[Terzaghi] Unsaturated Soil Mechanics (2007)
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Transcript of [Terzaghi] Unsaturated Soil Mechanics (2007)
-
Unsaturated Soil Mechanics in Engineering
Professor Delwyn G. FredlundUniversity of Saskatchewan
Saskatoon, SK, Canada
GeoFrontiers 2005, Austin, TexasGeo-Institute, ASCE
January 23-26, 2005
-
John Wiley & Sons, 1943
Karl Terzaghi elevated Soil Mechanics from an Art to a Science
Effective Stress, ( uw), for describing mechanical behavior of saturated soils
Chapter 14 Capillary Forces(Also Chapter 15)
Biot (1941) addressed consolidation of unsaturated soils
Concepts from Agriculture (Baver, 1940)
Chiu Chung Fai1.Terzaghi 2.3.""4. Biot (1941) 5.(Baver, 1940)
-
Unsaturated Soil Mechanics Problems Described in Theoretical Soil
Mechanics by K. Terzaghi (1943)
Chiu Chung FaiTerzaghi
-
Unsaturated Soil Mechanics in Engineering
Introduction Challenges to Implementation Description of the Stress State Fundamental Constitutive Relations Role of the Soil-Water Characteristic Curve Use of SWCC in the Constitutive Relations Solution of a Series of PDEs Modeling Unsaturated Soils Problems
Chiu Chung Fai1.2.3.4.5.6.7.8.
-
Objectives
To illustrate the progression from theories and formulations to practical engineering protocols for solving a variety of unsaturated soil mechanics problems (e.g., seepage, shear strength and volume change), through use of direct and indirect characterization of unsaturated soil property functions
To describe the Challenges Faced and the Solutions Generated in moving towards the Implementation of Unsaturated Soil Mechanics
Chiu Chung Fai1.2.
-
Gradual Emergence of Unsaturated Soil Mechanics
1950s: Independent measurement of pore-air and pore-water pressure through use of high air entry ceramic disks
1960s: Laboratory testing of unsaturated soils 1970s: Constitutive relations proposed and tested
for uniqueness for unsaturated soils 1980s: Solving formulations for classic Boundary
Value Problems 1990s: Establishing procedures for determination of
unsaturated soil property functions 2000+: Implementation into routine engineering
practice
Chiu Chung Fai1950s 1960s 1970s 1980s 1990s 2000+
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #1: To discover appropriate Stress State
Variables for describing the physical behavior of unsaturated soils
Solution #1:
?
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #2: To develop devices that could measure a wide range of negative pore-water pressures (i.e., high matric suctions)
Solution #2:
?
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #3: To develop (and test for uniqueness)
constitutive relations suitable for describing unsaturated soil behavior
Solution #3:
?
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #4: To overcome the excessive costs
associated with the determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions)
Solution #4: ?
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #5: To solve nonlinear partial differential
equations for unsaturated soils without having convergence difficulties during the iterative solution process
Solution #5:
?
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #6: To promote and teach unsaturated soil
mechanics at universities and in engineering practice
Solution #6:
?
Chiu Chung Fai
-
Regional distribution of unsaturated soils
Local vertical zones of
unsaturated soils
GROUNDWATER TABLE- Water filling the voids- Air in a dissolved state
S
A
T
U
R
A
T
E
D
S
O
I
L
-
Zones of Unsaturation Defined by a Soil-Water Characteristic Curve, SWCC
0
5
10
15
20
25
30
35
0.1 1.0 10. 100. 1000. 10,000 100,000 1000,000
G
r
a
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m
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w
a
t
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n
t
e
n
t
,
w
(
%
)
Soil suction (kPa)
Air entry value
Boundary effect zone
Transition zone
Residual zone
Residualcondition
Inflection point
Chiu Chung Fai(SWCC)
-
Unsaturated Soil REV as a Four Phase System
-Two Phases that deform and come to rest under a stress gradient (SOLIDS)
-Soil structure
-Contractile skin
-Two phases that continuously flowunder a stress gradient (FLUIDS)
-Water
-Air
Soil particlesAir
Contractile skin
Water
REV = Representative Elemental Volume
Chiu Chung Fai()()
-
Structure and Stresses in the Contractile Skin
Hyperbolic Tangent Function
Thickness:1.5 to 2 water molecules or about 5A (Israelachvili, 1991; Townsend and Rice, 1991)
thickness of thecontractile skin
Liquid water density
Water vapor density
Air Watert90/10
PBN
Surface tension = 75 mN/m; Equivalent stress = 140,000 kPa
