A Quasi-Analytical Model for Soil Solute Movement under Plant Water UseL. Connell* and R. Haverkamp

ABSTRACTSolute accumulation in surface soils through capillary rise transport,

driven by evaporation, is a serious management issue. In particular,for soils under a saline shallow water table, salt buildup can have aserious detrimental impact on agricultural productivity. For vegetatedsurfaces, evaporation is the sum of water loss directly from the surfaceand that taken up by plants for transpiration. We developed a proce-dure for the prediction of solute migration in soils under plant wateruse in a shallow water-table environment. For this situation, theadvection-dispersion equation is shown to be linear with nonconstantcoefficients. To solve this equation, the root zone is divided into a seriesof layers and for each layer the governing equation is approximated bya constant coefficient form with layer-averaged values for properties.We derived a solution to this equation in Laplace space, which iscoupled to its neighbors by requiring the flux and concentration beconstant across the layer boundaries. At each time level, a matrixsystem is posed for the equation coefficients and concentration resolvedby numerical inversion from Laplace space. The good agreementbetween model predictions and solutions obtained from a finite-elementanalysis indicates that the procedure presented is of high accuracyand could offer computational savings over purely numerical proce-dures. The multilayer approach allows the representation of nonuni-form plant water use functions, dispersivities, soil properties, andinitial conditions.

SOLUTE TRANSPORT by capillary rise under an evaporat-ing regime leads to the accumulation of solutes near

to the surface (Hassan and Ghaibeh, 1977; Fritton etal., 1967; Elrick et al., 1994). For solutes that havedetrimental impacts on surface soil quality and agricul-tural production, this accumulation is of concern. Inaddition, once solutes are near the surface, they canbecome available to surface waters, and be transportedto streams with negative effects on stream river quality(Walker, 1995).

In many areas under irrigation, the water table hasrisen to close to the soil surface. For example, in theirrigation areas of northern Victoria, Australia, thegroundwater table is within 2 m of the surface (Walker,1995). In these shallow water table areas, salinizationof land is a serious management problem. The mainmechanism leading to soil salinization is the transportof salt by capillary rise of soil moisture from the ground-water table under evaporative conditions.

Lindstrom et al. (1967) presented a solution for soluteaccumulation under bare soil conditions for a semi-infinite saturated soil. Elrick et al. (1994) consideredsolute accumulation under semi-infinite, unsaturated con-ditions. For this problem the governing advection disper-sion equation for unsaturated transport was transformed

L. Connell, Cooperative Research Centre for Catchment Hydrology, Dep.of Civil Engineering, Monash Univ., Clayton, Victoria 3168, Australia;and R. Haverkamp, Laboratoire d'Etude du Transferts en Hydrologie etEnvironnement (LTHE/IMG, UJF, CNRS URA 1512), BP 53X, 38041Grenoble, France. Received 5 June 1995. "Corresponding author (Luke.Connell @ eng. monash. edu. au).

Published in Soil Sci. Soc. Am. J. 60:1350-1367 (1996).

into a linear, constant coefficient equation through anintegral transform and assumption of steady-state mois-ture movement conditions. Barry and Sposito (1989) useda series of integral transforms to reform the unsaturatedtransport equation into a heat equation form with movingboundaries, and used the approach of Cannon (1984)to derive solutions. However, in a shallow water-tableenvironment, semi-infinite theory will not be able toaccurately represent the interactions that occur betweenthe problem domain boundaries. In addition, the workpresented above does not consider the effect of plantwater use. For vegetated soils, the amount of water takenup from the profile by plants will be much greater thanthat lost through bare surface evaporation.

While the governing equation for unsaturated transportcan be readily solved using purely numerical methods,the disadvantage of these procedures is that they can becomputationally demanding. To solve this problem intime requires not only a spatial discretization but a tempo-ral one as well. In applications that involve solvingtransport over large areas, such as with catchment model-ing, the computational burden associated with numericalmethods can be restrictive. Analytical or quasi-analyticalsolutions offer an alternative and can often provide accu-rate, stable solutions at a lower computational burden.The main disadvantage of analytical solutions are therestrictive assumptions introduced in redefining the prob-lem into a form that can be solved.

