2- and 3-D Analytical Solutions to CDE

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2- and 3-D Analytical Solutions to CDE. Equation Solved:. 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. ‘Instantaneous’ Source. Solute mass only M1, M2, M3

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### 2- and 3-D Analytical Solutions to CDE

Equation Solved:
• 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.
‘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!)
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

• 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