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This work presents the mathematical method to design complex trajectories for three-dimensional (3D) wells using spline in tension as coordinate functions. 3D spline-intension trajectories are obtained for various end conditions: free end, set end, free inclination/set azimuth, and set inclinationlfree azimuth. The resulting trajectories are smooth...
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... unit principal normal vector n is defined by n = K = n V ,n N ,n E In terms of components, the various elements in Eq. The tool-face angle is the angle that the principal normal to the trajectory makes with the intersection of the vertical plane containing the tangent to the trajectory and the plane perpendicular to this tangent, as shown in Fig. 11. This intersection defines an unit vector h, called the high side vector, given by ...
Citations
... Indeed, the trajectory cannot follow exactly the succession of waypoints through segment lines due to physical constraints: the drone cannot turn directly and the motor control implies some uncertainties in the trajectory. To simulate a realistic trajectory, different trajectory models can be used as for instance splines [12]. For testing purpose, a simple trajectory may be computed as set of consecutive segments and semi-circles (cf. Figure 1). ...
One key objective of Cyber-Physical System (CPS) simulation is to evaluate different CPS configurations regarding a certain user objective. First, simulation of CPS necessitates frameworks to handle heterogeneity of CPS components (the software and hardware system control, the behavior of the CPS itself and its physical environment). Then, to build simulators, designers use paradigms like FMI (Functional Mock-Up Interface) that proposes a data-driven generic interface facilitating the integration of heterogeneous models. However, in order to facilitate simulation configuration, an approach is required to drive modeling of parametric features and operational conditions. In this paper, we present CARES, a component-based and model-driven approach to facilitate CPS simulation. CARES is applied to evaluate an Autonomous Underwater Vehicle (AUV) navigation function by simulation. The proposed models integrate both the principles of a generic simulation (integration of Component Based Software Engineering CBSE concepts and FMI paradigm) and domain specific aspects through a component-based architecture style. From a design model, a code generator builds the structural (Java or C++) code of the simulator. The generated code relies on a given run-time library for its execution and its structure facilitates integration of domain-specific code. The experiments show the effectiveness of the approach to build simulators for evaluation of different AUV configurations.
... The model is completely described by Sampaio(2007). The three coordinates of the trajectories are described by hyperbolic functions of the form y(u)= A+B u +C sinh (λ u )+ D cosh (λ u ) ...
... The model is completely described by Sampaio(2017). The trajectory 3D function is described by the 3D cubic function ...
The process of designing a 3D wellbore trajectory, in general, takes into consideration only the expected initial and final coordinates, some “drillable” curvatures that can be delivered by the current technologies, and any of the various 3D planning models, each with a particular number of defining parameters. Some of the questions usually disregarded are: which model to take to design the trajectory, which values to take for the defining parameters, what to measure in the process of design, and what should be optimized. The objective of this work is to provide a systematic approach to 3D trajectory design based on torque and drag performance. For this purpose, it is unquestionable that the torque and the drag caused by the trajectory curvature, which are, considering all other variables the same, determined by the trajectory model and its parameters, dominate the discussion. Other things like trajectory length, borehole diameter, tubular good geometry, although influential, are irrelevant in the decision process because if they affect one model, they affect all others. Therefore, the use of an efficient, accurate, and general T&D model is of fundamental importance, and then we are left with the duty of measuring the adequate cost (or loss, or objective) function and optimizing this function. In this endeavor, a fast and accurate 3D stiff analytical T&D solution that allows analyzing a large number and wide range of parameters is of fundamental importance. Armed with such tool, and with the guidelines resulting from this work, the well designer can quickly determine the best trajectory and parameters that optimize the borehole construction and yet reaching the fundamental purpose of the well to be designed. Instead of requiring an extensive experience and/or creative (non-replicating) capacity of the well designer, the process delivers a systematic approach to trajectory design, based on the relevant objective parameters (e. g., minimum T&D, minimum equipment wear, reduced casing, cementing, hole cleaning and pipe sticking difficulties, and so far). To reach this goal a reasonable, but not thorough, understanding of the causes and effects of torque and drag is necessary in order to effectively play with the trajectory parameters. The causes and consequences of wellbore tortuosity is particularly discussed. In the process, several types of trajectory common in the industry are used and compared. To effectively compare the various models, it is assumed that all trajectories (based on the same initial and final conditions) have the same length (measured depth). This is not a necessary condition because different trajectory construction may require different amount of curvature control, which affects its cost. The T&D model itself is not covered because it has been discussed in another publication; however, a brief discussion is presented in the Annex 2. Any appropriate model serves the purpose although the requirement of being fast, accurate, 3D, and using a stiff model is necessary. Flexibility and efficiency on how the data are entered and changed are also important to a successful, efficient analysis.
