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