Mechanism of Cuttings Removing at the Bottom Hole by Pulsed Jet

Abstract

Vibration drilling technology induced by hydraulic pulse can assist the bit in breaking rock at deep formation. Simultaneously, the pulsed jet generated by the hydraulic pulse promotes removal of the cuttings from the bottom hole. Nowadays, the cuttings removal mechanism of the pulsed jet is not clear, which causes cuttings to accumulate at the bottom hole and increases the risk of repeated cutting. In this paper, a pressure-flow rate fluctuation model is established to analyze the fluctuation characteristics of the pulsed jet at the bottom hole. Based on the model, the effects of displacement, well depth, drilling fluid viscosity, and flow area of the pulsed jet tool on the feature of instantaneous flow at the bottom hole are discussed. The results show that the pulsed jet causes flow rate and pressure to fluctuate at the bottom hole. When the displacement changes from 20 L/s to 40 L/s in a 2000 m well, the pulsed jet generates a flow rate fluctuation of 4–9 L/s and pressure fluctuation of 0.1–0.5 MPa at the bottom hole. With the flow area of the tool increasing from 2 cm² to 4 cm², the amplitude of flow rate fluctuation decreases by 72.5%, and the amplitude of pressure fluctuation decreases by more than 60%. Combined with the fluctuation feature of the flow field and the water jet attenuation law at the bottom hole, the force acting on the cuttings under the pulsed jet is derived. It is found that flow rate fluctuation improves the mechanical state of cuttings and is beneficial for cuttings tumbled off the bottom hole. This research provides theoretical guidance for pulsed jet cuttings cleaning at the bottom hole.

1. Introduction

With the consumption of conventional fossil energy, unconventional oil and gas have become more and more important. In addition to hydraulic fracturing and horizontal wells, increasing the rate of penetration (ROP) is essential for the efficient development of unconventional oil and gas resources [1,2,3]. As an efficient rock-breaking method, the water jet has gradually been applied in mining and oil drilling [4]. With in-depth study of the water jet, it is found that pulsed jet shows a better rock-breaking effect than continuous jet.

There are two methods to modulate a continuous jet into a pulsed jet. One is to change the instantaneous flow mechanically, by adjusting the flow area to generate pressure fluctuations and pulsed jet. The other is to design the channel structure based on principles of hydroacoustics to modulate continuous jets into a pulsed jet [5]. In the 1980s, the pulsed jet nozzle was developed and applied to the drill bit [6]. The pulsed jet nozzle can form a small flow area and then form a high jet impact force, which is helpful for wellbore cleaning [6,7]. Compared to the cone-straight nozzle, the erosion effect of the pulse nozzle is significantly improved. Through investigation of the pulsed jet nozzle, it is found that the pulsed jet nozzle can increase instantaneous dynamic pressure at the bottom hole, which is conducive to cuttings migration at the bottom hole. Due to the small size of the nozzle, it generates a pulsed jet with relatively weak energy at the bottom hole. Therefore, many scholars began to develop pulsed jet tools. Tempress Technology Company of the United States developed the HydroPulse negative pressure pulse tool [8]. Through field experiments, it was found that the ROP could be increased by 33% to 200%. The principle of the tool is to control the valve switch by the drilling fluid, which causes the fluid to flow discontinuously. The discontinuous flow forms a pulsed jet at the bottom hole. The Waltech Company designed a negative pressure pulse tool based on change of fluid flow direction through the Coanda principle [9,10]. It generates a pulsed jet at the bottom hole to enhance the ROP. The hydraulic pulsed cavitating jet-assisted drilling tool was invented by the China University of Petroleum [11]. The tool has been popularized and widely applied in oil and gas fields in China and abroad. It has achieved an obvious effect on improving the ROP [12,13].

Predecessors have carried out some research on rock-breaking properties of the pulsed jet. Experimental results indicate that pulsed jet has a stronger eroding capacity than continuous jet [14,15]. In order to visualize the pulsed jet flow field, shadowgraph technology, combined with particle image velocimetry (PIV) method, is used to capture the flow field feature of the pulsed jet [16]. In order to obtain the fluctuation feature of the pulsed jet at the bottom hole, a numerical model is established, which is based on unstable flow in the wellbore [17,18]. In addition, the fluctuation of the pressure at the bottom hole is important for an oil well. On the one hand, it provides many pieces of information related to the formation in the well test [19,20,21]. On the other hand, it improves the magnitude of the cutting at the bottom in drilling.

Many scholars have carried out abundant research on the rock-breaking and ROP enhancing effect of the pulsed jet. However, research on cuttings cleaning of the pulsed jet at the bottom hole is still relatively lacking. In this paper, a pressure-flow rate fluctuation model at the bottom hole is established. The method of characteristics is adopted to solve the model. Based on the feature of pressure-flow rate fluctuation and water jet attenuation law at the bottom hole, the force acted on the cuttings at the bottom hole is analyzed. The effects of different displacements, well depths, drilling fluid viscosities, and flow areas of pulse jet tools on instantaneous flow and pressure at the bottom hole in pulse jet drilling are analyzed. In addition, the influence of different bottom hole pressure differences and cutting sizes on the force of cuttings is analyzed. The paper attempts to describe the cuttings clearing mechanism at the bottom hole of the pulsed jet with a quantitative method.

