Equivalent Circulating Density Prediction Model for High-Temperature and High-Pressure Extended-Reach Wells in the Yingqiong Basin

Abstract

The deep formations of the Yingqiong Basin are situated in a high-temperature and high-pressure (HTHP) environment, characterized by a narrow formation pressure window and consequently high operational risks. Accurate prediction of Equivalent Circulating Density (ECD) is crucial for wellbore stability control and well control safety during the drilling of extended-reach wells (ERWs) in this block. Existing calculation methods fail to account for the errors in well depth and true vertical depth (TVD) measurements caused by drill string buckling, which affect the ECD calculation. Therefore, to achieve precise ECD control, this study addresses the HTHP characteristics of ERWs in the Yingqiong Basin. It takes into consideration the variations in drilling fluid performance parameters, the influence of cuttings, and the well depth/TVD measurement errors induced by drill string buckling. On this basis, the traditional ECD calculation model is modified, and a set of ECD calculation models tailored to ERWs in the Yingqiong Basin is established. This model aims to meet the requirements for fine ECD control in drilling operations within the block and reduce operational risks. By comparing the error rates between the prediction results of the traditional ECD calculation model and those of the proposed model in this study, using the on-site measured ECD data from Well LD10-X-X in the Yingqiong Basin, the results demonstrate that for HTHP ERWs in the Yingqiong Basin, incorporating the well depth and TVD measurement errors caused by drill string buckling can enhance the accuracy of ECD prediction.

1. Introduction

As one of the three internationally recognized offshore HTHP blocks, the deep formations of the Yingqiong Basin in the South China Sea feature an extremely narrow pressure-density window (a scenario where the difference between formation fracture pressure and formation pore pressure is close to or less than the cyclic pressure loss during drilling operations). Under such stringent pressure-density window conditions, drilling operations face extremely high engineering risks: if the equivalent circulation density (ECD) of drilling fluid is too low, it cannot balance the formation pore pressure, inducing well control accidents such as well kick and blowout; if the ECD is too high, it may exceed the formation fracture pressure, causing formation leakage and well loss. Therefore, the development of accurate ECD prediction technology to achieve precise control of ECD during drilling is an indispensable key link in the drilling operation risk control system of this block, directly determining the safety and economy of drilling operations [1].

The accuracy of ECD prediction is affected by the coupling of multiple factors, among which the influence of downhole temperature and pressure on drilling fluid performance parameters is particularly significant. The HTHP environment in the deep formations of the Yingqiong Basin changes the core performance parameters of drilling fluid, such as density and flow index, thereby affecting the ECD calculation results [2,3,4]. Thus, the law of downhole temperature and pressure distribution in ERWs is a fundamental prerequisite for accurate ECD prediction. Relevant research has been conducted by existing scholars: Wang Xirun analyzed the influencing factors (depth, circulation time, static time, etc.) of measurement-while-drilling temperature and wireline logging formation temperature and established a single-well formation temperature calculation model for the Yingqiong Basin in the South China Sea using the extrapolation method, which can be used to predict the distribution characteristics of HTHP formation temperature with well depth [5]. Muhsan Ehsan et al. proposed a formation pressure prediction method based on a limited dataset; by establishing the relationship between continuous wavelet transform and geomechanical properties using a supervised machine learning model at the well location, they predicted the uncertainty of formation pressure and accurately forecasted the variation in formation pressure with depth [6]. Vahid Dokhani et al. addressed the problems of ECD calculation in HTHP and deepwater drilling, including the assumptions of constant drilling fluid density and single well type adaptation in traditional models, and proposed a numerical model for ECD that integrates heat transfer, fluid flow, and drilling fluid PVT properties. This model comprehensively considers the influence of operational parameters such as rate of penetration and fluid loss [7]. Gerald Ekechukwu et al. targeted the issues of traditional ECD determination methods (low accuracy of mathematical modeling, high cost of downhole measurement) and the dependence of some machine learning models on downhole parameters and proposed an ECD prediction method based on the extreme gradient boosting (XGBoost) algorithm, which focuses on the influence of drilling fluid inlet density, hook load, and standpipe pressure on ECD prediction [8]. Khaled Z. Abdelgawad et al. aimed at the problems that ECD determination in HTHP and deepwater drilling relies on expensive downhole sensors and that tools have temperature and pressure operation limitations and proposed an ECD prediction method based on an artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) for predicting ECD in deepwater HTHP processes [9]. Gao Yongde et al. used drilling data from a deepwater HTHP well in the South China Sea, combined with a PVT analyzer and rotational viscometer, to study the response characteristics of the equivalent static density (ESD) of deepwater water-based drilling fluid to temperature and pressure, and established an ECD calculation model for deepwater HTHP wells that simultaneously considers the influence of temperature and pressure on drilling fluid parameters, as well as the influence of subsea pressure boosting on the wellbore flow field and temperature field [10].

