Bauhaus-Universit ät

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Mechanic

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Deriving hydraulic conductivity function from soil column tests

Yvonne Lins, Maria Datcheva & Tom Schanz

DFG Fo 444 - TP 4 “Experimentelle und theoretische Untersuchungen teilgesättigter Reibungsmaterialien”

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sContent

• Equipment

column test device & modified pressure plate apparatus

• Presentation of results

SWCC & unsaturated hydraulic conductivity function

• Comparison of results

• Numerical analysis

Inverse Modelling with Comes Geo

• Conclusions

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sModified pressure plate device

Schematic sketch and photos of modified pressure plate apparatus

Porous stone

Soil sample

Ceramic disk

Water reservior

Dial gage

Air pressure supply ua

Water pressure supply uwWater pressure supply uw

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sColumn test device

305

67

10

0

10

5 0

90

10

x 5

0

80

50

TDR 5

TDR 4

TDR 3

TDR 2

TDR 15

Tim

e D

om

ain

e R

efl

ec

tom

etr

y

se

ns

ore

s

5 T

en

sio

me

ters

T 5

T 4

T 2

T 3

T 1

water reservior

dial gage

Layout of the test device

Mini Tensiometer T5 (UMS)

TDR Mini Buriable Waveguide (Soilmoisture Equipment Corp.)

Installation of sensors

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sCalibration TDR & Tensiometer

1.23143,01048.11081.1)( 2334 ++⋅−⋅= −−aaaa KKKKθ

1332.24557.01057.61029.2)( 2334 ++⋅−⋅= −−aaaa

KKKKθ

053.01092.2105.5103.4)( 22436 −⋅+⋅−⋅= −−−aaaa KKKKθ

Topp et al. (1980)

Lins (BUW)

y = -0,859x + 3,0075

y = -0,8122x + 1,624

y = -0,8499x + 1,7018

y = -0,8135x + 0,9386

y = -0,7961x - 0,7532

-30

-25

-20

-15

-10

-5

0

5

0 5 10 15 20 25 30 35

mV

Po

ten

tial [c

m]

T60

T61

T62

T63

T64

0

5

10

15

20

25

30

0 10 20 30 40 50

Vol. water content [-]

Ap

pa

ren

t d

iele

ctr

ic c

on

sta

nt

[-]

void ratio e=0.75

void ratio e=0.97

Topp et al. (1980)

Calibration TDR Calibration tensiometers

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Unsaturated hydraulic conductivity by using statistical model

Modified pressure plate device

mn

ae

C

+

⋅=Θ

ψ

ψ

ln

1)(

SWCC by using Fredlund & Xing equation

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.1 1 10

Matric suction [kPa]

Un

sa

tura

ted

hy

dra

ulic

co

nd

uc

tivity

[1

0

-4 m

/s]

Drying band

Loose specimenWetting band

Loose specimen

ψ aev

ψ wev

k s

0

10

20

30

40

50

0.1 1 10 100

M atric suction [kPa]

Vo

lum

etr

ic w

ate

r c

on

ten

t [%

]

Fredlund and Xing (1994)

initial void ratio e0 = 0.89

drying (average value)

wetting (average value)

Drying band

Wetting band

ψ aev =1.2kPa

ψ w ev =1.8kPa

θ r =0.02; ψ r =2.8kPa

θ s =0.47

∫

∫

−

−

⋅Θ=s

r

r

d

d

k

s

q

r θ

θ

θ

θ

ζψ

ζζθ

ζψ

ζζθ

θ

2

2

)(

)(

)(

)(

)(

Θ - normalized volumetric water content

ψ - suctionC(ψ) - Correction functiona, n, m - Parameters

θ - volumetric water content

θr - residual volumetric water content

θs - residual volumetric water content

ζ - dummy integration variable

q - parameter

kr - relative hydraulic conductivity,

Childs & Collis George (1950)

Fredlund & Xing (1994)

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Column deviceExample of Measurements

0

10

20

30

40

50

0.1 1 10 100

Matric suction [kPa]

Vo

lum

etr

ic w

ate

r c

on

ten

t [%

]

Layer 1 (top) - drying

Layer 1 (top) - wetting

Layer 2 - drying

Layer 2 - wetting

Layer 3 - drying

Layer 3 - wetting

Layer 4 - drying

Layer 4 - wetting

Layer 5 (bottom) - drying

Layer 5 (bottom) - wetting

Loose specimen

Initial void ratio = 0.89

0

10

20

30

40

50

0 100 200 300 400

Time [min]

Volu

metr

ic w

ate

r conte

nt [%

]

TDR1 (top)

TDR2

TDR3

TDR4

TDR5 (bottom)

Drying process-4

-3

-2

-1

0

1

2

3

4

0 100 200 300 400

Time [min]

Pore

- w

ate

r p

res

su

re [k

Pa

]

T1 (top)

T2

T3

T4

T5 (bottom)

Matric suction

Hydrostatic pressure

Drying process

SWCC

TDR & Tensiometermeasurements linked to SWCC

Tensiometer

TDR

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Column device

Influence of flow rate

SWCCs by using Fredlund & Xing equation resulting from different flow rates

0

10

20

30

40

50

60

0.1 1 10 100

Matric suction [kPa]

Vo

lum

etr

ic w

ate

r co

nte

nt

[%]

layer 1 (top)

layer 2

layer 3

layer 4

Initial drying

1st and 2nd drying1st and 2nd wetting

Flow rate ≈ 30 ml/min

Flow rate ≈ 100 ml/min

Fredlund and Xing (1994)

Initial void ratio e0 = 0.89

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.1 1 10

Matric suction [kPa]

