Optimization of Mechanical and Hydraulic Drilling Parameters for Maximizing Rate of Penetration Using Response Surface Methodology

https://doi-001.org/1025/17610567079108

Abdelkader Khentout¹, Rym Khettabi², Samira Chouicha³, Fatiha Chelgham^1,4

¹Drilling and Mechanics of Oil Fields Department, University of Kasdi Merbah Ouargla, Algeria

²Mechanical Engineering Department, University of KasdiMerbah Ouargla, Algeria

³Drilling and Mechanics of Oil Fields Department, University of Kasdi Merbah Ouargla, Algeria

⁴Kasdi Merbah Ouargla University, VPRS Laboratory, B.P. 511, 30000, Ouargla, Algeria

Email: khentoutabdelkader@gmail.com

Received: 11/01/2025  ;  Accepted: 25/08/2025

Abstract

Optimizing the Rate of Penetration (ROP) is a critical factor for enhancing productivity and reducing costs in well drilling operations. This study aims to identify the optimal operating conditions for achieving a maximum ROP by systematically investigating the influence of key drilling parameters. Response Surface Methodology (RSM) was employed as a powerful tool for process modelling and optimization, examining the individual and interactive effects of three crucial operational variables: Weight on Bit (WOB), Rotation Speed (RPM), and Mud Flow Rate (Q).

Experimental data were collected and analysed to develop a precise statistical mathematical model describing the relationship between these parameters and the ROP. Analysis of the results demonstrated that the developed model possesses high predictive capability and significant statistical accuracy, making it suitable for determining the optimal combination of parameters for a rational and efficient operational performance. This study highlights the practical importance of statistical optimization techniques, such as RSM, in drilling engineering to enhance productivity while ensuring operational efficiency.

Keywords: Drilling Optimization, Response Surface Methodology (RSM), Rate of Penetration (ROP) Modelling, Weight on Bit (WOB), Rotation Speed (RPM), Mud Flow Rate, Experimental Design.

1. Introduction

Oil is considered a fundamental element in the global economic dynamics, significantly impacting various sectors worldwide. To increase production, enhancing the performance of the drilling process is essential. Increasing the Rate of Penetration (ROP) plays a critical role in improving oil-drilling efficiency. A higher ROP allows for faster drilling, thus reducing costs, optimizing resources, and minimizing risks, while also enhancing the overall productivity of the operation. Operators employ various strategies such as optimizing drilling parameters, using more powerful drill rigs, and improving drill bits to maximize ROP while maintaining high safety standards. This research primarily focuses on analysing the impact of drilling parameters, such as Weight on Bit (WOB), Rotation Speed (RPM), and Flow Rate (Q), on the rate of penetration. Drilling parameters are factors that affect the ROP, which are categorised into two main groups: mechanical parameters related to the type and shape of the tool, weight, and rotation speed, and hydraulic parameters such as flow rate, pressure, and the characteristics of the drilling fluid. Understanding the relationships between these parameters enables the development of both theoretical and practical concepts for better controlling drilling operations. In this context, the current research aims to clarify the empirical relationships between various factors influencing the rate of penetration using the Response Surface Method (RSM), specifically Box-Behnken experimental designs. Box-Behnken designs were chosen for their efficiency, statistical robustness, and ease of interpretation when exploring and optimizing processes or systems[1].

Since the global recognition of the critical demand for hydrocarbons in both utilization and substantial financial investment, oil exploration and exploitation have become central factors in driving technological advancements and profit expansion. Moreover, it is widely acknowledged that the oil and gas sector is increasingly focusing on optimizing drilling process designs to lower operational expenses while enhancing operational efficiency [2]. Rotary blasting hole drills are extensively employed worldwide in surface mineral extraction for waste removal purposes. The precise estimation of the penetration rate for rotary drill rigs is highly significant within the context of rock drilling, particularly in the fields of geology and petroleum technology [3.4]. Accurately estimating the penetration rate is essential in the process of mine construction. The assessment of total drilling expenses can be achieved through the use of predictive formulas [5]. Additionally, predictive formulas can be employed to identify the most suitable type of drilling rig for specific situations. Rotary tricone bits, including tungsten carbide inserts, are widely favored as the primary drilling tools for deep holes with substantial diameters in extensive surface mining processes[6]. Over time, exploration rates have increased due to the adoption of more powerful drills and enhanced management of operational factors. This, in turn, has led to higher mining output and reduced drilling costs.

