Optimization of drilling parameters based on Copula function method for the Chepai area

. In areas with complex stratigraphic lithology, the relationship between the penetration rate and drilling parameters should be fully considered to optimize the drilling process and improve drilling efficiency. The most frequently utilized methods for performing parameter optimization through correlation analysis are the correlation coefficient, principal component analysis, and grey correlation. The correlation coefficient method solely evaluates the extent of linear correlation between two variables, it cannot be applied to the non-linear connection between penetration rate and drilling parameters. The application of principal component analysis may produce inaccurate experimental findings due to the intricate and poorly co-varying nature of drilling parameters. The grey correlation method can lead to the substantial bias in the results because of the vast quantity of data analysed. Based on the vast quantity of data, using the copula function, the big data analysis method analyses the nonlinear relationship between penetration rate and drilling parameters. It constructs a united distribution function expression to determine the optimal parameter selection criteria. The in-situ drilling data from dozens of wells in the Chepaizi area are collected and optimized six types of parameters. The optimal parameter combination is determined. Following field investigation, there was a noteworthy increase of 34.83% in the average penetration rate.


Introduction
Chepaizi area is located in Junggar Basin.There are four main lithologies of Carboniferous in the area, andesite, basalt, tuff and volcanic breccia.Andesite and basalt are the main lithologies in this area.The lithology of this formation is hard andpoor drillability.The penetration rate is slow.In the process of exploration and development in the oilfield, the drilling parameter data involved are very large and complex, and are usually characterized by nonlinearity [1] .So optimization of drilling parameters is very necessary and urgent.The optimization of drilling parameters means that scientific optimization method is adopted to select reasonable drilling parameters under certain objective conditions in order to achieve the best technical and economic indicators in the drilling process according to the influence law of various factors on the drilling process combined different parameters.

Copula function basic theory and analysis
The Copula function is a united distribution function that was first proposed in 1959 as the famous Sklar theorem. [2]() Fx , () Gx has its corresponding marginal distribution, and

( ) (
) By Sklar's theorem, this function C can connect the marginal distribution function with the united distribution function for the variable, so it is also called "connection function".With the deepening of research, the Copula function is widely used in the field of correlation analysis with its unique advantages.

Correlation level for Copula functions
It is essential to study the correlation between multiple variables for selecting the most appropriate Copula function.In Copula function, the correlation coefficient is used as a correlation level to characterize the strength of correlation between different parameters [3] .

( , ) xy
are independently and identically distributed random vectors, the correlation is described by the value of Kendall's rank correlation coefficient  [4] , which reflects the degree of consistency in the trend of change among random variables.

Classification of copula functions
The Copula functions which are commonly used are divided into two categories, elliptic family Copula functions and Archimedean family Copula functions.The common elliptic Copula functions are Gaussian (normal) Copula function and t-Copula function.Their distribution functions are characterized by symmetric tail correlations [5] .
1 1 1 1 1 ( ,..., ,... ) ( ),..., ( ),..., ( ) The common Archimedean Copula functions include the Gumbel-Copula function, the Clayton-Copula function and the Frank-Copula function [6] .Both the Gumbel-Copula and Clayton-Copula are appropriate for positively correlated asymmetric correlations.The Gumbel-Copula function is frequently used when the random variables are concentrated in the upper tail.However, the Clayton Copula function describes the correlation in the lower tail [7] .The Frank Copula function has a wider range of applicability and can describe positive and negative correlations.However it is unsuitable for asymmetrical correlations.The distribution functions are, ) 3 Copula function construction

Determining the marginal distribution function
The respective marginal distributions describe the randomness of the variables, while the Copula function represents the coupling characteristics among them [8] .Thus, to construct the most favourable Copula model, one must analyse the marginal distribution features of the data and determine the most appropriate function.The nonparametric method employs the classical kernel density estimation technique to directly tackle the marginal distribution of the sample data [9,10] .The kernel density estimator [11] is, ( ) Ashrafi et al. [12] (2019) employed a genetic algorithm to select the most impactful drilling speed parameters, narrow the field down to eight, which included weight-on-bit, pump pressure, and pore pressure.A significant amount of data was collected and organized to determine that density, rotational speed, penetration rate, weighton-bit, pump pressure and pump flow rate as the drilling parameters to be studied [13] .

