Geofluids

Volume 2018, Article ID 7321961, 14 pages

https://doi.org/10.1155/2018/7321961

## A Damage Constitutive Model for the Effects of CO_{2}-Brine-Rock Interactions on the Brittleness of a Low-Clay Shale

^{1}Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring, Ministry of Education, Central South University, Changsha 410083, China^{2}School of Geosciences and Info-physics, Central South University, Changsha 410083, China^{3}Key Laboratory of Hubei Province for Water Jet Theory & New Technology, Wuhan University, Wuhan 430072, China^{4}Deep Earth Energy Lab, Department of Civil Engineering, Monash University, Melbourne 3800, Australia

Correspondence should be addressed to Jingqiang Tan; moc.liamg@gnaiqgnijnat

Received 14 November 2017; Revised 24 February 2018; Accepted 25 March 2018; Published 30 May 2018

Academic Editor: Julie K. Pearce

Copyright © 2018 Qiao Lyu 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

CO_{2} is a very promising fluid for drilling and nonaqueous fracturing, especially for CO_{2}-enhanced shale gas recovery. Brittleness is a very important characteristic to evaluate the drillability and fracability. However, there is not much relevant research works on the influence of CO_{2} and CO_{2}-based fluids on shale’s brittleness been carried out. Therefore, a series of strength tests were conducted to obtain the stress-strain characteristics of shale soaked in different phases of CO_{2} including subcritical or supercritical CO_{2} with formation of water for different time intervals (10 days, 20 days, and 30 days). Two damage constitutive equations based on the power function distribution and Weibull distribution were established to predict the threshold stress for both intact and soaked shale samples. Based on the results, physical and chemical reactions during the imbibition cause reductions of shales’ peak axial strength (20.79%~61.52%) and Young’s modulus (13.14%~62.44%). Weibull distribution-based constitutive model with a damage threshold value of 0.8 has better agreement with the experiments than that of the power function distribution-based constitutive model. The energy balance method together with the Weibull distribution-based constitutive model is applied to calculate the brittleness values of samples with or without soaking. The intact shale sample has the highest value of 0.9961, which is in accordance with the high percentage of brittleness minerals of the shale samples. The CO_{2}-NaCl-shale interactions during the imbibition decrease the brittleness values. Among the three soaking durations, the minimum brittleness values occur on samples with 20 days’ imbibition in subcritical and supercritical CO_{2} + NaCl solutions and the reductions of which are 2.08% and 2.49%, respectively. Subcritical/supercritical CO_{2} + NaCl imbibition has higher effect on shale’s strength and Young’s modulus than on the brittleness. The low-clay shale still keeps good fracture performance after imbibition.

#### 1. Introduction

The increasing trend of greenhouse gas emissions contributes significantly to global warming. CO_{2} (carbon dioxide), one of the main greenhouse gases, has been investigated by many researchers which aim to reduce its concentration by means of CCUS (CO_{2} capture, utilization, and sequestration) project [1–5]. Because of the unique properties, CO_{2} presents a great feasibility in the utilizations of drilling industry [6, 7] and nonaqueous fracking for shale gas recovery [8–10]. During the drilling and fracturing process, CO_{2} together with ground water or other fluids will interact with rock and change rock’s mechanical properties [11–14].

Shale is a kind of a compacted sedimentary rock which accounts for 75% of drilled formations [15]. Therefore, the investigation of shale’s mechanical properties after fluid-rock interactions is very important for drilling and fracturing industry. Many scholars have done a brilliant research about the variation of shale’s strength and Young’s modulus after the adsorption of fluids, such as water [16, 17], saline solutions [18, 19], CO_{2} [20–22], CO_{2}-dissolved water [11], and CO_{2}-dissolved brine [23, 24]. However, the studies of the brittleness of shale after fluid saturation are limited. As brittleness is a very important mechanical characteristic which can be used to predict the rock’s failure features and obtain some mechanical properties such as sawability [25], drillability [26], TBM penetration rate [27], and fracture toughness [28], it is of great significance to give more attention to the variation of shale brittleness after fluid-rock interactions.

