A Survey on the Shortcomings of the Current Rate of Penetration Predictive Models in Petroleum Engineering

Abstract

In drilling engineering, the rate of penetration (ROP) is a prevalent indicator to evaluate the energy efficiency of drilling operations. Nowadays, ROP prediction has become more critical since the production from deeper hydrocarbon resources is unprecedentedly escalating. So far, a wealth of theoretical and practical investigations has been conducted to develop ROP models; however, the existing models have not been adequately updated with the new technological advancements or geological restrictions. This research strives to integrate the latest advancements, restrictions, and future requirements in ROP prediction. To do this, the existing empirical and data-driven ROP models are elaborated and compared. From the conducted research, it was deduced that four uncontrollable factors, including the rock permeability, wellbore inclination, temperature, and rock hardness, have not been adequately considered in ROP models. Moreover, although data-driven ROP models deliver more accurate results than the empirical models, the determination of the number and type of the input parameters is still challenging. To tackle this issue, it is recommended to develop a formation-based classification system of input parameters for future investigations. This inclusive review can be adopted by the companies and engineers involved in drilling operations to update and reform their drilling strategies.

1. Introduction

How can drilling engineers predict the rate of penetration (ROP) more accurately? As the need for the exploitation of deeper hydrocarbons is increasingly growing, are existing ROP models still adequately reliable? What factors have been neglected in existing ROP models? How can ROP models be improved? These questions are constantly exchanged between drilling engineers as they are dealing with the minimizing of total budget and time spent for drilling operations. To access the subsurface natural resources, e.g., oil, gas, groundwater, minerals, etc., drilling is known as the most expensive option [1]. To reduce the cost and time of drilling, usually, drilling engineers optimize the mechanical specific energy (MSE) or the ROP. The MSE demonstrates the required energy for breaking the rock, while the ROP indicates the pace of drilling operations [2]. In this review article, existing ROP models are assessed, together with their shortcomings and future requirements.

From the mathematical perspective, the ROP is defined as the ratio of the rock’s drilled length to the drilling time. The ROP can be affected by a wide spectrum of miscellaneous factors [3]. Those factors are classified as the controllable and uncontrollable factors. The adjustable factors, e.g., revolutions per minute (RPM), mud-flow rate (Q), torque (T), and weight on bit (WOB), can be changed and controlled by the operators [4]. By contrast, some other factors cannot be changed due to technological limitations, or geological conditions. For instance, wellbore trajectory, pore pressure, rock strength, and equivalent circulation density (ECD), are examples of uncontrollable factors [5].

Since the ROP directly influences the entire drilling cost and time, many researchers and companies have had special concerns to model this parameter. To tackle this, the impact of the different contributing factors on the ROP have been investigated for decades; however, ROP optimization is a sophisticated task as the relations between the ROP and different contributing factors are often commonly complicated [6,7,8,9,10,11,12]. Therefore, several attempts were made by different researchers to develop high-performance, predictive models to achieve optimal results.

The initial concentration of those studies was on the development of empirical models derived from the experimental tests. The first attempt to optimize drilling parameters was performed by [13]. In that study, an experimental formula correlating the ROP with the bit life was established. The factors of RPM, WOB, and drilling depth were included in the corresponding ROP model.

Maurer was the next researcher who offered a mathematical relation to predict the ROP for the roller-cone bits [14]. Galle and Wood used a number of graphs and diagrams to develop a method for determining the best association of WOB and RPM for a special kind of bit [15]. Bingham modified Maurer’s model, and proposed an experimental model; however, the drilling depth was not considered in the model, and it was also restricted to low values of RPM and WOB [16]. At the same time, Teale introduced an empirical correlation linking the ROP to the MSE [17]. In that model, the ROP was used to determine the necessary energy for drilling of a unit volume of rock. After Teale, Eckle performed several micro-bit experiments, and investigated the impact of overbalance pressure reduction on the ROP improvement [18].

One of the most significant investigations dates back to the empirical ROP model suggested by Bourgoyne and Young [3]. In such a mathematical model, several drilling parameters were included, and, therefore, it was widely applied as a practical optimization method to adjust the real-time parameters of drilling operations [19]. The next two models belonged to Warren who introduced his first model in [20], and, then, declared a modification of the previous model in [21]. The first model, which was called the “perfect-cleaning model”, included the rock strength, bit diameter, RPM, and WOB. The second model, known as the “imperfect-cleaning model”, was based on the first model; however, it included the cuttings removal term. In better words, mud properties, e.g., density and viscosity, and the effect of jet impact force were incorporated in ROP modeling. Improvement and development of empirical ROP models was then followed by several researchers [22,23,24,25,26,27].

Apart from the empirical models, some researchers strived to predict the ROP using artificial intelligence (AI) algorithms. In fact, numerous data are collected daily during the drilling activities. However, such data are full of uncertainties and hidden correlations that can be well-handled by AI analytical tools [28]. Recently, ROP prediction by the AI techniques is one of the most reliable methods delivering acceptable results [29,30,31,32]. Some valuable works related to AI-based ROP models can be found in [33,34,35,36,37,38,39,40,41,42,43].

ROP models are also applicable in the mining industry. There have been many investigations for ROP prediction in mines [44,45,46,47,48,49,50,51,52,53]. In the mining industry, drilling operations are executed for implementation of coring boreholes or blasting holes. Comparing the oil/gas wells with mining boreholes, it can be claimed that the ROP is more critical in oil/gas drilling operations as the destination depth is far deeper than that of mining activities. Apart from the mining and oil/gas industry, drilling operations are also an indispensable part of some other engineering applications, such as water-well drilling [54,55], coal-bed methane extraction, and extraterrestrial applications [56,57]. Thus, optimization of the ROP is much-needed to guarantee the success and profitability of those drilling applications.

Drilling engineering is a multidisciplinary area in which several science branches, including geology, petroleum engineering, mechanical engineering, metallurgy engineering, electronics, robotics, computer engineering, etc., come together to make drilling operations more energy-efficient, affordable, and time-efficient [58]. The objective of this research is to integrate the previous and recent advancements in ROP prediction for recognition of current limitations and future requirements pertinent to different aspects of ROP modeling. Therefore, in this study, from miscellaneous angles including the contributing factors, types (empirical and AI-based) of models, and potential shortcomings, ROP models are assessed.

The current research reveals that uncontrollable factors have not been adequately incorporated into the existing ROP models. To address this issue, four uncontrollable factors, including rock permeability, wellbore inclination, temperature, and rock hardness, are suggested to be added to future ROP models. The novelty of the current research lies in the detection of such uncontrollable factors to improve ROP models. To find the relation between those factors and ROP variation, both laboratory tests and AI approaches can be utilized. Moreover, since the uncontrollable factors mostly originated from uncertainty in the geological properties of the subsurface formations, development of a formation-based classification system of input parameters is proposed. For instance, the rock abrasiveness can be applied in sandstone ROP models, not for salt rocks. On the contrary, the rock creep can be included in salt rocks ROP models, not for sandstone rocks. Through this, for different formations, the minimum number of effective geological parameters can be included in ROP models.

