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Soil Science Society of America Journal 63:1493-1499 (1999)
© 1999 Soil Science Society of America

DIVISION S-1-SOIL PHYSICS

Numerical Analysis of the Effect of the Lower Boundary Condition on Solute Transport in Lysimeters

Markus Flurya, Marylynn V. Yatesb and William A. Juryb

a Department of Crop and Soil Sciences, Washington State University, Pullman, WA 99164 USA
b Department of Soil and Environmental Sciences, University of California, Riverside, CA 92521 USA

flury{at}mail.wsu.edu


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theory
 Results and discussion
 Conclusions
 REFERENCES
 
Field lysimeters are often used to assess environmental behavior of agrochemicals. Most lysimeters used to date have a free-draining lower boundary where leaching out of the lysimeter occurs by gravity alone. In this case, the lower boundary of a lysimeter is open to the atmosphere, and consequently, leachate is collected only if the bottom of the lysimeter becomes water saturated. In a field soil, such local water saturation does not occur. The objective of this study was to evaluate the effect of the lower boundary condition on chemical leaching. Numerical simulations were used to compare solute transport in field soils and in lysimeters. Simulations were carried out in homogeneous sandy and loamy soils under steady-state, unsaturated water flow conditions. Water flow was described by the Richards equation and solute transport by the advection–dispersion equation. The effect of linear and nonlinear and instantaneous and kinetic sorption was investigated. The results showed that for a conservative solute the differences between field soil and lysimeter increase as the coarseness of the soil increases. Decreasing water flux increases the difference between field soil and lysimeter. In general, solute transport in the lysimeter is characterized by a slower mean velocity, a larger spreading, and smaller concentration values. For solutes subject to linear equilibrium sorption, the sorption mechanism compensates for the effects of the lower boundary condition. The larger the sorption coefficient, the less the difference between lysimeter and field soil. However, large differences are found in the case of strongly convex nonlinear sorption isotherms.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theory
 Results and discussion
 Conclusions
 REFERENCES
 
LYSIMETERS ARE OFTEN USED to assess the leaching behavior of pesticides under field conditions. A lysimeter is a large soil block surrounded by a casing, with the lower boundary shaved off from the parent soil and usually exposed to atmospheric pressure (Bergström, 1990). This exposure results in a hydraulic barrier for water flow. The soil at the lower boundary has to be saturated with water before drainage outflow can occur. To overcome the problem of water saturation, at the bottom of the lysimeter a suction can be applied with porous ceramic plates, pipes, or fiberglass wicks (Bergström, 1990; Nordmeyer and Aderhold, 1994; Young et al., 1996). However, for lysimeters with a large surface area, the use of suction devices is impractical and often problematic (Bergström, 1990). Consequently, most large lysimeters have a drainage system open to atmospheric pressure. We will limit our discussion to this type of lysimeter.

Lysimeters are intended to represent field conditions much better than laboratory columns and have been widely used to investigate fate and behavior of chemicals in soils (Bergström, 1990; Hance and Führ, 1992; Winton and Weber, 1996). Assuming that there is an undisturbed soil block in the lysimeter, the only difference between the lysimeter and the field soil is the lower boundary condition. It is unclear to what extent water and solute transport are affected by this difference in boundary conditions.

Several comparative studies between field soils and lysimeters have been reported in the literature. The comparisons included the temporal variation of temperature (Kubiak et al., 1988; Pütz et al., 1992; Nordmeyer and Aderhold, 1994), water content (Kubiak et al., 1988; Pütz et al., 1992), P fluxes (Magid et al., 1992), and pesticide concentrations in soil (Kubiak et al., 1988; Weber and Keller, 1994; Jene et al., 1998). Colman (1946) and Dowdell and Webster (1980) compared gravity-drained lysimeters with suction-controlled lysimeters. Colman (1946) used porous fired clay plates at the bottom to apply suction, and reported considerable differences in drainage rate, drainage quantity, water content, and water potential between the two types of lysimeters. Dowdell and Webster (1980) applied the suction with porous ceramic candles installed at the bottom of the lysimeter. They found no significant differences in amount of drainage water and NO3 concentrations, but the suction-controlled lysimeter showed continuous water outflow for longer periods of time. Recently, major experimental efforts have been made to compare field soils and lysimeters in respect to pesticide leaching. Jene et al. (1998) studied Br and benazolin (4-chloro-2-oxobenzothiazolin-3-ylacetic acid) movement in a field soil instrumented with a dense grid of suction cup samplers, and compared the results with those obtained from lysimeters. The lysimeters were located 20 km away from the field site, but had otherwise the same soil characteristics. The suction applied to the suction cups in the field soil was adjusted to the potential measured with tensiometers installed adjacent to the suction cups. Cumulative outflow of water and bromide was larger in the lysimeters than in the field soil, but Jene et al. (1998) attributed this result to differences in evapotranspiration between the sites. In contrast to water and Br-, the authors did not find great differences in pesticide leaching between field soil and lysimeters. A comprehensive experimental comparison between field soils and lysimeters is currently ongoing (Pütz et al., 1998), but no experimental results are available at present. The experimental evidence as to whether, and to what degree, a suction-free lysimeter adequately represents field conditions is not conclusive to date.

