Constant mean velocity in x direction!

Constant mean velocity in x direction!

• 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.

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

Solute mass only

• 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

• Solute mass flux

• M1, M2, M3 = dM1,2,3/dt

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

1-D

• 1-D

%Hunt 1978 2-D dispersion solution Eqn.14.

• %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

%Hunt 1978 3-D dispersion solution Eqn.10.

• %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

Same equation (mean x velocity only)

• 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.

x increasing downward

• x increasing downward

Semi-infinite source

• Semi-infinite source

Finite rectangular source

• Finite rectangular source

Finite Circular Source

• Finite Circular Source

Finite Cylindrical Source

• Finite Cylindrical Source

Finite Parallelepipedal Source

• Finite Parallelepipedal Source

M3 = r2 (x1 – x2) Co (=1, small, high C)

• M3 = r2 (x1 – x2) Co (=1, small, high C)

• Co = 1/[r2 (x1 – x2)] = 106  for r = x= 0.01

Finite Parallelepipedal Source

• Finite Parallelepipedal Source