Breakdown Pressure Prediction of Tight Sandstone Horizontal Wells Based on the Mechanism Model and Multiple Linear Regression Model

Abstract

Accurately predicting the breakdown pressure in horizontal sections is essential when designing and optimizing fracturing jobs for horizontal wells in tight gas reservoirs. Taking the Sulige block in the Ordos Basin as an example, for different completion methods combined with indoor rock experience data and well data, a new method for predicting breakdown pressure based on a linear regression model is proposed. Based on the Hossain horizontal well stress field model, this paper established a calculation model of breakdown pressure under different completion methods by using experimental and well data. The average error between the calculation results and the actual breakdown pressure at the initiation point is 3.67%. A Pearson correlation analysis was conducted for eight sensitive factors of horizontal well stress, which showed that the maximum horizontal principal stress, minimum horizontal principal stress, tensile strength, and elastic modulus had strong linear correlations with breakdown pressure. In this study, multiple linear regression was used to establish the prediction model of breakdown pressure under different completion conditions, and the calculation method of the prediction model was optimized. The model was verified using the relevant data for four horizontal wells. The average relative error between the prediction model and the actual breakdown pressure was 4.33–6.30%, indicating that the breakdown pressure obtained by the new prediction model was similar to the actual conditions. Thus, the prediction model is reasonable and reliable.

1. Introduction

Hydraulic fracturing is a widely used simulation technology in unconventional reservoirs. The Sulige block in the Ordos Basin is a typical representative of a tight sandstone gas reservoir in China. It has low permeability, porosity, and pressure and strong heterogeneity. Staged horizontal well fracturing technology is crucial for the successful development of oil and gas reservoirs. Reservoir reconstruction measures must be taken, such as hydraulic fracturing. The accurate calculation of formation breakdown pressure is a key to successful tight gas horizontal well fracturing [1,2,3,4,5]. At present, the methods for calculating breakdown pressure both nationally and internationally are generally divided into three categories. The first is based on Hossain et al.’s work, establishing the calculation model of a mine’s unlined share distribution [6]. Considering the influence of certain factors, e.g., fracturing fluid leak-off, completion method, temperature field variation, casting pipes, measurement sheets, and crack-induced stress, some studies perfected the model of the stress field around horizontal wells [7,8,9]. However, the implicit models required complex iterative calculations, and no feasible equation was given. Next, the second category is based on Hubbert and Willis’s formula (H-W) [10] and Haimson and Fairhurst’s formula (H-F) [11], and shear failure criteria established based on the theory of elasticity. The explicit model of breakdown pressure can be obtained using these criteria, but its application conditions are limited. H-W and H-F can only predict rock breakdown pressure under impermeability and high permeability, respectively, but cannot predict rock breakdown pressure in intermediate transition states. The shear failure criterion applies to strata with a small elastic modulus and high permeability looseness [12]. Finally, the third category is a fracture model based on Griffith’s energy balance theory and its improved model based on Hardy’s stress-intensity factor theory. The model assumption is more in line with the actual hydraulic fracturing process, but it is difficult to measure the required initial crack scale and other parameters. Therefore, it is mainly applied in theoretical analysis and a few numerical models [13,14,15].

Based on the above problems, to obtain a feasible breakdown pressure calculation equation and predict the breakdown pressure continuity profile, this study used the Sulige block as the research object. Firstly, a perfect circular stress field model was constructed based on Hossain et al.’s breakdown model and Chen Mian’s breakdown model; the calculation model of breakdown pressure under different completion conditions was established and solved by combining indoor rock mechanics experiment data and logging data and comparing them with the actual breakdown pressure error on site. Then, the dataset’s learning samples were established, Pearson correlation was used to analyze the correlations of influencing factors and linear correlation coefficients between each factor, and the breakdown pressure was obtained. Finally, the linear regression method was used to establish the functional relationship between the breakdown pressure and in situ stress and rock mechanical parameters. The implicit model of iterative calculation was transformed into a feasible explicit model of the breakdown pressure calculation, providing an important decision-making basis for fracturing design and the construction of horizontal wells.

