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csh_lecture8_analytical - CWR 6537 Subsurface Contaminant...

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CWR 6537 Subsurface Contaminant Hydrology 1 ADVECTIVE-DISPERSIVE SOLUTE TRANSPORT EQUATION : ANALYTICAL SOLUTIONS A. Lecture Goal: To understand the concepts that surround the solution of the advective-dispersive solute transport equation for “ideal” solutes by examining several boundary and initial conditions; dimensionless solution approaches; and time vs. space perspectives. B. Advective-Dispersive Transport of “Ideal” Solutes: An "Ideal" solute is nonreactive (no adsorption-desorption, or ion-exchange with the solid matrix) and conservative (no degradation or precipitation; no gains owing to dissolution etc.) The partial differential equation for describing one-dimensional transport of ideal solutes during steady water flow in saturated (or uniformly unsaturated) homogeneous porous media is: x c v - x C D = t C 2 2 (1) where C = liquid-phase solute concentration (M·L -3 ·T -1 ); D = (longitudinal) hydrodynamic dispersion coefficient along the x direction, (L 2 ·T -1 ); and v = average pore-water velocity along the x direction, (L). Note: the concentration term, C, is referenced here to the aqueous-phase concentration, but for convenience, we have dropped the subscript w. We have also dropped the subscript m from dimension x; however, we are still using the bulk media frame of reference. Finally, we have adopted the use of the average pore-water velocity rather than the actual velocity distribution, f(v). Thus, D here represents the hydrodynamic dispersion term (i.e, the combined effects of molecular diffusion and mechanical dispersion). What changes in Eq (1) would be needed for solute transport during steady water flow in a homogeneous porous medium that was variably saturated ? C. Initial and Boundary Conditions: A variety of initial and boundary conditions may be specified now for solving Eq.[1]. For example, the porous medium may be:
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CWR 6537 Subsurface Contaminant Hydrology 2 (i) Infinite medium(- < x < + ); recall the diffusion case (ii) Semi-Infinite medium (0 x < + ); as in a very long column (iii) Finite medium (0 x L); as is a short laboratory column How do we decide what is a short column or a very long column?? Similarly, the initial solute concentration, C (x,0), may be: (i) C (x,0) = f(x); solute concentration varying with distance (ii) C (x,0) = C i ; constant solute concentration with distance (iii) C (x,0) = 0; no solute initially in the medium Finally, various boundary conditions can also be specified. The concentration at x = 0 (i.e., the surface), C (0,t), may be specified as: (i) C (0,t) = f(t); time-varying solute concentration at x=0 (ii) C (0,t) = C o ; constant solute concentration at x=0 These same conditions can also be stated at x= L, the bottom boundary, (i.e., C(L,t) or C(+ ,t)). The surface or bottom boundary condition can also be specified as flux boundary condition.
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