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290 ANALYSIS OF SEEPAGE AND CONTAMINANT TRANSPORT <br />the dispersion coefficient parallel to the direction of fluid motion is <br />DI=aIx v <br />where DI =longitudinal dispersion coefficient <br />a, =longitudinal dispersivity <br />v = average groundwater velocity <br />Dispersion transverse to the direction cf flow is also assumed propor- <br />tional to :he average flow velocity but is less than the longitudinal disper- <br />sion. The ratio of longitudinal to transverse dispersivity is generally in [he <br />Hoge 3-10. Several investigators have assumed on the basis of empirical <br />observation: <br />ar = 3.3 ar <br />where ar is transverse dispersivity (Bredehoeft and Pinder, 1973; Konikow <br />and Bredehoeft, 1974). Freeze and Cherry (1979) note that numerous labora- <br />tory studies have shown longitudinal dispersivity to range from 10-6 to 2 x <br />10-2m, yet field values from tracer [gists or back calculation of actual con- <br />taminant plumes consistently show ar to be on the order of meters to [ens of <br />meters. As a result, realistic values of dispersivity must be obtained by field <br />measurements or from the few reported investigations in similar geologic <br />environments (Freeze and Cherry, 1979; Pinder, 1973). <br />Complete dispersion results in complete mixing of the contaminants with <br />groundwater, which was described previously in lumped parameter ;erns. <br />For the same amount of contaminant introduced to an aquifer, high-velocity <br />aquifers, such as karstic limestone and alluvial gravels, wih produce larger <br />contaminant plumes at lower concentration than aquifers of low velocity, <br />which will tend to retain the plume in a smaller and more concentrated form. <br />For constant groundwater velocity, the retardation coefficient is related <br />to [he distribution coefficient obtained from geochemical laboratory iltvesti- <br />gations as follows: <br /> <br /> <br />Rd= ~ =i+ nKe <br />where v =average groundwater velocity <br />v~ =the velocity of fluid that has a concentration 50% of [ha[ intro <br />duced into the aquifer <br />p =bulk density of the aquifer solids <br />rs =porosity <br />Kd =distribution coefficient <br />Freeze and Cherry note that when the distribution coefficient is 1 mg~l,- <br />wtl] range from 5 to Il for most materials. Therefore, for distribut <br />