Summer School – Cagliari, 22 -23 giugno 2014 · Summer School – Cagliari, 22 -23 giugno 2014. Module 1 - 2 Prof. Alessandro Tarantino, University of Strathclyde, UK Tarantino,

Module 1 - 1

Prof. Alessandro Tarantino, University of Strathclyde, UK

Introduzione al comportamento idraulico dei terreni parzialmente saturi -principi fondamentali, riflessi sul comportamento idro-meccanico e

sulle prestazioni di opere in vera grandezza

Prof Alessandro Tarantino

University of Strathclyde, Glasgow, UK([email protected] )

Summer School – Cagliari, 22-23 giugno 2014

Module 1 - 2


Tarantino, A. 2010 Basic concepts in the mechanics and hydraulics of

unsaturated geomaterials

New Trends in the Mechanics of Unsaturated Geomaterials LyesseLaloui (ed.), 3-28. ISTE – John Wiley & Sons.

Module 1 - 3


Surface tension and capillary systems

Module 1 - 4


Cohesion and surface tension

Cohesion = Attraction force between molecules of the same type

At the surface, the resultant force is directed downward

The gas-liquid interface behaves like a membrane subject to a uniform tensile stress

liquid

gas

This stress is termed surface tension

Module 1 - 5


Adhesion

Liquid

Liquid

Adhesion < Cohesion Adhesion < Cohesion

Contact angle (measured throughthe

liquid)

θ<90°

Adhesion = Attraction force between molecules of different type

θ>90°

Solid

sur

face

Supe

rfici

e so

lida

The liquid ‘wets’ the surface The liquid does not wet

Module 1 - 6


Effect of curvature of the liquid-gas interface

θ

R

water

air

r

T

RT

rTuu aw

2cos2−=−=−

θ

ua = air pressure [F/L2]uw = water pressure [F/L2]θ = contact angleT = surface tension [F/L]r = radius of capillary tube [L]R = radius of curvature of spherical cup [L]

θπππ cos222 rTruru wa +=

if θ < 90°

The air pressure is partly sustained by the meniscus

The water pressure is lower than air pressure

Mechanical equilibrium

Module 1 - 7


Rise in capillary tube

uw<0uw=0

uw=0uw=0

rThu ww

θγ cos2−=−=

h

If θ<90°, the liquid enter the cavities in the solid surface ⇒ the liquid is said to wet the surface

ua=0

Module 1 - 8


Hysteresis of the contact angle

θ

θr θa

θr = receding angle

θa = advancing angle

θr=θmin

θmax=θa

In a capilary tube, the contact angle ranges from θa to θr

Module 1 - 9


Hysteresis of the contact angle

PRESSUREWATER

Module 1 - 10


Evaporation from a capillary tube

θ=θrθ>θr

θ=θr

1 2 3 4

uw=0 uw<0 uw= -2T cosθr /rr

uw= -2T cosθr /r

Module 1 - 11


Water retention of a capillary tube

θ=θr

1 2 3 4

Vw / V

- uw

V

1 2 3

4

θ=θr

θ=θr

θ>θr

Module 1 - 12


Tensile stress of wate in the capillary tube

θ = 0

T = 0.073 N/m (20°C)

ua = 100 kPa (absolute pressure)

4 cosw a

Tu ud

θ− = −

siltsand clay

The absolute water pressure may be negative, i.e. The water is subject to a tensile stress

d grain (mm) 2 0.075 0.002

d pore (mm) 0.2 0.0075 0.0002

uw-ua (kPa) -1.4 -38 -1440

uw (kPa) +98.6 +62 -1340

4b

pore

Tsd

=

Module 1 - 13


Evaporation from a system of capillary tubes

a

c

b

ra

rc

rb c

c

b

b

a

a

rrrθθθ coscoscos

==

wcwbwa uuu ==

La = Lb = Lc

ra = 2 rb = 4 rc

Meechanical equilibrium

Geometry

Module 1 - 14


Water retention of a sytem of capillary tubes

1 2 3 4 5

Vw / V

- uw

1 2

34

5

S

Module 1 - 15


Increasing water tension in a capillary tube

1 2 3

wrTh

γθ 1cos2

=

wrTh

γθ 1cos2

=

Module 1 - 16


Increasing water tension in a system of capillary tubes

Z

Vw / V1

S

Module 1 - 17


Capillary effects in soils

particellaacqua interstiziale

uw < 0 The contact angle of water with the particle surface is less than 90°

The meniscus is concave toward the air side and pore water presure is negative

Particles are stuck together by surface tension and negative pressure

-uw

T

Module 1 - 18


Soil as a system of capillary tubes

Module 1 - 19


Water retention behaviour

Module 1 - 20


Water retention relationship

Relationship between the degree of saturation (or water content) and suction

This relationship illustates the different state of water in the soil

It is determined in the laboratory by subjecting soil specimens to drying and wetting cycles

It is rarely determined in the field

Module 1 - 21


Soil water retention

S

ln (s)

1

Saturated soilQuasi-saturated soil

Partially saturated soil

Residual saturation

Module 1 - 22


Saturated state

S = 1

uw < 0

Suction is generated by the curving of menisci at the boundary

Soil is saturated, air is dissolved in water

Module 1 - 23


Quasi-saturated state

0.85-0.90 < S < 1

uw < 0

Suction is generated by the curving of menisci at the boundary and cavities form in the pore water

