An improved 3.5° single-bend positive displacement mud motor and its parameter optimization for deep short-radius open-hole sidetracking

In order to improve the sidetracking efficiency for the recovery of old wells in the middle and later stage of oilfield, an improved 3.5° single-bend positive displacement mud motor (PDM) is designed. This novel single-bend PDM first uses the structure of 3.5° large angle PDM + cushion block, which can meet the sidetracking requirements of high build-up rate and short directional section in large curvature short-radius sidetracking wells and can significantly improve the build-up ability. Furthermore, based on the mechanical analysis and finite element numerical simulation, its parameter optimization is carried out. The optimal main parameters are determined as follows: the structural bending angle is 3.5°, the outer diameter is 95 mm, the height of the cushion block is 14 mm, the distance between the cushion block and the bending point is 120 mm, the minimum curvature radius is 35.50 m, and the weight-on-bit (WOB) should be less than 30kN. The initial section build-up rate of directional sidetracking can be increased up to 52.67°/30 m ~ 53.29°/30 m. The research results have been applied in Tahe oilfield. It proves that 3.5° single-bend PDM can sidetrack successfully at one time and increase deviation, and the overall angle change rate of bottom hole can reach 45.00°/30 m, which can meet the requirements of short-radius sidetracking with curvature radius of 40 m. The successful application of open-hole sidetracking technology with 3.5° single-bend PDM is of great significance to the production increase and efficiency improvement of old oilfields.


Introduction
With the continuous development of old oilfields, the well condition of oil and water wells in old oilfields is deteriorating, which seriously restricts the improvement of oil recovery. In order to enhance oil recovery, infill wells and short-radius sidetracking horizontal wells can be used (Zhang et al. 2017). Comparing these two methods, the former has the disadvantages of long well construction period and high cost, while the latter can effectively communicate with new reservoirs through wellbore extension, which can not only save drilling cost, but also improve reservoir connectivity efficiency, enhancing oil recovery and economic benefits (Zhang et al. 2019). Sidetracking short-radius horizontal wells is an effective means to tap the latent power and increase reserves and production in old areas (Liu et al. 2013).
The sidetracking slim hole short-radius horizontal well technology is the important drilling technology developed rapidly in 1980s (Li 2020). The commonly used drilling tools for short-radius horizontal wells can be divided into two types, namely surface-driven flexible drilling system and downholedriven articulated elbow chain motor system. In the mid-1970s, the Eastman Whipstock company of the USA developed a short-radius deflecting drilling system through eight years of research and test, which is widely used in the Knparnk oilfield of Canada. Since 1983, Eastman Christenden company has made further improvement and has sidetracked more than 300 short-radius horizontal wells in recent years. The articulated positive displacement mud motor (PDM) developed by the Petroleum Exploration and Development Research Institute of Sinopec has the deflection rate of 90°-150°/30 m and the curvature radius of not greater than 18 m, which can change the wellbore from vertical to horizontal within 28 m wellbore length (Chen et al. 2010). Shengli Oilfield has carried out the scientific research of the sidetracking for short-radius horizontal well, the core of which is designing the articulated PDM, and it can realize the average deflection rate of 33°/30 m (Jiang 2004). The Drilling Institute of Daqing Oilfield adopts the steering elbow to control the direction, and the rotary table drives the articulated flexible drill pipe to drill. The buildup rate can reach 69.9°/30 m, and the curvature radius is not greater than 25 m (Yang et al. 2014). The depth of sidetracking point of short-radius horizontal wells in Tahe oilfield is generally about 5500 m (Yu 2009), its vertical distance from the target formation is less than 70 m, and the average curvature radius can reach 40 m or even smaller. This kind of short-radius horizontal well is deep and has large curvature, which makes it difficult to control the well trajectory, select the deflecting tools and predict the deflecting ability (Chen et al. 2007;he et al. 2003;Lu et al. 2017). On the other hand, after long-term production, the old well has high hole enlargement rate (Ma et al. 2007). All these will increase the difficulty of sidetracking. Taking a sidetracking horizontal well in Tahe oilfield as an example, the original sidetracking point is 5720 m, and there are undeveloped reservoirs under this sidetracking point at a vertical depth of 40 m in different azimuths. The downhole temperature reaches about 120° and the hole size of the original open-hole section is 165.1 mm. After 7 years of production, the measured hole diameter is 203.8 mm, and the hole enlargement rate reaches 23%.
According to the specific situation of Tahe oilfield, deepwell short-radius sidetracking technology is proposed to reduce cost and increase production in Tahe oilfield. To achieve the sidetracking success of the deep short-radius open hole, the improved 3.5° positive displacement mud motor (PDM) with cushion block is proposed. Then based on the mechanical analysis and finite element numerical simulation, its key parameters are calculated and optimized. It can realize high build-up rate and one-time sidetracking, which will provide an effective technical scheme for subsequent production and efficiency of old well.

