Wellbore Pressure Modeling for Pumping and Tripping Simultaneously to Avoid Severe Pressure Swab

Abstract

A pumping-while-tripping method is proposed to mitigate pressure swabs during tripping out in wells with a narrow mud density window and extended reach. In the proposed tripping-out process, the fluid circulation is started by using a special pump from a customized circulation line before tripping is initiated. During the tripping out, drilling fluid is circulated in the wellbore simultaneously while the drilling string is moving. A model to simulate the dynamic pressure changes in this process is developed based on the Navier–Stokes (N-S) equations and a damped free vibration system. The model was initially developed for Herschel–Bulkley (H-B) fluid; however, it can be applied to other fluid models by eliminating the non-existing terms. An analysis was conducted to investigate the effect of tripping velocity and circulation pumping rate on the pressure changes. The results show that pumping-while-tripping is effective in mitigating the pressure swab during tripping out, which is especially useful for extended-reach wells. It can also help to increase tripping out velocity and save tripping time for drilling operations.

1. Introduction

As more and more extended-reach drilling (ERD) is being conducted, the mud density window left for drilling operations becomes much narrower. The pressure surge/swab caused by tripping becomes significant as the measured depth of the wells increases. In drilling operations, even short durations of drilling outside of the safe mud window has led to costly well complications [1,2]. To maintain the wellbore pressure within the drilling window while tripping drill string, running casing, and cementing, the surge/swab pressures must be controlled precisely [3,4,5].

Existing approaches to determine the pressure swab and surge include steady-flow models and dynamic models. The steady-flow model focuses on the enduring equilibrium state at the wellbore bottom over an extended period, while the dynamic model takes into account the system’s instantaneous variations occurring within a brief timeframe. The primary distinction between the two models lies in the temporal scale and the approach adopted for handling pressure fluctuations. Burkhardt, Fontenot and Clark, and Schuh present good representations of the steady-flow model. Burkhardt first proposed a theoretical description of surge pressure and analyzed the influence of different parameters (such as the rheological model of drilling fluid, mud properties, inner-pipe end types, geometry, and tripping velocity) on surge/swab pressure during tripping [5]. Schuh provided an approximate numerical model for power-law fluids [6]. Fontenot and Clark improved the equations proposed by presenting a comprehensive and general approach, including both power-law and Bingham plastic fluids, to calculate bottom hole pressure while tripping and implemented them into a computer program [5,6,7]. Bing et al. studied the steady-state surge/swab pressure of the laminar flow of an H-B fluid [8]. In their models, the drilling fluid is assumed to be incompressible so that the drilling fluid is perfectly displaced by the drill string motion. The pressure losses caused by a moving inner pipe through a concentric annulus are used for predicting swab/surge pressures. An experimental study by Hussain and Sharif shows that the fluctuation pressure decreases with the increase in eccentricity [9]. Srivastav verified this rule later and developed a generic model to simulate the fluctuating pressure reduction caused by eccentricity via a concentric annulus modeling method [10]. As the research continues, more influence factors on surge/swab pressure have been taken into account, such as fluid compressibility, the elasticity of formation and drill string, axial vibration of the string, the effect of temperature on fluid rheology, and inflow of reservoir fluid, which improves the accuracy of model calculation [11,12]. Freddy presented a new steady-state model that can account for fluid and formation compressibility and pipe elasticity in H-B fluid [13]. Comparisons of the model predictions with the measurements showed a satisfactory agreement. Vadim’s model accounts for the mud compressibility, the elasticity of the drill string and the wellbore walls, and the string’s longitudinal vibrations [14], which shows that the amplitude and wave velocity of pressure decrease if elasticity effects are taken into account. Robello provided a coupled model not only considering the effects of fluid inertia, compressibility, wellbore elasticity, axial elasticity of the pipe and temperature-dependent fluid properties but also included a reservoir fluid influx [15], which shows the effect of the downhole pressures owing to mixed fluids under fluid influx results not only in lower wellbore pressures at the bottom of the well but also at the shoe. To investigate the effects of pipe speed (i.e., tripping speed), fluid properties, and borehole geometry on surge and swab pressures under laboratory conditions, Freddy performed a series of tests in varying the tripping speed and developed a model to calculate surge and swab pressures under steady-state flow conditions [16]. Experimental results and model predictions confirm that the trip speed, fluid rheology, annular clearance and pipe eccentricity significantly affect the surge and swab pressures. Tang presented a new model that takes the effects of the velocity of the drilling string on the boundary conditions of surge/swab pressure into consideration [17]. Meng established an elastodynamics model and a poroelastodynamics model, provided a numerical solution by the implicit finite difference method, and verified the analytical solution achieved from the Laplace transform and Crump inversion [18], which filled the gap in calculating dynamic wellbore stress under tripping conditions.