Meanwhile, for ECD prediction in ERWs, the influence of annular pressure loss and cuttings must be considered, and many scholars have conducted corresponding research. Hao Xining et al., based on the characteristics of the dual-string coiled tubing drilling process, considered the influence of parameters such as temperature, pressure, and cuttings concentration; established an ECD calculation model for dual-gradient drilling wellbore with dual-string coiled tubing; and analyzed the influence of factors such as sudden drilling fluid displacement and drilling fluid density on wellbore ECD [11]. Luo Hongbin et al. established a deepwater drilling ECD prediction model that considers the influence of temperature and cuttings as well as the lifting effect of subsea pressure boosting pipelines and also took into account the influence of displacement on the ECD value [12]. Chen Yuwei used a transient cuttings transport model to simulate the transport process of cuttings in the wellbore and analyzed the influence law of cuttings on wellbore ECD combined with drilling data [13].

Due to the large well depth of ERWs, downhole drilling tools are prone to buckling [14], and scholars have carried out relevant research on the influence of drill string buckling on ECD. Kong Linghao et al. considered the equivalent eccentricity of drill strings in directional wells and horizontal wells to represent the influence of drill string buckling on wellbore annular pressure loss, established a simplified annular pressure loss model under drill string buckling conditions, and verified it by comparison with ANSYS simulation [15]. Zhang Jinkai et al., based on computational fluid dynamic (CFD) theory, simulated the annular flow of Herschel–Bulkley fluid with different eccentricities and obtained the influence law of drill string eccentricity on the flow field characteristics and frictional pressure loss characteristics of annular drilling fluid under drill string buckling conditions [16]. Regarding the influence of drill string buckling on ECD prediction, current research mostly focuses on the influence of the eccentricity effect caused by drill string buckling on drilling fluid pressure loss. However, the factor of well depth and TVD measurement errors caused by drill string buckling has not been incorporated into the ECD prediction model.

Therefore, to achieve the goal of precise ECD control in the narrow pressure-density window operations of the Yingqiong Basin while targeting the HTHP formation characteristics of ERWs in this area, this study optimizes and improves the traditional ECD calculation model. It comprehensively incorporates the influence of HTHP on drilling fluid density, consistency coefficient, and flow behavior index; the additional pressure effect caused by cuttings in the wellbore; and the annular flow interference caused by drill string eccentricity. Furthermore, it focuses on introducing the well depth and TVD measurement errors caused by drill string buckling, which have not been considered before. Finally, an ECD calculation model suitable for HTHP ERWs in the Yingqiong Basin is constructed to meet the engineering demand for precise ECD control in drilling operations of this block.

2. ECD Prediction Model for HTHP Extended-Reach Wells

To establish an ECD prediction model suitable for the geological and operational conditions of the Yingqiong Basin, the following assumptions are made [11,12,13,14,15]:

• The physical properties of the drilling fluid are assumed to remain approximately constant within each computational time step. The fluid is considered incompressible and in a steady-state flow condition.
• The well trajectory of an ERW consists of vertical, build-up, hold, and drop sections. The build-up and drop sections are designed using the circular arc method.
• In sections where drill string buckling occurs, the drill string is assumed to contact the borehole wall.
• The rheological behavior of the drilling fluid under HTHP conditions can be described using the Herschel–Bulkley model [16,17].

2.1. Wellbore Temperature Field Model

As shown in Figure 1, the wellbore thermal system in an ERW during drilling fluid circulation can be divided into five subsystems [18]: the drill string fluid system, the drill string wall system, the annular fluid system, the formation system, and the drill bit system. The temperature distribution in each of these subsystems is governed by the energy conservation equation, presented as Equation (1). By applying appropriate boundary and initial conditions, the temperature distribution throughout the ERW can be determined using numerical methods [2,3,4].

During the drilling fluid circulation process, the interaction between heat and work is mainly reflected in two aspects: On the one hand, the high-speed flowing drilling fluid in the drill string rubs against the pipe wall, and the drill bit does work to break rocks. These processes convert mechanical energy into thermal energy, increasing the temperature of the drill string fluid system, the drill bit system, and the annular fluid system. On the other hand, there is a temperature difference between the annular fluid and the formation, leading to heat exchange. At the same time, the drill string wall also dissipates heat to the surrounding environment. These heat transfer processes offset part of the thermal energy generated by work, and finally, the entire temperature field system reaches a dynamic energy balance.

