soil matric potential – capillarity and more

CE/ENVE 320 – Vadose Zone Hydrology/Soil PhysicsCE/ENVE 320 – Vadose Zone Hydrology/Soil PhysicsSpring 2004Spring 2004

Copyright © Markus Tuller and Dani Or 2002-2004Copyright © Markus Tuller and Dani Or 2002-2004

Soil Matric Potential – Capillarity and More

Hillel, pp. 38 - 47 7

Copyright© Markus Tuller and Dani Or2002-2004

Tensiometer

The matric potential results from interactive capillary and adsorptive forces between the water and the soil matrix, which in effect bind water in the soil and lower its potential energy below that of bulk water.

The value of m ranges from zero, when the soil is saturated to often very low negative numbers when the soil is dry.

The Matric Potential

The matric potential per unit of weight is defined as the vertical distance between a porous cup in contact with the soil and the water level in a manometer connected to the cup [Hanks, 1992]

T = z +m + p + s +....


Interfacial processes:

Liquid-vapor surface tension

Contact angle and surface wettability

Geometrical constraints:

Curved interfaces and capillarity

Capillary rise

Capillarity in angular pores

More interfacial processes:

Surface forces and liquid film adsorption

Factors Affecting Soil Matric Potential


At interfaces (e.g., water-solid or water-air) water molecules are exposed to different forces than molecules within the bulk water.

Molecules inside the liquid are attracted by equal cohesive forces to form hydrogen bonds at all sides.

Molecules at the air water interface feel a net attraction into the liquid because the density of water molecules at the air side is much lower and all hydrogen bonds are towards the liquid.

Surface Tension

Surface acts like a membrane, having a tendency to contract. Like in a stretched spring, energy is stored in form of SURFACE TENSION

Surface tension is expressed as energy per unit area (=force per length).

Water at 20oC = 7.27 * 10-2 N/mEthyl Alcohol = 2.2 * 10-2 N/mMercury = 0.43 N/m


Surface tension values (Adamson, 1990)


Surface tension depends on temperature (linear decrease with temperature increase).

Thermal expansion reduces the density of the liquid. That means that cohesive forces at the surface as well as inside the liquid phase become smaller (= reduction of surface tension).

The decrease in surface tension is accompanied by an increase in vapor pressure (= increase of liquid molecules in the gaseous phase).

Temperature effects on surface tension


Measurement of Surface Tension

The method is simple and measures the detachment force(the surface tension multiplied by the periphery 2*2R) A platinum ring flamed before use and torsion wire (force) are used.Errors due to internal and planar curvatures require some modifications.

R4WW ringtot

pWW platetotWilhelmy slide (1863) p is the perimeter of a thin slide – no corrections are needed!

The Ring Method (du Nouy 1919)


Contact Angle and Wettablity

When liquid is placed in contact with a solid in the presence of gas (three phase system), the angle measured from the solid-liquid (S-L) interface to the liquid gas interface (L-G) is the CONTACT ANGLE .

For a drop resting on a solid surface under equilibrium, the vector sum of forces acting to spread the drop is equal to the opposing forces. This force balance is summarized in Young’s Equation:

0 GSSLLG cos

LG

SLGS

cos

LGGSSL AAF cos)(



When liquid is attracted to the solid surface (adhesion) more than to other liquid molecules (cohesion), the contact angle is small, and the surface said to be wettable.

When cohesive forces are dominating, the contact angle is large and the solid repels the liquid.

for water on glass is commonly taken 0o

for mercury on glass is 148o


Hydrophilic and Hydrophobic Surfaces

(a) Wettable silt soil surface ( ~ 0o). (b) Treated water-repellant silt soil surface (= 70o) (Bachmann et al., 2000).


Curved interfaces and Capillarity

Often forces that tend to spread a liquid (interactions with solids or gas pressure in a bubble) are balanced by surface tension that tends to minimize interfacial area, resulting in a curved liquid-gas interface.

Particularly in porous media, the liquid-gas interface shape reflects the “need” to form a particular contact angle with solids on the one hand, and a tendency to minimize interfacial area within a pore.

A pressure difference forms across the curved interface, where pressure at the concave side of an interface is larger by an amount determined by interfacial curvature and surface tension.

