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 solut e travel time tau and horizontal displacement eta from spatial covariance f unctions of the pore water velocity u in an unsaturated heterogeneous soil profile using a Lagrangian framework. The effects of four simplifications t hat are generally used to derive tau and eta statistics were evaluated: (1) first-order approximation of the stochastic flow equation, (2) first-order expansion of the inverse vertical pore water velocity 1/u(2), (3) identica l distributions of u and of solute particle velocity w, and (4) vertical so lute trajectories. Alternatives that comprehend numerical solutions of the stochastic flow equation to derive distributions of u and 1/u(2), using a f lux-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 tau and eta derived in a Lagrangian framework were compared with the stati stics derived from two types of transport simulations in generated, two-dim ensional heterogeneous soil profiles: simulations (1) for uniform solute fl ux 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-s at, was relatively large, sigma(2) In K-sat = 2.55, but it was based on con ductivity measurements in a loam soil and found to be realistic for predict ing solute transport in this soil. For the UBVP simulations the best predic tions 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.