Water-molecule distribution across the air-water interface(modified from Kyklema, 2000)
Chiu Chung Fai-(Kykelma, 2000)
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Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #1: To discover appropriate Stress State
Variables for describing the physical behavior of unsaturated soils
Solution #1: Designation of independent Stress State Variables based on multiphase continuum mechanics principles
(x - ua)
(z - ua)
(y - ua)(ua - uw)
(ua - uw)
zxzy
xz
xy
yzyx(ua - uw)
Chiu Chung Fai
-
Definition of stress state at a point in an unsaturated soil
(z - ua)
(y - ua)(ua - uw)
(ua - uw)
zxzy
xz
xy
yzyx(ua - uw)
(x - ua)
Defines the stress state at a point in a continuum
State variables are independent of soil properties
Derivation of the Stress State is based on the superposition of equilibrium stress fields for a multiphase continuum
-
State Variable Stage (Unsaturated Soils)
Net Total Stress Tensor
( ) ( ) ( )
x a yx zx
xy y a zy
xz yz z a
uu
u
( )( ) ( )
u uu u
u u
a w
a w
a w
0 00 00 0
Matric Suction Stress Tensor
X - direction
Y - direction
Z - direction
Stress Tensors form the basis for a Sciencebecause we live in a 3-D Cartesian coordinate world
Chiu Chung Fai
-
Variations in Stress State Description
= ( ua) + (ua uw) = effective stress = parameter related to saturation
*ij = ij [S uw + (1 S) ua ] ijij = total stress tensor, ij = Kroneker delta or substitution tensor,
*ij = Bishops soil skeleton stress (Jommi2000)
S = degree of saturation
Above proposed equations are constitutive relations
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #2:- To develop devices that can measure a wide range of negative pore-water pressures (i.e., high matric suctions)
Solution #2:- New instrumentation such as the high suction tensiometersand indirect thermal conductivity suction sensors provide viable techniques for the laboratory and the field
Chiu Chung Fai
-
Monitoring for Verification Purposes
Measurements of movement: same as for saturated soils
Measurement of water content: TDR Technology Measurement of matric suction:
Direct measurement techniques Low range tensiometers (< 90 kPa) High range direct tensiometers (< 1200 kPa)
Presently primarily for laboratory use Indirect measurement techniques
Thermal conductivity sensors
Chiu Chung Fai1.2.(TDR)3. - (< 90 kPa) (< 1200 kPa) -
-
Monitoring of Water Content
Measures the dielectric constant for the soil around the rods. Dielectric constant varies with the water content of the soil
TDR ThetaProbe, ML2x manufactured by AT Delta Devices, U.K.
Chiu Chung FaiTDR
-
Monitoring of Matric Suction
Measures the thermal conductivity of a standard ceramic that varies in water content with the applied matric suction
Chiu Chung Fai
-
Matric Suction Versus Time - 1.0 m to 1.3 m Depth Range
0.0
25.0
50.0
75.0
100.0
125.0
150.0
175.0
200.0
15-Sep-00 5-Oct-00 25-Oct-00 14-Nov-00 4-Dec-00 24-Dec-00
Time (Days)
M
a
t
r
i
c
S
u
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t
i
o
n
(
k
P
a
)
T 1-3
T 2-8
T 3-11
T 4-14
T 5-16
T 4-14
T 1-3
T 2-8
T 3-11
T 5-16
In Situ Matric Suction measurements using Thermal Conductivity sensors at 1.0 to 1.3 m below roadway
Equalization
Frost
Time (Days)
M
a
t
r
i
c
s
u
c
t
i
o
n
(
k
P
a
)
Chiu Chung Fai1 - 1.3 m
-
Silicone rubber grommet
Rubber membrane
Latex rubber, to seal the rubbermembrane and grommet
Mini suction probe
O - ring5 bar high air-entry ceramic disk
Direct, high suction sensor used to measure suctions greater than one atmosphere on the side of a triaxial specimen (Meilani, 2004)
Pore air pressure control
O - ringCoarse corundum disk
Filter paper
Specimen
Top cap
Water in the compartment is pre-pressurized to destroy cavitation nuclei
Chiu Chung Fai100 kPa(Meilani, 2004)
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #3: To develop (and test for uniqueness)
constitutive relations suitable for describing unsaturated soil behavior
Solution #3:
Matric suction(ua - uw)
Voidratio
e
Net nor
mal stre
ss
( - ua)
amat- Constitutive relations for
saturated soils needed to be extended to embrace the effect of changing degrees of saturation
Chiu Chung Fai
-
Fundamental Constitutive Relations for Unsaturated Soils
Constitutive Behaviors in Classic Soil Mechanics: Seepage Shear strength Volume-mass changes: Void ratio, water content
changes
Other topics in soil mechanics: Heat flow (Freeze-Thaw and Evaporation) Air flow Contaminant transport
Each constitutive relationship requires a nonlinear soil property function; therefore, Unsaturated Soil Mechanics might be referred to as NONLINEAR SOIL MECHANICS!
