1. Mathematical Model boundary conditions governing equation &
initial conditions

2. Initial
conditions
Zheng &
Bennett

3. Boundary Conditions no mass flux specified concentration
Specified mass flux
Zheng & Bennett

4. Simplest 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.

5. Co
Aquifer is infinitely extensive

6. Zheng & Bennett

7. Ogata and Banks solution in a spreadsheet
Short Form:
valid only for
small values of D
when x = v t; C/Co= ½
Spreadsheet can be accessed from the course homepage.

8. ADE with 1D flow: longitudinal dispersion & retardation
With instantaneous pulse input source at the boundary….

9. University of Illinois codes on the web
Find the link on the Geology 727 course homepage

10. ADE with 1D flow & 3D dispersion,
retardation & 1st order reaction term
Uniform 1D flow;
3D dispersion;
no sink/source term;
retardation;
1st order irreversible reaction ( is the same for the dissolved phase and the sorbed phase)
Patch source solutions:
In an infinitely extensive aquifer: Illinois codes
In a confined (vertically bounded) aquifer: ATRANS

12. ATRANS

13. ATRANS
Zheng & Bennett

14. ATRANS
Boundary Conditions

15. ATRANS Example in Zheng & Bennett
3D dispersion: Source is 0.5 m thick (occupies 5% of the aquifer thickness) and 2 m wide (in the y direction).
= 10 m
2D dispersion: Source is 10 m thick (occupies 100% of the aquifer thickness) and 2 m wide (in the y direction).
1D dispersion: Source is 10 m thick and infinitely wide in the y direction.
Zheng & Bennett

16. for 3D dispersion problem in Z&B
Source configuration
for 3D dispersion problem in Z&B
X = 10
Z = 5
centerline Z
Z = 0
Z = 10

17. Observation point is at x = 30 m.
Breakthrough curves
3D dispersion: Source is 0.5 m thick (occupies 5% of the aquifer thickness) and 2 m wide (in the y direction).
2D dispersion: Source is 10 m thick (occupies 100% of the aquifer thickness) and 2 m wide (in the y direction).
1D dispersion: Source is 10 m thick and infinitely wide in the y direction.
Homework Problem
Aquifer is 100 m thick.
Source occupies 20%
of the aquifer. 1D
20 m 2D
5 m wide 3D
Observation point is at x = 30 m.
Observation point is on the centerline at x= 10 m

18. 1D 3D

19. ADE with 1D flow & 3D dispersion,
retardation & 1st order reaction term
Uniform 1D flow;
3D dispersion;
no sink/source term;
retardation;
1st order irreversible reaction ( is the same for the dissolved phase and the sorbed phase)
Patch source solutions:
In an infinitely extensive aquifer: Illinois codes
In a confined (vertically bounded) aquifer: ATRANS

20. Homework Problem Z
Z = 100
centerline
Z = 50
X = 30
Z = 0

21. Problem Set #1 Aquifer is 100 m thick.
Source occupies 20% of the aquifer. z
20 m
5m wide y

22. Zheng & Bennett Fig. 5.6 Aquifer is 10 m thick.
Source is 0.5 m thick.
Source occupies 5% of the aquifer.
Homework Problem
Aquifer is 100 m thick.
Source occupies 20%
of the aquifer.
20 m
0.5 m
2 m wide
5 m wide
Observation point is at x = 30 m.
Observation point is
at x= 10 m

23. Homework Problem Z
centerline
Z = 50
X = 30

24. for 3D dispersion problem in Z&B
Source configuration
for 3D dispersion problem in Z&B
X = 10
Z = 5
centerline Z
Z = 0
Z = 10