A Damage Constitutive Model for the Effects of CO 2-Brine-Rock Interactions on the Brittleness of a Low-Clay Shale

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


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][2][3][4][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][9][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][12][13][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 fluidrock 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][21][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 curvebased 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 energybased method was applied to obtain the influence of the difference of soaking condition and soaking time on shale's brittleness.

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.

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 σ is axial stress, E is Young's modulus, ε is axial strain, and D is damage variable.
In the statistical model, we define the total number of microunit as N, the failed units at a certain loading as n.Then, damage variable is calculated as The number of failed units can be calculated by the following equation, where P ε is the distribution function of microstrength, which can be replaced by axial strain ε, and ε 1 is the axial strain at a certain loading.The probability density function for power function distribution and Weibull distribution are as follows: where m and ε 0 are distribution parameters.

Geofluids
Considering the fact that damage variable is zero when the compression is in the elastic stage, a threshold value of axial strain λ is adopted in the model.The value of λ 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:  3 Geofluids In view of the extreme point at the stress-strain curve, the two constitutive models should fulfil the following equation: where σ max the axial stress at failure point and ε max 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 S oabd means the elastic energy during compression; the area of S oacd means the energy stored in the rock during compression; the area of S abc means the energy consumed in the compression [46].The lower the value of S abc is, the more brittle the rock is.If S abc = 0, the rock will be a kind of ideal brittle material.The elastic energy of rock before failure is The stored energy before failure is The brittleness index (BI) is defined as the ratio of stored energy and elastic energy, which is Combined the constitutive models and BI equation, the final equation of BI with power function distribution is shown in (15) and Weibull distribution is shown in (16).

Experimental Methodology and Results
3.1.Sample Preparation.Shale samples used in this study were Longmaxi shale obtained from Sichuan Basin, China, as shown in Figure 2 [11].The shales in these areas are characterized as having high thickness, suitable burial depth, and effective fracture development [47].Samples were cored from a slightly weathered outcrop sample.XRD tests were  5 Geofluids The soaking pressure was controlled by a Model 500D pump with a precision of 1 kPa.The temperature was offered by an electrical cover with a precision of 1 °C.After imbibition, CO 2 in the high-pressure chamber was drained very slowly to keep the pressure dropped smoothly.Then, samples were taken out and placed in the air for 24 hours to eliminate high pore pressure.Before the compression tests, all shale samples were painted with white and black paintings to make sure that the lateral surface can be recognized by ARAMIS digital cameras.All the UCS tests were conducted in a Shimadzu AG 9300 kN compression machine with a loading rate of 0.10 mm/min.The axial and lateral strains of tested samples were recorded by a 3D ARAMIS digital equipment.

Experimental Results
. The UCS values and Young's modulus of tested samples were listed in Table 3.For each group, only one sample's stress-strain curve was presented in this study for analysis and brittleness calculation.
The axial stress-axial strain curves for samples soaked with subcritical CO 2 and NaCl are shown in Figure 3(a).It can be seen that, for shale samples soaked in subcritical According to the previous study [11], the change of shale's mechanical properties is because of the physical and chemical reactions during the imbibition.Therefore, it is necessary to manifest the effect of such physical and chemical reactions on shale's brittleness, which is of great importance for formation fracturing.When the soaking time is the same, the values of UCS and Young's for samples soaked in supercritical condition are smaller than that soaked in subcritical condition.Whether the difference of soaking condition will influence shale's brittleness is also need to be studied.

