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Where were we?Where were we?The physical properties of porous mediaThe three phasesBasic parameter set (porosity, density)

Where are we going today?Hydrostatics in porous media!

The physical properties of porous mediaThe three phasesBasic parameter set (porosity, density)

Where are we going today?Hydrostatics in porous media!

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Hydrostatics in Porous MediaHydrostatics in Porous MediaWhere we are going with hydrostaticsSource of liquid-solid attractionPressure (negative; positive; units)Surface tensionCurved interfacesThermodynamic description of interfacesVapor pressurePressure-Water Content relationshipsHysteresis

Where we are going with hydrostaticsSource of liquid-solid attractionPressure (negative; positive; units)Surface tensionCurved interfacesThermodynamic description of interfacesVapor pressurePressure-Water Content relationshipsHysteresis

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Filling all the spaceFilling all the spaceConstraint for fluids f1, f2, ...fn

Sum of space taken up by all constituents must be 1

Constraint for fluids f1, f2, ...fn

Sum of space taken up by all constituents must be 1

Solid PhaseVolume fraction

Fluid PhaseVolume Fraction

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Source of AttractionSource of AttractionWhy 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.

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 influenceForces range of influence

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Which forces do we worry about?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 watergas/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)

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 watergas/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 potentialTerminology for potentialtensionmatric potentialsuctionWe will use pressure head of the system.

Expressed as the height of water drawn up against gravity (units of length).

tensionmatric potentialsuctionWe will use pressure head of the system.

Expressed as the height of water drawn up against gravity (units of length).

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Units of measuring pressureUnits of measuring pressureAny 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)

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)

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What about big negative pressures?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.

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.

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Surface TensionSurface TensionA 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).

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).

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Surface TensionSurface TensionTo 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

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

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Typical Values of Typical Values of Dependent upon gas/liquid pairDependent upon gas/liquid pair

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Temp. dependence of air/water Temp. dependence of air/water

Cellular Automata Simulation of WaterCellular 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)

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)

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The Geometry of Fluid InterfacesThe Geometry of Fluid InterfacesSurface tension stretches the liquid-gas surface into a taut, minimal energyconfiguration balancingmaximal solid/liquid contact

with

minimal gas/liquid area.

(from Gvirtzman and Roberts,

WRR 27:1165-1176, 1991)

Surface tension stretches the liquid-gas surface into a taut, minimal energyconfiguration balancingmaximal 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 SpaceGeometry of Idealized Pore Space

Fluid resides in the pore space generated by thepacked particles.

Here the pore spacecreated by cubic andrombohedral packingare illustrated.

(from GvirtzmanAnd Roberts, WRR27:1165-1176, 1991)

Fluid resides in the pore space generated by thepacked particles.

Here the pore spacecreated by cubic andrombohedral packingare illustrated.

(from GvirtzmanAnd Roberts, WRR27:1165-1176, 1991)

Illustration ofthe geometry of wetting liquid on solidsurfaces of cubic andrhombohedralpackings ofspheres

(from GvirtzmanAnd Roberts, WRR27:1165-1176, 1991)

Illustration ofthe geometry of wetting liquid on solidsurfaces of cubic andrhombohedralpackings ofspheres

(from GvirtzmanAnd Roberts, WRR27:1165-1176, 1991)

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Let’s get quantitativeLet’s get quantitativeWe 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.

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 RelationshipDerivation of Capillary Pressure Relationship

r2

r1

1

2

S1

S2

Looking at an infinitesimal patch of a curved fluid/fluid interface

Cross Section

Isometric view

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Static means balance forcesStatic 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)

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)r2

r1

1

2

S1

S2

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Derivation cont.Derivation cont.

Since a very small patch, d2 is very smallSince a very small patch, d2 is very small

Laplace’s Equation!

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Where we were…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

• 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 CaseSpherical CaseIf 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

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

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Introduce Reduced RadiusIntroduce Reduced Radius

For general case where r1 is not equal to r2, define reduced radius of curvature, R

Which again gives us

For general case where r1 is not equal to r2, define reduced radius of curvature, R

Which again gives us

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Positive or Negative?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).

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).