Introduction

The wellbore trajectory refers to the path taken by a well to reach the underground target area from the surface wellhead position, also known as the wellbore trajectory1,2,3,4,5,6,7,8. In drilling technology, optimizing wellbore trajectory is crucial for safety, cost efficiency, and adapting to complex geological conditions, yet current models primarily focus on geological and engineering parameters rather than comprehensive multi-factor evaluation9,10,11,12,13,14,15,16,17. Together, they ensure the accuracy and safety of drilling operations. The optimization method of wellbore trajectory is crucial in petroleum engineering and geological engineering, as it helps determine the optimal wellbore path to improve drilling efficiency, safety, and economic benefits18,19. Currently, drilling trajectory optimization mainly combines geological data and drilling engineering parameters, and uses mathematical methods for optimization20,21,22,23. It cannot comprehensively evaluate wellbore trajectory based on construction difficulty, pipe stress, and drilling cost. Despite advances in well trajectory optimization, existing models lack an integrated approach that quantitatively balances trajectory complexity, drill string stress, and economic factors.

In order to solve the current trajectory optimization problem that only considers construction feasibility and geological objectives, this paper proposes a new trajectory optimization model that comprehensively considers factors such as construction difficulty, economy, and wellbore cleanliness. This model can provide a theoretical method for more comprehensive and scientific optimization of wellbore trajectories. And taking oilfield a as an example, the entire trajectory optimization scheme usage process was explained. Through analysis, it was shown that the method proposed in this paper can comprehensively optimize the wellbore trajectory. Subsequently, embedding specific drilling data of oilfield oil through machine learning can increase the robustness of the model.

Materials and methods

Development of the well trajectory selection model

In order to quantify the construction difficulty of drilling trajectories, this article comprehensively considers factors such as azimuth rate, inclination rate, and well depth, proposes the definition of trajectory difficulty coefficient D, and provides a calculation formula. In the formula, the larger the azimuth rate and slope rate, the greater the well depth, and increasing the number of inclines or twists will make the trajectory profile more complex. Correspondingly, on-site construction will increase in both period and difficulty. The specific calculation method is shown in Formula 1. The higher the trajectory difficulty coefficient, the greater the construction difficulty. \(\:{\beta\:}_{j}\)=0(When the j section is only a pure inclination section), and \(\:{\alpha\:}_{j}\)=0(when the j section is only a pure twist azimuth section).

$$\:D=\sum\:_{\text{j}=1}^{\text{j}=\text{m}}{({\upalpha\:}}_{\text{j}}\frac{\varDelta\:{\text{I}}_{\text{j}}}{\varDelta\:{\text{I}}_{\text{m}\text{a}\text{x}}}+{{\upbeta\:}}_{\text{j}}\frac{\varDelta\:{{\uptheta\:}}_{\text{j}}}{\varDelta\:{{\uptheta\:}}_{\text{m}\text{a}\text{x}}})+{\upgamma\:}\frac{{{\uptheta\:}}_{\text{m}\text{a}\text{x}}}{{{\uptheta\:}}_{\text{B}\text{m}\text{a}\text{x}}}\:+\text{h}\frac{{\text{L}}_{\text{m}\text{a}\text{x}}}{{\text{L}}_{\text{B}\text{m}\text{a}\text{x}}}$$
(1)

When analyzing the stress state of the drill string, torque, slide force, and effective stress are generally considered. The larger these forces, the more complex the stress state of the drill string, and the higher the probability of drill string failure. Different trajectories have an impact on the stress of the drill string, so this article will also consider the stress state of the drill string when selecting trajectories.The force on the drill string refers to the calculation of the torque, slide force, and effective tension of the drill string at the depth of completion under the same drilling tool combination and drilling parameter conditions for different trajectory schemes. The calculation conditions can include all working conditions such as lifting, lowering, rotating at the bottom of the well, sliding drilling, etc., or key pipe column working conditions. Here, the evaluation coefficient F for drill string force is defined, and the calculation formula is shown in Formula 2.

$$\:F={a}_{t}\ast\:Tr+{a}_{s}\ast\:S+{a}_{e}\ast\:E+......{a}_{0}\ast\:O$$
(2)

The coefficients \(\:{\text{a}}_{\text{t}\text{r}}\)、\(\:{\text{a}}_{\text{s}}\)、\(\:{\text{a}}_{\text{e}}\)、\(\:{\text{a}}_{0}\) are weight coefficients determined by users according to their needs, and \(\:{\text{a}}_{\text{t}\text{r}}\)+\(\:{\text{a}}_{\text{s}}\)+\(\:{\text{a}}_{\text{e}}\)+……\(\:{\text{a}}_{0}\)=1 The force values of Tr, S, E, O, etc. are absolute values and should be dimensionless according to Formula 3.

