fluid statics, hydrostatics and pressure

The pressure P at a depth h below the surface of a liquid open to the atmosphere is greater then the atmospheric pressure by an amount rgh

1. 1
Where were we?
The physical properties of porous
media
The three phases
Basic parameter set (porosity, density)
Where are we going today?
Hydrostatics in porous media!

2. 2
Hydrostatics in Porous Media
Where we are going with hydrostatics
Source of liquid-solid attraction
Pressure (negative; positive; units)
Surface tension
Curved interfaces
Thermodynamic description of interfaces
Vapor pressure
Pressure-Water Content relationships
Hysteresis

3. 3
Filling all the space
Constraint for fluids f
1, f
2, ...f
n
Sum of space taken up by all
constituents must be 1
Solid Phase
Volume fraction
Fluid Phase
Volume Fraction

4. 4
Source of AttractionSource of Attraction
Why doesn’t water just fall out of soil?
Four forces contribute, listed in order of decreasing strength:
1.Water is attracted to the negative surface charge of mineral
surfaces (Van der Waals attraction).
2.The periodic structure of the clay surfaces gives rise to an
electrostatic dipole which results in an attractive force to the water
dipole.
3.Osmotic force, caused by ionic concentration near charged
surfaces, hold water.
4.Surface tension at water/air interfaces maintains macroscopic units
of water in pore spaces.

5. 5
Forces range of influence

6. 6
Which forces do we worry about?
First 3 forces short range (immobilize water)
Surface tension affects water in bulk; influential in
transport
What about osmotic potential, and other non-
mechanical potentials?
In absence of a semi-permeable membrane, osmotic
potential does not move water
gas/liquid boundary is semi-permeable
high concentration in liquid drives gas phase into liquid
low gas phase concentration drives gas phase diffusion
due to gradient in gas concentration (Fick’s law)

7. 7
Terminology for potential
tension
matric potential
suction
We will use pressure head of the
system.
Expressed as the height of water
drawn up against gravity (units of
length).

8. 8
Units of measuring pressure
Any system of units is of equal theoretical
standing, it is just a matter of being
consistent
(note - table in book also has mmHg)

9. 9
What about big negative pressures?
Pressures more negative than -1 Bar? Non-physical?
NO.
Liquid water can sustain negative pressures of up to 150 Bars
before vaporizing.

Thus:
Negative pressures exceeding -1 bar arise commonly in porous
media
It is not unreasonable to consider the fluid-dynamic behavior of
water at pressures greater than -1 bar.

10. 10
Surface Tension
A simple thought experiment:
Imagine a block of water in a container which can be split in two.
Quickly split this block of water into two halves. The molecules on
the new air/water surfaces are bound to fewer of their neighbors.
It took energy to break these bonds, so there is a free surface
energy. Since the water surface has a constant number of
molecules on its surface per unit area, the energy required to
create these surfaces is directly related to the surface area
created. Surface tension has units of energy per unit area (force
per length).

11. 11
Surface Tension
To measure surface tension: use sliding wire.
For force F and width L
How did factor of 2 sneak into [2.12]?
Simple: two air/water interfaces
In actual practice people use a ring
tensiometer
Force
L

12. 12
Typical Values of 
Dependent upon gas/liquid pair

13. 13
Temp. dependence of air/water 

14. 14
The Geometry of Fluid Interfaces
Surface tension stretches the liquid-gas surface into a taut, minimal
energy
configuration
balancing
maximal
solid/liquid
contact
with
minimal
gas/liquid area.
(from Gvirtzman and Roberts,
WRR 27:1165-1176, 1991)

15. 15
Geometry of Idealized Pore Space
Fluid resides in the pore
space generated by the
packed particles.
Here the pore space
created by cubic and
rombohedral packing
are illustrated.
(from Gvirtzman
And Roberts, WRR
27:1165-1176, 1991)

16. Illustration of
the geometry
of wetting
liquid on solid
surfaces of
cubic and
rhombohedral
packings of
spheres
(from Gvirtzman
And Roberts, WRR
27:1165-1176, 1991)

17. 17
Let’s get quantitative
We seek and expression which
describes the relationship between
the surface energies, system
geometry, and fluid pressure.
Let’s take a close look at the shape
of the surface and see what we find.

18. 18
Derivation of Capillary Pressure Relationship
r
2
r
1

1

2
S
1
S
2
Looking at an infinitesimal patch of
a curved fluid/fluid interface
Cross Section
Isometric view

19. 19
Static means balance forces
How does surface tension manifests itself in a porous
media: What is the static fluid pressures due to surface
tension acting on curved fluid surfaces?
Consider the infinitesimal curved fluid surface with radii
r
1 and r
2. Since the system is at equilibrium, the forces
on the interface add to zero.
Upward (downward the same)
r
2
r
1

1

2
S
1
S
2

20. 20
Derivation cont.
Since a very small patch, d
2 is very small
Laplace’s Equation!

21. 21
Where we were…
• Looked at “saddle point” or
“anticlastic” surface and computed the
pressure across it
•Came up with an equation for
pressure as a function of the radii of
curvature

22. 22
Spherical Case
If both radii are of the same sign and magnitude
(spherical: r
1 = - r
2 = R)
CAUTION: Also known as Laplace’s equation.

Exact expression for fluid/gas in capillary tube of
radius R with 0 contact angle

23. 23
Introduce Reduced Radius
For general case where r
1
is not equal to r
2
,
define reduced radius of curvature, R
Which again gives us

24. 24
Positive or Negative?
Sign convention on radius
Radius negative if measured in the
non-wetting fluid (typically air), and
positive if measured in the wetting
fluid (typically water).