Brittleness Calculation and Discussion
4.1.Models' Calculation and Validation.A complete stressstrain curve contains five stages: crack closure stage, elastic stage, stable crack propagation stage, unstable crack propagation stage, and postpeak stage [48,49].The statistical damage constitutive models cannot correctly present the crack closure and postpeak stages.Therefore, the axial stress-axial strain curves in this study have been transformed to eliminate the crack closure stage.The transformation is made by inversely extending the line in the elastic stage and obtaining the intersection point with x-axis, then making a leftward shift of the curve dabc to let the point d move to the origin coordinate and a new curve oef is obtained, as shown in Figure 4. Finally, the new axial stress-axial strain curves for intact and soaked samples are shown in Figure 5.For each group (see Table 2), only one sample is chosen to do the calculation.The peak axial stress, the axial strain at failure point, Young's modulus, and the axial strain with different threshold values are listed in Table 4.   10) and (11), we can obtain the values of the two distribution parameters: m and ε 0 , as shown in Table 5.In order to manifest the influence of threshold value on the variations between experimental results and constitutive modelling results, the axial stress-axial strain relations of the intact sample and samples soaked in subcritical/supercritical CO 2 for 10 days are calculated based on the power function distribution and Weibull distribution, which are shown in Figures 6-8, respectively.
According to Figures 6-8, we can see that stress-strain curves obtained by power function distribution-based constitutive model have very steep trends around the failure points, while the curves calculated by Weibull distribution-based constitutive model present gradual trends around the failure points, which are closely related to the experimental results.It can be deduced that Weibull distribution can better present the stress-strain relations of shale samples than the power function distribution.Before the unstable crack propagation stage, Weibull distribution-based constitutive model presents the same stress-strain relations for the three threshold values.After that, the influence of threshold values on the stressstrain curves occurs.More specifically, the highest threshold  8 Geofluids value (λ = 0 9) leads to the overestimated results of stressstrain relationship at the unstable crack propagation stage; the lowest one (λ = 0 7) causes the underestimated results of stress-strain relationship, while the moderate threshold value (λ = 0 8) presents good stress-strain relationship for the three kinds of samples.Therefore, in this study, the Weibull distribution-based constitutive model with a threshold value of 0.8 is chosen to calculate the brittleness values of samples with/without imbibition.The experimental and model calculated stress-strain curves for samples soaked in subcritical CO 2 + NaCl solution and supercritical CO 2 + NaCl solution with 20 and 30 days are shown in Figures 9 and 10, respectively.It is clear that the Weibull distribution-based constitutive model with a damage threshold value of 0.8 could obtain excellent stressstrain relationship for intact and soaked shale samples.Compared to the experimental results, the constitutive model can only acquire good stress-strain results before the failure point.This is true that the constitutive model is infeasible for the postpeak stage.Weibull distribution-based damage constitutive equations for samples with different soaking conditions are listed in Table 6.

Brittleness Calculation.
Weibull distribution-based damage constitutive equations for samples with different soaking conditions are substituted into (16).The quad integration method is applied to calculate the brittleness index of shale samples.The BI values of the intact sample and soaked samples are shown in Table 7. Variations of BI values versus soaking time are shown in Figure 11.
According to Table 7 and Figure 11, all the shale samples have ultra-high BI values which vary from 0.9713 to 0.9961.More specifically, the intact shale sample has the highest BI value of 0.9961.BI values show minor reductions for all soaked samples when compared to intact samples.When the imbibition times are 10 days, 20 days, and 30 days, the B I values are 0.9843, 0.9754, and 0.9851 for samples absorbed in subcritical CO 2 + NaCl solutions and 0.9757, 0.9713, and    9 Geofluids 0.9835 for samples absorbed in supercritical CO 2 + NaCl solution, respectively.The minimum BI values for the two kinds of soaking solutions are occurred when the imbibition time is 20 days.When extending the soaking time to 30 days, the BI values increase.Meanwhile, the supercritical condition creates a higher reduction of the BI value than that of subcritical condition when the soaking time is the same.4.3.Discussion.The high value of BI for the intact shale sample is in accordance with the XRD results that shale samples in this study have a high percentage of brittleness minerals.As the loading direction is parallel to the bedding, the sample will reach to its failure point at a sudden and the consumed energy during the compression is very small.Therefore, the compression of the intact shale sample is close to an ideal brittle failure process.Figure 12 shows the lateral strain distribution of the intact shale sample at failure point.It can be seen that before the failure, a large crack has been formed through the sample which is parallel to the beddings.The strain variations for other places are very small, which means the stress is concentrated on the main load-bearing beddings.When the load-bearing beddings are broken, the shale sample fails.Therefore, the stable crack propagation stage and the unstable crack propagation stage are very short which result in a very high BI value.
The reductions of brittleness for soaked shale samples are mainly caused by the chemical reactions during the imbibition.According to the authors' previous work, the pH of    10 Geofluids water soaked with subcritical and supercritical CO 2 under the same pressure and temperature will be smaller than 3 [11].
The acid fluid will lead to the dissolution and precipitation of some minerals, such as K-feldspar, carbonates, some clay minerals, and iron-containing minerals.The chemical reactions will create pores and enlarge cracks which decrease shale's strength and Young's modulus.The soaked shale samples will become more ductile, and the brittleness values decrease.Figure 13 presents the lateral strain distributions of the soaked samples.Compared to the strain distribution of the intact sample in Figure 12, soaked shale samples have more macroscopic cracks on the surfaces when samples reach their failure points.This difference is also caused by pores and cracks created by chemical reactions.Samples soaked in supercritical CO 2 + NaCl solution present lower B I values than that soaked in subcritical CO 2 + NaCl solution when the soaking time is the same.This is because of the difference of soaking pressure and the phase of CO 2 .Higher confining pressure will promote the penetration of fluids into the shale and will make the chemical reactions more  11 Geofluids sufficiently.Compared to subcritical CO 2 , supercritical CO 2 has no capillary force and lower viscosity, which will make it easy to flow into shale samples.As the phase of CO 2 is also related to the pressure, it is difficult to manifest which one has the dominant effect on the difference of BI values.The increase of BI values of shale samples soaked in subcritical/ supercritical CO 2 + NaCl solutions from 20 days to 30 days is probably caused by the fact that the increasing speed of precipitation in the last 10 days increases.The crystals appeared in cracks and decrease the ductile of the shale which lead to the increase of brittleness.Overall, the imbibition of shale samples in subcritical/supercritical CO 2 + NaCl solutions has larger effect on shale's strength and Young's modulus than on the brittleness.The fracture performance of this kind of low-clay shale is still good enough after imbibition.
In this paper, because of the limitation of the theory and experiments, a much more precise results of brittleness values for shale samples soaked in different conditions with different times cannot be obtained.Firstly, the crack closure stage is ignored during the modelling and brittleness calculation process.It will influence the brittleness values of shale samples.Meanwhile, samples soaked in subcritical/supercritical CO 2 + NaCl solutions with different times will have different damage threshold values.However, this paper only uses one threshold value (λ = 0 8) for all statistical damage constitutive equations.More importantly, shale gas exploitation and CCUS are usually last for decades or even longer.Therefore, a much longer imbibition process is of a great significance to manifest the brittleness variation of shales caused by CO 2 capture and sequestration and CO 2 -enhanced shale gas exploitation.

