Swirr from RQI

Summary

The reservoir quality index, Reservoir quality index, combines Permeability and Effective porosity into one measure of pore throat size, and Irreducible water saturation is fitted to it as a power law. Use it when permeability and porosity are both available and the fit to core is better against RQI than against permeability alone.

Inputs and outputs

	Item	Units
Input	Permeability	mD
Input	Effective porosity	v/v
Input	Swirr-RQI coefficient a	v/v at 1 µm
Input	Swirr-RQI exponent b	dimensionless
Output	Reservoir quality index	µm
Output	Flow zone indicator	µm
Output	Irreducible water saturation	v/v

Equations

The reservoir quality index, in micrometres, from permeability in millidarcies and porosity as a fraction:

\[ \CpRQI = 0.0314\,\sqrt{\frac{k}{\phie}} \]

The normalized porosity and the flow zone indicator are:

\[ \CpPhiZ = \frac{\phie}{1 - \phie} \qquad \CpFZI = \frac{\CpRQI}{\CpPhiZ} \]

Swirr is a power law of RQI with a negative exponent, limited to the interval 0 to 1:

\[ \Swirr = \CpSwRqiA\,\CpRQI^{\CpSwRqiB} \]

Symbol	Variable	Units	Typical range
\(k\)	Permeability	mD	0.0001 to 10000
\(\phi_e\)	Effective porosity	v/v	0 to 0.35
\(RQI\)	Reservoir quality index	µm	0.01 to 5
\(\phi_z\)	Normalized porosity	v/v	0.05 to 0.7
\(FZI\)	Flow zone indicator	µm	0.1 to 10
\(a_{q}\)	Swirr-RQI coefficient a	v/v at 1 µm
\(b_{q}\)	Swirr-RQI exponent b	dimensionless	-0.8 to -0.2
\(S_{wirr}\)	Irreducible water saturation	v/v	0.05 to 0.5

Single-value calculator

Behavior

Swirr falls as permeability rises, and at a given permeability it is higher in the more porous rock, because the same permeability in a more porous rock means finer throats. At 10 mD, porosities of 0.10, 0.18 and 0.26 give Swirr of 0.318, 0.358 and 0.385, and at 1 mD they give 0.504, 0.567 and 0.610. With a = 0.2 and b = -0.4 Swirr is already 0.798 at 0.1 mD for a porosity of 0.10 and exceeds 1 for still poorer rock, so the result is capped at 1.

Parameter guidance

The constant 0.0314 is the square root of the conversion between millidarcies and square micrometres, about 0.987 x 10^-3, and is not a parameter to adjust. The two fitting constants come from a log-log cross-plot of core Swirr against RQI. The pair 0.2 and -0.4 is illustrative. RQI and FZI are also used to define hydraulic flow units: points with a similar FZI share a pore geometry and can be fitted together. Using FZI to group the data before fitting usually improves the Swirr relation. The permeability must come from core or a transform that does not use Swirr.

Worked example

Three rocks with different permeability and porosity, and the Swirr each implies with a = 0.2 and b = -0.4:

import math
a, b = 0.2, -0.4
print(f"{'k (mD)':>7} {'phie':>6} {'RQI':>7} {'phiz':>7} {'FZI':>7} {'Swirr':>7}")
for k, phie in ((0.5, 0.10), (10.0, 0.18), (300.0, 0.25)):
    rqi = 0.0314 * math.sqrt(k / phie)
    phiz = phie / (1 - phie)
    print(f"{k:7.1f} {phie:6.2f} {rqi:7.3f} {phiz:7.3f} {rqi / phiz:7.3f} {min(1.0, a * rqi ** b):7.3f}")

Output

 k (mD)   phie     RQI    phiz     FZI   Swirr
    0.5   0.10   0.070   0.111   0.632   0.579
   10.0   0.18   0.234   0.220   1.066   0.358
  300.0   0.25   1.088   0.333   3.263   0.193

Assumptions and limitations

Swirr is a single power law of RQI within a hydraulic flow unit. A section with several flow units needs a fit for each.
The porosity is effective porosity and the permeability is in the same sense as the core data used in the fit.
Permeability is accurate and independent of Swirr.
Core Swirr was measured at a pressure comparable to the column height of the field.

QC checks

RQI is between about 0.01 and 5 micrometres for most reservoirs.
The log-log fit of Swirr against RQI has a better correlation than the fit against permeability alone. If it does not, the extra variable is not helping.
Swirr from this method is of the same size as Buckles and Swirr-permeability for the same rock.
The result is not capped at 1 in the reservoir intervals.

Going Deeper

The reservoir quality index and the flow zone indicator are due to Amaefule and co-authors, who derived them from the Kozeny-Carman equation and used them to separate hydraulic flow units. Taking the logarithm of the Kozeny-Carman relation gives a straight line of slope one between log RQI and log normalized porosity for each unit, with FZI as the intercept. Here the same index is used as the predictor of Swirr. Where FZI-based flow unit clustering is available, it can replace rock typing by core description. The SwH Analysis topic covers the use of RQI and FZI in saturation-height models, on its RQI/FZI page.

References

Amaefule, J.O., Altunbay, M., Tiab, D., Kersey, D.G. and Keelan, D.K., 1993. Enhanced reservoir description: using core and log data to identify hydraulic (flow) units and predict permeability in uncored intervals/wells. SPE 26436, SPE Annual Technical Conference and Exhibition, Houston, TX.

Python reference implementation

Python reference implementation

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