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