Nonlinear Dynamic Characteristic Analysis of a Landing String in Deepwater Riserless Drilling

+e landing string is an important component of deepwater riserless drilling systems. Determination of the dynamic characteristics of the landing string plays an essential role in its design for ensuring its safe operation. In this paper, a dynamic model is developed to investigate the dynamic response characteristics of a landing string, where a landing string in a marine environment is modeled as a flexible slender tube undergoing coupled transverse and axial motions.+e heavingmotion of the drilling platform is taken as the upper boundary condition and the motion of the drilling bit caused by the interaction between the rock and the bit as the lower boundary condition. A semiempirical Morison equation is used to simulate the effect of the load imposed by the marine environment. +e dynamic model, which is nonlinearly coupled and multibody, is discretized by a finite element method and solved by the Newmark technique. Using the proposed model, the dynamic responses of the displacement, axial force, and moment in the landing string are investigated in detail to find out the influences of driving depth of surface catheter, platform motion, bit movement, and marine environment on the dynamical characteristics of the landing string.


Introduction
Riserless drilling is a new deepwater drilling technology in which the drilling platform no longer bears the interaction load induced by the ocean waves and the ocean currents which are imposed on the riser and the mass buoyancy block.Compared with the traditional deepwater drilling method with riser, this technique has many remarkable advantages, such as reducing the requirement of storage capacity of drilling platform, greatly shortening the construction period, saving the cost of drilling, and improving the safety of drilling operation [1].However, there are still some thorny problems in the technology.One salient issue of them is the dynamic load response in the landing string used in a jet technique for lowering and lifting the surface catheter and the BHA (bottom hole assembly) (Figure 1).Due to the lacking of protection from riser as well as the exposure to the ocean environment directly, the landing string bears very complex patterns of external loads, such as working load, marine environmental load, drilling platform movement, and their combinations.erefore, the internal force and deformation of the landing string are often complex and changeable, making the pipe string prone to be instability, fracture, and fatigue failure and causing serious accidents and economic loss [2].us, in order to ensure the safety and reliability of deepwater riserless drilling operation, it is very necessary to find out the dynamic characteristics of the landing string, including vibration modes, critical speed of the bit, deformation, and internal load response under the actions of environmental load and platform movement [3][4][5].
Many works, using theoretical analyses [6,7], numerical simulations [8], and experimental investigations [9,10], have been conducted to investigate the dynamics behaviour of on-land drilling string to determine the change regulations of the friction, torsion, and axial force under various working conditions.ese research results guided the operation of on-land drilling operation.However, there is a clear difference between offshore drilling and on-land drilling.For the former, the traditional drilling string is in contact with the riser in motion.For the latter, the drilling string is in direct contact with the fixed casing.erefore, the dynamic behaviour of offshore drilling pipe string is more complex than that of on-land drilling pipe string [11].Most of the research studies on the mechanics analysis of marine pipe string focused on the dynamic behaviour of the riser [12][13][14][15].Some works [16,17] investigated the interaction between the completion pipe string (drilling pipe or test pipe) and the riser.
In comparison with the conventional deepwater drilling with a riser, the riserless landing pipe system is obviously different in structure and load [18].At first, the upper end of the pipe string is fixed directly to the drilling ship, so the bending deformation of the pipe string cannot be reduced by applying the top tensioning force as that used in conventional deepwater riser.Secondly, since the landing string is not constrained by the riser in the radial direction, it is prone to buckle under the disturbance of the transverse load as the axial tension in the pipe is small.erefore, the landing string is more prone to damage induced by large deformation and complex vibration [19] and has attracted the attentions of many scholars [20][21][22].
An approach to the design of deepwater riserless landing strings was proposed by Adams [23], which focuses on the selection of pipe joints and tool joints.e design, manufacture, testing, and operation of landing strings have also been discussed by Breihan et al. [24], Jellison et al. [25], and Cantrell et al. [26].However, in these studies, dynamic analysis did not get enough attention.Sathuvalli et al. [27] pointed out that deepwater landing strings can usually withstand great axial loads, so slip collapse is the main factor to restrict the operational capability of such strings.ey also presented a new method for calculating the slip collapse load.Bradford et al. [28] carried out five different types of experiments on slip collapse.Everage et al. [29] and Zheng et al. [30] investigated methods for calculating the longitudinal vibration load on a landing string induced by platform heave motion during lowering of the casing.Samuel et al. [31] presented a simulation approach for hook load, torque, and drag calculations and investigated various parameters' influence on these load, such as wellhead offset from the rig center, wellbore inclination, curvature, wellbore torsion, angle of entry into the wellhead, wave forces, and ocean loop currents.e dynamic problems involved in these studies are either for the longitudinal vibration of the string or for the transverse vibration.Han and Benaroya [32] pointed out that the effects of longitudinal and transverse coupling vibration (LTCV) are not negligible for slender columns and investigated the LTCV behaviour of a tension leg platform.e LTCV behaviour of a beam simply supported at both ends was examined by Xing and Liang [33]; Hu and Yang [34]; Ghayesh and Farokhi [35]; and Zhao and Wu [36].Liu et al. [37] established a LTCV model to investigate the effects of the main parameters such as marine load, platform motion, and suspending length on the dynamic behaviour of a deepwater hard suspension riser under emergency evacuation conditions.
Besides the effect caused by the nonlinear LTCV of a pipe string or a beam discussed by Han and Benaroya [32]; Zhao and Wu [36]; and Liu et al. [37], the landing string in riserless drilling is affected simultaneously by the combined actions of the platform, drilling bit, ocean environment, driving depth, and so on.erefore, its vibrational behaviour is strongly nonlinear.However, up to now, there has been very limited research on this issue.is paper presents a nonlinear dynamic model of a deepwater landing string under the combined action of platform movement, drill movement, and marine load, taking into account the LTCV and driving depth effects.Based on the model, the nonlinear vibration modes and mechanical response mechanism of a landing string are investigated in detail, aimed at providing a theoretical basis for the design, construction, and operation of deepwater landing strings in riserless drilling.

