Quantitative Analysis and Modeling of Transient Cuttings Transport Impact on Drill String Mechanics in Extended Reach Drilling

Abstract

Cuttings beds in horizontal wells significantly affect the frictional torque and drag along the drill string; however, their quantification and modeling have been relatively underexplored. To gain deeper insights into the impact mechanisms of the cuttings bed distribution on drilling mechanics, this study establishes a model linking the cuttings bed height with variations in axial and tangential forces on the drill string through experimental investigations. By integrating this model with previously developed transient cuttings transport and torque–drag models, a coupled transient hole cleaning and drill string mechanics model is constructed. This comprehensive model simulates the dynamic distribution of cuttings along the entire well trajectory and its influence on the drill string torque and drag. The results reveal that accumulated cuttings significantly reduce the weight on bit (WOB), increase the drill string torque, and cause problems related to a high equivalent circulation density (ECD). For long horizontal sections, the key to achieving effective hole cleaning lies in optimizing the design of the tripping circulation time to ensure that all cuttings are removed from the wellbore. Using the proposed coupled model, a methodology is developed to minimize the tripping circulation time by solving optimization problems within a constrained 2D domain, providing scientific guidance for drilling operations. The findings demonstrate that dynamically managing the cuttings distribution in the wellbore can significantly mitigate issues arising from insufficient hole cleaning, thereby ensuring drilling safety and efficiency. This study provides a scientific foundation for the optimized design of long horizontal well drilling operations and highlights the critical role of cuttings management in enhancing hole cleaning performance and mitigating drilling risks.

1. Introduction

Extended reach wells (ERWs) and long horizontal lateral wells are commonly used in the development of unconventional and offshore oil and gas fields [1]. Long horizontal laterals increase the exposure of reservoirs and thus increase the overall production rate of oil/gas from a single well, which is critical to produce oil/gas economically for unconventional players, e.g., essentially all (99%) hydrocarbon production from the Marcellus has been from horizontally drilled wells [2,3,4]. In off-shore fields, extended reach drilling allows more wellheads to be grouped together on one platform and makes it easier and cheaper to produce those wells. Therefore, the production range of a single platform is extended, and the cost of moving the drilling platform and production facilities can be saved. In some fields, ERWs are also used to access reservoir formations that are difficult to reach through vertical drilling [5].

• As the conditions of newly founded oil and gas reservoirs become more challenging, the demand for longer ERWs increases. However, there are several challenges that restrict further increasing the length of horizontal laterals.
• As horizontal laterals become longer, drilling equipment is pushed to its mechanical and endurance limits. Drill string buckling is the most recognized challenge during drilling ERD wells [6]. General industry perception is that when drill strings or casing strings exceed conventional helical-buckling criteria, they cannot be operated safely because the risk of failure or lockup is too high [7,8]. In rotary drilling, buckling can also create high vibrations that may lead to failure of the BHA tools.
• Friction forces on the drill string increase as the length of the horizontal lateral increases [9], which may lead to insufficient weight and torque on the drill bit, especially during sliding drilling mode.
• With increasing lateral length, pressure loss in the wellbore increases. As a result, the difference between the static and dynamic equivalent circulation density (ECD) increases, requiring a broader pressure window for drilling. However, the safe wellbore pressure window is mostly constant or changes slightly due to the insignificant change in true vertical depth during horizontal drilling. Thus, the difficulty of maintaining the wellbore pressure within the safe pressure window increases, which is especially challenging in depleted reservoirs.

In order to improve the understanding of downhole conditions and resolve problems that hinder drilling long horizontal laterals, numerous studies have been conducted from different aspects.

From the buckling aspect, the focus is the critical compressive load beyond which tubular is no longer stable and deforms into either a sinusoidal or helical shape. It is commonly accepted that helical buckling is much more problematic than sinusoidal buckling for drilling operations [7]. Lubinski is one of the pioneers to study the buckling behavior of pipes for oil and gas drilling [10,11]. After that, many studies have been conducted to improve the understanding and modeling of the buckling phenomenon, including both experimental and theoretical works [12,13,14]. The effects of different parameters on buckling have also been studied, including the effects of the wellbore geometry [15], wellbore curvature [16,17], tool joints [18], and torque and torsion [19]. It is found that pipe rotation may decrease the critical load for helical buckling by approximately 50% [20]. Although some evidence showed that tubular may be tripped into the wellbore even in a buckling state within safe limits [7], drilling engineers generally try to avoid buckling in practical operations, by modifying the drill string design, changing the static friction force into dynamic friction force, and adding lubricants to reduce friction on a drill string [21]. However, there are few studies that have investigated the effect of hole cleaning on drill string buckling, although it is commonly accepted that poor hole cleaning increases the friction force on a drill string and moves the state of a compressed drill string closer to buckle [8,22].

The root causes of insufficient weight on the bit for horizontal drilling are essentially the same as buckling, which is the increase in axial friction force along a drill string. Tubular mechanics have been studied in the industry for decades, and the most commonly used models include the soft string models and the stiff string models. One of the first soft drilling models was proposed by Johancsik [23] and improved by different researchers later on [24,25,26,27,28]. The soft string models generally assume that contact between the drill string and wellbore is continuous and that the drill string trajectory is the same as the wellbore trajectory, which is not always the case in practical applications. Realizing the shortcomings of the soft string models, a number of stiff string models were developed to improve the understanding of the drill string status under complicated situations. Ho [29] was the first one to combine the soft string model with a stiff model to account for the stiffness of the BHA tools. A number of different stiff models were proposed later [30,31,32,33,34,35], and most of these models are based on FEM approaches. However, both soft string models and stiff string models only focus on tubular itself and ignore the effects of cuttings in the wellbore, which can be of significant impact for extended reach drilling. Cayeux et al. [36] proposed a dynamic drill string mechanical model coupled with a transient hydraulic model to analyze the results of friction tests, which is an important step towards the modeling of actual downhole conditions for the drill string. However, there are still few works discussing the effect of hole cleaning on drill string mechanics, which are critical for drilling extended reach and long horizontal lateral wells.

