UCS Correlations

Summary

Unconfined compressive strength, Unconfined compressive strength, is measured on core but is needed everywhere, so it is estimated from logs with empirical relations to slowness, porosity or modulus. The published relations are numerous, each fitted to one lithology and one region, and on the same rock they can differ by a factor of three. They give a continuous curve that must be calibrated to core strength tests before it is used to size a mud weight.

Inputs and outputs

Equations

Four representative forms, each with its own units and its own lithology. Slowness \(\dtc\) is in µs/ft, total porosity \(\phit\) is a fraction, static modulus \(\gmEs\) is in GPa, and strength comes out in MPa.

Sonic, sandstone (an exponential form fitted to coal-measure sandstones, Bowen Basin, Australia):

Porosity, clean well-consolidated sandstone of porosity below about 0.3:

Sonic, shale (a power law fitted to North Sea Tertiary shales; \(304.8/\Delta t\) is the velocity in km/s):

Static modulus, sandstone, with \(\gmEs\) in GPa:

Single-value calculator

Behavior

Strength falls steeply as the rock slows down. The sandstone sonic relation gives 198 MPa at 50 µs/ft, 67 at 80, 33 at 100 and 11 at 130 µs/ft, a fall of a factor of 18 over the plotted range, which the log axis makes visible as a nearly straight line, because the form is exponential. The shale power law falls from 154 to 9 MPa over the same range, and is lower than the sandstone form by about 42 percent at 80 µs/ft. At the default inputs (80 µs/ft, porosity 0.15, static modulus 25 GPa) the four relations give 67, 90, 39 and 105 MPa. That spread, a factor of nearly three, is not an error in the code: the forms were fitted to different rocks. The porosity form is the most sensitive to its input: strength is 227 MPa at a porosity of 0.02, 135 at 0.10 and 9 at 0.30.

Parameter guidance

Pick by lithology first, then by data. Choose the relation derived for the rock you have, with the input you trust most: sonic if the sonic is good, porosity if the porosity is better than the sonic, modulus if static modulus is calibrated. The form for sandstone is not for shale and neither is for carbonate.

Calibrate. The only safe approach is to fit a scalar or a regression to core UCS (or to triaxial tests corrected to unconfined) in the same facies, and to show the scatter. A sonic relation scaled to core is an interpolation tool; an uncalibrated one is only a ranking.

Units. Check each form's units before use. Slowness in µs/ft, velocity in km/s or m/s, and porosity as a fraction or percent differ between papers, and the constants change by orders of magnitude with the wrong unit.

Test type. UCS from triaxial tests extrapolated to zero confinement, from true uniaxial tests, and from scratch tests are not identical. Record what the core data are.

Other forms. A limestone and dolomite form with an exponential in porosity, \(135.9\,e^{-4.8\phi}\) MPa (about 84 MPa at a porosity of 0.1), is often quoted; I could not verify it and it is not in the calculator.

Worked example

The four relations at 80 µs/ft, a porosity of 0.15 and a static modulus of 25 GPa:

import math
dt, phit, Es = 80.0, 0.15, 25.0
forms = {
    'sonic, sandstone (1200 exp(-0.036 dt))':  1200 * math.exp(-0.036 * dt),
    'porosity, clean sandstone':               254 * (1 - 2.7 * phit) ** 2,
    'sonic, North Sea shale':                  0.77 * (304.8 / dt) ** 2.93,
    'static modulus, sandstone':               2.28 + 4.1089 * Es,
}
for name, ucs in forms.items():
    print(f"{name:42s} {ucs:6.1f} MPa = {ucs * 145.038:7.0f} psi")
vals = list(forms.values())
print(f"spread: {min(vals):.0f} to {max(vals):.0f} MPa, ratio {max(vals) / min(vals):.1f}")

Output

sonic, sandstone (1200 exp(-0.036 dt))       67.4 MPa =    9770 psi
porosity, clean sandstone                    89.9 MPa =   13042 psi
sonic, North Sea shale                       38.8 MPa =    5624 psi
static modulus, sandstone                   105.0 MPa =   15229 psi
spread: 39 to 105 MPa, ratio 2.7

Assumptions and limitations

• The rock resembles the set the relation was fitted to in lithology, age, burial, cementation and region. This is rarely checkable without core.
• The input log is good: slowness and porosity are free of washout and cycle skip effects, and fluid effects on the sonic are small.
• Strength is a function of a single log variable. In reality, grain contacts, cement, clay content and fabric all matter, which is why scatter on the order of 20 to 40 percent about the fitted line is normal.
• The sample was tested in conditions comparable to the in-situ rock. Lab strength on dried, unloaded core is usually different from that of the saturated, confined rock.
• UCS is the right strength parameter. For wellbore stability, strength under confinement and the friction angle matter as well, see the next page.

QC checks

• Strength compares with core UCS in the same facies. Plot predicted against measured, with the slope, the offset and the scatter.
• Strength is lower in porous, high-slowness rock and higher in tight rock. A relation that gives the wrong direction has the wrong input units.
• Strength does not exceed what is physically reasonable for the lithology: a few MPa for unconsolidated sand, about 200 MPa for tight quartzite.
• Estimates from two different relations (for example sonic-based and porosity-based) are within about a factor of two. A larger spread means one of the inputs, or the relation, is wrong for this rock.
• The downstream breakout prediction is checked against caliper or image-log breakouts where available. That is the best test of a strength curve.

Going Deeper

Empirical UCS relations began with the observation that strength, velocity and porosity are all controlled by the same things: how much pore space there is and how well the grains are bonded. Compilations such as the one by Chang, Zoback and Khaksar list tens of relations by lithology and location; reading that list shows how little consensus there is. Relations in terms of the dynamic modulus or velocity have scatter that does not come from measurement error but from the missing variables of cementation and fabric. Recent work trains regression or machine-learning models on a basin's core data, which can do better in-basin and is worse outside it. For wellbore stability the important property is often not the peak strength of intact rock but the strength of a rock containing bedding planes or natural fractures, which can be much lower and is not captured by any log relation.

References

1. Chang, C., Zoback, M.D. and Khaksar, A., 2006. Empirical relations between rock strength and physical properties in sedimentary rocks. Journal of Petroleum Science and Engineering, 51(3–4), 223–237.
2. McNally, G.H., 1987. Estimation of coal measures rock strength using sonic and neutron logs. Geoexploration, 24(4–5), 381–395.
3. Vernik, L., Bruno, M. and Bovberg, C., 1993. Empirical relations between compressive strength and porosity of siliciclastic rocks. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 30(7), 677–680.
4. Horsrud, P., 2001. Estimating mechanical properties of shale from empirical correlations. SPE Drilling & Completion, 16(2), 68–73.
5. Bradford, I.D.R., Fuller, J., Thompson, P.J. and Walsgrove, T.R., 1998. Benefits of assessing the solids production risk in a North Sea reservoir using elastoplastic modelling. SPE 47360, SPE/ISRM Eurock 98.

Python reference implementation

Related methods

• Dynamic Elastic Moduli
• Brittleness Indices
• Static vs Dynamic Moduli
• Friction Angle, Biot Coefficient and Cohesion
• Breakout and Breakdown Pressures