SOIL HYDRAULIC FUNCTIONS - Jan Hopmans

Copyright © Markus Tuller, 2006 Copyright © Markus Tuller & Marcel Schaap, 2001-2007

SOIL HYDRAULIC FUNCTIONS Marcel Schaap

SWES, University of Arizona

Adapted by Jan W Hopmans, UC Davis


Potential

Energy (ψ) (energy per mass)

J/kg=m2/s2

ψ

h=P/(ρwg) P=hρwg P=ψρw ψ=P/ρw

ψ=hg

h=ψ/g

Pressure (P) (energy per volume)

Pa=N/m2=J/m3=kg/(ms2)

Height (h) (water column)

m (or cm)

Potential

h

P


Capillary pressure Caused by surface tension, σ, of pore water Also “van der Waals” forces (films) Laplace’s Law:

P = 2σ cosα /r [Pa] h = 2σ cosα /(ρwgr) [m] Ψ = 2σ cosα /(ρwr) [J/kg] α=0° for water in most soils σ =0.075 N/m for water P decreases with smaller r (beware of the sign for P) α

r

Matric Potential

2 cos

w

rg h

σ αρ

=


Soils are complex porous media

● Complex solid &

Liquid phase geometry

● Difficult to deal with

● Numerical methods

1 mm

0

100 0

20

80 20

40

60 40

60

40 60

80

20 80

100 0 100

Clay [%] Silt [%]

Sand [%]

S lS

sL L siL

Si

scL

cL sicL

siC sC

C

Wildenschild, OSU

Lebron, Stanford


Soil pore structure ● Early conceptual models for the liquid distribution

in partially saturated porous media are based on the "bundle of cylindrical capillaries" (BCC) representation of pore space geometry (Millington and Quirk, 1961; Mualem, 1976).

● The BCC representation postulates that at a given

matric potential a portion of interconnected cylindrical pores is completely liquid filled, whereas larger pores are completely empty.

Soil Sample

Actual Pore

Equivalent Capillary


The Soil Water Characteristic Curve (SWC) ● The Soil Water Characteristic SWC curve describes relationship between

soil water content (θv or θ) and matric potential (hm or h) under equilibrium conditions.

● The SWC is an important macroscopic soil property that is affected by to pore size distribution and pore interconnectedness, which are strongly affected by texture and structure.

● The SWC is a primary hydraulic property essential for modeling unsaturated water flow in porous materials.

● The SWC function is highly nonlinear, difficult to measure accurately & in-situ, and highly variable in field soils.


Measurement of SWC: Tempe Cell The pressure flow cell (Tempe cell) is usually applied for the pressure (matric potential) range from 0 to -10 m.

Close to saturation soil water retention is strongly influenced by soil structure and the natural pore size distribution. Therefore undisturbed core samples are preferred over repacked samples.


Hysteretic Behavior of SWC

Soil Water Content and Matric Potential are not uniquely related and depend on the path of saturation or desaturation.

SWC can be either obtained by desaturation of an initially saturated sample by applying suction or pressure (DRYING or DRAINAGE CURVE), or by gradually wetting of an initially oven-dry sample (WETTING or IMBIBITION CURVE).

The two procedures often produce different SWC curves - the water content of the drying branch is typically higher than water content of a wetting curve at the same potential. This phenomenon is known as HYSTERESIS Essentially caused by the pore-scale structure


Hysteresis in Microtomography Images Wildenschild and Hopmans, 2002, J of Hydrology

drainage ”saturated”

wetting 6 mm diameter (405x406x350 @ 17 µm/pixel)

Source: Wildenschild


Saturated Hydraulic Conductivity

• Transport coefficient for flow through saturated soil.