Chiu Chung Fai1.2.3.-()4. (-)5.6.
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Water Seepage Constitutive Relations
Yg
uhw
w += Driving potential for water flow is hydraulic head, h
x wxdhv kdx
=
dydhkv wyy =
z wzdhv kdz
=
Darcys law (1856) for flow in the x-, y-, and z-direction
Coefficient of permeability, kw is a function of matric suction; therefore, the flow law is nonlinear and subject to hysteresis
Chiu Chung Fai1.2.Darcy3.(kw)4.5.
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Shape of the water permeability function for glass beads tested by Mualem (1976 )
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
0.1 1 10 Soil suction (kPa)
Drying
Wetting
C
o
e
f
f
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n
t
o
f
p
e
r
m
e
a
b
i
l
i
t
y
(
m
/
s
)
Wetting
Drying
DryingWetting
Soil suction (kPa)
Chiu Chung Fai(kw)Mualem (1976)
-
The SWCC for the glass beads showing hysteresis during drying and wetting
0
20
40
60
80
100
0.1 1 10 Soil suction (kPa)
D
e
g
r
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e
o
f
s
a
t
u
r
a
t
i
o
n
,
%
DryingWetting
Wetting
Drying
Soil suction (kPa)Hysteresis in the SWCC produces hysteresis in the Permeability function
Chiu Chung FaiSWCC
-
Water Storage in an Unsaturated Soil
0.1 1.0 10 100 1000 10000 100000 1E+6Soil suction, kPa
W
a
t
e
r
c
o
n
t
e
n
t
,
%
0
10
20
30
40Soil-Water Characteristic Curve, SWCC
Also has a hysteretic effect
Soil suction, kPa
W
a
t
e
r
s
t
o
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a
g
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m
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d
u
l
u
s
,
k
P
a
-
1
0
2
4
8
0.1 1.0 10 100 1000 10000 100000 1E+6
Water storage function is the slope of the SWCC; Required for transient seepage analyses
Chiu Chung Fai1.SWCC2.
-
Air Flow Constitutive Relations
Driving potential for air flow is Pore-air pressure, uadx
dukv aaxax =
dydu
kv aayay =
dzdukv aazaz =
Ficks law for flow in the x-, y-, and z-direction
Coefficient of permeability, ka is a function of matric suction; therefore, the flow law is nonlinear and subject to hysteresis
Observation: Soil properties for unsaturated soils become nonlinear functions and are hysteretic in character
Chiu Chung Fai1.2.Fick3.(ka)4.5.
-
Shear Strength Constitutive Relations
' '( ) tan ( ) tan bn a a wc u u u = + + Linear form of the extended Mohr-Coulomb shear strength equation
Fredlund, Morgenstern and Widger, 1978
' '1( ) tan ( )n a a wc u u u f = + +
f1 = function showing the rate of increase in shear strength with matric suction
Chiu Chung FaiMohr-Coulomb
-
Extended Mohr-Coulomb failure surface (Fredlund, Morgenstern and Widger, 1978)
(ua-uw)
S
h
e
a
r
s
t
r
e
n
g
t
h
,
Net normal stress, ( - ua)
Shear strength versus suction is nonlinear and affected by hysteresis
Air entry value
c
Chiu Chung Fai Mohr-Coulomb (Fredlund, Morgenstern & Widger, 1978)1.2.
-
Multistage direct shear test results
on compacted glacial till (Gan et
al., 1988)
Soil-Water Characteristic Curve
for glacial till
0102030405060708090
100
1 10 100 1000 10000 100000 1000000Suction, (kPa)
D
e
g
r
e
e
o
f
S
a
t
u
r
a
t
i
o
n
,
S
(
%
)
0 kPa, e = 0.517
25 kPa, e =0.514
OPTIMUM INITIAL WATER CONTENT
SPECIMEN
AEV = 60 kPa
AEV = 60 kPa
S
h
e
a
r
s
t
r
e
n
g
t
h
,
( kP
a
)
200
250
150
100
50
00 100 200 300 400 500
c = 10 kPa(f - ua)f tan = 34.6 kPa
(f - ua)f = 72.6 kPa = 25.5
Matric suction, (ua-uw) (kPa)
Chiu Chung Fai1.SWCC2.(Gan et al., 1988)
-
Reference compression curves for a Saturated Soil
v
=
(
1
+
e
)
e
Log()Ln(p)p0
Cs
0Cc
Effective mean stress Effective vertical stress
S
p
e
c
i
f
i
c
v
o
l
u
m
e
V
o
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d
r
a
t
i
o
Preconsolidationpressure
Yield stress
ss CC 434.0
)10ln(=
cc CC 434.0
)10ln(=
Effective mean stress Effective vertical stress
Elasto-plastic form
Classic soil mechanics form
Chiu Chung Fai
-
10000
1000100
101 0.1
0.01
Log net me
an stress (k
Pa)1e+061
0000010000
1000100
1010.
