Investigation of cuttings transport in directional and horizontal drilling wellbores injected with pulsed drilling fluid using CFD approach

Tunnelling and Underground Space Technology 90 (2019) 183–193

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Investigation of cuttings transport in directional and horizontal drilling wellbores injected with pulsed drilling fluid using CFD approach ⁎⁎

Boxue Panga, Shuyan Wangb, , Cailei Lua, Wenjian Caia, Xiaoxue Jianga, Huilin Lua, a b

T

⁎

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China School of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Horizontal and directional drilling Cuttings transport CFD model Pulsed drilling fluid Liquid-solid flow

Efficient cuttings transportation is a significant issue in wellbore drilling. The pulsed jet drilling technology is known to improve the rate of penetration (ROP), but lacks targeted research into its effect on cuttings transport. We presented a CFD model based on Eulerian-Eulerian method to investigate the cuttings transport characteristics in horizontal and directional wellbores injected with pulsed drilling fluid. Kinetic theory of granular flow was used to formulate the stress tensor of cuttings particles, and sliding mesh method was used to achieve the rotation of the drill pipe. The wave propagation of the flow of cuttings particles was analyzed from the variation of amplitude and frequency of the inlet velocities of pulsed drilling fluids. Effects of drilling fluid rheological properties and hole and pipe diameters on borehole cleaning in traditional and pulsed drilling were studied. Numerical results indicated that the pulsed drilling fluid contributes to borehole cleaning because it significantly reduces the cuttings concentration in the moving bed zone and increases the velocities of cuttings in the fixed bed zone. Increasing amplitude and frequency of the inlet velocity of pulsed drilling fluid also leads to a decrease of the height of cuttings bed. The pulsed drilling fluid produces higher turbulent kinetic energy and lower turbulent dissipation rate comparing with the case with a constant drilling fluid velocity. Furthermore, pulsed drilling was also proved to be universally applicable to the improvement of cuttings transport within a wide range of drilling fluid rheological parameters and hole and pipe diameters.

1. Introduction During drilling boreholes with horizontal and highly inclined sections, drilling cuttings tend to settle down to the lower wall of the borehole due to the radial component of gravity, building up cuttings bed, which is detrimental to drilling. Efficient cuttings transportation is one of the key issues in directional well drilling (Siamak et al., 2015; GhasemiKafrudi and Hashemabadi, 2016; Moraveji et al., 2017) and horizontal directional drilling (HDD) (Shu and Ma, 2016; Shu and Zhang, 2018; Zeng et al., 2018; Yan et al., 2018). The former refers to the important early stage of the exploration and development of oil and gas and other underground resources, while the latter is generally used to install underground pipelines with large diameters. In this paper we investigate the cuttings transportation in directional and horizontal well drilling in petroleum industry. To improve cuttings transport, researchers (Pang et al., 2018a, 2018b) have conducted extensive investigations and they concluded that the transportation of cuttings in borehole is dominated by many variables including wellbore structure,

⁎

cuttings size, drill pipe rotation, rheology and flow rate of drilling fluid, etc. All experimental and numerical studies have shown that drilling fluid flow rate affect cuttings transport significantly. The flow rate of drilling fluid in the borehole can be constant or fluctuating. The pulsed jet drilling technology is an application of mud pulse, which has become an effective way to improve the drilling rate (Wang et al., 2011). Fig. 1 shows the schematic diagram of drilling fluid continuous wave generation. The mud pulse generator is usually installed in the drilling fluid flow path in front of the drill bit. The pulsed drilling fluid is formed as the drilling fluid flows through the mud pulse generator, the amplitude and frequency of which are generally adjusted by the rotary valve. Comparing with the continuous jet, the pulsed jet improves drilling efficiency (Bizanti, 1990). Research indicated that the pressure of the pulsed jet will be at least four times higher than the continuous jet at the same velocity (Foldyna et al., 2004). Fluid through the pulsed jet formed by the self-excited oscillation nozzle provides a large instantaneous energy, which can reduce the chip hold down effect, create

Corresponding author. Corresponding author. E-mail addresses: [email protected] (S. Wang), [email protected] (H. Lu).

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https://doi.org/10.1016/j.tust.2019.05.001 Received 21 October 2018; Received in revised form 25 April 2019; Accepted 3 May 2019 Available online 09 May 2019 0886-7798/ © 2019 Elsevier Ltd. All rights reserved.

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successful modeling tool in studies of cuttings transport (Rooki et al., 2013; Sun et al., 2014; Siamak et al., 2015; GhasemiKafrudi and Hashemabadi, 2016; Epelle and Gerogiorgis, 2018; Pang et al., 2019). 2.1. Governing equations In present study, we prepared a three dimensional CFD model to investigate the flow of cuttings particles and drilling mud through the horizontal borehole with drill pipe rotation based on the two-fluid model. The solid phase and the liquid phase are strongly coupled by the interphase forces between them. For simplification, we assume that (1) the drilling fluid phase is an incompressible non-Newtonian liquid described by power-law or Herschel-Bulkley model; (2) the cuttings are spherical particles with a uniform particle size; (3) there is no interfacial mass transfer between the cuttings phase and the liquid phase (Pang et al., 2018a). The continuity equation for both phases (i = s for solid phase or l for liquids) is expressed as:

Fig. 1. Schematic diagram of drilling fluid continuous wave generation.

a partial pressure in the bottom-hole, and help to remove cuttings (Chuanlin et al., 2007). Several types of downhole drilling tools using pulsed jets have been developed, such as a shaped charge pulsed nozzle (Wang et al., 1999), a negative-pressure-pulse tool (Kolle and Marvin, 1999), an adjustable frequency pulse jet generating tool (Cui et al., 2012) and a screw-type pressure intensifier (Wang et al., 2012). Cleaning and breaking is enhanced and the penetration speed is improved due to the jet pulsation (Fu et al., 2012). The tests in field demonstrated that the mud pulse reduces the required energy to remove the unit volume of the rock, and it can improve the rate of penetration by 10.1–31.5% (Li et al., 2009, 2010). Most of the above mentioned researches concerning mud pulse have mainly focused on the efficiency of hydraulic rock breaking and the rate of penetration (ROP), but lack of quantitative analysis of the improvement on cuttings transport. The high velocity periodic self-excited oscillation pulsed jet acts on the bottom rock after being accelerated by the nozzle in the drilling bit. The annulus fluids and cuttings would also be accelerated in the transportation, and the cuttings transport ratio will be affected by frequency and amplitude of the pulsation of pressure or velocity of drilling fluid. Even more impressively, there may be a potential to enhance cuttings transport in borehole by applying the velocity pulse or pressure pulse which have already been widely used in drilling engineering in many aspects. In this paper, we investigated the cuttings transport characteristics in horizontal and directional wells injected with pulsed drilling fluid by means of CFD method based on Eulerian-Eulerian two-fluid model. Kinetic theory of granular flow is used to account for the collisional interactions of cutting particles, and the interactions between the cuttings and the walls of drill pipe and borehole. The effects of amplitude and frequency of pulsed drilling fluids on cuttings transport are analyzed. The wave propagation of drilling fluids and cuttings is also analyzed with the change of amplitude and frequency of pulsed drilling fluids through the horizontal borehole.

∂ (εi ρi ) + ∇ ·(εi ρi vi ) = 0 ∂t

(1)

where εi, vi and ρi are volume fraction, velocity vector and density, respectively. The momentum conservation equation for the liquid phase is given in Eq. (2) with the interphase momentum transfer:

∂ (εl ρ l v l ) + ∇ ·(εl ρ l v l v l ) = εl ∇ ·τ l + εl ρ l g − εl ∇p − β (v l − vs) + Fvm,l ∂t (2) where g is the gravitational acceleration, β is the drag coefficient and Fvm,l is the virtual mass force of drilling fluid. τl is the stress tensor for drilling fluid and can be expressed as τl = η(D)·D, where η is the apparent viscosity and D is the rate-of-deformation tensor. For power law model, η = K·γn−1, and η = τ0/γ + K·(γ/γc)n−1 for Herschel-Bulkley model, where γ is the shear rate of drilling fluid, τ0 is the yield stress, K is the consistency coefficient, and n is the flow behavior index. The turbulent viscosity of drilling fluid is calculated using the shearstress transport k-ω model, where k and ω are the turbulent kinetic energy and the specific dissipation rate (Wilcox, 1998). The transport equations of k and ω are expressed by Eqs. (3) and (4) for low Reynolds number modification (Menter, 1994):

∂ ∂ ∂ ⎛ ∂k ⎞ (ρ k ) + (ρ kv li ) = ⎜Γk ⎟ + Gk − Yk ∂t l ∂x i l ∂x j ⎝ ∂x j ⎠

(3)

∂ ∂ ∂ ⎛ ∂ω ⎞ (ρ ω) + (ρ ωv li ) = ⎜Γω ⎟ + Gω − Yω + Dω ∂t l ∂x i l ∂x j ⎝ ∂x j ⎠

(4)

where G is the generation due to mean velocity gradients, Г is the effective diffusivity, Y is the dissipation due to turbulence, and D is the cross-diffusion term. To formulate collisional interactions between cuttings particles, we use kinetic theory of granular flow (Gidaspow, 1994) for closure. The momentum balance for cuttings particles is given as:

2. CFD modeling for cuttings-drilling fluid flow Experimental and numerical studies are two major research methods in multiphase fluid mechanics. The experiments are considered to give more intuitive and reliable results, and they are the first choice if conditions permit. However, it is difficult and costly to achieve experimental measurements in many complex geometries and conditions. The computational fluid dynamics method shows good flexibility for testing under extreme physical models and operating parameters without expensive hardware replacement, and it provides valuable information about the flow field such as the local concentration and turbulence parameters of each phase, which are difficult or even impossible to obtain through experiments. Since Bilgesu et al. (2002), as the first group of researchers, introduced the CFD method into the study of cuttings transport, the CFD model is constantly improving and developing. Now the CFD method has been proved to be an efficient and

∂ (εs ρs vs) + ∇ ·(εs ρs vs vs) ∂t = −εs ∇p + ∇ ·ps I + ∇ ·τs + εs ρs g + β (v l − vs) + Fvm,s

(5)

where the cuttings stress tensor τs is computed from:

{

τs = ξs ∇ ·vs I + μs [∇vs + (∇vs)T] −

2 (∇ ·vs) I 3

}

(6)

in which ξs and μs are bulk viscosity and shear viscosity of solid phase, respectively. They are expressed as:

ξs = 184

4 θ εs ρ d s g0 (1 + e ) s 3 s π

(7)

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Fig. 2. Computational grid of the horizontal annulus.

