5.1 Wettability

Transcription

5.1 Wettability

Chapter 5 – Multiphase Phenomena
5.1 Wettability
Water and oil (or gas) in reservoirs coexist in an immiscible state (i.e., the water
phase does not mix miscibly with the hydrocarbon phase). There is a natural and strong
interfacial tension between the two fluids that keeps them separate, regardless of how small
the individual droplets may be. A common example of this immiscible nature is a household
salad dressing made of oil and vinegar.
In all reservoirs connate water is immiscible with the oil or gas, but chemicals can be
injected into the reservoir to reduce interfacial tension and make the water phase miscible
with the oil. There are advantages in doing this, and it is a form of enhanced oil recovery.
The oil and gas phases in reservoirs also generally behave immiscibly. However, at certain
pressures, temperatures, and compositions, they may become miscible. Throughout this
chapter, the three fluids, or phases (oil, gas, and water) are considered to be immiscible.
On all interfaces between solids and fluids and between immiscible fluids, there is a
surface free energy resulting from natural electrical forces. The forces cause molecules of the
same substance to attract one another through cohesion, and molecules of unlike substances
to attract one another through adhesion. Interfacial tension results from these molecular
forces causing the surface of a liquid to form the smallest possible area and act like a
membrane in tension.
Wettability can be defined as the ability of a fluid phase to preferentially wet a solid
surface in the presence of a second immiscible phase. In the reservoir context, it refers to the
state of the rock and fluid system; i.e., whether the reservoir is water or oil wet. Three
possible states of wettability in oil reservoirs exist as shown in Figure 5.1. The arrows
represent the tangent to the angle between the water droplet and the rock surface. The water
droplet is surrounded by the oil phase.
Wettability is generally classified into three categories: (1) The reservoir is said to be
water wet; that is, water preferentially wets the reservoir rock, when the contact angle 
between the rock and water is less than 90, (2) neutral wettability case would exist at a
contact angle of 90, and (3) oil wet occurs at a contact angle greater than 90.
5. 1
Figure 5.1 Three possible states of wettability in oil reservoirs.
Other lesser known types of wettability are:
Neutral or intermediate wettability – no preference is shown by the rock to either
fluid; i.e., equally wet.
Fractional wettability – heterogeneous wetting; i.e., portions of the rock are strongly
oil wet, whereas other portions are strongly water wet. Occurs due to variation in
minerals with different surface chemical properties. Silicate water interface is acidic,
therefore basic constituents in oils will readily be absorbed resulting in an oil-wet
surface. In contrast, the carbonate water interface is basic and will attract and absorb
acid compounds. Since crude oils generally contain acidic polar compounds, there is
a tendency for silicate rocks to be neutral to water-wet and carbonates to be neutral to
oil-wet.
Mixed wettability – refers to small pores occupied by water and are water-wet, while
larger pores are oil-wet and continuous. Subsequently, oil displacement occurs at
very low oil saturations resulting in unusually low residual oil saturation.
Figures 5.2 and 5.3 represent microscopic views of water-wet and oil-wet systems,
respectively.
Figure 5.2 Microscopic fluid saturation distribution in a water-wet rock [Pirson, 1963]
5. 2
Figure 5.3 Microscopic fluid saturation distribution in a oil-wet rock [Pirson, 1963]
The contact angle is a measure of the wettability of the rock-fluid system, and is
related to the interfacial energies by Young’s equation,
 os   ws   ow cos
(5.1)
where:
os
=
interfacial energy between oil and solid, dyne/cm;
ws
=
interfacial energy between water and solid, dyne/cm;
ow
=
interfacial energy, or interfacial tension, between oil and water, dyne/cm;


contact angle at oil-water-solid interface measured through the water phase,
deg.
Figure 5.4 identifies the variables in Equation 5.1.
Figure 5.4 Relationship of oil-water-solid interfacial tensions and contact angle
5. 3
5.2 Measurement of Contact Angle
A simple test to determine contact angle (and thus wettability) is known as the sessile drop
method. A drop of water is placed on a smooth, mineral surface suspended horizontally
below the surface of the oil (See Figure 5.5). A photograph is taken of the system for
accurate measurement of the contact angle. The contact angle is measured through the denser
phase (water).
A variation to this method is to suspend the plate horizontally in the water and place a drop
of oil at the bottom of the plate (Figure 5.5. d, e, f). The contact angle is measured through
the water phase and the same analysis is applied.
Figure 5.5 Measurement of contact angles for water-oil systems
A modification of the sessile drop method was introduced to measure water-advancing or
receding contact angles. Figure 5.6 illustrates two polished quartz plates mounted
horizontally with a small gap between them; one plate is fixed and the other can be moved
smoothly with a screw. A drop of oil is placed between the plates and allowed to age until
the contact angle no longer changes; then the mobile plate is moved, creating the advancing
or receding contact angle. This angle changes gradually and eventually reaches a stable value
after a few days.
5. 4
Figure 5.6 Method used to measure advancing/receding contact angles
These measurement techniques are most useful for measuring wettability effects of pure
fluids on clean, smooth mineral surfaces and thus provide qualitative information. However,
they do not represent reservoir rocks, because the plates do not account for surface
roughness, the large variety of minerals, presence of thin layers of organic materials, and
presence of polar substances in the oils. To obtain the average wettability of a core requires
the following tests.
Amott Wettability Test
The Amott test for wettability is based on the natural imbibition and forced displacement of
oil and water from cores. The procedure is:
(1) The test begins at the residual oil saturation; therefore, the fluids are reduced to Sor by
forced displacement of the oil.
(2) The core is immersed in oil for 20 hours, and the amount of water displaced by
spontaneous imbibition of oil, if any, is recorded as Vwsp.
5. 5
(3) The water is displaced to the residual water saturation (Swi) with oil, and the total amount
of water displaced (by imbibition of oil and by forced displacement) is recorded as Vwt.
(4) The core is immersed in brine for 20 hours, and the volume of oil displaced, if any, by
spontaneous imbibition of water is recorded as Vosp.
(5) The oil remaining in the core is displaced by water to Sor and the total amount of oil
displaced (by imbibition of water and by forced displacement) is recorded as Vot.
The Amott wettability index is defined as the displacement by oil ratio (o = Vosp/Vot) less the
displacement by water ratio(w = Vwsp/Vwt).
V
V
osp
wsp
Iw 

