Coupled Surface–Subsurface Solute Transport Model for Irrigation Borders and Basins

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 is used to model infiltration significant than is due to turbulent diffusion, hence can be readily into and water flow through the subsurface. The coupling of the absorbed in the dispersion term ͑Taylor 1954; Elder 1959; Fischer surface and subsurface flow models is internal and iterative. Dur- 1967͒; and ͑3͒ no significant loss/addition of N-fertilizer occurs ing a time step, the coupled model calculates flow depth, unit flow through chemical reactions and through such physical pathways rate, infiltration, and changes in soil water contents at each com- as volatilization during the course of a typical surface irrigation putational node. Description of the governing equations of water process, thus the N-fertilizer can essentially be considered con- flow, pertinent initial/boundary conditions, numerical solutions, servative. and the approach used to couple the surface and subsurface flow While the third assumption, above, can be considered reason- modules can be obtained from Zerihun et al. ͑2003, 2005a͒, hence able insofar as nonvolatile N-fertilizers are considered, the second detailed additional treatment of these issues is avoided here. assumption is generally valid provided the first assumption holds ͑e.g., Holly 1975, 1985; Rutherford 1994͒. The first assumption, that is the description of dispersion as a gradient-diffusion pro- Surface Solute Transport Equation cess, is valid only in the channel segment where differential advection is in equilibrium with turbulent diffusion. This is a seg- The motion of a neutrally buoyant conservative solute in a one- ment of the channel where deviations of local time-averaged dimensional free-surface turbulent flow field is subject to the solute concentrations from the cross-sectional mean are suffi- combined influence of time-averaged velocities, turbulent veloc- ciently low that the solution can be considered well mixed in a ity fluctuations, and molecular diffusion ͑Cunge et al. 1980͒. Pure cross section ͑Fischer 1967; Rutherford 1994͒. Equilibrium be- advection, due to time-averaged velocities, translates a parcel of tween differential advection and turbulent diffusion can be estab- fluid over space without inducing a change in solute concentra- lished only after the passage of an initial advection dominated tion. However, deviations of local time averaged velocities from period following the injection of a fertilizer solution ͑Taylor cross-sectional means lead to differential advection over a cross 1954; Elder 1959; Fischer 1967͒. The length of the advection section. Turbulent velocity fluctuations, due to deviations of local dominated period ͑or the channel reach͒, which establishes the instantaneous velocities from time averaged values, give rise to threshold for the limit of applicability of Eqs. ͑1͒ and ͑2͒, depends turbulent diffusion, a process that counters the effect of differen- on flow dynamics and channel geometry and can be approximated tial advection by transferring mass between zones of varying ve- using equations proposed by Elder ͑1959͒ and Fischer ͑1967, locities ͑Holly 1975; Rutherford 1994͒. The interaction of differ- 1973͒. Since the initial advective period is proportional to the ential advection with turbulent diffusion, and to a much lesser square of the channel width ͑Fisher 1967͒, this is particularly extent with molecular diffusion, produces a mixing process com- important in borders and basins ͑channels with significantly

