Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 7948612, 12 pages

http://dx.doi.org/10.1155/2016/7948612

## Dimension Analysis-Based Model for Prediction of Shale Compressive Strength

^{1}School of Petroleum and Natural Gas Engineering, Southwest Petroleum University, Chengdu 610500, China^{2}Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China^{3}School of Sciences, Southwest Petroleum University, Chengdu 610500, China^{4}Chongqing Mineral Resources Development Co., Ltd., Chongqing 40042, China

Received 17 January 2016; Accepted 22 May 2016

Academic Editor: Fernando Lusquiños

Copyright © 2016 Xiangyu Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The compressive strength of shale is a comprehensive index for evaluating the shale strength, which is linked to shale well borehole stability. Based on correlation analysis between factors (confining stress, height/diameter ratio, bedding angle, and porosity) and shale compressive strength (Longmaxi Shale in Sichuan Basin, China), we develop a dimension analysis-based model for prediction of shale compressive strength. A nonlinear-regression model is used for comparison. A multitraining method is used to achieve reliability of model prediction. The results show that, compared to a multi-nonlinear-regression model (average prediction error = 19.5%), the average prediction error of the dimension analysis-based model is 19.2%. More importantly, our dimension analysis-based model needs to determine only one parameter, whereas the multi-nonlinear-regression model needs to determine five. In addition, sensitivity analysis shows that height/diameter ratio has greater sensitivity to compressive strength than other factors.

#### 1. Introduction

Shale compressive strength is an important parameter that reflects the shale brittleness, and thus it is used as a comprehensive index to evaluate the stability of shale well boreholes. Most shale types have flaky bedding which is obviously anisotropic, which affects the performance of the rock strength. Thus, the physical properties of shale strongly affect the shale compressive strength. A common way to evaluate the effect of rock properties on rock strength is to carry out triaxial compressive experiments in the laboratory and then to analyze the relation between factors and rock strength. Knowledge of the shale anisotropy is vital for borehole instability issues [1, 2]. A total of three triaxial tests with different sample orientations (i.e., 0, 45, 60, and 90°), including Brazilian tests and CT scans, were used to investigate how the bedding angle affects the shale strength [3]. In particular, Lyu et al. [4] present an experimental study of the effects of bedding planes on the mechanical properties of Chinese shale samples (Sichuan Basin, China). Hudson et al. [5] carried out uniaxial compression tests of marbles of different sizes to obtain the rock strength with the aspect to the height/diameter ratio. It was found that an increase of the height/diameter ratio reduced the rock strength until it reached a relatively stable value. Hoek [6] suggests that this reduction in strength is due to the increased probability that failure of rock grains will occur as the specimen size increases. Pells [7] found that 150 mm specimens have strength of around 85% of that of 50 mm specimens. Darlington et al. [8] found that the multifractal scaling law (MFSL) model most closely predicts the strength-size relationship in rock and cementitious materials. Li and Aubertin [9] showed that increases in porosity within the rock reduce the compressive, tensile, and shear strengths of the intact rock. Similar results were also obtained by Chang et al. [10], Horsrud [11], and Lashkaripour and Dusseault [12]. It is comprehensively accepted that an increase of the confining stress improves the rock compressive strength; for example, Liang et al. [13] found that the elastic modulus of bedded salt rock increases with an increase in the confining stress. Lal [14] focused especially on the similar relation between confining stress and compressive strength. In summary, it is considered that the main factors of porosity, geometry, bedding angle, and confining stress affect the shale compressive strength. Therefore, considering the effect of only one factor on the shale strength may not reflect the true shale strength. Currently, rock strength prediction mainly depends on a multi-nonlinear-regression model [15, 16]. This type of method requires the determination of many parameters and has less physical meaning. In our study, we consider the shale specimens from the Longmaxi Shale in Sichuan Basin, China, by carrying out a correlation analysis between the main factors (confining stress, height/diameter ratio, bedding angle, and porosity) and the shale compressive strength. A dimension analysis-based model is developed to predict the shale compressive strength. A multi-nonlinear-regression model based on the same data is used for comparison.

#### 2. Shale Specimen and Experiments

##### 2.1. Introduction of Shale Origin and Shale Specimens

The Sichuan Basin is a prolific hydrocarbon region and is currently China’s largest gas-producing region. The Longmaxi region in this study is located at the margin of the southern Sichuan Basin. The Sichuan Basin, located in the west of the Yangtze metaplatform tectonically, is large and tectonically stable. The strata are of the South China type, with a complete regional sedimentary rock exposed from a Presinian system to a Quaternary system, whose sedimentary cover thickness is approximately 6000 to 12,000 m from the Paleozoic to Cenozoic. The Longmaxi Formation (shale) distribution is determined based on the depositional environment and the subsequent erosional events related to the tectonic history. The Longmaxi black shale is the most organically rich part of the Lower Silurian. The Longmaxi Formation has a thickness ranging from 229.2 to 672.5 m. The stratigraphy consists of black shale, black and dark/grey shale, dark grey shale, and silty mudstone, mainly comprising carbonaceous and clay shale. Shale samples are from the “Longmaxi Shale” in Sichuan Basin, China (Figure 1(a)). There are 93 shale specimens. All of the shale specimens have a diameter of 25 mm and a height between 34.5 and 54 mm. The bedding angles considered are 0, 15, 30, 45, 60, 75, and 90°, respectively. The range of porosity is from 1.7 to 5.4% (Figures 1(b)–1(e)). The shale samples generally belong to black shale, also called carboniferous shale. The variety of shale contains abundant organic matter, pyrite, and sometimes carbonate nodules or layers as well as, in some locations, concentrations of copper, nickel, uranium, and vanadium. Black shale is of interest commercially since it is a potentially valuable source of synthetic crude oil and plays an important role in oil-shale formation. In this study, X-ray diffraction (XRD) analysis was used to analyze the mineral components of shale (a total of 20 samples). The relative contents of minerals in the composition and the clay mineral content are shown in Figure 2. A comparison of the mineral compositions of the “Longmaxi” and “North American” shale formations is shown in Figure 3 (the statistical data for Bossier Shale, Ohio Shale, and West Texas Woodford/Barnett are from [17–21]).