# Study on Local Brittleness of Rock Based on Multiple Linear Regression Method: Case Study of Shahejie Formation

Authors: Meng Cai; Yang Wang; Wanchun Zhao; Xiaodong Shi; Teng Li
Journal: Geofluids
 (2023)

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## Abstract

Rock brittleness index is an important parameter that affects the volume fracturing effect of an unconventional reservoir. The higher the brittleness index, the larger the communication area of fracture network formed under volume fracturing. However, rock brittleness has strong heterogeneity, which seriously affects the propagation of volume fracture network. Most existing rock brittleness characterization methods start from macro brittleness and cannot analyze the influence of heterogeneity on reservoir brittleness. The mineral composition analysis method can characterize the heterogeneity of a reservoir by analyzing the influence of mineral composition on brittleness, but this is limited to qualitative analysis on different minerals, resulting in large errors. In this study, the relationship between macro mechanical properties and micro mineral composition of rocks was analyzed using multiple linear regression analysis and a large number of experimental core parameters on the Shahejie Formation to understand the quantitative characterization of rock brittleness distribution in a heterogeneous reservoir. The results showed that quartz, plagioclase, analcite, siderite, and pyrite were high brittle minerals while clay minerals were low brittle minerals. This method provides a new focus for the analysis of brittle distribution of heterogeneous reservoirs and an additional method to study rock fracture law.