Copula function model fit optimization
To achieve precise modelling of drilling data using distinct function models , it is necessary to fit the Copula function to the preferred degree [14] .Various methods for optimising the model fit it, and the adjustment can be measured through the computation of the Euclidean squared distance [15] .
Table 1 Euclidean squared distance and function optimization results

Copula function model characterization
According to the results of Copula function fit optimization in Table 1, the distribution function and density function expressions for different drilling parameters are obtained [16] .Furthermore, the distribution function and density function for each parameter are illustrated in the cited manner.

Rank-Correlation levels
The Kendall correlation coefficient and Spearman correlation coefficient [17] were utilised to compare the correlation magnitude of every drilling parameter, in accordance with the correlation guide mentioned earlier.The results were subsequently via the formula.
Based on the analysis of Table 2 data, it can be concluded that for the Carboniferous formation in the Chepaizi area.the five drilling parameters correlate in descending order is pump flow rate, pump pressure, density, rotational speed, and weight-on-bit.

Optimal parameter level determination
The analysis of the various drilling parameters has been processed, and their characteristics have been evaluated objectively.Table 3 outlines the reasoned division of values alongside the corresponding penetration rate ranges.As depicted in Fig. 1, with the change of pump pressure, the penetration rate shows a continuous upward trend.Therefore, under appropriate conditions, the pump pressure should be increased as much as possible in order to submit the penetration rate.Accordingly, referring to the table 3, the optimal level of the remaining parameters is the flow rate of 40-45L/s, density of 1.5-1.6 g/cm 3 , and rotating speed of 60-75 r/min.

Fig. 1 Penetration rates corresponding to different levels of parameters 4 Example analysis
The Chepaizi area comprises three open well depths, each requiring specific drilling tool combinations.Based on the optimal parameter combinations in the previous section, two experimental wells were selected to record various data during the drilling process, analyze the drilling efficiency, and compare with the neighboring wells in the block.

Experimental Well 1
One drill bit of 381mm 15KS1952S was utilized for the first drilling, with a pure drilling time of 15 hours and a penetration rate of 53.7m/h.Three Φ241.3mm drill bits and one Φ215.9mmdrill bit were used in the second drilling stage, the drilling trip was 3414m, the pure drilling time was 497 hours, and the penetration rate was 6.86m/h.

Experimental 2 wells
In the first stage, a Φ381mm drill bit was utilized for drilling up to the depth of 706m.The pure drilling lasted for 15.5h, and penetration rate was 45.55m/h.In the second stage, a Φ241.3mmdrill bit was used to drill to a depth of 2007m, the pure drilling lasted for 66.3 hours with the penetration rate of 19.62m/h.Φ215.9mm drill bit was utilized to drill to the depth of 3300m, and successfully completed the drilling process.The pure drilling lasted for 200.2h with the penetration rate of 5.86m/h.Among them, the penetration rate and single trip footage of the two experimental wells in Carboniferous formation are shown in Tables 4 and Table 5.The drilling data of the drilled wells in the Carboniferous layer section of the Chepaizi area are collected and plotted as bar graphs for statistical analysis, as shown in Fig. 2 and Fig. 3.   From the comparison of data in Table 4 and Table 5, it is apparent that in the Carboniferous layer, the average mechanical speed of the two experimental wells is 5.73 m/h, which is 34.83% higher than the past wells, and the single trip footage is also significantly improved.Additionally, the process of drilling in the Carboniferous layer obtained the penetration rates of the two experimental wells have been significantly improved by adopting the optimal drilling parameters in the previous section.

Conclusion
(1) In the Carboniferous formation in the Chepaizi area, the rock hardness is large and the drillability is poor, which leads to the difficulty of improving penetration rate.Using the big data analysis method, various parameters in the drilling process were comprehensively considered and the most suitable copula function was constructed to obtain the magnitude of the influence of different parameters on the penetration rate, which are, in descending order, pump flow rate, pumping pressure, mud density, rotating speed, and weight-on-bit.
(2) Based on an extensive analysis of data from a significantnumber of drilled wells, the ideal parameter combinations for the Carboniferous formation were determined.These combinations include a pump pressure greater than 23MPa, a flow rate of 40 to 45L/s, a density from 1.5 to 1.6g/cm 3 , and a rotating speed of 60 to 75r/min.
(3) The ideal parameter combinations were employed in the target layers of the two experimental wells, and according to the practical results, the average penetration rate was increased by 34.83% in the Carboniferous layer, and the enhancement effect was remarkable.
condition as follow.

Table 2
Calculation of correlation level values

Table 3
Table of levels for different parameter ranges

Table 5
Drilled well drilling data for the block