Shear Log Modeling

Purpose

A dipole sonic tool measures the shear slowness directly, but most wells do not have one. Many calculations need it: the dynamic elastic moduli and Poisson's ratio used in geomechanics and brittleness, the stress and strength calculation of the wellbore stability model, AVO and seismic modeling, fluid substitution and the Vp/Vs ratio used as a lithology and fluid indicator. This step produces a synthetic shear log, Shear slowness, from the compressional slowness (and where possible the lithology), and checks that the result is physically plausible.

Position in the workflow

Upstream. A clean, depth-matched, repaired compressional sonic from Stage 1, converted to Compressional velocity in km/s. If the well has mineral or lithology volumes from the inversion step, they can be used to choose or weight the regression. Measured shear logs in offset wells are the calibration data.

Downstream. The modeled shear velocity feeds:

the dynamic Young's modulus, Poisson's ratio and shear modulus on the geomechanics pages,
brittleness and the horizontal stress calculation, and thereby wellbore stability, and
synthetic seismograms, impedance and reflectivity, and fluid substitution.

Error propagation. Moduli depend on the square of velocity. A 5% error in Vs is a 10% error in the shear modulus, and a larger error in Poisson's ratio, which depends on Vp/Vs: at Vp = 3.39 km/s, a Vs of 1.75 km/s gives a ratio of 1.94 and Poisson's ratio of 0.32, while a Vs 5% higher (1.83 km/s) gives a ratio of 1.85 and 0.29. Brittleness indices and stress built on these moduli carry the error with them.

Key concepts

Empirical Vp-Vs relations. Compressional and shear velocity both depend on the same rock framework, so over a population of similar rock a regression of one on the other is tight enough to be useful. The relations are fitted to data of a certain lithology, saturation and velocity range and are valid only there.

Lithology dependence. At the same Vp, sandstone, limestone, dolomite and shale have different Vs. Lithology-specific lines, as in Greenberg-Castagna, capture this, and the volume fractions of the minerals are used to mix them.

Fluid effects. Compressional velocity falls when gas replaces water in the pores, but shear velocity is almost unchanged (the density change has a small effect). A regression fitted to water-saturated rock therefore underestimates Vs in a gas zone. The fix is a fluid substitution, based on Gassmann's equations, to put the measured Vp back to brine before applying the regression; this is not covered here.

Units. The published regressions are in km/s. Slowness in µs/ft converts with Vp (km/s) = 304.8 / DT, and the shear slowness is 304.8 / Vs. Using ft/s or µs/ft in a km/s relation is the commonest error on this step.

Physical limits. For a positive Poisson's ratio Vp/Vs must exceed 1.41, and a ratio of 1.5 to 2.5 covers most rock.

Method selection guide

Method	Inputs	Use when	Strengths	Weaknesses
Greenberg-Castagna	Vp; lithology fractions	Mineral or lithology volumes exist; mixed clastic and carbonate; brine-saturated	Lithology-specific, mixes smoothly with volumes	Needs lithology; wet rock only; coefficients to be confirmed from the paper
Castagna	Vp; choice of line	A shale-dominated clastic section, or a quick baseline	Simple; widely known; mudrock line for shale	One line for all clastics; limestone and dolomite lines for carbonate
Brocher	Vp in 1.5 to 8 km/s	Regional models or unknown lithology over a wide velocity range	One smooth curve over soft sediment to basement	Not lithology-specific; less accurate in a clastic reservoir
Eskandari	Vp	Only as a comparison in carbonate until calibrated locally	Illustrates a locally fitted quadratic	Coefficients unverified; local, and not monotonic above 7 km/s
Carroll	Vp	Stiff rock, or as a comparison	Simple power law through the origin	Coefficients unverified; too stiff for shale and soft rock

Decision guidance

If a measured shear log exists in an offset well in the same formation, fit a local regression to it and use it in preference to any published one.
With lithology volumes, use Greenberg-Castagna, weighted by the fractions.
In a shale-dominated clastic section without volumes, use the Castagna mudrock line.
In carbonate, use the limestone or dolomite line of Castagna or Greenberg-Castagna. Use the Eskandari form only after local calibration.
For a wide velocity range with no lithology information, use Brocher.
Treat gas and condensate zones separately; do not trust a wet-rock regression there.

