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ORIGINAL RESEARCH

Numerical Analysis of Solute Transport from Trickle Sources in a Combined Desert Soil–Imported Soil Flow System

^a Dep. of Environmental Physics and Irrigation, Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, The Volcani Center, Bet Dagan 50250, Israel
^b Present address: AMEC Earth & Environmental Ltd., 160 Traders Blvd. East, Suite 110, Mississauga, ON, Canada, L4Z 3K7
^* Corresponding author (vwrosd{at}agri.gov.il).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Received 14 March 2007.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theoretical Considerations Results and Discussion Conclusions REFERENCES

 
One of the alternative solutions to the problems associated with the agricultural use of stony desert soils is to apply imported soil material into trenches that are aligned along the crop rows. The purpose of the present study was twofold: first, to analyze the effect of two imported soil materials available in the Arava Valley of Israel on the movement and spread of water and solute originating from multiple trickle line laterals, taking into account the trench geometry, the discharge of the trickle line laterals, the pattern of the plant root distribution, and the spatial heterogeneity of the hydraulic properties of the local soil; and, second, to extend the analyses to hypothetical imported soil materials characterized by different sets of hydraulic parameters (saturated conductivity, K_s, and the van Genuchten soil parameters and n, related to the soil's pore size distribution). Results of the present study suggest that in the case of relatively low-conductive imported soil materials, a wide and shallow trench may leach the solute below a horizontal control plane more efficiently than a narrow and deep trench, while the opposite is true for high-conductive imported soil materials. Combinations of the threshold values of K_s and above which a narrow and deep trench is more efficient in solute leaching than a wide and shallow trench calculated here suggest that, for a given n, a relatively large threshold value of K_s may compensate for a relatively small threshold value of , particularly when n is relatively large.

Abbreviations: BTC, breakthrough curve • CDE, convection–dispersion equation • CP, control plane

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theoretical Considerations Results and Discussion Conclusions REFERENCES

 
Desert areas are often associated with climatic conditions that offer considerable advantages for agricultural production. In many cases, however, desert soils are very saline and often contain considerable amounts of gypsum and exchangeable Na and appreciable amounts of stones (Russo, 1983a,b, 1985). The latter poses a considerable difficulty for agricultural operations (e.g., preparation of the seedbed, seeding, and planting); about 50% of the cultivated area (2800 ha) in the central part of the Arava Valley of Israel, therefore, is based on coating of the local soils with a 0.4-m layer of imported sandy soils (6 x 10⁶ kg soil ha^–1). Because of the substantial cost of the coating procedure on the one hand, and the diminishing sources for these imported soils on the other hand, further development of agriculture in this region must rely on practices that utilize much smaller amounts of the imported soil material. Because most of the cash crops (vegetables) in the central part of the Arava Valley are grown in rows and are irrigated by surface drip irrigation systems, one of the alternative practical solutions to the aforementioned soil problem is to apply imported soil material into trenches aligned along the crop rows. This may considerably (by 80%) reduce the amount of imported soil material required for agricultural production.

For a local soil with a given spatial distribution of its hydraulic properties and for given characteristics of the crop, weather, and the drip irrigation system, the hydraulic properties of the imported soil material as well as the geometry of the trench might affect water and solute movement and spread in the root zone. Knowledge of the latter is essential for the design, management, and scheduling of surface drip irrigation systems, with the aim of maximizing the crop yield. Under the conditions of desert areas such as the Arava Valley of Israel (saline water, high soil salinity, high evaporation rate, and insignificant annual precipitation), to a large extent, the achievement of the latter goal is determined by the capability to control the salinity in the root zone during the growing season.

Our interest lies in solute transport occurring at the field scale, which, in turn, is much larger than the laboratory (core) scale. One of the distinctive features of porous formations at the field scale is the spatial heterogeneity of their properties that affect flow and transport (e.g., Nielsen et al., 1973; Russo and Bresler, 1981; Jones and Wagenet, 1984; Russo and Bouton, 1992; Russo et al., 1997). This spatial heterogeneity is generally irregular and occurs on a scale that is not captured by laboratory samples; it has a distinct effect on water flow and solute transport, as has been observed in field experiments (e.g., Schulin et al., 1987; Butters et al., 1989; Ellsworth et al., 1991; Flury et al., 1994; Forrer et al., 1999), and demonstrated by simulation (e.g., Russo, 1991; Russo et al., 1994, 1998, 2001, 2006; Tseng and Jury, 1994; Harter and Yeh, 1996; Roth and Hammel, 1996; Foussereau et al., 2001). The spatial heterogeneity in the soil hydraulic properties, therefore, must be considered in any attempt to quantify solute transport occurring at the field scale.

The general objective of the present study was twofold: first, to investigate the effect of two imported soil materials available in the central part of the Arava Valley on the movement and spread of water and solute originating from multiple trickle line laterals, taking into account the geometry of the trenches, the discharge of the trickle line laterals per unit length of the ponding strip (i.e., the source strength), the pattern of the plant root distribution, and the spatial heterogeneity of the hydraulic properties of the local soil; and second, to extend the analyses to hypothetical imported soil materials characterized by different sets of the relevant hydraulic parameters.