The Proposed Nonlinear Dynamic Model
In this section, the Hamilton variational principle is used to establish the nonlinear vibration control equation of the landing string, based on the energy analysis of system.

Basic Assumptions.
According to the structure and working conditions of the deepwater drilling string system  Shock and Vibration shown in Figure 1, the following assumptions are made to derive the dynamic governing equations of the system: (1) e material property is homogeneous and isotropic, and the elasticity remains linear during movement and deformation.(2) e influence of the joints on the bending rigidity of the landing string is not taken into account.
(3) e effect of structural damping on the model of the riser is neglected.(4) e directions of ocean currents and the deformation of the drilling string are assumed to lie in the same plane.(5) e landing string is connected rigidly with the platform and the surface catheter.e transverse displacement of the surface catheter is restricted.(6) e deformation of the surface catheter and BHA is ignored.(7) e surface catheter keeps vertical during driving.

Energy Equation of the System.
e total length of the landing string is denoted by L. e elements of the string are labeled by their locations in the undeformed configuration x.
e reference and current configurations of the midline of the string are shown in Figure 2. e reference configuration is indicated by the vertical straight dashed line and the current configuration by the curved dashed line.Similar coordinate system can also be found in Han and Liu's work [32,37].
According to Han and Benaroya [32], the Green strain based on Kirchhoff's hypothesis can be written as where u and υ are the deflections of the midline in the x and y directions, respectively; y is the transverse distance from the midline to the point of interest on the cross section in the reference frame.Assuming a symmetrical cross section, u is also the average deflection of the beam element.
According to the definition of strain energy and (1), the strain energy can be given as follows and its detailed derivation is presented in Appendix A: where the primes indicate derivatives with respect to x. e total kinetic energy of the system consists of two parts: the kinetic energy of the landing string and the kinetic energy of the combined catheter plus BHA.
e kinetic energy of the landing string is given by where the dots indicate derivatives with respect to time.
e total kinetic energy of the landing system can be expressed as follows and its detailed derivation can be found in Appendix B: where ρI 0 _ υ ′ 2 is the Rayleigh rotational term, which is the kinetic energy due to the rotation of the cross section and is usually small compared with the kinetic energy due to translation, ρA 0 ( _ Shock and Vibration 3

Control Equation and Boundary Conditions via
Hamilton's Principle.Hamilton's principle is used to formulate the governing dynamical equations of the landing string system.e Lagrangian integrated over time, and the virtual work done by the transverse forces f(x, t) and axial forces p(x, t) is given by Using Hamilton's principle, control equation and boundary condition can be formulated as and their detailed derivation can be found in Appendix C. Boundary conditions: where u boat (t) is the heaving displacement of the drilling platform.e hydrodynamic force is simulated by the Morison equation whose concise form is given by Han and Benaroya [32] as

Model Solution and Verification
In this section, the control equation is discretized by the finite element method and solved by the Newmark technique.e validity of the model is tested using the ANSYS software.

