Appendix D: SIMPLIFIED ESTIMATION FOR DEPTH OF PENETRATION

The 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  =  h_adv + h_disp        (D.1)

where H [L] is the depth of penetration, h_adv [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 h_adv is the depth that a particle would be transported under the influence of vertical advection 

where V_z [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 

h_adv  =   V_tt              (D.3) 

However, 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 B [L] is the saturated aquifer thickness, z [L] is the depth from the top of the water table and V_zo [L/T] is the maximum vertical seepage velocity. V_zo can be estimated from the net vertical recharge rate. 

Figure D.1 Variation in the vertical seepage velocity with depth

 

dz/dt  =  V_z(z)              (D.5)

Rearrange terms in equation (D.5) and integrate to depth h_adv 

Substitute equation (D.4) into equation (D.6) and integrate to get 

Solve for h_adv from equation (D.7) 

The time of travel t [T] can be estimated as the time it takes for a particle to be advected horizontally under an impoundment of length L [L] 

where V_x [L/T] is the horizontal seepage velocity. V_x is assumed to be a constant. 

Prickett, Naymik, and Lonnquist (1981) estimate the magnitude of the effect of the effect of dispersion on particle transport as: 

where a_L and a_L [L] are the longitudinal and vertical dispersinities; V [L/T] is the magnitude of the seepage velocity; and D_long and 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 H greater than the aquifer thickness B is not used. If the computed H is greater than B, set H equal to B. 

References 

• 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.