Copyright Β© the author(s). This work is licensed under a Creative Commons Attribution 4.0 International License. Improved Oil and Gas Recovery DOI: 10.14800/IOGR.1369 Received May 9, 2025; revised June 10, 2025; accepted July 23, 2025. *Corresponding author: babusahmin1966@yahoo.ca 1 Drilling Optimization through Rig Hydraulics Using a Mathematical Model Mohammad Na’il Shamsuddin and Bashir Busahmin*, Universiti Teknologi Brunei, Bandar Seri Begawan, Brunei Darussalam Abstract Hydraulics play a crucial role in the advancement and enhancement of drilling operations. The utilization of the Burgoyne and Young models for predicting the rate of penetration (ROP) and for the refinement of hydraulic parameters underscores the critical importance of jet velocity, jet impact force (JIF), and equivalent circulation density (ECD). It is of utmost importance to optimize flow rates, as they have a direct and significant impact on both the impact force and the equivalent circulating density of the jets. Consequently, this paper aims to thoroughly examine the fundamental interrelations among various parameters, including ECD, weight on bit (WOB), revolutions per minute (RPM), and flow rate, as well as to assess and evaluate how hydraulic optimization contributes to the improvement of drilling efficiency. The initial rate of penetration has experienced an increase of tenfold, with the smallest recorded increase being 1.65 times, indicating that ROP has been effectively optimized as a hydraulic parameter. The findings of this study suggest a substantial enhancement in the rate of penetration due to hydraulic optimization, underscoring the significance of flow rate in achieving optimal drilling outcomes. This research highlights the necessity of a well-planned and executed hydraulic strategy to maximize drilling efficiency and minimize operational challenges. By understanding and controlling the interplay between hydraulic parameters and drilling performance, operators can achieve significant improvements in drilling speed and overall operational effectiveness. Introduction The drilling operation represents a significant financial investment in the process of hydrocarbon extraction from beneath the surface. Therefore, enhancing drilling efficiency by optimizing parameters such as weight on bit is crucial, as it affects the rate of penetration. However, it is important to note that increasing the weight on bit does not always correlate with improved penetration rates. In softer formations, additional weight can lead to higher torque, while in harder formations, it may adversely affect the bit life (Al-Mahasneh 2017). Increasing weight on bit (WOB) sometimes resulted in the reduction of rate of penetration because of the bit foundering effect and that means increasing WOB breaks the rock cuttings and enhances rate of penetration, rate of penetration (ROP) up to the founder point and WOB causes an excess damage to the bit and lowers the ROP (Islam and Hossain 2021). Whereas hydraulic optimization mainly discussed the optimizing of pump pressure while maintaining high enough flow rate to carry the rock cuttings to the surface (Mustafa et al. 2021; Song et al. 2022). One way to optimize the rig hydraulic is to improve the bit design, which means optimizing the nozzle diameter while minimizing the frictional pressure drop to obtain an optimum jet impact force (JIF) (Ramsey 2019). In addition, it was suggested by different scholars (Longwa Milia 2008; Gulraiz and Gray 2020; Yavari 2023) that the optimum pressure drop across a polycrystalline diamond compact (PDC) bit should be 46% of the pump pressure, therefore higher ROP is achieved when the pressure drop is higher than 50% of the total pump pressure, and hydraulic horsepower is higher than 65% of the standpipe pressure, another property should be considered is mailto:dike.chukwuebuka@futo.edu.ng Improved Oil and Gas Recovery 2 the effect of thixotropy on the drilling hydraulics, where the changes in shear rate results in the fluctuations in the annular pressure losses. Additionally, geo-mechanical properties of the formation play a significant role such as torque and drag of the drill string, drilling vibrations, hole cleaning, to achieve more accurate ROP from the estimation models (Li 2023). A comprehensive understanding of the performance and wear characteristics