Static and Dynamic Three-Dimensional Bottomhole Assembly Computer Models

Abstract

Summary. This paper presents two three-dimensional (3D) mathematical models to predict the directional behavior of bottomhole assemblies (BHA's). The first model describes BHA dynamic, stepwise, transient behavior. Displacements and lateral forces, computed for each step, account for friction against the borehole wall. The second model computes a static BHA equilibrium whereby simplified friction forces are assumed. The static lateral forces are found to be an average of the highly varying ones computed by the dynamic model, but the static computer run is much faster. The static model is therefore used iteratively to compute an oriented dogleg severity that balances lateral forces. This is used to derive the directional behavior of the BHA. Comparison and calibration against real field cases are performed. Introduction The ORPHEE 3D static and ORPHEE 3D dynamic programs are two BHA analysis finite-element computer programs developed by Elf Aquitaine under an Assn. de Recherche sur les Techniques d'Exploitation du Petrole (ARTEP) research project. Both models are a 3D extension to the two-dimensional (2D) program ORPHEE 2D, presented previously. A 2D model permits prediction of only the inclination directional behavior. A 3D model permits prediction of the azimuthal directional behavior and therefore offers the possibility of prediction in space (rather than in a vertical plane only) of the future trajectory of a BHA, which is one of the fundamental problems a driller faces. Such a model permits a better selection of the BHA components, and thus expensive trips that once were necessary to find the proper BHA components can be avoided. Since Lubinski's first work in the 1950's, it has been known that the directional behavior of a BHA can be computed qualitatively from the side force on the bit. In this paper we distinguish two components of the side force (see Fig. 1).1. The inclination side force is the force perpendicular to the bit axis and is included in the vertical plane that contains the bit axis. This enables a qualitative prediction of the build/drop tendency of a BHA.2. The azimuthal side force is the horizontal force perpendicular to the bit axis. This enables a qualitative prediction of the left/right tendency of a BHA. First, we present the dynamic model that simulates the displacement of the BHA while rotating in the borehole, taking into account the friction when in contact with the borehole wall. This model enables computing the side forces vs. time. Second, we present the static model that determines the static equilibrium position of the BHA in the borehole. This model incorporates a quasidynamic friction at the contact points and is thus capable of providing side forces approaching the average of the side forces from the dynamic model but with a noticeably shorter calculation time, which is of paramount importance for easy application on a rig. Because of this readiness, the static model is used in an iterative process to calculate the inclination- and azimuth-equilibrium curvatures of a BHA. These calculations permit quantitative prediction of the directional behavior, whereas side forces give only qualitative information. The parameter analysis of the static model that follows shows the prime influence of hole enlargement on inclination and azimuthal behavior and the friction-coefficient influence on azimuth only. In the case of a building or holding assembly, a concept of critical hole enlargement is developed that explains the difference in azimuthal behavior between turbodrilling and rotary drilling. Finally, the model adjustment, still in the process of finalization, is presented. This pinpoints a relation between hole enlargement and rate of penetration (ROP) that is valid for a given field. The friction-coefficient analysis is currently being carried out. For the studied field, this coefficient has proved to be constant at 0.2. ORPHEE 3D Dynamic ORPHEE 3D dynamic determines the 3D displacement of the BHA in the hole, as well as the side forces vs. time, using the following data as a basis: BHA components; well data, including inclination and azimuthal measurements and hole size; drilling parameters, including weight on bit (WOB), mud weight, and rotation speed; and friction coefficient between pipes and borehole wall. The model is based on the finite-element method and is a transient dynamic model that calculates the deformed shape of the BHA and the side forces in a timestep manner. (See the Appendix for greater detail about the mathematical section.)Figs. 2 through 4 show example results. The case in question is a 40-m [130-ft] slick assembly composed of 24.13-cm [9 1/2-in. drill collars in a rectilinear well of 44.45-cm [17 1/2-in.] diameter, inclined by 1.04 rad [60 degrees] to the vertical. The rotation speed is 180 rev/min and the friction coefficient is 0.2. Fig. 2 shows the displacement of the BHA axis at 5, 10, 15, and 20 m [16, 33, 49, and 66 ft] from the bit. Note that the circles have radii equal to the annular space, which is equal to the difference between borehole radius and outside drill collar radius. The annular space denotes the maximum lateral displacement of the BHA axis. The algorithm is initialized by assuming that the BHA is centered on the borehole axis at instant t = 0. Under the effect of its own weight, the BHA starts dropping toward the lower generatrix of the well. This displacement corresponds to the vertical line distinct on each of the diagrams. The bit is supposed to be centered on the wellbore axis. Consequently, the contact of the drill collars with the wall occurs on late a certain distance from the bit. At 5 m [16 ft], the BHA follows a sinuous and complex trajectory without coming in contact with the wall. At and above 10 m [33 ft], contact with the wall takes place. Dynamic Modeling of Contact. Macroscopically, one may consider two discrete forms in the contact phenomenon: a continuous contact of the pipe with the wall, over which rolling occurs, with or without slippage; and successive shocks of the pipe on the wall. To simplify modeling the contact phenomenon, we made three assumptions:contact is modeled solely by successive shocks of pipes on the wall,shocks are of zero duration, andcontinuous contact (rolling of pipe over the wall) is simulated by low-intensity shocks in succession. These assumptions accelerate the problem's solution because they avoid the necessity of modeling a shock during its entire duration, which would lead to extremely small timesteps. For modeling a shock, one reverses the speed perpendicular to the wall and modifies the tangential and axial rotation speeds as a function of the friction coefficient (see the Appendix). A bounce "with spin" is thus obtained, which permits modeling the build tendency of the drill collars on the right side of the hole under the effect of rotation. SPEDE P. 160^