Shared parameter picking

Unit conversion. All methods share the conversion from slowness in µs/ft to km/s: Vp = 304.8/DT. Do it once and keep Vp in km/s for the whole step.

Lithology fractions. One set of sandstone, limestone, dolomite and shale fractions, from the mineral inversion or the volume of shale and carbonate curves, used for all lithology-weighted methods and shown with the result.

Calibration to measured shear. Where an offset well has a measured shear log, fit the modeled slowness to the measured one in the same lithologies, as a scale and a shift (measured = scale x model + shift), or refit the regression itself. Apply it to all methods in the same way. Calibrate on one set of wells and check on another.

Zones. Use separate lines or calibrations for formations with different lithology or compaction, rather than one for the whole well.

Splice with measured data. Where a measured shear log exists, keep it, and use the model to fill the gaps. Flag the modeled intervals so downstream steps know which values are measured.

Recommended default approach

Absent other information, a careful generalist would:

Convert the repaired compressional slowness to Vp in km/s.
Choose the lines by lithology: the sandstone, shale, limestone and dolomite lines of Greenberg and Castagna, weighted by lithology volumes where available, or the mudrock line in a shale-dominated clastic section.
Compare with measured shear logs in offset wells and fit a scale and shift if the difference is systematic.
Check Vp/Vs and Poisson's ratio against the physical limits and the lithology.
Replace the model with the measured shear log where one exists, and flag the modeled intervals.
Handle gas zones separately, with fluid substitution if it is available.

Combining methods

There are two kinds of combination. The first is a mix of lithology lines by volume, with the Voigt, Reuss or Hill average described on the Greenberg-Castagna page. The second is a splice of measured and modeled shear, where the measured value is used wherever it is valid. Averaging different published regressions (for example the mudrock line and Brocher) is not recommended: they are derived from different populations, and the average is neither of them. When several are tried, keep the spread as a measure of uncertainty and choose the one that matches the measured shear in the offset wells.

QC of results

A good result:

has Vp/Vs between about 1.5 and 2.5 and a Poisson's ratio between 0.1 and 0.4 in most intervals,
has Vp/Vs that is higher in shale than in clean sand or carbonate, and lower in a gas zone,
reproduces a measured shear log in an offset well to within a few percent, and
follows the sonic, with no spikes, steps or reversals that the compressional sonic does not have.

Signs of a bad result: Poisson's ratio below zero or above 0.45, a Vs that is greater than Vp / 1.41, a systematic offset from the measured log, and a lithology change that produces a jump in the modeled curve.

Common pitfalls

Using km/s relations with slowness in µs/ft or velocity in ft/s.
Applying a clastic relation (mudrock line) in carbonate, or the reverse.
Using a wet-rock regression in a gas or light-oil zone without a fluid correction.
Extrapolating a regression outside the velocity range it was fitted to, as with the Brocher range of 1.5 to 8 km/s or the Eskandari form above 7 km/s.
Importing coefficients from another basin without calibration, or using coefficients that were not checked against the original paper.
Driving the model with an uncorrected sonic, with cycle skips and washouts.
Treating a modeled shear log as a measurement, then computing moduli and stresses from it without uncertainty.
Mixing lithology lines with fractions that do not add to one or that refer to bulk volume including porosity.

Going Deeper

Empirical Vp-Vs relations date from the 1960s to the 1980s, from lab and well data, and became more specific to lithology in the 1980s with the mudrock line and in the early 1990s with the lithology-specific lines. Physically based alternatives use effective-medium models, such as Gassmann's equations and the Kuster-Toksoz or differential effective medium models, to compute both Vp and Vs from mineralogy, porosity, pore shape and fluid, which handles fluid effects and porosity directly. More recent approaches fit machine-learning models to measured shear logs in the field with several logs as predictors. These models are tied to the data they were trained on in the same way as the polynomial regressions are. The recurring difficulty is that a single regression absorbs differences in porosity, pore type, cementation and fluid into one curve, so its error is largest in the rocks for which it is most needed.