Irregular Eccentric Wellbore Cementing: An Equivalent Circulation Density Calculation and Influencing Factors Analysis

Abstract

In the field of cement, if the formation cannot be given sufficient pressure to maintain stability during construction, pressure control failure may occur, leading to the leakage of liquids and gasses from the formation to the wellbore. In addition, irregular wellbore diameter and casing eccentricity are important factors that are easily overlooked and affect the prediction of ECD (Equivalent Circulation Density) calculation. This results in major accidents and ecological disasters, further impacting the global environment. This study focuses on a well in the eastern oilfields of China, and based on a rheological experiment of high temperature and high pressure, an irregular eccentric wellbore model is established according to the measured wellbore diameter and eccentricity data to calculate the ECD of the whole cementing process. Then, a data set is constructed and analyzed using the random forest method to quantitatively evaluate influencing factors such as displacement, rheology, density, and eccentricity on the bottomhole and wellbore ECD. Results find that the density of cement slurry and drilling fluid has the most significant impact on the maximum ECD, with the impact reaching 0.3142 and 0.2902, respectively, and the main factors that affect the minimum ECD are the density and rheological changes in the drilling fluid, reaching 0.7014 and 0.2846. These research findings will contribute to the precise control of wellbore pressure during cementing operations, further ensuring the safety of cementing operations, and laying a technical foundation for the automation and intelligentization of subsequent cementing operations.

1. Introduction

Facing the intensifying issue of climate change, it is widely recommended that the impact of human activities on nature be reduced and the destruction inflicted upon the natural world be mitigated [1,2]. In the field of oil drilling, if the formation cannot be given sufficient pressure to maintain stability during construction, pressure control failure may occur, leading to the leakage of liquids and gasses from the formation to the surface through the wellbore [3]. For example, the Gulf of Mexico Oil Spill in 2010 covered 2500 square kilometers of the sea with crude oil, and the adverse impact on the ecological environment of the Gulf of Mexico is yet to disappear [4]. Subsequent expert analysis identified the direct cause of the incident as pressure control failure during the cementing process in well construction. According to statistics, the recent construction records of 39 oil wells in an oilfield in Eastern China found that more than 70% of the oil wells encountered complex accidents such as well leakage, invasion, and blowout caused by a pressure imbalance [5].

To ensure the safety of cementing operations in well cementing, four-string or five-string casing designs and open-hole completion are currently in use. This also puts forward higher requirements for the accuracy of wellbore pressure prediction and control results during cementing. Existing studies on cementing ECD (Equivalent Circulating Density) mainly focus on quantifying the influence rules of downhole high temperature and high pressure on wellbore fluid rheology based on experiments to improve the accuracy of the model calculation. Some researchers have conducted an experimental study on the rheology of water-based drilling fluids under high temperature and high pressure using the Fann high-temperature, high-pressure viscometer and conducted a qualitative analysis [6]. Another author proposed a comprehensive model for predicting the shape viscosity, yield value, and apparent viscosity by synthesizing multiple published models [7]. Other authors conducted an in-depth study on the influence of pressure on the yield stress, apparent viscosity, and fluidity index of water-based drilling fluids based on the high-temperature, high-pressure rheometer [8]. Researchers enhanced the prediction accuracy and further broadened the application range of the prediction model by using a non-parametric nonlinear regression model to fit the experimental data [9]. Other researchers have explored how the eccentricity of annuli significantly affects the velocity profile and frictional pressure losses in extended-reach wells and slimholes [10]. Authors developed a numerical model to simulate the laminar flow of yield-power-law fluids in eccentric annular geometries, which has an excellent agreement for the case of non-Newtonian fluids [11]. For ECD calculation models, the parameters that most significantly affect the precision of the results are the flow rate, fluid rheology, fluid density, wellbore diameter, and eccentricity. The aforementioned calculation models and commercial software all evolved from the classical drilling ECD calculation model framework, focusing mostly on the variations in flow rate, rheology [12,13,14], and density [15]; however, the wellbore diameter and eccentricity are typically calculated using fixed values throughout, which, ignoring the variations in wellbore diameter and eccentricity, would lead to discrepancies between the calculated results and field-measured values [16], as shown in Figure 1.

