Dynamic Elastic Moduli

Summary

The dynamic elastic moduli of an isotropic rock follow from the Compressional velocity, the Shear velocity and the bulk density: the Dynamic shear modulus, the Dynamic bulk modulus, the Dynamic Young's modulus and the Dynamic Poisson's ratio. They are the input to every other geomechanics calculation, but they are measured at sonic frequency and small strain and are not the moduli the rock shows when it is drilled or fractured.

Inputs and outputs

Equations

Convert the two slownesses to velocities in km/s. For slowness in µs/ft the factor is 304.8, because 1 ft is 0.3048 m; for slowness in µs/m the factor is 1000:

With density in g/cm³ and velocity in km/s, every product of density and velocity squared comes out directly in GPa, because 1 g/cm³ is 1000 kg/m³ and 1 km/s is 1000 m/s. The shear and bulk moduli of an isotropic elastic solid are:

Young's modulus and Poisson's ratio follow from them:

Poisson's ratio depends only on the velocity ratio \(\gmVpVs\), through \(\nu = (R^{2} - 2)/(2(R^{2} - 1))\) with \(R = \gmVp/\gmVs\). To get moduli in million psi, divide GPa by 6.894757; in the same units the shear modulus is \(13474.45\,\rhob/\gmDts^{2}\) for slowness in µs/ft.

Single-value calculator

Behavior

The plot holds the compressional slowness at 70 µs/ft and the density at 2.55 g/cm³ and sweeps the shear slowness. As the shear wave slows, the shear modulus falls steeply, from 23.7 GPa at 100 µs/ft to 5.9 GPa at 200 µs/ft, because it goes with the square of the velocity. Young's modulus falls with it, from 48.3 to 16.9 GPa. The bulk modulus moves the other way, from 16.8 to 40.5 GPa, because less of the compressional stiffness is taken up by the shear term. Poisson's ratio rises from 0.020 to 0.430 over the same range. The sweep starts at 100 µs/ft because below about 99 µs/ft, with this compressional slowness, Poisson's ratio is negative, and below 81 µs/ft the bulk modulus is negative, which no real rock has. A result in that region means a bad shear slowness, a bad compressional slowness, or a cycle skip.

Parameter guidance

The method has no tuning parameters, only inputs, so the guidance is about the inputs.

Slowness and units. Convert both slownesses to the same unit before doing anything else. A common failure is a compressional slowness in µs/ft with a shear slowness in µs/m, which gives a Vp/Vs ratio near 0.3 and nonsense moduli. A slowness of 70 µs/ft is 229.7 µs/m. If the shear log is a model rather than a measurement, see Shear Log Modeling.

Density. Use the bulk density from the repaired and environmentally corrected log, not the raw curve. In hole sections with washouts the density and the sonic are both unreliable, so the moduli there should be masked, not smoothed.

Which compressional slowness. Use the same compressional log everywhere in the calculation. A depth-shifted or filtered sonic gives moduli that do not line up with the density.

Anisotropy. Shales are anisotropic. A vertical-well sonic measures slowness along the bedding normal, so the moduli computed here are the vertical ones. See the limitations below.

Worked example

A compressional slowness of 70 µs/ft, a shear slowness of 100 µs/ft and a density of 2.55 g/cm³, then the same rock with the slownesses given in µs/m:

rho = 2.55
dtc, dts = 70.0, 100.0                       # us/ft
vp, vs = 304.8 / dtc, 304.8 / dts              # km/s
G = rho * vs**2
K = rho * (vp**2 - 4 / 3 * vs**2)
E = 9 * K * G / (3 * K + G)
nu = (3 * K - 2 * G) / (2 * (3 * K + G))
print(f"Vp = {vp:.3f} km/s, Vs = {vs:.3f} km/s, Vp/Vs = {vp / vs:.3f}")
print(f"G = {G:.2f} GPa, K = {K:.2f} GPa, E = {E:.2f} GPa, nu = {nu:.4f}")
print(f"E in Mpsi = {E / 6.894757:.2f},  G from 13474.45*rho/dts^2 = {13474.45 * rho / dts**2:.3f} Mpsi")
# same rock, slowness in us/m
dtc_m, dts_m = dtc / 0.3048, dts / 0.3048
vp_m, vs_m = 1000 / dtc_m, 1000 / dts_m
print(f"us/m inputs {dtc_m:.1f}, {dts_m:.1f} give Vp = {vp_m:.3f}, Vs = {vs_m:.3f}: same velocities")
# check Poisson's ratio from the velocity ratio alone
R = vp / vs
print(f"nu from Vp/Vs = {(R**2 - 2) / (2 * (R**2 - 1)):.4f}")

Output

Vp = 4.354 km/s, Vs = 3.048 km/s, Vp/Vs = 1.429
G = 23.69 GPa, K = 16.76 GPa, E = 48.31 GPa, nu = 0.0196
E in Mpsi = 7.01,  G from 13474.45*rho/dts^2 = 3.436 Mpsi
us/m inputs 229.7, 328.1 give Vp = 4.354, Vs = 3.048: same velocities
nu from Vp/Vs = 0.0196

Assumptions and limitations

• The rock is isotropic and linearly elastic. In a shale the vertical and horizontal velocities differ by 10 to 30 percent or more, and the isotropic formulas then give a single apparent modulus rather than a true one.
• The sonic velocities are the formation velocities. Invasion, washouts, cycle skips and shear slowness from a monopole tool in a slow formation (where the shear wave is not refracted) all violate this.
• The moduli are dynamic: they are measured at kilohertz frequency and microstrain amplitudes. They are higher than the static moduli that govern drilling and fracturing, see Static vs Dynamic Moduli.
• The moduli are for the saturated rock, with the fluid in the pores. Dry-rock moduli need a fluid substitution that is not covered here.

QC checks

• Vp/Vs lies between about 1.45 and 2.5 for most clastics and carbonates, and Poisson's ratio between 0.1 and 0.4. Values outside that range need an explanation.
• Bulk modulus is positive everywhere: a negative value means the shear slowness is too slow for the compressional slowness (Vs/Vp above 0.866).
• Moduli follow the lithology: stiffer in tight carbonate and cemented sandstone, softer in shale and porous sandstone. A curve that does not track the gamma ray or porosity points to a data problem.
• Moduli do not spike at washouts or at depth-shift mismatches between the sonic and density curves. Compare with the caliper.
• Where core exists, ultrasonic velocities on dry or saturated plugs, converted with the same formulas, give a check on the log values after allowing for frequency and stress state.

Going Deeper

The formulas are the textbook relations of linear isotropic elasticity, which need only two independent constants. A third relation, between P-wave modulus M = rho Vp squared, G and K, is M = K + 4G/3, which is where the bulk modulus expression comes from. Anisotropic media need up to five constants (in the transversely isotropic case) and a dipole sonic in a deviated well measures a mix of them. The more important distinction for engineering is between dynamic and static moduli: the first is measured with a small, fast, essentially undrained disturbance, and the second under the large, slow, drained loading that a wellbore or a fracture imposes. The rest of this step is largely about carrying the first into the second.

References

1. Mavko, G., Mukerji, T. and Dvorkin, J., 2009. The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media, 2nd edition. Cambridge University Press.
2. Fjær, E., Holt, R.M., Horsrud, P., Raaen, A.M. and Risnes, R., 2008. Petroleum Related Rock Mechanics, 2nd edition. Elsevier.

Python reference implementation

Related methods

• Brittleness Indices
• Static vs Dynamic Moduli
• UCS Correlations
• Friction Angle, Biot Coefficient and Cohesion
• Shear Log Modeling
• Porosity