Solute transport in divergent radial flow through heterogeneous porous media

Abstract

Radial flow takes place in a heterogeneous porous formation of random and stationary log-conductivity Y(x), characterized by the mean 〈Y〉, the variance σ²_Y and the two- point autocorrelation ρ_Y which in turn has finite and different horizontal and vertical integral scales, I and I_v, respectively. The steady flow is driven by a head difference between a fully penetrating well and an outer boundary, the mean velocity U being radial. A tracer is injected for a short time through the well envelope and the thin plume spreads due to advection by the random velocity field and to pore-scale dispersion. Transport is characterized by the mean front r=R(t) and by the second spatial moment of the plume S_rr. Under ergodic conditions, i.e. for a well length much larger than the vertical integral scale, S_rr is equal to the radial fluid trajectory variance X_rr.

The aim of the study is to determine X_rr(t) for a given heterogeneous structure and for given pore-scale dispersivities. The problem is more complex than the similar one for mean uniform flow. To simplify it, the well is replaced by a line source, the domain is assumed to be infinite and a first-order approximation in σ²_Y is adopted. The solution is still difficult, being expressed with the aid of a few quadratures. It is found, however, that it can be derived quite accurately for a sufficiently small anisotropy ratio e=I_v/I by retaining only one term of the velocity two-point covariance. This major simplification leads to simple calculations and even to analytical solutions in the absence of pore-scale dispersion.

To compare the results with those prevailing in homogeneous media, apparent and equivalent macrodispersivities are defined for convenience.

The major difference between transport in radial and uniform flow is that the asymptotic, large-time, apparent macrodispersivity in the former is smaller by a factor of 3 than in the latter. For a three-dimensional point source the reduction is by a factor of 5. This effect is explained by the rapid change of the mean velocity during the period in which the velocities of two particles injected at the source become uncorrelated.

In contrast, the equivalent macrodispersivity tends to its value in uniform flow far from the well, where the flow is slowly varying in space.