2c
There is a famous analytical solution to this form of the ADE with a continuous line source boundary condition. The solution is called the Ogata & Banks solution.
Figure from Freeze & Cherry (1979)
Dispersivity () is a measure of the heterogeneity present in the aquifer.
A very heterogeneous porous medium has a higher dispersivity than a slightly heterogeneous porous medium.
Continuous point source
Average linear velocity
Uniform：均匀的
Instantaneous（瞬时的） point source
center of mass
Figure from Freeze & Cherry (1979)
Instantaneous Point Source
(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.)
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.
Effects of dispersion on the concentration profile
no dispersion dispersion
t1
t2 t3 t4
(Freeze & Cherry, 1979, Fig. 9.1)
(Zheng & Bennett, Fig. 3.11)
Effects of dispersion on the breakthrough curve
Figure from Freeze & Cherry (1979)
Macroscopic Dispersion (caused by the presence of heterogeneities（异质性）) Plug：栓子；dilution：稀释
Homogeneous aquifer
Heterogeneous aquifers
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)
Dx Lv D*
D D0
*
D* is the effective molecular diffusion coefficient [L2T-1]

is the tortuosity（扭转） factor [-]
1
Assume 1D flow
Case 2
and a point source
Dispersion in a 3D flow field
z
global
z’
local
x’

Kxx Kxy Kxz
x
K’x 0
0
K=
Kyx Kyy Kyz
Kzx Kzy Kzz
0
0
K’y
0
0
K’z
[K] = [R]-1 [K’] [R]
h h h qx Kxx Kxy Kxz x y z h h h qy Kyx Kyy Kyz x y z h h h qz Kzx Kzy Kzz x y z
Instantaneous Point Source
Gaussian
Figure from Wang and Anderson (1982)
Breakthrough Curve （浓度比值和时间的曲线）
long tail
Concentration Profile （浓度比值和距离的曲线）
Microscopic or local（局部的） scale dispersion
(e.g., a point source in a uniform flow field)
vx = a constant vy = vz = 0
f = fA + fD Mass Balance: Flux out – Flux in = change in mass
Definition of the Dispersion Coefficient in a 1D uniform flow field
vx = a constant vy = vz = 0
Dx = xvx + Dd
Dy = yvx + Dd
Dz = zvx + Dd
where x y z are known as dispersivities（弥散度）. Dispersivity is essentially（本质上） a “fudge （蒙混） factor” to account for the deviations of the true velocities from the average linear velocities calculated from Darcy’s law. Rule of thumb: y = 0.1x ; z = 0.1y
3. No density effects
Density-dependent flow requires a different governing equation. See Zheng and Bennett, Chapter 15.
Figures from Freeze & Cherry (1979)
(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.
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
Tracer：示踪剂
Advective 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).
Simpler form of the ADE
c c D 2 v x t x
Uniform 1D flow; longitudinal dispersion; No sink/source term; no chemical reactions Question: Is this equation valid for both point and line source boundaries?
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.