CFD modeling of the flow behavior around a PDC drill bit: effects of nano-enhanced drilling fluids on cutting transport and cooling efficiency

ABSTRACT In the present study, hydraulic and thermal performances of nano-enhanced drilling muds have been studied across the complex geometry of a rotating polycrystalline diamond compact (PDC) drill bit using computational fluid dynamics techniques. Toward this goal, a robust converging procedure is applied to generate an advanced tetrahedral/prism grid structure using special techniques. We also utilize a combination of the realizable k-ԑ turbulence model with enhanced wall treatment and Eulerian-Lagrangian discrete phase model (DPM) to investigate the flow field and particles tracking across the bit. The results reveal that the rheological properties enhancement obtained by adding even a small quantity of nano-fluids leads to a remarkable increase in the cutting transport efficiency. According to our results, CuO and ZnO nano-enhanced muds show up to 127% and a 68% rise over the cutting transport ratio of the base fluid at the bit rotational speed of 30 rpm. Besides, the predicted temperature on tip of the bit cutters indicates that increasing the thermal conductivity of nano-enhanced muds can lead to effective cooling only in conjunction with improving rheological characteristics. CuO NWBMs show up to a 16.7% reduction in the temperature of the front surface of bit cutters over the base fluid.


Introduction
In a drilling operation, the bit hydraulic and cooling have great importance and carry significant costs for industries. In practice, a great amount of frictional heat is generated through the bit cutters and the rock interaction in polycrystalline diamond compact (PDC) drill bits. Thus, an efficient evacuation of this generated heat by drilling fluid is crucial to prevent drilling failure due to the thermal degradation and accelerated wear of cutters. The surface temperature of the bit cutters depends on rock formation composition and down-hole formation temperature along with the bit operating conditions, such as rotary speed, bit penetration rate, weight on bit, and cooling fluid properties. At temperatures above 350°C, thermal softening and degradation occur which are responsible for thermally accelerated wear of the PDC bit cutters (Ayop et al., 2020;Glowka, 1989). Furthermore, drilling fluids have a major role in the cutting evacuation and the bottom hole cleaning process. Failure of drilling fluids to perform these major functions (bit cleaning and cutter cooling) causes a reduction in drilling rate, redundant energy consumption, reduced bit life, and CONTACT Bahram Dabir drbdabir@aut.ac.ir therefore, excessive drilling costs (Bourgoyne et al., 1986;Glowka, 1983). The rheological and thermal properties of drilling fluids exert a major influence on flow characteristics, hydraulic performance, and cooling efficiency of the drill bit (Tomren et al., 1986;Walker & Li, 2000). In recent years, it has been found that nanoparticles can be used as an effective additive for drilling fluids as they can improve heat transfer and bit cooling via their enhanced thermal conductivity compared to the base fluids. Besides, lots of research studies have been carried out to indicate the improved rheological properties of nano-enhanced drilling fluids and their better performance in the bottom hole cleaning process (Agarwal et al., 2011;Amanullah et al., 2011;Lee et al., 2009;Mohammadzadeh et al., 2016;Quintero et al., 2014;Samsuri & Hamzah, 2011;Vryzas & Kelessidis, 2017). The main contributions of these works are briefly described below: Agarwal et al. (2011) did an experimental investigation on the rheological properties and the stability of the invert emulsion oil-based drilling muds containing nanosilica and nanoclays. They showed that the stabil-ity and gel formation capacity of the fluid are improved by using nano-particles as emulsion stabilizers in place of polymer surfactants. The formulation of multiple nanobased drilling fluids has been described by Amanullah et al. (2011). Their primary test results show suitable rheological properties including gelling characteristics for newly developed nano-based drilling muds, and these enhanced properties enable fulfilling the functional task of drilling muds during the operation and after cessation of drilling. Samsuri and Hamzah (2011) investigated the performance of water-based muds, enhanced with Multiwall Carbon Nanotubes (MWNTs), on the lifting capacity of drilled cuttings. Their results show improved viscosity and enhanced lifting capacity for drilled cuttings when the amount of MWNTs increases.
Related to the role of nanoparticles in improving the viscosity of drilling fluid, Lee et al. (2009) studied the application of magnetic nanoparticles for tunability of the drilling fluid viscosity. Besides, Baghban et al. (2019) developed a comprehensive model for estimating the relative viscosity of nano-enhanced fluids based on variables of temperature, density, viscosity of the base fluid, volume fraction, and diameter of nanoparticles. Moreover, William et al. (2014) conducted some experiments to study the enhancement of the rheological and thermal properties of nano-drilling fluids resulting from adding CuO and ZnO nano-particles to water-based drilling muds at several temperatures. Their results show a 35% enhancement in the thermal properties of nano-water-based muds compared to the base fluid.
On the other hand, behavior modeling of nano-fluids using computational fluid dynamic (CFD) techniques is known to be extremely beneficial, especially in applications that experimental analysis is difficult or even impossible. In this direction, for example, Ghalandari et al. (2019) conducted a CFD simulation of the nano-fluids inside a root canal and obtained beneficial information and an appropriate insight into the flow characteristics during irrigation in the root canal. Mohammadzadeh et al. (2016) applied a CFD model to study the effects of nano-particles as viscosity modifiers on the cutting transport capacity of the fluid within the wellbore. Vryzas and Kelessidis (2017) performed a comprehensive review of the nano-based drilling fluids. They studied various nano-particles used to improve the rheological and filtration characteristics of drilling fluids. They demonstrated that a significant improvement in the rheological behavior and the thermal conductivity of drilling fluids can be achieved using nano-particles in drilling fluids.