The purpose of our study was to evaluate the effect of the lower boundary condition on solute transport by using numerical simulations to compare solute transport in field soils and lysimeters. The field soils were characterized by a unit-gradient lower boundary condition, and the lysimeters by a seepage boundary condition. Simulations were performed under steady-state, unsaturated water flow with two textural classes of soils, a sand and a loam. Solutes with different sorption properties were used. The reaction processes considered were linear and nonlinear and instantaneous and kinetic sorption. Effects of different distribution coefficients, nonlinearity factors, and sorption rate coefficients were investigated.


   Theory
TOP
 ABSTRACT
 INTRODUCTION
 Theory
Results and discussion
 Conclusions
 REFERENCES
 
Transport Equations
We confine the discussion to a homogeneous soil, and assume that water transport can be described by the one-dimensional Richards equation:

(1)
where {theta} is the volumetric water content (L3 L-3), h is the matric potential (L), K(h) is the unsaturated hydraulic conductivity (L T-1), t is time (T), and z (L) is the vertical coordinate, taken positively upward. The water retention characteristic {theta}(h) and the unsaturated hydraulic conductivity function K(h) are given by the Mualem–van Genuchten parametrization (van Genuchten, 1980):


(2)
and


(3)

(4)
where {theta}r and {theta}s are the residual and saturated water contents (L3 L-3), respectively, Ks is the saturated hydraulic conductivity (L T-1), and {alpha} (L-1) and m are parameters.

The solute transport is described by the advection–dispersion equation:

(5)
where C is the solute concentration in solution (M L-3), S is the sorbed solute concentration (M M-1), {rho} is the soil bulk density (M L-3), D is the dispersion coefficient (L2 T-1), and Jw is the volumetric water flux (L T-1). The dispersion coefficient is given as (Bear, 1972):

(6)
where {lambda} is the dispersivity [L], Dm is the aqueous molecular diffusion coefficient of the solute in water [L2 T-1], and {tau} is the tortuosity factor. The tortuosity factor is calculated from the volumetric water content {theta} and the saturated water content {theta}s according to Millington and Quirk (1961) as . Solute sorption is assumed to be governed by first-order kinetics with a nonlinear Freundlich isotherm:

(7)
where K is the sorption coefficient (L3 M-1), n is a dimensionless nonlinearity factor (-), and ß is the rate coefficient (T-1).

Initial and Boundary Conditions
We consider steady-state, unsaturated water flow with a specified flux q at the soil surface. The initial condition for the matric potential h0 is the steady-state matric potential at the specified water flux. These initial potentials were obtained by simulating the water flow until steady-state was reached. Solutes are collected at a given depth L. We assume that the water table in the field soil is well below the observation depth. The conditions at the lower boundary correspond to a unit-gradient condition for the field soil (McCord, 1991) and to a seepage or zero-potential condition for the lysimeter. For the field soil, we consider a semi-infinite system with boundary conditions imposed at . The initial and upper boundary conditions for the water flow are then given as:

(8)

(9)
and the lower boundary conditions are:

(10)

Solute was added at the soil surface as a flux-type boundary condition, and the soil was initially void of solutes. Field soil and lysimeter have the same initial and upper boundary conditions for solute transport:

(11)

(12)


(13)

At the lower boundary, water exits the system, and the solutes in solution leave the system with the water flow. A zero-gradient condition was employed for this boundary (Danckwerts, 1953; Pearson, 1959; Bear, 1979):


(14)

For the finite lysimeter system, the zero-gradient boundary condition implies that at z = -L concentrations are continuous and no dispersive flux occurs. This type of boundary condition together with that as shown in Eq. [13] has been shown to be the correct choice for collecting solute outflow from a finite system, provided the Peclet number, , is larger than about five (van Genuchten and Parker, 1984). Estimated Peclet numbers in our simulations were in the range of 23 to 164.