Machine learning methods such as random forest, the simulated annealing (SA) algorithm, and support vector regression (SVR) have been widely used in the field of drilling and cementing, mainly for predicting pressure and optimizing conventional calculation models. Some authors using a combination of the SA algorithm and SVR propose an effective monitoring method for bottom well pressure, and the model integrates hydrostatic column pressure, annulus pressure loss, and surface back pressure data, allowing for accurate monitoring without traditional Pressure While Drilling (PWD) instruments [17]. Other authors considered that traditional models often fall short in both accuracy and computational efficiency and evaluated several machine learning algorithms—Bayesian Neural Network (BNN), random forest (RF), Artificial Neural Network (ANN), and Support Vector Machine (SVM) to improve predictions of pressure loss in concentric and eccentric annuli. The results indicate that RF and BNN provided the best accuracy, with mean absolute errors of 2.57% and 3.2%, respectively [18]. Another researcher found the use of computational fluid dynamics (CFD) and machine learning to model pressure loss and mass transfer in membrane channels, focusing on spacer-filled membranes [19].

In response to the deficiencies in the existing research, this study focuses on a well in the eastern oilfields of China, and based on the rheological experiment of high temperature and high pressure, the irregular eccentric wellbore model was established according to the measured wellbore diameter and eccentricity data to calculate the ECD of the whole cementing process. Then, the data set was constructed and analyzed using the random forest method to quantitatively evaluate the influence of various factors such as displacement, rheology, density, and eccentricity on bottomhole and wellbore ECD, identifying the main influencing factors. The research findings contribute to the precise control of wellbore pressure during cementing operations, further ensuring the safety of cementing operations, reducing the risk of environmental pollution, and laying a technical foundation for the automation and intelligentization of subsequent cementing operations.

2. Study on the Irregular Eccentric Shaft Model

2.1. Physical Parameters of the Wellbore Model

2.1.1. Well Diameter

Through research on related hydrodynamic parameter calculation models and simulation software, the method generally used is to first measure the well diameter and then take the average to calculate the well diameter parameter in the barehole section of the cementing casing model [11]. However, when calculating the ECD for cementing, the precision of the well diameter parameter not only affects the accuracy of the calculations for the annular return velocity and circulating frictional resistance but also influences the model’s judgment of the real-time upper and lower interface positions of different injected fluids at each moment. Therefore, in order to prevent a significant deviation between the model calculation results and the measured results, which would lead to complex accidents such as cementing leakage, the annulus open-hole section cannot be regarded as a uniform ring of the same size from top to bottom. Instead, it should be regarded as a combination of rings of different diameters.

Research has taken a directional well 7X27x2 in an oilfield in Eastern China as an example, and the well depth is 4980 m, the three-opening drill bit size is 215.9 mm, and the outer diameter of the casing is 139.7 mm. Based on the barehole section of the well logging data, this paper calculated parameters of average wellbore model and an irregular wellbore model which are shown in Figure 2. Among them, multifacetedly considering the comprehensive algorithm structure and computation time, the irregular wellbore model adopts a five-layer wellbore model with each layer’s wellbore diameter being the average measured diameter of that section. The comparison of wellbore diameters between the two models is shown in Figure 3.

2.1.2. Eccentricity

Considering the avoidance of complex strata such as high-pressure water and karst caves, the designed wellbore trajectory may include turns and twists. In a wellbore that extends for thousands of meters, the casing is likely to deviate from the center, especially in the barehole section where the uneven rock wall usually leads to an increase in the casing eccentricity. Research referring to the data from well 7X27x2 predicts the annular eccentricity at different depths in the model on the basis of the professional software CEMPRO+ version 2.7.8 of casing, as shown in Figure 4.

2.2. Physical Parameters of Wellbore Fluids

2.2.1. Density Under High Temperature and High Pressure

In this paper, a multivariate nonlinear regression analysis method is employed to standardize the experimental data of water-based drilling fluids from experimental and related research [20,21]. After fitting, the relationships between the density of water-based drilling fluids, isolation fluids, and temperature and pressure are obtained, as shown in the following equations.

$$drillingfluid:{\rho }_D={\displaystyle \frac{{\rho }_d}{1.63}}\left(-1.119\times \left(T-T_0\right)+0.29\times \left(P-P_0\right)-3.61\times {10}^{-6}\right)$$

(1)

$$spacer:{\rho }_S={\displaystyle \frac{{\rho }_s}{1.015}}\left(-0.616\times \left(T-T_0\right)+0.598\times \left(P-P_0\right)-3.65\times {10}^{-6}\right)$$

(2)

where the following variables are used:

• ρ_D is the predicted drilling fluid density, kg/m³;
• ρ_d is the test drilling fluid density, kg/m³;
• ρ_S is the predicted spacer density, kg/m³;
• ρ_s is the test spacer density, kg/m³;
• T is the downhole temperature, °C;
• T₀ is the test temperature, °C;
• P is the downhole pressure, MPa;
• P₀ is the test pressure, MPa.

