Inverting Dynamic Elastic Moduli of a Granular Pack to Get Shear Modulus of the Grain

Abstract

SUMMARYElastic moduli of rocks derived from its powder is a new concept and can be applied in practical geophysics studies. To develop this concept, we make ultrasonic velocity measurement on granular packs of quartz sand. We calculate dynamic elastic moduli from that measurement and invert afterwards to find the shear modulus of quartz. The inversion technique follows Extended Walton Model that relies on the grain's contact surface condition between infinitely rough and perfectly smooth. We use different coordination numbers from previous studies (for different samples) in the process of forward modelling and inversion. Forward models have good match with the laboratory measurements both in bulk and shear moduli of the granular pack. Our overall inverted results for the shear modulus are stable and close to actual shear modulus of quartz. However, the coordination numbers that has better match in forward modelling a little bit overestimates shear modulus. On the contrary, the coordination numbers that predicts the higher effective moduli of the pack is giving closest result. As the experiment set up and procedure are simple and robust, this technique can be extended and run in very rigorous situation such as at hard rock drilling rig site to get the elastic properties of the penetrated rocks in real time, where the effective elastic moduli of a grain can be represented as a statistical averaging of elastic moduli of hard rock minerals. This information can be helpful for planning and monitoring the ongoing drilling procedure. It can also be a replacement of solid cores that are missing or damaged for elasticity study.

Elastic properties of sands, Part 1: Micro computed tomography image analysis of grain shapes and their relationship with microstructure

“…Ultrasonic measurement is performed during the last loading phase. Using similar samples, setup and apparatus, Ahmed et al (2016) demonstrated that during unloading after a preceding loading phase, sample length does not recover its previous length. In that context, we can infer that slowly removing the top piston does not significantly change grain arrangement.…”

Elastic properties of sands, Part 2: Implementation of contact‐based model to determine the elasticity of the grains from ultrasonic measurements

“…Following the recommendations from Ahmed et al . (), we set key parameters that include the loading−unloading sequence and top stress level. We start our measuring sequence with a couple of loading−unloading cycles up to 2.52 MPa, and then apply a long loading tail up to 12.60 MPa (Fig.…”

“…Also, we maintain one scale to measure the change in length of the sample along with the increasing stress to calculate the corresponding volumetric change. Following the recommendations from Ahmed et al (2016), we set key parameters that include the loading− unloading sequence and top stress level. We start our measuring sequence with a couple of loading−unloading cycles up to 2.52 MPa, and then apply a long loading tail up to 12.60 MPa (Fig.…”

“…They adopt an arbitrary factor of non-spherical grain shape and poly-dispersity to modify the coordination number-pressure function derived from numerical simulation by Makse et al (2004). Ahmed et al (2016) use an extended Walton model (e.g. Jenkins et al 2005) to calculate the shear modulus of the grain using coordination numbers from existing literature and a constant fraction of no slip contact, α.…”