Hole cleaning has been studied by the industry for more than five decades. Most of the previous studies focus on the prediction of the cuttings bed height under fully developed conditions for drilling horizontal or deviated wells [37]. In practical drilling, the process of penetration into the formation is not continuous, and it is often interrupted by other operations, such as connecting the drill pipe and circulating drilling fluid to clean the well [38]. Several studies have been conducted to investigate transient hole cleaning [39,40,41,42,43,44,45,46,47,48,49,50,51]. However, the focus of these studies is still limited to the height of the cuttings bed, and only a few studies looked into the effect of cuttings on wellbore pressure loss [37,52,53,54,55,56]. In reality, the reason we care about hole cleaning during drilling deviated or horizontal wells is that packed cuttings in the inclined parts of a wellbore have significant effects on the drill string and wellbore pressure, which leads to a high torque/axial force along a drill string, high ECD, and wellbore pack-off. The importance of these effects is commonly recognized by the industry; however, there are a lack of studies that quantify the effect of cuttings on the drill string. As the horizontal departures of new drilled wells increase, drilling problems associated with hole cleaning and drill string mechanics, e.g., stuck pipe or buckling, become more serious. Demand for ways to quantify the effect of dynamic hole cleaning conditions on the drill string becomes urgent for long lateral drilling.

In this study, the innovation of this paper mainly lies in the proposal of a dynamic coupling model that integrates the mutual effects of hole cleaning and drill string mechanics while considering the non-uniform distribution of the cuttings bed through a transient model. This provides a new theoretical framework and technical approach to address the complex issues encountered in extended reach wells and long horizontal lateral drilling. Cases studies are conducted to demonstrate the importance of coupling hole cleaning and forces acting on the drill string together during drilling extended reach and long horizontal lateral wells.

2. Dynamic Hole Cleaning

The concept of transient hole cleaning monitoring is more widely accepted in the industry as more and more extended reach and long horizontal lateral wells are being drilled [57]. Traditional fully developed steady-state hole cleaning analysis reveals the worst-case scenario for hole cleaning, which only focuses on the theoretical maximum cuttings concentration, cuttings transport ratio, or cuttings bed height. It is usually used as guidance to choose drilling operational parameters during the planning stage. Compared to steady-state hole cleaning, transient hole cleaning approaches reveal details about the status and number of cuttings along an entire wellbore during the whole drilling process. For practical drilling, the processes of penetration and circulation are applied alternatively, and the cuttings cluster generated during the penetration process can be removed out of the wellbore during the circulating process. In vertical or slightly inclined wells, the lag time between cuttings and fluid flow from bit to surface is low, and newly generated cuttings clusters can be circulated out of the wellbore completely within 1 to 2 bottom ups (BUs, which is the time for drilling fluid flows from bit to surface). However, the lag time increases with increasing tilt angle. For extended reach and long horizontal lateral wells, the lag time increases significantly compared to vertical wells. In cases where the circulation time is not long enough to clean all the newly generated cuttings during the circulation stage following the drilling stage, an unevenly distributed cuttings bed can form a series of cuttings clusters in the wellbore, as illustrated in Figure 1.

The transient cuttings transport modeling approach is required to capture the details of the unevenly distributed cuttings bed during practical drilling. This study employs a generalized transient solid–liquid, two-phase flow model developed in a previous work [46] to obtain the dynamic cuttings distributions in a wellbore. In the generalized model, the flow patterns of solid–liquid mixtures are classified into several different patterns to cover all cuttings transport possible scenarios, as shown in Figure 2. The formulations of the model are based on transient mass and moment conservations, and the equations can be adjusted automatically according to the change in the flow patterns. Thus, the model can handle the change in flow patterns in both the time domain and space domain. By using this approach, the process of cuttings transport from bit to surface can be tracked continuously during the entire drilling process while considering the dynamic change in drilling operational parameters. Details of the model are shown in Appendix A.

3. Coupling of Transient Hole Cleaning and Drill String Model

It is commonly recognized that accumulated cuttings beds in the deviated or horizontal sections of a wellbore can increase the friction force on the drill string, which leads to changes in torque and axial load. Experimental studies show that the torque on the drill string increases significantly as the cuttings bed builds up in the wellbore [46]. To consider the effect of the cuttings bed on the forces along the drill string, a new torque and drag model is proposed by modifying the traditional soft string model. The effect of the cuttings bed on the drill string can be illustrated by using Figure 3. For a given section of the drill string, the traditional approach to obtain the axial force (as shown by F_t in Figure 4) is shown in Figure 3a, where W_c is the unit contact force, W_bp is the unit buoyant weight, R is the radius of curvature, and β is the rotation angle. To simulate the influence of accumulated cuttings, an additional frictional force (F_fw) was added at the location of the cuttings bed, which is related to the contact force (F_cw) between the cuttings bed and the wellbore, the friction coefficient (f_ws) between the cuttings bed and the drill string, and the depth of the drill string. The detailed calculation method for this additional force is shown in Figure 3b. In addition, in the figure, F_p is the axial force at the lower end of the elemental section, F_fp is the axial force exerted by the drilling fluid on the elemental section, and F_r is the axial force of the elemental section. To model the exact effect of the cuttings bed on the drill string, detailed positions of the drill string are required for the entire wellbore, which needs complicated finite element simulations to be conducted. For practical applications, we propose to use one term to represent the complex interactions between the cuttings bed and the drill string in the soft string model, which can be estimated by using experimental data.