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 / 397 ͒ ␣ ␭ longer initial advective period compared to furrows ,if persion coefficient cannot be expressed as the product − xv .An N-fertilizer solution is injected directly into the stream, at a single example is the expression for the mechanical dispersion coeffi- point, on the upstream end of the channel. cient for a 2D plain flow proposed by Elder ͑1959͒. In a typical open channel flow condition, mechanical dispersion is considered to have a much larger influence on solute trans- Subsurface Solute Transport Equation port than molecular diffusion ͑Holly 1975͒; hence the molecular diffusion coefficient can readily be absorbed by the dispersion The subsurface component of the solute transport module is based coefficient. However, in surface irrigation processes involving on HYDRUS-1D ͑Simunek et al. 1998͒. HYDRUS is capable of closed-end channels, very low velocity flows or ponded condi- simulating the transport as well as fate ͑reaction, sorption/ tions occur during the later stages of an irrigation event. In this ͒ desorption, volitilization, and dissolution/precipitation of chemi- phase of an irrigation/fertigation event, molecular diffusion can cals in the unsaturated zone on a seasonal basis. However, the be the most influential or the only physical mechanism of solute focus in the current study is the transport of N-fertilizer material transfer. By providing a flexible formulation that separates the in overland flow and its distribution in the crop root zone imme- contributions of flow dynamics and molecular diffusion to the diately following a fertigation event. This is a small enough time longitudinal hydrodynamic dispersion coefficient, D , Eq. ͑4͒ al- scale during which the N-fertilizer material can essentially be x lows the changes in the relative importance of the solute transport treated as a conservative chemical without significant loss of ac- mechanisms in the course of a surface fertigation process to be curacy. In the context of the model reported here, HYDRUS accounted for automatically. The formulation in Eq. ͑4͒ is also solves the 1D advection–dispersion equation that describes the advantageous, because it allows longitudinal dispersion to be transport of a conservative solute in a variably saturated porous medium characterized using a single parameter at the input and still enable the dispersion coefficient to remain variable along the flow do- q C ͒ main as a function of flow velocity, which is convenient and͑ץ Cץ ץ ͒ ␪C͑ץ z ͩ␪ z ͪ z z ͑ ͒ = Dz − 3 physically meaningful at the same time. zץ zץ zץ tץ ␪ ͑ ͒ where =volumetric water content - ; Cz =subsurface solute ͑ 3͒ ͑ ͒ concentration g/m ; qz =local Darcian flow rate m/min ; ͑ ͒ Numerical Solution of Surface Solute z=vertical dimension m ; and Dz =subsurface dispersion Transport Equation ͑ 2 ͒ ␪ coefficient m /min . Note that qz and are obtained from the subsurface water flow model. The solution of the advection–dispersion equation can be plagued by numerical problems. While the diffusion term in Eq. ͑2͒ can be discretized effectively ͑accurately and stably͒ using a wide variety Longitudinal Dispersion Coefficients D and D „ x z… of numerical schemes, the satisfactory discretization of the advec- The solution of Eq. ͑2͒ requires that the longitudinal dispersion tion term is a considerable challenge to existing numerical meth- ͑ coefficient, D , be known. D can be considered to be composed ods Sauvaget 1985; Leonard 1991; Karpick and Crockett 1997; x x ͒ of two components; the mechanical dispersion coefficient and Komatsu et al. 1997; Manson et al. 2001 . In advection dominated molecular diffusion ͑e.g., Bear 1972͒ mass transport, the numerical treatment of the advection term often leads to artificial diffusion that is much larger than the ␣ ␭ ␧ ͑ ͒ Dx = xv + 4 physical diffusion and to nonphysical oscillations, especially near the vicinity of large concentration gradients. A classic method for ͑ ͒ where the first term in Eq. 4 represents the mechanical disper- solving the advection equation ͑a hyperbolic partial differential ␣ ͑ sion coefficient; x =longitudinal dispersivity dimension depen- equation͒ is the method of characteristics ͑Hoffmann and Chiang ␭͒ ␭ dent on the value of ; =dimensionless constant; v=mean 1993͒. The differing requirements, with respect to effective nu- cross-sectional velocity ͑m͒; and ␧=molecular diffusion merical schemes, between the advection and dispersion equations ͑m2 /min͒, which is a function of the solute and ambient fluid. For meant that solution accuracy and efficiency can be enhanced by low velocity flows like those encountered in porous media, me- decoupling ͑in the mathematical sense͒ the advective transport chanical dispersion is generally expressed as the product of dis- from the mechanism of dispersion. The resulting pair of equations persivity ͑dependent on the property of the porous media͒ and are solved in two separate but consecutive steps, using numerical velocity, where ␭Ϸ1 and ␣ will have the dimension of length. x techniques most appropriate to each subproblem. This procedure, The subsurface component of the model reported here, HYDRUS-1D, uses a similar expression to Eq. ͑4͒, with ␭=1, to commonly known as the split-operator method, is widely used in relate subsurface longitudinal dispersion, D , with the Darcian solute transport modeling in a free-surface turbulent flow field z ͑ velocity, subsurface longitudinal dispersivity, and molecular dif- Holly 1975; Sauvaget 1985; Karpik and Crockett 1997; Garcia- ͒ fusion. Abbassi et al. ͑2003͒ used ␭=1 in their surface solute Navarro et al. 2000; Manson et al. 2001 . With the split-operator ͑ ͒ transport model as well. However, in open channel flows ͑includ- method, Eq. 2 is treated as a combination of two subsystems: ͑ ͒ ing surface irrigation streams͒, where much larger flow velocities pure advection Dx =0 described by are common, ␭ values exceed 1.0 and can be as high as 2.0 ͑e.g., Fisher 1967, 1973͒, in which case ␣ will have the dimension of Cץ Cץ x time or some combination of distance and time. Referring to the + v =0 ͑5͒ ץ ץ ␣ ͒ ͑ form of the Dx equation of Fischer 1967, 1973 x can be viewed t x as a parameter that encapsulates the effects of channel geometry and cross-sectional velocity distribution on the longitudinal dispersion coefficient. In some cases, however, the mechanical dis- and pure diffusion, expressed as

398 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 Fig. 1. Initial and boundary conditions. I, II, III, and IV refer to solute inﬂux conﬁgurations and pertinent boundary conditions ͑Fig. 4͒.

C Sauvaget 1985; Schoel and Holly 1991; Karpik and Crockettץ ץ Cץ h = ͩhD ͪ ͑6͒ ͒ -x 1997; Garcia-Navarro et al. 2000 . At the heart of a semiץ x xץ tץ Lagrangian advection scheme is the method of characteristics. where Eq. ͑5͒ is, first, solved for the intermediate solute concen- The method of characteristic recasts Eq. ͑5͒ as a statement of the trations, Ca, using an appropriate numerical scheme, then the ad- invariance of solute concentration in a fluid element with time vected concentrations, Ca, are diffused longitudinally through the ͑ ͒͑ dC numerical solution of Eq. 6 Holly 1975; Karpick and Crockett =0 ͑7͒ 1997; Garcia-Navarro et al. 2000; Manson et al. 2001͒. Pertinent dt ͑ ͒ ͑ ͒ initial and boundary conditions for Eqs. 5 and 6 are given in along a trajectory Fig. 1. The split-operator approach incurs an error irrespective of the accuracy with which the individual steps are computed dx ͑ ͒ = v͑x,t͒͑8͒ Karpik and Crockett 1997 . However, the splitting error is com- dt pensated for by the higher levels of computational efficiency at- ͑ ͒ tainable for the numerical solutions of the split subproblems than Integration of Eq. 7 between tn and tn+1 along the trajectory is the case for the unsplit problem ͑Karpik and Crockett 1997͒.A between points D and A in Fig. 2 yields discussion on the split-operator method as implemented in the C͑x ,t ͒ = C͑x ,t ͒ ⇔ Cn+1 = Cn ͑9͒ model reported herein is presented below. j+1 n+1 D n j+1 D where j=distance index; n=time index; and xD =distance of the ͑ ͒ departure point of the characteristic trajectory, xD ,tn , leading to Solution of Advection Step ͑ ͒ ͑ ͒ the arrival point, xj+1 ,tn+1 , from the inlet end of the channel m . ͑ ͒ Highly accurate semi-Lagrangian integration schemes that com- The departure points, xD ,tn , generally do not coincide with grid bine the method of characteristics with higher order ͑ജ2͒ inter- points at which solute concentrations are known ͑Fig. 2͒, thus the ͑ ͒ n polation schemes are widely used to solve Eq. 5 for the inter- concentration at a departure point, CD, need to be interpolated mediate concentration distributions ͑Holly and Preissmann 1977; from known concentrations at adjacent grid points. The calcula-

