Permeability

Purpose

Permeability, Permeability, answers how easily the rock lets fluid through. It is the property that decides whether a zone is worth completing, how much rate a well gives, and whether a cutoff-based net pay is a producible interval or only a porous one. No log measures it directly. Every curve on this step is an estimate from porosity and other logs, calibrated to core or tests, and is only as good as that calibration.

This step covers the log-based equations that use porosity and irreducible water saturation, and the porosity-permeability transforms fitted directly to core.

Position in the workflow

Upstream. Swirr comes first. Wyllie-Rose, Timur, Tixier, Coates and Heslop all take an irreducible water saturation as input, so compute Irreducible water saturation on the Swirr step before any of them, and from a method that does not itself use permeability (Buckles, Foil, Lucia or NMR). The other inputs are porosity (Total porosity or Effective porosity) from the Porosity step and core permeability for calibration. A porosity-permeability transform needs only porosity and core.

Downstream. Permeability feeds:

the cutoffs, where a permeability limit often defines net reservoir,
permeability-thickness (kh) and flow-capacity calculations in volumetrics,
hydraulic flow unit and rock typing, and relative permeability, and
reservoir models and simulation.

Error propagation. All of these models are steep power laws. With a porosity exponent of 4.5 a 10% relative error in porosity is a 54% error in permeability (1.1 to the power 4.5 is 1.54). With Swirr to the power -2 (Timur, Wyllie-Rose), a Swirr of 0.30 in place of 0.25 lowers permeability by 31%, and the Coates ratio lowers it by 40%. Using the log water saturation in place of Swirr is worse: with Sw of 0.30 in a hydrocarbon zone where Swirr is 0.20, a Timur-type model reads 56% too low, and at Sw of 0.60 in a transition zone it reads 89% too low. Permeability errors are on a log scale, and a factor of 2 to 3 is the usual limit of what is achievable.

Key concepts

Permeability is estimated, not measured. Logs respond to porosity, saturation and mineralogy, which control permeability only indirectly. The equations use porosity, which sets the pore volume, and Swirr, which is a proxy for pore size, to get a result. They are correlations.

Empirical constants. The multipliers and exponents in Wyllie-Rose, Timur, Tixier, Coates and Heslop come from fits to particular core sets. They are not universal, and the same equation with different constants can differ by a factor of 10 or more. Treat the published values as a starting point to be recalibrated to core for each rock type.

Swirr, not Sw. The models use the irreducible water saturation. Using the log water saturation makes permeability depend on how much hydrocarbon is in the zone, which is not a property of the rock.

Log-scale quantity. Permeability spans many decades and is close to lognormal. Compare, plot and regress it as log permeability, and average it with the mean that matches the flow geometry (arithmetic along layers, harmonic across, geometric as a central value).

Scale. Core plugs measure centimetres of rock, logs measure feet, and a test measures hundreds of feet. They give different values, and the log result is smoother than the plug data.

Method selection guide

Method	Inputs	Use when	Strengths	Weaknesses
Wyllie-Rose	Total porosity, Swirr	Clastics with a credible Swirr and core to calibrate	Simple template, explicit constants	Constants vary widely between sources
Coates	Effective porosity, Swirr (or NMR FFI/BVI)	NMR available, or a good effective-basis Swirr	Same form as the NMR permeability; excludes clay-bound water	Very sensitive to Swirr at high Swirr
Timur	Total porosity, Swirr (percent units)	Sandstone with a credible Swirr	Widely used, fit to sandstone core	Unit trap (percent); sandstone only
Tixier	Total porosity, Swirr	A quick check, fluid known	Simple, fluid-specific multiplier	Sixth-power porosity dependence; attribution unclear
Heslop	Effective porosity, Swirr	Only as a calibrated template	Robust to Swirr errors	No verified source; weak Swirr response
Porosity-permeability transform	Porosity; core by rock type	Core data in the rock type	Calibrated to the reservoir, gives uncertainty	Needs enough core; no pore-size information

Decision guidance

If there is core for the rock type, start with the porosity-permeability transform. It is the benchmark that the other models have to beat.
If Swirr is reliable (for example from NMR or a saturation-height model), try Coates or Timur and calibrate the multiplier to core. A gain over the transform is the evidence that Swirr is adding information.
If there is no core, treat any of the equations as a range, not a number. Vary the constants and carry the range.
In carbonates, none of the sandstone constants applies. Use a rock-fabric based relation (see Carbonate Analysis) or a transform by rock type.
Do not use the log Sw in place of Swirr.

