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1-, 2-, and 3-D Analytical Solutions to CDE

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.

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1-, 2-, and 3-D Analytical Solutions to CDE

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  1. 1-, 2-, and 3-D Analytical Solutions to CDE

  2. Equation Solved: • Constant mean velocity in x direction!

  3. Resident and Flux Concentrations

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

  5. ‘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:

  6. ‘Continuous’ Source • Solute mass flux • M1, M2, M3 = dM1,2,3/dt • Injection at origin of coordinate system (a point!)

  7. Instantaneous and Continuous Sources • 1-D

  8. 2-D Instantaneous Source

  9. 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

  10. 2-D Instantaneous Source Solution Dyy Dxx t = 51 t = 25 t = 1 Back dispersion Extreme concentration

  11. 3-D Instantaneous Source

  12. 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

  13. 3-D Instantaneous Source Solution Dzz Dyy Dxx t = 1 t = 25 Back dispersion t = 51 Extreme concentration

  14. 3-D Continuous Source

  15. 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.

  16. z z y y x x Coordinate systems • x increasing downward r

  17. z y x Boundary Conditions • Semi-infinite source -∞ -∞

  18. z y x Boundary Conditions • Finite rectangular source b -a a -b

  19. z y x Boundary Conditions • Finite Circular Source r = a

  20. Initial Conditions • Finite Cylindrical Source z y r = a x1 x2 x

  21. Initial Conditions • Finite Parallelepipedal Source z b y a x1 x2 x

  22. 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

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

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