Capillary Pressure

H. MahdiyarShiraz University, Aban-1391

1

Definition:

PC = Curvature × Surface Tension

The Higher Curvature,

The Higher Capillary Pressure

2

Thus:

In the smaller pores,

The Higher Curvature of the Interface

»»» The Higher PC

3

AS Sw Decreases:

Wetting Phase will be drawn into

The smaller PoresResulting in:

The Increase in PC

4

5

The lower Sw,

The Higher Curvature in smaller pores,

The Higher PC

At Swi : w phase : 1-In very small pores (infinite PC)

2- Spreads over the solid Surface (Thin Film)

6

Dipping a Lump of Sugar in a Cup of Tea

Hydrophilic Sugar: Tea Flows Upward Spontaneously to reach a

New Equilibrium State:Pn = Pw + PC

Hydrophobic Sugar: Tea Would not move into the cube at the 1st

contact. Cube must be Pushed Down into the Tea.7

Controlling Forces:

1- Viscous Force (Friction): Always acting against the flow depending on Velocity

2- Capillary force: ?

Drainage: Against the Flow

Imbibition: In Favor of Flow

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2rfc

2ruf

Entry or Threshold Pressure

• Required (Pn - Pw ) to enable:

n-Phase to invade a porous medium saturated with w-Phase (Sw = 1).

Entry Pressure is related to the largest pore diameter.

9 max

2rCosPd

:Thus• Drainage begins only when an entry pressure pn

− pw > 0 is applied. Related to the largest pore.

• Imbibition occurs spontaneously, does not require an entry pressure.

W-phase goes to the smallest pores with highest Pc. Pc increases as Sw decreases.

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In Drainage:

The Smaller pore, the Higher required Pn

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The Influence of Varying Pore Diameter

rCosPP wn

2

X6 : An Unstable equilibrium location.Interface jumping from X 6 to X5 is called a

Haines jump corresponding to a locally abrupt .change in saturation12

The Influence of Varying Pore Diameter

In Imbibition:

The Larger pore, the Higher required Pw

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The Influence of Varying Pore Diameter

rCosPP nw

2

HYSTERSIS:Different Behavior of the Process

depending on its direction.

The Difference Occurs because:The Small Pores Control Drainage.The Large Pores Control Imbibition.

»»» for a certain Sw

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imbibitionCDrainageC PP

Hysteresis Loop: The Path of the full Cycle

Scanning Curves: Shortcuts across the Hysteresis Loop.

Example of Hysteresis: Driving between home and workTwo paths together: Hysteresis loopScanning Curve: Halfway

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Fluid Entrapment

e.g: Drying a porous Solid by Passing air through it.

(some water remains in the small pores and is only released due to evaporation)

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Entrapped Saturation

• Irreducible saturation for w- Phase

• Residual Saturation for n-Phase

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Irreducible Saturation, Swi At Swi

--Some w-phase molecules are strongly adsorbed onto the solid surface.-- Capillary w-phase is held tightly in the Corners and Crevices.

So: w-Phase will no longer flow in response to a gradient in a hydraulic head.

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At Swi Actually: w-Phase is Hydraulically connected

{coating the solid surface with a thin film}

However effective conductivity is negligible.

Thus:In Practice: w-Phase is Hydraulically Disconnected.

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Trapped-Phase volume:

The Volume Which will no-longer Respond to a gradient in Hydraulic Potential.

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TrappedFree SSS

Entrapment in Imbibition

• More Complicated

• Two Primary Mechanisms:

1- Snap-off2- Bypassing

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Snap-Off

• W-Phase tends to flow along the pore wall as it displaces the n-Phase.

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Snap-Off

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..

diamThroatPorediamBodyPore

ARRatioAspect

The Higher AR, The More Snap-Off

The Smaller θ, The More Snap-Off

Bypassing

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Pore doublet

Residual Saturation depends on:• Pore Network Geometry

• Fluid Properties, Surface Tension Density Ratio Viscosity Ratio

• Potential Gradient (Flow)

• Incomplete Displacement

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In another View

• Sor depends on the Magnitude of Viscous and Capillary Forces.

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If Capillary force is dominantThe Higher Capillarity Effect, The Higher Sor

Capillary Number

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ww

c

uN

For Nc < 10-4

Capillary Forces Dominate Over Viscous Forces

2ruf

2rfc

Why?

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r

klu

essureCapillarydropessureViscous

forceCapillaryforceViscous

2PrPr

Carmen-Kozeny Eq.:

232.02 1054

rkrk

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rlN

rlu

forceCapillaryforceViscous

C

500

102 3

For common porous Medium:

41020

CNforceCapillary

forceViscousrl

Thus if Nc < 10-4

Capillary Forces Dominate Over Viscous Forces

In an ordinary water flooding:

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Sm

daymUw

5101

paScpw3101

mn31050

7102 CN

Effects of Flow Properties

when Nc < 10-4

Sor Can be Decreased by the Increase in Nc

The Increase in Water velocity, Viscosity

The decrease in Surface Tension

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wn

wwc

uN

Effect of Incomplete Displacement

• The Higher Initial Sn, The Higher Snr

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In incomplete drainage smaller pores are not occupied by the n-Phase resulting in

smaller Snr , If Nc < 10-4