To pursue these objectives, we analyzed the flow and the transport problem by means of physically based flow and transport models and a stochastic presentation of the spatially heterogeneous hydraulic properties of the local soil. Generally, the analysis should take into account interactive ions pertinent to irrigation with mixed salts solutions (e.g., Russo, 1986, 1988; Russo et al., 2004). In this study, however, to simplify matters, we concentrated on the transport of a single passive solute (Cl^–), which, in turn, is the dominant ion in the irrigation water. Furthermore, we would like to emphasize that the approach undertaken is a "single realization" approach (e.g., Ababou, 1988; Russo, 1991; Polmann et al., 1991; Russo et al., 1994, 1998, 2005; Tseng and Jury, 1994) that may be used to quantify the macroscopic spreading of a single solute mass from particular site-specific applications, particularly when point measurements are available for conditioning.

This study was a numerical experiment that may provide detailed information on the consequences of characteristics of soil, crop, weather, and the irrigation system for the movement and spreading of water and solute under realistic conditions. At the price of reduced generality, this approach circumvents most of the stringent assumptions of analytical studies and facilitates analysis of simplified yet realistic situations at a fraction of the cost of physical experiments. We concentrated on a specific case of drip irrigation of vegetables in the central part of the Arava Valley of Israel. We believe, however, that the scope of the study is of broad interest and has implications for similar situations involving agricultural production in desert areas. The significant contribution of the this study stems from the application of state-of-the-art quantitative tools to a practical problem that, to the best of our knowledge, has not been treated quantitatively in the past.

	Theoretical Considerations

TOP ABSTRACT INTRODUCTION Theoretical Considerations Results and Discussion Conclusions REFERENCES

 
The Physical Domain and Simulated Scenarios
Using a Cartesian coordinate system (x₁, x₂, x₃), where x₁ is directed vertically downward, a vertical slice of the actual three-dimensional flow domain that extends across 2 and 6 m along the x₁ and x₂ axes, respectively, was considered. The upper boundary of the two-dimensional vertical modeling domain corresponds to an irrigated field that consists of N_r = 4 beds of bell pepper (Capsicum frutescens L. cv. Maor) aligned parallel to the x₃ coordinate. Each bed includes a trench aligned along the axis of the bed and filled with a given imported soil material. It is assumed here that the boundary z(x₂) between the local soil and the imported soil, associated with the kth bed, is described by the parabola

[1]

where x_k2 is the coordinate location of the center of the kth bed, z_m is the maximum depth of the trench (i.e., the vertical distance to its bottom at x₂ = x_k2), x_m is the lateral extent of the trench at the soil surface (i.e., at x₁ = 0), and k = 1 to N_r.

Each bed consists of double plant rows spaced 0.2 m apart and the distance between the centers of the beds is d = 1.5 m. Each plant row is irrigated with a single drip line lateral whose emitters are spaced very close to each other such that their ponding areas overlapp very rapidly; consequently, one can assume that the total ponding area of the emitters along the drip line lateral has the form of an infinite strip of width ₂(t), which, in turn, is oriented parallel to the x₃ axis. A conceptual scheme of the modeling domain and the related boundary conditions, which are addressed below, is given in Fig. 1 .

View larger version (17K): [in this window] [in a new window]   FIG. 1. A conceptual scheme of the vertical two-dimensional modeling domain. Open circles indicate positions of the drip line laterals (and the crop rows) at the soil surface, while the short thick lines below them schematically represent the ponded strips. Thick solid curves indicate the boundaries between the local soil and the imported soil (Eq. [1]), while the vertical dashed lines represent the averaging domain (Eq. [10] and [11]) in the vicinity of the two inner beds. Vertical exaggeration 2x.

[2]

where t is time; = (x,t) is the pressure head, = (x,t) is the volumetric water content; K_ii, with i = 1,2, are the principal components of the hydraulic conductivity tensor, taken as a symmetrical tensor of Rank 2 with zero off-diagonal components; and S_w = S_w(x,t) is a sink term representing water extraction by plant roots.

Similarly, the CDE that governs two-dimensional transport of a passive solute in the vertical x₁x₂ plane is

[3]

where c = c(x,t) is the resident solute concentration, expressed as mass per unit volume of the soil solution; u_i (i = 1,2) are the components of the Eulerian velocity vector, and D_ij (i,j = 1,2) are the components of the pore-scale dispersion tensor, given (Bear, 1972) as

[4a]

where _L and _T are the longitudinal and the transverse pore-scale dispersivities, _ij is the Kronecker delta, |u| = (u₁² + u₂²)^1/2, and D_m is the effective molecular diffusion coefficient, given (Jury et al., 1983) as

[4b]

where D_lw is the liquid diffusion coefficient in water, and _s is the saturated volumetric water content.

Water is infiltrated into the soil from each of the drip line laterals through the ponding strips that develop along the drip line laterals by an imposed time-dependent discharge, uniform within the ponding strips, [F(t) = Q for t_j' < t < t_j'', F(t) = 0 elsewhere], where Q is the source strength, i.e., the emitter discharge per unit length of the ponding strip, t^I = t_j'' – t_j' is the duration of an irrigation event, tⁱ = t_j' – t_j–1'' is the time interval between successive events, j = 1 to N_I, N_I is the number of events, and t^I + tⁱ is the duration of an irrigation cycle. Water may leave the flow system by extraction by plant roots and by evaporation and drainage through the upper and lower horizontal faces of the flow domain; no-flow conditions are assumed here for the vertical faces of the flow domain. In each irrigation, Cl^– with concentration C₀ invades the soil via the ponding strips along the drip line laterals; there is no solute transport across the upper horizontal face of the flow domain outside the ponding strips. In addition, a zero-solute flux boundary is assumed here for the vertical faces of the flow domain, while a zero-gradient boundary is assumed for the lower horizontal face of the flow domain.

