Fundamentals of Fluid Flow in Porous Media
Empirical Correlations of Relative Permeability
As a result of the difficulties and cost involved in measuring relative permeability values, empirical correlations and calculations are often employed in order to estimate the values. This is typically done in areas where no core data is available, or if economics dictate that running laboratory permeability tests is not feasible. Estimating relative permeability values through calculations is extremely fast, however the accuracy of the results is debatable. There are numerous methods that are available to estimate 2-phase relative permeability curves. Since it is such an important topic, numerous individuals have devoted their lives to developing reasonable methods to estimate relative permeability values. Two of the more common methods will be discussed here. These are the well-known Corey relations (an entirely theoretical approach to the problem) and the empirical Hornarpour correlations.
The often-used Corey relations are actually an extension of equations developed by Burdine et al. (1953), for normalized drainage effective permeability. The equations shown here are the original Burdine equations modified for relative permeability calculations:
- krw = wetting phase relative permeability
- krn = non-wetting phase relative permeability
- kro = non-wetting phase rel. perm. at irreducible wetting phase saturation
- Sw* = normalized wetting phase saturation
- λ = pore size distribution index
- Sm = 1 – Sor (1 – residual non-wetting phase saturation)
- Sw = water saturation
- Siw = initial water saturation
The major difference between the Burdine solutions and the equations shown here is in the non-wetting phase equation, eq. (2‑127). The kro term is added to account for the fact that the non-wetting phase solution must be at irreducible wetting phase saturation. The other modification is the Sm term proposed by Corey in order to represent the point where the non-wetting phase first begins to flow. This is known as the critical saturation point. What this means, is that for a period at the beginning of the non-wetting phase curve, there exists a period where there is no connectivity. At the critical saturation, a minimum number of pores are connected, at which point flow is possible and the first relative permeability value can be determined. The Sm term describes the saturation at which flow is first possible and is necessary in order to calculate realistic relative permeability values.
Determining Pore Size Distribution Index
The λ value (pore size distribution index) seen in equations (2‑126) and (2‑127) is critical in calculating relative permeability. The actual number represents how uniform the pore size is in the sample/reservoir. A low value of λ (i.e. 2) indicates a wide range of pore sizes, while a high value represents a rock with a more uniform pore size distribution. Using a λ value of 2 in equations (2‑126) and (2‑127) results in the well-known Corey equations. This value is considered a general value, and is thought to represent a wide range of pore sizes. Since a λ value of 2 is so general, it is often used when nothing else is known about the reservoir. Using a λ value of 2.4 or infinity results in Wyllie’s equation for 3 rock categories