PET524 4 Multiphase

Chapter 5 –  5 –  Multiphase  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  can

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 90, (2) neutral wettability case would exist at a contact angle of 90 90, and (3) oil wet occurs at a contact angle greater than 90 90.

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Chapter 5 –  5 –  Multiphase  Multiphase Phenomena

Figure 5.1 Three possible states of wettability in oil reservoirs.

Other lesser known types of wettability are:   no preference is shown by the rock to either Neutral or intermediate wettability  –  no fluid; i.e., equally wet.  heterogeneous wetting; i.e., portions of the rock are strongly Fractional wettability  –  heterogeneous oil wet, whereas other portions are strongly water wet. Occurs due to variation in minerals with different different surface chemical properties. Silicate water interface interface is acidic, therefore basic constituents in oils will readily be absorbed resulting in an oil-wet surface. In contrast, the carbonate carbonate water interface is basic and will attract 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.  refers to small pores occupied by water and are water-wet, while Mixed wettability  –  refers 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]

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Chapter 5 –  5 –  Multiphase  Multiphase Phenomena

Figure 5.1 Three possible states of wettability in oil reservoirs.

Other lesser known types of wettability are:   no preference is shown by the rock to either Neutral or intermediate wettability  –  no fluid; i.e., equally wet.  heterogeneous wetting; i.e., portions of the rock are strongly Fractional wettability  –  heterogeneous oil wet, whereas other portions are strongly water wet. Occurs due to variation in minerals with different different surface chemical properties. Silicate water interface interface is acidic, therefore basic constituents in oils will readily be absorbed resulting in an oil-wet surface. In contrast, the carbonate carbonate water interface is basic and will attract 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.  refers to small pores occupied by water and are water-wet, while Mixed wettability  –  refers 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]

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Chapter 5 –  5 –  Multiphase  Multiphase Phenomena

Figure 5.3 Microscopic fluid saturation distribution in a oil-wet rock [Pirson, 1963]

The contact angl  is a measure of the wettability of the rock-fluid system, and is angl e  related to the interfacial energies by Young’s equation, equation,      os ws

cos      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

Chapter 5 –  5 –  Multiphase  Multiphase Phenomena 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 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 s ystems

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. 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.

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Chapter 5 –  5 –  Multiphase  Multiphase Phenomena

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 smooth mineral surfaces 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 obta in 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 S or 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

Chapter 5 –  Multiphase Phenomena (3) The water is displaced to the residual water saturation (S wi) 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).

 I  w



V  osp V  ot 



V  wsp V  wt 

  o   w

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

Chapter 5 –  Multiphase Phenomena 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:

 I  w

  A    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

Chapter 5 –  Multiphase Phenomena 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, nd

(6) final displacement of water by oil (2  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.

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Chapter 5 –  Multiphase Phenomena

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.

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Chapter 5 –  Multiphase Phenomena

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  2

force down = r   h  Capillary pressure, Pc, is the force per unit area; therefore  P  c  P  c

2

2

  force up /  r    force down /  r  

( 5.2)

2  cos  r 

where: r

= pore radius; cm



= interfacial tension; dynes/cm



 contact angle, deg.

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Chapter 5 –  Multiphase Phenomena 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:

 P  c



2  r   gh        w air  2  r 





   gh  

(5.3)

where: h

= height of capillary rise, ft;

Pc

= capillary pressure, psi;

w

= water, or wetting phase, density, slugs/ft  or kg/m

air 

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

3

3

3 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).  P  c

  P nw  P w  

(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

Chapter 5 –  Multiphase Phenomena  P  cow

  P o  P w  

(5.5)

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

  P  g   P 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  g   P o   cog 

(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

Chapter 5 –  Multiphase Phenomena 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.  P  c(res)

  P c(lab)

  cos res     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. 

