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Transport in Permeable Media

TPM

Leo Pel, Henk Huinink, David Smeulders, Thomas Arends, Hans van Duijn

Faculty of Applied Physics Mechanical Engineering

Eindhoven University of Technology The Netherlands

[email protected]

5 ECTS 2018

Examination : Oral

Transport in porous media 3MT130




TPM Surface tensions

Curved surface

Pressure difference

Unsaturated

Saturated




TPM




TPM

Surface tensions

Curved surface

Pressure difference

wnc rp γ2

−=




TPM

trxµγ2

=rg

hργ2

max =

Small pores

• slow absorption

• but very high

Large pores

• fast absorption

• but low




TPM

Liquid ‘fast’ Vapour ‘slow’

Same macroscopic pressure: suction




TPM

( )z

KDt ∂

∂+∇∇=

∂∂ )()( θθθθ

Richards equation

First order in time and second order in space; require 1. initial condition and 2. boundary conditions Outcome: θ as function x and t




TPM

Some Human Activities that Can Contaminate Groundwater




TPM

Radioactive contaminants




TPM

Movie Eric Doehne www.getty.edu/conservation/science

Madame John’s Legacy 1788

Cultural Heritage




TPM






TPM

Debris from salt weathering (6 months)



New Orleans




TPM

Transport of components saturated non-saturated




TPM

The laws of Fick First law:

Second law:

cDtc 2∇=∂∂

Concentration peaks are chopped

02 <∇ c

Concentration valleys are filled up

02 >∇ c

1831-1879

1th : Diffusion

∂∂

∂∂

=∂∂

=∂∂

xCD

xxq

tC x

cTDtc 2∇=∂∂

Porous medium




TPM

cDtc 2∇=∂∂

cTDtc 2∇=∂∂

Porous medium

Path gets longer: tortuosity




TPM

cDtc 2∇=∂∂

cTDtc 2∇=∂∂

Porous medium

Measure the diffusivity by NMR

Observation time:

Time Length 1 10-6 31 nm 1 10-3 1 µm

1 30 µm




TPM

R. W. Mair et all, Phys Rev Let 1999

Example of diffusion measurement by NMR




TPM

t∂∂θ

qin qout 0. =∇+∂∂ q

tθ

∂∂

∂∂

=∂∂

=∂∂

xCD

xxq

tC

porx

There is more: Ad/desorption on pore wall Cs(c)

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TPM




TPM

t∂∂θ

qin qout 0. =∇+∂∂ q

tθ

tS

xCD

xxq

tC

porx

∂∂

+

∂∂

∂∂

=∂∂

=∂∂

Sink term due to binding

http://images.google.com/imgres?imgurl=http://media.ebaumsworld.com/mediaFiles/picture/558765/783109.jpg&imgrefurl=http://www.ebaumsworld.com/pictures/view/783109/&usg=__KSJFzOup1qHczkp25j4b2WbidCk=&h=533&w=400&sz=27&hl=en&start=27&um=1&itbs=1&tbnid=4w69UlJ3vUiwYM:&tbnh=132&tbnw=99&prev=/images%3Fq%3Dbeer%2Bgut%26start%3D21%26um%3D1%26hl%3Den%26sa%3DN%26rlz%3D1T4ADBR_enUS309%26ndsp%3D21%26tbs%3Disch:1




TPM

Consider changes in the mass of solute by adsorbing onto the solid soil matrix, given by ρbs, where ρb is the soil bulk density and s is the adsorbed concentration in terms of mass of solute per mass of soil

( )

∂∂

∂∂

=∂+∂

xcD

xtsc

effbρ

Binding




TPM

( )

∂∂

∂∂

=∂+∂

xcD

xtsc

effbρ

Binding

( )

∂∂

∂∂

=∂

∂xcD

xtRc

eff csR bρ+=1With

Retardation factor




TPM

( )

∂∂

∂∂

=∂+∂

xcD

xtsc

effbρ

( )

∂∂

∂∂

=∂

∂xcD

xtRc

eff csR bρ+=1With

Simplest case Kcsb =ρ

∂∂

+∂∂

=∂∂

xc

KD

xtc eff

1

KR +=1

Diffusion gets ‘slower’




TPM Other mechanism?