The newly proposed model follows the assumptions of the traditional soft string model, as follows:

• The drill string is considered as a soft string, and the bending stiffness is neglected;
• The drill string is in continuous contact with the wellbore.

For the sake of simplicity, the commonly used Cartesian coordinates with unit vectors $\widehat{i},\widehat{j},\widehat{k}$ are transformed into the Frenet–Serret coordinates with unit vectors $\widehat{t},\widehat{n},\widehat{b}$, as shown in Figure 4. The relation between the direction cosines of Cartesian and Frenet–Serret coordinate systems is represented by Equations (1)–(3).

$$\widehat{\mathrm{t}}=\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\phi }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\vartheta }\right)\widehat{\mathrm{i}}+\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\phi }\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\vartheta }\right)\widehat{\mathrm{j}}+\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\phi }\widehat{\mathrm{k}},$$

(1)

$$\widehat{\mathrm{n}}=\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\phi }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\vartheta }\right)\widehat{\mathrm{i}}+\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\phi }\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\vartheta }\right)\widehat{\mathrm{j}}-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\phi }\widehat{\mathrm{k}},$$

(2)

$$\widehat{\mathrm{b}}=-\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\vartheta }\widehat{\mathrm{i}}+\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\vartheta }\widehat{\mathrm{j}},$$

(3)

where φ is the inclination angle and ϑ is the azimuth angle.

Starting from the free body diagram of one drill string element and following a similar approach as in existing works [19,49], the force and moment equilibrium equations for a rotating drill string can be obtained. The modifications are the additional forces from the cuttings bed, as shown in Equations (4)–(8). For example, for the cases of tripping in, the additional resistance force F_cb is applied opposite to the axial direction to represent the effect of the cuttings bed on the axial force. For a rotating drill string, the additional resistance force F_cb would be tangential to the drill string wall curve and pointing in the opposite direction of the rotating direction.

$${\displaystyle \frac{d{F}_t}{ds}}+w_{bp}cos\phi t_z=0,$$

(4)

$$F_{t}k{+w}_{bp}{n}_z+w_{c}cos\theta +\mu w_{c}sin\theta +{\mathrm{F}}_{cb}cos\theta +\mu {\mathrm{F}}_{cb}sin\theta =0,$$

(5)

$${\displaystyle \frac{d{M}_t}{ds}}-\mu \left(w_c+{\mathrm{F}}_{cb}\right)r_p=0,$$

(6)

$$w_{bp}{b}_z-w_{c}sin\theta +\mu w_{c}sin\theta -{\mathrm{F}}_{cb}cos\theta +\mu {\mathrm{F}}_{cb}sin\theta =0,$$

(7)

$$M_t\kappa =0,$$

(8)

where F_t is force in the tangent direction (axial force); w_bp is the unit weight of the drill string; w_c is the contact force; n_z and b_z are the normal and binomial Frenet–Serret unit vectors in the z direction, respectively; θ is the direction of the normal contact force component in the $\widehat{n}-\widehat{\mathrm{b}}$ plane; F_cb is the additional force caused by the cuttings bed; $k$ is the pipe curvature; $\mu$ is the friction coefficient; and $M_t$ is the magnitude of moment (torque) required for pipe rotation, the product of the tangential force, friction coefficient $\mu$, and rotation radius, where the tangential force direction aligns with the tangent of the drill string’s motion.

The unit contact force and θ can be obtained by Equations (9) and (10).

$$w_c=\sqrt{{\displaystyle \frac{{{(F}_{t}k{+w}_{bp}{n}_z)}^2+{\left(w_{bp}{b}_z\right)}^2}{1+u^2}}},$$

(9)

$$\theta =\arctan \mu -arctan\left({\displaystyle \frac{w_{bp}{b}_z}{F_{t}k{+w}_{bp}{n}_z}}\right)$$

(10)

The algorithm to solve the modified soft string model is similar to the standard torque and drag model, except that at each element, the additional resistance force F_cb needs to be calculated based on the local cuttings bed height.

The dynamic hole cleaning model discussed in the previous section is run by taking the drilling operational parameters as inputs and predicts the dynamic cuttings bed distribution along the wellbore. The modified soft string model takes the predicted cuttings bed distribution results as inputs, then calculates the local F_cb at each discretized point, and at last solves for the torque and drag along the string. A key point in this approach is obtaining the relation between F_cb and the cuttings bed height, which is discussed in the following experimental investigation section.

