The Longitudinal Dispersion Coefficient of Soils as Related to the Variability of Local Permeability

Abstract

Solute (NaCl) miscible displacement experiments are performed on long disturbed soil columns to determine the hydrodynamic longitudinal dispersion coefficient and correlate it with the variability of the local permeability. The solute concentration, averaged over several cross-sections along the soil column, is monitored by measuring the electrical resistance between rod electrodes. The measured solute concentration breakthrough curves are fitted simultaneously with the one-region and two-region analytical models of the 1-D advection–dispersion equation to estimate the longitudinal dispersion coefficient, D _L, as a function of Peclet number, Pe, for common groundwater flow velocities (2 < Pe < 50). Macroscopic simulations of miscible displacement in 2-D porous media described by a periodic permeability field with low, moderate and high variability are employed to evaluate the predictability of the one-region and two-region models, and the sensitivity of the dispersion coefficients and flow velocities estimated from soil column displacement tests to the variance of local permeability. When the variability of the local permeability becomes high, the one-region model fails, while the two-region model is capable of reproducing satisfactorily the breakthrough curves, and providing reliable values of dispersion coefficients. The two mean pore velocities estimated by the two-region model represent, on average, a fast and a slow mean velocity of the dispersion front, whereas their difference is a measure of the transient evolution of the width of the equi-concentration dispersion front.

Appendix

In soil S2, the grain (or pore) size distribution is very wide (Table 1) and hence, a great variety of mean pore velocities and hydraulic conductances (permeabilities) arises. In order to compute the volume-based distribution of local permeabilities from the weight-based grain-size distribution, the modified Kozeny equation (Dullien 1979)

$$ k_{i} = \lambda \frac{{d^{2}_{{{\text{g}}i}} \phi ^{3} }} {{\phi {\left( {1 - \phi } \right)}^{2} }} $$

(24)

was used, where λ is a parameter to estimate, and k _i , d _gi are the mean permeability and mean grain diameter of class, i. Considering that f _i (i = 1, 2, 3, 4) are the frequencies for the grain diameters d _gi (Table 1), h _i (i = 1, 2, 3, 4) are the frequencies for the local permeabilities k _i , and using Eq. 24 it is obtained

$$ h_{i} = \frac{{d_{{{\text{g}}i}} }} {{{\left\langle {d_{{\text{g}}} } \right\rangle }}}f_{i} $$

(25)

The average permeability defined by

$$ {\left\langle k \right\rangle } = {\sum\limits_{i = 1}^n {k_{i} h_{i} } } $$

(26)

is not necessarily equal to the overall permeability. Instead, methods of the statistical physics of disordered media are required for calculating the overall permeability from the distribution of local permeabilities. An approach based on the effective medium approximation (EMA), as applied to a 3-D cubic lattice, was used. The disordered medium consists of cubic cells of equal dimensions with permeabilities following the distribution function. This medium is replaced by an effective one consisting of cells of identical permeability k _m so that the overall perturbation caused on the flow field vanishes. Evidently, the foregoing condition is not valid for a very wide distribution of local permeabilities. The EMA is expressed by the algebraic relation

$$ {\sum\limits_{i = 1}^n {{\left( {\frac{{k_{i} - k_{m} }} {{k_{i} + 2k_{m} }}} \right)}h_{i} } } = 0 $$

(27)

By setting the effective permeability, k _m , equal to the experimentally measured value of the overall permeability of S2 (k _exp = 21.8 Da) and solving Eq. 27 we obtained the parameter λ = 6.3165 × 10⁻⁴ and the local permeability distribution shown in Table 9.

Table 9 Local permeability distribution of soil S2