Gas phase is discontinuous, liquid phase is continuous

Module 1 - 24


Partially saturated state

0-0.1 < S < 0.85-0.90

uw < 0

Suction is generated by the curving of menisci in the pores, are saturated parts (bulk water) and part where menisci form at the interparticle contact

Gas phase is continuous, liquid phase is continuous

Module 1 - 25


Residual state

S < 0-0.1

uw < 0

Suction is generated by the curving of menisci in the pores, and menisci form at the interparticle contact

Gas phase is continuous, liquid phase is discontinuous

Module 1 - 26


Retention curve parameters

S

ln (s)

1

Sr

sb sr

sb = air-entry value

sr = residual suction

Sr = residual degree of saturation

Module 1 - 27


Air-entry suction It is the suction where the soil desaturates

As a first approximation, the soil can be assumed saturated for suction lower than the air-entry value, sb

For an order of magnitude of the air-entry suction :

The air-entry suction essentialy depend on the pore size

d grain (mm) 2 0.075 0.002

d pore (mm) 0.2 0.0075 0.0002

s b (kPa) 1.4 38 1440

siltsand clay

4b

pore

Tsd

=

Module 1 - 28


Example No. 1

Clay subjected to cycles of drying and wetting

Very little evaporation is sufficient to curve menisci and lower water pressure to the air entry value, say uw=-1000 kPa

Little precipitation is sufficient to cancel meniscus curvature, and restore zero pore water pressure uw=0 kPa

As the soil is saturated, the effective stress increases of 1000 kPa, as if were placing an embankment 50 m high

Little evaporation and precipitation are sufficent to induce a cyclic stress change equivalent to the placement and removal of an embankment 50 m high

The effective stress redices of 1000 kPa, as if we were removing an embankment 50 m high

Module 1 - 29


Vertical cut in silt

As the soil is saturated, the shear strength ctiterion can be written as follows:

τ = (σ-uw) tan φ’ = σ tan φ’ + (-uw) tan φ’ = σ tan φ’ + capparent

Assuming φ’=25 °, risulta capparent = 47 kPa

Hγ=20 kN/m3

In the classical dry soil mechanics, H=0 if c’=0

Assuming tha H=2c / γ we obtain H= 4.7 m!

Example No. 2


Module 1 - 30


Example No. 3

11 =

−=

αφ

γγη

tan'tanw

Infinite slope in sand

As the soil is saturated, the facor of safetu can be written as follows:

Assuming η=1, we obtain αmax= 50° !

αφ’ = 35°

γ = 20 kN/m3

H=1 m Water table at the ground surface.

°= 19maxα

( )ααγ

φαφ

γγη

cos sin 'tan

tan'tan

Huww −

+

−= 1


Module 1 - 31


Residual suction

It is the suction beyond which degree of saturation does not change significantly

This suction is controlled by the smaller pores, in turn controlled by the grain size distribution

S

ln (s)

1

0

20

40

60

80

100

0.001 0.01 0.1 1 10sb1

sb2

The higher is the coefficient of uniformity (well graded GSD), the higher is residual suction

Module 1 - 32


Hydralic hystheresis

S

ln (s)

1

Hp: incompressible soil skeleton

Main drying curve

Main wetting curve

Scanning curves’

Module 1 - 33


Intepretation of hysteresis using the capillary model

wrTh

γθ 1cos2

1=

r1

r2

0≅h

r2

At the same applied suction, the degree of saturation is greater along a drying path

Module 1 - 34


Water content

w

ln (s)

Water content is easier to determine and water retention curve is qualitatively similar to that in terms of degree of saturation but …

Module 1 - 35


Saturated overconsolidated clay

w

ln (p’)

Specimens taken in the field and saturated in the lab are typically overconsolidated

S

ln (s)

A ⇒ sample in the fieldA

B≡ C

D

E

B ⇒ sample saturated in the lab

BCDEF ⇒ retention curve

C ⇒ air-entry value ?

D ⇒ air-entry value

ln (s),sc

Module 1 - 36


Effect of void ratio on WRC (1)

Main drying curves

S

ln (s)

1

θr θr

Module 1 - 37


Effect of void ratio on WRC (2)

0.00 0.20 0.40 0.60 0.80 1.00Water ratio, ew

0.01

0.1

1

10

100

1000

Sucti

on (M

Pa)

e ≈ 0.63

e ≈ 0.92

p y

qu d o

a(Ss) ≈ 300 MPa

Vapo

r equ

ilibriu

mte

chniq

ue (v

apor

phas

e co

ntro

l)

Air o

verp

ress

ure

tech

nique

(liqu

idph

ase

cont

rol)

b(Ss, path)

e wm

≈ 0.40

(Romero and Vaunat 2000)

Module 1 - 38


Effect of void ratio on WRC (3)(Tarantino and Tombolato 2005)