Notably, most of the studies in the literature on CFD simulation of the drilling process are focused on the hydraulic performance and cutting transport efficiency in the annulus (Akhshik et al., 2015;Bilgesu et al., 2002;Epelle & Gerogiorgis, 2017, 2018a, 2018bHeydari et al., 2017;Nazari et al., 2010;Ofei et al., 2014;Pang et al., 2018;Rooki et al., 2014). Epelle and Gerogiorgis (2017) presented a CFD modeling to simulate the cutting removal in the annulus by using the Eulerian-Eulerian multiphase flow model under steady-state, laminar and isothermal conditions. They showed that fluid velocity, annular eccentricity, and hole inclination have major effects on cuttings' transport efficiency. In a subsequent study of Epelle and Gerogiorgis (2018a) the velocity and volume fraction profiles of the cuttings in the annulus were investigated under turbulent and transient conditions. In another work, they performed a CFD analysis on the effects of particle sphericity on the cuttings transport in the annular bends (Epelle & Gerogiorgis, 2018b). They demonstrated that particle deposition is more likely to occur at the inclined to vertical bend, and observed that non-spherical particles are conveyed with higher velocities.
On the other hand, among a limited number of works that have investigated the flow field around the drill bit, it has been assumed that the drilling fluid has a Newtonian behavior (Garcia-Gavito & Azar, 1994;Kirencigil, 2017;Moslemi & Ahmadi, 2014;Watson et al., 1997). Besides, an isothermal condition has been considered in the previous investigations, and for some of them, PDC cutters have been neglected for simplifying the geometry. These simplifying assumptions have notable effects on hydraulic performance and flow pattern predictions using CFD techniques (Garcia-Gavito & Azar, 1994;Glowka, 1983;King et al., 1990;Moslemi et al., 2015;Moslemi & Ahmadi, 2014;Watson et al., 1997). To the best of our knowledge, despite of the importance of cooling efficiency of drilling fluid on the bit face, no attention has been paid to heat transfer aspects in previous CFD simulations around the drill bit.
In this direction, Moslemi and Ahmadi (2014) and Moslemi et al. (2015) performed a CFD analysis on the flow behavior around the stationary drill bit under the isothermal condition and Newtonian behavior assumption. With these assumptions, they investigated the flow behavior and cutting transport efficiency at various drill bit hydraulic design parameters such as the fluid jet velocity, nozzles' arrangement and diameter, and the bit body profile. Watson et al. (1997) presented experimentally validated results for the flow pattern around a stationary PDC drill bit at room temperature and pressure and using water as the drilling fluid.
Among new articles related to this topic, Minakov et al. (2021) and Al-Yasiri et al. (2020) conducted a numerical investigation to study the effects of adding nanoparticles to the drilling fluid on the cutting transport efficiency. They showed that adding SiO 2 nano particles results in a rise in cutting transport capacity by 2.7 times. Busch and Johansen (2020) investigated the effects of rotation of the drill pipe along with the lateral motion of the drill string on the cutting transport capacity using CFD techniques and dynamic mesh capabilities of Ansys Fluent software. Zhu et al. (2021) proposed the pulsed drilling fluid injection to improve cutting transport capacity in the inclined wellbores and estimated the optimal pulse parameters. Shao et al. (2019) applied a CFD-DEM model to investigate non spherical cuttings transport efficiency during coal rock drilling.
To summarize the challenges and major difficulties associated with the investigation of cutting transport and cooling efficiency of drilling fluids during the process, we remind that among the few research works that investigated the flow behavior around the drill bit, to the best of our knowledge, they have dropped complexity of the geometry and assumed that the drilling fluids behave like a Newtonian fluid. In the present work, to overcome these important challenges, first we generate a high-quality mesh structure that can make a converged solution even under non-Newtonian behavior of the fluid. Moreover, the converged solution obtained in the isothermal condition is used as the initial solution to obtain a converged solution under the condition of variation of temperatures. Finally, a mesh-independent converged solution is obtained under non-isothermal condition for the hydraulic behavior cooling efficiency of non-Newtonian drilling fluids around the real geometry of a typical PDC drill bit.
In this study, we aim to propose a novel and reliable procedure for CFD modeling of fluid behavior around a typical PDC drill bit considering the complications associated with the fluid nature and the geometrical structure. The CFD technique is used to investigate the improvement in hydraulic performance analysis and cooling efficiency of nano-enhanced drilling fluids. For this purpose, we investigate the effects of this enhancement on thermal and rheological characteristics of ZnO and CuO nano-enhanced water-based mud (NWBM) on the bit cutter's cooling and cutting transport ratio for a stationary and rotating drill bit, at the rotational speed of 10-70 rpm. The contributions of this paper are described as follows: First, we study the theory behind the model and present the rheological and physical properties of ZnO and CuO nan-enhanced drilling muds. Next, we detail the description of preparing geometry and present a step-by-step procedure for the mesh generation, which makes the results reproducible. Then, boundary conditions and solution setup are presented. Finally, the numerical considerations that ensure the convergence and reliability of the solution are described. We highlight that the presented robust procedure for CFD modeling of flow around the drill bit provides valuable information and insights regarding the efficiency of the bit cooling and hole cleaning process before performing extra expensive and time-consuming experiments or suffering in real scale field operations. We note that the experimental verification of conclusions of this paper is extremely interesting, and we leave this direction as a future study.

