Patent Publication Number: US-2023145859-A1

Title: Real-time well trajectory projection using stochastic processes

Description:
FIELD OF THE INVENTION 
     The present disclosure relates to a system and methods for projecting the trajectory of a drilling assembly in a subsurface formation, and more specifically to systems and methods for generating stochastic trajectory projections in real-time to predict the movement of a bottom hole assembly coupled to a drill string across a depth horizon and thereby improve control of the bottom hole assembly across the depth horizon. 
     BACKGROUND 
     Boreholes drilled into subsurface formations may enable recovery of desirable fluids, including, without limitation, hydrocarbons, using any number of different techniques. In drilling operations, typical drilling processes may be relatively complex and involve considerable expense. Many of these drilling operations may be done manually with experienced operators running the drilling platform. There are continual efforts to improve safety, improve fluid recovery, and lower costs associated with subsurface drilling and advancements in computerized and automated systems in drilling processes may support these efforts. 
     Model-based control methods are now widely utilized to control the trajectory of borehole placement during exploration of and extraction operations in subsurface formations. Due to the complexity and uncertainty in drilling operations, it is challenging to find effective models for control. High-fidelity models have been established in the past, but often cannot be used for real-time dynamic control of subsurface drilling operations as these high-fidelity models are generally high dimension and computationally expensive, thus cannot be used in real-time. Reduced physics-based models have also been developed. These reduced physics-based models are simpler and may provide more confidence for a short range that may be suitable for real-time control if they are updated frequently using the measurements from subsurface equipment. However, due to uncertainties in the bit-rock interactions, drilling parameter changes, sensor noise or malfunctions, downhole vibrations, and model/system discrepancies, reduced physics-based models with deterministic parameters may not be sufficient for real-time control of drilling operations. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the present embodiments and advantages thereof may be acquired by referring to the following description taken in conjunction with the accompanying drawings, in which like reference numbers indicate like features. 
         FIG.  1    depicts an exemplary drilling system. 
         FIG.  2    depicts an exemplary flow diagram illustrating a method for determining one or more stochastic trajectory projections and confidence regions for a bottom hole assembly based at least in part on one or more of system model parameters, steering inputs, bottom hole assembly initial conditions, working mode selection and settings, and a desired number of trajectory projections. 
         FIG.  3    depicts an exemplary well plan and stochastic trajectory confidence regions for a borehole across a depth horizon. 
         FIGS.  4 - 5    depict exemplary stochastic trajectory projections and confidence regions for a bottom hole assembly across a depth horizon.  FIGS.  6   a - b    depicts two exemplary stochastic trajectory projections and confidence regions for a bottom hole assembly across a depth horizon. 
         FIG.  7    depicts an exemplary flow diagram for generating stochastic trajectory projections for a bottom hole assembly. 
         FIG.  8    depicts a schematic diagram of an information handling system for use with or in a wellbore environment, according to one or more aspects of the present disclosure. 
     
    
    
     While embodiments of this disclosure have been depicted and described and are defined by reference to exemplary embodiments of the disclosure, such references do not imply a limitation on the disclosure and no such limitation is to be inferred. The subject matter disclosed is capable of considerable modification, alteration, and equivalents in form and function, as will occur to those skilled in the pertinent art and having the benefit of this disclosure. The depicted and described embodiments of this disclosure are examples only and are not exhaustive of the scope of the disclosure. 
     DETAILED DESCRIPTION 
     The present disclosure relates to a system and methods for projecting the trajectory of a drilling assembly in a subsurface formation, and more specifically to systems and methods for generating stochastic trajectory projections in real-time to predict the movement of a bottom hole assembly coupled to a drill string across a depth horizon and thereby improve control of the bottom hole assembly across the depth horizon. 
     The system and method disclosed herein uses a stochastic trajectory projection module. The stochastic trajectory projection module may use a plurality of inputs to generate a projected trajectory and confidence regions for a bottom hole assembly across a depth horizon in a subsurface formation. The stochastic trajectory projection module may use one or more stochastic models, including, without limitation, Monte Carlo simulation methods, to simulate and project the future trajectories. The stochastic trajectory projection module disclosed herein supports two modes, where selection of the mode and any corresponding settings may be dependent on any prior data analyses, including without limitation higher fidelity models, or knowledge of one or more of the subsurface formation and the equipment of the drilling system. 
     The stochastic trajectory projection module further enables real-time probabilistic projections for trajectories and corresponding confidence regions based on system model parameters, steering inputs, and bottom hole assembly initial conditions. 
     The system and method disclosed herein provide a unique way to project the borehole trajectories. This enables drilling personnel and steering control systems to plan ahead and improve steering decisions, resulting in improved well placement and, thereby, improving fluid recovery and lowering costs associated with subsurface drilling operations. As discussed herein, real-time trajectory projections enable feedback during drilling operations that enables an operator to refine the steering inputs to the drilling operation equipment during measurement-while drilling (MWD) or logging-while-drilling (LWD) operations. 