 o   w
V
V
ot
wt
A preferentially water wet core exhibits a positive displacement of oil ratio and a value of
zero for a displacement by water ratio. The result is a positive Iw, approaching one for a
strongly water wet sample. Cores that are preferentially oil-wet exhibit the reverse; i.e., a
zero displacement of oil ratio and a positive displacement of water ratio; thus the resulting
index is negative, approaching (-1) for strongly oil wet. Neutral wettability is characterized
by an Amott index near zero.
The 20-hr time limit is arbitrary and can be too short for the slow spontaneous imbibition
process. This would lead to underestimated displacement ratios and erroneous conclusions to
the sample wettability. One method to overcome this problem is to periodically measure the
fluid displaced and terminate the test when stable equilibrium is attained.
USBM Wettability Test
An alternative for determining wettability index is known as the United States Bureau of
Mines (USBM) method.
This method uses the hysteresis loop from capillary pressure
curves. The capillary pressure curves are obtained by alternately displacing water and oil
5. 6
from small cores using a centrifuge. The areas under the capillary pressure curves represent
the thermodynamic work required for the respective fluid displacements (see Figure 5.7).
Displacement of a non-wetting phase by a wetting phase requires less energy than
displacement of a wetting phase by a non-wetting phase. Therefore, the ratios of the areas
under the capillary pressure curves (between Swi and Sor) are a direct indicator of the degree
of wettability. The logarithm of the area ratio of oil displacing water (A1) to water displacing
oil, (A2) is used as a convenient scale for the wettability index:
A 
I w  log 1 
A 
 2
Figure 5.7 USBM method of determining wettability for a water wet sample.
Increasing positive values indicate a preference to water wet; i.e., A1 progressively becomes
greater than A2. Negative values of the index indicate an oil-wet preference (A2 > A1).
Combined USBM-Amott Wettability Test
A final procedure combined both the USBM and Amott methods and provides both the
USBM wettability index and the Amott ratio. Figure 5.8 illustrates this combined method. At
each point where the capillary pressure is equal to zero, the sample is immersed in the
displacing fluid for 20 hours and the amount of fluid imbibed is recorded and used to
determine the Amott ratios. For the combined test, the capillary pressure data are plotted
5. 7
versus the average saturation (not the saturations at inlet face of the core). Thus, the
procedure has six steps:
(1) Sample is 100% water saturated,
(2) Oil displaces water to Swi, (drainage cycle)
(3) spontaneous imbibition of brine,
(4) Water displaces oil to Sor, (imbibition cycle)
(5) spontaneous imbibition of oil,
(6) final displacement of water by oil (2nd drainage cycle).
Figure 5.8 Schematic of test procedure for combined USBM-Amott test
Example data are shown in Table 5.1 using the combined approach. Note for strongly waterwet samples that are 100% saturated by water, spontaneous imbibition of oil does not occur.
Therefore, the displacement of water by oil ratio for the Amott test is zero for the three water
wet-cores. The USBM wettability index is greater than one, indicating very strong water
wettability. However for the mixed wettability systems, spontaneous imbibition is observed,
therefore both the USBM index and the Amott index shows that these cores are near neutral
wettability.
5. 8
Displacement
of water
Displacement
of oil
Amott Index
USBM Index
Sample No.
Water wet samples
0
0
0
0.92
0.80
0.92
0.80
1.6
1.2
0.89
1
2
3
0.3
0.3
0.3
1
2
3
Mixed wettability samples
0.24
0.24
0.24
0.015
0.073
0.065
-0.225
-0.167
-0.175
Table 5.1 Wettability tests using the combined USBM-Amott wettability procedures
The wettability of reservoir rocks is difficult to determine. In situ measurements in
the reservoir have not yet been developed, and laboratory measurements are altered by:

mineral exposure to fluids

chemistry of the fluids

saturation history of the sample
Careful laboratory experiments show that contact with clean air alone can significantly alter
the wettability of a system.
Consequently, careful restoration of samples is important in
determining the correct wettability of the rock-fluid system.
5.3 Capillary Pressure
The pores in typical reservoir rock are microscopic. The small pore size combined
with the interfacial tension between the reservoir's immiscible fluids results in capillary
pressure that is a major factor in establishing fluid distributions. In fact, porous reservoir rock
can be considered to be a complex assortment of small capillary tubes.
Capillary pressure theory - Capillary pressure is the pressure difference across the
curved interface formed by two immiscible fluids in a small capillary tube. Figure 5.9 shows
capillary tubes immersed in wetting and nonwetting liquids. The capillary rise in a wetting
liquid can be demonstrated with a glass capillary tube in water. Capillary depression in a
non-wetting liquid can be demonstrated by a glass tube in mercury.
5. 9
Figure 5.9 Capillary tubes in wetting and Nonwetting liquids
Capillary pressure in a tube can be calculated from the interfacial tension between the
fluids, contact angle between the rock and fluid, and the capillary tube radius (referring to
Figure 5.9), as follows. At equilibrium in the capillary tube, a force balance equation can be
written as;
force up = force down
force up = 2r  cos 
force down = r2 h 
Capillary pressure, Pc, is the force per unit area; therefore
2
2
Pc  force up / r  force down / r
2 cos 
Pc 
r
where:
r
= pore radius; cm

= interfacial tension; dynes/cm

contact angle, deg.
5. 10
( 5.2)
5.2.1
Capillary pressure in the reservoir
Laboratory capillary pressure curves require corrections for the following effects:
1. Closure – the effect of surface irregularities of the core plug
2. Microporosity leading to “double curves”.
The microporosity results in a
different capillary entry point at a lower water saturation.
3. Confining stress – impacts the magnitude of porosity and permeability
4. Presence of clays – alters the effective water saturation
5. Wettability and interfacial tension differences between lab and reservoir fluids
Of these, the last one will be discussed in detail. Practical applications typically convert
capillary pressure to height above the FWL thus developing a saturation profile. The height
at which water will stand above the FWL is directly proportional to the capillary pressure.
Further development of Equation 5.2 leads to:
2


r gh  w   air
Pc 
  gh
2
r
(5.3)
where:
h
= height of capillary rise, ft;
Pc
= capillary pressure, psi;
w
= water, or wetting phase, density, slugs/ft3 or kg/m3
air
= air, or nonwetting phase, density, slugs/ft3 or kg/m3
Examining Equation 5.2, the larger the pore radius, the lower the capillary pressure.
From Equation 5.3 the lower the capillary pressure, the lower the height of water rise in the
reservoir. Low capillary pressure and low irreducible water saturations are associated with
reservoir rocks that have large pores, such as coarse-grained sand, coarse-grained oolitic
carbonates, and vuggy carbonates. It naturally follows that high capillary pressure and high
water saturations are associated with fine grained reservoir rocks.
In reservoirs, capillary pressure is the difference between the nonwetting-phase
pressure (Pnw) and the wetting-phase pressure (Pw).
Pc  Pnw  Pw
(5.4)
Capillary pressure is always positive and in a normal water-wet oil reservoir can be
expressed as the difference in pressure between the oil and water phases.
5. 11
Pcow  Po  Pw
(5.5)
Similarly, in a water-wet gas reservoir, or gas cap,
P
P P
cgw
g
w
(5.6)
and where oil and gas occur together, the oil phase usually behaves as the wetting phase
relative to the gas phase, therefore,
P
P P
cog
g
o
(5.7)
Density differences among oil, gas, and water as well as their immiscibility cause capillary
pressure and gradients in reservoirs, as shown in Figure 5.10.
Figure 5.10 Capillary pressure in reservoirs
Laboratory measurements of capillary pressure would ideally be conducted with
actual reservoir oil and water at reservoir temperature. However, not only are there
difficulties in obtaining representative reservoir oil and water samples, but there are
laboratory handling difficulties that discourage their use. Years of experience show that other
fluids can be substituted for reservoir oil and water in the laboratory measurements, and the
results can be adjusted to account for pertinent differences. Table 5.2 lists fluid combinations
used in laboratory measurements of capillary pressure and reservoir values.
5. 12
SYSTEM
CONTACT ANGLE,
deg.
INTERFACIAL TENSION,
dynes/cm
Laboratory
Air-water
Oil-water
Air-Mercury
Air-oil
Reservoir
0
30
140
0
Water-oil
Water-gas
30
0
72
48
480
24
30
50*
Table 5.2 Typical interfacial tension and contact angle constants
* Pressure and temperature dependent, reasonable value to depth of 5000 ft.
Laboratory results using fluids other than reservoir fluids at temperatures that differ from
reservoir temperature are corrected for use in reservoir calculations as follow.
 cos  res
Pc(res)  Pc(lab )
 cos  lab
(5.8)
where:
Pc(res)
Pc(lab)
res
lab
res
lab
= capiIlary
pressure, corrected to reservoir conditions and reservoir fluids, psig;
capillary
pressure
measured in the laboratory, psig;
=
= interfacial tension between reservoir fluids at reservoir conditions, dyne/cm;
= interfacial tension between laboratory fluids at laboratory conditions, dyne/cm;
= contact angle for reservoir conditions and fluids, deg;
= contact angle for laboratory conditions and fluids, deg.
Example 5.1
Correct the laboratory-measured capillary pressure data to reservoir conditions and determine
the saturation distribution.
Laboratory  (air-water)
(air-water)
Reservoir  (oil-water)
 (oil-water)
w = 65 lb /cu ft
Solution
= 72 dyne/cm
= 0
= 24 dyne/cm
= 30
o = 53 lb /cu ft.
From Eq. (5.8) the conversion from lab to reservoir capillary pressure can be determined.