2. Rock Mechanical Parameters and In Situ Stress Parameter Calculation

2.1. Tensile Strength Calculation

There was a large error between the tensile strength value calculated by logging and the actual rock strength parameters in this area. Generally, the tensile strength parameters obtained by indoor rock experiments must be used to calibrate the logging data and establish a calibration relationship in line with this area, as follows [16]

$${\sigma }_t=\frac{0.0045E_d\left(1-V_{cl}\right)+0.008V_{cl}}{K}$$

(1)

The static splitting tensile strength of dense sandstone is an important mechanical property in the tensile strength [17]. The dense sandstone cores used in the experiment were taken from two wells in the block. The tensile strength was tested using the Brazilian splitting test disc. The equation for tensile strength when the Brazilian splitting test disc specimen was split was as follows [18]

$${\sigma }_t=\frac{2P}{\pi dt}$$

(2)

The experimental results are shown in

Table 1. Test results for tensile strength of rock.

Well	Depth (m)	Failure Load (N)	Tensile Strength (MPa)
Well A	3410.40	1623.00	2.08
	3574.30	2326.00	4.67
	3667.40	3788.00	7.46
	3866.20	2016.00	4.27
	3945.70	3359.00	7.05
Well B	3346.50	6485.00	11.56
	3562.40	2152.00	4.63
	3698.40	1769.00	3.24
	3869.10	2135.00	4.23
	3995.50	3658.00	7.50

. By comparing the tensile strength calculated by logging with the tensile strength obtained by the indoor rock experiment, the conversion coefficient $K$ = 10.38 could be obtained

$$K=\frac{\left[0.0045E_d\left(1-V_{cl}\right)+0.008V_{cl}\right]\times \pi dt}{2P}$$

(3)

2.2. Dynamic and Static Conversion of Elastic Modulus and Poisson’s Ratio

Generally, there are two common techniques, static and dynamic, for measuring the elastic modulus and Poisson’s ratio [19].

The dynamic elastic modulus and Poisson’s ratio were calculated using longitudinal and transverse wave time difference and density data. The equation was as follows [20]

$$\{\begin{array}{l}{E}_d=\frac{\rho v_s^2\left(3v_P^2-4v_s^2\right)}{v_p^2-v_s^2}\\ {\mu }_d=\frac{v_p^2-2v_s^2}{2\left(v_p^2-v_s^2\right)}\end{array}$$

(4)

The empirical conversion equation for the dynamic and static elastic modulus and Poisson’s ratio of tight sandstone in the Sulige area under different temperatures and confining pressure levels was as follows. The experimental temperature ranged from 22 °C to 100 °C, and the confining pressure ranged from 5 MPa to 60 MPa [21]

$$\{\begin{array}{l}{E}_s\left(P,T\right)=2.492E_d\left(P,T\right)-1.786ln\frac{P}{5}+0.151\left(T-22\right)-78.372\\ {\mu }_s\left(P,T\right)=-0.165{\mu }_d\left(P,T\right)+0.133ln\frac{P}{5}+g\left(T-22\right)+0.196\end{array}$$

(5)

2.3. In Situ Stress Calculation

The direction of present in situ stress was obtained from borehole breakouts and drilling-induced fractures from imaging logs; the value of present in situ stress was derived from a rock acoustic emission experiment, hydraulic fracturing, and acoustic logging data [22,23,24,25,26].

Overlying formation pressure was obtained by integrating the formation density as the tectonic stress existed in all directions and was unequal. Based on the small dip angle of the formation in the Sulige block, previous researchers obtained the following relationship, representing the level of in situ stress based on years of research [27]

$$\{\begin{array}{l}{\sigma }_v={{\displaystyle \int }}_0^{h}g\rho \left(h\right)dh\\ {\sigma }_H=\left(\frac{{\mu }_s}{1-{\mu }_s}+\beta \right)\left({\sigma }_v-p_p\right)+p_p\\ {\sigma }_h=\left(\frac{{\mu }_s}{1-{\mu }_s}+\gamma \right)\left({\sigma }_v-p_p\right)+p_p\end{array}$$

(6)

According to the on-site fracturing construction curve, the breakdown pressure and instantaneous shut-in pressure could be obtained to determine the horizontal in situ stress and its technical stress coefficient for the Sulige block. The instantaneous pump stop surface pressure plus the liquid column pressure of fracturing fluid in the well must be equal to the pressure required to act on the pressed vertical fracture and maintain the fracture opening state, which is balanced with the minimum horizontal in situ stress. Therefore, it can be determined that

$$\{\begin{array}{l}{\sigma }_h=P_s+{\rho }_{o}gh\\ {\sigma }_H=3{\sigma }_h-P_b-\alpha P_p+{\sigma }_t\end{array}$$

(7)

Then, ${\sigma }_h$ and ${\sigma }_H$ determine the tectonic stress coefficients $\beta$ and $\gamma$, respectively,

$$\{\begin{array}{l}\beta =\frac{{\sigma }_H-p_p}{{\sigma }_v-p_p}-\frac{{\mu }_s}{1-{\mu }_s}\\ \gamma =\frac{{\sigma }_h-p_p}{{\sigma }_v-p_p}-\frac{{\mu }_s}{1-{\mu }_s}\end{array}$$

(8)

By determining the static Poisson’s ratio, tectonic stress coefficient, overlying formation pressure, and formation pore pressure, horizontal in situ stress can be calculated.

