Post on 31-Dec-2015
description
2- and 3-D Analytical Solutions to CDE
Equation Solved:
• Constant mean velocity in x direction!
t
CR
z
CD
y
CD
x
CD
x
Cv zzyyxx
2
2
2
2
2
2
•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:
)(x
dx
xHd
)()()( afdxaxxf
‘Continuous’ Source• Solute mass flux
– M1, M2, M3 = dM1,2,3/dt
• Injection at origin of coordinate system (a point!)
Instantaneous and Continuous Sources
• 1-D
tD
vtx
tD
MC
xxxx
i 4exp
2
21
tD
vtxerfc
D
vx
tD
vtxerfc
D
vx
v
Dxv
M
C
xxxx
xxxxxxc
22exp
22exp
2
2exp1
2-D Instantaneous Source
tD
y
tD
vtx
DDt
MC
yyxxyyxx
i 44exp
4
222
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
tD
y
tD
vtx
DDt
MC
yyxxyyxx
i 44exp
4
222
2-D Instantaneous Source Solution
Dyy
Dxx
Back dispersion Extreme concentration
t = 1t = 25
t = 51
3-D Instantaneous Source
tD
z
tD
y
tD
vtx
DDDt
MC
zzyyxxzzyyxx
i 444exp
8
222
33
3
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
tD
z
tD
y
tD
vtx
DDDt
MC
zzyyxxzzyyxx
i 444exp
8
222
33
3
3-D Instantaneous Source SolutionDzz
Dxx
Back dispersion
Extreme concentration
t = 1t = 25
t = 51
Dyy
3-D Continuous Source
tD
vtRerfc
D
Rv
tD
vtRerfc
D
Rv
DDR
Dxv
M
C
xxxx
xxxx
zzyy
xxc
22exp
22exp
8
2exp3
zz
xx
yy
xx
D
Dz
D
DyxR 222
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
b
a
Boundary Conditions
• Finite Circular Source
x
z
y
r = a
Initial Conditions
• Finite Cylindrical Source
x
z
y
r = a
x1
x2
Initial Conditions
• Finite Parallelepipedal Source
x
z
y
x1
x2
b
a
Comparing with Hunt
• M3 = r2 (x1 – x2) Co (=1, small, high C)
• Co = 1/[r2 (x1 – x2)] = 106 for r = x= 0.01 x
z
y
r =
a
x 1 x 2
Wells?• Finite Parallelepipedal Source
x
z
y
x1
x2
b
a