Swirr from Foil

Summary

The Foil method fits bulk volume water against height above the free water level with a power law, and takes Irreducible water saturation as the bulk volume water at the maximum column height divided by the porosity. Use it when the data for the fit come from a well-defined height-saturation set, and the result is wanted without a full capillary pressure model.

Inputs and outputs

	Item	Units
Input	Total porosity	v/v
Input	Height above free water level	ft
Input	Height-BVW coefficient a	v/v at h = 1 ft
Input	Height-BVW exponent b	dimensionless
Output	Bulk volume water at irreducible saturation	v/v
Output	Irreducible water saturation	v/v

Equations

Bulk volume water falls with height above the free water level as a power law with a negative exponent:

\[ \CpBvwIrr(h) = \CpFoilA\,\CpHeight^{\CpFoilB} \]

Swirr is the bulk volume water at the maximum column height \(h\), divided by the total porosity and limited to the interval 0 to 1:

\[ \Swirr = \frac{\CpFoilA\,\CpHeight^{\CpFoilB}}{\phit} \]

Symbol	Variable	Units	Typical range
\(\phi_t\)	Total porosity	v/v	0 to 0.40
\(h\)	Height above free water level	ft	10 to 1500
\(a_h\)	Height-BVW coefficient a	v/v at h = 1 ft
\(b_h\)	Height-BVW exponent b	dimensionless	-0.3 to -0.05
\(BVW_{irr}\)	Bulk volume water at irreducible saturation	v/v	0.005 to 0.10
\(S_{wirr}\)	Irreducible water saturation	v/v	0.05 to 0.5

Single-value calculator

Behavior

Swirr falls as the height rises, because a taller hydrocarbon column displaces more water, and the exponent controls how fast. At a porosity of 0.20 and a coefficient of 0.10, Swirr at 1500 ft is 0.347, 0.241 and 0.116 for exponents of -0.05, -0.10 and -0.20, and at 10 ft it is 0.446, 0.397 and 0.315. A steeper exponent therefore means a stronger height dependence, and a result that is more sensitive to the column height you pick.

Parameter guidance

The coefficient \(a_h\) and exponent \(b_h\) come from a straight-line fit of the logarithm of bulk volume water against the logarithm of height, using core capillary pressure data converted to height or log saturations in thick clean zones well above the free water level. The pair 0.10 and -0.10 in the calculator is illustrative, not a calibrated value. Height above free water level is a choice: it is the maximum height of the hydrocarbon column in the field or compartment, not the height of the sample. Capillary pressure to height conversion and the fit itself are part of the SwH Analysis topic, in particular its Foil and Buckles page.

Worked example

A porosity of 0.20 with a = 0.10 and b = -0.10, evaluated at several heights:

a, b, phit = 0.10, -0.10, 0.20
print(f"{'h (ft)':>7} {'BVW':>7} {'Swirr':>7}")
for h in (10, 100, 500, 1500):
    bvw = a * h ** b
    print(f"{h:7d} {bvw:7.4f} {min(1.0, bvw / phit):7.3f}")

Output

 h (ft)     BVW   Swirr
     10  0.0794   0.397
    100  0.0631   0.315
    500  0.0537   0.269
   1500  0.0481   0.241

Assumptions and limitations

Bulk volume water follows a single power law in height over the range used. It fails at the base of the column, where saturation is near 1 and the power law is not valid.
The porosity is total porosity here and the fit was made on the same basis.
Swirr is evaluated at the maximum height of the column. In a thin column or a transition zone it overstates what a given height of hydrocarbon has reached.
The fit is for one rock type. A single pair for a mixed section produces a Swirr that is controlled by porosity alone.

QC checks

The fitted line passes through the data on a log-log plot of bulk volume water against height, with scatter that can be explained by rock type.
Swirr falls with height in the reservoir, is between 0 and 1, and is below the water saturation from the resistivity model in hydrocarbon-bearing zones.
The maximum column height used is consistent with the free water level and with the hydrocarbon contact picked in the well.
Compare with the Buckles value for the same rock: at the maximum column the two should be of the same size.

Going Deeper

A power law of water volume against height is the simplest consequence of a pore-throat size distribution that is itself a power law, which is how fractal and Brooks-Corey capillary pressure models arise. Fitting bulk volume water rather than saturation removes much of the porosity trend, and so tends to collapse points from different porosities onto one line. I have not been able to tie the name Foil to a published paper; some software uses it for this height-BVW fit. Where a model is available, saturation-height functions on the SwH Analysis topic are the better route.

References

References will be added once verified.

Python reference implementation

Python reference implementation

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