μs =

10ρs d s πθs 4 4 θ [1 + g0 εs (1 + e )]2 εs ρ g d s (1 + e ) s + 96(1 + e ) εs g0 5 5 s 0 π

2.2. Computational conditions and geometric modeling (8)

Due to the high shear rates caused by drilling fluid pulsed velocity and drill pipe rotation, the power-law model is acceptable to describe rheological properties of drilling fluids. The consistency coefficient and the power-law index of drilling fluid are based on experimental measurements (Kelessidis et al., 2006) and are specified as 0.9448 Pa·sn and 0.4097 respectively. The form of drilling fluid pulse is sinusoidal continuous wave. At the annulus entrance boundary, the pulsed inlet velocity of drilling fluid takes the following form:

where ds is the particle diameter, g0 is the radial distribution function, e is the coefficient of restitution, and θs is the granular temperature. ps is the cuttings pressure and computed from:

ps = εs ρs θs + 2g0 εs2 (1 + e ) ρs θs

(9)

The granular temperature in the viscosity and pressure formula is simulated using the conservation equation of solid phase fluctuating energy:

u f,in = u 0 + uA sin(2πf )

3⎡∂ (εs ρs θs) + ∇ ·(εs vs ρs θs)⎤ = (−ps I + τs): ∇vs + ∇·(k s ∇θs) − γs + ϕls 2 ⎣ ∂t ⎦

where f is the fluctuating frequency, u0 is the time-averaged velocity, and uA is the amplitude of fluctuation. uA is smaller than u0 so that the fluctuating component makes up a relatively small fraction of the total drilling fluid velocity. The maximum instantaneous velocity is u0 + uA, while the minimum instantaneous velocity is u0-uA in a period. This brings the following advantages over the conventional constant velocity of drilling fluid: (1) helps to enlarge turbulence of drilling fluid; (2) the controllability of the flow rate of drilling fluid is improved because only part of the flow rate needs to be changed. The velocity magnitude of drilling fluid at any given moment depends not only on u0, but on the magnitudes of f and uA also. Therefore, the optimum performance of the transportation of cuttings may be achieved through an optimum combination of these parameters. The computational domain in present work is a horizontal annulus consisting of a borehole and an eccentrically rotating pipe. The computational grid of the horizontal annulus is shown in Fig. 2 and the dimensions of the annulus are shown in Table 1. In order to achieve the rotation of the drill pipe in the wellbore, we use the sliding mesh method due to its advantage of saving computing resources while ensuring the accuracy of calculations. The mesh of the annular geometry was divided into two flow zones, namely the inner moving mesh zone at the same rotational speed as the drill pipe wall and the static outer zone. The connections between the inner rotating zone and the outer static zone were achieved by one pair of interfaces between them. The feed concentration of cuttings phase are taken from the experimental measurements (Sifferman et al., 1974) and the inlet velocity

(10) where ks is the thermal conductivity of particles, γs is the dissipation of fluctuating energy, and ϕls is the exchange of fluctuating energy between liquid and particles. The last term on the right hand side of Eq. (5) represents the virtual mass force of cuttings particles. When particles are accelerated through the interaction of fluid phase, the inertia of the fluid mass encountered by the accelerating particles exerts the virtual mass force on particles (Drew and Lahey, 1993):

Fvm,s = 0.5εs ρl

d (vl − vs ) dt

(11)

which is proportional to the density of fluid phase. Due to the low density of gas phase, the virtual mass force effect is often ignored in gasparticles flow. However, for liquid-particles flow, the primary phase density is not much smaller than the particles. Researchers (Ding et al., 1995; Kendoush et al., 2007; Guskov, 2012; Wang et al., 2017) confirmed that the virtual mass force should not be ignored in liquid-particles flow. Therefore for the pulsed drilling fluids-cuttings flow, the virtual mass force effect is considered in present study. Based on the Newton’s third law, the virtual mass force of drilling fluid of the last term on the right hand side of Eq. (2) is:

Fvm,l = −Fvm,s

(12)

To achieve a smooth transition in the case that the cuttings concentration is less than 0.2, we calculated the drag force between liquid phase and solid phase by means of Huilin-Gidaspow model (Lu et al., 2003), which can be shown as:

β = φβErgun + (1 − φ) β Wen & Yu φ=

1 arctan[262.5(εs − 0.2)] + 2 π

βErgun = 150

β Wen

& Yu

=

εs2 μl εl ds2

+ 1.75

εs ρ l |v l − vs | ds

3Cd εl εs ρ l |v l − vs | −2.65 εl 4d s

(17)

Table 1 Summarized input parameters in simulations.