We derived a quasi-analytical solution for solute trans-port for a finite soil column with plant water use, boundedby a lower surface representing the water table and anupper surface from which evaporation occurs.

THEORETICAL BACKGROUNDUnsaturated solute transport in porous media can be de-

scribed by the advection dispersion equation:

dt dx dx dx [1]

where c is the solute concentration, D is the dispersivity, Ris decay or adsorption-desorption, and x is positive downward(de Marsily, 1986). The first term on the right-hand side ofEq. [1] relates the contribution of dispersive processes and thesecond term convective. The unsaturated volumetric moisturecontent, 0, and the soil moisture flux, q, may vary with timeand depth.

During evaporation, solute will accumulate at the soil sur-face. For this situation Eq. [1] is subject to

= 0 [la],=o

The influence of a shallow water table can be described by

c(L,t) = Ciwith the initial condition

c(x,0)=g(x)1350

CONNELL & HAVERKAMP: MODEL FOR SOIL SOLUTE MOVEMENT UNDER PLANT USE 1351

where g(x) is a function describing the form of the initialcondition. In Eq. [Ib], c\ is the concentration of solutes in thegroundwater and L is the depth of the groundwater table.

A key assumption in the analysis presented here is that thesolute transport and thus also root water uptake relationshipcan be approximated as vertical one-dimensional processes.While for an individual plant the root distribution has a three-dimensional structure, if there is a uniform surface coveringof plants, individual root systems are intermingled and rootwater uptake can be considered in terms of a horizontal spatialaverage. Thus, for this situation lateral variations are notconsidered important, and water and solute transport is drivenby gradients that only occur in the vertical direction.

Important variables in Eq. [1] are the moisture movementproperties, 9 and q. In the presence of plant water use, thecontinuity equation for moisture within the soil can be writtenas

.dt .,dx[2]

where S, is the root water uptake.By expanding terms and using Eq. [2], Eq. [1] can be

rewritten as

e- = -dt dx

[3]

With the same transformation as used by Barry and Sposito(1989), that is,

Q = Q(x,t)dx [4]Jo

the time derivative of the concentration can be written as

Zr Zf) Af Af[5]dt dt dQ dt

where c(x,t) = f(Q,t) and

— = q 0 - q - \ Sr(x,t)dx,dt Jo

where qa is the surface moisture flux, a function of time.With Eq. [5] and [6], Eq. [3] can be rewritten as

[6]

dt dQ dQ

-\q0- ^ + -^F-f mThe Q space transformation replaces the moisture flux, q(x,t),with the surface moisture flux, qa(f). However, when there is

fx(Q) _ _plant water use, the term Jo Sr (x,t)dx is introduced into theequation, as in Eq. [7].

In the absence of plant water use, a common approximationhas been to assume that 62(;t,r) in Eq. [7] can be replaced byits average (Bond and Smiles, 1983; Barry and Sposito, 1989;Elrick et al., 1994). With this assumption, for a constantsurface moisture flux and no plant water use, a constant-coefficient equation is obtained. Elrick et al. (1994) used thisas the basis for a relation to describe solute accumulation fora bare surface soil in a semi-infinite domain. Barry and Sposito(1989) showed how this equation could be transformed intoa moving boundary problem for a nonconstant surface moistureflux.

By replacing Q2(x,i) and Q(x,t) by their spatial and temporalaverages, <92> and <9>, Eq. [7] has the following form:

X(Q) Sr

ar^ael*where Da = <62>D.

MODEL DERIVATIONThe Laplace transform of Eq. [8], when qa is constant, and

Da and S, are invariant with time, is

STdx - [9]

where s is the transform variable.To solve Eq. [9], the water uptake term, Sr, must be resolved.

The assumption behind Eq. [9] is that 5r is constant in time,which means that Eq. [9] is valid only for periods wheretranspiration can be approximated, or averaged, by a constantin time. If S, is constant throughout the rooting depth, Eq. [9]becomes a nonconstant-coefficient, nonhomogeneous ordinarydifferential equation. However, a spatially constant S, is nota realistic assumption for most field conditions. Another diffi-culty is that the dispersivity is a function of pore water velocity.Therefore, in the presence of plant water use, the dispersivitybecomes a function of position.