 P  c(res)

  P c(lab)

24 cos 30 72 cos 0



 0.289 P c(lab)

From Eq. (5.3) the height above the free water level can be estimated. 5. 13

Chapter 5 –  Multiphase Phenomena

h

 P   P  *144 c  c    65  53

 12.0 * P c(res)

Table 5.3 shows the results of laboratory capillary pressure measurements using air and water, corrections to reservoir oil and water, and calculated elev ations 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

Chapter 5 –  Multiphase Phenomena 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

Chapter 5 –  Multiphase Phenomena 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

Chapter 5 –  Multiphase Phenomena

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

Chapter 5 –  Multiphase Phenomena

Pressure gauge Sample Displacement read

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.

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Chapter 5 –  Multiphase Phenomena

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

Chapter 5 –  Multiphase Phenomena  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

Chapter 5 –  Multiphase Phenomena  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

Chapter 5 –  Multiphase Phenomena 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 5.19 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).

5. 22

Chapter 5 –  Multiphase Phenomena

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 S wor . (Curve 5 of Figure 5.20)

Figure 5.20 Example capillary pressure curves from centrifugal data. Cu rves 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

Chapter 5 –  Multiphase Phenomena

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: ( P c ) i

 L   1.096 x10 6   N 2    r e   L   2   

(5.19)

where (Pc)i   N r e 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: S  i

 S   ( P c )i *

d S  d ( P  ) c 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 r i/r e 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

Chapter 5 –  Multiphase Phenomena hyperbolic least squares solution has been developed to remove the noise in the data. The resulting equation for inlet capillary pressure is: ( P c ) i



 A   BS  1  C S 

 

(5.21)

and for the derivative: d ( P  ) c i d S 



 B   AC    2 1  C S 

(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.

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 3 V p = 1.73 cm k = 144 md Centrifuge arm = 8.6 cm  = 0.9988 gm/cc  = 0.17

5. 25

Chapter 5 –  Multiphase Phenomena

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

Chapter 5 –  Multiphase Phenomena 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

Chapter 5 –  Multiphase Phenomena Of these the lambda-fit   normally works best (shell recommendation). It fits the wetting saturation Sw to the capillary pressure P c 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  P  k  c  J ( S  )    w   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. 28

Chapter 5 –  Multiphase Phenomena

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  )  cos res w          k    res w o





where,

5. 29



(5.24)

Chapter 5 –  Multiphase Phenomena 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

Chapter 5 –  Multiphase Phenomena

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

Chapter 5 –  Multiphase Phenomena The minimum capillary pressure to displace the wetting phase in a capillary tube is given by,



2  cos  

  (5.25) r  Poiseuille’s equation for flow rate in a single tube of radius, r, is: 4  r   p q   (5.26) 8  La 2 The volume of a capillary tube is V = r  La, thus substituting in Eq. (5.26) results in, 2 Vr   p q   (5.27) 2 8  L a Rearrange Eq. (5.25) in terms of radius, and substituting into Eq. (5.27),   cos 2  p V  q    (5.28) 2 2 2  L  P  a c Consider the porous media as “n” capillary tubes of equal length but random radius, then Poiseuille’s equation becomes,   p m (5.29) q  ni r i4   8  La i 1 and the volume, m 2 V i    La  ni r i   (5.30) i 1 The resulting total flow rate is  P c

q

  cos 2  p 2  L2 a

m V   i   2 i 1 P ci

The macroscopic flow rate in porous media can be described by Darcy’s Law. kA  p q      L Thus combining Eqs. (5.31) and (5.32) results in a relationship of k =f(V p, Pc).   cos 2 L m V i k      2 2 2 AL i 1 P ci a

(5.31)

(5.32)

(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, k  

  cos 2 2 

5. 32

m S      i   2 i 1 P ci

(5.34)

Chapter 5 –  Multiphase Phenomena 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  2 k   10.24  cos        (5.35) 2 S  0  P c where k  Pc

= md = dynes/cm = psi

Comparing Eq. (5.35) with the Carmen-Kozeny equation reveals that the Kozeny factor, k z = 1/  . The S pv can be expressed as: S   pv

