05 Capillary Pressure

Capillary Pressure

- List four uses of capillary pressure data. - Define hysteresis.

- Sketch capillary pressure curves for typical drainage and imbibition processes. - Explain the relation between capillary pressure data and reservoir fluid saturation. - Define oil-water and gas-oil transition zones.

- Convert capillary pressure lab data to reservoir conditions. - Define the J-function.

- List four different methods for measuring capillary pressure in the lab.

Uses of Capillary Pressure Data:

- Determine initial water saturation in the reservoir. - Determine fluid distribution in the reservoir.

- Determine residual oil saturation for water flooding applications. - Determine pore size distribution index.

- May help in identifying zones or rock types. - Input for reservoir simulation calculations.

Capillary pressure measurements determine the initial water saturation. This is the saturation at which the increase in capillary pressure does not affect the saturation.

Capillary pressure data can also determine the vertical fluid distribution in the reservoir by establishing the relation between the capillary pressure and height above the free water level. Imbibition capillary pressure measurements determine the residual oil saturation in water flooding operation.

We can infer the pore size distribution index, λ, from capillary pressure data. This index can be used to calculate relative permeability using industry correlations.

Capillary pressure curves are similar for the same rock type. The shape also gives indication about the rock permeability.

Capillary pressure curves are used to initialize simulation runs and in flow calculations between grid blocks.

Capillary Pressure Concept:

A

B

ρ

ρ

1

2

3

P

= 0

Pressure

Depth

ρ

ρ

Oil and water pressure gradients are different because their density is different. At level 2, pressure in both the water and oil phases is the same.

At any level above 2, such as level 3, water and oil pressures are different. This difference in pressure is called the capillary pressure.

Capillary Pressure Definition:

- The pressure difference existing across the interface separating two immiscible fluids. - It is usually calculated as:

Pc = pnwt - pwt

One fluid wets the surfaces of the formation rock (wetting phase) in preference to the other (non-wetting phase).

Gas is always the non-wetting phase in both oil-gas and water-gas systems. Oil is often the non-wetting phase in water-oil systems.

Example:

Define capillary pressure in the following systems: - Water-gas system.

- Water-wet water-oil system. - Oil-gas system.

Solution:

- water-gas system: Pc = pg - pw

- water-wet water-oil system: Pc = po - pw

- oil-gas system: Pc = pg - po

Relation between Capillary Pressure and Fluid Saturation:

Free Water Level

Pc Pd Water-oil contact Hd Hei g h t Ab ove Free W a te r L e v e l (F eet) 0 50 100 Sw (Percent) 0 50 100 Sw (Percent) 0

For uniform sands, reservoir fluid saturation is related to the height above the free-water level by the following relation:

(

)

h

p

p

P

=

−

=

Δ

ρ

−

ρ

144

Fluid Distribution in Petroleum Reservoirs:

Gas & Water Gas density = ρg

Oil, Gas & Water

Oil & Water Oil density = ρo Water Water density = ρw ‘A’ h₁ h₂ ‘B’

Free Oil Level

Free Water Level

Capillary pressure difference

between

oil and water phases in core ‘A’

P

= h

g (

ρ

-

ρ

)

Capillary pressure difference

between

gas and oil phases in core ‘B’

P

= h

g (

ρ

-

ρ

)

Free water level (surface) is the level at which water-oil capillary pressure is zero, i.e. the pressures in both water and oil phases are equal.

Above the free water level, oil and water can coexist. Above this level, there usually exists another level (or surface) called the water-oil contact, WOC. Below the WOC, only water can be produced. Above the WOC, both oil and water can be produced. At some point above the WOC, water will reach an irreducible value and will no longer be movable.

VCL d ec 0 1 0 VC L VCL 1 ILD C oh m m 0 .2 2 00 R HO CN u s/f 1 .65 2. 65 C NL L SS 0 .6 0 D TC u s/f 1 35 55 RH OC N CN L LSS G AS u s/f 1 0 BVW d e c 1 0 O IL 1 0 BVW G 1 0 SW d e c 1 -1 1 07 50 1 08 00

This processed log section shows the fluid distribution in the reservoir. Gas and oil are clearly identified from water.