One approach to solving Eq. [9] that does not impose afunctional form on 5r is to divide the root zone into a series

M2) _of layers. Within each layer Sr and qa — jo 5r dx are approxi-mated by their average values for the layer and/(<2,0) = g,a constant for the layer. The general solution to Eq. [9] forthese conditions is

/' = + ai + a>2 exp(MG') [10]

where

M = V + ~ S'r)

64= V-v'2 + - Sj)

and

V = q'0 -

2Da

Sidxdx/(xi

The superscripted variables in Eq. [10] refer to their valuefor the rth layer, except for Si, which is the layer averagevalue for 5r/<9>. The transform variable, Q, can be rewrittenas

Q f = \ Q(x,t)dx [11]

for the rth layer.Each layer must be linked with its neighbors in order to

build up a solution for the soil column. This linkage shouldensure that the derived solution is continuous across the layerboundaries as well as guaranteeing that the solute mass balancewithin the soil column is maintained. The conditions on layer

1352 SOIL SCI. SOC. AM. J., VOL. 60, SEPTEMBER-OCTOBER 1996

connectivity are imposed on the solution through the twounknown coefficients, ai and 0.2, for each layer.

Therefore, at the boundary of layer i with the layer aboveit, i - I,

and/'-'=/

— i — 1 — / r 1 TI<?c = <?c [12]

where <fc is the transformed solute flux for the ith layer at theboundary with the i — 1 layer. The variable q'c is written as

[13]*--«*L+vr|«Substituting Eq. [13] into the flux condition in Eq. [12] leadsto

d/'dQ Q=& dQ

+ [V-1 - v']/' |G=C, = 0 [14]

Note that in Eq. [14] since/'""1 = /' it is replaced by/1'.Substituting Eq. [10] into [14], differentiating, and collecting

terms leads to

a'i[v'~' — v1 + D'ab\] d2 [v1"1 — v' + .DIM] =- (v1'-1 - vi)g'/(s-Si

r) [15]where Eft = exp(frf • 2')-

The concentration condition leads to the following equation:

/'-' - /'' = arlE{nl + a>2-lE& - a{ - a2 = 0 [16]When Eq. [15] and [16] for each layer are combined with theboundary conditions, a system of linear algebraic equations isderived where the model coefficients, a\ and ai, are the un-knowns.

At the soil surface, Condition [la] can be written in Q andLaplace transform spaces as

C=o[17]

Substitution of Eq. [10] and [17] leads to the following expres-sion:

='

a\(Dlb\ - g,) - -SI)0 ~ Dlb\

[18]

Introducing this into Eq. [15] and [16] for the first layer definesthe first equation in the matrix system.

Below the root zone, S, = 0 and Eq. [10] reduces to

[19]

where

- 2Awhere g™ refers to the flux at the top of sub-root-zone layer.

Substitution of Eq. [19] into Condition [Ic] leads to thefollowing expression:

[20]where Q' is the value of Q" at the lower boundary.

COMPARISON WITH NUMERICALRESULTS

The above results allow the solute concentration withrespect to depth to be estimated within Laplace space.As a first step, the unknown coefficients in the equationsfor each layer, that is Eq. [10] or for depths below theroot zone, Eq. [19], are resolved by solving the linearsystem of equations posed by Eq. [15] and [16] for eachlayer, with Eq. [18] and [20] incorporating the effectsof the surface boundary conditions. Numerical inversionof Eq. [10] or [19] from Laplace space allows the concen-tration with respect to depth and time to be resolved.

In order to ascertain the accuracy of the above proce-dure, a finite element routine (Connell et al., 1993) wasused to generate solutions as a basis for comparison.Water movement was described by the steady -state formof Richards' equation, which can be written as

K(h) (?-)!-*\dx ]\[21]

where h is the matric potential and K(h) the unsaturatedhydraulic conductivity. Solute transport was describedby Eq. [3].