Conclusions
The damage statistical constitutive model together with the energy-based approach was applied to calculate the brittleness of shale samples soaked in subcritical/supercritical CO 2 + NaCl solutions with different times.Some remarkable conclusions can be obtained as follows.
Physical and chemical reactions during the imbibition in subcritical/supercritical CO 2 + NaCl solutions cause reductions of shales' peak axial strength (20.79%~61.52%)and Young's modulus (13.14%~62.44%).The two values of soaked samples decrease with increasing saturation time.With the same damage threshold value, the Weibull distribution-based model presents better stress-strain results than the power function distribution-based model.For intact and soaked shale samples, the damage threshold value of 0.8 is appropriate to describe the stress-strain relations; lower or higher threshold values will cause the underestimation or overestimation of the peak point on the stress-strain curve.
The energy balance method together with the Weibull distribution based-constitutive model is applied to calculate the brittleness values of samples with or without imbibition.The intact shale sample has the highest BI value of 0.9961.The CO 2 -NaCl-shale interactions during the imbibition decrease the brittleness values.The reductions of BI values vary between 1.10% and 2.49%.A 20-day imbibition in subcritical and supercritical conditions results in the lowest BI values among the three soaking durations.Samples in supercritical CO 2 + NaCl solution have lower BI values than that in subcritical CO 2 + NaCl solution when the soaking time is the same.
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.
t r a l S i c h u a n p a l e o u p l i

Figure 2 :
Figure 2: A sketch map of Longmaxi shale in Sichuan Basin, China.

Figure 5 :
Figure 5: The transformed axial stress-axial strain curves without a crack closure stage.(a) Intact samples and samples with subcritical CO 2 + NaCl solution.(b) Intact samples and samples with supercritical CO 2 + NaCl solution.

8 Figure 6 : 8 Figure 7 :
Figure 6: Experimental and constitutive modelling axial stress-axial strain curves of the intact sample.

Figure 9 :
Figure 9: Experimental and constitutive model calculated stressstrain curves for samples soaked in subcritical CO 2 for 20 and 30 days.

Figure 10 :
Figure 10: Experimental and constitutive model calculated stressstrain curves for samples soaked in supercritical CO 2 for 20 and 30 days.

8 Figure 8 :
Figure 8: Experimental and constitutive modelling axial stressaxial strain curves of the sample soaked in supercritical CO 2 for 10 days.

Figure 11 :Figure 12 :
Figure 11: The variation of BI values with soaking time.
30 days in supercritical CO 2 + NaCl solution

Table 2 :
The soaking arrangement of shale samples.
85%), which means that the shale formation has good fracture performance.The samples were cored parallel to bedding; based on the ASTM standards, owing to a <10% coring success rate of the shale limited samples was available.The coring process may have created some microfractures.Samples used in the study were selected carefully from all cored samples to ensure good quality.The size of the samples was Φ30 mm × 60 mm.The experiments were conducted at the 3G Deep Earth Lab, Monash University, Australia.3.2.Testing Arrangement.Samples were firstly divided into seven groups.Each group had two samples.Six groups were soaked in subcritical/supercritical CO 2 with NaCl solution for different times (10 days, 20 days, and 30 days).The group with no imbibition (intact) was set as the control group.The concentration of NaCl solution was 20% by weight.The soaking arrangement was listed in Table2.If the compression results of one group have large deviations, additional tests will be done until the results are acceptable.

Table 3 :
Values of uniaxial compressive strength (UCS) and Young's modulus (E) for all tested samples.

Table 4 :
The mechanical values of shales obtained from stress-strain curves.

Table 5 :
Distribution parameters of intact and soaked samples.

Table 6 :
The Weibull distribution based damage constitutive equations of all samples.

Table 7 :
The brittleness index (BI) of intact samples and soaked samples.