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Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells
Journal of Natural Gas Science and Engineering xxx (2015) 1e9
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells Zhanghua Lian*, Qiang Zhang, Tiejun Lin, Fuhui Wang State Key Lab of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 July 2015 Received in revised form 22 September 2015 Accepted 5 October 2015 Available online xxx
The combination of gas drilling and horizontal well has been considered as an effective technique for the exploitation of low permeability reservoirs to protect reservoir, enlarge drainage area and increase production. Given the currently inadequate understanding about drill string dynamic characteristics in gas drilling of horizontal wells, a theoretical model of drill string dynamics is established in this paper. The nonlinear dynamics equations are derived to study the motion state of drill string. Meanwhile, an experimental apparatus is developed according to similarity principle, and the kinetic characteristic of drill string is investigated based on the simulation experiment. Particular attention is focused on the lateral vibration which results from the impact and frictional interaction with wellbore constraint. The effect of weight on bit and rotary speed on drill string motion pattern is also discussed based on experimental results. Finally, the buckling and contact of drill string are analyzed through finite element simulation study. The results indicates that the contact force between wellbore and drill string is relatively large and helical buckling of drill string can be caused without the lubrication and damping effects of drilling fluid in gas drilling. The work presented in this paper can provide theoretical foundation and technological basis for drill string dynamics analysis and drilling parameter optimization in horizontal wells drilled with gas. © 2015 Elsevier B.V. All rights reserved.
Keywords: Drill string dynamics Horizontal gas drilling Similarity theory Lateral vibration Buckling and contact
1. Introduction Gas drilling is an underbalanced drilling technology using air, nitrogen or natural gas as circulating medium. As a widely-applied technique, gas drilling has shown its superiority in drilling rate improvement, reservoir discovery and formation protection (Lian et al., 2012). The combination of gas drilling and horizontal well technique can effectively protect reservoir, enlarge drainage area and increase production of oil and gas wells (Sun et al., 2008; Han and Yan, 2009). Therefore, drilling horizontal well with gas has recently been considered as an innovative technique for the exploitation of low permeability reservoirs. However, when gas is used as circulating medium, some problems related to friction and vibration of horizontal drill string happen frequently. These problems seriously restrict the effect of trajectory control and drilling speed improvement. Researches on drill string dynamics have been conducted from various perspectives. Lubinski (1950) presented systematic
* Corresponding author. E-mail address: [email protected] (Z. Lian).
analyses on the stress and deformation of drill string and the critical conditions of drill string buckling were also investigated based on the theory of elastic stability. His work laid the foundation for the study of drill string mechanics. Dykstra (1996) established the dynamical equations of drill string based on Hamilton principle and his research provide a reference for further study on dynamic characteristics in both vertical and curved borehole. Menand et al. (2008) presented how the drill string rotation affected the critical buckling load. Through the comparison of an advanced model for drill string mechanics with an experimental setup enabling to reproduce the buckling in wellbore, they found that the critical helical buckling load of a rotating pipe is 50% of the one obtained from non-rotating pipe. Ertas et al. (2013) developed a general drill string mechanics model to analyze the axial and torsional vibrations and provide vibration indices. Huang et al. (2015) proposed an automatic generalized quasi-static model of drill string system and provided an inversion model to revise the model parameters. To analyze the drill string dynamics in horizontal wells, different models were established and various experiments were carried out. Omojuwa et al. (2012) established the theoretical model for friction, torque, buckling and vibration of drill string in horizontal
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Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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wells; he worked out the distribution of axial force and torque along drill string using the dynamic equations. Wilson and Heisig (2015) presented a detailed analysis of the fully coupled, threedimensional, nonlinear behavior of drill string under induced vibrations and specific focus was given to the dynamic characteristics of drill string in horizontal wells. However, in the previous models and experiments, drill string dynamics in gas drilling of horizontal wells hasn't been fully studied. In this paper, a nonlinear dynamics model was established to investigate the motion behavior of drill string. Meanwhile, according to similarity theory, an experiment was designed to study the kinetic characteristic of drill string in gas drilling of horizontal wells. A finite element model was also established and the buckling and contact of drill string was analyzed. 2. Theoretical model of drill string dynamics The dynamics of drill string belongs to the category of rotor dynamics which mainly discusses the vibration, equilibrium and stability of rotary systems (Zhang, 1900). The drill string is generally simplified as a segmented, homogeneous and uniform-section beams bearing variable loads. 2.1. Assumption The assumptions in the derivation are shown as follows: 1. Drill string is regarded as elastic beam with homogeneous geometric characteristics and material properties. 2. The stiffness of threaded connections and local notches is neglected. 3. The effect of temperature on material properties is neglected.