     In one or more aspects of the present disclosure, a borehole environment may utilize an information handling system to control one or more operations associated with the borehole environment. For purposes of this disclosure, an information handling system may include any instrumentality or aggregate of instrumentalities operable to compute, classify, process, transmit, receive, retrieve, originate, switch, store, display, manifest, detect, record, reproduce, handle, or utilize any form of information, intelligence, or data for business, scientific, control, or other purposes. For example, an information handling system may be a personal computer, a network storage device, or any other suitable device and may vary in size, shape, performance, functionality, and price. The information handling system may include random access memory (RAM), one or more processing resources such as a central processing unit (CPU) or hardware or software control logic, ROM, and/or other types of nonvolatile memory. Additional components of the information handling system may include one or more disk drives, one or more network ports for communication with external devices as well as various input and output (I/O) devices, such as a keyboard, a mouse, and a video display. The information handling system may also include one or more buses operable to transmit communications between the various hardware components. The information handling system may also include one or more interface units capable of transmitting one or more signals to a controller, actuator, or like device. 
     For the purposes of this disclosure, computer-readable media may include any instrumentality or aggregation of instrumentalities that may retain data and/or instructions for a period of time. Computer-readable media may include, for example, without limitation, storage media such as a sequential access storage device (for example, a tape drive), direct access storage device (for example, a hard disk drive or floppy disk drive), compact disk (CD), CD read-only memory (ROM) or CD-ROM, DVD, RAM, ROM, electrically erasable programmable read-only memory (EEPROM), and/or flash memory, biological memory, molecular or deoxyribonucleic acid (DNA) memory as well as communications media such wires, optical fibers, microwaves, radio waves, and other electromagnetic and/or optical carriers; and/or any combination of the foregoing. 
     Illustrative embodiments of the present disclosure are described in detail herein. In the interest of clarity, not all features of an actual implementation may be described in this specification. It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions may be made to achieve the specific implementation goals, which may vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of the present disclosure. 
     Throughout this disclosure, a reference numeral followed by an alphabetical character refers to a specific instance of an element and the reference numeral alone refers to the element generically or collectively. Thus, as an example (not shown in the drawings), widget “l a” refers to an instance of a widget class, which may be referred to collectively as widgets “ 1 ” and any one of which may be referred to generically as a widget “ 1 ”. In the figures and the description, like numerals are intended to represent like elements. 
     To facilitate a better understanding of the present disclosure, the following examples of certain embodiments are given. In no way should the following examples be read to limit, or define, the scope of the disclosure. Embodiments of the present disclosure may be applicable to drilling operations that include but are not limited to target (such as an adjacent well) following, target intersecting, target locating, well twinning such as in SAGD (steam assist gravity drainage) well structures, drilling relief wells for blowout wells, river crossings, construction tunneling, as well as horizontal, vertical, deviated, multilateral, u-tube connection, intersection, bypass (drill around a mid-depth stuck fish and back into the well below), or otherwise nonlinear boreholes in any type of subsurface formation. Embodiments may be applicable to injection wells, and production wells, including natural resource production wells such as hydrogen sulfide, hydrocarbons or geothermal wells; as well as wellbore or borehole construction for river crossing tunneling and other such tunneling boreholes for near surface construction purposes or borehole u-tube pipelines used for the transportation of fluids such as hydrocarbons. Embodiments described below with respect to one implementation are not intended to be limiting. 
       FIG.  1    depicts an exemplary drilling system  100 . As depicted, borehole  102  may extend from a wellhead  104  into a subsurface formation  106  from a surface  108 . As depicted in  FIG.  1   , in one or more embodiments, borehole  102  may extend generally vertically into the subsurface formation  106 . Alternatively, in one or more embodiments, borehole  102  may extend at an angle through subsurface formation  106 , such as horizontal and slanted boreholes. For example and without limitation, although  FIG.  1    depicts a vertical or low inclination angle well, in one or more embodiments, a high inclination angle or horizontal placement of the well and equipment may be possible. In one or more embodiments, borehole  102  may comprise any one or more of horizontal, vertical, slanted, curved, and any other types of borehole geometries and orientations. Borehole  102  may be cased or uncased. in one or more embodiments, borehole  102  may include a metallic member, wherein the metallic member may be a casing, liner, tubing, or other elongated steel tubular disposed in borehole  102 . While  FIG.  1    generally depicts a land-based system, it should be noted that like systems may operate in subsea locations as well. 
