1. Relative permeability
Definition of relative permeability
Laboratory method of measuring relative permeabilities
Effect of wettability
Factors affecting relative permeability
Correlations
Averaging
It is expected that students will be able to:
define relative permeability of rock to a particular fluid for multi-phase fluid flow system
draw relative permeability curves
describe relative permeability measurements using Steady State and Un-Steady State Flow techniques
explain the effect of rock wettability on relative permeabilities and draw the relevant graph
explain factors affecting relative permeability
estimate relative permeabilities using published correlations, such as Stone, Corey, and NPC.
calculate and draw curves for relative permeability.

2. Relative permeability
Definition
Give the accurate definition of relative permeability..
Determination of relative permeability in lab.
Introduce the steady state and unsteady state methods, their strength and weaknesses.
Factors influencing relative perm
Discuss the effect of wettability, drainage process, imbibition process and connate water.
Correlations
Discuss and apply the correlations to estimate relative perm.

3. Theory
Concept
If two or more fluid flow through a porous media, each fluid will flow according to Darcy’s Law.
Saturated with water, oil and gas

4. Theory

5. Theory
Effective permeability is the ability of a porous media to flow a fluid in the presence of one or more other fluids.
The effective permeability to the fluid depends on
the amount or saturation of that fluid in the porous media
Wettability characteristics of the porous media
Saturation history

6. Theory Relative permeability
The effective permeability is usually presented in term of relative permeability.
Relative permeability is the ratio of the effective permeability to a base permeability.
For example,
The ratio of effective permeability compared to absolute permeability
or,
The ratio of effective permeability to the effective permeability of non-wetting phase at irreducible wetting phase saturation.

7. Relative Permeability Curve

8. Relative Permeability Curve

9. Relative Permeability Curve

10. Relative Permeability Curve
Point 1 - on the wetting phase relative permeability curve shows that a small saturation of nonwetting phase will drastically reduced the relative permeability of the wetting phase (the nonwetting phase occupies the larger pore spaces, and it is in the large pore spaces that flow occurs with the least difficulty.
Point 2 - on the nonwetting phase relative permeability curve shows that the nonwetting phase begins to flow at relatively low saturation of the nonwetting phase. The saturation of the oil at this point is called critical oil saturation Soc.

11. Relative Permeability Curve
Point 3 - on the wetting phase relative permeablity curve shows that the wetting phase will cease to flow at a relatively large saturation (the wetting phase preferentially occupies smaller pore space, where capillary forces are the greatest). The saturation of the water at this point is refered to as the irreducible water saturation Swir (ketepuan air tak terkurang)
Point 4 - on the nonwetting phase relative permeability curve shows that, at low saturations of the wetting phase, change in the wetting phase saturation have only small effect on the magnitude of the nonwetting phase relative permeability curve ( at low saturations the wetting phase fluid occupies the small pore spaces which do not contribute materially to flow, and therefore changing the saturation in the small pore spaces has relatively small effect on the flow of the nonwetting phase).

12. Relative Permeability Curve
Write your own note on Gas-oil relative permeability curves

13. Effect of Res. Parameters on kr
2 reservoir parameters considered:
1. Saturation history
drainage
imbibition
2. Rock wettability
water wet rock
oil wet rock

14. Effect of Res. Parameters on kr
Saturation history
2 types of saturation history
Drainage process
- Porous rocks is initially saturated with wetting fluid. The wetting fluid was then displaced with non-wetting fluid. This process, displacement of wetting phase by non-wetting phase, is called drainage process.
Example – A water-wet rock that was saturated with water. Oil is then injected into the rock and displacing the water. The oil was non-wetting relative to water.
Imbibition process
- Porous rocks is initially saturated with non-wetting fluid. The non-wetting fluid was then displaced with wetting fluid. This process, displacement of non-wetting phase by wetting phase, is called imbibition process.
Example – An water-wet rock was saturated with oil. Water is then injected into the rock and displacing the oil. The oil was non-wetting relative to water.

15. Effect of Res. Parameters on kr
Saturation history
Hysteresis: refers to irreversibility or path dependence.
Drainage relative permeability curve is higher than the imbibition curve for non-wetting phase.

16. Effect of Res. Parameters on kr
Rock wettability
Several important differences between oil-wet curves and water-wet curves are generlly noted:
a. The water saturation at which oil and water permeabilities are equal (intersection point of curves) will generally be greater than 50% for water-wet system and less than 50% for oil-wet system.
b. The connate water saturation for a water-wet system will generally be greater than 20%, whereas, for oil wet-systems, it will normally be less than 15%.
c. The relative permeability to water at maximum water saturation (residual oil saturation) will be less than about 0.3 for a water-wet system, but will be greater than 0.5 for oil-wet systems.