• Symbol: Ks

• Poiseuille's law for a round tube:

8

4πR ΔPQ =ηL

Q: flow rate m3/s

R: tube radius (m)

∆P: pressure difference (Pa or kg/[m s2])

η: viscosity (Pa s or kg/[m s])

L: length of the tube


● So, for a bundle of capillary tubes we have to sum over the number of tubes Ni with a particular radius Ri

8

4i i

ti

NπR ΔPQ =ηL∑

Saturated Hydraulic Conductivity, Ks

84

t i ii

πΔPQ = N RηL ∑

● This is a simplification, soils are not bundles of capillaries!

● But, it illustrates that larger pores have a dominant contribution.

● The transport coefficient of this medium is then (Darcy’s Law)

t

sQ L = KAΔH

−

● Where A is the sample area.

(m3/s)

(m/s)


Guidelines for Ks


Unsaturated Hydraulic Conductivity

8

4i i

ti

NπR ΔHQ =ηL∑

• largest pores are filled with air (soil water char.)

• Poisseuille's law: Darcy's Law:

• For a bundle of capillary tubes most of the flow through the largest pores

• Elimination of large pores would lead to a

dramatic reduction in K(h).

ts

Q L = KAΔH

−(m3/s) (m/s)


0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

0.1 1 10 100 1000 10000

- Capillary head (cm)

Con

duct

ivity

(cm

/day

)

Figure 1. Unsaturated conductivity data versus capillary head for 423 samples of the UNSODA database.

K(h

) dat

a (c

m/d

ay)

K(h) data from UNSODA (423 soils)


0 0.1 0.2 0.3 0.4 0.5 1

10

102

103

104

Water content (cm3/cm3)

-Mat

ric h

ead

(cm

)

Loam

Sand

0 0.2 0.4 0.6 0.8 1

10-5

10-3

0.1

10

103

Relative saturation, Se

log

K (

cm/d

ay)

Loam

Sand

0 10 102 103 104 105

10-5

10-3

0.1

10

103

-Matric head (cm)

log

K [

cm/d

ay]

Loam

Sand

Note that at higher suctions (more negative pressures) the conductivity of loam is higher than that of sand

Ks Retention

Unsat-K-Se

Unsat-K-h

Examples of Hydraulic Characteristics


THE SOIL WATER CHARACTERISTIC Parametric Models


Parametric Models

• Measuring soil hydraulic properties is laborious and time consuming and expensive. Usually there are only a few data pairs available from measurements. • For modeling and analysis (characterization and comparison of different soils) it is beneficial to represent the hydraulic relationships as continuous parametric functions. • Commonly used parametric models are the van Genuchten and Brooks & Corey relationships. Other models are Kosugi lognormal and Campbell model. • We like to use specific equations rather than splines (or other mathematical ways to describe/interpolate data) because the equation parameters can be interpreted or communicated.

● No “true” SWC model defined based on (microscopically ) observed pore size distributions.


Characteristic Points

Saturated water content

Residual water content


Characteristic Points

Saturated water content

θs ≤ porosity (φ)

Often:

θs ≈ 0.9 φ

Due to air trapping

Residual water content

θ : water content [m3/m3]

θs : saturated water content

θr : residual water content

Se : relative saturation

Se= (θ- θr)/ (θs- θr) [0..1]

θ


Van Genuchten and Brooks & Corey Models

Air Entry

or

Bubbling pressure

Inflection point



Inflection point

Slope


( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++

4 parameters 4 or 5 parameters

θr residual water content

θs saturated water content

hb air entry pressure

λ pore distribution param.

( )( / )

s rr

b

hh h λ

θ θθ θ −= +

α inverse air entry pressure

hb ≈ 1/α n pore distribution param. λ≈n-1 m often m=1-1/n

( ) shθ θ=

h > hb

h ≤ hb

Brooks-Corey Van Genuchten


Inflection point (second derivative is zero)

0,max)( =

=h

dhdSd

orvalueimumhasdhdShC

e

e

)1(1:

,01

2

2

∞≈≈=−=

=

−

normmhenVanGenucht

hatdh

Sd

m

i

ie

αα


( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++


θr residual water content

θs saturated water content

hb air entry pressure

λ pore distribution param.