24 cos 30
Pc(res)  Pc(lab )
 0.289 Pc(lab )

72 cos 0
From Eq. (5.3) the height above the free water level can be estimated.
5. 13
P
P *144
h c  c
 12.0 * P
c(res)

65  53
Table 5.3 shows the results of laboratory capillary pressure measurements using air and
water, corrections to reservoir oil and water, and calculated elevations above zero Pc, and
Figure 5.11 is the plot of height and capillary pressure vs. water saturation.
Sw, %
100
90
80
70
60
50
45
40
35
30
Lab measured pc, airwater (psig)
2
3
4
5
6
7
8
10
27
75
Reservoir pcow oilwater (psig)
0.578
0.867
1.16
1.45
1.73
2.02
2.31
5.8
7.8
21.7
Table 5.3 Laboratory Pressure-Measurement Results
Figure 5.11 Example of Laboratory Pressure Measurement
5. 14
Height above free
water level, ft
6.9
10.4
13.9
17.4
20.8
24.2
27.7
70
94
260
5.2.2 Entry Pressure
The wetting phase (usually water) is preferentially attracted to the rock. The
nonwetting phase (usually oil) does not enter the rock as easily as water. In fact, the oil phase
will not enter the rock without being forced in by increasing pressure. The capillary pressure
required to force the first droplet of oil into the rock is called the entry pressure. As pressure
is increased above the entry pressure, more oil enters the rock, thereby reducing the water
saturation. Note in Figure 5.10 that the free water level is the elevation where
Figure 5.12 Conventional capillary pressure curve
capillary pressure between the oil and water is zero; but the OWC, where oil saturation first
appears, occurs at a shallower depth where the capillary entry pressure is reached. Naturally,
the first oil to enter the rock does so in the largest pore. Oil enters the smaller pores as
capillary pressure increases at elevations farther above the OWC. Figure 5.12 shows entry
pressure, fluid pressures, capillary pressure, and fluid saturations for a typical oil and water
system that is water wet.
5.2.3 Hysteresis - Imbibition versus Drainage
Figure 5.13 shows that the capillary pressure relationship as a function of saturation
depends on the direction of saturation change. The drainage curve is the relationship
followed during normal migration and accumulation of oil in the reservoir. Originally the
reservoir was completely filled with water (water saturation of 100%). The nonwetting oil
phase first enters the reservoir at entry pressure. As the capillary pressure increases, so does
5. 15
the oil saturation. The arrows on the drainage curve show the direction of saturation change.
By convention, because wetting-phase saturation is decreasing, it is called a drainage
process. Water saturation continues to reduce until it reaches irreducible water saturation,
Swi.
For this discussion, assume that the oil reservoir was at irreducible water saturation at
discovery. The imbibition curve is the path followed if water is injected into the rock starting
at irreducible water saturation. This is a common laboratory procedure, and is the case during
waterflooding an oil reservoir. In Figure 5.13, the arrows show the direction of saturation
change. By convention, because wetting-phase saturation is increasing (water is being
imbibed into the rock), the process is called imbibition. The difference in paths between the
drainage an imbibition curves is called hysteresis.
Residual oil saturation - The imbibition curve in Figure 5.13 terminates at water
saturation less than 100% (not all of the oil is displaced from the rock). The amount of oil
trapped in an immobile state at the end of the imbibition process is called the residual oil
saturation, Sor.
Figure 5.13 Imbibition and drainage capillary pressure curves illustrating hysteresis
5.2.4 Permeability Effects
As discussed earlier, the larger the pores, the lower the capillary pressure and
irreducible water saturation. Figure 5.14 shows the effects of permeability or grain-size
distribution on capillary pressure relationships. Table 5.4 summarizes important permeability
effects.
5. 16
Entry pressure
Irreducible water saturation
Slope of transition zone
curve
Grain size distribution
Grain and pore size
HIGH PERMEABILITY
Lower
Lower
Lower
LOW PERMEABILITY
Higher
Higher
Higher
Narrow
Large
Wide
Small
Table 5.4 Permeability Effects
Figure 5.14 Effect of Permeability on Capillary Pressure
5.2.5 Capillary Pressure Measurement
There are three common methods for measuring capillary pressure as a function of water
saturation in porous media: porous plate, mercury injection, and centrifuge.
Mercury Injection
A dry core sample is placed in the sample chamber of the mercury injection apparatus. The
test sample is evacuated and Mercury is injected in measured increments into the sample
while the pressure required for injection of each increment is recorded (Figure 5.15). The
volume of mercury injected into the core is noted. The procedure is repeated for a number of
intervals until a pressure of about 800 psi is attained.
5. 17
Fig. 22 Mercury injection equipment
Pressure
gauge
Sample
Displacement reading
Pump
Mercury
Figure 5.15 Schematic of mercury injection apparatus
The volume of mercury injected corresponds to the non-wetting phase volume, because
mercury is a non-wetting fluid, whereas the mercury vapor corresponds to a wetting phase.
The incremental pore volumes of mercury injected are plotted as a function of the injection
pressure to obtain the injection capillary pressure curve (Figure 5.16, curve 1). When the
volume of mercury injected reaches a limit with respect to pressure increase (Simax), a
mercury withdrawal capillary pressure curve can be obtained by decreasing the pressure in
increments and recording the volume of mercury withdrawn (Figure 5.16, curve 2). A limit
will be approached where mercury ceases to be withdrawn as the pressure approaches zero
(Swmin). A third capillary pressure curve is obtained if mercury is re-injected by increasing
the pressure incrementally from zero to the maximum pressure at Simax (Figure 5.16, curve
3).
The closed loop of the withdrawal and re-injection curves (2 and 3, Figure 5.16) is the
characteristic capillary pressure hysteresis loop. Mercury is a non- wetting fluid; therefore,
the hysteresis loop exhibits a positive pressure for all saturations-that is, the hysteresis loop is
above the zero pressure line.
5. 18
Figure 5.16 Example mercury-air capillary pressure curves. Note threshold pressure and
hysteresis loop.
Capillary pressure curves for rocks have been determined by mercury injection and
withdrawal because the method is simple to conduct and rapid. The data can be used to
determine the pore size distribution, to study the behavior of capillary pressure curves, and to
infer characteristics of pore geometry. In addition, mercury injection capillary pressure data
of water-oil systems are in good agreement with the strongly water-wet capillary pressure
curves obtained by other methods.
The mercury injection method has two disadvantages:
1. after mercury is injected into a core, it cannot be used for any other tests because the
mercury cannot be safety removed, and
2. mercury vapor is toxic, so strict safety precautions must be followed when using
mercury.
Nevertheless, because it offers a rapid method for obtaining capillary pressure data on
irregular-shaped samples (any shape can be used), it remains as one of the standard
petrophysical tests.
5. 19
Porous Diaphragm Method
A second method is the use of a porous diaphragm, which relies on the selection of a suitable
porous disk to provide a barrier that excludes the passage of the non-wetting fluid and
permits the passage of the wetting fluid. Porcelains can be made with low permeability (< 5
mD) and very uniform pore openings; therefore, it is the most frequently used material for
the porous diaphragm. When water-wet systems are used, the diaphragm is saturated with
water and a core is placed on it in good capillary contact. Then, the pressure exerted in the
non-wetting phase (gas or oil) is recorded for incremental displacements of water. Saturating
the disk with the non-wetting phase and recording the pressure required to displace
incremental volumes of the non-wetting fluid also may reverse the displacement.
A schematic diagram of the Ruska Diaphragm Pressure Cell is presented in Figure 5.17.
Figure 5.17 Schematic of a Ruska diaphragm pressure cell
A core is saturated with water containing salts to stabilize the clay minerals, which tend to
swell and dislodge when in contact with freshwater. The saturated core is then placed on a
porous disk, which also is saturated with water. The porous disk has finer pores than does the
rock sample. (The permeability of the disk should be at least 10 times lower than the
permeability of the core.) The pore sizes of the porous disk should be small enough to
5. 20
prevent penetration of the displacing fluid until the water saturation in the core has reached
its irreducible value.
The pressure of the displacing fluid is increased in small increments. After each increase of
pressure, the amount of water displaced is monitored until it reaches static equilibrium. The
average water saturations of the core can be calculated by dividing the volume of fluid
displaced by the pore volume. The capillary pressure is plotted as a function of water
saturation as shown in Figures 5.18. If the pore surfaces are preferentially wet by water, a
finite pressure (the threshold pressure, PD) will be required before any of the water is
displaced from the core. The displacement process can be reversed, resulting in curve 2 in
Figure 5.18. In this case, for a preferentially water wet sample, water will imbibe into the
core and displace the oil towards the residual oil saturation.
Figure 5.18 Example of capillary pressure curve for a water-wet system.
The drawback of this method is the long period of time required for completion of the test
(sometimes several weeks), because one must wait for completion of the incremental fluid
displacement after each increase of pressure.
Centrifuge Method
The third method is the centrifuge method. Cores are placed in specially designed holders
equipped to collect either water or oil in a calibrated portion of the coreholder. A centrifuge
5. 21
is then used to displace one of the fluids by centrifugal force (Figure 5.19). The angular
velocity of the centrifuge (revolutions per minute) is increased in increments, and the amount
of fluid displaced at each incremental velocity is measured. The capillary pressure is
calculated from the centrifugal force.
Fig. 23 Centrifuge
measurement
rotor
rotation
sample
rotor
axis
water
produced in
collection tube
Figure
5.19plays
Capillary
measurement
by centrifuge
Centrifugal
force
rolepressure
of gravitational
force
Only the
initial displacement
of fluid
irreducible force
saturation was discussed for the
Increasing
rotation speed
--->toincreasing
preceding
methods.
Mercury
---->
increasing
Pc injection may be used for displacement of air, in which case
the capillary pressure is generally plotted as a function of mercury saturation; or it may be
increasing
water
used forGives
mercury
displacement
of aproduction
wetting fluid --->
suchS
aswwater, and in this case the capillary
pressure is plotted against the wetting phase saturation. The porous diaphragm method may
be used for displacement of a wetting phase by a gas, or the core (saturated with the wetting
fluid) may be covered with a non-wetting phase and gas pressure used to cause displacement
of the wetting phase by the non-wetting phase. All of these yield Pc curve 1.
The centrifuge may be used to obtain three capillary pressure displacement curves (Figure
5.20):
a. Initial displacement of a fluid to irreducible saturation. For example, assuming the core is
initially saturated with water, oil is used to displace the water from Sw = 100% to irreducible
saturation. (Curve 1 of Figure 5.20).
b. Displacement of the oil with water to the water saturation equivalent to the residual oil
saturation. (Curve 3 of Figure 5.20).
5. 22
c. Repeat the displacement of the water to irreducible water saturation once more by starting
with the core saturated with water and oil at saturation Swor. (Curve 5 of Figure 5.20)
Figure 5.20 Example capillary pressure curves from centrifugal data. Curves 2 and 4 are
estimated because they typically cannot be determined by centrifuge.
The centrifugal force affecting the core varies along the length of the core. Thus the
capillary pressure and the water saturation also vary along the entire core length. (Figure
5.21) The capillary pressure at any position in the core is the difference in hydrostatic
pressure between the two phases. At the outlet end of the core the capillary pressure is
assumed to equal zero for all centrifuge speeds, based on the core remaining at 100% water
saturation.
5. 23
Figure 5.21 Schematic illustrating the variation of pressure and water saturation as a function
of core length.
At the inlet end, (Pc)i, is defined as:
L