Numerical Simulations
The flow and transport equations along with the initial and boundary conditions were solved numerically with the CHAIN-2D code (imnek and van Genuchten, 1994). The code is based on finite elements according to the Galerkin method, and the time derivatives in the solute transport equation were approximated by a Crank-Nicholson finite differences scheme. The spatial domain was discretized into elements of 1-cm length. Two different soils were used in the simulations: a sand and a loam. The Mualem–van Genuchten parameters for these soils are taken from Carsel and Parrish (1988) and are shown in Table 1 . The hydraulic functions are depicted in Fig. 1 . The parameters for solute transport, the dispersivity {lambda} and the molecular diffusion coefficient Dm, are assumed to be the same in all simulations: {lambda} = 0.5 cm and Dm = 1.25 x 10-5 cm2 s-1. The molecular diffusion coefficient is for Cl at infinite dilution in water at 25°C (Cussler, 1984). The sorption coefficients of the solutes used in the simulations varied from K = 0 to 2 mL g-1. For comparison, the herbicide atrazine (2-chloro-4-ethylamino-6-isopropylamino-s-triazine) has reported K values of 0.2 to 2.46 mL g-1 (Tomlin, 1994).


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  		Table 1 Parameters of hydraulic functions (Eq. [2] to [4]) used in the simulations (data taken from Carsel and Parrish, 1988).{dagger}

 


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  		Fig. 1 Hydraulic properties of the soils used in the simulations. Parameters of the hydraulic functions are given in Table 1

 
Simulations were performed under steady-state, unsaturated water flow with three different fluxes q: 2, 0.5, and 0.1 cm d-1. The depth L of the lysimeter was assumed to be 1.2 m. This depth was chosen according to guidelines issued by the European Community for testing fate and behavior of chemicals in the environment (European Community, 1995). For the field soil, the lower boundary conditions (Eq. [10] and [14]) were approximated in the simulations by using a soil profile of 10-m depth. Test runs showed that this depth was adequate to accurately simulate solute transport at the 1.2-m depth. Solute concentrations were calculated with time at the lower boundary of the lysimeter and at the corresponding depth in the field soil. All concentrations reported are relative concentrations, normalized by the input mass of solutes.

With the parameters chosen, solute spreading is dominated by dispersion at the highest flux rate and by diffusion at the lowest flux rate. The effect of the different spreading mechanisms on the simulation results was tested by a series of model calculations in the lysimeter and field soil at different flow rates. Breakthrough curves obtained by using the molecular diffusion coefficient for Cl were compared with curves obtained by neglecting molecular diffusion. No effect was observed for the two higher flow rates. However, for the lowest flow rate, the lysimeter breakthrough curves obtained with the two diffusion coefficients showed deviations, but these were orders of magnitude smaller than the differences between the lysimeter and field soil.


   Results and discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Results and discussion
Conclusions
 REFERENCES
 
Water Content and Matric Potential Distributions
For steady-state water flow, the volumetric water contents and the matric potentials in the soil remain time invariant. Figure 2 shows the distributions of the volumetric water contents within the lysimeter. Because of the seepage boundary condition, the lower boundary of the lysimeter is in all cases saturated, and accordingly the matric potential is zero. Figure 2a shows the effect of the flow rate on the steady-state water content, illustrated with the loamy soil. The smaller the flow rate, the drier the soil becomes, and the larger the gradient d{theta}/dz near the lower boundary. The effect of the soil type is depicted in Fig. 2b, where water content profiles are plotted for a given flow rate of q = 0.1 cm d-1. In the sandy soil, the capillary fringe is {approx}5 cm in height, whereas in the clay loam soil the capillary rise extends upward to the soil surface. In contrast to the lysimeter, the water contents and matric potentials in the field soil with the unit-gradient boundary condition are constant throughout the soil profile, and there is no gradient along the vertical coordinate.



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  		Fig. 2 Steady-state volumetric water content in the lysimeter. (a) Water contents for different flow rates illustrated with the loamy soil, (b) water contents for a constant flow rate of q = 0.1 cm d-1 in sandy and loamy soil

 
Nonsorbing Solutes
Figure 3 shows the breakthrough of a nonsorbing solute, Br- or Cl- for example, at a depth of 1.2 m in the field soil and the lysimeter at different flow rates. The difference in solute breakthrough between field soil and lysimeter is more pronounced in the sand than in the loam soil. The two soils show the same characteristic features of solute transport between the field and the lysimeter simulations, even though these features are more distinct in the sand soil. In general, the differences between field soil and lysimeter increase as the flow rate decreases. In the following, we will discuss the differences in terms of the mean pore water velocity, the variance of the breakthrough curve, and the concentration values.