Water flow : advection




TPM

ADVECTION • Chemical transport due to bulk movement of the fluid. • The fastest form of chemical transport in porous

media. • Concentration decreases in the direction of fluid

movement.

xCU

tC

∂∂

−=∂∂CUq −=

Darcy law liquid

ww JUthatNoteJU >→=θ

Darcy law liquid




TPM

−

∂∂

∂∂

=∂∂ CU

xCD

xtC

effθθ

−

∂∂

∂∂

=∂∂ CU

xCD

xtC

eff

Saturated porous medium

Non-saturated porous medium: Ion transport only in the liquid

Transport can only be liquid of component




TPM




TPM

29

Reasons for Spreading: mechanical dispersions

Some solute mass travels faster than average, while some solute mass travels slower than average

Completely dependent on flow




TPM

Completely dependent on flow




TPM




TPM Dispersion

Mechanical dispersion - caused by motion of the fluid

Longitudinal dispersion – along the streamline

Transverse dispersion – perpendicular to flow path

Flow Direction




TPM




TPM Experiment




TPM

Advection, Diffusion, Dispersion




TPM

−

∂∂

∂∂

=∂∂ CU

xCD

xtC

effθθ

−

∂∂

∂∂

=∂∂ CU

xCD

xtC

eff

Saturated porous medium

Non-saturated porous medium: Ion transport only in the liquid

Deff= diffusion + tortuosity + dispersion




TPM

NaCl

Wind

damage




TPM

Damage to rising damp in city of Venice

2004 2007




TPM

How are ions moving?

Characterize the transport?




TPM drying

surface

airflow

Wick action conceptual model

supply

Drying front at surface (drying externally limited)

= Liquid velocity is function of drying rate

= position drying front

moisture flow

u=constant

See also sharp front model C. Hall




TPM drying

surface

airflow


supply

moisture flow

advection




TPM drying

surface

airflow


supply

moisture flow

accumulation > crystallization

For NaCl max concentration = 6M

advection




TPM drying

surface

airflow


supply

moisture flow

advection

(neglect adsorption)

diffusion

accumulation leveling off

competition




TPM

Peclet number

DLUPe =

liquid velocity

length of the sample

diffusion coefficient of Na in porous medium

:U:L:D

competition advection diffusion

1>Pe1<Pe

accumulation uniform distribution

see e.g.H.P. Huinink et al, Phys. Fluids 14, 1389 (2002)




TPM

effC cD uCt x x

∂ ∂ ∂ = − ∂ ∂ ∂

Diffusion Advection + = flux

Initial profiles

Advection diffusion equation for transport




TPM

effC cD uCt x x

∂ ∂ ∂ = − ∂ ∂ ∂

Initial profiles

airflow supply

moisture flow

advection

q=0 Ions can not leave

q= uCo continuous supply

Boundary conditions




TPM

effC cD uCt x x

∂ ∂ ∂ = − ∂ ∂ ∂ Diffusion Advection

Boundary conditions: Top : q=0 Bottom : q= uCo

+ = flux

Simple solution:

Initial profiles

• Exponential decay

• Width peak =4D/U (e-4~0)




TPM

effC cD uCt x x

∂ ∂ ∂ = − ∂ ∂ ∂ Diffusion Advection

Boundary conditions: Top : q=0 Bottom : q= uCo

After reaching the solubility limit> crystallization

C*=6 for NaCl




TPM

Na lower sensitivity longer

measurement time

Signal proportional

to

moisture content

or

Na content

Pulsed NMR signal (spin-echo experiment)

Information on

water and ion

in pores Amplitude spin-echo S~Gρ [1-exp(-TR/T1)] exp(-TE/T2) G = relative sensitivity (for 1H G=1, 23Na=0.1) ρ = density of nuclei

T1 = spin lattice relaxation

TR = repetition time experiment

T2 = spin-spin relaxation time

TE = spin-echo time

see,e.g.,E.L. Hahn,Phys. Rev., 80, 580-594 (1950)




TPM Experimental setup

pump

NMR measurement

Measurements

- NMR moisture profile

- NMR Na profile

(only free ions: no crystals)

1m NaCl reservoir

electrical level control NaCl

Step motor

0% RH air flow

epoxy coating

evaporation shield top

bottom

100

mm




TPM

( )0

,1

2 .eff

C x t xerfC D t

= −

C CDt x x

∂ ∂ ∂ = ∂ ∂ ∂

9 20.8 10 /D m s−= ×

0 1 2 3 4 5 6 7 8

x 10-5

0

0.5

1

1.5

2

2.5

3

3.5

4

x/sqrt(t) [m s-0.5]

Na

conc

entra

tion

[m]

No airflow > only diffusion




TPM Results with airflow




TPM Results

No 6M ????