Under the same conditions of other factors, the wellbore cleanliness, drilling cycle, and drilling cost of different trajectory schemes are different. When optimizing the trajectory, the trajectory scheme with good wellbore cleanliness, short drilling cycle, and low drilling cost should be considered.The wellbore cleaning condition is an evaluation of the wellbore cleaning condition at the depth of drilling under the same drilling tool combination and drilling parameter conditions for different trajectory schemes. The quantitative indicator is the thickness of the rock debris bed C, and the larger the thickness of the rock debris bed, the worse the wellbore cleaning condition.The drilling cycle T is an estimation of different trajectory plans based on the same daily footage, moving speed, drilling rig dismantling speed, intermediate completion speed, and cementing speed. The longer the cycle, the longer the construction time required for the trajectory plan.The drilling cost M is an estimation of the operating cost for different trajectory schemes under the same operating fee standard. The higher the drilling cost, the higher the construction cost for the trajectory scheme.

Data processing and model implementation

Due to inconsistent units of indicators such as drill string stress (torque, effective tension, slide force), wellbore cleanliness, drilling cycle, drilling cost, etc., it is necessary to dimensionalize each indicator using the Formula 3.

$$\:{Z}_{i}^{0} = \frac{{Z}_{i}}{\text{m}\text{a}\text{x}\left\{{\text{Z}}_{1}, {\text{Z}}_{2}, {\text{Z}}_{3}......{\text{Z}}_{\text{n}}\right\}}$$
(3)

The present invention is used for optimizing the trajectory of a single well. For different trajectory schemes, the difficulty coefficient of the trajectory, the stress on the drill string, the cleanliness of the wellbore, the drilling cycle, and the drilling cost can be calculated separately. The corresponding normalized values of the indicators are the characteristic values of each indicator of the trajectory scheme.

For the optimal selection of trajectory schemes for multiple wells, the method for determining the characteristic values of the schemes should be determined. In the present invention, if there are m wells (m ≥ 2) in one trajectory scheme, the principle for selecting the characteristic values of each indicator of the trajectory scheme is as follows,

\(\:\text{t}\text{r}\text{a}\text{j}\text{e}\text{c}\text{t}\text{o}\text{r}\text{y}\:\text{d}\text{i}\text{f}\text{f}\text{i}\text{c}\text{u}\text{l}\text{t}\text{y}{\text{D}}_{\text{i}}^{0}=\text{m}\text{a}\text{x}({\text{D}}_{1}^{0},{\text{D}}_{2}^{0},......{\text{D}}_{\text{m}}^{0}\))

\(\:\text{d}\text{r}\text{i}\text{l}\text{l}\text{i}\text{n}\text{g}\:\text{s}\text{t}\text{r}\text{i}\text{n}\text{g}\:\text{s}\text{t}\text{r}\text{e}\text{s}\text{s}\:{\text{F}}_{\text{i}}^{0}=\text{m}\text{a}\text{x}({\text{F}}_{1}^{0},{\text{F}}_{2}^{0},......{\text{F}}_{\text{m}}^{0}\))

\(\:\text{r}\text{o}\text{c}\text{k}\:\text{d}\text{e}\text{b}\text{r}\text{i}\text{s}\:\text{b}\text{e}\text{d}\:\text{t}\text{h}\text{i}\text{c}\text{k}\text{n}\text{e}\text{s}\text{s}{\text{C}}_{\text{i}}^{0}=\text{m}\text{a}\text{x}({\text{C}}_{1}^{0},{\text{C}}_{2}^{0},......{\text{C}}_{\text{m}}^{0}\))

\(\:\text{d}\text{r}\text{i}\text{l}\text{l}\text{i}\text{n}\text{g}\:\text{c}\text{y}\text{c}\text{l}\text{e}{\text{T}}_{\text{i}}^{0}={\text{T}}_{1}^{0}+{\text{T}}_{2}^{0}+......{\text{T}}_{\text{m}}^{0}\)

\(\:\text{d}\text{r}\text{i}\text{l}\text{l}\text{i}\text{n}\text{g}\:\text{c}\text{o}\text{s}\text{t}{\text{M}}_{\text{i}}^{0}={\text{M}}_{1}^{0}+{\text{M}}_{2}^{0}+......{\text{M}}_{\text{m}}^{0}\)

This article proposes optimal values for trajectory schemes, calculated using Formula 4.