     As depicted in  FIG.  1   , a drilling platform  110  may support a derrick  112  having a traveling block  114  for raising and lowering drill string  116 . Drill string  116  may include, but is not limited to, drill pipe and coiled tubing, as generally known to those skilled in the art. A kelly  118  may support drill string  116  as it may be lowered through a rotary table  120 . A drill bit  122  may be attached to the distal end of drill string  116  and may be driven either by one or more of a downhole motor and rotation of drill string  116  from surface  108 . For example and without limitation, drill bit  122  may include one or more of roller cone bits, PDC bits, natural diamond bits, hole openers, reamers, coring bits, and the like. As drill bit  122  rotates, it may create and extend borehole  102  that penetrates various subsurface formations  106 . A pump  124  may circulate drilling fluid through a feed pipe  126  through kelly  118 , downhole through interior of drill string  116 , through orifices in drill bit  122 , back to surface  108  via annulus  128  surrounding drill string  116 , and into a retention pit  132 . 
     With continued reference to  FIG.  1   , drill string  116  may begin at wellhead  104  and may traverse borehole  102 . Drill bit  122  may be attached to a distal end of drill string  116  and may be driven, or example and without limitation, either by a downhole motor and/or via rotation of drill string  116  from surface  108 . Drill bit  122  may be a part of bottom hole assembly  130  at the distal end of drill string  116 . Bottom hole assembly  130  may further include tools for look-ahead resistivity applications. As will be appreciated by those of ordinary skill in the art, bottom hole assembly  130  may be a measurement-while drilling or logging-while-drilling system. 
     Bottom hole assembly  130  may comprise any one or more of tools, transmitters, and receivers to perform downhole measurement operations. For example and without limitation, bottom hole assembly  130  may comprise one or more of any number of assemblies for one or more of measurement, communication, energy storage, and the like. For example and without limitation, bottom hole assembly  130  may comprise measurement assembly  134 . In one or more embodiments, measurement assembly  134  may comprise at least one transducer  136   a,  which may be disposed at the surface of measurement assembly  134 . While  FIG.  1    depicts a single transducer  136   a,  in one or more embodiments, there may be any number of transducers disposed on measurement assembly  134 . References to and illustrations showing any one or more of transducers  136   a - c  may be applicable to any transducers disclosed herein. Without limitation, transducers may be referred to herein as a transceiver and transducer  136   a  may be disposed within measurement assembly  134 . In one or more embodiments, measurement assembly  134  may further comprise four other transducers that may be disposed ninety degrees from each other. In one or more embodiments, any number of transducers may be disposed along bottom hole assembly  130  at any degree from each other. In one or more embodiments, transducer  136   a,  and any other transducer, may function and operate to generate an acoustic pressure pulse that travels through one or more borehole fluids. In one or more embodiments, transducers  136   a  may further sense and acquire the reflected pressure wave which is modulated (for example and without limitation, reflected as an echo by the borehole wall). During measurement operations, the travel time of the pulse wave from transmission to recording of the echo may be recorded. In one or more embodiments, the acquired information may be used to determining for example and without limitation, a radius of the borehole, which may be derived by the fluid sound speed. By analyzing the amplitude of the echo signal, the acoustic impedance may also be derived. In one or more embodiments, transducers  136   a  may be made of piezo-ceramic crystals, magnetostrictive materials, or any other materials that generate an acoustic pulse when activated, either electrically or otherwise. In one or more embodiments, transducers  136   a  may also include backing materials and matching layers. In one or more embodiments, transducers  136   a  and assemblies housing transducers  136   a  may be removable and replaceable, for example and without limitation, in the event of damage or failure. 
     In one or more embodiments, bottom hole assembly  130  may be one or more of coupled to and controlled by information handling system  138 , which may be disposed on surface  108 . In one or more embodiments, information handling system  138  may be disposed down hole in bottom hole assembly  130 . Processing of information recorded may occur at one or more of down hole and on surface  108 . Processing occurring downhole may be transmitted to surface  108  to be one or more of recorded, observed, and further analyzed. In one or more embodiments, information recorded on information handling system  138  that may be disposed down hole may be stored until bottom hole assembly  130  may be brought to surface  108 . In one or more embodiments, information handling system  138  may communicate with bottom hole assembly  130  through a communication line (not shown) disposed in or on drill string  116 . In one or more embodiments, wireless communication may be used to transmit information back and forth between information handling system  138  and bottom hole assembly  130 . Information handling system  138  may transmit information to bottom hole assembly  130  and may receive as well as process information recorded by bottom hole assembly  130 . In one or more embodiments, a downhole information handling system (not shown) may include suitable circuitry, for example and without limitation, a microprocessor, for estimating, receiving, and processing signals from bottom hole assembly  130 . Downhole information handling system (not shown) may further comprise one or more of additional components, including, without limitation, memory, input devices, output devices, interfaces, and the like. In one or more embodiments, while not shown, bottom hole assembly  130  may include one or more additional components, including, without limitation, analog-to-digital converters, filters, and amplifiers, among others, that may be used to process the measurements of bottom hole assembly  130  before they may be transmitted to surface  108 . In one or more embodiments, raw measurements from bottom hole assembly  130  may be transmitted to surface  108 . 
