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Transport in Permeable Media

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

Faculty of Applied Physics Mechanical Engineering

Eindhoven University of Technology The Netherlands

l.pel@tue.nl

5 ECTS 2018

Examination : Oral

Transport in porous media 3MT130

TPM Surface tensions

Curved surface

Pressure difference

Unsaturated

Saturated

Surface tensions

Curved surface

Pressure difference

wnc rp γ2

trxµγ2

hργ2

Small pores

• slow absorption

• but very high

Large pores

• fast absorption

• but low

Liquid ‘fast’ Vapour ‘slow’

Same macroscopic pressure: suction

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

Some Human Activities that Can Contaminate Groundwater

Radioactive contaminants

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

Madame John’s Legacy 1788

Cultural Heritage

Debris from salt weathering (6 months)

New Orleans

Transport of components saturated non-saturated

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

∂∂

=∂∂

cTDtc 2∇=∂∂

Porous medium

cDtc 2∇=∂∂

cTDtc 2∇=∂∂

Porous medium

Path gets longer: tortuosity

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

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

Example of diffusion measurement by NMR

t∂∂θ

qin qout 0. =∇+∂∂ q

∂∂

=∂∂

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

t∂∂θ

qin qout 0. =∇+∂∂ q

∂∂

=∂∂

Sink term due to binding

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

∂∂

=∂+∂

effbρ

Binding

∂∂

=∂+∂

effbρ

Binding

∂∂

∂xcD

eff csR bρ+=1With

Retardation factor

∂∂

=∂+∂

effbρ

∂∂

∂xcD

eff csR bρ+=1With

Simplest case Kcsb =ρ

∂∂

+∂∂

=∂∂

xtc eff

KR +=1

Diffusion gets ‘slower’

TPM Other mechanism?

Water flow : advection

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.

∂∂

−=∂∂CUq −=

Darcy law liquid

ww JUthatNoteJU >→=θ

Darcy law liquid

∂∂

=∂∂ CU

effθθ

∂∂

=∂∂ CU

Saturated porous medium

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

Transport can only be liquid of component

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 Dispersion

Mechanical dispersion - caused by motion of the fluid

Longitudinal dispersion – along the streamline

Transverse dispersion – perpendicular to flow path

Flow Direction

TPM Experiment

Advection, Diffusion, Dispersion

∂∂

=∂∂ CU

effθθ

∂∂

=∂∂ CU

Saturated porous medium

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

Deff= diffusion + tortuosity + dispersion

damage

Damage to rising damp in city of Venice

2004 2007

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

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)

effC cD uCt x x

∂ ∂ ∂ = − ∂ ∂ ∂

Diffusion Advection + = flux

Initial profiles

Advection diffusion equation for transport

effC cD uCt x x

∂ ∂ ∂ = − ∂ ∂ ∂

Initial profiles

airflow supply

moisture flow

advection

q=0 Ions can not leave

q= uCo continuous supply

Boundary conditions

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)

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

Na lower sensitivity longer

measurement time

Signal proportional

moisture content

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

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

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

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

No airflow > only diffusion

TPM Results with airflow

TPM Results

No 6M ????

Stone sample saturated with 1 M NaCL solution

position

Limestone sample saturated with 1 M NaCl solution

position

resolution

TPM Results

1D resolution: average over slice

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

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

TPM Conservators: poulticing

General idea of poulticing

poultice substrate

transport Water absorption

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

General idea of poulticing by diffusion

poultice substrate

diffusion Water absorption

poultice substrate

diffusion

TO KEEP THE DIFFUSION GOING

(maintain sink)

RENEW POULTICE VERY OFTEN

poultice substrate

diffusion

TO KEEP THE DIFFUSION GOING

(maintain sink)

RENEW POULTICE VERY OFTEN

∂∂

=∂∂ 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?

Poulticing side

(where salt comes out)

Time in days

Salt concentration

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)

Equations of moisture and ion transport

∂∂

=∂∂

xt cθθ

∂∂

=∂∂ CU

effθθ

moisture

So two couple non-linear partial differential equations

+ boundary conditions

We do not learn anything !!!!