The structure of this research has been designed as follows: in Section 2, the different contributing factors in ROP modeling are described. Then, in Section 3.1, the different empirical ROP models are elaborated and compared with each other. Afterwards, in Section 3.2, the focus shifts to AI-based studies performed for ROP prediction/optimization. In that section, the type of input data, the merits, and demerits of those AI-based models are also described and contrasted. Then, in Section 4, an inclusive discussion on the key results and recognized shortcomings is presented. Finally, the article ends with a concise conclusion depicting the main results, propositions, and implications of this research.

2. Contributing Factors on the ROP

To enhance and develop ROP models, it is necessary to investigate the factors affecting this parameter. The most important contributing factors on the ROP can be divided into two main groups: uncontrollable and controllable factors [33]. Figure 1 depicts these factors. The main uncontrollable factors stem from the uncertainty in the formation characteristics. No adequate attention has been given to these factors in available ROP models for oil/gas drilling operations. However, in the mining industry, it seems that researchers have incorporated more physical and mechanical properties of formations into ROP models. On the opposite side, the number of controllable factors affecting the ROP are more than the uncontrollable ones. Therefore, in the real field, the controllable factors are usually changed to enhance the ROP. The main controllable factors included in the available oil/gas ROP models are “rig operating conditions”, “drilling fluid properties”, and “bit hydraulics”. In the mining industry, less investigation has been dedicated to the parameters of rig operating conditions, including the WOB, RPM, bit type, and torque. Those factors are elaborated in what follows.

2.1. Formation Characteristics

There are several formation characteristics influencing the bit-penetration rate. One of the most determining characteristics is the shear strength of the rock. For a standard compression test, the following formula can be utilized to calculate the rock shear strength [59]:

$${\tau }_0=\frac{{\sigma }_1}{2}\cos \theta$$

(1)

where ${\tau }_0$ (psi) represents the rock shear strength, ${\sigma }_1$ (psi)represents the rock compressive strength, and θ depicts the internal friction angle. There are several rock-failure criteria to predict the rock shear strength. Some examples are the Hoek–Brown, Drucker–Prager, Mohr–Coulomb, and Modified Lade. Those criteria are based on some assumptions which may affect the predicted value of the shear strength of the rock.

Moreover, the elemental composition of rocks has an impact on the ROP values. For example, rocks containing a high percentage of hard minerals, e.g., SiO₂ and Fe₂O₃, have less drillability in comparison to the rocks containing clay minerals. Abrasiveness is another important factor affecting the ROP. For instance, the common igneous rocks, such as granite and diorite, are more abrasive than the sedimentary formations, such as sandstone and shale. Such high abrasiveness gives rise to a premature, fast dulling of the bit’s teeth [59].

Rock permeability is another parameter that has a remarkable influence on the ROP. In fact, permeability allows the drilling fluid to move into the subsurface layers, thereby equalizing the pressure differential at the bottom-hole.

Poroelastic properties of rocks also play a significant role in the interaction between the rock and the bit, thereby affecting the ROP [60]. Such poroelastic properties are derived from the presence of pore pressure within the pores of rocks. The main poroelastic parameters include the porosity, pore pressure, Biot’s coefficient, and Skempton’s coefficient [61]. During the drilling operations, the coupling between the pore pressure, mud flow, WOB, and the in situ stresses cause deformations around the wellbore. Such coupling is of paramount significance in terms of wellbore instability problems [62], lost circulation issues, wellbore deviation from the planned trajectory, etc. It should be noted that the poroelastic parameters are dependent on the temperature of the surrounding formations [63,64].

2.2. Mud Properties

Muds play an integral role in every drilling operation. Some vital functions of mud include the cuttings removal, prevention of pore fluid influx into the wellbore, and preservation of the wellbore stability. Therefore, providing an accurate drilling fluid system is quite crucial for successful drilling. To appropriately characterize the drilling fluid, calculation of the fluid flow and its pressure within the wellbore is of paramount significance. The science of wellbore fluid mechanics is applied to investigate the influence of mud characteristics on the drilling operations. In other words, wellbore fluid mechanics describes the dominant equations on the fluid flows in the wellbore to predict their performance during the interaction with the surrounding formations, and pore fluids at different depth and temperature conditions.

Regarding the mud, the chemical composition, viscosity, density, and solid content of the mud are of paramount significance. For instance, when it comes to the chemical composition, water-based drilling fluids may interact with shale formations, thereby gradually washing them. Consequently, such circumstances may bring about wellbore instability issues. Concerning mud viscosity, it is a critically important fluid characteristic in the ROP optimization. The mud viscosity is a determining factor in the cleaning and transportation of the rock cuttings towards the ground surface.

Furthermore, the solid content along with the mud density regulate the differential pressure in the bottom-hole. Drilling fluids with high density create a positive differential pressure around the bit (overbalanced drilling). The correlation between the overbalanced drilling and the ROP for different bit types has been shown in Figure 2. As it can be seen, such correlation is a straight line when the values of log (R/R₀) are presented as a function of the overbalance pressure. The parameters of R and R₀ represent the ROP and the initial ROP, respectively. On the other hand, when the mud density is less than the formation pressure, underbalanced drilling occurs. Based on this figure, when drilling is performed under underbalanced conditions, the ROP values increase [59].

Bourgoyne and Young offered the following relation between the overpressure (overbalance) and the ROP (ft/h) [3]:

$${\log }_{10}\left(\frac{ROP}{RO{P}_0}\right)=-m\left(P_{bh}-P_f\right)$$

(2)

where $P_{bh}$ (psi) demonstrates the circulating bottomhole pressure $P_f$ (psi) depicts the pore fluid pressure $RO{P}_0$ (ft/h) is the ROP at zero overbalance, and m is the tangent of the straight line in Figure 2. The parameter of $P_{bh}$ can be calculated as:

$$P_{bh}=0.052{\rho }_{c}D$$

(3)

where ${\rho }_c$ (Ib/gal) represents the ECD, and D (ft) is the total depth. Moreover, the pore fluid pressure is calculated as:

$$P_f=0.052g_{p}D$$

(4)

where $g_p$ (Ib/gal) represents the gradient pore pressure. Equations (2) and (4) can be rewritten as:

$${\log }_{10}\left(\frac{ROP}{RO{P}_0}\right)=-0.052mD\left({\rho }_c-g_p\right)$$

(5)

$${\log }_{10}\left(\frac{ROP}{RO{P}_0}\right)=-a_{4}D\left({\rho }_c-g_p\right)$$

(6)

where $a_4$ is the overbalance exponent. The above-mentioned equations are beneficial to investigate the influence of the mud density on the ROP. Thus, Equation (6) is rewritten as:

$$\frac{RO{P}_2}{RO{P}_1}=e^{2.304a_{4}D\left({\rho }_1-{\rho }_2\right)}$$

(7)

where, $RO{P}_1$ and $RO{P}_2$ represent the ROP values related to the ${\rho }_1$ and ${\rho }_2$ respectively. Note that ${\rho }_1$ (Ib/gal) is the initial mud weight while ${\rho }_2$ (Ib/gal) is the secondary one [3].

2.3. Bit Operating Conditions

The parameters pertinent to the bit working conditions also affect the ROP values. Such parameters include the bit type, bit tooth wear, RPM, WOB, and torque.