2. Modeling of Pressure-Flow Rate Fluctuation at the Bottom Hole

2.1. Modeling

Figure 1 shows the conventional drilling process. In a drilling process, the drill fluid flows through the kelly, drill pipe, and collar to the bit in sequence. After the fluid flows through the nozzles on the bit, the jet is formed at the bottom hole. The jet cleans the cuttings at the bottom hole and assists the bit in breaking the rock. Then, the fluid flows outside of the well head through the annulus, carrying the cuttings. For pulsed jet drilling, a pulsed jet generator is often installed above the drill bit to modulate the continuous flow into a pulsed jet at the bottom hole. Due to complex drilling conditions, there are some reasonable assumptions in the model:

(1)

It is assumed that the wellbore is vertical.

(2)

It is assumed that the size of the drill pipe is identical. The flow characteristics at the joint are ignored.

(3)

It is assumed that the drilling string contains only the drill pipe, not the collars and other bottom hole assembly.

(4)

The difference between the inner and outer diameter of the pulse jet generator and the drill pipe is ignored.

(5)

The flow in the wellbore is one-dimensional. The radial flow in the drill pipe and annulus is ignored.

Figure 1. Schematic of drilling.

Figure 1. Schematic of drilling.

Figure 2 shows a schematic of the fluid flow in the wellbore during the drilling process. Based on the control volume 1—1—2—2 shown in Figure 2, the mass conservation equation and momentum equation in the drill pipe are established, respectively, as shown in Equations (1) and (2).

$$\frac{\partial }{\partial x}\left(\rho v{A}_p\right)+\frac{\partial }{\partial t}\left(\rho A_p\right)=0$$

(1)

$$\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}+\frac{1}{\rho }\frac{\partial p}{\partial x}+2\frac{{\tau }_p}{\rho r_{pi}}-g=0$$

(2)

where $A_p=\pi r_{pi}{}^2$ is the flow area of the drill pipe, v is the fluid velocity, ρ is the density of the fluid, ${\tau }_p$ is the shear stress on the inner wall of the drill pipe, $r_{pi}$ is the radius of the drill pipe.

Expanding Equation (1) and introducing the full derivative equation, Equation (3) can be derived:

$$\frac{1}{A_p}\frac{d{A}_p}{dt}+\frac{1}{\rho }\frac{d\rho }{dt}+\frac{\partial v}{\partial x}$$

(3)

The first term and the second term on the left side of Equation (3) are respectively related to the variation rate of the pipe section and the density of the fluid, which are caused by the pressure variation of the elastic wave. The variation rates are relative to the compressibility coefficient of the fluid and the expansion coefficient of the flow channel:

$$\alpha =\frac{\frac{dV}{V}}{dp},\beta =\frac{\frac{d{A}_p}{A_p}}{dp}$$

(4)

where $\alpha$ is the compressibility coefficient of the fluid and $\beta$ is the expansion coefficient of the flow channel.

According to the total derivative of p, substitute Equation (4) into (3):

$$\left(\alpha +\beta \right)\frac{dp}{dt}+\frac{\partial v}{\partial x}=0\iff v\frac{\partial v}{\partial x}+\frac{\partial p}{\partial t}+\frac{1}{\alpha +\beta }\frac{\partial v}{\partial x}=0$$

(5)

The fluid compressibility coefficient and the flow channel expansion coefficient affect the propagation velocity of the pressure wave C in the pipe as follows:

$$\rho C^2=\frac{1}{\alpha +\beta }$$

(6)

Combining Equations (5) and (6), the fluid continuity equation in the pipeline under the condition of pressure fluctuation is obtained:

$$v\frac{\partial v}{\partial x}+\frac{\partial p}{\partial t}+\rho C^2\frac{\partial v}{\partial x}=0$$

(7)

In Equation (2), $2\frac{{\tau }_p}{\rho r_{pi}}$ is the pressure loss caused by friction between the fluid and the inner wall of the drill pipe, which is related to the flow velocity and the pipe diameter, which is equal to $\frac{f_{a}v\left|v\right|}{2D}$, where $f_a$ is the Darcy-Weisbach friction factor, and D is the inner diameter of the drill pipe.

For unsteady flow, the friction factor contains the effect of steady flow and fluid fluctuations [22]:

$$f_a=f_q+\frac{kD}{v\left|v\right|}\left[\frac{\partial v}{\partial t}+C\mathrm{sign}\left(v\right)\left|\frac{\partial v}{\partial x}\right|\right]$$

(8)

where k is the Brunone friction coefficient, and sign is the sign function.

$$k=\frac{\sqrt{C*}}{2}$$

(9)

where C* is the Vardy shear attenuation coefficient [23]:

$$\left\{\begin{array}{l}C*=0.00476 \mathrm{Laminar} \mathrm{flow}\\ C*=\frac{7.41}{{\mathrm{Re}}^{\log \left(\raisebox{1ex}{$14.3$}\!\left/ \!\raisebox{-1ex}{${\mathrm{Re}}^{0.05}$}\right.\right)}} \mathrm{Turbulence} \mathrm{flow}\end{array}\right.$$

(10)

