Block‐effective macrodispersion in variably saturated heterogeneous formations

Abstract

[1] Numerical models are frequently used in order to quantify the effect of the spatial heterogeneity in the formation hydraulic properties on field-scale solute transport. Because of practical constraints it is common to discretize the flow domain into relatively large numerical cells, with characteristic length scales, ' 1 , ' 2 , ' 3 , comparable with those of the formation heterogeneity, I 1 , I 2 , I 3 , leading to a loss in the velocity variability, and, concurrently, in solute spreading. We analyzed block-effective dispersion coefficients, D 0 ij (i, j = 1, 2, 3), required to compensate for this loss in three-dimensional, variably saturated heterogeneous formations under conditions of steady state, gravity-dominated, unsaturated flow. The results that the principal components of D 0 ij are controlled by the ratio ' 0 = ' 2 /I 2 = ' 3 /I 3 , that they may reach their ergodic limits if ' 0 is sufficiently large, and that their tendency to their asymptotic limits with increasing scaled travel time, t 0 , is inversely related to ' 0 , are in agreement with previous results for saturated flow. New findings suggest that under unsaturated flow conditions, both the time required to reach the asymptotic limits of D 0 ij and the length scale needed to reach the ergodic limits of D 0 ij depend on soil stratification, soil texture, and mean soil water saturation and differ for the longitudinal and the transverse components of D 0 ij .

Simulation of transport in three‐dimensional heterogeneous unsaturated soils with upscaled dispersion coefficients

“…The second is a more complicated, transient, nonmonotonic unsaturated flow regime, originating from periodic influx, uniform across the entry zone (II-type BC). The former relatively simple flow regime is pertinent to the flow conditions assumed in the derivation of the upscaled conductivity [Russo, 2003a] and dispersion [Russo, 2003b] tensors. The later, more complicated flow regime is pertinent to realistic situations involving periodic rain/irrigation events and water uptake by plant roots.…”

“…[24] Expressions for the first-order approximation of the stationary Cu ij (x) for a steady state flow in an unbounded, three-dimensional, partially saturated formation with mean flow parallel to the x 1 axis, have been presented previously [Russo, 1998[Russo, , 2003b. Note that for unsaturated flow conditions, the calculation of Cu ij (x) requires the calculations of the cross-covariances between perturbations of the flowcontrolled attributes and those of the relevant formation properties.…”

“…[44] The relatively simple flow regimes originating from the surface boundary condition (17a) are pertinent to the flow conditions assumed in the derivation of the upscaled dispersion tensor [Russo, 2003b] and therefore are suitable to test the capability of the first-order, upscaling methodology to compensate for the loss of subgrid-scale variations of the velocity field. In the case of continuous infiltration under a given prescribed pressure head at the soil surface, y 0 , (equation (17a)) the distribution of the velocity vector, u, is controlled mainly by the spatial distribution of the unsaturated conductivity, K, and the head gradient vector, J, and to a less extent, by the spatial distribution of the water content, q(y 0 ), which, in turn, are controlled by the magnitude of y 0 and by the spatial distribution of the soil properties K s , a VG , n, q s and q r , [Russo et al, 1994a].…”

“…[23] Considering average uniform flow U = (U 1 , 0, 0), where U is the mean velocity vector, in an unbounded, three-dimensional, partially saturated domain with a stationary velocity covariance tensor, assuming that w is a parallelepiped-shaped block with sides ' 1 , ' 2 and ' 3 , parallel to the coordinate axes (x 1 , x 2 , x 3 ), and using first-order approximation in log conductivity variance, taking into account that the streamwise dimension of w, ' 1 , has negligible effect upon D ij (t, w) [Dagan, 1991], and neglecting pore-scale dispersion, a first-order approximation of the upscaled (block-effective) dispersion tensor, D ij (t, w), (i, j = 1, 2, 3), for any size of block w, is given [Russo, 2003b] by:…”

“…[5] The purpose of the present investigation is twofold: first, to test the capability of the first-order, upscaling methodology to compensate for the loss of subgrid-scale variations of the velocity field under relatively simple flow conditions similar to those assumed in the derivation of the upscaled dispersion tensor [Russo, 2003b]; and second, to assess the applicability of the upscaling methodology to more general situations, free of most of the aforementioned restrictions, that involve complex transient flow regimes originating from periodic rain/irrigation events and water uptake by plant roots. To this end, use is made of a numerical approach, which combines a statistical generation method to produce realizations of the heterogeneous formation properties, with an efficient numerical method to solve the PDEs that govern flow and transport in heterogeneous, variably saturated formations.…”