2. Lecture (2)Lecture (2)
1. After how many years the contaminant reaches a river or a
water supply well?
2. What is the level of concentration at the well?
3. Layout of the LectureLayout of the Lecture
• Transport Processes in Porous Media.Transport Processes in Porous Media.
• Derivation of The Transport Equation (ADE).Derivation of The Transport Equation (ADE).
• Methods of Solution.Methods of Solution.
• Effect of Heterogeneity on Transport:Effect of Heterogeneity on Transport:
Laboratory Experiments (movie).Laboratory Experiments (movie).
4. Transport ProcessesTransport Processes
1)1) Physical :Physical :
Advection-Diffusion-DispersionAdvection-Diffusion-Dispersion
2) Chemical:2) Chemical:
Adsorption- Ion Exchange- etc.Adsorption- Ion Exchange- etc.
3) Biological:3) Biological:
Micro-organisms ActivityMicro-organisms Activity
(Bacteria&Microbes)(Bacteria&Microbes)
4) Decay:4) Decay:
Radioactive Decay-Natural Attenuation.Radioactive Decay-Natural Attenuation.
6. Advection (Convection)Advection (Convection)
adv
J Cq=
Advective Solute Mass Flux:
.q = K− ∇Φ
is the advective solute mass flux,
is the solute concentration, and
is the water flux (specific discharge) given by Darcy's law:
C
q
adv
J
7. Molecular DiffusionMolecular Diffusion
Diffusive Flux in Bulk: (Fick’s Law of Diffusion)
is the diffusive solute mass flux in bulk,
dif
o oJ = - D C∇
dif
oJ
is the solute concentration gradient,C∇
is the molecular diffusive coefficient in bulk.oD
Random Particle motion
Time
t1
t2
t3
t4
8. Molecular Diffusion (Cont.)Molecular Diffusion (Cont.)
dif
effJ = - D C∇
τ
O
eff
D
D =
Diffusive Flux in Porous Medium
is the effective molecular diffusion
coefficient in porous medium,
effD
τ is a tortuosity factor ( = 1.4)
0.7eff oD D:
9. Mechanical DispersionMechanical Dispersion
dis
J = - Cε ∇.D
Depressive Flux in Porous Media (Fick’s Law):
is the depressive solute mass flux,
is the solute concentration gradient,
is the dispersion tensor,
is the effective porosity
dis
J
ε
C∇
D
xx xy xz
yx yy yz
zx zy zz
D D D
D D D
D D D
=
D
[after Kinzelbach, 1986]
Causes of Mechanical Dispersion
10. Hydrodynamic DispersionHydrodynamic Dispersion
( ) ( ) i j
ij efft ij l t
v v
= |v|+ + -D D
|v|
α δ α α
_hydo dis
J = - Cε ∇.D
Hydrodynamic Depressive Flux in Porous Media (Fick’s Law):
The components of the dispersion tensor in isotropic soil
is given by [Bear, 1972],
is Kronecker delta, =1 for i=j and =0 for i≠ j,ijδ
are velocity components in two perpendicular directions,i jv v
is the magnitude of the resultant velocity,v 2 2 2
i j kv v v v= + +
is the longitudinal pore-(micro-) scale dispersivity, andlα
tα is the transverse pore-(micro-) scale dispersivity
ijδ ijδ
11. Hydrodynamic Dispersion (Cont.)Hydrodynamic Dispersion (Cont.)
In case of flow coincides with the horizontal x-direction
all off-diagonal terms are zeros and one gets,
0 0
0 0
0 0
xx
yy
zz
D
D
D
=
D
xx effl
yy efft
zz efft
= |v|+D D
= |v|+D D
= |v|+D D
α
α
α
, 0.5
, 0.0157
3.5 Random packing
is the grain diameter
l l p l
t t p t
p
c d c
c d c
d
α σ
α σ
σ
= =
= =
=
12. Dispersion Regimes at Micro-ScaleDispersion Regimes at Micro-Scale
D
VL
Pe
eff
cc
=
Peclet Number:
Advection/Dispersion
Perkins and Johnston, 1963
17. Derivation of Transport Equation inDerivation of Transport Equation in
Rectangular CoordinatesRectangular Coordinates
Flow In – Flow Out = rate of change within the control volume
18. Solute Flux in the x-directionSolute Flux in the x-direction
( )
( )
( )
in adv dis
x x x
adv dis
out adv dis x x
x x x
J J J y z
J J
J J J x y z
x
∂
∂
= + ∆ ∆
+
= + + ∆ ∆ ∆
19. Solute Flux in the y-directionSolute Flux in the y-direction
( )
( )
( )
in adv dis
y y y
adv dis
y yout adv dis
y y y
J J J x z
J J
J J J y x z
y
∂
∂
= + ∆ ∆
+
= + + ∆ ∆ ∆
20. Solute Flux in the z-directionSolute Flux in the z-direction
( )
( )
( )
in adv dis
z z z
adv dis
out adv dis z z
z z z
J J J y x
J J
J J J z y x
z
∂
∂
= + ∆ ∆
+
= + + ∆ ∆ ∆
21. From Continuity of Solute MassFrom Continuity of Solute Mass
( )solute
in out
M
J J C x y z
t t
∂ ∂
ε
∂ ∂
− = = ∆ ∆ ∆∑ ∑
Where ε is the porosity, and
C is Concentration of the solute.