4. Experimental Investigation of Effect of Cuttings on Drill String

The Wellbore Flow and Intelligent Drilling Test (WFIDT) facility at Yangtze University was used to conduct experimental tests on the effects of cuttings on the force acting on the drill string. Graphical information of the facility is shown in Figure 5. It contains an annulus test section, a circulation and solid handling module, a hoisting module, a pipe rotating and axial moving module, and a data collection and control module. The test section is composed of a transparent outer pipe to simulate the wellbore and a metal inner pipe to simulate the drill string. The transparent outer pipe allows observation of the height of the cuttings bed and the friction testing process. The length of the test sections is 15 m, and there are several size combinations for the annulus geometry, which include 88.9 × 139.7 mm, 50.8 × 101.6 mm, and 25.4 × 50.8 mm. Driven by two different motors, the inner pipe can rotate and move in the axial direction inside the wellbore. The maximum rotation speed is 150 RPM, and the maximum moving distance is 20 mm (compressed or stretched in the axial direction by the axial motor). The torque and axial forces on the drill string can be recorded by using the sensors arranged at both ends of the drill string during a test. The inclination angle of the test section can vary between horizontal and vertical by using a hoisting system. The circulation and solid handling module include a mud pump, a solid injection system that injects cuttings into a fluid flow, and a solid collection system that separates the cuttings from the fluid. During a test, the cuttings movement and fluid flow can be directly observed from the test section. Pressure and temperature data can be obtained from a number of sensors distributed along the test section.

A test to investigate effect of cuttings on the drill string include the following steps: (i) start the system and circulate the system with pure drilling fluid to clean the wellbore and flow lines; (ii) rotate the drill string at a desired speed and record the steady torque values at both ends; (iii) stop the rotation, move the drill string in the axial direction, and record the axial load at both ends; (iv) inject cuttings into the system and conduct a normal cuttings transport test until the cuttings distribution reaches steady state; (v) record the cuttings bed height; (vi) rotate the drill string at a desired speed with cuttings in the wellbore and record the steady torque value at both ends; (vii) stop rotation, move the drill string in the axial direction with cuttings in the wellbore, and record the axial load at both ends; and (viii) flush the system. Six parameters are measured from a test, which include the torque value during rotation without cuttings, axial load during axial motion without cuttings, cuttings bed height, pressure drop, torque value during rotation with cuttings, and axial load during axial motion with cuttings. The same experimental procedure is carried out under different conditions of the cuttings bed height. Each experiment is repeated three times to improve the accuracy of the test results. The experiment is conducted at room temperature and atmospheric pressure.

In addition to the newly conducted tests results, existing data from the literature were also used in the analysis [21,54,58,59]. The matrix for all available data used to develop the correlations in the sections is shown in

Table 1. Test matrix for the analyzed data.

Name	Value	Unit
Mean fluid velocity	0.0015~0.031	m/s
ROP	5~30	m/h
Pipe rotation speed	0~100	RPM
Wellbore diameter	51~203	Mm
Drill pipe diameter	25~114	Mm
Inclination angle	30~90	degree
Test section length	15~25	M

. All data were obtained from tests conducted using water-based drilling fluids, where the behavior index is 0.814, and the consistency index is 0.065 pa·sⁿ. Quartz sand was used as the cuttings, with a density of 2667 kg/m³ and a particle size range of 2–5.66 mm.

The changes in torque and axial load for conditions with and without cuttings under the same test conditions are analyzed, which were used to develop correlations in the following sections.

4.1. Effect of Cuttings Bed on Torque

For the purpose of convenience, the cuttings bed height and its effect on torque are normalized by using Equations (11) and (12), respectively.

$$\tilde{h}={\displaystyle \frac{h_{bed}}{D_w}},$$

(11)

$$TCR={\displaystyle \frac{T_{\tilde{h}}-T_{clean}}{T_{clean}}}$$

(12)

where h_bed is the cuttings bed height, D_w is the well diameter, $\tilde{h}$ is the normalized bed height, $T_{\tilde{h}}$ is the measured torque value at the corresponding normalized cuttings bed height $\tilde{h}$, T_clean is the torque value without cuttings for the corresponding wellbore configurations, and TCR is the torque change ratio, which is caused by the cuttings bed. After normalization, the relationship between the dimensionless bed height $\tilde{h}$ and the torque change ratio TCR is shown in Figure 6.

From Figure 6, it can be seen that the relationship between the normalized bed height and the torque change ratio roughly follows an exponential function. Thus, an exponential function correlation is developed based on the data set, which can be expressed by Equation (13).

$$TCR=\alpha {\tilde{h}}^{\beta },$$

(13)

where α = 20 and β = 2.5 are two constants.

Because the tests were conducted in facilities that simulate a part of a straight hole, the torque value is proportional to two factors: the contact force between the wellbore and the drill pipe, and the friction coefficient between the wellbore and the drill pipe. For the results shown in Figure 6, the contact force remains constant throughout the tests, and the change in torque is only caused by the change in the friction coefficient. Because the parameters are dimensionless, Equation (13) can be converted to represent the relationship between the normalized bed height and the change in the additional circumferential friction coefficient caused by the cutting bed between the wellbore and the drill pipe, which is shown in Equation (14).

$$f_{CR}=\kappa \tilde{h}^\eta ,$$

(14)

where κ = 6.78 and η = 0.88 are two constants. f_CR is the additional circumferential friction coefficient caused by the cuttings bed, which is defined by Equation (15).

$$f_{CR}={\displaystyle \frac{f_{\tilde{h}}-f_{clean}}{f_{clean}}},$$

(15)

where $f_{\tilde{h}}$ is the friction coefficient with normalized cuttings bed height, $\tilde{h}$, and $f_{clean}$ is the friction coefficient without cuttings. For torque calculation in a complex well with cuttings beds, Equation (16) can be applied to the torque and drag model discussed in the previous section with the given local bed heights, which can be obtained from the transient cuttings transport model. It needs to be pointed out that $f_{CR}$ is an equivalent friction coefficient change ratio that includes all the changes in the friction force caused by the cuttings bed. It is defined by using the ratio between the friction force with cuttings beds and the original contact force for no cuttings conditions. Thus, it is a representation of the changes in friction forces caused by the cuttings bed instead of the actual friction coefficient.