The first dynamic model was developed by Lubinski [19]. The model emphasized the importance of drilling fluid compressibility in swab–surge calculations. Lubinsiki’s dynamic approach is based on Bergeron’s method in the 1930s for solving complicated engineering problems of plane wave propagation. Different from traditional methods, this method is used to find the relationship between pressure and flow rate by imaginary observers moving upstream and downstream with the velocity of propagation. Lal corrected a number of deficiencies in Lubinski’s model and investigated the effects of pipe acceleration as well as the formation elasticity and borehole expansion [20]. Mitchell added the effect of pipe axial elasticity to the dynamic surge analysis [21]. The results of the dynamic models were verified by field data, and the verifications showed that a dynamic model is capable of accurately simulating the swab–surge during drilling. Gjerstadd and Time developed a set of simplified flow equations for H-B fluid in laminar Couette Poiseuille flow for a dynamic surge/swab model, replacing the complex part of the solution with a simpler approximation [22]. Zhang presented a new transient swab/surge model, which considers drill string components, wellbore structure, formation elasticity, pipe elasticity, fluid compressibility, fluid rheology, and the flow between wellbore and formation [23].

Typically, there is no pumping during the tripping operations. For tripping in process, tripping should be stopped approximately every 5000 feet, and “break circulation” needs to be run to fill the empty volume inside the drill string. Moreover, pumping is useful in reducing the pressure swab during tripping out. With fluid circulation during tripping out, the tripping speed can be increased without increasing the pressure swab; thus, significant tripping time can be saved. Furthermore, in choosing the mud weight, the hydra static pressure can be designed closer to the pore pressure without worrying about the pressure swab, which leads to a larger margin in the drilling window [24]. Thus, pumping is required for the tripping out process in back reaming, tight hole conditions and situations where swab pressure is concerned. For tripping out without circulation, there would be a pressure swab; for tripping out with circulation, there would be either a pressure surge or pressure swab depending on the tripping speed and the pumping flow rate [25]. If fluid is circulating during tripping out, the traditional models for surge/swab calculations are no longer applicable, and new methods are needed for the pressure prediction of the situations of pumping and tripping simultaneously [26,27,28,29,30].

This work focuses on the effects of fluid circulation on pressure change during tripping out. The model is developed based on the N-S equations for H-B fluids and can be applied to the other three major fluid models (Bingham plastic, power law, and Newtonian) by eliminating the relevant terms. The H-B model is primarily characterized by three essential parameters: the flow behavior index, yield point, and consistency index. The model can be simplified to the Bingham plastic model by setting the “flow behavior index” to 1 and the “yield point” to 0. The main idea of this approach is to calculate the pressure gradient through the study of the fluid flow velocity profile in the wellbore by moving the drill string. The relationships between flow rate, pressure, and drill string tripping velocity are obtained through N-S equations for H-B fluid. Numerical methods are used in solving the model. The effects of tripping speed, tripping velocity profile, and fluid circulation flow rate on the wellbore pressure during tripping and circulating operations are analyzed.

2. Model

During the process of tripping, it is imperative to account for all potential scenarios within the model, encompassing both the internal flow within the drill string and the circulation within the annulus formed between the wellbore and the moving drill string. The approach involves the incorporation of the H-B model within the framework of the N-S equations. This is followed by the establishment of the N-S equations based on appropriate assumptions and simplified geometrical properties. Subsequently, the application of boundary conditions enables the derivation of an expression delineating frictional loss as a function of both tripping velocity and circulation rate. The absence of an analytical solution for the N-S equations pertaining to the H-B model necessitates the utilization of numerical integration techniques for computation purposes.

To simplify the problem, the following assumptions are made in the formulation of the model.

• The fluid is incompressible;
• Only the axial direction flow is considered;
• The flow is fully developed;
• No slip condition of the fluid flow at the wall;
• Drilling string is assumed to be concentric with the wellbore.

2.1. Flow Inside Drill String

The velocity distribution of fluid flow inside the drill string in the context of the H-B model is depicted in Figure 1. This distribution exhibits two plausible scenarios: one where the fluid and the drill string are moving in different directions (Figure 1a), and the other where both the fluid and the drill string move in the same direction (Figure 1b). In the context of drilling applications, the occurrence of scenario (b) is infrequent. This study specifically investigates scenario (a), which manifests during the tripping process involving fluid circulation.