3. Establishment and Verification of the Breakdown Pressure Calculation Model under Different Completion Conditions

In this paper, the following assumptions are made in order to analyze and calculate the stress distribution of the rock around the wellbore and the perforation hole:

(1)

Rock is a poroelastic medium, and the flow of fluid in the medium satisfies the assumption.

(2)

Changes in rock mechanical properties caused by the physical and chemical interaction between fracturing fluid and rock are not considered.

(3)

Both the inside of the wellbore and the inside of the borehole are subjected to uniform and consistent fluid pressure.

4. Mathematical Model of Borehole Stress Distribution in an Open Hole Completion Horizontal Well

Using the coordinate transformation diagram of horizontal well axis orientation and in situ stress, as shown in Figure 1, it can be determined that, in the coordinate system ($x$, $y$, $z$), the components of normal and shear stress around the horizontal well’s wellbore were as follows

$$\{\begin{array}{l}{\sigma }_{zz}={\sigma }_v\\ {\sigma }_{yy}={\sigma }_h{\sin }^2\beta +{\sigma }_h{\cos }^2\beta \\ {\sigma }_{xx}={\sigma }_h{\sin }^2\beta +{\sigma }_h{\cos }^2\beta \\ {\tau }_{xy}=0\\ {\tau }_{yz}=\left({\sigma }_h-{\sigma }_h\right)cos\beta sin\beta \\ {\tau }_{xz}=0\end{array}$$

(9)

Based on the Hossain model, considering that the rock could fracture and the fluid could then propagate through the fracture, the stress field distribution at the borehole wall during horizontal well open hole completion was obtained by adding a fracturing fluid seepage effect and the additional stress generated by the rock thermal effect using the superposition principle [6]

$$\{\begin{array}{l}{\sigma }_r=P_w-\delta \varphi \left(P_w-P_p\right)-\frac{{\alpha }_{T}E\mathsf{\Delta }T}{1-2v}\\ {\sigma }_{\theta }=-P_w+\left({\sigma }_{xx}+{\sigma }_y\right)-2\left({\sigma }_{xx}-{\sigma }_{yy}\right)\cos 2\theta -\frac{{\alpha }_{T}E\mathsf{\Delta }T}{1-2v}-\delta [\frac{\alpha \left(1-2v\right)}{\left(1-v\right)}-\varphi ](p_w-p_p)\\ {\sigma }_z=-c{P}_w+{\sigma }_{zz}-v\left[2\left({\sigma }_{xx}-{\sigma }_{yy}\right)\cos 2\theta \right]-\frac{{\alpha }_{T}E\mathsf{\Delta }T}{1-2v}-\delta [\frac{\alpha \left(1-2v\right)}{\left(1-v\right)}-\varphi ](P_w-P_p)\\ {\tau }_{r\theta }=0\\ {\tau }_{\theta z}=2{\tau }_{yz}cos\theta \\ {\tau }_{rz}=0\end{array}$$

(10)

4.1. Mathematical Model of Stress Distribution around the Borehole Completion Horizontal Well

The stress distribution around the horizontal well was much more complex than that of the open hole completion well. To prevent hole wall collapse, casting dimension sheeting was adopted for the horizontal well section; thus, the influence of the casting pipe and cement sheet on the distribution of circular stress was considered [28]

$$\{\begin{array}{l}{\sigma }_{rc}=-P_w\frac{\left[2R_i^2\left(1+v_c\right)\left(1-v_c\right)\right]}{E_C\left(R_o^2-R_i^2\right)}/[\frac{1+v}{E}+\frac{R_o^2\left[\left(R_i^2+\left(1-2v_c\right)\right)\left(1+v_c\right)\right]}{E_C\left(R_o^2-R_i^2\right)}]\\ {\sigma }_{\theta c}=P_w\frac{\left[2R_i^2\left(1+v_{\mathrm{c}}\right)\left(1-v_c\right)\right]}{E_C\left(R_o^2-R_i^2\right)}/[\frac{1+v}{E}+\frac{R_o^2\left[\left(R_i^2+\left(1-2v_c\right)\right)\left(1+v_{\mathrm{c}}\right)\right]}{E_C\left(R_o^2-R_i^2\right)}]\end{array}$$