     Any suitable technique may be used for transmitting signals from bottom hole assembly  130  to surface  108 , including, without limitation, wired pipe telemetry, mud-pulse telemetry, acoustic telemetry, and electromagnetic telemetry. While not shown, bottom hole assembly  130  may include a telemetry subassembly that may transmit telemetry data to surface  108 . At surface  108 , pressure transducers (not shown) may convert the pressure signal into electrical signals for a digitizer (not shown). The digitizer may supply a digital form of the telemetry signals to information handling system  138  via a communication link  140 , which may be a wired or wireless link. The telemetry data may be analyzed and processed by information handling system  138 . 
     As depicted in  FIG.  1   , communication link  140  (which may be wired or wireless, for example) may be provided that may transmit data from bottom hole assembly  130  to an information handling system  138  at surface  108 . Information handling system  138  may comprise one or more of a personal computer  141 , a video display  142 , a keyboard  144  (i.e., other input devices.), and non-transitory computer-readable media  146  (e.g., optical disks, magnetic disks) that can store code representative of the methods described herein. In addition to, or in place of processing at surface  108 , processing may occur downhole. As discussed below, methods may be utilized by information handling system  138  for stochastic trajectory projection of the bottom hole assembly  130  of drilling system  100 . 
       FIG.  2    depicts an exemplary flow diagram illustrating a method for determining one or more stochastic trajectory projections and confidence regions for a bottom hole assembly  130  based at least in part on one or more of system model parameters, steering inputs, bottom hole assembly initial conditions, working mode selection and settings, and a desired number of trajectory projections. In one or more embodiments, the stochastic trajectory projection module  210  for the steering model of drilling system  100  may be described in Equation ( 1 ), with a sequence of steering inputs  222  for upcoming drilling footage. 
         {dot over (x)}=f ( x,u,p )   (1)
 
     where x represents the initial conditions of the bottom hole assembly  224 , which may include one or more of inclination, azimuth, build rate, walk rate, true vertical depth and similar values; u represents steering inputs  222 , which may include one or more of steering ratios and tool face angles and which may be provided as a sequence of inputs; and p denotes the system model parameter probability distributions  220 . In one or more embodiments, the steering inputs  222  may be derived using a dynamic control scheme such as model predictive control. In one or more embodiments, the steering inputs  222  may be quantitative values specified by the drilling personnel or the control system. In one or more embodiments, the probability distributions of p can be developed in real-time using one or more parameter data sets. In one or more embodiments, a parameter data set may comprise any one or more of a real-time (or online) calibration method or data analytics, non-real-time (or offline) calibration method or data analytics, any one or more models of varying degrees of fidelity, and the experience of one or more persons skilled in the art of drilling or control systems. In one or more embodiments, system model parameter set p may depicted by Equation (3). In one or more embodiments, system model parameter set p may be directly obtained from an online identification method. In one or more embodiments, an identification method or system identification may refer to one or more methods of using one or more of measurements and known external influence to determine one or more system model parameters. In one or more embodiments, a known external influence may comprise one or more system inputs. In one or more embodiments, the term “online” may be used to denote a real-time method or system in which a model controller for the bottom hole assembly is operating simultaneously with and controlling the bottom hole assembly. An online method or system enables identification of one or more new values for system model parameter set p as one or more of new measurements and inputs are obtained. In one or more embodiments, the new measurements and inputs may enable improved controller performance by refining and updating prior measurements and inputs during one or more drilling operations. In one or more embodiments, the term “offline” may be synonymous with a method or system that is not operating in real-time. In one or more embodiments, the system model parameter probability distributions  220  may be assumed to follow a normal distribution, as is assumed in Equations (3)-(5). In one or more embodiments, the system model parameters probability distributions  220  may be any alternative type of distribution. 
     In one or more embodiments, the system model parameters probability distributions  220  may be one or more of the elements of a steering model. For example and without limitation, the steering model may be used to estimate the position of a drill bit  122  (depicted in  FIG.  1   ) and attitude of the drill bit may be represented by the following depth-based second order differential equation: 
       τ{umlaut over (θ)}=−{dot over (θ)}+ K   act   u+K   bias , initial conditions: θ 0 , {dot over (θ)} 0    (2)
 
     identified as T is a depth constant, K act  is the magnitude of the bottom hole assembly  130  turning capability, K bias  represents both the inherent steering tendency of bottom hole assembly  130  as well as any external forces on bottom hole assembly  130 , θ 0  is the initial angle (inclination or azimuth), and θ 0  is the initial curvature (build rate or walk rate). The dot notation in this equation represents a derivative with respect to distance, not time. Without limitation, θ is also used to represent a vector of the system model parameter probability distributions  220 . In one or more embodiments, the system model parameter probability distributions  220  may be generated using prior experience, knowledge of the subsurface formation and the equipment of the drilling system, prior analyses, and the like. 