where ui, vi and wi are the displacement of node i in x-, y- and zdirection respectively; qxi, qyi and qzi are the rotation angles of node i in x-, y- and z-direction respectively; variables with subscript “j” represent the DOF of node j. The nodal force vector of drill string element is expressed as
T fRe g ¼ Pi ; Qyi ; Qzi ; Mli ; Myi ; Mzi ; Pj ; Qyj ; Qzj ; Mlj ; Myj ; Mzj
(2)
where Pi is the axial force of node i; Qyi and Qzi are the shear forces of node i in y- and z-direction respectively; Mli is the torque of node i; Myi and Mzi are the bending moments of node i in xey plane and xez plane respectively; variables with subscript “j” represent the loads of node j. The generalized displacement vector of drill string element is depicted by
fue g ¼ ½Nfde g
(3)
where [N] is the shape function matrix. For three-dimensional beam element, it can be expressed as
3 Nu 6 Nv 7 7 ½N ¼ 6 4 Nw 5 Nq 3 2 N1 0 0 0 0 0 N2 0 0 0 0 0 6 0 N3 0 0 0 N4 0 N5 0 0 0 N6 7 7 ¼6 4 0 0 N3 0 N4 0 0 0 N5 0 N6 0 5 0 0 0 0 N2 0 0 0 0 0 N1 0 2
(4) where N1 ¼ 1 x/l, N2 ¼ x/l, N3 ¼ 1 3x2/l2 þ 2x3/l3, N4 ¼ z 2x2/ l þ x3/l2, N5 ¼ 3x2/l2 2x3/l3, N6 ¼ x2/l þ x3/l2
2.2. Element displacement vector 2.3. Equation of motion A simplified drill string element is shown in Fig. 1. The relevant parameters of the drill string element are divided into two categories, namely nodal DOF (degree of freedom) data and load parameters. In the finite element analysis, nodal displacements are generally taken as the basic unknown variables. The nodal displacement vector of drill string element is expressed as
T fde g ¼ ui ; vi ; wi ; qxi ; qyi ; qzi ; uj ; vj ; wj ; qxj ; qyj ; qzj
(1)
Lagrange equation must be satisfied for the discretized drill string system (Dykstra, 1996):
d vðT UÞ vðT UÞ ¼ fRe g dt v d_ e vfde g
where T and U are the element kinetic energy and potential energy, respectively.
T¼ yj (Myj)
1 2
Z
rfu_ e gT fu_ e gdV
(6)
Ve
vj (Qyj)
U¼
x y
1 2
Z
fεgT fsgdV
Ve yi (Myi)
Z
fue gT fPA gdA
Ae T
j
vi (Qyi)
fue g fPe g
wj (Qzj ) zj
i wi (Qzi) zi (Mzi)
z Fig. 1. Drill string element.
(5)
(Mzj )
Z
fue gT fPV gdV
Ve
(7)
where {PA}, {PV} and {Pe} are body force, surface force and nodal force vectors, respectively. Then substituting Eqs. (6) and (7) into Eq. (5), the motion equation of drill string element at local coordinate system is expressed as (Zhu et al., 2012)
½Me € de þ ½Ce d_ e þ ½Ke fde g ¼ fRe g
(8)
where [M]e, [C]e and [K]e are the mass, damping and stiffness matrixes, respectively.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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3
The relationship of nodal displacements at local coordinate system and global coordinate system is denoted by
0 de ¼ ½Tfde g
(9)
where [T] is the matrix for coordinate transform, fd0e g is the nodal displacement matrix at global coordinate system. After coordinate transformation, the motion equation of drill string system is obtained by modeling the drill string as an assemblage of much shorter beam elements (Fu and Shi, 1996)
0 0 ½M 0 € d þ ½C 0 d_ þ ½K 0 fd0 g ¼ fR0 g
(10)
0 0 where {d0 }, fd_ g and f€ d g are the generalized nodal displacement, velocity and acceleration matrixes; [M0 ], [C0 ] and [K0 ] are the mass, damping and stiffness matrixes of drill string system at global coordinate; {R0 } is the generalized external force vector. These equations are the basic drill string dynamics equations for both mud drilling and gas drilling. However, when drilling with gas, the damping effect of drilling fluid on drill string motion state can be neglected as the viscosity and density of gas are relatively low. Moreover, the above derivation does not consider the constraint of wellbore, which can make global stiffness matrix become singular (Liu, 2008). To address this problem, gap-element stiffness matrix [KG] is applied in this paper and the new dynamics equation can be expressed by Eq. (11).
0 ½M 0 € d þ ð½K þ ½KG Þfd0 g ¼ fR0 g
(11)
The established nonlinear model is the basis for finite element analysis of drill string dynamics. However, these equations are difficult to solve when drill string length reaches up to thousands of meters. Therefore, experimental study and numerical simulation are two effective methods for the study of drill string dynamics. 3. Experimental study of horizontal drill string dynamics 3.1. Similarity principle According to similarity principle, an experiment apparatus was designed to simulate the process of horizontal drilling with gas. As shown in Fig. 2, the horizontal drill string is simplified as revolving shaft with even-distributed mass. The dynamics differential equations of horizontal drill string are expressed as (Shi et al., 2006)
! 8 > v4 x v2 x v4 x v3 x v2 x > 2 > þ 2w 2 þm 2 ¼0 EI 4 þ P 2 mr > > 2 2 < vz vz vz vt vz vt vt ! > > v4 y v2 y v4 y v3 y v2 y > > > EI 4 þ P 2 mr2 2w 2 þm 2 ¼0 : 2 2 vz vz vz vt vz vt vt
(12)
where E is the elasticity modulus of drill string; I is the moment of inertia; mr2 is the mass moment of inertia; r is the turning radius; P is the average axial force; w is the self-rotating angular velocity. Based on Eq. (12), the relevant variables in the experiment are expressed with subscript “m” and the similarity ratios of these variables are denoted by Eq. (13).
8 mm Pm Em > > ; CP ¼ ; CE ¼ Cm ¼ > > m P E > > > < Im lm tm CI ¼ ; Cl ¼ ; Ct ¼ > I l t > > > > w r r > > : Cw ¼ m ; Cr ¼ m ; Cr ¼ m w r r
(13)
Fig. 2. Sketch of horizontal drill string.
