Elastic properties of an unconsolidated sand are largely dependent on the elastic properties of its constituent grain and the micro‐structure that defines how the grains are arranged within themselves. Coordination number, that is the average number of contacts a grain has with its neighbours, and contact surface area are the two parameters closely related to the microstructure. Moreover, grain shapes and sorting also have substantial influence on these parameters. To calculate these parameters and find any potential relationships with the shape factors, we acquire high‐resolution micro computed tomography images of four mechanically compacted unconsolidated dry sand samples that are of different shape factors and sorting indices. After a comprehensive voxel‐based data processing, we calculate shape factors such as sphericity and roundness of each grain in all samples. Using own algorithm, we then calculate the coordination number and contact surface area. Results show that samples of well‐sorted and higher spherical and rounded grains have higher coordination number and contact surface area than the samples of poorly sorted and lower spherical and rounded grains. Among the poorly sorted samples, coordination number is largely dependent on the fraction of larger grain sizes present in the sample. Inside any given sample, grains of lower sphericity tend to have higher coordination numbers. Moreover, more spherical and rounded grains have greater contact surface area with their neighbours.

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.

The prediction of effective elastic properties of a granular medium using ultrasonic data based on contact models has been studied widely in both laboratory experiments and numerical simulations. In contrast, a calculation of the elastic properties of the constituent grains using similar data by inverting the equations from those models is a rather new concept. To do so, we have developed a controlled experiment technique that includes a uniaxial compaction test and measures ultrasonic velocities of four unconsolidated quartz sand samples with different sorting and grain shapes. We observe that both P‐ and S‐wave velocities are significantly influenced by the microstructure or internal arrangement of the grains. Well sorted and more spherical and rounded samples show higher velocities than the poorly sorted and less spherical and rounded samples. A microstructural parameter – namely coordination number – we have calculated from high resolution micro computed tomography images provides a good match between the model and dynamic effective bulk moduli of the sand pack. Combining this coordination number with a frictional parameter calculated from the measured velocity ratios has been very effective to fit the model with the dynamic effective shear moduli. Using these two key parameters along with the experiment results in the contact model we have been able to obtain the elastic parameters of the quartz sand grains in the sample. Elastic parameters obtained thus are very close to the actual values of the quartz grains found in the literature. This technique can be useful in hard rock mineral exploration where missing core samples or an absence of well logs can be replaced by laboratory measurements of powders to find the elasticity or velocities of the rocks. Moreover, the elastic properties of the solid phase calculated using this technique can be used as input parameters for the fluid substitution and rock physics characterization of unconsolidated reservoir sands.

The elastic properties of crystalline rocks can be estimated from ultrasonic measurements on the powders of crushed rocks produced by the drilling process. To determine the elastic properties of grains from properties of powder packs, we study the dependence of their ultrasonic wave velocities on pressure. From this dependency, using the Hertz–Mindlin theory, we can calculate the effective ratio of the grain shear modulus to one minus the Poisson ratio. The Hertz–Mindlin theory requires the knowledge of grain coordination number as a function of pressure, which can be obtained using an empirical relation based on published numerical simulations. Previous work has shown that this approach gives an accurate prediction of the effective bulk modulus of glass beads but produces a significant discrepancy for sand. This discrepancy may be attributed to the angularity of sand grains. To overcome this problem, we introduce a shape factor into the empirical relation for the coordination number. This new shape factor allows us to reconcile the rock physics model with laboratory measurements. We show that the shape factor varies from 1 to 2.5 for different grain shape angularity and sorting (grain size distributions). The modified theory allows us to estimate a combination of elasticity parameters of the grains from the measured dependence of P‐wave velocity in the pack on the pressure.

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