10.01Log soil suction (kPa)
0
0
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
V
o
i
d
r
a
t
i
o
V
o
i
d
r
a
t
i
o
VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)
Void ratio, e
Residual value
YieldYield
Soil suctionNet tot
al stress
-
VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)
Residual value
10000
1000100
101 0.1
0.01
Log net mean
stress (kPa)
1e+06100000
100001000
10010
10.10.01Log soil suction (kPa)
0
0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
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Residual value
Air entry value
YieldYield
Water content, w
Soil suction Net total stres
s
SWCC
-
10000
1000100
101 0.1
0.01
Log net me
an stress (k
Pa)1e+061
0000010000
1000100
1010.
10.01Log soil suction (kPa)
0
0
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
D
e
g
r
e
e
o
f
s
a
t
u
r
a
t
i
o
n
(
S
)
D
e
g
r
e
e
o
f
s
a
t
u
r
a
t
i
o
n
(
S
)
Air entry value
Residual value
VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)
Degree of saturation, S
Soil suction Net total stres
s
-
VolumeMass Constitutive Surfaces for
Beaver Creek Sand (Pham, 2004)
10000
1000100
101 0.1
0.01
Log net mean
stress (kPa)1e+061
0000010000
1000100
1010.
10.01Log soil suction (kPa)
0.3
0.3
0.34625
0.34625
0.3925
0.3925
0.43875
0.43875
0.485
0.485
0.53125
0.53125
0.5775
0.5775
0.62375
0.62375
0.67
0.67
V
o
i
d
r
a
t
i
o
(
e
)
V
o
i
d
r
a
t
i
o
(
e
)
10000
1000
10010
1 0.10.01
Log net me
an stress
(kPa)
1e+061000
0010000100
010010
10.10.0
1Log soil suction (kPa)
0
0
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
D
e
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s
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a
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i
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D
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f
s
a
t
u
r
a
t
i
o
n
10000
1000100
101 0.1
0.01
Log net me
an stress (
kPa)
1e+061000
0010000100
010010
10.10.01Log soil suction (kPa)
0
0
0.05
0.05
0.1
0.1
0.15
0.15
0.2
0.2
0.25
0.25
0.3
0.3
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n
t
Void ratio, e
Water content, w
Degree of saturation, S
SWCC
AEV
Basic volume-mass equation S e = w Gs
Chiu Chung FaiBeaver Creek (Pham, 2004)
-
10000
1000100
101 0.1
0.01
Log net me
an stress (
kPa)
1e+061000
001000010
0010010
10.10.0
1Log soil suction (kPa)
0
0
0.05
0.05
0.1
0.1
0.15
0.15
0.2
0.2
0.25
0.25
0.3
0.3
G
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w
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c
o
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t
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n
t
Residual value
Air entry value
VolumeMass Constitutive Surfaces for Beaver Creek Sand (Pham, 2004)
Water content, w
Soil suctionNet tot
al stress
SWCC
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #4: To overcome the excessive costs associated with the
determination (i.e., measurement) of unsaturated soil properties (i.e., nonlinear functions)
Solution #4:
0.0
10
20
30
40
1E-1 1E+0 1E+1 1E+2 1E+3 1E+4Suction (kPa)
Predicted from grain-sizeExperimental
W
a
t
e
r
c
o
n
t
e
n
t
1E+5 1E+6
- Indirect, estimation procedures have been developed to obtain unsaturated soil property functions based on Soil-Water Characteristic Curves
Chiu Chung FaiSWCC
-
Role and Measurement of the Soil-Water Characteristic Curve, SWCC
-Soil-Water Characteristic Curves, SWCC define the relationship between the amount of water in a soil and soil suction (i.e., matric suction and total suction)
- SWCC has been successfully used to estimate all unsaturated soil property functions
* ASTM Standard D6836-02 (2003)
0
4
8
12
16
20
0.1 1 10 100 1000Soil suction (kPa)
G
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t
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t
(
%
)
Specimen 1Specimen 2Specimen 3
Air Entry Value
Residual ValueSand
- Water permeability - Air permeability - Shear strength - Thermal flow - Incremental
elasticity
Chiu Chung Fai(SWCC)1.SWCC2.SWCC - - - -
-
Measured drying and wetting curves on processed silt (Pham, 2002)
0
5
10
15
20
25
0.1 1 10 100 1000Soil suction (kPa)
G
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t
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n
t
(
%
)
Specimen 1Specimen 2Specimen 3
AEV = 10 kPa
WEV = 4.5 kPa
Residual = 120 kPa
Residual = 62 kPa
Chiu Chung FaiSWCC (Pham, 2002)
-
Pressure Plate Apparatus to Measure Void Ratio and Water Content While Applying Total Stress and Matric Suction
Manufactured by GCTS, Tempe, AZ
Air supply
High air entry disk
Chiu Chung Fai
-
Fifteen bar Pressure Plate equipment manufactured by
GCTS, U.S.A.
