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)

Chiu Chung Fai

附注

饱和土的压缩曲线

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