Numerical Laplace inversion of Eq. [10] and [19] wasperformed using the Stehfest algorithm (Stehfest, 1970)and the matrix system posed by Eq. [15] and [16] forthe multilayer system was solved using LU decomposi-tion with back substitution as implemented in the Numeri-cal Recipes routines LUDCMP.FOR and LUBKSB.FOR(Press et al., 1986). The soil properties in the examplespresented here were for Yolo light clay (Haverkamp etal., 1977), where

K(h) = 5.51728124.6+ l/zl1-7 7 cm/h,

—— — TTT739 + \h\4 + °-124 cm3 water/cm3 soil [22]

where 0 is the volumetric water content.The examples presented here are based on a uniform

root distribution of 20-cm depth. The root zone is dividedinto layers and only one layer is employed for the zonebelow the root zone. In this region, there is no plantwater use and the water flux is constant, thus the govern-ing equation exact. For the calculations presented below,a constant plant water use and surface water flux wasimposed on the soil column for the duration of theprediction period. In the root zone, 13 layers were used,and thus 14 altogether.

In the first case a plant water use, Sr, of 1 X 10~4

cm/h with an evaporative surface flux of 1 X 10~4 cm/hwere imposed. For the conditions described above andfor a 24-h period, this yields a total evaporative demand

CONNELL & HAVERKAMP: MODEL FOR SOIL SOLUTE MOVEMENT UNDER PLANT USE 1353

1.9 T

t=1000hNumerical

Quasi-analytical

0.920 25 30 35 40

Q (as defined by Eq [4])Fig. 1. Comparison between quasi-analytical and finite element nu-

merical solutions for solute concentration with respect to the trans-form variable Q (as given by Eq. [4]). For this example D = 0.1cm2/h, q,, x 10-" cm/h, Sr = 1 x 10-4 cm3 water/cm3

soil-h, L = 100 cm, g = 1 mg/cm3, ci = 1 mg/cm3, and the rootingdepth is 20 cm.

of 0.504 mm/d. Figure 1 compares finite element solu-tions with the quasi-analytical procedure presented herefor three times and for a constant root water uptakethroughout the root zone. In Fig. 2 the water contentand matric potential profiles are presented. The excellentagreement between the results presented in Fig. 1 demon-strates the accuracy of the approximate procedure usedin the quasi-analytical solution. The accuracy of theapproach presented above will be related to the accuracyof the approximation introduced through the multilayerprocedure's use of element averages for the water contentand soil water flux. Under higher rates of plant wateruse, the rate of change in the soil water flux will begreater and therefore the procedure's accuracy moresensitive to the layer-average approximation. Increasingthe number of layers would improve the representationof the water content and flux variation.

Figure 3 presents results at three time intervals wherethe root water uptake, 5r, is twice the magnitude of thatused in the calculations presented in Fig. 1. The totalwater use for this example is thus 1.008 mm/d. Whilethere is some divergence between the two approachesat the 750-h level, the agreement is very good for theother, earlier, times presented. The divergence could bethe result of a number of effects: numerical error in theLaplace inversion, numerical error in the finite elementsolution, or error associated with the multilayer approxi-mation.

The root water uptake, 5r, determines the rate ofchange of the soil water flux within the soil. An importantapproximation with the quasi-analytical procedure is thedivision of the root zone into layers and the assumptionof a constant water flux within each layer. This allowsthe root water uptake to vary with respect to depth,in a piecewise fashion. This allows a more realisticapproximation to the actual water uptake behavior thanbeing constant with respect to depth, as would be requiredfor an analytical solution. For a large ST the rate of

.a -100IQ. -150

m2 -200

-2500.50

Jj 0.45

80.40 -

0.35

0.30

0.2520 40 60

Depth (cm)80 100 120

Fig. 2. The variation in volumetric moisture content and matric poten-tial with respect to depth for the results presented in Fig. 1.

change in the soil water flux will be more dramatic andthe multilayer approximation requiring thinner, and asa result, more layers to maintain accuracy. Figure 4presents an example of the effect of layer thicknesson accuracy. In this figure, the difference between theconcentration profile calculated using 24 layers and thatcalculated using eight and four layers is given. In thisexample the root water uptake is a constant with depth.Error does increase as layer width increases but notdramatically, and the method still has acceptable accuracy

3.9 T

t = 750 h

Numerical

Quasi-analytical

10 15 20 25 30 35

Q (as defined by Eq [4])

—i40

Fig. 3. Comparison between quasi-analytical and finite element nu-merical solutions for solute concentration with respect to the trans-form variable Q (as given by Eq. [4]). For this example D = 0.1cm2/h, q0 = -2 x 10"4 cm/h, Sr = 2 x 10~4 cm3 water/cm3

soil-h, L = 100 cm, c, = 1 mg/cm3, and rooting depth is 20 cm.

