Capillary Rise

1. Capillary Pressure
Objectives:
- List the 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 the 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 application.
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 and blocks.
Capillary Pressure Concept:
Water exists at all levels below 2, and both water and oil exist at all levels above 2
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 1

2. Capillary Pressure Definition:
• The pressure difference existing across the interface separating two immiscible fluids.
• It is usually calculated as
Pc = Pnwt - Pwf
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 =ρg-ρw
Water-wet water-oil system:
Pc =ρo-ρw
oil-gas system:
Pc =ρg-ρo
Relation between Capillary Pressure and Fluid Saturation
For uniform sands, reservoir fluid saturation is related to the height above the free-water level by the following
relation:
Capillary Pressure 2

3. Fluid Distribution in Petroleum Reservoirs:
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 VVOC, 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

4. 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 me 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
Ibm/ft3 Calculate the length of the transition zone in the following two situations:
A. oil density = 59.3 fbm/ft3
B. oil density = 43.7 ibm/ft3
This example assumes the rock quality is the same in both situations.
Solution:
Rearranging the relation between capillary pressure and height:
A. Heavy oil
B. Light oil
Capillary Tube Model - Air-Water System:
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.
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5. 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 = Interfacial 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, ft 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.
Water rise in capillary tube depends on the density difference of fluid
Combining the two relations:
Capillary Pressure 5

6. Capillary Pressure - Oil-Water System:
From a similar derivation, the equation for capillary pressure for an oil/water system is:
Converting Laboratory Capillary Pressure Data to Reservoir Conditions:
Basic Equations:
Subscripts L and R refer to laboratory and reservoir conditions, respectively.
Setting rL = rR and combining equations yields:
Therefore, capillary pressure at reservoir conditions is given by:
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 cynes/cm. θHg = 140°
Reservoir Data: σow = 24 dynes/cm. θow = 20°
Note. The reservoir data are very difficult to obtain. The reservoir data above are representative values based
upon industry literature.
Capillary Pressure 6

7. Solution;
Steps:
• Solve the equation that relates lab capillary pressure data to reservoir capillary pressure data for the
conditions we have.
• Obtain laboratory capittaiy pressure data from the curve for Sample 28.
• Convert the lab numbers to reservoir capillary pressure.
Drainage and Imbibition Capillary Pressure Curves:
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 fiutd 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 m which the saturation of the wetting phase increases and the nonwetting phase
saturation decreases.
• Mobility of wettirg phase increases as wetting phase saturation increases.
Capillary Pressure 7

8. Effects of Rwrvoff Properties on Capillary Pressure:
- Capillary pressure characteristics in reservoir are affected by:
• Variations in permeability
• Grain size distribution
• Sataration "history
• Contact angle
• Interfacial tension
• Density difference between fluids
Effect of Permeability:
• Curves shift to the nght (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 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 8

9. Effect of Saturation History:
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:
• 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:
Capillary Pressure 9

10. Low interracial tension indicates higher tendency of phases to mix together.
With higher interfacial tension, transition zone is expected to be larger.
■
Effect of Density Difference:
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 pressure reservoir.
• The Leverett J-function was developed in an attempt to convert ail capillary to a universal
curve.
Definition of Leverett J-Function:
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.
Capillary Pressure 10

11. Example of J-Function for West Texas Carbonate
10.00
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
Jc
Jmatch
Jn1
Jn2
Jn3
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
Water saturation, fraction
J-function
The cata represented by plus signs (+} are from an original capillary pressure experiments. The outer
data were for three new experiments, performed to cetermine 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 lithoiogies.
• J-function usually does not predict an accurate correlation for different lithoiogies.
• 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 %
Capillary Pressure 11

12. Solution:
In the table above.we have no multiplied through by the conversion factor 0.22
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 =100md.
Φ =10%
Solution :
Laboratory Methods for Measuring Capillary Pressure:
• Porous diaphragm method
• Mercury injection
• Centrifuge method
• Dynamic method
Capillary Pressure 12