Power Law

Summary

The power-law, or Corey-type, relative permeability relations raise the normalized mobile saturation of each phase to its own exponent: Water relative permeability is a power of Normalized mobile water saturation and Oil relative permeability is a power of its complement, with the Corey exponent for water and Corey exponent for oil free to differ. Use them when the curve shape is fitted to core data or to a wettability, and there is no pore size distribution index to constrain it.

Inputs and outputs

Equations

With the same normalized saturation as on the Brooks-Corey page,

limited to the interval 0 to 1, the two relative permeabilities are

Limits. At \(\SeNorm = 0\), \(\krWat = 0\) and \(\krOil = \krOilEnd\); at \(\SeNorm = 1\), \(\krWat = \krWatEnd\) and \(\krOil = 0\). With exponents above 1 both curves are convex near their zero ends, and the curves cross once.

Link to Brooks-Corey. The water curve of the Brooks-Corey model is a power law with the exponent \(\nCw = 2/\lambda + 3\), so \(\lambda\) = 2 gives \(\nCw\) = 4. The oil curve of the Brooks-Corey model is not a pure power law. A power-law exponent is a free parameter, and Brooks-Corey ties it to the pore size distribution.

Single-value calculator

Behavior

The exponent controls how quickly a phase becomes mobile. The plot is on a log axis and shows the water curve: a larger water exponent lowers it at every saturation below the endpoint, and the difference grows towards low saturation. At a water saturation of 0.30, water kr is 0.0099, 0.0018 and 0.00006 for exponents of 2, 3 and 5, and at 0.50 it is 0.0893, 0.0487 and 0.0145. At 0.70 the curves are close together, at 0.248, 0.225 and 0.186, and they meet at the endpoint of 0.30. For an oil exponent of 3, oil kr is 0.548 at 0.30, 0.094 at 0.50 and 0.0008 at 0.70. With both exponents at 3 and a water endpoint of 0.3, the curves cross at a water saturation of 0.529, where both are 0.064; a lower water endpoint moves the crossing to a higher water saturation.

Parameter guidance

Exponents. Fit them to measured relative permeability from special core analysis, by a straight-line fit of the logarithm of kr against the logarithm of the normalized saturation. The default of 3 for both is illustrative. Values from 2 to 4 for oil and from 2 to 6 for water are commonly quoted for sandstone; they are not a substitute for core. In water-wet rock the water exponent tends to be higher and the water endpoint lower, and in oil-wet rock the reverse, so the two should be chosen together.

Endpoints. Take them from the same core data. A water endpoint of 1 with a very low residual saturation is not realistic; see the Brooks-Corey page for the same discussion.

Irreducible and residual saturation. From the Irreducible and Residual Saturation page. They set the window of the exponents.

When to use it. Use the power law when core data exist or when the two phases need different shape parameters (for example for a mixed-wet rock), and the Brooks-Corey form when the pore size distribution index is known and no relative permeability measurement is available.

Worked example

Corey curves with Swirr = 0.20, Sor = 0.25, a water endpoint of 0.30, and exponents of 3 for both phases, then the crossing saturation by bisection, and a check that the Brooks-Corey water curve for \(\lambda\) = 2 equals a power law with an exponent of 4:

swirr, sor, nw, no, krw0, kro0 = 0.20, 0.25, 3.0, 3.0, 0.30, 1.0
d = 1 - swirr - sor
def curves(sw):
    se = min(1.0, max(0.0, (sw - swirr) / d))
    return se, krw0 * se ** nw, kro0 * (1 - se) ** no
print(f"{'Sw':>5} {'Se':>6} {'krw':>8} {'kro':>8}")
for sw in (0.20, 0.30, 0.50, 0.70, 0.75):
    se, krw, kro = curves(sw)
    print(f'{sw:5.2f} {se:6.3f} {krw:8.4f} {kro:8.4f}')
lo, hi = swirr, 1 - sor
for _ in range(60):
    mid = (lo + hi) / 2
    se, krw, kro = curves(mid)
    lo, hi = (lo, mid) if krw > kro else (mid, hi)
se, krw, kro = curves(lo)
print(f'crossing at Sw = {lo:.3f}, Se = {se:.3f}, kr = {krw:.4f}')
lam, se = 2.0, 0.6
print(f'Brooks-Corey water, lambda 2, Se 0.6: {se ** ((2 + 3 * lam) / lam):.5f}; power law n = 4: {se ** 4:.5f}')

Output

   Sw     Se      krw      kro
 0.20  0.000   0.0000   1.0000
 0.30  0.182   0.0018   0.5477
 0.50  0.545   0.0487   0.0939
 0.70  0.909   0.2254   0.0008
 0.75  1.000   0.3000   0.0000
crossing at Sw = 0.529, Se = 0.599, kr = 0.0645
Brooks-Corey water, lambda 2, Se 0.6: 0.12960; power law n = 4: 0.12960

Assumptions and limitations

• Each phase has a single power law, with no curvature in the log-log relation of kr and normalized saturation. Real curves may need a different exponent at low and high saturation.
• The exponents are constant within the rock type, and Swirr, Sor and the endpoints are known.
• The curves are for one flow direction (drainage or imbibition) and one wettability.
• The endpoints are independent of the exponents. In measurements, they are correlated with wettability, and with each other.
• A pure power law has no relation to the pore size distribution, so nothing ties it to the capillary pressure curve.

QC checks

• Check the endpoints: water kr is 0 at Swirr and the endpoint at 1 - Sor, oil kr is the endpoint at Swirr and 0 at 1 - Sor.
• The crossing saturation and the endpoint of water kr agree with the wettability of the rock: low water endpoint and crossing above 50% of the mobile range for water-wet rock.
• On a log-log plot of kr against the normalized saturation, the measured data fall on a straight line of slope equal to the exponent. Curved data need a different model.
• The fit exponents are inside the usual ranges. An exponent above 6 or below 1.5 points to a bad fit or an endpoint error.
• Compare with Brooks-Corey using the pore size distribution index from capillary pressure. A large difference between the two on the same rock is worth understanding.

Going Deeper

The form was used by Corey in 1954 for gas-oil systems, with an exponent of 4 for oil, and has been generalised since, with the exponents as free parameters, in the form often called Corey or power-law relative permeability. It is the common choice for history matching, because only a few parameters (two exponents and two endpoints) have to be adjusted and the curves are smooth. In reservoir simulation work, tables from core measurements are used where they exist, and Corey curves where they do not or where the measured curves need to be smoothed and extended. Craig's rules of thumb relate the curves to wettability: in strongly water-wet rock the water endpoint is below about 0.3, the connate water above 20 to 25%, and the curves cross above a water saturation of 50%; in oil-wet rock the water endpoint is above 0.5 and the crossing is below 50%. These come from memory and should be checked before they are used as a screening rule. A log-derived relative permeability curve is a model with these exponents and the saturation from the log, and does not measure the exponents.

References

1. Corey, A.T., 1954. The interrelation between gas and oil relative permeabilities. Producers Monthly, 19(1), 38–41.
2. Craig, F.F., 1971. The Reservoir Engineering Aspects of Waterflooding. SPE Monograph Series Vol. 3, Society of Petroleum Engineers, Dallas.
3. Brooks, R.H. and Corey, A.T., 1966. Properties of porous media affecting fluid flow. Journal of the Irrigation and Drainage Division, ASCE, 92(2), 61–88.

Python reference implementation

Related methods

• Brooks-Corey
• Irreducible and Residual Saturations
• Fractional Flow