1 / 26

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.

estellee
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related