As a mixture, there is a significant difference in properties between different cement systems. Additionally, chemical reactions persist after the system is prepared, making accurate density measurement challenging. Therefore, in the model, the density of the cement is based on the design value.

2.2.2. Rheology Under High Temperature and High Pressure

In this study, we choose the Chandler 7600 High Temperature and Pressure Rheometer (Oklahoma City, OK, USA) to measure the rotation shear stress values of reused drilling fluids [22], isolation fluids, and cement slurry at 60, 90, 120, and 150 °C, which are simulated using multivariate nonlinear regression based on the Herschel–Bulkley rheological model. The fitting formula is shown in the following equation, as shown in Figure 5 and Figure 6.

$$\left\{\begin{array}{l}\tau ={\alpha }_1+{\alpha }_2\times \left(T-T_0\right)+{\alpha }_3\times \left(P-P_0\right)+ \left({\alpha }_4+{\alpha }_5\times \left(T-T_0\right)+{\alpha }_6\times \left(P-P_0\right)\right)\times {\gamma }^{{\alpha }_7}\\ n={\alpha }_7\\ k={\alpha }_4+{\alpha }_5\times \left(T-T_0\right)+{\alpha }_6\times \left(P-P_0\right)\\ {\tau }_0={\alpha }_1+{\alpha }_2\times \left(T-T_0\right)+{\alpha }_3\times \left(P-P_0\right)\end{array}\right.$$

(3)

where the following variables are used:

• α₁…α₇ are the fitting coefficients, dimensionless;
• τ is the shear stress, Pa;
• τ₀ is the dynamic shear stress, Pa;
• n is the fluidity index, dimensionless;
• k is the consistency coefficient, Pa·sn.

The function relationship between temperature, pressure, and n, k, and τ is obtained by fitting, and the results are shown in

Table 1. Fitting flow equation of the drilling fluid, spacer, and cement slurry.

Fluid	Parameters	The Fitted Equation	R2
Drilling fluid	k	k = 0.623 − 0.006 × (T − T0) − 0.004 × (P − P0)	0.982
	n	n = 0.769
	τ	tau = 1.784 + 0.015 × (T − T0) + 0.017 × (P − P0)
Spacer	k	k = 0.69 − 0.001 × (T − T0)	0.998
	n	n = 0.573
	τ	tau = 18.514 − 0.013 × (T − T0)
Cement slurry	k	k = 0.992 − 0.003 × (T − T0)	0.999
	n	n = 0.739
	τ	tau = 1.698 + 0.005 × (T − T0)

.

3. Study on the ECD Model of the Irregular Eccentric Wellbore

3.1. Temperature Field

3.1.1. Wellbore Model

The thermal analysis within the wellbore can focus on three regions, namely the drill string, the annulus, and the formation [23,24]. The following assumptions are considered to simplify the problem:

• The radial temperature gradient within the drilling fluid is neglected;
• The vertical heat conduction along the drill string and the formation is neglected;
• The thermal conductivity of the formation is assumed to be constant.

The model of the bottomhole temperature field in cementing mainly includes three parts of heat exchange. Firstly, there is a vertical thermal convection along the drill pipe within the drill string. Secondly, there is a heat exchange between the drill pipe and the annulus. Lastly, there is a thermal conduction within the formation and a thermal convection within the annulus. According to the cementing construction plan, the fluid temperature in the drill pipe and annulus can be close to the formation temperature after two weeks of full drilling fluid circulation.

