No abstract

This paper extends the approach proposed by to analyze the axial and torsional vibrations of drilling systems that are excited by the particular boundary conditions at the drag bit-rock interface, by basing the formulation of the model on a continuum representation of the drillstring rather than on a characterization of the drilling structure by a two degree of freedom system. These boundary conditions account for both cutting and frictional contact at the interface. The cutting process combined with the quasi-helicoidal motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring can be detected. In particular, stickslip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters.

Abstract. This paper studies the dynamical response of a rotary drilling system with a drag bit, using a lumped parameter model that takes into consideration the axial and torsional vibration modes of the bit. These vibrations are coupled through a bit-rock interaction law. At the bit-rock interface, the cutting process introduces a state-dependent delay, while the frictional process is responsible for discontinuous right-hand sides in the equations governing the motion of the bit. This complex system is characterized by a fast axial dynamics compared to the slow torsional dynamics. A dimensionless formulation exhibits a large parameter in the axial equation, enabling a two-time-scales analysis that uses a combination of averaging methods and a singular perturbation approach. An approximate model of the decoupled axial dynamics permits us to derive a pseudoanalytical expression of the solution of the axial equation. Its averaged behavior influences the slow torsional dynamics by generating an apparent velocity weakening friction law that has been proposed empirically in earlier work. The analytical expression of the solution of the axial dynamics is used to derive an approximate analytical expression of the velocity weakening friction law related to the physical parameters of the system. This expression can be used to provide recommendations on the operating parameters and the drillstring or the bit design in order to reduce the amplitude of the torsional vibrations. Moreover, it is an appropriate candidate model to replace empirical friction laws encountered in torsional models used for control.

This Note studies the self-excited stick-slip oscillations of a rotary drilling system with a drag bit, using a discrete model which takes into consideration the axial and torsional vibration modes of the bit. Coupling between these two vibration modes takes place through a bit-rock interaction law which accounts for both frictional contact and cutting processes at the bit-rock interface. The cutting process introduces a delay in the equations of motion which is ultimately responsible for the existence of self-excited vibrations, exhibiting stick-slip oscillations under certain conditions. To cite this article: T.

Summary Knowledge of rock properties is essential to predict and optimize the performance of oil and gas reservoirs by means of the reduction of the uncertainty pertaining to standard subsurface issues such as the mechanical integrity of the borehole (Tiab and Donaldson 1996; Moos et al. 2003), the risk of sanding (Tronvoll et al. 2004), and the geometry and efficiency of hydraulic fractures. These properties are evaluated by combining different field-measurement techniques (wireline logs, results of well tests, seismic surveys) and laboratory-test results (Archie 1942, 1950; Serra 1986; Bassiouni 1994). When cores are available, empirical models are built from correlations derived between well logs and laboratory measurements to estimate rock properties in noncored wells. The validity of these empirical models is often limited to specific litho-facies (see reviews by Chang et al. 2006; Blasingame 2008; Khaksar et al. 2009), which makes the identification of lithofacies a necessity before applying the model for predictions in uncored wells (Massonnat 1999). Because of the heterogeneity of rocks (Haldorsen 1996), with characteristic length scales commonly smaller than the resolution of wireline logs or even the core-plug size, the robustness of correlations is determined by how plug samples capture the dispersion in rock properties over the lithofacies under consideration. The correlation between a very localized core-plug measurement and a low-resolution wireline log with inherent low-pass filtering properties raises issues related to the upscaling of a property from one length scale (few centimeters for core plugs) to another (up to 1 m for wireline log). As an illustration, consider the high-resolution, continuous profile X, where the variations of the measured property are quantified for length scales smaller than typical plug sizes. We filter this data to produce the profiles X5 and X50 (the subscript stands for the length scale in centimeters at which the signal is averaged out) with lower spatial resolutions similar to the plug and the well-log resolutions, respectively (Fig. 1). The resulting crossplot, shown in Fig. 2, of X5 vs. X50 exhibits a cloud of points in which the dispersion is governed by the properties of the signal (the degree of heterogeneity or the frequency content) and the difference between the two resolution length scales. Two linear-fit optimizations were carried out with the low-resolution-data X50 and the high-resolution-data X5 as the dependent variables, respectively. It is interesting to note that these linear fits yield different results, with a slope of 0.96 in the first case and 0.69 in the second case. This is a mathematical artifact caused by the minimization process inherent in the search for the best linear fit, which is most commonly a minimization of the vertical distance between the representative data points and the best-fit line. On the basis of this result, it should always be advisable to select the high-resolution data (plug) as the dependent variable. Discrete sampling (i.e., plugging) and the dispersion caused by the difference in resolution scales of two measurements are two important root causes of the errors often seen in correlations between two variables. The examples shown in Fig. 2 illustrate how the correlations derived from several sampling schemes can deviate from the expected one-to-one relation between the two variables. To circumvent these issues, petrophysicists usually select large quantities of plugs to build representative statistical data sets, with the hope that they are large enough to attenuate the effects listed previously. However, extensive plugging strategies imply longer lead times and higher costs, and are therefore not always viable (e.g., in the cases of rock-mechanics testing or special-core-analysis programs). As an illustration, consider the modeling of the variations of rock strength, one of the key geomechanical properties along a well trajectory. Such an exercise relies heavily on correlations derived between well logs and laboratory tests (uniaxial or triaxial compressive tests), because there is no wireline log providing a direct measure of a mechanical property related to strength. In their comprehensive review of existing literature, Khaksar et al. (2009) listed approximately 40 models designed to derive strength properties from wireline logs. The authors showed that the relevance of these as empirical is limited to specific rock types. A broader application of these models would require the considerations of additional complexity such as the coexistence of several facies within the same data set or the impact of diagenesis on petrophysical variability within one facies. The elements of reflection introduced previously all suggest that a continuous measure of a physical property such as the strength profiles generated from the scratch test, which provides some useful elements for the mapping of rock heterogeneity, could partially fill the gap between measurements on plugs and well logs and help with the optimization of the selection of plug samples. In the main sections of this paper, we first describe briefly the scratch test and outline the key intrinsic benefits of the test. We then discuss how standard and special core analysis could benefit most from all the features of the scratch test when introduced at a very early stage of the work flows. In particular, we illustrate with some examples how rock-strength profiles averaged to the relevant length scale can be correlated with other petrophysical properties either measured on core plugs or inferred from well logs.