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Relative Permeability
Petroleum University of Technology
Abadan Faculty of Petroleum Engineering
RESERVOIR ROCK PROPERTIES
Introduction
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Introduction
Absolute permeability: is the permeability of a porous medium saturated with a
single fluid (e.g. Sw=1).
Except for gases at low pressure , the permeability f rock is a property of the rock
and not the fluid that flows through it.
Commonly, reservoirs contain 2 or 3 fluids
•Water-oil systems
•Oil-gas systems
•Water-gas systems
•Three phase systems (water, oil, and gas)
To evaluate multiphase systems, must consider the effective and relative
permeability
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Ac
water Test
L oil Test
qw 0.3 bbl / day qo 0.1 bbl / day
w 1 cp o 3 cp
p 30 psi p 30 psi
qw w L q o o L
k 413 md k 413 md
0.001127Ac p 0.001127Ac p
As the historical review showed, all efforts were made to extend the validity of
the Darcy-law to multiphase filtration. If this is possible, then the following
formulas may be set up:
SO=30% SO=30%
Ac
SW =70% SW =70%
L
p 30 psi
qw 0.18 bbl / day
qo 0.012 bbl / day
w 1 cp
o 3 cp
p 30 psi
qw w L q o o L
kw 284 md ko 50md
0.001127Ac p 0.001127Ac p
•Effective permeability: is a measure of the conductance of a porous medium for
one fluid phase when the medium is saturated with more than one fluid.
•Effective permeabilities: (ko, kg, kw)
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qw 0.18 bbl / day SO=30% SO=30%
Ac
qo 0.012 bbl / day SW =70%
SW =70%
w 1 cp
o 3 cp L
p 30 psi
p 30 psi
k w 284 md
k rw 0.6
k 413
k 50 md
k ro o 0.12
k 413
Relative permeability is a dimensionless measure of the permeability of a fluid
phase as it flows through porous rock in the presence of another fluid phase.
•Relative Permeability is the ratio of the effective permeability of a fluid at a given
saturation to some base permeability.
•The flowing water –oil ratio at reservoir conditions depends on viscosity ratio and
relative permeability ratio; i.e. , at MOBILITY RATIO.
qw k w Ac p k o Ac p
qo w L o L
qw k w k o k rw w w
M
qo w o k ro o o
k k 248 50
M w o
w o 1 3
M 14.9
•At 70% water saturation and 30% oil saturation, the water is flowing at 14.9 times
the oil rate.
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Typical Curves for Oil-water Relative
Permeability at a Water-wetted System
The end point saturations determine the movable saturation range and is directly
related to amount of recoverable oil.
Importance of Drainage and Imbibition Relative
Permeability
Drainage curves are important for:
• solution gas drive (since oil and water are generally wetting relative to gas),
• for gravity drainage (gas displaces drained oil),
• gas injection processes,
• oil or gas displacing water (in tertiary recovery processes).
Imbibition curves are relevant to:
• waterflood calculations,
• water influx,
• oil displacing gas (e.g. oil moving into a gas cap).
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Definitions of End-Point Saturations
A relative permeability is zero at a given saturation. For the displacing phases
these saturations are called critical water saturation (Swc) and critical gas
saturation (Sgc), respectively. They are the saturations at which the displacing
water or gas phase begins to flow.
When the water saturation becomes low enough, the water becomes disconnected
and forms pendular rings. Once disconnected, the wetting phase can no longer
flow and hence the remaining fluid is at immobile wetting phase saturation.
Definitions of End-Point Saturations
Residual Non-wetting Phase Saturation: For the displaced phase the minimum
saturation is residual saturation. The residual non-wetting phase saturation occurs
because small blobs of non-wetting phase become trapped in the pores, and once
disconnected from each other they can no longer flow.
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Factors Affecting Relative Permeabilities
• Fluid saturations
• Geometry of the pore spaces and pore size distribution
• Wettability
• Fluid saturation history (i.e., imbibition or drainage)
Typical Two-phase Flow Behavior
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Point 1
Point 1 on the wetting phase relative permeability shows that a small saturation of
the non-wetting phase will drastically reduce the relative permeability of the wetting
phase. The reason for this is that the non-wetting phase occupies the larger pore
spaces, and it is in these large pore spaces that flow occurs with the least difficulty.