An oil-water transition zone of about 7 feet can be also seen on the log.

Transition zone height is partially a function of the density difference between the two fluids. For a given rock quality, the transition zone will be smaller for larger density differences (e.g. gas and oil) and larger for smaller density differences (e.g. heavy oil and water). This relationship is evident from the relation between capillary pressure and the height above the free-water level. Example:

Capillary pressure was measured to be 5 psia at connate water saturation. Water density is 66.5 lbm/ft3. Calculate the length of the transition zone in the following two situations:

A) oil density = 59.3 lbm/ft3 B) oil density = 43.7 lbm/ft3

This example assumes the rock quality is the same in both situations. Solution:

Rearranging the relation between capillary pressure and height:

(

)

P

h

ρ

ρ

−

×

=

144

(

66

.

5

59

.

3

)

100

ft.

5

144

₌

−

×

=

h

(

66

.

5

43

.

7

)

31

.

6

ft.

5

144

₌

−

×

=

h

Capillary Tube Model – Air-Water System:

Water

Air

θ

Δ

We can imagine the porous media as a collection of capillary tubes. This model is useful for providing insight into how fluids behave in the reservoir pore spaces.

Water rises in a capillary tube when that tube is placed in a beaker of water. Likewise, water (the wetting phase) fills small pores leaving larger pores to non-wetting phases.

The height of water is a function of:

- The adhesion tension between the air and water. - The radius of the tube.

- The density difference between fluids.

aw aw

g

r

h

ρ

θ

σ

Δ

=

Δ

2

cos

The above relation can be derived from balancing the upward and downward forces. Upward force is due to adhesion tension and downward force is due to the weight of the fluid.

Δh = Height of water rise in capillary tube, cm.

σaw = Surface tension between air and water, dynes/cm.

θ = Air/water contact angle, degrees. r = Radius of capillary tube, cm.

g = Acceleration due to gravity, 980 cm/sec2.

Δρaw = Density difference between water and air, gm/cm3.

Contact angle, θ, is measured in the denser phase (water in this case).

Air

Water

p

Δ

h

p

p

p

Water rise in capillary tube depends on the density difference of fluids. pa2 = pw2 = p2 pa1 = p2 - ρa g Δh pw1 = p2 - ρw g Δh Pc = pa1 - pw1 = ρw g Δh - ρa g Δh = Δρaw g Δh

Combining the two relations:

r

P

g

r

g

P

g

P

h

g

r

h

θ

σ

ρ

θ

σ

ρ

ρ

ρ

θ

σ

cos

2

cos

2

cos

2

=

Δ

=

Δ

Δ

=

Δ

Δ

=

Δ

Capillary Pressure – Oil-Water System:

From a similar derivation, the equation for capillary pressure for an oil/water system is:

r

P

θ

σ

cos

2

=

Pc = Capillary pressure between oil and water.

σow = Interfacial tension between oil and water, dyne/cm.

θ = Oil/water contact angle, degrees. r = Radius of capillary tube, cm.

Converting Laboratory Capillary Pressure Data to Reservoir Conditions: Basic Equations:

L L L cL

r

P

=

2

σ

cos

θ

r

P

=

2

σ

cos

θ

Pc = capillary pressure, psia

σ = interfacial tension, dynes/cm θ = contact angle, degrees r = radius of pore throat, cm

Subscripts L and R refer to laboratory and reservoir conditions, respectively. Setting rL = rR and combining equations yields:

cR R R cL L L R L

P

P

r

r

=

→

2

σ

cos

θ

=

2

σ

cos

θ

Therefore, capillary pressure at reservoir conditions is given by: cL L L R R cR

P

P

θ

σ

θ

σ

cos

cos

=

Convert the laboratory capillary pressure data for sample 28 in the attached capillary pressure curve (obtained using mercury injection method) to reservoir conditions for a formation containing oil and water.

