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

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 = U/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 V 0 , 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.