... These are irremediably determined by the end conditions. A solution for this limitation was obtained by Sampaio [4]. The cubic functions were replaced by spline-in-tension functions, which are functions defined in terms of hyperbolic functions, and have the characteristic of providing a parameter called tension. ...
... The proposed model for 3D well trajectory in this work makes use of both second-and third-order Bézier curves as defined above. 4 To obtain a 3D well trajectory, independently of the model used, several data are absolutely necessary, namely: ...
... In general, the control points are not collinear with the starting and the ending points. Figure A.4 depicts the concept in discussion. Three intermediate points, I 1 , I 2 , and I 3 , are first--order Bézier points of the segments SC S , C S C E , and C E E respectively, and the intermediate points I 4 and I 5 are first-order Bézier points of the segments I 1 I 2 and I 2 I 3 . We also recognize I 4 and I 5 as second--order Bézier points of SC S C E and C S C E E, respectively. ...
A relatively simple, general, and very flexible method to design complex, three-dimensional hole trajectories can be obtained by using a 3D extension for Bezier curves. This approach offers superior results in terms of coding, use, and flexibility compared to other methods using double-arc, cubic functions, spline-in-tension functions, or constant curvature. The mathematics is surprisingly simple, and the method can be used to obtain trajectories for any of the four typical end conditions in terms of inclination and azimuth, namely: free-end, set-end, set-inclination/free azimuth, and free-inclination/set-azimuth. The resulting trajectories are smooth, with continuous and smooth change of curvature and toolface, better exploiting the expected delivery of modern rotary steerable deviation tools, particularly the point-the-bit and the push-the-bit systems. With the relevant parameters at any point of the trajectory (curvature and toolface angle) an automated
system can steer the hole toward the defined targets in a smooth fashion. The beauty of the method is that the description of the trajectory is obtained with one single expression that handles the three space coordinates, instead of working with three separate coordinate functions. It uses a generalization of the well known 2D Bezier curve. The concept is easy to understand, and implementation even using spreadsheets is straightforward. Besides, the conditions at both ends (coordinates and inclination/azimuth for set ends)
the trajectory curve has up to two independent parameters. By playing suitably with these parameters, one can obtain a curve that favors the reduction of drag and torque during drilling, tripping, and casing running. In addition to the formulation for trajectory calculation, the paper presents the expressions to calculate the inclination, azimuth, curvature, and toolface at any point along the trajectory. Proper numerical examples illustrate the various end-conditions. The method can be used during the hole planning cycle as well as during the hole drilling for automatic and manual steerage.
... The limitation is due to the use of the cubic spline interpolation method; depending on the degree of spacing between points there will be a degree of curvature added to the solution. This can be remedied using a different interpolation method such as spline in tension (Sampaio 2007). Further evidence of the higher Wellbore Profile Energy can also be seen in the top view near the starting point and the vertical section near the target; these issues can be remedied with a second pass via adjustment or applying a linearity pattern constraint in these segments (Stojmenović et al. 2008 ...
A key challenge in developing brown fields is identifying a strategy that enables placement of horizontal wells in a field riddled with existing, depleted wells. These wells have drained multiple reservoirs in proximity to current target intervals, resulting in altered in-situ pressures that may impose additional technical and economic drilling risks. This work presents a new technique which optimizes well path design by combining hybrid nature-based metaheuristics with spline curvature to navigate around depleted zones. The proposed method is validated by testing on synthetic and actual well cases.
