The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers

Abstract

Transport of an inert solute in a heterogeneous aquifer is governed by two mechanisms: advection by the random velocity field V(x) and pore-scale dispersion of coefficients D(dij). The velocity field is assumed to be stationary and of constant mean U and of correlation scale I much larger than the pore-scale d. It is assumed that D(dij) = α(dij)U are constant. The relative effect of the two mechanisms is quantified by the Peclet numbers Pe(ij) = UI/D(dij) = I/α(dij), which as a rule are much larger than unity. The main aim of the study is to determine the impact of finite, though high, Pe on (C) and σ(c)/2, the concentration mean and variance, respectively. The solution, derived in the past, for Pe = is reconsidered first. By assuming a normal X probability density function (p.d.f.), closed form solutions are obtained for (C) and σ(c)/2. Recasting the problem in an Eulerian framework leads to the same results if certain closure conditions are adopted. The concentration moments for a finite Pe are derived subsequently in a Lagrangean framework. The pore-scale dispersion is viewed as a Brownian motion type of displacement X(d) of solute subparticles, of scale smaller than d, added to the advective displacements X. By adopting again a normal p.d.f. for the latter, explicit expressions for (C) and σ(c)/2 are obtained in terms of quadratures over the joint p.d.f. of advective two particles trajectories. While the influence of high Pe on (C) is generally small, it has a significant impact on σ(c)/2. Simple results are obtained for a small V0, for which trajectories are fully correlated. In particular, the concentration coefficient of variation at the center tends to a constant value for large time. Comparison of the present solution, obtained in terms of a quadrature, with the Monte Carlo simulations of Graham and McLaughlin [1989] shows a very good agreement.