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## Body

## 1. Introduction

With the deepening of exploration and development of conventional oil and gas reservoirs, it is more difficult to exploit conventional oil and gas. In recent years, the exploration and development of unconventional oil and gas resources has been increasing and has become an important substitute resource. Unconventional oil and gas resources are widely distributed, rich in geological reserves, locally enriched, and not controlled by structures. Thus, reservoirs have the characteristics of strong heterogeneity, low porosity, and low permeability, which makes them difficult to exploit. Cipolla et al. and Wang et al. [1, 2] have studied fractures in unconventional reservoirs that are developed with complex sizes. They are rich in fracture systems with different sizes from micro nano to millimeter and mostly exist in the form of micro fractures. Most are filled with calcareous and clay minerals, with poor connectivity, as shown in Figure 1.Figure 1
Multistage micro scale natural fracture system of an unconventional reservoir (after the polishing of argon ions).At present, fracture network volume fracturing is the primary technology for efficient development of unconventional oil and gas [3–5]. In the process of fracturing, natural fractures expand continuously, and brittle rocks produce shear slip, forming a fracture network with staggered natural fractures and artificial fractures to increase the reconstruction volume and improve the initial production and final recovery. Therefore, rock brittleness in a reservoir is a critical index to evaluate the fracturing effect of a reservoir. Previous research indicates that the higher the brittleness of rock, the more complex the fracture network formed after compression and the better the connectivity [6]. Zhao et al. [7, 8] found that mutation jump occurs when a rock mass breaks in a brittle area; fractures are formed instantaneously as the single fracture absorbs energy for a short time, and the energy released by mutation easily promotes the bifurcation and evolution of a “short fracture.” The plastic rock mass expands in a slowly varying and asymptotic manner during fracture, and the fractures continue to absorb energy. In this case, a single fracture absorbs energy for a long time, and the energy released by the slow change easily promotes the continuous expansion of a “long fracture,” as shown in Figure 2.Figure 2
Schematic diagram of fracture evolution in different brittle regions of a rock mass.Statistics show that there are more than 30 representative brittleness measurement methods [9]. Through induction and summary, the brittle plastic evaluation of rock is mainly expressed from six aspects: total stress-strain curve [10–16], mechanical properties [17–19], rock hardness [20–25], rock mineral composition [26–29], rock strength [30–33], and energy release relationship [34, 35]. These methods show the significance of evaluating rock brittleness and urge researchers to explore a reasonable brittleness measurement method as the characterization of rock brittleness has great complexity and limitations. The brittleness index is summarized as shown in Table 1.Table 1
The determination result of core mineral compositions and mechanical parameters.
Serial number
Mineral type and (%) content
Clay minerals (%)
Elastic modulus (GPa)
Poisson’s ratio
Quartz
Potassium feldspar
Plagioclase
Calcite
Dolomite
Analcite
Siderite
Pyrite
1
16.30
3.10
26.80
7.30
20.20
3.20
2.10
2.50
18.50
30.23
0.293
2
19.70
3.60
7.30
8.90
9.50
29.10
1.80
1.90
18.20
32.79
0.283
3
14.80
3.80
16.60
4.40
21.70
12.10
1.40
2.80
22.40
24.65
0.300
4
33.90
3.20
13.80
0.00
19.80
8.90
0.00
0.00
20.40
26.98
0.286
5
21.10
4.00
15.90
5.20
15.10
13.60
1.50
2.00
21.60
26.74
0.279
6
19.50
3.70
12.50
1.10
25.40
14.00
0.00
0.00
23.80
24.88
0.279
7
17.70
5.40
27.30
0.00
15.30
3.30
0.00
1.50
29.50
22.56
0.276
8
9.60
4.30
18.50
0.00
8.80
38.10
3.50
0.00
17.20
22.56
0.281
9
10.60
7.00
11.70
2.00
26.70
16.00
3.00
1.80
21.20
26.27
0.267
10
14.00
5.30
24.20
11.00
18.00
5.00
1.30
2.40
18.80
20.93
0.281
11
25.30
4.10
6.90
2.40
5.30
28.20
5.60
0.00
22.20
22.10
0.286
12
1.80
2.70
16.40
1.10
41.30
10.90
15.10
0.00
10.70
29.53
0.271
13
12.10
7.30
33.40
0.00
18.90
4.00
8.70
2.00
13.60
31.63
0.262
14
19.60
6.50
9.60
2.00
13.90
30.90
0.00
0.00
17.50
30.00
0.271
15
12.40
5.80
6.30
1.70
5.40
42.90
1.90
0.00
23.60
28.84
0.286
16
4.10
14.80
36.00
3.90
24.10
0.00
0.00
4.20
12.90
29.53
0.281
17
6.80
11.40
49.30
0.00
13.50
0.00
2.20
1.50
15.30
26.28
0.281
18
8.20
6.70
41.50
1.90
17.70
4.80
3.10
0.00
16.10
28.60
0.273
19
9.50
10.10
11.30