The nature-based metaheuristic method employed is a modified firefly algorithm with hybrid implementations of mutation and annealing. It considers a potential well's starting coordinates, target coordinates, possible obstructions, subsurface stress distribution, an RSS tool's dogleg limitation range, kick off depth limitation, and required length of lateral section to optimize overall wellbore length, all of which can directly be linked to the economics behind drilling a well.
The functionality of the designed algorithm is examined with both synthetic data and publically available field data. Further complexity is added in the model by including geomechanical stresses in the model when available. Comparisons of overall wellbore length, wellbore orientation, and wellbore profile energy are also provided for a case in the Wattenberg basin derived from public data. Using sparse information, the algorithm was able to automatically design entire well paths in a relatively short period for all cases and the final solutions resembled industry solutions based on minimum design constraint. The uniqueness of the work is highlighted by the algorithm's ability to converge towards optimal solutions which can help the operator shift work load from well design to more critical tasks.
... Сплайнові методи розрахунку траєкторії [4,5] використовують значення в декількох точках вимірювань (від чотирьох). В точках спряження ділянок забезпечується існування похідних, кривина таких ліній мінімальна. ...
... These are irremediably determined by the end conditions. A solution for this limitation was obtained by Sampaio [4]. The cubic functions were replaced by spline-in-tension functions, which are functions defined in terms of hyperbolic functions, and have the characteristic of providing a parameter called tension. ...
... Skew lines are lines that have no intersections but are not parallel.4 This paper uses a common coordinate system typical in directional drilling in which horizontal axes are parallel to the N/S and E/W directions, and the third axis is the vertical direction (positive downwards). ...
A relatively simple, general, and very flexible method to design complex, three-dimensional hole trajectories
can be obtained by using a 3D extension for Bézier curves. This approach offers superior results in terms of coding, use, and flexibility compared to other methods using double-arc, cubic functions, spline-in-tension functions, or constant curvature.
The mathematics is surprisingly simple and the method can be used to obtain trajectories for any of the four
typical end conditions in terms of inclination and azimuth, namely: free-end, set-end, set-inclination/free azimuth, and free-inclination/set-azimuth.
The resulting trajectories are smooth, with continuous and smooth change of curvature and toolface, better
exploiting the expected delivery of modern rotary steerable deviation tools, particularly the point-the-bit and the
push-the-bit systems. With the relevant parameters at any point of the trajectory (curvature and toolface angle) an automated system can steer the hole towards the defined targets in a smooth fashion.
The beauty of the method is that the description of the trajectory is obtained with one single expression that
handles the three space coordinates, instead of working with three separate coordinate functions. It uses a generalization of the well-known 2D Bézier curve. The concept is easy to understand, and implementation even using spreadsheets is straightforward.
Besides the conditions at both ends (coordinates, and inclination/azimuth for set ends) the trajectory curve has
up to two independent parameters. By playing suitably with these parameters, one can obtain a curve that favors the reduction of drag and torque during drilling, tripping, and casing running.
In addition to the formulation for trajectory calculation the paper presents the expressions to calculate the inclination, azimuth, curvature, and toolface at any point along the trajectory. Proper numerical examples illustrate the various end-conditions. The method can be used during the hole planning cycle as well as during the hole drilling for automatic and manual steerage.
... The major drawback of search methods is that the time complexity typically increases significantly with the resolution of the search space discretisation and the incorporation of kinematic constraints. Spline methods [1], [11], [12] use geometric splines that satisfy the desired constraints to represent the resulting trajectory. Obstacle avoidance can be achieved by specifying intermediate goal locations. ...
Fluid motion planners are a type of artificial potential field (APF) motion planners that use the differential equations of fluid flow to determine the desired trajectory. The fluid flow approach in motion planning can efficiently produce natural-looking trajectories. However, the differential equations used in previous studies are restricted to motion planning in 2-D environments. In this paper, the fluid flow approach is extended to a motion planning framework for 3-D mobile robots that avoids spheroidal obstacles. Compared with existing APF approaches, kinematic constraints in both speed and curvature are also considered. Possessing the efficiency of 2-D fluid motion planners, the proposed approach is able to plan natural-looking reference trajectories for nonholonomic 3-D mobile robots. The approach is demonstrated through various 3-D example scenarios. The work can be considered as a fundamental framework for 3-D fluid motion planning, where additional kinematic constraints and more complex scenarios can be incorporated.
