A number in millimetres that is not a measurement
A borehole-image interpretation report for a single vertical carbonate well arrives with a column of fracture apertures quoted in millimetres. It sits next to the dip, the azimuth, and the conductive-or-resistive label for each pick, and it reads exactly like the others: a per-fracture number, printed to two decimals. But the aperture is not on the same footing as the dip. The dip and the polarity come off the geometry of the sinusoid on the image. The aperture comes off an electrical inversion, and the report that quotes it also tells you, in its own methods section, that the inversion is unreliable exactly where a large fraction of carbonate fractures live.
The equation is the one Luthi and Souhaite published for electrical borehole scans [1]. It reads the excess current a conductive fracture pulls into the tool and converts it to a width:
Here W is the aperture, A is the additional conductance across the fracture, R_m is the mud resistivity, R_xo is the flushed-zone resistivity, and c and b are coefficients tuned to the specific tool. In the setting it was built for, a conductive fracture in low-contrast rock, it works. The problem is the two resistivity terms. When the surrounding rock is resistive and the contrast between the fracture fill and the matrix is high, small errors in R_xo move W a long way, and the vendor report we worked from says so plainly: the Luthi-Souhaite aperture is flagged unreliable in high-resistivity-contrast zones. That caveat is written into the deliverable the operator paid for, not one we added.
What the tool measures well, and what it only infers
It is worth separating the two kinds of output a borehole-image picker produces, because they do not fail together. One kind is geometric and countable. The position of a sinusoid on the image, its dip and azimuth, and whether the fracture reads as conductive (dark, current-absorbing) or resistive (bright, current-blocking) all come straight off the resistivity contrast the tool was designed to render. Those survive being automated. The same vertical-well report that worries us on aperture is confident on class: it counts one continuous conductive fracture, 335 discontinuous conductive fractures, three continuous resistive, and 67 discontinuous resistive, and it segments 19 faults into resistive, conductive, and possible sets. That census is the robust product of the tool.
The other kind of output is a physical magnitude recovered by inversion. Aperture is the clearest case. Nothing in the image is a width in millimetres; the width is computed from the current, through coefficients, against two resistivities. It inherits every weakness of the inputs. And it carries a second, separate failure that has nothing to do with resistivity contrast: image logs exaggerate the aperture of thin fractures. Aghli and co-authors, working the fractured carbonate Asmari Formation with electrical image logs, state it directly. Fractures thinner than 0.1 mm can be reported as if they were an inch wide, because the current-spreading footprint of a hairline crack on the electrode array is far larger than the crack itself [4]. So the aperture number fails in two independent ways at once: it is unstable where the rock is resistive, and it is inflated where the fracture is thin. A hairline in a resistive zone, which is a common carbonate case, is trapped by both.
Why we stopped trying to report aperture
When we built the automated fracture picker for this carbonate programme, the temptation was to match the vendor report field for field, aperture included, so the two deliverables would line up in a spreadsheet. We decided not to. An automated picker earns its keep by being consistent and auditable across hundreds of wells. Reporting a quantity that the source physics disowns in resistive zones, and that is known to be inflated below 0.1 mm, would hand every downstream reader a number that looks like a measurement and behaves like a guess. The honest move is to report the outputs the tool supports and name the fracture-width chain as an inference the picker deliberately does not make.
The figure below is the argument in one frame.
Read the three tracks from top to bottom. Fracture density and the conductive-or-resistive class hold their reliability across the whole resistivity-contrast axis, from the conductive left to the resistive right. Those are the two teal tracks, and they stay flat and high because nothing about high contrast breaks a count or a polarity. The aperture track is the single orange one, and it is the only element in the picture that argues. It starts reliable on the conductive left, where Luthi-Souhaite was designed to work, and then diverges from truth as contrast rises, because the R_xo term in the equation runs away in resistive rock. Drag the lever to set a true fracture aperture and the second failure appears on its own: below 0.1 mm the reported width inflates, and a true 0.03 mm hairline is read out at millimetres, the inch-wide artefact Aghli and co-authors described [4]. The two failures are drawn as two axes of the same trap because that is how they combine in a real carbonate well.
The density-first vocabulary a picker can stand behind
If aperture magnitude is off the table, the report still has to say something quantitative about how fractured the rock is, and here the literature is on firm ground. Berg's P32 method estimates fracture abundance as an areal intensity, fracture area per unit rock volume, from image-log observations, with a connectivity measure defined as intersections over whole fractures and a percolation threshold of 2.0 [3]. That threshold is a real, transferable number: below it the fracture set is disconnected, above it a connected network percolates. An automated picker produces exactly the inputs P32 needs, the per-fracture geometry and the count, the same way in every well, which is the property the vendor's per-interval manual pass cannot promise. Reporting P32 and a connectivity figure reports something the tool measures, not something it inverts.
Where aperture does have to enter, it should enter as a class-level statistic feeding a density-scaled permeability chain, not as a per-fracture width. Marun's own permeability step uses the relation
with K in millidarcies and w the aperture in inches, and anchors it against clean-sandstone zones whose permeability runs 80 to 400 md [2]. The point of pinning it to that band is not to trust any single w, but to bound the permeability contribution of a fracture population and check it against a known reservoir range. A permeability estimate carried by a fracture-density distribution and a bounded aperture band is defensible in a way that a column of per-fracture apertures, each computed through an equation that is unreliable half the time, is not.
