1. Solutions to the Advection-Dispersion Equation

2. Road Map to Solutions We will discuss the following solutions
Instantaneous injection in infinite and semi-infinite 1-dimensional columns
Continuous injection into semi-infinite 1-D column
Instantaneous point source solution in two-dimensions (line source in 3-D)
Instantaneous point source in 3-dimensions
Keep an eye on:
the initial assumptions
symmetry in space, asymmetry in time

3. Recall the Governing Equation
What have we assumed thus far?
Dispersion can be expressed as a Fickian process
Diffusion and dispersion can be folded into a single hydrodynamic dispersion
First order decay
What do we need next?
More Assumptions!

4. Adding Sorption
Thus far we have addressed only the solute behavior in the liquid state.
We now add sorption using a linear isotherm
Recall the linear isotherm relationship where cl and cs are in mass per volume of water and mass per mass of solid respectively
The total concentration is then

5. Retardation factor We have Which may be written as
where we have defined the retardation factor R to be

6. Putting this all together
The ADE with 1st order decay & linear isotherm
What do we need now? More assumptions!
 constant in space (pull from derivatives + cancel)
D’ constant in space (slide it out of derivative)
R constant in time (slide it out of derivative)
Use the chain rule: the divergence of
a scalar =gradient; divergence of a constant is zero

7. Applying the previous assumptions
The divergence operators turn into gradient operators since they are applied to scalar quantities.
What does this give us? The new ADE

8. Looking at 1-D case for a moment
To see how this retardation factor works, take t* = t/R, and  = 0. With a little algebra,
The punch line:
the spatial distribution of solutes is the same in the case of non-adsorbed vs. adsorbed compounds!
For a given boundary condition and time t*, the solution is unique and independent of R

9. 1-D infinite column Instantaneous Point Injection
Column goes to +and -
Area A
steady velocity u
mass M injected at x = o and t=0 (boundary condition)
initially uncontaminated column. i.e. c(x,0) = 0 (initial condition)
linear sorption (retardation R)
first order decay ()
velocity u
x = 0

10. Gaussian, symmetric in space, 2 = Dt/R Exponential decay of pulse
Features of solution:
Gaussian, symmetric in space, 2 = Dt/R
Exponential decay of pulse
Except for decay, R only shows up as t/R
Upstream
solutes
Peak at 1.23 hr
Center
of Mass
2 hr
Spatial Distribution
Temporal Distribution

11. Putting this in terms of Pore Volumes
If solute transport is dominated by advection and dispersion, then the process is really only dependent on total water displacement (the two transport processes scale linerarly with time).
This would not be true if there were mass transfer between the mobile solution and liquid within the particles, or if diffusion were significant.

12. Pore Volume form of Solution
Want concentration at x=L, in vs. pore-volumes passed through the system, P.
 Key substitutions:
Coefficient n relates time and volume: nt=P
At P=1, ut=L. This implies that u=Ln.
This gives us:
Now how do we get the n out?

13. Pore Volume Form of Solution
Note that D/n = DL/u = luL/u = lL
With this the result is entirely written in terms of the pore volumes and the media dispersivity, as we desired!

14. What about Retardation?
No Problem, just put it in as before:

15. 1-D semi-infinite Instantaneous Point Injection
How do we handle a surface application?
Use the linearity of the simplified ADE
Can add any two solutions, and still a solution
By uniqueness, any solution which satisfies initial and boundary conditions is THE solution
Boundary and initial conditions

16. Semi-infinite solution
Upward pulse and downward.
Only include region of x > 0 in domain
Solution symmetric about x = 0, therefore slope of dc(x=0,t)/dx = 0 for all t, as required
Compared to infinite column, c starts twice as high, but in time goes to same solution

17. Continuous injection, 1-D
Since the simplified ADE is linear, we use superposition. Basically get a continuous injection solution by adding infinitely many infinitely small Gaussian plumes.
Use the complementary error function: erfc

18. Plot of erfc(x)

19. Solution for continuous injection, 1-D

20. Plot of solution •
 = 0.1, R = 1, = 0.02, u = 1.0, and m = 1

21. 1-D, Cont., simplified
With no sorption or degradation this reduces to

22. 2-D and 3-D instantaneous solutions
Note:
- Same Gaussian form as 1-D
- Note separation of longitudinal and transverse dispersion

23. Review of Assumptions Assumption Effects if Violated
 constant in space -R higher where lower
-Velocity varies inversely with 
D constant in space -Increased overall dispersion due
to heterogeneity
D independent of scale - Plume will grow more slowly at
first, then faster.
Reversible Sorption - Increase plume spreading and
overall region of contamination
Equilibrium Sorption - Increased “tailing” and spreading
Linear Sorption - Higher peak C and faster travel
Anisotropic media - Stretching & smearing along beds
Heterogeneous Media - Greater scale effects of D and ALL
EFFECTS DISCUSSED ABOVE