Influence of bit design on the stability of a rotary drilling system

Nonlinear Dynamics - This paper extends the RGD model originally proposed by Richard et al. (J Sound Vib 305(3):432–456, 2007, https://doi.org/10.1016/j.jsv.2007.04.015) to investigate...     

Eulerian formulation of constrained elastica

The formulation of the constrained elastica problem proposed in this paper is predicated on two key concepts: first, the deformed elastica is described by means of the distance from the conduit axis; second, the problem is formulated in terms of the Eulerian curvilinear coordinate of the conduit rather than the natural curvilinear coordinate of the elastica. This approach is further implemented within a segmentation algorithm, which transforms the global constrained elastica problem into a sequence of analogous auxiliary problems that result from dividing the conduit and the elastica into segments limited by contacts. Each auxiliary segment entails solving a segment of elastica subject to isoperimetric constraints corresponding to the assumed positions of the segment ends along the conduit. This new formulation resolves in one stroke a series of issues that afflict the classical Lagrangian approach: (i) the contact detection is reduced to checking whether a threshold on the distance function is violated, (ii) the isoperimetric conditions are transformed into regular boundary conditions, instead of being treated as external integral constraints, (iii) the method yields a well-conditioned set of equations that does not degenerate with decreasing flexural rigidity of the elastica and/or decreasing clearance between the conduit and the elastica.     

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.     

A Remark on the Poroelastic Center of Dilation

In the original derivation due to Dougall (Proc. Edinb. Math. Soc. 16(1):82–98, [22]), the center of dilation is constructed from the superposition of orthogonal force dipoles. However, it can also be built by combining dislocation dipoles. While the two representations are equivalent in elasticity, the different nature of the force and dislocation dipoles leads to different transient behaviors in poroelasticity, when the response evolves from undrained to drained. Indeed, we show that the poroelastic solution of a center of dilation is obtained by superposing the corresponding elastic solution and either an instantaneous fluid sink or an instantaneous fluid source, depending on whether the center of dilation is treated as a singular stress or displacement discontinuity. A simple explanation to the instantaneous injection or withdrawal of a finite volume of fluid from/to the singularity is given by adopting an exploded view of the poroelastic center of dilation. The exploration of the drained, undrained and transient responses of this singularity also requires to clarify the close relationship between an instantaneous fluid source and a continuous fluid dodecapole, an object constructed by collapsing onto a point three orthogonal pairs of fluid dipoles.     

Inner mechanics of three-dimensional black holes

We investigate properties of the inner horizons of certain black holes in higher-derivative three-dimensional gravity theories. We focus on Ba?ados-Teitelboim-Zanelli and spacelike warped anti-de Sitter black holes, as well as on asymptotically warped de Sitter solutions exhibiting both a cosmological and a black hole horizon. We verify that a first law is satisfied at the inner horizon, in agreement with the proposal of Castro and Rodriguez [arXiv:1204.1284]. We then show that, in topologically massive gravity, the product of the areas of the inner and outer horizons fails to be independent on the mass, and we trace this to the diffeomorphism anomaly of the theory.     

A Reexamination of the Classical PKN Model of Hydraulic Fracture

This article reexamines the classical PKN model of hydraulic fracture Perkins and Kern (J. Pet. Tech. Trans. AIME, 222:937–949 (1961)) and Nordgren (J. Pet. Tech. 253:306–314 (1972)) using novel approaches, which have recently been developed to tackle this class of problems that are characterized by a moving boundary and strong non-linearities in the governing equations. First, we demonstrate, using scaling arguments only, that a PKN hydraulic fracture has two limiting time asymptotic behaviors: storage-dominated at small time, and leak-off-dominated at large time. Next, we investigate the multiscale nature of the tip asymptotics and its implication for the construction of a robust and efficient numerical algorithm. In particular, we show that in the storage-dominated regime the tip aperture w behaves according to w ~ x ^1/3 (where x is the distance from the tip), and in the leak-off-dominated regime according to w ~ x ^3/8. However, the solution in the leak-off-dominated regime has a boundary layer structure at the tip, with w ~ x ^3/8 acting as an intermediate asymptote that matches an inner solution with a w ~ x ^1/3 asymptote to the outer (global) solution. Finally, we describe an efficient numerical algorithm with the multiscale tip asymptotics embedded in the tip element, which is used to compute the transient solution that connects the small- and large-time asymptotes.     

Crack tip behavior in near-surface fluid-driven fracture experiments

This Note presents experimental results for the near-tip fracture opening of fluid-driven fractures. The effect of fluid viscosity, quantified by a dimensionless parameter, was varied between tests. The tip region closely followed the classical square-root behavior from linear elastic fracture mechanics when the viscosity parameter was small. Conversely, when the viscosity parameter was of order one and the lag between the fluid-filled region and the fracture front accounted for less than 30% of the fracture, the tip region behaved according to a known intermediate asymptotic solution which results from the solid/fluid coupling. To cite this article: A.P. Bunger et al., C. R. Mecanique 333 (2005).     

Instability regimes and self-excited vibrations in deep drilling systems

This paper analyzes the stability of the discrete model proposed by Richard et al. (2004 [1], 2007 [2]) to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω₀ of the drillstring imposed at the rig. On the one hand, if Ω₀ is larger than a critical velocity Ω_c, the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω₀ is smaller than Ω_c, the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω_c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.