4.2. Effect of Cuttings Bed on Axial Friction Force

For cases where the drill string moves axially, the relationship between the axial friction force and the dimensionless bed height $\tilde{h}$ can be represented by a linear relationship, as shown in Figure 7, which is consistent with the findings of other researchers [60].

The additional axial friction force from the cuttings bed can be obtained using Equation (16).

$$F_t=k\tilde{h}+c,$$

(16)

where k = 20.743 and c = 3.314 are two constants.

Following the definition in Equation (15), the relationship between the dimensionless bed height and additional equivalent sliding friction coefficient caused by the cuttings bed can be expressed by Equation (17), which is applied to the drill string model discussed in the previous section to consider the effect of cuttings on the axial friction force.

$$f_s=a\tilde{h}+b,$$

(17)

where a, b are empirical coefficients. It needs to be pointed out that a number of parameters can change in practical drilling applications compared to the experimental tests. The purpose of the experimental study was to find a general formation that can be used in the modified drill string model. The correlations developed from the experimental data may not be applicable for all situations. For conditions that were not covered by the test matrix, the fitting parameters can vary significantly, but the trend should be the same. In other words, Equations (14) and (17) are still valid, but the equivalent coefficients may be different for different scenarios. More new data with new types of drilling fluids and pipes would be valuable to make the model applicable for a broader range of situations. The model can also be improved by fitting the model with field data, which was performed in the following case study, and the procedures to tune the model are listed as follows:

• Collect benchmark data for the standard drill string model, which represents the friction forces on the drill string under the no cuttings scenario (the data measured when there is little cutting accumulated in the wellbore, e.g., after a thorough circulation or wiper trip was performed). The torque benchmark data can be obtained by selecting the surface torque value under off-bottom rotations, and the axial force benchmark data can be obtained from the hookload data under tripping in or pulling out.
• Construct the object function for standard drill string model tuning to obtain the friction coefficients for the specific well under no cuttings conditions:

$$\begin{gathered} \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}Z=F_{sd}\left(f_{clean}\right)-M_{nc}, \\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}{f}_{min}<f_{clean}<f_{max}, \end{gathered}$$

(18)

where Z is a two-component vector that includes the torque and axial force [torque, axial force], $F_{sd}\left(f_{claen}\right)$ is the standard drill string model, $M_{nc}$ is the benchmark data to tune the standard drill string model, $f_{min}$ and $f_{max}$ are the minimum and maximum possible friction coefficients, and $f_{clean}$ is the friction coefficient for the no cuttings condition, which can be a vector with more than one single number (e.g., coefficients for axial and radial directions, and for casing and open hole conditions).

3.

Collect tuning data for the coupled model, which represents the friction forces on the drill string with cuttings in the wellbore. The torque and axial force values can be obtained by using the same method in step 1, and the difference is that the data should be collected under conditions where there are cuttings in the wellbore. It should be noted that the data collection in this step should be conducted after a certain length of new well section was drilled after step 1, and all the operational parameters (e.g., ROP, pumping flow rate, rotation speed) should be collected to analyze the downhole cuttings distributions, which is discussed in the next step.

4.

Run the transient hole cleaning model by using the operational data collected in step 3 as inputs, and estimate the downhole cuttings distribution for the given condition.

5.

Repeat steps 3 and 4 several times to obtain multiple data points for regression.

6.

Construct the object function for hole cleaning coupled drill string model tuning to obtain the friction coefficients:

$$\begin{gathered} \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}Z=F_{hc}\left(f_{CR},f_{clean},C_D\right)-M_c, \\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}{f}_{CRmin}<f_c<f_{CRmax}, \end{gathered}$$

(19)

where Z is a two-component vector that includes the torque and axial force [torque, axial force], $F_{hc}\left(f_{CR},f_{clean},C_D\right)$ is the hole cleaning coupled drill string model, $M_c$ is the benchmark data collected in step 3, $f_{CRmin}$ and $f_{CRmax}$ are the minimum and maximum possible coefficients, and $C_D$ is the cuttings distribution result along the wellbore obtained in step 4.

7.

Solve the objective functions in steps 2 and 6 by using proper optimization algorithms.

5. Case Study

In this section, the model developed in the previous sections is applied to practical drilling activities to demonstrate the effects of the dynamic cuttings distribution on the drill string. The well information for case analysis is shown in Figure 8. In this example, the kick-off depth is about 400 m, and the inclination angle reaches 85° at about 900 m and is kept constant thereafter up to 5845 m. The azimuth remains unchanged at 112.35°. The ratio of the horizontal displacement to the true vertical depth (HD/TVD) is 4.27. The bottom hole assembly (BHA) components are listed in

Table 2. BHA components.

Hole Size (mm)	Name	Outer Diameter (mm)	Weight per Meter (Kg/m)
φ311.15	Drill pipe	139.7	35.37
	Heavyweight Drill pipe	139.7(5)	89.44
	Jar	203.2(1)	229.71
	NMDC	203.2(1)	227.33
	LWD	209.55(1)	212.55
	MWD	209.55(1)	210.00
	Mud motor	228.6(1)	200.25
	Drill Bit	311.15(1)	397.34
φ215.9	Drill pipe	127	33.13
	Heavyweight Drill pipe	139.7(2)	77.43
	Jar	165.1(1)	136.60
	Drill collar	165.1(1)	132.24
	NMDC	171.45(1)	143.92
	LWD	171.45(1)	150.00
	MWD	171.45(1)	150.00
	Mud motor	171.45(1)	127.30
	Drill Bit	215.9(1)	133.93

, and the drilling fluid information is shown in

Table 3. Fluid Properties.

