Average linear velocity
True velocities
We will assume that dispersion follows Fick’s law, or in other words, that dispersion is “Fickian（费克方程）”. This is an important assumption;it turns out that the Fickian assumption is not strictly valid（有效的） near the source of the contaminant.
How about Fick’s law （见下一张PPT） where D is the effective d of diffusion? diffusion coefficient.
c2 c1 FDiff DdA x
Fick’s law describes diffusion of ions on a molecular scale as ions diffuse from areas of higher to lower concentrations.
3. No density effects
Density-dependent flow requires a different governing equation. See Zheng and Bennett, Chapter 15.
Figures from Freeze & Cherry (1979)
c2 c1 fD Dx x
D is the dispersion coefficient. It includes the effects of dispersion and diffusion. Dx is sometimes written DL and called the longitudinal（纵向的） dispersion coefficient.
Tracer：示踪剂
Advective flux
fA = qxc
c2 c1 fDx Dx ( ) x
Dispersive fluxes
c 2 c1 fDy Dy ( ) y
c2 c1 fDz Dz ( ) z
Dx represents longitudinal dispersion (& diffusion); Dy represents horizontal transverse （水平横波）dispersion (& diffusion); Dz represents vertical transverse dispersion (& diffusion).
(i.e., a medium with connected pore space or a densely fractured medium with a single network of connected fractures)
2. Miscible flow（混相流动）
(i.e., solutes dissolve in water; DNAPL’s（重非轻亲 水相液体） and LNAPL’s （轻非轻亲水相液体） require a different governing equation. See p. 472, note 15.5, in Zheng and Bennett.)
(Zheng & Bennett, Fig. 3.8.)
Transverse：横向
We need to introduce a “law” to describe dispersion, to account for（解释） the deviation （偏差） of velocities from the average linear velocity calculated by Darcy’s law.
Dispersive Transport & Advection-dispersion Equation (ADE)
Advection only
C0
Advection & Dispersion
C0
C (viC ) xi t
v = q/θ
Assuming particles travel at same average linear velocity v=q/θ
Dx Lv* is the effective molecular diffusion coefficient [L2T-1]

is the tortuosity（扭转） factor [-]
1
Assume 1D flow
Case 2
and a point source
In fact, particles travel at different velocities v>q/θ or v<q/θ
Derivation（推导） of the Advection-Dispersion Equation (ADE) Assumptions
1. Equivalent（当量） porous medium (epm)
porosity
c2 c1 fD D x
where D is the dispersion coefficient.
Case 1
Advective flux Assume 1D flow
porosity
h2 h1 fA qxc [K ]c vxc x
Dispersive flux
Adective flux
h1 h2
Darcy’s law:
h2 h1 Q KA s
q = Q/A
advective flux fA = q c f = F/A
How to quantify the dispersive flux?
h1 h2 fA = advective flux = qc f = fA + fD