(13)

(14)

(15)

(16) 185

Symbol

Property

Simulation value

Units

Dh Dd e ωd ρf u0 uA f dp ρs εin

Hole diameter Pipe diameter Eccentricity (e = E/(Dh/2 − Dd/2)) Drill pipe rotation speed Mud density Time-averaged velocity Amplitude of velocity pulsation Frequency of velocity pulsation Particle diameter Particle density Injected particle volume fraction

203.2 101.6 0.5 100 1437.6 1.016 0.1, 0.3, 0.5 1.0, 2.0, 10.0 3.0 2550 4

mm mm – rpm kg/m3 m/s m/s Hz mm kg/m3 %

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is consistent with the drilling fluid. The exit is set to continuous outflow boundary conditions for drilling fluids and cuttings particles. The reference pressure is set to 300,000 Pa at the exit. At the walls, apply noslip velocity boundary conditions to the liquid phase and Johnson and Jackson boundary conditions (Johnson and Jackson, 1987) to the solid phase. In this study, we set the coefficient of restitution for cuttings particles to be 0.9 since it is a widely accepted value in experimental (Han et al., 2009, 2010) and numerical studies (Akhshik et al., 2015) of conventional drilling, and in which the numerical simulation agrees well with the experiment. While the specularity coefficient is an empirical parameter that measures the fraction of relative tangential momentum transferred as a result of particle-wall collision (FLUENT, 2011; Huang and Kuo, 2018), and its value ranges between zero (smooth wall) and unity (rough wall). In contrast to the restitution coefficient, the specularity coefficient is generally not measured directly from experiments, and usually determined by empiricism (Sommerfeld and Huber, 1999; Geng et al., 2016). As a relatively small intermediate specularity coefficient (Fede et al, 2016; Huang and Kuo, 2018), the value of 0.1 was chosen to describe the conversion of tangential momentum due to the collisions between the cuttings particles and the wellbore wall. The set of governing equations described above is solved via the commercial CFD code FLUENT 14.0 (FLUENT, 2011) on each cell throughout the annulus. A constant time step of 1 × 10−5 s for satisfactory convergence is used in all simulations. Time-averaged statistics of simulation results are computed covering several periods of the oscillation of pulsed drilling fluid from 80 to 110 s.

Fig. 3. Compared cuttings volume fraction between simulations and experimental measurements (Han et al., 2010) at different borehole inclination angles. (0.4% CMC solution + 4% sand, ωdrillpipe = 200 rpm, uin = 0.89 m/s).

3. Results and discussion 3.1. Model validation Numerical simulation facilitates the modeling of complex boundary conditions including pulse jets, and educes predictive models that optimize the operating conditions. Effective numerical methods can make up for the shortcomings in the experimental measurement, however, the use of experimental results to verify the validity of CFD models is essential for the study of numerical methods. Since there is still lack of detailed experimental data specifically for cuttings transport in pulsed drilling, simulations are performed and compared with the experiments conducted by Han et al. (2010) in the wellbore injected with constant velocity drilling fluid and with the drill pipe rotating at a constant speed of 200 rpm. Detailed experimental configuration and parameters are found in the literature (Han et al., 2010). Fig. 3 illustrates the time history of the simulated concentration of the retained cuttings throughout the wellbore and the corresponding experimental measurements at different borehole inclination angles, and Fig. 4 depicts the 3D view of retained cuttings concentration distribution along the wellbore. The cuttings particles are continuously injected into the wellbore from the inlet at a constant concentration of 4%. At the beginning of the calculation, the cuttings gradually deposited to the lower side of the wellbore, and the concentration of cuttings is gradually increased in the annulus. After a certain calculation time, the flow rate of injected and discharged cuttings reaches equilibrium and the total cuttings concentration retained in the annulus no longer changes. With the increase of the inclination angle of the wellbore, the cuttings remaining in the annulus increases, cuttings transport efficiency deteriorates, and the stratification of cuttings concentration is more obvious. The simulation results agree well with the experimental data, with an average error of 2.57%. The error mainly comes from the inevitable orbital motion of the drill pipe in the experiment and the given rheology of the drilling fluid, which is determined only by the flow rate of drilling fluid and the rotating speed of drill pipe. Furthermore, it is recommended to conduct more verification of CFD models for nonNewtonian fluids and experimental studies of cuttings transport in wellbore injected with pulsed drilling fluid.

Fig. 4. Distribution of retained cuttings concentration throughout the borehole at different inclination angles. (0.4% CMC solution + 4% sand, ωdrillpipe = 200 rpm, uin = 0.89 m/s).

3.2. Instantaneous velocity and pressure fluctuations The pulsed inlet velocity of drilling fluid is specified as sinusoidal continuous wave which relates with two parameters of amplitude and frequency. To study effects of pulsing amplitude and frequency on the flow of drilling fluids and cutting particles in the annulus, the pulsed inlet velocity of drilling fluid with three different pulsing amplitudes (0.1, 0.3 and 0.5 m/s) and three different frequencies (10.0, 2.0 and 1.0 Hz) at the same average velocity of 1.016 m/s are simulated, and the variations of relative pressure and inlet velocity of drilling fluid are shown in Fig. 5 with times. It is worthy to first look at a case in which no pulsation of inlet velocity of drilling fluid is applied, and the inlet velocity is constant. The induced pressure of mixture does not vary with times. For the pulsed inlet velocity of drilling fluid, the inlet velocity of drilling fluid consists of an active phase and an inactive phase. In active phase, the instantaneous velocity of drilling fluid is larger than the time-averaged velocity. And in inactive phase, the situation is reversed. Numerical simulations show that the inlet pulsing velocity of drilling fluid provides a synchronized pressure fluctuation. The pulsing inlet velocity of drilling fluid causes the increase in pressure in the inactive phase and the decrease in pressure in the active phase, and enhances 186