Through derivation, the governing equations of fluid flow, considering pressure fluctuation, are obtained:

$$\left\{\begin{array}{l}\frac{\partial p}{\partial t}+v\frac{\partial p}{\partial x}+\rho c^2\frac{\partial v}{\partial x}=0\\ \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}+\frac{1}{\rho }\frac{\partial p}{\partial x}-g+\frac{fv\left|v\right|}{2D}+k\left[\frac{\partial v}{\partial t}+\mathrm{sign}\left(v\right)c\left|\frac{\partial v}{\partial x}\right|\right]=0\end{array}\right.$$

(11)

2.2. Model Solving

The governing equations in this paper are composed of the mass conservation equation and the momentum equation of the fluid. It is difficult to obtain the exact solution of the equation by the analytical method. Therefore, the numerical solution method is adopted. Through comparison, it is found that the mass conservation equation and the momentum equation can be transformed into a typical hyperbolic differential equation, and the partial differential equation can be transformed into an ordinary differential equation, using the method of characteristics (MOC). Compared with the direct finite difference equation, the MOC method has high solution accuracy and is easy to program [17,18].

According to the MOC method, the mass conservation equation and the momentum equation are linearly combined by the coefficient λ:

$$\begin{array}{r}\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}+\frac{1}{\rho }\frac{\partial p}{\partial x}-g+\frac{fv\left|v\right|}{2D}+k\left[\frac{\partial v}{\partial t}+sign\left(v\right)c\left|\frac{\partial v}{\partial x}\right|\right]+\\ \lambda \left(\frac{\partial p}{\partial t}+v\frac{\partial p}{\partial x}+\rho c^2\frac{\partial v}{\partial x}\right)=0\end{array}$$

(12)

The total derivatives of velocity and pressure to time are introduced as:

$$\left\{\begin{array}{l}\frac{dp}{dt}=\frac{\partial p}{\partial t}+\frac{\partial p}{\partial x}\frac{dx}{dt}\\ \frac{dv}{dt}=\frac{\partial v}{\partial t}+\frac{\partial v}{\partial x}\frac{dx}{dt}\end{array}\right.$$

(13)

It is found that the terms on the left side of Equation (12) can be combined into a form of total derivatives of velocity and pressure, as follows:

$$\left\{\begin{array}{l}\left(1+k\right)\frac{\partial v}{\partial t}+\left[v+k\mathrm{sign}\left(v\frac{\partial v}{\partial z}\right)c+\lambda \rho c^2\right]\frac{\partial v}{\partial z}=\left(1+k\right)\left[\frac{\partial v}{\partial t}+\frac{B}{1+k}\frac{\partial v}{\partial z}\right]\\ \frac{1}{\rho }\frac{\partial p}{\partial z}+\lambda \frac{\partial p}{\partial t}+\lambda v\frac{\partial p}{\partial z}=\lambda \left[\frac{\partial p}{\partial t}+\frac{D}{\lambda }\frac{\partial p}{\partial z}\right]\end{array}\right.$$

(14)

where $B=v+k\mathrm{sign}\left(v\frac{\partial v}{\partial z}\right)c+\lambda \rho c^2$, $D=\frac{1}{\rho }+\lambda v$.

When the coefficient λ satisfies Equation (15), the linear combination of the mass conservation equation and the momentum equation is transformed into an ordinary differential equation, shown as Equation (16).

$$\rho c^2{\lambda }^2+\left[E-\left(1+k\right)v\right]\lambda -\left(1+k\right)\frac{1}{\rho }=0$$

(15)

where $E=v+k\mathrm{sign}\left(v\frac{\partial v}{\partial x}\right)c$.

$$\left(1+k\right)\frac{dv}{dt}+\lambda \frac{dp}{dt}-g+\frac{fv\left|v\right|}{2D}=0$$

(16)

The value of λ can be calculated from Equation (15):

$${\lambda }^{\pm }=\frac{\left(1+k\right)v-E\pm \sqrt{{\left[E-\left(1+k\right)v\right]}^2+4\rho c^2\left(1+k\right)\frac{1}{\rho }}}{2\rho c^2}$$

(17)

In order to solve the equation, a grid of space x and time t is established, as shown in Figure 3. The red dotted line LN is the forward characteristic line, corresponding to the previous characteristic equation in Equation (18); the blue dotted line RN is the backward characteristic line, corresponding to the latter characteristic equation in Equation (18).

$$\left\{\begin{array}{l}{\frac{dx}{dt}}^{+}=\frac{1}{\rho {\lambda }^{+}}+v\Rightarrow {\frac{dt}{dx}}^{+}=\frac{\rho {\lambda }^{+}}{1+\rho {\lambda }^{+}v}\\ {\frac{dx}{dt}}^{-}=\frac{1}{\rho {\lambda }^{-}}+v\Rightarrow {\frac{dt}{dx}}^{-}=\frac{\rho {\lambda }^{-}}{1+\rho {\lambda }^{-}v}\end{array}\right.$$

(18)

As shown in Figure 3, since point N satisfies the two characteristic line equations of LN and RN, the equations, including the pressure and velocity of points L, N, and R, are established based on Equation (16). The first equation in Equation (19) corresponds to the characteristic line LN, and the second equation corresponds to the characteristic line RN.