Regarding the bit wear, it is noticeable that, due to the tooth wear, most bits drill more slowly as drilling time passes. Abrasion and chipping mechanisms decrease the tooth length continuously. Hard facing erodes the tooth in a way that encourages self-sharpening tooth wear. Although such action maintains the tooth sharpness, the shorter tooth length is not compensated by this. The tooth of PDC bits and rolling-cutter bits constructed of tungsten carbide fail via breaking rather than abrasion [2,15,65,66,67].

Another important factor in ROP optimization is the torque. It is stated that torque becomes significantly more sensitive to the variations of WOB as the bit penetrates further into the deeper hard rocks [68].

Regarding the RPM, Figure 3 illustrates the relation between the RPM and the ROP (red curve). For the low values of RPM (segment ab), there is a direct, linear relationship between the RPM and the ROP. The green dashed line illustrates such linear relation. Point b is known as the foundering point. After point b, the rise in the RPM causes a nonlinear increase in the ROP with a slower pace (segment bc). Such reduction is due to the inefficient bottom-hole cleaning and mud characteristics, such as buoyancy factor [59].

Regarding the WOB, Figure 4 depicts the change of the ROP versus the WOB (red curve). Note that no tooth wear is assumed in this figure. Point a is known as the threshold formation stress. Before point a is surpassed, no substantial penetration rate is obtained. For the low values of bit weight (segment ab), the ROP builds up linearly with rising values. The moment at which the rock-failure condition switches from scraping to shearing is known as point b. After point b, as the WOB increases, the ROP heightens linearly (segment bc); however, this segment has a steeper slope than the segment ab, thereby indicating a higher drilling efficiency. Beyond point c, additional bit weight enhances the ROP marginally (segment cd). Point d is the foundering point. After point d, in some instances, the ROP drops off at excessively large WOB values (segment de). The reason is that for the intensely high WOBs, the bottom-hole cleaning and cutting removal mechanisms become inefficient due to the large volume of rock cuttings produced in the bottom-hole. Such a situation also prevents the mud from escaping from the nuzzles, thereby leading to further inefficient bottom-hole cleaning [37,59].

2.4. Bit Hydraulics

Optimization of bit hydraulics for acquisition of high ROP values has been the subject of much debate [69,70]. Commonly, the mentioned factors include the Reynolds number, mud-flow rate, impact force of the jet, nuzzle velocity, and bit hydraulic horsepower. Many researchers believe that the bit’s foundering point is affected by the hydraulic pressure attained at the bit. Eckel proposed the underlying relation based on the Reynolds number [18]:

$$N_{R_e}=K_s\frac{{\rho }_{f}Q{d}_{nz}}{{\mu }_a}$$

(8)

where $N_{Re}$ stands for the Reynolds number function, $K_s$ represents the scaling coefficient, ${\rho }_f$ (Ib/gal) indicates the density of mud, Q (gal/min) represents the fluid-flow rate, $d_{nz}$ (in) indicates the diameter of nozzle, and ${\mu }_a$ (cp)) represents the mud’s apparent viscosity at 10⁴ s⁻¹. Figure 5 displays the impact of the Reynolds number function and WOB on the ROP [18]. For the whole range of the Reynolds numbers investigated, the penetration rate rises as the Reynolds number function increases. As it can be observed, the corresponding curve moves upward as the WOB is raised. It is noteworthy that Eckle did not investigate the effect of the foundering point on the ROP.

2.5. Personal Efficiency

Any drilling issue can result in extremely high overall well expenses; as a result, ongoing training for all drilling crew is crucial to achieve high ROP values [37,71].

2.6. Rig Efficiency

Selection of a proper rig for the drilling operation is of paramount importance. The selected rig can influence the whole drilling cost, time, and specified energy. Moreover, appropriate maintenance is required to keep the rig in an operable condition. The simplicity of the rig is also beneficial for the crew to take correct, rapid actions during the drilling operation.

3. The ROP Models

A number of empirical and data-driven models have been established to predict or optimize the ROP. The empirical models were mainly developed using real field drilling data or laboratory tests. The linear regression technique was mostly used to formulate the ROP relationship with other drilling parameters. On the opposite side, the data-driven models forecast the ROP values using AI techniques [72]. Both types of ROP models are elaborated in what follows.

3.1. Empirical ROP Models

3.1.1. Review on Empirical ROP Models

Maurer created the first empirical model for ROP prediction [14]. The main components of that model were the rock strength, WOB, RPM, and drill bit diameter. The following equation describes Maurer’s model:

$$ROP=\frac{K}{S^2}{\left(\frac{WOB}{d_{bit}}-\frac{WO{B}_0}{d_{bit}}\right)}^{2}RPM$$

(9)

In this equation, K shows the proportionality coefficient, $WOB$ (Ib) represents the weight on the bit, $d_{bit}$ (in) demonstrates the bit diameter, RPM (rev/min) is the rotational speed of the bit, WOB₀ (Ib) depicts the threshold weight on the bit, and S represents the rock compressive strength.

Galle and Woods proposed the second ROP model [15]. The corresponding formula was:

$$ROP\propto {\left(\frac{1}{0, 928125h_f{}^2+6h_f+1}\right)}^{b_7}$$

(10)

In this equation, $h_f$ (in) depicts the fractional bit tooth dullness, and $b_7$ shows an exponent (suggested to be selected as 0.5).

Bingham proposed a different model to forecast the ROP [16]. The model was expressed as:

$$ROP=K {\left(\frac{WOB}{d_{bit}}\right)}^{b_5}RPM$$

(11)

where $b_5$ is the WOB exponent. In addition, K included the rock strength from Maurer’s model.

One of the most practical empirical models to forecast the ROP was developed by Bourgoyne and Young [3]. This ROP model was defined by the following equation:

$$ROP=f_1\times f_2\times f_3\times f_4\times f_5\times f_6\times f_7\times f_8$$

(12)

where

$$f_1=ex{p}^{2.303\times a_1}$$

(13)

$$f_2=ex{p}^{2.303\times a_2\times \left(10000-TVD\right)}$$

(14)

$$f_3=ex{p}^{2.303\times a_3\times TV{D}^{0.69}\times \left(g_p-9.0\right)}$$

(15)

$$f_4=ex{p}^{2.303\times a_4\times TVD\times \left(g_p-ECD\right)}$$

(16)

$$f_5={\left[\frac{\left(\frac{WOB}{d_{bit}}\right)-{\left(\frac{WOB}{d_{bit}}\right)}_t}{4-{\left(\frac{WOB}{d_{bit}}\right)}_t}\right]}^{a_5}$$

(17)

$$f_6={\left(\frac{RPM}{60}\right)}^{a_6}$$

(18)

$$f_7=ex{p}^{-a7\times h_f}$$

(19)

$$f_8={\left(\frac{F_{Jet}}{1000}\right)}^{a_8}$$

(20)

In Equations (12)–(20), $a$₁, $a$₂, $a$₃…, and $a$₈ are determined using real drilling data. Moreover, TVD (ft) displays true vertical depth, $g_p$(Ib/gal) indicates the gradient of pore fluid pressure, (WOB/d_bit)_t stands for the parameter of the threshold WOB pertinent to one inch of the bit diameter when the bit starts drilling (10³ lbf/in), $F_{Jet}$ (Ib) demonstrates the fluid motion force beneath the bit, and (WOB/d_bit) represents the WOB pertinent to one inch of bit diameter. It is also important to note that Osgouei enhanced this model in 2007 by adding the hole-cleaning term for directional and horizontal wellbores [68]. Such an improved model is applicable for both roller-cone bits and PDC bits [68].

