Constant mean velocity in x direction!


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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




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