associated with several types of drilling bits is crucial for enhancing drilling efficiency and minimizing operational costs. PDC bits are subjected to both axial and torsional impacts during the drilling process (Huang and Li 2018). Research indicated that as the duration of impact increases, the average torque tends to decrease, and higher impact speeds further enhance rock-breaking efficiency. Additionally, roller cone bits are susceptible to damage and wear, especially at the shoulder of the rubber ring and the sealing face of the metal ring, primarily due to elevated shear stress encountered during drilling activities. Consequently, the wear of roller cone bits has a direct effect on the overall efficiency of the drilling operation (Darwesh et al. 2020.). The rate of penetration is influenced by multiple factors, including WOB, rotary speed, depth, formation strength, and hydraulic conditions, leading to the development of various mathematical models aimed at predicting the rate of penetration (Tanko 2020). Among these, the Burgoyne and Young model stands out as one of the most comprehensive for forecasting the rate of penetration for roller cone bits (Li et al. 2023; Huang and Li 2018). Darwesh et al. (2020) have suggested normalization factor values of 11200, 9, 4, and 1000 for the true vertical depth (TVD), equivalent pore pressure, weight on bit per inch of bit diameter, and jet impact force, respectively. Additionally, the concept of mechanical specific energy (MSE), initially was introduced (Tanko 2020) and has gained significant traction in enhancing drilling efficiency and has proven effective in real-time estimation of optimal WOB while varying revolutions per minute (RPM) to penetrate specific formation intervals using positive displacement motors (PDM). This paper aims to enhance drilling performance by optimizing hydraulic parameters, including flow rate, jet impact force, and equivalent circulating density, through a mathematical modeling approach. Methodology Drilling data was extracted from wells drilled in South-West Queensland Australia. Different wells were drilled with roller cone bit and PDC bit. Key parameters are as follows: ROP, drill string design, TVD, WOB, rotary speed, bit record, flow rate (Q), bit diameter (db) and mud weight (Mw) are recorded during drilling operations. Table 1 presents the drilling data extracted from wells drilled in South-West Queensland Australia. This paper analyzed the data to improve the drilling efficiency by optimizing the hydraulic system while considering the effect of drilling parameters on ROP, including pore pressure gradient (pg), equivalent circulating density (ECD), bit wear, and jet impact force (Fj). Table 1β€”Drilling data from South-West, Queensland, Australia. ROP, ft/hr TVD, ft WOB, klb Rotary, RPM Q, gpm db, in ρg, ppg ECD, ppg Bit Tooth wear. JIF, lbf 71.8 718.5 12.9 115 358 10.625 9.1 9.1 0.25 205.83 166.7 4416.0 6.3 132 327 7.875 9.1 9.3 0.125 365.96 147.6 720.1 13.5 118 367 10.625 9.1 9.1 0.125 273.12 195.2 4586.6 9.5 130 389 7.875 9.0 9.2 0.25 512.19 350.4 2103.0 6.0 120 370 12.25 9.0 9.0 0.125 347.73 537.7 18986.0 6.6 134 397 8.50 9.0 9.2 0.125 355.65 206.7 2519.7 7.9 132 499 10.625 9.0 9.1 0.25 619.84 74.1 7637.7 5.7 84 368 7.875 9.0 9.2 1 337.11 Improved Oil and Gas Recovery 3 ROP Prediction. The Burgoyne and Young model were employed in this study to forecast and enhance the hydraulic parameters on ROP. The coefficients aj and xj were determined through the application of multiple regression analysis. Subsequently, the data was utilized to calculate the parameters xj. ln(𝑅𝑂𝑃) = π‘Ž1 + π‘Ž2π‘₯2 + π‘Ž3π‘₯3 + π‘Ž4π‘₯4 + π‘Ž5π‘₯5 + π‘Ž6π‘₯6 + π‘Ž7π‘₯7 + π‘Ž8π‘₯8 ,........................................................(1) or x5 = ln ( Cr 1000WOB db βˆ’0.02Ξ”Pb( dbβˆ’1 db ) ( 1000WOB db ) c ) π‘“π‘œπ‘Ÿ 𝑃𝐷𝐢 𝑏𝑖𝑑𝑠,............................................................................................(2) whereby TVD is the true vertical depth, ft; pg is the pore pressure gradient, ppg; and pc is equivalent circulating density, ppg; Cr represents the dimensionless weight distribution between the 12ΒΌβ€³ drill bit and 13Β½β€³ under- reamer. In the absence of the under-reamer, Cr is equal to 1. Initially, the Cr was 0.942, but it was later decreased to 0.02 due to being excessively high (Tanko 2020). Ξ”Pb is a bit pressure drop, psi; db denotes the bit diameter; WOB indicates the weight, klb; N represents the rotary speed. x6 = ln ( N 100 ) for roller cone bits,...................................................................................................................