Fig. 2. Computational grid and characteristic trajectory for numerical solution of advection equation

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 / 399 n+1 ␣3 ␣2 ␣ ͑ ͒ tion of Cj+1 involves three steps. In the first step, the location of CD = c1 + c2 + c3 + c4 12 the foot of the trajectory on t is determined through integration n ␣ ͑ ͒ ͑ ͒ ͑ ͒ n where = xj+1 −xD / xj+1 −xj and c1, c2, c3, and of Eq. 8 , then the concentration at the departure point, CD,is interpolated between known concentrations at adjacent grid c4 =coefficients dependent on concentrations and their spatial ͑ ͒ ͑Cn+1 Cn+1͒ t ͑ ͒ Cn+1 derivatives at adjacent grid points, j and j+1 Fig. 2 . Note points, j , j+1 ,on n Fig. 2 , and finally j+1 is determined ␣ using Eq. ͑9͒. that is equal to the grid Courant number, Cr. Using concentrations and their spatial derivatives at nodes j and j+1 1. Determine location of departure point of trajectory: Assum- ͑ ͒ ing a linear velocity variation over a time and space step, Eq. along tn, Eq. 12 can be given as ͑ ͒ 8 can be integrated numerically to yield a second-order C = a Cn + a Cn + a Cn + a Cn ͑13͒ approximation of the location of the departure point of the D 1 j 2 j+1 3 xj 4 xj+1 n n 2 ;͒␣x͉͑ ͒; a =␣ ͑3−2ץ/Cץ͉= where C =C͑x ,t ͒; C ͒ ͑ back trajectory xD ,tn j j n xj xj,tn 1 ␣2͑ ␣͒͑ ͒ ␣͑ ␣͒2͑ a2 =1−a1; a3 = 1− xj+1 −xj ; and a4 =− 1− xj+1 vD + vA ͒ x = x − ⌬t ͑10͒ −xj . D j+1 2 n n+1 ͑ ͒ 3. Determine Cj+1 using Eq. 9 : Given the concentration at the n n+1 channel inlet and the concentrations and their derivatives at Referring to Fig. 2, vD =vD and vA =vj+1 . Since the rate of change of velocity both in time and distance is generally low all grid points at the beginning of a time step, say tn; the in surface irrigation applications, the preceding assumption on concentrations at all nodes at tn+1 can be calculated through a recursive application of the procedure described above. This velocity is valid. Both xD and vD are unknown, thus xD cannot ͑ ͒ completes the calculation of concentration in the advection be directly calculated from Eq. 10 only. In light of the as- ഛ Ͻ step for the case xj xD xj+1. sumption that velocity varies linearly within a distance step, ͑ ͒ The use of Eq. 13 to calculate CD presupposes that the con- vD can be expressed in terms of the velocities at adjacent n n n centration derivatives, Cxj and Cxj+1, are known in addition to Cj nodes n and Cj+1. While the concentrations at neighboring grid points can i+1 ␺ ͑ ␺͒ vD = vj+1 + 1− vj be used to calculate the spatial derivatives through finite differencing, accuracy can be enhanced by advecting the concentration where ͑11͒ derivatives themselves along a characteristic in the same way as the concentrations ͑Holly and Preissmann 1977; Holly and Ko- ͑ i ͒ matsu 1983͒. The advection equation for the concentration deriva- xD − xj ␺ = tive, C , can be obtained by differentiating Eq. ͑5͒. The corre- x − x x j+1 j sponding characteristic equation is ͑ ͒ ͑ ͒ ͑ Eqs. 10 and 11 can be solved iteratively for xD and vD e.g., vץ dC ͒ Sauvaget 1985; Staniforth and Cote 1991 by the following x =−C ͑14͒ ץ x ͒ ͑ 1 ͒ ͑ scheme: 1 find an initial estimate for xD, xD, using Eq. 10 after dt x setting the initial estimate for v , v1 =v ; ͑2͒ determine a revised D D A which is valid along a trajectory defined by Eq. ͑8͒. The same estimate of v , vi+1͑xi ͒, using Eq. ͑11͒͑where i=iteration index͒; D D D procedure as the one used for the calculation of concentration is and ͑3͒ check if ͉ i+1 − i ͉ഛ0.01 i . If, in the current iteration, the vD vD vD to be used in the solution of the advection equation for concen- i ͑ i ͒ prescribed tolerance is exceeded, set i=i+1 and calculate xD vD ͑ ͒ ͑ ͒ ͑ ͒ tration derivatives, Eqs. 14 and 8 . The departure point for the using Eq. 10 , then repeat steps 2 and 3 above. If, on the other characteristic trajectory has already been determined from the ͉ i+1 i ͉ഛ i hand, vD −vD 0.01vD, then xD and vD are known and the it- preceding step ͑calculation of concentration͒. Holly and Preiss- eration is to be stopped. As can be seen from Fig. 2, the preceding mann ͑1977͒ set C , the coefficient of the right hand side of Eq. ഛ Ͻ x procedure presupposes that xj xD xj+1, which means the grid ͑14͒,toC ͑Fig. 2͒ and approximated the derivatives through ͓͑ ͒⌬ ͔ ͑ ⌬ ͒ xD Courant number, Cr= vD +vA t / 2 x , is less than or equal to finite differencing to obtain a solution to Eq. ͑14͒. Sauvaget Ͼ 1. If, on the other hand, Cr 1, the method becomes unstable ͑1985͒ integrated Eq. ͑14͒ numerically over incremental time ͑ ͒ Holly and Preissmann 1977 , thus a different approach will be steps based on the assumption that the product on the right hand ͑ ͒ used to integrate Eq. 8 and that will be discussed later. side of Eq. ͑14͒ varies linearly in a time increment. In the model 2. Calculation of concentration at the departure point of the reported here, only velocity is assumed to vary linearly along a trajectory: Unlike velocities, concentrations can show rapid characteristic over incremental time and distance steps. With this changes both in time and space, hence linear interpolation assumption in place, Eq. ͑14͒ can be integrated, after separation schemes are unsatisfactory. Linear interpolation leads to nu- of variables; which results in an expression for Cn+1 merical diffusion and severe damping at low Courant num- xj+1 bers ͑Holly and Preissmann 1977; Sauvaget 1985͒. Higher n+1 ͑ ⌬ ͒͑͒ Cxj+1 = CxD exp − vg tn 15 order ͑ജ2͒ interpolation schemes using concentrations at more than two points ͑e.g., Martin 1975͒ often lead to non- where vg =average spatial velocity gradient along a characteristic ⌬ physical oscillations and difficulties to impose boundary con- over tn. Given the assumption that velocity variation within a ditions ͑Holly and Preissmann 1977; Sauvaget 1985͒. Holly time and space grid can be assumed linear, vg is constant in a time ͑ ͒ ͑ ͒ ͑ ͒ and Preissmann ͑1977͒ proposed a compact and accurate step and can be given as: vg = vA −vD / xj+1 −xD Fig. 2 .An Hermite cubic interpolation polynomial, that uses concentra- interpolating function for the concentration gradient at the depar- ͑ tions and corresponding derivatives at two consecutive grid ture point of the back trajectory, CxD, can be constructed Holly ͒ points, to estimate the concentration at the departure point of and Preissmann 1977 the characteristic trajectory ͑Fig. 2͒. The interpolating poly- C = b Cn + b Cn + b Cn + b Cn ͑16͒ nomial proposed by Holly and Preissmann ͑1977͒, and used xD 1 j 2 j+1 3 xj 4 xj+1 ␣͑␣ ͒ ͑ ͒ ␣͑ ␣ ͒ in the model reported here, can be expressed as a function of where b1 =6 −1 / xj+1 −xj ; b2 =−b1; b3 = 3 −2 ; and ␣ ͑␣ ͒͑ ␣ ͒ a dimensionless distance b4 = −1 3 −1 . The procedure described above is stable and