( Pc ) i  1.096 x10  6 N 2  re   L
2

(5.19)
where
(Pc)i

N
re
L
= inlet capillary pressure, kPa
= density difference in gm/cc
= revolutions per minute
= outer radius of core measured from center of rotation, cm
= core length, cm
The water saturation measured is the average water saturation in the core at the time of
measurement.
However, the capillary pressure from Equation (5.19) requires an inlet
saturation. An approximation for the inlet saturation is:
Si  S  ( Pc ) i *
dS
d ( Pc ) i
(5.20)
For Equation (5.20), it is assumed that the length of core is negligible with respect to the
radius of rotation of the centrifuge, a valid assumption for ri/re ratios > 0.7.
To apply equation (5.20), the derivative term must be evaluated. Using experimental data to
determine the derivative will amplify the error inherent in experimental data; therefore, a
5. 24
hyperbolic least squares solution has been developed to remove the noise in the data. The
resulting equation for inlet capillary pressure is:
( Pc ) i 
A  BS
1  CS
d ( Pc ) i

(5.21)
and for the derivative:
dS
B  AC
1  CS 2
(5.22)
where A, B, and C are regression constants.
The following table compares the limitations and advantages of the various capillary pressure
Fig.
24 Measuring
measurement
techniques.
Capillary Pressure
Equilibrium
Air/Hg
Centrifuge
5 weeks
1day
3 days/run
30/60
7000/14500
80/160
At stress?
Yes
Yes
No
On cuttings?
No
Yes
No
Sample damaged?
No
Yes
Weak only
Unconsolidated
Yes
Yes
Yes?
Equilibrium reached?
Yes
Yes
Nearly
Clay correction
required
No
Yes
No
Costs
Expensive
Cheap
Medium
Additional
Information
Imbibition
RI
Imbibition
Imbibition
Wettability
Duration
Max Height (m gas/oil)
Example 5.2
Experimental data for an air displacing water capillary pressure experiment is shown in Table
1. Additional information is:
L
= 2.0 cm
d
= 2.53 cm
Vp
= 1.73 cm3
k
= 144 md
Centrifuge arm
= 8.6 cm

= 0.9988 gm/cc

= 0.17
5. 25
Prepare the inlet saturation vs Pc curve for this experiment.
Air displacing water
N, rpm
Vdis(wtr)
rpm
cc
1300
0.30
1410
0.40
1550
0.50
1700
0.60
1840
0.70
2010
0.75
2200
0.80
2500
0.90
2740
1.00
3120
1.05
3810
1.10
4510
1.20
5690
1.25
Table 1. Centrifuge data
5. 26
5.2.6 Averaging of Capillary Pressure data
As capillary pressure data are obtained on small core samples which represent a
minor portion of a reservoir, it is necessary to combine all the capillary data to classify a
particular reservoir.
Since core data from a reservoir typically indicates variability of
permeability and porosity, these parameters should be included in the averaging technique.
For example, the figure below relates water saturation to permeability and height above an
oil-water contact in the reservoir. This information is useful in estimating hydrocarbons in
place.
Figure 5.22 Correlation of connate water saturation vs permeability and height.
Various mathematical equations are available to fit a laboratory measured capillary pressure
curve for the purpose to produce a saturation-height function.
• Averaging curve fit parameters (e.g. a, b,  vs. )
5. 27
Of these the lambda-fit normally works best (shell recommendation). It fits
the wetting saturation Sw to the capillary pressure Pc using three fit constants,
a, b and , according to: Sw = a.Pc- + b
• Interpolation within data set
• Leverett-J
• Neural networks
• Regression (linear, non-linear, multi-variate)
In 1941, Leverett published an approach correlating height of water saturation in the
reservoir for different values of capillary pressure, permeability, porosity, interfacial tension,
and contact angle. The J-function is a dimensionless capillary pressure, defined as
Pc
J (S ) 
w
 cos 
k

