From acoustic impedance to the elastic rock

Recap: acoustic impedance and the reflection coefficient

From §1.1: the acoustic impedance of a rock is the product of its bulk density and its P-wave velocity:

$Z_p = \rho \cdot V_p$

At any boundary between two rocks, the fraction of the seismic energy that reflects back at normal incidence (zero offset) is the reflection coefficient:

$R_0 = \dfrac{Z_2 - Z_1}{Z_2 + Z_1}$

where $Z_1$ is the impedance of the layer above and $Z_2$ is the impedance of the layer below. $R_0$ ranges between −1 and +1; in real sedimentary rocks it is rarely larger than ±0.3.

What this formula says is profound: the reflection at a boundary depends ONLY on the impedance contrast, not on the absolute impedance values. A 10% impedance step from 5000 to 5500 produces the same reflection as a 10% step from 10000 to 11000. This is why seismic interpretation works at all — the same algorithms apply at any depth, in any basin.

The missing piece: shear waves

Acoustic impedance describes how a rock responds to compression — how stiff it is when squeezed. But rocks can also be sheared (twisted, bent at constant volume), and they have a separate stiffness for that mode: the shear modulus $\mu$. Shear stiffness governs the speed of S-waves (shear waves):

$V_s = \sqrt{\mu / \rho}$

Two important properties of S-waves:

• S-waves are slower than P-waves in any rock that supports them — typically $V_s \approx V_p / 1.5$ to $V_p / 2.5$.
• S-waves do not propagate in fluids (water, oil, gas have $\mu = 0$). This is the key fact that makes shear behaviour fluid-sensitive: when fluid replaces fluid in the pore space, the rock's shear modulus barely changes, but its bulk modulus changes a lot. We will exploit this in §5.3 (Gassmann substitution).

The shear impedance is the analog of acoustic impedance:

$Z_s = \rho \cdot V_s$

And the ratio $V_p / V_s$ is one of the most useful single numbers in quantitative interpretation. Different lithologies have characteristic Vp/Vs ratios: most shales sit near 2.0; brine sands around 1.7–1.9; gas sands often below 1.6. We will see why in §5.2.

The full elastic description

An isotropic elastic rock is fully described by THREE independent quantities at any point:

• P-wave velocity Vp (m/s)
• S-wave velocity Vs (m/s)
• Bulk density ρ (g/cm³)

Everything else — acoustic impedance, shear impedance, Vp/Vs ratio, bulk modulus, shear modulus, Lamé parameters, Young's modulus, Poisson's ratio — is computed from those three. So if a rock-physics model gives you $(V_p, V_s, \rho)$ as a function of porosity, mineralogy, fluid type, and saturation, you can predict every seismic-relevant quantity. That is exactly the workflow Part 5 will build out.

Try the widget below. It lets you pick any two rocks from a small library and see their three elastic numbers, the derived impedance and Vp/Vs values, and the reflection coefficient at the boundary between them.

Exercise — see the rock physics

• The widget starts with medium shale on top and gas-saturated sandstone below — the textbook gas bright-spot recipe. Notice the strong negative reflection (R ≈ −0.10): a clear trough on conventional seismic, the kind of anomaly DHI workflows look for.
• Switch the BOTTOM layer to brine-saturated sandstone. The reflection becomes much weaker, often near zero. This is why brine-bearing sands are often invisible against shale on amplitude displays — the impedance contrast is small.
• Now switch the BOTTOM to oil-saturated sandstone. The reflection is intermediate between gas and brine — still negative, but less dramatic. Oil reduces impedance less than gas does.
• Switch the TOP to limestone and the BOTTOM to shale. The reflection flips sign: a peak rather than a trough. This is the classic carbonate-on-shale response — limestones are typically much harder than overlying shales, so the reflection at the carbonate top is a strong positive.
• Try salt as the BOTTOM under various overburdens. Salt usually shows as a strong positive reflection at its top because its impedance is high relative to most clastic overburdens. This is one reason salt domes are visually prominent in seismic surveys.
• Compare the Vp/Vs ratios of the various rocks. Notice gas-saturated sands have Vp/Vs near 1.5 while shales sit near 2.0 — this Vp/Vs gap is the basis for AVO analysis (§5.4) and for many quantitative-interpretation workflows.

Why one number isn't enough

A common beginner question: if reflection coefficient is just $(Z_2 - Z_1)/(Z_2 + Z_1)$, why do we need Vs and ρ separately? Why not just talk about Z?

The answer: at zero offset, you are right — only Z matters. But seismic data is rarely just zero-offset. Real surveys record reflections at many incidence angles, and the way the reflection changes with angle (the AVO response, §5.4) depends on Vs and ρ separately. Two rock pairs that produce identical zero-offset reflections can have completely different AVO behaviour, and that difference is often what tells you whether the bright spot is a hydrocarbon indicator or just a lithology change.

Also, even at zero offset, two rocks can have similar Z but very different Vs or ρ individually — say, a brine sand and a shale of the same impedance. They look identical on stack but differ in AVO. The framework we build over the rest of Part 5 will make these distinctions concrete.

Where the values come from

The rocks in the widget library use middle-of-the-range values from standard published tables. Real rocks vary considerably:

• Compaction: a young soft shale at 1 km depth might have Vp = 2200 m/s; the same shale buried to 5 km is Vp = 4000 m/s or higher. Density rises from ~2.2 to ~2.6 g/cc.
• Mineralogy: a clean quartz sand has different elastic properties than a feldspathic or lithic sand of identical porosity. Carbonates vary even more, depending on the mineral composition (calcite, dolomite, aragonite).
• Pore fluid: gas dramatically reduces both Vp and ρ; oil reduces them less; brine barely changes them at typical pressures.
• Pressure: increasing effective stress closes microcracks and stiffens rocks — Vp and Vs both rise.

Production rock-physics work calibrates the published averages against well logs from the specific basin and reservoir under study. The values in the widget are reasonable starting points, but no two reservoirs have exactly the same elastic-property distribution — that is why every quantitative interpretation begins with a well-tie.