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Rock acoustic velocities and pressure
Rock moduli (compressibility) and elastic velocities are strongly influenced by pressure. With increasing effective pressure, compliant pores within a rock will deform, contract, or close. The rock becomes stiffer, and, as a result, velocities increase.
Effect of pressure
Two examples are shown in Fig. 1. The typical behavior is rapid increase in velocity, with increasing pressure at low pressures, followed by a flattening of the curve at higher pressures. Presumably, compliant pores and cracks are closed at higher pressure, and velocities asymptotically approach a relatively constant velocity. This specific behavior at high pressures leads to the simple velocity-porosity transforms and probably is responsible for our ability to use sonic tools as in-situ porosity indicators with little regard to local pressures.
Poorly consolidated sands
where f is approximately linear. This type of relation is particularly useful for poorly consolidated sands.
Although the absolute pressure dependences shown in Fig. 1a vs 1b are in significant contrast, for most sandstones, relative changes are more consistent. By normalizing the velocities to those at high pressure (40 MPa), we get a much more consistent behavior (Fig. 2).
where Pe is the effective pressure.
For carbonates, the explicit pressure dependence given in Tables 2 and 3 allow the pressure dependence to be evaluated.
The pressure dependence for carbonate Vp from Rafavich et al. is shown in Fig. 3. Note that pressure sensitivity increases with increasing porosity. These types of relations permit velocity changes associated with pressure changes in the reservoir to be modeled.
Fig. 3 – Generalized compressional velocities dependence on pressure seen in carbonates by Rafavich et al. Pressure dependence is a function of porosity.
It is important to note that all these relations involve either differential pressure (Pd) or effective pressure (Pe). Pore pressure (Pp) counters the influence of confining pressure (Pc), so the difference between these two controls rock properties. This has been expressed simply in the Terzaghi relation for the pressure dependence of a given porous material property S,
This kind of behavior has been seen in numerous cases, as in Fig. 4. This is one reason why properties such as density, resistivity, and velocity can decrease with increasing depth when "overpressure" or when increased pore pressure is encountered. Changes in reservoir pore pressure will have a similar influence. More precisely, it is the effective pressure that controls properties rather than just the differential. However, the magnitude of effective pressure is often found to be close to the simpler differential pressure.
Fig. 4 – Compressional velocities through a water-saturated oil-wet sandstone sample at various confining and pore pressures. When confining and pore pressures are varied together to give constant differential pressure, the velocity stays almost constant (after Wylie et al.).
|Vp||=||compressional velocity, m/s|
|Vp||=||compressional velocity, m/s|
|Pc||=||confining pressure, MPa|
|Pd||=||differential pressure, MPa|
|Pe||=||effective pressure, MPa|
|Pp||=||pore pressure, MPa|
- Gassmann, F. 1951. Elastic waves through a packing of spheres. Geophysics 16 (4): 673–685. http://dx.doi.org/10.1190/1.1437718.
- Duffy, J. and Mindlin, R.D. 1956. Stress-strain relations and vibrations of a granular medium, No. 24, 584–593. New York: Columbia University.
- Eberhart-Phillips, D., Han, D.-H., and Zoback, M.D. 1989. Empirical relationships among seismic velocity, effective pressure, porosity, and clay content in sandstone. Geophysics 54 (1): 82–89. http://dx.doi.org/10.1190/1.1442580.
- 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.
- Terzaghi, K. and Peck, R.B. 1948. Soil Mechanics in Engineering Practice. New York: John Wiley & Sons.
- Wyllie, M.R.J., Gregory, A.R., and Gardner, G.H.F. 1958. An Experimental Investigation of Factors Affecting Elastic Wave Velocities in Porous Media. Geophysics 23 (3): 459. http://dx.doi.org/10.1190/1.1438493.
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