Wide range of applied suctions Applies total stresses Measures water and total
volume change Measure diffused air Test individual specimens Null-type initial suction Drying and wetting modes
Chiu Chung Fai15 bar 1.2.3.4.5.6.7.
-
Equations to Best-Fit SWCC Data
Fredlund and Xing (1994)
Correction FactorAir entry value
Rate of desaturation
Asymmetry Variable
= Soil suction
fm
fn
f
s
ae
wCw
]})({ln[)()( +=
)1000000(1ln[
)1ln(1)(
r
rC
++
=
Numerous equations have been proposed:-Brooks & Corey (1964)- van Guenuchten (1980)
0
4
8
12
16
20
0.1 1 10 100 1000Soil suction (kPa)
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t
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(
%
)
Specimen 1Specimen 2Specimen 3
Wetting
Drying
Chiu Chung FaiSWCC
-
Hysteresis in the Soil-Water Characteristic Curve
Hysteretic SWCC Models will eventually be available for geotechnical usage
Presently, the Geotechnical Engineer must decide which curve to use: Select wetting curve or drying curve based on
process being simulated Hysteresis loop shift at point of inflection:
Sands: 0.15 to 0.35 Log cycle Average: 0.25 Log cycle
Loam soils: 0.35 to 0.60 Log cycle Average: 0.50 Log cycle
Estimation Values
Chiu Chung FaiSWCC1.SWCC2.
-
Model measurements of water content and matric suction showing the SWCC relationship from water contents and
matric suctions during wetting and drying simulations (Tami et al, 2004)
Section MSection B Section T
1 100.0
0.1
0.2
0.3
0.4
I&II-D
III-WIV&V-D
IV&V-W
III-D
1 10
I&II-D
III-WIV&V-D
IV&V-W
III-D
1 10
I&II-D
III-WIV&V-D
IV&V-W
III-D
Soil suction (kPa)
V
o
l
u
m
e
t
r
i
c
w
a
t
e
r
c
o
n
t
e
n
t
,
w
Suctions with Tensiometers Water content with TDR
Chiu Chung FaiSWCC (Tami et al., 2004)
-
Approaches that can be used to obtain the SoilWater Characteristic curves
Determination of Soil-Water Characteristic Curves, SWCC
Laboratory measurement of
water content versus suction
SWCC predictions from grain
size distribution
SWCC predictions
from grain size & Atterberg
limits
Dataset mining for
typical SWCC
Pressure plate <
1500 kPa
Vacuum desiccators > 1500 kPa
Numerous models
Parameters for
numerous models
Soils with similar grain size or soil
classification
Decreasing accuracy
Chiu Chung FaiSWCC1.2.3.Atterberg4.
-
Soil-Water Characteristic Curve computed from a Grain Size Distribution Curve
0.0
10
20
30
40
0.1 1 10 100 1000 10000Suction (kPa)
Predicted from grain-sizeExperimental
W
a
t
e
r
c
o
n
t
e
n
t
1E+5 1E+6
020
40
60
80
100
0.0001 0.001 0.01 0.1 1 10 100Pe
r
c
e
n
t
p
a
s
s
i
n
g
(
%
)
Particle size (mm)
ExperimentalFit - curve
Fredlund et al,1997
Chiu Chung FaiSWCC
-
Incorporation of SWCC into the Constitutive Relations for Unsaturated Soils
Unsaturated soil property functions rely on the saturated soil properties PLUS the soil-water characteristic curve, SWCC
Therefore, MUST have an indication of the SWCC
Unsaturated soil property functionsrender the solution of a problem nonlinear
Chiu Chung FaiSWCC1.SWCC2.SWCC3.
