Summary
Experimental studies indicate that when effective stress increases, compressional wave velocity in porous rocks increases. Reservoir pressure reduction, resulting from hydrocarbon production, increases effective stress. For a rock with a given porosity the sonic log may show decreasing values as the pressure in the reservoir decreases. This in turn may lead to underestimation of the actual porosity of the reservoir rocks in low pressure reservoirs. The range of such underestimation for liquid saturated reservoirs may not be significant, but since the influence of effective stress on velocity increases as fluid saturation changes to gas, porosity underestimation by conventional velocity-porosity transforms for gas bearing rocks may increase. Examples are taken from partially depleted gas reservoirs in the Cooper basin, South Australia. The stress dependent nature of velocity requires that the in situ pressure condition should be considered when the sonic log is used to determine the porosity of gas producing reservoir rocks.
Introduction
Knowledge of the elastic velocities in porous media is of considerable interest in many research fields including rock mechanics, geological engineering, geophysics, and petroleum exploration. In petroleum exploration this concept mainly concerns the relationship between reservoir rock characters and the acoustic velocity. Porosity estimation is one of the most common applications of acoustic velocity data in hydrocarbon wells. There are numerous empirical equations to convert sonic travel time (ts) to porosity. It is well known that the P-wave velocity (vp), for a rock with a given porosity, is also controlled by several other factors such as pore filling minerals, internal and external pressures, pore geometry, and pore fluid saturation, etc.1 These factors may have significant effect on measured ts and thus on porosity interpretation from the sonic log.
Several investigators (see Refs. 2-4) have studied the effect of clay content and the type and saturation of pore fluids on acoustic velocity and the sonic log derived porosity in reservoir rocks. In contrast, the in situ pressure condition has rarely been considered as a parameter in the commonly used velocity-porosity equations. This paper addresses the influence of effective stress on the elastic wave velocities in rocks and its implications on porosity determination from the sonic log in hydrocarbon bearing reservoirs. Examples from the literature and a case study in a gas-producing reservoir are used to highlight the importance of the issue.
Effective stress is the arithmetic difference between lithostatic pressure and hydrostatic pressure at a given depth. It may normally be considered equivalent to the difference between confining pressure (pc) and pore pressure (pp).5 Experimental studies indicate that as effective stress increases, vp increases.6 This increase depends on the rock type and pore fluid. The change in vp due to effective stress increase is more pronounced when the pore fluid is gas.7 Current sonic porosity methods do not account for the variation of vp due to pressure change in hydrocarbon producing fields.
Effective Stress Versus Velocity
Wyllie et al.6 measured ultrasonic P-wave velocity as a function of effective stress in water saturated Berea sandstone. They showed that at constant confining pressures vp increases with decreasing pore pressure, and for constant effective stress, the vp remains constant. Similar relationships between effective stress and P-wave velocity have also been reported by other researchers.7–10 King,9 and Nur and Simmons7 reported a more pronounced stress effect on vp when air replaces water.
Experimental results indicate that confining and pore pressures have almost equal but opposite effects on vp. Confining pressure influences the wave velocities because pressure deforms most of the compliant parts of the pore space, such as microcracks and loose grain contacts. Closure of microcracks increases the stiffness of the rock and increases bulk and shear moduli. Increases in pore pressure mechanically oppose the closing of cracks and grain contacts, resulting in low effective moduli and velocities. Hence, when both confining and pore pressures vary, only the difference between the two pressures has a significant influence on velocity8 that is
Δ p = p c − p p , ( 1 )
where ?p is differential pressure. The more accurate relationship may be of the form of
p e = p c − σ p p , ( 2 )
where pe is effective stress and ? is the effective pressure coefficient. The value of ? varies around unity for different rocks and is a function of pc11 Eq. 2 indicates that for ? values not equal to unity, changes in a physical property caused by changes in confining pressure may not be exactly canceled by equivalent changes in pore pressure. Experimentally derived ? values for the water saturated Berea sandstone by Christensen and Wang10 show values less than 1 for properties that involve significant bulk compression (vp), whereas a pore pressure increment does more than cancel an equivalent change in confining pressure for properties that significantly depend on rigidity (vs).
Semiglobal viscoacoustic full-waveform inversion
