Lucia m

Summary

Separate vugs add porosity but no connected conduction path, so they raise the resistivity of the rock above what a constant cementation exponent predicts. Lucia's relation lets the Lucia cementation exponent rise with the separate-vug fraction Separate-vug fraction, from about 1.76 with interparticle porosity only. Use it in Archie's equation for vuggy carbonates in place of a constant m, with the vug porosity taken from the difference between total and sonic porosity.

Inputs and outputs

	Item	Units
Input	Total porosity	v/v
Input	Separate-vug porosity	v/v
Input	True formation resistivity	ohm·m
Input	Formation water resistivity	ohm·m
Input	Saturation exponent	dimensionless
Input	Cementation exponent	dimensionless
Output	Separate-vug fraction	v/v
Output	Lucia cementation exponent	dimensionless
Output	Water saturation	v/v
Output	Archie water saturation	v/v

Equations

The separate-vug fraction is the vug porosity as a share of total porosity:

\[ \fSV = \frac{\phiSV}{\phit} \]

Lucia's relation for the cementation exponent of rock with separate vugs is linear in that fraction:

\[ \mLucia = 2.14\,\fSV + 1.76 \]

It is used in Archie's equation with \(a = 1\):

\[ \Sw = \left(\frac{\Rw}{\phit^{\,\mLucia}\,\Rt}\right)^{1/n} \]

limited to 0 to 1. For comparison the calculator also gives the saturation with a constant \(m\) (the output named \(S_{w,A}\)). The vug porosity is limited to the range 0 to the total porosity. The relation is for separate vugs. Touching vugs and fractures are not covered.

Symbol	Variable	Units	Typical range
\(\phi_t\)	Total porosity	v/v	0 to 0.40
\(\phi_{sv}\)	Separate-vug porosity	v/v	0 to 0.15
\(\phi_{sv}/\phi_t\)	Separate-vug fraction	v/v	0 to 0.6
\(m_L\)	Lucia cementation exponent	dimensionless	1.76 to 3.9
\(R_t\)	True formation resistivity	ohm·m	0.2 to 2000
\(R_w\)	Formation water resistivity	ohm·m	0.02 to 2
\(m\)	Cementation exponent	dimensionless	1.6 to 2.5
\(n\)	Saturation exponent	dimensionless	1.6 to 2.5
\(S_w\)	Water saturation	v/v	0 to 1
\(S_{w,A}\)	Archie water saturation	v/v	0 to 1

Single-value calculator

Behavior

With no separate vugs, \(m\) is 1.76, which is below the common value of 2, so the saturation is lower than with \(m = 2\): at a porosity of 0.15, a resistivity of 20 ohm·m and a water resistivity of 0.05 ohm·m, 0.265 against 0.333. As the vug fraction rises, \(m\) rises, and so does the saturation, because more of the high resistivity is read as water in a rock that conducts less well than the constant-\(m\) rock. At vug porosities of 0.03, 0.06 and 0.09 (fractions of 0.2, 0.4 and 0.6) \(m\) is 2.19, 2.62 and 3.04, and the saturation is 0.398, 0.598 and 0.897. It reaches 1 at a vug fraction of about 0.8 and above. The three curves for resistivities of 10, 20 and 40 ohm·m have the same shape, shifted downward as the resistivity rises (at a vug porosity of 0.06, 0.846, 0.598 and 0.423). The effect is large: ignoring the vugs at a fraction of 0.4 gives a saturation of 0.333 where Lucia's \(m\) gives 0.598.

Parameter guidance

Total porosity is the neutron-density porosity. Separate-vug porosity is total porosity minus sonic porosity, as on Sonic-Derived Secondary Porosity. It is a noisy quantity, since it is the difference of two logs, so smooth it or limit it before use. Rw and n are as for Archie and are covered in Water Saturation. The constant m given as a comparison is the one you would otherwise use, from Cementation and Saturation Exponents. The saturation here is a resistivity-based Sw; the capillary approach to the same rock is on Sw from Lucia.

Worked example

A rock with a total porosity of 0.15 and a vug porosity of 0.06 (40% of the pore space), Rt 20 ohm·m, Rw 0.05 ohm·m, n = 2, compared with a constant m of 2. The last lines show the limiting case with no vugs:

phit, sv, rt, rw, n, m = 0.15, 0.06, 20.0, 0.05, 2.0, 2.0
frac = sv / phit
m_l = 2.14 * frac + 1.76
sw = lambda mm: min(1.0, (rw / (phit ** mm * rt)) ** (1 / n))
print(f"vug fraction = {frac:.3f}, Lucia m = {m_l:.3f}")
print(f"Sw with Lucia m = {sw(m_l):.3f}, Sw with m = {m:g}: {sw(m):.3f}")
print(f"no vugs: m = {2.14 * 0 + 1.76:.2f}, Sw = {sw(1.76):.3f}")
print("vug fraction -> m: " + ", ".join(f"{f:.1f}: {2.14 * f + 1.76:.2f}" for f in (0, 0.2, 0.4, 0.6, 1.0)))

Output

vug fraction = 0.400, Lucia m = 2.616
Sw with Lucia m = 0.598, Sw with m = 2: 0.333
no vugs: m = 1.76, Sw = 0.265
vug fraction -> m: 0.0: 1.76, 0.2: 2.19, 0.4: 2.62, 0.6: 3.04, 1.0: 3.90

Assumptions and limitations

The vugs are separate: connected only through the interparticle pore network, so they carry no current of their own. Touching vugs and fractures lower m and are not covered.
The linear relation, fitted to a particular set of carbonate data, holds for the rock being studied. The intercept of 1.76 is for interparticle porosity only.
Total and sonic porosity are both right. The vug fraction is the difference of two logs and carries the errors of both. Gas, shale and bad hole also move it.
Archie's equation holds with a = 1, a constant n and a known Rw.
The formation is clean. Shale is not accounted for.

QC checks

m is 1.76 in rock with no vugs and rises with the vug fraction. Values above about 3 are rare: check the vug porosity.
Sw with Lucia m is above the constant-m (m = 2) Sw where the vug fraction is above about 0.11 (where 1.76 + 2.14 f = 2) and below it at lower fractions.
The vug porosity curve is not systematically positive in non-vuggy rock; a bias means a wrong matrix slowness or porosity.
Compare with the capillary Sw of Lucia at the same depth. A large persistent difference points to the rock fabric number, free water level or vug porosity.

Going Deeper

In a rock with only interparticle porosity the current follows the pore network and m is close to the sandstone range. Separate vugs are conductive volume in parallel with nothing: they do not shorten the current path, but the current must go around them, so the rock is more resistive than its total porosity suggests, and a higher m reproduces that. Lucia fitted a straight line between m and the vug fraction. Touching vugs and fractures connect the pore network and do the opposite. The caveat of the approach is the vug porosity: it comes from the difference of two logs, and its uncertainty (a few porosity units) is large compared with the vug porosities of interest. The same rock can instead be described by a capillary saturation-height relation that needs no resistivity, which is why the two are compared.

References

Lucia, F.J., 1983. Petrophysical parameters estimated from visual descriptions of carbonate rocks: a field classification of carbonate pore space. Journal of Petroleum Technology, 35(3), 629–637.
Lucia, F.J., 2007. Carbonate Reservoir Characterization: An Integrated Approach, 2nd edition. Springer-Verlag, Berlin Heidelberg.

Python reference implementation

Python reference implementation

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