3.1.2. Governing Equation

(1)

Inside the casing:

The heat exchange of the fluid inside the casing includes the thermal transfer of fluid flow during the pumping process (axial), the forced convection heat transfer between the fluid and the casing’s inner wall due to fluid–solid coupling (radial), and the energy (viscous dissipation energy) generated by the fluid friction pressure drop.

$$Q_c-{\rho }_{1}q{C}_1{\displaystyle \frac{\partial T_c}{\partial Z}}+\pi r_{ci}{h}_{ci}(T_w-T_c)={\rho }_1{C}_1\pi r_{ci}^2{\displaystyle \frac{\partial T_c}{\partial t}}$$

(4)

where the following variables are used:

• Q_c is energy generated by friction pressure drop of drilling fluid per unit length of casing, W/m;
• ρ₁ is the density of the working fluid inside the casing, kg/m³;
• q is the discharge rate of the drilling fluid, m³/s;
• C₁ is the specific heat capacity of the drilling fluid, W/(m·K);
• T_c is the temperature of the drilling fluid inside the casing, °C;
• T_w is the temperature of the casing column, °C;
• r_ci is the inner diameter of the casing, m;
• h_ci is the convective heat transfer coefficient between the fluid inside the casing and the casing, W/(m·K).

(2)

Casing string:

$${\rho }_2{C}_2{\displaystyle \frac{\partial T_w}{\partial t}}={\lambda }_2{\displaystyle \frac{{\partial }^2{T}_w}{{\partial }_z^2}}+{\displaystyle \frac{2r_{co}{h}_{co}}{r_{co}^2-r_{ci}^2}}\left(T_a-T_w\right)+{\displaystyle \frac{2r_{ci}{h}_{ci}}{r_{co}^2-r_{ci}^2}}\left(T_c-T_w\right)$$

(5)

where the following variables are used:

• ρ₂ is the density of the casing string, kg/m³;
• C₂ is the specific heat capacity of the casing string, J/(kg·K);
• λ₂ is the thermal conductivity of the casing, W/(m·K);
• r_ci is the inner diameter of the casing, m;
• r_co is the outer diameter of the casing, m;
• h_ci is the convective heat transfer coefficient between the fluid inside the casing and the casing itself, W/(m²·K);
• h_co is the convective heat transfer coefficient between the annular fluid and the casing, W/(m²·K);
• T_a is the temperature of the annular fluid, °C.

(3)

Annular fluid:

The heat exchange of the annular fluid includes the thermal transfer of fluid flow during the pumping process, the forced convection heat transfer formed by fluid–solid coupling between the fluid and casing outer wall and the formation in the radial direction, and the energy (viscous dissipation energy) produced by the fluid friction pressure drop. The heat conduction in the axial direction and the natural convection heat transfer are neglected.

$${\rho }_{1}q{C}_1{\displaystyle \frac{\partial T_a}{\partial z}}+2\pi r_b{h}_b\left(T_{f1}-T_a\right)+2\pi r_{co}{h}_{co}\left(T_w-T_a\right)+Q_a={\rho }_1{C}_1\pi \left(r_b^2-r_{co}^2\right){\displaystyle \frac{\partial T_a}{\partial t}}$$

(6)

where the following variables are used:

• T_f1 is the temperature of the wellbore wall, °C;
• r_b is the radius of the barehole annulus, m;
• r_co is the outer diameter of the casing, m;
• h_b is the convective heat transfer coefficient between the formation and the drilling fluid in the annulus, W/(m·K);
• h_co is the convective heat transfer coefficient between the annular drilling fluid and the casing, W/(m·K);
• Q_a is the heat generated by the frictional pressure drop of the drilling fluid in the annulus per unit length and per unit time, W.

(4)

The formation and the wellbore wall:

$${\displaystyle \frac{{\rho }_e{c}_e}{{\lambda }_e}}{\displaystyle \frac{\partial T_e}{\partial t}}={\displaystyle \frac{{\partial }^2{T}_e}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_e}{\partial r}}+{\displaystyle \frac{{\partial }^2{T}_e}{\partial z^2}}$$

(7)

$$T_h={\displaystyle \frac{{\lambda }_e{T}_e+T_a{r}_{h}kf\left(t\right)}{r_{h}kf\left(t\right)+{\lambda }_e}}$$

(8)

where the following variables are used:

• r is the radial distance, m;
• ρ_e is the density of the formation rock, kg/m³;
• c_e is the specific heat capacity of the formation rock, J/(kg·K);
• λ_e is the thermal conductivity of the formation, W/(m·K).