Boundary and Initial Conditions
For the flow, the boundary and initial conditions for the N-drip-line lateral system (N = 2N_r) are

[5a]

[5b]

[5c]

[5d]

[5e]

where E is the evaporation rate; G_k, k = 1 to N, is given by G_k = _k2, and _k2 = _k22 – _k21 is the time-dependent width of the ponded strip that develops along the kth drip line lateral, which, in turn, is oriented along the x₃ axis.

For the transport, the boundary and initial conditions for the Cl^– resident concentration, c, for the N-drip-line lateral system are

[6a]

[6b]

[6c]

[6d]

[6e]

Characterization of Water Extraction by Plant Roots
It is assumed here that, locally, the rate of water extraction by plant roots is proportional to the unsaturated conductivity and to the difference between the total pressure head at the root–soil interface, _r, and the reduced water pressure head of the soil, + , where is the osmotic pressure head of the soil solution. According to this approach (e.g., Nimah and Hanks, 1973; Feddes et al., 1974; Bresler, 1987), the sink term S_w in Eq. [2] is

[7a]

where R_e(x,t) is the root effectiveness function, given by R_e(x,t) = R_ek(x) for x₂ within the range b_k (defined below), and R_e(x,t) = 0 elsewhere, R_ek is the root effectiveness function associated with the kth plant row, given by the Gaussian bivariate semilognormal spatial distribution (Coelho and Or, 1996) as

[7b]

where r_k = |x_k2 – x₂|, x_k2 is the coordinate location of the kth plant row at the soil surface (x₁ = 0); µ_h, µ_v, and _h² and _v² are the mean values and the variances, respectively, of R_ek(x) in the horizontal direction and the log-transformed vertical direction, respectively; β is a scaling parameter; and b_k = x_k2 ± (3_h²)^1/2 is the so-called range of R_ek (along the x₂ axis), which defines the lateral extent of the influence of the roots of the kth plant row. To simplify matters, the growth stage of the plants is ignored in this study, and R_ek is considered as time invariant.

Characterization of the Flow and the Transport Parameters
It is assumed here that the local K() and () relationships for both the local and the imported soils are described by the van Genuchten (1980) parametric expressions. Ignoring local hysteresis and local anisotropy, considering the pressure head, , as the dependent variable, they read:

[8a]

[8b]

where = ( – _r)/( _s – _r) is the effective water saturation; _s and _r are the saturated and residual volumetric water contents, respectively; K_s is the saturated conductivity; and m are parameters related to the soil pore size distribution, and n = 1/(1 – m).

It is assumed further that for the imported soil materials, each parameter in Eq. [8] denoted by p(x), is a deterministic constant, independent of x. On the other hand, for the local soil, each parameter in Eq. [8] is a second-order stationary, statistically anisotropic, random space function characterized completely by a constant mean, p(x), independent of the spatial position, and a covariance function, C_pp(x,x'), that, in turn, depends on the separation vector, = x – x', and not on x and x' individually. For the local soil, a three-dimensional, exponential covariance is adopted here for p(x), i.e.,

[9]

where ' = (₁/I_p1, ₂/I_p2,₃/I_p3) is the scaled separation vector, ' = |'|; _p² and I_p = (I_p1,I_p2,I_p3) are the variance and correlation scales of p(x), respectively. Axisymmetric anisotropy is adopted for C_pp(), i.e., I_pv = I_p1 and I_ph = I_p2 = I_p3 are the characteristic length scales of p(x) in the vertical (longitudinal) direction and the horizontal plane, respectively. Regarding the local soil, for the various soil parameters of Eq. [8], mean values, µ_p, and variances, _p², were adopted from Russo (1986) based on measured hydraulic properties of Zofar soil (Russo, 1983a,c); their correlation length scales were assumed as equal to those of logK_s, adopted from Russo et al. (1997), i.e., I_ph = I_h = 0.8 m and I_pv = I_v = 0.2 m. Two imported soil materials available in the Arava Valley were considered in this study. The first one is a sandy soil, which traditionally has been used for coating the local soils; in this case, the soil parameters of Eq. [8] for the Arava II sandy soil were adopted from Shani and Or (1995). The second soil material is tuff (Scoria, granulated volcanic ash), which traditionally has been used as a container medium in greenhouses; in this case, the soil parameters of Eq. [8] for a red tuff were adopted from Wallach et al. (1992). In addition, hypothetical imported soil materials characterized by different combinations of the parameters of Eq. [8] (i.e., K_s = 3.6, 7.2, 15.5, 24.2, and 33 m d^–1; = 3.2, 6.4, 12.8, 19.2, and 25.6 m^–1; and n = 1.8, 3.6, and 7.2) were considered here. Values of _L = 2 x 10^–3 m and _T = 1 x 10^–4 m (Perkins and Johnston, 1963) were adopted for the local soil and for the actual and hypothetical imported soil materials.