The energy conservation equation is:

$${\displaystyle \frac{\partial \left(\rho CT\right)}{\partial t}}=-\nabla \left(\rho Cv\right)+\nabla \left(\lambda \nabla T\right)+\Delta$$

(1)

where ρ is the density of the drilling fluid, g/cm³; C is the specific heat, J/(kg·°C); v is the velocity vector, m/s; $\lambda$ is the thermal conductivity, W/(m·°C); T is the temperature, °C; $\Delta$ is the additional heat source, J; and t is time, s.

2.2. ESD Model

For HTHP wells, numerous scholars have investigated the relationship between drilling fluid density and variations in temperature and pressure [19,20,21]. High temperatures at the bottom of the well cause the drilling fluid to expand, resulting in an increase in volume and a decrease in density. Meanwhile, the high temperatures also reduce the viscosity of the drilling fluid, which in turn leads to a decrease in (ECD). The commonly used correlation between drilling fluid density and temperature–pressure conditions can be expressed as:

$$\rho \left(p,T\right)={\rho }_0\mathit{exp}\left[\Gamma \left(p,T\right)\right]$$

(2)

where ρ (p, T) is the drilling fluid density at temperature T and pressure p, kg/m³; ρ₀ represents the initial density of the drilling fluid before being affected by the HTHP conditions of the formation, kg/m³.

$$\Gamma \left(p,T\right)={\xi }_P(p-p_0)+{\xi }_{pp}(p-p_0{)}^2+{\xi }_T(T-T_0)+{\xi }_{TT}(T-T_0{)}^2+{\xi }_{pT}(p-p_0)(T-T_0)$$

(3)

where p₀ represents the surface pressure, Pa; T₀ is the surface temperature, °C; The parameters ${\xi }_P$, ${\xi }_P$, ${\xi }_{pp}$, ${\xi }_{TT}$, ${\xi }_{pT}$, are fluid characteristic constants that can be determined using a multivariate nonlinear regression method.

During drilling, the density of the drilling fluid varies with temperature and pressure due to expansion and compression effects, meaning that it is not constant. To more accurately represent the variation in static hydrostatic pressure in the wellbore, the concept of ESD is introduced. ESD denotes the equivalent density corresponding to the hydrostatic pressure at any point along the wellbore and is a function of the drilling fluid density and the fluid column height [22]. It can be defined as:

$$ESD={\displaystyle \frac{P-P_0}{gH}}$$

(4)

where P₀ is the wellhead pressure (MPa), $g$ is the gravitational acceleration (9.81 m/s²), and H is the TVD (m).

Using an iterative numerical approach, an ESD model was developed to describe drilling fluid behavior during circulation. Suppose the hydrostatic fluid column height is H, which is discretized into n computational nodes, each with a step size of Δh = H/n. Based on the wellbore temperature field model established in Section 2.1, the temperature variation in the drilling fluid along the well depth during circulation can be obtained as:

$$T_i=T\left(h_i\right)(i=\mathrm{1,2},\cdots ,n)$$

(5)

Assuming that within a fluid element of length Δh, the temperature, pressure, and density of the drilling fluid are uniform, the iterative formula for calculating the hydrostatic pressure at the bottom of the well is:

$$P_{i+1}=P_i+△P_i=P_i+g△h_i{\rho }_0\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right](i=\mathrm{1,2},\cdots ,n)$$

(6)

With boundary condition:

$$P\left(h=0\right)=P_0,T\left(h=0\right)=T_0$$

(7)

After n iterations, the hydrostatic pressure at the bottom of the annulus is given by:

$$P-P_0=g{\rho }_0\sum _{i=1}^n(△h_i\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])$$

(8)

Thus, the ESD model for the drilling fluid can be expressed as:

$$ESD={\displaystyle \frac{P-P_0}{gH}}={\displaystyle \frac{{\rho }_0}{H}}\sum _{i=1}^n(△h_i\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])$$

(9)

In practical calculations, the TVD H depends on the length of the drill string inserted into the well. In vertical sections, H corresponds directly to the length of the drill string. In build-up, drop-down, and hold sections, H can be derived from the drill string length using geometric conversion formulas [23].

However, in ERWs, due to significant well depth and complex trajectories, the drill string is prone to buckling. Once buckling occurs, the actual drill string length differs from the unbuckled length, resulting in a deviation between the measured drill string length and the TVD.

Figure 2a shows the overall schematic diagram of drill string buckling during drilling, and Figure 2b presents a comparative schematic diagram of local buckling (Figure 2b(I)) and non-buckling sections (Figure 2b(II)) of the buckled drill string. Within the wellbore section with the same horizontal projection length, the drill string undergoes bending deformation after buckling, and its actual axial length is significantly longer than that in the straight state before buckling. In on-site drilling operations, if the length of the drill string run into the well is still used to represent the TVD H without considering the axial elongation of the drill string caused by buckling, the calculated TVD H will be greater than the actual TVD of the wellbore. However, the calculation of ESD is based on the accurate TVD; deviations in TVD will directly lead to inaccurate calculation of the drilling fluid hydrostatic pressure. Due to the overestimated calculated TVD, the hydrostatic pressure will also be overestimated, which in turn causes the ESD calculation result to deviate from the true value. Therefore, it is necessary to modify Formula (9).