CFD model description
For predicting the hydraulic performance of a typical PDC drill bit and flow field simulation around the drill bit, ANSYS-FLUENT software is used and Discrete Phase Model (DPM) as the particle tracking method is deployed for investigation of the bit cleaning efficiency of different fluids with various rheological properties. For this purpose, the cutting transport ratio (C t ) is defined as the ratio of the particle's average velocity to the fluid average velocity at a particular cross section of the flow (Bourgoyne et al., 1986;Sifferman et al., 1974). This cross section is considered to be the outlet of the computational domain in the present work. A larger C t corresponds to a better bit cleaning efficiency. Realizable k − ε turbulence model with enhanced wall treatment along with the discrete phase model (DPM) for particle tracking are used here. In this model, Lagrangian trajectory calculations considering inertia of the discrete phase, hydrodynamic drag, and the gravity force are conducted for particles as a dispersed phase (Fluent, 2017a).

Flow field equations
Conservation equations for mass and momentum in conjunction with heat transfer equation and transport equations for turbulence modeling are solved numerically using finite volume formulation (Fluent, 2017b).
Continuity and momentum equations. General form of the continuity equation for steady state flow without mass transfer between phases is as follows: ρ is the fluid density and v stands for the velocity vector. Conservation of momentum under the steady state condition is described by: where p is the static pressure and = τ is the stress tensor which is described by the rheological model. ρ g and F represent the gravitational body force and external body forces, respectively.
Energy equation. The steady state energy equation in the fluid domain has the following form where k eff is the effective thermal conductivity (summation of molecular and turbulent conductivity), and J j is the diffusion flux of species j. S h represents volumetric heat sources.
where sensible enthalpy h for incompressible flows is defined as The pressure work and kinetic energy are neglected for solving incompressible flows. In solid regions, the steady state energy equation has the following form: Convective energy transfer resulting from the rotational motion of the solids is represented on the left-hand side of Equation (6). The velocity field v is determined from the motion specified for the solid zone. For a stationary bit, this term does not exist. Turbulence modeling. For closure of equations of motion under turbulent condition, realizable k − ε model, proposed by Shih et al. (1994), is applied for turbulence modeling of the flow around the stationary drill bit. This model considers the effect of swirl on turbulence and accounts for both high Reynolds number and low Reynolds number effects and includes strong streamline curvature, vortices, and rotation effects. For better convergence in presence of rotation, RNG k − ε (Orszag et al., 1993) is applied for turbulence modeling around the rotating drill bit. The enhanced wall treatment is applied for near-wall modeling which is a combination of a two-layer model with enhanced wall functions. This model is able to resolve coarse and fine near-wall meshes in the viscous sublayer, intermediate, and log-law regions, simultaneously (Fluent, 2017b). Therefore, the first layer mesh height is not restricted by the limitations imposed by near-wall modeling method.