$$\left\{\begin{array}{l}\left(1+k_N\right)\left(v_N-v_L\right)+{\lambda }_N{}^{+}\left(p_N-p_L\right)+\left(\frac{f{v}_N\left|v_N\right|}{2D}-g\right)△t=0\\ \left(1+k_N\right)\left(v_N-v_R\right)+{\lambda }_N{}^{-}\left(p_N-p_R\right)+\left(\frac{f{v}_N\left|v_N\right|}{2D}-g\right)△t=0\end{array}\right.$$

(19)

The pressure and velocity relationship between the point N and points L and R can be derived from Equation (19), as shown in Equation (20). In addition, in each time step, the pressure and velocity at points L and R can be calculated from the values of points W and E using linear interpolation. Therefore, an explicit relationship of the governing equations at every point in Figure 3 is obtained using the MOC method.

$$\left\{\begin{array}{l}{p}_N=\frac{\left(1+k_N\right)\left(v_R-v_L\right)-{\lambda }_N{}^{+}{p}_L+{\lambda }_N{}^{-}{p}_R}{\left({\lambda }_N{}^{-}-{\lambda }_N{}^{+}\right)}\\ v_N=v_L-{\lambda }_N{}^{+}\frac{\left(1+k_N\right)\left(v_R-v_L\right)-{\lambda }_N{}^{+}{p}_L+{\lambda }_N{}^{-}{p}_R}{\left({\lambda }_N{}^{-}-{\lambda }_N{}^{+}\right)\left(1+k_N\right)}+\frac{{\lambda }_N{}^{+}{p}_L}{\left(1+k_N\right)}\\ -\left(\frac{f{v}_N\left|v_N\right|}{2D}-g\right)\frac{△t}{\left(1+k_N\right)}\end{array}\right.$$

(20)

2.3. Initial and Boundary Conditions

Figure 4 shows the boundary conditions and the distribution of spatial nodes during the model solution process. Based on the assumption mentioned above, the boundary conditions include the drill string inlet (mud pump outlet), the pulse jet tool and drill bit, and the annular outlet (mud tank).

The boundary condition at the drill string inlet is the outlet of the mud pump. The mud pump is set to a constant displacement Q₀. The pressure at the inlet of the drill string changes with pressure fluctuations in the wellbore. Based on the MOC method, the pressure at the inlet can be calculated as follows:

$$\left\{\begin{array}{l}{Q}_1^{j+1}=Q_0\iff v_1^{j+1}=\frac{Q_0}{A_a}\\ p_R=p_1^j-\frac{1+\rho {\lambda }_1^{j+1}{}^{{}^{-}}{v}_1^{j+1}}{\rho {\lambda }_1^{j+1}{}^{{}^{-}}}\frac{△t}{△x}\left(p_2^j-p_1^j\right)\\ v_R=v_1^j-\frac{1+\rho {\lambda }_1^{j+1}{}^{{}^{-}}{v}_1^{j+1}}{\rho {\lambda }_1^{j+1}{}^{{}^{-}}}\frac{△t}{△x}\left(v_2^j-v_1^j\right)\\ p_1^{j+1}=p_R-\frac{\left(1+k_1^{j+1}\right)\left(v_1^{j+1}-v_R\right)+\left(\frac{f{v}_1^{j+1}\left|v_1^{j+1}\right|}{2D}-g\right)△t}{{\lambda }_1^{j+1}{}^{{}^{-}}}\end{array}\right.$$

(21)

The boundary condition at the tool and the bit are relative to the pressure drop and instantaneous flow rate. The pressure drop of the tool is equal to the pressure difference between the points $\frac{depth}{\Delta x}+1$ and $\frac{depth}{\Delta x}+3$, which can be obtained by the MOC method, as shown in Figure 4. Based on the relationship between the pressure drop and flow rate at the tool and bit, the fluid velocity of the pulsed jet tool can be calculated:

$$v_{tool}=\frac{-B+\sqrt{B^2-4AC}}{2A}$$

(22)

where

$$\begin{array}{l}A=\frac{f_a{\left(\frac{S}{A_a}\right)}^2△t}{2D_a{\lambda }_a{}^{{}^{-}}}-\frac{f_p{\left(\frac{S}{A_p}\right)}^2△t}{2D_p{\lambda }_p{}^{{}^{+}}}-\frac{1}{2}\rho \left(1-\frac{S}{A_p}\right)\left(\frac{3}{2}-\frac{S}{A_p}\right)-\frac{1}{2}\rho {\left(\frac{S}{C{A}_n}\right)}^2\\ B=\frac{\left(1+k_a\right)\frac{S}{A_a}}{{\lambda }_a{}^{{}^{-}}}-\frac{\left(1+k_p\right)\frac{S}{A_p}}{{\lambda }_p{}^{{}^{+}}}\\ C=p_L-p_R+\frac{\left(1+k_p\right)v_L}{{\lambda }_p{}^{{}^{+}}}+\frac{g△t}{{\lambda }_p{}^{{}^{+}}}-\frac{\left(1+k_a\right)v_R}{{\lambda }_a{}^{{}^{-}}}+\frac{g△t}{{\lambda }_a{}^{{}^{-}}}\end{array}$$

(23)

where the subscripts a and p correspond to the annuls and pipe, respectively. S is the flow area of the tool. The instantaneous pressure and flow rate of the tool and bit can be calculated from the velocity at the tool.

