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DIVISION S-1-SOIL PHYSICS

Generalized Transfer Function Model for Solute Transport in Heterogeneous Soils

Department of Renewable Resources, University of Wyoming, Laramie, WY 82071-3354 USA

renduo{at}uwyo.edu

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Results and discussions Conclusions REFERENCES

 
The convection-dispersion (CDE) equation and stochastic–convective models are the most commonly used process representations for predicting solute transport in the field. The convection–dispersion equation assumes that the solute is perfectly mixing in the lateral direction, whereas the stochastic–convective model assumes that the solute moves at different velocities in isolated stream tubes without lateral mixing. However, solute transport in heterogeneous porous media cannot always be conceptualized as being either a convective–dispersive or a stochastic–convective process. In this study, a generalized transfer function model (GTF) was proposed to describe various solute transport processes in heterogenous soils. The model is similar to the convective lognormal transfer function model, but two parameters, _µ and , are introduced to characterize the depth-dependency of the mean (µ) and standard deviation () of the logarithm of travel time, respectively. The GTF can describe well the two extremes of solute dispersion, the convective–dispersive and stochastic–convective processes, and transport processes between the two extremes. In addition, the GTF can be used to characterize other solute transport processes in heterogeneous soils, such as those in which the mean of travel time increases with depth nonlinearly, and those in which the dispersivity is a scale-dependent function of the travel distance with any power values.

Abbreviations: CDE, convection–dispersion equation • cdf, cumulative travel time distribution function • CLT, convective lognormal transfer function model • ECLT, extended convective lognormal transfer function model • ETFM, extended transfer function model • GTF, generalized transfer function model • pdf, probability density function

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Results and discussions Conclusions REFERENCES

 
THE CONVECTION–DISPERSION EQUATION is the most common model to describe solute movement through soils (van Genuchten and Wierenga, 1976). Solute movement in soils is influenced by the variation of the pore–water velocity. A number of solute transport models have been developed for the classical, one-dimensional CDE to represent velocity variations (Bresler and Dagan, 1981; Parker and van Genuchten, 1984). As used in the stochastic approach (Gelhar and Axness, 1983), a macroscopic mean transport equation with effective transport parameters (the dispersion coefficient and the pore-water velocity) can be applied to describe solute movement in heterogeneous soils. The CDE assumes a high degree (Fickian) mixing among flow paths. Therefore, the CDE is generally used for solute transport when the water velocity and solute concentration are relatively uniform within the modeled region among flow paths. It is often applied to describe the transport process with well developed lateral mixing.

An alternative to represent the process of solute transport is the use of transfer functions based on a probability density function (pdf) of the travel time of solute molecules and on the concept of stochastic–convective transport (Jury, 1982; White et al., 1986; Jury and Scotter, 1994). Transfer functions have been applied to model complex systems in a simple way by characterizing the output flux as a function of the input flux. By a field experiment, Butters et al. (1989) found that the convective lognormal transfer function model (CLT), one of the stochastic-convective models, described solute transport well within the top 3 m of soil. However, the CLT was a poor predictor at deeper depths where layering occurred. It was also shown by Roth et al. (1991) that the CLT gave poor predicted results for solute movement in a layered soil. Zhang (1995) predicted solute transport using both the CLT and CDE. Compared with experimental data collected in two 12.5-m soil columns, the CLT was found to provide better predictions than the CDE. Vanderborght et al. (1996) evaluated different procedures to derive model parameters of the CLT. Based on an analysis of steady state Cl^- transport through two heterogeneous field soils, Jacques et al. (1998) concluded that at both sites the breakthrough curves at different depths were better described by the CLT than by the CDE.

The pdf characterizes the distribution of possible travel times that a solute molecule might experience in moving from the inlet to the outlet. The transformation of an arbitrary input signal into an output signal for a linear system is achieved by means of the impulse response function, which defines the response of the system to a narrow pulse input at the inlet (Jury and Roth, 1990). Transfer functions, such as the CLT, are usually derived and applied by assuming that travel time of solute molecules is a linear function of travel distance. However, in complicated soil systems, the assumption may not be adequate to characterize chemicals moving through porous media.