$$\:Y={a}_{D}\ast\:D+{a}_{F}\ast\:F+{a}_{C}\ast\:C+{a}_{T}\ast\:T+{a}_{M}\ast\:M$$
(4)

The trajectory difficulty coefficient, drill string stress, wellbore cleanliness, drilling cycle, and drilling cost were assigned weight coefficients based on operational priorities, ensuring a balanced selection approach.The coefficients of \(\:{\text{a}}_{\text{D}}\), \(\:{\text{a}}_{\text{F}}\), \(\:{\text{a}}_{\text{C}}\), \(\:{\text{a}}_{\text{T}}\), \(\:{\text{a}}_{\text{M}}\) are determined by the user according to their needs as weight coefficients, and \(\:{\text{a}}_{\text{D}}+{\text{a}}_{\text{F}}+{\text{a}}_{\text{C}}+{\text{a}}_{\text{T}}+{\text{a}}_{\text{M}}\)=1.The selection of weights should take into account the actual situation of the oilfield, including the influence of geological factors, the priority level of the first level manager during the development stage, and the weight coefficients of the same factor are not fixed. We recommend using our trajectory selection method to iteratively check whether each weight coefficient meets the actual situation, in order to determine the weight coefficient that is more suitable for the oilfield itself.

The larger the values of trajectory difficulty coefficient, drill string stress, rock debris bed thickness, drilling cycle, and drilling cost, the worse the trajectory scheme. Therefore, the trajectory scheme corresponding to the minimum value of trajectory scheme selection should be the optimal scheme.

Case study: oilfield A

The A oilfield has two single target wells, Well 1 and Well 2, which can be designed as vertical or two-dimensional directional wells. The design parameters and requirements are shown in Table 1. The maximum drilling dogleg in the history of the oilfield is 4.5 °/30m, the maximum well inclination angle is 45 °, and the maximum well depth is 3000 m. The directional wells in the oilfield are three-stage system, with a ground elevation of 371.22 m and a filling height of 7.5 m. If batch drilling is used for construction, the drilling rig guide rail height is 0.4 m, the slot spacing is 6 m, set \(\:{a}_{D}={a}_{F}={a}_{C}={a}_{T}={a}_{M}\)=0.2, and the force on the drill string only considers the torque, slide force, and effective tension of the drill string. The calculation conditions include lifting, drilling, spinning at the bottom of the well, and sliding drilling.\(\:{a}_{tr}\)=\(\:{a}_{s}\)=\(\:{a}_{e}\)=1/3, \(\:{{\upalpha\:}}_{1}\)=\(\:{\upgamma\:}\)=\(\:\text{h}\)=1/3.

Table 1 Parameters and requirements for trajectory design of well 1 and well 2 for four schemes in A oilfield.
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The Landmark Wellplan module was used to simulate drilling conditions, ensuring uniform parameter settings across all trajectory schemes.According to the wellhead location, two wells can be deployed in four different wellhead deployment schemes, as shown in Table 2. The trajectory design process and detailed parameters of the four schemes can refer to the supplementary file. Scheme 1 and Scheme 2 are cluster well schemes, with Well 1 and Well 2 as the main centers of the well site. The distance between the single well slots is designed to be 6 m. Scheme 3 is a cluster well scheme with both wells being directional wells, with the well site located at the point where target points A and B are connected. Scheme 4 is a single well scheme with two well site locations, where Well 1 and Well 2 are both vertical wells. Scheme 1,Scheme 2 and Scheme 3 incorporate cluster wells, whereas Scheme 4 employs a single-well approach, influencing stress distribution and cost factors.

Table 2 Wellhead deployment plan of well 1 and well 2 for four schemes in A oilfield.
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Results

Trajectory difficulty and drill string stress analysis

Taking Well 1 in Scheme 1 as an example, calculate the difficulty coefficient of the well trajectory according to Formula 1.

$$\begin{aligned}{D}_{11} & = \sum\:_{\text{j}=1}^{\text{j}=\text{m}}{({\upalpha\:}}_{\text{j}}\frac{\varDelta\:{\text{I}}_{\text{j}}}{\varDelta\:{\text{I}}_{\text{m}\text{a}\text{x}}}+{{\upbeta\:}}_{\text{j}}\frac{\varDelta\:{{\uptheta\:}}_{\text{j}}}{\varDelta\:{{\uptheta\:}}_{\text{m}\text{a}\text{x}}})+{\upgamma\:}\frac{{{\uptheta\:}}_{\text{m}\text{a}\text{x}}}{{{\uptheta\:}}_{\text{B}\text{m}\text{a}\text{x}}} \\ & \quad + \text{h}\frac{{\text{L}}_{\text{m}\text{a}\text{x}}}{{\text{L}}_{\text{B}\text{m}\text{a}\text{x}}} =\frac{1}{3} \times \frac{4}{4.5} + \frac{1}{3} \times \frac{80.17}{45} \\ & \quad + \frac{1}{3} \times \frac{2565.26}{3000} =1.027\end{aligned}$$