Wellbore Size (mm)	Mud Type	Density (g/cm3)	Yield Point (Pa)	Plastic Viscosity (mPa·s)
φ311.15	Synthetic	1.18	19.5	24
φ215.9	Synthetic	1.13	12	15

.

The rate of penetration (ROP) during the drilling process was 40 m/h, and the well was circulated for about 45 min after each stand was drilled. The operational parameters during a typical drilling cycle are shown in Figure 9. In this case study, we investigated the period during drilling the sections between ~4000 and 4200 m MD. It is assumed that the borehole has been completely cleaned before drilling this section because it is the start of the 215.9 mm wellbore section, which follows the 244.48 mm casing. It needs to be pointed out that the model was tuned to match the field measurements by adjusting the coefficients in the correlations for the friction factor, and details can be found in a previously published paper. The purpose of this case study was to demonstrate the value of the coupled model in practical drilling instead of validating the correlations developed in the previous section.

5.1. Dynamic Hole Cleaning Results

Before drilling the new wellbore section, there were no cuttings in the wellbore because a new casing was set in place. As the bit moved forward to drill the new formation, cuttings were generated and transported towards the wellhead from the bottom hole. Because of the alteration between penetration and circulation (the wellbore was cleaned by circulating with one bottom up (BU) after each stand was drilled), new drilled cuttings were feed into the annulus from the bit periodically. Therefore, the distribution of cuttings in the wellbore followed a wavy pattern, as shown in Figure 10. This figure shows the cuttings concentration along the entire wellbore at 45 min, 135 min, 225 min, 315 min, 405 min, and 495 min after the start of drilling the new section at 4000 m MD. These six time points represent the completion of drilling the first, second, third, fourth, fifth, and sixth stands, respectively.

At t = 45 min, the first cuttings bed wave was generated by cuttings from drilling the first stand. At this time, the cuttings bed was between ~3700 and 4000 m MD, and the maximum cuttings concentration was around 23% (equivalent to 80 mm cuttings bed height) in this section of the wellbore. There was a small amount of cuttings ahead of the cuttings bed wave, which were the cuttings suspended in the drilling fluid because the suspended cuttings traveled much faster than the cuttings bed. During circulation after drilling the first stand, the cuttings bed wave was pushed upward and eroded. Another cuttings bed wave was generated during drilling the second stand and then flushed upward once the penetration process was interrupted, as shown in Figure 10 at t = 135 min. Similarly, a cuttings bed wave was generated during drilling each of the following stands. As the cuttings bed waves were flushed towards the wellhead, the height of the waves diminished because the rate of cuttings feed into a specific wave (mainly from the cuttings in suspension mode) was less than the cuttings generation rate at the drill bit. The cuttings bed waves moved in the wellbore in a way similar to the motion of sand dunes in a desert. Particles were picked up at the upwind direction and deposited back at the other end of the dune. The moving speed of the cuttings bed waves decreased as their height decreased because the in situ flow velocity decreased. Thus, the newly generated waves could catch up with the older waves after a certain amount of time and formed a longer cuttings bed. Therefore, significant amounts of cuttings were accumulated in the wellbore after drilling for a long period of time, as shown at t = 495 min in Figure 10.

The reason for the accumulation of cuttings was that the deposited cuttings bed was not fully cleaned out of the wellbore during the circulation periods, either because the flow rate was not high enough or the circulation time was not long enough.

From a practical point of view, the flow rate used during drilling was constrained by a number of factors. However, to clean the well efficiently, the flow rate should be no less than the minimum required flow rate to suspend and transport the deposited cuttings when ROP is zero. In this scenario, increasing the circulation time became a practical choice to clean the well. One of the most important engineering problems during drilling long lateral wells is to optimize the circulation time, circulation frequency, and ROP. The constraints for the optimization include the effect of the cuttings bed on the ECD and the drill string, which are discussed in the following sections.

5.2. Effect of Dynamic Cuttings Distribution on ECD

It is generally accepted that cuttings can change the flow pressure loss in the annulus significantly, and details can be found in a previous study [54]. In ERD drilling, one of the constraints is that the ECD should not exceed the safe mud weight window. In practical drilling, the ECD increases as the horizontal lateral increases, while the safe mud weight window stays almost constant. There are two factors that contribute to the increase in ECD: the increase in well length and the cuttings. The transient cuttings transport model employed in this study can simulate both dynamic cuttings distributions (as shown in Figure 10) and the detailed effects of cuttings on the annulus pressure loss, as shown in Figure 11. It needs to be pointed out that to highlight the pressure changes caused by circulation and cuttings, the static pressure caused by the pure drilling fluid was eliminated from the overall pressure gradient.

By comparing the wellbore pressure gradient with and without considering the effect of cuttings in Figure 11 (the red solid line and blue dashed line), it can be seen that unevenly distributed cuttings in the horizontal lateral caused important changes in the pressure gradient. If there were no cuttings in the wellbore, the pressure gradient should be constant. At the locations with cuttings beds or dunes, the actual pressure gradients with cuttings considered increased accordingly. The cuttings-induced pressure loss led to significant changes in the ECD, as shown in Figure 12.