The CDE and the stochastic–convective model represent two extreme processes for solute dispersion. In the CDE, perfect lateral mixing is assumed, whereas the stochastic–convective process assumes that the solute moves at different velocities in isolated stream tubes without lateral mixing. These models are limited to describing solute transport characterized by either a linear or a quadratic increase in the travel time variance with depth (Liu and Dane, 1996). However, solute transport in heterogeneous porous media cannot always be conceptualized as being either a convective–dispersive or a stochastic–convective process. Therefore, it is necessary to develop a general model that can describe not only the two extremes, but also other transport processes. Liu and Dane (1996) first discussed the partially mixing concept and proposed an extended transfer function model (ETFM) using an adjustable constant representing the degree of horizontal solute mixing. The ETFM is useful to describe transport processes between the CDE and the CLT. Nevertheless, the model is restricted to application within a certain depth larger than the calibration depth. In addition, the travel time of solute may increase with depth nonlinearly, for example due to changing water content or changing pore-water velocity in soil profiles. We also need models to describe solute transport characterized by the travel time variance increasing with depth at a rate higher than a quadratic rate. Neuman's (1990) universal scaling relationship of dispersivity in geologic media shows such solute transport processes with the travel time variance vs. depth increasing faster than a quadratic rate. In this paper we develop a GTF model to characterize various solute transport processes in heterogeneous soils. Examples are applied to test the GTF.

	Theory

TOP ABSTRACT INTRODUCTION Theory Results and discussions Conclusions REFERENCES

 
In a one-dimensional steady flow field, solute transport may be described with the CDE.

(1)

where C is the solute flux concentration, t is time, x is distance, D is the dispersion coefficient, and V is the mean pore-water velocity. For the CDE with a pulse input, its pdf of the travel time, also called the Fickian pdf, is written as follows (Jury and Roth, 1990)

(2)

Based on the impulse response function, Jury and Roth (1990) derived the CLT function model in the form of

(3)

where µ and are the mean and standard deviation of lnt. The parameters of the CLT are determined by

(4)

where µ_l and µ_z are the mean values at a calibration distance l and a distance of interest z from the inlet, respectively, and _l and _z are the standard deviations of lnt at these distances. The parameters of the CLT are derived using the following assumption about the pdf for the stochastic-convective model (Jury and Roth, 1990)

(5)

To extend the CLT, we modify the hypothesis of Jury and Roth (1990) as follows. In a statistical homogeneous porous medium, the probability that a solute molecule added at t = 0 at z = 0 will arrive at a distance x in time less than or equal to t, is the same as the probability that it will arrive at a depth l in time less than or equal to t(l/z). The exponent depends on transport processes as well as soil properties, and is introduced to describe transport processes in which the travel time of solute may increase with depth nonlinearly, for example due to changing water content or changing pore-water velocity in soil profiles. Such phenomena have been observed in field experiments (Zhang et al., 2000). In terms of the travel time pdf and the cumulative distribution function (cdf), the hypothesis for the stochastic–convective model can be expressed mathematically as

(6)

The travel time moment has the general expression

(7)

Combining Eq. [3] and [6] we can derive the travel time pdf of the extended CLT (ECLT) as follows:

(8)

or

(9)

where µ_l and µ_z are the mean values of lnt at distances l and z from the inlet, respectively, and _l and _z are the standard deviations of lnt at these distances. Equation [9] has the same form as Eq. [3] with the parameters

(10)

Obviously the CLT is a special case of the ECLT with .

