Fluid substitution for thinly interbedded sand-shale sequences using the mesh method

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Gassmann’s fluid substitution model is intended for a monomineralic, homogeneous porous rock, in which pore pressures induced by applied loads can equilibrate throughout the pore space. These assumptions are violated when Gassmann’s equations are applied to measurements that represent effective medium averages over subresolution layers of alternating sand and shale. The conventional procedure for treating this problem has been to first downscale; i.e., estimate the properties of the fine-scale sand and shale endmembers from the coarse-scale measurements, apply Gassmann’s fluid substitution to the sand only, and then Backus average back to the original scale. This procedure, however, is very sensitive to errors in estimated sand fraction and shale properties and becomes particularly unstable at small sand fractions. A new method for fluid substitution combines rock-physics models for dispersed and interbedded sand-shale systems, which are often approximated with Reuss or lower Hashin-Shtrikman interpolations between endmembers quartz mineral, clean sand, and shale. When expressed as P-wave compliance versus porosity, these trends become approximately linear. The Backus average of the normal incidence P-wave compliance of thinly layered mixtures of various sand-shale facies is also a linear trend with porosity. As a result, the upscaled fluid substitution change of compliance of any point within the dispersed or layered sand-shale system is approximately proportional to the fluid-substituted change of compliance of the clean sand endmember, scaled by the ratio of effective porosity to clean sand porosity. The result is a fluid substitution procedure that operates directly at the measurement scale, without the need to downscale the measurements, while still changing fluid in the sand layers only.