Furthermore, the parameter of bit wear was presented by utilization of some specific assumptions. The tooth wear can be calculated as [3]:

$$\frac{dh}{dt}=\frac{H_3}{{\tau }_H}{[\frac{RPM}{100}]}^{H_1}\times \left\{\frac{{\left(\frac{WOB}{d_{bit}}\right)}_{max}-4}{\left(\frac{WOB}{d_{bit}}\right){ }_{max}-\frac{WOB}{d_{bit}}}\right\}\times \left\{\frac{1+\frac{H_2}{2}}{1+H_{2}h}\right\}$$

(21)

where h (ft) is the depth and t (h) is time. The coefficients of $H_1, H_2, \mathrm{and} H_3$ are dependent on the type of bit. Moreover,${\tau }_H$ is a coefficient representing the abrasiveness of the rock. To calculate the bearing wear, the following relationship was proposed:

$$\frac{d{B}_{bw}}{dt}=\frac{1}{{\tau }_b}\times \frac{RPM}{100}\times {[\frac{WOB}{4d_{bit}}]}^b$$

(22)

where $B_{bw}$ is the bearing wear proportion over the bearing’s entire lifetime, ${\tau }_b$ (h) is the lifetime of the tooth under standard circumstances, and $b$ is an empirical coefficient.

Reza and Alcocer in Ref. [66], used the Buckingham p-theorem to express a nonlinear drilling model for deep-drilling operations. In their model, the ROP was expressed through the following equation:

$$\frac{ROP}{RPM{d}_{bd}}=K[\frac{RPM{d}_{bd}{}^2}{v}{]}^a[\frac{RPM{d}_{bd}{}^3}{Q}{]}^b[\frac{E{d}_{bd}}{WOB}{]}^c[\frac{\Delta p{d}_{bd}}{WOB}{]}^d$$

(23)

where $K$ shows the proportionality constant, $d_{bd}$(in) represents the diameter of bearing, $v$ (cp) represents the kinematic viscosity of the mud, $Q$ (gal/min) demonstrates the mud-flow rate, $\Delta p$ (psi) is the differential pressure between the mud pressure and pore pressure, and $E$(psi) represents the formation hardness. It is noteworthy that the variables of K, a, b, c, and d were calculated using regression analysis. Finally, the following ROP equation was obtained:

$$\frac{ROP}{RPM{d}_{bd}}=0.33[\frac{RPM{d}_{bd}{}^2}{v}{]}^{-0.43}[\frac{RPM{d}_{bd}{}^3}{Q}{]}^{-0.68}[\frac{E{d}_{bd}}{WOB}{]}^{-0.91}[\frac{\Delta p{d}_{bd}}{WOB}{]}^{-0.15}$$

(24)

In addition, the general relationship for calculation of bit dulling was proposed as:

$$\frac{h_f}{RPM{d}_{bit}}=0.001[\frac{Q}{RPM{d}_{bit}{}^3}{]}^{0.56}[\frac{WOB}{E{d}_{bit}{}^2}{]}^{0.26}{[\frac{d_{bit}}{Q}]}^{-0.03}$$

(25)

where $h_f$ is the fractional dullness of the bit tooth. On the other side, the general relationship for bit-bearing life was expressed as:

$$\frac{B_{bw}}{RPM}=0.05[\frac{T{h}_t{d}_{bd}}{RPM WOB}{]}^{0.51}[\frac{v}{RPM{d}_{bd}{}^2}{]}^{0.26}{[\frac{Q}{RPM{d}_{bd}{}^3}]}^{-0.5}$$

(26)

where $T$ (°F) is the temperature of the bottom-hole setting and $h_t$ (Btu/°F) is the coefficient of heat transfer.

The “perfect-cleaning model” was proposed by Warren [20]. The model had variables of bit diameter, rock strength, RPM, and WOB. The corresponding formula was:

$$ROP=1/( \frac{a{S}^2{d}_{bit}^3}{RP{M}^{b}WO{B}^2}+\frac{b}{RPM{d}_{bit}})$$

(27)

where a and b represent two dimensionless coefficients and S (psi) represents the rock’s strength. The applicability of this model was constrained as it did not consider the cuttings removal. To address this issue, Warren proposed the “imperfect-cleaning model”. The improved model included the cuttings removal which is influenced by the density and viscosity of the mud, as well as the impact force of the jet. Consequently, this model became more applicable than the former one. The model is represented by the following equation:

$$ROP=1/(\frac{a{S}^2{d}_{bit}^3}{RP{M}^{b}WO{B}^2}+\frac{b}{RPM{d}_{bit}}+\frac{c{d}_{bit}{\gamma }_f\mu }{F_{jet}})$$

(28)

where c is a dimensionless constant, ${\gamma }_f$ stands for the fluid-specific gravity, and $\mu$ (cp) indicates the viscosity of the drilling fluid. Moreover, $F_{jet}$ (Ib) shows the impact of the jet force.

Warren’s imperfect-cleaning model was modified by Hareland and Rampersad [24]. They incorporated the impact of bit wear in the above equation by proposing a wear function,$W_f$, as:

$$ROP=W_f/[f_c\left(P_e\right)\left( \frac{a{S}^2{d}_{bit}^3}{RP{M}^{b}WO{B}^2}+\frac{b}{RPM{d}_{bit}}+\frac{c{d}_{bit}{\rho }_f\mu }{F_{jet}}\right)]$$

(29)

$$W_f=1-\frac{\Delta BG}{8}$$

(30)

where $a$, b, and c are dimensionless constants, $f_c\left(P_e\right)$ represents the chip hold-down function, and $\Delta BG$ is the variation of the bit-tooth wear. The parameter of $\Delta BG$ is obtained through:

$$\Delta BG=W_c[{\displaystyle \sum }_{i=1}^{n}WO{B}_{i}RP{M}_i{S}_i{\left(A_{r_{abr}}\right)}_i]$$

(31)

where $W_c$ represents the bit-wear constant. Moreover, $A_{r_{abr}}$ indicates rock relative abrasiveness and S (psi) represents the rock confined compressive strength obtained through the following relationship:

$$S=S_0\left[1+a_s P_e{}^{b_s}\right]$$

(32)

where $S_0$ (psi) represents the unconfined strength of rock. In addition, the constants of $a_s$ and $b_s$ (md) are dependent on the rock permeability.