The pressure at the annulus outlet is 0 Pa, which is atmospheric pressure. The velocity is obtained by the MOC method, as shown in Equation (24).

$$\left\{\begin{array}{cc}\hfill & p_{2\frac{depth}{\Delta z}+3}^{j+1}=0\hfill \\ \hfill & v_L=v_{2\frac{depth}{\Delta z}+2}^j-\frac{1+\rho \lambda {{}_{2\frac{depth}{\Delta z}+3}^{j+1}}^{{}^{+}}{v}_{2\frac{depth}{\Delta z}+3}^{j+1}}{\rho \lambda {{}_{2\frac{depth}{\Delta z}+3}^{j+1}}^{{}^{+}}}\frac{\mathrm{\Delta t}}{\mathrm{\Delta z}}\left(v_{2\frac{depth}{\Delta z}+2}^j-v_{2\frac{depth}{\Delta z}+3}^j\right)\hfill \\ \hfill & p_L=p_{2\frac{depth}{\Delta z}+2}^j-\frac{1+\rho \lambda {{}_{2\frac{depth}{\Delta z}+3}^{j+1}}^{{}^{+}}{v}_{2\frac{depth}{\Delta z}+3}^{j+1}}{\rho \lambda {{}_{2\frac{depth}{\Delta z}+3}^{j+1}}^{{}^{+}}}\frac{\mathrm{\Delta t}}{\mathrm{\Delta z}}\left(p_{2\frac{depth}{\Delta z}+2}^j-p_{2\frac{depth}{\Delta z}+3}^j\right)\hfill \\ \hfill & v_{2\frac{depth}{\Delta z}+3}^{j+1}=v_L-\frac{\lambda {{}_{2\frac{depth}{\Delta z}+3}^{j+1}}^{{}^{+}}\left(p_{2\frac{depth}{\Delta z}+3}^{j+1}-p_L\right)+\left(\frac{f{v}_{2\frac{depth}{\Delta z}+3}^{j+1}\left|v_{2\frac{depth}{\Delta z}+3}^{j+1}\right|}{2D_a}+g\right)\mathrm{\Delta t}}{1+k_{2\frac{depth}{\Delta z}+3}^{j+1}}\hfill \end{array}\right.$$

(24)

At the initial time, the flow rate in the drill pipe and annulus is set as Q₀. Moreover, the pulsed jet tool is in a state of minimum flow area.

Figure 5 is the flowchart of solving the model. First, it is necessary to input the basic parameters and the initial and boundary conditions. Then, the step size for space and time is selected. The pressure and flow rate at points L and R in Figure 3 are calculated using the interpolation method. Based on the MOC method, the pressure and flow rate of each point at the next time step are calculated. Due to the update of the values of each point, it is necessary to evaluate the convergence of the pressure and flow rate. Finally, the results are obtained when the iteration of all the points is finished.

3. Cuttings Cleaning Model with Pulsed Jet

During the drilling process, the PDC bit teeth break the rock and generate fractures at the bottom hole. In addition, the pulsed jet vibration drilling technic promotes cracks by the vibration of the bit at the bottom hole [24,25]. Under the water jet, the cuttings escape from the bottom hole into the annulus and are carried out of the wellbore. For deep wells and ultra-deep wells, huge drilling fluid column pressure creates a chip hold-down effect, which prevents cuttings from breaking away from the bottom hole, resulting in repeated cutting of the drill bit, aggravation of the bit wear and reduction of the ROP. The pulsed jet can generate pressure and flow rate fluctuations at the bottom hole, and can reduce instantaneous bottom hole pressure and alleviate the chip hold-down effect [12,26].

In order to quantify the cuttings cleaning ability of the pulsed jet, it is necessary to establish a mechanical model of the cuttings under the pulsed jet at the bottom hole. Based on the analysis mentioned above, it assumes that the cuttings are spherical. The pressure gradient and collision between the cuttings are ignored. Figure 6 shows the force for a single particle at the bottom hole. As shown, the cuttings are to be held down in the fractures at the bottom hole. According to the assumptions, the drill fluid column pressure p_c combined with the pore pressure generates the chip hold-down effect on the cuttings. The forces acting on the cuttings include gravity G, buoyancy F_b (Equation (25)), pressure force F_c and F_p (Equation (26)), and the drag force of the water jet F_D (Equation (27)). As shown in Figure 6, the nozzle forms a certain angle with the bottom hole. The drag force can be divided into horizontal and vertical directions, corresponding to F_Dh and F_Dv, respectively. According to Newton’s third law, cuttings particles are also subjected to support force F_N and friction force F_f from the bottom hole.

$$\left\{\begin{array}{l}G=\frac{1}{6}\pi d_p{}^3{\rho }_{p}g\\ F_b=\frac{1}{6}\pi d_p{}^3{\rho }_{l}g\end{array}\right.$$

(25)

$$\left\{\begin{array}{l}{F}_c=p_a{s}_p\\ F_p=p_p{s}_p\end{array}\right.$$

(26)

$$F_D=\frac{1}{6}\pi d_p{}^3{\rho }_p\frac{\overrightarrow{u}-{\overrightarrow{u}}_p}{{\tau }_r}$$

(27)

where d_p is the diameter of the cuttings, ${\rho }_p$ and ${\rho }_l$ are the density of cuttings and fluid, ${\tau }_r$ is the relaxation time, which can be calculated from Equation (28) [27]:

$${\tau }_r=\frac{{\rho }_p{d}_p{}^2}{18\mu }\frac{24}{C_D{\mathrm{Re}}_p}$$

(28)

where $\mu$ is the viscosity of the fluid, $C_D$ is the drag coefficient, ${\mathrm{Re}}_p$ is the Reynolds number of the particle.

$${\mathrm{Re}}_p=\frac{{\rho }_l{d}_p\left|\overrightarrow{u}-{\overrightarrow{u}}_p\right|}{\mu }$$

(29)

where $\overrightarrow{u}$ is the fluid velocity at the bottom, which can be calculated from the water jet attenuation law, and ${\overrightarrow{u}}_p$ is the velocity of the cuttings, which is zero at the bottom hole.