Un

sa

tura

ted h

yd

rau

lic c

on

du

ctiv

ity [

10

-4

m/s

]

layer 1 (top)

layer 2

layer 3

layer 4

Initial drying

1st and 2nd drying1st and 2nd wetting

Flow rate ≈ 30 ml/min

Flow rate ≈ 100 ml/min

Fredlund et al. (1994)

Initial void ratio e0 = 0.89

Unsaturated hydraulic conductivity resulting from different flow rates by using statistical model

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Column device

Statistical model vs. Instantaneous Profile Method

0.001

0.01

0.1

1

10

1 10

Matric suction [kPa]

Uns

at.

hy

dra

ulic

co

nd

uc

tivity

[1

0-4

m/s

]

first drying

first drying

Instantaneous Profile Method:

Statistical Model:

Unsaturated hydraulic conductivity resulting from statistical model and instantaneous profile method

i

w

w

dzdhtA

Vk

1)( ⋅

∆⋅

∆−=ψ

∆Vw - amount of water flowing past point i at time interval ∆t

A - cross sectional area

dhi/dzi - hydraulic head gradient

z - vertical coordinate

hw - hydraulic head

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Modified pressure plate versus column deviceSWCC

SWCC from small scale and large scale tests

0

10

20

30

40

50

60

0.1 1 10 100

Matric suction [kPa]

Vo

lum

etr

ic w

ate

r c

on

ten

t [%

]

Fredlund and Xing (1994)

Initial void ratio e0=0.89

Flow rate ≈ 30 ml/min

Flow rate ≈ 100 ml/min

Results small scale tests: Drying band

Wetting band

Results large scale tests:Initial, 1st and 2nd drying

1st and 2nd wetting

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Unsaturated hydraulic conductivity from small scale and large scale tests for

Modified pressure plate versus column deviceHydraulic conductivity function

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.1 1 10

Matric suction [kPa]

Un

sa

tura

ted

hy

dra

ulic

co

nd

uc

tivity

[1

0

-4m

/s] Fredlund et al. (1994)

Initial void ratio e0 = 0.89

Flow rate ≈ 30 ml/min

Flow rate ≈ 100 ml/min

Results small scale tests:

Drying band

Wetting band

Results large scale tests:

Initial, 1st and 2nd drying

1st and 2nd wetting

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Numerical analysiscoupled hydro-mechanical analysis of column drying test

using the FE code Comes-Geo (University of Padua)

FE with 5 degrees of freedom in each node:

water pressure - ; gas pressure - ; capillary pressure -

Darcy‘s law

relative permeability

SWCC

linear elastic material

effective stress

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Numerical analysisInverse analysis procedure

Optimizerk, a, n, m, α

Comes Geo Experimental results

pc, S, t pc, S, t

Objective Function

No Minimum

Yes

End

Scheme of inverse analysis procedure

a p

1.5

5.5 0.0015

0.0003

F(x)

1.0

0.0

Particle swarm optimizer (Meier 2006)

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-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300

Time [min]

Pre

ssu

re h

ea

d [

kP

a]

Numerical analysisDrying of loose specimen: scenarios 1- 3

-4

-3

-2

-1

0

1

2

3

0 50 100 150 200 250 300

Time [m in]

Pre

ss

ure

he

ad

[k

Pa

]

0

0,2

0,4

0,6

0,8

1

0 50 100 150 200 250 300

Time [min]

De

gre

e o

f S

atu

ration

[-]

Numerical versus experimental results: fit using only Tensiometer readings (left) and only TDR readings (right)

0

0,2

0,4

0,6

0,8

1

0 50 100 150 200 250 300

Time [min]

De

gre

e o

f S

atu

ratio

n [

-]

Numerical and experimental results: fit using both Tensiometer readings and TDR readings

1 2

3a 3b

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Numerical analysisComparison predicted & observed values

1

2

3

Predicted versus observed values for the different scenarios

Pre

dic

ted

valu

es

Observed values

Degree of Saturation Capillary pressure

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 -3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 -3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

Optimized parameters.

Methods k [m2] a [1/Pa] n m α

1.Tensiometers only 2.53e-12 7.403e-4 3.016e+1 8.062e-2 2.098

2. TDRs only 1.455e-11 3.62e-4 3.019e+1 3.687e-1 4.146

3. TDRs and tensiometers 1.878e-11 4.903e-4 2.849e+3 2.783e-1 4.478

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Numerical analysisSWCC for 3 scenarios

SWCC: numerical and experimental results in comparison

0

10

20

30

40

50

60

0.1 1 10 100

Matric suction [kPa]

Vo

lum

etr

ic w

ate

r co

nte

nt

[%]

Results large scale tests

Initial void ratio e0=0.89

Results small scale tests

Results simulation

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Summary & conclusions

We study the hydraulic behavior of unsaturated granular material in the case of Hostun sand.

New experimental devices were developed, a modified pressure plate apparatus and a column type device.

SWCC & hydraulic conductivity function were established for loose specimen under variable loading path directions utilizing steady state & transient method.

SWCC & hydraulic conductivity function for Hostun sand takes place in a narrow range of suction.

Significant effect of hysteresis was found for loose specimen in the drying and wetting curves

of SWCC.

Experiments carried out under steady state & transient method show similar behavior regarding SWCC and hydraulic conductivity function.

Inverse analysis of drying cycle of sand column test results was performed in different scenarios.

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Mechanic

sComes-Geo: Governing Equations

1. Equation of mass balance of the dry air

2. Equation of mass balance of the liquid

3. Enthalpy balance equation for the whole media

4. Linear momentum balance equation for the whole media

5. Solid mass conservation equation

6. The balance equations 1-5 are completed by an appropriate set of constitutiveand state equations (slide 12)

Formulation of the coupled problem is in