TPM

Stone sample saturated with 1 M NaCL solution

position

Con

cent

ratio

n 1




TPM


position

Con

cent

ratio

n

1




TPM

Limestone sample saturated with 1 M NaCl solution

position

Con

cent

ratio

n

1




TPM


position

Con

cent

ratio

n

1




TPM


position

Con

cent

ratio

n

1

resolution




TPM Results

1D resolution: average over slice




TPM Results: model fit data




TPM Results: model fit data

Max concentration = 6 m

Decay width ~ 80 mm




TPM Integral of concentration

No crystallization

crystallization at 6 m

linear increase ucot




TPM



How to clear a wall (painting) of the salt?




TPM Conservators: poulticing




TPM

General idea of poulticing

poultice substrate

transport Water absorption




TPM

poultice substrate What are the mechanisms ???? - time scales?

- efficiency?

- poresize dependence?

General idea of poulticing




TPM Working principle of poulticing

Diffusion

Diffusion of ink in glass of water




TPM

General idea of poulticing by diffusion

poultice substrate

diffusion Water absorption




TPM

General idea of poulticing by diffusion

poultice substrate

diffusion

TO KEEP THE DIFFUSION GOING

(maintain sink)

RENEW POULTICE VERY OFTEN




TPM

2

2

xD

tC

∂∂

=∂∂ C

Desalination by diffusion process:

D (m2s-1) water Bentheimer fired clay brick

NaCl 1.1 10-9 0.4 10-9 0.8 10-9

Na2SO4 1.1 10-9 0.4 10-9 0.85 10-9

In the order of 1 10-9 m2s-1

TIME SCALE?




TPM

Poulticing side

(where salt comes out)

Time in days

Salt concentration




TPM

Diffusion Pros • If enough time, can have 100 % efficiency • No pore size dependency

Cons • Slow ( 80% in 10 days for first 25 mm) • Renew poultice very often • Sample wet (long time, bio degradation)) • At end, dry sample (salt damage)




TPM

Equations of moisture and ion transport

∂∂

∂∂

+

∂∂

∂∂

=∂∂

xCD

xxD

xt cθθ

θ

−

∂∂

∂∂

=∂∂ CU

xCD

xtC

effθθ

moisture

salt

So two couple non-linear partial differential equations

+ boundary conditions

We do not learn anything !!!!

The competition




TPM

drying

surface

airflow

How do we get salt effloresence???




TPM

drying

surface

airflow

moisture flow

Saline drying conceptual model




TPM

Saline drying conceptual model drying

surface

advection airflow

moisture flow




TPM


surface

airflow

moisture flow

accumulation > crystallization




TPM


surface

advection


airflow

moisture flow

accumulation




TPM


surface

advection


airflow

moisture flow

diffusion

accumulation leveling off

competition




TPM

Peclet number

DLUPe =

liquid velocity

length of the sample

diffusion coefficient of Na in the pores

:U:L:D

competition advection diffusion

1>Pe1<Pe

accumulation uniform distribution

see e.g.H.P. Huinink et al, Phys. Fluids 14, 1389 (2002)




TPM

Drying experiment

The initially saturated sample is sealed at all sides, except for the top

The sample is moved by step motor

The one-dimensional resolution ~ 1 mm

The measurement of a profile takes ~ 3 hours

Webcam for visual inspection

NMR only free Na ions are measured: no crystals

Pel et al, Applied Physics Letters 81, 2893-2895 (2002)