The velocity profile for case (a) has two distinct regions: I. the sheared region flows near the pipe wall, II. the plug zone, where the shear rate is zero and the flow velocity is constant.

The expression of viscosity for the H-B model is:

$${\mu }_{\mathrm{a}\mathrm{p}\mathrm{p}}=\frac{{\tau }_{\mathrm{y}}}{\gamma }+K{\left(\gamma \right)}^{m-1}$$

(1)

where ${\mu }_{\mathrm{a}\mathrm{p}\mathrm{p}}$ is the apparent viscosity, cP, ${\tau }_{\mathrm{y}}$ is the yield point, psi, $\gamma$ is the shear rate, s⁻¹, $K$ is the fluid consistency index, lb-sⁿ/ft², and $m$ is the fluid behavior index.

The model derivation commences with the N-S equations. Initially, Equation (1) is substituted into the N-S equations in cylindrical coordinates. Subsequently, the equations are simplified based on assumptions, and boundary conditions are applied to determine the fluid flow velocity distribution within the pipe. Finally, integrating the velocity distribution across the cross-sectional area of the drill string yields the pressure gradient as a function of the net flow rate and the drill string’s movement velocity. Detailed steps of the derivation are provided in Appendix A.

After some derivations, the fluid flow velocity profile for region (I) and (II) can be expressed as Equation (2):

$$v_{\mathrm{I}}={\int \left(A_{1}r+\frac{A_2}{r}-A_3\right)}^{\frac{1}{m}}dr+C_2$$

(2)

where

$$A_1=-\frac{1}{2K}\frac{\mathsf{\Delta }P}{\mathsf{\Delta }L}$$

(3)

$$A_2=\frac{C_1}{rK}$$

(4)

$$A_3=\frac{{\tau }_y}{K}$$

(5)

where $C_1$ and $C_2$ are constants from the integration; they can be calculated by applying the boundary conditions.

The radius of the plug flow region is:

$$a=\frac{{2\tau }_{\mathrm{y}}}{\Delta P/\Delta L}$$

(6)

The boundary conditions for the velocity profile are:

$$\mathrm{At}r=a,\frac{dv}{dr}=0$$

(7)

$$\mathrm{At}r=R_p,v=V_p$$

(8)

where $a$ is the radius of the plug flow region, $R_p$ is the inner radius of the pipe, ft, and $V_p$ is the pipe movement velocity, ft/s.

The net flow rate inside the pipe can be expressed as:

$$Q_{\mathrm{p}}=2\pi {\int }_0^{Rp}v\left(r\right)rdr=2\pi \left[{\int }_0^a{v}_{II}\left(r\right)rdr+{\int }_a^{Rp}{v}_I\left(r\right)rdr\right]$$

(9)

where $v_{II}$ is the plug region flow velocity, $v_{II}=v_I\left(a\right)$.

The analytical solution for the velocity profile in region I is notably intricate and impractical for real-world applications. Utilizing numerical integration methods becomes necessary to attain the velocity profile. By employing the Simpson’s rule, the integration presented in Equation (2) can be reformulated as:

$${\int }_{r_0}^{r_2}\left(A_{1}r+\frac{A_2}{r}-A_3\right)dr\approx \frac{h}{3}\left(F\left(r_0\right)+4F\left(r_1\right)+F\left(r_2\right)\right)$$

(10)

where $h$ is the numerical integration step, and

$$F\left(r\right)=A_{1}r+\frac{A_2}{r}-A_3$$

(11)

The step-by-step procedure to calculate the pressure gradient by using this approach is shown below:

• Assuming a pressure gradient, $\Delta P/\Delta L$;
• Calculate $A_1$ in Equation (2);
• Calculate the plug flow region radius by using Equation (6);
• Calculate $C_1$ in Equation (2) by using the boundary condition: At $r=a,$ $\frac{dv}{dr}=0$;
• Assuming the plug flow region velocity $v_{II}$;
• Integrate the velocity profile from $r=a$ to $r=R_p$ by using Equation (10), to obtain the velocity at the inner wall of the pipe $V_{p\_assumption}$;
• Compare $V_p$ with $V_{p\_assumption}$, if the difference is larger than tolerance, then go back to step 5 and change the guessed flow region velocity $v_{II}$ and repeat step 6 and 7. Otherwise, continue to step 8;
• Calculate the net flow rate in the pipe, $Q_{assumption}$, by integrating Equation (9) numerically;
• Compare $Q_{assumption}$ with the real flow rate $Q_{real}$, if $Q_{assumption}$ > $Q_{real}$, go back to step 1 and decrease $\Delta P/\Delta L$; if $Q_{assumption}$ < $Q_{real}$, go back to step 1 and increase $\Delta P/\Delta L$, until convergence is obtained.

