1 2 and 3 d analytical solutions to cde
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1-, 2-, and 3-D Analytical Solutions to CDE. Equation Solved:. Constant mean velocity in x direction!. Resident and Flux Concentrations. Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85. ‘Instantaneous’ Source.

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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

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


Equation solved

Equation Solved:

  • Constant mean velocity in x direction!


Resident and flux concentrations

Resident and Flux Concentrations


1 2 and 3 d analytical solutions to cde

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


Instantaneous source

‘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:


Continuous source

‘Continuous’ Source

  • Solute mass flux

    • M1, M2, M3 = dM1,2,3/dt

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


Instantaneous and continuous sources

Instantaneous and Continuous Sources

  • 1-D


2 d instantaneous source

2-D Instantaneous Source


2 d instantaneous source matlab

2-D Instantaneous Source (MATLAB)

  • %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


2 d instantaneous source solution

2-D Instantaneous Source Solution

Dyy

Dxx

t = 51

t = 25

t = 1

Back dispersion

Extreme concentration


3 d instantaneous source

3-D Instantaneous Source


3 d instantaneous source matlab

3-D Instantaneous Source (MATLAB)

  • %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


3 d instantaneous source solution

3-D Instantaneous Source Solution

Dzz

Dyy

Dxx

t = 1

t = 25

Back dispersion

t = 51

Extreme concentration


3 d continuous source

3-D Continuous Source


Stanmod 3dade

StAnMod (3DADE)

  • 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.


Coordinate systems

z

z

y

y

x

x

Coordinate systems

  • x increasing downward

r


Boundary conditions

z

y

x

Boundary Conditions

  • Semi-infinite source

-∞

-∞


Boundary conditions1

z

y

x

Boundary Conditions

  • Finite rectangular source

b

-a

a

-b


Boundary conditions2

z

y

x

Boundary Conditions

  • Finite Circular Source

r = a


Initial conditions

Initial Conditions

  • Finite Cylindrical Source

z

y

r = a

x1

x2

x


Initial conditions1

Initial Conditions

  • Finite Parallelepipedal Source

z

b

y

a

x1

x2

x


Comparing with hunt

z

y

r = a

x1

x2

x

Comparing with Hunt

  • M3 = qpr2 (x1 – x2) Co (=1, small, high C)

  • Co = 1/[pr2 (x1 – x2)] = 106p for r = Dx= 0.01


Wells

z

b

y

a

x1

x2

x

Wells?

  • Finite Parallelepipedal Source


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