The four moduli and the Vp/Vs ratio

The four moduli, geometrically

Imagine a small cube of rock at depth, deformed in different ways:

• Bulk modulus $K$: squeeze the cube equally from all sides. $K$ is its resistance to a change of volume: a high-$K$ rock is hard to compress. The rocks of Figure 5.2 run from 4.7 GPa in the coal to 72 GPa in the anhydrite; brine is near 2.5 GPa, and gas at reservoir pressure only a few hundredths of a GPa.
• Shear modulus $\mu$: push the top face sideways, so the cube changes shape at constant volume. $\mu$ is the resistance to that change of shape. A fluid cannot resist shear: its $\mu$ is zero, which is why fluids carry no S-waves. The library runs from 1.8 GPa in the soft shale to 34 GPa in the dolomite.
• Lamé’s first parameter $\lambda$: with $\mu$ it writes Hooke’s law for an isotropic solid, $\sigma_{ij} = \lambda\,\varepsilon_{kk}\,\delta_{ij} + 2\mu\,\varepsilon_{ij}$, so $\lambda$ is the stress a change of volume produces in every direction, on top of the shear part; $\lambda = K - \tfrac{2}{3}\mu$. Times the density it is one axis of Goodway’s $\lambda\rho$ against $\mu\rho$ crossplot, plate (b) of the figure and a workhorse of the AVO analysis of Section 5.4.
• Young’s modulus $E$: pull a bar along its length, leaving it free to thin sideways; $E$ is the stress per unit of stretch. It is the modulus of introductory mechanics and of geomechanics (fracture design, wellbore stability), and less used in seismic work.

Poisson’s ratio $\nu$ (some texts write $\sigma$) is not a modulus but a ratio: how much the bar thins sideways for each unit it stretches. Ordinary rocks fall between 0 and 0.5, and a fluid, with $\mu = 0$, sits at the limit of 0.5.

$K$ and $\mu$ are the two primary moduli for seismic. Every other quantity here, $\lambda$, $E$, $\nu$ and both velocities, follows from $K$, $\mu$ and $\rho$.

From moduli to velocities

For an isotropic elastic medium the wave equation gives

The P-wave velocity depends on both the bulk and the shear stiffness, because a P-wave changes both the volume and the shape of the rock it passes through; the S-wave velocity depends on the shear stiffness alone. Density sits under both square roots, but a denser rock is not necessarily slower: in most rocks the moduli rise faster than the density does.

Turned round, the velocities and the density give the moduli:

With velocities in km/s and density in g/cm³ the moduli come out directly in GPa. The brine sand of Figure 5.2, with $V_{\mathrm{P}} = 3.30$ km/s, $V_{\mathrm{S}} = 1.81$ km/s and $\rho = 2.30$ g/cm³, has $\mu = 2.30 \times 1.81^2 = 7.54$ GPa and $K = 25.05 - 10.05 = 15.00$ GPa.

So any pair of ($V_{\mathrm{P}}$, $V_{\mathrm{S}}$), ($K$, $\mu$) or ($\lambda$, $\mu$), with the density, carries the full elastic description of an isotropic rock. Communities pick different pairs: rock physicists and reservoir engineers work in $K$ and $\mu$; AVO analysts often in $\lambda\rho$ and $\mu\rho$ (Goodway et al. 1997), which come straight from the P and S impedances an inversion delivers, $\lambda\rho = Z_{\mathrm{P}}^2 - 2Z_{\mathrm{S}}^2$ and $\mu\rho = Z_{\mathrm{S}}^2$; machine-learning workflows often in $V_{\mathrm{P}}$ and $V_{\mathrm{P}}/V_{\mathrm{S}}$.