Particles' motion
Euler-Lagrange approach of discrete phase model is applied for tracking the particles' motion in the flow field around the drill bit. Particle-particle interaction is neglected during particles' motion simulation, since cutting particles occupy a low volume fraction in the computational domain (Fluent, 2017b;Moslemi & Ahmadi, 2014). Besides, the discrete phase model applied for particle tracking in this study is a 'one-way coupling' in which particles' motion does not affect the flow field, and the discrete and continuous phases do not exchange momentum and heat. The value of solid volume fraction during this set of computational analyses, for various mud samples at different rotational speeds, is observed to be 2.73-6.84%. As the volume fraction of the cuttings in the computational domain around the drill bit is less than 10%, applying the one-way coupling is justified (Fluent, 2017b).
The trajectory of particles in the discrete phase model is calculated by integrating the force balance on the particles which is written in a Lagrangian reference frame as follows (Fluent, 2017b).
The first term on the right-hand side of Equation (7) represents the drag force on the particle and the second one stands for the gravity term. u p is the particle velocity and u is the fluid velocity. μ represents the molecular viscosity of the fluid, ρ is the fluid density, and ρ p is the particle density. d p is the diameter of particles and Re is the relative Reynolds number defined as (Fluent, 2017b): Drag coefficient, C D , for smooth spherical particles can be calculated as: a 1 , a 2 , and a 3 are constants given by Morsi and Alexander (1972). F represents additional forces acting on the particles which are important at certain conditions. Virtual mass force ( − → F v ) is one of these forces which is essential to accelerate the fluid surrounding the particle. This force is quantified as: The other force resulting from the pressure gradient ( − → F p ) in the fluid is given by: Also, for the rotating drill bit, another force term arises due to the flow modeling in the moving reference frame. These forces for rotation around the z-axis, for a given rotational speed (rpm), are defined below in the x and y directions ( The thermophoretic force is neglected in this study because it is only important for small particles suspended in a gas fluid which is not the case here. Furthermore, lift and Brownian forces are not considered as they matter for sub-micron particles, and are negligible for particles under investigation since they are mostly larger than one millimeter.
In the present work, the random walk model is applied to calculate the dispersion of particles resulting from the turbulence in the continuous phase based on the fluctuations of instantaneous turbulent velocity.

Fluid rheology and physical properties
Drilling fluids are non-Newtonian fluids showing high viscosity in the low-shear values, high yield stress, and strong shear thinning character; making them suitable for efficient transportation of drill cuttings during the operation and suspending them effectively in the stationary phases of the process. The Herschel-Bulkey model, which incorporates yield point into the power-law model, is often used to describe the rheological behavior of oil well drilling fluids (Briscoe et al., 1994;Kelessidis et al., 2006;Kelessidis et al., 2009;Khataniar et al., 1994;Kök et al., 2000;Rooki et al., 2012;Sayindla et al., 2017). The Herschel-Bulkley model considers both the yield point and shear thinning behavior of drilling fluids and is the best-fit model for the rheological behavior of the drilling mud. The Herschel and Bulkley (1926) model is given by: Which τ 0 , K, and n represent the yield stress, fluid consistency index, and fluid behavior index, respectively. These A water-based mud with density 1020kg/m 3 , specific heat C p = 4186J/kg · K, and thermal conductivity k th = 0.485W/m · K is used as the base drilling fluid. The rheological and thermal properties of the base fluid and nano-enhanced samples, presented in Table 1, are taken from the experimental results performed by William et al. (2014). The nano-enhanced mud samples, i. e., C.1 to C.3 and C.4 to C.6, contain 1% (by volume) of nanofluids of ZnO and CuO with concentrations of 0.1, 0.3, and 0.5 wt%, respectively.

Geometry
The geometrical model of a typical 244mm diameter PDC drill bit with 6 blades and 6 equal-size nozzles of diameter 22mm, which includes 20 diamond cutters precisely located on steel blades of the bit is developed and modified for CFD simulation through the computeraided design methods. The front and side views of the PDC bit are illustrated in Figure 1. The bottom-hole diameter is considered 248mm with about 2mm offset from the blades to realistically model the cuttersformation engagements. The height of the wellbore under investigation is considered 0.6m. Geometrical characteristics, fluid flow properties, and other simulation input parameters are summarized in Table 2. We note that in presence of rotation, the hole diameter is increased to 256mm to prevent the engagement between the rotating parts of the bit and the stationary bottom hole wall. Besides, in presence of rotation, the flow rate of the mud is increased to 570gpm to prevent reduction of flow velocity and the reverse flow due to larger cross section of the flow.
The drill bit body faces including blades and internal walls are thermally coupled interfaces between the bit body and the drilling fluid, see Figure 1. Various parts of the geometry including mud inlet, outlet, external walls  (bottom hole walls), and 180°rotational periodic boundaries are depicted in Figure 2. The pipe and annulus sections connected to the bit are added to prevent the reverse flow in the outlet. The engagement part between the cutters and the bottom-hole wall on top of each cutter is shown in Figure 3. Heat flux generated due to the cutter-formation interaction is uniformly distributed on the front section of this part, and is specified as S1 in Figure 3. The interface parts (between the fluid and cutters) in the front and side faces of each cutter, and the interface between cutters (diamond solid) and the bit body blades (steel solid), are illustrated in Figure 3 as S2 and S3, respectively.

