Unsaturated Soil Mechanics in Engineering · Unsaturated Soil Mechanics Problems Described in “Theoretical Soil Mechanics” by K. Terzaghi (1943)
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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, (σ – 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)
Unsaturated Soil Mechanics Problems Described in “Theoretical Soil
Mechanics” by K. Terzaghi (1943)
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
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
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
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:
?
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:
?
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:
?
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: ?
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:
?
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:
?
Regional distribution of unsaturated soils
Local vertical zones of
unsaturated soils
GROUNDWATER TABLE- Water filling the voids- Air in a dissolved state
SATU
RA
TED
SO
IL
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
Gra
vim
etric
wat
er c
onte
nt,w
(%)
Soil suction (kPa)
Air entry value
Boundary effect zone
Transition zone
Residual zone
Residualcondition
Inflection point
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
Structure and Stresses in the Contractile Skin
Hyperbolic Tangent Function
Thickness:1.5 to 2 water molecules or about 5°A (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)
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
(σx - ua)
(σz - ua)
(σy - ua)(ua - uw)
(ua - uw)
τzx
τzy
τxz
τxy
τyzτyx
(ua - uw)
Definition of stress state at a point in an unsaturated soil
(σz - ua)
(σy - ua)(ua - uw)
(ua - uw)
τzx
τzy
τxz
τxy
τyzτyx
(ua - uw)
(σx - 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
( )( )
( )
σ τ ττ σ ττ τ σ
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
Variations in Stress State Description
σ’ = (σ – ua) + χ (ua – uw)σ’ = effective stressχ = parameter related to saturation
σ *ij = σij – [S uw + (1 – S) ua ] δ ij
σij = total stress tensor, δij = Kroneker delta or substitution tensor,
σ *ij = Bishop’s soil skeleton stress (Jommi2000)
S = degree of saturation
Above proposed equations are constitutive relations
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
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
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.
Monitoring of Matric Suction
Measures the thermal conductivity of a standard ceramic that varies in water content with the applied matric suction
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)
Mat
ric
Suct
ion
(kPa
)
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)
Mat
ricsu
ctio
n (k
Pa)
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
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 normal stress
(σ - ua)
amat- Constitutive relations for
saturated soils needed to be extended to embrace the effect of changing degrees of saturation
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!
Water Seepage Constitutive Relations
Yg
uh
w
w +=ρ
Driving potential for water flow is hydraulic head, h
x wxdhv kdx
= −
dydhkv wyy −=
z wzdhv kdz
= −
Darcy’s 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
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
Coe
ffici
ent o
f per
mea
bilit
y (m
/s)
Wetting
Drying
DryingWetting
Soil suction (kPa)
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)
Deg
ree
of s
atur
atio
n, %
DryingWetting
Wetting
Drying
Soil suction (kPa)Hysteresis in the SWCC produces hysteresis in the Permeability function
Water Storage in an Unsaturated Soil
0.1 1.0 10 100 1000 10000 100000 1E+6Soil suction, kPa
Wat
er c
onte
nt, %
0
10
20
30
40Soil-Water Characteristic Curve, SWCC
Also has a hysteretic effect
Soil suction, kPa
Wat
er s
tora
ge
mod
ulus
, kPa
-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
Air Flow Constitutive Relations
Driving potential for air flow is Pore-air pressure, uadx
dukv a
axax −=
dydu
kv aayay −=
dzdu
kv aazaz −=
Fick’s 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
Shear Strength Constitutive Relations
' '( ) tan ( ) tan bn a a wc u u uτ σ φ φ= + − + −
Linear form of the extended Mohr-Coulomb shear strength equation
Fredlund, Morgenstern and Widger, 1978
' '1( ) tan ( )n a a wc u u u fτ σ φ= + − + −
f1 = function showing the rate of increase in shear strength with matric suction
Extended Mohr-Coulomb failure surface (Fredlund, Morgenstern and Widger, 1978)
φ’
(ua-uw)
Shea
r str
engt
h,τ