To assure the kinematic similarity, the corresponding coefficients of Eq. (12) in the model and experiment must be equal. Then similarity criterion is expressed as:
cE cI cE cI ¼ ¼ ct cw ¼ 1 cP c2l cm c4l cw
(14)
In the experiment apparatus, these parameters (drill string length, drill string diameter and wellbore diameter, etc.) with dimension of length had almost the same similarity ratio, namely cl ¼ 1/8, as shown in Table 1. The drill string material used in the experiment was the same as that in field application and cE ¼ cr ¼ 1. Therefore, the experimental parameters can be determined based on similarity criterion. WOB (weight on bit) and rotary speed are important parameters affecting the motion state of drill string and their similarity ratios are cP ¼ c2l ¼ 1/64 and cw ¼ 1/cl ¼ 8. 3.2. Experimental apparatus and procedure The experimental apparatus comprises six main subsystems: cement base, drill string system, wellbore system, drill bit, measurement system and power system, as shown in Fig. 3. Steel tube was used as drill string in the experiment and M8 tool joint was welded on the drill string. In order to observe the motion and deformation of drill string in wellbore, organic glass tube served as wellbore, as shown in Fig. 4. In the experiment, drill string can move and rotate in the wellbore while the wellbore system was fixed. Cone bit was used in the experiment and its material was highspeed tool steel whose Rockwell hardness can reach up to HRD58-
Table 1 Similarity ratio of parameters with dimension of length. Parameters
Actual value
Experimental value
Similarity ratio
Wellbore diameter Wellbore length Annulus clearance Drill string length Drill string diameter Drill string thickness
152.40 mm 200.00 m 31.75 mm 200.00 m 88.90 mm 11.40 mm
20.00 mm 25.00 m 4.00 mm 25.00 m 12.00 mm 1.50 mm
7.62 8.00 7.94 8.00 7.41 7.60
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 3. Experiment apparatus for drill string dynamics.
62 after quenching treatment. Drill string was connected with drill bit using M8 thread, as shown in Fig. 5. Power system was mainly composed of electromotor (1.5 kW, 940 r/min), Ф50 1000 mm air cylinder (0e1 MPa), speed control box and pressure regulating valve. In the experiment, speed control box can realize frequency control and change the rotary speed of drill string. Pressure regulating valve can adjust the movement speed of air cylinder to load different WOB. Before the experiment, the experimental apparatus was well prepared, and the measurement system was examined. The drill string was first pushed by air cylinder to simulate the tripping process in horizontal section. The axial force distribution was measured by axial force testers. Then, electromotor was started and the drill string rotated to simulate horizontal drilling with gas. This process was recorded with special attention paid to the rotation of drill bit and bending of drill string. After drill string motion was stable, various parameters were measured by different testers. After that, rotary speed and WOB were changed and these parameters were measured again. The experiments were conducted at rotary speed from 50 rpm to 500 rpm and WOB from 0.2 MPa to 0.6 MPa.
4. Numerical simulation The contact and buckling of drill string were difficult to observe or measure in the experiment. To clarify the motion state of drill string while horizontal drilling with gas, especially the contact and buckling of drill string, FEA (finite element analysis) was conducted. The assumptions in numerical simulation are shown as follows: 1. 2. 3. 4.
Reaming and undergauge borehole are neglected. Drill pipes share the same inner and outer diameters. Drill bit, formation rock and borehole wall are rigid bodies. The heat induced by the friction between drill bit and formation rock is neglected and effect of temperature on material properties is neglected. 5. The effect of gas and cuttings on drill string motion is not considered. Well A2 was selected to carry out the FEA of horizontal gas drilling. Well A2 is a horizontal well in West China, with measured depth (MD) of 1968 m and horizontal length of 240 m, as shown in Fig. 6. The horizontal section of this well was drilled with gas.
Fig. 4. Drill string and wellbore in the experiment.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 5. Drill bit system.
5. Results and discussion 5.1. The effect of rotary speed on drill string motion
Fig. 6. Well structure of A2.
To compare the FEA results with experimental data, drill string length in the FE model was 200 m. The drill string dimension was 88.9 mm 9.35 mm with steel grade of S135. Fig. 7 is the established drill string dynamics model of horizontal gas drilling. Rigid element and solid element were used for wellbore and drill string respectively. In the simulation process, WOB was 50 kN and rotary speed was 45 rpm.
Fig. 8(a) and (c) show the change of WOB with time under the condition of different rotary speeds. We can see that the change of WOB is a typical sine curve. The experimental data was then processed with FFT (Fast Fourier Transform) to obtain the frequency of WOB, as shown in Fig. 8(b) and (d). The comparison between the frequency of WOB and drill sting is given in Table 2. Fig. 9 illustrates the relationship between the frequency of WOB and rotary speed. Non-dimensional frequency is defined as the ratio of WOB frequency to the rotation frequency of drill sting. As observed, the frequency of WOB increases linearly with rotary speed and overlaps with rotation frequency of drill sting. As shown in Fig. 9(b), non-dimensional frequency is quite close to 1.0 at different rotary speed, i.e., the frequency of WOB is approximately equal to the rotation frequency of drill sting. Fig. 10 presents the results of lateral vibration acceleration. The accelerations of X-direction (left-right direction of the wellbore) and Y-direction (up-down direction of the wellbore) are approximately equal, though there exists a phase difference. In the horizontal drilling process, contact and impact between drill string and wellbore are inevitable, which can lead to high lateral vibration acceleration. As shown in Fig. 10(a)e(d), the lateral vibration acceleration of drill string is between 1.0 g and 1.0 g (here “g” represents for local acceleration of gravity). But when rotary speed reaches up to 450 rpm and 500 rpm, the maximum amplitude is 2.0 g, as shown in Fig. 10(e) and (f). It is concluded that severe impact has happened between drill string and wellbore at that condition.
Fig. 7. Solid model and mesh model for drill string dynamics.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 8. The change of WOB and its fluctuation frequency.