Mesh generation
The computational fluid domain consists of passages across pipe inlet, nozzles, flow around the drill bit, and returning fluid to the outlet through the annulus. Grid generation for such a complex geometry containing cutters and the narrow gap between cutter blades and the bottom hole wall is not a trivial task. The grid system including hybrid prim/tetra grids should handle the flow in the boundary layer and have an acceptable quality to prevent the divergence during the numerical solution. For this purpose, a precise procedure is applied to generate an efficient and robust grid structure around the PDC drill bit by using the capabilities of ICEM-CFD package in ANSYS workbench.
For obtaining a high-quality mesh with sufficient size and density, the global scale factor and maximum element size are set to 0.000125 and 64, respectively. The local maximum element sizes for different parts are assigned according to Table 3, which corresponds to the part names given in Figures 1 and 2. A volume mesh is generated using the top-down approach by the robust (Octree) method. Since it is difficult to smooth prism elements, it is desirable to start prism meshing with the best possible quality of the triangular surface mesh. Therefore, the created surface elements in the previous step are smoothed using Laplace smooth method, which significantly improves the mesh. In the next step, a bottom-up approach is applied to fill the volume mesh properly.    Thermal boundary condition C1-C20 S1 Heat flux = 2.6362e+7 w/m 2 S2 Thermally coupled interface wall (between fluid and cutters) S3 Thermally coupled interface wall (between bit body and cutters) S4 Heat flux = 0 After filling the volume with high-quality tetra grid structures, prism layers are created for capturing boundary layer flows. Prisms should preserve enough quality in sharp angles along with smooth transition between prism layers and between prisms and adjacent tetras. To do so, a single layer of prisms is generated to prevent collision between prism layers in narrow gaps, and this layer is then divided and redistributed into 3 layers. Once the mesh is generated, after several steps of smoothing, the final mesh consists of 4,157,000 tetra/prism elements in the fluid domain with minimum mesh quality of 0.04 which is acceptable for such complex geometry. The schematic view of the created mesh is illustrated in Figure 4.
Moreover, the created mesh in the solid domain of the bit body is illustrated in Figure 5. The mesh structure in this region consists of tetra elements notably coarser than the fluid domain. The number of elements in the bit body region is about 768,000 tetra elements. Furthermore, a hexahedral mesh including about 40,000 elements is generated in each cutter which is shown in Figure 3.

Boundary conditions and solution set up
Flow and thermal boundary conditions are presented in Tables 3 and 4. Surface boundaries are specified in Figures 1-3. Conjugate heat transfer is solved to consider the coupled heat transfer in the interfaces between the solid and fluid regions. The thermally coupled interfaces provide equal heat transfer and temperature on the surfaces at each cell zone. The estimated value for the heat flux exerted to the surface of cutters in contact with rock surface (S1 in Table 4) has been extracted from the work by Che et al. (2015) on the heat transfer analysis in PDC cutters in rock turning processes.
In the first series of numerical tests, the heat transfer equation is inactive, and under the isothermal condition, the flow field calculation is performed which is an initial solution for the conjugate heat transfer problem. For simulation of rotating drill bit, the rotational speed is increased step-by-step from 10 to 70 rpm. The SIMPLE algorithm (Semi-Implicit Method for Pressure-Linked Equations) for pressure-velocity coupling (Patankar & Spalding, 1972) is used. Also, firstorder upwind spatial discretization is used for momentum, turbulent, and energy equations; and for pressure, PRESTO (PREssure STaggering Option) method is applied.
After obtaining a converged non-isothermal solution; for particle tracking, an uncoupled discrete phase method (DPM) is applied. Particles are injected uniformly with zero initial velocity from the bottom hole surface, and reflect boundary condition is used when a particle hits any wall. Particles' density, diameter, and flow rate are presented in Table 2.

Numerical considerations
A mesh-independence study of the pressure drop across the bit for the base fluid is performed. For this purpose, adaptive mesh refinement based on the velocity gradients is investigated. The results of the mesh size reliability analysis are presented in Table 5. As can be seen, the refined mesh elements in the fluid domain indicate lower than 1% change in the pressure drop across the bit. Therefore, the generated mesh using the above parameters provides a mesh-independent solution.
For convergence assessment, scaled residuals of continuity, velocity components, turbulent parameters, and energy equations are monitored. In the converged solution, scaled residuals reach 0.001 for continuity, and below 10e-4 for velocity and turbulent parameters, and below 10e-7 for the energy equation. These values for convergence criteria indicate the success of the generated grid structure to capture all the complexities associated with the geometry of the bit and guarantee the reliability of the solution. Furthermore, for tracking the solution at steady state condition, mass flow rate leaving the outlet, static pressure at the inlet, along with average temperature on front surfaces of some cutters are monitored. The results show that after about 2000 iterations, steady state solution is obtained and no considerable variation in the mentioned parameters is observed.  Moreover, the total mass balance and heat balance in each domain are checked showing that the total mass balance error is about 2.7e-7 kg/s, and total heat balance error is about 103 j/s (lower than 0.006% of the inlet heat to the domain) that confirm the reliability and convergence of the solution. The simulations are performed on a core i7-6800K CPU 3.40 GHz processor (6 cores and 12 threads) with 32.0 GB RAM.
In this setting, the steady state solution is obtained after about 6 h for each case. This simulation execution time does not change dramatically in different scenarios. This is because the main factor influencing the simulation time is the number of cells in the grid structure, which is fixed, and we only change the properties of drilling fluid in this sufficiently fine grid structure.

