geohydrology ii (2)
TRANSCRIPT
Lecture (2)Lecture (2)
Transport Processes in Porous Media
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?
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).
Transport ProcessesTransport Processes
1)1) Physical : Physical : Advection-Diffusion-DispersionAdvection-Diffusion-Dispersion2) 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.
Physical ProcessesPhysical Processes
1. Advection
2. Molecular Diffusion
3. Mechanical Dispersion
4. Hydrodynamic Dispersion
Advection (Convection)Advection (Convection)
advJ Cq
Advective Solute Mass Flux:
.q = K
is the advective solute mass flux,
is the solute concentration, and
is the water flux (specific discharge) given by Darcy's law:
Cq
advJ
Molecular DiffusionMolecular Diffusion
Diffusive Flux in Bulk: (Fick’s Law of Diffusion)
is the diffusive solute mass flux in bulk,
difo oJ = - D C
difoJ
is the solute concentration gradient,C is the molecular diffusive coefficient in bulk.oD
Random Particle motion
Time
t1
t2
t3
t4
Molecular Diffusion (Cont.)Molecular Diffusion (Cont.)
difeffJ = - D C
O
effD
D
Diffusive Flux in Porous Medium
is the effective molecular diffusion coefficient in porous medium,
effD
is a tortuosity factor ( = 1.4)
0.7eff oD D
Mechanical DispersionMechanical Dispersion
disJ = - C .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
disJ
CD
xx xy xz
yx yy yz
zx zy zz
D D DD D DD D D
D
[after Kinzelbach, 1986]
Causes of Mechanical Dispersion
Hydrodynamic DispersionHydrodynamic Dispersion
i jij efft ij l t
v v = | v | + + - D D| v |
_hydo disJ = - C .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,ijare velocity components in two perpendicular directions,i j v vis the magnitude of the resultant velocity,v 2 2 2
i j kv v v v is the longitudinal pore-(micro-) scale dispersivity, andl
t is the transverse pore-(micro-) scale dispersivity
ij ij
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 00 00 0
xx
yy
zz
DD
D
D
xx effl
yy efft
zz efft
= | v | + D D = | v | + D D = | v | + D D
, 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
Dispersion Regimes at Micro-ScaleDispersion Regimes at Micro-Scale
D
VLPe
eff
cc
Peclet Number:Advection/Dispersion
Perkins and Johnston, 1963
Chemical ProcessesChemical Processes
• Sorption & De-sorption.Sorption & De-sorption.
• Ion Exchange.Ion Exchange. • Retardation.Retardation.
Adsorption IsothermsAdsorption Isotherms )(CfS
mbCS CKS d
21 kCkS
4
3
1 kCk
S
Freundlich (1926)
Langmuir (1915, 1918)
Biological ProcessesBiological Processes
•Biological Degradation and Natural Attenuation.
•Micro-organisms Activity.
•Decay. C
dtCd
)(
is the decay coefficient
Transport Through Porous MediaTransport Through Porous Media
Derivation of Transport Equation in Derivation of Transport Equation in Rectangular CoordinatesRectangular Coordinates
Flow In – Flow Out = rate of change within the control volume
Solute Flux in the x-directionSolute Flux in the x-direction
( )
( )( )
in adv disx x x
adv disout adv dis x xx x x
J J J y z
J JJ J J x y zx
Solute Flux in the y-directionSolute Flux in the y-direction
( )
( )( )
in adv disy y y
adv disy yout adv dis
y y y
J J J x z
J JJ J J y x z
y
Solute Flux in the z-directionSolute Flux in the z-direction
( )
( )( )
in adv disz z z
adv disout adv dis z zz z z
J J J y x
J JJ J J z y xz
From Continuity of Solute Mass From Continuity of Solute Mass
( )solutein out
MJ J C x y zt t
Where is the porosity, andC is Concentration of the solute.
From Continuity of Solute MassFrom Continuity of Solute Mass
( ) ( ) ( )
( )( )
( )( )
( )( )
( )
adv dis adv dis adv disx x y y z z
adv disadv dis x xx x
adv disy yadv dis
y y
adv disadv dis z zz z
J J y z J J x z J J y x
J JJ J x y zx
J JJ J y x z
y
J JJ J z y xz
C x y zt
By canceling out termsBy canceling out terms
( )( ) ( )adv disadv dis adv disy yx x z zJ JJ J J J z y x
x y z
(C x y zt
)
( )( ) ( )
( )
adv disadv dis adv disy yx x z z
J JJ J J Jx y z
Ct
Assuming Advection and Hydrodynamic Assuming Advection and Hydrodynamic DispersionDispersion
,
,
,
adv disx x x xx xx
adv disy y y yy yy
adv disz z z zz zz
CJ = Cq J = - D C - DxCJ = Cq J = - D C - Dy
CJ = Cq J = - D C - Dz
. .
. .
. .
Solute Transport Through Porous Media by Solute Transport Through Porous Media by advection and dispersion processesadvection and dispersion processes
( )
y yyx xx z zz
CC CCq - DCq - D Cq - Dyx z
x y z
Ct
.. .
( ) ( ) ( )
Hyperbolic Part
x y z
Parabolic Part
xx yy zz
C v C v C v Ct x y z
C C CD D Dx x y y z z
General Form of The Transport EquationGeneral Form of The Transport Equation
/
( ')
Dispersion DiffusionAdvection Source SinkChemical reaction
Decay
ij ii j i
C C S C C W v C + Q C Dt x x x
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,
iji
j
K = - v
x
where Kij is the hydraulic conductivity tensor, and is the porosity of the medium.
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
Methods of SolutionMethods of Solution
1) Analytical Approaches:1) Analytical Approaches:2) Numerical Approaches:2) Numerical Approaches:
i)i) Eulerian Methods:(FDM,FEM).Eulerian Methods:(FDM,FEM).ii) Lagrangian Methods:(RWM).ii) Lagrangian Methods:(RWM).iii) Eulerian-Lagrangian Methods: iii) Eulerian-Lagrangian Methods: (MOC).(MOC).
Pulse versus Continuous InjectionPulse versus Continuous Injection
Concentration Distribution in case of Pulse and Continuous Injections in a 2D Field [after Kinzelbach, 1986].
tV4)Y-(y+
tV4)t V-X-(x-
tV4 tV4H) /(M =t)y,C(x,
xt
2o
xl
2xo
xtxl
o
exp d
tV4Y-y
+tV4
t V-X-x-
t V4
H M =ty,x,Ct
xt
2o
xl
2xo
tlx
o
0 )()(
)()((
exp1)(/)(
Flow
t = 0
f
t = Flowing Time
Var(X)
The spread of the front is a measure of the heterogeneity
Random WalkRandom Walk
Analytical versus Random WalkAnalytical versus Random Walk
Scale dependent dispersivity Scale dependent dispersivity
Experimental Set upExperimental Set up
Experiment No. 1Experiment No. 1
Experiment No. 2Experiment No. 2