(11)

As shown in Figure 2, taking hydraulic perforation as a micro-horizontal well, the hydraulic perforation well core was the vertical intersection of two holes of different sizes. According to the stress concentration at the base of the perforation (the well core intersects with the perforation), the equation of perforating circular stress could be obtained [6]

$$\begin{gathered} {\sigma }_{{\theta }^{\prime }}=-2P_w\left(1+\cos 2{\theta }^{\prime }\right)+\left({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_z\right)+2\left({\sigma }_{xx}+{\sigma }_{yy}-{\sigma }_z\right)\cos 2{\theta }^{\prime } \\ -2\left({\sigma }_{xx}-{\sigma }_{yy}\right)\cos 2\theta \left(1+2\cos 2{\theta }^{\prime }\right)-4{\sigma }_z\sin 2{\theta }^{\prime }-\frac{{\alpha }_{I}E\mathsf{\Delta }T}{1-2v} \\ -2\delta [\frac{\alpha \left(1-2v\right)}{\left(1-v\right)}-\varphi ]\left(p_w-p_p\right)\left(1+\cos 2{\theta }^{\prime }\right) \end{gathered}$$

(12)

The stress field produced by the original in situ stress, the stress field produced by the casting pipe thickness, the stress field produced by the perforation, the fracturing fluid seepage effect, and the stress field produced by the rock thermal effect were superimposed according to the superposition principle to obtain the stress distribution on the perforation wall in the formation area

$$\{\begin{array}{l}{\sigma }_r=P_w-\delta \varphi \left(P_w-P_p\right)-\frac{{\alpha }_{T}E\Delta T}{1-2v}\\ \hspace{1em}\hspace{1em}-P_w\frac{[2R_i^2\left(1+v_c\right)\left(1-v_c\right)]}{E_C\left(R_o^2-R_i^2\right)}/[\frac{1+v}{E}+\frac{R_o^2[\left(R_i^2+\left(1-2v_c\right)\right)\left(1+v_{\text{c}}\right)]}{E_C\left(R_o^2-R_i^2\right)}]\\ {\sigma }_{{\theta }^{\prime }}=-2P_w\left(1+\text{cos}2{\theta }^{\prime }\right)+\left({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_z\right)+2\left({\sigma }_{xx}+{\sigma }_{yy}-{\sigma }_z\right)\text{cos}2{\theta }^{\prime }\\ \hspace{1em}\hspace{1em}-2\left({\sigma }_{xx}-{\sigma }_{yy}\right)\text{cos}2\theta \left(1+2\text{cos}2{\theta }^{\prime }\right)-4{\sigma }_z\text{sin}2{\theta }^{\prime }-\frac{{\alpha }_{T}E\Delta T}{1-2v}\\ \hspace{1em}\hspace{1em}-2\delta [\frac{\alpha \left(1-2v\right)}{\left(1-v\right)}-\varphi ]\left(p_w-p_p\right)\left(1+\text{cos}2{\theta }^{\prime }\right)\\ \hspace{1em}\hspace{1em}+P_w\frac{[2R_i^2\left(1+v_c\right)\left(1-v_c\right)]}{E_C\left(R_o^2-R_i^2\right)}/[\frac{1+v}{E}+\frac{R_o^2[\left(R_i^2+\left(1-2v_c\right)\right)\left(1+v_c\right)]}{E_C\left(R_o^2-R_i^2\right)}]\\ {\sigma }_z=-c{P}_w+{\sigma }_{zz}-v[2\left({\sigma }_{xx}-{\sigma }_{yy}\right)\text{cos}2\theta ]-\frac{{\alpha }_{T}E\Delta T}{1-2v}-\delta [\frac{\alpha \left(1-2v\right)}{\left(1-v\right)}-\varphi ]\left(P_w-P_p\right)\\ {\tau }_{r\theta }=0\\ {\tau }_{\theta z}=2{\tau }_{yz}\text{cos}\theta \\ {\tau }_{rz}=0\end{array}$$

(13)

