Relative Permeability

Purpose

Relative permeability says how much each fluid can flow when two share the pore space: the fraction of the base permeability that remains for water and for oil, as a function of water saturation. It answers whether a zone will produce hydrocarbon, water or both, and it is the input of any flow calculation, from a water cut estimate to a reservoir simulation. Core measurements (steady-state and unsteady-state displacement tests, special core analysis) are the reference. A log-based curve is a model driven by saturation: it does not replace the measurement, but it can be built at every depth, and it makes the link between the petrophysical saturation and flow explicit.

Position in the workflow

Upstream. The curves need saturation (Water saturation from the Water Saturation step, or from a saturation-height model), irreducible water saturation (Irreducible water saturation) from the Swirr topic, and a residual oil saturation, which can come from the flushed-zone saturation (Flushed-zone water saturation, see Rxo and Sxo) or from core. For the Brooks-Corey form, the pore size distribution index comes from capillary pressure data or from the saturation-height fit.

Downstream. The curves and the fractional flow feed:

a water cut and water-free production flag for each interval, used in Cutoffs and in completion decisions,
the mobile hydrocarbon volume, which is the connection from saturation to movable volumes, and
reservoir simulation, as curves or as end points that scale the simulator's tables.

Error propagation. The end points are the largest source of error. The normalized saturation is rescaled between Swirr and 1 - Sor, so a window of 0.55 changes by 0.05 when one end point is off by 0.05, about 9%, and the curve shifts by the same fraction of its range. The exponents then act on that error: the water curve is a power of 4 near Se = 1 in the Brooks-Corey form, and a 9% shift of Se is a change of about 40% in water kr.

Key concepts

End points. The irreducible water saturation and the residual oil saturation bound the mobile window, and the endpoint values of the relative permeabilities scale the curves. They are covered on Irreducible and Residual Saturation.

Normalized saturation. All the models use a saturation scaled between the two end points, which is 0 where water stops flowing and 1 where oil stops flowing.

Wettability. The shape of the curves and the position of their crossover depend on which fluid wets the rock. The equations in this topic are for a water-wet rock.

Two families of curves. Brooks-Corey ties both curves to one pore size distribution index. Power-law curves have a free exponent for each phase.

Fractional flow. The curves and the viscosity ratio give the share of the flow that is water. See Fractional Flow.

Hysteresis and direction. Drainage and imbibition curves differ, and the models here are single curves. Their direction of use (hydrocarbon filling or water displacing) should be stated.

Log versus core. The log result is a model of the curve at the logged saturation. It is calibrated by special core analysis, not derived from logs alone.

Method selection guide

Method	Inputs	Use when	Strengths	Weaknesses
Brooks-Corey	Sw, Swirr, Sor, pore size distribution index, endpoints	The pore size index is known from capillary pressure or a saturation-height fit	One parameter, consistent with the capillary pressure model	Same index for both phases; no hysteresis; the oil form depends on the wetting assumption
Power law	Sw, Swirr, Sor, two exponents, endpoints	Core curves exist and can be fitted, or the two phases need different shapes	Flexible, simple to fit and history match	No link to the pore structure; fits only the range of the data
Irreducible and residual saturation	Swirr source, Sxo or core Sor	Always: the end points of every curve	Defines the mobile window and flags immobile zones	Sor from Sxo is an approximation; Swirr depends on the drive
Fractional flow	Both curves, viscosities	A water cut or a water-free flag is wanted from a saturation	Direct link to production behaviour; shock-front recovery	Neglects gravity and capillary pressure; the curves dominate the result

Decision guidance

Fix the end points first. They matter more than the choice of curve.
If special core analysis curves exist, fit a power law to them, and use that. They are the reference.
If there is no measurement and a capillary pressure fit exists, use Brooks-Corey with the index from that fit.
If there is neither, use a power law with generic exponents and carry a range of exponents and end points into the result.
For a water-free or water-cut screen, compute the fractional flow with the end points and viscosity ratio, and calibrate the thresholds to production tests.
For an oil-wet or mixed-wet rock, do not use the water-wet equations unchanged.

