One-dimensional transport models that predict field-scale averaged solute fluxes are often used to estimate the risk of nonpoint source groundwater contamination by widespread surface-applied chemicals. However, within-field variability of soil hydraulic properties leads to lateral variation in local solute fluxes. When this smaller scale variability is characterized in a geostatistical sense, stochastic three-dimensional flow and transport equations can be used to predict field-scale averaged transport in terms of geostatistical parameters. We discuss the use of stochastic equations for the parameterization of equivalent one-dimensional models predicting averaged solute fluxes. First, we consider the equivalent one-dimensional convection dispersion model and the equivalent dispersivity, which characterizes the spreading of laterally averaged concentrations or solute fluxes. Second, we discuss the parameterization of a stream tube model to predict local transport variables (i.e., distributions of local concentrations and local arrival times) These local transport variables are shown to be important for predicting nonlinear local transport processes and useful for inversely inferring the spatial structure of soil properties. Stochastic flow and transport equations reveal a dependency of equivalent model parameters on transport distance and flow rate, which reflects the importance of smaller scale heterogeneities on field-scale transport. Approximate solutions of stochastic flow and transport equations are obtained for steady-state and uniform flow. The effect of transient flow conditions on transport is discussed. Throughout the paper we refer to experimental and numerical data that confirm or contradict results from stochastic flow and transport equations.

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