1354 SOIL SCI. SOC. AM. J., VOL. 60, SEPTEMBER-OCTOBER 1996

0.5 --

g.

¥-0.5

IQ

-1.5

20 30 40

- 8 layers4 layers

Q (as defined by Eq [4])Fig. 4. The effect of layer thickness on model accuracy behavior. The

results plotted are the relative difference between a concentrationprofile calculated using 24 layers and profiles calculated using eightand four layers. For example, for the eight-layer results, (CM —c8)/cM x 100. For this example D = 0.1 cnrVh, q, = -1 x 10-"cm/h, Sr = 2 x 10-" cm3 water/cm3 soil-h, L = 100 cm, Ci = 1mg/cm3, and the rooting depth is 20 cm.

for the least number of layers considered. This hasimplications for the computational burden associated withthe method: a small number of layers will ensure thatthe computational overheads are minimized. In fact thenumber of layers would be chosen to ensure satisfactoryrepresentation of the plant water use behavior or disper-sivity with depth, rather than being necessary to maintainaccuracy.

CONCLUSIONSAn important conclusion of this work is that the

multilayer approach developed in this study can yieldaccurate solutions to linear, nonspatially constant co-efficient differential equations. For the cases considered,the procedure was found to be stable with respect to layerthickness and offers significant flexibility with respect tothe spatial behavior of the key variables within the rootzone. For example, the layering allows the root wateruptake, initial solute concentration, and dispersivity tobe nonuniform between layers and, for more accuraterepresentation, within layers represented by fairly simplefunctional forms that still allow the differential equationto be solved. An extension of this would be to problemsrelated to solute adsorption, where different parts of thesoil profile could have different properties.

The quasi-analytical procedure presented here has anumber of advantages over purely numerical methods.One important advantage is the lower computationalburden associated with calculating a solution at a pointin time. Since the procedure does not step forward froman initial condition, a solution at a particular time is

calculated directly. The number of iterations requiredfor the numerical Laplace inversion is usually <10. Whenthis is combined with the demonstrated stability withrespect to layer thickness, significantly less effort isrequired to calculate a concentration profile at a particulartime interval.

While the method has some advantages over purelynumerical routines, a disadvantage is the requirement ofa constant moisture flux at the surface of each layer.This is introduced by the transformation to Q space andrewriting the governing differential equation into a morereadily solvable form. Effectively, this means that thesoil moisture flux must be constant with respect to timewithin in the root zone. However, this is not the casebelow the root zone, where the only requirement is thatthe flux at the layer boundary be constant.

The governing advection-dispersion equation was sim-plified into the form given in Eq. [8] by replacing thewater content with its average. This reduced accuracysomewhat but not significantly. The multilayer solutionprocedure allowed the variation in moisture content withrespect to depth to be approximated within the root zone.For the cases considered, there was still good agreementbetween the two alternative approaches to solving theproblem. In one example, some divergence did occur atlong times. However, this could have been the result ofinaccuracies in either, or both, procedures.

The examples considered here were for a root wateruptake that is constant throughout the plant root zone.A more realistic representation would be that the wateruptake has an exponential form. In this situation theeffect of using layer-averaged properties would meanthat this exponential behavior could only be representedin a piecewise fashion. Thus, depending on the rate ofchange of root water uptake, error would be introducedinto the analysis. However, this could be managedthrough the layer width; introducing more layers wouldmean a better piecewise approximation to the exponentialbehavior and thus greater accuracy in the solution pro-cedure.

We have presented an alternative to purely numericalmethods to solve solute transport within a soil wherethere is a shallow water table and plant water use. Thequasi-analytical procedure presented has some of theadvantages of purely numerical procedures (such asallowing properties to vary within the root zone) whilehaving a lower computational burden.

ELLSWORTH & BOAST: SPATIAL STRUCTURE OF TRANSPORT VARIABILITY 1355

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Hassan, F.A., and A.S. Ghaibeh. 1977. Evaporation and salt move-ment in soils in the presence of water table. Soil Sci. Soc. Am.J. 41:470-478.

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