The mean and variance of the travel time are related to the mean and variance of lnt at depth z by (Parkin et al., 1988)

(11)

The coefficient of variation (CV) of the travel time is expressed by

(12)

The mean and variance of the travel time at depth z can be estimated based on the pdf

(13)

For the CDE, from the Fickian pdf (Eq. [2]), the means, variances, and CVs of the travel time at depths z and l are related by

(14)

For the CLT (Eq. [3] and [4]), the relationships of the means, variances, and CVs of the travel time at depths z and l are

(15)

In the same way, for the ECLT, the relationships of the means, variances, and CVs of the travel time are

(16)

For the ETFM (Liu and Dane, 1996), the relationships of the means, variances, and CVs of the travel time at depths z and l are

(17)

Observing the relationships of the means, variances, and the coefficient of variation of the travel time at depths z and l for the CDE (Eq. [14], the CLT (Eq. [15]), the ECLT (Eq. [16]), and the ETFM (Eq. [17]), we propose the following time moment relationships for a GTF:

(18)

Here ₁ and ₂ are parameters of the time moments. As mentioned before the ECLT, the parameter ₁ is introduced to describe transport processes in which the mean of solute travel time increases with depth nonlinearly (Zhang et al., 2000). The generalization of variance using the parameter ₂ can be justified based on Neuman's (1990) universal scaling relationship and on transport in perfectly stratified media with local scale mixing across the layer boundaries (Dagan, 1989). Equation [18] can be used to characterize transport in self-affine media, and the parameters of ₁ and ₂ may be related to the fractal dimension of self-affine media (Wheatcraft and Tyler, 1988).

For the GTF, we propose a pdf, having the same form as Eq. [3], but the mean and standard deviation of lnt in the forms of

(19)

and

(20)

where and _µ are parameters, introduced to characterize various transport processes. For example, in the transport process of the CDE, the variance of lnt decreases with the travel distance and positive values of should be used. In the transport processes of the CLT and ECLT, the variance of lnt is constant within the travel distance and . In some cases of the scale-dependent transport process, the variance of lnt increases with the travel distance (Neuman, 1990) and negative values of should be used. As shown below, the GTF can be used to describe the CDE, CLT, ECLT, ETFM, and other transport processes in heterogeneous soils.

The parameters of lnt, and _µ, are related to the parameters of the time moments, ₁ and ₂, as follows. Combining Eq. [11], [12], and [20] yields

(21)

Combining Eq. [21] and the relationship of CVs for the GTF (Eq. [18]), we have

(22)

or

(23)

For the CDE, and , Eq. [23] becomes

(24)

For the CLT, , and for the ECLT, , thus from Eq. [22] we have

(25)

Combining the relationships of the means of Eq. [11] and Eq. [18] gives

(26)

or

(27)

Substituting Eq. [19] and [20] into Eq. [27] yields

(28)

For the CDE, Eq. [28] becomes

(29)

For the CLT, ₁ = 1 and = 0; therefore, _µ = 1. Similarly, for the ECLT, _µ = and = 0. The general relationship of the mean of lnt (Eq. [19]) now is reduced to Eq. [4] and [10], respectively, for the CLT and ECLT. For ETFM, ₁ = 1 and ₂ = a (0.5 a 1; Liu and Dane, 1996), from which and _µ can be calculated using Eq. [23] and [28], respectively.

The time moments and the parameters of and _µ for the CDE, CLT, ECLT, ETFM, and GTF are summarized in Table 1 . Figure 1a and 1b , respectively, show the relationships of _µ and as functions of l/z and the variance of lnt at depth l (²_l) for the CDE (₁ = 1 and ₂ = 0.5) and Fig. 2a and 2b show the relationships for an example of the GTF (₁ = 0.5 and ₂ = 1 in this example). For the CDE, _{µ 1} (= 1) for small values of ²_l and/or l/z, whereas _µ 2₁ - ₂ (= 1.5) for large values of ²_l and/or l/z. For small values of ²_l and/or l/z, ₁ - ₂ (= 0.5), whereas for large values of ²_l and/or l/z, 0. This relationship of can also be explained as follows. From the Taylor expansion of the left hand of Eq. [22] with small ²_l and/or small l/z, we have