3.1.3. Initial and Boundary Conditions

• Formation temperature: the initial temperature of the drilling fluid in the drill string and the annulus and each element in the formation;
• Casing inlet boundary: The inlet temperature of the drilling fluid can usually be measured in practice. T(z = 0, t) = T_in, T_in is known;
• Annulus inlet boundary: at the bottom of the well (z = H_max), the temperature of the fluid inside the drill string is approximately equal to that in the annulus;
• Formation boundary: T_e(z,r→∞,t) is equal to the geothermal gradient.

3.2. ECD Calculation Model

3.2.1. Hydrostatic Pressure Calculation

From the wellhead to the bottom, based on different layers, the changed annular radius R_i is iteratively calculated via ECD with the functional expression as follows:

$$EC{D}^{\left(R_i\right)}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\rho }_{\left(T_i\right)}gh}+{\displaystyle \sum _{i=1}^n{P}_{f\left(T,R_i\right)}+P_c}}{gH}}$$

(9)

where the following variables are used:

• i is the layer, dimensionless;
• R_i is the layer’s annular radius, m;
• h is the well layer’s depth, m;
• P_c is the wellhead back pressure, Pa;
• ρ_(Ti) is the fluid density at the layer’s temperature, kg/m³;
• P_f(T,Ri) is the frictional pressure drop at the layer’s radius and temperature, Pa.

3.2.2. Frictional Pressure Drop Calculation

In field cementing operations, the annulus below the wellbore is often eccentric, with the downhole fluid flowing through the eccentric annulus [11]. An eccentric annular flow model is established, where the inner and outer radii of the annulus are R₁ and R₂, respectively, with the center of the inner circle being O₁ and the outer circle being O₂, as shown in Figure 7. By establishing a polar coordinate system (α, r) with the center of the inner circle as the origin and the radius direction as the ray, the calculation Formula (10) for the annular gap R_i can be obtained.

$$R_i=\left(1+\cos \alpha \right)\left(R_2-R_1\right)-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{R_2-R_1}{R_2}}\right){\sin }^2\alpha$$

(10)

where the following variables are used:

• α is the angle, °;
• R₁ is the annular inner radius, m;
• R₂ is the annular outer radius, m.

When α = 0, the annular gap is at its maximum; when α = π, it is at its minimum. The Herschel–Bulkley model with three parameters is more representative of the cement slurry, and its constitutive relationship is shown in Equation (11).

$$\tau =k{\gamma }^n+{\tau }_0$$

(11)

This Equation is established based on the dynamic resistance balance of the fluid flow in the annulus. The frictional pressure drop in the Herschel–Bulkley model can be derived and represented as follows:

$$\left\{\begin{array}{l}{P}_f={\left({\displaystyle \frac{1+2n}{3n}}{\displaystyle \frac{6u}{R_{i}G}}\right)}^n\left({\displaystyle \frac{2hk}{R_i}}\right)+\left({\displaystyle \frac{1+2n}{1+n}}\right){\displaystyle \frac{2h\tau N}{R_{i}G}}\\ G=1+{\displaystyle \frac{1+n}{4n}}\left({\displaystyle \frac{3+{\left(R_1/R_2\right)}^2}{1+\left(R_1/R_2\right)}}+{\displaystyle \frac{1}{n}}\right){\left({\displaystyle \frac{e}{R_2-R_1}}\right)}^2\\ N=1+{\displaystyle \frac{1+n}{4n}}\left({\displaystyle \frac{3+{\left(R_1/R_2\right)}^2}{1+\left(R_1/R_2\right)}}+{\displaystyle \frac{1-n}{n}}\right){\left({\displaystyle \frac{e}{R_2-R_1}}\right)}^2\end{array}\right.$$

(12)

where

• P_f is friction pressure drop, Pa;
• e is the eccentric distance, m.

Based on another researcher’s experimental data [25], the friction calculation of the model is proven to be accurate. The length of experimental device annulus is 9 m, and the diameters of the inside and outside pipes are 54.42 mm and 127 mm, respectively. The comparison of model calculations and experimental results is shown in

Table 2. Comparison of model calculations and experimental results.