( )( / )

s rr

b

hh h λ

θ θθ θ −= +

α inverse air entry pressure

hb ≈ 1/α n pore distribution param. λ≈n-1 m often: m=1-1/n

( ) shθ θ=

h < hb

h > hb



( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++


( )( / )

s rr

b

hh h λ

θ θθ θ −= +

( ) shθ θ=

h > hb

h ≤ hb


1)/11(

,1,,)())(1(

−=−==

==+

nnnmn

andh

orhhb

mn

λ

ααα λ

If n-value is relatively large:


Parametric models

● The unknown parameters -free parameters- {θr, θs, hb, λ} or {θr, θs, α, n} can be obtained by fitting the models to measured data pairs

● There are various computer codes available (e.g., RETC) for estimation of free model parameters. A simple procedure is the application of solver tools that are part of most spreadsheet software packages.

● In both models MATRIC POTENTIALS are expressed as positive (absolute) quantities.

( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++

h > hb

Brooks-Corey van Genuchten

( )( / )

s rr

b

hh h λ

θ θθ θ −= +


Parametric models

● The unknown parameters -free parameters- {θr, θs, hb, λ} or {θr, θs, α, n} can be obtained by fitting the models to measured data pairs

● There are various computer codes available (e.g., RETC) for estimation of free model parameters. A simple procedure is the application of solver tools that are part of most spreadsheet software packages.

● In both models MATRIC POTENTIALS are expressed as positive (absolute) quantities.

( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++

be

hSh

λ

=

h > hb

Brooks-Corey van Genuchten

( )( / )

s rr

b

hh h λ

θ θθ θ −= +

[1 ( ) ]n meS hα −= +h > hb

1eS = h ≤ hb



Inflection point

Slope

(VG n or BC λ)

“Air Entry”

hb or 1/α


1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

0 0.1 0.2 0.3 0.4 0.5

Water content (cm3/cm3)

Pres

sure

, h (c

m) n=1.2

n=1.5n=2n=3n=5

• Parameter values “capture” the shape of the WRC, facilitating calculations and communication

Variation in “n” for van Genuchten

λ≈n-1 Sand

Clay


Campbell (1974)

● Campbell (1974) is similar to the Brooks-Corey equation, but with θr=0:

( )( / )

s

b

hh h λ

θθ = h > hb

( )( / )

s rr

b

hh h λ

θ θθ θ −= +

Brooks-Corey

( ) shθ θ= h ≤ hb

Campbell

( ) shθ θ= h ≤ hb

h > hb


Physically-based models

● All parametric SWC models are empirical: they are successful because they describe observed θ-h data well (for certain soils)

● No “true” SWC model defined based on (microscopically ) observed pore size distributions.

● However, it is often observed that soil particle sizes are “log-normally” distributed.

● Assuming that:

this implies a log-normal size distribution of pore-volume pores are cylindrical

● We can define what a SWC should look like.

● Arya and Paris (actually distribution-free, not in this lecture)

● Kosugi model


Kosugi’s SWC model

● Skipping over some derivations, the log-normal size distribution of pore volume is:

● Using capillary equation (Laplace’s law with cylindrical pores) :

erfc : complementary error function :

rm : median pore radius hm : median soil water matric head

σ : standard deviation of log-transformed pore size distribution or

matric head distribution

This image cannot currently be displayed.

rAhrArwh lnlnlnor //cos2 −=== gρασ

( ) ( )

−−−

= 22

2lnlnexp2

lnσσπ

θθmhh

hrshf

( ) ( )

−−−

= 22

2lnlnexp2

lnσσπ

θθmrr

rrsrg


Kosugi’s SWC model

This gives us pore volume as a function of the pore radius r. Integration over all pore sizes (and using Laplace’s law with cylindrical pores) gives us:

This image cannot currently be displayed.