(3) or x6 = ln ( N 160 ) for PDC bits,...............................................................................................................................(4) Then, parameters for bit wear and jet impact force are: x7 = βˆ’h,............................................................................................................................................................(5) x8 = ln ( Fj 1000 ),...................................................................................................................................................(6) where N represents the rotational speed in revolutions per minute, and Fj denotes the jet impact force in pounds- force. Hydraulics and ROP Optimization. The hydraulic parameters in the Burgoyne and Young model, Eqs. 7 and 8 include the equivalent circulating density (𝑝𝑐) and the jet impact force (𝐹𝑗), Fj = ρQvn 1930 = ρQ2 6016An ,...........................................................................................................................................(7) ρc = ρ + Pa 0.052Γ—TVD ,...........................................................................................................................................(8) The calculation of annular pressure, π‘ƒπ‘Ž, involves the utilization of the Darcy-Weisbach Equation, considering the Reynolds’ number and Chen’s correlation friction factor for the flow within the annulus. It is crucial to ensure that the carrying capacity of the cutting surpasses the slip-velocity, while optimizing the flow rate. 𝑣𝑠𝑙𝑖𝑝 = 1.54 √ 𝑑𝑐(π‘π‘βˆ’π‘€π‘€) 𝑀𝑀 ,...................................................................................................................................(9) π‘£π‘šπ‘–π‘› = 2.5 Γ— 𝑣𝑠𝑙𝑖𝑝,..........................................................................................................................................(10) π‘„π‘šπ‘–π‘› = π‘£π‘šπ‘–π‘› 2.448(𝑑2 2βˆ’π‘‘1 2) ..........................................................................................................................................(11) The slip velocity (vslip), minimum annular velocity (vmin) and minimum flow rate (Qmin) are determined based on Eqs. 9 to 11, accordingly. It is essential for the flow rate in the annulus to exceed the minimum flow rate to effectively transport the cuttings to the surface. The ROP is calculated using the Burgoyne and Young Improved Oil and Gas Recovery 4 model, and the flow rate is optimized through the Qoptimum model. Subsequently, the optimized total nozzle area (An, min) is established based on the jet impact force. Finally, the flow rate is cross-checked with the minimum flow rate to ensure proper hole cleaning. Results and Discussion The data related to Pearson's correlation analysis is presented in Figure 1. By examining the correlation matrix, relationships among various drilling parameters were identified. The ROP shows a significant correlation with TVD, RPM, WOB, and bit wear (h). While the Burgoyne and Young model initially assumed these parameters to be independent (Li et al. 2023), it is important to note that TVD, RPM, WOB, bit diameter (db), mud weight (Mw), pore pressure gradient (pg), and flow rate (Q) are all considered independent variables. The jet impact force is influenced by flow rate, mud weight, and pressure drop. Given the low to moderate correlation among the model parameters, it is reasonable to conclude that these parameters are independent of one another. Figure 1β€”Pearson correlation matrix heatmap for the drilling data. As shown in Figure 2, the ROP generally decreases with increasing depth due to the increasing formation strength. However, one data point deviates from this trend, likely due to other factors that have a greater influence on ROP, as previously discussed Furthermore, there is a clear relationship between ROP and revolutions per minute, with higher rotational speeds leading to faster drilling rates as illustrated in Figure 3 illustrate that ROP initially increases with an increase in WOB, peaking at approximately seven kilo pounds. However, beyond