PET524 4 Multiphase

Chapter 5 – Multiphase Phenomena

5. 1

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. 2

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. 3

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,

cosowwsos

(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. 4

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. 5

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. 6

(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).

wowt

V

wspV

otV

ospV

wI

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. 7

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:

2

1logA

A

wI

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. 8

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 water-

wet 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. 9

Displacement

of water

Displacement

of oil

Amott Index USBM Index Sample No.

Water wet samples

0 0.92 0.92 1.6 1

0 0.80 0.80 1.2 2

0 0.89 3

Mixed wettability samples

0.24 0.015 -0.225 0.3 1

0.24 0.073 -0.167 0.3 2

0.24 0.065 -0.175 0.3 3

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. 10

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 = 2r cos

force down = r2 h

Capillary pressure, Pc, is the force per unit area; therefore

rcP

rdownforcerupforcec

P

cos2

2/

2/

( 5.2)

where:

r = pore radius; cm

= interfacial tension; dynes/cm

contact angle, deg.


5. 11

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:

gh

r

airwghr

cP

2

2

(5.3)

where:

h = height of capillary rise, ft;

Pc = capillary pressure, psi;

w = water, or wetting phase, density, slugs/ft3 or kg/m

3

air = air, or nonwetting phase, density, slugs/ft3 or kg/m

3

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).

wP

nwP

cP (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. 12

wP

oP

cowP (5.5)

Similarly, in a water-wet gas reservoir, or gas cap,

wP

gP

cgwP (5.6)

and where oil and gas occur together, the oil phase usually behaves as the wetting phase

relative to the gas phase, therefore,

oP

gP

cogP (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. 13

SYSTEM CONTACT ANGLE,

deg.

INTERFACIAL TENSION,

dynes/cm

Laboratory

Air-water 0 72

Oil-water 30 48

Air-Mercury 140 480

Air-oil 0 24

Reservoir

Water-oil 30 30

Water-gas 0 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.

lab

reslabc

Presc

P

cos

cos

)()( (5.8)

where:

Pc(res) = capiIlary pressure, corrected to reservoir conditions and reservoir fluids, psig;

Pc(lab) = capillary pressure measured in the laboratory, psig;

res = interfacial tension between reservoir fluids at reservoir conditions, dyne/cm;

lab = interfacial tension between laboratory fluids at laboratory conditions, dyne/cm;

res = contact angle for reservoir conditions and fluids, deg;

lab = 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) = 72 dyne/cm

(air-water) = 0

Reservoir (oil-water) = 24 dyne/cm

(oil-water) = 30

w = 65 lb /cu ft o = 53 lb /cu ft.

Solution

From Eq. (5.8) the conversion from lab to reservoir capillary pressure can be determined.

)(289.0

0cos72

30cos24)()( labc

Plabc

Presc

P

From Eq. (5.3) the height above the free water level can be estimated.


5. 14

)(*0.12

5365

144*

rescPc

Pc

Ph

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, % Lab measured pc, air-

water (psig)

Reservoir pcow oil-

water (psig)

Height above free

water level, ft

100 2 0.578 6.9

90 3 0.867 10.4

80 4 1.16 13.9

70 5 1.45 17.4

60 6 1.73 20.8

50 7 2.02 24.2

45 8 2.31 27.7

40 10 5.8 70

35 27 7.8 94

30 75 21.7 260

Table 5.3 Laboratory Pressure-Measurement Results

Figure 5.11 Example of Laboratory Pressure Measurement


5. 15

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. 16

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. 17

HIGH PERMEABILITY LOW PERMEABILITY

Entry pressure Lower Higher

Irreducible water saturation Lower Higher

Slope of transition zone

curve

Lower Higher

Grain size distribution Narrow Wide

Grain and pore size Large 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. 18

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.

Fig. 22 Mercury injection equipment

Sample

Mercury

Pump

Displacement reading

Pressure

gauge


5. 19

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. 20

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. 21

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. 22

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.

Figure 5.19 Capillary pressure measurement by centrifuge

Only the initial displacement of fluid to irreducible saturation was discussed for the

preceding methods. Mercury 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

used for mercury displacement of a wetting fluid such as water, 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).

rotor

axis

rotor

sample

rotation

water

produced in

collection tube

Centrifugal force plays role of gravitational force

Increasing rotation speed ---> increasing force

----> increasing Pc

Gives increasing water production ---> Sw

Fig. 23 Centrifuge

measurement


5. 23

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. 24

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:

LL

erNxicP

2

2610096.1)( (5.19)

where

(Pc)i = inlet capillary pressure, kPa

= density difference in gm/cc

N = revolutions per minute

re = outer radius of core measured from center of rotation, cm

L = 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:

icPd

SdicPSiS

)(*)( (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. 25

hyperbolic least squares solution has been developed to remove the noise in the data. The

resulting equation for inlet capillary pressure is:

SC

SBAicP

1)( (5.21)

and for the derivative:

21

)(

SC

ACB

Sd

icPd

(5.22)

where A, B, and C are regression constants.

The following table compares the limitations and advantages of the various capillary pressure

measurement techniques.