These relationships between interfacial curvature and pressure difference are given by the Young-Laplace equation.

P = Pliq-Pgas When the interface curves into the gaseous phase (water droplet in air)

P = Pgas-Pliq When the interface curves into the liquid (air bubble in water, water in a small glass tube)

Curved Liquid-Vapor Interfaces


Derivation of the Young-Laplace Equation

dAPdV ])[()]]([[ 2233 43

4rdrrrdrrP

rP

2)()( rdrdrrP 243

3

4 2

Neglecting terms of order dr2 and higher


Interfaces and capillary pressures

For pendular rings between spherical particles (sand grains) the pressure difference is given as:

21 R

1

R

1P


Interfaces and capillary pressures

(c)


The Capillary Rise Model

● When a small cylindrical capillary is dipped in a water reservoir a meniscus is formed in the capillary reflecting balance between contact angle and minimum surface energy.

● The smaller the tube the larger the degree of curvature, resulting in larger pressure differences across the air-water interface.

● The pressure in the water is lower than atmospheric pressure (for wetting fluids) causing water to rise into the capillary until this upward capillary force is balanced by the weight of the hanging water column (equilibrium).


ghrcosr2 w2

rg

cos2h

w

Vertical force balance:

Upward force(capillary pull)

Downward force(weight of water)

The Capillary Rise Model


Capillary Rise – Example 1

Problem Statement:Calculate the height of capillary rise in a glass capillary tube having a radius of 35 µm. The surface tension of water is assumed to be 72.7 mN/m.

rg

cos2h

w

]m[423.0105.3100081.9

)0cos(0727.02h

5

Solution:We use the capillary rise equation with =0o, g=9.81 m/s2, and w=1000 kg/m3; recall that cos(0)=1:

The capillary rise eq. can be simplified by combining constants to yield:

m

kg

ms

m

1

s

mkg

mm

kg

s

m

m

N2

2

32

]m[r

84.14]m[h



Problem Statement:

What is the height of rise in clean glass capillaries having diameters 0.1, 0.5, 1.0, and 2.0 mm when they are tipped vertically into a pool of pure water? What would be the effect on capillary rise if the capillaries were tilted at 30o from the surface (draw a sketch)? What would be the effect on the volume of water held in the capillaries?

Solution:

For capillary rise in vertical capillaries we use the capillary rise equation:

rg

cos2h

w

where is surface tension of the water-air interface (0.0728 N/m at 20oC), is the water-glass contact angle (zero for pure water on clean glass), w is the density of water (998.2 kg/m3 at 20oC), g is the acceleration of gravity (9.81 m/s2), and r is the capillary radius

Diameter [mm] Radius [m] h1 [m]

0.1 0.00005 0.297

0.5 0.00025 0.059

1.0 0.0005 0.030

2.0 0.002 0.015



Solution - Continued:

For the vertical case the capillary force is in equilibrium with the gravitational force.

rg

cos2hghrcosr2

w1w1

2

When we tilt the capillaries at an angle the capillary force stays the same, but is now in equilibrium with only the vertical component of the gravitational force.

sinrg

cos2hsinghrcosr2

w2w2

2

Diameter [mm] Radius [m] h2 [m]

0.1 0.00005 0.595

0.5 0.00025 0.119

1.0 0.0005 0.060

2.0 0.002 0.030 Combining the equations derived above yields:

12o1

2 h2h:30forsin

hh

Since the height of capillary rise for 30o tilted capillaries doubles the volume of liquid also doubles.

CE/ENVE 320 – Vadose Zone Hydrology/Soil PhysicsCE/ENVE 320 – Vadose Zone Hydrology/Soil PhysicsSpring 2004Spring 2004

Copyright © Markus Tuller and Dani Or 2002-2004Copyright © Markus Tuller and Dani Or 2002-2004

Capillarity (and Adsorption) in Soils


Adsorption and Capillarity in Soils

The complex geometry of the soil pore space creates numerous combinations of interfaces, capillaries, wedges, and corners around which water films are formed resulting in a variety of air water and solid water contact angles.

Water is held within this complex geometry due to capillary and adsorptive surface forces.

