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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Surface tension and capillary systems
Module 1 - 4
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Hysteresis of the contact angle
PRESSUREWATER
Module 1 - 10
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Water retention of a sytem of capillary tubes
1 2 3 4 5
Vw / V
- uw
1 2
34
5
S
Module 1 - 15
Prof. Alessandro Tarantino, University of Strathclyde, UK
Increasing water tension in a capillary tube
1 2 3
wrTh
γθ 1cos2
=
wrTh
γθ 1cos2
=
Module 1 - 16
Prof. Alessandro Tarantino, University of Strathclyde, UK
Increasing water tension in a system of capillary tubes
Z
Vw / V1
S
Module 1 - 17
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Soil as a system of capillary tubes
Module 1 - 19
Prof. Alessandro Tarantino, University of Strathclyde, UK
Water retention behaviour
Module 1 - 20
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Soil water retention
S
ln (s)
1
Saturated soilQuasi-saturated soil
Partially saturated soil
Residual saturation
Module 1 - 22
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Very little evaporation is sufficient to curve menisci and lower water pressure to the air entry value, say uw=-100 kPa
Module 1 - 30
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Very little evaporation is sufficient to curve menisci and lower water pressure to the air entry value, say uw=-10 kPa
Module 1 - 31
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Hydralic hystheresis
S
ln (s)
1
Hp: incompressible soil skeleton
Main drying curve
Main wetting curve
Scanning curves’
Module 1 - 33
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Effect of void ratio on WRC (1)
Main drying curves
S
ln (s)
1
θr θr
Module 1 - 37
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Effect of void ratio on air-entry and air-occlusion value
(Karube and Kawai, 2001)
Module 1 - 40
Prof. Alessandro Tarantino, University of Strathclyde, UK
Soil water retention mechanisms and the concept of suction
Module 1 - 41
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Capillary mechanisms
uw = 0
particlepore water
uw < 0Owing to capillary tension, pore water pressure is negative
Module 1 - 43
Prof. Alessandro Tarantino, University of Strathclyde, UK
Osmotic mechanisms
+- - - - - - - - - -
+ + + + + + + + ++
++
++
++
++
+ + + + + + + + + ++
++
++
++
++
Clay particle
+
+
+
+
+
+
-
--
- -
-
Free water
+++
+
++
++
+
+
++
+
--
--
--
-
--
--
--
+
+
+ +-
---
+ -+ -
- - - - - - - - - -
Module 1 - 44
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Rusty in basic thermodynamics ?….. get out our thermodynamics book !
Module 1 - 51
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Liquid-vapour equilibrium for curved interface (3)
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Liquid-vapour equilibrium for curved interface (4)
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
acqueous solution, pw
+
++
+
+
++
+
- --
-
--
-
vapour, pv
air, pa
Module 1 - 57
Prof. Alessandro Tarantino, University of Strathclyde, UK
Liquid-vapour equilibrium for aqueous solution (3)
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Liquid-vapour equilibrium for aqueous solution (4)
Raoult’s law
( )0 1v v sp p x= −
Module 1 - 59
Prof. Alessandro Tarantino, University of Strathclyde, UK
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 :
acqueous solution, pw
+
++
+
+
++
+
- --
-
--
-In particular:
(Negative) liquid pressure
Solute concentration
Module 1 - 60
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Principle of suction measurement
Module 1 - 62
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Unsaturated hydraulic conductivity
Module 1 - 69
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Influence of S on unsaturated permeability of compacted clay
∝k S3
Module 1 - 75
Prof. Alessandro Tarantino, University of Strathclyde, UK
Hysteresis of the permeability function
Module 1 - 76
Prof. Alessandro Tarantino, University of Strathclyde, UK
Effect of the shape of the permeability function on sample drying
k/ksat
ln (s)
evaporation
‘impermeable’ outer shell
Faster drying
Module 1 - 77
Prof. Alessandro Tarantino, University of Strathclyde, UK
Water flow in boundary value problems:
Stability of vertical cuts
Module 1 - 78
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
σh=0
Stable vertical cuts in ‘cohesionless’ soils
The real situationStable because of suction
σ’’
τ
φ’
σ’’h σ’’vσh σv
Effective
Total
Pyroclastic ‘cohesionless’ silty sand
σv
uw
H=10-20m
-
+
τ = σ’’ tan (φ’) = [σ+sSrM ] tan (φ’)
Module 1 - 81
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
SaturatedUnsaturated
Reconstituted silt(Geiser et al. 2006)
Μ
1
0 400 800 1200σ+sSr (kPa)
0
200
400
600
800
τ (k
Pa)
SaturatedUnsaturated
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
σh=0
Stable vertical cuts in ‘cohesionless’ soilsunder hydrostatic conditions
Pyroclastic ‘cohesionless’ silty sand
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Suction has beneficial effect on shear strength and, hence, stability
What about loss of suction associated with rainfall?