The competition

drying

surface

airflow

How do we get salt effloresence???

drying

surface

airflow

moisture flow

Saline drying conceptual model

Saline drying conceptual model drying

surface

advection airflow

moisture flow

surface

airflow

moisture flow

accumulation > crystallization

surface

advection

airflow

moisture flow

accumulation

surface

advection

airflow

moisture flow

diffusion

accumulation leveling off

competition

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)

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)

Na concentration

profiles at various

drying times

drying surface

0 days

drying surface

0 days1

Na concentration

profiles at various

drying times

drying surface

NaCl max concentration 6 M

crystallization

0 days13

Na concentration

profiles at various

drying times

drying surface

0 days13

Pe~0.7

Na concentration

profiles at various

drying times

drying surface

0 days13

Na concentration

profiles at various

drying times

drying surface

0 days13

Na concentration

profiles at various

drying times

0 days

accumulation

leveling off

Pe number is simple indication

0 5 10 150.0

SavgCavg

Pe < 1Pe > 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

poultice substrate

advection

General idea of poulticing by advection

Water absorption airflow

poultice substrate

General idea of poulticing by advection

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

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

drying

surface

airflow

Widest pores first rPc

φγ cos2≈

Capillary pressure

Desalination phase

PORES POULTICE SMALLER THAN SUBSTRATE

poultice substrate

Advection

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)

0 50 100 150 2000

Plaster Bentheimer sandstone

0 10 20 30 40 500.0

25 h0 h

3 /m3 )

x (mm)

rplaster< rBentheimer

moisture

drying surface

0 10 20 30 40 500.0

25 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

0 10 20 30 40 500.0

25 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

0 10 20 30 40 500.0

25 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

0 50 100 150 2000

0 10 20 30 40 500.0

25 h0 h

3 /m3 )

x (mm)

moisture

0 10 20 30 40 500.0

0.8a Plaster Bentheimer sandstone

103 (m

x (mm)0 50 100 150

)t (h)

Na-content

drying surface

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150

)t (h)

Na-content

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150

)t (h)

Na-content

Start of crystallization

at surface

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150

)t (h)

Na-content

Efficiency high

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150

)t (h)

Na-content

DLUPe =

Peclet number

Water velocity ???

Mass conservation

0. =∇+∂∂ q

tθ 0)( =∇+

∂∂ U

``)()(

1)( dxxtx

x∫∂

∂= θθ

From measured moisture profiles

0 10 20 30 40 500

m2 /s)

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

t (h) 0 10 25 75

Bentheimer sandstoneplaster

x (mm)

Peclet number as function position

0 10 20 30 40 500.0

Plaster Calcium-silicate brick

0 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

rcalcium-silicate< rplaster

drying surface

0 10 20 30 40 500.0

0 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

0 10 20 30 40 500.0

0 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

0 10 20 30 40 500.0

0 h0 h

3 /m3 )

x (mm)0 50 100 150 200

moisture

0 10 20 30 40 500.0

0.6a Plaster Calcium-silicate brick

103 (m

x (mm)0 50 100 150 200

nt (µ

)t (h)

Na-content

drying surface

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150 200

nt (µ

)t (h)

Na-content

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150 200

nt (µ

)t (h)

Na-content

Backward flow of salt from plaster into brick

0 10 20 30 40 500

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

0 10 20 30 40 500.0

103 (m

x (mm)0 50 100 150 200

nt (µ

)t (h)

Na-content

Efficiency low

salt remains in substrate

ion chromatography

MAX DESALINATION

PORES POULTICE SMALLER THEN SUBSTRATE

Conclusion

Performance =

Poultice property

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

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

can not be moved out again

‘Gedanken Experiment’

sample

Fired clay brick

Permeability ~ 10-8 ms-1

Advection domimant

Concrete

Permeability ~ 10-13 ms-1

Diffusion domimant

Limitations of advection based poulticing ???

Influence osmotic pressure

Macroscopic pressure = capillary pressure + osmotic pressure

Water activity (pure water aw=1)

Effective pore size changes

TPM Extreme example

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