400 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 Cץ -provides excellent accuracy as it introduces little damping or dis ͑ ͒ ␣Ͻ = C ͑20͒ x xץ persion phase error for 1 and it yields exact solutions for ␣=1 ͑Holly and Preissmann 1977͒. Formally, the solution becomes unstable for ␣Ͼ1. However, experience with models Eqs. ͑19͒ and ͑20͒ can be solved subject to pertinent initial and based on the semi-Lagrangian scheme shows that the solution boundary conditions ͑Fig. 1͒. Using the Preissmann implicit finite remains stable even for values of ␣ that exceed unity ͑Staniforth difference scheme ͑Cunge et al. 1980͒, Eqs. ͑19͒ and ͑20͒ can be and Cote 1991; Wallis and Manson 1997͒. To ensure uncondi- discretized as tional stability, a modification proposed by Sauvaget ͑1985͒, for ␺ ͑1−␺͒ ␣Ͼ1, is incorporated into the model presented here. Sauvaget ͫ ͑hn+1 + hn+1͒ + ͑hn + hn͒ͬ ␣Ͼ ͑ Ͻ ͒ 2 j+1 j 2 j+1 j proposed a stable scheme, for 1 i.e., xD xj , by extending Ј the back trajectory only up to point D in segment MN, instead of Cn+1 − Ca Cn+1 − Ca Љ ϫͫ␾ j+1 j+1 ͑ ␾͒ j j ͬ point D on tn and interpolating over time instead of distance + 1− ͑ ͒ ⌬t ⌬t Fig. 2 . The values of CDЈ and CxDЈ can be interpolated from the Cn, Cn+1, Cn , and Cn+1, where C represents the derivative of n+1 n+1 a a j j tj tj t n Cxj+1 − Cxj Cxj+1 − Cxj n ͑ ͒ +1␺ ͑ ͒n͑ ␺͒ ͑ ͒ ␣ഛ +1 − hDx j − hDx j 1− =0 21 concentration with respect to time. As is the case for 1, Cj+1 ⌬ ⌬ n+1 x x and Cxj+1 are set equal to CDЈ and CxDЈ, respectively. Using a ␥ Hermitic interpolating polynomial with a dimensionless time, n+1 n+1 a a ␺ ͑ ͒ ͑ ͒ ͑ ͒ Cj+1 − Cj Cj+1 − Cj n+1 n+1 = tn+1 −tDЈ / tn+1 −tn Fig. 2 , as its argument and recognizing ␺ + ͑1−␺͒ − ͑C + C ͒ ͓ ͑ ͔͒ ⌬x ⌬x 2 xj+1 xj that Ct =−vCx Eq. 5 ; CDЈ and CxDЈ can be expressed as ͑1−␺͒ ͑ a a ͒ ͑ ͒ Ј n Ј n+1 Ј n Ј n+1 ͑ ͒ − Cxj+1 + Cxj =0 22 CDЈ = a1Cj + a2Cj + a3Cxj + a4Cxj 17 2 a ͑ and where Cj =concentration at the foot of the trajectory or at the end of the advection phase in the current time step͒ ͑g/m3͒, a 1 Cxj=concentration gradient at the end of the advection phase in ͑ Ј n Ј n+1 Ј n Ј n+1͒͑͒ 3 n CxDЈ =− b C + b C + b C + b C 18 the current time step ͑g/m ͒; ͑hD ͒ =average of the product of v 1 j 2 j 3 xj 4 xj x j DЈ flow depth and longitudinal dispersion coefficient at nodes j and j+1 at t ; and ␺ and ␾=time and space weighing coefficients. where aЈ=␥2͑3−2␥͒; aЈ=1−aЈ; aЈ=−vn␥2͑1−␥͒͑t −t ͒; n 1 2 1 3 j n+1 n Values of ␺=0.67 and ␾=0.5 are used in the model described aЈ=vn+1␥͑1−␥͒2͑t −t ͒; bЈ=6␥͑␥−1͒/͑t −t ͒; bЈ=−bЈ, 4 j n+1 n 1 n+1 n 2 1 here ͑e.g., Cunge et al. 1980͒. Rearranging terms, Eqs. ͑21͒ and bЈ=−vn␥͑3␥−2͒; and bЈ=−vn+1͑␥−1͒͑3␥−1͒. The time at the 3 j 4 j ͑22͒ can be expressed as departure point, tDЈ, and corresponding velocity, vDЈ, can be calculated iteratively using the same procedure as the one used for ␦ n+1 ␦ n+1 ␦ n+1 ␦ n+1 ͑ ͒ a Cxj + b Cj + c Cxj+1 + d Cj+1 = R 23 determining xD. The equation for tDЈ can be obtained by integrat- ͑ ͒ ing Eq. 8 over incremental time and space steps and solving the ␦ n+1 ␦ n+1 ␦ n+1 ᐉ␦ n+1 ͑ ͒ e Cxj + f Cj + g Cxj+1 + Cj+1 = M 24 resulting expression for tDЈ. vDЈ can be linearly interpolated over n n+1 a time step between vj and vj . This completes the calculation of where the advection stage for a single time step. This phase is followed ␦ n+1 n+1 a ␦ n+1 n+1 a by the solution of the diffusion equation, Eq. ͑6͒. Cj = Cj − Cj , Cxj = Cxj − Cxj and Advection in Tip-Cell During Advance In the current development, advection in the tip cell during ad- ␺͑hD ͒n+1 a = x j vance is calculated using the procedure described for the condi- ⌬x tion when ␣Ͼ1. Since the velocity over the advancing tip cell is assumed constant in a zero-inertia model ͑Zerihun et al. 2005a͒, ␤ the tip velocity can be considered the same as the velocity at the b = ⌬ penultimate node. This assumption can be used to determine the 2 t location of the departure point for the characteristic trajectory in the tip cell. c =−a