3.60
35.90
4.50
1.60
2.70
20.80
23.49
0.310
20
37.50
2.00
19.20
14.40
4.10
4.50
1.30
4.60
12.40
23.95
0.262
21
19.80
5.90
10.30
2.00
10.80
29.80
4.80
0.00
16.60
19.77
0.286
22
6.10
8.40
46.80
2.40
13.70
1.70
4.90
0.00
16.00
19.53
0.290Currently, the following two methods are widely used at construction sites to calculate the reservoir brittleness distribution. The rock mineral analysis method is one such method proposed by Jarvie et al. [26]. By measuring the contents of quartz, carbonate rock, and clay minerals in the rock, the ratio between quartz and the total mineral content is calculated as the brittleness of the rock, as shown in
(1)BRIT=VquartzVtotal×100,where BRIT is the rock brittleness index and V is the mineral volume.Wang and Gale [27] found that the appearance of dolomite also increases the brittleness of shale. Thus, dolomite is also considered a brittle mineral. It is believed that the higher the content of quartz and dolomite, the higher the brittleness of a reservoir. Additionally, the higher the content of clay minerals, the lower the brittleness of a reservoir and the less distinct development characteristics of natural fractures. In 2013, when studying the Lower Cambrian, Upper Ordovician Lower Silurian, and Upper Permian shale of Southern Paleozoic, Chen and Xiao [28] noted that the mineral composition of Southern Paleozoic shale is relatively complex, and the brittleness index calculation has gradually developed from unclear qualitative analysis to accurate quantitative analysis. The mineral types affected by brittleness have been gradually screened and refined, but the proportion of different minerals still cannot be analyzed. In the same year, Diao [29] compared the rock mechanical brittleness evaluation method with the mineral composition brittleness evaluation method and, combined with the experimental results, found that the volume deformation during rock fracture decreased with the increase of Young’s modulus and increased with the increase of Poisson’s ratio. Therefore, the rock brittleness increases with the increase of Young’s modulus and decreases with the increase of Poisson’s ratio. In order to highlight the rock with strong brittleness, the brittleness index calculation method of the combination method of elastic parameters and mineral composition is proposed as follows:
(2)BRIT=Vquartz×Equartz/λquartzVquartz×Equartz/λquartz+Vcalcite×Ecalcite/λcalcite+Vclay×Eclay/λclay×100,where V is the mineral volume, E is Young’s modulus, and λ is Poisson’s ratio.These methods combine the mechanical properties of different minerals as a weight analysis to characterize the degree of rock brittleness and highlight that different minerals occupy different brittleness proportions, which is of great research value.Another commonly used rock brittleness calculation method is the elastic parameter method proposed by Grieser and Bray [36] in 2007, as shown in equation (3). Using this method, the higher the elastic modulus of rock, the lower the Poisson ratio and the stronger the rock brittleness, as shown in Figure 3. This method defines the rock brittleness index using rock mechanical parameters, which are generally accepted and adopted in the oil field.
(3)BRIT=12E−EminEmax−Emin+λ−λmaxλmin−λmax×100,where Emax is the maximum elastic modulus of rock, Emin is the minimum elastic modulus of rock, λmax is the maximum Poisson ratio of rock, and λmin is the minimum Poisson ratio of rock.Figure 3
Intersection diagram of influence of elastic modulus and Poisson’s ratio on brittleness [17].The elastic parameter method, combined with the existing reservoir data for analysis, can integrate the effects of different factors in the study area, such as reservoir in situ stress distribution, cohesion, and fracture development degree. However, this method requires a large number of samples in the analysis process, can only analyze the mechanical properties of the reservoir as a whole, and cannot finely quantify the small differences in brittleness between different regions.Through the analysis of the rock mineral compositions and the mechanical properties from unconventional reservoirs, we used multiple linear regression analysis method to research the relationship between shale mineral composition and the rock elastic modulus and the rock Poisson ratio. A new method based on the elastic parameter method and the mineral constituent method was considered and compared with existing formations. This method combined the rock mineral compositions and the rock mechanical properties, and it played a critical role in representing the reservoir brittleness that impacts fracturing and oil recovery.