Results and discussion
In this section, the results of non-isothermal turbulent flow simulation of non-Newtonian drilling fluid around the PDC drill bit are discussed using CFD approach in ANSYS-Fluent with DPM particle tracking method. The outcomes of this research are presented for a water-based mud enhanced with ZnO and CuO nano-particles and the procedure remains helpful for any type of drilling fluids and different drill bit structures. We first validate the results for pressure drop across the bit. Afterwards, the cutting transport ratio and temperature distribution on the face of the bit cutters are studied for the stationary drill bit, and then the results are investigated in the presence of rotation.

Pressure drop around the PDC bit
The pressure drop across the drill bit is mainly a consequence of the drilling fluid acceleration through the bit nozzles. This pressure drop is expressed by the following well-known equation widely used in hydraulic calculations of drill bits (Bourgoyne et al., 1986;Eckel & Bielstein, 1951;Guo & Liu, 2011).
In this equation, p b is expressed in kPa and other parameters are in the SI unit system. As can be seen in this equation, pressure drop across drill bit is a function of the cross-sectional area of the bit nozzles, A T , drilling fluid density, ρ, and mud flow rate, q. C d is the bit nozzle discharge coefficient determined experimentally by several researchers. Value of 0.95 has been recommended for C d in the pressure drop calculation across the drill bits (Eckel & Bielstein, 1951). The pressure drop across the PDC drill bit investigated in the present work is calculated numerically through the CFD simulation of mud flow samples (C.0-C.6) around the drill bit. According to Equation (15), bit pressure drop is not affected by rheology, and then, it is expected that the bit pressure drop is the same for all mud samples in this study. In Figure 6, the results of numerical simulations for different mud samples are compared to the values predicted by  for the pressure drop. The average relative error of 3.5% from the equation value for the bit pressure drop is observed during the numerical simulations which is an indication of the accuracy of CFD calculation results and validation against the well-known equation for drill bit pressure drop determination.

Effects of nano-particles on the cutting transport ratio
Cutting transport ratio (C t ) is widely used as a measure for evaluating drilling fluid hydraulic performance (GhasemiKafrudi & Hashemabadi, 2015;Sifferman et al., 1974;Tomren et al., 1986). In fact, the cutting transport ratio shows how fast the cutting particles are evacuated and is an indication of the cleaning efficiency of the drilling mud. A better bottom hole cleaning will result in increasing the Rate of Penetration (ROP) and it has been shown that increasing the ROP causes decreasing the drilling cost (Bourgoyne et al., 1986;Moslemi & Ahmadi, 2014). The cutting transport ratio is calculated by ensemble averaging of fluid and cuttings velocity in the flow direction.

Cutting transport ratio enhancement in the stationary drill bit
The cutting transport ratio for nano water-based mud samples, including ZnO and CuO nanoparticles, predicted through the CFD simulations around the stationary drill bit are illustrated in Figure 7. These results are presented in terms of the rheological properties of the NWBM, including the fluid behavior and consistency indices (n and K).
According to Figure 7, rheology has a notable effect on the cutting transport ratio, such that C t increases with the reduction of fluid behavior index and with an increase in the fluid consistency index. In other words, based on this set of results we observe that when adding nanoparticles causes a reduction in the fluid behavior index (n) and increases the fluid consistency index (K), it causes an enhancement in the cutting transport ratio and subsequently improves the carrying capacity for the efficient bit cleaning. As shown in Figure 7, sample C.6 among CuO nanofluids and sample C.1 among ZnO nanofluids, which have the minimum n and maximum K among  the mud samples, show the highest value of C t by 30.1% for C.6 and 22% for C.1 improvement over the cutting transport ratio of the base fluid.
The lower values of n result in a more shearthinning and non-Newtonian behavior in the fluid. On the other hand, annular viscosity increases by enhancement of the consistency index (K) values, which consequently improves the hole-cleaning capacity. Nanoparticles make these rheological properties tuneable. In fact, nano-particles thanks to their very small size along with their extremely high surface to volume ratio enhance the rheological properties of bentonite drilling muds. This enhancement can be attributed to the formation of links between embedded nano-particles in the pore structures on the surface of clay particles. This consequently will promote a stronger complex network in the solution and result in rheological enhancement of the drilling mud (Rafati et al., 2018;Vryzas & Kelessidis, 2017).

