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Rock acoustic velocities and porosity
The bounding relations defined in Rock moduli boundary constraints can be applied directly to rock acoustic velocities. Some dolomites with vuggy pores may approach the Voigt bound. Highly fractured rocks may approach the Reuss bound. However, there is often a great difference between these idealized bounds and real rocks. For sandstones, we would expect to begin with quartz velocity at zero porosity and have decreasing velocity with increasing porosity. By combining Eqs. 1 and 2 for moduli in Eq. 3, we can plot expected velocity bounds, as in Fig. 1a. Observed distributions for sandstones are also plotted, and we see a systematic discrepancy with the upper (Voigt) bound. At high porosities, grains separate, and the mixture acts as a suspension. The majority of rocks have an upper limit to their porosity usually termed "critical porosity," Φc (Yin et al. and Nur et al.). At this high porosity limit, we reach the threshold of grain contacts and grain support (Han et al.).
where MR is the lower Reuss effective modulus. The average value between these two limits is often used in property estimation and is termed the Voigt-Reuss-Hill relation
Fig. 1 – (a) Compressional velocity vs. porosity in sandstones: laboratory data with elastic bounds (modified from Marion). (b) Sandstone velocities divided into classes.
Brine-saturated sandstone velocities can be separated into classes based on their velocity-porosity relations (Fig. 1b).
- Very clean sandstones (Class I) decrease in a simple linear trend from the 6 km/s velocity of quartz as porosity increases.
- Most consolidated rocks (Class II) have somewhat lower velocities, still decreasing with increasing porosity.
- Poorly cemented sands (Class III) approach the lower Reuss bound for velocity.
- Pure suspensions are dominated by the modulus of water (Class IV) and are almost independent of the porosity. However, such suspensions are rare.
- Another important class is dominated by fractures (Class V). Fractures have a far greater effect on velocity than might be expected for their low porosity, and may approach the Reuss bound.
Measured velocity-porosity relationship
Numerous systematic investigations into the relationship of velocity, porosity and lithology (usually clay content) have been conducted. The results of Vernik and Nur for brine-saturated sandstones are shown in Fig. 2 for compressional and shear velocities, respectively. Very clean sands (clean arenites) show the linear decrease from quartz velocity. However, even small amounts of clays will substantially lower the trend. Increasing clay content will then continue to lower velocities.
Fig. 2 – (a) Brine-saturated sandstone compressional velocities as a function of porosity. Sandstones are segregated into lithologic type, and show decreasing velocity as rocks depart from pure quartz content (after Vernik and Nur). (b) Brine-saturated sandstone shear velocities as a function of porosity. Behavior is similar to the Vp trends seen in (a) above (after Vernik and Nur).
Numerous examples of general porosity/velocity/clay content relations for sandstones are given in Table 1 a and b (symbol definitions for these relations are in Table 1c). These types of relations have proved very useful in giving velocities under general conditions, providing the overall effects of clay, and establishing the relation of compressional to shear velocity (Vp/Vs ratios). Vp−Vs relations are extremely important, because shear logs are relatively rare, yet shear velocities are critical in determining seismic direct hydrocarbon indicators such as reflection Amplitude-Versus-Offset (AVO) trends (Castagna et al.).
Measured data for carbonates are less abundant. A systematic investigation of samples from several wells was reported by Rafavich et al. A plot of their results for carbonate Vp as functions of porosity and composition is shown in Fig. 3. They collected detailed information on fabric and texture as well as porosity and mineralogy. Performing regressions on their extensive data set produced the relations given in Table 2a. The coefficients associated with these equations are given in Table 2b. Note that the relations are dependent on the effective pressure.
Fig. 3 – Carbonate compressional velocity as functions of porosity and mineralogy. The compositions are plotted by number according to the Anhydrite-Calcite-Dolomite triangular diagram in the lower-left portion of the figure (from Rafavich et al.).
Table 2A - Rafavich et al. carbonate Vp andVs relations.