2.2. Flow in the Annulus

The fluid flow in the annulus is more complex than inside the drill string. The three potential scenarios depend on the tripping out velocity and the pumping flow rate: (1) $Q_{pumping}$ > $Q_{displaced}$, the direction of the average flow velocity is the same as the tripping direction; (2) $Q_{pumping}$ = $Q_{displaced}$, the average flow velocity is zero; (3) $Q_{pumping}$ < $Q_{displaced}$, the direction of the average flow velocity is opposite with the tripping direction. The flow velocity profiles for different scenarios are shown in Figure 2.

In Figure 2, the drill string is moving out. Assume there is no slip at the wall, so the fluid velocity is the same as the drill string velocity at the wall of the drill string, and the fluid velocity is zero at the wall of the wellbore. In this figure, profile (a) is the case that $Q_{pumping}$ < $Q_{displaced}$, profile (b) is the case that $Q_{pumping}$ = $Q_{displaced}$, and profile (c) is the case that $Q_{pumping}$ > $Q_{displaced}$.

The approach to calculate the velocity profile for H-B fluid in the annulus is similar to the approach used inside the drill string. Because of the complexity of the geometry and the boundary conditions, the concentric annulus is expressed by an equivalent narrow slot.

Start from the N-S equations, after a series of derivations, detailed steps of the derivation are provided in Appendix B. In Figure 2a,b, the velocity profile for each region can be expressed as:

Region I:

$${\tilde{v}}_I={\pi }_1\left[{\left({\tilde{y}}_1-\tilde{y}\right)}^b-{\left({\tilde{y}}_1\right)}^b\right];0\le \tilde{y}\le {\tilde{y}}_1$$

(12)

Region III:

$${\tilde{v}}_{III}={1-\pi }_1\left[{\left(1-{\tilde{y}}_1\right)}^b-{\left(\tilde{y}-{\tilde{y}}_2\right)}^b\right];{\tilde{y}}_2\le \tilde{y}\le 1$$

(13)

where

$${\pi }_I=\frac{m}{m+1}\left(\frac{H}{V_p}\right){\left(\frac{\Delta P}{\Delta L}\frac{K}{H}\right)}^{\frac{1}{m}}$$

(14)

and

$$b=\frac{n+1}{n};{\tilde{v}}_1=\frac{v_1}{V_p};{\tilde{v}}_2=\frac{v_2}{V_p};{\tilde{y}}_1=\frac{y_1}{H};{\tilde{y}}_2=\frac{y_2}{H};$$

(15)

In Figure 2c, the velocity profile for each region can be expressed as:

Region I:

$${\tilde{v}}_I={\pi }_1\left[{\left({\tilde{y}}_1\right)}^b-{\left({\tilde{y}}_1-\tilde{y}\right)}^b\right];0\le \tilde{y}\le {\tilde{y}}_1$$

(16)

Region III:

$${\tilde{v}}_{III}={1+\pi }_1\left[{\left(1-{\tilde{y}}_1\right)}^b-{\left(\tilde{y}-{\tilde{y}}_2\right)}^b\right];{\tilde{y}}_2\le \tilde{y}\le 1$$

(17)

The thickness of the plug region can be expressed as:

$${\tilde{y}}_2-{\tilde{y}}_1={\pi }_2=\frac{2{\tau }_0/H}{\Delta P/\Delta L}$$

(18)

The fluid flow velocities at $y=y_1$ and $y=y_2$ should be the same. By combining Equations (15)–(17), the following can be obtained:

$${\left(1-{\tilde{y}}_1-{\pi }_2\right)}^b-{\left({\tilde{y}}_1\right)}^b-\frac{1}{{\pi }_1}=0$$

(19)

Similarly, In Figure 2a,b, combine Equations (9), (10), and (18) to obtain

$${\left(1-{\tilde{y}}_1-{\pi }_2\right)}^b-{\left({\tilde{y}}_1\right)}^b+\frac{1}{{\pi }_1}=0$$

(20)

The total flow rate is the sum of the flow rate in each region:

$$\begin{gathered} Q_t=2\pi W{\int }_0^{H}v\left(y\right)dy \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=2\pi W\left[{\int }_0^{y_1}{v}_I\left(y\right)dy+{\int }_{y_1}^{y_2}{v}_{II}\left(y\right)dy+{\int }_{y_2}^H{v}_{III}\left(y\right)dy\right] \end{gathered}$$

(21)

The total flow rate is the sum of the pumping flow rate, $Q_{pumping}$, and the fluid volume displaced by drill string in unit time, $Q_{displaced}$. The $Q_{displaced}$ is positive for tripping in and negative for tripping out.