Osgouei suggested the final prominent empirical model, in Ref. [68]. The Bourgoyne and Young model served as the foundation of his model. Osgouei added three factors, specifically, $f_9$, $f_{10}$, and $f_{11}$, which represent the borehole cleaning terms in the horizontal, directional, and vertical wellbores for both the roller-cone and the PDC bits. In fact, he accounted for the impact of the hole-cleaning factor on the values of the ROP. The ROP model improved by Osgouei is:

$$ROP=f_1\times f_2\times f_3\times f_4\times f_5\times f_6\times f_7\times f_8\times f_9\times f_{10}$$

(33)

where

$$f_1=e^{a_1}$$

(34)

$$f_2=e^{a_2\times (8800-TVD)}$$

(35)

$$f_3=e^{a_3\times TV{D}^{0.69}\times \left(g_p-9\right)}$$

(36)

$$f_4=e^{a_4\times TVD\times \left(g_p-ECD\right)}$$

(37)

$$f_5={\left[\frac{\frac{WOB}{d_{bit}}}{{\frac{WOB}{d_{bit}}|}_t}\right]}^{a_5}$$

(38)

$$f_6={\left(\frac{RPM}{RP{M}_c}\right)}^{a_6}$$

(39)

$$f_7=e^{-a7 h_f}$$

(40)

$$f_8={\left(\frac{F_{Jet}}{F_{Jc}}\right)}^{a_8}$$

(41)

$$f_9={\left(\frac{A_{bed}/A_{well}}{0.2}\right)}^{a_9}$$

(42)

$$f_{10}={\left(\frac{V_{Actual}}{V_{Critical}}\right)}^{a_{10}}$$

(43)

$$f_{11}={\left(\frac{C_c}{100}\right)}^{a_{11}}$$

(44)

The constants of $a$₁, $a$₂, ₃, …, and $a$₁₁ are estimated through real drilling data. Moreover, ${RPM}_c$ (rev/min) stands for the critical rotary speed, which must be estimated by considering the characteristics of the drilling string, bit type, and field data. Additionally, the parameter of $F_{Jc}$ is dependent on the type of bit, the mud’s characteristics, and pump pressure. In addition, h_f displays the fractional tooth dullness. The parameter of $V_{Actual}$ (ft/s) denotes the volume of clippings, $V^{critical}$ (ft/s) is the critical volume of the cuttings removal, and $C_c$ is the constant of the cuttings concentration. The parameters of $A_{bed}$ (ft²) and $A_{well}$(ft²) demonstrate the area of the cuttings bed and the wellbore, respectively.

3.1.2. Analysis of the Empirical ROP Models

The most well-known empirical ROP models were elaborated in the previous section. As mentioned, different researchers had strived to improve ROP models by adding new contributing factors to them. This is why empirical ROP models underwent continuous improvements from the early Maurer model to Osgouei’s model.

Table 1. Factors considered in existing empirical ROP models.

Reference		Controllable Factors										Uncontrollable Factors
	WOB	RPM	${\mathit{d}}_{\mathit{b}\mathit{i}\mathit{t}}$	${\mathit{h}}_{\mathit{f}}$	${\mathit{F}}_{\mathit{J}\mathit{e}\mathit{t}}$	TVD	$\mathit{\mu }$	B	ρf	Q	Pm	S	Ab	E	Pp	T	Pe	${\mathit{C}}_{\mathit{c}}$	Awell
Maurer, 1962 [14]	×	×	×									×
Galle and Woods, 1963 [15]				×
Bingham, 1965 [16]	×	×	×									×
Bourgoyne and Young, 1974 [3]	×	×	×	×	×	×		×	×		×	×	×		×
Warren (perfect model), 1981 [6]	×	×	×									×
Warren (imperfect model), 1984 [7]	×	×	×		×		×		×			×
Reza and Alcocer, 1986 [64]	×	×		×			×	×		×	×			×	×	×
Hareland and Rampersad, 1994 [24]	×	×	×	×	×		×		×			×	×				×
Osgouei, 2007 [5]	×	×	×	×	×	×	×	×	×		×	×	×		×			×	×

WOB: weight on the bit, RPM: rotational speed, d_bi: diameter of bit, $h_f$: bit-tooth wear, $F_{Jet}$: fluid motion force beneath the bit, TVD: true vertical depth, $\mu$: mud viscosity, B: bearing wear, ${\rho }_f$: mud density, Q: flow rate, $P_m$: mud pressure, S: rock strength, Ab: rock abrasiveness, E: rock hardness, $P_p$: pore pressure, T: temperature, Pe: rock permeability, $C_C$: cutting properties, $A_{well}$: wellbore cross-sectional area.

summarizes the different controllable and uncontrollable factors applied in those empirical ROP models. Based on Table 1, the total frequency of those controllable and uncontrollable factors was calculated (Figure 6). Furthermore, the frequency of each individual factor in ROP models was quantified. The results are depicted in Figure 7.

Based on Figure 6, only 27% of the investigated factors belong to the uncontrollable category. Hence, in comparison to the controllable factors, the uncontrollable factors are required to be further included in future ROP models.

In addition, according to Figure 7, it can be seen that WOB and RPM are the controllable factors with highest frequency. Furthermore, rock strength (S) is the most frequent uncontrollable factor applied in the models. By contrast, flow rate (Q), rock hardness (E), cutting properties (Cc), temperature (T), and permeability (Pe) are the parameters with the least repetition in the models. Amongst these parameters, only the flow rate is controllable while the rest are uncontrollable factors. Thus, the influence of these low-frequent parameters needs to be further investigated in future studies.

Apart from the contributing factors, the applicability of each ROP model is also of paramount importance. Such applicability can be considered from the perspectives of bit type and wellbore direction.

In drilling engineering, bits are commonly divided into two main groups: roller-cone bits and fixed-cutter bits. Figure 8 illustrates those types of bits. The ROP can be remarkably affected by the type of bit. For instance, the initial ROP of roller-cone bits is commonly the highest, especially for bits with large offset and long teeth. These kinds of bits are suitable for soft formations since the ROP decreases after the bit penetrates deeper rocks, especially into the harder layers [37]. On the contrary, the fixed-cutter bits destroy the rocks through a wedging-type fragmentation mechanism. Thus, the bottom-cutting angle and number of blades have predominant effects on the ROP per each revolution.

The second applicability is wellbore direction. In the past, most of the wellbores were drilled vertically; however, due to the emergence of new technologies, such as hydraulic fracturing, the drilling companies tend to drill wellbores directionally. Such directional wellbores access more hydrocarbon-bearing formations, thereby allowing the companies to increase total oil and gas production. For instance, from 2010 to 2021, as a consequence of widespread directional drilling in the US, the total number of completed wellbores decreased by 66%, and average drilling length declined by 30%; however, the production of crude oil increased by two times [73]. In 2021, only 19% of US wellbores were drilled vertically, while 81% of the wellbores had directional trajectories.

Table 2. Applicability of empirical ROP models in terms of bit type and well direction.

Reference	Bit Type		Well Direction
	Roller-Cone Bit	Fixed-Cutter Bit	Vertical	Directional
Maurer, 1962 [14]	×		×
Galle and Woods, 1963 [15]	×		×
Bingham, 1965 [16]		×	×
Bourgoyne and Young, 1974 [3]	×		×
Warren (perfect model), 1981 [6]	×		×
Warren (imperfect model), 1984 [7]	×		×
Reza and Alcocer, 1986 [64]	×	×	×
Hareland and Rampersad, 1994 [24]		×	×
Osgouei, 2007 [5]	×	×	×	×

demonstrates the applicability of different ROP models in terms of bit type and wellbore direction. According to this table, the majority of empirical ROP models have been developed for roller-cone bits. Moreover, only two models, specifically, Reza and Alcocer’s model and Osgouei’s model, can be applied for roller-cone bits and fixed-cutter bits. Regarding the wellbore direction, only Osgouei’s model is applicable for both vertical and directional wells. Therefore, any future investigation must consider the applicability of the proposed ROP models for directional wellbores.