-
Seepage Constitutive Relations in Terms of SWCC
References for the Soil-Water Characteristic CurvePermeability Models
van Genuchten (1980) Brooks & Cory (1964)
Mualem (1976)
Burdine (1953) nnmnn
rk 22
])(1[])(1[)(1)(
++=
5.0
21
])(1[}])(1[)(1{)( n
mnn
rk +
+=
32)()( =rkn
m 21 =
nm 11 =
kr = kw/ksat
Chiu Chung FaiSWCC
-
Seepage Constitutive Relations in Terms of SWCC
References for the Soil-Water Characteristic CurvePermeability Models
Fedlund and Xing (1994) Campbell (1974)
Child and Collis George (1950)
= b
aev
yy
sy
by
y
y
r
dyee
e
dyee
e
k
)ln(
'
)ln(
'
)()(
)()()(
b
aevrk
22
)(=
b = Ln (1000000) () = Soil water contenty = Dummy variable of integration representing
the logarithm of integration
kr = kw/ksat
-
Usage of several functions to predict permeability functions from the SWCC for a particular soil and a suggested lower limit for the permeability function
1.E-171.E-161.E-151.E-141.E-131.E-121.E-111.E-101.E-091.E-081.E-071.E-061.E-051.E-04
0.1 1 10 100 1000 10000 100000 1E+06
Experimental data
Overall Kw + Kv
Fredlund and Xing
VaporKv
Campbell
Brooks and Corey
Van Genuchten - Mualem
Van Genuchten- Burdine
Soil suction (kPa)
C
o
e
f
f
i
c
i
e
n
t
o
f
p
e
r
m
e
a
b
i
l
i
t
y
(
m
/
s
)
Chiu Chung FaiSWCC
-
Shear Strength Constitutive Equation Written in Terms of SWCC
Vanapalli et al. (1996)rs
rn
=
'tan'tan)(' nan uc ++=Shear strength
Net normal stress on the failure plane
Angle of internal frictionIntercept of the Mohr-
Coulomb failure envelope on the shear stress axis
Matric suction
SWCC
Chiu Chung FaiSWCC
-
Stress Analysis (for Shear Strength Problems) Constitutive Relations in Terms of SWCC
sd
=
Shear strength
net normal stress on the failure plane
Angle of internal frictionIntercept of the Mohr-
Coulomb failure envelope on the shear stress axis
'tan)('tan)(' dan uc ++=
Fitting parameter used for obtaining a best-fit between the measured and predicted value
Fredlund et al. (1996)
Matric suction
SWCC
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #5: To solve nonlinear partial differential equations for
unsaturated soils without having convergence difficulties during the iterative solution process
Solution #5: Adaptive mesh (grid) generation techniques in
computer technology facilitates convergence
0 10 20 30 40 50 60 70 80 90 100 110 120 130 14001020304050
Chiu Chung Fai
-
Solving a Boundary Value Problem
Element for which a PartialDifferential Equation, PDE, must be derived
Boundary Value Must be Supplied
Boundary
Boundary
Boundary
Typical Boundary Conditions
Flux Head - for seepageForce Displacement - for stress
Boundary
Utilize general purpose PDE Solvers to solve partial differential equations for saturated-unsaturated soil system
Chiu Chung FaiPDE-
-
Problem Solving Environments, PSEs, for Soil Mechanics Partial Differential Equations, PDEs
All classic areas of soil mechanics can be viewed in terms of the solution of a Partial Differential Equation
Water flow through porous soils (Saturated or Unsaturated)
Air flow through unsaturated soils Stress analysis for slope stability, bearing capacity
and earth pressure Stress-Deformation volume change and distortion
Incremental elasticity Elasto-plastic models
Chiu Chung Fai1.2.()3.4.5.-
-
Partial Differential Equation for Saturated-Unsaturated Water Flow Analysis
thm
yh
yk
yhk
xh
xk
xhk w
wwyw
y
wxw
x =
++
+
22
2
2
2
Head variable to be solved
Water coefficient of permeability (function of soil suction)
Time Water storage (function of soil suction)
Chiu Chung Fai-
-
Partial Differential Equation for Unsaturated Air Flow Analysis
Pore-air pressure(primary variable to be solved)
Air coefficient of permeability (function of soil suction)
Time Air storage and compressibility
(function of soil suction)
tu
RTgmuS
ee
yu
yk
xu
xk
yuk
xuk aawaaaaaaaaaa
+=
+
+
+
22
2
2
2
1
Chiu Chung Fai
-
Partial Differential Equation for Saturated-Unsaturated Stress-Deformation Analysis
0441211 =
+
+
+
xv
yuD
yyvD
xuD
xX
44 12 11 0tu v u vD D D
x y x y x y + + + + =
Y
D11, D12, D44 = Combination of E and which are function of soil suction and net total stresses
Stress-deformation analyses have a degrees of freedom in each of the Cartesian coordinate directions
Chiu Chung Fai-D11, D12D44
-
Convergence of Nonlinear Partial Differential Equations
Convergence is the single most pressing problem facing modelers
Most successful solutions have involved Adaptive Grid Refinement methods, AGR (Oden, 1989; Yeh, 2000)
Mesh is dynamically upgraded during the solution based on error estimates
AGR becomes extremely important when solving the nonlinear PDEs associated with Unsaturated Soil Mechanics
Chiu Chung Fai1.2., AGR(Oden,1989;Yen,2000)3.4.PDEsAGR
-
Two-dimensional seepage analysis through an earthfill dam with a clay core.