d = b Solution of Diffusion Step The equation that describes pure diffusion, Eq. ͑6͒, can be solved ␺ ͑ e =− effectively by a wide variety of numerical schemes Holly 1975; 2 Sauvaget 1985; Karpik and Crockett 1997; Komatsu et al. 1997͒. The Preismann implicit ﬁnite difference scheme can be used to ␺ discretize Eq. ͑6͒. The method has the advantage of being uncon- f =− ͑25͒ ditionally stable and can handle boundary conditions effectively. ⌬x Eq. ͑6͒ can be written as an equivalent pair of ﬁrst-order linear differential equations ͑Sauvaget 1985͒ g = e

␺ ץ Cץ h = ͑hD C ͒͑19͒ ᐉ = x x x ⌬xץ tץ

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 / 401 Fig. 3. Structure of coefﬁcient matrix and other details related to Eq. ͑26͒

␺ ͑1−␺͒ the incremental changes in the unknown at each computational ␤ = ͑hn+1 + hn+1͒ + ͑hn + hn͒ 2 j+1 j 2 j+1 j node ␦ n+1 n+1␦ n+1 n+1 ͑ ͒ Cj = Sj Cxj + Tj 27 ␺͑ ͒n+1 ͑ ␺͒͑ ͒n where S and T=coefficients that are calculated for each node as a hDx 1− hDx j ͑ a a ͒ j ͑ a a ͒ a R =− Cxj − Cxj+1 − Cxj − Cxj+1 C ⌬x ⌬x function of the elements of the coefficient matrix m, evaluated on the basis of the values of the variables at the end of a preceding time and advection step and S and T values calculated for the ͑Ca + Ca ͒ ͑Ca − Ca͒ node immediately upstream ͑Cunge et al. 1980͒. The double- M = xj+1 xj − j+1 j 2 ⌬x sweep procedure involves two sweeps through the solution domain: forward sweep starts from the upstream end using a known ͑ ͒ ͓ ͔n+1 At any given time step, applying the pair of linear algebraic boundary condition Fig. 1 and calculates the Sj ,Tj array as equations ͓Eqs. ͑23͒ and ͑24͔͒ to each computational cell and it sweeps through the solution domain to the downstream end. imposing pertinent boundary conditions ͑Fig. 1͒ yields a system The back sweep begins from the downstream end using a known of 2N linear equation with 2N unknowns, where N represents the downstream boundary condition and calculates the unknowns ͓␦ ␦ ͔n+1 ͓␦ ͔n+1 number of computational cells. During each time step, a system Cj , Cxj back to the upstream boundary. Once the Cj of linear equations of the following form is solved are determined, nodal concentrations are updated in accordance ͓ ͔n+1 ͓ ͔a ͓␦ ͔n+1 with the function Cj = Cj + Cj . Note that computation a ␦ n+1 n ͑ ͒ Cm x = F 26 of advection is highly sensitive to changes in Cx, hence the advective step in the next time step is to be based on Cx values a obtained in the advection calculation in the current time step. This where Cm =coefficient matrix whose elements are evaluated based on the values of the variables at the end of the advection phase completes the solution for the surface solute transport process in a ⌬ ␦ n+1 time step. during tn, x =vector of the incremental changes in the un- ͑␦ ␦ ͒ ⌬ a knowns Cj , Cxj in the diffusion phase during tn, and F =vector whose elements are the right hand side of the system of Calculation of Solute Mass in Infiltration linear equations, evaluated at the end of the advection phase dur- and Surface Runoff ⌬ ing tn. The structure of the coefficient matrix and other details related to Eq. ͑26͒ are provided in Fig. 3. Eq. ͑26͒ is solved using At any given time in the course of a fertigation event, the mass of ͑ ͒ n+1 ͑ ͒ the double-sweep algorithm Cunge et al. 1980 . The double- solute stored in the soil at a node, Msj g/m , can be calculated sweep algorithm assumes that a linear relationship exists between as follows:

402 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 Fig. 4. Four solute influx configuration ͑I=solute is applied throughout irrigation application time; II=solute application begins at start of irrigation, but concentration drops to zero before inflow is stopped; III=fertilizer application begins after start of irrigation and continues until inflow is stopped; and IV=fertilizer application begins some time after start of irrigation and solute concentration drops to zero before inflow is stopped͒

Cn+1 + Cn the application of fertilizer continues until inflow is cutoff ͑i.e., Mn+1 = Mn + ͑Zn+1 − Zn͒ͩ j j ͪ ͑28͒ ͒ sj sj j j 2 solute application Configurations I and III, Fig. 4 , concentration at the inlet ͑unlike solute flux͒ cannot be assumed zero immedi- where Z=cumulative infiltration at a node ͑m3 /m͒. At any given ately following inflow cutoff ͑Figs. 1 and 4͒. While the back- time during a fertigation event, the mass of solute loss due to ground concentration in the irrigation water can be used as an n+1 ͑ ͒ tailwater runoff, MsL g/m , can be given as initial condition for fertilizer application Configurations III and IV, in Configurations I and II, it can be set to zero ͑Fig. 1͒. However, at the conceptual level, at least, it is important to note qn+1Cn+1 + qnCn n+1 n ͩ L L L L ͪ⌬ ͑ ͒ that for solute influx Configurations I and II, assuming a zero MsL = MsL + tn 29 2 initial concentration at any given node is equivalent to stating that the node is covered with a thin film of pure water just ahead of The subscript L represents values of variables at the downstream end of a free-draining channel. the arrival of the advancing tip at the node. Finally, the advection step involves the solution of the auxiliary problem, Eq. ͑14͒, which requires the specification of the spatial derivatives of concentration at the upstream boundary during each time step and at

Numerical Solution of Subsurface Solute t=0. Cx at the upstream boundary is approximated through finite Transport Equation differencing using the Preissmann scheme ͑see Cunge et al. 1980͒. The initial condition for the auxiliary problem is set as The transport of solute in the crop root zone is simulated using C =0, Holly and Preissmann ͑1977͒ have shown that the effect of ͑ ͒ x HYDRUS-1D Simunek et al. 1998 . HYDRUS-1D has been such an approximation on solution accuracy is insignificant. widely used in applications ranging from water flow to solute and Numerical solution of the diffusion equation, for the surface heat transfer in the vadose zone ͑e.g., Scott et al. 2000; Simunek transport process, requires that boundary conditions be specified et al. 2000, Ventrella et al. 2000͒. Pertinent initial and boundary conditions are described in Fig. 1. The details of the numerical at both ends. Concentration at the upstream end and the spatial solution implemented in HYDRUS-1D can be obtained from derivative of concentration at the downstream end need to be ͑ ͒ Simunek et al. ͑1998͒. specified Fig. 1 . Known solute concentration distribution at the inlet end provides the upstream boundary condition and the spa-

tial derivative, Cx, at the downstream end is assumed to remain unchanged during the diffusion step. Initial and Boundary Conditions For the subsurface model, boundary conditions to the solute transport problem are in the form of solute flux at the top bound- Numerical solution of the advective step requires specification of ary and its spatial derivative at the bottom boundary ͑Fig. 1͒.At an upstream boundary condition and initial conditions. The boundary and initial conditions, for Eq. ͑5͒, are specified in terms the bottom boundary, free drainage is assumed, and the simulation ͑ of solute concentration. Measured solute breakthrough curves at is conducted such that the active zone of the subsurface i.e., the the basin/border inlet provide the upstream boundary condition region where significant water flow and solute transport occurs͒ is ͑Fig. 1͒. For a hypothetical simulation, however, the specification far from this boundary. The initial conditions can be constant of inlet concentration after inflow cutoff could be a challenge concentration or can vary with depth and distance along the depending on the fertilizer application configuration ͑Fig. 4͒.If length of border or basin ͑Fig. 1͒.

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 / 403 Fig. 5. Flow chart showing coupling of surface and subsurface modules