As shown in Figure 3, during the drilling process, when the well depth increases to a certain extent, if the axial pressure generated by the accumulated weight of the lower drill string exceeds the critical value for sinusoidal buckling, low-amplitude sinusoidal buckling will occur, though the buckling deformation is relatively gentle at this point. In the horizontal well section, the drill string is subject to stronger radial constraints from the wellbore wall, and axial pressure is more likely to concentrate. This not only triggers sinusoidal buckling rapidly but also further develops into high-amplitude helical buckling. At this time, the drill string in the vertical well section consists of three independent segments: the non-buckled drill string, the sinusoidally buckled drill string, and the helically buckled drill string. The unbuckled portion has a length L₀, equal to the well section length. The sinusoidally buckled section can be divided into u subsections according to different amplitudes A_i (i = 1, …, u). The unit-period line length of a sinusoidally buckled drill string segment is $L_{\mathrm{si}}^{\prime }$:

$$L_{\mathrm{si}}^{\prime }=2\left[A_{si}\sqrt{1+A_{\mathrm{si}}^2}+\mathit{ln}(A_{si}+\sqrt{1+A_{\mathrm{si}}^2})\right]$$

(10)

The corresponding vertical projection depth ΔS_i for the segment with amplitude A_i is given by:

$$\Delta S_i={\displaystyle \frac{L_{si}}{L_{\mathrm{si}}^{\prime }}}\Delta S_i^{\prime }$$

(11)

where $\Delta S_i^{\prime }$ is the helical pitch (m) of the sinusoidally buckled segment, and $L_{\mathrm{si}}^{\prime }$ is the unit-period line length (m). The values of $\Delta S_i^{\prime }$ and $L_{\mathrm{si}}^{\prime }$ can be determined from the model derived by Gao and Miska [24], which established the geometric relationship between axial load and sinusoidal buckling of the drill string. $L_{si}$ is the total length (m) of the sinusoidally buckled drill string with amplitude A_i.

The total TVD of the sinusoidally buckled drill string sections with different amplitudes A_i (i = 1, …, u) can b expressed as:

$$L_s=\sum _{i=1}^u\Delta S_i$$

(12)

The helically buckled drill string can be divided into v segments according to the pitch ${\Delta h}_j$ (j = 1, …, v), with the unit-period length of a helically buckled segment defined as $L_{hj}^{\prime }$:

$$L_{hj}^{\prime }=\sqrt{(\pi D{)}^2+\Delta {h^{\prime }}_j^2}$$

(13)

where $L_{hj}^{\prime }$ is the unit-period length of a helically buckled segment, m, D is the helix radius, m, assumed equal to the wellbore radius, and $\Delta h_j^{\prime }$ is the pitch, m. The pitch $\Delta h_j^{\prime }$ can be determined based on the model proposed by Lubinski [25], which establishes the relationship between the axial load on the drill string and the geometrical parameters of the helically buckled section.

The corresponding TVD of each helically buckled segment with pitch $\Delta h_j^{\prime }$ is calculated as:

$$\Delta h_j={\displaystyle \frac{L_{hj}}{L_{\mathrm{hj}}^{\prime }}}\Delta h_j^{\prime }$$

(14)

where $L_{hj}$ is the total length of the helically buckled drill string segment with pitch $\Delta h_j^{\prime }$, m. The total TVD of all helically buckled segments is then:

$$L_h=\sum _{j=1}^v\Delta h_j$$

(15)

As illustrated in Figure 3, the total vertical section length of the wellbore is:

$$L_v=L_0+L_s+L_h$$

(16)

where $L_v$ denotes the total length of the drill string considering buckling; $L_0$ denotes the length of the buckled drill string section, $L_s$ denotes the length of the low-amplitude sinusoidal buckling drill string section, and $L_h$ denotes the length of the high-amplitude helical buckling drill string section; the schematic diagram of the latter three is shown in Figure 3.

According to the design assumptions, the build-up and drop sections are designed using the arc method. Based on the results of the drill string mechanics model and field safety analysis, the drill string in the build-up and drop sections does not buckle due to the geometric constraints of the wellbore trajectory [15]. Therefore, the ESD model in these sections does not need to account for length measurement errors induced by buckling.