0 200 400 600 800 1000Matric suction, s: kPa

0.4

0.6

0.8

1

Deg

ree

of s

atur

atio

n, S

r

e<0.850.85<e<0.950.95<e<1.051.05<e<1.151.15<e<1.25e>1.25 1.36

1.35

1.271.221.26

1.261.10

1.12

1.13

1.171.17

1.091.09 1.04

0.981.03

1.23

1.121.11

1.021.03

0.990.971.14

1.08

1.001.021.011.02

1.001.02 0.93

0.93

0.870.98

0.910.92 0.82

0.860.86

0.79 0.820.63

e=0.7

e=0.8

e=0.9

e=1.0

e=1.3

e=1.1

Wetting paths (undrained compression)

Module 1 - 39


Effect of void ratio on air-entry and air-occlusion value

(Karube and Kawai, 2001)

Module 1 - 40


Soil water retention mechanisms and the concept of suction

Module 1 - 41


Suction through the liquid phase

An unsaturated soil is capable of drawing water through the liquid phase.

This property is termed matric suction

Terreno non saturo

Module 1 - 42


Capillary mechanisms

uw = 0

particlepore water

uw < 0Owing to capillary tension, pore water pressure is negative

Module 1 - 43


Osmotic mechanisms

+- - - - - - - - - -

+ + + + + + + + ++

++

++

++

++

+ + + + + + + + + ++

++

++

++

++

Clay particle

+

+

+

+

+

+

-

--

- -

-

Free water

+++

+

++

++

+

+

++

+

--

--

--

-

--

--

--

+

+

+ +-

---

+ -+ -

- - - - - - - - - -

Module 1 - 44


Electrostatic mechanisms

+- - - - - - - - - -

++

+

+

+++ +

- - - - - - - - - -HH

O

HH

O

(Hydrogen bonding)

Hydration of exchangebale cations

- - - - - - - - - -H H

O

H H

O

H H

O

Oxygen plane

Hydroxile plane

Module 1 - 45


Suction generated by the solid matrix(matric suction)

MEDIUM AND HIGH DEGREE OF SATURATION

Capillary mechanisms

LOW DEGREE OF SATURATION OF HIGH CLAY FRACTION

Osmotic and electrostatics mechanisms

Module 1 - 46


Suction through gas phase

Unsaturated soil

An unsaturated soil is capable of drawing water through the gas phase.

This property is also termed suction

Module 1 - 47


Liquid-vapour equilibrium (pure water - flat interface)

water, pw

vapour, p°v

pw = p°v p°v = p°v (T)

c

b

a

liquid

vapour

solid

p p

T

at T = 20°p°v ≅ 2.3 kPa

p°v is a function of temperature only

Phase diagram of water

Module 1 - 48


Liquid-vapour equilibrium (pure water + air - flat interface)

water, pw

vapour, p°v

air, pa

pw = p°v+ pa ≅ pa

p°v ≅ p°v (T)

In presence of air, the relationship p°v = p°v (T) remains practically unchanged. p°v shall be regarded as partial pressure

Module 1 - 49


Liquid-vapour equilibrium (free water - curved interface)

water, pw

pv < p°v

The curvature of the gas-liquid interface reduces the (partial) vapour pressure

vapour, pv

air, pa

Owing to the meniscus, liquid pressure is negative

pw < pa = 0

Why ?

(p°v= vapour pressure over flat surface)

Module 1 - 50


Rusty in basic thermodynamics ?….. get out our thermodynamics book !

Module 1 - 51


Liquid-vapour equilibrium for curved interface (1)

water, p0w

vapour, p°v

air, pa

water, pw

vapour, pv

air, pa

Reference state, 0 Current state, 1

w vµ µ=

Equilibrium is controlled by the chemical potential µ of the species water in liquid and gas phase

Module 1 - 52


Liquid-vapour equilibrium for curved interface (2) CHEMICAL POTENTIAL

h Ts u pv Tsµ ≡ − = + −Enthalpy Entropy

Temperature

Internal energy

(Extensive variables are per unit mass)

PressureVolume

First principle

Second principle

du q pdvδ= −

q Tdsδ = d vdp sdTµ = −

Heat

Module 1 - 53



1 1 1 10 0 0 0 w v w w v

v

RTd d v dp dpp

µ µ= ⇒ =∫ ∫ ∫ ∫

Assuming:i) water vapour follows the ideal gas law ii) isothermal transformation (dT=0), iii) liquid water is incompressible

Bring the system, reversibly and isothermally, from reference state 0 to current state 1 (liquid under negative pressure).

1 10 0w vd dµ µ=∫ ∫

0 0

1 1

w v

w v

µ µµ µ

=

=

Module 1 - 54



Integration leads to:

( )00

expv ww w

v

p v p pp RT

= − −

10 100 1000 10000 100000Capillary suction, pw

0 - pw (kPa)

1

0.9

0.8

0.7

0.6

0.5

Air r

elat

ive h

umid

ity, p

v/pv0

Suction

Module 1 - 55


Liquid-vapour equilibrium for aqueous solution (1)(acqueous solution - flat interface)

pv < p°v

Dissolved ions reduces the (partial) vapour pressure

acqueous solution, pw

+

++

+

+

++

+

- --

-

--

-

vapour, pv

air, pa

Why ?

