1 2 and 3 d analytical solutions to cde
Download
Skip this Video
Download Presentation
1-, 2-, and 3-D Analytical Solutions to CDE

Loading in 2 Seconds...

play fullscreen
1 / 26

1-, 2-, and 3-D Analytical Solutions to CDE - PowerPoint PPT Presentation


  • 95 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' 1-, 2-, and 3-D Analytical Solutions to CDE' - naif


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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
equation solved
Equation Solved:
  • Constant mean velocity in x direction!
slide5
Hunt, B., 1978, Dispersive sources in uniform ground-water flow, ASCE Journal of the Hydraulics Division, 104 (HY1) 75-85.
instantaneous source
‘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
‘Continuous’ Source
  • Solute mass flux
    • M1, M2, M3 = dM1,2,3/dt
  • Injection at origin of coordinate system (a point!)
2 d instantaneous source matlab
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
2-D Instantaneous Source Solution

Dyy

Dxx

t = 51

t = 25

t = 1

Back dispersion

Extreme concentration

3 d instantaneous source matlab
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
3-D Instantaneous Source Solution

Dzz

Dyy

Dxx

t = 1

t = 25

Back dispersion

t = 51

Extreme concentration

stanmod 3dade
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

z

z

y

y

x

x

Coordinate systems
  • x increasing downward

r

boundary conditions

z

y

x

Boundary Conditions
  • Semi-infinite source

-∞

-∞

boundary conditions1

z

y

x

Boundary Conditions
  • Finite rectangular source

b

-a

a

-b

boundary conditions2

z

y

x

Boundary Conditions
  • Finite Circular Source

r = a

initial conditions
Initial Conditions
  • Finite Cylindrical Source

z

y

r = a

x1

x2

x

initial conditions1
Initial Conditions
  • Finite Parallelepipedal Source

z

b

y

a

x1

x2

x

comparing with hunt

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
wells

z

b

y

a

x1

x2

x

Wells?
  • Finite Parallelepipedal Source
ad