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!)

Instantaneous and Continuous Sources

2-D Instantaneous Source

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

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

StAnMod (3DADE) 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.

Coordinate systems x increasing downward x z y x z y r

Boundary Conditions Semi-infinite source x z y -∞ -∞

Boundary Conditions Finite rectangular source x z y b -a a -b

Boundary Conditions Finite Circular Source x z y 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

Comparing with Hunt M3 = qpr2 (x1 – x2) Co (=1, small, high C) z y r = a x1 x2 M3 = qpr2 (x1 – x2) Co (=1, small, high C) Co = 1/[pr2 (x1 – x2)] = 106 p for r = Dx= 0.01

Wells? Finite Parallelepipedal Source x z y x1 x2 b a