Similarly, the difficulty coefficients of other wells can be obtained.\(\:{\text{D}}_{12}\)=0.148, \(\:{\text{D}}_{21}\)=0.149, \(\:{\text{D}}_{22}\)=1.082, \(\:{\text{D}}_{31}\)=0.636, \(\:{\text{D}}_{32}\)=0.664, \(\:{\text{D}}_{41}\)=0.149, \(\:{\text{D}}_{42}\)=0.148.

Using the Landmark software wellplan module, the torque, slide force, and effective tension of each well in four different schemes were calculated. During the simulation process, the drilling combination, drilling parameters, and hydraulic parameters were set to be consistent. The simulation conditions included A tripping in, B tripping out, C slide drilling, D rotating on bottom and E rotating off bottom.

The analysis results of drill string torque show that the drill string does not rotate during the tripping and lowering conditions of each well, and the torque is 0. For straight wells, the drill string is centered under the conditions of rotating, sliding, and rotating lifting off the bottom of the well. For directional wells, the torque provided by the ground is provided by the drill string in the sliding drilling condition. Compared with straight wells, directional wells have greater torque under the conditions of rotating and rotating lifting off the bottom of the well. For directional wells, the torque received by the drill string under the conditions of rotating at the bottom of the well is greater than that under the conditions of rotating lifting off the bottom of the well. In Scheme 1 and Scheme 2, the drilling dogleg of Well 1 is larger, and the drill string torque is greater than that of Well 1 and Well 2.

Slide force refers to the lateral force exerted on the drill string during directional drilling, increasing drag and potential obstruction risks .The slide force analysis results show that the vertical well is considered as an ideal state in various working conditions, with the drilling tool centered and the slide force on the drilling tool being 0. Slide force occurs in directional well drill strings, and under ideal conditions, it remains consistent across all operations. In Scheme 1, the dogleg of well 1 and Scheme 2, well 2 is larger, and the slide force on the drill string is greater than that in Scheme 3.

The results of effective tension analysis show that under the sliding drilling condition, the effective tension on the drill string is the highest. The effective tension of the vertical well is basically the same in all conditions, while the effective tension of the directional well is higher. The effective tension of the drilling and sliding drilling conditions is significantly higher than that of the other schemes. The drill string of well 1 in scheme 1 and well 2 in scheme 2 has already buckled (Table 3 ).

Table 3 Drill string stress parameters of well 1 and well 2 for four schemes in oilfield A.
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Wellbore cleaning and cost analysis

Using the Landmark software Wellplan module, the wellbore cleaning effect of four directional wells was evaluated based on the thickness index of the rock debris bed. During the simulation process, the drilling combination, drilling parameters, and hydraulic parameters were set to be consistent. The simulation results showed that there were rock debris beds in well 1 and well 2 in scheme 1 and scheme 2, respectively, with poor wellbore cleaning conditions. The rock debris bed thicknesses were 25.88 mm and 20 mm, respectively. Through analysis, it was found that the dogleg of rock debris beds in 2 wells in scheme 2 was larger than that in scheme 3, and under the same drilling displacement conditions, it was easier to form rock debris beds ( Fig. 1).

Fig. 1
Fig. 1
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Analysis of rock debris carried by directional wells for Four Schemes. (a)Well 1 for Scheme 1, (b) Well 1 for Scheme 3, (c) Well 2 for Scheme 2, (d) Well 2 for Scheme 3.

According to the drilling design, well site construction, measurement, and comprehensive drilling contracts of Oilfield A, the drilling costs of four schemes were calculated. Contingency costs cover unexpected drilling expenditures, such as unforeseen geological challenges or equipment failures.The calculation results showed that Scheme 2 had the highest drilling cost, Scheme 4 has the lowest total cost,, and the cluster well scheme had more advantages in well site construction, transportation and feeding costs, and supervision costs. While drilling design and construction costs were similar across all schemes, total costs varied significantly due to differences in well depth. There were significant differences among different schemes, with Scheme 2 having the highest comprehensive drilling cost and Scheme 4 having the lowest comprehensive drilling cost(Fig. 2 ).

Fig. 2
Fig. 2
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Drilling cost bar chart for four schemes.