1



S 1 S  2   cos    i 2 S  0  P ci For mercury injection, Eq. (5.35) can be further reduced to, S 1 dS  k   14,260      2 S  0  P c

 

(5.36)

(5.37)

where the contact angle for air-mercury is assumed to be 140º and the interfacial tension is 480 dynes/cm. 2

A plot of Pc and 1/Pc  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

Chapter 5 –  Multiphase Phenomena

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 dV p is the total volume of pores having a radius between the pore entry radius, r i and r i-dr. Differentiating and rearranging Eq. (5.38) results in,

  r 2   dP    dr     2  cos   c    

(5.39)

Substituting Eq. (5.39) into (5.38),  D (ri)

dS   2  cos     V   w       p dP    r 2   c

(5.40)

or V  p dS   D(ri)   P c   w  r  dP c

(5.41)

To determine the P.S.D. first calculate the p ore radius at given values of capillary pressure. r i



2  cos   P ci

 

(5.42)

 Next, evaluate the distribution function, D(r i), 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

Chapter 5 –  Multiphase Phenomena

  m  ,   n   o    i    t   u    b    i   r    t   s    i    d   e   z    i   s   e   r   o    P

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  n S   r   p i   k   (5.43)   7 i 1 2 2 8(9.87 x10 )  x  r  i i where r i 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

Chapter 5 –  Multiphase Phenomena

r  i



r  i ( j 1)

 r i

j

2

2 Term  S  r   air  i

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

Chapter 5 –  Multiphase Phenomena 

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

Chapter 5 –  Multiphase Phenomena 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: k eff 

= effective permeability to each fluid phase (oil, gas, or water), md;

k abs

= 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

Chapter 5 –  Multiphase Phenomena 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 oilat 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 k  brine = 115 md, k air = 120 md and k o @ 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: k o is effective permeability to oil, md; k w is effective permeability to water, md; and k g is the effective permeability to gas, md. Consequently, we typically denote relative permeability to a particular phase also with subscripts, where k ro is relative permeability to oil; k rw is relative permeability to water, k rg relative permeability to gas. Therefore, to be more specific, the relative permeabilities to each of the three fluids can be written as, k  ro k  rw k  rg







k  o k  k  w   k  k  g

(5.46)

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 k ro and k rw both equal to 1.0 at 100% of their respective saturations, in actual practice, this is frequently not the

5. 39

Chapter 5 –  Multiphase Phenomena case. In normal water-wet systems; k rw  seldom reaches 1.0 even at Sw =100%; usually  because of water- sensitive clays in the core. Also k ro 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, k ro 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 k rw 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

Chapter 5 –  Multiphase Phenomena 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. 41

Chapter 5 –  Multiphase Phenomena 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, k rw almost follows the same curve (i.e., there is very little hysteresis between the k rw  imbibition and drainage processes). However, the k ro imbibition curve deviates noticeably from the drainage curve. There is significant hysteresis as k ro  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 k rg = 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

Chapter 5 –  Multiphase Phenomena 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%, k ro = .14 and k rw = .09 Therefore, using Eq. (5.46),

5. 43

Chapter 5 –  Multiphase Phenomena k   k  * k   (0.14)(214 md )  30.0 md . o ro k   k  * k   (0.09)(214)  19.3md . w rw

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)

Chapter 5 –  Multiphase Phenomena  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, kk  AP ro  0.001127 o  o Bo L   kk  AP rw q  0.001127 w  w Bw L q

(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

r e

= 660 ft

o

= 3 cp

r w

= 0.5 ft

w

= 1 cp

Find: a. The oil and water flow rates at initial reservoir state @ S wi.  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  h p o  r     141.2 ln e o r     w 



k  (10)(500) o 141.2(3) ln 660



 .5

5. 45

 1.642k o

Chapter 5 –  Multiphase Phenomena Since the permeability reference is air, k o = k*k ro=214*.75=160.5 md. Substitution for effective permeability results in qo = 299 bopd. The water flow rate equal to zero since k rw = 0.  b. From Example 5.2, relative permeability at 50 % saturation are k rw = 0.09 and k ro = 0.14. The subsequent flow rates are: kkro

