Static vs Dynamic Moduli

Summary

Static moduli, measured on core in slow loading, are lower than the Dynamic Young's modulus from logs, often by a factor of two in soft rock and by a little in stiff rock. Published relations convert one to the other, but each was fitted to a particular rock and modulus range and the answers differ widely. Treat any conversion as a starting form to be calibrated to core, never as a general law.

Inputs and outputs

Equations

Two forms are in common use. A linear relation in GPa:

and a power law, with both moduli in GPa:

The ratio of static to dynamic modulus is \(\gmEsRatio\). One published linear fit is \(a_E = 1.05\) and \(b_E = -3.16\) GPa for stiff rock (about 25 to 100 GPa dynamic). A power-law form with \(c_E\) of about 0.10 to 0.15 and \(p_E\) of about 1.39 to 1.49 has been reported for a different rock set. The calculator's defaults are these two published-style forms, so that the page can show how far they disagree.

For Poisson's ratio no general conversion exists. The usual practice is to take the static value equal to the dynamic one, or to fit a straight line to core, and to clip the result to the physical range 0 to 0.5.

Single-value calculator

Behavior

With the default constants the two forms give very different answers. At a dynamic modulus of 10 GPa the linear form gives 7.3 GPa and the power form 2.8 GPa; at 40 GPa they give 38.8 and 21.0 GPa; at 100 GPa, 101.8 and 79.4 GPa. The linear form is nearly equal to the dynamic modulus (ratio 0.97 at 40 GPa) and, because its slope is above 1, would exceed it above 63 GPa, which is unphysical for most rocks. The power form falls below the 1:1 line everywhere in the plotted range, with a ratio of 0.53 at 40 GPa. The spread between the forms at one dynamic modulus, almost a factor of two at 40 GPa, is the main message: the choice of relation is worth more than any other input to the conversion, and it can only be made with core data.

Parameter guidance

There are no universal constants. The parameters here are placeholders for a calibration. Fit static modulus from triaxial or uniaxial tests against the dynamic modulus from the log at the same depth, with the same lithology, and report the fit with its scatter and the modulus range it covers.

Stress state. Static moduli measured on core depend on the confining stress and on the stress path, and soft rock stiffens with confinement. Compare tests at an effective stress representative of the reservoir.

Range. Do not use a relation outside the modulus range it was fitted over. A relation quoted for 25 to 100 GPa tells you nothing at 10 GPa, where the linear and power-law defaults here return 7.3 and 2.8 GPa, a factor of 2.6 apart.

Shales. Static-dynamic differences in shale are large and scarce in the literature, and anisotropy makes them worse. If there is no core, state that the static modulus is an assumption and run the stress model at a range of values.

Worked example

A dynamic Young's modulus of 40 GPa converted with the linear and power forms, and the crossing point of the linear form with the 1:1 line:

Ed = 40.0                                   # GPa, dynamic
a, b = 1.05, -3.16                          # linear form, GPa
c, p = 0.10, 1.45                           # power form, GPa
es_lin = max(0.0, a * Ed + b)
es_pow = c * Ed**p
print(f"linear: Es = {a} x {Ed:g} + ({b}) = {es_lin:.2f} GPa  (Es/Ed = {es_lin / Ed:.2f})")
print(f"power:  Es = {c} x {Ed:g}^{p} = {es_pow:.2f} GPa  (Es/Ed = {es_pow / Ed:.2f})")
print(f"linear form equals Ed at Ed = {b / (1 - a):.1f} GPa; above that Es > Ed")
# van Heerden style a, b ranges: spread of the power form at 40 GPa
lo, hi = 0.097 * Ed**1.388, 0.152 * Ed**1.485
print(f"power form with the published-style a, b ranges: {lo:.1f} to {hi:.1f} GPa at Ed = 40 GPa")

Output

linear: Es = 1.05 x 40 + (-3.16) = 38.84 GPa  (Es/Ed = 0.97)
power:  Es = 0.1 x 40^1.45 = 21.04 GPa  (Es/Ed = 0.53)
linear form equals Ed at Ed = 63.2 GPa; above that Es > Ed
power form with the published-style a, b ranges: 16.2 to 36.4 GPa at Ed = 40 GPa

Assumptions and limitations

• A single smooth relation links dynamic and static modulus. Scatter in such fits is large, often 20 to 30 percent, and the relation changes with porosity, clay, cementation and fracturing.
• The conversion applies the same way along the whole interval. Different facies need different calibrations.
• Static moduli measured on core are representative of the in-situ rock. Core damage, unloading and dehydration all reduce them.
• The dynamic modulus is correct. Errors in the sonic and density flow straight into the static estimate, and a power law exaggerates them.

QC checks

• The static modulus does not exceed the dynamic modulus. If it does, the relation is outside its range.
• The static modulus tracks the core static data with no systematic bias. Plot calibrated-to-core and report the scatter.
• The ratio Es/Ed increases with stiffness: soft rock shows a ratio of 0.3 to 0.7 and stiff, low-porosity rock close to 1. A ratio that does not behave this way means the relation is wrong for the rock.
• Run the downstream stress and strength calculations with at least two conversions, and see which results change. If the outcome does not depend on the choice, the choice does not matter.

Going Deeper

Static moduli are lower than dynamic ones for several reasons that act together: the dynamic wave passes with strain amplitudes of about 1e-7 or less, and the static test imposes 1e-3; microcracks and compliant grain contacts close under the small strain and open under the large one; fluid in the pores has no time to flow in the dynamic case (undrained) but does in the static one (drained); and the dynamic modulus is an elastic one while the static modulus includes some inelastic deformation. The gap is therefore largest in porous, cracked or poorly consolidated rock and smallest in tight crystalline rock, which is the reason the published relations for high modulus (above 25 GPa) are nearly linear and close to 1:1, while those for soft rock are power laws with a steep gap. The compilations by Fjaer and co-authors and by Mavko and co-authors list many more relations than are reproduced here, and the lesson from reading them is the same: each fits its own data.

References

1. Christaras, B., Auger, F. and Mosse, E., 1994. Determination of the moduli of elasticity of rocks. Comparison of the ultrasonic velocity and mechanical resonance frequency methods with direct static methods. Materials and Structures, 27(4), 222–228.
2. van Heerden, W.L., 1987. General relations between static and dynamic moduli of rocks. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 24(6), 381–385.
3. 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

• Dynamic Elastic Moduli
• Brittleness Indices
• UCS Correlations
• Friction Angle, Biot Coefficient and Cohesion