Calculate reservoir capillary pressure data for mercury saturations of 70, 60, 50, 40, 30, 20, and 10 percent.

Laboratory Data: σHg = 480 dynes/cm, θHg = 140o

Reservoir Data: σow = 24 dynes/cm, θow = 20

o

Note: The reservoir data are very difficult to obtain. The reservoir data above are representative values based upon industry literature.

0 400 800 1200 1600 2000 0 20 40 60 80

Mercury Saturation, percen t pore sp ace

In ject io n P ressu re, p s ia

Solution: Steps:

- Solve the equation that relates lab capillary pressure data to reservoir capillary pressure data for the conditions we have.

- Obtain laboratory capillary pressure data from the curve for Sample 28. - Convert the lab numbers to reservoir capillary pressure.

Mercury Saturation (SHg) % PcL psia PcR psia 70 1,320 80.5 60 820 50.0 50 560 34.2 40 410 25.0 30 310 18.9 20 240 14.6 10 200 12.2

( )

( )

( )

( )

P

P

P

P

061

.

0

140

cos

480

20

cos

24

cos

cos

=

=

=

θ

σ

θ

σ

Drainage and Imbibition Capillary Pressure Curves:

Drainage (1) Imbibition (2) S_i S_m

S

P

0

0.5

1.0

The drainage curve is always higher than the imbibition curve. Si = Initial or irreducible wetting phase saturation.

Sm = critical non-wetting phase saturation.

Pd = entry pressure or displacement pressure.

The entry (displacement) pressure is defined as the pressure required to force the non-wetting fluid through an initially wetting-phase-saturated sample.

Drainage Process:

- Fluid flow process in which the saturation of the nonwetting phase increases.

- Mobility of nonwetting fluid phase increases as nonwetting phase saturation increases. Imbibition Process:

- Fluid flow process in which the saturation of the wetting phase increases and the nonwetting phase saturation decreases.

- Mobility of wetting phase increases as wetting phase saturation increases. Class Exercise: 50 0 45 40 35 30 25 20 15 10 5 (1) (2) S_w P_c , psi 0 0.25 0.5 0.75 1.0

The above figure is the result of core flood experiments on a water-wet rock. Answer the following questions:

1- What process does curve 1 represent? 2- What process does curve 2 represent? 3- What is the irreducible water saturation? 4- What is the residual oil saturation? 5- What is the displacement pressure? Solution: 1- Drainage process. 2- Imbibition process. 3- Sw = 0.25 4- Sor = 0.13 5- Pd = 15 psi

Effects of Reservoir Properties on Capillary Pressure:

- Capillary pressure characteristics in reservoir are affected by: - Variations in permeability

- Grain size distribution - Saturation history - Contact angle - Interfacial tension

- Density difference between fluids Effect of Permeability: Decreasing Permeability A B C 20 16 12 8 4 0 0 0.2 0.4 0.6 0.8 1.0

Water Saturation

Capillary Pressure

- Curves shift to the right (i.e., larger water saturations at a given value of capillary pressure) as the permeability decreases.

- Displacement pressure increases as permeability decreases.

- Minimum interstitial water saturation increases as permeability decreases. Effect of Grain Size Distribution:

Well-sorted Poorly sorted

Capillary p

ressure, psia

- Well-sorted grain sizes:

- Majority of grain sizes are the same size. - Minimum interstitial water saturation is lower. - Displacement pressure is lower.

- Poorly sorted grain sizes:

- Significant variation in range of grain sizes. - Minimum interstitial water saturation is higher. - Displacement pressure is higher.

Effect of Saturation History:

Water saturation, % C a pillary pressure, psia Imbibition Drainage

For the same saturation, capillary pressure is always higher for the drainage process (increasing the non-wetting phase saturation) than the imbibition process (increasing the wetting phase saturation).