Module 1 - 56


Liquid-vapour equilibrium for aqueous solution (2)

water, p0w

vapour, p°v

air, pa

Reference state, 0 Current state, 1

w vµ µ=

Equilibrium is controlled by the chemical potential µ of the species water in liquid and gas phase


+

++

+

+

++

+

- --

-

--

-

vapour, pv

air, pa

Module 1 - 57



Ideal water vapour

Bring the system, reversibly and isothermally, from reference state 0 to current state 1 (liquid under negative pressure).

1 10 0w vd dµ µ=∫ ∫

0 0

1 1

w v

w v

µ µµ µ

=

=

( ) ( )00

, , ln vv v v v

v

pp T p T RT

pµ µ

= +

( ) ( )0 , ln 1w w w sp T RT xµ µ= + − Ideal (diluted) aqueous solution

Molar fraction of solute

Module 1 - 58



Raoult’s law

( )0 1v v sp p x= −

Module 1 - 59


Liquid-vapour equilibrium (acqueous solution - curved interface)

pv < p°v

pv = pv (T, Xs, pw) < p°w (T)

vapour, pv

air, pa

Owing to dissolved ions and water tensile stresses :


+

++

+

+

++

+

- --

-

--

-In particular:

(Negative) liquid pressure

Solute concentration

Module 1 - 60


Suction mechanism through gas phase

An unsaturated soil draws water through the gas phase because of the gradient in vapour pressure.

(Total) suction is generated by the combined action of negative pressure (solid matrix) and dissolved ions

Unsaturated soil

pv p°v

Module 1 - 61


Principle of suction measurement

Module 1 - 62


Direct measurement of matric suction

sm/γw

sm = ua - uw

The water level in the piezometer moves downward to generate a (negative) pressure to counterbalances the suction exerted by soil water

The difference between the external air pressure, ua, and the water pressure in the instrument, uw, is referred to as matric suction

(sm = - uw if ua=0)

The term ‘matric’ points out that suction exerted by soil water is, in turn, due to the action of the matrix (capillary, osmotic, and electrostatics mechanisms)

Module 1 - 63


Osmotic (solute) suction

sm/γw

semipermeable membrane

st/γw

Water pressure in the instrument further decreases to counterbalance the attraction exerted by ions dissolved in the pore water

The acqueous solution in the pores and the pure water in the instrument form an osmotic system.

Osmotic suction so depends on the difference on concentration gradient between the pore water and the pure water in the instrument

Total suction st also account for the action of ions dissolved in the pore water

so/γw

Pure water

Module 1 - 64


Need for indirect measurement of suction

sm/γw

Water in the measuring system is subjetced to cavitation

cavity

A trick it to increase the ambient air pressure to translate instrument water pressure into the positive range (axis-translation technique)

sm/γw

Pa>0

Pa/γw

Module 1 - 65


Suction measurement through vapour equilibrium

pv < p°v Another trick is to measure the partial pressure of water vapour in equilibrium with the soil water.

Vapour pressure is related to the negative pore water pressure by the psycrometric law

( )RHvRT

pp

vRTs

mv

v

mt lnln 0 −=

−=

Water in the measuring system (vapour phase) is pure water whereas ions are dissolved in the pore water. Owing to the concentration gradient, the total suction is actually measured

It is not possible to differentiate between the osmotic and matric component of suction

Dry bulbWet bulb

Module 1 - 66


Indirect measurement of total suction

( )0 0exp wv v w w

vp p p pRT

= − −

0ln lnv

tw v w

pRT RTs RHv p v

= =

Total suction is measured indirectly by:

1) Measuring the relative humidity RH in equilibrium with the soil water2) Relating the relative humidity RH using the theoretical relationship

developed previuosly (known as psychrometric law)

(Psychrometric law)

Module 1 - 67


Some comments

o The psychrometric law is NOT the thermodynamic definition of suction as opposed to the ‘enginnering’ defintion given by the difference between between ambient-air pressure and pore-water pressure.

It is the theoretical relationship to transform the vapour pressure in the equivalent liquid pressure that can generate such a vapour pressure

o The discriminating factor between total and matric suction measurement is the nature of the liquid in the measurement system, pure water or aqueos solution respectively.

o The solute suction is often referred to as osmotic suction, which should not be mistaken by the omostico mechanism generating the matric suction.

Module 1 - 68


Unsaturated hydraulic conductivity

Module 1 - 69


Water flow equation

( ) grad ww

w

uv - k u z γ

= +

div 0 v tθ∂

+ =∂

Darcy’s law

Mass balance equation (no water vapour flow)

( )w div grad w ww

w w

u uk u zu tθγ

γ ∂∂

= + ∂ ∂

Water flow equation (Richard’s equation)

Module 1 - 70


Flow through a saturated round capillary tube

Poiseuille’s law for laminar fluid flow

q = flow ratev = average flow velocityi = hydraulic gradientAw = wetted areaη = kinematic viscosityγ = unit weight of the permeantR = tube radiusRH = hydraulic radius

q

D=4RH

A A

section A_A

Aw = wetted area

2

flow channel cross section area 4wetted perimeter 4

wH

w

DA DRP D

π

π= = = =

k

2

2act Hw

q gv R iA η

= = ⋅

Pw = wetted perimeter

Module 1 - 71


Saturated hydraulic conductivityHydraulic conductivity

(Kinematic viscosity of water at 20°C η=10-6 m2/s)