Finite Element Scheme for Nonlinear Governing
Equations.We use a linear Lagrange function and a cubic Hermitian function to approximate the transverse displacement u and longitudinal displacement v of a beam element: where Using (10) and the Galerkin weighted residual method, the finite element scheme for the nonlinear governing equation ( 7) is given as follows:  Shock and Vibration where D is the generalized coordinate vector; K, M, and F are, respectively, the stiffness, mass, and load matrices, which, respectively, consist of element stiffness, mass, and load matrices, k e , m e , and f e shown in Appendix D. Using Newmark's integral method in Appendix E, (12) yields the following equation: e system responses, including the displacements, velocities, and accelerations at the instant i + 1, can be obtained by substituting the solution of ( 13) into (E.1) and (E.2) in Appendix E.

Verification of the Nonlinear Dynamic Model.
A longitudinal-transverse coupling dynamic analysis code (LTND) based on the above model is developed.e validity of the model and the code is tested using the ANSYS software, which does not take into account the LTCV effect (Figure 3).e basic parameters of the example are shown in Table 1.
e longitudinal and transverse displacement responses of the middle section of the standpipe under two kinds of working conditions are simulated, respectively, by the coupling method and ANSYS.ANSYS has adopted BEAM188 element, and case 1 and case 2 have divided 10 elements and 100 elements, respectively.e greatest difference between the two cases is the slenderness ratio, which is much larger in case 2. It is worth noting that as the slenderness ratio increases, the LTCV effect becomes more obvious [32].
Figures 4(a) and 5(a) show that, for longitudinal vibration, the results obtained with the coupling method are close to those obtained using ANSYS.For the larger slenderness ratio, the vibrational amplitude obtained with the coupling method is greater than that from ANSYS.Figures 4(b) and 5(b) show that, for the smaller slenderness ratio, the transverse vibrational response obtained with the coupling method is close to that from ANSYS.However, when the slenderness ratio is increased, the difference between the results becomes very large.ese comparisons show that LTCV behaviour depends significantly on the slenderness ratio, with transverse vibrations being more strongly affected by this ratio than longitudinal vibrations.

LTCV Dynamic Characteristic Analysis
Using the developed LTND code, the LTCV characteristics of a practical deepwater riserless landing string system are investigated.e structure and environment parameters are presented in Table 2. e motion of a bit is simulated by the following formula [38]: where where ω � nπ/10; x 0 is the initial displacement, in this paper x 0 � 0. e lateral displacement of the surface casing is constrained, and the longitudinal friction can be calculated by the following formula: e calculation formulas for formation pressure and friction factor can be found in literature [39,40], respectively, and are listed as follows:

Vibration Modal Analysis.
From the submatrix of the stiffness matrix of the nonlinear system, k 2 , k 3 , and k 4 , it can be seen that the stiffness matrix is time-variant.For an arbitrarily small-time quantum, the frequencies and modes are assumed not to change with time and their calculation equations are given as which shows that the frequencies and modes in the nonlinear system in this paper are also time-variant, where φ is the vector of mode shape.Using (18), the first four order frequencies and modes of the deepwater landing string have been calculated and, respectively, are shown in Figures 6 and 7.
Figure 6 shows that the frequency of transverse vibration varies greatly with time and increases with the increase of order.e mode diagram in Figure 7 reflects the same trend.

Shock and Vibration
At the initial stage of the vibration, the frequency of each order rises sharply and then changes up and down at a certain value.e main reason is that at the beginning of the vibration, the longitudinal tension as well as the stiffness of the landing string increases sharply due to the string's flexes.Later, the landing string vibrates in a certain bending shape, and the axial force and stiffness change periodically.
To further illustrate the effect of LTCV on the vibration frequency of a landing string, the vibration frequency without considering LTCV is also shown in Figure 6.It is demonstrated that the frequencies taking into account the LTCV are much larger than those that do not consider coupling.
e increase and changeability of the vibration frequency make the landing string more inclined to resonate with the platform and bit.

Dynamic Response Analysis.
In this section, the influence of LTCV, shear flow velocity, platform motion, drilling 8 Shock and Vibration bit speed, and driving depth on the dynamic response analysis characteristic of a landing string is analyzed.