Generation of the Flow and Transport Parameters Field
In the present study, we considered a vertical slice of the three-dimensional, spatially heterogeneous local soil, and thus we generated a hypothetical two-dimensional flow domain in the vertical x₁x₂ plane. The flow domain, which extends over 2 and 6 m along the x₁ and x₂ axes, respectively, was discretized into uniform numerical cells, , measuring l₁ = 0.01 m and l₂ = 0.02 m in the vertical and horizontal directions, respectively. Using the mean values, µ_p, the variances, _p² (Table 1 ), the aforementioned values of the correlation length scales, I_pv and I_ph, and the covariance Eq. [9], the turning bands method (Tompson et al., 1989), in combination with the multivariate normal distribution function method (Mood and Graybill, 1963) and a bilinear interpolation scheme were used to generate cross-correlated realizations of logK_s, log, n, _s, and _r for each of the 200 x 300 numerical cells of the flow domain, taking into account the linear cross-correlation coefficients between the various soil properties (Table 2 ).

View this table: [in this window] [in a new window]   TABLE 1. Hydraulic parameters of the van Genucthen model (Eq. [8]), hydraulic conductivity (Ks), parameters related to soil pore size distribution [, m, and n = 1/(1 – m)], and saturated and residual volumetric water content (s and r), for the local desert soil and two different imported soil materials.

View this table: [in this window] [in a new window]   TABLE 2. Estimates of the linear cross-correlation coefficients between the various local soil parameters.

For a given discretization of the flow domain, using Eq. [8], two-dimensional tables of relative conductivity, K_ij^r = K/K_s, and relative water content, _ij^r = ( – _r)/(_s – _r) were constructed as functions of _i' = (i – 1) and n_j = 1.1 + (j – 1)_n, where ' = ||, i = 1 to n, j = 1 to n_n, n = 50,000, n_n = 80, = 0.01, and _n = 0.05. Using a bilinear interpolation scheme, these tables were used to calculate values of K^r and = ^r(_s – _r) + _r for each cell of the flow domain by means of the generated realizations of , n, _s, and _r and the simulated pressure head, . Assuming local isotropy, the principal components of the hydraulic conductivity tensor in Eq. [2] are given by K_ii = K_s(x)K^r(,x), i = 1,2. Hydraulic conductivity between cells, the so-called interblock conductivity, was estimated from the generated realization of K_s(x) and the calculated K^r(,x), using the asymptotic weighting scheme proposed by Zaidel and Russo (1992).

Implementation
The three-dimensional numerical code of Russo et al. (1998) was adopted here to solve Eq. [2] and [3], subject to Eq. [5] and Eq. [6], respectively. In this code (with the x₃ coordinate being suppressed), the "mixed" form of Richards' equation governing two-dimensional water flow subject to the appropriate boundary and initial conditions was approximated by a fully implicit Euler scheme with a truncation error of O(t,x₁²,x₂²). The resulting system of nonlinear algebraic equations with respect to the pressure head, , was solved iteratively by applying the so-called modified Picard method (Celia et al., 1990). Picard (external) iterations were applied for both capillary and gravity terms and the resulting system of linear algebraic equations was solved by the polynomial preconditioned conjugate gradient method (e.g., Hageman and Young, 1981). The scheme is convergent and unconditionally stable for the linear diffusion equation, is suitable for flow problems in which the main discretization errors are spatial, and does not degenerate for saturated flow conditions. This scheme, therefore, is appropriate for flow problems in spatially heterogeneous porous media, in which both steep gradients and saturated regions can be developed.

Inasmuch as the width of the ponded strips along the drip line laterals (Eq. [5a]) is not known a priori, the numerical code was combined with an iterative procedure. In this procedure, for each time step during an irrigation event (i.e., when F(t) = Q), assuming zero depth for the ponded strips, their width that allows the fulfillment of the equality in Eq. [5a] was determined by appropriate adjustments. The iterative procedure continues until a prescribed error tolerance is met or until the number of iterations has exceeded a predetermined limit.

The three-dimensional numerical code of Russo et al. (1998) was adopted here to solve the CDE (Eq. [3]) governing solute transport in the two-dimensional flow domain. Suppressing the x₃ coordinate, the CDE was approximated by a numerical scheme, similar to the third-order total-variation-diminishing (TVD) scheme implemented in the MT3D code (Zheng and Wang, 1999), which, in turn, is based on the ULTIMATE algorithm proposed by Leonard (1979, 1988). Both the TVD scheme and our scheme use a high-order flexible (i.e., adjusting to the concentration gradients) upstream approximation of the interface concentrations involved in the computation of the advective fluxes. Both schemes are shown to virtually eliminate oscillations while minimizing numerical dispersion for advection-dominated problems.

Parameters of the time-invariant root effectiveness function (Eq. [7b]) pertinent to bell pepper were estimated based on measured root distribution data (S. Kramer, personal communication, 2005), i.e., b = 1, m_h = 0, m_v = –1.75, _h² = 0.04 m², and _v² = 0.16. Water extraction by plant roots was implemented by a maximization iterative approach (Neuman et al., 1975). In this approach, the rate of transpiration per unit area of the soil surface at time t is maximized subject to two requirements: (i) the actual rate of transpiration is not allowed to exceed the potential rate of transpiration, T_p; and (ii) the total pressure head at the root–soil interface, _r, is not allowed to fall below a critical value, _c, equivalent to the so-called wilting point of the soil–plant system (e.g., _c = –150 m). Class A pan evaporation data from the field site at Zofar (Russo, 1983b) were used to estimate potential evapotranspiration rates, ET_p(t). To simplify the problem, potential soil evaporation, E_p(t), was set to E_p(t) = ET_p(t)/10 (ET_p = T_p + E_p).