Today, deep drilling practices hold significant importance and are widely promoted within the oil and gas industry. However, this technique is not without its challenges, primarily due to the substantial depth involved and the complex process of tool replacement, compounded by anomalies encountered within formation layers. These factors often lead to inconsistent results, causing mechanical issues that ultimately reduce the tool’s penetration depth. In this context, there is a shared interest among industry experts and academics in designing and developing novel drilling techniques to improve drilling operation performance [7.8]. Enhancing drilling operation efficiency and achieving superior performance levels require the optimization of various drilling parameters, including the weight of the drill bit, the rotational speed of the drilling apparatus, the rock’s resistance, and the properties of the drilling mud. This optimization primarily revolves around achieving the highest drilling rate while minimizing costs and the mass of the rock drillable indicator [9.10]. Much attention has been given to improving the quality of the drilling process. Garnier and Van Lingen [11] focused on specific phenomena that could affect drilling operations. Response Surface Methodology (RSM) is one of the most effective approaches for understanding and modelling such phenomena. RSM aims to systematically and efficiently explore the correlation between input factors and response variables in order to optimize procedures, products, or systems while minimizing the need for extensive experimentation and resources [12]. RSM is regarded as a crucial component of experimental design for developing new processes and improving their performance. This methodology was also developed to enhance products and systems, with the goal of optimizing the load component and reducing process response instability [13]. In general, RSM consists of a collection of statistical and mathematical techniques that are highly effective in analysing and addressing problems where multiple factors influence the response variable. Its goal is to improve this response[14.15]. The objective of RSM is to determine the optimal empirical design with the fewest possible design repetitions. Its use in empirical design dates back to the late 1990s [16]. This technique has been used by numerous researchers, such as Panagiotis et Angelos [17]. To investigate how the process parameters of fiber laser percussion drilling influence the geometric characteristics of 1.0 mm thick Inconel 718, experiments were conducted using RSM by Moradi and Mohazabpak [18]. The primary aim of this study is to develop mathematical simulations to predict the propulsion force and cutting torque in the context of drilling operations. Salehnezhad et al. [19] utilized RSM to optimize and improve the properties of drilling mud. By using the box-Behnken design within the RSM framework, Zhang [20] conducted several laser drilling experiments. The goal of these experiments was to determine the specific energy of rock by varying three key empirical factors: laser power, irradiative time, and spot diameter. Alakbari et al. [21] introduced new statistical empirical correlations for prediction through the application of RSM. RSM was used to establish mathematical relationships between factors and responses, as well as to clarify the interactions among variables. Surekha et al. [22] attempted to examine the effect of aluminum powder on the electrical discharge machining (EDM) of EN-19 alloy steel. Using surface response modeling, a relationship was established between the responses and the operational factors of the procedure.

In recent years, advanced methods have been developed to optimize drilling parameters, focusing on increasing the Rate of Penetration (ROP) and reducing operational costs. A recent study demonstrated the use of machine learning algorithms to analyse field data and optimize drilling parameters such as Weight on Bit (WOB), Rotation Speed (RPM), and Flow Rate (Q), resulting in a reduction of prediction error for the ROP from 18.72% to 10.56%. [23]

Additionally, Response Surface Methodology (RSM) with Box-Behnken design was applied to optimize drilling mud properties, helping to improve fluid stability and reduce fluid loss during drilling. [24]

These studies highlight the importance of integrating modern techniques, such as machine learning and advanced experimental design, to improve drilling operations and enhance their efficiency.