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  		Fig. 3 Breakthrough curves of a nonsorbing solute (K = 0) in the field (solid line) and the lysimeter (dashed line) for different flow rates. Note the different scales of the axes

 
For a constant flow rate q, the mean transport velocity of solutes is faster in the field than in the lysimeter. The lysimeter has a larger water content than the field soil, particularly at the lower boundary, and for a steady-state water flow this translates to a slower pore water velocity V in the lysimeter. The smaller the flow rate q, the more pronounced become the differences in water content in field and lysimeter, and thus the greater become the differences in the pore water velocity V.

The variance of the breakthrough curves is larger in the lysimeter than in the field soil. This increased temporal spreading of solutes in the lysimeter is accompanied by smaller concentration values. It is notable that in the sand soil, at a flow rate of 0.1 cm d-1, the maximum concentration measured in the lysimeter is only about half of the maximum concentration in the field soil. The increased spreading of the breakthrough curve for the seepage boundary condition is due to the larger volumetric water content and the smaller pore water velocity in the lysimeter compared with the field soil. The variance (VAR) of the breakthrough curve for the advection–dispersion equation for constant coefficients and a narrow solute input is (Jury and Roth, 1990):

(15)
which shows that for a constant dispersivity {lambda} and decreasing pore water velocity V, the variance increases. D is the hydrodynamic dispersion coefficient, {tau} is the tortuosity, Dm the molecular diffusion coefficient in water (see Eq. [6]), and z is the depth where the breakthrough curve is measured.

While the discussed features of the breakthrough curves are true for both soils, Fig. 3 shows that the finer the soil texture, the smaller the differences in solute transport between field soil and lysimeter. Whereas the sand soil is susceptible to the lower boundary condition, solute transport in the loam soil is less affected by the lower boundary, an observation which can be inferred from the water content distribution in the two systems.

Sorbing Solutes
Linear Equilibrium Sorption
The effect of the lower boundary condition on the transport of a solute subject to linear equilibrium sorption is depicted in Fig. 4 and 5 . An increasing distribution coefficient causes the breakthrough curves to approach each other. Whereas for no sorption there are more pronounced deviations between lysimeter and field soil in the sandy soil than in the loamy soil, the effect of the soil type diminishes for increasing sorption (Fig. 4). For a distribution coefficient of 2 mL g-1, the breakthrough curves in the field soil and the lysimeter are almost identical for the sandy as well as for the loamy soil. To quantitatively assess these differences, the mean solute travel time, the variance of the solute travel time, and the maximum concentrations of the breakthrough curves were calculated for different sorption coefficients. The ratio of mean travel times between lysimeter and field soil is denoted as {eta}, the ratio of the travel time variances as {zeta}, and the ratio of maximum concentrations as {kappa}. Figure 5 shows the ratios, {eta}, {zeta}, and {kappa}, plotted vs. sorption coefficient for the sandy and the loamy soil. Mean and variance of the solute travel time are larger in the lysimeter than in the field soil, and correspondingly the contrary is true for the maximum concentrations. Two interesting points can be mentioned. First, the effect of the sorption is most pronounced at low K values, indicated by the steep slope of the ratio curves, and as the sorption coefficient increases its effect decreases. Second, the sandy soil is more susceptible toward the sorption coefficient than the loamy soil.



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  		Fig. 4 Comparison of the transport of a solute with different distribution coefficients K in the sandy and loamy soil (linear equilibrium sorption). The water flux was in all cases q = 0.5 cm d-1. Note the different scales of the axes

 


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  		Fig. 5 Effect of linear equilibrium sorption on the difference in solute transport between lysimeter and field soil. {eta} is the ratio of the mean travel times in lysimeter and field soil, {zeta} is the ratio of the variances of the travel time, and {kappa} is the ratio of the maximum concentrations of the breakthrough curves. The water flux was in all cases q = 0.5 cm d-1

 
It is clearly illustrated that the sorption compensates for the effect of the lower boundary condition. The reason for this compensation effect is that the mobility of a sorbing chemical is dependent on the volumetric water content {theta}. For equilibrium sorption, the mobility can be expressed by the retardation coefficient: . For a constant K and bulk density {rho}, the retardation coefficient is inversely proportional to the volumetric water content {theta}. Since the lysimeter has a larger water content than the field soil, particularly near the lower boundary, a chemical experiences less sorption in the lysimeter than in the field soil. However, the mean pore water velocity is faster in the field soil than in the lysimeter, resulting in a compensatory effect. The larger the K value, the more alike were the breakthrough curves of field soil and lysimeter.