Discussion
The lesson is about matching what you report to what your instrument actually measures. A borehole-image tool measures resistivity contrast, and from contrast it gives position, orientation, polarity, and count with high fidelity. It gives aperture only through an electrical inversion that its own vendor documentation flags as unreliable in resistive zones, and that the applied literature shows is inflated for sub-0.1 mm fractures. An automated picker built to run across a large well inventory should lean on the first list and treat the second as a clearly-labelled inference at most. Reporting fracture density and conductive-or-resistive class is not a retreat from quantification; it is choosing the quantities that survive contact with resistive carbonate and hairline fractures, which is most of what these reservoirs contain.
This also sharpens what automating the interpretation should mean. It does not mean reproducing every field of the human deliverable, because some of those fields were fragile when a human produced them too. The interpreter reading the Luthi-Souhaite output in a resistive zone faced the same instability; the automated picker only makes it impossible to hide behind a confident-looking column. The right target is the robust subset, produced identically every time, with the fragile magnitude dropped or demoted to a bounded population statistic.
Limitations
The reliability tracks and the exaggeration multiplier in the figure are illustrative renderings, not measured error curves. We did not run a controlled study that inverts aperture across a graded resistivity-contrast series and plots the true error, and such a study, if the paired core existed, would test the claim more rigorously than a schematic can. The sourced anchors are the Luthi-Souhaite equation and its unreliability caveat in high-resistivity-contrast zones, the sub-0.1 mm exaggeration, the K = 5e7 * w^2 relation with clean-sandstone permeability of 80 to 400 md, and Berg's P32 method with a connectivity percolation threshold of 2.0; the shape of the decay in the plot is chosen to make the mechanism legible, not fitted. The K = 5e7 * w^2 relation and the 80 to 400 md band come from a clean-sandstone setting, and their transfer to a carbonate reservoir is an approximation the population-level use is meant to tolerate rather than a calibrated carbonate law. Finally, this argues for what an automated picker should report and against reporting per-fracture aperture as if it were a measurement; it is not a claim that aperture is never recoverable, only that it is not recoverable reliably enough, from image logs alone, to belong in an automated per-fracture deliverable.
What to carry from this piece
- A borehole-image fracture aperture is not a measurement but an electrical inversion through the Luthi-Souhaite relation W = c*A*Rm^b*Rxo^(1-b), and the vendor's own interpretation report flags it as unreliable in high-resistivity-contrast zones.
- Aperture fails in two independent ways: the R_xo resistivity term runs away where the rock is resistive, and image logs exaggerate any fracture thinner than 0.1 mm, so a hairline crack can be reported an inch wide. In a resistive zone a thin fracture is trapped by both.
- The same tool measures fracture position, orientation, conductive-or-resistive class, and count robustly, because those come off resistivity contrast directly rather than through an inversion. One report counted 336 conductive and 70 resistive fractures with confidence while its apertures were disowned by its own physics.
- An automated picker should report fracture density (Berg's P32 areal intensity with a connectivity percolation threshold of 2.0) and conductive/resistive class, which survive high contrast and hairline widths, rather than per-fracture aperture magnitude.
- Where permeability is needed, carry it through a density-scaled chain (K = 5e7 * w^2, bounded against clean-sandstone permeability of 80 to 400 md) at the population level, not a per-fracture width computed through an equation that is unreliable half the time.
References
[1] Luthi, S. M., and Souhaite, P. Fracture apertures from electrical borehole scans. Geophysics, 55(7), 821-833, 1990. Derives the electrical relation W = cARm^b*Rxo^(1-b) that converts the excess current flowing into a conductive fracture into an aperture, and notes its sensitivity to the resistivity terms it depends on. https://library.seg.org/doi/10.1190/1.1442896
[2] Marun, M. Estimating the reservoir permeability and fracture density using image logs in the Asmari Formation. Petroleum Science and Technology, 31(10), 1048-1056, 2013. Applied study relating fracture aperture to permeability via the crack-permeability relation K = 5e7 * w^2 (K in md, w in inches), with clean-sandstone zones whose permeability runs 80 to 400 md. https://doi.org/10.1080/10916466.2010.540609
[3] Berg, S., and co-authors. Methods of estimating fracture abundance and size from image logs. GeoConvention, 2020. P32 areal-intensity methodology for fracture density and size from image logs, with a connectivity measure defined as intersections over whole fractures and a percolation threshold of 2.0. https://geoconvention.com/wp-content/uploads/abstracts/2020/57700-methods-of-estimating-fracture-abundance-and-size.pdf
[4] Aghli, G., Soleimani, B., Moussavi-Harami, R., and Mohammadian, R. Reservoir heterogeneity and fracture parameter determination using electrical image logs and petrophysical data in a carbonate reservoir. Petroleum Science, 17, 51-69, 2020. Derives fracture aperture, density and spacing from electrical image logs in the fractured carbonate Asmari Formation, and cautions that image logs exaggerate the aperture of fractures thinner than 0.1 mm, which can appear an inch wide. https://doi.org/10.1007/s12182-019-00413-0
