Foil and Buckles Models

Summary

Two simple models describe water saturation as a function of height without a capillary pressure curve. The height power law fits Bulk volume water to a power law of Height above free water level and divides by total porosity to get Water saturation from height-BVW power law, and the Buckles model holds bulk volume water at a constant, Bulk volume water at irreducible saturation, so that Sw is that constant divided by porosity. Use the height power law where core or thick clean zones give a bulk volume water profile, and Buckles above the transition zone.

Inputs and outputs

Equations

Height power law. Bulk volume water is a power law of the height above the free water level, and saturation follows from porosity on the same basis (total here), limited to the interval 0 to 1:

The exponent is negative, so bulk volume water falls with height, and \(\SwHgt = 1\) close to the free water level, where the power law exceeds the porosity. This is the same equation as the Swirr from Foil page; there it is evaluated at the maximum column height to give the irreducible saturation, and here it is evaluated at every height.

Buckles. Bulk volume water is a constant for the rock type above the transition zone, so Sw is that constant over porosity:

as on the Swirr from Buckles page.

Where the two meet. The height power law falls to the Buckles number at

Below \(\hBuck\) the power law gives more water than the Buckles constant, so the zone is still in the transition zone; above it, the power law continues to fall while Buckles holds at the constant.

Single-value calculator

Behavior

On the log height axis the power law is a straight line of bulk volume water but a curve for Sw, since Sw is capped at 1 near the free water level. With a = 0.2, b = -0.25 and a porosity of 0.20, Sw is 1.00 at 1 ft, 0.562 at 10 ft, 0.376 at 50 ft, 0.316 at 100 ft, 0.240 at 300 ft and 0.178 at 1000 ft. The Buckles Sw is a constant 0.200. The two cross at 625 ft: below it the power law is the wetter of the two, above it it is drier. The exponent controls the shape. At 100 ft, Sw is 0.501, 0.316 and 0.158 for b of -0.15, -0.25 and -0.40, and the crossing height with the Buckles constant is 45 688, 625 and 56 ft. A gentle exponent never reaches the Buckles constant within any realistic column, while a steep one reaches it quickly.

Parameter guidance

Coefficients. Take \(a_h\) and \(b_h\) from a straight-line fit of the logarithm of bulk volume water against the logarithm of height, with heights from capillary pressure data converted as on the Capillary Pressure page, or from log Sw in thick clean zones well above the free water level. The values 0.2 and -0.25 in the calculator are illustrative. A fit on log Sw is biased towards the part of the zone that is near the contact, so check it above the transition zone.

Buckles number. Read it from a Buckles plot of bulk volume water against porosity for zones at irreducible saturation. Typical ranges by grain size are given on the Swirr from Buckles page. One value is for one rock type; the porosity basis (effective for \(\mathrm{BVW}_{irr}\) here) has to match the log.

Porosity. The calculator takes total porosity for the power law and effective porosity for Buckles, as on the two Swirr pages. In clean rock the two are the same, as in the defaults.

Choosing between them. Use Buckles where the zone is at irreducible saturation, that is, above the height \(h_B\). Use the power law across the transition zone and the whole column when a height profile is wanted. A permeability-based model (Permeability, RQI and FZI) is the better choice when permeability varies strongly, as these two models see porosity only. The related Swirr from RQI and Swirr from permeability pages cover the irreducible endpoint.

Worked example

A rock with total and effective porosity of 0.20, the height power law with a = 0.2 and b = -0.25, and a Buckles number of 0.04. The table gives Sw from the power law against the Buckles constant, and the crossing height:

a, b, c, phi = 0.2, -0.25, 0.04, 0.2
print(f"{'h (ft)':>7} {'BVW':>7} {'Sw power':>9} {'Sw Buckles':>11}")
for h in (1, 10, 50, 100, 300, 1000):
    bvw = a * h ** b
    print(f'{h:7.0f} {bvw:7.4f} {min(1.0, bvw / phi):9.3f} {min(1.0, c / phi):11.3f}')
hb = (c / a) ** (1 / b)
print(f'crossing height h_B = ({c:g}/{a:g})^(1/{b:g}) = {hb:.0f} ft')
print(f'BVW at h_B = {a * hb ** b:.4f}')

Output

 h (ft)     BVW  Sw power  Sw Buckles
      1  0.2000     1.000       0.200
     10  0.1125     0.562       0.200
     50  0.0752     0.376       0.200
    100  0.0632     0.316       0.200
    300  0.0481     0.240       0.200
   1000  0.0356     0.178       0.200
crossing height h_B = (0.04/0.2)^(1/-0.25) = 625 ft
BVW at h_B = 0.0400

Assumptions and limitations

• The height power law has no lower limit: it keeps falling with height and can go below the true irreducible saturation. Use it only up to the highest height in the data it was fitted to.
• Porosity is the only rock property that enters. Permeability, pore throat size and rock type are not represented, so each rock type needs its own coefficients.
• The free water level is known, and the fit and the application use the same one. A wrong free water level changes every height by the same amount, which is a large relative error near the contact.
• Buckles assumes bulk volume water is constant at irreducible saturation within a rock type. Where grain size changes with porosity, the constant drifts.
• Sw and porosity are on the same basis (total or effective), and the clay-bound water is accounted for the same way in both.

QC checks

• Plot bulk volume water against height on log-log axes for the calibration data. The points should lie on a straight line above the transition zone.
• Sw from the power law is 1 at the free water level and falls with height. It is not below Swirr by a large margin in the upper part of the column.
• The crossing height \(h_B\) is within the column. If it is far above the highest height of the data, the Buckles number and the power law are not consistent.
• Compare with log Sw in thick clean intervals. The residual should not trend with depth.

Going Deeper

A constant bulk volume water is the large-height limit of any saturation-height model with a finite Swirr, because Sw tends to Swirr and so bulk volume water tends to \(\phi\,S_{wirr}\). The Buckles number is therefore the end of the curve, and a model of the whole curve needs more than a constant. A power law in height is the simplest such model, and has the same shape as a Brooks-Corey capillary pressure curve in the part of the curve where \(S_w - S_{wirr}\) is a power of the height. With \(\mathrm{BVW} \approx \phi\,[S_{wirr} + (1 - S_{wirr})(h_e/h)^{\lambda}]\) a pure power law is only reached when Swirr is small compared with the water that is still draining, and the exponent then equals \(-\lambda\) (my interpretation, not sourced). The physically based alternative is the J-function model, which gives the same shape from capillary pressure data and adds permeability; the power law is the shortcut when only porosity and a height profile are available. I could not trace the height power law, as used here, to a named publication, so the coefficients are treated as a fitted empirical form.

References

1. Buckles, R.S., 1965. Correlating and averaging connate water saturation data. Journal of Canadian Petroleum Technology, 4(1), 42–52.

Python reference implementation

Related methods

• Capillary Pressure and the Leverett J-Function
• Permeability-Based and RQI/FZI Models
• Hydraulic Flow Units and Winland R35
• Free Water Level and Column Height
• Swirr from Foil
• Swirr from Buckles