The algorithm for this method is shown as below:

• Calculate the total flow rate in the annulus, $Q_t$, based on the pumping flow rate and the tripping velocity, then, obtain the direction of the average flow rate;
• Choose the velocity profile from Figure 2, and determine the equations that can represent the velocity profile;
• Guess a pressure gradient based on the direction of the average flow velocity;
• Transfer all the parameters dimensionless by using Equations (12) and (13);
• Calculate the thickness of the plug region by using Equation (18);
• Calculate ${\tilde{y}}_1$ by solving Equations (19) or (20) (these two equations are for different velocity profiles) numerically (Newton’s method is a good choice);
• Calculate the velocity profile for each region (Equations (9) and (10) or Equations (15) and (16));
• Integrate the velocity profile and calculate the assumptive total flow rate in the annulus, $Q_{t\_assumption}$;
• Compare the real flow rate, $Q_t$, and assumptive flow rate, $Q_{t\_assumption}$, if the difference is larger than tolerance, go back to step 3 and change the pressure gradient, and then repeat steps 3–8. Otherwise, the system becomes converged and outputs the pressure gradient.

2.3. Non-Converging Region

It needs to be mentioned that for cases when the direction of the average velocity is the same as the direction of pipe movement, there is a possibility that the velocity profile does not have a plug flow region even if the fluid has yield power law properties. The fluid flow is mainly caused by the movement of the drill pipe rather than pressure gradient. In such circumstances, if the velocity profile depicted in Figure 2 is continuously applied, convergence might not be achieved during the iterative process. Therefore, alternative functions are required to handle this specific region.

The first step is to detect the non-converging region and find the boundaries for this region. The non-converging happens when the pressure gradient is close to zero. Therefore, the way to detect the non-converging region is to start from a relatively large pressure gradient to calculate the according flow rate. Decrease the pressure gradient gradually until the function cannot find a converging flow rate, and then, record the first non-converging value as the boundary.

The pressure gradient can be negative or positive in the wellbore. Therefore, when detecting non-convergent regions, both positive and negative aspects need to be considered to identify two boundaries.

Once the boundaries are detected, a square function is plugged into the non-converging region to represent the relationship between the net flow rate and the pressure gradient.

2.4. Turbulent Model

The calculation of pressure loss in the turbulent zone during tripping is predicated upon the regular pressure loss characteristics exhibited in turbulent flow. Addressing the impact of drill string movement on pressure loss within the turbulent regime involves considering the relative flow velocity between the fluid flow and the moving drill string. For example, the absolute velocity of the fluid flow is U, and the drill string is moving at velocity V; thus, the relative velocity between the fluid flow and drill string is U + V. The equation to calculate the friction loss inside a drill string for a stationary wall is:

$$\frac{DP}{DL}=\frac{\rho f}{2gd}\frac{Q}{A}\left|\frac{Q}{A}\right|$$

(22)

where d is the drill pipe inner diameter.

If the drill pipe is moving at velocity $V$, the following equation can be used:

$$\frac{DP}{DL}=\frac{\rho f}{2gd}\left(\frac{Q}{A}+V\right)\left|\frac{Q}{A}+V\right|$$

(23)

The friction loss in annulus with both walls stationary can be expressed by:

$$\frac{DP}{DL}=\frac{\rho }{2g\left(d_o^2-d_i^2\right)}\left\{d_i{f}_i\left(\frac{Q}{A}\right)\left|\frac{Q}{A}\right|+d_o{f}_o\left(\frac{Q}{A}\right)\left|\frac{Q}{A}\right|\right\}$$

(24)

where $d_i$ is the inner diameter of the annulus, and $d_o$ is the outer diameter.

In the event of drill string movement, the above equation can be modified for the annulus with a moving inner wall and a stationary outer wall, and the frictional losses can be expressed as:

$$\frac{DP}{DL}=\frac{\rho }{2g\left(d_o^2-d_i^2\right)}\left\{d_i{f}_i\left(\frac{Q}{A}+V\right)\left|\frac{Q}{A}+V\right|+d_o{f}_o\left(\frac{Q}{A}\right)\left|\frac{Q}{A}\right|\right\}$$

(25)

2.5. Effect of the Inertia

The surge/swab component of inertial pressure fluctuations arises due to the tendency of the mud column to resist changes in motion. It can be expressed by the following:

For the closed-end strings:

$$\frac{\Delta P}{\Delta L}=\frac{\rho a_p{D}_p^2}{g\left(D_w^2-D_p^2\right)}$$

(26)

For the open-end strings:

$$\frac{\Delta P}{\Delta L}=\frac{\rho a_p\left(D_p^2-D_i^2\right)}{g\left(D_w^2-D_p^2+D_i^2\right)}$$

(27)

where $\rho$ is the density of the drilling fluid, ppg, $a_p$ is the acceleration of the string, ft/s², $D_p$ is the outer diameter of the string, ft, $D_i$ is the inner diameter of the string, ft, and $D_w$ is the diameter of the well, ft.