(30)

or

(31)

View larger version (15K): [in this window] [in a new window]   Fig. 1 Relationships of (a) µ and (b) as functions of l/z and the standard deviation of lnt at depth l (l) for the CDE (1 = 1 and 2 = 0.5). The values of l are shown on the curves

View larger version (15K): [in this window] [in a new window]   Fig. 2 Relationships of (a) µ and (b) as functions of l/z and the standard deviation of lnt at depth l (l) for the GTF (1 = 0.5 and 2 = 1). The values of l are shown on the curves

For any values of ₁ and ₂, the range of _µ is between ₁ and 2₁ - ₂. If ₁ > ₂, _{µ 1} for small values of ²_l and/or l/z, whereas _µ 2₁ - ₂ for large values of and/or l/z. If ₁ < ₂, _{µ 1} for small values of ²_l and/or large l/z, whereas _µ 2₁ - ₂ for large values of ²_l and/or small l/z. If ₁ = ₂ (e.g., the CLT and ECLT), _µ = ₁ = 2₁ - ₂ for any values of ²_l and l/z. For any values of ₁ and ₂, the range of is between 0 and ₁ - ₂. If ₁ > ₂, (₁ - ₂) 0 and the variance of lnt decreases with the travel distance, such as in the transport process of the CDE. If ₁ = ₂, = 0 and the variance of lnt is constant within the travel distance, such as in the transport processes of the CLT and ECLT. If ₁ < ₂, (₁ - ₂) 0 and the variance of lnt increases with the travel distance, such as in the scale-dependent transport processes (Neuman, 1990; Zhang et al., 1994) to be discussed below.

	Results and discussions

TOP ABSTRACT INTRODUCTION Theory Results and discussions Conclusions REFERENCES

 
Convection–Dispersion Transport
To show applications of the GTF to approximate the convection–dispersion process, we used the Fickian pdf (Eq. [2]) to generate data. For example, using Eq. [2] with V = 1 and D = 5 to generate data at l = 50 and fitting Eq. [3] with the CDE data through an optimization procedure (Marquardt, 1963), we obtained µ_l = 3.816 and . With Eq. [24], [19], [29], and [20], _z and µ_z can be calculated at other depths. For instance, we calculated _z values at z = 30, 60, and 100 as 0.543, 0.398, and 0.313, respectively, and µ_z values of 3.251, 4.012, and 4.553 at these depths. Using the values of _z and µ_z, we predicted the CDE at the depths with the GTF. Compared with the CDE in Fig. 3 , the prediction of the GTF is excellent for all the depths of z > l as well as z < l.

View larger version (20K): [in this window] [in a new window]   Fig. 3 Prediction of the convection–dispersion equation (CDE) (V = 1 and D = 5) at different depths using the generalized transfer function model (GTF)

(32)

Defining the Peclet number

(33)

we have

(34)

Combining Eq. [34] with Eq. [11], we have

(35)

For convection-dominated problems, Pe is large and from Taylor expansion we obtain

(36)

For the CDE, V_z = V_l = V and D_z = D_l = D; therefore, we have

(37)

Comparing the results of _µ and for small values of l/z (large values of z/l) in Fig. 1, we can use _µ = 1 and = 0.5 for convection-dominated problems. As an example, we fitted data generated by Eq. [2] with V = 10 and D = 1 at l = 10 and obtained _l = 0.141 and µ_l = -0.00947. Using Eq. [19] and [20] with _µ = 1 and = 0.5, we calculated _z = 0.199, 0.0997, 0.0705, respectively, at z = 5, 20, 40, and µ_z = -0.684, 0.684, 1.377 at these depths. Using the parameter values and the GTF, we estimated the CDE at the depths and matched the generated CDE data extremely well (Fig. 4) .