LTCV Effect.
To investigate the effect of LTCV on the dynamical behaviour of the landing string, a 1000 m long landing string in a shear flow with 1.2 m/s maximum velocity at sea level and 0.036 m/s minimum velocity at the seabed is considered.e velocity of the shear flow is approximately considered to change linearly.Other parameters are shown in Table 2.As can be seen from Figure 8, longitudinal-transverse coupling has little effect on the amplitude of longitudinal vibrations compared with the uncoupled case.However, when coupling is taken into account, the first-order frequency of longitudinal vibrations obviously decreases, and, as can be seen from Figure 9, the amplitude and the first-order frequency of transverse vibrations are both significantly affected.erefore, in the dynamic analysis of a deepwater landing string, the influence of nonlinear coupling of vertical and horizontal motions cannot be ignored.

Effect of Shear Flow Velocity.
e dynamic behaviour of a landing string of length 1500 m is analyzed to investigate the influence of shear velocity on vibration of the pipe string.Figure 10 (where V u is the flow velocity at sea level and V d that at the seabed) shows that the shear velocity has little influence on the longitudinal displacement and frequency of the tube, and the longitudinal vibrational displacement and frequency are controlled mainly by the motion of the platform.Figure 11 shows that the shear velocity has a strong influence on the transverse displacement of the tube: the   Shock and Vibration higher the shear velocity, the greater is the transverse displacement at the center of the tube.As shown in Figure 12, the maximum dynamic tensile force appears in the upper and middle parts of the landing string, whereas Figure 13 shows that the maximum dynamic compressive force appears in the lower part.e two axial forces decrease with increasing shear velocity.Figure 14 shows that the maximum dynamic moment increases with increasing water depth.e main reason for this is that the tension in the tube gradually decreases with increasing water depth owing to the influence of the pipe weight, and thus, bending gradually becomes more and more significant.

Effect of Platform Motion.
e influence of platform motion on the dynamic behaviour of the landing string is analyzed under the condition that other parameters remain unchanged.As can be seen from Figures 15 and 16, the longitudinal vibration at the center of the pipe string is controlled mainly by the longitudinal motion of the platform.
e longitudinal vibration of the platform has no obvious influence on the transverse displacement but does have a significant influence on the higher-order vibrational frequencies of transverse vibration.Figures 17-19 show that longitudinal vibration of the platform will increase the maximum dynamic axial force and bending moment of the whole landing string, especially its lower part.erefore, platform motion has an important influence on the safety of the landing string and the working efficiency of the drilling bit.

Effect of Drilling Bit Speed.
e influence of drilling bit speed on the dynamic behaviour of the landing string is investigated under the condition that other parameters show that the bit speed has no significant effect on the displacement response of the landing string.Figures 22 and 23 show that the higher the drill speed, the greater is the dynamic axial force on the pipe string.Figure 24 shows that the bit speed has no significant influence on the maximum dynamic bending moment of the whole pipe string.

Effect of Driving Depth.
e influence of driving depth on the dynamic behaviour of the landing string is analyzed under the condition that other parameters remain unchanged.
e depth from the platform in the section contains the driving depth and the length of the landing string.Figures 25  and 26 show that the increase of the driving depth has no significant influence on the vibration displacement in the middle of the landing string, whereas Figures 27-29 show that the maximum dynamic axial force and moment are influenced significantly by the driving depth.
e maximum dynamic axial force and moment increase with the increase of driving depth; the main reason is that the increasing driving depth increases the longitudinal friction force on the pipe section in the formation section.Figures 27-29 also show that the driving depth has the greater influence on the dynamic internal forces in the middle-upper part of the landing string than those in the lower part.

Conclusions
(1) A longitudinal-transverse coupling dynamic analysis code (LTND) for deepwater landing string is Shock and Vibration developed and tested by comparing its simulation result with that from finite element software ANSYS.e comparison and analysis show that the code has good calculation accuracy, and the LTCV vibration is obvious for the pipe with large slenderness ratio.Deepwater landing strings are typical members with large slenderness ratio.
e vibration model and displacement response analysis of an actual landing string show that the LTCV effect is significant and cannot be ignored to a landing string, especially for its transverse vibration.
(2) e effects of shear flow, platform motion, bit moment, and driving depth on the dynamic response of the actual landing string are investigated, by examining the displacement, dynamic axial force, and dynamic moment.It is shown that the axial force in the pipe string is significantly influenced by platform motion, shear flow velocity, and bit movement.e greater the amplitude of platform motion or the drill speed, the greater is the maximum dynamic axial force.By contrast, the higher the shear flow velocity, the smaller is the maximum dynamic axial force.It is also found that the driving depth has the greater influence on the dynamic internal forces in the middle-upper part of the landing string than those in the lower part.underwater wellhead.When a threat such as a typhoon is imminent, this joint is automatically disconnected by moving the platform to increase the axial tension near the joint to a certain value.However, it is important that this emergency joint remains connected under normal drilling conditions.erefore, under normal conditions, the axial tension must be less than the set value, which requires particular measures to be taken, such as using a heave compensation device to limit the movement of the platform and controlling the drilling speed of the bit.e general expression for the strain energy is