     In one or more embodiments, model parameter set p may be described by a multivariate normal probability distribution using one or more of the model parameter set&#39;s mean and variances or covariances. The multivariate normal distribution of an n-dimensional parameter vector P=(P, P 2 , . . . P n ) may be written as: 
       P˜N(μ, K pp )   (3)
 
     where μ is an n-dimensional mean vector: 
       μ= E[P ]=( E[P   1   ], E[P   2   ], . . . , E[P   n ])   (4)
 
     and where K pp  is an n×n covariance matrix: 
         K   pp   =E [( P   i −μ i ) ( P   j −μ j )]  (5)
 
     such that 1≤i and j≤n. That is, in one or more embodiments, the probability distribution for K act  may have one or more interactions with K bias . In one or more embodiments, the multivariate distribution may be any alternative type of distribution. 
     As depicted in  FIG.  2   , the stochastic trajectory projection module  210  may be run with the specified input parameters to calculate the projected trajectories. Disclosed herein are two working modes  226 . In the first working mode, a single model is used to project the trajectory of the bottom hole assembly throughout the entire depth horizon. A first set of model parameters may be selected from the system model parameter probability distributions  220  input to the stochastic trajectory projection module  210 . The first set of model parameters may then used in combination with the steering inputs  222  throughout the entire depth horizon to generate the stochastic trajectory projections for the bottom hole assembly. The first working mode may be referred to as a one-model mode. In one or more embodiments, the one-model mode may be useful as it may provide a more responsive solution because these calculations may be performed more quickly. The one-model mode may also be beneficial for shorter prediction horizons. 
     In the second working mode, multiple different models are used at different depths within the projected. For a first predetermined depth interval (or length of the borehole), a first set of model parameters may be selected from the system model parameter probability distributions  220  input to the stochastic trajectory projection module  210 . The first set of model parameters may then used in combination with the steering inputs  222  to generate the stochastic trajectory projections for the bottom hole assembly for the first predetermined depth interval. After projecting the stochastic trajectory projections for the bottom hole assembly for the first predetermined depth interval, a second set of model parameters is selected from the system model parameter probability distributions  220  input to the stochastic trajectory projection module  210  and used to generate the stochastic trajectory projections for the bottom hole assembly for the second predetermined depth interval. This process is repeated until the final trajectory for the bottom hole assembly, which is the combined stochastic trajectory projections, traverses the entire depth horizon. The second working mode may be referred to as a multi-model mode. In one or more embodiments, the multi-model mode may reduce the size of the confidence regions of the stochastic trajectory projections. In one or more embodiments, the multi-model mode may provide more confident predictions, across the depth horizon than a one-model model because, for example, the multi-model mode supports updates to the model parameter set across the depth horizon. 
     The selection of the mode and other settings, including, without limitation, the depth interval for stochastic trajectory projections may be dependent on one or more factors, including, without limitation, any prior data analyses or knowledge of one or more of the subsurface formation and the equipment of the drilling system. In one or more embodiments, the number of stochastic trajectory projections for the entire depth horizon may be provided as an input to the stochastic trajectory projection module  210 . 
     In one or more embodiments, a number of stochastic trajectory projections over the entire depth horizon may be generated, where the number of stochastic trajectory projections  228  may be represented by N and where N is a positive integer. In one or more embodiments, N may be 100 or more. In one or more embodiments, a median or mean value for the N stochastic trajectory projections may be calculated. In one or more embodiments, any outliers in the stochastic trajectory projections may be identified and eliminated before the median or mean value is calculated for the N stochastic trajectory projections. 
     In one or more embodiments, the N stochastic trajectory projections  228  may be used to generate a vector including N data points, where each data point corresponds to particular position within the subsurface formation. In one or more embodiments, the position within the subsurface formation may comprise a depth point, horizontal point, or any other identifier for the location of the position within the subsurface formation. Quantiles between the cumulative probabilities of q1 and q2 of the elements in the vector can be calculated where q1, q2 are in the interval of [0, 1] and q1&lt;q2. With quantiles derived at each depth point, a confidence region is established in which the abovementioned resulting trajectory is at the center. For example, a 95% confidence region means a pair of 2.5% and 97.5% quantiles need to be calculated and the remaining data points in the vector are used to determine the confidence region. Multiple confidence regions or quantiles can be obtained and plotted and presented in the same or separate displays. 