Effect of Contact Angle:

Decreasing θ_R θ_{R = 30°} θ_{R = 60°} θ_{R = 0°} Capillary Pressure θ_{R = 80°} 20 16 12 8 4 0 0 0.2 0.4 0.6 0.8 1.0 Water Saturation

- Contact angle less than 90o indicates water-wet characteristics (refer to wettability notes).

- Curves shift to the right (i.e., larger water saturations at a given value of capillary pressure) as the contact angle decreases.

- Displacement pressure increases as contact angle decreases.

- Minimum interstitial water saturation increases as contact angle decreases. Effect of Interfacial Tension:

Water Saturation

Height

Above F

ree Water Level

High Tension

Low Tension

0 1.0

Low interfacial tension indicates higher tendency of phases to mix together. With higher interfacial tension, transition zone is expected to be larger. Effect of Density Difference:

Water Saturation

Small Density Difference

Large Density Difference He ig ht Ab ov e Fre e Wate r L e v e l 0 1.0

Smaller density difference between fluids results in a larger transition zone (refer to Exercise 1). Averaging Capillary Pressure Data Using the Leverett J-Function:

- A universal capillary pressure curve is impossible to generate because of the variation of properties affecting capillary pressures in reservoir.

- The Leverett J-function was developed in an attempt to convert all capillary pressure data to a universal curve.

Definition of Leverett J-Function:

( )

φ

θ

σ

k

P

Sw

J

cos

22

.

0

=

J(Sw) = Leverett J-function, dimensionless.

Sw = water saturation, fraction.

Pc = capillary pressure, psia.

σ = interfacial tension, dynes/cm. θ = contact angle, degrees. k = formation permeability, md.

φ = porosity, fraction.

Example of J-Function for West Texas Carbonate

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

W ater s aturation, frac tion

J-fu n c ti o n Jc Jm atc h Jn1 Jn2 Jn3

The data represented by plus signs (+) are from an original capillary pressure experiments. The other data were for three new experiments, performed to determine the reliability of the original data.

The black line represents an average J-function that was initially used in a reservoir simulation study for the field. Ultimately, the rock type was broken into smaller units, and five different capillary pressure relationships were used to represent the ranges of carbonate rock quality observed in the reservoir.

Uses of Leverett J-Function:

- J-function is useful for averaging capillary pressure data from a given rock type from a given reservoir.

- J-function can sometimes be extended to different reservoirs having same lithologies. - J-function usually does not predict an accurate correlation for different lithologies.

- If J-functions are not successful in reducing the scatter in a given set of data, then this suggests that we are dealing with different rock types.

Example: Calculation of Leverett J-Function:

Using the average capillary pressure data from the table, calculate and plot the Leverett J-function. Use the following rock and fluid properties:

σ = 72 dynes/cm. k = 47 md. θ = 0o . φ = 19.4 % A v e r a g e d A ir /B r in e C a p illa r y P r e s s u r e D a t a Pc p s ia Sw % 1 9 8 .3 2 9 8 .3 4 9 6 .8 8 5 9 .0 1 5 3 6 .3 3 5 2 5 .4 5 0 0 1 5 .3 Solution:

Calculated

J-Functions

S

%

0.22*

J(S

)

98.3

0.22

98.3

0.43

96.8

0.86

59.0

1.73

36.3

3.24

25.4

7.57

15.3

108.1

In the table above, we have not multiplied through by the conversion factor 0.22.

120 100 80 60 40 0 20 0 20 40 60 80 100 0.2 2 * J (S w ) S_w , %

Example: Estimating Pc from the J-Function:

Estimate capillary pressures from Leverett J-function calculated in the previous example for a different core sample.

Properties of core sample: k = 100 md. φ = 10 % Solution:

Estimated Capillary

Pressures for the 100-md

Permeability Core Sample

Sw,

%

Pc,

psia

98.3

2.27

98.3

4.45

96.8

8.91

59.0

17.91

36.3

36.90

25.4

78.18

15.3

1118.63

Laboratory Methods for Measuring Capillary Pressure: - Porous diaphragm method

- Mercury injection - Centrifuge method - Dynamic method