Determining Young's modulus of granite using accurate grain-based modeling with microscale rock mechanical experiments
Introduction
Granitic rock formations are commonly associated with hot dry rock (HDR) reservoirs for the exploitation of geothermal energy and are considered for hosting geologic nuclear waste repositories. An important part of site investigation and design is the accurate measurement of mechanical properties. Conventionally, the mechanical properties of rocks are measured by macroscale rock mechanics experiments (macro-RMEs) including uniaxial/triaxial compression tests suggested by the International Society for Rock Mechanics (ISRM) and the American Society for Testing and Materials (ASTM).1 The rock samples are usually standard cylinders 100 mm in height and 50 mm in diameter, and were prepared from available intact sections of the core. Drilling and recovery of such rock samples in deep reservoirs are expensive and time-consuming and suffer from technical difficulties. Due to high in situ stress, cores achieved by drilling are frequently fragmented,2 especially core disking, which can occur during deep underground drilling. The host rock around faults is usually highly fractured in a damage zone with a core of fragmented rocks in clay gouge.3 Thus, significant challenges and limitations exist in conventional rock mechanical testing for any site investigation.
Based on the elastic wave of minerals,4 the elastic modulus of rock can be evaluated by the Voigt-Reuss-Hill average algorithm,5 which is widely used to understand the lithology of deep earth formations. The Voigt (upper) and Reuss (lower) bounds are the averaged elastic properties of a polyphase and polycrystalline aggregate assuming homogeneous strain or stress, respectively. The average value provides a convenient estimation for the properties of an aggregate rock without a preferred orientation and with grains locked together.6 However, the Voigt-Reuss-Hill average algorithm oversimplifies the variability of rocks because it ignores the grain interactions and grain shapes. This causes the elastic properties to scatter widely between the upper and lower theoretical bounds.7 Further mean-field homogenization schemes have been developed that consider the inclusion geometries based on the Eshelby solution8 of the ellipsoidal inhomogeneity embedded in an infinite medium. These approaches include the dilute scheme,9 self-consistent approximation,10 differential scheme,11 and Mori-Tanaka approach.12 Unfortunately, none of these analytical models can describe the complex microstructure well and accurately determine the elastic parameters of crystalline rocks. The microstructure (e.g., interphases, pores, and grain geometry) of rocks plays a dominant role in their mechanical response and affects the applicability of these homogenization methods.13
To overcome the aforementioned challenges and limitations of core-scale experiments (macro-RME) and analytical homogenization methods, microscale rock mechanics experiments (micro-RME) have been developed to determine rock mechanical properties. First, the technology of micro-RME can test the mechanical properties of specimens that can be quite small and arbitrarily shaped, thus overcoming the difficulties in obtaining core samples for conventional macro-RME. Second, highly heterogeneous and anisotropic properties can be evaluated through their dependency on the distribution of microscale properties. With these advantages, micro-RME technology provides a viable cost-effective approach to testing the mechanical properties of rock samples related to any subsurface science and engineering project.
Nanoindentation test recently attracted great interest in the area of rock engineering.14 Bobko et al.15 investigated the mechanical stiffness, anisotropy, and strength properties of shale using nanoindentation tests. Akono et al.16 investigated sandstone in CO2-saturated brine and used the statistical nanoindentation method to determine the relationship between the mineral compositional changes at the microscale and the constitutive behavior at the macroscale. Zeng et al.17 proposed an improved energy-based model to determine the fracture energy and toughness of shale. Based on a continuous stiffness model and big data statistical nanoindentation tests, Zhang and his coworkers developed an analytical method for measuring the cross-scale mechanical properties of rocks,13,18 which has been successfully used to investigate the water softening properties of shale.19 Xu et al.20 investigated the influence of water softening on the elastic and failure properties of shale obtained from weak interlayers of a rock slope. Using numerical models, the results of nanoindentation tests were successfully applied to model the failure process of slopes with weak shale interlayers. Voltolini et al.21 visualized the indentation of samples as a function of axial load by using X-ray imaging. Then, a series of 3D datasets were used to analyze local strain fields, which provided a unique opportunity to characterize the mechanical properties for predicting proppant and fracture closure evolution.
Atomic force microscopy (AFM) has been widely used to characterize the nanomechanical properties of biological materials22 but has only recently been applied to cement23 and natural rocks.24, 25, 26 Liu et al.27 demonstrated that AFM is a robust and quantitative method to investigate the mechanical properties of shale, which can be considered a nanocomposite material. Graham et al.28 investigated the mechanical response of calcareous shales over length scales from 10 nm to 100 μm by combining AFM and nanoindentation test. This combination allowed for the consideration of microcracks and achieved a better prediction of Young's modulus.
In recent decades, numerical techniques have been commonly used to help understand complicated mechanisms from a grain-scale perspective.29,30 Some researchers investigated the relationship between microscale and macroscale properties using traditional grain-based modeling (GBM).31, 32, 33 In conventional GBM, some microscale mechanical properties are measured by micro-RME directly, and the method of numerical calibration with macro-RME results is typically used to roughly estimate other properties that are difficult to measure experimentally. Without extensive data on the microstructure, the geometry of rock-forming minerals is usually simplified using Voronoi meshes based on the statistical grain size distribution. As shown in Table 1, Peng et al.34 investigated the mechanical behavior of Bukit Timah granites using GBM. The microstructure was represented using a Voronoi mesh, and the microscale mechanical properties were back-calculated by PFC2D numerical calibration against macro-RME results. With GBM, Peng et al.32 revealed that when confining pressure increases gradually during compression tests, shear cracks along grain boundaries occur while the failure mode changes from splitting to shear. Zhou et al.35 developed a three-dimensional (3D) GBM with Voronoi gridding in which the microscale mechanical properties were estimated using numerical calibration.
