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The role of friction on heave-induced motion of offshore drilling systems: modelling, analysis, and numerical experiments

Luigi Romano, Ole Morten Aamo

arXiv:2610.22189v1physics.app-phcs.SYeess.SYmath.DS

TL;DR

The paper asks how distributed friction affects heave-induced disturbances in offshore drilling, where coupled hydraulic and structural waves matter. It develops and analyzes a hydraulic-structural ODE–PDE model with friction memory, finding distinct roles for hydraulic and frictional dynamics in amplification, well-posedness, and stability.

  • Problem

    The interaction between distributed friction and hydraulic-structural wave propagation remains poorly understood, although heave-induced disturbances affect drilling-system responses.

    Source: §1 · pp. 1–2

  • Method

    The paper develops a coupled hydraulic-structural ODE–PDE model with distributed nonlinear friction-memory dynamics, proves well-posedness results, derives a small-gain stability criterion, and conducts numerical studies.

    Source: §1 · p. 2, §2 · p. 3, §5 · p. 28

  • Results

    Hydraulic effects primarily govern well-posedness and hydraulic amplification, while distributed friction has comparatively limited influence on existence and uniqueness but can affect stability margins and structural responses.

    Source: §5 · p. 29

  • Takeaways & Limitations

    Offshore drilling dynamics cannot be accurately described using purely hydraulic models because structural wave propagation and distributed friction also shape drillstring stability, velocity transmission, and stress loading.

    Source: §5 · p. 29

    The bound in Eq. (57a) is not a true ISS estimate for the full system because the lumped-state part does not decay over time.

    Source: §3.2.2 · p. 14

Abstract

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This paper investigates the influence of friction dynamics on heave-induced disturbance propagation using a coupled hydraulic-structural drilling model formulated as a semilinear interconnection of ordinary and hyperbolic partial differential equations (ODE-PDEs), and capturing spatially distributed memory effects in the drillstring-wellbore interaction. First, local and global existence and uniqueness results are established, and sufficient conditions for boundedness and input-to-state stability are deduced via a small-gain analysis. Subsequently, extensive numerical investigations are carried out, including harmonic excitation studies, nonlinear distortion analysis, spatial amplification investigations, mechanism-isolation experiments, parameter sensitivity analysis, and broadband simulations under realistic offshore sea-state excitation. The simulation results indicate that, whilst the hydraulic response is governed primarily by wave propagation phenomena, compressibility, and boundary reflections, the structural dynamics display substantially greater sensitivity to distributed friction effects. Furthermore, drillstring-wellbore friction is shown to have a relatively limited influence on well-posedness but a potentially significant impact on stability margins. These findings highlight the importance of accounting for coupled hydraulic-structural dynamics and distributed friction mechanisms when analysing, monitoring, and controlling offshore drilling systems.

1 Introduction

The paper examines how distributed friction affects heave-induced disturbances in offshore drilling, addressing a gap in understanding its interaction with hydraulic and structural wave propagation. It combines a coupled model and theoretical analysis with numerical studies to distinguish the roles of hydraulic and frictional mechanisms.

1 Introduction

  • Heave-driven pressure and structural waves can affect drilling performance, fatigue loading, equipment degradation, and, in severe cases, well integrity and operational safety.Prior work also identifies downhole pressure fluctuations that may remain largely invisible from surface measurements.Source: §1 · pp. 1–2
  • The interaction between distributed friction dynamics and hydraulic-structural wave propagation remains poorly understood in models that generally use static or quasi-static friction.The paper notes that drillstring studies show friction can affect vibration amplitudes, damping, stress transmission, and stability margins.Source: §1 · p. 2
  • The paper develops a semilinear ODE–PDE drilling model with nonlinear friction-memory dynamics and establishes well-posedness results and a small-gain stability criterion.Numerical studies examine frequency response, nonlinear distortion, spatial amplification, mechanism isolation, parameter sensitivity, and broadband offshore excitation.Source: §1 · p. 2
  • Hydraulic effects primarily govern well-posedness and hydraulic amplification, while distributed friction influences stability margins and structural responses.Source: §1 · p. 3

Notation

  • The notation uses R for real numbers, R>0 and R≥0 for positive and nonnegative reals, and 1_n for the n-dimensional all-ones vector.Source: Notation · p. 3

2 Model

The model couples hydraulic transmission lines, drillstring deformation, and distributed friction dynamics. Boundary conditions connect these dynamics to rig motion and distinguish open/closed-pipe and bit configurations.

