geohydrology ii (2)
TRANSCRIPT
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Lecture (2)Lecture (2)
Transport Processes in Porous Media
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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?
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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).
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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.
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Physical ProcessesPhysical Processes
1. Advection
2. Molecular Diffusion
3. Mechanical Dispersion
4. Hydrodynamic Dispersion
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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
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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
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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
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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
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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
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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
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Dispersion Regimes at Micro-ScaleDispersion Regimes at Micro-Scale
D
VLPe
eff
cc
Peclet Number:Advection/Dispersion
Perkins and Johnston, 1963
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Chemical ProcessesChemical Processes
• Sorption & De-sorption.Sorption & De-sorption.
• Ion Exchange.Ion Exchange. • Retardation.Retardation.
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Adsorption IsothermsAdsorption Isotherms )(CfS
mbCS CKS d
21 kCkS
4
3
1 kCk
S
Freundlich (1926)
Langmuir (1915, 1918)
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Biological ProcessesBiological Processes
•Biological Degradation and Natural Attenuation.
•Micro-organisms Activity.
•Decay. C
dtCd
)(
is the decay coefficient
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Transport Through Porous MediaTransport Through Porous Media
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Derivation of Transport Equation in Derivation of Transport Equation in Rectangular CoordinatesRectangular Coordinates
Flow In – Flow Out = rate of change within the control volume
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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
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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
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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
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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.
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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
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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
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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
. .
. .
. .
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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
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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.
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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
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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).
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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)(/)(
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Flow
t = 0
f
t = Flowing Time
Var(X)
The spread of the front is a measure of the heterogeneity
Random WalkRandom Walk
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Analytical versus Random WalkAnalytical versus Random Walk
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Scale dependent dispersivity Scale dependent dispersivity
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Experimental Set upExperimental Set up
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Experiment No. 1Experiment No. 1
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Experiment No. 2Experiment No. 2