## 2. Multiple Linear Regression Model

The multiple linear regression analysis was used when a phenomenon was affected by multiple factors to analyze the relationship between the dependent variable and the factors. According to the coefficients in the observation sample estimation model, the regression model can make predictions and analysis through statistical tests to estimate coefficients and the regression equation. The multiple regression model included many independent variables, which all influenced the dependent variableY. In order to find out the effect of one independent variable on Y, the other independent variables were assumed to be constant. Therefore, the regression coefficients of the multiple linear regression model were partial regression coefficients, indicating the influence of one independent variable on the mean value of Y when the other variables in the model remained constant.The general multiple linear regression model is shown as follows:(4)Y=β0+β1Z1+β2Z2+⋯⋯+βjZj+ε,where βjj=1,2,⋯,k is regression coefficient, Y represents the dependent variable, Zjj=1,2,⋯,k means the kth independent variable which has an obvious effect on Y (k≥2), and ε is the random error term which inflects the comprehensive influence of different errors.The multiple sample regression function isY^i=β^0+β^1z1i+β^2z2i+⋯⋯+β^kzki. Regression residual ε equals Yi−Yi.Since there aren periods of observations, this model includes n functions. The matrix form of these functions is shown as follows:
(5)Y1Y2⋮Yn=1z11z21⋯zk11z12z22⋯zk2⋮⋮⋮⋮⋮1z1nz2n⋯zknβ0β1β2⋮βk+ε1ε2⋮εn,where Y1Y2⋯YnT is the observation vector of dependent variable, β0β1β2⋯βkT is the entire regression coefficient vector, and ε1ε2⋯εnT is the random error term vector.If the regression function of multivariate samples isY^i=β^0+β^1Z1i+β^2Z2i+⋯⋯+β^jZji, the regression residual calculation method εi=Yi−Y^i reaches the minimum.According to the principle of least squares, the sum of residual squaresRSS is defined as
(6)RSS=∑i=1nεi2=∑i=1nYi−Y∧i2.WhenRSS satisfies the following relationship, the sample regression function Y^i is the actual regression function of the model.
(7)∂RSS∂β^0=0,∂RSS∂β^1=0,⋯⋯∂RSS∂β^j=0.The formula for calculating regression functionY^i of samples is obtained as follows:
(8)∑i=1nβ^0+β^1Z1i+β^2Z2i+⋯⋯+β^jZji=∑i=1nYi,∑i=1nβ^0+β^1Z1i+β^2Z2i+⋯⋯+β^jZjiZ1i=∑i=1nYiZ1i,⋯⋯∑i=1nβ^0+β^1Z1i+β^2Z2i+⋯⋯+β^jZjiZji=∑i=1nYiZji.The model hasj+1 unknown quantity and j+1 equation, which can be used to solve the unknown quantity and calculate the global regression parameter vector in the multivariate sample regression function.The progress of setting up a multiple linear model included subjective elements. Additionally, we must consider that the theory and experience which the model is based on may be incorrect and that the model may not conform to the existing relationship between variables. To ensure that the relationship was consistent with the basic stipulations of the multiple linear regression analysis, clear analysis object, and guarantee the validity and characteristics of the regression analysis, this research usedR2 as an index to test the goodness of fit. Here, R20≤R2≤1. When R2 was larger, it revealed that the degree of explanation to dependent variable is higher; thus, the regression equation fits better to the observations and vice versa. The computational formula is shown as follows:
(9)R2=ESSTSS=1−RSSTSS=1−∑Yi−Y∧i2∑yi2,where ESS means the regression sum of squares of fitting results; RSS means the residual sum of squares of fitting results, which shows the effect of random error; and TSS is the square sum of total departure of fitting results.The calculation formulas of rock elastic modulus and Poisson’s ratio obtained by the multiple linear regression analysis areE¯ and ν¯, respectively. Thus, the studied sample can be divided into multiple study areas by discrete analysis, and the mineral composition of different areas can be analyzed, respectively, as shown in Figure 4. The values were assigned according to the measured mechanical parameters, and each grid block represented a different element that characterized the local mechanical parameters. The calculation method of Figure 4 can evaluate the mechanical parameters, mineral composition, and brittleness characteristics of the total rock through the calculation of local elements.Figure 4
Sketch map of core discretization.Each study area can be located by using (a, b, c). The elastic modulus E¯a,b,c and Poisson’s ratio ν¯a,b,c in different study areas were calculated according to equation (4) after mineral composition analysis. The results of the regression equation were finally substituted into equation (3), and the expression method of brittleness distribution under the heterogeneous condition of the study sample was determined as follows:
(10)BRITa,b,c=12E¯a,b,c−EminEmax−Emin+ν¯a,b,c−νmaxνmin−νmax×100.Equation (10) is the expression of rock brittleness distribution under heterogeneous conditions. The distribution equation can quantify the influence of different mineral components on the overall rock brittleness and can accurately describe the brittleness distribution law of a heterogeneous reservoir, which provides a basis for the volume fracturing construction method.