Equipotential lines
Optimized mesh for saturated-unsaturated seepage analysis
Chiu Chung Fai
-
Problem illustrating the solution of a 3-dimensional, saturated-unsaturated seepage PDEOptimized, automatically generated finite element mesh
Modeling of a waste tailings pond
Chiu Chung Fai-
-
Stress analysis PDE combined with the Dynamic Programming procedure to compute the factor of safety
Distance0 20 40 60 80
0
5
10
15
20
25
30
DP Search Bounda ry
Finite Element S hea r Stre ss
DP Ge ne ra te dCritica l S lip SurfaceFOS = 1.3 Shape and location of the slip
surface are a part of the solution
Chiu Chung FaiPDE
-
Prediction of Heave or Collapse of a Soil
Requires the solution of a saturated-unsaturated seepage model and a stress-deformation model
Saturated-Unsaturated Seepage Model
Saturated-Unsaturated Stress-Deformation
Model
Coupled Uncoupled
Pseudo-coupled
Computes changes in matric suction Computes deformations
Chiu Chung Fai1.--2.
-
Scenario of Edge Lift for a Flexible Impervious Cover
Flexible cover 0
1
2
3
0 3 6 9 12
Distance from centre of cover or slab, m
CL
Constant suction = 400 kPa
Flux = 0
D
e
p
t
h
,
m
Flux = 0
Infiltration, q SVFlux
Boundary conditions and initial conditions must be specified both seepage and stress-deformation
Chiu Chung Fai-
-
Concrete slab0
1
2
3
0 3 6 9 12
CL
D
e
p
t
h
,
m
Distance from center, m
SVFluxand
SVSolid
Can have one optimized Adaptive Mesh generated for seepage model and another for the stress-deformation model
Chiu Chung Fai-
-
Matric Suction at Ground Surface after One Day of Infiltration for Various Infiltration Rates
SVFlux
0
100
200
400
0 2 4 6 8 10 12
Distance from centre of cover, m
M
a
t
r
i
c
s
u
c
t
i
o
n
,
k
P
a
300
500
CL
Initial
q = 10 mm/day
q = 20q = 30
q = 40q = 50q = 60
Specifiedzero suction
Distance under slab
Chiu Chung Fai
-
Vertical Displacements at Ground Surface after One Day of Infiltration
Distance under slab
0
10
20
25
0 2 4 6 8 10 12
Distance from centre of cover, m
H
e
a
v
e
,
(
m
m
)
5
15
specified zero suction
q = 10q = 20
q = 30q = 40
q = 50
q = 60 mm/day
CL
SVSolid
Chiu Chung Fai
-
Challenges to the Implementation of Unsaturated Soil Mechanics
Challenge #6: To promote implementation of unsaturated soil
mechanics into engineering practice Solution #6:
- Educational materials and visualization tools have been produced to better teach and understand unsaturated soil mechanics
Chiu Chung Fai
-
Concluding Remarks
Unsaturated Soil Mechanics needs to be first understood from the standpoint of the Constitutive equations describing soil behavior
Constitutive Equations can be written in terms of the SWCC for the soil which are then known as Unsaturated Soil Property Functions, USPF
Direct and Indirect procedures are available for the assessment of the SWCC
It is always possible to obtain an estimate of the required Unsaturated Soil Property Functions for geotechnical engineering applications
Chiu Chung Fai1.2.SWCC(USPF)3.SWCC4.(USPF)
-
Karl Terzaghi deserves credit not only for the
fundamentals of saturated soil behavior but also for
the fundamentals of unsaturated soil behavior
Geo-Institute, Austin, TexasJanuary 23-26, 2005 Thank You
Unsaturated Soil Mechanics in EngineeringUnsaturated Soil Mechanics Problems Described in Theoretical Soil Mechanics by K. Terzaghi (1943)Unsaturated Soil Mechanics in EngineeringObjectivesGradual Emergence of Unsaturated Soil MechanicsChallenges to the Implementation of Unsaturated Soil MechanicsChallenges to the Implementation of Unsaturated Soil MechanicsChallenges to the Implementation of Unsaturated Soil MechanicsChallenges to the Implementation of Unsaturated Soil MechanicsChallenges to the Implementation of Unsaturated Soil MechanicsChallenges to the Implementation