The hold section has a straight spatial trajectory with a constant inclination angle, ${\alpha }_{\mathrm{inc}}$. For horizontal wells, this inclination angle is 90°. The mechanical properties of the drill string in the hold section differ from those in the vertical section, and the critical buckling load is also different [15]. If the drill string buckles in the hold section, the TVD increment $\Delta H_{inc}$ can be expressed as:

$$\Delta H_{\mathit{inc}}=L_{\mathit{inc}}\mathit{cos}{\alpha }_{\mathit{inc}}=(L_0^{inc}+L_s^{inc}+L_h^{inc})\mathit{cos}{\alpha }_{\mathit{inc}}$$

(17)

where $L_0^{inc}$, $L_s^{inc}$, and $L_h^{inc}$ denote the lengths of the unbuckled, sinusoidally buckled, and helically buckled sections, respectively, in the hold section. The computation of $L_s^{inc}$ and $L_h^{inc}$ follows the same method as in the vertical section.

For an ERW with a arc segments and b hold sections, the TVD H in Equation (9) can be modified to H_b as:

$$H_b=L_v+\sum _{k=1}^a\Delta H_{\mathit{arck}}+\sum _{l=1}^b\Delta H_{\mathit{incl}}=L_0+L_s+L_h+\sum _{k=1}^a\Delta H_{\mathit{arck}}+\sum _{l=1}^b\left(L_{0l}^{inc}+L_{sl}^{inc}+L_{hl}^{inc}\right)\mathit{cos}{\alpha }_i$$

(18)

where $\Delta H_{\mathit{arc}}$ denotes the TVD of a spatial arc segment, which can be calculated following the method in Han (2007) [23].

The computation procedure of $H_b$ considering buckling-induced errors is as follows: first, input the wellbore structure, well trajectory, and downhole tool parameters into the drill string mechanics model [25] to determine whether buckling occurs, and if so, where sinusoidal or helical buckling happens. If no buckling occurs, the inserted drill string length equals the well depth, and the terms $L_s$, $L_h$, $L_s^{inc}$, and $L_h^{inc}$ are zero. If buckling occurs, the axial load of the drill string is calculated, and the geometric parameters of sinusoidal and helical buckling are determined. These parameters are substituted into Equations (11) and (14) to compute the vertical section depth. The same procedure applies to inclined-straight sections, and finally, the depths of build-up and drop sections are added to obtain the current drill bit TVD.

The corrected ESD model considering the TVD error is:

$$ESD={\displaystyle \frac{P-P_0}{g{H}_b}}={\displaystyle \frac{{\rho }_0}{(L_0+L_s+L_h+{\sum }_{k=1}^a\Delta H_{\mathrm{arck}}+{\sum }_{l=1}^b\left(L_{0l}^{inc}+L_{sl}^{inc}+L_{hl}^{inc}\right)\mathit{cos}{\alpha }_i)}}\sum _{i=1}^n(△h_i\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])$$

(19)

where $△h_i$ is the depth increment at the drill bit position, obtained by dividing the well length L into n segments:

$$△h_i={\displaystyle \frac{L}{n}}={\displaystyle \frac{L_0+{\sum }_{k=1}^a\Delta L_{\mathrm{arck}}+{\sum }_{l=1}^b{L}_{0l}^{inc}}{n}}$$

(20)

However, similar to the relationship between the TVD H and H_b, if the well depth measurement error caused by drill string buckling is not considered, and the length of the drill string run into the wellbore is directly used as the equivalent well depth, the calculated well depth value will be larger, resulting in an inaccurate ESD value. Therefore, it is also necessary to correct the well depth L by accounting for drill string buckling. The corrected well depth at the drill bit position is denoted as L_b, and the correction result is as follows:

$$△h_i^{\prime }={\displaystyle \frac{L_b}{n}}={\displaystyle \frac{L_0+L_s+L_h+{\sum }_{k=1}^a\Delta L_{\mathrm{arck}}+{\sum }_{l=1}^b\left(L_{0l}^{inc}+L_{sl}^{inc}+L_{hl}^{inc}\right)}{n}}$$

(21)

If no buckling occurs, $L_s$, $L_h$, $L_{sl}^{inc}$, and $L_{hl}^{inc}$ are zero.

Finally, the ESD model considering both drill string buckling-induced depth and TVD errors is:

$$ESD={\displaystyle \frac{P-P_0^{\prime }}{g{H}_b}}={\displaystyle \frac{{\rho }_0}{H_b}}\sum _{i=1}^n(△h_i^{\prime }\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])$$

(22)

2.3. ECD Prediction Model

The ECD of drilling fluid can be defined as the sum of the ESD, the equivalent density of annular frictional pressure losses, and the equivalent density caused by cuttings accumulation. The expression for ECD is as follows:

$$ECD=ESD+{\displaystyle \frac{P_f}{gH}}+{\displaystyle \frac{P_d}{gH}}$$

(23)

where P_f is the annular frictional pressure loss (Pa), P_d is the additional bottom-hole pressure due to drill cuttings (Pa), and H is the height of the static drilling fluid column, m. After establishing the ESD prediction model, it is also necessary to develop a model for calculating the annular pressure losses during circulation. The frictional pressure drop depends on the rheological properties of the drilling fluid, the wellbore geometry, and the flow velocity of the drilling fluid.