View larger version (19K): [in this window] [in a new window]   Fig. 4 Prediction of the convection–dispersion equation (CDE) (a convection-dominated problem: V = 10 and D = 1) at different depths using the generalized transfer function model (GTF)

View larger version (16K): [in this window] [in a new window]   Fig. 5 Comparison of predicted results of the convection–dispersion equation (CDE) using the generalized transfer-function model (GTF) and the extended transfer function model (ETFM) with a = 0.5 (Liu and Dane, 1996)

(38)

in which _z is a scale-dependent dispersivity function. As shown from experimental studies (Pickens and Grisak, 1981), _z may be a linear function of travel distance, that is,

(39)

where b is a constant.

More generally, the expression for the dispersivity is given by (Butters et al., 1989):

(40)

Substituting the CV of the GTF (Eq. [18]) into Eq. [40] gives

(41)

For the CDE (₂ - ₁ = -0.5) the dispersivity is constant with the travel distance. For the CLT and ECLT (₂ - ₁ = 0), the dispersivity is a linear function of the travel distance as shown above (Eq. [39]) and by Jury and Sposito (1985).

For solute transport through the saturated zone, Neuman (1990) found

(42)

in which z is the mean travel distance. To satisfy Neuman's (1990) universal scaling relationship (Eq. [42]) we have

(43)

To test Eq. [43], we applied the GTF with ₂ - ₁ = 0.25 to generate data, fitted the CDE with the generated data for the parameters of V and D, and calculated = D/V. Table 2 lists some results of the computation, which was based on l = 20, µ_l = 1.0, and _l = 0.5 with two sets of ₂ - ₁ = 0.25: ₁ = 0.25, ₂ = 0.5, and ₁ = 1.25, ₂ = 1.5. For the two sets of ₁ and ₂, the values of ₁ - ₂ are negative and equal. Therefore, the results of are negative and increases with the travel distance. The both sets resulted in the same values of and at each depth. For the first set of ₁ and ₂, V increases with the travel distance because ₁ < 1, while V decreases with the travel distance for the second set because ₁ > 1. Nevertheless, the results of are almost identical for the two sets of ₁ and ₂, and approximated by a scale-dependent function:

(44)

with a coefficient of determination (r²) of 0.99999.

View this table: [in this window] [in a new window]   Table 2 Computation of the scale-dependent dispersivity as a function of 1 and 2. Other parameters used include l = 20, µ1 = 1.0, and l = 0.5

Solute Transport in Heterogeneous Soil Columns
To apply the GTF to predicting solute transport processes in soils, we need to estimate the parameters µ_z and _z at any depth z. If the transport process is defined by the CDE or the CLT, the parameters at any depth can be estimated based on pdf or cdf data at one location l, as in the examples discussed above. If the transport process is unknown, in general, we need pdf or cdf data at two locations (z₁ and z₂) to determine µ_z and _z at other depths. The procedure to determine the parameters is as follows:

1. Based on pdf or cdf data at locations of z₁ and z₂, calculate µ₁, µ₂, ₁, and ₂ by fitting the data with Eq. [3].
   
2. If ₁ = ₂, the transport process is the ECLT, then _z = ₁ and
   

   (45, 3)

   If _{1 2}, the transport process is the GTF and (i) determine ₁ - ₂ using
   

   (46)

   (ii) determine _µ using Eq. [45]; (iii) determine ₁ using Eq. [28]; (iv) determine (z) using Eq. [23] and _µ(z) using Eq. [28]; (v) determine µ_z using Eq. [19] and _z using Eq. [20].
   

