Transcript Document

Derivation of the
Advection-Dispersion Equation (ADE)
Assumptions
1. Equivalent porous medium (epm)
(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.)
3. No density effects
Density-dependent flow requires
a different governing equation. See
Zheng and Bennett, Chapter 15.
Figures from Freeze & Cherry (1979)
Derivation of the
Advection-Dispersion Equation (ADE)
Darcy’s law:
h1
h2  h1
Q   KA
s
h2
q = Q/A
advective flux
fA = q c
f = F/A
How do we quantify the
dispersive flux?
h1
h2
fA = advective flux = qc
f = fA + fD
FDiff
How about
Fick’s law of diffusion?
c 2  c1
  DdA
s
where Dd is the effective
diffusion coefficient.
Dual Porosity
Domain
Figure from Freeze & Cherry (1979)
Fick’s law describes diffusion of ions on a
molecular scale as ions diffuse from areas of
higher to lower concentrations.
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.
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.
porosity
c 2  c1
fD   D
s
where D is the dispersion coefficient.
Porosity ()
Mathematically, porosity functions as a kind of
units conversion factor.
for example:
qc=vc
Later we will define the dispersion coefficient
in terms of v and therefore we insert  now:
c 2  c1
fD   D
s
Case 1
and a line source
Assume 1D flow
Case 1
porosity
Advective flux
Assume 1D flow
h2  h1
fA  qxc  [K
]c  vxc
x
Dispersive flux
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 dispersion coefficient.
Assume 1D flow
Case 2
and a point source
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 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
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)
vx = a constant
vy = vz = 0
f = fA + fD
c2  c1
fDx   Dx(
)
x
c 2  c1
fDy   Dy(
)
y
c2  c1
fDz   Dz (
)
z
Mass Balance:
Flux out – Flux in = change in mass
Porosity ()
There are two types of porosity in transport problems:
total porosity and effective porosity.
Total porosity includes immobile pore water, which contains
solute and therefore it should be accounted for when
determining the total mass in the system.
Effective porosity accounts for water in interconnected pore
space, which is flowing/mobile.
In practice, we assume that total porosity equals effective
porosity for purposes of deriving the advection-dispersion eqn.
See Zheng and Bennett, pp. 56-57.
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. 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: y = 0.1x ; z = 0.1y
ADE for 1D uniform flow
and 3D dispersion
 2c
 2c
 2c
c
c
Dx 2  Dy 2  Dz 2  v

x
t
x
y
z
No sink/source term; no chemical reactions
Question: If there is no source term, how does
the contaminant enter the system?
Simpler form of the ADE
 2c
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 sources?
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.
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
Instantaneous
Point Source
Gaussian
Figure from Wang and Anderson (1982)
Breakthrough
curve
Concentration
profile
long tail
Microscopic or local scale dispersion
Figure from Freeze & Cherry (1979)
Macroscopic Dispersion
(caused by the presence of heterogeneities)
Homogeneous aquifer
Heterogeneous
aquifers
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.
Dispersion in a 3D flow field
z
global
local
z’
x’

K’x 0
0
Kyx Kyy Kyz
0
K’y
0
Kzx Kzy Kzz
0
0
K’z
Kxx Kxy Kxz
K=
x
[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
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
fDx
c
c
c
 Dxx
 Dxy
 Dxz
x
y
z
fDy
c
c
c
 Dyx
 Dyy
 Dyz
x
y
z
fDz
c
c
c
 Dzx
 Dzy
 Dzz
x
y
z
In a 3D flow field it is not possible to simplify the dispersion
tensor to three principal components. In a 3D flow field,
we must consider all 9 components of the dispersion tensor.
The definition of the dispersion coefficient is more
complicated for 2D or 3D flow. See Zheng and Bennett,
eqns. 3.37-3.42.
Recall, that for
1D uniform flow:
Dx = xvx + Dd
Dy = yvx + Dd
Dz = zvx + Dd
General form of the ADE:
Expands to 9 terms
Expands to 3 terms
(See eqn. 3.48 in Z&B)