For each of the actual and hypothetical imported soil materials, flow and transport simulations were performed for the aforementioned two different geometries of the trenches. In each simulation, the initial pressure head, _i, was selected as the pressure head corresponding to K(,x)/ K_s(x) = 0.001. In each flow and transport simulation, four crop beds spaced 1.5 m apart, each consisting of double plant rows spaced 0.2 m apart with one drip line lateral per row, were taken into account. The benchmark value for the source strength was taken as Q = 1500 cm² h^–1, equivalent to 6.25 mm h^–1. Each simulation proceeded until half of the total mass of solute applied into the flow system had been leached below a horizontal control plane (CP) located at a soil depth of 0.75 m. Daily irrigations (with amounts equal to the estimated daily amount of evapotranspiration) and with Cl^– concentration C₀ = 12.5 mol_c m^–3 were considered. The total estimated amount of evapotranspiration for the maximum simulation period (68 d) was 375 mm, while precipitation was essentially null.

	Results and Discussion

TOP ABSTRACT INTRODUCTION Theoretical Considerations Results and Discussion Conclusions REFERENCES

 
Hydraulic Properties of the Imported Soils
Hydraulic conductivity, K, and water content, , as functions of pressure head, , associated with the two imported soil materials and the local soil, are depicted in Fig. 2 . It is clearly demonstrated in this figure that the decrease in both and K with increasing is faster in the imported soils (particularly in the tuff soil) than in the local soil (i.e., both dK/d and d/d increase as both n and increase). Note that at saturation (i.e., when = 0), the values of both K and associated with the imported soil materials are larger than their counterparts associated with the local soil. Since the former decrease with increasing faster than the local soil, above a critical pressure head, = _c (i.e., the pressure head at which the values of K associated with the imported soil materials are equal to the mean value of K associated with the local soil), however, they are smaller than their counterparts associated with the local soil. Analysis of Eq. [8] using the aforementioned (K_s, , n) parameter sets associated with the hypothetical imported soil materials suggests that the critical pressure head, _c, mainly depends on , and to a lesser extent on n and K_s; it increases with decreasing and n and with increasing K_s. Consequently, the value of _c associated with the tuff soil is smaller than its counterpart associated with the sandy soil (Fig. 2).

View larger version (14K): [in this window] [in a new window]   FIG. 2. Average constitutive relations for the local and the two imported soil materials: (a) logarithmic conductivity and (b) water content as functions of pressure head.

We refer to the local soil first. Figure 3 displays contour lines of the simulated pressure head and the Cl^– concentration distributions in the upper 0.6 and 2 m, respectively, of the soil profile in the vicinity of the two inner beds (see Fig. 1) at t = 30 d, just after the cessation of an irrigation event, for source strength Q = 1500 cm² h^–1. Note that the wriggles in these figures are a consequence of the local variations in the hydraulic parameters of the local soil and not of the grid discretization. Figure 3 (top) shows that because of the relatively low values of both K_s and and the resultant appreciable lateral pressure head gradients, the lateral extent of the ponded area in the vicinity of the double drip line laterals is relatively large, the interaction between the double drip line laterals is significant, and the flow occurs along both horizontal and vertical axes. This is also apparent in the restricted solute displacement and the appreciable lateral solute spread depicted in Fig. 3 (bottom).

View larger version (76K): [in this window] [in a new window]   FIG. 3. Contours of the simulated pressure head, (top, in meters) and the Cl– concentration, c (bottom, in molc m–3) distributions in the vertical x1x2 plane of the local soil in the vicinity of the two inner beds. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength, Q = 1500 cm2 h–1. Vertical exaggeration 0.7x and 0.3x for and c, respectively.

Simulated pressure head distributions associated with the two imported soil materials in the vicinity of the two inner beds are illustrated in Fig. 4 and 5 . These figures demonstrate the effect of both the imported soil material and the geometry of the trench on the pattern of the pressure head distribution in the vicinity of the double drip line laterals. The texture of the imported soil materials (particularly that of the tuff soil, Fig. 5) is coarser than that of the local soil. Consequently, in the combined local soil–imported soil flow system, gravity has a stronger effect; the lateral extent of the ponded area in the vicinity of the double drip line laterals is relatively small, the interaction between the double drip line laterals is relatively small, and the flow is more aligned along the vertical axis. Both water flow (Fig. 5) and solute migration (Fig. 5 and 6 ) are prevalently vertical, particularly if the imported soil is tuff and the trench is narrow and deep (Fig. 5, top, and Fig. 7 , top, respectively).

View larger version (55K): [in this window] [in a new window]   FIG. 4. Contours of the simulated pressure head (in meters) distribution in the vertical x1x2 plane of the soil in the vicinity of the two inner beds, for the imported sandy soil material applied into narrow and deep (top) and wide and shallow (bottom) trenches. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength, Q = 1500 cm2 h–1. Vertical exaggeration 0.7x.

View larger version (92K): [in this window] [in a new window]   FIG. 5. Contours of the simulated pressure head (in meters) distribution in the vertical x1x2 plane of the soil in the vicinity of the two inner beds for the imported tuff soil material applied into narrow and deep (top) and wide and shallow (bottom) trenches. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength, Q = 1500 cm2 h–1. Vertical exaggeration 0.7x.

View larger version (64K): [in this window] [in a new window]   FIG. 6. Contours of the Cl– concentration (in molc m–3) distribution in the vertical x1x2 plane of the soil in the vicinity of the inner beds for the imported sandy soil material applied into narrow and deep (top) and wide and shallow (bottom) trenches. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength Q = 1500 cm2 h–1. Vertical exaggeration 0.3x.