This result is, at least qualitatively, in agreement with experimental findings from Jene et al. (1998), who reported large differences in water outflow and Br- leaching between field soil and lysimeters, but no differences in pesticide leaching.

Nonlinear Equilibrium Sorption
Nonlinearity of the sorption isotherm does not affect the difference between lysimeter and field soil, except for strongly convex isotherms. Figure 6a shows that the response of the ratio {kappa} is very sensitive for small values of the isotherm exponent n. In the case of our simulations, where , a large deviation between lysimeter and field soil occurred when the n value was < 0.8. A convex isotherm leads to a self-sharpening solute front in the soil profile. Since the lysimeter has a larger water content near the bottom, the aqueous concentration of the travelling front will decrease. Due to the nonlinearity of the isotherm, this leads to increased sorption, and thus to an enhanced retardation of the solutes. This is in contrast to the case of linear sorption, where sorption is independent of concentration. Strong convex nonlinearity in the sorption isotherm can lead to large differences between lysimeter and field soil, as demonstrated in Fig. 6a.



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  		Fig. 6 Effect of (a) nonlinear equilibrium sorption (S = KCn) and (b) nonequilibrium sorption [C/t = ß(KC - S)] on the ratio {kappa} of maximum concentrations in the sandy soil. The parameters n and ß are the nonlinearity constant of the sorption isotherm and the nonequilibrium rate coefficient, respectively. The water flux for these simulations was q = 0.5 cm d-1

 
Linear Nonequilibrium Sorption
The effect of nonequilibrium sorption was investigated for a linear sorption isotherm . The sorption rates ß in a nonequilibrium sorption process may vary between zero and infinity. In the limit , the sorption becomes an equilibrium process with the reaction being described by the sorption isotherm . In the limit ß = 0, solutes do not interact with the sorbent and are completely mobile. To demonstrate the effect of nonequilibrium, we chose K = 2 mL g-1, so that we can cover a large range of breakthrough curves. Simulations were carried out with rate coefficients varying across several orders of magnitudes, and the results are shown in Fig. 6b. The semi log plot depicts two distinct plateaus for the ratio {kappa} of the maximum concentrations. These plateaus correspond to the cases when the first moments of the breakthrough curves are located close to the two limiting cases of no and equilibrium sorption. The change in {kappa} is associated with the change of the first moments of the breakthrough curves between the two limiting cases. Note that the change of {kappa} would appear gradual when plotted on a linear scale.


   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Results and discussion
 Conclusions
REFERENCES
 
Under the assumptions of a homogeneous soil and a steady-state, unsaturated water flow, our numerical simulations allow the following conclusions with respect to solute transport in field soils and lysimeters:

  1. The coarser the soil, the more pronounced the differences between field soil and lysimeter. The smaller the water flux, the larger the differences between field soil and lysimeter.
  2. The differences in solute transport are manifested in the pore water velocity, the variance of the breakthrough curve, and the maximum concentration values. Compared with the field soil, the lysimeter has a slower mean pore water velocity, a larger solute spreading, and smaller concentration values.
  3. Linear equilibrium sorption compensates for the effects of the lower boundary condition. The larger the distribution coefficient, the more similar solute transport in field soil and lysimeter become.
  4. The influence of sorption is most pronounced for small sorption coefficients K and decreases as the sorption coefficients increase. Sandy soils are more susceptible toward the effect of sorption than loamy soils.
  5. The effects of nonlinear equilibrium sorption do not differ much from the linear equilibrium case, except for strongly convex sorption isotherms. Under such conditions, sorption increases near the bottom of the lysimeter due to the increased water content, and lysimeter breakthrough curves are much retarded compared with the field soil.
  6. For nonequilibrium sorption, the sorption rate coefficient can be considered a factor weighting the effects of zero and equilibrium sorption. The weighting is strongly nonlinear, reflecting the effect of the rate coefficient on solute breakthrough curves.

Due to the simplified assumptions made about soil homogeneity and steady-state water flow, our simulations do not allow a quantitative assessment of the difference between field soil and lysimeter, and the conclusions made are of a qualitative nature. Nevertheless, the behavior of the two systems can be demonstrated in an unequivocal way, since the two systems in our simulations are identical except for the lower boundary condition. In view of the frequent use of lysimeters to assess transport of pesticides in soil, the results from this study show the complex behavior of sorbing chemicals in a simple system like the one chosen and the potentially large differences in solute leaching that may occur between a lysimeter and a field soil.

Received for publication April 13, 1998.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Results and discussion
 Conclusions
 REFERENCES
 




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