Table 2 Frequency of WOB and drill string at different rotational speed. Rotary speed (rpm)
Self-rotating frequency of drill sting (Hz)
Frequency of WOB (Hz)
Non-dimensional frequency
50 100 150 200 250 300 350 400 450 500
0.78 1.60 2.60 3.20 4.20 4.90 6.10 6.50 7.40 8.50
0.83 1.67 2.50 3.33 4.17 5.00 5.83 6.67 7.50 8.33
0.94 0.96 1.04 0.96 1.01 0.98 1.03 0.98 0.99 1.02
According to above analyses, lateral vibration acceleration is relatively low when rotary speed is 300 rpme400 rpm. However, when rotary speed increases to 450 rpme500 rpm, significant increase of lateral vibration acceleration is observed, which can result
in resonance, fatigue and other serious downhole incidents. So rotary speed of 450 rpme500 rpm should be avoided in the experiment. Consequently, for the given well structure and drill string assembly, the proposed rotary speed is 300 rpme400 rpm.
Fig. 9. The relationship between WOB frequency and rotary speed.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 10. The lateral vibration acceleration of drill string at different rotary speed.
Fig. 11. Lateral displacement of drill string at different WOB and rotary speed.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 12. Von Mises stress contour of horizontal drill string at different time.
According to similarity principle, the proposed rotary speed range is 37.5 rpme50 rpm for the actual operation of horizontal gas drilling. 5.2. The effect of WOB on drill string motion Because of the inherent characteristics of drill bit and various forms of drill string vibration, WOB is not constant in most cases. It is proved that appropriate WOB fluctuation can help rock breaking
but excessive fluctuation will bring about the impact failure of bit teeth and bearings. In consequence, appropriate WOB is essential for drilling operation. This paper mainly discusses the effect of WOB on drill string motion. Comparison of the two displacement curves in Fig. 11 indicates that Y displacement is a regular sine curve while X displacement is irregular. In addition, X displacement is always positive while Y displacement is negative. Combing with the motion trajectory of drill string, we can find that the swaying of drill string is at the right
Fig. 13. Contact state at different time.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 14. Contact force between horizontal drill string and wellbore.
lower part of wellbore. At rotary speed of 200 rpm, with the increase of WOB, the fluctuation amplitude of lateral displacement is 1.2 mm, 1.5 mm and 2.0 mm respectively. The amplitude of lateral displacement increases with the increase of WOB, indicating more violent swaying. When WOB is 0.4 MPa, with the increase of rotary speed, the fluctuation amplitude of lateral displacement is 1.2 mm, 1.5 mm and 2.2 mm respectively. The increase of rotary speed leads to larger frequency and amplitude of drill string swaying. Therefore, WOB and rotary speed are two important factors affecting drill string motion in horizontal wells.
equal to the rotation frequency of drill sting. According to similarity principle, the recommended rotary speed range is 37.5 rpme50 rpm in the horizontal gas drilling of given well structure. The swaying of drill string is at the right lower part of wellbore and the variation of WOB has little effect on swaying range. In addition, a finite element model is established to study the buckling and contact of horizontal drill string. Numerical simulation results demonstrate that the contact force between wellbore and drill string is relatively large and helical buckling may be caused in horizontal drilling using gas.
5.3. Buckling and contact of drill string
Acknowledgments
Based on the established finite element model, numerical simulation was carried out to study the contact and buckling of drill string. When drill string just starts to move (1 s) against gravity and friction, stress is concentrated at the pushing end. Near-bit drill string starts to move at 3 s, at that moment the stress of near-bit drill string is obviously lower than other sections. When the whole drill string obtain translational velocity and rotary velocity, stress is concentrated at the drill string away from drill bit. Bending deformation of drill string is mainly caused by friction, axial force and other factors. After 10 s of simulation, the rotary motion brings about large torque on drill string and buckling has taken place. As shown in Fig. 12, sinusoidal buckling happens at near-bit drill string while the drill string away from bit is controlled by helical buckling. When the velocity of drill string becomes stable, continuous contact is observed between drill string and wellbore. The combination of Figs. 13 and 14 shows that the magnitude and direction of contact force varies over time. At 20 s, the average contact force of horizontal drill string is 18 kN, larger contact force is also observed at some nodes, indicating that severe buckling and pipe sticking can be caused in the horizontal drilling operation.
The authors thank the National Natural Science Foundation of China (No. 51574198 and No. 51504207) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20135121110005), for their contributions to this paper.
