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Waste injected in deep geologic formations mixes with formation fluids.  This results in a plume defined by lines of equal concentration.  The size of the plume depends on the injection rate, duration of injection, height of the receiving formation, formation porosity and coeffcient of dispersivity.

In the form below, enter values for the calculation and the analytical solution are presented.  We use the Hoopes and Harleman solution for radial dispersion. This page allows evaluation to concentrations down to 1 x 10-30. The solution include a Fortran program to evaluate the Error Function.  Other solutions such as Excel are limited to 1. x 10-15.

The model for fluid displacement without dispersion is defined as:



            ro = advective radius to position of undispersed injectate front (ft), 

            V = cumulative injectate volume (ft3),

            H = injection interval thickness (ft), and

            f = formation porosity.

The effect of dispersion are added to the advective displacement and calculated using the approximate solution provide by Bear, 1972 and Hoopes and Harleman, 1967 as:



           C = dimensionless concentration,

r = radial distance from well (ft),

α = dispersivity (ft), and

erfc = complementary error function.


Hoopes, J. A. and D. R. F. Harleman, 1967. Dispersion in radial flow from a recharge well, Journal of Geophysical Research, 72(14), 3595-3607.

Bear, J., 1972. Dynamics of Fluids in Porous Media, American Elsevier, New York.


            Additional Deep Well References