this point, ROP decreases due to the founder effect. This effect occurs when the bit encounters deeper layers with reduced cutting efficiency, likely due to damage, as supported by reference (Islam and Hossain 2021). The flow rate follows a similar trend, rising with ROP before declining. In terms of hydraulic parameters, both jet impact force and hydraulic horsepower contribute to increased ROP. However, jet impact force begins to decline at around 450 pounds per foot, whereas hydraulic horsepower continues to rise alongside ROP. Improved Oil and Gas Recovery 5 Figure 2β€”(a) Rate of penetration against true vertical depth; (b) Rate of penetration against rotary speed. Figure 3β€”(a) Rate of penetration against weight on bit; (b) Rate of penetration against flow rate; (c) Rate of penetration against jet impact force; (d) Rate of penetration against hydraulic horsepower. The calculations for optimal nozzle area, ECD, and jet impact force were performed using the equations outlined in the methodology section. The results are presented in Table 2. Additionally, minimum flow rates were analyzed to ensure that the mud flow rates are sufficient for transporting rock cuttings to the surface. Table 2β€”Optimized hydraulic parameters. Optimum Bit Pressure Drop Optimum Flow Rate Optimum Nozzle Area Optimum ECD Optimum Jet Impact Force Minimum Flow Rate 2400 1259 0.744 9.35 3223.82 189 2400 1214 0.717 9.35 3106.32 189 2400 1010 0.593 9.09 2569.92 264 2400 573 0.337 9.07 1458.40 191 Improved Oil and Gas Recovery 6 The anticipated penetration rate was calculated through multiple regression analysis, with the evaluation of the Burgoyne and Young model coefficients presented in Table 3. The findings revealed that RPM and bit wear exerted the greatest influence on ROP, followed by hydraulics and weight on bit. Furthermore, the results indicated a weak formation strength based on the coefficient range of a1, while coefficients a6, a7, and a8 highlighted the importance of focusing on rotary speed, bit wear, and jet impact force. At higher depth, normal compaction, under-compaction, and differential pressure are worth optimizing. Table 3β€”Coefficient of Burgoyne & Young model. Factors Coefficients Values Formation Strength a1 24.34 Normal Compaction a2 -0.0013 Under-compaction a3 -0.0620 Differential Pressure a4 0.0085 Weight on Bit a5 0.7686 Rotary Speed a6 -9.67 Bit Tooth Wear a7 3.91 Jet Impact Force a8 1.53 The impact of flow rate on ROP is illustrated in Figure 4. As the flow rate rises, the parameters x4 and x8, representing the equivalent circulating density and jet impact force, are impacted. Specifically, an increase in jet impact leads to a higher ROP, whereas a rise in equivalent circulating density results in a lower rate of penetration. Despite this, ROP experiences an increase with flow rate due to the heightened jet impact force. However, it decreases because of the differential pressure between equivalent circulating density and the high pore pressure gradient. Figure 4β€”Rate of penetration versus flow rate. Figure 5 depicts the correlation between the ROP and the jet impact force. Elevating the jet impact force results in an increase in the ROP. Nevertheless, boosting the jet impact force necessitates a rise in the flow rate, which could potentially lead to a decrease in the ROP once it reaches the optimal level. Therefore, it is crucial to optimize the jet impact force by reducing the total nozzle area to the maximum bit pressure drop. Furthermore, the enhancement in the ROP is attributed to the reduction in the total nozzle area and becomes more pronounced at higher jet impact forces. Improved Oil and Gas Recovery 7 Figure 5β€”Rate of penetration against jet impact force at different total nozzle areas (An). The ROP refers to how quickly the drill bit advances through the formation, while ECD represents the effective mud density inside the wellbore, accounting for both the static mud weight and the dynamic effects of fluid circulation. Therefore, Figure 6 presents the relationship between rate of penetration and equivalent circulating density, where ROP increases with ECD, is primarily due to optimized differential pressure, improved jet impact force, and more efficient cuttings removal. However, controlling ECD within optimal limits is crucial to maintaining drilling efficiency