6. Conclusions A theoretical model is established to study the nonlinear drill string dynamics of horizontal wells drilled with gas. An experiment apparatus is designed and developed based on the actual structure of drill string in horizontal gas drilling and the principle of similarity. The experimental results shows that the change of WOB is a typical sine curve and the frequency of WOB increases linearly with rotary speed. Moreover, the frequency of WOB is approximately
References Dykstra, M.W., 1996. Nonlinear Drill String Dynamics. University of Tulsa, Tulsa. Ertas, D., Bailey, J.R., Wang, L., Pastusek, P.E., 2013. Drillstring Mechanics Model for Surveillance Root Cause Analysis and Mitigation of Torsional and Axial Vibrations. SPE 163420. Fu, J., Shi, T., 1996. Dynamic Analysis of BHA with Bent Housing. SPE 37046. Huang, W., Gao, D., Wei, S., Li, X., 2015. A generalized quasi-static model of drill string system. J. Nat. Gas Sci. Eng. 23, 208e220. Han, C., Yan, T., 2009. An analysis of the drill string vibrating during gas drilling of horizontal wells. Nat. Gas. Ind. 29 (7), 45e47. Lian, Z., Lin, T., Meng, Y., 2012. The Basic Theory and Application of Gas Drilling. Petroleum Industry Press, Beijing. Liu, W., 2008. Research on Drillstem Mechanics Analysis and Extension Capacity of External of Extended Reach Well. Daqing Petroleum Institute, Daqing. Lubinski, A., 1950. A study of the buckling of rotary drilling strings. Drill. Prod. Pract. 178e214. Menandm, S., Sellami, H., Akowanou, J., 2008. How Drillstring Rotation Affects Critical Buckling Load. SPE 112571. Omojuwa, E., Osisanya, S., Ahmed, R., 2012. Computation of Axial Load and Torque Distribution in Tubulars Used for Extended-reach Horizontal Wells Using a Holistic Design Philosophy. SPE 161076. Sun, H., Qiao, K., Hu, C., Ji, F., Liang, M., 2008. Gas drilling practices in the horizontal well Guang'an 002-H8. Nat. Gas. Ind. 28 (4), 61e63. Shi, Y., Wan, Y., Wu, C., 2006. Setting up a simulate device on motion behavior of bottomdhole assembly according to similitude principles. J. Guangxi Univ. (Nat. Sci. Ed.) 31, 159e162. Wilson, J.K., Heisig, G., 2015. Investigating the Benefits of Induced Vibrations in Unconventional Horizontals via Nonlinear Drill String Dynamics Modeling. SPE 173049. Zhu, X., Jia, Y., Tong, H., 2012. Vibrating characteristics of drill string in Gas drilling and its control measures. Acta Pet. Sin. 33 (2), 293e297. Zhang, W., 1900. Theoretical Basis for Rotor Dynamics. Science Press, Beijing.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
Contents lists available at ScienceDirect
Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells Zhanghua Lian*, Qiang Zhang, Tiejun Lin, Fuhui Wang State Key Lab of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 July 2015 Received in revised form 22 September 2015 Accepted 5 October 2015 Available online xxx
The combination of gas drilling and horizontal well has been considered as an effective technique for the exploitation of low permeability reservoirs to protect reservoir, enlarge drainage area and increase production. Given the currently inadequate understanding about drill string dynamic characteristics in gas drilling of horizontal wells, a theoretical model of drill string dynamics is established in this paper. The nonlinear dynamics equations are derived to study the motion state of drill string. Meanwhile, an experimental apparatus is developed according to similarity principle, and the kinetic characteristic of drill string is investigated based on the simulation experiment. Particular attention is focused on the lateral vibration which results from the impact and frictional interaction with wellbore constraint. The effect of weight on bit and rotary speed on drill string motion pattern is also discussed based on experimental results. Finally, the buckling and contact of drill string are analyzed through finite element simulation study. The results indicates that the contact force between wellbore and drill string is relatively large and helical buckling of drill string can be caused without the lubrication and damping effects of drilling fluid in gas drilling. The work presented in this paper can provide theoretical foundation and technological basis for drill string dynamics analysis and drilling parameter optimization in horizontal wells drilled with gas. © 2015 Elsevier B.V. All rights reserved.
Keywords: Drill string dynamics Horizontal gas drilling Similarity theory Lateral vibration Buckling and contact
1. Introduction Gas drilling is an underbalanced drilling technology using air, nitrogen or natural gas as circulating medium. As a widely-applied technique, gas drilling has shown its superiority in drilling rate improvement, reservoir discovery and formation protection (Lian et al., 2012). The combination of gas drilling and horizontal well technique can effectively protect reservoir, enlarge drainage area and increase production of oil and gas wells (Sun et al., 2008; Han and Yan, 2009). Therefore, drilling horizontal well with gas has recently been considered as an innovative technique for the exploitation of low permeability reservoirs. However, when gas is used as circulating medium, some problems related to friction and vibration of horizontal drill string happen frequently. These problems seriously restrict the effect of trajectory control and drilling speed improvement. Researches on drill string dynamics have been conducted from various perspectives. Lubinski (1950) presented systematic
* Corresponding author. E-mail address: [email protected] (Z. Lian).
analyses on the stress and deformation of drill string and the critical conditions of drill string buckling were also investigated based on the theory of elastic stability. His work laid the foundation for the study of drill string mechanics. Dykstra (1996) established the dynamical equations of drill string based on Hamilton principle and his research provide a reference for further study on dynamic characteristics in both vertical and curved borehole. Menand et al. (2008) presented how the drill string rotation affected the critical buckling load. Through the comparison of an advanced model for drill string mechanics with an experimental setup enabling to reproduce the buckling in wellbore, they found that the critical helical buckling load of a rotating pipe is 50% of the one obtained from non-rotating pipe. Ertas et al. (2013) developed a general drill string mechanics model to analyze the axial and torsional vibrations and provide vibration indices. Huang et al. (2015) proposed an automatic generalized quasi-static model of drill string system and provided an inversion model to revise the model parameters. To analyze the drill string dynamics in horizontal wells, different models were established and various experiments were carried out. Omojuwa et al. (2012) established the theoretical model for friction, torque, buckling and vibration of drill string in horizontal
http://dx.doi.org/10.1016/j.jngse.2015.10.005 1875-5100/© 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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wells; he worked out the distribution of axial force and torque along drill string using the dynamic equations. Wilson and Heisig (2015) presented a detailed analysis of the fully coupled, threedimensional, nonlinear behavior of drill string under induced vibrations and specific focus was given to the dynamic characteristics of drill string in horizontal wells. However, in the previous models and experiments, drill string dynamics in gas drilling of horizontal wells hasn't been fully studied. In this paper, a nonlinear dynamics model was established to investigate the motion behavior of drill string. Meanwhile, according to similarity theory, an experiment was designed to study the kinetic characteristic of drill string in gas drilling of horizontal wells. A finite element model was also established and the buckling and contact of drill string was analyzed. 2. Theoretical model of drill string dynamics The dynamics of drill string belongs to the category of rotor dynamics which mainly discusses the vibration, equilibrium and stability of rotary systems (Zhang, 1900). The drill string is generally simplified as a segmented, homogeneous and uniform-section beams bearing variable loads. 2.1. Assumption The assumptions in the derivation are shown as follows: 1. Drill string is regarded as elastic beam with homogeneous geometric characteristics and material properties. 2. The stiffness of threaded connections and local notches is neglected. 3. The effect of temperature on material properties is neglected.