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Fig. 6. Flow pattern of cuttings in the horizontal annuli injected with a constant inlet fluid velocity and a pulsing inlet velocity with amplitude of 0.5 m/s and frequency of 2.0 Hz.

annulus is significantly reduced under the action of pulsed drilling fluid. The pulsed drilling fluid significantly improves hole cleaning in horizontal borehole. In the fixed bed zone, the moving bed zone and the suspension zone, three points a (0, −65, 3000), b (60, −10, 3000) and c (0, 100, 3000) are taken and the variations of cuttings flow parameters are monitored. Fig. 7 illustrates the variations of volume fraction of cuttings with time at points a, b, and c of the annuli injected with constant or pulsed drilling fluid velocity. Because of the accumulation of solids, the cuttings concentration in the lower part of the annulus is the highest, while it is the lowest in the upper part of the annulus, maintaining at the order of magnitude of the feed concentration of cuttings. Due to the influence of the pulsed flow rate of the drilling fluid, the cuttings concentration also fluctuates with time. In the three zones, the concentrations of cuttings in the annulus injected with pulsed drilling fluid are lower than that in the annulus injected with the constant drilling fluid velocity. It is worth noting that the decrease in cuttings concentration is the greatest at point b of the moving bed zone. Fig. 8 shows the variations of axial velocity of cuttings with time at

Fig. 5. Profile of inlet velocity pulse of drilling fluid and its induced pressure fluctuation in the annulus. (a) Different pulsing amplitudes; (b) different pulsing frequencies.

the pressure oscillation in the borehole. At the same frequency of inlet velocity, the maximum induced pressure increases with the increase of amplitude of inlet velocity of drilling fluid. The amplitude of the pressure fluctuation decreases dramatically with the increase of inlet velocity pulse period (i.e., the decrease of frequency). 3.3. Flow pattern of cuttings in boreholes injected with constant or pulsed drilling fluid velocity Fig. 6 illustrates the 3D view of cuttings volume fraction and velocity magnitude along the horizontal borehole injected with a constant inlet fluid velocity or a pulsing inlet velocity. Both time-averaged inlet velocities of drilling fluid are 1.016 m/s. As can be seen, the cuttings gradually settle to the lower wall of the horizontal annulus under the action of gravity components, and as a consequence, a fixed cuttings bed forms along the borehole. Both bed height and velocity distribution tend to a relatively stable state in a few meters from the entrance in these simulations, consistent with experiments (Osgouei, 2010). Furthermore, due to the rotation of drill pipe, both the core zone distribution of cuttings concentration on the lower side of the annulus and that of cuttings velocity on the upper side reveal swaying phenomena (Tomren et al., 1986). In the upper, middle and lower zones of the borehole, flow parameters of cuttings are significantly different, which can be used to define cuttings flow as three zones. They are the fixed bed zone in the lower part of the borehole, the cuttings suspension zone in the upper part of the borehole and the moving bed zone between them, similar to the experimental conclusion (Doron and Barnea, 1996). Comparing with the annulus injected with the drilling fluid with a constant inlet velocity, the concentration of cuttings retained in the

Fig. 7. Variations of volume fraction of cuttings with time at point a, point b and point c of the annuli injected with constant or pulsed drilling fluid velocity. 187

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Fig. 10. Variations of granular temperature of cuttings with time at point a, point b and point c of the annuli injected with constant or pulsed drilling fluid velocity.

Fig. 8. Variations of axial velocity of cuttings with time at point a, point b and point c of the annuli injected with constant or pulsed drilling fluid velocity.

fluctuation velocity of particles. In the present work, θs characterizes the intensity of the oscillations of the cuttings. Fig. 10 shows the variations of granular temperature of cuttings with time at points a, b, and c of the annuli injected with constant or pulsed drilling fluid velocity. Due to the lower particle concentration in the cuttings suspension zone (point c), interactions between the cuttings are very weak, leading to lower granular temperatures. While in the moving bed zone (point b), the drastic rolling and collision of cuttings particles increase the granular temperature sharply. In the fixed bed zone (point a), even if the particle concentration is high, the lower mean free path restricts the interaction between the particles, making the granular temperature lower than that in the moving bed zone. The statistical results show that the velocity pulses of drilling fluids significantly increase the granular temperature in the fixed bed zone. In the moving bed zone, although the average granular temperature does not increase significantly, the pulsed drilling fluid enhances the pulsation of the granular temperature and increases its extremum. Fig. 11 shows the variations of turbulent kinetic energy and its dissipation rate with time at points a, b, and c of the annuli. Due to the dense accumulation of cuttings particles, the turbulent kinetic energy and its dissipation rate are significantly lower in the fixed bed zone (point a) than that in the moving bed zone (point b) and in the suspension zone (point c). The turbulent dissipation rate in the moving bed zone is the highest because the energy of the drilling fluid is transferred to the energy required for the intense interaction between the particles. In the moving bed zone and the suspension zone, the pulsed drilling fluid increases the turbulent kinetic energy and reduces the dissipation rate. The pulsation of inlet velocity of drilling fluid enhances the oscillation of flow parameters in the annulus, which introduces high interaction forces between the drilling fluid and the cuttings. As a consequence, the pulsed drilling fluid carries more cuttings out of the borehole.