• Material and Methods
  • Materials Used in the Study

The experimental tests conducted in this study were performed using the Simulators Company, a petroleum-drilling simulator from the National Algerian Drilling Company (ENAFOR). This simulator is specifically designed to replicate the structure and functionality of a conventional drilling rig. It includes traditional drilling controls, analog instruments, and a manual brake system, all integrated with 3D graphical representations. This configuration provides a realistic simulation environment that allows for the testing of various drilling conditions without the need for actual field operations. The Simulators Companyoffers a versatile platform for conducting different experiments, enabling the examination of drilling parameters and the optimization of drilling processes in a controlled setting.

To explain and identify the relationship between the different factors and the response ROP, we use the Response Surface Methodology (RSM).

• Response Surface Methodology (RSM) and Box–Behnken Design

Response Surface Methodology (RSM) is a statistical tool used for optimizing processes involving multiple variables by modeling the relationship between input factors and the response variable. The key objective of RSM is to explore the optimal levels of input variables that result in the best outcome for a system. It employs a series of designed experiments to establish mathematical models for the response variable, often using quadratic polynomials. A typical RSM model can be represented as follows:

Where factors (such as temperature, pressure, etc.) are manipulated according to the design matrix to study their effects on the response. The efficiency of the Box–Behnken design in reducing the number of experimental runs while providing comprehensive information on the interactions between variables makes it a powerful tool for optimizing complex processes. By fitting a second-order (quadratic) model to the experimental data, it helps in understanding the response surface and identifying the optimal combination of factors for the desired outcome.

Recent studies have demonstrated the versatility of the BBD in process optimization. For example, Perveen et al. (2024) employed BBD to optimize the synthesis conditions of Schiff bases and dihydropyrimidinones, achieving higher yields under optimal conditions [25]. Similarly, Shao et al. (2024) applied BBD to optimize the formulation of activated lithium slag composite cement, improving its mechanical properties [26]

3. Data Analysis and Processing

Analysis of Variance (ANOVA) is a statistical technique used to compare the means of different groups. It is used to test whether there are statistically significant differences between the means of more than two populations. If the variance is significant, it suggests that the explanatory variable (parameters) has a significant effect on the dependent variable (response).

The results of the analysis are summarised in Table3.

Table 3.Analysis of Variance for ROP

Source	Sum of Squares	df	Mean Square	p-valeur
Model	34.56	9	3.84	< 0.0001
A-WOB	30.81	1	30.81	< 0.0001
B-RPM	2.53	1	2.53	< 0.0001
C-Q	0.2450	1	0.2450	0.0131
AB	0.7225	1	0.7225	0.0008
AC	0.0000	1	0.0000	1.0000
BC	0.1600	1	0.1600	0.0322
A²	0.0322	1	0.0322	0.2703
B²	0.0533	1	0.0533	0.1677
C²	0.0059	1	0.0059	0.6237
Residual	0.1575	7	0.0225
Total	34.72	16

The results presented in the table indicate that the model is well-fitted. This is evident from the significantly lower sum of squared residuals (0.1575) compared to the total sum of squares due to regression (34.72). Therefore, the influence of parameters not included in the model on the ROP behavior is minimal relative to the effect of the model parameters.

The p-value of 0.0001 for the model confirms its statistical significance. Parameters with a p-value below 0.05 are considered statistically significant. In this case, the significant terms of the model include A, B, C, AB, and BC. This outcome demonstrates that the experiments yielded reliable results, indicating that the model is well-adjusted and appropriately reflects the underlying data.

4. Goodness of Fit Statistics

Table 4 presents the statistical indices used to evaluate the quality of the fit of the developed mathematical model.

Table 4. Statistical Indices

R²	0.9955
Adjusted R²	0.9896
Predicted R²	0.9274
Adequate Precision	43.8960

The predicted R² value of 0.9274 closely aligns with the adjusted R² of 0.9896, indicating that the difference between them is less than 0.2, which suggests a good model fit.
Adequate Precision is a measure of the signal-to-noise ratio, with a ratio greater than 4 being desirable. The value of 43.896 indicates an adequate signal quality, which implies that the model is suitable for use in the design space.