(5.23)
where:
J(Sw) = Leverett's J-function;
k
= rock permeability, md;

= rock porosity, fraction;
Cos = lab values.
During core analysis, capillary pressure curves are usually measured for several
different reservoir rock samples. Naturally, each rock sample that has a different pore-size
distribution, permeability, and porosity will yield a different capillary-pressure curve.
Leverett's J-function is used to normalize the group of varying capillary curves to obtain a
single relationship between J(Sw) and water saturation. The following figures illustrate the
variation in capillary pressure curves and the reduction to a single J-function curve.
5. 28
Figure 5.23 Set of capillary pressure curves for the D sands
Figure 5.24 Resulting J-function curve for the D-sands
Once the curve of J(Sw) versus water saturation curve is determined, it can be used to
obtain the height of any water saturation above the free water level
h
J ( S w ) cos  res
 w   o 

k  res
where,
5. 29
(5.24)
h
= height of a particular water saturation above the free water level, ft;
(w - o)
= density gradient difference between oil and water, psi/ft;
The procedure for using Leverett's approach is:
(1) calculate J(Sw) for each capillary pressure point using Equation 5.23,
(2) plot J(Sw) versus Sw and draw a smooth curve through the points,
(3) calculate h from Equation 5.24 for each Sw, for any set of k and ;
(4) plot h versus Sw.
Figure 5.25 is a typical plot of J(Sw) versus Sw obtained from Equation 5.23. The
results of Equation 5.24 are plotted in Figure 5.26 for different values of permeability. In this
manner, common variations in permeability and their effects on water saturation are
accounted for.
Figure 5.25 Leverett’s J-Function vs. Water Saturation
5. 30
Figure 5.26 Variation in Saturation Height for Different Permeabilities
5.2.7 Pore Size Distribution
As shown previously, capillary pressure is a function of wettability and the radius of
the capillary tube. However, porous media is composed of interconnected capillaries of
various pore throat sizes. Therefore, a capillary pressure curve represents this variation, from
large pore throats at low capillary pressures to smaller pore throats as capillary pressure
increases. The minimum value will occur at the irreducible water saturation. Computations
of pore size distributions and/or permeability have been obtained from mercury injection,
capillary pressure data.
Purcell (1949) used the concept of pore size distribution without directly evaluating
the distribution to estimate permeability. The concept is based on analogy with CarmenKozeny’s equation.
Recall from the Carmen-Kozeny, k=f(porosity, PSD).
Capillary
pressure, Pc = f(wettability, saturation), where saturation is a function of pore geometry.
Thus a Pc – Sw curve relates the pore size penetrated by the non-wetting fluid at a given
capillary pressure.
5. 31
The minimum capillary pressure to displace the wetting phase in a capillary tube is given by,
2 cos 
Pc 
r
Poiseuille’s equation for flow rate in a single tube of radius, r, is:
r 4 p
q
8La
The volume of a capillary tube is V = r2La, thus substituting in Eq. (5.26) results in,
Vr 2 p
q
8L2a
Rearrange Eq. (5.25) in terms of radius, and substituting into Eq. (5.27),
 cos  2 p  V
q
2L2a
Pc2
(5.25)
(5.26)
(5.27)
(5.28)
Consider the porous media as “n” capillary tubes of equal length but random radius, then
Poiseuille’s equation becomes,
p m
4
(5.29)
q
 nr
8La i 1 i i
and the volume,
m
(5.30)
Vi  La  ni ri2
i 1
The resulting total flow rate is

 cos  2 p
q

2L2a
m Vi

i 1 Pci2
(5.31)
The macroscopic flow rate in porous media can be described by Darcy’s Law.
kA p
q
(5.32)
 L
Thus combining Eqs. (5.31) and (5.32) results in a relationship of k =f(Vp, Pc).
 cos  2 L  m Vi
k
(5.33)

2
2
2 ALa
i 1 Pci
The volume of each tube as a fraction of the total void volume can be expressed as a
saturation, Si = Vi/VT. Substituting Si in Eq. (5.33) and porosity, results in,
k
 cos  2  
2
5. 32
m Si

i 1 Pci2
(5.34)
Purcell introduced a “lithology factor”, , to account for the deviation of the actual pore
space from the simple capillary tube model.
This factor was determined by adjusting
calculated permeabilities from Eq. (5.34) to match observed air permeabilities. For 27
sandstone samples, the average lithology factor was 0.216, with a minimum of 0.085 to a
maximum of 0.363.
Introducing conversion factors, Eq. (5.34) can be expressed in a more applicable form.
S 1 dS
(5.35)
k  10.24 cos  2   
2
S  0 Pc
where
k
= md

= dynes/cm
Pc
= psi
Comparing Eq. (5.35) with the Carmen-Kozeny equation reveals that the Kozeny factor, k z =
1/. The Spv can be expressed as:
1
S pv 
(5.36)
S 1 S
 cos  2   i
2
S  0 Pci
For mercury injection, Eq. (5.35) can be further reduced to,
S 1 dS
k  14,260  
(5.37)
2
S  0 Pc
where the contact angle for air-mercury is assumed to be 140º and the interfacial tension is
480 dynes/cm.
A plot of Pc and 1/Pc2 vs wetting phase saturation is shown in Figure 5.27. The integral can
be evaluated by either numerical integration of the area below the curve, or by developing a
regression fit equation for the function and then analytically integrating.
5. 33
Figure 5.27 Graphical presentation of capillary pressure data for calculating permeability
Burdine (1950) developed a method to determine pore size distribution from
capillary pressure data and calculate permeability from this distribution. The distribution
function, D(ri), is defined as:
dV p  D(ri) dr  V p dS w
(5.38)
where dVp is the total volume of pores having a radius between the pore entry radius, ri and
ri-dr. Differentiating and rearranging Eq. (5.38) results in,
 r2 
dP
dr  
 2 cos   c


(5.39)
dS
 2 cos  
 V  w
D(ri)  
 p dP
 r2 
c
(5.40)
V p dS
D(ri)  Pc 
 w
r dPc
(5.41)
Substituting Eq. (5.39) into (5.38),
or
To determine the P.S.D. first calculate the pore radius at given values of capillary pressure.
ri 
2 cos 
Pci
(5.42)
Next, evaluate the distribution function, D(ri), using Eq. (5.40), where the slope was
calculated previously. Figure 5.28 is an example of a typical distribution curve. The area
under the curve at a particular radius represents the fraction of the volume with pores larger
than the given radius.
5. 34
Pore size distribution, m2
Pore radius, microns
0
Figure 5.28 Typical distribution function
Permeability is estimated by analogy with the bundle of capillary tubes model,
4
100V p
n S  ri
(5.43)
k

 7 i 1 2 2
8(9.87 x10 )
xi  ri
where ri is pore entry radius in cm, k is in Darcies, and xi is a dividing factor to account for
the complex geometry of porous media.
It was empirically derived by comparing computed
permeabilities with measured ones (see Figure 5.29). As seen in the figure, the dividing
factor approaches a value of 4.00 for permeabilities greater than 100 md, and therefore is
assumed constant for high permeabilities. Also note, the high degree of uncertainty for low
permeabilities, suggesting this factor is inadequate in these cases.
Figure 5.29 Dividing factor correlation developed by Burdine.
5. 35
r
i
( j 1)
r 
i
2
r
i
j
2
Term  S air  ri
5.2.8 Rock type and Pc Curves
As mentioned previously, the capillary pressure curve is a strong function of pore
radius and hence, rock type. Subsequently, the saturation-height function is also dependent
on rock type. The figure below illustrates the impact of various rock types (Pc curves) on the
saturation-height response. The result is multiple oil-water contacts and pay zones.
Figure 5.30 Schematic illustrating saturation-height dependency on rock type
A comparison of saturations between capillary pressure derived from a core sample
and log-derived water saturation is shown in the following figure. Either one can be used to
calibrate the other. Causes of errors in capillary pressure curves are:

