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 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. 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 (Fick’s 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 257:1664-1666)

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

17. Illustration ofthe geometry of wetting liquid on solidsurfaces of cubic andrhombohedralpackings ofspheres • (from Gvirtzman • And Roberts, WRR • 27:1165-1176, 1991)

18. 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.

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 Laplace’s 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 Laplace’s 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).