A similar set of measurements by Wang et al. are shown in Fig. 4. For carbonates, the data can be quite scattered, but can still show the general velocity decrease with increasing porosity. These results were summarized in a set of relations (Table 3) again showing pressure dependence. Their data, however, includes measurements made with samples not only brine-saturated, but hydrocarbon-saturated and after simulated reservoir floods. They demonstrate that the overall velocity and impedance changes were strongly dependent on the imposed sequence of flooding. The ability to observe a particular reservoir process will be more complicated than simply completely substituting fluids into the rocks.
Fig. 4 – (a) Dry carbonate velocities. Although there is considerable scatter, perhaps because of the heterogeneous nature of the porosity, a systematic decrease in both compressional and shear velocities with increasing porosity is obvious (from Wang et al.). (b) Carbonate velocities when saturated with a light refined oil. Compressional velocities are higher and shear velocities slightly lower. The porosity dependence is similar to the dry case (from Wang et al.).
Table 3 - Carbonate comressional and shear velocity relations (Wang et al.)
|M||=||molecular weight, g/mole|
|MA, MB||=||modulus of component a, b, etc., GPa or MPa|
|MO||=||reference oil molecular weight, g/mole|
|MR||=||Reuss bound modulus, GPa or MPa|
|MV||=||Voigt bound modulus, GPa or MPa|
|MVRH||=||Voigt-Reuss-Hill bound modulus, GPa or MPa|
|Vp||=||compressional velocity, m/s|
|μ||=||shear modulus, GPa or MPa|
|ρ||=||density, kg/m3 or g/cm3|
- Marion, D.P. 1990. Acoustic, mechanical, and transport properties of sediments and granular materials. PhD dissertation, Stanford University, Palo Alto, California.
- Yin, H., Nur, A., and Mavko, G. 1993. Critical porosity - a physical boundary in poroelasticity. Proc., 34th US Symposium on Rock Mechanics, Madison, Wisconsin, USA, 27-30 June.
- Nur, A., Mavko, G., Dvorkin, J. et al. 1998. Critical porosity: A key to relating physical properties to porosity in rocks. The Leading Edge 17 (3): 357-362. http://dx.doi.org/10.1190/1.1437977.
- Han, D.-H. 1986. Effects of Porosity and Clay Content on Acoustic Properties of Sandstones and Unconsolidated Sandstones. PhD dissertation. 1986. . PhD dissertation, Stanford University, Stanford, California (October 1986).
- Vernik, L. and Nur, A. 1992. Petrophysical classification of siliciclastics for lithology and porosity prediction from seismic velocities. AAPG Bull. 76 (9): 1295-1309.
- Castagna, J.P., Batzle, M.L., and Kan, T.K. 1993. Rock physics—the link between rock properties and AVO response. In Offset-Dependent Reflectivity—Theory and Practice of AVO Analysis, ed. P. Castagna and M.M. Backus, No. 8, 124–157. Tulsa, Oklahoma: Investigations in Geophysics series, Society of Exploration Geophysicists.
- Rafavich, F., Kendall, C., and Todd, T. 1984. The relationship between acoustic properties and the petrographic character of carbonate rocks. Geophysics 49 (10): 1622-1636. http://dx.doi.org/10.1190/1.1441570.
- Wang, Z., Hirsche, W.K., and Sedgwick, G. 1991. Seismic Velocities In Carbonate Rocks. J Can Pet Technol 30 (2): 112. PETSOC-91-02-09. http://dx.doi.org/10.2118/91-02-09.
Noteworthy papers in OnePetro
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read
Domenico, S.N. 1984. Rock lithology and porosity determination from shear and compressional wave velocity. Geophysics 49 (8): 1188-1195. http://dx.doi.org/10.1190/1.1441748.
Freund, D. 1992. Ultrasonic compressional and shear velocities in dry clastic rocks as a function of porosity, clay content, and confining pressure. Geophys. J. Int. 108 (1): 125-135. http://dx.doi.org/10.1111/j.1365-246X.1992.tb00843.x