In this paper, the Morsi and Alexander drag model [28] is adopted to calculate the drag force of the cuttings:

$$C_D=a_1+\frac{a_2}{{\mathrm{Re}}_p}+\frac{a_3}{{\mathrm{Re}}_p{}^2}$$

(30)

where a₁, a₂, a₃ are the coefficients corresponding to the particle Reynolds number, shown in

Table 1. The values of coefficient for spherical drag model. Adapted with permission from Elsevier, 2022 [29].

$\mathbf{The} \mathbf{Range} \mathbf{of}$Rep	a1	a2	a3
Rep < 0.1	0	24.0	0
$0.1 <$Rep < 1.0	3.69	22.73	0.0903
$1.0 <$Rep < 10.0	1.222	29.1667	−3.8889
$10.0 <$Rep < 100.0	0.6167	46.5	−116.67
$100.0 <$Rep < 1000.0	0.3644	98.33	−2778
$1000.0 <$Rep < 5000.0	0.357	148.62	−4.75 × 104
$5000.0<$Rep < 10,000.0	0.46	−490.546	57.87 × 104
$10,000.0 <$Rep < 50,000.0	0.5191	−1662.5	5.4167 × 106

.

As shown in Figure 7, all of the forces are decomposed into the F₂ along the oe direction and the force F₁ perpendicular to the oe direction. Since F₂ and oe are collinear, they cannot generate the moment to point e. When F₁ > 0 and F₂ > 0, the cuttings particles begin to tumble off the bottom hole. When F₂ < 0, the cuttings break away directly from the bottom hole.

Therefore, a mechanical model of the critical state, when cuttings particles just leave the bottom hole is analyzed. In the critical state, the support force F_N and the friction force F_f are equal to 0. Then, component force F₁ and F₂ can be obtained:

$$\left\{\begin{array}{l}{F}_1=F_{Dh}\frac{\sqrt{d_p{}^2-d_f{}^2}}{d_p}+\left(F_b+F_p-G-F_c-F_{Dv}\right)\frac{d_f}{d_p}\\ F_2=F_{Dh}\frac{d_f}{d_p}+\left(G+F_c+F_{Dv}-F_b-F_p\right)\frac{\sqrt{d_p{}^2-d_f{}^2}}{d_p}\end{array}\right.$$

(31)

4. Results and Discussion

4.1. The Fluctuation Characteristics of Pressure and Flow Rate in the Wellbore

The fluctuation of pressure and flow rate in the wellbore with the 127 mm drill pipe is calculated. During the solution, the space step is set as 10 m. The time step is calculated from the Courant-Friedrichs-Lewy condition to keep the solution stable [30].

Figure 8 shows the pressure and flow rate fluctuation propagation in the wellbore, including the state at 0 s, 0.05 s, 0.1 s, 0.5 s, and 1 s. It can be found that at the initial moment, the fluid flows without fluctuation. As the pulsed jet generator starts to work, pressure fluctuations are created at the tool, which induces the flow rate fluctuation. Within the drill pipe, the pressure and flow rate fluctuations propagate along the drill pipe to the wellhead. In addition, at the rear end of the tool, the fluctuations of pressure and flow rate are transmitted through the nozzle into the annulus and propagate to the wellhead. It is found that the drilling process is stable flow without the tool. When the pulse jet tool is installed at the bottom hole, the stable flow is modulated into a pulsed jet with fluctuations.

Figure 9 shows the pressure-flow fluctuation curve at the bottom hole and the tool. It is indicated that in the initial flow stage, the fluctuation of fluid flow is unstable. With time, the fluctuation of pressure and flow rate at the bottom and tool become stable. For a 2000 m vertical well, with an average flow rate of 20 L/s, the fluctuation of pressure and flow rate in the wellbore become stable after 50 s.

Figure 10 shows the fluctuations of pressure and flow rate at different times in the wellbore. It can be found that when fluctuation becomes stable in the wellbore, the flow rate at different positions fluctuates up and down near the initial flow rate. At the same time, the flow rate near the tool fluctuates more violently, and the flow rate fluctuates smoothly near the wellhead. Due to the limitation of the constant flow boundary condition at the drill pipe inlet, the flow rate is always maintained at 20 L/s at the inlet. From Figure 10b, it is found that the initial pressure in the drill pipe is the highest. As the pulse jet generator works stably, the flow area is always higher than that at the initial moment. The pressure drop of the pulse jet tool is always lower than that at the initial moment. This results in the situation mentioned above. In addition, comparing the pressure fluctuations in the drill pipe and the annulus, it is indicated that the pressure fluctuation in the drill pipe is slightly stronger than that in the annulus. Due to the constant pressure boundary condition at the annulus outlet, the outlet pressure is maintained as atmospheric pressure during calculation.