Following the procedure, we used data from a laboratory experiment to test the GTF for solute transport in heterogeneous soils. The laboratory experiment, with 1250-cm-long, horizontally placed soil columns, was conducted to investigate solute transport in heterogeneous media (Zhang et al., 1994). Two square columns having cross-sectional areas of 10 by 10 cm² were set up in the experiment, a homogeneous sandy soil column and a heterogeneous column. The small cross-section area relative to the large length of the columns was designed to promote predominantly one-dimensional solute transport. Because of the difficulty in packing such a long soil column, the homogeneous soil column was not really homogeneous. The heterogeneous column contained a wide range of soil materials including clay, fine-, medium-, and coarse-grained sands, gravel, and pebbles. High heterogeneity was formed by lenses and layers with various shapes and sizes of soils. On average, the particle size of the heterogeneous column decreased gradually from coarse at the inflow end to relatively fine at the outlet. A tracer experiment was carried out after establishing a steady-state saturated water flow. Concentration of the tracer NaCl at various time was measured with electrical conductivity probes inserted at 50- and 100-cm intervals, respectively, in the homogeneous and heterogeneous columns. The concentration data were normalized by dividing with the input concentration (C₀).

With a step input, the time function of the normalized concentration (C/C₀) is essentially the cumulative travel time distribution function (cdf) and is characterized by the integration form of Eq. [2]:

(47)

For the homogeneous column, we fitted Eq. [47] to the data at locations of z₁ = 50 and z₂ = 1200 cm, and calculated µ₁ = 4.368, µ₂ = 7.669, ₁ = 0.0646, and ₂ = 0.0903. From the parameter values and the procedure, we obtained ₁ - ₂ = -0.10 and ₁ = 1.04. Since the difference between ₁ and ₂ was small, it was assumed that (z) = 0 and _µ(z) = 1 ( ₁). That is, the transport process was characterized well by the CLT. Using _z = _l = 0.0646 and µ_z values estimated with Eq. [4], we predicted cdfs at other depths. The examples in Fig. 6 show good agreement between the data and the predicted results of the GTF. The estimated results of the CDE (based on the parameters at l = 50 cm: D = 0.0662 cm² min^-1, V = 0.6335 cm min^-1) are also presented in Fig. 6. As the distance increases, the estimated results of the CDE become less accurate, mainly attributable to the assumption of a constant dispersion coefficient. In fact, the data show that the dispersion coefficients increase with distance, ranging from D = 0.0662 cm² min^-1 at z = 50 cm to D = 2.745 cm² min^-1 at z = 1200 cm.

View larger version (19K): [in this window] [in a new window]   Fig. 6 Comparison of experimental data in the homogeneous soil column and predicted results of the generalized transfer function model (GTF) and convection–dispersion equation (CDE)

View larger version (22K): [in this window] [in a new window]   Fig. 7 Comparison of experimental data in the heterogeneous soil column and predicted results of the generalized transfer function model (GTF)

	Conclusions

TOP ABSTRACT INTRODUCTION Theory Results and discussions Conclusions REFERENCES

The GTF can characterize the transport processes of the CDE, CLT, ECLT, and ETFM. For the CDE, ₁ = 1 and ₂ = 0.5, from which _µ and can be determined. For the CLT, ₁ = ₂ = 1 and _µ = 1, and for the ECLT, ₁ = ₂ = and _µ = . For the CLT and ECLT, = 0. For the ETFM, ₁ = 1 and ₂ = a (0.5 a 1), and _µ and are determined based on ₁ and a specified a value.

As shown from the examples, the GTF was successfully applied to predict the convective–dispersive and stochastic–convective processes. The GTF can be used to estimate transport processes in which the travel time increases with depth nonlinearly. The GTF can also be used to describe transport processes in which the dispersivity is a scale-dependent function of the travel distance with any power values, including the processes characterized by the ETFM (Liu and Dane, 1996) with power values between 0 and 1, and those by the Neuman's (1990) universal scaling relationship with a power value = 1.5.

	ACKNOWLEDGMENTS

 
This research was partly funded by the Wyoming Agricultural Experimental Station. We are grateful to W.A. Jury for useful discussions.

Received for publication September 30, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Results and discussions Conclusions REFERENCES

 

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