4.2. Calculation Model and Error Analysis of Breakdown Pressure

Without considering the influence of natural weak structural planes, the rock along the tensile initiation and the breakdown pressure were determined according to the tensile fracture criterion iteration [28]; that is, when the tensile stress of the rock at the shaft wall reaches and exceeds the tensile strength of the rock, the rock will break and form cracks

$$\{\begin{array}{l}{\sigma }_{max}\left({\theta }_0\right)=\frac{1}{2}[\left({\sigma }_{{\theta }_0}+{\sigma }_z\right)-\sqrt{{\left({\sigma }_{{\theta }_0}-{\sigma }_z\right)}^2+4{\tau }_{{\theta }_{0}z}^2}]-\alpha P_p\ge {\sigma }_t\\ {\sigma }_{max}\left({\theta }_0^{’}\right)=\frac{1}{2}[\left({\sigma }_{{\theta }_0^{’}}+{\sigma }_z\right)-\sqrt{{\left({\sigma }_{{\theta }_0^{’}}-{\sigma }_z\right)}^2+4{\tau }_{{\theta }_{0}z}^2}]-\alpha P_p\ge {\sigma }_t\end{array}$$

(14)

Since the established model contains the breakdown pressure, it must be solved numerically using an iterative method. The calculation steps for breakdown pressure under different completion conditions are shown in Figure 3.

According to the breakdown pressure calculation models under different completion methods, the MATLAB software was used for programming and calculation.

The wellhead breakdown pressure was obtained from the surface construction pressure curve, and the bottom hole breakdown pressure was obtained by combining the fracturing fluid column pressure and friction.

As shown in Figure 4, the surface construction pressure curve was used to obtain the wellhead breakdown pressure; the bottom hole breakdown pressure was obtained by combining the fracturing fluid column pressure, friction along the way, and perforation friction.

In tight formation, the formation breakdown pressure was the highest pumping pressure in the hydraulic fracturing construction curve, indicating that the injected fluid pressure overcame the high wellbore stress and rock tensile strength caused by stress concentration [29].

According to the established breakdown pressure calculation model and the basic data in

Table 2. Basic data.

Parameter	Numerical Value	Parameter	Numerical Value
Horizontal well azimuth angle (°)	90.00	Formation temperature increment (°C)	3.07
Thermal expansion coefficient (°C−1)	0.000045	Correction coefficient of packer	0.90
Permeability coefficient—open hole	0.70	Permeability coefficient—casting pipe	0.00
Inner diameter of casting pipe (mm)	97.18	Outer diameter of casting pipe (mm)	114.30
Casting pipe elastic model (MPa)	206,000.00	Poisson’s ratio of casting pipe	0.30
Perforation azimuth angle (°)	60.00	Formation pore pressure (MPa)	29.30

, based on the calculation flow chart, MATLAB software was used for programming calculations (replaced by the iterative method). Taking Well A of open hole completion and Well B of casing pipe completion as examples, the rock mechanical and in situ stress parameters of different initiation points were added to the model to calculate the breakdown pressure at the initiation point.

Figure 5 shows that the relative error of the breakdown pressure for Well A of open hole completion was 0.97–6.30%, and the average error was 4.62%. Figure 6 shows that the relative error of the breakdown pressure of Well B completed by the casting pipe was 0.17–6.91%, and the average error was 4.09%. Therefore, the above model can accurately predict the breakdown pressure at the initiation point and guide the fracture design.

In this model, breakdown pressure was an implicit function. Iterative calculations were needed to obtain the continuous breakdown pressure profile. To obtain a feasible explicit model for breakdown pressure calculation, the breakdown pressure obtained from the above model was used as a learning sample. Through the correlation analysis of the influencing factors of breakdown pressure, a multiple linear regression model was established to obtain the calculation equation of continuous breakdown pressure.

5. Correlation Analysis and Establishment of Multiple Linear Expression Model

5.1. Correlation Analysis of Factors Affecting Breakdown Pressure

In the field of statistics, the Pearson correlation coefficient is often used to measure the linear correlation between two variables. The Pearson correlation coefficient was between −1 and +1; −1 indicated that the two variables were completely negatively correlated, +1 indicated that the two variables were completely positively correlated, and 0 indicated that the two variables were completely independent. The closer it was to 1 or −1, the stronger the linear correlation between the two variables. The Pearson correlation was determined using the following equation [30]

$${\rho }_{x,y}=\frac{\mathrm{cov}\left(x,y\right)}{{\sigma }_x{\sigma }_y}$$

(15)

Correlation analysis of the factors affecting breakdown pressure was conducted to form a sample set of the breakdown pressure and rock tensile strength, elastic modulus, Poisson’s ratio, vertical stress, maximum horizontal principal stress, minimum horizontal principal stress, poor elastic coefficient, and effective porosity, and we conducted correlation analysis according to Equation (15). The correlation analysis results concerning the influential factors regarding the breakdown pressure under different completion methods are shown in

Table 3. Correlation analysis results for breakdown pressure of open hole completion Well A.