2.6. Effect of the Gel

The formulations to calculate the pressure required to break the mud gel and initiate fluid circulation have been given by Melrose [31].

Inside the drill string:

$$\frac{\Delta P}{\Delta L}=\frac{4\zeta }{D_i}$$

(28)

In the annulus:

$$\frac{\Delta P}{\Delta L}=\frac{4\zeta }{\left(D_w-D_p\right)}$$

(29)

where $\zeta$ is the gel of drilling fluid, lb/100 ft². Once the gel is broken, the effect of the gel would be insignificant; the pressure gradient would be mainly determined by viscous and inertia.

2.7. Model Validation

Crespo conducted an experiment to determine the fluctuating pressure of H-B fluid in a concentric annulus. The fundamental parameters used in the experiment are presented in

Table 1. Rheology of test fluids.

Test Fluids	K (lbf/100 ft2 × sn)	n	${\mathit{\tau }}_0$ (lbf/100 ft2)	Pipe OD (Inch)	Well (Inch)
0.44% xanthan gum	0.0075	0.52	0.072	0.417	0.893

Pipe OD means pipe outer diameter (OD).

, and Figure 3 illustrates the comparison between the results of our model and the experimental finding [17].

Based on the comparative results depicted in Figure 3, it is evident that the model’s calculated outcomes align well with the experimental results in the non-pumping condition. However, after initiating pumping, the model’s calculated results are comparatively lower than the experimental findings conducted by Crespo. This discrepancy is primarily attributed to the reduction in surge pressure caused by the pumping operation.

3. Sensitivity Analysis

The analysis below was run for a practical case in which there is a drill bit at the end of the drill string. If there is no pumping, it is treated as a closed-end scenario since the nozzles on the drill bit are extremely small, and the flow through the nozzles can be neglected. If there is pumping, the fluid will flow through the nozzle and then flow back to the surface through the annulus. All the scenarios described below are conducted on a vertical well of 5000 feet depth with a diameter of 8.5 inches by 4.5 inches. The fluid parameters and base geometry parameters are shown in

Table 2. Fluid rheology parameters.

Tal_y (lbf/100 ft2)	K (lbf/100 ft2 × sn)	m	Density (ppg)	Gel Strength (lbf/100ft2)
				10 s	10 min	30 min
16.89	0.709	12.7	12.7	27	33	50

and

Table 3. Geometry parameters.

Pipe ID (Inch)	Pipe OD (Inch)	Well (Inch)	Length (Feet)
3.826	4.5	8.5	5000

Pipe ID means pipe inner diameter (OD).

.

3.1. Effect of Tripping Velocity

Equivalent circulating density (ECD) is a parameter that evaluates the impact of fluid circulation on the bottom hole static pressure during drilling operations. The effects of tripping out speed on ECD at different circulation flow rates for the well given in Table 3 are shown in Figure 4. The ECD decreases as the tripping velocity increases, which is caused by the pressure swab. At lower tripping speeds, there is a relatively higher ECD, whereas at higher speeds, the ECD decreases due to the influence of pressure swabbing. This relationship requires precise control during drilling operations to ensure that the ECD remains within a safe range, simultaneously enhancing drilling efficiency. When there is no fluid circulation (0 gpm) during tripping, the ECD is always lower than the fluid density at static conditions, and there is a significant decrease in ECD as the tripping velocity increases to 200 feet/min. When the pumping flow rate is 160 gpm, at a tripping velocity lower than 100 ft/min, the ECD is higher than the equivalent static density (ESD) because of the pressure surge caused by the over-pumping; at a tripping speed larger than 100 ft/min, the pumping flow rate is not enough to balance the pressure swab caused by tripping, so the ECD is lower than the ESD. The term ESD refers to the static density of the fluid in the wellbore under specified conditions, excluding the effects of fluid circulation or drilling operations.