where ui, vi and wi are the displacement of node i in x-, y- and zdirection respectively; qxi, qyi and qzi are the rotation angles of node i in x-, y- and z-direction respectively; variables with subscript “j” represent the DOF of node j. The nodal force vector of drill string element is expressed as
T fRe g ¼ Pi ; Qyi ; Qzi ; Mli ; Myi ; Mzi ; Pj ; Qyj ; Qzj ; Mlj ; Myj ; Mzj
(2)
where Pi is the axial force of node i; Qyi and Qzi are the shear forces of node i in y- and z-direction respectively; Mli is the torque of node i; Myi and Mzi are the bending moments of node i in xey plane and xez plane respectively; variables with subscript “j” represent the loads of node j. The generalized displacement vector of drill string element is depicted by
fue g ¼ ½Nfde g
(3)
where [N] is the shape function matrix. For three-dimensional beam element, it can be expressed as
3 Nu 6 Nv 7 7 ½N ¼ 6 4 Nw 5 Nq 3 2 N1 0 0 0 0 0 N2 0 0 0 0 0 6 0 N3 0 0 0 N4 0 N5 0 0 0 N6 7 7 ¼6 4 0 0 N3 0 N4 0 0 0 N5 0 N6 0 5 0 0 0 0 N2 0 0 0 0 0 N1 0 2
(4) where N1 ¼ 1 x/l, N2 ¼ x/l, N3 ¼ 1 3x2/l2 þ 2x3/l3, N4 ¼ z 2x2/ l þ x3/l2, N5 ¼ 3x2/l2 2x3/l3, N6 ¼ x2/l þ x3/l2
2.2. Element displacement vector 2.3. Equation of motion A simplified drill string element is shown in Fig. 1. The relevant parameters of the drill string element are divided into two categories, namely nodal DOF (degree of freedom) data and load parameters. In the finite element analysis, nodal displacements are generally taken as the basic unknown variables. The nodal displacement vector of drill string element is expressed as
T fde g ¼ ui ; vi ; wi ; qxi ; qyi ; qzi ; uj ; vj ; wj ; qxj ; qyj ; qzj
(1)
Lagrange equation must be satisfied for the discretized drill string system (Dykstra, 1996):
d vðT UÞ vðT UÞ ¼ fRe g dt v d_ e vfde g
where T and U are the element kinetic energy and potential energy, respectively.
T¼ yj (Myj)
1 2
Z
rfu_ e gT fu_ e gdV
(6)
Ve
vj (Qyj)
U¼
x y
1 2
Z
fεgT fsgdV
Ve yi (Myi)
Z
fue gT fPA gdA
Ae T
j
vi (Qyi)
fue g fPe g
wj (Qzj ) zj
i wi (Qzi) zi (Mzi)
z Fig. 1. Drill string element.
(5)
(Mzj )
Z
fue gT fPV gdV
Ve
(7)
where {PA}, {PV} and {Pe} are body force, surface force and nodal force vectors, respectively. Then substituting Eqs. (6) and (7) into Eq. (5), the motion equation of drill string element at local coordinate system is expressed as (Zhu et al., 2012)
½Me € de þ ½Ce d_ e þ ½Ke fde g ¼ fRe g
(8)
where [M]e, [C]e and [K]e are the mass, damping and stiffness matrixes, respectively.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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The relationship of nodal displacements at local coordinate system and global coordinate system is denoted by
0 de ¼ ½Tfde g
(9)
where [T] is the matrix for coordinate transform, fd0e g is the nodal displacement matrix at global coordinate system. After coordinate transformation, the motion equation of drill string system is obtained by modeling the drill string as an assemblage of much shorter beam elements (Fu and Shi, 1996)
0 0 ½M 0 € d þ ½C 0 d_ þ ½K 0 fd0 g ¼ fR0 g
(10)
0 0 where {d0 }, fd_ g and f€ d g are the generalized nodal displacement, velocity and acceleration matrixes; [M0 ], [C0 ] and [K0 ] are the mass, damping and stiffness matrixes of drill string system at global coordinate; {R0 } is the generalized external force vector. These equations are the basic drill string dynamics equations for both mud drilling and gas drilling. However, when drilling with gas, the damping effect of drilling fluid on drill string motion state can be neglected as the viscosity and density of gas are relatively low. Moreover, the above derivation does not consider the constraint of wellbore, which can make global stiffness matrix become singular (Liu, 2008). To address this problem, gap-element stiffness matrix [KG] is applied in this paper and the new dynamics equation can be expressed by Eq. (11).
0 ½M 0 € d þ ð½K þ ½KG Þfd0 g ¼ fR0 g
(11)
The established nonlinear model is the basis for finite element analysis of drill string dynamics. However, these equations are difficult to solve when drill string length reaches up to thousands of meters. Therefore, experimental study and numerical simulation are two effective methods for the study of drill string dynamics. 3. Experimental study of horizontal drill string dynamics 3.1. Similarity principle According to similarity principle, an experiment apparatus was designed to simulate the process of horizontal drilling with gas. As shown in Fig. 2, the horizontal drill string is simplified as revolving shaft with even-distributed mass. The dynamics differential equations of horizontal drill string are expressed as (Shi et al., 2006)
! 8 > v4 x v2 x v4 x v3 x v2 x > 2 > þ 2w 2 þm 2 ¼0 EI 4 þ P 2 mr > > 2 2 < vz vz vz vt vz vt vt ! > > v4 y v2 y v4 y v3 y v2 y > > > EI 4 þ P 2 mr2 2w 2 þm 2 ¼0 : 2 2 vz vz vz vt vz vt vt
(12)
where E is the elasticity modulus of drill string; I is the moment of inertia; mr2 is the mass moment of inertia; r is the turning radius; P is the average axial force; w is the self-rotating angular velocity. Based on Eq. (12), the relevant variables in the experiment are expressed with subscript “m” and the similarity ratios of these variables are denoted by Eq. (13).
