Derivation of theAdvection-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 requiresa different governing equation. SeeZheng and Bennett, Chapter 15.

Figures from Freeze & Cherry (1979)

Derivation of theAdvection-Dispersion Equation (ADE)

s

hhKAQ

12

Darcy’s law:

s

h1

h2

q = Q/A

advective flux fA = q c

s

h1

h2

f = F/A

s

h1

h2fA = advective flux = qc

f = fA + fD

How do we quantify thedispersive flux?

s

ccADF 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 PorosityDomain

Figure from Freeze & Cherry (1979)

We need to introduce a “law” to describedispersion, to account for the deviation ofvelocities from the average linear velocitycalculated by Darcy’s law.

Average linear velocityTrue velocities

We will assume that dispersion followsFick’s law, or in other words, that dispersionis “Fickian”. This is an important assumption;it turns out that the Fickian assumption is notstrictly valid near the source of the contaminant.

s

ccDfD

12

where D is the dispersion coefficient.

porosity

Mathematically, porosity functions as a kind of units conversion factor.

Porosity ()

for example:

q c = v c

Later we will define the dispersion coefficientin terms of v and therefore we insert now:

s

ccDfD

12

Assume 1D flow

qx

s

and a line source

Case 1

cvcx

hhKcqf xxA

][12

Advective flux

x

ccDf xD

12

Dispersive flux

Assume 1D flow

qx

s

D is the dispersion coefficient. It includesthe effects of dispersion and diffusion. Dx is sometimeswritten DL and called the longitudinal dispersion coefficient.

porosityCase 1

Assume 1D flow

qx

s

and a point source

Case 2

fA = qxcAdvective flux

Dx represents longitudinal dispersion (& diffusion);Dy represents horizontal transverse dispersion (& diffusion);Dz represents vertical transverse dispersion (& diffusion).

)(12

x

ccDf xDx

)(12

z

ccDf zDz

Dispersive fluxes )(12

y

ccDf yDy

Figure from Freeze & Cherry (1979)

Continuous point source

Instantaneous point source

Averagelinearvelocity

center of mass

Figure from Wang and Anderson (1982)

InstantaneousPoint Source

transversedispersion

longitudinal dispersion

Gaussian

Derivation of the ADE for1D 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 ()

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 effectiveporosity for purposes of deriving the advection-dispersion eqn.See Zheng and Bennett, pp. 56-57.

Definition of the Dispersion Coefficientin a 1D uniform flow field

vx = a constantvy = 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 = 0.1x ; z = 0.1y

t

c

x

cv

z

cD

y

cD

x

cD 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?

t

c

x

cv

x

cD

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)

InstantaneousPoint Source

Gaussian

Breakthroughcurve

Concentrationprofile

long tail

Figure from Freeze & Cherry (1979)

Microscopic or local scale dispersion

Macroscopic Dispersion (caused by the presence of heterogeneities)

Homogeneous aquifer

Heterogeneousaquifers

Figure from Freeze & Cherry (1979)

Dispersivity () is a measure of the heterogeneity present in the aquifer.

A very heterogeneous porous mediumhas a higher dispersivity than a slightlyheterogeneous porous medium.

Dispersion in a 3D flow field

x

z

x’

z’

global local

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 =

z

hK

y

hK

x

hKq

z

hK

y

hK

x

hKq

z

hK

y

hK

x

hKq

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

z

cD

y

cD

x

cDf

z

cD

y

cD

x

cDf

z

cD

y

cD

x

cDf

zzzyzxDz

yzyyyxDy

xzxyxxDx

In a 3D flow field it is not possible to simplify the dispersiontensor 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 = xvx + Dd

Dy = yvx + Dd

Dz = zvx + Dd

Recall, that for1D uniform flow:

General form of the ADE:

Expands to 9 terms

Expands to 3 terms

(See eqn. 3.48 in Z&B)