## Appendix D: SIMPLIFIED ESTIMATION FOR DEPTH OF PENETRATIONThe depth of penetration of a solute plume that is developing under a surface impoundment can be estimated by separating the contribution of advection and dispersion during solute transport
h
(D.1)_{disp}where L] is the vertically advected component of the penetration depth and h [_{disp}L] is the vertically dispersed component of the penetration depth.
The advected depth
where L/T] is the vertical seepage velocity and t
[T] is time of travel. If the vertical seepage velocity is a constant with depth, then V
(D.3) _{t}tHowever, under impoundments, the vertical seepage velocity varies linearly with depth, with a maximum value at the top of the water table and zero at the bottom of the aquifer. A numerical solution for a surface impoundment was performed using SEFTRAN, with the vertical velocity variation under the impoundment plotted in Figure D.1. This variation can be modeled mathematically as:
where L/T] is the maximum vertical seepage velocity. V can be estimated from the net vertical recharge rate. _{zo}
As written, equation (D.2) cannot be integrated since V_{z} is not an explicit function of time. Consider the following differential equation for the vertical seepage velocity
Rearrange terms in equation (D.5) and integrate to depth Substitute equation (D.4) into equation (D.6) and integrate to get
Solve for
The time of travel
where L/T] is the horizontal seepage velocity. V
is assumed to be a constant. _{x}Prickett, Naymik, and Lonnquist (1981) estimate the magnitude of the effect of the effect of dispersion on particle transport as:
where a
[_{L}L] are the longitudinal and vertical dispersinities; V [L/T] is the magnitude of the seepage velocity; and
D and _{long}D
[_{vert}L] are the longitudinal and vertical dispersed distances that correspond to one standard deviation of random transport. If the effect of the horizontal seepage velocity is assumed to be much larger than that of the vertical, then the dispersed depth is estimated from equation (D.11) as: Hence, the total depth of penetration is the sum of the vertically advected and dispersed components. Substitute equations (D.8) and (D.12) into equation (D.1) to obtain the total estimated depth of penetration
The solution to equation (D.13) needs to be checked when evaluating any particular case so that a value of
- Prickett, T., T. Naymik and C. Lonnquist, 1981. A random-walk solute transport model for selected groundwater quality evaluations. Bulletin 65 Illinois State Water Survey, Department of Energy and Natural Resources, Champaign, Illinois. 103 pages.
- USEPA, 1990. Background Document for EPA's Composite Model for Landfills (EPACML), Prepared by Woodward-Clyde Consultants for USEPA, Office of Solid Waste, Washington D.C.
Last modified: Oct 15, 1999 VG Model / Samuel Lee / VADOSE.NET |