## Dispersion

Dispersion is the result of two processes, molecular diffusion and mechanical mixing.

Molecular diffusion is the process whereby ionic or molecular constituents move under the influence of their kinetic activity in the direction of their concentration gradients. Under this process, constituents move from regions of higher concentration to regions of lower concentration; the greater the difference, the greater the diffusion rate. Molecular diffusion can be expressed by Fick's law as dC

dx where:

F = mass flux per unit area per unit time Df = diffusion coefficient C = contaminant concentration dC/dx = concentration gradient

Fick's law was derived for chemicals in unobstructed water solutions. When this law is applied to porous media, the diffusion coefficient should be smaller because the ions follow longer paths between solid particles and because of adsorption. This application yields an apparent diffusion coefficient D* represented by

where w is an empirical coefficient less than 1. Perkins and Johnston (1963) suggest an approximate value of 0.707 for w. Bear (1979) suggests that w is equivalent to the tortuosity of the porous medium with a value close to 0.67. Values of D* for major ions can be obtained from Robinson and Stokes (1965).

Mechanical mixing is the result of velocity variations within the porous medium. The velocity is greater in the center of the pore space between particles than at the edges. As a result, the contaminant spreads gradually to occupy an ever-increasing portion of the flow field. Mechanical mixing dispersion can occur both in the longitudinal direction of the flow as well as in the transverse direction. According to Bachmat and Bear (1964), the mechanical mixing component of dispersion can be assumed proportional to the seepage velocity as

where:

D11 = longitudinal mechanical mixing component of dispersion

D22 = transversal mechanical mixing component of dispersion aL = longitudinal dispersivity aT = transversal dispersivity

V = average linear pore water velocity

Finally the hydrodynamic dispersion coefficients can be written as

The dispersivity coefficients aL and aT are characteristic of the porous medium. Representative values of dis-persivity coefficients can be determined from breakthrough column tests in the laboratory or tracer tests in the field (Anderson 1979).

Figure 9.13.1 shows how dispersion can cause some of the contaminant to move faster than the average ground-water velocity and some of the contaminant to move slower than the average groundwater velocity. The front of the contaminant plume is no longer sharp but rather smeared. Therefore, when dispersion is also considered,

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