8 mm Pm Em > > ; CP ¼ ; CE ¼ Cm ¼ > > m P E > > > < Im lm tm CI ¼ ; Cl ¼ ; Ct ¼ > I l t > > > > w r r > > : Cw ¼ m ; Cr ¼ m ; Cr ¼ m w r r
(13)
Fig. 2. Sketch of horizontal drill string.
To assure the kinematic similarity, the corresponding coefficients of Eq. (12) in the model and experiment must be equal. Then similarity criterion is expressed as:
cE cI cE cI ¼ ¼ ct cw ¼ 1 cP c2l cm c4l cw
(14)
In the experiment apparatus, these parameters (drill string length, drill string diameter and wellbore diameter, etc.) with dimension of length had almost the same similarity ratio, namely cl ¼ 1/8, as shown in Table 1. The drill string material used in the experiment was the same as that in field application and cE ¼ cr ¼ 1. Therefore, the experimental parameters can be determined based on similarity criterion. WOB (weight on bit) and rotary speed are important parameters affecting the motion state of drill string and their similarity ratios are cP ¼ c2l ¼ 1/64 and cw ¼ 1/cl ¼ 8. 3.2. Experimental apparatus and procedure The experimental apparatus comprises six main subsystems: cement base, drill string system, wellbore system, drill bit, measurement system and power system, as shown in Fig. 3. Steel tube was used as drill string in the experiment and M8 tool joint was welded on the drill string. In order to observe the motion and deformation of drill string in wellbore, organic glass tube served as wellbore, as shown in Fig. 4. In the experiment, drill string can move and rotate in the wellbore while the wellbore system was fixed. Cone bit was used in the experiment and its material was highspeed tool steel whose Rockwell hardness can reach up to HRD58-
Table 1 Similarity ratio of parameters with dimension of length. Parameters
Actual value
Experimental value
Similarity ratio
Wellbore diameter Wellbore length Annulus clearance Drill string length Drill string diameter Drill string thickness
152.40 mm 200.00 m 31.75 mm 200.00 m 88.90 mm 11.40 mm
20.00 mm 25.00 m 4.00 mm 25.00 m 12.00 mm 1.50 mm
7.62 8.00 7.94 8.00 7.41 7.60
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 3. Experiment apparatus for drill string dynamics.
62 after quenching treatment. Drill string was connected with drill bit using M8 thread, as shown in Fig. 5. Power system was mainly composed of electromotor (1.5 kW, 940 r/min), Ф50 1000 mm air cylinder (0e1 MPa), speed control box and pressure regulating valve. In the experiment, speed control box can realize frequency control and change the rotary speed of drill string. Pressure regulating valve can adjust the movement speed of air cylinder to load different WOB. Before the experiment, the experimental apparatus was well prepared, and the measurement system was examined. The drill string was first pushed by air cylinder to simulate the tripping process in horizontal section. The axial force distribution was measured by axial force testers. Then, electromotor was started and the drill string rotated to simulate horizontal drilling with gas. This process was recorded with special attention paid to the rotation of drill bit and bending of drill string. After drill string motion was stable, various parameters were measured by different testers. After that, rotary speed and WOB were changed and these parameters were measured again. The experiments were conducted at rotary speed from 50 rpm to 500 rpm and WOB from 0.2 MPa to 0.6 MPa.
4. Numerical simulation The contact and buckling of drill string were difficult to observe or measure in the experiment. To clarify the motion state of drill string while horizontal drilling with gas, especially the contact and buckling of drill string, FEA (finite element analysis) was conducted. The assumptions in numerical simulation are shown as follows: 1. 2. 3. 4.
Reaming and undergauge borehole are neglected. Drill pipes share the same inner and outer diameters. Drill bit, formation rock and borehole wall are rigid bodies. The heat induced by the friction between drill bit and formation rock is neglected and effect of temperature on material properties is neglected. 5. The effect of gas and cuttings on drill string motion is not considered. Well A2 was selected to carry out the FEA of horizontal gas drilling. Well A2 is a horizontal well in West China, with measured depth (MD) of 1968 m and horizontal length of 240 m, as shown in Fig. 6. The horizontal section of this well was drilled with gas.
Fig. 4. Drill string and wellbore in the experiment.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 5. Drill bit system.
5. Results and discussion 5.1. The effect of rotary speed on drill string motion
Fig. 6. Well structure of A2.
To compare the FEA results with experimental data, drill string length in the FE model was 200 m. The drill string dimension was 88.9 mm 9.35 mm with steel grade of S135. Fig. 7 is the established drill string dynamics model of horizontal gas drilling. Rigid element and solid element were used for wellbore and drill string respectively. In the simulation process, WOB was 50 kN and rotary speed was 45 rpm.
Fig. 8(a) and (c) show the change of WOB with time under the condition of different rotary speeds. We can see that the change of WOB is a typical sine curve. The experimental data was then processed with FFT (Fast Fourier Transform) to obtain the frequency of WOB, as shown in Fig. 8(b) and (d). The comparison between the frequency of WOB and drill sting is given in Table 2. Fig. 9 illustrates the relationship between the frequency of WOB and rotary speed. Non-dimensional frequency is defined as the ratio of WOB frequency to the rotation frequency of drill sting. As observed, the frequency of WOB increases linearly with rotary speed and overlaps with rotation frequency of drill sting. As shown in Fig. 9(b), non-dimensional frequency is quite close to 1.0 at different rotary speed, i.e., the frequency of WOB is approximately equal to the rotation frequency of drill sting. Fig. 10 presents the results of lateral vibration acceleration. The accelerations of X-direction (left-right direction of the wellbore) and Y-direction (up-down direction of the wellbore) are approximately equal, though there exists a phase difference. In the horizontal drilling process, contact and impact between drill string and wellbore are inevitable, which can lead to high lateral vibration acceleration. As shown in Fig. 10(a)e(d), the lateral vibration acceleration of drill string is between 1.0 g and 1.0 g (here “g” represents for local acceleration of gravity). But when rotary speed reaches up to 450 rpm and 500 rpm, the maximum amplitude is 2.0 g, as shown in Fig. 10(e) and (f). It is concluded that severe impact has happened between drill string and wellbore at that condition.