points a, b, and c of the annuli injected with constant or pulsed drilling fluid velocity. As can be seen, the cuttings velocity is the highest in the cuttings suspension zone (point c), and the lowest in the fixed bed zone (point a). In the annulus injected with a constant drilling fluid velocity, the velocity of cuttings in the three zones varies significantly, and it is almost zero in the fixed bed zone. In the annulus injected by pulsed drilling fluid, the velocity difference of cuttings in the three zones decreases. Pulsed drilling fluids significantly increase the velocity of cuttings in the fixed bed zone. Spectral analysis of turbulent oscillations is commonly used in the study of fluid turbulent flow using the Fourier transforms. Thus, we can estimate the power spectrum of velocity of particles using the fast Fourier transform (FFT) technique. Fig. 9 shows the power spectrum density from the instantaneous velocity of particles as a function of frequency. The spectrum densities decay with the increase of frequency. While at low frequencies the oscillations of cuttings velocities play an important role. The magnitudes of the fluctuations are different at these three points. It gives a highest power spectrum density at point c and a lowest value at point a. However, the trends are similar. From the figures, the diagram highlighted a dominant frequency at about 1.94 Hz which is close to the frequency of the inlet velocity. This indicates the oscillation of velocity of cuttings is dominated by the pulsed inlet velocity of drilling fluid. Similar to the thermal temperature in the kinetic theory of gases, the granular temperature θs is defined as θs = 〈Cs Cs〉/3, where Cs is the

3.4. Impact of rheological properties of drilling fluids on cuttings transport in conventional drilling and pulsed drilling Rheological properties of drilling fluids have apparent influence on flow characteristics and thus affect cuttings transport in the annulus. The three parameters Herschel-Bulkley model which is expressed as τ = τ0 + KHBγn is considered to be more suitable for describing drilling fluids in more conditions (Kelessidis et al., 2005). Compared with the power law model, the H-B model has better applicability under low shear rates due to the existence of yield stress τ0. It is of research value to separate effects of different rheological parameters on cuttings transport, but it is difficult to vary a certain rheological parameter of drilling fluids independently in experimental studies. In order to study

Fig. 9. Power spectrum density at point a, point b and point c of the annuli as a function of frequency. 188

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Fig. 12. The histograms of cuttings transport ratio for different drilling fluids in conventional drilling and pulsed drilling.

and 2.0 Hz, respectively. As the rheological parameters increase, the apparent viscosity of the drilling fluid increases, and the cuttings transport ratio increases regardless of conventional drilling or pulsed drilling. What’s more, for all the drilling fluids calculated, the cuttings transport ratio in pulsed drilling is higher than that in conventional drilling, which indicates that pulsed drilling is universally applicable to the improvement of cuttings transport within a wide range of drilling fluid rheological parameters. However, for drilling fluids with higher consistency coefficient (S5) or flow behavior index (S7), the improvement in cuttings transport by pulsed drilling is not significant.

Fig. 11. Variations of turbulent kinetic energy and its dissipation rate with time at point a, point b and point c of the annuli injected with constant or pulsed drilling fluid velocity. (a) Turbulent kinetic energy; (b) turbulent dissipation rate.

3.5. Impact of wellbore and drill pipe diameters on cuttings transport in conventional drilling and pulsed drilling

Table 2 Rheological parameters for drilling fluids and corresponding simulated retained cuttings volume fraction. Drilling fluid samples

S1 S2 S3 S4 S5 S6 S7

Rheological parameters

Retained cuttings volume fraction

τ0 (Pa)

K (Pa·sn)

n

Conventional drilling

Pulsed drilling

0.1747 3 6 0.1747 0.1747 0.1747 0.1747

0.9448 0.9448 0.9448 2 4 0.9448 0.9448

0.4097 0.4097 0.4097 0.4097 0.4097 0.6 0.8

0.09626 0.07376 0.06355 0.07182 0.04889 0.07651 0.04906

0.07503 0.05754 0.04966 0.05684 0.04634 0.06123 0.04825

To investigate the effect of hole and pipe diameters on cuttings transport, five sets of annulus configurations are modeled in this section, and the detailed size information is listed in Table 3, all taken from the API (American Petroleum Institute) standard. The hole diameters of configurations P1, P2 and P3 are the same, while the diameter ratios of the hole to the pipe of configurations P2, P4 and P5 are approximately equal to 2. The drilling fluid used in this section is S6, where τ0 = 0.1747 Pa, K = 0.9448 Pa·sn, and n = 0.6. Same as the previous section, the amplitude (uA) and frequency (f) of the pulsed drilling fluid velocity are 0.5 m/s and 2.0 Hz, respectively. Fig. 13 depicts the simulated contours of cuttings concentration in conventional drilling and pulsed drilling for different hole and pipe diameters, and Fig. 14 shows the corresponding histograms of cuttings transport ratio. It can be seen from the configurations P1, P2 and P3

the independent effects of the three rheological parameters of drilling fluids on hole cleaning while controlling variables, seven sets of drilling fluids based on S1 (Drilling fluid S1: real rheological parameters taken from the experiment (Kelessidis et al., 2006)) were simulated, and the detailed rheological parameters are listed in Table 2, which are within the wide range of parameters often encountered in drilling engineering (Horton et al., 2005; Pang et al., 2018a). Fig. 12 shows the histograms of cuttings transport ratio for different drilling fluids in conventional drilling and pulsed drilling. The cuttings transport ratio (CTR) is expressed as the ratio of the injection concentration of cuttings (0.04 in present study) to the total concentration of cuttings retained in the annulus. A greater value of CTR means more cuttings are removed from the wellbore. The amplitude (uA) and frequency (f) of the pulsed drilling fluid velocity in this section are 0.5 m/s