The frictional pressure drop represents the pressure loss caused by the interaction between the drilling fluid and the wellbore wall during flow. A boundary layer is formed along the wall of the flow conduit. The viscosity characteristics of the drilling fluid cause variations in velocity perpendicular to the flow direction, leading to momentum loss and flow resistance. The pressure drop caused by this process is proportional to the length of the flow path, the density of the drilling fluid, and the square of its velocity, and inversely proportional to the pipe diameter.

The annular pressure loss without considering drill string buckling can be calculated as:

$$P_f={\displaystyle \frac{2f\rho v^2}{D_H-D_p}}L$$

(24)

where f is the friction factor (dimensionless), $\rho$ is the drilling fluid density, g/cm³, L is the well depth, m, v is the drilling fluid velocity, m/s, $D_H$ and $D_p$ are the outer and inner diameters of the annulus, mm, respectively. The calculation of the friction factor f depends on the flow regime, as detailed in references [16,17].

Equation (24) is suitable for calculating turbulent annular pressure losses when no drill string buckling occurs. However, once the drill string buckles, two main effects must be considered:

• In Equation (24), ($D_H-D_p$) assumes that the drill string is centered within the borehole, which is valid for a non-buckled string. When buckling occurs, whether in a sinusoidal or helical form, the drill string becomes eccentric, as shown in Figure 4b (Among them, Figure 4b(I) indicates that the drill string is not eccentric, and Figure 4b(II) shows the eccentricity of the drill string), and the eccentricity must be considered.
• The well depth H in Equation (24) is typically determined by the length of the drill string run into the hole. During drilling, if no buckling occurs, the drill string length can be equated to the true well depth. However, in ERWs, buckling is likely due to the long wellbore, and the actual string length may exceed the borehole trajectory length, as shown in Figure 4a. Using the drill string length as a substitute for true well depth under buckling conditions leads to inaccuracies in the calculated annular pressure loss.

Therefore, Equation (24) must be modified. To account for the eccentricity induced by drill string buckling, a correction factor R is introduced:

$$P_f={\displaystyle \frac{2f\rho v^2}{D_H-D_p}}RL$$

(25)

The calculation method for the eccentricity correction factor R is provided in references [16,17].

Although Equation (25) includes the influence of drill string eccentricity on annular pressure losses, it does not account for the difference between the actual buckled string length and the true well depth. Based on the analysis in Section 2.2, the well depth L in Equation (25) can be corrected as L_b:

$$P_f={\displaystyle \frac{2f\rho v^2}{D_H-D_p}}R{L}_b={\displaystyle \frac{2f\rho v^2}{D_H-D_p}}R\left(L_0+L_s+L_h+\sum _{k=1}^a\Delta L_{\mathrm{arck}}+\sum _{l=1}^b\left(L_{0l}^{inc}+L_{sl}^{inc}+L_{hl}^{inc}\right)\right)$$

(26)

Substituting Equations (22) and (26) into Equation (23), the ECD model considering the measurement errors of well depth and TVD caused by drill string buckling is obtained:

$$ECD={\displaystyle \frac{{\rho }_0}{H_b}}\sum _{i=1}^n(△h_i^{\prime }\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])+{\displaystyle \frac{2{\rho }_0}{g{H}_b}}\sum _{i=1}^n{\displaystyle \frac{f_i{v_i}^2{R}_i{L}_{\mathrm{bi}}\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right]}{\left(D_{Hi}-D_{\mathrm{pi}}\right)}}+{\displaystyle \frac{P_d}{g{H}_b}}$$

(27)

During managed pressure drilling, the annular fluid is essentially a solid–liquid two-phase flow. The presence of cuttings introduces an additional bottom-hole pressure $P_d$, which is closely related to the cutting concentration. The expression for $P_d$ is given as:

$$p_d=\left({\rho }_c-{\rho }_0\mathit{exp}\left[\Gamma \left(P,T\right)\right]\right)C_{a}g{H}_b$$

(28)

Based on the solid–liquid two-phase flow theory, the cuttings concentration in the annulus can be expressed as a function of annular return velocity:

$$C_a={\displaystyle \frac{R}{\left[{\left(\frac{D_1}{D_2}\right)}^2-{\left(\frac{D_3}{D_2}\right)}^2\right]\left(v_1-v_2\right)}}$$

(29)

where ${\rho }_c$ is the cuttings density, kg/m³, $C_a$ is the cuttings concentration, R is the rate of penetration, m/s, $D_1$, $D_2$, and $D_3$ are the borehole, drill pipe, and bit diameters, m, respectively, $v_1$ is the annular return velocity, m/s, and $v_2$ is the average slip velocity of cuttings, m/s.