One primary objective of drone simulation is to evaluate diverse drone configurations and contexts aligned with specific user objectives. The initial challenge for simulator designers involves managing the heterogeneity of drone components, encompassing both software and hardware systems, as well as the drone’s behavior. To facilitate the integration of these diverse models, the Functional Mock-Up Interface (FMI) for Co-Simulation proposes a generic data-oriented interface. However, an additional challenge lies in simplifying the configuration of co-simulation, necessitating an approach to guide the modeling of parametric features and operational conditions such as failures or environment changes.
The paper addresses this challenge by introducing CARES, a Model-Driven Engineering (MDE) and component-based approach for designing drone simulators, integrating the Functional Mock-Up Interface (FMI) for Co-Simulation. The proposed models incorporate concepts from Component-Based Software Engineering (CBSE) and FMI. The NAVIDRO architectural style is presented for designing and configuring drone co-simulation. CARES utilizes a code generator to produce structural glue code (Java or C++), facilitating the integration of FMI-based domain-specific code. The approach is evaluated through the development of a simulator for navigation functions in an Autonomous Underwater Vehicle (AUV), demonstrating its effectiveness in assessing various AUV configurations and contexts.
Directional drilling is a complex operation that is difficult to execute with precision and repeatability. It is challenging to accurately model and predict the results of drilling actions, which leaves much to directional driller's experience. To reduce the impact of uncertainties on the drilling actions and to improve drilling efficiency, an automatic computational framework is proposed for directional drilling to optimize the drilling path and associated actions to drill this path. The proposed framework has low computational overhead and is suitable for real-time application. The framework consists of a nonlinear wellbore propagation model coupled with a genetic algorithm (GA). Using this framework, optimal drilling actions are found automatically with the intent of maximizing the value of the well. A new well path propagation model is developed that maintains an optimal trade off between modeling accuracy and computational overhead. For orientation, a quaternion formulation is used instead of the industry standard Euler angles to avoid singularity problems. The well path optimization problem is formulated by defining suitable cost functions including functions representing production loss, wellbore quality, drilling time, completion cost, and state and input constraints that represent the physical limitations of the directional drilling system. Using this approach, the solution space has trillions of candidate well paths, which makes an exhaustive search impractical for real-time directional drilling advisory and control. In order to address this issue, a GA methodology is applied. Solving the optimization problem generates sets of drilling instructions (e.g slide and rotate instructions for downhole motor drilling)that can be applied to drill practically optimal wells that maximize the value of the well. For most cases, a solution is found in under a minute. Even though the solution could be a local optimal since the optimization problem is non-convex, the optimization framework shows excellent practical results when applied to actual field cases.
The amount of uncertainty related to directional drilling makes it challenging to accurately model and predict the results of drilling actions, leaving much to human know-how and interpretation. Additionally, few path planning methods in the literature consider the directional steering tool being used which results in a loss of optimality when sliding and rotating instructions are fitted on a geometric optimal path. The formulation of the optimization problem varies greatly between rotary steerable systems (RSS) and mud-motor configurations. Additional cost functions and constraints are present for mud-motor use, which significantly increases the problem complexity. A slide drilling guidance system is proposed to combat this issue and to help automate directional drilling. The guidance system leverages three main modules. The first is a computationally efficient, non-linear wellbore propagation model. The second is a set of cost functions that aims to quantitatively represent the actual value of the well, representing production loss, drilling time, completion cost, and wellbore quality. The last module is a Genetic Algorithm (GA) solver that generates sets of optimal drilling instructions. The guidance system is built into a software package that utilizes an intuitive, easily-accessible Graphical User Interface (GUI) to be an effective advisory tool for the directional driller. The software is currently being implemented into the Real Time Drilling (RTD) system by an operator.