Sidetracking project
For sidetracking technology, there are two widely used construction methods: casing sidetracking and open-hole sidetracking (Zhang et al. 2015;Ma et al. 2020). During the implementation of open-hole sidetracking, different sidetracking methods can be adopted for different downhole specific working conditions, including open-hole inclinator sidetracking, conventional open-hole sidetracking, etc. (Sun 2016). In Tahe oilfield, the conventional open-hole sidetracking construction method with cement backfill is adopted in a sidetracking well. This method not only avoids the disadvantage of difficult control of suspended sidetracking, but also has the advantages of simpler construction, less tool adjustment, and more cost savings than the downward incliner sidetracking.
The selection of sidetracking points is a control work that affects the success of sidetracking. Its determination should not only consider the geological conditions and avoid the poor geological conditions, but also combine the directional drilling and completion construction after sidetracking to reduce the difficulty of subsequent construction. Furthermore, it is necessary to choose the right orientation for sidetracking in old wells, so as to reduce the workload of twist orientation in the whole drilling process and improve the drilling efficiency.
Before tripping in sidetracking drilling assembly, the pressure bearing test of cement plug should be carried out to ensure the quality of cement plug and meet the requirements of sidetracking and old well packer.

Well structure design
The sidetracking point is 5730 m, the top boundary of Yijianfang group is 5724.5 m, the upper casing is 193.7 mm, the casing is connected back to the wellhead, the casing diameter is 165.1 mm, and the casing shoe is 5704.78 m. The original well deviation point is 5720 m, the designed sidetracking point is 5730 m, the well deviation is 8.17°, and the designed build-up rate is 38.53°/30 m. The designed well depth is 5876.47 m (inclined)/5774 m (vertical). The well structure is shown in Fig. 1.

Technical difficulties in initial construction stage of Sidetracking
There are many construction technical difficulties in the initial stage of sidetracking, as follows: (1) After two acid fracturing, one acidizing and three rounds of water injection in 5704-5770 m well section, the original formation structure of the well section is damaged to a certain extent, and there are risks such as low build-up rate and unstable wellbore in the initial stage of sidetracking. Optimizing PDM and deflecting tool is necessary to strengthen technical support.
(2) The sidetracking point of 5730 m has a certain deviation of 8.17°. It is an important prerequisite to ensure the success of sidetracking to observe the changes of various drilling parameters during plug sweeping.
According to the actual working conditions of Tahe oilfield and the technical difficulties, combined with the existing sidetracking technical scheme and further research, the optimal sidetracking scheme and supporting technology suitable for the specific situation of Tahe oilfield are proposed.

Design basis for structural parameters of single-bend PDM
Although there are many factors that affect the build-up rate of drilling tools, the structural parameters of curved joint PDM are dominant (Zhang et al. 2007). Given the expected build-up rate of the drilling tool, any structural parameter can be obtained according to the build-up rate relation. In this paper, the calculation methods of the structural bending angle ( ), the length of the first span ( L 1 ) and the clearance between the lower stabilizer and the wellbore ( ) are given.
The calculation formula of structural bending angle is: The calculation formula of the first span length is: in which The calculation formula of clearance is as follows: In particular, when = 0 , there is where k is the geometric build-up rate of the drilling tool, °/30 m; is the structural bending angle, °; L 1 is the distance from the lower stabilizer to the bit, m; L 2 is the distance from the bending elbow to the lower stabilizer, m; L 3 is the distance from the bending elbow to the upper stabilizer, m; L T is the total length of the drilling tool from the upper stabilizer to the bit, m; L s is the length of drilling tool between stabilizers, m; is the influence factor of bending angle position, dimensionless; is the clearance between the lower stabilizer and the wellbore.

