## 1 Introduction  
Drilling is one of the most important operations and also essential in ensuring access to some specific energy-related natural resources, including oil and gas, hydrogen, and geothermal systems [1; 2; 3]. In this context, the pre-salt hydrocarbon reserves existent in the South Atlantic Ocean have stood out. These formations, carbonate rocks, are considerably hard and abrasive, implying a low drillability, which has historically ranged from 0.5 to 8 m/h [4; 5]. In addition, the scenarios of ultra-deep water, deep reservoir (above 7000 m in some locations), and up to 300 km from shore make it challenging and expensive [6].  
Since the 1950s, drilling optimization has been a major focus of the drilling-engineering-related industry [7], and, most of the time, the efforts focus on improving the rate of penetration (ROP) [8]. The ROP analysis reflects the drilling speed, being considered an economic, crucial, and key performance indicator to evaluate drilling activities [2; 9; 10; 11]. ROP depends on a number of factors and on the applied drilling mechanics parameters: geological formation and related properties of drilling rotation speed, applied weight on bit, implicit achievable torque, drilling fluid flow rate, type and size of the drill bit, differential pressure on the bottom of the hole, and customized drilling mud, among others [12; 13]. Some of these parameters are related directly (or even indirectly) to formation properties and also present difficulties in obtaining accurately measured downhole, adding complexity and difficulties in developing mathematical models to accurately and consistently represent the effective ROP when penetrating the rock through the drilling operations [7; 8].  
An additional parameter that should be analyzed when targeting operational efficiency is the mechanical specific energy (MSE) [14]. The concept of MSE was first widely introduced by [PERSON], in 1965. MSE can be understood as the related energy implied for penetrating a unit volume of rock [15; 16; 17] and has been used in the industry as a qualitative reference for drilling performance evaluation [18; 19]. High MSE values represent inefficiency, expressing, among others, heat and vibration, excessive wear, and also an increased probability of downhole tool failure, which reflects a scenario susceptible to more operational time, eventual workover, and maintenance [14]. MSE values should be as low as possible, ideally close to the unconfined compressive strength (UCS) of the formation rock being drilled [20].  
The drilling industry's interest in the MSE concept has grown over the past decade; however, it is not yet a common practice to have it in the loop of the drilling optimization and efficiency improvement [17]. While any type of optimization will save time and money, the industry has been slow to adapt, in part due to the isolated nature of drilling operations [20]. Thus, the general goal of this research is to optimize the drilling process, which can be applied not only to hydrocarbon exploration but also to geothermal and natural hydrogen exploration. The specific objectives are to (i) analyze the influence of drilling mechanical parameters (WOB, RPM, FLOW, and TOR) on the drilling process; (ii) employ the traditional method of analyzing drilling parameters as a function of depth, i.e., a two-dimensional (2D) analysis; (iii) explore alternative approaches of optimization, such as the response surface method (RSM) and multi-objective optimization techniques; (iv) implement the single and multi-objective optimization strategies to identify the ideal combination between ROP and MSE; and (v) compare the effectiveness of traditional versus alternative approaches for analyzing drilling parameters.