SONIC LOG DATA HELP DETERMINE FORMATION STRENGTH

Nathan Stein
Sans Co.
Boston

A set of equations helps improve the reliability of using acoustical wave velocity (dynamic) data to obtain quick, acceptable estimates for the Mohr-Coulomb (static) strengths of different types of rock.

Good estimates of formation strength from offset well sonic log data can help in the planning of bit programs and well stimulation procedures.

Dynamic shear modulus values were calculated from Mohr-Coulomb strength plots for five limestone samples.

Care must be exercised in applying this method with log data. The terminal stress decreases as rock strength increases. The true vertical stress should be used in the calculations until the terminal vertical stress is exceeded.

If the field-measured stress conditions exceed 6,300 psi for friable sands, 1,500 psi for well-cemented sands, and 500 psi for limestones, then the calculations should be based on the terminal stress values rather than true stress.

The use of sonic log data to estimate rock strength has two pluses:

The measurements relate to the in situ strength of undamaged rock.
Strength surveys can be made quickly and inexpensively if the measurements can be properly converted to standard strength values. Previously available methods for using sonic data to predict the standard strength values of rock may be unreliable.

The calculation procedure presented in this article uses the results of theoretical analyses of laboratory rock strength tests and acoustical wave velocity data to improve the reliability of using sonic log data.

The traditional Mohr-Coulomb strength plots for rocks can be related to sonic data obtained either in the laboratory or from sonic logs.

ACOUSTICAL WAVE VELOCITY

Wyllie, et al., measured compressional acoustic wave velocity values for different rock samples over a wide range of applied stresses.' One of the objectives was to determine how to use acoustical wave velocity log data to calculate the porosity of formations penetrated by a well.

Wyllie developed a time-average equation which seemed to be most accurate when there was sufficient stress applied to the rock sample to reach a near-constant velocity value, called the terminal velocity (Fig. 1).

In Fig. 1, note the initial relatively rapid rise in velocity values with increasing stress.

The velocity values become level at high stresses for the stronger rock. The level velocity values are terminal velocities, and the stress value where the level velocity values begin is the terminal stress.

Because high stress levels were required to close flaws or fractures in the samples tested, an approximate safe level of 6,000 psi was used. Terminal velocity was not reached in friable material because the rock grains crushed when stress was increased.'

Well log data were used to calculate the terminal stress for a friable sand.' That analysis indicated that a theoretical stress of 6,300 psi was required to reach terminal velocity and to apply the time-average equation to calculate a valid porosity value.1

VELOCITY PLOTS

The original velocity plots may reflect the matrix properties of the rock samples and not the flaws or fractures . 2

The following model description may explain these results, which are based on rock behavior in a series of failure tests.

Triaxial equipment was used for a series of tests in which vertical stress was increased until the rock samples failed. Each test was conducted at a different level of lateral stress.

In Fig. 2, the slope of the straight line portion of each test plot is defined as the modulus for that lateral stress condition.

Dobrin reported that another way to calculate modulus values is with acoustical wave velocity data.' There is a relationship between acoustical wave velocity and slope of the straight-line portion in a plot of strength test data.

The changing slopes for tests with low lateral stress levels correspond to the curved portion of Wyllie's plots of acoustical wave velocity where stress levels were low.

At higher lateral stresses, the modulus value in the strength test plots becomes constant.

This condition is analogous to Wyllie's terminal velocity.

Peaks in the strength test plots continue to increase at greater lateral stress levels even though the modulus value remains constant. Rock strength parameters based on acoustical wave velocities must be evaluated at stress levels less than terminal conditions, in which both strength and velocity values vary with stress.

STRESS REGION

The rock properties in the stress region less than terminal conditions are the basis for converting rock strengths between dynamic and static values. Dynamic values are based on acoustical velocity data; static values are based on rock failure tests in a press. The following are three characteristics of these rock properties:

The relationship between shear modulus and stress is a straight line.
In studies with friable sands at stress levels less than terminal conditions, a straight-line plot of shear modulus vs. vertical stress was used to calculate rock properties. 4-6 This plot was developed from compressional acoustic wave data, assuming the average bulk modulus value in the sands remained constant.
This procedure is analogous to working with partial differentials in which only the change in shear modulus with stress is considered. The procedure obtained good results because the predicted fracture pressures agreed with empirical data reported in the literature.
There is a common point (a,O) on the shear modulus vs. stress on the Mohr-Coulomb plots (Fig. 3).
The horizontal axis of the Mohr-Coulomb strength plot is the stress vertical to the plane of failure in rock samples subjected to compression strength tests at different lateral stress conditions.
The vertical axis plots the shear stress along the plane of failure.
The strength of cohesion, or cementation, is the value on the Mohr-Coulomb line at zero vertical stress. A strength test measures the stress conditions when the sample fails. The negative vertical stress is significant. If the effect of the negative vertical stress (a) is equal to the strength of cementation then the shear strength is zero.
The test plots in Fig. 2 confirm that the shear modulus coordinates are the same as these shear strength coordinates, assuming the curves were obtained in compressional tests with triaxial equipment.
The slopes for that case would be Young's modulus values. No compression would be required for the sample to fail if shear strength is zero.
Therefore, the test plot of interest would lie on the horizontal axis, and the value of Young's modulus for a horizontal line is zero. Also, the shear modulus should approach zero when Young's modulus approaches zero, in accord with Equation 1. The assumptions are that the bulk modulus has a finite value and that the rock is homogeneous and isotropic.
The stress required to reach terminal velocity decreases as the strength of the rock increases. From these data, estimates of terminal stress thresholds for rocks of different strengths are plotted in Fig. 4.

SHEAR MODULUS

Wuerker reported data for a limited number of limestone samples used to convert the Mohr-Coulormb strength plots to shear modulus values (Table 1).

The following iterative procedure is used to calculate shear modulus values using porosity fraction, bulk density, and the Mohr-Coulomb equation data for each sample:

Calculate the travel time (the reciprocal of velocity) of a compressional acoustic wave going through the rock at 8 terminal stress (Equation 2). (It is assumed the sample pores are filled with brine.)
Calculate the combined dynamic modulus at the terminal stress (Equation 3).4
Calculate the combined dynamic modulus at s = 0 psi and any assumed reasonable value for PSI (Equation 4).
Estimate the bulk modulus, K, from R from Table 2. The correlation can be found in the literature. 4 5
Calculate the "a" value for a plot of shear modulus vs. stress. Compare this value with the "a" value calculated from the Mohr-Coulomb strength plot (Equations 5 and 6).
The elastic identity R (4/3)G + K is used, assuming the rock is homogeneous and isotropic. The rock matrix is assumed to be rigid, and the bulk modulus is assumed to remain constant with changing stress. Only R or G values may change with stress (as in the procedure for partial differential equations).
If the two values for "a" from Equations 5 and 6 are different, a new value for PSI is assumed in the third step. This iterative procedure should be continued until the two "a" values match.
Calculate Go = (3/4)(Ro-K). The angle of the G vs. s plot is (3/4)tan PSI.

Table 3 lists the calculated values of G and the shear modulus values for five limestone samples reported by Wuerker.7

The original work did not note the stress conditions when the dynamic modulus values were measured. It is likely that some stress was applied because these values were greater than the calculated G. values by an average algebraic difference of 11%.

Fig. 5 is a plot of dynamic modulus values calculated with the iterative procedure for limestone samples. using a format convenient for converting modulus data into equivalent Mohr-Coulomb equations.

The x-axis is the tangent of the shear modulus vs. stress plot. The y-axis is the Mohr-Coulomb strength plot at a vertical stress equal to 0 psi.

In log measurements at stress levels less than terminal stress, Fig. 5 may be used to determine the Mohr-Coulomb strength plot.9

In log measurements at depths with stresses greater than the terminal stress, the actual stress level is not needed.

The well log data at the rock depth may be considered to be at terminal property conditions. A value for tan P will be assumed to obtain the terminal stress from Fig. 4. The calculation procedure described in this article may then be applied.

REFERENCES

Wyllie, M.R.J., Gregory, A.R., and Gardner, G.H.F., "An Experimental Investigation of Factors Affecting Elastic Wave Velocities in Porous Media," Geophysics, Vol. 23, No.3, 1958.
Stein, N., "Porosity of Friable Sand Using Acoustic Wave Velocity", Presented at the Third International Symposium on Borehole Geophysics for Minerals, Geotechnical, and Ground Wave Applications, Las Vegas, Oct. 2-5, 1989.
Dobrin, M.B., Introduction to Geophysical Prospecting, McGraw-Hill Book Co., New York, 1952.
Stein, N., "Mechanical Properties of Friable Sands from Conventional Log Data," journal of Petroleum Technology, July 1976.
Stein, N., "Estimate Formation Strength Using Log Data," "World Oil, November 1987.
Stein, N., "How To Calculate Fracture Pressures From Well Logs," Petroleum Engineer, August 1968.
Wuerker, R.G., "Annotated Tables of Strength and Elastic Properties of Rocks," Petroleum Transactions Reprint Series, No. 6, Drilling, 1962.
"Log Interpretation Principles," Schlumberger Ltd., Vol. 1, 1972.
Stein, N., "Resistivity and density logs key to fluid pressure estimates," OGJ, Apr. 8, 1985.