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PNG 405 (Fall 2019)
Class Notes Lecture 14
Date: 10/15/2019
Taken from
Previous year’s notes (PNG 405 Fall 2017) by Prof. Karpyn; and
Dandekar, Petroleum Reservoir Rock and Fluid Properties, Second Edition, (2012)
Blunt, Multiphase Flow in Permeable Media: A Pore-Scale Perspective, (2017)
INTERFACIAL TENSION
Interfacial tension is the energy per unit area of the surface (or interface) between the two
phases.
It is also referred to as the energy penalty of breaking the intermolecular interactions within the
two phases and instead creating an interface between them.
In the figure below, the energy penalty for creating the interfaces is A < B < C, therefore,
𝜎𝑜𝑎 < 𝜎𝑤𝑎 < 𝜎𝑚𝑎
Where, 𝜎𝑜𝑎 is the interfacial tension between oil and air, 𝜎𝑤𝑎 is the interfacial tension between
water and air, 𝜎𝑚𝑎 is the interfacial tension between Mercury and air.
A
B
C
Air
Air
Air
Oil
Water
Mercury
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This is because intermolecular forces within the oil phase are much weaker. Oil is largely nonpolar and oil molecules experience weak Vander Waal’s forces which are relatively easier to
break. Water on the other hand is polar and water molecules experience stronger bonding such
as Hydrogen-bonding, and are therefore difficult to break which requires more energy. Lastly,
Mercury has the strongest boding (metal bonding) which would require the most energy to break
its bond.
SURFACE TENSION
It refers to the energy per unit area of a surface between a fluid (or solid) and its vapor phase in
thermodynamic equilibrium, with no other components present.
In a petroleum system, we have a complex mixture of many components in the phases
therefore, interfacial tension is used instead of surface tension
Adhesive forces
Unlike cohesive forces, adhesive forces show attraction of molecules in one phase to molecules
of another phase.
Case 1. When adhesion forces are stronger than cohesive forces
Phase 1
Phase 2
Adhesive
force
Phase 1
Water rises in
the capillary
Cohesive
force
Phase 2
Cohesive
force
Interface
Here, the force of adhesion is stronger – it produces a concave curvature. Phase 2 wets the
solid surface, because it has a greater affinity to the solid
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Case 2. When adhesion forces are weaker than cohesive forces
Phase 1
Phase 2
Interface
Adhesive
force
Cohesive
force
Resultant
force
Cohesive
force
Here, the force of cohesion is stronger – it produces a convex curvature. Phase 1 wets the solid
surface, because it has a greater affinity to the solid
Case 3. When adhesion forces are comparable to cohesive forces
Interface
Adhesive
force
Cohesive
force
Resultant
force
Cohesive
force
Here, the force of cohesion and adhesion are similar – it produces a flat curvature. Both phases
have equal affinity to the solid
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Wettability
“It is the relative ability of a fluid to spread on a solid surface in the presence of another fluid.”
(Anderson, 1986)
The knowledge of reservoir wettability is critical because it affects other reservoir rock and fluid
properties such as capillary pressure and relative permeability. These properties are impacted
because wettability affects how fluids are distributed in the porous medium
The adhesion force determines which fluid will preferentially wet the solid surface (rock)
Young’s equation
𝜎12
Phase 1
Phase 2
𝜃
𝜎1𝑠
𝜎2𝑠
Solid
On balancing the forces in the horizontal direction,
𝜎2𝑠 + 𝜎12 𝑐𝑜𝑠𝜃 = 𝜎1𝑠
Therefore,
𝑐𝑜𝑠𝜃 =
𝜎1𝑠 − 𝜎2𝑠
𝜎12
The difference in the interfacial tensions of the fluids with the solid phase is called the adhesion
tension (AT)
𝐴 𝑇 = 𝜎1𝑠 − 𝜎2𝑠
From Young’s equation,
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𝐴 𝑇 = 𝜎12 𝑐𝑜𝑠𝜃
Example cases – Oil/water systems (Water is the denser phase)
Case 1. Water-wetting
𝜎𝑜𝑤
Oil
Water
𝜃
𝜎𝑜𝑠
𝜎𝑤𝑠
Solid
𝑐𝑜𝑠𝜃 =
𝜎𝑜𝑠 − 𝜎𝑤𝑠
𝜎𝑜𝑤
Here,
𝜃 < 90𝑜 => 𝑐𝑜𝑠𝜃 𝑖𝑠 + 𝑣𝑒
Meaning that 𝜎𝑜𝑠 > 𝜎𝑤𝑠 because water has a higher affinity to the solid and therefore the penalty
paid in the creation of water/solid interface is less
Case 2. Oil-Wetting
Oil
𝜎𝑜𝑤
𝜃
𝜎𝑜𝑠
𝜎𝑤𝑠
Water
Solid
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𝑐𝑜𝑠𝜃 =
𝜎𝑜𝑠 − 𝜎𝑤𝑠
𝜎𝑜𝑤
Here,
𝜃 > 90𝑜 => 𝑐𝑜𝑠𝜃 𝑖𝑠 − 𝑣𝑒
Meaning that 𝜎𝑜𝑠 < 𝜎𝑤𝑠 because oil has a higher affinity to the solid and therefore the penalty
paid in the creation of oil/solid interface is less
Types of Wettabilities
•
Completely water-wet (𝜃 = 0𝑜 )
Oil
Water film
Solid
•
Water-wet (0𝑜 < 𝜃 < 60 − 75𝑜 )
𝜎𝑜𝑤
Oil
𝜃
𝜎𝑜𝑠
𝜎𝑤𝑠
Water
Solid
•
Neutral-wet (60 − 75𝑜 < 𝜃 < 105 − 120𝑜 )
Oil
𝜎𝑜𝑤
𝜃
𝜎𝑜𝑠
𝜎𝑤𝑠
Water
Solid
•
Oil-wet (60 − 75𝑜 < 𝜃 < 105 − 120𝑜 )
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Oil
𝜎𝑜𝑤
𝜃
𝜎𝑜𝑠
𝜎𝑤𝑠
Water
Solid
•
Completely oil-wet (𝜃 = 180𝑜 )
Oil
𝜃
𝜎𝑜𝑠 𝜎𝑜𝑤
𝜎𝑤𝑠
Water
Solid
Intermediate wettability (different from neutral wettability)
•
Fractional wettability – some pores are oil-wet, some others are water-wet due to
variability in the mineral composition of the rock
•
Mixed-wet – small pores are water-wet but larger pores are oil-wet
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CAPILLARY PRESSURE
Rise of fluid in a capillary
Consider a water/air system. Water rises due to the adhesion tension.