3.1.3. Shortcomings and Limitations of Existing Empirical ROP Models

In the previous sections, different empirical ROP models, together with their required factors, were elaborated. Moreover, the frequency of each influential factor in existing ROP models was quantified (Figure 7). As mentioned earlier, four uncontrollable factors, specifically, rock permeability (Pe), temperature (T), cuttings concentration coefficient (Cc), and hardness (E), have not been sufficiently incorporated into the ROP models. Hence, in the current section, a brief debate is given to clarify why and how those factors can be deployed to improve future ROP models:

• Rock permeability represents the capacity of rock to allow fluid to move through the rock pores. In nature, such fluid is water, brine, oil, etc. During drilling operations, the bit cutters break the rock, and, simultaneously, the mud pressure prevents the pore fluid from flowing into the well space. In a microscopic perspective, the bit pushes the pore fluid back through two mechanisms [74]. In the first mechanism, the mud pushes back the pore fluid directly, thereby driving it to flow backwards. This mechanism is dominantly governed by the bit-rotation speed. In the second mechanism, the pores are compressed by bit cutters, and, consequently, the pore fluid is squeezed away from the bottom hole. This mechanism is mainly governed by rock diffusivity which is directly related to the rock permeability. Therefore, since the rock permeability changes, the ROP is altered [74]. Thus, it is suggested that more investigations on the relations between rock permeability and the ROP are conducted to improve the previous empirical models or to develop new ones;
• The second less-frequent factor is the coefficient of cuttings concentration which can be utilized as an indicator of the cuttings volume accumulated at the bottom hole. This factor is especially determining in the ROP during drilling of directional wells. As mentioned in the previous section, with the increasing demands for production from unconventional reservoirs, directional (horizontal and inclined) drilling has markedly increased; however, some features of past ROP models were considered only for vertical wells. Thus, it is essential to account for those features in ROP models developed for directional wellbores. The coefficient of cuttings concentration is such an essential factor.

During the drilling of directional wells, cuttings may be excessively deposited on the lower side of the wellbore. This leads to formidable changes, such as reduced weight transmitted to the bit, and intense friction between the drill pipe and wellbore wall. To consider this effect, the WOB must be modified in ROP models. For this purpose, the following formula is suggested [75]:

$$WO{B}_d=WO{B}_s\times {\mathrm{e}}^{-\mathsf{\mu }\times \mathsf{\theta }}$$

(45)

where $WO{B}_d$ (Ib) represents the modified WOB for directional drilling, μ indicates the coefficient of sliding friction, $WO{B}_s$ (Ib) is the measured WOB at the surface, and θ (radian) is the wellbore inclination angle. Generally, the $WO{B}_d$ is less than the $WO{B}_s$. If this difference is not applied for the ROP models developed for directional drilling, the accuracy of the model may decline to a great extent.

• The third parameter is rock hardness. Rock hardness can be defined as the rock resistance to drilling. In other words, rock hardness is reciprocal of drillability. Some researchers have linked hardness with drilling speed; other researchers related the hardness to the amount of energy required for cutting a unit volume of rock [76]. Rock hardness is mainly dependent on the hardness of the minerals, grain size, grain shape, grain distribution, and cementation material. The silica content of the rock greatly affects the rock hardness. Although other resistive features, such as compressive strength and rock abrasiveness, have been adopted more frequently in ROP models, the inclusion of rock hardness into ROP models seems to be necessary;
• Temperature is another factor influencing the penetration rate. The bit brecks the rock under a thermo–hydro–mechanical condition. During the drilling operation, the bit penetrates the deeper formations with different thermal conditions. The heat changes the poroelastic properties of the rocks as well as the characteristics of the pore fluid [63,64]. One of those important poroelastic parameters is Biot’s coefficient. When the temperature changes, Biot’s coefficient varies and has an impact on the effective stress applied on the rock at the bottom hole. As a matter of consequence, the rock compression or shear strength changes, thereby influencing the ROP. Hence, the impact of temperature should be regarded in ROP models, especially using inclusion of Biot’s coefficient as a temperature-dependent factor.

3.2. Data-Driven ROP Models

Empirical ROP models may deliver inaccurate results since they cannot take into account all contributing factors [77,78,79,80,81,82]. Therefore, a number of researchers made efforts to deploy AI techniques in ROP modeling works [83,84]. During drilling operations, a significant amount of data is gathered daily. Because of this, some petroleum engineers and researchers turned to AI methods to forecast the ROP. The AI algorithms allowed the engineers to thoroughly evaluate the gathered data and produce valuable insights. Such AI techniques are used in a variety of applications related to drilling issues [85,86,87,88,89,90,91].

So far, several AI approaches have been utilized in ROP modeling. Table 3 summarizes the outstanding data-driven ROP models established by different researchers. As it can be seen, the majority of those studies have applied the ANN approach. Moreover, a wide spectrum of input parameters has been deployed in AI-based models.

ANNs resemble how the brain uses its biological neurons to process information to learn features [92,93]. Every ANN model possesses at least three structural layers: input layer, hidden layer (intermediate), and output layer. The neurons are the fundamental building blocks found within each layer. Fundamentally, for making connections between the layers, firstly, the transferring functions are applied, and, then, suitable algorithms are deployed to train the model [94]. The connections between the neurons are accompanied by weights and biases [95]. The AAN algorithm has two main steps: training and testing. The input parameters are transferred by the hidden layer to the output layer. Then, the testing step is conducted to compare the predicted ROP values with the real field ROP data. The main disadvantage of the ANN algorithm is that it creates “black box” models, meaning that there is no way to express how the results were obtained. Another demerit is that a large number of variables must be set in the neural network to prevent potential overtraining [96].

The first ANN study for predicting the rate of penetration was provided by Bilgesu et al. [97]. Then, Anemangely et al. devised two hybrid ANN models to develop an applicable ROP model [33]. Another ANN-based model was developed by Elkatatny [34]. The results of such research were found to be highly accurate. Jahanbakshi developed an ANN model to predict the ROP in oil and gas wells. The model was reported as a strongly reliable estimator capable of predicting precise results. Furthermore, some researchers applied other ANN architectures, such as back propagation neural networks (BPNN) [29] and the extreme learning machine (ELM) [98,99], for ROP prediction. The ELM approach was reported as a more accurate estimator in comparison to the ANN model [98,99].