Net normal stress, (σ - ua)
Shear strength versus suction is nonlinear and affected by hysteresis
Air entry value
c’
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)
Deg
ree
of S
atur
atio
n, S
(%)
0 kPa, e = 0.517
25 kPa, e =0.514
OPTIMUM INITIAL WATER CONTENT
SPECIMEN
AEV = 60 kPa
AEV = 60 kPa
Shea
r str
engt
h, τ
(kPa
)
200
250
150
100
50
00 100 200 300 400 500
c’ = 10 kPa(σf - ua)f tan φ’= 34.6 kPa
(σf - ua)f = 72.6 kPa φ’ = 25.5
Matric suction, (ua-uw) (kPa)
Reference compression curves for a Saturated Soil
v =
(1+e
)
e
Log(σ)Ln(p)p0
λ
κ
Cs
σ0
Cc
Effective mean stress Effective vertical stress
Spec
ific
volu
me
Void
ratio
Preconsolidationpressure
Yield stress
ss CC 434.0
)10ln(≈=κ
cc CC 434.0
)10ln(≈=λ
Effective mean stress Effective vertical stress
Elasto-plastic form
Classic soil mechanics form
Limiting or Boundingrelationships for Ko
loading of a typical clayey silt soil
Residual
e
wGs
Air Entry value
106
Cc
Cs
e0
Cc
Cs
Air Entry value
Residual
Log (p-ua)
Log (soil suction)
Log (soil suction)
Log(p-u a)
Yield
Virgin compression
line
p0-ua
Void ratio
Water content
Volume-mass constitutive relations for an unsaturated soil take the form of two, 3-dimensional surfaces with distinct breaks that identify changes in soil behaviorNew points:
- Air entry value- Residual point
Volume–Mass Constitutive Surfaces
for Regina Clay –Preconsolidated at 200
kPa (Pham, 2004)
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)1e+0610000010000100010010
10.10.01Log soil suction (kPa)
0
0
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
Deg
ree
of s
atur
atio
n (S
)
Deg
ree
of s
atur
atio
n (S
)
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)1e+0610000010000100010010
10.10.01Log soil suction (kPa)
0
0
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
Void
ratio
Voi
d ra
tio
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)
1e+061000001000010001001010.10.01Log 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
Gra
vim
etric
wat
er c
onte
nt
Gra
vim
etric
wat
er c
onte
nt
Basic volume-mass equation S e = w Gs
Void ratio, e Degree of saturation, S
Water content, w
SWCC
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)1e+0610000010000100010010
10.10.01Log soil suction (kPa)
0
0
0.5
0.5
1
1
1.5
1.5
2
2
2.5
2.5
Voi
d ra
tio
Void
ratio
Volume–Mass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)
Void ratio, e
Residual value
YieldYield
Soil suctionNet total stress
Volume–Mass 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+0610000010000100010010
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
Gra
vim
etric
wat
er c
onte
nt
Gra
vim
etric
wat
er c
onte
nt
Residual value
Air entry value
YieldYield
Water content, w
Soil suction Net total stress
SWCC
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)1e+0610000010000100010010
10.10.01Log soil suction (kPa)
0
0
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
Deg
ree
of s
atur
atio
n (S
)
Deg
ree
of s
atur
atio
n (S
)
Air entry value
Residual value
Volume–Mass Constitutive Surfaces for Regina Clay Preconsolidated at 200 kPa (Pham, 2004)
Degree of saturation, S
Soil suction Net total stress
Volume–Mass Constitutive Surfaces for
Beaver Creek Sand (Pham, 2004)
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)1e+0610000010000100010010
10.10.01Log 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
Voi
d ra
tio (e
)
Void
ratio
(e)
10000
1000
10010
1 0.10.01
Log net mean stress (kPa)
1e+061000001000010001001010.10.01Log soil suction (kPa)
0
0
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
Deg
ree
of s
atur
ation
Deg
ree
of s
atur
ation
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)
1e+061000001000010001001010.10.01Log 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
Gra
vim
etric
wat
er c
onte
nt
Gra
vim
etric
wat
er c
onte
nt
Void ratio, e
Water content, w
Degree of saturation, S
SWCC
AEV
Basic volume-mass equation S e = w Gs
10000
1000100
101 0.1
0.01
Log net mean stress (kPa)
1e+061000001000010001001010.10.01Log 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
Gra
vim
etric
wat
er c
onte
nt
Gra
vim
etric
wat
er c
onte
nt
Residual value
Air entry value
Volume–Mass Constitutive Surfaces for Beaver Creek Sand (Pham, 2004)
Water content, w
Soil suctionNet total 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
Wat
er c
onte
nt
1E+5 1E+6
- Indirect, estimation procedures have been developed to obtain unsaturated soil property functions based on Soil-Water Characteristic Curves
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)
Gra
vim
etric
wat
er c
onte
nt (%
)Specimen 1Specimen 2Specimen 3