TPM

NaCl

Na concentration

profiles at various

drying times

drying surface

0 days





TPM

drying surface

0 days1

Pe~3

NaCl

Na concentration

profiles at various

drying times





TPM

drying surface

NaCl max concentration 6 M

crystallization

0 days13

NaCl

Na concentration

profiles at various

drying times





TPM

drying surface

0 days13

6

Pe~0.7

NaCl

Na concentration

profiles at various

drying times





TPM

drying surface

0 days13

6

9

NaCl

Na concentration

profiles at various

drying times





TPM

drying surface

0 days13

6

9

12,15

NaCl

Na concentration

profiles at various

drying times





TPM

0 days

13

6

9

12,15

Pe>1

Pe<1

accumulation

leveling off

Pe number is simple indication





TPM

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

0

1

2

3

4

5

SavgCavg

Savg

Pe < 1Pe > 1

S

avgC

avg (

mol

l-1)

S avg (-

)

time (days)

Savg is the average (water) saturation of the sample (drying curve)

Savg Cavg represents the total amount of dissolved NaCl

I II III

I: Pe ~ 3 accumulation

II: Pe ~ 0.7 leveling off

III: homogeneous at 6 M




TPM

poultice substrate

advection

General idea of poulticing by advection

Water absorption airflow




TPM

poultice substrate

General idea of poulticing by advection




TPM

Demands on poultice Step 1) Water is absorbed from poulice into substrate

Step 2) Reverse of water flow, i.e., from substrate into poultice

poultice substrate

Absorption

poultice substrate

Advection




TPM

Maximum height > capillary pressure

wnc rp γ2

=

Capillary pressure

Conclusions

1) A porous material will absorb water

2) Small porous will absorb water from larger pores

=

Water wants to stay

in small pores




TPM Demands on poultice

Step 1) Water is absorbed from poulice into substrate

poultice substrate

Absorption

Poulice : Reservoir pores larger than largest pores in substrate

reservoir pores

substrate substrate poultice

pore size




TPM

drying

surface

airflow

Widest pores first rPc

φγ cos2≈

Capillary pressure

Desalination phase

PORES POULTICE SMALLER THAN SUBSTRATE

poultice substrate

Advection




TPM

Calcium-silicate brick

r ∼ 12 nm

Bentheimer sandstone r ∼ 30 µm

Plaster (lime:cement:sand = 4:1:10 (v/v)) r ∼ 0.5 µm

rcalcium-silicate< rplaster< rBentheimer

Ph.D thesis J. Petković TU-Eindhoven (2005)




TPM

0 50 100 150 2000

500

1000

1500

V (m

m3 )

t (h)

Plaster Bentheimer sandstone

0 10 20 30 40 500.0

0.1

0.2

0.3a

150 h


10 h

25 h

75 h

25 h0 h

0 h

θ (m

3 /m3 )

x (mm)

rplaster< rBentheimer

moisture


drying surface




TPM

0 10 20 30 40 500.0

0.1

0.2

0.3a

150 h


10 h

25 h

75 h

25 h0 h

0 h

θ (m

3 /m3 )

x (mm)0 50 100 150 200

0

500

1000

1500

V (m

m3 )

t (h)



moisture





TPM

0 50 100 150 2000

500

1000

1500

V (m

m3 )

t (h)


0 10 20 30 40 500.0

0.1

0.2

0.3a

150 h


10 h

25 h

75 h

25 h0 h

0 h

θ (m

3 /m3 )

x (mm)


moisture





TPM

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8a Plaster Bentheimer sandstone

25 h

10 h

0 h

150 h

75 h

0 h

10 h

25 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150

0

1000

2000

3000

4000

Na c

onte

nt (m

mol

)t (h)



Na-content


drying surface




TPM

0 10 20 30 40 500.0

0.2

0.4

0.6


25 h

10 h

0 h

150 h

75 h

0 h

10 h

25 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150

0

1000

2000

3000

4000

Na c

onte

nt (m

mol

)t (h)



Na-content





TPM

0 10 20 30 40 500.0

0.2

0.4

0.6


25 h

10 h

0 h

150 h

75 h

0 h

10 h

25 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150

0

1000

2000

3000

4000

Na c

onte

nt (m

mol

)t (h)



Na-content


Start of crystallization

at surface




TPM

0 10 20 30 40 500.0

0.2

0.4

0.6


25 h

10 h

0 h

150 h

75 h

0 h

10 h

25 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150

0

1000

2000

3000

4000

Na c

onte

nt (m

mol

)t (h)



Na-content


Efficiency high




TPM

0 10 20 30 40 500.0

0.2

0.4

0.6


25 h

10 h

0 h

150 h

75 h

0 h

10 h

25 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150

0

1000

2000

3000

4000

Na c

onte

nt (m

mol

)t (h)



Na-content





TPM

DLUPe =

Peclet number

Water velocity ???