2 Model

  • The model collects pressure, flow, axial stress, and velocity fields into an eight-state vector and couples them with friction variables in a nonlinear ODE–PDE system.The derivation of the equations is described as analogous to prior work and omitted for brevity.Source: §2 · p. 3

2.1 Coupled drillstring and fluid dynamics

  • Hydraulic PDEs describe pressure and volumetric-flow evolution in the drillstring, annulus, and below the bit, with viscous drags acting on the fluid.The model defines the relevant pipe and annulus drags and their associated friction factors.Source: §2.1 · pp. 3–5
  • The structural PDEs describe drillstring stress and velocity, with contact friction and viscous drag included in the dynamics.Source: §2.1 · p. 5

2.2 Friction dynamics

  • The FrBD model represents drillstring-wellbore contact friction through bristle-like internal states and distributed memory dynamics.Its Kelvin–Voigt formulation is described as physically derived and passive for any parameter choice.Source: §2.2 · pp. 6–7

2.3 Boundary conditions

  • Boundary conditions prescribe coupling among the hydraulic and structural variables and use a configuration switch to distinguish open/closed-pipe and bit cases.Rig velocity, external pressure, and bit force are treated as exogenous inputs.Source: §2.3 · p. 7

3 System in Riemann variables and theoretical analysis

This section reformulates the system in Riemann variables and establishes theoretical conditions for well-posedness, boundedness, and input-to-state stability. It examines how friction and pipe geometry affect stability criteria, with the stated ISS estimate applying only to PDE variables under an input interpretation of the ODE states.

3.1 ODE–PDE system in Riemann variables

  • The Riemann-variable formulation yields a semilinear hyperbolic ODE–PDE interconnection; its boundary map relates incoming and outgoing characteristics to the input.For open and closed pipe configurations, the boundary map has a linear expression; the bit configuration may not admit a closed-form expression in Riemann variables.Source: §3.1 · p. 9

3.2.1 Well-posedness

  • Under the stated regularity and Lipschitz assumptions, the system has a unique local generalised solution; additional linear-growth conditions on the PDE source extend it globally.The global result excludes finite-time blow-up. Its conditions are met for sufficiently vertical wells in laminar flow with open, closed, or regularised bit-pipe boundary conditions.Source: §3.2.1 · pp. 10–11
  • Drillstring-wellbore friction is not itself a principal obstacle to global well-posedness; the main obstacle is nonlinear feedback between tangential friction forces and state-dependent normal load.The global result requires no additional assumption on the ODE source F, consistent with friction’s intrinsically dissipative character.Source: §3.2.1 · p. 11

3.2.2 Stability analysis

  • The friction subsystem is dissipative and cannot generate forces that grow unboundedly with input velocity; a small-gain criterion then establishes global boundedness and ISS in the PDE variables.The ISS result applies under the stated bounds and spectral-radius condition, for inputs and initial conditions satisfying the theorem’s assumptions.Source: §3.2.2 · pp. 12–14
  • The small-gain result requires no fading-memory assumption on the ODE subsystem, but its parameter-vector existence condition is difficult to remove and may be checked numerically.The stability analysis indicates friction dynamics may alter stability even without oscillating normal loads, unlike the limited role friction plays in well-posedness.Source: §3.2.2 · p. 14
  • The bound (57a) is not a true ISS estimate under the standard definition, but can be viewed as ISS for PDE variables when ODE states are treated as inputs.Source: §3.2.2 · p. 14
  • Friction may reduce the stability margin available under the sufficient stability condition.Violating the theorem’s requirements does not imply instability; it means only that this criterion is inconclusive.Source: §3.2.2 · p. 15
  • Avoiding abrupt variations in pipe cross-sectional areas can reduce interconnection gains and help reduce the risk of instability.The sufficient conditions also guide pipe geometry design; the relevant coefficients depend on transport velocities and cross-sectional areas.Source: §3.2.2 · p. 15

4 Numerical Experiments

The numerical experiments combine a discretized hydraulic-structural model with harmonic and broadband excitation to examine disturbance propagation across pipe configurations, friction mechanisms, and operating conditions. Across these tests, hydraulic responses are strongly shaped by fluid transport and resonance, while structural responses show substantial wave-driven amplification and sensitivity to friction.

4.1.1 Numerical implementation; 4.4 Synthetic offshore excitation

  • The model was discretized by the method of lines on a uniform 100-point spatial mesh, producing 10Nz coupled ODEs; simulations then examined harmonic and synthetic sea-state excitations.The broadband input used a JONSWAP spectrum with Hs = 4 m and Tp = 12 s, weighted by an idealized rig-heave response amplitude operator.Source: §4.1.1 · p. 15, §4.4 · p. 26

4.1.2 Nominal hydraulic and structural response

  • Under small harmonic excitation, hydraulic variables approach periodic oscillations while drillstring velocity tracks rig motion and stress oscillates strongly; the closed configuration produces wider stress oscillations.The hydraulic variables are governed primarily by transport and compressibility, whereas structural variables retain stronger memory of boundary excitation and wave propagation.Source: §4.1.2 · p. 17