Multi-criteria evaluation and optimal scheme

The maximum values of torque, slide force, and effective tension are taken for four working conditions: lifting, lowering, rotating at the bottom of the well, and sliding drilling.According to Formula 3, the torque, slide force of the drill string, and effective tension are dimensionless. The evaluation coefficient of the drill string force is calculated based on Formula 2, and the calculation results are shown in Table 4.

Table 4 Calculation of drill string stress of well 1 and well 2 for four schemes in oilfield A.
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Calculate the thickness of the rock debris bed at the completion depth for different trajectory schemes under the same drilling tool combination and drilling parameter conditions.Predict drilling cycles for different trajectory plans based on the same daily footage, moving speed, drilling rig dismantling speed, intermediate completion speed, and cementing speed. Predict the drilling costs for different trajectory schemes under the same charging standards.The original values of thickness of the rock debris bed, drilling costs, and drilling cycle and the normalized numerical results calculated according to Formula 3 are listed in Table 5.

Table 5 Calculation results of rock debris bed thickness, drilling cost, and drilling cycle of well 1 and well 2 for four schemes in oilfield A.
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Based on the above calculations, the evaluation coefficient values for each single well in the four trajectory schemes can be obtained, as shown in Table 6. This embodiment optimizes the trajectories of multiple wells simultaneously, and determines the characteristic values of the four schemes according to the method of determining the characteristic values of the trajectory schemes.

Table 6 Dimensionless evaluation indicators and characteristic values for four schemes in oilfield A.
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According to Formula 4, the optimal values for different trajectory schemes are calculated as\(\:{\text{Y}}_{1}=\)1.27, \(\:{\text{Y}}_{2}=\)1.35, \(\:{\text{Y}}_{3}=\)0.79, \(\:{\text{Y}}_{4}=\)0.66, \(\:{\text{Y}}_{4}\)<\(\:{\text{Y}}_{3}\)<\(\:{\text{Y}}_{1}\)<\(\:{\text{Y}}_{2}\). Therefore, the trajectory scheme should be selected as Scheme 4, where Well 1 and Well 2 are single well and vertical well schemes, respectively.

Discussion

In order to compare the calculation results of different schemes considering different factors, we compared the four schemes in the radar chart (Fig. 3). From Fig. 3, we can see that Scheme 4 forms the smallest enclosed area, and has significant advantages in trajectory difficulty, drill string stress, and wellbore cleaning. However, Scheme does not have an advantage in terms of drilling cycle, as the two single wells have increased relocation time. The drilling cost of Scheme 4 is relatively low, while the construction cost of Scheme 4 well site is high. However, the overall drilling cost is relatively low because the drilling footage of Plan 4 has been reduced as a whole. Overall, Scheme 4 was selected due to its lower complexity and reduced drill string stress, despite slightly higher relocation time.

Fig. 3
Fig. 3
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Comparison chart of trajectory optimization factor analysis for four schemes.

The trajectory optimization scheme provided in this article breaks through the shortcomings of incomplete consideration of factors in current trajectory optimization. However, the trajectory optimization scheme provided in this article only uses the height of the rock debris bed as an indicator for wellbore cleaning, and cannot rule out the possibility that the wellbore cleaning ability may vary among different trajectory schemes when the rock debris bed height is 0. In addition, different drilling parameters such as displacement, speed, and drilling pressure can also affect the trajectory optimization results.

Conclusions

  1. 1.

    Vertical well designs (Scheme 4) exhibited lower trajectory difficulty, drill string stress and drilling depth, reducing operational risks, lowest total cost. However, the drilling cycle is the largest among the four plans due to single-well construction.

  2. 2.

    Due to the dogleg of directional wells 1 and 2, rock debris beds will be generated during the actual drilling process in schemes 1 and 2. Except for schemes 1 and 2, which correspond to directional wells 1 and 2, there is no rock debris bed in any other single well, and the wellbore cleanliness is good.

  3. 3.

    According to the comprehensive selection method of wellbore trajectory established in this article, the optimal values of different trajectory schemes are obtained. Scheme 4, single well and vertical well schemes, are the optimal schemes. The current wellbore trajectory optimization method takes the shortest drilling cycle as the sole basis for optimization. Therefore, in this example, Scheme 3 will be selected as the optimal trajectory plan based on the shortest cycle, which will cover up the disadvantages of Scheme 3 such as drill string stress and construction difficulty. The comprehensive selection method of wellbore trajectory established in this article provides theoretical support for trajectory selection.In future research, this model should be introduced into machine learning to establish a trajectory database, fully explore data information, and make trajectory selection more accurate and model stability.