Cutting transport ratio enhancement in the rotating drill bit
Effect of bit rotation on the cutting transport ratio and the corresponding maximum enhancement over the base fluid for mud samples ZnO and CuO nanofluids are shown in Figure 8. As shown in the figure, increasing the bit rotational speed results in a reduction in the cutting transport ratio which can be attributed to the fact that by increasing the bit rotational speed and subsequently intensified turbulency, the particles are trapped around the bit and consequently the cutting transport efficiency decreases. On the other hand, bit rotation creates vibration movement, which prevents cuttings to settle. As we see, in rotational speeds below 30 rpm, the vibration and turbulency created by the rotational motion of the bit have a dominant effect in transporting particles, and in higher rotational speeds, particles' trapping is dominant which causes a reduction in the cutting transport ratio.
This behavior is consistent with the results presented by Keshavarz Moraveji et al. (2017) and Ytrehus et al. (2018) for the effect of the rotational speed of the pipe on particles' transportation across the annular region. They showed that increasing the pipe rotational speed in low-angle wells causes a reduction in the cutting transport efficiency mainly due to the particles' trapping in the annulus. Besides, the particles' trapping is more likely to occur around the drill bit due to the complexity of the geometry, which is obvious through the presented results in this article.
Effects of rheology enhancement on the cutting transport ratio of NWBMs, as a result of adding nanoparticles to the base drilling fluid, in terms of fluid behavior index (n) and fluid consistency index (K) at different bit rotational speeds are illustrated in Figures 9-12. As shown in these figures, at all rotational speeds, C t increases with the reduction of the fluid behavior index and with the increase of the fluid consistency index. This behavior was also reported for the stationary drill bit in the previous section and is confirmed for rotating drill bit at different rotational speeds (10-70 rpm).
As can be seen in Figure 9-12, in the rotating drill bit similar to the stationary one, sample C.6 among CuO nanofluids and sample C.1 among ZnO nanofluids, which have the minimum fluid behavior index (n) and maximum fluid consistency index (K) show the highest values of C t at all rotational speeds. In addition, we note that CuO nano-enhanced water-based muds generally show higher values of C t . In Figure 8, the enhancement percentage of the cutting transport ratio relative to the base fluid for mud samples C.1 and C.6 has been added to the graphs. The maximum enhancement of the cutting transport ratio is observed at rotational speed 30 rpm with an increase of 127% for C.6 and 68% for C.1 over the base fluid. This remarkable increase in the cutting transport capacity of nanoenhanced drilling fluids shows that nanoparticles can effectively decrease the drilling costs and optimize the process.

Cooling efficiency enhancement in the stationary drill bit
The average temperatures of heated surfaces on tip of the cutters in contact with the rock surface are shown in Figure 13 for ZnO and CuO NWBMs, for the stationary drill bit. Here, we explored the temperature of 10 different cutters, which are presented in Figure 1. Each cutter has a surface that experiences the highest temperature due to the contact with the rock. These surfaces are presented as S1 in Figure 3 and the average temperature of them is presented in Figure 13. According to Figure 13(a), C.1 and C.3 show a lower temperature in comparison with other ZnO NWBMs. Moreover, C.3 shows a bit better performance in the cutters' cooling. This observation can be attributed to the effects of rheology along with the thermal conductivity of the drilling fluid on the cooling of cutters.
Based on our results, for temperature and heat transfer analysis, not only the thermal conductivity is important but also the rheological properties (n and K) play a significant role; as C.1 sample with the most proper rheological properties (maximum C t as discussed previously) results in a cutters' temperature very close to C.3 sample which has the maximum thermal conductivity, see Table 1. This claim can also be supported by the results obtained for bit cutters' cooling using CuO NWBMs presented in Figure 13(b). As can be seen, the minimum temperature in all cutters is related to the mud sample C.6 which shows the best cutting transport efficiency owing to its proper rheological properties (characterized by the lowest value of n and highest value of K) in addition to its maximum value of the thermal conductivity; see Table 1.
It can be concluded that increasing the thermal conductivity of nano-enhanced water-based drilling muds can lead to an effective bit cooling only in conjunction with improving the rheological characteristics. The rheology has a dominant effect even in the thermal behavior of drilling muds during the bit cooling process. In this regard, we observe that the mud samples C.2 and C.5 show an adverse effect on the bit cutters' temperature compared to C.1 and C.4, respectively, although they have higher values of thermal conductivity. Therefore, Figure 10. Effect of rheology enhancement on cutting transport ratio at bit rotation speed 30 rpm. Figure 11. Effect of rheology enhancement on cutting transport ratio at bit rotation speed 50 rpm.
the increased thermal conductivity does not guarantee an improvement of the bit cooling efficiency; and rheology should be considered as an effective parameter along with the thermal properties.
As shown in Figure 13, the NWBM sample C.3 can reduce the temperature of the front surface of bit cutters from 4.8% to 8.4% over the base fluid. In addition, the mud sample C6, shows a temperature reduction from 6.9% to 11.9% over the base fluid for the bit cutters cooling process.
It is important to note that adding nano-particles at higher concentrations may adversely affect the performance of the drilling fluids due to the formation of agglomerated nanoparticles (Mahmoud et al., 2016;Rafati et al., 2018). However, the effectiveness of various concentrations of nano-particles could be estimated by checking the criteria of having a lower value of n and a higher value of K along with an increased thermal conductivity, to achieve a more efficient bit cooling capacity.