Figure 12 illustrates the changes in the ECD during the period of drilling the six stands. The yellow line and the blue line represent the ECD with and without considering the effects of cuttings, respectively. The red dots are the measured ECD from field data. The pink line represents the changes in the well measured depth (MD), which refers to the vertical axis on the right side of the figure. As the well depth increased, the two ECD values increased at different rates. The effect of the ECD considering cuttings increased faster than the ECD without considering cuttings. During the circulation periods (the time periods when the well depth was constant), the effect of the ECD without considering cuttings was kept constant, and the ECD with cuttings considered decreased as new cuttings generation stopped and the cuttings were transported out of the wellbore. If the circulation was sufficient and all the accumulated cuttings were cleaned away, the yellow line would decrease to the same value as the blue line. As more stands were being drilled, the difference between these two ECD values increased because the amount of cuttings accumulated in the wellbore increased. The results in Figure 12 are helpful to practical drilling applications in two ways: (i) they could be used to monitor the actual wellbore pressure and keep it within the safe window (especially in narrow mud weight window conditions), and (ii) they could be used to guide the circulation time and frequency to ensure the cuttings inside the wellbore are not imperiling normal drilling from the ECD perspective.

5.3. Effect of Dynamic Cuttings Distribution on Drill String

As discussed in previous sections, another important factor that limits long lateral drilling are the friction forces on the drill string (including the torque and axial force), which are also closely related to cuttings distributions. The torque and axial force gradients along the drill string during the penetration process at t = 495 min are shown in Figure 13. In the shallow parts of the well (MD < 2000 m), the effects of cuttings on the drill string were negligible because the cuttings concentrations were low and mainly suspended in drilling fluids. In the well sections with deposited cuttings beds, the friction forces on the drill string increased proportionally with the cuttings concentration compared to the no cuttings scenario, which was reflected in the differences between the axial force gradients and torque gradients for conditions with and without considering cuttings effects. As discussed in the previous sections, the reasons for increasing the friction forces as the cuttings concentration increases mainly include two aspects: (i) the averaged friction coefficients caused by the cuttings bed around the drill pipe increase because the contact area between the pipe and cuttings bed increases; and (ii) the cuttings bed constrains the movement of the drill string and adds additional contact forces, e.g., part of the cuttings bed’s weight is borne by the drill string.

In practical drilling applications, one straightforward way to reflect and monitor forces on the drill string is to use surface torque and actual WOB during penetration. High torque could lead to severe damage to the drill string or stuck pipe incidents, and low WOB could lead to a decrease in the ROP. The changes in surface torque and WOB during the drilling of the six stands are shown in Figure 14. As shown in Figure 14, if cuttings were not considered, the surface torque increased slightly during the penetration process (because the well depth increased) and remained constant during the circulation process (the sharp changes between circulation and penetration were the torque from the drill bit, which can be neglected during circulation because the bit was off bottom). For conditions where the effect of cuttings was considered, the surface torque increased significantly during the penetration process as the cuttings accumulated in the wellbore. During the following circulation process, the surface torque decreased as the cuttings were removed out of the wellbore, which was consistent with the field data. The difference between the initial surface torque values at the beginning of drilling each stand for scenarios with and without considering cuttings grew larger as the well went deeper. This was another indication that the wellbore was not circulated enough to remove all the accumulated cuttings. From this figure, it can be seen that the increase in surface torque caused by cuttings was significant, and the upper limit for surface torque could be reached much earlier than in ideal scenarios where there was no cuttings accumulation.

Another commonly encountered obstacle during drilling long lateral wells is insufficient WOB, which is also closely related to dynamic cuttings distributions. Figure 14 also shows the simulation of WOB by assuming that the hook load applied at the surface was constant. If there were no cuttings in the wellbore, the theoretical WOB would decrease slightly as the well depth increased, and it should not decrease significantly after drilling the six stands. However, insufficient WOB and low ROP problems were reported during actual drilling, which can be explained by the results for the scenario with considering the effect of cuttings on the drill string. With the same initial WOB at the beginning (t = 0 min), the actual WOB that considers the effect of cuttings decreased significantly after each stand was drilled. Although some recovery was gained after circulation, it was significantly smaller than the theoretical value. The difference grew as the well depth increased because more cuttings accumulated in the wellbore.

For the insufficient WOB problem, one straightforward solution is to apply more weight on the drill string. If we kept the actual WOB the same as the initial value (42.5 kN) during the drilling of the six stands, the axial force along the entire drill string could be represented by Figure 15. Because more weight was applied to compensate for the friction loss caused by cuttings, the neutral points moved upward as the well depth increased and the drill string became closer to the buckling criteria. As shown in the small embedded graph within Figure 15, within the depth interval of 350 m to 400 m in the wellbore, the green curve exceeds the critical limit for sinusoidal buckling, while the blue curve surpasses the critical limit for helical buckling. Sinusoidal buckling could occur at 225 min (three stands were drilled), and helical buckling could occur at 495 min (six stands were drilled).

Therefore, the cuttings dynamic distribution could impact the friction forces on the drill string significantly, which leads to an increase in torque, a decrease in WOB, and an increase in buckling risk. Similar to the effect of cuttings on the ECD, these problems can be avoided or delayed if the cuttings in the wellbore are managed properly by optimizing the drilling parameters. In the following section, a parameter optimization method is demonstrated by using the proposed coupled model.

5.4. Drilling Parameter Optimization

In the following case, we looked into four parameters that can be easily changed during practical drilling (mean fluid velocity (equivalent to the pumping flow rate), ROP, circulation time, and circulation frequency) and converted it into a constrained optimization problem.

