Analysis of Eshelby-Cheng's model in anisotropic porous cracked medium: An ultrasonic physical modeling approach

Many effective medium theories are designed to describe the macroscopic properties of a medium (the rock, or reservoir in this case) in terms of the properties of its constituents (the background matrix of the rock and the inclusions, for our scenario). A very well known effective medium theory is the Eshelby-Cheng model, which was studied by us in previous work, being tested for the case where the background medium was weakly-anisotropic and porous. The analysis was done testing elastic velocities and Thomsen parameters - as a function of crack density for fixed values of aspect ratio - predicted by the model with data acquired from synthetic rock samples. In this work, we aim to complete the analysis of the Eshelby-Cheng model capabilities when applied to rocks with porous and vertical transversely isotropic (VTI) backgrounds, testing the model for the elastic velocities as functions of aspect ratio - for fixed values of crack density - against experimental data. The data used to test the model were obtained from 17 synthetic rock samples, one uncracked and 16 cracked, the latter divided into four groups of four samples each, each group with cracks having the same aspect ratio, but with the samples having different crack densities. In these samples, ultrasonic pulse transmission measurements were performed to obtain the experimental velocities used to test the model. As was not possible to acquire data for velocity as a function of aspect ratio for fixed values of crack density, we performed interpolations of the experimental data to estimate these velocities. Eshelby-Cheng model effective velocities and Thomsen parameters were calculated using three formulations proposed for the crack porosity: one proposed by Thomsen, the second one proposed in our previous work (which depends only on the crack density) and the third one proposed in this work (which depends on the crack porosity and the aspect ratio, just like Thomsen's proposal). The comparisons between elastic velocities and Thomsen parameters - as function of crack aspect ratio, for fixed values of crack density - predicted by the model and estimated from the data via interpolation showed that the third formulation produced better fittings (lower root-mean-square errors) between model and experimental data for all ranges of aspect ratio and crack density.