Shared parameter picking

Irreducible water saturation. From the Swirr topic or from the lower end of the saturation-height model, on the same porosity basis as Sw. One value or curve per rock type.

Residual oil saturation. From core where it exists, otherwise from \(1 - S_{xo}\) in the invaded zone or a constant for the formation. Record which one was used.

Endpoints of the curves. The water relative permeability at residual oil and the oil relative permeability at irreducible water come from core. They scale the curves and are shared by the Brooks-Corey and power-law forms.

Rock type. The end points and the shape of the curves vary with rock quality, so use the rock types or flow units of the saturation-height model.

Wettability and direction. State which one applies. The equations here assume water-wet rock.

Viscosities. Reservoir values of water and oil (or gas) viscosity, shared with any fractional-flow check and with the fluid properties of the other steps.

Saturation source. Log Sw or the saturation-height model. The same source should be used in every interval.

Recommended default approach

Absent other information, a careful generalist would:

Decide the wettability and the rock types, and the porosity basis.
Take Swirr from the Swirr topic or the saturation-height model, and Sor from core or from the flushed zone, and check that the mobile window is positive and of sensible size.
Take the endpoints and, if there are measured curves, the exponents from special core analysis.
Without measured curves, use Brooks-Corey with the index from the capillary pressure fit, or a power law with exponents of about 2 to 4, and say that the curves are generic.
Compute kr for water and oil from the log or modelled saturation, and the fractional flow from the viscosities.
Compare with production tests: water-free intervals should have low fractional flow, and water-producing ones high.
Carry a range of end points and exponents into any conclusion that depends on the curves.

Combining methods

Do not blend Brooks-Corey and power-law curves. They are alternative parametrisations of the same curve, and a mean of the two has no physical meaning and may not meet the end points. Pick one family per rock type: measured curves fitted by a power law where there are measurements, and Brooks-Corey where the curve is tied to capillary pressure data. Use the other as a sensitivity check. If they differ widely on the same rock, the shape parameters are not consistent with the capillary pressure data, and that is worth investigating. Where core curves exist for only some rock types, use them for those and a generic curve for the rest, and mark the difference.

QC of results

A good result:

has water kr equal to 0 at Swirr and oil kr equal to 0 at 1 - Sor, with the endpoint values at the other ends,
has curves that cross at a saturation consistent with the wettability,
uses end points that agree with core and with the Swirr and flushed-zone results,
gives a fractional flow near 0 in zones that produce water-free and near 1 in zones that produce water, and
has a mobile window of a sensible size, normally 0.3 to 0.7.

Signs of a bad result: a negative or very narrow mobile window, a normalized saturation of 1 in a zone that produces oil, curves that do not meet the end points, an exponent far outside the usual range, and a water cut that does not match tests.

Common pitfalls

Using a water-wet equation for an oil-wet or mixed-wet rock.
Using the wetting-phase form for the non-wetting phase, or the reverse.
Taking Swirr and Sor on a different porosity basis from Sw.
Using \(1 - S_{xo}\) as Sor in a zone that was not fully flushed, or in gas.
Treating the endpoints as 1 without checking core.
Using one curve for several rock types.
Fitting exponents to a narrow range of data and using them over the whole window.
Neglecting gravity and capillary pressure in the fractional flow of a thick column or a transition zone.
Treating a log-based curve as a measurement.

Going Deeper

Relative permeability began with Wyckoff and Botset's observation in 1936 that two fluids flowing together impede each other, and with Leverett's and Buckley's work in the early 1940s, which turned the curves into a displacement theory. Corey in 1954 and Brooks and Corey in the 1960s supplied the closed-form power-law and pore-size-distribution models used since. (The dates and names in this paragraph are from memory and are not verified.) The unresolved difficulty is that relative permeability depends on wettability, saturation history and pore structure in ways that no equation captures, and that a log can only supply the saturation. Practice has therefore moved to rock-type-specific curves from special core analysis, with logs used to assign the rock types and the saturations, and the closed-form equations used to fill gaps and to scale the end points.