Coupling of Surface and Subsurface Components Summary and Conclusions of Solute Transport Model A coupled surface–subsurface solute transport model is developed Unlike the water flow model ͑Zerihun et al. 2005a͒, in the solute for use in surface fertigation management. The surface solute transport model the interaction between the surface and subsur- transport component of the coupled model is based on a numeri- face components is unidirectional ͑Fig. 5͒—i.e., data flows only cal solution of the area averaged advection–dispersion equation from the surface module to the subsurface module and no data are performed using the split-operator approach. The subsurface returned to the surface module. A driver program was written for transport process is simulated using HYDRUS-1D, which also use as an interface between the surface solute transport model and solves the 1D advection–dispersion equation. A driver program is HYDRUS-1D. In order to enhance computational efficiency the used for the internal coupling of the surface and subsurface trans- driver program was designed to bypass the user interface of port modules. A companion paper presents the results of model HYDRUS and access the executable file of its computational mod- evaluation and applications of the model in fertigation and salin- ule directly. A special version of HYDRUS, supplied by Dr. Jirka ity management. Simunek of the U.S. Soil Salinity Laboratory, was used for this purpose. The driver program interacts with HYDRUS-1D using a series of ASCII files stored in separate folders for each computational node. Acknowledgments At any given time step, the solution begins with the surface solute transport component passing an array of the nodal surface The writers are grateful to the USDA-NRI competitive grants solute concentration and infiltration rate values to the subsurface program for funding the research reported in this paper. solute transport module ͑Fig. 5͒. Changes in the subsurface solute concentration profile during a time step at all the computational nodes are calculated sequentially starting from the node at the inlet end of the border or basin. At each computational node, Notation HYDRUS solves the 1D advection–dispersion equation for the subsurface solute concentration profile at the end of a time step. The following symbols are used in the paper: Appropriate initial and boundary conditions are described above C ϭ concentration; and in Fig. 1. Note that at each node the vertical solute concen- Ca ϭ concentration at end of advective step; a ϭ tration profile at the end of the preceding time step is used as an Cm coefficient matrix whose elements are initial condition to the current time step. Following the calcula- evaluated based on values of variables at end ⌬ tion of the subsurface concentration profiles at all of the compu- of advection phase during tn; ϭ tational nodes, program control is transferred to the water flow Ci initial concentration; ͑ ͒ ϭ model Fig. 5 . This completes the hydraulic and solute transport Cx concentration derivative; ϭ simulation over a time step. The solution then proceeds to the Cz subsurface solute concentration; ϭ next time step, where first water flow is simulated followed by the Dx longitudinal dispersion coefficient; ϭ two-step surface solute transport computation, which in turn is Dz subsurface dispersion coefficient; followed by a simulation of the subsurface transport process. This Fa ϭ vector whose elements are right hand side of procedure is repeated until the entire fertigation process is simu- system of linear equations, evaluated at end of ⌬ lated. advection phase during tn;