( )

−=σ2lnlnerfc2

1ln mhhheS


• Empirical models (Gardner, see Hillel p209)

• Pore size distribution models (Mualem, Burdine)

PSD-models are theoretical models that use SWC equations to provide “closed-form” expressions for K(h), or K(Se). See van Genuchten (1980), Mualem and Dagan(1978), and Raats (1992).

• SWC is used to provide the pore-size distribution.

• pore are filled with water, depending on their size and matric potential and only water filled pores contribute to hydraulic conductivity

• pores of different sizes are connected to each other

Functions for Hydraulic Conductivity?


• Mualem

• Burdine

Pore Size Distribution Models

21

01

1

0

( )

eS

el

e s e

e

h dSK S K S

h dS

−

−

=

∫

∫

12

01

2

0

( )

eS

el

e s e

e

h dSK S K S

h dS

−

−

=

∫

∫

01

0

( )

eS

el

e s e

e

h dSK S K S

h dS

γη

η

−

−

=

∫

∫

General form

s

er K

SKK )(=


/( 1) 1 1/ 2( ) {1 [1 ] }L n n ne s e eK S K S S − −= − −

Mualem-van Genuchten

( ) 1[1 ( ) ]

re n m

s r

hSh

θ θθ θ α

−= =

− +m=1-1/n

( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++

21

01

1

0

( )

eS

el

e s e

e

h dSK S K S

h dS

−

−

=

∫

∫and

gives

van Genuchten Mualem


/( 1) 1 1/ 2( ) {1 [1 ] }L n n ne s e eK S K S S − −= − −

Mualem-van Genuchten

( ) 1[1 ( ) ]

re n m

s r

hSh

θ θθ θ α

−= =

− +

Saturated Hydraulic Conductivity

Tortuosity/pore connectivity (“0.5” or “-1”?)

Pore-size distribution

m=1-1/n

( )[1 ( ) ]

s rr n mh

hθ θθ θ

α−

= ++

21

01

1

0

( )

eS

el

e s e

e

h dSK S K S

h dS

−

−

=

∫

∫and


0 0.1 0.2 0.3 0.4 0.5 1

10

102

103

104

Water content [cm3/cm3]

Pre

ssur

e he

ad [c

m]

Loam

Sand

0 0.2 0.4 0.6 0.8 1

10-5

10-3

0.1

10

103

Relative saturation, Se

log

K [

cm/d

ay]

Loam

Sand Ks

1) Measure Ks

2) Measure WRC

3) Fit VG parameters to WRC

4) Apply MVG to get K(h, Se)

Using MVG to predict Unsat-K


2 2/( ) le s eK S K S λ+ +=

Mualem/Burdine-Brooks Corey

21

01

1

0

( )

eS

el

e s e

e

h dSK S K S

h dS

−

−

=

∫

∫gives

Mualem

( )( / )

s rr

b

hh h λ

θ θθ θ −= +

( ) shθ θ=

h > hb

h ≤ hb

be

hSh

λ =

h > hb

1eS = h ≤ hb

12

01

2

0

( )

eS

el

e s e

e

h dSK S K S

h dS

−

−

=

∫

∫Burdine

1 2/( ) le s eK S K S λ+ +=gives


Mualem/Burdine - Kosugi

● Kosugi’s model (no air-entry parameter)

● See Table 3.3.4-2 in “Parametric_models.pdf” for more information (Kosugi, Hopmans and Dane, Methods of Soil Analysis, 2002)”

{ }21( ) ( )le s e eK S K S Q Q S σ− = + Mualem

1( ) ( ) 2le s e eK S K S Q Q S σ− = + Burdine

( )

−=σ2lnlnerfc2

1ln mhhheS


Estimating Hydraulic Functions Pedotransfer Functions (PTF)

SOIL HYDRAULIC FUNCTIONS - Jan Hopmans

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