The accurate evaluation of brittleness is of great importance to drilling and fracturing efficiency. However, expressions to define rock brittleness are various [29, 30]. Considering all the proposed definitions, brittleness can be calculated by three categories: (1) stress-strain curve-based approach. This approach can be used to quantify the brittleness based on the values from the curve, such as strain [31], Young’s modulus and Poisson’s ratio [32], and energy balance [33]. (2) Strength tests and impact or hardness test approach. Based on some simple laboratory tests, some values like strength [34], degree of impact [35], and hardness [36] show accurate calculating results of rock brittleness [26, 31]. (3) Mineral composition approach. The content of brittle minerals [37], porosity [38], and grain size distribution [39] have been proved to have close relationships to rock brittleness.

In this study, brittleness variations of shale samples which adsorbed in sub/supercritical CO_{2} + NaCl fluids with different times were investigated by conducting a series of uniaxial compressive strength (UCS) tests. The statistical damage constitutive model together with the energy-based method was applied to obtain the influence of the difference of soaking condition and soaking time on shale’s brittleness.

#### 2. Theoretical and Empirical Equations for Brittleness Index Calculation

The stress-strain-based calculation, which consists of a strain-based method, Young’s modulus-based method, Poisson’s ratio-based method, and energy-based method, is commonly used in brittleness evaluation [29]. As the compression process of a rock presents a balance between energy storage and consumption, the energy-based method is chosen for the brittleness evaluation in this study. The key issue for the energy-based method is to obtain the relationship between stress and strain. The statistical damage constitutive model is a well-known method to establish the stress-strain relations which considers the anisotropy of rocks. Therefore, in this study, the brittleness of shale samples is calculated by energy-based method together with the statistical damage constitutive model.

##### 2.1. Statistical Damage Constitutive Models of Rocks under Uniaxial Tests

There are four common distributions which have been used in rock mechanics: power function distribution [40], Weibull distribution [41], normal distribution [42], and normal logarithmic distribution [43]. According to Wu et al. [30], the power function distribution and Weibull distribution are chosen to model the stress-strain curves of shale samples.

Based on the strain equivalent principle [44] and the Hooke’s law, the relationship between stress and strain can be described as follows:

The damage variable is calculated from (1), where is axial stress, is Young’s modulus, is axial strain, and is damage variable.

In the statistical model, we define the total number of microunit as , the failed units at a certain loading as . Then, damage variable is calculated as

The number of failed units can be calculated by the following equation, where is the distribution function of microstrength, which can be replaced by axial strain , and is the axial strain at a certain loading.

The probability density function for power function distribution and Weibull distribution are as follows: where and are distribution parameters.

Considering the fact that damage variable is zero when the compression is in the elastic stage, a threshold value of axial strain is adopted in the model. The value of varies between 0.7 and 0.9 times of axial strain at failure point [45]. This can be described as follows:

Therefore, the probability density function for the two distributions is rewritten as follows:

Combined the previous equations, the statistical damage constitutive models based on power function distribution and Weibull distribution can be obtained as follows:

In view of the extreme point at the stress-strain curve, the two constitutive models should fulfil the following equation: where the axial stress at failure point and the axial strain at failure point.

Then, the two distribution parameters can be calculated by (10) for power function distribution and (11) for Weibull distribution.

##### 2.2. Energy Method for Brittleness Calculation

The statistical damage constitutive models could only obtain accurate stress-strain relations in the prepeak stage. Therefore, we only use the energy before the peak point to calculate the brittleness in this study. The energy balance during compression is shown in Figure 1. The area of means the elastic energy during compression; the area of means the energy stored in the rock during compression; the area of means the energy consumed in the compression [46]. The lower the value of is, the more brittle the rock is. If , the rock will be a kind of ideal brittle material.