Transcript No Slide Title

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!
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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
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Filling all the space
Constraint for fluids f1, f2, ...fn

+
s
n
fi
=1
[see 2.11]
i=1
Solid Phase
Volume fraction
Fluid Phase
Volume Fraction
Sum of space taken up by all
constituents must be 1
3
Source 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.
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Forces range of influence
Force
Attraction law
Van der Waals
Electrostatic Dipole
Osmotic (double
layer)
Surface Tension
1/r7
1/r2 - ion
1/r6 - dipole
1/r
1/r
Range of
influence
1Å
5Å
100 Å
1 mm
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Which forces do we worry about?
First 3 forces short range (immobilize water)
Surface tension effects water in bulk; influential in
transport
What about osmotic potential, and other nonmechanical 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)
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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).
7
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 is more up-to-date)
to\from
Kilo Pascal
bar
lb./in2
cm water
kilo Pascal
1
100.000
6.89476
0.0980665
bar
0.010000
1
0.0689476
lb./in2
0.1450377
14.50377
1
0.00098066 0.0142233
ATM
mm Hg
101.325
1.01325
14.69595 1,033.23 1
0.001315789
0.1333224 0.00133322 0.0193368 1.3595
5
cm water
10.1972
1,019.72
70.3072
1
ATM
0.0098692
0.986923
0.068046
9.6782x10-4
4
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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 fluiddynamic behavior of water at pressures greater
than -1 bar.
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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).
1unit
1unit
2 units
(a )
(b)
(c )
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Surface Tension
To measure surface tension: use sliding wire.
For force F and width L
Force
L
F
 = 2L
[2.12]
How did factor of 2 sneak into [2.12]?
Simple: two air/water interfaces
In actual practice people use a ring
tensiometer
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Typical Values of 
Dependent upon gas/liquid pair
Substa nce
Wa ter
Acetone
Formula
H 2O
Benzene
C 3H 6O
C 6H 6
C a rbon Tet.
CCl4
M ercury
Hg
C onta cting
Temp (oC)

a ir
a ir
20
20
(dynes/ cm)
72.75
23.7
a ir
w a ter
va por
w a ter
w a ter
20
20
20
20
20
28.85
35.00
26.95
45
375
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Temp. dependence of air/water 
Surface Tension (dynes/cm)
80
75
70
65
60
Slope = -0.17 dy ne/(cm-deg. C), or 0. 25%/ deg. C at 20 deg. C
55
-10
0
10
20
30
40
50
60
70
80
90
100
Temperature (deg. C)
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Cellular Automata Simulation of Water
The process of
minimizing
surface energy is
facilitated by
semi-vapor
phase molecules
which “feel”
proximal liquid.
(from Koplik and
Banavar, 1992,
Science 257:16641666)
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)
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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)
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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)
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.
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Derivation of Capillary Pressure Relationship
Looking at an infinitesimal patch of
a curved fluid/fluid interface
 2
p
r2
S1
1
p2

r1
S2
1
S 2
S 2 sin()
Cross Section
Isometric view
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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
r1 and r2. Since the system is at equilibrium, the forces
on the interface add to zero.
 2
r2
Upward (downward the same)
2
Fup = p2S1S2 + 2 S1 sin  2 
S1
r1
S2
[2.13]
 1
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Derivation cont.
Since a very small patch, d2 is very small
2
sin  2  -
2
2
S2
=2 r
2
[2.14]
F n et = 0 = u p wa rd force s - d own wa rd force s
S1
= p 2S1S2 + 2 S2 2 r
2
S2
- p 1S2S1 - 2 S1 2 r
[2.15]
1
which simplifies to
p2 - p1 =

 
 - 
r1 r2 
[2.17]
Laplace’s Equation!
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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
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Spherical Case
If both radii are of the same sign and
magnitude (spherical: r1 = - r2 = R)
2
P =
R
[2.18]
CAUTION: Also known as Laplace’s
equation.
Exact expression for fluid/gas in capillary
tube of radius R with 0 contact angle
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Introduce Reduced Radius
For general case where r1 is not equal to r2,
define reduced radius of curvature, R




 + 
=
R
r1
r2 
R=
 r1r2 


r1 + r2
[2.19]
[2.20]
Which again gives us
2
p = R
[2.18]
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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).
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