Cooling efficiency enhancement in the rotating drill bit
The average temperatures of heated surfaces on tip of the cutters for the rotating drill bit are shown in Figure 14 for CuO NWBMs at different bit rotational speeds. As can be seen, at all rotational speeds, the maximum temperature reduction over the base fluid occurs for mud sample C6. Besides, mud sample C.4 shows a better cooling compared to mud sample C.5, indicating that our previous discussions regarding the effects of rheological properties along with the thermal conductivity on the bit cutters' cooling are still valid in presence of the rotation.
In addition, it is observed that the highest percentage of temperature reduction varies between 15.7% and 16.7% at bit rotational speeds from 10 to 70 rpm. It is concluded that the bit rotational speed does not dramatically affect the cooling of cutters. This result is attributed to the fact that the rotational speed of the bit is not considerable compared to the jet velocity of inlet flow through the nozzles.
The temperature distribution on the bit body and cutter surfaces for cooling via the base fluid C.0, and CuO nano water-based mud sample C.6 is illustrated in Figure  Figure 15. Contours of temperature on the surface of drill bit, rotating at speed 30 rpm: (a) base fluid C.0 as drilling mud, (b) CuO NWBM C.6 as drilling mud.
15 for the drill bit rotating at speed 30 rpm. As can be seen, nano drilling fluid C.6 shows temperature contours concentrated in lower values in comparison with the base fluid C.0. For nano drilling fluid C.6, this temperature reduction is up to 16.7% over the base fluid.
The temperature reduction predicted in our work thanks to the enhanced properties of the nano-drilling fluids is an indication of their effective role in preventing thermally accelerated wear of the cutters. The thermal wear reduction consequently improves the rate of penetration, reduces the bit imbalance and down-hole vibrations, and increases the long life of the bit cutters (Glowka, 1989).

Conclusions
The hydraulic and thermal performance of water-based mud samples, including ZnO and CuO nanoparticles, as rheology enhancing agents and thermal conductivity modifiers, have been studied numerically using computational fluid dynamics (CFD) techniques across a typical PDC drill bit. This paper has taken into account the challenges of CFD simulation of mud behavior across the drill bit, including complex geometry of the bit, non-Newtonian behavior of the drilling fluid, dynamic effects of the rotation, and temperature variation due to the heat generation in the bit cutters at rock-contact regions. The grid structure for the computational domain was developed by a special procedure to ensure generating the prism layers along with preserving the mesh quality. In addition to the robust procedure proposed to successfully simulate the flow behavior around the PDC drill bit, the following conclusions can be drawn from the results obtained from this study: • The hydraulic performance analysis of the PDC drill bit revealed that an improvement in the rheological properties of the drilling mud by adding nanofluids results in a remarkable increase in the cutting transport efficiency. We observe that the CuO and ZnO nano-enhanced water-based muds (NWBM) show up to 127% and 68% increase over the cutting transport ratio of the base fluid, respectively, at the bit rotational speed 30 rpm. • The mud samples with minimum fluid behavior index (n) and maximum fluid consistency index (K) result in a better hole cleaning efficiency. Therefore, the effectiveness of various concentrations of nano-particles should be examined by checking these criteria for n and K. • Increasing the thermal conductivity of NWBMs leads to an effective bit cooling only in conjunction with improving the rheological characteristics. In this regard, CuO NWBMs show up to 16.7% reduction in front surface temperature of the bit cutters over the base fluid. Improving the cooling efficiency of the drilling fluid can decrease the thermal wear of the PDC bit cutters and consequently improve the drill bit performance and its lifetime.

Future work recommendations
The CFD model and grid structure proposed in this paper can be a reliable prediction tool and a cost-effective approach for understanding the flow behavior around the drill bit, a successful and efficient design of drill bits, and optimization of the drilling process. The procedure applied in this work can be used as a road map to study the flow behavior and the temperature distribution around other structural drill bits. This is because we addressed all problems associated with the complexity of geometry and grid generation process, which is an essential step toward a comprehensive CFD modeling around the drill bits. The procedure proposed in this article can be integrated to a flow simulator in the annulus to address the whole drilling process from the injecting point of the drilling mud to the returning position on the surface. Besides, the results of the present work and the procedure of CFD simulation around the complex geometry of the drill bit are a roadmap for future simulations around other drill bit structures with different drilling muds. The investigation of a variety of drilling fluid densities is an interesting topic that is beyond the scope of this article and is a very promising future work.

Disclosure statement
No potential conflict of interest was reported by the author(s).