Point 2
Point 2 on the non-wetting phase relative permeability curve shows that the non-
wetting phase begins to flow at the relatively low saturation of the non-wetting phase.
The saturation of the oil at this point is called critical oil saturation Soc.
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Point 3
Point 3 on the wetting phase relative permeability curve shows that the wetting
phase will cease to flow at a relatively large saturation. This is because the wetting
phase preferentially occupies the smaller pore spaces, where capillary forces are
the greatest. The saturation of the water at this point is referred to as the irreducible
water saturation Swir
Point 4
Point 4 on the non-wetting phase relative
permeability curve shows that, at the
lower saturations of the wetting phase,
changes in the wetting phase saturation
have only a small effect on the
magnitude of the non-wetting phase
relative permeability curve. The reason
for the phenomenon at Point 4 is that at
the low saturations the wetting phase
fluid occupies the small pore spaces
which do not contribute materially to
flow, and therefore changing the
saturation in these small pore spaces
has a relatively small effect on the flow
of the non-wetting phase.
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The Jamin Effect
The sum of the relative permeabilities for all phases is almost always less than
unity because of interference among phases sharing flow channels. There are a
number o f reasons for this interference.
1. The part of the pore channels available for flow of a fluid may be
reduced in size by the other fluids present in the rock.
2. The immobilized droplets of one fluid may completely plug some
constrictions in a pore channel through which another fluid would
otherwise flow.
3. Some pore channels may become effectively plugged by adverse
capillary forces if the pressure gradient is too low to push an interface
through a constriction.
4. The trapping of a group of globules that are clustered together and
cannot be moved, since the grain configuration allows fluid to flow
around the trapped globules without developing a pressure gradient
sufficient to move them.
Effect of Saturation on Kr
At low saturation so f the fluid that preferentially tends to wet the grains of a rock,
the wetting phase forms doughnut-shape rings around the grain contact points.
These are called pendular rings. The rings do not communicate with each other
and pressure cannot be transmitted from one pendular ring to another.
Sometimes such a distribution may occupy an appreciable fraction of the pore
space. The amount depends upon the nature and shape of individual grains,
distribution, as well as degree and type of cementation
Above the critical wetting-phase saturation, the wetting phase is mobile through a
tortuous path under a pressure differential and as the wetting-phase saturation
increases, the wetting phase relative permeability increasesas well. The wetting-
phases saturation distribution in this region is called funicular and up to a point,
the relative permeability to the wetting phase is less than the relative permeability
to the non-wetting phase due to the adhesion force between the solid surface and
wetting fluid, and the greater tortuosity of the flow path for the wetting phase. The
non-wetting phase moves through the larger pores within this range of saturation,
but as the saturation of the wetting phase further increases, the non-wetting
phase breaks down and forms a discontinuous phase at the critical non-wetting
phase saturation.This is called an insular state of non-wetting-phase saturation
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Effect of Wettability on Kr
• Water flows more freely
• Higher residual oil saturation
WATER-OIL-GAS System
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BROOKS and COREY
Drainage Relative Permeability
Imbibition Relative Permeability
During imbibition we have to worry about the trapped non-wetting phase,
which does not contribute to flow.
Applications of Relative Permeability Functions
• Reservoir simulation
• Flow calculations that involve multi-phase flow in reservoirs
• Estimation of residual oil (and/or gas) saturation
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Relative Permeability Measurement Methods
Steady State Method
In the steady state method, two fluids are injected simultaneously at a fixed ratio
until the produced ratio is equal to the injected ratio. Core saturations have to be
measured at each equilibrium and new fluid ratio is applied. This is repeated until
the relative permeabilities are determined. Most tests are started with the core
sample saturated with 100% wetting phase, and the test is then a desaturation
test (drainage).
Unsteady State Method
The procedure for performing an unsteady state test is relatively simple and fast
First the core is saturated with 100% water and then the sample is desaturated by
injecting oil until no more production of water is obtained. Water production is
recorded and Swi calculated. Effective oil permeability is then measured at Swi. Oil
is displaced by a predetermined constant rate of water, oil permeability and
pressure drop across the core will be recorded. Alternatively, oil is displaced by
keeping the differential pressure across the core constant with varying rate of fluid
flow. With the recording of cumulative water injection, pressure drop and produced
oil volume, it is possible to calculate relative permeabilities by theory developed by
Welge (1952)
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