22 2 5 2

26

9.8 1 3 10 32 32 10

mg sk D D D

m sms

η −

= = = ⋅ ⋅

d grain (mm) 2 0.075 0.002

d pore (mm) 0.2 0.0075 0.0002

k (m/s) 6·10-3 2·10-6 6·10-9

siltsand clay

Module 1 - 72


water

air

solid

wetted area, Aw

solid area, As

total area, A

void area, Av

wetted perimeter, Pw

Flow through a unsaturated round capillary tube

2

2act w w w

w

v A A Agv iA P Aη

⋅ = = ⋅ ⋅

As for the saturated tube:

0 0

11

1

w r

w s

eA S Ae

P A S ASe

= + = = +

S0 = specific surfaceSr = degree of saturatione = void ratio

Module 1 - 73


Hydraulic conductivity of unsaturate geomaterials

Kozeny-Carman equation3

320

12 1 r

g ev S iS eη

= ⋅ +

33 3

20

12 1 r sat r

g ek S k SS eη

= = ⋅ +

Module 1 - 74


Influence of S on unsaturated permeability of compacted clay

∝k S3

Module 1 - 75


Hysteresis of the permeability function

Module 1 - 76


Effect of the shape of the permeability function on sample drying

k/ksat

ln (s)

evaporation

‘impermeable’ outer shell

Faster drying

Module 1 - 77


Water flow in boundary value problems:

Stability of vertical cuts

Module 1 - 78


σh=0

Vertical cuts in ‘cohesionless’ soils:the ‘dry’ approach

σ’

τ

φ’

σ’h σ’v

Total = Effective

σv

uw

+

Zero pore pressure

Hmax=0

Vertical cuts cannot be stable

Hmax

LOWER BOUND THEOREM OF PLASTICITY

Module 1 - 79


Stable vertical cuts in ‘cohesionless’ soils(De Vita et al. 2008, IJEGE)

Giugliano near Naples, Italy(courtesy of Prof. De Vita, University of Naples Federico II)

Dry approach (c’=0)

Pyroclastic ‘cohesionless’ silty sand

Hmax = 0 !!!

H = 10-12m

Module 1 - 80


σh=0

Stable vertical cuts in ‘cohesionless’ soils

The real situationStable because of suction

σ’’

τ

φ’

σ’’h σ’’vσh σv

Effective

Total


σv

uw

H=10-20m

-

+

τ = σ’’ tan (φ’) = [σ+sSrM ] tan (φ’)

Module 1 - 81


Shear strength criteria for unsaturated materials(incorporating suction and degree of saturation)

( ) ( )tan ' or k kr rsS q M p sSτ σ φ= + = +

( ) ( )tan ' or rM rMsS q M p sSτ σ φ= + = +

= w wmrM

wm

e eSe e

−−

Tarantino and Tombolato (2005)

Vanapalli et al. (1996):

Intra-aggregate water(not contributing to strength)

⇒ Microstructural water ratio, ewm(Romero and Vaunat 2000)

• Two criteria different conceptually but equivalent in terms of modelling capability• In principle, only one test sufficient to characterise unsat shear strength (k or ewm)

Inter-aggregate water(contributing to strength)

Effective degree of saturation

Degree of saturation of macropores

(Tarantino and El Mountassir 2012)

Module 1 - 82


Validation for non-aggregated geomaterials (ewm=0, k=1)

0 200 400 600 800 10001200p+sSr (kPa)

0

200

400

600

800

q (

)

SaturatedUnsaturated

Compacted silt(Capotosto and Russo 2011)

0 40 80 120 160p+sSr (kPa)

0

50

100

150

200

250

q (k

Pa)

Saturateds=6 kPas=12 kPas=20 kPa

Natural silty sand(Papa et al. 2008)

Μ1

Μ1

0 400 800 1200p+sSr (kPa)

0

400

800

1200

1600


Reconstituted silt(Geiser et al. 2006)

Μ

1

0 400 800 1200σ+sSr (kPa)

0

200

400

600

800

τ (k

Pa)


Reconstituted clayey sandy silt (Boso 2005)

tan φ'1

0 200 400 600 800 1000S (kP )

0

400

800

1200

1600

saturateds=5 kPas= 20 kPas=75 kPa

Natural and reconstituted silty sand(Cattoni et al. 2007)

1Μ

0 200 400 600 800+ S (kP )

0

200

400

600

800

1000

q (k

Pa)

Saturateds=100 kPas=200 kPa

Compacted Silt (Thu et al. 2006)

Μ1

( ) ( ) or r rq M p sS q M p sS= + = +

p + sSr

Devia

tor s

tress

, q

No. 1 No. 2 No. 3

No. 4 No. 5 No. 6

(Tarantino and El Mountassir 2012)

Module 1 - 83


Pyroclastic soils(Stanier and Tarantino 2013)

∆τ = tan φ’ · sSr if s ≤ sr

∆τ = tan φ’ · sr Sr (sr) if s > sr

(φ’ = 37°)

Water retention function

Module 1 - 84


σh=0

Stable vertical cuts in ‘cohesionless’ soilsunder hydrostatic conditions


σv

uw

Hc

-

+

Hw

0

2

4

6

8

10

12

14

0 10 20 30 40 50

Hc (m)

Hw (m)

Module 1 - 85


Suction has beneficial effect on shear strength and, hence, stability

What about loss of suction associated with rainfall?