without causing wellbore stability issues. Figure 6β€”Rate of penetration against equivalent circulating density. Table 4 demonstrates that the optimized Jet impact force has a considerable influence on the ROP. On average, there is an increase of ten times the original ROP and a minimum of 1.65 times the original ROP. However, the actual ROP value could be lower than expected due to other factors that may affect the ROP when increasing the jet impact force or the accuracy of the model. Table 4β€”The actual ROP and optimized ROP. Actual ROP Optimized ROP Percentage Increased (%) 71.85 1236.04 1620% 147.64 1709.97 1058% 350.39 2301.25 557% 206.69 548.30 165% Improved Oil and Gas Recovery 8 Conclusions The research has demonstrated the efficacy of hydraulic optimization in enhancing drilling efficiency by concentrating on the improvement of the rate of penetration through the optimization of jet impact force and equivalent circulating density. Additionally, the study noted an increase in rate of penetration when applying the Burgoyne and Young model. The findings suggest that factors, including depth, weight on bit, revolutions per minute, flow rate, equivalent circulating density, and jet impact force, require optimization. The effect of flow rate on both equivalent circulating density and jet impact force is vital for boosting drilling efficiency. On average, rate of penetration experienced a tenfold increase compared to the baseline, with a minimum enhancement of 1.65 times the original rate of penetration. In summary, this paper highlights the critical role of flow rate in influencing rate of penetration, emphasizing the necessity for further investigation and optimization of flow rate management strategies to attain optimal drilling performance. Nomenclature ANN =Artificial Neural Network ECD =Equivalent Circulating Density GA =Genetic Algorithm GPM =Gallon per Minute Hhp =Hydraulic Horsepower JIF =Jet Impact Force MLR =Multiple Linear Regression MSE =Mechanical Specific Energy PDC =Polycrystalline Diamond Compact RF =Random Forest ROP =Rate of Penetration RPM =Rotary per Minute SPP =Standpipe Pressure TVD =True Vertical Depth WOB =Weight on Bit Ab =Bit Area aj =model coefficients An =Nozzle Area An min =Minimum Nozzle Area Cd =Dimensionless Nozzle Discharge Coefficient Cr =Dimensionless Weight Split d1 =Drill Pipe Outer Diameter d2 =Hole Inner Diameter db =Bit Diameter dc =Cutting Diameter de =Equivalent diameter πœ– =Relative Roughness f =Friction factor Fj =Jet Impact Force h =Fractional Bit Tooth Wear Hhp =Hydraulic Horsepower m =Flow Rate Exponent Improved Oil and Gas Recovery 9 Mw =Mud Density N =Rotary per Minute 𝜌 =Mud Density Pa =Annulus Pressure Pb =Bit Pressure Drop Δ𝑃𝑏 =Bit Pressure Drop πœŒπ‘ =Cutting Density πœŒπ‘” =Pore pressure gradient PLOT =Leak-off Test Pressure Pp =Pump Pressure Q =Flow Rate Qmin =Minimum Flow Rate Qoptimum =Optimum Flow Rate T =Bit Torque va =Annular velocity vmin =Minimum Velocity vn =Nozzle Velocity vslip =Slip Velocity (π‘Šπ‘‚π΅/𝑑)𝑑 =Threshold Weight on Bit per inch bit diameter xj =Model parameters Conflict of Interest The authors declare that there is no conflict of interest. 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Improved Oil and Gas Recovery 10 Bashir Busahmin, SPE, EI, is a Senior Assistant Professor of Energy Engineering at Universiti Teknologi Brunei (UTB), where he has served on the faculty for over 11 years. His research interests focus on petroleum engineering, with particular emphasis on technical drilling engineering, formation wettability, and reservoir characterization. He holds an M.Sc. in Petroleum Engineering from AGH University of Science and Technology in Krakow, Poland, as well as dual Ph.D. degrees in Petroleum Engineering from AGH University and the University of Calgary, Alberta, Canada. Dr. Busahmin is a member of the Society of Petroleum Engineers (SPE) and a registered Engineering Institution (EI) professional. Mohammad Na’il Shamsuddin is a graduate in Petroleum Engineering from Universiti Teknologi Brunei (UTB), and a member of the Society of Petroleum Engineers (SPE). His academic background and training focus on upstream oil and gas operations, with particular interest in drilling optimization. He continues to engage in professional development and research activities in the petroleum engineering field.