Shock and Vibration
Using the expressions for the second Kirchhoff stress and the Green strain, we obtain Expanding the integrand, we obtain On integrating over the area, the terms that are odd functions of y (the second and fifth terms) disappear because of the symmetry of the cross section, and y 2 becomes the area moment of inertia I 0 about the neutral axis.e remaining terms are functions of x and t only, that is, (2) can be obtained.

B. Derivation of KE
According to Kirchhoff's hypothesis, we can write where u 1 , u 2 , and u 3 are the displacement field functions corresponding to the coordinate systems x, y, and z.
Replacing the axial displacements u 1 and u 2 in (B.1) using (3), we can write where, on integration over the cross section, the term with factor y has disappeared and the term with factor y 2 has become I 0 as before.e kinetic energy of the combined catheter plus BHA is given by

C. Formulation of Hamilton's Principle
e variation of ( 5) is given by Integrating by parts and noting that the variations in time at the two endpoints are assumed to be zero, we obtain (℘ + W) dt � 0, and thus, the nonlinear coupled control equations of the landing string system and its boundary condition are obtained.

E. Newmark's Integral Method
Newmark's integral method is used to solve (12), in which the displacement and velocity at the instant t + Δt(i + 1) can be expressed, respectively, as where Δt is the time step length and δ and β are parameters related to the precision and stability and under normal circumstances take the values δ � 0.5 and β � 0.25.Using (D.2), the velocity and acceleration at the instant i + 1 can be expressed approximately by the responses at the instant i and the displacement at the instant i + 1:

E:
Young's modulus A 0 : Cross-sectional area of the drilling string V 0 : Volume of the deformed landing string I 0 : Area moment of inertia of the cross section around the neutral axis L: Length of drilling string ρ: Material density C A : Added mass coefficient C D ′ : Drag coefficient for an oscillating cylinder in still water ρ f : Density of the surrounding fluid A f : Cross section of the displaced volume l e : Length of one finite element m: Mass of drilling string with unit length M CBHA : Mass of the combined catheter plus BHA r outer : Outer radius of the drill string f(x, t): Transverse force of unit length on the drill string p(x, t): Axial force of unit length on the drill string u boat (t): Heave displacement of the drilling platform ω: Angular frequency of the system ω n : Natural frequency of the drill string ω d : Natural frequency of damped system k: Equivalent stiffness of drill string m: Equivalent mass of drill string ξ: Damping ratio of the system n: Drilling speed c: System viscosity coefficient P f : Formation pressure acting on surface casing A sc : Area of the surface casing subjecting formation pressure F f : Casing formation friction factor P(H e ): Overlying formation pressure H e : Vertical depth of a sea ρ(H e ): Formation density H z : Total vertical depth including sea section and formation section G 0 : Pressure gradient of overlying formation G n : Static fluid gradient of sea section.

Figure 1 :
Figure 1: Schematic diagram of a landing string with the riserless drilling technique.

Figure 2 :
Figure 2: Reference and current configurations of the midline in a landing string system.

Figure 3 :
Figure 3: Diagram of the forced riser.

Figure 4 :
Figure 4: Time history responses of (a) longitudinal and (b) transverse displacements of the middle section of the standpipe in case 1.

Figure 5 :Figure 6 :Figure 7 :
Figure 5: Time history responses of (a) longitudinal and (b) transverse displacements of the middle section of the standpipe in case 2.

Figure 11 :
Figure 11: Effect of shear velocity on transverse vibration.

AppendixA.
Derivation of PE e second Kirchhoff stress, ignoring the Poisson effect, is given by  σ xx � Eε xx .(A.1)

Figure 27 :Figure 28 :Figure 29 :
Figure 27: Effect of driving depth on maximum dynamic axial tensile force.

Table 1 :
Parameters of the standpipe.

Table 2 :
Parameters of landing string and drilling system. 