214 * .14

q

 4.927  4.927 o o

q

 4.927  4.927 w w

kkrw

3

 49 bopd

214 * .09 1

 95 bopd

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 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, S w



W   W   V  p *  o  sat  dry 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

Chapter 5 –  Multiphase Phenomena 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



SL  SLr 4 1 S

 

(5.52)

Lr 

4 2  SL  SLr    SL  SLr       k   1   * 1   rg   Sm  SLr      1  SLr      

(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.  R 

k     B rg  o o   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)

Chapter 5 –  Multiphase Phenomena The relative permeability ratio (k rg/k ro) 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 k rg/k ro 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 k rw/k ro 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 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 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

Chapter 5 –  Multiphase Phenomena 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 k rw and k row  from the two-phase water-oil relative permeability curves;

2.

Sg is used to obtain k rg and k rog from the two-phase gas-oil curves;

3.

Three-phase relative permeabilities are obtained.as k rw is used directly from step 1, k rg is used directly from step 2, and the three-phase k ro is calculated from k  ro

 k row  k rw    k rog   k rg       k rw  k rg            

(5.55)

where: k ro

= three-phase relative permeability to oil;

k rw

= two-phase and three-phase relative Permeability to water,

k rg

= two-phase and three-phase relative Permeability to gas;

k row

= two-phase relative permeability to oil from step 1;

k rog

= two-phase relative permeability to oil from step 2; If Equation 5.55 calculates a negative value for k ro it should be ignored and k ro = 0. A

 basic assumption in Stone's model is that k rw and k rg are the same functions in the three-phase system as they are in two-phase systems. Only k ro  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

Chapter 5 –  Multiphase Phenomena 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 (k rg/k ro or k rw/k ro). For k rg/k ro curves, the steps are: 1. For each curve, Sg is read at selected values of k rg/k ro 2. Sg 's from the set of curves are averaged at the selected k rg/k ro; 3. K rg/k ro is plotted versus the average S g, 4. A smooth curve is drawn through the data to obtain the averaged set of k rg/k ro. Normalization of curves . This method requires a value for Sor , which can be difficult to

obtain. If a reliable determination of S or  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, * S  w



S   S  w wi   1  S   S  or  wi

where: Swi = irreducible water saturation, fraction;

5. 50

(5.56)

Chapter 5 –  Multiphase Phenomena *

Sw = 0 for S w = Swi; *

Sw = 1.0 for Sw = 1 - Sor  3. The normalized oil saturation in the gas- oil system is, * S  o



S   S  o or  S   S  oi or 



S  oi

 S  g   S or 

S  oi

 S 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 S wi  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/k ro curves, the steps are: 1. Sg is read at selected values of k rg/k ro for each curve; 2. Sg 's from the set of curves are averaged at the selected k rg/k ro, 3. Values of Sg for a given k rg/k ro are plotted versus irreducible water saturation for each core sample; 4. Straight lines are drawn through the data for each value of k rg/k ro  as shown in Figure 5.21. 5. Average water saturation is determined for the reservoir or a zone of interest; 6. For each k rg/k ro  value in Figure 5.37 Sg is read at the average Sw  determined in step 5; 7. k rg /k ro values are plotted versus Sg values, and a smooth curve is drawn through the data. The result is a single k rg/k ro 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

Chapter 5 –  Multiphase Phenomena

Figure 5.37 Variation of S g and Swi with values of k rg/k ro (Relative perm. averaging by correlation with S wi)

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

Chapter 5 –  Multiphase Phenomena

Figure 5.38 Relationship between capillary pressure and relative permeability

5. 53

Chapter 5 –  Multiphase Phenomena

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

5. 54