Capillary Pressure

1. Capillary Pressure
1

2. 2
:Definitio
n
PC=Curvature ×Surface Tension
TheHigherCurvature
,
TheHigherCapillary Pressure

3. 3
:Thus
In the smallerpores,
The Higher Curvature of the
Interface
»»» The Higher PC

4. 4
Sw
:AS Decreases
Wetting Phase will be drawn into
Thesmaller Pores
Resulting in:
TheIncreasein PC

5. A5
t Swi:
ThelowerSw,
TheHigherCurvatureinsmallerpores,
TheHigherPC
w phase: 1-Inverysmallpores(infinitePC)

6. 6

7. 7
Dipping a Lumpof Sugarin aCupof
Tea
HydrophilicSugar:
T
ea FlowsUpward Spontaneously to reacha
New EquilibriumState:
Pn = Pw+PC
Hydrophobic Sugar:
T
eaWouldnot moveinto the cubeat the 1st
contact.
Cubemustbe PushedDowninto the Tea.

8. ControllingForces:
1- Viscous Force (Friction):Always acting against the flow
dependingon Velocity
2- Capillary force:?
Drainage: Against theFlow
Imbibition: In Favor ofFlow
8

fc =
r 2
r2
 u
f

9. Entry or ThresholdPressure
• Required (Pn - Pw) to enable:
n-Phase to invadea porous medium
saturated with w-Phase (Sw=1).
Entry Pressure isrelatedto the largest pore
diameter.
9
max
P =
2Cos
r
d

10. Thus: 10
• Drainage beginsonly when an entry
pressure pn−pw>0 is applied. Related to
the largest pore.
• Imbibition occursspontaneously, does not
require an entry pressure.
W-phase goesto the smallest pores with
highest Pc. Pc
increases as S
wdecreases.

11. In Drainage:
TheSmallerpore, the Higher required Pn
TheInfluence of Varying PoreDiameter
r
11
2Cos
Pn  Pw +

12. X6
: AnUnstable equilibrium location.
Interface jumping from X6
to X5
iscalled a
Hainesjump corresponding to a locally abrupt
.change in saturation
The Influenceof Varying Pore
Diameter 12

13. In Imbibition: Pw  Pn
TheLargerpore, the Higher required Pw
TheInfluence of Varying PoreDiameter
r
13
2Cos
−

14. HYSTERSIS:
Different Behavior of the
Process depending on its
direction.
14
TheDifference Occurs because:
The Small Pores Control Drainage.
The LargePores Control Imbibition.
»»» for a certainSw
imbibition
Drainage
C C
(P )  (P )

15. HysteresisLoop: ThePath of the full
Cycle
15
Scanning Curves:Shortcuts acrossthe Hysteresis
Loop.
Example of Hysteresis: Driving between home andwork
Two paths together: Hysteresis loop
Scanning Curve: Halfway

16. Fluid
Entrapment
16
e.g: Drying aporous Solid by
Passingair throughit.
(some water remains in the small pores
and is only releaseddue to evaporation)

17. Entrapped
Saturation
17
• Irreducible saturationfor w- Phase
• Residual Saturation for n-Phase

18. Irreducible Saturation,
Swi
18
At Sw
i
--Somew-phase moleculesare strongly
adsorbedonto the solidsurface.
-- Capillaryw-phase isheld tightly inthe
Cornersand Crevices.
So: w-Phase will no longerflowin responseto a
gradientin ahydraulichead.

19. At Sw
i
19
Actually: w-Phase is Hydraulically connected
{coating the solid surface with athinfilm}
However effective conductivity is
negligible.
Thus:
In Practice: w-Phase is Hydraulically Disconnected.

20. :Trapped-Phase
volume
20
TheVolumeWhich will no-longer Respondto a
gradientinHydraulicPotential.
S
= S
+ S
Free Trapped

21. 21
Entrapment in
Imbibition
• More Complicated
• Two Primary Mechanisms:
1 Snap-off
2 Bypassing

22. Snap-
Off
22
• W-Phasetends toflow along the pore wall as
it displacesthen-Phase.

23. Snap-
Off
23
Pore Throat diam.
Pore Body diam.
Aspect Ratio = AR =
TheHigherAR,TheMore Snap-Off
TheSmallerθ, TheMore Snap-Off

24. Bypassing 24
Poredoublet

25. :Residual Saturation depends
on
25
• Pore Network Geometry
• Fluid Properties, SurfaceTension
Density Ratio
Viscosity Ratio
• Potential Gradient (Flow)
• Incomplete Displacement

26. In another
Vie
w
26
• So
rdepends on the Magnitudeof Viscous
and Capillary Forces.
If Capillary force isdominant
TheHigher Capillarity Effect, The Higher Sor

27. Capillary
Numbe
r
27

uw w
Nc =
For Nc<10-4
Capillary Forces Dominate Over Viscous Forces
r2

u
f
r2
c
f 


28. ?Why 28
r
k
2
ul
Capillary Pressure
=
Viscous − Pr essure drop
=
Capillary − force
Viscous − force
Carmen-KozenyEq.:
⎯

⎯
=
⎯
0
.
2
→k =10−3
r2
r2
45
k =


29. r
Capillary − force  210−3
r
 Viscous − force
=
u 
l
29
C
= N 500
l
:For common porousMedium
l
= 20 
Viscous −
force r Capillary −
force
Thus if Nc<10-4
C
= N 104
Capillary Forces Dominate Over Viscous Forces

30. :In an ordinary water
flooding
30
day S
w
U 1
m
 10−5 m
w  1cp10 paS
−3
m
n
  5010−3
−7
 NC  210

31. Effects of FlowProperties 31
when Nc<10-4
So
rCan be Decreased by the Increase in Nc
The Increase in Water velocity, Viscosity
wn
w w
c
u 
N =


32. Effect of Incomplete
Displacemen
t
32
• The Higher InitialSn, The Higher Sn
r
In incompletedrainage smaller pores are not
occupied by the n-Phase resulting in
smaller Snr , If Nc<10-4

33. THANK YOU
33