As well as the ANN approach, the SVM algorithm has been successfully adopted in ROP modeling tasks. SVM is a supervised machine-learning algorithm which uses an ε-insensitive loss function. Bodaghi et al. compared the performance of SVM with the BPNN algorithm in ROP prediction. It was found that the SVM model was more accurate than the BPNN model [100]. Abdulmalek et al. used the SVM technique for ROP prediction [37]. Furthermore, other SVM-based approaches, such as the least-squares support-vector regression (LS-SVR) model, have been applied in ROP modeling. In another research, Ahmed at al. compared the performance of ANN, ELM, SVM, and LS-SVR algorithms in ROP prediction for two on-shore wells in the Niger Delta. It was reported that the LS-SVR and SVM models showed better performance than the ANN and ELM counterparts; however, the testing time required for the LS-SVR and SVM algorithms was much higher than the ANN and ELM approaches [99].

Apart from the neural networks and SVM-based algorithms, some researchers have utilized hybrid approaches to predict the ROP in their works. As a newly conducted research, Sobhi attempted to establish an ROP predictive model based on the previous empirical ROP models [35]. In such research, several algorithms (fminsearch, fsolve, fminunc, lsqcurvefit, lsqnonlin ant colony optimization (ACO), and multiple regression) and different objective functions were applied. Moreover, Mantha and Samuel combined several AI methods with the statistical regression technique to estimate the ROP [36]. Furthermore, Hegde et al. applied some hybrid models for ROP modeling [38]. In addition, Yavari et al. employed neuro-fuzzy technique to analyze ROP data [39]. In a recent study, Gan et al. combined the SVM, mutual information analysis, and wavelet filtering to create a hybrid ROP model [40].

Selecting an appropriate AI algorithm for the purpose of ROP prediction can be a challenging task. In better words, there is no standard procedure or reference to choose the best AI algorithm for ROP prediction. For instance, although the linear regression is computationally simple, it is not reliable in case of the presence of delimited parameters [101]. Other examples are decision trees, which exhibit noticeable accuracy even in the presence of outliers and noise; however, their results may be affected by unwanted overfitting.

As shown in Table 3, the number and type of input parameters are very inconsistent in different researches. Thus, even if the most optimal AI algorithm is selected, the determination of the optimal number of input parameters remains still challenging. Moreover, the type of input parameters can remarkably affect the accuracy of the results predicted by the applied AI technique [102]. It is noteworthy that the drilling operations are executed in geologically different formations, such as sandstone, shade, limestone, etc. Hence, the in situ lithology can closely determine the selection of number and type of the input parameters. As the geological parameters may exhibit intense fluctuations with depth, the presence of outliers or noisy data is inevitable. Therefore, the capability of the selected AI algorithm in prevention of such impacts must be taken into account. Generally, the statistical regression techniques are more affected by noisy data compared to the neural networks. Nevertheless, neural networks are more sensitive to noisy data when compared to SVM-based models.

Even more than this, the user-entered ratio of training data to the testing data may affect the results [102]. The optimal ratio may be obtained by the trial-and-error process which is time-consuming and inconsistent, depending on different cases.

Table 3. Some major studies related to ROP prediction by AI techniques.

Reference	AI Technique	Input Parameters
Bilgesu, 1997 [97]	ANN	drillability; formation abrasiveness; rotary time; bearing wear; torque; tooth wear; WOB; pump rate; RPM.
Moran, 2010 [103]	ANN	formation strength; formation abrasiveness; bit weight; RPM; drilling fluid weight; rock gense.
Jahanbakhshi, 2012 [29]	ANN	mud type; pressure differential; hydraulic power of the bit; hydraulics; bit wear; depth; RPM; pump pressure; WOB; formation density; bit type; ECD; 10 min gel strength of drilling fluid; wellbore diameter; early gel strength of drilling fluid; drillability; drilling fluid’s yield point; permeability; drilling fluid’s plastic viscosity; porosity.
Amar and Ibrahim, 2012 [79]	ANN	pore fluid pressure; RPM; depth; ECD; Reynolds number function; tooth wear; WOB.
Alarfaj et al., 2012 [104]	ANN	Reynolds number; depth; WOB; RPM; tooth wear; gradient of pore pressure; ECD.
Cui et al., 2014 [105]	ANN	apparent viscosity; unconfined compressive strength; mud density; RPM; bit geometry; WOB; bit type; gross hours drilled; drillability constant.
Bodaghi, 2015 [100]	SVM	viscosity; mud weight; tooth wear; pump rate; bit geometry; formation; deviation of well; RPM; depth.
Mantha and Samuel, 2016 [36]	Hybrid system	RPM; flow rate; WOB.
Shi et al., 2016 [99]	ANN, ELM	mud properties; formation; RPM; geomechanical characteristics; WOB; hydraulics.
Amer, 2017 [106]	ANN	WOB; mud weight; bit gense; mineralogy; IADC codes; drill-pipe pressure; bit size; mud pump; bit condition; torque; depth; RPM; TVD.
Abdulmalek et al., 2018 [37]	SVM	yield point; WOB; solid; RPM; funnel viscosity; flow rate; plastic viscosity; standpipe pressure; mud density; torque.
Yavari et al., 2018 [39]	Hybrid system	WOB and RPM.
Anemangely, 2018 [33]	Hybrid system	RPM; WOB; shear wave slowness; compressional wave slowness; flow rate.
Ahmed et al., 2018 [99]	ANN, ELM, SVM, and LS-SVR	depth; flow rate; WOB; RPM; torque; standpipe pressure; mud weight; bit diameter.
Hegde et al., 2018 [38]	Hybrid system	RPM; UCS; flow rate; WOB.
Gan et al., 2019 [40]	ANN	seismic velocity; depth; torque; WOB; RPM; drillability; depth; mud density.
Elkatatny, 2021 [107]	Hybrid system	UCS; WOB; drill-pipe pressure; RPM; flow rate; torque; mud density; gamma ray; bit design; total flow area.
Lawal, 2021 [108]	ANN	density; porosity; point load index.
Mahdi, 2021 [98]	ANN	flow rate; WOB; RPM; bit diameter; standpipe pressure.
Sobhi et al., 2022 [35]	Hybrid system	depth; Reynolds number; WOB; tooth wear; ECD; RPM.

Table 3. Some major studies related to ROP prediction by AI techniques.