Inlet Velocity (m/s)	Model Calculation (Pa)	Experimental Result (Pa)	Deviation (%)
0.18	24,043.35	21,858.351	9.09
0.3	27,829.73	26,313.37	5.45
0.4	30,270.05	29,501.259	2.54
0.5	32,334.30	32,301.236	0.1

.

3.3. Validation of Computational Results

The ECD calculation model is used to simulate and calculate the ECD in the wellbore and at the bottom of the well during the cementing process in the 7X 7x2 oilfield in Eastern China. The displacement scheme is shown in

Table 3. The displacement scheme.

Item Number	Type of Fluid	Injection Flow Rate (m3/min)	Injection Volume (m3)
1	Spacer	1.8	30
2	Leading cement	1.0	18.0
3	Tailing cement	1.0	34.0
4	Displacement fluid 1	1.2	4.8
5	Displacement fluid 2	1.2	43

, and the wellbore and related fluid parameters are shown in

Table 4. The wellbore parameters and related fluid parameters.

Item	Parameter	Value	Parameter	Value
Annulus parameters	Drill bit size, m	0.2159	Inlet mud temperature, °C	29.4
	Pipe size, m	0.1397	Geothermal gradient, °C/100 m	4.9
	Pipe depth, m	4971	Wellbore depth, m	4980
fluid parameters	Rheology of fluid	Refer to Table 1
	Density of drilling fluid and spacer, g/cm3	Refer to Equations (1) and (2)
	Density of cement, g/cm3	Leading: 2.05/tailing: 1.90
	Displacement and replacement time	Refer to Table 3

.

The bottomhole ECD and total well ECD can be calculated using calculation models that consider conventional, coupled temperature–pressure and irregular wellbore diameter factors, respectively. The calculation results are shown in Figure 8. In the figure, compared with the conventional model and the model that only considers temperature and pressure, the new model’s calculation results consider the temperature, pressure, and radius changes. They can more accurately calculate the change in the displacement interface position caused by the irregular wellbore.

4. Based on the Random Forest Algorithm for Irregular Wellbore Diameter Weight Analysis

4.1. Random Forest Model and Related Parameters

The database is constructed using the calculation results of the previous model; then, the model is established using the random forest method to quantitatively analyze the influence of factors such as displacement, rheological changes, different liquid densities, eccentricity, and well diameter changes on ECD [26]. Based on the random forest method combined with the bagging algorithm and the decision tree algorithm, the model’s output is determined via either a majority vote or by averaging the outcomes from multiple decision trees. This approach addresses the issue of insufficient predictive accuracy associated with individual decision trees, bolstering the model’s generalization capabilities.

The database is constructed using the computational outcomes derived from standard cementing ECD models, ECD models that account for thermoelastic coupling, and ECD models that factor in irregular wellbore diameters. It encompasses the complete set of results following the variation in parameters such as well depth, displacement timing, flow rate, eccentricity, rheological properties, density, and the irregularity of the wellbore diameter. The model coefficients are defined as n_estimators set to 200, max_samples set to the full data set, and max_features ranging from 3 to 5.

4.2. The Maximum/Minimum ECD Multifactor Weight Analysis During Cementing Operations

The control variables include the injection displacement rate, drilling fluid density, spacer density, cement slurry density, eccentricity, consideration of expanding diameter, and consideration of physical property changes in high-temperature and high-pressure fluids. The data labels and classification are shown in

Table 5. Data labels and classification.

Labels	Classification	Labels	Classification
Drilling fluid density	1.4/1.5/1.6	Eccentricity	0.4/0.8
Spacer density	1.6/1.7/1.8	Irregular wellbore diameter	Presence = 1/absence = 0
Cement slurry density	1.8/1.9/2.0
Flow rate	0.5/1.2/1.8	Rheology	Presence = 1/absence = 0

. A total of 90 groups of four-factor three-level and three-factor two-level experimental schemes were constructed to perform simulation calculations for the maximum/minimum ECD in cementing operations.

According to the comprehensive weight analysis of the factors influencing the maximum ECD of the annulus using the random forest method, the model parameters were set with estimators at 200, a random state at 42, and a test set proportion at 0.3 with an R² of 0.8922 and an MSE of 2.84 × 10⁻⁴. The predictive results of the test set compared with the actual values are shown in Figure 9, indicating a good fit for the model. The significance of the factors is presented in

Table 6. The significance of influencing factors and average SHAP values for maximum ECD in cementing operations.