Air Entry Value
Residual ValueSand
- Water permeability - Air permeability - Shear strength - Thermal flow - Incremental
elasticity
Measured drying and wetting curves on processed silt (Pham, 2002)
0
5
10
15
20
25
0.1 1 10 100 1000Soil suction (kPa)
Gra
vim
etric
wat
er c
onte
nt (%
)
Specimen 1Specimen 2Specimen 3
AEV = 10 kPa
WEV = 4.5 kPa
Residual = 120 kPa
Residual = 62 kPa
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
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
Equations to Best-Fit SWCC Data
Fredlund and Xing (1994)
Correction FactorAir entry value
Rate of desaturation
Asymmetry Variable
ψ = Soil suction
fm
fn
f
s
ae
wCw
]})({ln[)()(
ψψψ
+×=
)1000000(1ln[
)1ln(1)(
r
rCψ
ψψ
ψ+
+−=
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)
Gra
vim
etric
wat
er c
onte
nt (%
)
Specimen 1Specimen 2Specimen 3
Wetting
Drying
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
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)
Volu
met
ric w
ater
con
tent
, θw
Suctions with Tensiometers Water content with TDR
Approaches that can be used to obtain the Soil–Water 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
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
Wat
er c
onte
nt
1E+5 1E+6
020
40
60
80
100
0.0001 0.001 0.01 0.1 1 10 100Perc
ent p
assi
ng (%
)
Particle size (mm)
ExperimentalFit - curve
Fredlund et al,1997
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
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) nn
mnn
rk 2
2
])(1[])(1[)(1)(
αψαψαψψ
++−
=−−
5.0
21
])(1[}])(1[)(1{)( n
mnn
rkαψ
αψαψψ+
+−=
−−
λαψψ 32)()( −−=rk
nm 21 −=
nm 11 −=
kr = kw/ksat
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(
'
)()(
)()()(
ψ
ψ
θθθ
θψθθ
b
aevrk
22
)(−−
=ψ
ψ
b = Ln (1000000) θ (ψ) = Soil water content
y = 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)
Coe
ffici
ent o
f per
mea
bilit
y (m
/s)
Shear Strength Constitutive Equation Written in Terms of SWCC
Vanapalli et al. (1996)rs
rn θθ
θθ−−
=Θ
'tan'tan)(' φψφστ nan uc Θ+−+=
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
Stress Analysis (for Shear Strength Problems) Constitutive Relations in Terms of SWCC
sd θ
θ=Θ
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)(' φψφστ κdan uc Θ+−+=
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
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
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
Partial Differential Equation for Saturated-Unsaturated Water Flow Analysis
thm
yh
yk
yhk
xh
xk
xhk w
wwyw
y
wxw
x ∂∂
γ−=∂∂
∂
∂+
∂∂
+∂∂
∂∂
+∂∂
22
2
2
2
Head variable to be solved
Water coefficient of permeability (function of soil suction)
Time Water storage (function of soil suction)
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 aaw
aaaaaaa
aa
a ∂∂
⎟⎠⎞
⎜⎝⎛ −
+−=⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+∂∂
+∂∂ ω
22
2
2
2
1
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 yγ
⎡ ⎤⎛ ⎞ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂+ + + + =⎢ ⎥⎜ ⎟ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎣ ⎦⎣ ⎦
Y –
D11, D12, D44 = Combination of E and µ 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
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
Two-dimensional seepage analysis through an earthfill dam with a clay core.
Equipotential lines
Optimized mesh for saturated-unsaturated seepage analysis
Problem illustrating the solution of a 3-dimensional, saturated-unsaturated seepage PDEOptimized, automatically generated finite element mesh
Modeling of a waste tailings pond
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
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
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
Dep
th, m Flux = 0
Infiltration, q SVFlux
Boundary conditions and initial conditions must be specified both seepage and stress-deformation
Concrete slab0
1
2
3
0 3 6 9 12
CL
Dep
th, m
Distance from center, m
SVFluxand
SVSolid
Can have one optimized Adaptive Mesh generated for seepage model and another for the stress-deformation model
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
Mat
ricsu
ctio
n, k
Pa
300
500
CL
Initial
q = 10 mm/day
q = 20q = 30
q = 40q = 50q = 60
Specifiedzero suction
Distance under slab
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
Hea
ve,(m
m)
5
15
specified zero suction
q = 10q = 20
q = 30q = 40
q = 50
q = 60 mm/day
CL
SVSolid
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
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
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
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