Mass conservation

0. =∇+∂∂ q

tθ 0)( =∇+

∂∂ U

tθθ

``)()(

1)( dxxtx

xUl

x∫∂

∂= θθ

From measured moisture profiles




TPM

0 10 20 30 40 500

2

4

6

8

10

12

|U| L

x 1

0-9 (

m2 /s)

D = 1 x 10-9 (m2/s)

t (h) 0 10 25 75

Bentheimer sandstoneplaster

75 h

25 h

10 h

0 h

x (mm)


Peclet number as function position





TPM

0 10 20 30 40 500.0

0.1

0.2

0.3a

120 h

Plaster Calcium-silicate brick

120 h

60 h

30 h

60 h

30 h

0 h0 h

θ (m

3 /m3 )

x (mm)0 50 100 150 200

0

500

1000

1500

V (m

m3 )

t (h)


moisture

rcalcium-silicate< rplaster


drying surface




TPM

0 10 20 30 40 500.0

0.1

0.2

0.3a

120 h


120 h

60 h

30 h

60 h

30 h

0 h0 h

θ (m

3 /m3 )

x (mm)0 50 100 150 200

0

500

1000

1500

V (m

m3 )

t (h)


moisture






TPM

0 10 20 30 40 500.0

0.2

0.4

0.6a Plaster Calcium-silicate brick

120 h

30 h

60 h

0 h

60 h

30 h

0 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150 200

0

1000

2000

3000

Na c

onte

nt (µ

mol

)t (h)


Na-content



drying surface




TPM

0 10 20 30 40 500.0

0.2

0.4


120 h

30 h

60 h

0 h

60 h

30 h

0 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150 200

0

1000

2000

3000

Na c

onte

nt (µ

mol

)t (h)


Na-content






TPM

0 10 20 30 40 500.0

0.2

0.4


120 h

30 h

60 h

0 h

60 h

30 h

0 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150 200

0

1000

2000

3000

Na c

onte

nt (µ

mol

)t (h)


Na-content



Backward flow of salt from plaster into brick




TPM

0 10 20 30 40 500

2

4

6

|U| L

x 1

0-9 (

m2 /s

)

D = 1 x 10-9 (m2/s)

t (h) 0 10 20 30 60

calcium-silicate brickplaster

x (mm)

rcalcium-silicate< rplaster Ph.D thesis J. Petković TU-Eindhoven (2005)

Peclet number as function position

leveling off




TPM

0 10 20 30 40 500.0

0.2

0.4


120 h

30 h

60 h

0 h

60 h

30 h

0 h

Na c

onte

nt x

103 (m

ol/m

3 )

x (mm)0 50 100 150 200

0

1000

2000

3000

Na c

onte

nt (µ

mol

)t (h)


Na-content



Efficiency low

salt remains in substrate




TPM

ion chromatography




TPM

ion chromatography

MAX DESALINATION

IF

PORES POULTICE SMALLER THEN SUBSTRATE




TPM

Conclusion

Performance =

Poultice property




TPM

Advection Pros • Fast • Object is dry at the end

Cons • Pore size dependent

- adapt poultice to substrate • Renew poultice in time (back diffusion) • Not all salt removed




TPM

Be aware

wetting = advection for ions

Accumulation of ions

drying = advection for ions

Accumulation of ions

= not moved

Diffusion dominant

Advection dominant

So salts are moved in

and

can not be moved out again




TPM

‘Gedanken Experiment’

sample

water

1 cm

Fired clay brick

Permeability ~ 10-8 ms-1

Advection domimant

Concrete

Permeability ~ 10-13 ms-1

Diffusion domimant

Limitations of advection based poulticing ???




TPM

Influence osmotic pressure

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TPM

Macroscopic pressure = capillary pressure + osmotic pressure

Water activity (pure water aw=1)

Effective pore size changes




TPM Extreme example




TPM

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