Sample Scale Liquid Retention and Liquid-Vapor Interfacial Area

Dani Or & Markus TullerDept. of Plants, Soils and Biometeorology

Utah State University, Logan, Utah

Outline for Section 2

• How can we use simplified Y-L expressions in a unit cell to

represent a population of pores (i.e., a soil sample)?

• Statistical representation of pore size distribution (normal, log-

normal, and Gamma)

• Relationships between psd and soil water characteristic curve

• Measurable soil attributes that provide constraints for psd

estimation

• A proposed estimation scheme

• Examples for various soils – capillary and film water content

• Liquid-vapor interfacial area – an important function for gas

exchange processes (e.g., bioremediation)

Statistical representation of soil pore size distribution

• From Brutsaert (1966) to Assouline (2000) many have proposed to represent soil pore size distribution by various statistical PDF’s such as: normal, log-normal, Gamma, and Weibul PDF’s.

• For example, Kosugi (1994) proposed a log-normal expression:

• There are some subtle differences between f(r) and f(r3) – psd of size usually pertains to pore radii and not volume distribution…

• Relationships between psd and SWC – water capacity (d/d) with f(r) based on the capillary rise equation.

2

20

[ ( / )]( ) exp with ( ) 0

22o o

o o o

n r rf r f r dr r

r

Statistical representation of soil pore size distribution

• Pore size (radii) distribution is calculated by taking the derivative of the SWC (d/dh); and

• By employing the capillary rise equation (r=a/h)

Pore Radius (m)

1 10 100

Freq

uen

cy (1/

m)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

Initial

Water Content (m3/m3)

0.35 0.40 0.45 0.50 0.55 0.60

- M

atri

c Po

tential

(m

)

0.01

0.1

1

10

Initial (model)5th Cycle (model)Initial (measurement)1st Cycle (measurement)5th Cycle (measurement)

Mat

ric

Po

ten

tial

- h

- [

m]

Water Content - - [m3 m-3]

ih

ii

hgr

2

i

2

i

Vi

rn

Upscaling from Pore-to Sample-Scale

WetDryL1

L2

L3

L4

L5

21 3

L6

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

2:withLExp!L)L(f

1

f(L)

Slits

L1 L2 L3 L4 L5 L6

Gamma Distribution for L A statistical approach using

Gamma distributed cell size is employed to represent a sample of a porous medium.

Gamma distribution – facilitates analytical solutions and preserves the observed skewness in psd.

In developing “upscaled” equations for liquid retention, one must keep track of portions of pore population at various filling stages (due to differences in their pore size).

Limits of Integration for the Upscaling Scheme

Lmax

Full Cells

Full Slits-Partially-Filled Pores

Partially-FilledSlits & Pores

Lmin

L2=

3 svl

6

A3

nC

L1

=

Filling StageFilling StageBoundary Cell SizeBoundary Cell Size

The application of Limits of Integrationin the Upscaling Scheme

full cells

cornerspore

full slits

filmspore

filmspore+slits

Measured Media Properties ProvideConstraints for Geometry and PSD

Measured soil specific surface area and the “dry end” of the SWC curve provide constraints for and parameters.

The “bubbling pressure” defines the largest pore to be considered.

The smallest pore size is bounded by slit-spacing.

Measured and Modeled Water Retention Curve

Degree of Saturation

0.0 0.2 0.4 0.6 0.8 1.0 1.2

- C

hem

ical

Po

ten

tial

[J/

kg]

10-2

100

102

104

106

AdsoroptionCapillarityNew ModelVG-ModelMeasurements

Millville Silt Loam

films corners/full pores

Measured and Modeled Water Retention Curve

Degree of Saturation

0.0 0.2 0.4 0.6 0.8 1.0 1.2

- C

he

mic

al P

ote

nti

al [

J/k

g]

10-2

100

102

104

106

AdsorptionCapillarityNew ModelVG-ModelMeasurements

Salkum

Measured and Predicted Liquid-Vapor Interfacial Area

Degree of Saturation

0.0 0.2 0.4 0.6 0.8 1.0

Liq

uid

-Vap

or

Inte

rfac

ial A

rea

[m2/m

3]

100

101

102

103

104

105

Menisci (SA=0.05 m2/g)

Films & Menisci (SA=0.05 m2/g)Karkare & Fort [1996]

Kim et al. [1997]

Menisci

Films & Menisci

Sand

An Overview of the Proposed Scheme