Labels	Importance	SHAP Value	Labels	Importance	SHAP Value
Flow rate	0.0111	0.001	Eccentricity	0.0111	0.0021
Drilling fluid density	0.2902	0.0266	High T/P rheology and density	0.1278	0.0156
Spacer density	0.1694	0.0072
Cement slurry density	0.3142	0.0159	Hole enlargement	0.0762	0.0109

and Figure 10. According to the results, it can be found that the most significant factors affecting the maximum ECD in the annulus are the density of the drilling fluid, spacer, and cement slurry. Secondary factors include the consideration of high-temperature and high-pressure fluid properties and wellbore irregularity, while eccentricity and displacement volume are non-significant.

In addition, the positive and negative impacts of factors on ECD need to be analyzed. SHAP (SHapley Additive exPlanations) is an interpretable model inspired by the Shapley value and represents additive explanations. It quantifies the impact of each factor on the prediction outcome. By utilizing the SHAP model for in-depth data analysis, the output value for an individual sample is 0.0793, and the influence of sample factors is shown in Figure 11. This figure shows the impact of each sample’s features on the model’s prediction results. The horizontal axis is the SHAP value, and the vertical axis is the actual value of the feature. The redder the color, the higher the positive impact the feature has on the prediction results, and the bluer the color, the higher the negative impact. Meanwhile, the wider the color area on the horizontal axis, the greater the impact of the feature value on the result. The figure reveals that the flow rate, density of the drilling fluid, spacer, cement slurry, and consideration of wellbore enlargement are positively correlated with the maximum ECD result, while eccentricity and the consideration of high-temperature high-pressure fluid properties are negatively correlated with the outcome.

We suggest the readers to refer to the previous analysis path to quantitatively analyze the factors affecting the minimum ECD. The R² for the minimum ECD model is 0.9982, with an MSE of 1.48 × 10⁻⁵. The predictive results of the test set compared with the actual values are shown in Figure 12, indicating a good fit for the model. The significance of the factors is presented in

Table 7. Significance of influencing factors and average SHAP values for the minimum ECD in cementing operations.

Labels	Importance	SHAP Value	Labels	Importance	SHAP Value
Flow rate	0.0065	0.0059	Eccentricity	0.0006	0.0009
Drilling fluid density	0.7015	0.0633	High T/P rheology and density	0.2846	0.0542
Spacer density	0.0007	0.0004
Cement slurry density	0.0003	0.0004	Hole enlargement	0.0058	0.0046

and Figure 13. The results reveal that the most significant factors affecting the minimum ECD in the annulus are the drilling fluid density and the consideration of high-temperature, high-pressure fluid properties. The secondary factors include flow rate and the consideration of wellbore irregularity, while the density of the spacer, cement slurry, and eccentricity are non-significant.

By utilizing the SHAP model for in-depth data analysis, the output value for an individual sample is 0.1297. The influence of sample factors on the result is shown in Figure 14. This figure reveals that the density of the drilling fluid and the flow rate are positively correlated with the minimum ECD result, while high-temperature and high-pressure properties and wellbore enlargement are negatively correlated with the result. The influence of other factors is not significant. This understanding is consistent with the engineering experience, further demonstrating the model’s good interpretability.

5. Conclusions

• In view of the narrow density window problem faced by deep well drilling and cementing operations, this study establishes a new calculation model for cementing ECD. This new model does not adopt the assumption of a uniform wellbore but instead establishes a multi-layer structure of the wellbore based on an irregular wellbore, further clarifying the impact of irregular well diameter factors on both the downhole and wellbore ECD. The comparison between the model calculation results and the conventional, temperature, and pressure models proves that an irregular well diameter significantly impacts the downhole and wellbore ECD and provides new ideas for the study of deep well ECD control methods.
• Based on the random forest method, a quantitative analysis of the calculation results of the new model found that the density of cement slurry and drilling fluid has the most significant impact on the maximum ECD, with the impact reaching 0.3142 and 0.2902, respectively. Moreover, the impact of spacer fluid density, rheological changes, and irregular well diameter cannot be ignored. The main factors that affect the minimum ECD are the density and rheological changes in the drilling fluid, reaching 0.7014 and 0.2846.