Module 1 - 86


Analysis of effect of rainfall on suction profile

Darcy’s law (under 1-D conditions)


Water flow equation (Richard’s equation)

𝑣𝑣 = −𝑘𝑘 𝑢𝑢𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕

𝑢𝑢𝑤𝑤𝛾𝛾𝑤𝑤

+ 𝜕𝜕

𝜕𝜕𝑣𝑣𝜕𝜕𝜕𝜕

+𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡

= 0

𝜕𝜕𝜕𝜕𝜕𝜕

𝑘𝑘 𝑢𝑢𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕


+ 𝜕𝜕 =𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡

𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡

= 𝑆𝑆𝑟𝑟𝜕𝜕𝑛𝑛𝜕𝜕𝑢𝑢𝑤𝑤

𝜕𝜕𝑢𝑢𝑤𝑤𝜕𝜕𝑡𝑡

+ 𝑛𝑛𝜕𝜕𝑆𝑆𝑟𝑟𝜕𝜕𝑢𝑢𝑤𝑤


𝜃𝜃 =𝑉𝑉𝑤𝑤𝑉𝑉

= 𝑛𝑛 � 𝑆𝑆𝑟𝑟

Volumetric water content

𝑆𝑆𝑟𝑟 = 1If then𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡

= −𝜕𝜕𝜀𝜀𝑣𝑣𝜕𝜕𝑡𝑡

𝑘𝑘 = 𝑘𝑘𝑠𝑠𝑠𝑠𝑠𝑠

Terzaghi’s consolidation equation

Module 1 - 87


Analysis of effect of rainfall on suction profile via numerical analysis

𝜕𝜕𝜕𝜕𝜕𝜕

𝑘𝑘 𝑢𝑢𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕


+ 𝜕𝜕 =𝜕𝜕𝜃𝜃𝜕𝜕𝑢𝑢𝑤𝑤


k

uw

θ

uw

+ Boundary and initial conditions

Module 1 - 88


The simplest analysis of effect of rainfall on suction profile

2

2sat w w

ww

k u ux t

uθγ

∂ ∂=

∆ ∂ ∂∆

Assumptions:i) hydraulic conductivity = constant = ksat (conservative assumption)ii) Water retention curve is linearised

Terzaghi consolidation equation

Water flow equation

Assumptions:iii) Initial condition: hydrostatic suction profile (conservative assumption)iv) Boundary condition: Ponded infiltration (conservative assumption)

Initially triangular excess pore-water pressure(with solution from undegraduate geotechnical textbooks)

Module 1 - 89


Linearising the water retention curve

Module 1 - 90


Example for pyroclastic soilHigh permeability, ksat=5⋅10-6 m/s

0 20 40 60 80 100Suction, s (kPa)

10

8

6

4

2

0

Dep

th, z

(m)

Hydrostatic

t = 1 day

t = 2 days

Suction reduces but still remains significant

Suction (kPa)

Dept

h (m

)

Module 1 - 91


Example for pyroclastic soilLow permeability, ksat=7⋅10-7 m/s

Suction is essentially affected by rainfall only at very shallow depths

-300 -200 -100 0Pore-water pressure, uw (kPa)

30

20

10

0

Dep

th, z

(m)

2 days of ponded infiltrationHydrostatic

Hw=10m

Hw=20m

Hw=30m

Negative pore-water pressure (kPa)

Dept

h (m

)

Module 1 - 92


An analytical solution for unsaturated water flow

Assumptions:

i) Exponential hydraulic conductivity and water retention function

ii) Initial condition from steady-state water flow

Yuan and Lu 2005

𝜓𝜓 𝐿𝐿, 𝑡𝑡

=1𝛼𝛼𝑙𝑙𝑛𝑛

𝛼𝛼𝐾𝐾𝑠𝑠𝐾𝐾𝑠𝑠𝛼𝛼𝑒𝑒𝑒𝑒𝑒𝑒 −𝛼𝛼𝐿𝐿 − 8𝑞𝑞1

𝛼𝛼𝐾𝐾𝑠𝑠�𝑛𝑛=1

∞𝑠𝑠𝑠𝑠𝑛𝑛2 𝜆𝜆𝑛𝑛𝐿𝐿

2𝛼𝛼 + 𝛼𝛼2𝐿𝐿 + 4𝐿𝐿𝜆𝜆𝑛𝑛21 − 𝑒𝑒𝑒𝑒𝑒𝑒 −𝐷𝐷 𝜆𝜆𝑛𝑛2 +

𝛼𝛼2

4𝑡𝑡

Solution :

𝐾𝐾 = 𝐾𝐾𝑠𝑠 𝑒𝑒𝑒𝑒𝑒𝑒 𝛼𝛼𝜓𝜓 𝜃𝜃 = 𝜃𝜃𝑠𝑠 𝑒𝑒𝑒𝑒𝑒𝑒 𝛼𝛼𝜓𝜓

Module 1 - 93


Water flow in boundary value problems:

Slope Stability

Module 1 - 94


Infinite slope

α

dx

W

T

N

W=γHdx

Wsinα

Wcosα

τmob=T / (dx/cosα)=W sinα / (dx/cosα) = γHsinαcosα

σn = N / (dx/cosα) = W cosα / (dx/cosα) = γHcos2α

H

Module 1 - 95


Factor of safety

( ) ( ) ( )wwwn

mob

res uH

uH

u ηηααγ

ταφ

ααγτφσ

ττη +=

∆+=

∆+== 0cossintan

'tancossin

'tan

Shear strength criterion

( )wnres uτφστ ∆+= 'tan( )( )

<>∆><∆

0 if 00 if 0

ww

ww

uuuu

ττ

Factor of safety

Module 1 - 96


Saturated slope

α

EquipotentialFlowline

Hcos2α

uw

αφ

γγ

ααγφ

αφ

ττη

tan'tan1

cossin'tan

tan'tan

−=

−+== ww

mob

res

Hu

( ) ( )'tan'tan'tan φφσφστ wnwnres uu −+=−=

+

Module 1 - 97


Partly dry slope

α hwcos2α

uw

αφ

γγ

ααγφ

αφ

ττη

tan'tan1

cossin'tan

tan'tan

−=

−+==

Hh

Hu www

mob

res


hw

+

Module 1 - 98


Saturated slope with negative pressures

α


Hcos2α

uw

ααγφ

αφ

γγ

ααγφ

αφ

ττη

cossin'tan

tan'tan1

cossin'tan

tan'tan

Hs

Hu bww

mob

res +

−=

−+==


+

-

sb/γw

Module 1 - 99


Unsaturated slope

α


uw

( )ααγ

ταφ

ττη

cossintan'tan

Huw

mob

res ∆+==


-

Module 1 - 100


Stability of saturated/unsaturated slopes

α

W

H

Hw

'tantan'tan1 φαφγγ

≤≤

− w

'tantan φα >

'tan1tan φγγα

−< w Unconditionally stable

Unstable for uw≥0

Stable for uw<0

Module 1 - 101


Landslides in presence of water table

T

N

α

z

uw

_

hW

H

∆l

( ) )tan()'tan( bwu φφστ −+=

( ) ( )Hh sin cos

tantan

'tan<

−+=

ααγφ

αφη

hu bw

Module 1 - 102


Factor of safety under hydrostatic conditions

y

soil (φ' < α)

bedrock

z' zH

y'

2

H*1

α

-uw 1

z

η

HH*

Module 1 - 103


Modelling water flow

( ) grad (h) u - Kv w=

0 div =∂Θ∂

+ t

v

Darcy’s law


( )[ ]

∂

Θ∂=

=∂∂

wu

hKth

w Cwith

grad divC

γ

Module 1 - 104


One-dimensional flow in infinite slope

( ) cossin '

'

'

'

C ww γuαz'α y'h zhK

zyhK

yth

++=

∂∂

∂∂

+

∂∂

∂∂

=∂∂

In y’, z’ reference system:

Since:

0 '

and 0'

2

2=

∂

∂=

∂∂

yh

yK

then:

'

'

C

∂∂

∂∂

=∂∂

zhK

zth

One-dimensional flow

Module 1 - 105


Low intensity rainfall

-2.0

-1.5

-1.0

-0.5

0.0

dept

h (m

)

hydrostaticafter 1hafter 3hafter 6hη = 1

0.96 1 1.04 1.08 1.12 1.16factor of safety η

-2.0

-1.5

-1.0

-0.5

0.0

dept

h (m

)

a)

b)

Module 1 - 106


High intensity rainfall

-2.0

-1.5

-1.0

-0.5

0.0

dept

h (m

)

hydrostaticafter 1hafter 2hη = 1

0.9 1 1.1 1.2 1.3factor of safety η

-2.0

-1.5

-1.0

-0.5

0.0

dept

h (m

)

a)

b)

Module 1 - 107


Intensity-duration threshold curve

0 5 10 15 20 25duration (h)

0.0

0.5

1.0

1.5

2.0

2.5

inten

sity (

x 10-

5 m/

s)

deep slides

shallow slides

Excellent qualitative agreement with experimental data (Wieczorek 1987)

Module 1 - 108


Shallow landslides in absence of water table

H

y'

y

zz'

H'h'

h

α

impervious bedrock

( ) ( )[ ] 'tanφχστ waa uuu −+−=

Module 1 - 109


Shear strength criterion (Khalili and Kabbaz 1998)

( ) ( )[ ] 'tanφχστ waa uuu −+−=

( )( ) ( ) ( )

( ) ( )

−<−=

−>−

−−

=−

bwawa

wawabwa

wa

uuuu

uuuuuuuu

1

b

55.0

χ

χ

with

Non-linear envelope

Module 1 - 110


Factor of safety

( ) ( ) ( ) ( )