Alteration of wettability by invasion of drilling mud filtrate

Biased sampling

Sample integrity

Effect of core cleaning on wettability
5. 36

Laboratory measurement and appropriate corrections for temperature, stress, and
clays

Averaging
Similarly, causes of errors in log calculations are:

tool calibrations and quality control

Invasion, thin bed and borehole effects

Application of the correct interpretation model

Assumptions or validity of m, n, or a
Also, the difference in “scale” will impact the degree of agreement between core-derived and
log-derived saturation-height curves.
Figure 5.31 Log vs core water saturation vs height for a Middle East carbonate
5. 37
5.3 Relative Permeability
Discussions of permeability in Chapter 3.0 were concerned only with the absolute
permeability of the rock. By definition, absolute permeability is the capacity of the rock to
transmit fluid, when that fluid is the only one present (when a particular fluid saturation is
100%). Absolute permeability is an intrinsic property of the rock and does not depend on
whether the fluid is oil, water, gas, or air. To determine absolute permeability correctly and
accurately, Chapter 3.0 pointed out that some phenomena must be accounted for, such as
water-sensitive clay swelling, particle movement and plugging, as well as Klinkenberg's
effect when using a gas.
Applications of relative permeability data are:
1. to model a particular process, for example, fractional flow, fluid distributions,
recovery and predictions
2. Determination of the free water surface; i.e., the level of zero capillary pressure or the
level below which fluid production is 100% water.
3. Determination of residual fluid saturations
When there is more than one fluid in the rock, (normally the case in real reservoirs),
permeability to each phase is less than the rock's absolute permeability. Permeability to each
phase is called effective permeability (in millidarcies, as is absolute permeability). The sum
of effective permeabilities of all fluids in the rock at any time is always less than the absolute
permeability:
n
 k eff  k abs
i
i 1
(5.44)
where:
keff
= effective permeability to each fluid phase (oil, gas, or water), md;
kabs
= absolute permeability. md;
n
= total number of phases present; in the laboratory, n can be 1 or more; in the
reservoir, n is generally 2 or 3.
The sum of effective permeabilities is less than the absolute permeability because the
fluids are immiscible, thus interfacial tension exists between the phases. The presence of
more than one fluid in the same pores causes additional resistance to flow for both fluids.
5. 38
Relative permeability is the ratio of the effective permeability for a particular fluid to
a reference or base permeability of the rock.
This reference could be the absolute
permeability to water, effective permeability to oil at Swi or air permeability.
k
eff
k 
r k
ref
(5.45)
As an example, consider the effective permeability equal to 50 md. at some saturation. The
core has measured base permeabilities of kbrine = 115 md, kair = 120 md and ko @ Swi = 85
md, respectively. The corresponding relative permeabilities are 0.43, 0.42, and 0.59.
The absolute permeability of the rock is generally written as k. The effective
permeability to a particular phase has subscripts as follows: ko is effective permeability to oil,
md; kw is effective permeability to water, md; and kg is the effective permeability to gas, md.
Consequently, we typically denote relative permeability to a particular phase also
with subscripts, where kro is relative permeability to oil; krw is relative permeability to water,
krg relative permeability to gas. Therefore, to be more specific, the relative permeabilities to
each of the three fluids can be written as,
k
k ro  o
k
kw
k

rw
k
k rg 
(5.46)
kg
k
Because effective permeability is always less than or equal to absolute permeability,
relative permeability varies between 0 and 1 and the sum of the relative permeabilities for all
phases in the same rock at the same time must be less than or equal to 1.
n
 k r  1.0
i 1 i
(5.47)
or
k
ro
k
rw
k
rg
 0 to 1
(5.48)
Although many references and some core analysis reports show kro and krw both equal
to 1.0 at 100% of their respective saturations, in actual practice, this is frequently not the
5. 39
case. In normal water-wet systems; krw seldom reaches 1.0 even at Sw =100%; usually
because of water- sensitive clays in the core. Also kro frequently does not reach 1.0 at So =
100% because of the effects of laboratory analytical procedures and water-sensitive clays.
5.3.1 Two-phase relative permeability
In this section, consider a two-phase system of oil and water. Also, consider the
system to be water wet. Figure 5.32 shows a typical set of relative permeability curves for
this case.
Figure 5.32 Typical water-wet, oil-water relative permeability curves
In the figure, kro is the nonwetting phase and has the curve shape characteristic of the
non- wetting phase (S-shaped). The typical shape of the wetting phase is that of the krw curve
(concave upward throughout). Preferential wettability is also shown by the crossover of the
two curves. In Figure 5.32, the curves cross at a water saturation greater than 50%, which
indicates a system that is probably water wet. If the curves had crossed at a water saturation
less than 50%, the system would probably be preferentially oil wet, and the shapes of the
curves would be reversed. It is common for laboratory-measured relative-permeability curves
to cross very near the 50% water saturation and the shape of both curves to be similarly
concave up. This would indicate intermediate wettability; however, no hard and fast rules can
be established. Each system must be investigated individually, and the discussion above is
5. 40
only a generalization. Residual oil saturation, irreducible water saturation, curve shapes, and
crossover points must all be considered when investigating wettability.
Figure 5.33 is an
example of an oil-wet system for comparison.
Figure 5.33 Typical oil-wet relative permeability curves
5.3.2 Hysteresis -- Imbibition Versus Drainage
The previous discussion of hysteresis in capillary pressure also pertains to relative
permeability. Figure 5.34 shows the characteristics of drainage and imbibition relativepermeability curves and the resulting hysteresis.
Figure 5.34 Relative permeability hysteresis, imbibition vs. drainage
5. 41
The drainage curve in Figure 5.34 represents the nonwetting phase (oil) displacing the
wetting phase (water) from the rock, which occurs during oil migration and accumulation in
the reservoir. The arrows on the curve show the direction of saturation change. For this
discussion, consider that drainage continues to the irreducible water saturation (Swi = 20% in
Figure 5.34). Now consider that water is injected into the reservoir (or laboratory rock
sample) starting at the irreducible water saturation. As water saturation increases, krw almost
follows the same curve (i.e., there is very little hysteresis between the k rw imbibition and
drainage processes). However, the kro imbibition curve deviates noticeably from the drainage
curve. There is significant hysteresis as kro goes to zero at a relatively high oil saturation,
called the residual oil saturation (Sor). In Figure 5.34, Sor appears to be 25%, which means
that waterflooding or natural water influx from an aquifer would displace oil down to a
minimum saturation of 25%. Because the water and oil are immiscible, oil saturation cannot
be reduced below the residual value.
The relative-permeability curves in Figure 5.32 were for the imbibition process. For
most reservoir engineering calculations, water is displacing oil, and the relative-permeability
curves must represent the imbibition process. Care must be taken to ensure that laboratory
measurements are conducted appropriately.
5.3.3 Gas Relative Permeability
During two-phase flow of gas and oil and two-phase flow of gas and water, the gas
acts as the nonwetting phase, and the gas relative-permeability curve normally has the typical
nonwetting-phase S-shape. The hysteresis effect in gas relative permeability is similar to that
in oil-water systems. Gas saturation and relative permeability can be reduced by water or oil
displacement down to an immobile saturation when krg = 0, called the residual or trapped-gas
saturation. This can occur at values of 15 to 40% gas saturation, representing a large amount
of gas that may be trapped in a water-drive gas reservoir.
Critical gas saturation is the value at which gas first begins to flow; i.e.,when gas is
evolving from solution in an oil reservoir when pressure has fallen below bubble-point
pressure during normal production. The first few bubbles of gas coming from solution in the
oil are immobile and remain in the pores in which they evolved. Gas saturation increases as
pressure continues to drop below the bubble point, until the gas phase becomes continuous
5. 42
among many pores. The gas mobilizes at a saturation called the critical or equilibrium
saturation (Sgc), which ranges between 0 and 10%, but generally is 2 to 5% in magnitude.
In summary;
Sgc
= critical gas saturation, when gas first becomes mobile, generally at Sg = 2 to
5%, always between 0 and 10% this would be measured during a drainage
process);
Sgr = Sgt
= residual or trapped gas saturation, when gas can no longer flow
because its saturation is being reduced during an imbibition process,
generally at values between 15 and 40%.
Figure 5.35 shows typical relative- permeability curves for two-phase gas and oil flow.
Figure 5.35 Typical Gas and Oil Relative-Permeability Curves
Example 5.3
Using the oil and water relative-permeability curves in Figure 5.36, determine the effective
permeabilities for oil and water at a water saturation of 50%. Absolute permeability of the
rock is 214 md.
Solution
Figure 5.20 shows that at Sw = 50%,
kro = .14 and krw = .09
Therefore, using Eq. (5.46),
5. 43
ko  kro * k  (0.14)(214 md )  30.0 md.
kw  krw * k  (0.09)(214)  19.3md.
Figure 5.36 Example – oil-water relative permeability curves
5.3.4 Effects of Relative Permeability on Flow Rates
Chapter 3.0 introduced Darcy's Law, which permits flow rate to be calculated from
reservoir conditions. Recall, for simple linear steady-state flow of incompressible fluids,
q  0.001127
where,
q
= flow rate, STB/D;
k
= absolute permeability of rock, md;
A
= cross-sectional area of flow, sq ft;
5. 44
kAp
BL
(5.49)
p
= pressure drop, psi;

