The four moduli and the Vp/Vs ratio

The four moduli, geometrically

Imagine a small cube of rock at depth. We can deform it in different ways:

• Bulk modulus K (or B): squeeze the cube uniformly from all sides. K is the resistance to volume change — a high-K rock is hard to compress. Units: Pa (Pascals); typical reservoir rocks are 10–40 GPa.
• Shear modulus μ (or G): twist the cube so opposite faces slide past each other at constant volume. μ is the resistance to shape change. Fluids cannot resist shear — their μ is zero, which is why they don’t carry S-waves. Reservoir rocks have μ between 5 and 30 GPa.
• Lamé first parameter λ: relates volumetric strain to lateral stress. Algebraically, $\lambda = K - \frac{2}{3} \mu$. Useful in the AVO analysis of §5.4 — the LMR (“lambda-rho, mu-rho”) crossplot is a workhorse of quantitative interpretation.
• Young’s modulus E: stretch the cube along one axis and measure the resulting elongation. E is the modulus you encounter in introductory mechanics (steel beams, etc.). Less central to seismic but important for geomechanics and fracture-related work.

K and μ are the two primary moduli for seismic. Everything else (λ, E, Poisson’s ratio σ, Vp, Vs) can be computed from K, μ, and ρ.

From moduli to velocities

For an isotropic elastic medium, the wave equation gives:

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**   and   $V_s = \sqrt{\dfrac{\mu}{\rho}}$

The P-wave velocity depends on BOTH bulk and shear stiffness (because a P-wave compresses AND shears the rock as it propagates). The S-wave velocity depends only on shear stiffness. Density appears in the denominator under both square roots, but the numerator effects usually dominate — a denser rock is not necessarily slower if its moduli rise faster than its density.

Solving for the moduli given the velocities and density:

$\mu = \rho V_s^{2}$     **

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So if you know any pair of (Vp, Vs), (K, μ), or (λ, μ) plus density, you have full elastic information about the rock. Different communities work in different parameterizations — reservoir engineers like K and μ; AVO analysts use λ·ρ and μ·ρ (the LMR crossplot space); machine-learning workflows often use Vp and Vp/Vs.

Vp/Vs ratio: the lithology / fluid diagnostic

The Vp/Vs ratio can be written entirely in terms of the modulus ratio K/μ:

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Density cancels. So Vp/Vs depends ONLY on the modulus ratio — it is a property of the rock’s stiffness type, not its bulk weight. This makes Vp/Vs a clean diagnostic. Two effects shift it dramatically:

• Fluid substitution. Replacing brine with gas drops K substantially (gas is much less stiff under compression than water) but barely touches μ (fluid μ = 0 in either case). So K/μ drops, and Vp/Vs drops. Gas sands typically have Vp/Vs near 1.5–1.6; brine sands at the same porosity sit at 1.7–1.9.
• Lithology. Carbonates have a very different K/μ balance than clastics; their Vp/Vs is typically near 1.9. Pure quartz sandstone is around 1.55 to 1.65 (low-Vp/Vs lithology by nature). Coal is anomalously low-K and shows Vp/Vs near 1.9. Salt sits near 1.65.

The combination is what makes Vp/Vs powerful: a low Vp/Vs anomaly inside a sand-shale sequence suggests gas (because shales sit at 2.0). A low Vp/Vs reading inside a carbonate sequence has different implications and may not point to gas at all.

Castagna’s mudrock line

In 1985 John Castagna published an empirical fit relating Vs to Vp for water-saturated clastic shales:

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This is the mudrock line. It is famous because it is a tight relationship for shales — published Vp/Vs values for shales scatter very little around it. So departures from the line are diagnostic:

• Below the mudrock line (lower Vs at given Vp): typically a lithology with higher μ/K ratio than shale — brine sand sits slightly below; gas sand sits well below.
• Above the mudrock line (higher Vs at given Vp): unusual for clastics, may indicate cementation or a special mineralogy.
• Off to the right at high Vp: carbonates and evaporites generally plot in their own clouds, far from the clastic mudrock line.

The mudrock line is one of the standard reference curves overlaid on every Vp–Vs crossplot in QI. The widget below draws it automatically when the axes are Vp and Vs (or Vp and Vp/Vs).

Exercise — read the rock library in elastic space

• The widget starts with Vp on the X axis and Vp/Vs on the Y. Notice the three shales (soft / medium / hard) form a vertical line near Vp/Vs = 2.0 — they have different velocities but the same Vp/Vs. This is the classic shale signature.
• Find the three sandstones (brine, oil, gas). Notice they sit at progressively lower Vp/Vs as you swap the fluid: brine ≈ 1.74, oil ≈ 1.63, gas ≈ 1.47. Same rock matrix; the gas drops Vp/Vs by half a unit.
• Look at where the carbonates plot (limestone, dolomite). Both have high Vp (5500–6400 m/s) and Vp/Vs near 1.9. Distinct cloud, well separated from the clastics.
• Switch the X axis to Acoustic impedance Zₚ. The lithologies sort by Z: shales 5–10, sands 5–8, carbonates 14–18. This is the AI-vs-Vp/Vs crossplot used in inversion-derived attribute interpretation.
• Switch the Y axis to Vs. Now the points form an approximately linear band — the mudrock line is overlaid as a dashed reference. The three shales sit almost exactly on the line; the brine sand sits slightly below; the gas sand is well below the line. Departure from the mudrock line is the visual diagnostic for non-shale lithology or non-brine fluid.
• Try the Poisson’s ratio axis. Notice: shales ≈ 0.33, gas sand ≈ 0.07, oil sand ≈ 0.20, brine sand ≈ 0.25. Low Poisson is the gas signature; carbonates near 0.30 are intermediate.

Why the mudrock line works

Castagna’s relation is not theoretical — it is empirical. But there is a physical reason it is so tight: shales are dominated by clay platelets aligned roughly horizontally, and as compaction increases all components of stiffness rise together in a predictable way. K and μ grow roughly proportionally, so Vp/Vs stays near 2.0 across a wide range of depths.

Sandstones break this pattern because their grain framework supports shear stiffness disproportionately compared to bulk stiffness. The result: Vp/Vs is lower for sands than for shales at any given Vp. Add gas (which shrinks K further by replacing water with a much-more-compressible fluid) and Vp/Vs falls further still.

The mudrock line therefore separates two physically distinct families of clastic rocks. It does not directly identify gas (gas is just one of several mechanisms that drops a sample below the line) but it identifies which samples are not behaving like shale, which is the first question in lithology interpretation.

Common pitfalls

• Mudrock line is calibrated for one geological setting at a time. Castagna’s original 1985 fit was for Texas Gulf Coast shales. Other basins (North Sea, Permian, etc.) have slightly different lines because of mineralogy, depth, and pore-pressure differences. Production QI workflows refit the line to the local data.
• Vp/Vs alone is not enough to identify gas. Coal has low K and shows low Vp/Vs (~1.9, but in the relevant range when interpreted carelessly). Dolomite has slightly low Vp/Vs (~1.83). Always combine Vp/Vs with other attributes.
• Density is needed for absolute-modulus interpretation. Vp/Vs gives you K/μ but not K and μ separately. To compute K and μ in GPa you also need ρ.
• Anisotropy. Real rocks are not perfectly isotropic, especially shales. Vp and Vs depend on the direction of propagation relative to the rock fabric. Production interpretation often uses two Vps (vertical and horizontal) and a Thomsen anisotropy parameter ε; the moduli get correspondingly more complex.