Influence Factor	Correlation Coefficient	Influence Factor	Correlation Coefficient
Tensile strength	0.93	Effective porosity	−0.49
Vertical stress	0.52	Maximum horizontal principal stress	0.96
Minimum horizontal principal stress	0.93	Porous elastic coefficient	−0.86
Elastic modulus	0.94	Poisson’s ratio	0.86

and

Table 4. Correlation analysis results of breakdown pressure of Well B after casting pipe completion.

Influence Factor	Correlation Coefficient	Influence Factor	Correlation Coefficient
Tensile strength	0.96	Effective porosity	−0.58
Vertical stress	0.64	Maximum horizontal principal stress	0.96
Minimum horizontal principal stress	0.98	Porous elastic coefficient	−0.79
Elastic modulus	0.93	Poisson’s ratio	0.79

:

From the linear correlation coefficients of various factors in Table 3 and Table 4, it can be observed that, under different completion methods, the effective porosity and poor elastic coefficients showed negative correlations with the breakdown pressure. The maximum horizontal principal stress, minimum horizontal principal stress, tensile strength, elastic modulus, Poisson’s ratio, and multi-elasticity coefficient had strong correlations with breakdown pressure. The Pearson’s correlation coefficient was up to 0.99. Figure 7 and Figure 8 show that most rock mechanical and in situ stress parameters had a strong linear relationship with breakdown pressure. Therefore, a multiple linear regression model was used to establish the equation for breakdown pressure.

5.2. Multiple Linear Regression Model Establishment

Multiple linear regression is a widely used predictive modeling technology that originates from statistical analysis. It is an important algorithm that combines the multiple linear regression model and statistics. It has the advantages of simple principles, a fast calculation speed, and high fitting accuracy. The multiple linear regression method is used to obtain the best-fit relationship between parameters and can generally be employed to establish the association between independent (input) and dependent (output) parameters [19]. Assuming that the random variable (explained variable) $\widehat{y}$ is affected by n independent variables (explanatory variables), its calculation equation can be expressed as follows [31]

$$\widehat{y}=w_0+w_1{x}_1+w_2{x}_2+\dots +w_n{x}_n$$

(16)

If Equation (16) is represented by a matrix, we can obtain the following

$$\left[\begin{array}{l}{\widehat{y}}_1\\ {\widehat{y}}_2\\ {\widehat{y}}_3\\ \dots \\ {\widehat{y}}_m\end{array}\right]=\left[\begin{array}{llllll}1& x_{11}& x_{12}& x_{13}& \dots & x_{1n}\\ 1& x_{21}& x_{22}& x_{23}& \dots & x_{2n}\\ 1& x_{31}& x_{32}& x_{33}& \dots & x_{3n}\\ 1& x_{m1}& x_{m2}& x_{m3}& \dots & x_{mn}\end{array}\right]\left[\begin{array}{l}{w}_0\\ w_1\\ w_2\\ \dots \\ w_n\end{array}\right]$$

(17)

A linear expression model was used to construct a prediction function to map the linear relationship between the input characteristic matrix $x$ and the label value $\widehat{ y}$. The aim of constructing the prediction function was to obtain the parameter vector of the model [32], define the loss function, and solve the parameter vector by minimizing the loss function or changing the loss function to transform the problem into an optimization problem. In this study, the loss function was used to solve the problem, which was defined as follows

$${\displaystyle \sum }_{i=1}^m{\left(y_i-{\widehat{y}}_i\right)}^2={\displaystyle \sum }_{i=1}^m{\left(y_i-X_{i}w\right)}^2$$

(18)

In the practical application of multiple linear expression analysis, to measure the data information acquired by the model and evaluate its regression effect, the correlation coefficient was defined as follows

$$R^2=1-\frac{{{\displaystyle \sum }}_{i=0}^m{\left(y_i-{\widehat{y}}_i\right)}^2}{{{\displaystyle \sum }}_{i=0}^m{\left(y_i-\overline{y}\right)}^2}$$

(19)

In $R^2$, the numerator is the difference between the real and predicted values, which is the total amount of information not captured by the established model. The denominator is the amount of information carried by the real label. Therefore, the closer $R^2$ is to 1, the higher the fitting degree of the regression model [33].