Table 3 Wellbore and drill pipe diameters and corresponding simulated retained cuttings volume fraction. Hole and pipe diameter pairs

P1 P2 P3 P4 P5

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Hole diameter Dh (mm)

Pipe diameter Dd (mm)

201.2 201.2 201.2 178.5 121.4

73.0 101.6 139.7 88.9 60.3

Retained cuttings volume fraction Conventional drilling

Pulsed drilling

0.09018 0.07651 0.05539 0.07703 0.07683

0.07694 0.06123 0.05195 0.06331 0.06863

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Fig. 15. Variations of volume-average cuttings concentration over time throughout the annulus for different amplitudes of pulsed drilling fluid velocities.

pipe diameters also. 3.6. Impact of amplitude and frequency of drilling fluid velocity on cuttings transport in pulsed drilling Fig. 15 shows the variations of volume-average cuttings concentration over time throughout the annulus as a function of the amplitude of pulsing drilling fluid velocity. The injection concentration of cuttings remains at a constant of 4%. Initially cuttings gradually settle to the bottom of the annulus and the accumulation of cuttings is formed. The exit flux of cuttings is lower than the inlet, so the total volume-average cuttings concentration throughout the annulus is gradually increased over time in the beginning. Within a few tens of seconds, the discharge of the cuttings tends to be stable, and the total concentration of cuttings throughout the annulus also tends to a stable value. The time-averaged cuttings concentration (εs) and standard deviation (σ) are calculated at the different amplitudes of pulsed drilling fluid. The volume average cuttings concentration throughout the annulus is reduced with the increase of the amplitude of drilling fluid velocity. The contour plots of cuttings concentration at the cross section of the annulus are shown in Fig. 16 with the change of amplitude of pulsed drilling fluid velocity. It can be seen that increasing the amplitude of velocity pulse reduces the height of cuttings bed. Fig. 17 shows the variations of volume-average cuttings concentration over time throughout the annulus as a function of the frequency of pulsing drilling fluid velocity, and Fig. 18 shows the contour plots of cuttings concentration. The concentration of solids oscillates over time because of the fluctuation of the inlet velocity of drilling fluid. The cuttings concentration decreases from 0.0841 with the frequency of 1.0 Hz to 0.0781 with the frequency of 10.0 Hz of pulsed drilling fluid velocity. As the amplitude and frequency of pulsing fluid velocity increase, the oscillation of fluid velocity produces high local accelerations of fluid and cuttings, and increases the interaction

Fig. 13. Simulated contours of cuttings concentration in conventional drilling and pulsed drilling for different hole and pipe diameters. (a) Dh = 201.2 mm; (b) Dh/ Dd ≈ 2.

Fig. 14. The histograms of cuttings transport ratio for different hole and pipe diameters in conventional drilling and pulsed drilling.

(Figs. 13a and 14), when the hole diameter is constant and the pipe diameter is increased, the cuttings transport is improved and the swaying phenomenon of the cuttings bed is more obvious. From P2, P4 and P5 (Figs. 13b and 14), we found that in the case of the same diameter ratio of the hole to the pipe, the cuttings transport ratio is not much different. Furthermore, for all the configurations calculated, the cuttings transport ratio in pulsed drilling is higher than that in conventional drilling, which indicates that pulsed drilling is applicable to the improvement of cuttings transport within a wide range of hole and

Fig. 16. Simulated contours of cuttings concentration in the horizontal annulus at different amplitudes of drilling fluid velocity. 190

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Fig. 17. Variations of volume-average cuttings concentration over time throughout the annulus for different frequencies of pulsed drilling fluid velocities.

Fig. 20. Variations of area-weighted average turbulent kinetic energy with time at different amplitudes and frequencies of pulsed drilling fluid velocity.

fluid velocity, the pulsed drilling fluid produces a high turbulent kinetic energy. With the increase of the amplitude of pulsed inlet velocity, the time-averaged value of turbulent kinetic energy is increased. However, the time-averaged turbulent kinetic energy at the different frequencies of the pulsed drilling fluid is approximately the same. The effect of the amplitude and frequency of the pulsed drilling fluid on the attenuation of the pulsed velocity of cuttings is shown in Fig. 21 from the inlet (black line) to the exit (red line) of the borehole. From simulated instantaneous axial velocity of cuttings, the time-averaged Fig. 18. Simulated contours of cuttings concentration in the horizontal annulus at different frequencies of drilling fluid velocity.

between them. The cuttings concentration throughout the annulus decreases. Fig. 19 shows the effects of amplitude and frequency of pulsed inlet velocity of drilling fluid on the area-weighted average turbulent dissipation rate in the borehole. The oscillation of turbulent dissipation rate is caused by oscillation of drilling fluid velocity. Comparing with results at the constant inlet fluid velocity, the energy dissipation is reduced using the pulsed drilling fluid. With the increase of amplitude and frequency of inlet velocity of drilling fluid, the energy dissipation is reduced in the borehole. Fig. 20 shows the effects of amplitude and frequency of pulsed drilling fluid velocity on the area-weighted average turbulent kinetic energy. Comparing with results at the constant inlet

Fig. 21. Variations of area-weighted average axial velocity of cuttings at inlet and exit of the annulus. (a) Different pulsing amplitudes; (b) different pulsing frequencies.