r
a
P
𝜃
Air
h
Q
R
Water
How far does the water rise?
Water rises until the total pulling force by the adhesion tension is balanced by the weight of the
column
Upward force due to capillarity,
𝐹𝑢𝑝 = 𝐴 𝑇 2𝜋𝑟
𝐴 𝑇 = 𝜎𝑎𝑠 − 𝜎𝑤𝑠 = 𝜎𝑎𝑤 𝑐𝑜𝑠𝜃
𝐹𝑢𝑝 = 𝜎𝑎𝑤 𝑐𝑜𝑠𝜃2𝜋𝑟
Downward force due to gravity,
𝐹𝑑𝑜𝑤𝑛 = 𝑚𝑔 = 𝜌𝑉𝑔 = 𝜌𝑤 𝜋𝑟 2 ℎ𝑔
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Balancing forces,
𝐹𝑢𝑝 = 𝐹𝑑𝑜𝑤𝑛
𝜎𝑎𝑤 𝑐𝑜𝑠𝜃2𝜋𝑟 = 𝜌𝑤 𝜋𝑟 2 ℎ𝑔
We get, the equilibrium height, h, for the rise of the fluid in the capillary
𝒉=
𝟐𝝈𝒂𝒘 𝒄𝒐𝒔𝜽
𝝆𝒘 𝒓𝒈
In the same capillary system, consider the pressures at the following points,
•
Pressure at point a, (Pa). This is equal to atmospheric pressure
•
Pressure at point P, (PW). Water pressure at point P
•
Pressure at point Q, (PQ). This is equal to atmospheric pressure, negligible hydrostatic
head of air
•
Pressure at point R, (PR). This is equal to PW + 𝜌𝑤 𝑔ℎ. That is, pressure of water at P and
the hydrostatic head of water
We know,
PQ = PR. These are pressure at the same horizontal level.
Therefore, PQ = Pa = PR
Therefore, Pa = Pw + 𝜌𝑤 𝑔ℎ
Therefore, Pa - Pw = 𝜌𝑤 𝑔ℎ
Capillary pressure (Pc) is defined as the difference in the non-wetting phase pressure and the
wetting phase pressure
𝑃𝑐 = 𝑃𝑛𝑜𝑛−𝑤𝑒𝑡𝑡𝑖𝑛𝑔 − 𝑃𝑤𝑒𝑡𝑡𝑖𝑛𝑔
For the air/water system, air being the non-wetting phase, wand water being the wetting phase
𝑃𝑐 = 𝑃𝑎 − 𝑃𝑤
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Therefore, for the air/water capillary rise system,
𝑃𝑐 = 𝜌𝑤 𝑔ℎ
Putting the equilibrium height, h, we get,
𝑃𝑐 =
2𝜎𝑎𝑤 𝑐𝑜𝑠𝜃
𝑟
Where, 𝑟 is the tube radius, 𝜎𝑎𝑤 is the air/water interfacial tension, 𝜃 is the wetting angle
(contact angle)
Note: As 𝑟 ↓ => ℎ ↑ => 𝑃𝑐 ↑
As 𝐴 𝑇 (𝜎𝑎𝑤 𝑐𝑜𝑠𝜃) ↑ => ℎ ↑ => 𝑃𝑐
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Q5. Below is a portion of a phase-partitioned cross-sectional slice from a 3D computed
microtomography scan of a glass bead pack post two cycles of flooding by two immiscible
fluids. Of the two fluids present (orange or yellow), which is the wetting phase? How do we
know this? If I told you the brine is yellow and the kerosene is orange, what is the wettability of
the beads used?
5
6
[mm]
10
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Imm!
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