Water flow

Air flow

THEORY

No suctiongradient but

still flow

Suctiongradient to

the right

Suctiongradient to

the left

Does Suction Gradient

Cause Flow?

Does water content

gradient cause Flow?

Coarse stone

Fine ceramic

Does water flow from A to B?, or B to A?

Permeability is fixed at a particular water content

Childs and Collis-George, 1950

PROOF: Darcy’s Law is valid

kw constant

For a saturated soil,k = k(e) or k(w)

A void of air has the same effect as a void filled with carbowax

SWCC - Soil-Water Characteristic Curve

Deg

ree

ofSa

tura

tion,

S

Soil Suction (kPa)0

1

Residual Saturation

Air-EntryValue

S = 1 1 > S > Sr S = Sr

Water can only flow where there is water

in the voids

Tortuosity dramatically changes permeability

Air entry value is ~ 4 kPa

SWCC - Soil-Water Characteristic Curve

Brooks and Corey 1964

Effective degree of saturation or (Normalized) degree of saturation

Two parameter equation

Assumes there is “no-flow” below residual w%

=

Becomes a straight line on a log-log plot

The Brooks and Corey (1964) equation for SWCC

The Brooks and Corey (1964) equation for Permeability

Becomes linear on a log scale)

Delta is related to the Pore Size Distribution Index

Summary of Brooks and Corey (1964) Equation for the Coefficient of Permeability

Se = (S – Sr ) / ( 1 – Sr )

Se = Measure of amount of water

S = Any degree of saturation

Sr = Residual degree of saturation

λ = Pore size distribution index

( ua – uw)b2+3λ

kw = ks --------------(ua – uw){ }

=

Pore size distribution for SWCC

Constant δ for Permeability

Brooks and Corey is a discontinuous function since it starts at the Air Entry Value of the soil

Gardner’s equation1958

‘a’ parameter bears an inverse relationship to the air entry value

Integration Forms for the SWCC and Permeability Function

• (Childs and Collis-George, 1950); assumed that the soil has a random distribution of pores of various sizes

• Used the summation of a series of terms from the statistical probability of interconnections between the pores

• SWCC was used as an indication of the configuration of the water-filled pores

• Permeability equation was derived based on the Poiseuille equation

Childs and Collis-George

(1950)

Log suction

Measured permeability

Log

Perm

eabi

lity

Childs and Collis-George

(1950)

Based on summation (or integration) along the SWCC

Variable “p” is a power applied to volumetric

water contentAssume p = 2.0

Air entry value= ~ 3 kPa

Log suction

Usual form for Permeability

function

Relationship Between Soil-Water Characteristic Curve

and the Coefficient of Permeability for sand and

a Clayey Silt

• The soil-water characteristic curve defines the amount of water in the soil

– Air entry value initiates a reduction in the coefficient of permeability

– Is possible for a sandy soil to have a lower permeability than a clayey soil

mw2

1

Saturation coefficient ofpermeability (clayey silt)

Matric suction, (ua -uw) (kPa)

Coe

ffici

ent o

f per

mea

bilit

y,k w

(m/s

)

Soil-water characteristic curve

Wat

er c

onte

nt, w

(%)

Start of desaturationfor

a clayey silt

100 200 300 400 500

10-5

10-6

10-7

10-8

10-9

10-10

100 200 300 400 500

0

00

10

20

30

40

50

Saturation coefficient ofpermeability (fine sand)

Matric suction, (ua -uw) (kPa)

desaturationfor a fine sand

Start of

Typical Gardner’s Empirical Permeability Functions Shown for a Sand and a Clayey Silt

• Permeability function is commonly plotted as a function of the logarithm of suction

• Most soils show a straight line as the soil desaturates towards residual conditions

• Gardner’s function is one of the simplest permeability functions with physical meaning to the “a” and “n” parameters

Clayey silt

Fine sand

1 10 100 1000

Clayey silt

kw=ks

1+a(ua- uw)n

10

10-10

Fine sand

Start of desaturation for a

fine sand

Permeability function

203040

50

1 10 100 1000

10-9

10-8

10-7

10-6

10-5

Gardner’s equation n clayey silt

n

Matric suction, (ua -uw) (kPa)

Coe

ffici

ent o

f per

mea

bilit

y,k w

(m/s

)W

ater

con

tent

, w (%

) Start of desaturation for a clayey silt

Matric suction, (ua -uw) (kPa)

sand

Commonly used Permeability Functions• van Genucthen (1980)

– SWCC

– k-functionwhere m = 1-1/n

• Fredlund and Xing (1994)– SWCC

– k-function

m

neS

+

=)(1

1αψ

2

2

1

111

++−

=−−

/mn

mnn

r ])([])([)(k

αψαψαψ

)(C)

A(eln

S

C

Bψ

ψ

+

=1

dy)e(e

)e(

dy)e(e

)uu(f)e(

ky

wln

b)wuauln( ys

yw

yw

ln)wuauln( y

wawy

w

r

θθθ

θθθ

′−

′−−

=

∫ −

∫ −

610

610

10-12

10-10

10-08

10-06

10-04

10-02

10+00

1 10 100Soil suction (kPa)

k r

Brooks and Corey

van Genuchten

0.0

0.1

0.2

0.3

0.4

0.0 0.1 1.0 10.0 100.0 1000.0Soil Suction (kPa)

van Genuchtenlab data

θ

Comparison of van Genuchten (1980) and

Brooks and Corey (1964)

Forms for the Permeability Function Based on the SWCC

• Childs and Collis-George (1950)– Summation

• van Genuchten (1980)• van Genuchten-Burdine (1953, 1980)• van Genuchten-Maulem (1976, 1980)• Fredlund, Xing and Huang (1994)

– Integration form• Rahardjo and Leong (2000)

– Closed form; raised SWCC to a power

Difficulties with Hysteresis of SWCC and Permeability

Is there one Permeability Function for drying and another function for wetting?

Difference at inflection point = 0.2 to 0.5 of a log

cycle

Difference at w% inflection point = 0.2 to 0.5 of a log

cycle

No hysteresis in the water content versus

permeability relationship

Must Live with Hysteresis in SWCC and Permeability

• Generally it is the Drying (or desorption) curve that is measured or estimated

• Sometimes the Wetting (or Adsorption) curve might be measured or estimated

• The Wetting Curve might be estimated as being shifted to the left by approximately (one half) log cycle at the inflection point

• Independent permeability functions can be determined for both the Drying and the Wetting processes

• Some rigorous permeability models have been proposed with scanning curves