Reference	AI Technique	Input Parameters
Bilgesu, 1997 [97]	ANN	drillability; formation abrasiveness; rotary time; bearing wear; torque; tooth wear; WOB; pump rate; RPM.
Moran, 2010 [103]	ANN	formation strength; formation abrasiveness; bit weight; RPM; drilling fluid weight; rock gense.
Jahanbakhshi, 2012 [29]	ANN	mud type; pressure differential; hydraulic power of the bit; hydraulics; bit wear; depth; RPM; pump pressure; WOB; formation density; bit type; ECD; 10 min gel strength of drilling fluid; wellbore diameter; early gel strength of drilling fluid; drillability; drilling fluid’s yield point; permeability; drilling fluid’s plastic viscosity; porosity.
Amar and Ibrahim, 2012 [79]	ANN	pore fluid pressure; RPM; depth; ECD; Reynolds number function; tooth wear; WOB.
Alarfaj et al., 2012 [104]	ANN	Reynolds number; depth; WOB; RPM; tooth wear; gradient of pore pressure; ECD.
Cui et al., 2014 [105]	ANN	apparent viscosity; unconfined compressive strength; mud density; RPM; bit geometry; WOB; bit type; gross hours drilled; drillability constant.
Bodaghi, 2015 [100]	SVM	viscosity; mud weight; tooth wear; pump rate; bit geometry; formation; deviation of well; RPM; depth.
Mantha and Samuel, 2016 [36]	Hybrid system	RPM; flow rate; WOB.
Shi et al., 2016 [99]	ANN, ELM	mud properties; formation; RPM; geomechanical characteristics; WOB; hydraulics.
Amer, 2017 [106]	ANN	WOB; mud weight; bit gense; mineralogy; IADC codes; drill-pipe pressure; bit size; mud pump; bit condition; torque; depth; RPM; TVD.
Abdulmalek et al., 2018 [37]	SVM	yield point; WOB; solid; RPM; funnel viscosity; flow rate; plastic viscosity; standpipe pressure; mud density; torque.
Yavari et al., 2018 [39]	Hybrid system	WOB and RPM.
Anemangely, 2018 [33]	Hybrid system	RPM; WOB; shear wave slowness; compressional wave slowness; flow rate.
Ahmed et al., 2018 [99]	ANN, ELM, SVM, and LS-SVR	depth; flow rate; WOB; RPM; torque; standpipe pressure; mud weight; bit diameter.
Hegde et al., 2018 [38]	Hybrid system	RPM; UCS; flow rate; WOB.
Gan et al., 2019 [40]	ANN	seismic velocity; depth; torque; WOB; RPM; drillability; depth; mud density.
Elkatatny, 2021 [107]	Hybrid system	UCS; WOB; drill-pipe pressure; RPM; flow rate; torque; mud density; gamma ray; bit design; total flow area.
Lawal, 2021 [108]	ANN	density; porosity; point load index.
Mahdi, 2021 [98]	ANN	flow rate; WOB; RPM; bit diameter; standpipe pressure.
Sobhi et al., 2022 [35]	Hybrid system	depth; Reynolds number; WOB; tooth wear; ECD; RPM.

4. Discussion

Over the past few decades, drilling engineers have dealt with ROP prediction to heighten drilling efficiency [109]. More complicated wells must be drilled since access to hydrocarbon reservoirs is getting harder due to the deeper formations and increasing geologically problematic conditions. In order to increase drilling efficiency, precise prediction of the ROP has become more critical. This allows drilling engineers to accurately evaluate associated expenditure, required time, and proper phasing of the drilling operations.

Since the relationships between the factors affecting the ROP are quite complicated, a number of empirical and data-driven ROP predictive models have already been developed. Each ROP model has its own advantages and disadvantages. Therefore, there is no unique ROP model which is applicable for all circumstances. The empirical ROP models were developed mainly based on field experience and observations. Hence, the working experience of researchers influenced the number and type of factors considered in the empirical ROP models. On the opposite side, although powerful AI techniques, such as ANN and SVM algorithms, formulate the ROP without prior assumptions, they are highly sensitive to the number and type of input parameters. Thus, one potential solution is to develop formation-classified ROP models in which the number and type of contributing factors are determined for specific rock classes. In the oil and gas industry, the main rock types are sedimentary formations, including sandstones, shales, limestones, dolomites, and evaporated rocks [110]. If such formation-based ROP models are developed, the ROP values can be accurately predicted using the minimum number of effective parameters. For instance, rock abrasiveness can be classified as a highly influencing factor in ROP predictive models for sandstones, but not for evaporative rocks. By contrast, rock creep is of paramount importance in ROP models for evaporative rocks, but not for sandstone formations.

Drilling a directional well costs from 1.4 to 3 times more than a vertical well. Hence, for directional drilling operations, the ROP prediction must be calculated more carefully. Among the empirical ROP models, only Osguei’s [68] considered the effect of wellbore inclination on the ROP. On the other hand, since the study refers to 15 years ago and drilling technology has advanced remarkably since then, the inclusion of wellbore inclination in current ROP models is of paramount urgency.

Future ROP models need further inclusion of uncontrollable factors, such as rock permeability, wellbore inclination, temperature, and rock hardness. In addition to these rock characteristics, the impact of discontinuities on the ROP can be further investigated. Fowell and Mcfeat-Smith in Ref. [111] discovered that when joint spacing reduces, the penetration rates rise. Moreover, Willbur in Ref. [112] proposed a classification to quantify the drillability of different rocks based on the rock composition and the characteristics of discontinuities. Also, Hoseinie et al. in Ref. [50] created a categorization system based on the texture and grain size, Mohs hardness, UCS, joint spacing, joint filling, and joint dipping to evaluate the drillability of different rock masses.

5. Conclusions

In this research, an inclusive survey on existing ROP models was conducted. The focus was on the elaboration of up-to-date investigations, current shortcomings, and future requirements. To do this, firstly, the contributing factors on the ROP were explained. Then, the different empirical and data-driven ROP models were elaborated, and their input parameters, assumptions, applicabilities, and drawbacks were described. Assessment of existing ROP models revealed that only 27% of the investigated contributing parameters were uncontrollable factors. Those uncontrollable factors are mostly related to the geological characteristics of the rocks. Furthermore, the frequency analysis of the input parameters showed that four uncontrollable factors, including rock permeability, wellbore inclination, rock hardness, and temperature, have not been adequately incorporated into the ROP predictive models. Among these factors, the wellbore inclination is more critical as the majority of the today’s wellbores are drilled directionally. Hence, the modification of previous ROP models for inclined wellbores appears to be much needed. For this purpose, different experimental and numerical modelings should be carried out to investigate the ROP variations for inclined wellbores. In such inclined wellbores, the bottom-hole cleaning parameters, such as the coefficient of cuttings concentration and modified directional WOB, must be studied.

Generally, empirical ROP models are limited by different assumptions. Moreover, they may need experimental coefficients which can affect the results. Comparing the empirical and AI-based ROP models, it can be said that the application of data-driven models can markedly enhance the accuracy of the predicted ROP values. These models consider no operational assumption in their computational processes. Furthermore, numerous controllable and uncontrollable variables can be imported to such models. However, different AI techniques, such as ANN, SVM, decision tree, etc., may display different accuracy levels, especially for different lithology conditions. As a matter of fact, lithology is remarkably influential in the quality of the data gathered during the drilling operations; in some lithology, the different geological parameters can be accompanied by different levels of noisy data and outliers. Thus, the sensitivity of different AI algorithms to noisy data and outliers must be considered during the development of data-driven ROP models.

The size and type of the input parameters can also impact the accuracy of data-driven ROP models. To address this issue, it is suggested to develop a formation-based classification system in which the type of the contributing factors is determined for drilling different formations. Such formation-based classification systems have already been developed for geomechanical investigations. For instance, the rock mass rating (RMR) system was proposed by Bieniawski for the excavation of tunnels in different rocks [113]. Another illustration is the Q-System developed by Bartun et al. for the design of underground excavations [114]. Both RMR and the Q-System are in the form of tables in which different geological conditions of rocks have been quantified. Therefore, drilling/geomechanics engineers can convert the qualitative geological properties into quantitative numbers. Those numbers are then used for classification of rocks.

The findings of this research are quite practical for the future investigations on ROP prediction. Moreover, it benefits the companies and organizations dealing with drilling operations in petroleum engineering, mining engineering, space engineering, and water-well drilling.