View larger version (66K): [in this window] [in a new window]   FIG. 7. Contours of the Cl– concentration (in molc m–3) distribution in the vertical x1x2 plane of the soil in the vicinity of the two inner beds for the imported tuff soil material applied into narrow and deep (top) and wide and shallow (bottom) trenches. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength Q = 1500 cm2 h–1. Vertical exaggeration 0.3x.

View larger version (19K): [in this window] [in a new window]   FIG. 8. Mean profiles of the (a) vertical and the (b) horizontal components of the velocity vector, horizontally averaged across a single drip line lateral in the left-hand-side inner bed for the local soil and the two imported soil materials applied into narrow and deep (N) and wide and shallow (W) trenches. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength Q = 1500 cm2 h–1.

Solute Concentration Profiles
Mean concentration profiles, c(x₁,t), obtained by averaging c(x,t) horizontally across the two inner beds, are depicted in Fig. 9 (at t = 30 d, just after the cessation of an irrigation event) for the local soil and the two imported soil materials. The results depicted in Fig. 9 suggest that, generally, the mean concentration profiles are asymmetric and may exhibit substantial values in the upper part of the soil due to the accumulation of solute between adjacent drip line laterals (Fig. 3 [bottom], 6, and 7). Furthermore, the results depicted in Fig. 9 suggest that, compared with the local soil, the imported soil materials, particularly the tuff soil material, decrease the secondary concentration peak at the soil surface, increase the spread of c(x₁,t) in the vertical direction, expedite its dilution, and increase the skewness of the mean concentration profiles, particularly when the trench is narrow and deep (Fig. 7 [top]).

View larger version (16K): [in this window] [in a new window]   FIG. 9. Mean concentration profiles, obtained by averaging the Cl– concentration c(x,t) horizontally across the two inner beds for the local soil and the two imported soil materials applied into (a) narrow and deep and (b) wide and shallow trenches. Results are depicted for t = 30 d, just after the cessation of an irrigation event, for a source strength Q = 1500 cm2 h–1; zm and xm are the trench depth and width, respectively.

Solute Displacement and Spread
Solute spread may be quantified in terms of integrated measures of solute transport, such as the moments of the spatial distribution of the solute concentration, c(x,t), at a given time. After Aris (1956), the time-dependent first two normalized moments of the spatial distribution of the concentration point values, c(x,t), for the two inner beds are given by

[10a]

[10b]

where

[10c]

is the dissolved mass of the solute, i = 1,2; x₂₂' = x₂₂ – d/2; x₃₂' = x₃₂ + d/2, x₂₂ and x₃₂ are the positions of the centers of the second and the third (inner) beds along the x₂ axis, respectively; d is the bed spacing; x₂₃' – x₂₂' is the horizontal extent of the averaging domain (bounded by the vertical dashed lines in Fig. 1); R_ci (i = 1,2) are the components of the centroid vector of the solute body; and S_ij'(t) (i,j = 1,2) are second spatial moments, proportional to the moments of inertia of the solute body. In other words, Eq. [10c], [10a], and [10b] provide measures of the mass, location, and spread of the solute body, respectively.

Compared with the imported sandy soil material, the imported tuff soil material enhances solute movement and spread in the longitudinal direction (expressed in terms of the longitudinal component of Eq. [10a], R_c1, and the longitudinal component of Eq. [10b], S₁₁(t) = S₁₁'(t) – S₁₁'(0), respectively). Note that in the case of the tuff soil, both R_c1 (not shown here) and S₁₁ (Fig. 10a and 10b ) are essentially independent of the trench geometry, while in the case of the sandy soil, a wide and shallow trench may considerably enhance solute movement (not shown here) and spread (Fig. 10a and 10b) in the longitudinal direction. As for the solute spread in the transverse direction (expressed in terms of the transverse component of Eq. [10b], S₂₂(t) = S₂₂'(t) – S₂₂'(0), [insets of Fig. 10a and 10b]), when the trench is narrow and deep (inset of Fig. 10a), solute spread in the transverse direction is slightly larger in the tuff soil than in the sandy soil. Note that in the case of the sandy soil, S₂₂ is essentially independent of the trench geometry, while in the case of the tuff soil, a narrow and deep trench may enhance solute spread in the transverse direction (compare the inset of Fig. 10a with the inset of Fig. 10b).

View larger version (21K): [in this window] [in a new window]   FIG. 10. Longitudinal and transverse (insets) components of the displacement covariance function (Eq. [10b]) associated with the two inner drip line laterals, as functions of time, for the local soil and the two imported soil materials applied into (a) narrow and deep and (b) wide and shallow trenches for a source strength Q = 1500 cm2 h–1.

Solute Leaching
With respect to suitability of an imported soil material to leach solutes, a relevant attribute is the solute breakthrough curve (BTC) monitored at a horizontal CP at an arbitrary vertical distance from the soil surface, L. The solute BTC is calculated from the flux-averaged concentration, c_f, defined (Kreft and Zuber, 1978) by c_f = (s)da/(u)da, where s and u are the solute flux and velocity vectors, respectively, with s = s and u = u being the mass of solute and the volume of water per unit time and unit area, respectively, moving through a surface element of unit normal and water content , and the integration is across a planar area a.