Combining the above, the final HTHP ECD prediction model suitable for the Yingqiong Basin ERWs can be expressed as:

$$ECD={\displaystyle \frac{{\rho }_0}{H_b}}\sum _{i=1}^n(△h_i^{\prime }\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])+{\displaystyle \frac{2{\rho }_0}{g{H}_b}}\sum _{i=1}^n{\displaystyle \frac{f_i{v_i}^2{R}_i{L}_{\mathrm{bi}}\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right]}{\left(D_{Hi}-D_{\mathrm{pi}}\right)}}+{\displaystyle \frac{△h^{\prime }}{H_b}}\sum _{i=1}^n{C}_{ai}({\rho }_{ci}-{\rho }_0\mathit{exp}\left[\Gamma \left(P_i,T_i\right)\right])$$

(30)

2.4. Prediction of Rheological Properties of Drilling Fluids Under High Temperature and High Pressure

When calculating the ECD of drilling fluids, it is necessary to determine parameters such as the Reynolds number, eccentricity factor, and friction factor in each wellbore section. These parameters vary with temperature and pressure. For ERWs in the Yingqiong Basin, the downhole environment is characterized by high temperature and high pressure. If the effects of temperature and pressure on drilling fluid properties are ignored, the calculated ECD may be inaccurate.

The Herschel–Bulkley model [16,17] is expressed as:

$$\tau ={\tau }_0+K{\gamma }^m$$

(31)

where ${\tau }_0$ is the yield stress, Pa, also called the dynamic shear stress; K is the consistency coefficient, Pa·sn; and m is the flow behavior index.

The rheological parameters of Herschel–Bulkley fluids under the influence of temperature and pressure were analyzed, and mathematical models describing the dependence of ${\tau }_0$, K, and m on temperature T and pressure P were established. By fitting these models to experimental data, the following relationships between Herschel–Bulkley fluid parameters and temperature and pressure were obtained:

$${\tau }_0=5.2866\times e^{1.35\times 10^{-2}(T-40)-0.89\times 10^{-7}(P-\mathrm{10,100})-0.32\times 10^{-4}(T-40{)}^2}$$

(32)

$$K=0.5021\times e^{-1.23\times 10^{-1}(T-40)+0.44\times 10^{-6}(P-\mathrm{10,100})+2.26\times 10^{-4}(T-40{)}^2}$$

(33)

$$n=0.7748\times e^{1.22\times 10^{-2}(T-40)-0.31\times 10^{-7}(P-\mathrm{10,100})-0.39\times 10^{-4}(T-40{)}^2}$$

(34)

Substituting Equations (32)–(34) into the calculations of Reynolds number, eccentricity factor, and friction factor for each wellbore section allows for correction of the ECD calculation model in Equation (30) to account for HTHP effects.

3. Field Application of the ECD Model

In this section, the LD10-X-X well in the Yingqiong Basin was selected as a case study. This well is an ERW and also operates under HTHP conditions [26]. With reference to the on-site drilling measured data of the same type of HTHP wells in the Yingqiong Basin, as well as the verified coefficient range for ERWs in the HTHP blocks of the South China Sea [27]. The axial friction coefficient was set to 0.1, and the circumferential friction coefficient was set to 0.25. The drilling parameters for the third spud of this well are as follows: weight on bit (WOB) 90 kN, average rate of penetration (ROP) 4.21 m/h, and rotary speed 120 r/min. According to the calculation results of the drill string mechanics model, the drill strings in the 3280–3589 m interval are in a state of sinusoidal buckling, while those in other intervals remain unbuckled.

By inputting the sinusoidal buckling error of the drill string at 3280–3589 m into the TVD and measured depth correction model, it can be seen that if drill string buckling is ignored, the measured depth L is 4217 m, and the TVD H is 2916 m; when drill string buckling is considered, the measured depth L_b is 4022 m, and the TVD H_b is 2904 m. It can be observed that the difference between H and H_b is small because the drill string buckling occurs in the horizontal section, and according to Equation (18), the drill string buckling-induced error in measured depth has a negligible impact on TVD.

By substituting the parameters from

Table 1. Parameters of the LD10-X-X Well.

Parameter	Value
Wellhead Temperature	30 °C
Geothermal Gradient	4.0 °C/100 m
Seawater Depth	87 m
Circulation Time	30 min
Bit Diameter	212.73 mm
Drill Pipe Outer Diameter	139.7 mm
Drill Pipe Inner Diameter	121.36 mm
Drill Collar Length	200 m
Drill Collar Outer Diameter	165.1 mm
Drill Collar Inner Diameter	121.36 mm
Mud Pump Displacement	1500 L/min
Drilling Fluid Thermal Conductivity	1.45 W/(m·°C)
Drilling Fluid Heat Capacity	2000 J/(kg·°C)

into both the ECD model considering drill string buckling-induced depth errors and the ECD model ignoring drill string buckling, the calculated results were compared with the field measurements. The comparison results are shown in

Table 2. Comparison of Measured and Calculated ECD Values.