Fig. 19. Variations of area-weighted average turbulent dissipation rate with time at different amplitudes and frequencies of pulsed drilling fluid velocity. 191

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Appendix A. Supplementary material

velocity of cuttings ue,ave and its amplitude ue,A at the exit are obtained. As shown in Fig. 21a, at the same average velocity u0 at the inlet, the high amplitude of the inlet pulsed drilling fluid results in a high average velocity ue,ave at the exit. The similar trends are found with the change of frequency of the pulsed drilling fluid, seeing in Fig. 21b. At the exit, the average velocity increases from 0.888 m/s to 0.904 m/s, and the amplitude decreases from 0.423 m/s to 0.404 m/s with the frequency of pulsed drilling fluid velocity from 1.0 Hz to 10.0 Hz. This indicates that increasing pulse frequency reduces the attenuation of the average velocity but increases the attenuation of the pulse velocity amplitude at the exit of the borehole. To obtain an analytical solution for the wave propagation, the propagation characteristics of cuttings and pulsed drilling fluids are predicted. For the homogeneous flow in a liquid-solid system, the sound speeds of liquid phase and solid phase are al and as, respectively. The wave speed of liquid-solid mixture can be expressed by (Gidaspow, 1994):

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tust.2019.05.001. These data include Google maps of the most important areas described in this article. References Akhshik, S., Behzad, M., Rajabi, M., 2015. CFD-DEM approach to investigate the effect of drill pipe rotation on cuttings transport behavior. J. Pet. Sci. Eng. 127, 229–244. Bilgesu, H.I., Ali, M.W., Aminian, K., Ameri, S., 2002. Computational Fluid Dynamics (CFD) as a tool to study cutting transport in wellbores. In: SPE Eastern Regional Meeting. Society of Petroleum Engineers. Bizanti, M.S., 1990. Jet pulsing may allow better hole cleaning. Oil Gas J. 88, 67–68. Chuanlin, T., Dong, H., Jianghong, P., 2007. Experimental study on the frequency characteristic of the self-excited oscillation pulsed nozzle. Acta Pet. Sin. 28 (4), 122. Cui, L., Zhang, F., Wang, H., Ge, Y., Zhuo, L., Li, H., 2012. 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−1/2

ε ⎡ ε ⎤ cm = ⎢ ⎜⎛ 2s + 2l ⎟⎞ (εl ρ l + εs ρs )⎥ a ρ a ρ l s s l ⎝ ⎠ ⎣ ⎦

(18)

There exists a wave speed of cuttings and drilling fluid mixture that is dependent on the cuttings concentration. At the low cuttings concentration, the wave speed of mixture depends upon the sound speed of drilling fluid. The wave speed of mixture at the cuttings volume fraction of 0.081 is on the order of 1000 m/s, which is close to the magnitude of the measurements at the range from 1400 m/s to 1500 m/s in the sedimentation of particles (Hampton, 1985). The phase shift of the average axial velocity of particles within 3.0 m is about 0.004 s at the inlet mean velocity and amplitude of 1.016 m/s and 0.5 m/s with the frequency of 2.0 Hz. The propagation velocity is approximately 750 m/ s, which is close to the calculated value from Eq. (18). 4. Conclusions The transport performance of cuttings driven by pulsed drilling fluids through the horizontal and directional boreholes is investigated based on the kinetic theory of granular flow. Effects of drilling fluid rheological parameters and hole and pipe diameters on cuttings transport in conventional and pulsed drilling are analyzed. Three zones of cuttings flow are observed from simulations, namely the fixed bed zone, the moving bed zone and the cuttings suspension zone. Moreover, the non-symmetric distributions of cuttings deposition due to the rotation of drill pipe are observed in the horizontal borehole. Comparing with the cuttings transport in the annulus injected with a constant drilling fluid velocity, the pulsed inlet velocities of drilling fluid produces oscillations of pressure, and significantly increase the velocity of cuttings in the fixed bed zone and reduce the cuttings concentration in the moving bed zone. With the increase of amplitude and frequency of the inlet velocity of pulsed drilling fluid, the granular temperature and turbulent kinetic energy are increased, and the height of cuttings bed is reduced. Present simulations indicate that the pulsation of drilling fluid contributes more cuttings out of borehole and it offers an effective approach to enhance borehole cleaning in horizontal and directional drilling. Funding This work was supported by the National Natural Science Foundation of China [grant numbers 11472093, 51421063 and 51776059]. Declaration of Competing Interest None. 192

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