The velocity ratio: a lithology and fluid diagnostic

The ratio of the two velocities depends on the ratio of the two moduli alone:

Density cancels. $V_{\mathrm{P}}/V_{\mathrm{S}}$ measures the kind of stiffness a rock has, not its weight, which makes it a clean diagnostic. Two things move it:

• Pore fluid. Gas in place of brine lowers $K$ sharply, because gas is far more compressible than water, and leaves $\mu$ alone, because neither fluid carries shear. $K/\mu$ falls, and $V_{\mathrm{P}}/V_{\mathrm{S}}$ with it. The brine, oil and gas sands of the library share one frame: $\mu$ is 7.54, 7.53 and 7.48 GPa, nearly equal (Gassmann’s equation in Section 5.3 makes them exactly equal), while $K$ falls from 15.00 to 11.10 to 6.10 GPa and $V_{\mathrm{P}}/V_{\mathrm{S}}$ from 1.82 to 1.68 to 1.47. Gas sands commonly read near 1.5 to 1.7; brine sands of the same porosity read higher.
• Lithology. The minerals set the baseline: from their moduli (as tabulated by Mavko et al. 2009), quartz alone has $V_{\mathrm{P}}/V_{\mathrm{S}} \approx 1.48$, calcite 1.92 and dolomite 1.86. So dry or gas-filled quartz sandstones read low, limestones near 1.9 (the library’s reads 1.90) and dolomites a little lower (1.83). Shales read high when soft and young (the library’s soft shale reads 2.44) and come down as they compact (the hard shale reads 1.81); halite reads 1.67 and coal 1.92.

So a low $V_{\mathrm{P}}/V_{\mathrm{S}}$ inside a sand and shale sequence points to a sand, and an unusually low one to gas. In a carbonate the same reading means something else and may not point to gas at all.

Castagna’s mudrock line

Castagna, Batzle and Eastwood (1985) fitted measurements of water-saturated mudrocks with a straight line:

This is the mudrock line. Water-saturated mudrocks scatter closely about it, so a departure from it says that a rock is not behaving like one. Because the line does not pass through the origin, the ratio it implies falls as mudrocks stiffen: 3.04 at 2.2 km/s, 2.18 at 2.9 km/s and 1.81 at 3.8 km/s. A departure is therefore best stated as a difference in $V_{\mathrm{P}}/V_{\mathrm{S}}$ at the rock’s own $V_{\mathrm{P}}$:

• At lower $V_{\mathrm{P}}/V_{\mathrm{S}}$ than the line, a higher $V_{\mathrm{S}}$ for the same $V_{\mathrm{P}}$: clean sands, and hydrocarbons. Brine sands lie a little off the line on this side, and oil and gas sands further off, gas furthest. In the opening view of Figure 5.2, $V_{\mathrm{P}}/V_{\mathrm{S}}$ against $V_{\mathrm{P}}$, these rocks plot below the line; with $V_{\mathrm{S}}$ across and $V_{\mathrm{P}}$ up, to its right.
• At higher $V_{\mathrm{P}}/V_{\mathrm{S}}$ than the line: rocks softer in shear than a mudrock of the same $V_{\mathrm{P}}$. The limestone, the dolomite, the anhydrite and the basalt of the library sit here, 0.29 to 0.48 above the line at their high $V_{\mathrm{P}}$, where its ratio has fallen to about 1.5, while the halite, at 4.5 km/s, sits almost on it: the line was fitted to mudrocks, and other rocks follow trends of their own.

Greenberg and Castagna (1992) published such trends for brine-saturated sandstone, limestone, dolomite and shale. Figure 5.2 draws the first three beside the mudrock line, which their shale line follows to within 0.16 km/s from 2 to 5 km/s; it draws them on any pair of axes built from the velocities alone, except $V_{\mathrm{P}}/V_{\mathrm{S}}$ against Poisson’s ratio, where every rock and every trend falls on one curve, and it measures every rock’s distance from the mudrock line.