4.2.1 Frequency-domain amplification characteristics

  • Hydraulic pressure and flow gains peak near T = 12 s, with higher pressure peaks in the closed configuration, while structural gains vary more smoothly across frequency.The hydraulic resonance may arise from constructive interaction of travelling and reflected waves; structural gains are attributed primarily to distributed elastic wave propagation.Source: §4.2.1 · p. 17

4.2.2 Nonlinear harmonic distortion; 4.2.4 RMS and peak amplification metrics

  • Hydraulic distortion varies nonlinearly near resonance, whereas structural distortion remains comparatively limited and nearly linear; resonant conditions also elevate RMS and peak responses.Structural peak amplification can remain substantial even when hydraulic amplification is moderate.Source: §4.2.2 · p. 20, §4.2.4 · p. 20

4.2.3 Spatial amplification of disturbances

  • Pressure amplification increases with depth while flow gains trend oppositely; structural velocity generally attenuates before increasing near z∗ = 0.8, and stress gains persist across much of the well.The velocity increase is attributed to rapid friction variation at z = 0.8; surface measurements alone are insufficient to predict downhole dynamic conditions.Source: §4.2.3 · p. 20

4.3.1 Mechanism-isolation analysis

  • Removing FrBD friction substantially changes drillstring velocity and stress, whereas replacing its dynamic law with a memoryless version causes comparatively small changes under the tested conditions.Hydraulic responses remain primarily governed by fluid dynamics, although FrBD dynamics also modify pressure and flow responses.Source: §4.3.1 · p. 23

4.3.2 Parameter sensitivity analysis

  • At the tested open-pipe condition, kfric dominates hydraulic sensitivity: increasing it lowers pressure and flow gains, while structural gains remain nearly unchanged and variations in ¯k0 and τ are small.The sensitivity study used z∗ = 0.8 and T = 12 s; FrBD stiffness and relaxation-time changes had negligible effects on fundamental gains.Source: §4.3.2 · p. 24

4.4 Synthetic offshore excitation

  • Broadband excitation produces intermittent amplification and large structural fluctuations; its spectral analysis also finds broad stress amplification and hydraulic resonances consistent with harmonic tests.Coherence remains close to unity across large portions of the spectrum, indicating responses are strongly driven by rig motion.Source: §4.4.1 · p. 26, §4.4.2 · p. 28

5 Conclusions

The paper develops and analyzes a coupled hydraulic-structural ODE–PDE model with distributed friction, then tests disturbance propagation numerically. The results distinguish hydraulic wave-driven amplification from structural sensitivity to friction and show that friction can affect stability margins more than well-posedness.

5 Conclusions

  • The model combines compressible fluid transport, drillstring elasticity, hydraulic drag, mud-clinging effects, and drillstring-wellbore friction in a semilinear ODE–PDE interconnection.The numerical campaign includes harmonic excitation, distortion, spatial amplification, mechanism isolation, parameter sensitivity, and broadband offshore simulations.Source: §5 · pp. 28–29
  • Distributed friction has a comparatively limited role in existence and uniqueness but can affect stability margins in the coupled system.The paper derives a conservative input-to-state stability criterion using small-gain arguments.Source: §5 · p. 29
  • Hydraulic responses depend primarily on fluid-wave propagation, compressibility, and boundary conditions, while structural responses are substantially more sensitive to distributed friction.Pressure fluctuations may increase toward lower well sections, and broadband sea-state simulations preserve amplification mechanisms found in harmonic analysis.Source: §5 · p. 29
  • The results suggest that purely hydraulic models cannot accurately describe offshore drilling dynamics; future work includes experimentally identified friction laws and broader boundary configurations.The authors also identify uncertainty quantification and reduced-order control-oriented formulations as future research directions.Source: §5 · p. 29

Funding declaration

The research was funded by the FASTEST project through the Swedish Research Council.

Funding declaration

  • The FASTEST project (Reg. no. 2023-06511), funded by the Swedish Research Council, supported this research.Source: Funding declaration · p. 29

A Technical results

The appendix contains proofs of the paper’s technical results, covering local and global well-posedness and stability.

A Technical results

  • Appendices A.1 and A.2 derive local and global well-posedness results for the ODE–PDE system, while Appendix A.3 addresses stability analysis.Source: A Technical results · p. 31

A.1 Local well-posedness of the ODE–PDE system (30)

The local well-posedness proof uses a Banach fixed-point argument and an exponential rescaling of the Riemann variables. The rescaling makes the boundary map a strict contraction while preserving equivalence with the original variables.