Design results of improved single-bend PDM
According to the design and construction requirements of a sidetracking well in Tahe oilfield, the structure of singlebend PDM is improved as shown in Fig. 2 to meet the high build-up rate of construction requirements. According to the structural analysis of single-bend PDM, the build-up rate of single-bend PDM is mainly determined by many factors, such as the size and position of structural bending angle, the size and position of stabilizer (cushion block) and weight-on-bit (WOB) (Sun et al. 2004). The control variable method (Zhang et al. 2016) is used to analyse the influence trend of various factors on the build-up rate of single-bend PDM. The same increasing variable factors include structural (5) L 1 = 60 k − L s bending angle and outer diameter of lower stabilizer (cushion block); the different increasing variable factors include the distance from lower stabilizer (cushion block) to bending angle. Especially, increasing WOB can increase build-up rate without considering bottom layer factor, otherwise, it will decrease. Based on the above principle, the single-bend PDM is improved, and the improvement results of each structural parameter are shown in Table 1. According to the modified three-point circle determination method and limit curvature method (Liu et al. 2004;Su, 1997;Li et al.;Tang et al. 2000), the prediction results have high accuracy. Combined with Wang et al. (1994), the incremental expression of build-up rate caused by eccentric cushion block was derived by using energy method, which can predict the limit build-up rate of 43.89-48.45 (°/30 m) of the improved 3.5° single-bend PDM  with considering formation factors, and the build-up rate can reach 52.67-53.29 (°/30 m) without considering formation factors. The actual build-up rate is limit build-up rate, which is less than simulated/designed value. The key lies in the influence of formation factors. Formation factors are very complex and changeable and have obvious regional characteristics. In order to simplify the calculation of numerical simulation, some assumptions are made.

Optimization of WOB
From the above factors, it can be seen that in the actual construction process, the specific influence of WOB on the 3.5° single-bend PDM build-up needs to be further studied. WOB provides the power for drilling, and appropriate WOB provides drilling speed. If the WOB is too small, the PDM cannot break the rock and the drill string cannot drill in the process of sidetracking; If the WOB is too large, it will cause the buckling of the drill string, and the friction resistance caused by the excessive buckling will increase, so that the WOB cannot be transmitted to bottom hole and the phenomenon of drag will appear. Therefore, the selection of appropriate WOB is the premise to ensure the success of sidetracking. Making WOB as the representative of drilling parameters, the appropriate value of the improved PDM is optimized by the numerical simulation method.

Model simplification
In the actual sidetracking operation, there is a small azimuth change in the well trajectory, which inevitably leads to a certain spiral bending in the geometric structure of the well, distortion of the geometric standard circle in the radial section, and non-ring shape in the longitudinal section of the well. In order to simplify the calculation process, the wellbore trajectory is simplified as a thin-walled hollow cylinder structure in the two-dimensional plane. In the process of drilling tool lowering, the tool surface and the center line of wellbore trajectory are in the same plane. The internal structure and material composition of each part of high-angle PDM are complex. If the internal structure and material properties of each part are considered, the calculation is large and the process is complex, which has little effect on the WOB of the whole drilling tool. Further, according to the outline size, each part of the drilling assembly is simplified into a three-dimensional thin-walled hollow string structure.
In addition, there are some assumptions as follows: (a) The wellbore is regular, and the hole enlargement and wellbore roughness are neglected.
(b) The influence of cuttings blockage and bit sticking is not considered in the calculation, (c) The influence of the fluid pressure fluctuation in the wellbore on the drill string is ignored.

Model establishment
In opening the window of sidetracking stage, the drilling assembly is as follows: 120.65mmPDC + 95mm3.5° single-bend PDM + single flow valve + 88.9 mm nonmagnetic pressure bearing drill pipe + 105mmm MWD sub + 88.9 mm non-standard drill pipe + 88.9 mm weighted drill pipe + 88.9 mm drill pipe + 114.3 mm drill pipe. According to the actual size and type of drill string, the three-dimensional model of the lower 20 m drill string is established as shown in Fig. 3.
With the help of ANSYS software, the numerical simulation analysis can be carried out. Firstly, according to the design trajectory parameters of a sidetracking well in Tahe oilfield as shown in Table 2, the wellbore model is established as shown in Fig. 4.
In the numerical model, the dynamic and implicit analysis step is used to simulate the tripping in of nonlinear drill string. The analysis of the contact force applies face-to-face contact, and its friction coefficient is 0.05 according to the performance of the drilling fluid in Tahe oilfield. The top surface of the drill pipe is loaded by point-surface coupling. The tripping in speed is 2 m/s. The outer side and bottom of the wellbore are set as full constraints.
Then the 3D stress element, namely C3D8R, is used to calculate the model. The wellbore and drill pipe parts in the model are divided by hex hexahedron mesh, and the cushion block part of single-bend screw is divided by TET tetrahedron mesh. The minimum unit size of wellbore is 10 mm and the maximum is 20 mm, and the minimum unit size of drill pipe is 10 mm and the maximum is 18 mm. The meshing results of wellbore and drilling string are shown in Fig. 5.