of Unsaturated Soil MechanicsZones of Unsaturation Defined by a Soil-Water Characteristic Curve, SWCCUnsaturated Soil REV as a Four Phase SystemStructure and Stresses in the Contractile SkinChallenges to the Implementation of Unsaturated Soil MechanicsDefinition of stress state at a point in an unsaturated soilVariations in Stress State DescriptionChallenges to the Implementation of Unsaturated Soil MechanicsMonitoring for Verification PurposesTDR ThetaProbe, ML2x manufactured by AT Delta Devices, U.K.In Situ Matric Suction measurements using Thermal Conductivity sensors at 1.0 to 1.3 m below roadwayDirect, high suction sensor used to measure suctions greater than one atmosphere on the side of a triaxial specimen (Meilani,Challenges to the Implementation of Unsaturated Soil MechanicsFundamental Constitutive Relations for Unsaturated SoilsWater Seepage Constitutive RelationsShape of the water permeability function for glass beads tested by Mualem (1976 )The SWCC for the glass beads showing hysteresis during drying and wettingWater Storage in an Unsaturated SoilAir Flow Constitutive RelationsShear Strength Constitutive RelationsExtended Mohr-Coulomb failure surface (Fredlund, Morgenstern and Widger, 1978)Soil-Water Characteristic Curve for glacial tillReference compression curves for a Saturated SoilLimiting or Bounding relationships for Ko loading of a typical clayey silt soilVolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)VolumeMass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)VolumeMass Constitutive Surfaces for Beaver Creek Sand (Pham, 2004)VolumeMass Constitutive Surfaces for Beaver Creek Sand (Pham, 2004)Challenges to the Implementation of Unsaturated Soil MechanicsRole and Measurement of the Soil-Water Characteristic Curve, SWCCManufactured by GCTS, Tempe, AZFifteen bar Pressure Plate equipment manufactured by GCTS, U.S.A.Equations to Best-Fit SWCC DataHysteresis in the Soil-Water Characteristic CurveModel measurements of water content and matric suction showing the SWCC relationship from water contents and matric suctions dApproaches that can be used to obtain the SoilWater Characteristic curvesSoil-Water Characteristic Curve computed from a Grain Size Distribution CurveIncorporation of SWCC into the Constitutive Relations for Unsaturated SoilsSeepage Constitutive Relations in Terms of SWCCSeepage Constitutive Relations in Terms of SWCCUsage of several functions to predict permeability functions from the SWCC for a particular soil and a suggested lower limit fShear Strength Constitutive Equation Written in Terms of SWCCStress Analysis (for Shear Strength Problems) Constitutive Relations in Terms of SWCCChallenges to the Implementation of Unsaturated Soil MechanicsSolving a Boundary Value ProblemProblem Solving Environments, PSEs, for Soil Mechanics Partial Differential Equations, PDEsPartial Differential Equation for Saturated-Unsaturated Water Flow AnalysisPartial Differential Equation for Unsaturated Air Flow AnalysisPartial Differential Equation for Saturated-Unsaturated Stress-Deformation AnalysisConvergence of Nonlinear Partial Differential EquationsTwo-dimensional seepage analysis through an earthfill dam with a clay core.Problem illustrating the solution of a 3-dimensional, saturated-unsaturated seepage PDEStress analysis PDE combined with the Dynamic Programming procedure to compute the factor of safetyPrediction of Heave or Collapse of a SoilScenario of Edge Lift for a Flexible Impervious CoverMatric Suction at Ground Surface after One Day of Infiltration for Various Infiltration RatesVertical Displacements at Ground Surface after One Day of InfiltrationChallenges to the Implementation of Unsaturated Soil MechanicsConcluding Remarks