Due to practical limitations of present measurement methods no distinction is made between adsorptive and capillary forces. All individual contributions are lumped into the matric potential.

10 m


Models for Capillarity in Porous Media

• Common conceptual models for water retention in porous media and matric potential rely on a simplified picture of soil pore space as a “bundle-of-capillaries”

• The key conceptual step is converting behavior in a complex pore to an equivalent (idealized) cylindrical capillary:

Soil sample

actualpore

equivalentcapillary


Models for Water Distribution in Soil Pore Space


• These models assume that soil pores behave similar to capillary tubes.

• Capillary rise or capillary forces that determine the matric potential are higher in soils with smaller pores

• The height of rise is inversely proportional to the radius of a tube or pore.

Soil Pores

]cm[r

15.0]cm[h


Capillary Rise

Water moves up due to capillarity from irrigation furrows.


Capillary Rise

Example of capillary rise in a soil having a water table.

Zone ofCapillary

Rise


Water film adsorption in “real” soils [Or and Tuller, 1999]

3svl

satm h6

A

p

pln

V

TR)h(

We assume that under “dry” conditions soil water is primarily in the form of films held by van der Waals surface forces.

0.0

0.5

1.0

1.5

2.0

0 1000 2000 3000 4000

Water vapor pressure [Pa]

Ad

sorb

ed w

ater

film

-th

ickn

ess

[nm

]

L-soil

Palouse

Palouse B

Royal

Salkum

Walla-Walla

Millville

calc. Asvl = -6E-20 [J]

calc. Asvl = -5E-19 [J]

Spec. Surface Area [m2/g](Campbell and Shiozawa)L-soil 27Palouse 97Palouse B 203Royal 45Salkum 51 Walla-Walla 70Millville sl 73


Capillary Considerations in Angular Pores Made Simple

[Mason and Morrow, 1991; Tuller et. al, 1999]

n

1i

i

i

n 360

)180(

2tan

1)(F

Scanning electron micrographs of soils [Blank and Fosberg, 1989]

)(F)(FrA2

2

w

where:

r r

r

pore

2

w A

)(FrS


POTENTIAL DIAGRAMS (Water Potentials under Equilibrium)

For a soil profile in equilibrium the components of total water potential can be represented by a potential diagram.

Equilibrium means total water potential equal everywhere in the system

Equilibrium means no flow – otherwise water moves from high to low T

For small or zero solute potential we define a HYDRAULIC POTENTIAL h:

hzmp


POTENTIAL DIAGRAMS UNDER EQUILIBRIUM

PROCEDURE:

1) Define a convenient reference level (e.g., water table, soil surface).

2) Draw a diagram of the system (use energy per unit of weight for potential components) and find the total potential for the system. REMEMBER THAT UNDER EQUILIBRIUM CONDITIONS THE TOTAL (HYDRAULIC) POTENTIAL IS CONSTANT EVERYWHERE!

3) Draw a 1:1 line for gravitational potential z versus depth through the reference point.

4) Use the equation h=m+z+p to identify values of all components. REMEMBER - m AND p ARE MUTUALLY EXCLUSIVE.



BUBBLING PRESSURE:

The largest pore in the pore size distribution is an important characteristic of the soil, because it drains first.

The pressure required to drain the largest pore in the system is called BUBBLING PRESSURE or AIR ENTRY VALUE and determines the onset of air entering saturated soil (Minimum pressure required to start desaturation).

Given that the equivalent pore is cylindrical and liquid filled according to the capillary rise equation the Bubbling pressure would be the pressure at the atmospheric side necessary to offset the negative pressure at the liquid side of the meniscus formed in the largest pore (see equation next slide).

The concept of Bubbling Pressure is also important for the design of porous materials that are required to remain saturated to a specific pressure (e.g., porous plates in Tempe cells and Pressure Plate devices, porous cups in tensiometers).



BUBBLING PRESSURE:

hg

cos2r

w

SOIL WATER CHARACTERISTIC

0.01

0.1

1

10

100

1000

0 0.1 0.2 0.3 0.4 0.5

Volumetric Water Content [m3/m3]

Ma

tric

Po

ten

tia

l [-m

]

soil matric potential – capillarity and more

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