Result analysis
When 3.5° single-bend PDM is tripped in the wellbore, the variation of its lateral displacement and corresponding stress nephogram with the tripping in depth is shown in Figs. 6 Fig. 3 Establishment of drill string model and 7. According to Figs. 6 and 7, with the increase in tripping in depth, the whole drilling assembly is bending continuously, and the bending stress also increases. With the increase in WOB and tripping in depth, there is the risk of backing pressure.
When the WOB is 20kN, driving the whole drilling tool into the hole, the stress distribution of drilling assembly is shown in Fig. 8, and the stress distribution of 3.5° singlebend PDM is also enlarged. It can be seen from the calculation results that under the WOB of 20kN, the support reaction force between drill string and wellbore is small. From the initial stage of drill string running to the stable deviation section, the stress fluctuation of string is relatively stable and the value is small. Part of the force mainly exists in the contact part between the bending part and the wellbore, which is also the main force that causes the bending and inclination of the drill string due to the support reaction force. The tripping in speed of the drill string is always stable and slow, but it will not lead to the sticking, that is, the drill string can trip in smoothly.
When the WOB is increased to 30kN, the stress distribution of drilling assembly is shown in Fig. 9. At this time, the support reaction force between the drill string and the wellbore increases, the initial tripping in speed is faster, and the stress on the drill string increases faster. The later tripping in speed of the drill string decreases significantly, and the drill string can still trip in, but it is less than the tripping in speed under the low WOB condition of 20kN. It is now difficult for the drill string to trip in, and the stress on the drill string rises sharply, reaching the initial value of instability. The stress nephogram of drilling assembly shows that the drill string has undergone large plastic deformation and occurs buckling, the transmission of WOB is limited, and the drill string is in a disadvantageous state at the beginning of tripping in.
Based on the above numerical simulation results, the drill string can be tripping in under the WOB of 20kN-30kN. If the WOB is higher than 30kN, the risk of buckling will occur when the drill string is tripping in, which may cause problems such as difficulty in passing the drill string.

Sidetracking field application
This sidetracking well is formally drilled from 5728 m, the sidetracking tool face is 20-30° smaller than the target azimuth (the original wellbore azimuth is 214° and the target azimuth is 176°), the time control is 3-4 h/m, and the sliding drilling is carried out. In order to protect the sandwich wall, the times of drilling tool movement are reduced as far as possible and the drilling is stable. During the drilling, the geological sand bailing monitoring is carried out. Each 0.5 m sand bailing is evaluated once. The cuttings content increases rapidly, reaching 10% in the first 1 m, 60% in the third 3 m, and more than 80% in the 5732 m, which is judged that the sidetracking is successful. After successful sidetracking, if the drilling tools are not pulled out of the hole and continue drilling with the deviation increasing, the phenomenon of backing pressure may occur in the drilling process. By controlling the WOB at 20-30kN, the possible deformation and bending of the bottom hole can be reduced, and the hole can be drilled to 5745 m smoothly. The bottom hole deviation and the change rate of full angle can be predicted, and the PDM can be replaced after pulling out. The calculation formula of overall angle change rate and curvature radius during drilling measurement  is as follows: where K ab is the overall angle change rate of well deviation between measuring points A and B, °/30 m; ΔL ab is the length of well section between measuring points A and B, m; a , b are the well deviation angle at measuring point A and B, °; Φ a , Φ b is the azimuth difference between measuring points A and B, °; R is the curvature radius between measuring points A and B, m.
In the process of this drilling survey, according to the well depth of 5728 m at the initial position, and increasing drilling deviation to the well depth of 5745 m, the well depth, well deviation, azimuth, overall angle change rate and bottom hole overall angle change rate measured at two points are shown in Table 3.
According to the calculation results, in the case of sidetracking deep short-radius open hole with high build-up rate, short directional section and low sidetracking success rate, 3.5° single-bend PDM with cushion block can be selected. The sidetracking overall angle change rate can (6)  short-radius sidetracking horizontal well in deep wells. However, in practice, the tool face should be adjusted to be consistent with the wellbore orientation in the full round trip operations. At the same time, further research is needed on the adaptability of this drill tool in the case of tripping in resistance, hole enlargement and actual geological conditions.

Ethical statements
The manuscript is submitted to only this one journal for consideration. The submitted work is original and has not been published elsewhere in any form or language (partially or in full). This study will not be split up into several parts to increase the quantity of submissions and submitted to various journals or to one journal over time.
Results are presented clearly, honestly, and without fabrication, falsification or inappropriate data manipulation (including image-based manipulation). Authors adhere to the discipline-specific rules for acquiring, selecting and processing data. The author has not plagiarized other people's data, text or theory and has given appropriate recognition to other reference works and has made a clear citation mark in the text. Fig. 9 Stress nephogram of drill string buckling in tripping in process when WOB is 30kN including the partial enlarged view of 3.5° singlebend PDM

Conflict of interest
On behalf of all the co-authors, the corresponding author states that there is no conflict of interest.
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