Convective-Dispersive Stream Tube Model for Field-Scale Solute Transport: I. Moment Analysis

Field-scale solute transport is typically difficult to model due to the complexity and heterogeneity of flow and transport in natural soils. The stream tube model attempts to stochastically describe transport across the field for relatively short travel distances by viewing the field as a series of independent vertical soil columns. This study investigates the stream tube model with the chemical equilibrium and nonequilibrium convection-dispersion equation (CDE) for local-scale transport. A bivariate (joint) lognormal probability density function was used for three pairs of random transport parameters: (i) the dispersion coefficient, D, and the pore-water velocity, v; (ii) the distribution coefficient for linear adsorption, &, and v; and (iii) the firstorder rate coefficent for nonequilibrium adsorption, a, and v. Expressions for travel time moments as a rsult of a Dirac input were derived to characterize field-scale transport according to the stream tube model. The mean breakthrough time for the field-scale flux-averaged concentration, 21, was found to be identical to that for the deterministic CDE. Variability in D has generally a minor effect on solute spreading compared with variability in v . Spreading of reactive solutes increased for negatively correlated v and Kd, even if the variability in K,, was relatively small, while nonequilibrium adsorption further increased spreading. If a was variable, a negative correlation between v and a enhanced the skewness of the breakthrough curve for&while spreading was independent of the correlation between (I and v. F I E L D-SCALE SOLUTE TRANSPORT has been the topic of considerable experimental and theoretical research because of concerns for the quality of the subsurface environment, which is especially threatened by downward movement of contaminants. Traditional deterministic modeling approaches, based upon the CDE for chemical transport and the Richards equation for water flow, work relatively well for homogeneous field soils and packed laboratory columns. Experimental investigations, however, have shown that flow and transport processes in most fields are heterogeneous (Biggar and Nielsen, 1976; Sudicky, 1986). Three approaches may be employed to describe fieldscale transport (Jury and Fltihler, 1992): (i) the traditional convective-dispersive model; (ii) a stochastic-continuum model that uses covariance functions for random localscale transport parameters (e.g., Dagan, 1984; Kabala and Sposito, 1991; Sposito and Barry, 1987); and (iii) a stochastic-convective stream tube model that views the field as a series of independent vertical soil columns (Dagan, 1993; Jury and Roth, 1990). These models are distinguished by the degree of lateral solute mixing (Jury and Fltihler, 1992). The stream tube model does not allow horizontal mixing, and the concentration for each tube represents a discrete value in the horizontal plane. N. Toride, Dep. of Agricultural Sciences, Saga Univ., Saga 840, Japan; and F.J. Leij, U.S. Salinity Lab., 450 West Big Spring Road, Riverside, CA 925074617. Contribution from the USDA-ARS Salinity Laboratory. Received 28 July 1994. *Corresponding author ([email protected]). Published in Soil Sci. Soc. Am. J. 60:342-352 (1996). 342 On the other hand, the CDE assumes perfect mixing, and the concentration across the horizontal plane is uniform for one-dimensional, vertical transport. During field-scale transport, it is likely that a transition occurs from a stochastic-convective to a convective-dispersive process. Dagan and Bresler (1979) and Bresler and Dagan (1979) first described downward movement of nonreactive solutes at the field scale with the stream tube model. They assumed a lognormal distibution for the saturated hydraulic conductivity. Amoozegar-Fard et al. (1982) demonstrated the effect of a random pore-water velocity, v, and dispersion coefficient, D, on field-scale concentrations with Monte Carlo simulation. Jury (1982) proposed a CLT, which neglects local-scale dispersion. Van der Zee and van Riemsdijk (1986, 1987) applied the stream tube model to reactive solutes, while Destouni and Cvetkovic (1991) introduced physical and chemical nonequilibrium in the local-scale transport model. Jury and Scotter (1994) discussed the application of the stochasticconvective model to boundary and initial value problems. They pointed out that stochastic stream tube models have not been used as widely as convective-dispersive and stochastic-continuum models due to the limited discussion of its theoretical foundation and a lack of procedures for its application. The purpose of this study is to further investigate field-scale transport with the stream tube model for reactive and nonreactive solutes. The effect of the variability in local-scale transport parameters on field-scale solute transport is demonstrated by obtaining field-scale mean concentrations with the chemical nonequilibrium or equilibrium formulation of the CDE for local-scale transport. Three pairs of random transport parameters are used, which are described with a bivariate lognormal pdf: the pore water velocity, v (cm d-‘), in combination with either the dispersion coefficient, D (cm2 d-l), the distribution coefficient for linear adsorption, Kd (cm3 g-l), or the first-order rate coefficient for nonequilibrium adsorption, a (d-l). The volumetric water content, 8 (cm3 cme3), and the soil bulk density, pb (g cmm3), are always assumed to be deterministic. Since we use a linear CDE for local-scale transport, the resulting stream tube model for field transport is also linear. Hence, we can derive moments of the travel time pdf to characterize field-scale solute distributions. Several typical examples of resident concentration profiles and field-scale BTC will be discussed with time moments. In the second part of this study (Toride and Leij, 1996), the stream tube model is applied to various types of boundary and initial value problems that may occur in the field. Abbreviations: BTC, breakthrough curve; CDE, convection-dispersion equation; CLT, convective lognormal transer function model; CV, coefficient of variation; pdf, probability density function. TORIDE AND LEIJ: CONVECTIVE-DISPERSIVE STREAM TUBE MODEL MOMENT ANALYSIS 343