Depth	Displacement	Density	Injection Temperature (°C)	Viscosity	Measured ECD	Calculated ECD Without Buckling Error (g/cm3)	Relative Error Without Buckling (%)	Calculated ECD with Buckling Error (g/cm3)	Relative Error with Buckling (%)
3745	1577	1.85	47.4	51	1.959	1.979	1.067	1.966	0.407
3748	1564	1.85	48.1	51	1.939	1.981	2.204	1.965	1.352
3760	1546	1.85	49.4	51	1.947	2.002	2.852	1.962	0.793
3768	1651	1.85	52.2	51	1.959	1.991	1.635	1.965	0.332
3837	1515	1.9	48.8	52	2.005	2.045	2.043	2.010	0.267
3860	1541	1.9	42.5	51	2.019	2.043	1.219	2.011	−0.385
3905	1605	1.9	42.4	50	2.021	2.036	0.785	2.013	−0.391
3932	1603	1.9	53.8	49	2.011	2.027	0.825	2.012	0.067
3979	1566	1.9	53.4	42	2.026	2.026	0.012	2.012	−0.679
3987	1547	1.94	52.7	44	2.05	2.078	1.377	2.049	−0.036
3994	1482	2.02	52.2	44	2.14	2.149	0.435	2.128	−0.519
4023	1316	2.05	49	45	2.16	2.196	1.693	2.147	−0.573
4054	1369	2.09	50.7	45	2.19	2.226	1.660	2.190	0.026
4055	951	2.11	48.5	45	2.195	2.184	−0.496	2.168	−1.188
4057	950	2.13	48.5	45	2.21	2.195	−0.658	2.188	−0.957
4067	710	2.19	46.4	45	2.245	2.232	−0.553	2.228	−0.713
4085	720	2.19	45.3	45	2.275	2.259	−0.691	2.229	−1.979
4097	1158	2.21	43.9	45	2.3	2.305	0.238	2.301	0.047
4098	1153	2.21	45.3	43	2.3	2.325	1.087	2.299	−0.032

and Figure 5 and Figure 6.

As shown in Figure 5, the ECD values calculated by the model ignoring drill string buckling are represented by the red solid circles, the ECD values calculated by the model considering drill string buckling are represented by the blue solid triangles, and the field-measured ECD values are represented by the black solid squares. Compared with the red solid circles, the blue solid triangles are closer to the black squares.

As shown in Figure 6, the black solid squares represent the relative error between the ECD values calculated by the model, ignoring drill string buckling and the measured ECD, with a maximum relative error of 2.325% and an average absolute error of 0.88%. The red solid circles represent the relative error between the ECD values calculated by the model considering drill string buckling and the measured ECD, with a maximum relative error of 1.48% and an average absolute error of 0.22%, demonstrating accuracy that meets field engineering requirements and is more precise.

4. Conclusions

(1)

Aiming at the formation characteristics of high temperature and high pressure (HTHP) and a narrow formation pressure window in the Yingqiong Basin, on the basis of the traditional model, an ECD prediction model suitable for ERWs in this area was constructed by comprehensively considering the effects of HTHP on drilling fluid density, consistency coefficient and flow behavior index; the additional pressure effects caused by cuttings in the wellbore; drill string eccentricity; and the well depth and TVD measurement errors caused by drill string buckling. This model more accurately represents the physical process of multi-factor coupling in actual drilling conditions.

(2)

Drill string buckling has a significant impact on the accuracy of ECD prediction, and this impact is reflected in two dimensions: On one hand, drill string buckling leads to drill string eccentricity, so it is necessary to introduce an eccentricity coefficient to correct the calculation of annular pressure loss. On the other hand, there is a deviation between the actual axial length of the buckled drill string and the length of the wellbore trajectory. If this deviation is ignored and the length of the run-in drill string is directly used as the equivalent well depth and TVD, the calculated values of ESD and annular pressure loss will deviate from the true values. However, the modified model can effectively eliminate such errors.

(3)

Taking the on-site data of Well LD10-X-X in the Yingqiong Basin as the verification object, the comparative analysis shows that: for the traditional model without considering the above error, the maximum relative error between the calculated values and the measured values reaches 2.325%, and the average absolute error is 0.88%, while for the ECD prediction model considering the drill string buckling error, the maximum absolute value of relative error is 1.979%, and the average absolute error is 0.22%.