With micro-RME, more accurate microstructure and microscale mechanical properties of crystalline rocks can be evaluated. Characteristic microstructures derived from digital images are taken into account within the field of digital rock physics to analyze rock mechanical properties.36 In a study by Mahabadi et al.,37 the Young's modulus and failure properties of rock-forming minerals were obtained by nanoindentation tests, while the microstructure was obtained using computerized tomography (CT) scanning. With micro-REM and finite-discrete element method (FDEM) modeling, Mahabadi et al.38 found that microscale heterogeneity of mechanical properties and microcracks at the grain scale can lead to a significant variation in stress distribution and local tensile stress. Meanwhile, oriented biotite flakes and microcracks have a significant impact on fracture propagation. In Abdelaziz et al.,39 the microstructure was also represented using Voronoi meshing, while the microscale elastic property was extracted from previous micro-RME results,37 and the microscale failure properties were estimated using numerical calibration. They demonstrated that the path of the macroscopic failure plane is the result of combining local stress conditions with the microstructure of the model. However, the interphases between different rock-forming minerals were ignored in previous work because the sizes of interphases are too small to be measured.
The goal of this work is to measure the entire microstructure and all microscale mechanical properties experimentally and apply those in accurate grain-based modeling (AGBM) to derive the Young's modulus of granite samples. The mineral composition and microstructure of the granite samples were measured using a TESCAN Integrated Mineral Analyzer (TIMA). The mechanical properties of rock-forming minerals and interphases were measured using nanoindentation and AFM, respectively. Finally, the AGBMs were generated based on micro-RME, and microscale simulations of uniaxial compression tests were carried out to determine the macroscale Young's modulus of the granites.
Section snippets
Determining macroscale mechanical properties of granites using micro-RME and upscaling modeling
The micro-RME system includes optical microscopy, nanoindentation, TIMA, and AFM. (1) An orthogonal polarization optical microscopy was used to observe the microstructures of mineral morphology, cleavage cracks within mineral grains, bedding, and micropores. (2) The identification and distribution of minerals was investigated by TIMA, which is an automated quantitative analysis system that can identify minerals quickly and accurately. (3) Nanoindentation was applied to measure the elastic
Principles of TIMA test
TIMA identifies rock-forming minerals based on scanning electron microscopy (SEM) in backscattered electron (BSE) imaging modes and energy-dispersive spectroscopy (EDS).41 The TIMA dot mapping mode imposes a BSE grid with a specific resolution (determined as pixel spacing) over the entire sample. The maximum accuracy of TIMA is 0.2 μm, which can also be used to analyze a variety of geomaterials such as clay minerals (<2 μm) and complex soil particles.42 Considering the large mineral crystals of
Preparation of rock specimens
In this work, granitic rock specimens, identified as Bukit Timah granite, were collected in the Mandai area of the central part of Singapore. This granite is a light gray, coarse-granied granodiorite. The macroscale properties of Bukit Timah granites have been investigated by many researchers.47, 48, 49 The macroscale mechanical properties of Bukit Timah granites are strongly heterogeneous, and they are significantly influenced by their rock-forming minerals.50
As shown in Fig. 4b, the granites
Mineral composition and microstructure of granitic samples
The present experiment took only 1.5 h by TIMA to test a 20 × 20-mm granite specimen, which was completed at the Chengpu geological Testing Co., Ltd, Langfang, China. Then, in grayscale images, the biotite, feldspar, and quartz were presented as white, gray and black, respectively, as shown in Fig. 5. With grayscale images, the mineralogical composition, mineral size, and mineral distribution can be analyzed easily by computers. This Bukit Timah granite consists of quartz (SiO2, volume approx.
Generating numerical models
In this section, an accurate grain-based numerical model of granite was generated based on the microstructure and mechanical properties derived from micro-RME to evaluate the elastic property of granite. AGBM further investigated the influence of the microstructure and mechanical properties of minerals on the macroscale properties of rock samples.
The digital images obtained by TIMA were transformed into computational numerical models. As stated in section 5.1, an image retrieved from fresh
Discussion
Bukit Timah granite with low volumetric porosity was investigated in this work. When the present method is extended to other types of rocks such as sandstone and weathered granite, the following issues should be addressed:
- (1)
Effect of microcracks and microvoids
It is known that the Young's modulus of rocks decreases with increasing numbers of microcracks and microvoids.61,62 In this work, the microstructure of samples was measured using the X-ray computed tomography (CT) technique and
Conclusions
To overcome the challenges of drilling and sampling intact and standard-sized rock samples, we proposed a novel methodology to determine the elastic properties of natural granitic rocks that can be arbitrarily sized and shaped. By conducting micro-RME, the spatial distribution and mechanical properties of rock constituent phases were characterized. With accurate microstructure and micromechanical parameters, AGBM was then developed to derive the macroscale Young's modulus of granite. The
Data availability
Datasets related to this article are made available by Zenodo, a public repository of research data, at the following link: https://zenodo.org/record/5729120.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant nos. 12172264 and 41941018) and Basic Research Program of Central Government Funds for Shenzhen Science and Technology Development (2021Szvup103).
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