When the pumping flow rate reaches 320 gpm, it consistently results in an ECD exceeding the ESD. As the rate of tripping the drill string out of the borehole increases, the amplitude of pressure surges gradually diminishes. It is readily discernible that, in this particular scenario, the flow rate of 320 gpm exceeds the optimal requirement for the tripping-out procedure. In typical field operations, the conventional tripping velocity stands at approximately 90 feet per minute. Hence, in situations where the drilling safety density window is narrower than the suction pressure without fluid circulation, it is recommended to reduce the fluid pumping flow rate to 160 gpm during the tripping-out phase.

3.2. Effect of Pumping Flow Rate

The effect of the pumping flow rate on ECD is shown in Figure 5. The augmentation in pumping flow rate results in an escalated fluid velocity and pressure at the wellbore bottom, potentially leading to an increase in ECD. Under a high pumping flow rate, a pressure swab may occur due to the phenomenon of rapid fluid accumulation at the wellbore bottom, causing transient elevations in ECD. It can be observed from this figure that if the pumping flow rate is very low (which is not practical in field operations), ECD remains nearly unchanged. After a certain point, the ECD increases with increasing the pumping flow rate, regardless of the tripping velocity. When increasing the pump flow rate to make the ECD value equal to the ESD value, this pumping flow rate is referred to as the “balanced pumping flow rate”. The balanced pumping flow rate increases as the tripping velocity increases. Once the circulation flow rate is larger than the balanced pumping flow rate, there would be a pressure surge during tripping out, and the ECD is larger than the ESD. The ECD keeps increasing as the pumping flow rate increases. In the cases of over-pumping, the pressure surge is more severe for a low tripping speed than a high tripping speed. At high tripping speed, a little bit of over-pumping has a slight effect on the ECD.

3.3. Effect of Inertia

The effect of inertia on the pressure change during tripping is not very significant [32]. The results are shown in Figure 6 for open-end cases and Figure 7 for closed-end cases. Normally, the acceleration is no more than 0.5 ft/s², which means the drill string velocity reaches 90 ft/min in 3 s. In this situation, the pressure change caused by the drill string acceleration is 5 psi for open-end cases and about 20 psi for the closed-end cases for given examples in Table 2 and Table 3, which is insignificant for a 5000 ft depth well.

In scenarios involving the continuous circulation of drilling fluid during the process of tripping the drill string out of the borehole, the fluid is inherently in motion, thus mitigating the influence of the acceleration of the drill string on the fluid. In succinct summation, it can be ascertained that the impact of inertial forces on the alteration of pressure dynamics during the tripping-out procedure remains negligible.

3.4. Effect of the Fluid Rheology

Alterations in fluid rheological parameters, such as yield point, fluid behavior index, and fluid consistency index, have the potential to modify the flow characteristics of the fluid within the wellbore, consequently impacting the ECD. The effects of fluid rheology parameters on the ECD during tripping out with pumping are shown in Figure 8, Figure 9 and Figure 10.

Figure 8 shows the effect of yield point on ECD at different tripping velocities. The pressure surge/swab is more severe for fluids with high yield points than fluids with low yield points. For fluids with high yield points, it takes more pressure to move it in the annulus, which leads to a lower ECD for inefficient pumping and a higher ECD for over-pumping. Therefore, accurately controlling the pumping flow rate is required for a high yield point fluid.

The effects of the fluid consistency index and fluid behavior index are shown in Figure 9 and Figure 10, separately. Increasing either of these two indexes leads to a higher effective viscosity, which causes a more severe pressure surge/swab.

4. Tripping out Case

In this section, the preceding research findings are practically applied to drilling operations to demonstrate the impact of pump initiation on bottom-hole pressure during the drilling process. The well information pertinent to the case study is visually represented in Figure 11. In this instance, the kick-off depth is approximately 10,269 ft, where the inclination angle reaches 86.1° at 13,794 ft and, subsequently, remains constant up to 20,080 ft. The azimuth angle remains unchanged at 10.97°.

After the drilling finished, the 4 ½ inch drill strings were pulled out of the hole at the tripping velocity profile shown in Figure 12.

The bottom hole pressure during tripping out of the first stands for this case is shown in Figure 13. At time zero, the drill bit is at the bottom hole. The blue plot is the bottom hole pressure during tripping without pumping, and the red plot is the bottom hole pressure with pumping at 150 gpm.

Without fluid circulation, there is a significant pressure swab after the tripping starts. After the tripping velocity becomes steady, the bottom hole pressure also becomes steady (in this case, a steady model was used and fluid compressibility was not considered). At the end of the tripping of one stand, the tripping velocity gradually decreases to zero, and the bottom hole pressure returns back to the hydra static pressure. It keeps constant at the hydra static pressure during the time to disconnect the drill string until the tripping of the second stand starts.