The rock mechanical parameters and in situ stress parameters at different depths of Wells A and B were added to the established breakdown pressure calculation model to obtain a series of breakdown pressures at different depths. After data processing and analysis, nine eigenvalues of 400 groups of sample points (200 groups of open hole completion and 200 groups of casing completion) were measured using Python, including the tensile strength, elastic modulus, Poisson’s ratio, vertical stress, maximum horizontal principal stress, minimum horizontal principal stress, porous elastic coefficient, effective porosity, and breakdown pressure, which establishes learning samples (in which breakdown pressure was used as the target output); 70% of the sample points in the dataset were used as training values and 30% of the sample points were used for verification. The breakdown pressure calculation and display models under different completion conditions were obtained through multiple linear regression analysis.

(1)

Open hole completion Well A breakdown pressure calculation and display model

$$\left[\begin{array}{l}{P}_{b1}\\ P_{b2}\\ P_{b3}\\ \dots \\ P_{bm}\end{array}\right]=\left[\begin{array}{lllllllll}1& {\sigma }_{t1}& {\varphi }_{e1}& {\sigma }_{v1}& {\sigma }_{H1}& {\sigma }_{h1}& {\alpha }_1& E_1& {\mu }_1\\ 1& {\sigma }_{t2}& {\varphi }_{e2}& {\sigma }_{v2}& {\sigma }_{H2}& {\sigma }_{h2}& {\alpha }_2& E_2& {\mu }_2\\ 1& {\sigma }_{t3}& {\varphi }_{e3}& {\sigma }_{v3}& {\sigma }_{H3}& {\sigma }_{h3}& {\alpha }_3& E_3& {\mu }_3\\ & & \dots & & & & & & \\ 1& {\sigma }_{tm}& {\varphi }_{em}& {\sigma }_{v\mathrm{m}}& {\sigma }_{Hm}& {\sigma }_{hm}& {\alpha }_m& E_m& {\mu }_m\end{array}\right]\left[\begin{array}{l}-20.475870\\ 7.2620531\\ -0.5523505\\ -2.4787826\\ 3.8979001\\ -0.1089418\\ 158.76982\\ -0.0054909\\ 46.499562\end{array}\right]$$

(20)

$$\begin{array}{ll}{P}_{bt}=& -20.475870+7.2620531\times {\sigma }_t-0.5523505\times {\varphi }_e\\ & -2.4787826\times {\sigma }_v+3.8979001\times {\sigma }_H-0.1089418\times {\sigma }_h\\ & +158.76982\times \alpha -0.0054909\times E+46.499562\times \mu \end{array}$$

(21)

(2)

Calculation and display model of breakdown pressure in Well B of casing pipe completion

$$\left[\begin{array}{l}{P}_{bt1}\\ P_{bt2}\\ P_{bt3}\\ \dots \\ P_{btm}\end{array}\right]=\left[\begin{array}{lllllllll}1& {\sigma }_{t1}& {\varphi }_{e1}& {\sigma }_{v1}& {\sigma }_{H1}& {\sigma }_{h1}& {\alpha }_1& E_1& {\mu }_1\\ 1& {\sigma }_{t2}& {\varphi }_{e2}& {\sigma }_{v2}& {\sigma }_{H2}& {\sigma }_{h2}& {\alpha }_2& E_2& {\mu }_2\\ 1& {\sigma }_{t3}& {\varphi }_{e3}& {\sigma }_{v3}& {\sigma }_{H3}& {\sigma }_{h3}& {\alpha }_3& E_3& {\mu }_3\\ & & \dots & & & & & & \\ 1& {\sigma }_{tm}& {\varphi }_{em}& {\sigma }_{vm}& {\sigma }_{Hm}& {\sigma }_{hm}& {\alpha }_m& E_m& {\mu }_m\end{array}\right]\left[\begin{array}{l}-156.68207\\ 2.2436979\\ 0.4586922\\ -0.2117677\\ 0.8039094\\ 2.0217019\\ 101.09569\\ -0.0022709\\ 196.69480\end{array}\right]$$

(22)

$$\begin{array}{ll}{P}_{bt}=& -156.68207+2.2436979\times {\sigma }_t+0.4586922\times {\varphi }_e\\ & -0.2117677\times {\sigma }_v+0.8039094\times {\sigma }_H+2.0217019\times {\sigma }_h\\ & +101.09569\times \alpha -0.0022709\times E+196.69480\times \mu \end{array}$$

(23)

Therefore, comparing the breakdown pressure profile obtained by considering the circular stress field model with that obtained by the multiple linear regression model in Figure 9, it can be observed that the correlation coefficients $R^2$ of the prediction models under different completion methods were 0.983 and 0.967, respectively. The equation had a high fitting degree, strong equation linearity, and parameter significance, and each regression model was in line with statistical significance. The regression effect was good and could be used to calculate the breakdown pressure.