     In one or more embodiments, one or more of data, analysis, experience, and knowledge may be used to select identify desired confidence regions, where smaller confidence regions generally indicate greater confidence that the actual trajectory of the bottom hole assembly will align with the stochastic trajectory projections. In one or more embodiments, a multi-model mode may generate narrower confidence regions, indicating increased confidence in the multi-model trajectory projections than in the stochastic trajectory projections generated by a one-model mode. For example and without limitation, a multi-model trajectory projection may generate narrower confidence regions by better representing the variations in subsurface operating conditions based on generation of multiple trajectory projections using multiple models from the system model parameter probability distributions  220 . In one or more embodiments, the number of models utilized may affect the area of confidence regions given the same distributions. For example and without limitation, over a given depth horizon, a one-model mode may use only a single model parameter set drawn from data set p while a multi-model mode may use, for example, ten parameter sets drawn from model parameter set p for the same depth horizon. Accordingly, it may be desirable to sample multiple models from data set p to attempt to generate a better representation of the probability distribution of model parameters. 
     In summary, as shown in  FIG.  1   , the stochastic trajectory projection module  210  may receive inputs specifying the model parameter probability distributions. In one or more embodiments, these inputs may be one or more established system model parameter probability distributions acquired directly from any identification method discussed herein. One or more sets of model parameters may be obtained by randomly sampling from the model parameters probability distributions  220 . After selecting the desired working mode and the desired number of trajectory projections, a projected trajectory may be calculated using the selected set or sets of model parameters, steering inputs  222 , and bottom hole assembly initial conditions  224 . This calculation process may be repeated according to the desired number of trajectory projections. As described above, the working mode (one-model or multi-model) and its settings may be specified to achieve a desired confidence of the stochastic trajectory projection. The outputs of the stochastic trajectory projection module  210  are the projected trajectory of the bottom hole assembly and the confidence region, which may be provided in real-time to, for example and without limitation, a trajectory controller  242  and a display  240 . In one or more embodiments, the trajectory controller  242  may comprise one or more of a system and a method for producing control outputs that may guide the bottom hole assembly along, or as closely as possible, the target well, which may be provided in a well plan, based at least in part on the outputs of the stochastic trajectory projection module  210 , including at least the projected trajectory  230  and projected confidence region  232 . 
     In one or more embodiments, the projected trajectory  230  and projected confidence region  232  may be used for a variety of purposes. In one or more embodiments, the projected trajectory  230  and projected confidence region  232  may be used to update future steering decisions, resulting in new steering inputs  222  to the stochastic trajectory projection module  210 . In one or more embodiments, an actual trajectory that poorly aligns with the stochastic projected trajectory  230  or lies outside the projected confidence region  232  may indicate changes in the drilling conditions and signal an operator to recalibrate and update the model parameter set drawn from data set p, for example and without limitation, by running the system identification again using the new measurements. In one or more embodiments, the projected trajectory  230  and projected confidence region  232  may be used to mitigate any possible risks of the borehole  102  interfering with other boreholes in a subsurface formation  106  or falling behind the target. For example and without limitation, an operator may provide new inputs to trajectory controller  242  if there is a significant overlap between the projected trajectory  230  and an area of possible collision. 
     In an additional example without limitation, an operator may provide new inputs to trajectory controller  242  such as applying additional steering power if the projected trajectory  230  shows a risk of deviating from a well plan for borehole  102  and, in one or more embodiments, the additional steering power can be repeatedly provided to the trajectory controller  242  if the projected trajectory  230  continues to show the borehole  102  deviating from the well plan. For example, if the actual borehole has begun to deviate from the well plan, the projected trajectory  230  and projected confidence region  232  for a given set of steering inputs  222  enables an operator or controller to determine that the projected trajectory  230  will return to or overlap with the well plan. 
       FIG.  3    depicts an exemplary well plan and stochastic trajectory confidence regions for a borehole across a depth horizon.  FIG.  3   a    illustrates an exemplary trajectory of a wellbore as segments  301 ,  303 ,  305 ,  307 , and  309  with well plan  330  across depth horizon  350 , while  FIG.  3   b    illustrates an exemplary trajectory of a similar wellbore as segments  321 ,  323 ,  325 ,  327 , and  329  with substantially the same well plan  330  across the same depth horizon  350 . In  FIG.  3   a   , the confidence region  311  is generated by stochastic trajectory projection module  210  using a one-model mode is shown overlapping segment  307 . An operator, using this exemplary figure, would have confidence that the projected trajectory of the bottom hole assembly will overlap with the trajectory of the borehole in the well plan. As illustrated in  FIG.  3   a   , a one-model mode would use the model parameter set drawn from data set p across the entire depth horizon. That is, the model parameter set to stochastically project segment  301  would be the same set used to stochastically project segments  303 ,  305 ,  307 , and  309 . In  FIG.  3   b   , the confidence region  331 , which is slightly narrower than confidence region  311 , is similarly shown overlapping segment  327  after the confidence region  331  is generated by stochastic trajectory projection module  210  using a multi-model mode. In contrast to  FIG.  3   a   , each of the segments in  FIG.  3   b    would use a model parameter set drawn from data set p than any other segment. That is, a first model parameter set drawn from data set p would be used to stochastically project the trajectory for segment  321 , a second model parameter set drawn from data set p would be used to stochastically project the trajectory for segment  323 , a third model parameter set drawn from data set p would be used to stochastically project the trajectory for segment  325 , a fourth model parameter set drawn from data set p would be used to stochastically project the trajectory for segment  327 , and a fifth model parameter set drawn from data set p would be used to stochastically project the trajectory for segment  329 . While confidence region  331  is shown as a narrower region than confidence region  311 , this is merely illustrative the circumstances of drilling operations, subsurface conditions, and other factors may result in some one-model mode stochastic projections generating narrower confidence regions than comparable multi-model mode stochastic projections. 