The pressure changes for tripping with fluid circulation (red plot in Figure 13) are different with no fluid circulation. In this case, the operation procedure is to turn on the fluid circulation before the tripping starts and turn off the fluid circulation after the tripping of one stand finishes. Therefore, there are pressure surges at the beginning and end of the tripping, which are caused by the fluid circulation. In fact, in this particular scenario, there is no need for concern regarding the pressure surge: the fluid circulation flow rate during tripping is less than the fluid circulation rate during drilling, which means the pressure surge here is much less than the pressure increase caused by the fluid circulation during drilling.

The pressure swab during the tripping out process with fluid circulation is significantly lower compared to without fluid circulation. When designing the well, it is possible to bring the hydrostatic pressure closer to the pore pressure. Consequently, a larger safety margin can be preserved within the drilling window, enabling further drilling in extended-reach drilling operations. Furthermore, with fluid circulation during tripping, the tripping speed can be increased without increasing the risk of fluid influx, which means the tripping time can be saved.

In Figure 14, the change in bottom hole pressure during the whole tripping out process under different operation conditions was plotted. The horizontal axis of this figure is the position of the drill bit during tripping. Assuming that the drill bit was positioned at the bottom hole at the start of the tripping operation, as the tripping goes on, the drill bit moves closer to the wellhead, and the total pressure swab caused by tripping decreases as the length of the drill string in the well decreases. From Figure 14, the following remarks can be concluded: (1) the pressure swab is significant only in deep wells; (2) higher tripping speed leads to a more severe pressure swab if other operation parameters are the same; (3) fluid circulation during tripping out can reduce the pressure swab; (4) insufficient pumping during tripping out at high speed can still lead to significant pressure swab (150 gpm, 200 ft/min); (5) over-pumping may lead to a pressure surge in the wellbore (260 gpm, 100 ft/min); (6) the severe pressure swab is mainly caused by the tight hole conditions (the pressure changing ratio is the highest in an 83/4 inch open hole).

Due to the relations between the pressure swab, drill bit position in the well, and the well geometries summarized above, tripping operational parameters can be adjusted during various phases of the tripping process to optimize the cost-effective retrieval of the drill string. For the case shown above, two tripping plans are presented herein (Figure 15 and Figure 16).

In Figure 15, it is assumed that the drill bit was positioned at the bottom hole at the start of the tripping operation, and the draw works have enough power to pull out the drill string at the given velocity. This plan is designed to trip out the drill strings at a constant high tripping velocity (200 ft/min). As shown in Figure 15, the whole tripping process has three periods: high pumping flow rate, low pumping flow rate, and no pumping. In this case, the pore pressure (green curve in figure) is about 890 psi lower than the hydrostatic pressure (red curve in figure). To keep the bottom hole pressure above the pore pressure during tripping, the pressure swab must be controlled within 890 psi. At the beginning, the clearance between the drill string and the wall of the well bore is narrow, and the length of the drill string inside the wellbore is very long. To avoid a severe pressure swab, the drilling fluid is circulated at 400 gpm. When the drill bit reaches the liner section of the well, the clearance becomes larger and the drill string length is shorter, the pumping flow rate can be lowered to 150 gpm. At last, the pumping is no longer needed at the last stage of the tripping.

The second tripping plan is shown in Figure 16. In this case, it is assumed that the winch lacks sufficient horsepower to pull the drill string out of the wellbore at a rate of 200 ft/min from the bottom. Therefore, at the beginning, the tripping velocity is 100 ft/min, and the pumping flow rate is 150 gpm. Increase the tripping velocity to 200 ft/min when the load of the drill string acting on the draw works is low enough. At last, stop the pumping when the drill bit position is very shallow.

5. Conclusions

From the studies above, the following conclusions can be obtained:

• In deep well or tight well conditions, there is a significant pressure swab during tripping out without fluid circulation. Pumping fluid properly during tripping out is able to decrease or diminish the pressure swab;
• With fluid circulation during tripping out, the tripping speed can be increased without increasing the pressure swab; thus, significant time can be saved during tripping;
• In choosing the mud weight, the hydra static pressure can be designed to be closer to the pore pressure without worrying about the pressure swab by circulating fluid flow during tripping out, which leads to a larger margin in the drilling window for drilling deeper or saves some casing work;
• By using the modeling results, tripping out operation parameters can be optimized based on the formation properties, well properties, and rig facilities.

Although this study has achieved some results, it is not without limitations. In the next phase of research, we intend to validate our findings through additional instances and consider other potential influencing factors. For instance, we plan to explore the impact of temperature on the ECD.