22. From Continuity of Solute MassFrom Continuity of Solute Mass
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
adv dis adv dis adv dis
x x y y z z
adv dis
adv dis x x
x x
adv dis
y yadv dis
y y
adv dis
adv dis z z
z z
J J y z J J x z J J y x
J J
J J x y z
x
J J
J J y x z
y
J J
J J z y x
z
C x y z
t
∂
∂
∂
∂
∂
∂
∂
ε
∂
+ ∆ ∆ + + ∆ ∆ + + ∆ ∆
+
− + + ∆ ∆ ∆
+
− + + ∆ ∆ ∆
+
− + + ∆ ∆ ∆
= ∆ ∆ ∆
23. By canceling out termsBy canceling out terms
( )( ) ( )
adv disadv dis adv dis
y yx x z z
J JJ J J J
z y x
x y z
∂∂ ∂
∂ ∂ ∂
++ +
− + + ∆ ∆ ∆
(C x y z
t
∂
ε
∂
= ∆ ∆ ∆ )
( )( ) ( )
( )
adv disadv dis adv dis
y yx x z z
J JJ J J J
x y z
C
t
∂∂ ∂
∂ ∂ ∂
∂
ε
∂
++ +
− + +
=
24. Assuming Advection and HydrodynamicAssuming Advection and Hydrodynamic
DispersionDispersion
,
,
,
adv dis
x x x xx xx
adv dis
y y y yy yy
adv dis
z z z zz zz
C
J = Cq J = - D C - D
x
C
J = Cq J = - D C - D
y
C
J = Cq J = - D C - D
z
ε ε
ε ε
ε ε
∂
∇ =
∂
∂
∇ =
∂
∂
∇ =
∂
. .
. .
. .
25. Solute Transport Through Porous Media bySolute Transport Through Porous Media by
advection and dispersion processesadvection and dispersion processes
( )
y yyx xx z zz
CC C
Cq - DCq - D Cq - D
yx z
x y z
C
t
∂ ε∂ ε ∂ ε
∂ ∂ ∂
∂
ε
∂
∂∂ ∂
÷ ÷ ÷∂∂ ∂ − + +
=
.. .
( ) ( ) ( )
Hyperbolic Part
x y z
Parabolic Part
xx yy zz
C
v C v C v C
t x y z
C C C
D D D
x x y y z z
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= − − −
+ + + ÷ ÷ ÷
644444474444448
644444444474444444448
26. General Form of The Transport EquationGeneral Form of The Transport Equation
[ ]
}
}
/
( ')
Dispersion Diffusion
Advection Source SinkChemical reaction
Decay
ij i
i j i
C C S C C W
v C + Q CD
t x x x
λ
ε ε
−
−
∂ ∂ ∂ ∂ −
= − − +
∂ ∂ ∂ ∂
6447448 64748 64748
where
C is the concentration field at time t,
Dij
is the hydrodynamic dispersion tensor,
Q is the volumetric flow rate per unit volume of the source or sink,
S is solute concentration of species in the source or sink fluid,
i, j are counters,
C’ is the concentration of the dissolved solutes in a source or sink,
W is a general term for source or sink and
vi
is the component of the Eulerian interstitial velocity in xi
direction
defined as follows,
ij
i
j
K
= -v
xε
∂Φ
∂
where
Kij
is the hydraulic conductivity tensor, and ε is the porosity of the medium.
27. Schematic Description of ProcessesSchematic Description of Processes
Figure 7. Schematic Description of the Effects of Advection, Dispersion, Adsorption, and Degradation on Pollution
Transport [after Kinzelbach, 1986].
Advection+Dispersion
Advection
Advection+Dispersion+Adsorption
Advection+Dispersion+Adsorption+Degradation
