Modeling of Cutting Rock: From PDC Cutter to PDC Bit—Modeling of PDC Bit

Summary In this paper, we integrated our polycrystalline diamond compact (PDC) cutter model (Chen et al. 2021) into a PDC bit model that can predict the weight on bit (WOB), torque on bit (TOB), and imbalanced side force on a bit under given drilling conditions. We first proposed a method to determine the actual cutting plane and depth of cut of each cutter on a PDC bit. Once the two parameters for each cutter are determined, the cutter model can then be applied to calculate the cutting force of each cutter. The final bit force and moment (i.e., WOB, TOB, and imbalanced side force) are calculated as the resultant force and moment of cutting forces of all cutters. The PDC bit model in this paper considers all bit design parameters, including bit matrix geometry, blade profile, cutter layout, and the inclination of each cutter. Furthermore, the bit model also considers some bottomhole assembly (BHA) parameters (e.g., bit tilt angle, location of first fulcrum point, and tool face/steering plane angle), which allows the bit model to simulate a bit under different drilling modes. The bit model is also validated by published test data and field applications. Finally, case studies are conducted, and the influence of bottomhole stresses, BHA parameters, and drilling modes on bit force and moment are discussed. A field application of the bit model is also provided. The bit model can be directly used for PDC bit design and simulation. In fact, this paper presents a general way to integrate a cutter model into a PDC bit model. Readers are also encouraged to apply this method to integrate their own cutter model into a PDC bit model.

Stochastic modeling for hysteretic bit–rock interaction of a drill string under torsional vibrations

This paper aims at constructing a stochastic model for the hysteretic behavior of the nonlinear bit–rock interaction of a drill string under torsional vibrations. The proposed model takes into account the fluctuations of the stick–slip oscillations observed during the drilling process. These fluctuations are modeled by introducing a stochastic process associated with the variations of the torque on bit, which is a function of the bit speed. The parameters of the stochastic model are calibrated with field data. The response of the proposed stochastic model, considering the random bit–rock interaction, is analyzed, and statistics related to the stability of the drill string are estimated.

A New Dynamic Model of Coupled Axial–Torsional Vibration of a Drill String for Investigation on the Length Increment Effect on Stick–Slip Instability

In this paper, a new model is proposed to study the coupled axial–torsional vibration of the drill string. It is assumed that rotary table angular speed is constant and equals to the nominal angular speed of the drill string. In addition, axial displacement of any point on the drill string is considered to be as the sum of rigid-body motion and elastic vibrations. The depth of cut is defined using instantaneous dynamic states instead of using the delayed model as presented in previous researches. A velocity-weakening function is introduced for modeling the behavior of the frictional component of the torque-on-bit (TOB) with respect to the bit angular speed. After discretizing vibration equations, stability analysis of the system is investigated by linearizing the nonlinear system around its steady-state response point. Considering nominal weight-on-bit (WOB) (W0) and nominal rotational speed (Ω) as the input parameters of the drilling, variation of maximum allowable value of (W0) is presented with respect to variation of Ω . It is shown that the maximum allowable value of W0 has an increasing–decreasing behavior with respect to Ω. The effect of drill string upper and lower part lengths is studied on the stability of the system, and practical results are presented both in the condition that W0 is constant and in the condition that the hook upward force is constant. It is shown that by increasing the drill string length, the system is more exposed to instability, and this must be considered in regulating the input parameters of drilling.

Estimation of Weight and Torque on Bit: Assessment of Uncertainties, Correction and Calibration Methods

The efficiency of a drilling operation is to a great extent governed by how well one is able to optimize the rate of penetration (ROP) throughout each stage of the operation. ROP optimization normally involves balancing drilling speed on the one hand with acceptable wear to the drill bit on the other. The bit lifetime is largely determined by the mechanical conditions at the bit-formation rock interface and the weight on bit (WOB) and torque on bit (TOB) provide important information related to the working condition of the bit. The accuracy of WOB and TOB measurements can thus become a determining factor for the overall drilling efficiency. Due to the low bandwidth of downhole mud pulse-based telemetry systems, the WOB and the TOB are generally derived from surface measurements, i.e. from the hook load and the top-drive torque. Field experience indicates that a WOB derived from surface measurements can be of limited accuracy, such as when surface measurements suggest a negative WOB even though the ROP is positive, or when high sampling rate and high precision downhole measurements confirm a large discrepancy between the memory recorded downhole data and the estimated values based on the measurements made at the level of the hoisting equipment and the top-drive. The reason for these inconsistencies is simply that there are numerous physical processes taking place between the bit and the surface measurements that are normally not accounted for when WOB and TOB are estimated. This paper reviews and analyses the sources of these deviations and models the physical processes in order to quantify the precision for which the WOB and the TOB can be ascertained using solely surface measurements. Methods are also proposed that compensate for certain side effects by utilizing real-time torque and drag and hydraulic calculations.

Drilling Action of Roller-Cone Bits: Modeling and Experimental Validation

This paper presents a new model of the drilling response of roller-cone bits. First, a set of relations between the weight-on-bit W, the torque-on-bit T, the rate of penetration V, and the angular velocity Ω is established in the spirit of the model developed for polycrystalline diamond compact (PDC) bits. In contrast to models that depend on a precise description of the bit, the drilling response is investigated by lumping the effect of the bit geometry into a few parameters and on averaging the drilling quantities (W,T,V,Ω) over at least one revolution of the bit. Within the framework of the model, quantitative information from drilling data related to rock properties, bit conditions, and drilling efficiency can be extracted. Finally, a series of laboratory tests at atmospheric pressure conducted with an in-house designed drilling rig, together with published experimental data, is used to evaluate the proposed model. The good match between the experimental results and the theoretical predictions are promising in regard to the potential use of this model to investigate the drilling response of roller-cone bits.

Stick-Slip and Bit-Bounce of Deep-Hole Drillstrings

A mathematical model for the drilling process is derived and solved numerically as an initial value problem. The equations of motion are nonlinear differential equations for longitudinal, lateral, and rotational motion of the pipe as well as for the rate of flow and pressure of the mud. The model comprises a mud (Moineau) motor which rotates the bit relative to the lower end of the pipe. The model accounts for buckling of the pipe due to excessive torque and longitudinal forces, as well as for the effect of hydraulic pressure on the deformed pipe. Weight on bit and torque on bit are computed from characteristic curves which are functions of the penetration of the bit into the rock and the angular velocity of the bit. Numerical simulations show self-excited oscillations of the drillstring, including bit take-off from the bottom hole and large amplitudes in the bit’s angular velocity. [S0195-0738(00)00602-6]