Shared parameter picking

Swirr source. One Swirr curve, from a method that does not use permeability (see Swirr), feeds every model here. Pick it once, so that models can be compared on the same input.

Porosity basis. Timur, Tixier and Wyllie-Rose are written for total porosity and Coates and Heslop for effective porosity. The Swirr has to be on the same basis; see Total vs Effective Sw.

Constants. The constants Permeability model multiplier P, Permeability model porosity exponent Q and Permeability model saturation exponent R are the same three roles in every model (a multiplier, a porosity exponent and a saturation exponent), with values that depend on the model. Calibrate the multiplier first, and change the exponents only with data to support it.

Rock typing. Every calibration is for one rock type or flow unit. Pick the zones and rock types first.

Limits. Limit the result to a physical range (a floor near the tightest measured core permeability and a cap at the highest) and work in log permeability.

Core preparation. Use core permeability at reservoir confining stress and corrected for Klinkenberg slip, at the same depth as the log after depth shift.

Recommended default approach

Absent other information, a careful generalist would:

Compute Swirr first, with a method that does not use permeability.
Define rock types and fit a porosity-permeability transform to core for each.
Compute one or two Swirr-based models (Timur or Coates) with the multiplier fitted to the same core.
Compare all of them with core on a log-log 1:1 plot, and keep the ones with the lowest bias and scatter.
Report the result with its uncertainty, a factor of about 2 to 3, and not as an exact number.
Check the result against any well test or kh from pressure transient analysis.

Combining methods

Where several calibrated estimates are available, combine them in log space: take the geometric mean of the estimates, which is the mean of the log permeability curves. Averaging permeability directly gives too much weight to the highest estimate. Estimates that share an input, such as Timur and Wyllie-Rose on the same Swirr, are not independent. A better combination is a regression of core log permeability on log porosity and log Swirr, which learns the weights and the constants together. Always calibrate before combining.

QC of results

A good result:

matches core permeability in cored intervals on a log-log 1:1 plot with no bias and with a scatter of a factor of 2 to 3,
rises with porosity and falls with Swirr in every zone,
does not depend on the log water saturation or on the fluid in the zone,
is consistent with a well test or with kh from pressure transient analysis, and
has a sensible range in each rock type, with no values above the highest core permeability that cannot be explained.

Signs of a bad result: a permeability that steps at a change of Swirr method, a permeability that falls in the pay interval because the log Sw was used, constants taken from a published example without calibration, and results of many thousands of mD.

Common pitfalls

Using the log water saturation in place of Swirr.
Computing Swirr from a permeability that was computed from Swirr.
Using constants from a paper or a software default without calibrating them to core.
Entering porosity as a fraction in an equation written for percent (Timur).
Mixing total and effective porosity bases between the model and the Swirr.
Using the oil multiplier of Tixier in a gas zone.
Fitting one transform through several rock types.
Regressing porosity on log permeability and then inverting.
Averaging permeability linearly where a log mean is needed, or the reverse.
Presenting a smooth log permeability curve as if it had the variance of the core.

Going Deeper

The equations that use irreducible water saturation go back to the 1940s and 1950s, when the electric log was the main tool and Swirr was the only log-derived measure of pore size. NMR changed this: it measures the pore-size distribution, and the Coates and SDR NMR permeability models use bound and free fluid volumes, or the mean relaxation time, directly. Alongside them the rock-typing school, with flow zone indicators and rock fabric numbers, argues that permeability is best predicted by classes of pore geometry, with a transform for each class. Machine-learning models fitted to core are the present extension. Whatever the model, the unresolved difficulty is the same: the permeability of a layered reservoir is controlled by its best layers, which the logs smooth out, and which the core, being sparse, often misses.