Where were we

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
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
Filling all the space

• Constraint for fluids f1, f2, ...fn
• Sum of space taken up by all constituents must be
  1

Solid Phase Volume fraction
Fluid Phase Volume Fraction
4
Source of Attraction

• Why doesnt 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
Forces range of influence
6
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
  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
  (Ficks law)

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
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)

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
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
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

12
Typical Values of ?

• Dependent upon gas/liquid pair

13
Temp. dependence of air/water ?
14
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
  2571664-1666)

15
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 271165-1176,
  1991)

16
Geometry of Idealized Pore Space

• Fluid resides in the pore space generated by
  thepacked particles.
• Here the pore spacecreated by cubic
  andrombohedral packingare illustrated.
• (from Gvirtzman
• And Roberts, WRR
• 271165-1176, 1991)

17

• Illustration ofthe geometry of wetting liquid
  on solidsurfaces of cubic andrhombohedralpacki
  ngs ofspheres
• (from Gvirtzman
• And Roberts, WRR
• 271165-1176, 1991)

18
Lets get quantitative

• We seek and expression which describes the
  relationship between the surface energies, system
  geometry, and fluid pressure.
• Lets take a close look at the shape of the
  surface and see what we find.

19
Derivation of Capillary Pressure Relationship
Looking at an infinitesimal patch of a curved
fluid/fluid interface
Cross Section
Isometric view
20
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.
• Upward (downward the same)

21
Derivation cont.

• Since a very small patch, d?2 is very small

Laplaces Equation!
22
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

23
Spherical Case

• If both radii are of the same sign and magnitude
  (spherical r1 - r2 R)
• CAUTION Also known as Laplaces equation.
• Exact expression for fluid/gas in capillary tube
  of radius R with 0 contact angle

24
Introduce Reduced Radius

• For general case where r1 is not equal to r2,
  define reduced radius of curvature, R
• Which again gives us

25
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).