A solute BTC monitored at a given horizontal CP is evaluated here by averaging the longitudinal components of the water flux and the solute flux vectors at x₁ = L over the two inner beds, i.e.,

[11]

where c_f = s₁/q₁, and q₁ and s₁ are the longitudinal (vertical) components of the water flux and solute flux vectors, respectively.

Another entity of interest is the accumulated mass of the solute that has crossed the CP from the beginning of the transport process to time t, M_L, obtained by integrating Eq. [11] with time, i.e.,

[12]

The solute BTC monitored at a given horizontal CP below the main root zone (L = 0.75 m) and the normalized accumulated mass of the solute that crossed the CP, M_L/M₀, where M₀ is the total mass of solute injected into the flow domain, are depicted in the insets of Fig. 11 as functions of time for the local soil and the two imported soil materials. The results presented in Fig. 11 suggest that the imported tuff soil material enhances the solute BTC and, consequently, leaches the solute below the horizontal CP more efficiently, particularly when the trench is deep and narrow. In the case of the imported sandy soil material, however, a wide and shallow trench is shown to enhance the solute BTC and, consequently, to leach the solute below the horizontal CP more efficiently than a narrow and deep trench.

View larger version (26K): [in this window] [in a new window]   FIG. 11. The solute breakthrough curve monitored at a given horizontal control plane (CP) at L = 0.75 m and (insets) the normalized accumulated mass of the solute that crossed the CP, as functions of time, for the local soil and the two imported soil materials applied into (a) narrow and deep and (b) wide and shallow trenches for a source strength Q = 1500 cm2 h–1.

The efficiency of the different imported soil material to leach solutes is quantified here by analyzing the amount of applied water, W, required to leach half of the total mass of solute applied into the flow system, i.e., W_0.5 = W(M_L/M₀ = 0.5). For the narrow and deep trench and for the wide and shallow trench, respectively, results of the simulations yield W_0.5 = 329 and 284 mm for the sandy soil and W_0.5 = 240 and 258 mm for the tuff soil (compared with W_0.5 = 355 mm for the local soil). Results of the simulations for the different hypothetical imported soil materials associated with different (K_s,,n) sets, i.e., W_0.5(K_s,,n), are displayed by the contour lines in Fig. 12 .

View larger version (43K): [in this window] [in a new window]   FIG. 12. Contours of the amount of applied water (in meters) required to leach half of the total mass of solute applied into the flow system, as functions of the soil parameters saturated hydraulic conductivity (Ks) and the van Genuchten shape parameter characterizing different hypothetical imported soil materials applied into (a) narrow and deep and (b) wide and shallow trenches. Results are depicted for n = 3.6 for a source strength Q = 1500 cm2 h–1.

View larger version (14K): [in this window] [in a new window]   FIG. 13. Combinations of threshold values of saturated hydraulic conductivity (Ks) and the van Genuchten shape parameter above which a narrow and deep trench is more efficient in solute leaching than a wide and shallow trench for selected values of the van Genuchten shape parameter n, for a source strength Q = 1500 cm2 h–1.

Results of the analyses for additional source strengths, Q, suggest that for the practical range of values of Q = 750–3000 cm² h^–1 (equivalent to irrigation rates varying from 3–12 mm h^–1), in the case of the imported tuff soil material the solute leaching efficiency is essentially independent of Q; in the case of the imported sand soil material, however, the solute leaching efficiency slightly increases with increasing Q, particularly when the trench is wide and shallow. For example, in the case of the narrow and deep trench (x_m = 0.4 m and z_m = 0.5 m), W_0.5 = 336, 329, and 312 mm for Q = 750, 1500, and 3000 cm² h^–1, respectively; in the case of the wide and shallow trench (x_m = 0.67 m and z_m = 0.3 m), W_0.5 = 314, 284, and 274 mm for Q = 750, 1500, and 3000 cm² h^–1, respectively. The latter stems from the fact that, in the case of the sand soil material, increasing Q is shown to increase the both u₁ and u₂ at relatively large soil depth, particularly when the trench is wide and shallow.

We also analyzed the effect of the pattern of the root distribution on the solute leaching efficiency. The additional root distributions used in the analyses are characterized by different combinations of the parameters of Eq. [7b] within the range m_v = –2 to –1.4, _v² = 0.09 to 0.25 and _h² = 0.01 to 0.09 m²; this parameter range represents root distribution functions whose centroids are located at soil depths within the range 0.19 to 0.27 m, while the spreads about the centroids are within the range ± 0.08 to 0.10 m in the vertical direction and ± 0.10 to 0.30 m in the horizontal direction. Results of the analyses suggest that, in the case of the imported sand soil material, for both trench configurations the solute leaching efficiency is essentially independent the of the extension of the roots in the vertical direction and increases as the extension of the roots in the horizontal direction decreases, particularly when the trench is deep and narrow. On the other hand, in the case of the imported tuff soil material, shallower and narrower root distributions are shown to increase the solute leaching efficiency, particularly when the trench is wide and shallow.

