Solute transport in a heterogeneous soil for boundary and initial conditions: Evaluation of first-order approximations

We compared four different approaches to derive the statistics of the solute travel time γ and horizontal displacement η from spatial covariance functions of the pore water velocity u in an unsaturated heterogeneous soil profile using a Lagrangian framework. The effects of four simplifications that are generally used to derive γ and η statistics were evaluated: (1) first-order approximation of the stochastic flow equation, (2) first-order expansion of the inverse vertical pore water velocity 1/u₂, (3) identical distributions of u and of solute particle velocity w, and (4) vertical solute trajectories. Alternatives that comprehend numerical solutions of the stochastic flow equation to derive distributions of u and 1/u₂, using a flux-weighted distribution of u to represent the distribution of w and using two dimensional covariance functions to represent the effect of horizontal deviations of the particle trajectories, were discussed. The statistics of γ and η derived in a Lagrangian framework were compared with the statistics derived from two types of transport simulations in generated, two-dimensional heterogeneous soil profiles: simulations (1) for uniform solute flux at the soil surface (uniform boundary value problem, UBVP) and (2) for a uniform initial concentration profile (uniform initial value problem, UIVP). The considered heterogeneity of the saturated hydraulic conductivity, K(sat), was relatively large, σ² ln K(sat) = 2.55, but it was based on conductivity measurements in a loam soil and found to be realistic for predicting solute transport in this soil. For the UBVP simulations the best predictions of the solute travel time and horizontal displacement statistics were obtained using the flux-weighted distribution of simulated u. For the UIVP Simulations the distribution of w was not stationary but changed from the nonweighted distribution of u for small travel depths to the flux-weighted distribution of u for larger travel depths.