Mathematical Model boundary conditions governing equation initial conditions

Mathematical Model boundary conditions governing equation & initial conditions boundary conditions

Initial conditions Zheng & Bennett

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

Simplest form of the ADE Uniform 1 D 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.

Co Aquifer is infinitely extensive

Zheng & Bennett

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.

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

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

ADE with 1 D flow & 3 D dispersion, retardation & 1 st order reaction term Uniform 1 D flow; 3 D dispersion; no sink/source term; retardation; 1 st order irreversible reaction ( is the same for the dissolved phase and the sorbed phase) Patch source solutions: 1. In an infinitely extensive aquifer: Illinois codes 2. In a confined (vertically bounded) aquifer: ATRANS

ATRANS

ATRANS Zheng & Bennett

ATRANS Boundary Conditions

ATRANS Example in Zheng & Bennett 3 D dispersion: Source is 0. 5 m thick (occupies 5% of the aquifer thickness) and 2 m wide (in the y direction). = 10 m 2 D dispersion: Source is 10 m thick (occupies 100% of the aquifer thickness) and 2 m wide (in the y direction). 1 D dispersion: Source is 10 m thick and infinitely wide in the y direction. Zheng & Bennett

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

Breakthrough curves Homework Problem Aquifer is 100 m thick. Source occupies 20% of the aquifer. 1 D 20 m 2 D 5 m wide 3 D Observation point is at x = 30 m. Observation point is on the centerline at x= 10 m 3 D dispersion: Source is 0. 5 m thick (occupies 5% of the aquifer thickness) and 2 m wide (in the y direction). 2 D dispersion: Source is 10 m thick (occupies 100% of the aquifer thickness) and 2 m wide (in the y direction). 1 D dispersion: Source is 10 m thick and infinitely wide in the y direction.

1 D 3 D

ADE with 1 D flow & 3 D dispersion, retardation & 1 st order reaction term Uniform 1 D flow; 3 D dispersion; no sink/source term; retardation; 1 st order irreversible reaction ( is the same for the dissolved phase and the sorbed phase) Patch source solutions: 1. In an infinitely extensive aquifer: Illinois codes 2. In a confined (vertically bounded) aquifer: ATRANS

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

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

Homework Problem Aquifer is 100 m thick. Source occupies 20% of the aquifer. Zheng & Bennett Fig. 5. 6 Aquifer is 10 m thick. Source is 0. 5 m thick. Source occupies 5% of the aquifer. 20 m 0. 5 m wide Observation point is at x = 30 m. 2 m wide Observation point is at x= 10 m

Homework Problem Z Z = 50 centerline X = 30

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