We would like to emphasize that the result that for high-conductive imported soil materials, a narrow and deep trench is more efficient in solute leaching than a wide and shallow trench, while the opposite is true for less conductive imported soil materials (Fig. 11 and 12), applies for the entire range of source strengths and root distribution patterns analyzed here. Results of the analyses for additional trench configurations characterized by different combinations of x_m and z_m, within the range of x_m = 0.40 to 0.67 m and z_m = 0.3 to 0.5 m suggest that, in the case of the imported sand soil material, when the trench is relatively narrow (x_m = 0.4m), the solute leaching efficiency is essentially independent of the trench depth; as the width of the trench increases, however, the solute leaching efficiency slightly increases with increasing trench depth at a rate that increases with increasing trench width. On the other hand, in the case of the imported tuff soil material, the solute leaching efficiency depends substantially on both the trench depth and width. The resultant W_0.5(z_m, x_m) for the imported tuff soil material is displayed by the contour lines in Fig. 14 for Q = 1500 cm² h^–1. The results depicted in this figure suggest that W_0.5 decreases with increasing z_m, and to a lesser extent, with increasing x_m. The latter stems from the fact that in the case of the imported tuff soil material, increasing trench depth is shown to increase both u₁ and u₂ at relatively large soil depth, particularly when the trench is narrow. This means that in the case of the imported tuff soil material, from the economical point of view, it is more efficient to increase the trench depth than to increase its width.

View larger version (27K): [in this window] [in a new window]   FIG. 14. Contours of the amount of applied water (in meters) required to leach half of the total mass of solute applied into the flow system as functions of the trench depth (zm) and width (xm). Results are depicted for the tuff soil material for a source strength Q = 1500 cm2 h–1.

	Conclusions

TOP ABSTRACT INTRODUCTION Theoretical Considerations Results and Discussion Conclusions REFERENCES

Results of the investigation suggest that, as expected, under surface drip irrigation for a given source strength, local soil, and crop and evaporative demand, the spread of water and solutes in the combined imported soil–local soil flow system are affected by both the texture of the imported soil material and the geometry of the trench. Regarding the problem of solute leaching efficiency (SLE), findings of the present study can be summarized as follows:

1. For a given source strength, Q, in the case of less conductive imported soil materials associated with appreciable capillary forces, a wide and shallow trench may leach the solute below a horizontal control plane more efficiently than a narrow and deep trench, while the opposite is true for high-conductive imported soil materials associated with small capillary forces.
   
2. For the range of Q equivalent to the practical range of irrigation rates, in the case of a high-conductive imported soil material, SLE is essentially independent of Q, while in the case of a less conductive soil material, SLE slightly increases with Q, particularly when the trench is wide and shallow.
   
3. For a given Q, in the case of a relatively low-conductive imported soil material, SLE is essentially independent of the trench depth, particularly when the trench is narrow. For a high-conductive imported soil material, however, SLE increases with increasing trench depth and to a lesser extent with increasing trench width. This suggests that, for a high-conductive imported soil material, from the economical point of view it is more efficient to increase the trench depth than to increase its width.
   
4. For a given Q, in the case of a relatively low-conductive imported soil material, SLE is essentially independent of the extension of plant roots in the vertical direction and increases as the extension of the roots in the horizontal direction decreases, particularly when the trench is deep and narrow. On the other hand, for a high-conductive imported soil material, shallower and narrower root distributions are shown to increase the solute leaching efficiency, particularly when the trench is wide and shallow.
   
5. Combinations of the threshold values of K_s and above which a narrow and deep trench is more efficient in solute leaching than a wide and shallow trench derived for hypothetical imported soil materials suggest that for a given n, a relatively large threshold value of K_s may compensate for a relatively small threshold value of , particularly when n is relatively large.
   

Finding 1 applies to a substantial range of source strengths (within the practical range of irrigation rates) and a considerable range of root distribution patterns that may represent different plant types. This finding, however, may not apply to steady-state flow conditions in which, for both high-conductive and less conductive imported soil materials, a narrow and deep trench may leach the solute below the horizontal CP more efficiently than a wide and shallow trench.

Inasmuch as our interest lies in solute transport occurring at the field scale, the spatial heterogeneity of the local soil must be considered; consequently, the combined flow system was modeled as a spatially homogeneous (imported soil)–spatially heterogeneous (local soil) flow system. This complication is needed since, for the time period considered in this study, the characteristic length scales of the transport are much larger than their counterparts associated with the trenches; consequently, the spatial heterogeneity of the hydraulic properties of the local soil might considerably affect flow and transport below the trenches.

The numerical experiments conducted in here provide detailed information on the consequences of soil and irrigation water characteristics for water flow and solute transport under quite realistic conditions—information that, in general, cannot be obtained in practice from field investigations. The conclusions drawn from the present study, however, should be considered with caution, inasmuch as the numerical results presented here are based on analyses of single realizations of the local soil properties. The "single realization" approach was chosen here because of the substantial computational effort involved in this study, which, in turn, stems from the relatively large number of combinations of soil parameters, trench configurations, source strengths, and root distributions considered here. Since a Monte Carlo approach requires a relatively large number of realizations, the practical choice was to adopt a "single realization" approach. Based on the results of previous analyses (Russo et al., 1998, 2006), however, we believe that the simulated results based on a "single realization" approach are sufficiently accurate to indicate appropriate trends.

Finally, the analyses presented here are restricted to a particular site-specific application in the Arava Valley of Israel with given characteristics of soil, crop, weather, and irrigation system. We believe, however, that the scope of the study is of broad interest and has implications to similar situations involving agricultural production in desert areas.

	ACKNOWLEDGMENTS

 
This is contribution 705/06 from the Institute of Soils, Water and Environmental Sciences, ARO, the Volcani Center, Bet Dagan, Israel. Research was supported in part by a grant from the Chief Scientist of the Ministry of Agriculture.

	REFERENCES

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