404 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 h ϭ flow depth; Garcia-Navarro, P., Playan, E., and Zapata, N. ͑2000͒. “Solute transport i ϭ rate of infiltration; modeling in overland flow applied to fertigation.” J. Irrig. Drain. j ϭ distance index; Eng., 126͑1͒, 33–40. n ϭ Gardner, B. R., and Roth, R. L. ͑1984͒. “Applying nitrogen in irrigation Msj mass of solute deposited in soil at jth node and nth time line; waters.” Nitrogen in crop production, R. D. Hauck, ed., American n ϭ Society of Agronomy, Crop Science Society of America, Soil Science MsL mass of solute loss due to tail-water runoff at nth time line; Society of America, Madison, Wis., 493–505. Hoffmann, K. A., and Chiang, S. T. ͑1993͒. Computational fluid dynam- n ϭ time index; ϭ ics for engineers, Vol. 1, Engineering Education Systems, Witchita, q flow rate; Kan. ϭ qz local Darcian flow rate; Holly, F. M. ͑1975͒. “Two dimensional mass dispersion in rivers.” ϭ S,T coefficients related to double-sweep algorithm; Hydrologic papers, Colorado State University Press, Fort Collins, t ϭ time; Colo. v ϭ mean cross-sectional velocity; Holly, F. M. ͑1985͒. “Dispersion in rivers and coastal waters—1. Physical x ϭ horizontal distance; principles and dispersion equations.” Developments in hydraulic ϭ xD distance of departure point of characteristic engineering—3, P. Novak, ed., Elsevier, New York. trajectory from inlet end of channel; Holly, F. M., and Komatsu, T. ͑1983͒. “Derivative approximation in the Z ϭ cumulative infiltration; two-point fourth order method for pollutant transport.” Proc., ASCE z ϭ vertical dimension; 1983 Hydraulics Division Speciality Conf., Cambridge, Mass. ␣ ϭ ͑ ͑ ͒ x longitudinal dispersivity dimension dependent Holly, F. M., and Preissmann, A. 1977 . “Accurate calculation of on value of ␭͒; transport in two dimensions.” J. Hydraul. Div., Am. Soc. Civ. Eng., ⌬ ϭ 103͑11͒, 1259–1277. tn nth time step size; ͑␦ ␦ ͒n+1ϭ Islam, M. R., and Chaudhry, M. H. ͑1997͒. “Numerical solution of Cj , Cxj array of changes in nodal solute concentrations and concentration derivatives; transport equation for applications in environmental hydraulics and hydrology.” J. Hydrol., 191, 106–120. ␦xn+1 ϭ vector of incremental changes in unknowns Jaynes, D. B., Rice, R. C., and Hunsaker, D. J. ͑1992͒. “Solute transport ͑␦ ␦ ͒n+1 ⌬ Cj , Cxj in diffusion phase during tn; during chemigation of level basin.” Trans. ASAE,35͑6͒, 1809–1815. ␧ϭmolecular diffusion; Karpik, S. R., and Crockett, S. R. ͑1997͒. “Semi-Lagrangian algorithm ␪ϭvolumetric water content; for two-dimensional advection-diffusion equation on curvilinear ␭ϭdimensionless constant; coordinate meshes.” J. Hydraul. Eng., 123͑5͒, 389–401. ␾ϭspace weighing coefficient; and Komatsu, T., Ohgushi, K., and Asai, K. ͑1997͒. “Refined numerical ␺ϭtime weighing coefficient. scheme for advective transport in diffusion simulation.” J. Hydraul. Eng., 123͑1͒, 41–50. Leonard, B. P. ͑1991͒. “The ULTIMATE conservative difference scheme to unsteady one-dimensional advection.” Comput. Methods Appl. References Mech. Eng.,88͑1991͒, 17–74. Manson, J. R., Wallis, S. G., and Hope, D. ͑2001͒. “A conservative semi- Abassi, F., Simunek, J., van Genuchten, M. Th., Feyen, J., Adamsen, F. J., Lagrangian transport for rivers with transient storage zones.” Water Hunsaker, D. J., Strelkoff, T. S., and Shouse, P. ͑2003͒. Resour. Res.,37͑12͒, 3321–3329. “Overland flow and solute transport: Model development and Martin, B. ͑1975͒. “Numerical representations which model properties field-data analysis.” J. Irrig. Drain. Eng., 129͑2͒, 71–81. of the solution to the diffusion equation.” J. Comput. Phys., 17, Abbott, M. B., Andersen, J. K., Havno, K., Jemsem, K. H., Kroszynski, 358–383. U. I., and Warren, I. R. ͑1982͒. “Research and development for Morita, M., and Yen, B. C. ͑2002͒. “Modeling of conjunctive two- unsaturated component of the European hydrologic system—Systeme dimensional surface-three-dimensional subsurface flows.” J. Hydraul. hydrologique European ͑SHE͒.” Engineering applications of compu- Eng., 128͑2͒, 184–200. tational hydraulics, M. B. Abbott and J. A. Cunge, eds., Vol. 1, Pit- Playan, E., and Faci, J. M. ͑1997͒. “Border fertigation: field experiments man, London, 40–70. and a simple model.” Irrig. Sci., 17, 163–171. Abbott, M. B., Bathurst, J. C., Cunge, J. A., O’Connell, P. E., and Rutherford, J. C. ͑1994͒. River mixing, Wiley, New York. Rasmussen, J. ͑1986͒. “An introduction to European hydrological Sauvaget, P. ͑1985͒. “Dispersion in rivers and coastal waters—2. system—Systeme hydrologique European ͑SHE͒, SHE 2: Structure Numerical computation of dispersion.” Developments in hydraulic of a physically-based distributed modeling system.” J. Hydrol., 87, engineering—3, P. Novak, ed., Elsevier, New York. 61–77. Schoel, G. A., and Holly, F. M. ͑1991͒. “Cubic-spline interpolation Bathurst, J. C., Wicks, J. M., and O’Connell, P. E. ͑1995͒. “Chapter 16: in Lagrangian advection computation.” J. Hydraul. Eng.,117͑2͒, The SHE/SHESED basin scale water flow and sediment transport 248–253. modeling system.” Computer models of watershed hydrology, Scott, R. L., Shuttleworth, W. J., Keefer, T. O., and Warrick, A. W. V. P. Singh, ed., Water Resources, Highlands Ranch, Colo., 563–594. ͑2000͒. “Modeling multiyear observations of soil moisture recharge Bear, J. ͑1972͒. Dynamics of fluids in porous media. Elsevier, New York. in the semiarid American Southwest.” Water Resour. Res.,36͑8͒, Boldt, A. L., Watts, D. G., Eisenhauer, D. E., and Schepers, J. S. ͑1994͒. 2233–2247. “Simulation of water applied nitrogen distribution under surge Simunek, J., Hopmans, J. W., van Genuchten, M. Th., and Nielsen, D. R. irrigation.” Trans. ASAE,37͑4͒, 1157–1165. ͑2000͒. “Horizontal infiltration revisited using parameter estimation.” Cunge, J. A., Holly, F. M., and Verwey, A. ͑1980͒. Practical aspects of Soil Sci., 165͑9͒, 708–717. computational river hydraulics. Pitman, Marshfield, Mass. Simunek, J., Sejna, M., and van Genuchten, M. Th. ͑1998͒. The Elder, J. W. ͑1959͒. “The dispersion of marked fluid in turbulent shear HYDRUS-1D software package for simulating the movement of water, flow.” J. Fluid Mech., 5, 544–560. heat, and multiple solutes in variably saturated media, version 2.0, Fischer, H. B. ͑1967͒. “The mechanics of dispersion in natural streams.” United States Salinity Laboratory, USDA-ARS, Riverside, Calif. J. Hydraul. Div., Am. Soc. Civ. Eng.,93͑6͒, 187–216. Staniforth, A., and Cote, J. ͑1991͒. “Semi-Lagrangian integration Fischer, H. B. ͑1973͒. “Longitudinal dispersion and turbulent mixing in schemes for atmospheric models—a review.” Mon. Weather Rev., open channel flow.” Annu. Rev. Fluid Mech., 5, 59–78. 119, 2206–2223.

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / SEPTEMBER/OCTOBER 2005 / 405 Taylor, G. I. ͑1954͒. “The disperson of matter in turbulent flow through and simplified solutions.” Proc., 2nd Int. Conf. on Irrigation and a pipe.” Proc. R. Soc. London, Ser. A, 223, 446–48. Drainage, Water for a Sustainable World—Limited Supplies and Ex- Ventrella, D., Mohanty, B. P., Simunek, J., Losavio, N., and panding Demand, United States Committee on Irrigation and Drain- van Genuchten, M. Th. ͑2000͒. “Water and chloride transport age, Denver. in a fine-textured soil: Field experiments and modeling.” Soil Sci., Zerihun, D., Furman, A., Warrick, A. W., and Sanchez, C. A. ͑2005a͒. 165͑8͒, 624–631. “A coupled surface-subsurface flow model for improved basin Wallis, S. G., and Manson, J. R. ͑1997͒. “Accurate numerical simulation irrigation management.” J. Irrig. Drain. Eng., 131͑2͒, 111–128. of advection using large time steps.” Int. J. Numer. Methods Fluids, Zerihun, D., Sanchez, C. A., Furman, A., and Warrick, A. W. ͑2005b͒. 24, 127–139. “A coupled surface-subsurface solute transport model for irrigation Zerihun, D., Furman, A., Sanchez, C. A., and Warrick, W. A. ͑2003͒. borders and basins, II. Model evaluation.” J. Irrig. Drain. Eng., “Calculation of recession in basins and closed-end furrows: problems 131͑5͒, 407–419.