# 1-, 2-, and 3-D Analytical Solutions to CDE - PowerPoint PPT Presentation

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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

### 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

• 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

• Solute mass flux

• M1, M2, M3 = dM1,2,3/dt

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

• 1-D

### 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

Dyy

Dxx

t = 51

t = 25

t = 1

Back dispersion

Extreme concentration

### 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

Dzz

Dyy

Dxx

t = 1

t = 25

Back dispersion

t = 51

Extreme concentration

### 3-D Continuous Source

• 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

### Coordinate systems

• x increasing downward

r

z

y

x

### Boundary Conditions

• Semi-infinite source

-∞

-∞

z

y

x

### Boundary Conditions

• Finite rectangular source

b

-a

a

-b

z

y

x

### Boundary Conditions

• Finite Circular Source

r = a

### Initial Conditions

• Finite Cylindrical Source

z

y

r = a

x1

x2

x

### Initial Conditions

• Finite Parallelepipedal Source

z

b

y

a

x1

x2

x

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

z

b

y

a

x1

x2

x

### Wells?

• Finite Parallelepipedal Source