5.3. Error Analysis

Table 5. Error analysis between actual breakdown pressure at different initiation positions and breakdown pressure predicted by linear regression.

Well	Crack Initiation Position (M)	Actual Breakdown Pressure in Site Construction (MPa)	Prediction of Breakdown Pressure by Linear Regression (MPa)	Relative Error (%)
Well A	4425.00	62.00	59.13	4.62
	4384.00	58.00	60.07	3.56
	4237.00	54.00	56.17	4.01
	4095.00	71.00	65.96	7.09
	4060.00	75.00	71.76	4.32
	3956.00	70.00	65.90	5.85
	3923.00	71.00	74.32	4.67
	3889.00	52.00	55.74	7.20
Well B	4667.00–4668.00	54.00	53.00	1.85
	4588.00–4589.00	51.00	54.16	6.20
	4508.00–4509.00	65.00	59.10	9.08
	4395.00–4396.00	62.00	56.80	8.39
	4145.00–4146.00	57.00	59.24	3.93
	4063.00–4064.00	55.00	50.40	8.36
	4005.00–4006.00	58.00	61.58	6.17
	3900.00–3901.00	63.00	59.98	4.79
	3845.00–3846.00	57.00	60.33	5.84
	3728.00–3730.00	48.00	52.25	8.85

shows that, when the multiple linear regression model was adopted, the relative error for the breakdown pressure of open hole completion Well A was 3.56–7.20%, and the average error was 5.15%. The relative error of the breakdown pressure of Well B completed by the casting pipe was 1.85–8.85%, and the average error was 6.35%. Therefore, the above model could replace the complex iterative algorithm for calculating the breakdown pressure and predicting the breakdown pressure efficiently and accurately.

To ensure the universality of the regression equation, two other wells in the Sulige block were selected for demonstration, namely, Well C of open hole completion and Well D of casting pipe completion. Figure 10 shows that the relative error of the breakdown pressure of Well C (Figure 10a) was 0.05–8.47%, and the average error was 4.34%. The relative error of the breakdown pressure of Well D (Figure 10b) was 2.52–9.40%, and the average error was 4.87%. This shows that the multiple linear expression model established in this paper could accurately predict the breakdown pressure and provide support for the fracturing design and construction of horizontal wells.

6. Conclusions

(1)

The rock mechanical and in situ stress parameters of the Sulige block could be calculated from logging data, laboratory experiments, and field fracturing construction curves.

(2)

Based on Hossain’s model, the horizontal well stress field distribution could be obtained by stacking each induced stress field through the superposition principle, without considering the influence of natural weak structural planes, according to the tensile fracture criterion iteration, in order to establish the breakdown pressure calculation model under different well completion conditions. The calculation results were compared with the actual fracture. The results show that the average relative error of open hole completion was 3.25% and the average relative error was 4.09%, which shows that the fracture calculation model based on horizontal well stress distribution was accurate and reliable.

(3)

Pearson correlation was used to analyze the correlation of factors affecting breakdown pressure. The results show that, whether open hole or cased hole, effective porosity and poor elastic coefficients were negatively correlated with breakdown pressure. Most of the factors had strong correlations with breakdown pressure, indicating that the linear algorithm had a good prediction effect on breakdown pressure.

(4)

The results of the multiple linear regression analysis show that the correlation coefficients $R^2$ for the open hole and cased hole prediction models reached 0.983 and 0.967, respectively, close to 1, indicating that the equation (as shown in Equations (20) and (21)) had a high fitting degree. The breakdown pressure predicted by the linear regression equation for four wells was compared with the actual breakdown pressure in the field. The average relative error results show that the breakdown pressure was 5.17% for Well A, 6.30% for Well B, 4.33% for Well C, and 4.87% for Well D, which shows that the complex prediction method for breakdown pressure based on the horizontal well stress field model and linear regression was accurate and reliable and could effectively predict breakdown pressure.