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

Fig. 24 Measuring Capillary Pressure

Equilibrium Air/Hg Centrifuge

Duration 5 weeks 1day 3 days/run

Max Height (m gas/oil) 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 No Yes Norequired

Costs Expensive Cheap Medium

Additional Imbibition Imbibition ImbibitionInformation RI Wettability


5. 26

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. 27

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. 28

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

kcP

wSJ

cos)( (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. 29

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

res

kow

reswSJ

h

cos)( (5.24)

where,


5. 30

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. 31

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 Carmen-

Kozeny’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. 32

The minimum capillary pressure to displace the wetting phase in a capillary tube is given by,

rcP cos2

(5.25)

Poiseuille’s equation for flow rate in a single tube of radius, r, is:

aL

prq

8

4 (5.26)

The volume of a capillary tube is V = r2La, thus substituting in Eq. (5.26) results in,

28

2

aL

pVrq

(5.27)

Rearrange Eq. (5.25) in terms of radius, and substituting into Eq. (5.27),

222

2cos

cP

V

aL

pq

(5.28)

Consider the porous media as “n” capillary tubes of equal length but random radius, then

Poiseuille’s equation becomes,

m

iirin

aL

pq

1

4

8

(5.29)

and the volume,

m

iirinaLiV

1

2 (5.30)

The resulting total flow rate is

m

i ciP

iV

aL

pq

1222

2cos

(5.31)

The macroscopic flow rate in porous media can be described by Darcy’s Law.

L

pkAq

(5.32)

Thus combining Eqs. (5.31) and (5.32) results in a relationship of k =f(Vp, Pc).

m

i ciP

iV

aAL

Lk

1222

2cos (5.33)

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,

m

i ciP

iSk

122

2cos

(5.34)


5. 33

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.

1

02

2cos24.10S

S cP

dSk (5.35)

where

k = md

= dynes/cm

Pc = psi

Comparing Eq. (5.35) with the Carmen-Kozeny equation reveals that the Kozeny factor, kz =

1/. The Spv can be expressed as:

1

02

2cos

1

S

S ciP

iSpvS

(5.36)

For mercury injection, Eq. (5.35) can be further reduced to,

1

02

260,14S

S cP

dSk (5.37)

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. 34

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:

wdSpVdrriDpdV )( (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,

cdPr

dr

cos2

2 (5.39)

Substituting Eq. (5.39) into (5.38),

cdP

wdS

pVr

riD

2

cos2)(

(5.40)

or

cdP

wdS

r

pV

cPriD )( (5.41)

To determine the P.S.D. first calculate the pore radius at given values of capillary pressure.

ciPir cos2

(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. 35

Pore radius, microns 0

Po

re s

ize d

istr

ibu

tio

n, m

2

Figure 5.28 Typical distribution function

Permeability is estimated by analogy with the bundle of capillary tubes model,

n

ii

ri

x

irS

x

pV

k

122

4

)7

1087.9(8

100 (5.43)

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. 36

2

)1( ji

rj

ir

ir

2

ir

airSTerm

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. 37

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. 38

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

1iabs

ki

effk (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. 39

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.

refk

effk

rk (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

gk

rgk

k

wk

rwk

k

ok

rok

(5.46)

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

1i

0.1i

rk (5.47)

or

1to0rg

krw

kro

k (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. 40

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. 41

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 relative-

permeability curves and the resulting hysteresis.

Figure 5.34 Relative permeability hysteresis, imbibition vs. drainage


5. 42

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 krw 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. 43

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. 44

.3.19)214)(09.0(*

.0.30)214)(14.0(*

mdkrw

kw

k

mdmdkro

ko

k

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,

BL

pkA001127.0q

(5.49)

where,

q = flow rate, STB/D;

k = absolute permeability of rock, md;

A = cross-sectional area of flow, sq ft;


5. 45

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,

Lw

Bw

PArw

kk001127.0

wq

Lo

Bo

PAro

kk001127.0

oq

(5.50)

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.

ok642.1

5.660ln)3(2.141

)500)(10(o

k

wr

er

lno

2.141

pho

k

oq


5. 46

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:

bopd951

09.*214927.4

w

kkrw927.4

wq

bopd493

14.*214927.4

o

kkro927.4

oq

5.3.5 Methods of Generating Relative-Permeability Data

It is best to obtain relative-permeability curves from laboratory measurements of two-

phase 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 pre-

determined 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,

wpV

opV

dryW

satW

wS

*

* (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. 47

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:

LrS1

4Lr

SL

S

rok

(5.52)

2

LrS1

LrS

LS

1*

4

LrS

mS

LrS

LS

1rg

k (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.

soR

gB

oB

g

o

rok

rgk

R

(5.54)

where:

R = measured producing GOR, scf/STB;

Rso = solution GOR, scf/STB,

B = formation volume factor;

= viscosity, cp.


5. 48

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 gas-

drive oil reservoirs where water saturation is immobile, two-phase oil and gas relative-

permeability 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 per-

meability. 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. Later investigators used various


5. 49

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 relative-

permeability 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

rgk

rwk

rgk

rogk

rwk

rowk

rok (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. 50

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 relative-

permeability 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,

wiS

orS

wiS

wS

wS

1

* (5.56)

where:

Swi = irreducible water saturation, fraction;


5. 51

Sw* = 0 for Sw = Swi;

Sw* = 1.0 for Sw = 1 - Sor

3. The normalized oil saturation in the gas- oil system is,

orS

oiS

orS

gS

oiS

orS

oiS

orS

oS

oS

* (5.57)

and

*1

*

oS

gS (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 krg/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. 52

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. As capillary pressure

increases the water saturation reduces to irreducible. 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. 53

Figure 5.38 Relationship between capillary pressure and relative permeability


5. 54

Figure 5.39 Relationship between capillary pressure and relative permeability [CoreLab,1983]


5. 55

Figure 5.40 Spatial variation of Pc and kr curves for a reservoir [Amyx, et al. 1960]

PET524 4 Multiphase

Documents

Transcript of PET524 4 Multiphase