Different mean drilling fluid velocities were applied to the case discussed in the previous section (stopping drilling and circulating for 45 min after drilling each stand), and the dynamic ECD results are shown in Figure 16 (all other parameters were unchanged). From this figure, it can be seen that the ECD for pure drilling fluid flow increased as the mean drilling fluid velocity increased. However, the relationship between the velocity and ECD for the scenarios considering the effects of cuttings was more complicated. At low velocity (1.1 m/s), the ECD increased quickly as the well depth increased, and the slope became less steep as the velocity increased, which was opposite to the scenarios without considering cuttings. The minimum ECD value at t = 540 min (circulated 45 min after the sixth stand was drilled) was from 1.3 m/s, and the maximum value was from 1.1 m/s. The high ECD values at low velocities (1.1 and 1.2 m/s) were caused by accumulated cuttings (which indicated that hole cleaning was not sufficient at those conditions). At 1.3 m/s, 45 min of circulation was able to remove all cuttings in the wellbore (the ECD with cuttings fell back to the value of the ECD without cuttings at the end of each circulation, as shown in the figure). A further increase in velocity was no longer helpful in decreasing the ECD because the hole cleaning was sufficient beyond 1.3 m/s, and it led to an increase in the ECD at 1.4 m/s.

Figure 17 shows the corresponding changes in WOB and surface torque at the same operational conditions. This figure reiterates the fact that the accumulation of cuttings can significantly reduce the effective WOB and increase the surface torque for long lateral drilling. These two parameters can recover to minimum values (values that did not consider the effect of cuttings) with sufficient circulation, and the corresponding minimum mean fluid velocity was 1.3 m/s. From the results in Figure 16 and Figure 17, it can be seen that the key point to avoid ECD- and drill string-related problems during long lateral drilling is to ensure sufficient hole cleaning.

Figure 16 and Figure 17 indicate that sufficient circulation (which means the removal of all accumulated cuttings during drilling the previous stand) is a critical point to avoid the escalation of ECD- and drill string-related problems, and it can be improved by using a sufficient flow rate (which is equivalent to the mean fluid velocity). In practical drilling applications, the pumping flow rate is constrained by equipment or wellbore formation conditions. In these scenarios, a similar outcome can be achieved by adjusting the ROP or circulation time. Figure 18 shows the minimum circulation time required to remove all cuttings in the wellbore after drilling one stand at different ROP and mean drilling fluid velocities. The first step of the optimization is to determine the maximum possible mean fluid velocity. Assuming that the objective of the optimization is to minimize the overall drilling time (which is a function of the ROP and circulation time), the optimized parameters can be obtained by minimizing Equation (20) in a constrained 2D domain defined by Figure 18, where l_w is the length of the well lateral to be drilled, $l_{stand}$ is the length of each stand, and CT is the minimum circulation time.

$$Minf(ROP,CT)=Min({\displaystyle \frac{l_w}{ROP}}+CT{\displaystyle \frac{l_w}{l_{stand}}}),$$

(20)

Another factor that can be adjusted is the circulation frequency, or the number of stands that can be drilled without stopping for circulation. Figure 19 and Figure 20 illustrate the maximum possible ECD and worst case drill string axial force distributions for different scenarios, respectively. Figure 19 indicates that the maximum ECD increased sharply as the circulation frequency decreased, and Figure 20 indicates that the drill string became more likely to buckle as the number of consecutive stands drilled increases; the position where the drill string is most likely to buckle was around 400 m MD. The number of stands that could be drilled continuously can be obtained by applying constraints from the safe mud weight window, maximum surface torque, and drill string buckling status (two stands to avoid sinusoidal buckling and four stands to avoid helical buckling in this case). Although the numbers can change based on cases and well depth, the method can be applied to other different cases, which is of significant value to practical drilling applications.

6. Conclusions

(1) Through an experimental system, the influence of cuttings beds on the mechanical behavior of drill strings was investigated. It was found that the cuttings bed height significantly impacts the torque and frictional resistance: the torque increase rate exhibits exponential growth with increasing cuttings bed height, while the frictional resistance increases linearly. Based on these mechanisms, a localized correlation formula was developed to describe the relationship between the cuttings bed height and changes in drill string forces. This provides a quantitative tool for accurately evaluating the impact of cuttings beds on the torque and drag during the drilling process.

(2) By combining the localized correlation formula, transient cuttings transport model, and soft string model, a dynamic coupled model was developed to simulate the uneven distribution of cuttings beds in the wellbore and their dynamic impact on the drill string torque and drag. The simulation results show that the variations in the axial force and torque gradient are closely related to the location of cuttings accumulation. The uneven distribution of the cuttings bed leads to a more than 30% increase in the fluctuation amplitude of the drill string torque gradient.

(3) Case validation indicates that the accumulation of cuttings beds significantly affects drilling parameters. When the cuttings accumulation exceeds 25% of the wellbore cross-sectional area, the weight on bit (WOB) decreases by 281.09%, and the surface torque increases by more than 11.29%. Additionally, high cuttings accumulation leads to an increase in the equivalent circulation density (ECD), raising the risk of drill string buckling and, in severe cases, potentially causing helical buckling.

(4) In actual extended reach drilling operations, it is crucial to account for the relationship between the dynamic distribution of cuttings beds and constraints such as the ECD limit, drill string buckling risk, and surface torque limitations. Optimizing the circulation frequency or continuous drilling length is essential to ensure drilling safety. This paper proposes a novel optimization method based on the objective of minimizing the drilling time. By solving within a constrained 2D domain, the method determines the optimal set of operational parameters, providing scientific support for safe and efficient drilling under complex wellbore conditions.