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

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

- Constant mean velocity in x direction!

- Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85.

- Solute mass only
- M1, M2, M3

- Injection at origin of coordinate system (a point!) at t = 0
- Dirac Delta function
- Derivative of Heaviside:

- Solute mass flux
- M1, M2, M3 = dM1,2,3/dt

- Injection at origin of coordinate system (a point!)

- 1-D

- %Hunt 1978 2-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

Dyy

Dxx

t = 51

t = 25

t = 1

Back dispersion

Extreme concentration

- %Hunt 1978 3-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

Dzz

Dyy

Dxx

t = 1

t = 25

Back dispersion

t = 51

Extreme concentration

- Same equation (mean x velocity only)
- Better boundary and initial conditions
- Leij, F.J., T.H. Skaggs, and M.Th. Van Genuchten, 1991. Analytical solutions for solute transport in three-dimensional semi-infinite porous media, Water Resources Research 20 (10) 2719-2733.

z

z

y

y

x

x

- x increasing downward

r

z

y

x

- Semi-infinite source

-∞

-∞

z

y

x

- Finite rectangular source

b

-a

a

-b

z

y

x

- Finite Circular Source

r = a

- Finite Cylindrical Source

z

y

r = a

x1

x2

x

- Finite Parallelepipedal Source

z

b

y

a

x1

x2

x

z

y

r = a

x1

x2

x

- M3 = qpr2 (x1 – x2) Co (=1, small, high C)
- Co = 1/[pr2 (x1 – x2)] = 106p for r = Dx= 0.01

z

b

y

a

x1

x2

x

- Finite Parallelepipedal Source