the over time, depth, and width, of the instantaneous mass conserva- flexibility to adapt to the specific needs of varying boundary con- tion equation ͑Holly 1985͒, or using a simpler approach based on ditions encountered in surface irrigation processes. A detailed the application of mass balance principles directly to an elemental summary of past attempts to develop coupled surface–subsurface volume in a 1D flow field where dispersion is fully developed as hydrologic models can be obtained from Morita and Yen ͑2002͒. a gradient-diffusion process In this study, the development of a fully coupled surface– Cץ ץ qC͒͑ץ hC͒͑ץ -subsurface solute transport model is presented. A coupled over ͑ ͒ + + Ci = ͩhD ͪ ͑1͒ xץ x xץ xץ tץ land and subsurface flow model Zerihun et al. 2005a provided the hydrodynamic basis for the solute transport model described here. The surface component of the solute transport module is where h=flow depth ͑m͒; C=concentration ͑g/m3͒; t=time based on the numerical solution of the 1D advection–dispersion ͑min͒; q=flow rate ͑m3 /min/m͒; x=horizontal distance ͑m͒; equation. The subsurface solute transport process is simulated i=rate of infiltration ͑m/min͒; and D =longitudinal dispersion co- ͑ ͒ x using HYDRUS-1D Simunek et al. 1998 , which also solves the efficient ͑m2 /min͒. Note that h, q, and i are obtained from the 1D advection–dispersion equation. A driver program is used to solution of the water flow model. The presence of the sink term, couple the surface–subsurface water flow and solute transport Ci, distinguishes Eq. ͑1͒ from the advection–dispersion equation modules. Solute flux into the crop root zone, calculated by the commonly used to model 1D open channel flow transport pro- overland flow and transport module, is used as the upper bound- cesses. Eq. ͑1͒ can be recast in its nonconservative form, a form ary condition for the subsurface solute transport model. Evalua- commonly used in the numerical solutions of the advection– ͑ tion of the model is presented in a companion paper Zerihun et dispersion equation, by expanding it and combining the resulting ͒ al. 2005b . equation with an expression for i, obtained from the continuity equation of the hydrodynamic model Cץ ץ Cץ Cץ Water Flow Model h + q = ͩhD ͪ ͑2͒ ץ x ץ ץ ץ A coupled surface–subsurface water flow model ͑Zerihun et al. t x x x 2005a͒ provided the hydrodynamic basis for the solute transport The following limiting assumptions apply to Eqs. ͑1͒ and ͑2͒: model described here. A zero-inertia model ͑Zerihun et al. 2003, ͑1͒ hydrodynamic dispersion can be modeled as a gradient diffu- 2005a͒ forms the surface hydraulic component of the coupled sion process using Fick’s law ͑Taylor 1954; Elder 1959; Fischer model and a numerical solution of the Richards equation 1967͒; ͑2͒ mass transport due to molecular diffusion is much less ͑HYDRUS-1D, Simunek et al. 1998͒