[1]:

• λ = 2 (cemented sandstones, oolotic and small-vug limestones)
• λ = 4 (poorly sorted unconsolidated sandstones)
• λ = infinity (well sorted unconsolidated sandstones)

Wyllie’s equations are used when there is some general knowledge of the geology of the reservoir.

The Corey and Wyllie equations are sufficient for approximation purposes, but in order to obtain a more accurate value of the pore size distribution index, λ can be determined empirically from capillary pressure data. The equation shown here was developed by Brooks and Corey (1964 & 1966), and relates capillary pressure to normalized wetting phase saturation:

Where,

• P_c = capillary pressure
• P_e = minimum threshold pressure
• S_w* = normalized water saturation, eq. (2‑128)

If capillary pressure data is available, eq. (2‑130) can be used to determine the pore size distribution index. A log-log plot of the capillary pressure vs the normalized water saturation should result in a straight line with a slope of –1/ λ and an intercept of P_e. This method of determining λ from experimental data is preferable to using the Wyllie or Corey relations, since the value obtained with eq. (2‑130) can actually be backed up with hard data.

Example 2‑9[2]

For a water wet reservoir with S_wi = 0.16 and residual gas saturation equal to 0.05, use the following capillary pressure data to find the relative permeability data.

Pc ( Sw )	Sw	Pc ( Sw )	Sw
0.5	0.965	8	0.266
1	0.713	16	0.219
2	0.483	32	0.191
4	0.347	300	0.16

Solution

Step 1: Calculate Normalized Water Saturation (S_w*), using eq. (2-130)

Pc ( Sw )	Sw	Sw*	Pc ( Sw )	Sw	Sw*
0.5	0.965	0.958	8	0.266	0.126
1	0.713	0.658	16	0.219	0.070
2	0.483	0.385	32	0.191	0.037
4	0.347	0.223	300	0.16	0.000

 

Step 2: Deterimne λ by Plotting LogP_c vs. LogS_w*

Recall eq.(2‑130), Slope of the graph is –1/ λ = -1.25, Therefore, λ = 0.8

Step 4: Calculating Non-Wetting Phase Relative Permeability at Irreducible Wetting Phase Saturation ( k_ro ),

Recall eq. (2‑129), k_r^o = 0.919 and S_m = 0.95 = 1-S_rg.

Step 5: Calculating Relative Perm Values, Recall equations (2‑126) and (2‑127) to find the relative permeability of the respective phases at various water saturations:

Then plot k_rg and k_rw vs. S_w in order to display the information in its most typical form:

Hornarpour Correlations

A different technique was approached by Hornarpour et al. in order to come up with correlations to estimate relative permeability. Instead of attempting to solve the problem theoretically, Hornarpour developed an entirely empirical solution to the problem. Data from numerous different fields around the world was gathered, and stepwise linear regression analysis was used to come up with mathematical descriptions to match the actual data. Data sets came from Canada, U.S., Alaska and the Middle East. The data sets were classified as either carbonate or non-carbonate and also broken up into wettability and property ranges. In this manner, equations were developed for numerous different reservoir conditions.

In order to use the Hornarpour correlation, a general understanding of the reservoir is required. Namely the fluids in the system, a rough description of the geology, the wettability and the range of rock properties and fluid saturations in the reservoir. This information is then used to determine which set of equations are to be utilized in calculating the wetting and non-wetting phase relative permeability.  Hornarpour’s paper[3] provides all the necessary details (tables and equations) that are required for this method.

Comparison of Empirical Methods

Out of the methods discussed here, the most accurate results are obtained from the method in which the pore size distribution index is determined experimentally, and is then utilized in equations (2‑126) and (2‑127) for drainage relative permeability. The other methods (Corey, Wylie and Hornarpor) rely a great deal on general estimates of the reservoir conditions. For example, in the Hornarpor correlations the only choices available for geology are either carbonate or non-carbonate. Since the description of the reservoir in these methods is so vague, it is uncertain how accurate the results obtained from them can be with respect to the actual reservoir conditions. Conversely, the pore size distribution index is determined experimentally and there is hard data to back up the results.

Other Methods

Since relative permeability is such an important topic, numerous individuals have devoted their lives to determining methods to estimate relative permeability values. As such, there are numerous different techniques and methods that are used to calculate relative permeability. For example, relative permeability can be calculated through field data and analysis of pressure data. Some of the other methods not discussed in detail include theoretical models such as Naar-Wygal’s and Naar-Henderson (1961), which are for imbibition processes. The Burdine equations discussed here are for drainage calculations only.

Overall Comparison of Methods

As discussed, there are many different ways, in which relative permeability values can be estimated, from rigorous laboratory measurements, to “quick and dirty” calculations. With all of these methods available, the question as to which method should be used becomes extremely important. Like most things in the reservoir-engineering world, all of the options have their advantages and disadvantages. The only way to make a sound decision is to see what is best for the given situation, and weigh the options accordingly. For example, if relative permeability data is required for a major discovery in the North Sea, then chances are good that you would decide to spend the money to get an excellent core sample in order to run a rigorous steady-state test on it. Conversely, if data for a mature oil field is required then a quick calculation or displacement test would be sufficient for your purposes.  As always, economics and practicality will be the overriding factor in deciding what sort of method to use in order to estimate relative permeability values.

References

[1] M.B. Standing (1975)

[2] International Petroleum Consultants, “Fundamentals of Relative Permeability”, Mobil Oil Indonesia Course Notes, 1985.

[3] Honarpour et al (1982)

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