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Mathematical Model

boundary conditions. boundary conditions. Mathematical Model. governing equation & initial conditions. Initial conditions. Zheng & Bennett. Boundary Conditions. no mass flux. specified concentration. Specified mass flux. Zheng & Bennett. Simplest form of the ADE.

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Mathematical Model

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  1. boundary conditions boundary conditions Mathematical Model 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. Short Form: valid only for small values of D when x = v t; C/Co= ½ Ogata and Banks solution in a spreadsheet 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

  11. ATRANS

  12. ATRANS Zheng & Bennett

  13. ATRANS Boundary Conditions

  14. 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

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

  16. 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

  17. 1D 3D

  18. 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

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

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

  21. 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

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

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

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