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)
shhKAQ
∆−
−=12
Darcy’s law: h1
h2
q = Q/A
advective flux fA = q c
h1
h2
f = F/A
h1
h2 fA = advective flux = qc f = fA + fD
How do we quantify the dispersive flux?
sccADF dDiff
∆−
−=12
How about Fick’s law of diffusion? 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.
Dual Porosity Domain
Figure from Freeze & Cherry (1979)
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.
sccDfD
∆−
−=12θ
where D is the dispersion coefficient.
porosity
Mathematically, porosity functions as a kind of units conversion factor.
Porosity (θ)
for example:
q c = v θ c
Later we will define the dispersion coefficient in terms of v and therefore we insert θ now:
sccDfD
∆−
−=12θ
Assume 1D flow
and a line source
Case 1
cvcxhhKcqf xxA θ=
∆−
−== ][ 12
Advective flux
xccDf xD
∆−
−=12θ
Dispersive flux
Assume 1D flow
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 Case 1
Assume 1D flow
and a point source
Case 2
fA = qxc Advective flux
Dx represents longitudinal dispersion (& diffusion); Dy represents horizontal transverse dispersion (& diffusion); Dz represents vertical transverse dispersion (& diffusion).
)( 12
xccDf xDx
∆−
−= θ
)( 12
zccDf zDz
∆−
−= θ
Dispersive fluxes )( 12
yccDf yDy
∆−
−= θ
Figure from Freeze & Cherry (1979)
Continuous point source
Instantaneous point source
Average linear velocity
center of mass
Figure from Wang and Anderson (1982)
Instantaneous Point Source
transverse dispersion
longitudinal dispersion
Gaussian
Derivation of the ADE for 1D uniform flow and 3D dispersion (e.g., a point source in a uniform flow field)
f = fA + fD
Mass Balance: Flux out – Flux in = change in mass
vx = a constant vy = vz = 0
Porosity (θ)
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 = 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
tc
xcv
zcD
ycD
xcD zyx
∂∂
=∂∂
−∂∂
+∂∂
+∂∂
2
2
2
2
2
2
ADE for 1D uniform flow and 3D dispersion
No sink/source term; no chemical reactions
Question: If there is no source term, how does the contaminant enter the system?
tc
xcv
xcD
∂∂
=∂∂
−∂∂
2
2Simpler form of the ADE
Uniform 1D flow; longitudinal dispersion; No sink/source term; no chemical reactions
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.
Question: Is this equation valid for both point and line source boundaries?
Effects of dispersion on the concentration profile
(Zheng & Bennett, Fig. 3.11)
no dispersion dispersion
(Freeze & Cherry, 1979, Fig. 9.1)
t1 t2 t3 t4
Effects of dispersion on the breakthrough curve
Figure from Wang and Anderson (1982)
Instantaneous Point Source
Gaussian
Breakthrough curve
Concentration profile
long tail
Figure from Freeze & Cherry (1979)
Microscopic or local scale dispersion
Macroscopic Dispersion (caused by the presence of heterogeneities)
Homogeneous aquifer
Heterogeneous aquifers
Figure from Freeze & Cherry (1979)
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.
Dispersion in a 3D flow field
x
z
x’
z’
global local
θ
Kxx Kxy Kxz
Kyx Kyy Kyz
Kzx Kzy Kzz
K’x 0 0
0 K’y 0
0 0 K’z
[K] = [R]-1 [K’] [R]
K =
zhK
yhK
xhKq
zhK
yhK
xhKq
zhK
yhK
xhKq
zzzyzxz
yzyyyxy
xzxyxxx
∂∂
−∂∂
−∂∂
−=
∂∂
−∂∂
−∂∂
−=
∂∂
−∂∂
−∂∂
−=
Dispersion Coefficient (D)
D = D + Dd
Dxx Dxy Dxz
Dyx Dyy Dyz
Dzx Dzy Dzz
D =
In general: D >> Dd
D represents dispersion Dd represents molecular diffusion
zcD
ycD
xcDf
zcD
ycD
xcDf
zcD
ycD
xcDf
zzzyzxDz
yzyyyxDy
xzxyxxDx
∂∂
−∂∂
−∂∂
−=
∂∂
−∂∂
−∂∂
−=
∂∂
−∂∂
−∂∂
−=
θθθ
θθθ
θθθ
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.
Dx = αxvx + Dd
Dy = αyvx + Dd
Dz = αzvx + Dd
Recall, that for 1D uniform flow:
General form of the ADE:
Expands to 9 terms
Expands to 3 terms
(See eqn. 3.48 in Z&B)