A.1 Local well-posedness of the ODE–PDE system (30)

  • The local well-posedness proof applies Banach’s fixed-point theorem after transforming the distributed states into Riemann variables.Source: §A.1 · p. 31

A.1.1 Exponential rescaling

  • For fixed scaling, the variable transformation is an isomorphism, so local well-posedness in transformed variables is equivalent to local well-posedness in the original variables.Source: §A.1.1 · p. 31
  • Exponential rescaling makes the transformed boundary map a strict contraction when the scaling parameter is chosen sufficiently large.The transformed map’s contraction constant can be made arbitrarily small.Source: §A.1 · p. 31, §A.1.1 · p. 32

A.1.2 Preliminary bounds

The preliminary estimates support a local fixed-point argument for the transformed ODE–PDE system. The map is shown to preserve a closed ball and become a strict contraction for sufficiently small time.

A.1.2 Preliminary bounds

  • The preliminary analysis bounds differences in the transformed nonlinear terms and uses continuity at zero to bound the forcing terms.These bounds prepare the analysis in Section A.1.3.Source: §A.1.2 · pp. 32–33

A.1.3 Fixed-point map

  • Banach’s fixed-point theorem is the basis for proving local existence and uniqueness through a map on the closed ball.The map is constructed from the characteristic formulas for the transformed system.Source: §A.1.3 · p. 33
  • For sufficiently small time, the map sends the ball into itself and is a strict contraction.The contraction estimate accounts for PDE components whose characteristics reach either the initial line or the boundary.Source: §A.1.3 · p. 34

A.1.4 Proof of Theorem 3.1

The proof establishes local well-posedness by constructing a unique fixed point, then rules out continuation failure unless the solution norm becomes unbounded.

A.1.4 Proof of Theorem 3.1

  • The transformed system has a unique generalized solution for sufficiently small time, and the original system is locally well-posed in the stated continuous-function space.The solution can be continued from the endpoint of that interval when the initial data satisfy the zeroth-order compatibility condition.Source: §A.1.4 · p. 35
  • If the maximal existence time is finite, the solution norm must diverge as that time is approached.A bounded sequence of norms would allow extension beyond the alleged maximal time.Source: §A.1.4 · p. 35

A.2 Global well-posedness for the ODE–PDE system (30)

The global well-posedness proof bounds the ODE and PDE components and applies Grönwall-Bellman’s inequality to exclude finite-time blow-up. The stability proof then treats the ODE subsystem first and extends the result to the interconnection using small-gain-like arguments.

A.2 Global well-posedness for the ODE–PDE system (30)

  • The global-existence argument uses transformed variables and Grönwall-Bellman’s inequality to exclude the blow-up alternative.Preliminary bounds on the functions involved are established first.Source: §A.2 · p. 35
  • Under Theorem 3.2’s assumptions, the solution remains bounded on every finite time interval, excluding finite-time blow-up and yielding a global solution.The estimate follows from bounds on the ODE and transformed PDE variables and Grönwall-Bellman’s inequality.Source: §A.2 · p. 37

A.3 Stability of the ODE–PDE interconnection

  • The stability proof first analyzes the ODE subsystem, then uses friction’s dissipative nature and small-gain-like arguments for the full interconnection.Source: §A.3 · p. 37

A.3.1 Proof of Lemma 3.1

The lemma’s proof uses a Lyapunov function to derive estimates for the ODE subsystem. It also makes explicit a coefficient that distinguishes this proof from cited prior work.

A.3.1 Proof of Lemma 3.1

  • Unlike the cited proofs [27] [28], this argument determines the coefficient Cf2 explicitly.Source: §A.3.1 · p. 37
  • The proof derives estimates establishing (42a), (43), and (42b) using bounds on the Lyapunov function.The argument notes that W is equivalent to a norm on the ODE state space.Source: §A.3.1 · pp. 38–39

A.3.2 Proof of Theorem 3.3

The proof transforms the PDE subsystem into homodirectional Riemann variables, then establishes a global boundedness and stability estimate. The appendices provide closed-form equilibria and boundary-map expressions for open and closed pipe configurations.

A.3.2 Proof of Theorem 3.3

  • Transforming the PDE subsystem into a homodirectional form enables closed-form solutions for Riemann variables sharing a transport velocity.The reformulated PDEs are treated as an interconnection of four sets of Riemann variables.Source: §A.3.2 · p. 39
  • The proof uses an isomorphism of C0([0, 1]; R2) to transform the variables before substituting them into the PDE dynamics and boundary conditions.It then integrates along characteristic lines to obtain the solution representation.Source: §A.3.2 · pp. 39–40
  • Theorem 3.3’s proof establishes global existence and boundedness, then deduces the estimate (57) for all t ≥ 0.The argument excludes finite-time blow-up by keeping the transformed state within its bounds.Source: §A.3.2 · p. 42

B.2 Boundary map for the open and closed pipe configurations

  • The appendix gives the complete boundary-map matrices H and B for both open- and closed-pipe configurations.The matrix B is the same as in Eq. (166).Source: §B.2 · pp. 43–45
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