4.2. Influence of Flow Rate

In order to obtain the displacement influence on pressure fluctuation and flow rate fluctuation at the bottom during drilling, cases in a 6000 m vertical well were carried out. The displacement range was 20 to 40 L/s. Figure 11 demonstrates the fluctuations of flow rate and pressure at the bottom hole and near the tool at different displacements. From Figure 11a, it is found that the fluid flow at the bottom hole fluctuates up and down near the corresponding displacement. The amplitude of fluctuation increases, but its fluctuation frequency does not change with increased displacement. Figure 11b indicates that the pressure fluctuation at the front end of the pulse jet generator is the most violent, followed by the rear end of the tool. When fluid passes through the bit nozzle, the pressure fluctuation decreases sharply. Comparing Figure 11a,b, the pressure fluctuation trend at the front end of the tool is opposite to the flow rate fluctuation trend. The maximum flow rate corresponds to the minimum pressure at the front end of the tool. The pressure fluctuation at the rear end of the tool is the same as its flow fluctuation. The reason is that the flow area changes with rotation of the impeller of the pulse jet tool. The smallest flow area corresponds to the minimum flow rate at the bottom hole. At the same time, the fluid tends to accumulate at the front end of the tool due to inertia, which causes the pressure to increase. Similarly, fluid inertia generates a suction at the rear end of the tool and the bottom hole, which causes the pressure to decrease.

In order to obtain the regularity of the fluctuation amplitude of flow rate and pressure with displacement, fluctuation amplitude is calculated, as shown in Figure 12. From the figure, it is found that the fluctuation at the tool is higher than that at the bottom hole. The fluctuation reaches a minimum after fluid flows through the nozzle. With flow displacement increasing from 20 L/s to 40 L/s, pressure fluctuation at the front end of the tool increases from 0.98 MPa to 2.18 MPa. The pressure fluctuation at the rear end changes from 0.49 MPa to 1.78 MPa. The pressure fluctuation at the bottom hole is enhanced from 0.23 MPa to 0.53 MPa. The flow rate fluctuation changes from 6.13 L/s to 14.32 L/s.

4.3. Influence of Well Depth

Viscosity of 1 mPa s, with a flow rate of 20 L/s, was selected to analyze the influence of well depth on pulsed jet fluctuation. The well depth was from 2000 m to 10,000 m. From Figure 13, it is found that with depth increasing, the instantaneous flow rate at the bottom hole decreases slightly. The degree of reduction is most obvious when instantaneous flow rate reaches maximum and minimum values.

Figure 14 shows the amplitude of flow rate and pressure with different well depths. With depth increasing, fluctuation amplitude of the flow rate decreases dramatically at first. When depth is over 4000 m, fluctuation amplitude of the flow rate decreases slightly with increase in well depth. Moreover, pressure fluctuation amplitude at the front end of the tool increases dramatically with increase of well depth from 2000 m to 4000 m. In contrast, amplitude decreases slightly when well depth is over 4000 m. Pressure amplitudes at the rear end of the tool and at the bottom hole remain stable with variation in well depth. The pressure amplitude at the bottom hole is about 0.23 MPa. The flow rate amplitude decreases from 6.6 L/s to 6.0 L/s when well depth increases from 2000 m to 10,000 m.

4.4. Influence of Drilling Fluid Viscosity

The amplitude of flow rate and pressure fluctuation with different viscosities is shown in Figure 15. In these cases, the well depth was 6000 m, and the displacement was 20 L/s. The other parameters were the same as in

Table 2. The basic parameters of the wellbore, drill pipe, and drill fluid.

Parameters	Values	Parameters	Values
Drill pipe inner diameter, mm	108.62	Casing elastic modulus, Pa	2.0 × 1011
Drill pipe outer diameter, mm	127	Casing Poisson’s ratio	0.29
Drill pipe elastic modulus, Pa	2.0 × 1011	Drilling fluid density, kg/m3	1000
Poisson’s ratio of drill pipe	0.3	Elastic modulus of drill fluid, Pa	2.2 × 109
Casing inner diameter, mm	257.18	The size of the nozzle, cm	1.4 cm × 5
Drill pipe inner diameter, mm	108.62	Casing elastic modulus, Pa	2.0 × 1011
Drill pipe outer diameter, mm	127	Casing Poisson’s ratio	0.29
Drill pipe elastic modulus, Pa	2.0 × 1011	Drilling fluid density, kg/m3	1000
Poisson’s ratio of drill pipe	0.3	Elastic modulus of drill fluid, Pa	2.2 × 109

. It can be seen from the figure that with increase of drilling fluid viscosity, fluctuation amplitude of pressure and flow at the tool and the bottom hole remain stable without obvious change. The pressure amplitude at the bottom hole is about 0.23 MPa. The flow rate amplitude is about 6.15 L/s. The results indicate that drilling fluid viscosity has an extremely slight effect on amplitude of pressure and flow rate at the bottom hole under pulsed jet.