= fluid viscosity, cp;
B
= formation volume factor to convert conditions from reservoir to stock tank,
RB/STB;
L
= length of flow system between two measured pressures, ft.
Equation (5.49) is applicable for only a single phase flowing in the porous media. If two
phases are simultaneously flowing then the flow rate is reduced proportionately by the
relative permeability. Therefore the oil and water flow rates are respectively,
q o  0.001127
kk ro AP
q w  0.001127
 o Bo L
kk rw AP
(5.50)
 w Bw L
Example 5.4
Consider steady state radial flow in an undersaturated oil reservoir with an active water
drive. A set of oil-water relative permeability curves were obtained from a core sample and
are given in Figure 5.36. Other information are:
h
= 10 ft
p
= 500 psi
re
= 660 ft
o
= 3 cp
rw
= 0.5 ft
w
= 1 cp
Find:
a. The oil and water flow rates at initial reservoir state @ Swi.
b.
Over time the water encroaches as oil is produced. At Sw = So = 50% what is the
oil and water flow rates?
Solution
a. At the initial reservoir state only a single phase is flowing, therefore the radial flow
equation can be used to determine production rate.
q 
o
k o hp
r


141.2 ln  e
o
rw 



k o (10)(500)

141.2(3) ln 660
 1.642k
o

.5
5. 45
Since the permeability reference is air, ko = k*kro=214*.75=160.5 md. Substitution for
effective permeability results in qo = 299 bopd. The water flow rate equal to zero since
krw = 0.
b. From Example 5.2, relative permeability at 50 % saturation are krw = 0.09 and kro = 0.14.
The subsequent flow rates are:
q o  4.927
kkro
214 * .14
 4.927
 49bopd
o
3
q w  4.927
kkrw
214 * .09
 4.927
 95bopd
w
1
5.3.5 Methods of Generating Relative-Permeability Data
It is best to obtain relative-permeability curves from laboratory measurements of twophase flow. Two techniques widely applied are steady state and unsteady state methods;
however, only the steady state method will be described here.
The steady state measurement technique begins with the core sample completely
saturated with one of the phases, typically the oil phase. After equilibrium is attained the
effective permeability is obtained. The next step is to inject two fluids into the core at a predetermined water-oil ratio. When the effluent WOR is equal to the injected WOR and the
pressure drop remains constant then steady state is achieved in the core. Recorded are the
pressure drop, produced volumes of oil and water and the time.
Also, this data must
correspond to an average saturation within the core; therefore additionally during each step
the core is removed and saturation determined.
Either resistivity or volume balance
measurements are applied to determine the saturation. By volume balance,
W W
V * 
sat
dry
p o
Sw 
V p * w
(5.51)
The core is placed back in the holder and the procedure repeated at a different WOR.
In many cases, special core analysis is not practical or possible to accomplish.
However, curves are often needed for engineering calculations when there are no laboratory
data for the reservoir rock of interest. In such cases, the engineer can resort to correlations
based an empirical measurements.
Gas-oil correlations - Arps and Roberts published sets of gas and oil relative-permeability
correlations in 1955.
They are helpful when no other data exists, especially when only
5. 46
rough estimates of maximum or minimum recovery are needed. In 1965, Knopp published a
similar set of gas and oil relative-permeability curves that are more generally applicable than
Arps and Roberts' curves. In 1954, Corey presented a set of equations for calculating gas and
oil relative permeabilities. His approach has proven to be quite popular in the absence of
measured data. Corey's equations are:
k ro


S  S 4
Lr
 L

S  S Lr
k rg  1  L
 S m  S Lr
1 S
Lr
 4 * 1   SL  SLr  2 




(5.52)
 1 S
 
Lr  


(5.53)
where:
SL
= total liquid saturation, So + Sw;
SLr
= total residual liquid saturation, Sor + Swi;
Sm
= 1 - Sgc.
Oil-water correlations - Many attempts have been made to develop correlations or equations
to estimate oil and water relative permeabilities in the absence of measured values. None of
the published techniques have proved to be reliable enough to be used for more other than
theoretical calculations. Smith discusses several of the approaches.
Relative-Permeability from production records - Both gas-oil and oil-water relative
permeability relationships can be estimated by observing a reservoir's producing
characteristics. For example, the producing gas/oil ratio (GOR) of a reservoir is related to
relative permeability as follows.
k rg  B
o o R
R
so
k  B
ro g g
where:
R
= measured producing GOR, scf/STB;
Rso
= solution GOR, scf/STB,
B
= formation volume factor;