       FIGS.  4 - 5    depict exemplary stochastic trajectory projections for a bottom hole assembly. The circles,  410 ,  420 ,  510 ,  520  represent control points (or depth points) at which new inputs, which may comprise one or more sets of model parameters, steering inputs  222 , bottom hole assembly initial conditions  224 , working mode selection and settings  226 , and a number of trajectory projections  228 , are provided to the stochastic trajectory projection module  210  before a new projected trajectory  230  and projected confidence region  232  are output to display  240  and trajectory controller  242 . See  FIG.  2   . For example, as depicted in  FIGS.  4 - 5   , a prediction depth horizon may be from a depth of approximately 6000 feet to a depth of approximately 6300 feet. As depicted in  FIG.  4   , the left circle  410  specifies an initial inclination of the bottom hole assembly, while, in  FIG.  5   , the left circle  510  specifies an initial azimuth of the bottom hole assembly. Using the process disclosed herein, stochastic trajectory projections for inclination and azimuth are achieved (depicted using dashed lines) with 95% confidence regions (depicted as the shaded area). In  FIGS.  4 - 4   , an exemplary multi-model mode using 10 models for a stochastic trajectory projection.  FIGS.  4 - 5    further depict a dashed-dot line  440  between the circles  410 ,  420 , and a dashed-dot line  540  between the circles  510 ,  520  depicting actual trajectories for inclination and azimuth of the bottom hole assembly conforming closely to the stochastic trajectory projections  430 ,  530  and within the confidence regions  450 ,  550  of the stochastic trajectory projection. As depicted in both  FIGS.  4 - 5   , the right circles  420 ,  520  are—about 90 feet deeper than the left circles  410 ,  510  and depict control points at which a new set of inputs may be provided to the stochastic trajectory projection module  210  to generate additional stochastic trajectory projections. 
     In one or more embodiments, a projected confidence region generated by the stochastic trajectory projection module  210  using the one-model mode may be larger, and in some cases substantially larger, than the projected confidence region generated by the stochastic trajectory projection module  210  using the multi-model mode. 
     In one or more embodiments, the stochastic trajectory projection module  210  may use one or more stochastic simulation methods including, without limitation, Monte Carlo simulation methods, to simulate and project the stochastic trajectory projections. In one or more embodiments, one or more additional inputs may be provided to the stochastic trajectory projection module  210 , including, without limitation, weight on bit, RPM, flow rate. These additional inputs may enable the stochastic trajectory projection module  210  to account for changes in one or more drilling parameters and may thereby improve the quality of the stochastic trajectory projections. 
       FIG.  6    depicts an exemplary stochastic trajectory projection and confidence regions for a bottom hole assembly across a depth horizon. Like  FIGS.  4 - 5   ,  FIG.  6   a - 6   b    show the projected trajectories  610 ,  630  and confidence regions  615 ,  620 ,  635 ,  640  generated by stochastic trajectory projection module  210 .  FIG.  6   a    illustrates a stochastic trajectory projection  610  and two confidence regions  615 ,  620  for the inclination of the bottom hole assembly across a depth horizon from approximately 9775 feet deep to approximately 10075 feet deep.  FIG.  6   b    illustrates a stochastic trajectory projection  630  and two confidence regions  635 ,  640  for the azimuth of the bottom hole assembly across the same depth horizon from approximately 9775 feet deep to approximately 10075 feet deep. Both  FIG.  6   a - 6   b    illustrate that using a multi-model mode may result in increased confidence in the stochastic trajectory projection generated by the stochastic trajectory projection module  210 , as shown by narrower confidence regions  615 ,  635  generated by using the multi-model mode in comparison with the wider confidence regions  620 ,  640  generated by using the one-model mode. 