## 3. Case Calculation and Analysis

In this study, the core mineral compositions were determined using an X-ray diffractometer on field test cores collected from the Shahejie Formation in Liaohe oil field. The core sample is shown in Figure5, while the experimental equipment instrumentation and process are shown in Figure 6. Using the uniaxial compression test, mechanical parameters like uniaxial compressive strength, elastic modulus, and Poisson’s ratio can be tested. The results of this experiment are shown in Table 1.Figure 5
Natural core samples of the study block (Leijia).Figure 6
Experiment with loading devices and processes.The core samples taken from Shahejie Formation were mainly composed of quartz, potassium feldspar, plagioclase, dolomite, and clay minerals. Some cores contained calcite, analcite, siderite, and pyrite, and other minerals were too low to be considered in this paper. The core elastic modulus was in the range of [19.53, 32.79], and Poisson’s ratio was in the range of [0.262, 0.310].The multiple linear regression analysis result is shown in Table2.Table 2
Multiple linear regression analysis results.
Regression coefficient
Initial coefficient
Quartz
Potassium feldspar
Plagioclase
Calcite
Dolomite
Analcite
Siderite
Pyrite
Clay minerals
Elastic modulusE
15.576
0.148
0.080
0.145
0.129
0.262
0.200
0
0
-0.171
Poisson’s ratioν
0.254715
-0.000204
0.001197
-0.000108
0.001218
0.000271
0
-0.000343
-0.002222
0.001069Substituting the results into equation (4), we can determine the relationship among the elastic modulus, Poisson’s ratio, and rock mineral compositions as follows:
(11)E=15.576+0.148xquartz+0.080xpotassiumfeldspar+0.145xplagioclase+0.129xcalcite+0.262xdolomite+0.200xanalcite−0.171xclay,(12)ν=0.254715−0.000204xquartz+0.001197xpotassiumfeldspar−0.000108xplagioclase+0.001218xcalcite+0.000271xdolomite−0.000343xsiderite−0.002222xpyrite+0.001069xclay.The multiple linear regression analysis results showed that under simulated confining pressure, quartz, potassium feldspar, plagioclase, calcite, and dolomite are positively correlated with the content of analcite and the rock elastic modulus, while the content of siderite and pyrite had almost no effect on the elastic modulus. Additionally, the clay content was negatively correlated with the rock elastic modulus of rock. Quartz, plagioclase, and siderite were negatively correlated with pyrite content and rock Poisson’s ratio, while analcite content had little effect on rock Poisson’s ratio. Potassium feldspar, calcite, and dolomite were positively correlated with clay content and rock Poisson’s ratio.In order to verify the reliability and accuracy of the calculation results, the analysis of the standardized residuals is shown in Figures7 and 8.Figure 7
Probability distribution map of elastic modulus.Figure 8
Probability distribution map of Poisson’s ratio.TheR2 of elastic modulus and Poisson’s ratio in the research were 0.889 and 0.720, respectively. The goodness of fit showed a higher relationship accuracy between the shale elastic modulus, Poisson’s ratio, and rock mineral compositions, which can be considered a valid result for practice. The calculation errors were mainly caused by two components: the core was not perfectly homogenous, and natural micro cracks may exist in the core. Both of which may have negatively influenced the experiment result.As discussed earlier, this study used the elastic parameter method to analyze the rock brittleness as equation (10) shows. This method can build a measurable and comprehensive relationship between mineral compositions and the rock brittleness, allowing mechanical parameters like elastic modulus and Poisson’s ratio to be calculated by mineral percentage. This is consistent with the reality and gives a more accurate calculation on the rock brittle characterization.
(13)BRIT=1215.576+0.148xquartz+0.080xpotassiumfeldspar+0.145xplagioclase+0.129xcalcite+0.262xdolomite+0.200xanalcite−0.171xclay−EminEmax−Emin+0.254715−0.000204xquartz+0.001197xpotassiumfeldspar−0.000108xplagioclase+0.001218xcalcite+0.000271xdolomite−0.000343xsiderite−0.002222xpyrite+0.001069xclay−νmaxνmin−νmax×100.Equation (13) is the rock local brittleness evaluation model of the Shahejie Formation. The model considered the influence of reservoir heterogeneity and natural fracture development on the reservoir and characterized the relationship between micro mineral distribution and macro brittle distribution in an unconventional reservoir. This model suggests that the reservoir brittleness characterization is accurate and conforms to the existing study block condition.Using the provided rock mechanical parameters and the calculation model obtained by regression, the brittleness index was calculated and compared to different calculation methods. The calculation results suggest that the proposed method has certain rationality and accuracy.Through verification analysis of the field example, Figure9 demonstrates that the average brittleness of this layer measured by the on-site determination of brittleness, mineral analysis method, and the new evaluation method is 63.2, 53.5, and 59.7, respectively. The rock brittleness index distribution measured using the new method is more consistent with the site brittleness distribution and is better than the mineral analysis results. Compared with the mineral analysis results, the new evaluation method can effectively reduce the calculation error caused by only considering quartz as a brittle mineral in the brittleness index calculation, so that the calculation results are representative of the existing site condition. The brittleness evaluation method proposed in this study reflects two obvious advantages; the first is to fully consider the dual influence of rock minerals and rock mechanical parameters and to reevaluate the essential characteristics of brittleness. In the existing application process, our evaluation of brittleness is often a local heart combined with the brittleness evaluation of the entire underground depth of the logging data, and due to the influence of a variety of factors, this brittleness evaluation error is often larger. The regression method proposed in this study serves as a viable new correction and will allow researchers to evaluate and verify rock brittleness.Figure 9
Calculation results of rock brittleness of different algorithms.
(a)
Mechanical parameter method
(b)
Mineral analysis method
(c)
Multiple linear regression method


## 4. Conclusion

(1)
In this study, the relationship between the rock elastic modulus, Poisson’s ratio, and mineral composition was quantitatively analyzed using a multiple linear regression analysis method and combined with the existing environment of an experimental reservoir. The results showed that quartz, plagioclase, analcite, siderite, and pyrite were positively correlated with rock brittleness, while clay minerals were negatively correlated with rock brittleness. Other mineral components need to be analyzed according to the site
(2)
Combined with the multiple linear regression equations between rock elastic modulus, Poisson’s ratio, and mineral composition, the local brittleness distribution of heterogeneous reservoir rock was established using the elastic parameter method. This method accurately describes the local rock brittleness distribution under strong heterogenous conditions shaped by the distribution of reservoir minerals, in a manner that is scientific and instructive



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