Fig. 7. Solid model and mesh model for drill string dynamics.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 8. The change of WOB and its fluctuation frequency.
Table 2 Frequency of WOB and drill string at different rotational speed. Rotary speed (rpm)
Self-rotating frequency of drill sting (Hz)
Frequency of WOB (Hz)
Non-dimensional frequency
50 100 150 200 250 300 350 400 450 500
0.78 1.60 2.60 3.20 4.20 4.90 6.10 6.50 7.40 8.50
0.83 1.67 2.50 3.33 4.17 5.00 5.83 6.67 7.50 8.33
0.94 0.96 1.04 0.96 1.01 0.98 1.03 0.98 0.99 1.02
According to above analyses, lateral vibration acceleration is relatively low when rotary speed is 300 rpme400 rpm. However, when rotary speed increases to 450 rpme500 rpm, significant increase of lateral vibration acceleration is observed, which can result
in resonance, fatigue and other serious downhole incidents. So rotary speed of 450 rpme500 rpm should be avoided in the experiment. Consequently, for the given well structure and drill string assembly, the proposed rotary speed is 300 rpme400 rpm.
Fig. 9. The relationship between WOB frequency and rotary speed.
Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005
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Fig. 10. The lateral vibration acceleration of drill string at different rotary speed.
Fig. 11. Lateral displacement of drill string at different WOB and rotary speed.
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Fig. 12. Von Mises stress contour of horizontal drill string at different time.
According to similarity principle, the proposed rotary speed range is 37.5 rpme50 rpm for the actual operation of horizontal gas drilling. 5.2. The effect of WOB on drill string motion Because of the inherent characteristics of drill bit and various forms of drill string vibration, WOB is not constant in most cases. It is proved that appropriate WOB fluctuation can help rock breaking
but excessive fluctuation will bring about the impact failure of bit teeth and bearings. In consequence, appropriate WOB is essential for drilling operation. This paper mainly discusses the effect of WOB on drill string motion. Comparison of the two displacement curves in Fig. 11 indicates that Y displacement is a regular sine curve while X displacement is irregular. In addition, X displacement is always positive while Y displacement is negative. Combing with the motion trajectory of drill string, we can find that the swaying of drill string is at the right
Fig. 13. Contact state at different time.
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Fig. 14. Contact force between horizontal drill string and wellbore.
lower part of wellbore. At rotary speed of 200 rpm, with the increase of WOB, the fluctuation amplitude of lateral displacement is 1.2 mm, 1.5 mm and 2.0 mm respectively. The amplitude of lateral displacement increases with the increase of WOB, indicating more violent swaying. When WOB is 0.4 MPa, with the increase of rotary speed, the fluctuation amplitude of lateral displacement is 1.2 mm, 1.5 mm and 2.2 mm respectively. The increase of rotary speed leads to larger frequency and amplitude of drill string swaying. Therefore, WOB and rotary speed are two important factors affecting drill string motion in horizontal wells.
equal to the rotation frequency of drill sting. According to similarity principle, the recommended rotary speed range is 37.5 rpme50 rpm in the horizontal gas drilling of given well structure. The swaying of drill string is at the right lower part of wellbore and the variation of WOB has little effect on swaying range. In addition, a finite element model is established to study the buckling and contact of horizontal drill string. Numerical simulation results demonstrate that the contact force between wellbore and drill string is relatively large and helical buckling may be caused in horizontal drilling using gas.
5.3. Buckling and contact of drill string
Acknowledgments
Based on the established finite element model, numerical simulation was carried out to study the contact and buckling of drill string. When drill string just starts to move (1 s) against gravity and friction, stress is concentrated at the pushing end. Near-bit drill string starts to move at 3 s, at that moment the stress of near-bit drill string is obviously lower than other sections. When the whole drill string obtain translational velocity and rotary velocity, stress is concentrated at the drill string away from drill bit. Bending deformation of drill string is mainly caused by friction, axial force and other factors. After 10 s of simulation, the rotary motion brings about large torque on drill string and buckling has taken place. As shown in Fig. 12, sinusoidal buckling happens at near-bit drill string while the drill string away from bit is controlled by helical buckling. When the velocity of drill string becomes stable, continuous contact is observed between drill string and wellbore. The combination of Figs. 13 and 14 shows that the magnitude and direction of contact force varies over time. At 20 s, the average contact force of horizontal drill string is 18 kN, larger contact force is also observed at some nodes, indicating that severe buckling and pipe sticking can be caused in the horizontal drilling operation.
The authors thank the National Natural Science Foundation of China (No. 51574198 and No. 51504207) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20135121110005), for their contributions to this paper.
6. Conclusions A theoretical model is established to study the nonlinear drill string dynamics of horizontal wells drilled with gas. An experiment apparatus is designed and developed based on the actual structure of drill string in horizontal gas drilling and the principle of similarity. The experimental results shows that the change of WOB is a typical sine curve and the frequency of WOB increases linearly with rotary speed. Moreover, the frequency of WOB is approximately
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Please cite this article in press as: Lian, Z., et al., Experimental and numerical study of drill string dynamics in gas drilling of horizontal wells, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.10.005