( ) ( ) ( ) cossin

'tan

tan'tan

cossin

'tan

tan'tan

b

b

55.045.0

www

wwbww

uuh-u

uuh

-u-u

−<−+=

−>−+=

ααγφ

αφη

ααγφ

αφη

Module 1 - 111


Modelling water flow

'

'

∂∂

∂∂

=∂∂

zhK

zthC

** *

* **

∂∂

∂∂

=∂

∂zhK

zthC

1-D flow in infinite slope

1-D flow in vertical direction

( ) ww zuu γα cos1* *w −+=

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]wwww

wwww

zuKuKuK

zuuu

γα

γαθθθ

cos1*

cos1*

***

***

−+=≡

−+=≡

Module 1 - 112


Case study: Upper Carinthia and Eastern Tyrol

Module 1 - 113


Water retention and conductivity function

0.1 1 10 100 1000 10000Suction (kPa)

0.000.050.100.150.200.250.300.35

Volum

etric

water

conte

nt θ

0.1 1 10 100 1000 10000Suction (kPa)

1E-016

1E-014

1E-012

1E-010

1E-008

1E-006

Hydra

ulic c

ondu

ctivit

y (m/

s)

From pedo-transfer function by:Vereecken et al. 1989 Vereecken 1990

Module 1 - 114


Antecedent evapotranspiration

1 10 100 1000 10000Log suction (kPa)

0

0.4

0.8

1.2

1.6

2

Elev

ation

z (m

)

initial (hydrostatic)after 6dafter 12dafter 12d and 16h

1 10 100Log suction (kPa)

0

0.4

0.8

1.2

1.6

2

Elev

ation

z (m

)

initial (hydrostatic)after 10d and 3hafter 60d

qflux=const. s=const.

Module 1 - 115


Rainfall infiltration

0 10 20 30 40 50Suction (kPa)

0

0.4

0.8

1.2

1.6

2El

evati

on z

(m)

critical suctioninitialafter 10hafter 20hafter 30hafter 36h after 36h 45'

i=5 mm/h

Module 1 - 116


Intensity-duration threshold curve

0.01 0.1 1 10 100 1000Rainfall duration (h)

0.1

1

10

100

1000

Rain

fall i

nten

sity

(mm

/h)

Numerical simulation Observed Data

Module 1 - 117


References

De Vita, P., Angrisani, A.C., Di Clemente, E., 2008. Engineering geological properties of the phlegraean pozzolan soil (Campania region, Italy) and effect of the suction on the stability of cut slopes. Italian Journal of Engineering Geology and Environment 2, 5–22.

Karube, D. & Kawai, K. (2001). The role of pore water in the mechanical behaviour of unsaturated soils. Geotech. Geol. Engng 19, 211–241.

Romero, E. & Vaunat, J. (2000). Retention curves in deformable clays. In Experimental evidence and theoretical approaches in unsaturated soils: Proceedings of an international workshop (eds A. Tarantino and C. Mancuso), pp. 91–106. Rotterdam: A. A. Balkema.

Stanier S and Tarantino A (2013). An approach for predicting the stability of vertical cuts in cohesionless soils above the water table. Engineering Geology, Geology 158, 98–108.

Tarantino, A. & Bosco, G. 2000. Role of soil suction in understanding the triggering mechanisms of flow slides associated with rainfall. In G.F. Wieczorek & N.D. Naeser (eds.), Debris-Flow Hazard Mitigation, Second International Conference on debris-flow hazards mitigation, 16-18 August 2000, Taipei, Taiwan: 81-88. Rotterdam: A. A. Balkema.

Tarantino, A. and Mongiovì, L. 2003. Numerical modelling of shallow landslides triggered by rainfall. International Conference on Fast Slope Movements, Sorrento, Italy, 491-495.

Tarantino, A., and Tombolato, S. 2005. Coupling of hydraulic and mechanical behaviour in unsaturated compacted clay. Géotechnique, 55(4), 307-317. Tarantino, A. 2010. Basic concepts in the mechanics and hydraulics of unsaturated geomaterials. New Trends in the Mechanics of Unsaturated Geomaterials Lyesse Laloui (ed.), 3-28. ISTE – John Wiley & Sons.

Tarantino, A. 2010. Field measurement of suction, water content and water permeability. New Trends in the Mechanics of Unsaturated Geomaterials Lyesse Laloui (ed.), 129-154. ISTE – John Wiley & Sons

Tarantino, A. 2010. Basic concepts in the mechanics and hydraulics of unsaturated geomaterials. New Trends in the Mechanics of Unsaturated Geomaterials Lyesse Laloui (ed.), 3-28. ISTE – John Wiley & Sons.

Tarantino A and El Mountassir G (2013). Making unsaturated soil mechanics accessible for engineers: preliminary hydraulic-mechanical characterisation & stability assessment. Engineering Geology, 165 , 89–104.

Yuan F and Lu Z 2005. Analytical Solutions for Vertical Flow in Unsaturated, Rooted Soils with Variable Surface Fluxes. Vadose Zone Journal 4:1210–1218 (2005).

Documents

Summer School – Cagliari, 22 -23 giugno 2014 · Summer School – Cagliari, 22 -23 giugno 2014. Module 1 - 2 Prof. Alessandro Tarantino, University of Strathclyde, UK Tarantino,