17. Relative Permeability Correlation
Two-phase relative permeability correlations
There are several correlations, among them are:
a. Wyllie and Gardner Correlation (1958)
b. Torcaso and Wyllie Correlation (1958)
c. Pirson's Correlation (1958)
d. Corey's Method (1954)

18. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
These correlations use the effective phase saturation as the correlating parameter.
So*, Sw*, Sg* = Effective oil, water and gas saturation
So, Sw, Sg = Oil, water and gas saturation
Swc = Connate (irreducible) water saturation

19. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
a. Wyllie and Gardner Correlation (1958)
Types of formation
kro
krw
Unconsolidated sand, well sorted
Unconsolidated sand, poorly sorted
Cemented sandstone, oolitic limestone

20. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
a. Wyllie and Gardner Correlation (1958) (cont.)
Types of formation
kro
krg
Unconsolidated sand, well sorted
Unconsolidated sand, poorly sorted
Cemented sandstone, oolitic limestone

21. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
a. Wyllie and Gardner Correlation (1958) (cont.)
If one relative permeability is available
System kr
Oil-water system
Gas-oil system

22. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
b. Torcaso and Wyllie Correlation (1958)
kro in a gas-oil system.
kro is calculated from the measurements of krg.

23. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
c. Pirson's Correlation (1958)
Process
Wetting phase
Nonwetting phase
Imbibition
Drainage

24. Relative Permeability Correlation
Two-phase relative permeability correlations (cont.)
d. Corey's Method (1954)
A simple mathematical expression for generating the relative permeability data for the oil-gas system. The approximation is good for drainage processes, i.e gas-displacing oil.

25. Example
Generate the relative permeability data for an unconsolidated well-sorted sand by using the Wyllie and Gardner method. Assume the following critical saturation values:
Soc = 0.3, Swc = 0.25, Sgc = 0.05
Solution
Drainage oil-water system
Drainage oil-gas system

26. Oil-water relative permeability

27. Oil-gas relative permeability

28. Example
Generate the relative permeability data for an unconsolidated well-sorted sand by using the Pirson's method for water-oil system. Assume the following critical saturation values:
Soc = 0.3, Swc = 0.25, Sgc = 0.05
Solution
Process
Wetting phase
Nonwetting phase
Imbibition
Drainage

29. Oil-water relative permeability (assume oil-wet system)

30. Example
Use Corey's approximation to generate the gas-oil relative permeability for a formation with connate water saturation 0.25.
Solution

31. Gas-oil relative permeability

32. NORMALIZATION AND AVERAGING
Results of relative permeability tests performed on several core samples of a reservoir rock often vary.
It is necessary to average the relative permeability data obtained on individual rock samples.
Prior to usage for oil recovery prediction, the relative permeability curves should first be normalized to remove the effect of different initial water and critical oil saturations.
The relative permeability can then be de-normalized and assigned to different regions of the reservoir based on the existing critical fluid saturation for each reservoir region.

33. NORMALIZATION AND AVERAGING
The most generally used method adjusts all data to reflect assigned end values, determines an average adjusted curve and finally constructs an average curve to reflect reservoir conditions.
These procedures are commonly described as normalizing and de-normalizing the relative permeability data.

34. NORMALIZATION AND AVERAGING
For a water-oil system:
Step 1. Select several values of Sw starting at Swc (column 1), and list the corresponding values of kro and krw in columns 2 and 3.
Step 2. Calculate the normalized water saturation S*w for each set of relative permeability curves and list the calculated values in column 4 by using the following expression:
where ;
Soc =Critical oil saturation
Swc = Connate water saturation
S*w = Normalized water saturation

35. NORMALIZATION AND AVERAGING
Step 3. Calculate the normalized relative permeability for the oil phase at different water saturation by using the relation (column 5):
where kro = relative permeability of oil at different Sw, (kro)Swc = relative permeability of oil at connate water saturation: kr*o = normalized relative permeability of oil.
Step 4. Normalize the relative permeability of the water phase by applying the following expression and document results of the calculation in column 6.
where (krw)Soc is the relative permeability of water at the critical oil saturation.

36. NORMALIZATION AND AVERAGING
Step 5. Using regular Cartesian coordinate, plot the normalized kro* and krw* versus Sw* for all core samples on the same graph.
Step 6. Determine the average normalized relative permeability values for oil and water as a function of the normalized water saturation by select arbitrary values of Sw* and calculate the average of kro* and krw* by applying the following relationships:
and
where n = total number of core samples, hi = thickness of sample i, ki = absolute permeability of sample i.

37. NORMALIZATION AND AVERAGING
Step 7. The last step in this methodology involves de-normalizing the average curve to reflect actual reservoir and conditions of Swc and Soc. These parameters are the most critical part of the methodology and, therefore, a major effort should be spent in determining representative values.
The Swc and Soc are usually determined by averaging the core data, log analysis, or correlations, versus graphs, such as: (kro)Swc vs. Swc, (krw)Soc vs. Soc, and Soc vs. Swc which should be constructed to determine if a significant correlation exists.
Often, plots of Swc and Sor versus log (k/Ф)0.5 may demonstrate a reliable correlation to determine end-point saturations as shown schematically in Figure 5-8.
When representative end values have been estimated, it is again convenient to perform the denormalization calculations in a tabular form as illustrated below:

38. NORMALIZATION AND AVERAGING
where (kro)Swc and (krw)Soc are the average relative permeability of oil and water at connate water and critical oil, respectively, and given by:

39. NORMALIZATION AND AVERAGING

40. Example