Reading Figure 5.2

1. The brine sand. The figure opens on it: $\mu = 7.54$ GPa, $K = 15.00$ GPa, $K/\mu = 1.99$ and $V_{\mathrm{P}}/V_{\mathrm{S}} = 1.82$, which is 0.15 below the 1.97 the mudrock line gives at its $V_{\mathrm{P}}$. It lies on the Greenberg–Castagna sandstone trend, which gives $V_{\mathrm{S}} = 1.80$ km/s at 3.30 km/s against its 1.81.
2. The shales. The soft, medium and hard shale read 2.44, 2.00 and 1.81: the ratio falls as the shale stiffens, as along the mudrock line. The hard shale sits on the line; the medium shale is 0.18 below it and the soft shale 0.60 below, where the line gives 3.04. Real shales scatter about the line, and the library’s soft shale is stiffer in shear than Castagna’s fit.
3. The fluids. Swap the brine for oil and then gas (the first exercise does it for you): $\mu$ hardly moves, $K$ falls from 15.00 to 11.10 to 6.10 GPa, and the ratio from 1.82 to 1.68 to 1.47, which puts the gas sand 0.79 below the mudrock line.
4. Plate (b). On Goodway’s plane the same swap moves the sand to the left: $\lambda\rho$ falls from 22.9 to 13.4 to 2.29 GPa·g/cm³, the lowest of the library, while $\mu\rho$ falls only from 17.3 to 15.3, and only because the rock gets lighter. Incompressibility sees the fluid; rigidity sees the frame.
5. Carbonates and evaporites. The limestone (1.90) sits on its Greenberg–Castagna trend and the dolomite (1.83) a little above its own, both above the mudrock line at their high $V_{\mathrm{P}}$. In (b) they stand with the anhydrite and the basalt at $\lambda\rho$ from 94 to 157, more than twice the hard shale’s 39 and four to seven times the brine sand’s 23.
6. Coal. Choose it and plot $V_{\mathrm{P}}/V_{\mathrm{S}}$ against $Z_{\mathrm{P}}$. Coal has the lowest impedance in the library, 3.22 km/s·g/cm³ against 5.74 for the gas sand, so on a stack it is as soft and bright as gas; but its ratio is an ordinary 1.92 and its $\mu\rho$, 2.82, is a fifth of the gas sand’s 15.3. Low $\lambda\rho$ with very low $\mu\rho$ is coal; low $\lambda\rho$ with ordinary $\mu\rho$ is gas.
7. Your own rock. Hold $V_{\mathrm{P}}$ and $\rho$ and drag $V_{\mathrm{P}}/V_{\mathrm{S}}$: $\rho V_{\mathrm{P}}^2 = K + \tfrac{4}{3}\mu$ stays fixed, so every step of the ratio moves stiffness between $K$ and $\mu$. Plotting $V_{\mathrm{P}}/V_{\mathrm{S}}$ against $Z_{\mathrm{P}}$ gives the crossplot of inversion-derived attributes in Section 5.6.

Why the mudrock line works

Castagna’s relation is empirical, but its shape has a physical reading. Along the line, $K/\mu = (V_{\mathrm{P}}/V_{\mathrm{S}})^2 - \tfrac{4}{3}$ falls from about 7.9 at 2.2 km/s to 1.9 at 3.8 km/s. Young mud is a loose, water-filled aggregate of clay whose bulk stiffness is mostly the water’s and whose shear stiffness is small; compaction and cement build the grain contacts that carry shear, so $\mu$ grows faster than $K$.

Sandstones lie off the line on the low side because their quartz framework is stiff in shear (quartz alone has $V_{\mathrm{P}}/V_{\mathrm{S}} \approx 1.48$), so at a given $V_{\mathrm{P}}$ a sand has a higher $V_{\mathrm{S}}$ than a mudrock. Gas, by lowering $K$ further, takes it further off.

The line therefore separates rocks that behave like water-saturated mudrock from rocks that do not. It does not identify gas: a clean, consolidated brine sand departs from it too, and in the library so does the coal, 0.92 below it at its low $V_{\mathrm{P}}$, further than the gas sand’s 0.79. What it answers is the first question of lithology interpretation: is this rock behaving like shale?

Common pitfalls

• The mudrock line is a regional fit. Castagna and his co-authors pooled measurements of water-saturated mudrocks; a basin with a different mineralogy, pressure or temperature has a slightly different line, and production QI work refits it to local logs.
• A low $V_{\mathrm{P}}/V_{\mathrm{S}}$ alone does not mean gas. A clean, well-cemented quartz sandstone reads low with brine in it. The opposite trap is coal: soft and bright like gas on impedance, with an ordinary ratio and very low rigidity. Combine the ratio, or $\mu\rho$, with the impedance.
• Absolute moduli need density. $V_{\mathrm{P}}/V_{\mathrm{S}}$ gives $K/\mu$, not $K$ and $\mu$; to put them in GPa you need $\rho$ as well.
• Anisotropy. Real rocks, shales above all, are not isotropic: their velocities depend on the direction of travel relative to the fabric. The two moduli are then replaced by a stiffness tensor, and interpretation uses vertical and horizontal velocities and Thomsen’s parameters such as $\varepsilon$.