4.5. Influence of Flow Area in the Tool

The change of the flow area in the pulse jet generator affects drop in tool pressure at different instantaneous flow rates. Simulations carried out about the flow area of the pulse jet generator changed from 2 cm² to 8 cm² in a 6000 m well. The displacement and viscosity were 20 L/s and 1 mPa·s, respectively. The other parameters were the same as in Table 2.

In order to explore the influence of tool flow area on instantaneous bottom hole pressure and flow rate fluctuations, the fluctuations of the pressure and flow rate were extracted, as shown in Figure 16 and Figure 17. By comparing the pressure fluctuations near the tool and at the bottom hole under different tool flow area conditions, it is found that when the minimum flow area of the tool is 2 cm², the fluctuations of the pressure and flow rate are the most severe. When the flow area is over 6 cm², the flow rate becomes slightly lower than 20 L/s.

Figure 18 shows the amplitude of flow rate and pressure with different flow areas of the tool. From the statistics, it can be found that tool flow area greatly influences its fluctuation amplitude. When the flow area increased from 2 cm² to 4 cm², the amplitude of flow rate fluctuation decreased by 70.5%, from 4.0 L/s to 1.1 L/s. The bottom hole pressure fluctuation amplitude decreased by 69.6%, from 0.23 MPa to 0.07 MPa. The results indicate that the pressure and flow rate fluctuations are greatly affected by the minimum flow area in the pulse jet generator. When the minimum flow area of the tool increases, the fluctuations of the pressure and flow rate decrease rapidly in the first stage. With further increase of the minimum flow area, the amplitudes of pressure and flow rate decrease slightly.

4.6. Cuttings Mechanical Characteristics Analysis at the Bottom Hole under the Pulsed Jet

Force acted on cuttings at the bottom hole under pulsed jet and continuous jet are compared.

Table 3. The basic parameter of a single-particle model of cuttings at the bottom hole.

Parameters	Values	Parameters	Values
Cuttings diameter, mm	2	Nozzle diameter, mm	14
Cuttings density, kg/m3	2700	Distance from the nozzle to bottom, mm	30
Pore pressure, MPa	58.86	Nozzle angle, °	20
Fracture width at the bottom hole, mm	0.5	Fluid density, kg/m3	1000

shows the basic parameters of the cases.

The final state of the cuttings can be obtained according to the component forces F₁ and F₂. Figure 19 and Figure 20 are the component force states of the cuttings under continuous jet and pulsed jet, respectively. Moreover, the corresponding instantaneous flow rate and pressure at the bottom hole are shown in the figures. From Figure 19, it is indicated that pulsed jet makes the component force F₁ fluctuate compared with continuous jet. When the value of F₁ is positive, it means that the cuttings will tumble away from the bottom hole. As seen in the figure, the instantaneous component force F₁ becomes positive, corresponding to maximum instantaneous flow rate and pressure. At this time, the cuttings will tumble away from the bottom hole. Component force F₂ is also analyzed in Figure 20. It is found that the minimum state of component force F₂ corresponds to the minimum instantaneous flow rate. In all cases simulated in this paper, the value of F₂ is always positive. By combining the results in the figures, it can be concluded that the pulsed jet generates the flow rate and pressure fluctuation at the bottom hole. Fluctuation improves the force acted on the cuttings and helps the cuttings tumble off the bottom hole.

In addition, all forces of the cuttings are calculated under the pulsed jet condition, as shown in Figure 21. Due to the flow fluctuation at the bottom hole, the pressure force of the drilling fluid column also fluctuates. Therefore, the pulsed jet can reduce the chip hold-down effect at the bottom hole. Moreover, the flow rate fluctuation caused by the pulsed jet generator induces frequent variation in relative velocity between fluid and cuttings. The pulsed jet causes fluctuations of the particle Reynolds number, drag coefficient, and drag force. Fluctuated drag force is beneficial for cuttings to break away from the bottom hole.

5. Conclusions

Pulsed jet drilling can generate the fluctuation of pressure and flow rate at the bottom hole, which is beneficial for cuttings cleaning. It is expected to improve the ROP during drilling in unconventional oil and gas formations. In this paper, a pressure-flow rate fluctuation model is established. Based on the model, the force acted on the cuttings is analyzed to reveal the mechanism of cuttings cleaning under pulsed jet at the bottom hole. The main conclusions are as follows:

(1)

For a 2000 m well, the pulsed jet can generate pressure fluctuation at the bottom hole from 0.23 MPa to 0.53 MPa, corresponding to flow rate amplitude from 6.13 L/s to 14.32 L/s with displacement range of 20 L/s to 40 L/s.

(2)

The flow rate amplitude at the bottom hole decreases dramatically with increase in well depth when well depth is less than 4000. When well depth is over 4000 m, the flow rate amplitude varies slightly with variation of well depth. Pressure amplitude at the bottom hole keeps stable with increase in well depth.

(3)

Fluid viscosity has a slight influence on flow fluctuation at the bottom hole. The amplitude of pressure and flow rate fluctuation is mainly affected by the flow area of the tool. When the flow area increased from 2 cm² to 4 cm², the amplitude of flow fluctuation decreased by 70.5%, and the amplitude of pressure decreased by more than 60%.

(4)

Pulsed jet generates fluctuation in the instantaneous drilling fluid column force acted on the cuttings and reduces the chip hold-down effect. In addition, the pulsed jet caused fluctuation of drag force to assist the cuttings in breaking away from the bottom hole.