Module 1 - 86
Prof. Alessandro Tarantino, University of Strathclyde, UK
Analysis of effect of rainfall on suction profile
Darcy’s law (under 1-D conditions)
Mass balance equation (no water vapour flow)
Water flow equation (Richard’s equation)
𝑣𝑣 = −𝑘𝑘 𝑢𝑢𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕
𝑢𝑢𝑤𝑤𝛾𝛾𝑤𝑤
+ 𝜕𝜕
𝜕𝜕𝑣𝑣𝜕𝜕𝜕𝜕
+𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡
= 0
𝜕𝜕𝜕𝜕𝜕𝜕
𝑘𝑘 𝑢𝑢𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕
𝑢𝑢𝑤𝑤𝛾𝛾𝑤𝑤
+ 𝜕𝜕 =𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡
𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡
= 𝑆𝑆𝑟𝑟𝜕𝜕𝑛𝑛𝜕𝜕𝑢𝑢𝑤𝑤
𝜕𝜕𝑢𝑢𝑤𝑤𝜕𝜕𝑡𝑡
+ 𝑛𝑛𝜕𝜕𝑆𝑆𝑟𝑟𝜕𝜕𝑢𝑢𝑤𝑤
𝜕𝜕𝑢𝑢𝑤𝑤𝜕𝜕𝑡𝑡
𝜃𝜃 =𝑉𝑉𝑤𝑤𝑉𝑉
= 𝑛𝑛 � 𝑆𝑆𝑟𝑟
Volumetric water content
𝑆𝑆𝑟𝑟 = 1If then𝜕𝜕𝜃𝜃𝜕𝜕𝑡𝑡
= −𝜕𝜕𝜀𝜀𝑣𝑣𝜕𝜕𝑡𝑡
𝑘𝑘 = 𝑘𝑘𝑠𝑠𝑠𝑠𝑠𝑠
Terzaghi’s consolidation equation
Module 1 - 87
Prof. Alessandro Tarantino, University of Strathclyde, UK
Analysis of effect of rainfall on suction profile via numerical analysis
𝜕𝜕𝜕𝜕𝜕𝜕
𝑘𝑘 𝑢𝑢𝑤𝑤𝜕𝜕𝜕𝜕𝜕𝜕
𝑢𝑢𝑤𝑤𝛾𝛾𝑤𝑤
+ 𝜕𝜕 =𝜕𝜕𝜃𝜃𝜕𝜕𝑢𝑢𝑤𝑤
𝜕𝜕𝑢𝑢𝑤𝑤𝜕𝜕𝑡𝑡
k
uw
θ
uw
+ Boundary and initial conditions
Module 1 - 88
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Linearising the water retention curve
Module 1 - 90
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Water flow in boundary value problems:
Slope Stability
Module 1 - 94
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Partly dry slope
α hwcos2α
uw
αφ
γγ
ααγφ
αφ
ττη
tan'tan1
cossin'tan
tan'tan
−=
−+==
Hh
Hu www
mob
res
( ) ( )'tan'tan'tan φφσφστ wnwnres uu −+=−=
hw
+
Module 1 - 98
Prof. Alessandro Tarantino, University of Strathclyde, UK
Saturated slope with negative pressures
α
EquipotentialFlowline
Hcos2α
uw
ααγφ
αφ
γγ
ααγφ
αφ
ττη
cossin'tan
tan'tan1
cossin'tan
tan'tan
Hs
Hu bww
mob
res +
−=
−+==
( ) ( )'tan'tan'tan φφσφστ wnwnres uu −+=−=
+
-
sb/γw
Module 1 - 99
Prof. Alessandro Tarantino, University of Strathclyde, UK
Unsaturated slope
α
EquipotentialFlowline
uw
( )ααγ
ταφ
ττη
cossintan'tan
Huw
mob
res ∆+==
( ) ( )'tan'tan'tan φφσφστ wnwnres uu −+=−=
-
Module 1 - 100
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Modelling water flow
( ) grad (h) u - Kv w=
0 div =∂Θ∂
+ t
v
Darcy’s law
Mass balance equation (no water vapour flow)
( )[ ]
∂
Θ∂=
=∂∂
wu
hKth
w Cwith
grad divC
γ
Module 1 - 104
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Shallow landslides in absence of water table
H
y'
y
zz'
H'h'
h
α
impervious bedrock
( ) ( )[ ] 'tanφχστ waa uuu −+−=
Module 1 - 109
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
Case study: Upper Carinthia and Eastern Tyrol
Module 1 - 113
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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
Prof. Alessandro Tarantino, University of Strathclyde, UK
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).