ANALYTICAL SOLUTIONS FOR THE COEFFICIENT OF VARIATION OF THE VOLUME-AVERAGED SOLUTE CONCENTRATION IN HETEROGENEOUS AQUIFERS

Under the assumption that local solute dispersion is negligible, a new general formula (in the form of a convolution integral) is found for the arbitrary k-point ensemble moment of the local concentration of a solute convected in arbitrary m spatial dimensions with general sure i nitial conditions. From this general formula new closed-form solutions in m = 2 spatial dimensions are derived for 2-point ensemble moments of the local solute concentration for the impulse (Dirac delta) and Ga ussian initial conditions. When integrated over an averaging window, t hese solutions lead to new closed-form expressions for the first two e nsemble moments of the volume-averaged solute concentration and to the corresponding concentration coefficients of variation (CV). Also, for the impulse (Dirac delta) solute concentration initial condition, the second ensemble moment of the solute point concentration in two spati al dimensions and the corresponding CV are demonstrated to be unbound. For impulse initial conditions the CVs for volume-averaged concentrat ions axe compared with each other for a tracer from the Borden aquifer experiment. The point-concentration CV is unacceptably large in the w hole domain, implying that the ensemble mean concentration is inapprop riate for predicting the actual concentration values. The volume-avera ged concentration CV decreases significantly with an increasing averag ing volume. Since local dispersion is neglected, the new solutions sho uld be interpreted as upper limits for the yet to be derived solutions that account for local dispersion; and so should the presented CVs fo r Borden tracers. The new analytical solutions may be used to test the accuracy of Monte Carlo simulations or other numerical algorithms tha t deal with the stochastic solute transport. They may also be used to determine the size of the averaging volume needed to make a quasi-sure statement about the solute mass contained in it.