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.
Solute mass only Solute mass only Injection at origin of coordinate system (a point!) at t = 0
Solute mass flux Solute mass flux Injection at origin of coordinate system (a point!)
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]) 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
Semi-infinite source
Finite rectangular source
Finite Cylindrical Source Finite Cylindrical 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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