The results of the analysis of variance and the evaluation of statistical indices demonstrate that the model is well-fitted, making it appropriate for accurately predicting the response (ROP).

5. Mathematical Modeling of ROP

The mathematical model provides a method to calculate the rate of penetration (ROP) for any given values of the three parameters within the scope of the study. The coded equation is useful for identifying the relative impact of the factors by comparing their coefficients, where high factor levels are coded as +1, and low levels are coded as -1. The quadratic response equation for ROP, calculated using DESIGN EXPERT 11 software, is expressed as:

ROP = 3.4 + 1.9625 × A + 0.5625 × B – 0.175 × C + 0.425 × AB – 1.24058e^-17 × AC – 0.2 × BC + 0.0875 × A² – 0.1125 × B² – 0.0375 × C².

Additionally, the model in real factors, which can be used to predict the response for different levels of each factor, requires the levels to be specified in the original units of the factors. However, this equation should not be used to determine the relative impact of each factor, as the coefficients are scaled to account for the units, and the intercept is not centered within the design space. The final model for ROP in real factors is:

ROP = -3.90966 – 0.0175 × WOB + 0.0704591 × RPM + 0.00278659 × Q + 0.00425 × WOB × RPM – 6.84785e^-20 × WOB × Q – 3.01659e^-0.5 × RPM × Q + 0.0035 × WOB² – 0.00028125 × RPM² – 3.41243e^-0.7 × Q²

6. Influence of Different Factors on ROP

The following study investigates the influence of various factors, including weight on the bit (WOB), rotation speed (RPM), and flow rate (Q), on the rate of penetration (ROP), as shown in Figures 2, 3, and 4.

8.Validation of Results

Regression is a widely recognized and commonly used statistical technique to establish a relationship between a dependent variable and one or more independent variables. When examining the relationship between a dependent variable and multiple independent variables, the application of regression requires a causal relationship between the variables included in the model. The regression line, which illustrates the predicted rate of penetration based on the observed (real) rate of penetration values, is shown in Figure 8.

The graph demonstrates that the scatter plot representing the measured response values is very close to the regression line, indicating a strong convergence between the two. This result confirms that the model has good descriptive quality and is highly useful for predicting the response behavior.

9.Optimization of Rate of Penetration (ROP)

The primary goal of this study is to maximize the rate of penetration (ROP) by analysing the variations in several key factors. To determine the optimal value of a multivariable function, it is essential to find the points where the partial derivatives with respect to each factornamely, weight on the bit (WOB), rotation speed (RPM), and mudflow rate (Q)equal zero. Table 5 displays the results obtained from solving the optimization equations using Design Expert 11 software. The optimal values for the parameters are as follows:

Conclusion

This study aimed to optimize the rate of penetration (ROP) by adjusting various parameters, such as weight on the bit (WOB), rotation speed (RPM), and mud flow rate (Q). To achieve this, we employed the Response Surface Methodology (RSM). The results of this study clearly demonstrate that the parameters examined WOB, RPM, and Q significantly influence the rate of penetration. Among these, the effect of WOB was found to be the most significant, while the combined effect of WOB and RPM had the greatest impact on ROP compared to other parameter combinations. The study confirms that the highest ROP (5.526 m/h) was achieved with the following optimal values: WOB = 14.338 TM, RPM = 91.080 rpm and Q = 1456.799 l/min.

The application of RSM validates the effectiveness and accuracy of the developed model, as evidenced by the strong correlation between the predicted and experimental data. This method allows for the precise prediction of ROP. Furthermore, incorporating additional parameters, such as rock type or tool geometry, can provide even more precise results in understanding the behavior of ROP and improving drilling performance.

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[26] Shao, W., et al. (2024). Experimental study based on Box–Behnken design and response surface methodology for optimization proportioning of activated lithium slag composite cement-based cementitious materials. Materials, 17(11), 2651. box- behnken experimental design in factorial experiments …

27‏/02‏/2013 — A BBD should consist of an equal number of replicates of all possible combinations of factor levels. In some experimen…

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