This paper describes the application of a directional drilling model to wellbore spiraling and compares it to field data. In this paper, the spiraling tendency of a Bottom Hole Assembly (BHA) is determined from a stability analysis of the delay differential equations that govern the propagation of a borehole. These propagation equations are derived from a novel mathematical model, constructed by combining (1) a bit/rock interaction law, which relates the forces acting on the bit to the penetrations of the bit in the rock per revolution; (2) kinematical relationships, which link the bit motion to the local borehole geometry; and (3) a model for the BHA, which expresses the forces and moments at the bit from the external loads and the deflections imposed by the stabilizers. Spatial delays associated with the positions of the stabilizers account for the feedback of the borehole as the stabilizers interact with the wellbore. The analytical form of the propagation equations makes it possible to carry out a stability analysis and determine whether borehole spiraling is expected. The coefficients of the propagation equations embody the characteristics of a particular drilling system; these include the BHA configuration, bit properties, and the active weight, Wa, a calculated reduced weight on bit that depends on the actual Weight On Bit (WOB), the cutter wearflats, and the rock strength. If the bit trajectory is unstable, then any perturbation in the borehole geometry is amplified gradually, eventually leading to the generation of a spiraled hole. The stability of the bit trajectory essentially is controlled by the magnitude of a dimensionless group relative to a critical value that depends on the BHA configuration. This group is a function of the lateral steering resistance of the bit, the bit wear, the rock strength, and the WOB. Thus, a BHA can be either stable or unstable depending on the selected bit, its state of wear, and the WOB. Predictions of the stability analysis are compared with field data from spiral holes pertaining to eight sections from four wells drilled with different bit types and BHA configurations. The paper shows that the propensity of a BHA to spiral can be predicted by the model by assuming reasonable values for parameters such as the lateral steering resistance and the part of the WOB transmitted by the cutter wearflats. This comparison suggests that the model can be used to optimize BHA designs and critical WOB levels that will mitigate the creation of spiral holes.
This paper describes the application of a directional-drilling model to the phenomenon of wellbore spiraling and compares its predictions with field data. The spiraling tendency of a bottomhole assembly (BHA) is determined from a stability analysis of the delay differential equations (DDEs) that govern the propagation of a borehole. These propagation equations are derived from a novel mathematical model, constructed by combining a bit/rock-interaction law, which relates the force and moment acting on the bit to its penetrations per revolution through the rock; kinematic relationships, which link the bit motion to the local borehole geometry; and a model for the BHA, which expresses the force and moment at the bit as a function of the external loads and the deflection imposed by the stabilizers. Spatial delays, associated with the positions of the stabilizers, account for the feedback of the borehole geometry as the stabilizers interact with the wellbore.The analytical form of the propagation equations makes it possible to perform a stability analysis and determine whether borehole spiraling is expected. The coefficients of the propagation equations embody the characteristics of a particular drilling system; these include the BHA configuration, bit properties, and the active weight W a , a reduced downhole weight on bit (WOB) that depends on the actual downhole WOB, the state of wear of the bit, and the rock strength. If the bit trajectory is unstable, then any perturbation in the borehole geometry is amplified gradually, eventually leading to the generation of a spiraled hole.The stability of the bit trajectory essentially is controlled by the magnitude of a dimensionless group, a function of the lateralsteering resistance of the bit, the active weight, and properties of the BHA, relative to a critical value that depends only on the BHA configuration.Predictions of the stability analysis are compared with field data from spiral holes pertaining to eight sections from four wells drilled with different bit types and BHA configurations. The paper shows that the propensity of a BHA to spiral can be estimated by the model by assuming reasonable values for parameters such as the lateral-steering resistance and the part of the WOB transmitted by the cutter wear flats. This ability means that the model can be used to optimize BHA designs and determine critical WOB levels, both of which will mitigate the creation of spiraled holes.
Summary The fundamental solution of a continuous line source, injecting fluid at a constant rate over the thickness of a poroelastic reservoir bounded by infinite impermeable elastic layers, is derived in this paper. This idealized problem has applications in hydrogeology and in petroleum engineering, as it can be used to assess the mechanical perturbations caused by injection or withdrawal of fluid in the subsurface through a vertical well. Construction of the solution takes advantage of the uncoupling of the pore pressure field, which, in this particular case, is given by the classical singular solution of the diffusion equation for an infinite line source. The mechanical fields then are determined by solving an elasticity‐like problem with a body force field proportional to the time‐dependent pore pressure gradient. On account of the problem symmetries, the Navier equations of elasticity reduce to two uncoupled partial differential equations for the radial and vertical (axial) displacement components, which are solved by a twofold application of Fourier and Hankel transforms. The solution exhibits different regimes at small, intermediate, and large times. When the diffusion radius, proportional to the square root of time, is smaller than or comparable to the thickness of the permeable layer, the induced deformation is confined to a region with a characteristic dimension of the same order as the diffusion radius. At large time, when the diffusion radius is large compared with the permeable layer thickness, the deformation rate in the reservoir is essentially oedometric (uniaxial). The different regimes of solutions are justified with a conceptual model based on identifying the evolving characteristics of complementary interior and exterior domain problems. The derived solution can serve as a valuable benchmark for coupled reservoir simulators. It also provides insights in to such problems as waterflooding, shearing at reservoir/cap rock interfaces, and stress redistribution around producing wells. Copyright © 2015 John Wiley & Sons, Ltd.
The paper investigates the influence of the design of push-the-bit rotary-steerable systems (RSSs) on the tendency to drill spiraled boreholes by analyzing the directional stability of the bit trajectory. In this model, differences in RSS designs are accounted for conceptually by assigning a lateral stiffness to the RSS pads. This simple device, which introduces a dependence of the force on the pads upon the deflection of the bottomhole assembly (BHA) relative to the borehole axis, enables exploration of the influence of the actuation mechanism, with the RSS behaving at the ends of the spectrum either as a soft or as a stiff node of the drilling structure.According to this analysis, low pad stiffness has little consequence on the general behavior of the system. However, as the pad stiffness increases, any perturbation in the borehole geometry sensed by the pads alters the drilling direction of the bit and triggers, under certain conditions, self-excited oscillations in the borehole geometry. By increasing the transverse rigidity of the BHA in the vicinity of the bit, stiff RSS pads thus enhance the propensity of a drilling structure to drill spiraled holes and generate, if the system is directionally unstable, borehole oscillations with pitch that corresponds to the distance between the bit and the pads. In contrast, a directionally unstable BHA equipped with RSS characterized by a low stiffness produces spiraled holes with a wavelength corresponding to the distance between the bit and the first stabilizer.
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