Prediction of Wellbore Trajectory Considering Bottomhole Assembly and Drill-Bit Dynamics
- J.D. Brakel (U. of Tulsa) | J.J. Azar (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling Engineering
- Publication Date
- June 1989
- Document Type
- Journal Paper
- 109 - 118
- 1989. Society of Petroleum Engineers
- 1.2.3 Rock properties, 1.5 Drill Bits, 1.6.6 Directional Drilling, 5.3.4 Integration of geomechanics in models, 1.4.1 BHA Design, 1.6.1 Drilling Operation Management, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 1.6 Drilling Operations
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This paper presents a numerical dynamic model to predict wellbore trajectory in both the vertical and horizontal planes. The model is three-dimensional (3D) and consists of the idealization of the bottomhole assembly (BHA) into a number of finite elements. Inertia properties are included to predict the transient dynamic behavior of the assembly during drilling. To ensure the correct boundary condition at the bit/rock interface, two rock/bit interaction models, one for a roller-cone bit and one for a polycrystalline-diamond-compact (PDC) bit, are developed. The time average of the rock/bit interaction force components is used to obtain an indication about the directional tendencies of the BHA. Examples of the rock/bit interaction forces for roller-cone and PDC bits and examples of the motion of a holding assembly for both bit types are given. The effects of hole inclination, weight on bit (WOB), rotary speed, and bit/stabilizer clearance are presented for an example of a building, holding, and dropping assembly. The deviation tendencies, as predicted by the model for several building and holding assemblies, are compared with field case data.
Directional drilling has been defined as the technology of deviating a wellbore along a planned course to a subsurface target. The drilling direction generally will be determined by the BHA (including the bit), the operating conditions, and the formation characteristics. BHA design and operation traditionally has been based on experience. Over the last decade, several computer programs were developed to allow a more scientific approach to deviation control. The majority of these models are static and yield only a prediction of the inclination response of a BHA. As Millheim and Apostal 11 point out, however, rotation of the drillstring should be considered to be able to predict both inclination and azimuth tendencies of an assembly accurately.
In most of the models currently available for analyzing BHA behavior, the effects of interaction between the bit and the formation are neglected. The bit is generally assumed to be centered in the hole and to act as a hinge-i.e, no moment exists between the bit and the formation. These shortcomings were recognized by Sandia Natl. Laboratories.
This paper describes a 3D model based on the finite-element method to simulate the transient dynamic behavior of a BHA during drilling. The effects of rock/bit interaction for roller-cone and PDC bits are included. The model can be used to study the directional tendencies of a BHA in both the vertical (inclination) and horizontal (azimuth) planes. Geologic effects are not included in the model.
The model consists of the idealization of the BHA into a number of finite elements. The rock/bit interaction models yield the boundary condition at the bit/rock interface.
BHA Model. To obtain a finite-element representation of the BHA, 3D-beam elements are used. The force-deflection equations for the beam element were derived with the theory of linearized elastic stability of beam columns. 13 To describe the transient dynamic behavior of the assembly during drilling, the following equation of motion is used:
Eq. 1 is solved with the Wilson-0 14 numerical integration method and a Gaussian elimination algorithm.
A special technique was developed to consider the constraint imposed by the wellbore wall. For each time increment, after calculation of the nodal displacements, a check is performed to determine whether all nodal points are still confined within the wellbore. If so, the calculation will continue for the next time increment. If not, the calculation for the time increment under consideration will be redone, whereby a force is applied to the nodes with radial displacements exceeding the corresponding radial clearances. The magnitude of this contact force is determined through an iteration process using the Newton-Raphson method such that the radial velocity of the considered node is reduced to zero. The final contact forces are used to calculate friction forces by means of a Coulomb friction coefficient. Friction effects occur when the rotating BHA contacts the wellbore wall.
Both extensional and torsional springs are connected to the top node of the BHA model to simulate the elastic response of the upstring part of the drillstring. At the bit node (the lowest node of the model) the rock/bit interaction forces are applied as shown in Fig. 1.
Rock/Bit Interaction Models. Two rock/bit interaction models, one for a PDC bit and one for a conventional roller-cone bit, were developed. Both yield a force input at the bit node of the BHA model.
The actual cutting action of a PDC bit is performed by the individual PDC cutters. The forces acting on a single cutter during the cutting process depend on rock properties, cutter geometry, and cutting depth. The following expression presented by Cheatham and Daniels was used to quantify the resultant cutting force, FR:
The area of cut, A,, is a function of cutting depth and of cutter placement on the bit body. It is calculated with a routine developed by Sandia Natl. Laboratories. 16 Once the forces acting on a single cutter are known, they are transformed to forces and moments about a point on the bit centerline (Fig. 2). This point will coincide with the bit node in the final BHA model. The contribution of all cutters yields the resultant rock/bit interaction forces at) out the bit node.
The forces acting on a single PDC cutter during the cutting process depend on the cutting depth-i.e., the position of the cutter with respect to the wellbore. Because the bit is assumed to be rigid, the cutter position can be related to the position of the bit node with respect to the wellbore. Consequently, the resultant rock/bit interaction forces for a PDC bit, F rb, are a function of the bit-node displacement
The dependence of the rock/bit interaction forces on bit-node displacement requires an iteration process to solve for the nodal dis placements of the BHA model.
The rock/bit interaction model for the roller-cone bit is similar to that of the PDC bit. For a roller-cone bit, the cutting action is performed by the individual teeth. To quantify the force acting on a tooth during penetration into the rock, a model presented by McLamore was used. As before, the penetration force acting on a tooth is a function of penetration depth. The penetration force is assumed to act in a direction parallel to the tooth centerline.
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