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Analytical solutions for the coefficient of variation of the volume-averaged solute concentration in heterogeneous aquifers

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Abstract

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 initial 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 Gaussian initial conditions. When integrated over an averaging window, these solutions lead to new closed-form expressions for the first two ensemble moments of thevolume-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 thesolute point concentration in two spatial dimensions and the corresponding CV are demonstrated to be unbound. For impulse initial conditions the CVs for volume-averaged concentrations axe compared with each other for a tracer from the Borden aquifer experiment. The point-concentration CV is unacceptably large in the whole domain, implying that the ensemble mean concentration is inappropriate for predicting the actual concentration values. The volume-averaged concentration CV decreases significantly with an increasing averaging volume. Since local dispersion is neglected, the new solutions should be interpreted as upper limits for the yet to be derived solutions that account for local dispersion; and so should the presented CVs for Borden tracers. The new analytical solutions may be used to test the accuracy of Monte Carlo simulations or other numerical algorithms that 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.

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Authors and Affiliations

  1. Department of Civil Engineering, Duke University, 27706, Durham, NC, USA

    Z. J. Kabala

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Kabala, Z.J. Analytical solutions for the coefficient of variation of the volume-averaged solute concentration in heterogeneous aquifers. Stochastic Hydrol Hydraul 11, 331–348 (1997). https://doi.org/10.1007/BF02427923

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