Capillary Pressure Instructional Objectives: - 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
ρo 3
Pc = 0
2
ρw
ρo
Depth
ρw
1 Pressure
Water exists at all levels below 2, and both water and oil exist at all levels above 2.
Capillary Pressure
1
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 (nonwetting 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
Height Above Free Water Level (Feet)
Relation between Capillary Pressure and Fluid Saturation:
Pc
Water-oil contact Pd 0 0
Capillary Pressure
50 100 Sw (Percent)
Hd 0
50 100 Sw (Percent)
Free Water Level
2
For uniform sands, reservoir fluid saturation is related to the height above the free-water level by the following relation:
Pc = po − pw =
Δh (ρ w − ρo ) 144
Fluid Distribution in Petroleum Reservoirs:
Capillary pressure difference between gas and oil phases in core ‘B’ Pc = h2g (ρo-ρg)
Gas & Water Gas density = ρg
‘B’ Oil, Gas & Water
h2
Free Oil Level ‘A’ Oil & Water Oil density = ρo
h1
Free Water Level
Water Water density = ρw
Capillary pressure difference between oil and water phases in core ‘A’ Pc = h1g (ρw-ρo) 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. Likewise, a free oil level and an oil-gas contact (OGC) may exist as shown in the figure above.
Capillary Pressure
3
0
VCL dec
1
0.2
ILD C ohmm
200
1.65
R HO CN us/f
2.65
1
G AS us/f
0
0
1
BVW dec
0
55
1
CN LLSS
1
C NLLSS 0
VC L
0.6
1
135
VCL
D TC us/f
O IL 0 BVW G
RH OC N
1
0 SW dec
-1
10750
10800
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:
h=
144 × Pc (ρ w − ρo )
A) Heavy oil:
Capillary Pressure
4
h=
144 × 5 = 100 ft. (66.5 − 59.3)
B) Light oil:
h=
144 × 5 = 31.6 ft. (66.5 − 43.7 )
Capillary Tube Model – Air-Water System:
Δh
θ
Air
Water
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.
Δh =
2 σ aw cos θ r g Δρ aw
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 σaw θ r g
Δρaw
= = = = = =
Height of water rise in capillary tube, cm. Surface tension between air and water, dynes/cm. Air/water contact angle, degrees. Radius of capillary tube, cm. Acceleration due to gravity, 980 cm/sec2. Density difference between water and air, gm/cm3.
Contact angle, θ, is measured in the denser phase (water in this case). This relation shows that the wetting phase (water) rise will be larger in small capillaries.
Capillary Pressure
5
pa1
Δh
pw1
Air
pa2 pw2 Water
Water rise in capillary tube depends on the density difference of fluids. pa2 pa1 pw1 Pc
= = = = = =
pw2 = p2 p2 - ρa g Δh p2 - ρw g Δh pa1 - pw1 ρw g Δh - ρa g Δh Δρaw g Δh
Combining the two relations:
Δh = Δh =
2 σ aw cos θ r g Δρ aw Pc
Δρ aw g
=
2 σ aw cos θ r g Δρ aw
Pc =
2 σ aw cos θ r
Pc
Δρ aw g
Capillary Pressure – Oil-Water System: From a similar derivation, the equation for capillary pressure for an oil/water system is:
Pc = Pc
σow θ r
= = = =
2 σ ow cos θ r
Capillary pressure between oil and water. Interfacial tension between oil and water, dyne/cm. Oil/water contact angle, degrees. Radius of capillary tube, cm.
Converting Laboratory Capillary Pressure Data to Reservoir Conditions: Basic Equations:
Capillary Pressure
6
2 σ L cos θ L rL 2 σ R cos θ R PcR = rR PcL =
= capillary pressure, psia = interfacial tension, dynes/cm = contact angle, degrees r = radius of pore throat, cm Subscripts L and R refer to laboratory and reservoir conditions, respectively. Pc
σ θ
Setting rL = rR and combining equations yields:
rL = rR →
2 σ L cos θ L 2 σ R cos θ R = PcL PcR
Therefore, capillary pressure at reservoir conditions is given by:
PcR =
σ R cos θ R PcL σ L cos θ L
Example: 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 = 20o Note: The reservoir data are very difficult to obtain. The reservoir data above are representative values based upon industry literature.
Injection Pressure, psia
2000
1600
1200
800
400
0 80
60
40
20
0
Mercury Saturation, percent pore space
Capillary Pressure
7
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) % 70 60 50 40 30 20 10
PcR =
PcL psia
PcR psia
1,320 820 560 410 310 240 200
80.5 50.0 34.2 25.0 18.9 14.6 12.2
(24) cos (20 ) P σ R cos θ R PcL = (480) cos (140 ) c σ L cos θ L
L
= 0.061 PcL Drainage and Imbibition Capillary Pressure Curves:
Drainage (1) Imbibition (2)
Pc
Pd
Si
0
Capillary Pressure
Sm
0.5 Sw
1.0
8
The drainage curve is always higher than the imbibition curve. Si Sm Pd
= Initial or irreducible wetting phase saturation. = critical non-wetting phase saturation. = 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 45 40 35 30
(1)
Pc , psi 25
(2)
20 15 10 5 0
0
0.25
0.5 0.75 1.0 Sw 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
Capillary Pressure
9
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: 20 Decreasing Permeability
Capillary Pressure
16
12 C B
A
8
4
0 0.2
0
0.4
0.6
0.8
1.0
Water Saturation -
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.
Capillary pressure, psia
Effect of Grain Size Distribution:
Well-sorted
Poorly sorted
Water saturation, %
Capillary Pressure
10
-
-
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.
Capillary pressure, psia
Effect of Saturation History:
Drainage
Imbibition
Water saturation, %
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: 20
Capillary Pressure
16 Decreasing θR
12
8
θR = 30°
θR = 0°
θR = 60°
4
θR = 80° 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).
Capillary Pressure
11
-
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.
Height Above Free Water Level
Effect of Interfacial Tension:
High Tension
Low Tension
0
1.0
Water Saturation Low interfacial tension indicates higher tendency of phases to mix together. With higher interfacial tension, transition zone is expected to be larger.
Height Above Free Water Level
Effect of Density Difference:
Small Density Difference
Large Density Difference 0
1.0
Water Saturation
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.
Capillary Pressure
12
-
The Leverett J-function was developed in an attempt to convert all capillary pressure data to a universal curve.
Definition of Leverett J-Function:
J (Sw) = J(Sw) Sw Pc
σ θ
k
φ
= = = = = = =
0.22 Pc σ cosθ
k
φ
Leverett J-function, dimensionless. water saturation, fraction. capillary pressure, psia. interfacial tension, dynes/cm. contact angle, degrees. formation permeability, md. porosity, fraction.
Example of J-Function for West Texas Carbonate 10.00
9.00 Jc Jm atch 8.00
Jn1 Jn2
7.00
Jn3
J-func tion
6.00
5.00
4.00
3.00
2.00
1.00
0.00 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
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 Jfunction. Use the following rock and fluid properties:
Capillary Pressure
13
σ θ
= =
72 dynes/cm. 0o.
k
φ
= =
47 md. 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 ta Pc Sw p s ia % 1 9 8 .3 2 9 8 .3 4 9 6 .8 8 5 9 .0 15 3 6 .3 35 2 5 .4 500 1 5 .3 Solution:
Calculated J-Functions 0.22* Sw J(Sw) % 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
0.22 * J (Sw )
100 80 60 40 20 0 0
Capillary Pressure
20
60 40 Sw , %
80
100
14
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 , P c, % 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
Capillary Pressure
15
