地下水污染物迁移模拟——对流弥散

ADE for 1D uniform flow and 3D dispersion
∂c ∂c Dx 2 + Dy 2 + Dz 2 − v = ∂x ∂t ∂x ∂y ∂z
No sink/source term; no chemical reactions
∂ 2c
∂ 2c
∂ 2c
Question: If there is no source term, how does the contaminant enter the system?
longitudinal dispersion transverse dispersion
Figure from Wang and Anderson (1982)
Derivation of the ADE for 1D uniform flow and 3D dispersion
(e.g., a point source in a uniform flow field)
3. No density effects
Density-dependent flow requires a different governing equation. See Zheng and Bennett, Chapter 15.
Figures from Freeze & Cherry (1979)
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 lenear velocity v=q/θ
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
c 2 − c1 fD = − Dxθ ∆x
D is the dispersion coefficient. It includes the effects of dispersion and diffusion. Dx is sometimes written DL and called the longitudinal dispersion coefficient.
porosity
c 2 − c1 fD = − Dθ ∆x
where D is the dispersion coefficient.
Case 1
Advective flux Assume 1D flow
porosity
h 2 − h1 fA = qxc = [− K ]c = vxθc ∆x
Dispersive flux
Simpler form of the ADE
ຫໍສະໝຸດ Baidu
∂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”. This is an important assumption; it turns out that the Fickian assumption is not strictly valid near the source of the contaminant.
∂ 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.
Advective flux
fA = qxc
c 2 − c1 fDx = − Dxθ ( ) ∆x
Dispersive fluxes
c 2 − c1 fDy = − Dyθ ( ) ∆y
c 2 − c1 fDz = − Dzθ ( ) ∆z
Dx represents longitudinal dispersion (& diffusion); Dy represents horizontal transverse dispersion (& diffusion); Dz represents vertical transverse dispersion (& diffusion).
Continuous point source
Average linear velocity
Instantaneous point source
center of mass
Figure from Freeze & Cherry (1979)
Instantaneous Point Source
Gaussian
fDx
∂c ∂c ∂c = −θD xx − θD xy − θD xz ∂x ∂y ∂z
Dispersion in a 3D flow field
z
global z’
local
x’
θ
Kxx Kxy Kxz K= Kyx Kyy Kyz Kzx Kzy Kzz
x
K’x 0 0 0 K’y 0
0 0 K’z
[K] = [R]-1 [K’] [R]
∂h ∂h ∂h q x = − K xx − K xy − K xz ∂x ∂y ∂z ∂h ∂h ∂h q y = − K yx − K yy − K yz ∂x ∂y ∂z ∂h ∂h ∂h q z = − K zx − K zy − K zz ∂x ∂y ∂z
vx = a constant vy = vz = 0
Dx = αxvx + Dd Dy = αyvx + Dd Dz = αzvx + Dd
where αx αy αz are known as dispersivities. Dispersivity is essentially a “fudge factor” to account for the deviations of the true velocities from the average linear velocities calculated from Darcy’s law. Rule of thumb: αy = 0.1αx ; αz = 0.1αy
Adective flux
h1 h2
Darcy’s law:
h 2 − 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
Dx = α L v + D*
D = τD0
*
D* is the effective molecular diffusion coefficient [L2T-1]
τ
is the tortuosity factor [-]
τ <1
Assume 1D flow
Case 2
and a point source
Instantaneous Point Source
Gaussian
Figure from Wang and Anderson (1982)
Breakthrough curve
long tail
Concentration profile
Microscopic or local scale dispersion
How about Fick’s law of diffusion?
c 2 − c1 FDiff = − DdA ∆x
where Dd is the effective diffusion coefficient.
Fick’s law describes diffusion of ions on a molecular scale as ions diffuse from areas of higher to lower concentrations.
(Zheng & Bennett, Fig. 3.8.)
We need to introduce a “law” to describe dispersion, to account for the deviation of velocities from the average linear velocity calculated by Darcy’s law.
Figure from Freeze & Cherry (1979)
Macroscopic Dispersion (caused by the presence of heterogeneities)
Homogeneous aquifer
Heterogeneous aquifers
Figure from Freeze & Cherry (1979)
(i.e., a medium with connected pore space or a densely fractured medium with a single network of connected fractures)
2. Miscible flow
(i.e., solutes dissolve in water; DNAPL’s and LNAPL’s require a different governing equation. See p. 472, note 15.5, in Zheng and Bennett.)
In fact, particles travel at different velocities v>q/θ or v<q/θ
Derivation of the Advection-Dispersion Equation (ADE) Assumptions 1. Equivalent porous medium (epm)
Dispersion Coefficient (D)
D = D + Dd D represents dispersion Dd represents molecular diffusion Dxx Dxy Dxz D = Dyx Dyy Dyz Dzx Dzy Dzz
In general: D >> Dd
Dispersivity (α) 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.
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