1. 1-, 2-, and 3-D Analytical Solutions to CDE

2. Equation Solved:
Constant mean velocity in x direction!

4. Resident and Flux Concentrations

5. Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1)

6. ‘Instantaneous’ Source
Solute mass only
M1, M2, M3
Injection at origin of coordinate system (a point!) at t = 0
Dirac Delta function
Derivative of Heaviside:

7. ‘Continuous’ Source Solute mass flux
M1, M2, M3 = dM1,2,3/dt
Injection at origin of coordinate system (a point!)

8. Instantaneous and Continuous Sources

9. 2-D Instantaneous Source

10. 2-D Instantaneous Source (MATLAB)
%Hunt D dispersion solution Eqn.14.
clear
close('all')
[x y] = meshgrid(-1:0.05:3,-1:0.05:1);
M2=1
Dyy=.0001
Dxx=.001
theta=.5
V=0.04
for t=1:25:51
data = M2*exp(-(x-V*t).^2/(4*Dxx*t)-y.^2/(4*Dyy*t))/(4*pi*t*theta*sqrt(Dyy*Dxx));
contour(x, y, data)
axis equal
hold on
clear data
end

11. 2-D Instantaneous Source Solution
Dyy
Dxx
t = 51
t = 25
t = 1
Back dispersion
Extreme concentration

12. 3-D Instantaneous Source

13. 3-D Instantaneous Source (MATLAB)
%Hunt D dispersion solution Eqn.10.
clear
close('all')
[x y z] = meshgrid(-1:0.05:3,-1:0.05:1,-1:0.05:1);
M3=1
Dxx=.001
Dyy=.001
Dzz=.001
sigma=.5
V=0.04
for t=1:25:51
data = M3*exp(-(x-V*t).^2/(4*Dxx*t)-y.^2/(4*Dyy*t)-z.^2/(4*Dzz*t))/(8*sigma*sqrt(pi^3*t^3*Dxx*Dyy*Dzz));
p = patch(isosurface(x,y,z,data,10/t^(3/2)));
isonormals(x,y,z,data,p);
box on
clear data
set(p,'FaceColor','red','EdgeColor','none');
alpha(0.2)
view(150,30); daspect([1 1 1]);axis([-1,3,-1,1,-1,1])
camlight; lighting phong;
hold on
end

14. 3-D Instantaneous Source Solution
Dzz
Dyy
Dxx
t = 1
t = 25
Back dispersion
t = 51
Extreme concentration

17. 3-D Continuous Source

18. StAnMod (3DADE) Same equation (mean x velocity only)
Better boundary and initial conditions
Leij, F.J., T.H. Skaggs, and M.Th. Van Genuchten, Analytical solutions for solute transport in three-dimensional semi-infinite porous media, Water Resources Research 20 (10)

19. Coordinate systems
x increasing downward x z y x z y r

20. Boundary Conditions
Semi-infinite source x z y -∞ -∞

21. Boundary Conditions
Finite rectangular source x z y b -a a -b

22. Boundary Conditions
Finite Circular Source x z y
r = a

23. Initial Conditions
Finite Cylindrical Source z y
r = a x1 x2 x

24. Initial Conditions
Finite Parallelepipedal Source z b y a x1 x2 x

25. Comparing with Hunt M3 = qpr2 (x1 – x2) Co (=1, small, high C)z y
r = a x1 x2
M3 = qpr2 (x1 – x2) Co (=1, small, high C)
Co = 1/[pr2 (x1 – x2)] = 106 p for r = Dx= 0.01

26. Wells?
Finite Parallelepipedal Source x z y x1 x2 b a