= viscosity, cp.
5. 47
(5.54)
The relative permeability ratio (krg/kro) can be estimated from Equation 5.54 if producing
GOR and PVT properties are known. This approach is often used to estimate gas and oil
relative- permeability ratios during a reservoir's producing life. Future producing GORs can
be predicted by extrapolating the past trend of krg/kro to future estimates of gas saturation.
Oil and water relative-permeability ratios can be related to observed producing
water/oil ratios by an equation similar to Equation 5.45, and the krw/kro ratio can be estimated.
However, this approach to water/oil ratios is less reliable and less often used for water/oil
systems than it is for gas/oil flow systems.
5.3.6 Three-Phase Relative Permeability
So far in this chapter, discussions have involved only two-phase flow: either oil and
gas, oil and water, or gas and water. In many natural reservoir situations, all three fluid
phases are flowing simultaneously. Analytical description of three-phase relative
permeability is extremely difficult.
The two-phase relative permeability curves presented earlier considered only two
fluids. For reservoirs under waterflood or active water drive above bubble-point pressure,
two-phase oil and water relative-permeability curves are sufficient. Also, in solution gasdrive oil reservoirs where water saturation is immobile, two-phase oil and gas relativepermeability curves are sufficient. In Figure 5.35, gas and oil relative permeabilities are
plotted versus total liquid saturation (i.e., irreducible water saturation, whatever its value, is
added to the variable oil saturation to arrive at total liquid saturation). For cases in which all
three fluid phases are mobile and flowing in the reservoir, the two-phase relative
permeability curves shown earlier cannot fully describe the phenomena.
The first reported results of research on the characteristics of three-phase relative
permeability were by Leverett and Lewis in 1941. They used only unconsolidated sand and
did not account for hysteresis and other effects that are now known to require special
treatment. Although their results were not accurate enough for use in engineering design for
specific reservoirs, they were the first to show the behavior of three-phase relative permeability. They demonstrated that significant three-phase flow should occur only in a rather
small saturation interval; and, because one phase would most likely be increasing in
saturation at the expense of one or both of the others, the difficulties of understanding and
describing three-phase flow should be short lived.
5. 48
Later investigators used various
consolidated cores and minimized or eliminated some of the effects that prevented accurate
measurements. However, none of the reported results proved to be of much use until Stone
published his method.
Stone's method was first published in 1970, then revised and improved in 1973. It is a
probability model that uses two sets of two-phase relative- permeability data to estimate
three-phase behavior. The two-phase data are water displacing oil and gas displacing oil.
Along with estimating three-phase relative-permeability behavior, the model estimates
residual oil saturation as a function of trapped gas saturation.
The steps for using Stone's method are:
1.
Sw is used to obtain krw and krow from the two-phase water-oil relativepermeability curves;
2.
Sg is used to obtain krg and krog from the two-phase gas-oil curves;
3.
Three-phase relative permeabilities are obtained.as krw is used directly from
step 1, krg is used directly from step 2, and the three-phase kro is calculated
from


k ro  k row  k rw  k rog  k rg    k rw  k rg 

 

(5.55)
where:
kro
= three-phase relative permeability to oil;
krw
= two-phase and three-phase relative Permeability to water,
krg
= two-phase and three-phase relative Permeability to gas;
krow
= two-phase relative permeability to oil from step 1;
krog
= two-phase relative permeability to oil from step 2;
If Equation 5.55 calculates a negative value for kro it should be ignored and kro = 0. A
basic assumption in Stone's model is that krw and krg are the same functions in the three-phase
system as they are in two-phase systems. Only kro is a function of both water and gas
saturations. Experimental observations show that these assumptions are valid for most
reservoir situations.
Stone's method is widely accepted and popular, especially in numerical reservoir
simulation models. Some modifications and improvement were published by Dietrich and
Bondor in 1976. Their paper is an excellent reference for three-phase relative-permeability
5. 49
research. It presents modifications to Stone's method based on two models that can be tested
to determine which is more reasonable and suitable.
5.3.7 Average Relative-Permeability Data
Relative-permeability data are generally measured and reported in a laboratory core
analysis of several core samples from a particular well. If data were measured on cores from
several wells, there could easily be ten or more different sets of relative-permeability curves.
In conventional engineering calculations (not numerical reservoir simulation) only one set of
relative-permeability curves can be accommodated. Therefore, the group of curves drawn
from the core analysis must be averaged to determine the most representative set.
There are numerous papers explaining how to determine the most representative relativepermeability curves from a group obtained from core analyses of several core samples. The
literature contains many references to methods that work well for particular reservoir
conditions. However, there are so many specific reservoir multiphase flow problems, and
such a variety of reservoir characteristics, that the number of useful averaging methods is too
large to review. Three more common and generally applicable methods are discussed here.
Simple average. This approach is only valid if the initial saturation of each core sample is
the same as the average initial saturation in the reservoir. The method is generally applied
only to relative-permeability ratio data (krg/kro or krw/kro). For krg/kro curves, the steps are:
1. For each curve, Sg is read at selected values of krg/kro
2. Sg 's from the set of curves are averaged at the selected krg/kro;
3. Krg/kro is plotted versus the average Sg,
4. A smooth curve is drawn through the data to obtain the averaged set of krg/kro.
Normalization of curves. This method requires a value for Sor, which can be difficult to
obtain. If a reliable determination of Sor cannot be made, this method is not recommended.
The steps are:
1. Saturation is redefined to range between 0 and 1;
2. Normalized water saturation in the water-oil system is,
*
Sw 
S w  S wi
1  S or  S wi
where:
Swi = irreducible water saturation, fraction;
5. 50
(5.56)
Sw* = 0 for Sw = Swi;
Sw* = 1.0 for Sw = 1 - Sor
3. The normalized oil saturation in the gas- oil system is,
S S S
* S  S or
oi
g
or
So  o

S S
S S
oi
or
oi
or
(5.57)
and
S
*
*
 1 S
g
o
(5.58)
The result is a set of curves that have been normalized to the same Swi and Sor. Simple
averaging at values of Sw or Sg can be used to generate one set of curves.
Correlation with irreducible water saturation. This method results in average curves for
any irreducible water saturation that may exist in the reservoir. For the example of k rg/kro
curves, the steps are:
1. Sg is read at selected values of krg/kro for each curve;
2. Sg 's from the set of curves are averaged at the selected krg/kro,
3. Values of Sg for a given krg/kro are plotted versus irreducible water saturation for
each core sample;
4. Straight lines are drawn through the data for each value of krg/kro as shown in
Figure 5.21.
5. Average water saturation is determined for the reservoir or a zone of interest;
6. For each krg/kro value in Figure 5.37 Sg is read at the average Sw determined in
step 5;
7. krg /kro values are plotted versus Sg values, and a smooth curve is drawn through
the data. The result is a single krg/kro curve as a function of Sg, recognizing and
honoring the irreducible water saturation of each core sample and the reservoir or
zone of interest.
5. 51
Figure 5.37 Variation of Sg and Swi with values of krg/kro (Relative perm. averaging by correlation with Swi)
5.4 Relative Permeability – Capillary Pressure Relationship
Both relative permeability and capillary pressure are strong functions of the pore size
distribution and fluid saturation. The first involves the pore geometry of the system, and the
second refers to the multiphase fluid effects. Figure 5.38 is a schematic of the saturation
dependency and interrelationship between these two concepts.
increases the water saturation reduces to irreducible.
As capillary pressure
At the same time, the relative
permeability to water decreases and to oil increases.
The effect of pore geometry can be seen in Figure 5.39. In this illustation, an increase
in permeability is a direct reflection of the pore size distribution. A lower irreducible water
saturation and capillary pressure curve result as permeability increases.
The relevance of these curves is shown in Figure 5.40. The position of the oil – water
contact, and therefore the productive interval is identified by the curves. Furthermore, the
effect of variable permeability (from low on the left to high in the right side of the diagram)
is included. And finally, the change in production over time for multiphase flow can be
predicted using these fundamental curves.
5. 52
Figure 5.38 Relationship between capillary pressure and relative permeability
5. 53
Figure 5.39 Relationship between capillary pressure and relative permeability [CoreLab,1983]
5. 54
Figure 5.40 Spatial variation of Pc and kr curves for a reservoir [Amyx, et al. 1960]
5. 55

5.1 Wettability

Transcription

Similar documents

ContourGlobal acquires Fibrominn litter

Hycal Energy Research Laboratories Ltd

08 - Rohrbasser - Electrophorese capillaire_projet PHARMELP

Using Selection Tools - Lasso Request Response Procedures

Start your career at Geotrace

GWD1000W GWD1000BLS Water Dispenser

GWD860W-4 GWD860BLS-4 Water Dispenser

Burrator Site Leaflet Click to

MuShield`s High Permeability Magnetic Shielding per ASTM A753