A Gassmann consistent rock physics template

Brian H. Russell, Laurence R. Lines

In this report, we discuss a new approach to the derivation of the rock physics template, or RPT. We first discuss pore space stiffness and use the Betti-Rayleigh reciprocity theorem to derive Gassmann's equation from the dry and saturated pore space stiffnesses. We then review the work of Russell and Smith (2007), which showed empirically that the dry rock pore space stiffness stays constant over a range of porosities for a constant pore pressure. This allows us to use both the pore space stiffness method and the Gassmann equations for estimating bulk and shear moduli as a function of both saturation and porosity. Using empirical measurements in sandstones, we compare this fit to the alternate approach proposed by degaard and Avseth (2003), which is based on Hertz-Mindlin contact theory. We show that our new method is both more intuitive and also produces a modulus ratio as a function of porosity which is closer to that derived in experimental studies. Figure 1 on the left below shows a comparison of the results from the two methods, where we have cross-plotted Vp/Vs ratio versus acoustic impedance. The curves have been calibrated at a porosity of 20%. Figure 2 on the right below shows a comparison of the results from the two methods plotted for the dry rock modulus ratio as a function of porosity, where the dashed line shows a constant ratio. Note that the pore space compressibility method gives a much better fit to the experimentally determined constant ratio, except at zero porosity, where both methods (pore space compressibility and Hertz-Mindlin) converge to the correct mineral modulus ratio.