       FIG.  7    depicts a flow diagram for generating stochastic trajectory projections for a bottom hole assembly. In step  710 , a set of system model parameters  220  are provided to the stochastic trajectory projection module  210 . In step  720 , one or more steering inputs  222  are collected and provided to the stochastic trajectory projection module  210 . In one or more embodiments, the one or more steering inputs  222  may be collected from the trajectory controller  242 . In one or more embodiments, the one or more steering inputs  222  may be collected from an operator, for example, an individual that has experience with drilling and production operations in or around the borehole environment  100 . In step  730 , a stochastic trajectory projection is generated based on a Monte Carlo method, for example, using the set of system model parameters from model parameter set p and Equation (2). In step  740 , the stochastic trajectory projection module  210  compares the total number of stochastic trajectory projections generated against the specified number of stochastic trajectory projections  228 . If the total number of stochastic trajectory projections generated by the stochastic trajectory projection module  210  does not meet or exceed the specified number of stochastic trajectory projections  228 , the method returns to step  730  and an additional stochastic trajectory projection is generated based on the Monte Carlo method for example, using Equation ( 2 ). Alternatively, if the total number of stochastic trajectory projections generated by the stochastic trajectory projection module  210  does meet or exceed the specified number of stochastic trajectory projections  228 , the method moves to step  750  and a projected trajectory  230  and a projected confidence region  232  for the bottom hole assembly  130  is calculated. In step  760 , one or more of the projected trajectory  230  and a projected confidence region  232  may be provided to one or more of the display  240  and the trajectory controller  242 . In step  770 , the projected trajectory  230  and a projected confidence region  232  may be used to update future steering decisions. For example and without limitation, future steering decisions may be provided to step  720  such that the future steering decisions may be used for additional stochastic trajectory projections generated by the stochastic trajectory projection module  210 . 
       FIG.  8    depicts a schematic diagram of example information handling system  800 , for example, for use with or in an associated borehole environment, for example without limitation the borehole environment  100  depicted in  FIG.  1   . The information handling system  138  of  FIG.  1    may take a form similar to the information handling system  800 . A processor or central processing unit (CPU)  801  of the information handling system  800  is communicatively coupled to a memory controller hub (MCH) or north bridge  802 . The processor  801  may include, for example a microprocessor, microcontroller, digital signal processor (DSP), application specific integrated circuit (ASIC), or any other digital or analog circuitry configured to interpret and/or execute program instructions and/or process data. Processor  801  may be configured to interpret and/or execute program instructions or other data retrieved and stored in any memory such as memory  803  or hard drive  807 . Program instructions or other data may constitute portions of a software or application, for example, application  858  or data  854 , for carrying out one or more methods described herein. Memory  803  may include read-only memory (ROM), random access memory (RAM), solid state memory, or disk-based memory. Each memory module may include any system, device, or apparatus configured to retain program instructions and/or data for a period of time (for example, non-transitory computer-readable media). For example, instructions from a software program or application  858  or data  854  may be retrieved and stored in memory  803  for execution or use by processor  801 . In one or more embodiments, the memory  803  or the hard drive  807  may include or comprise one or more non-transitory executable instructions that, when executed by the processor  801 , cause the processor  801  to perform or initiate one or more operations or steps. The information handling system  800  may be preprogrammed or it may be programmed (and reprogrammed) by loading a program from another source (for example, from a CD-ROM, from another computer device through a data network, or in another manner). 
     The data  854  may include treatment data, geological data, fracture data, microseismic data, mud candidate data, borehole imager measured data, inversion-estimated imaging properties, or any other appropriate data. The one or more applications  858  may include one or more machine learning models, applications for one or more of down-sampling measured data, calculating misfits or to minimize cost functions, to perform petrochemical inversions, to solve for formation permittivity, to align measured data based on depth, azimuth, resolution, or any other measurement, extrapolating permittivity, scaling coefficients to match borehole imager measurements with dielectric tool measurements, calculate dispersion curves of permittivity, calibrating coefficients, or any other appropriate applications. In one or more embodiments, a memory of a computing device includes additional or different data, application, models, or other information. In one or more embodiments, the data  854  may include treatment data relating to fracture treatment plans. For example, the treatment data may indicate a pumping schedule, parameters of a previous injection treatment, parameters of a future injection treatment, or one or more parameters of a proposed injection treatment. Such one or more parameters may include information on flow rates, flow volumes, slurry concentrations, fluid compositions, injection locations, injection times, or other parameters. The treatment data may include one or more treatment parameters that have been optimized or selected based on numerical simulations of fracture propagation. In one or more embodiments, the data  854  may include one or more signals received by one or more transducers  136   a - c  of  FIG.  1   . 
     The one or more applications  858  may comprise one or more software programs or applications, one or more scripts, one or more functions, one or more executables, or one or more other modules that are interpreted or executed by the processor  801 . For example, the one or more applications  858  may include a fracture design module, a reservoir simulation tool, a hydraulic fracture simulation model, or any other appropriate function block. The one or more applications  858  may include machine-readable instructions for performing one or more of the operations related to any one or more embodiments of the present disclosure. The one or more applications  858  may include machine-readable instructions for generating a user interface or a plot, for example, depicting fracture geometry (for example, length, width, spacing, orientation, etc.), pressure plot, hydrocarbon production performance. The one or more applications  858  may obtain input data, such as treatment data, geological data, fracture data, measurement data, or other types of input data, from the memory  803 , from another local source, or from one or more remote sources (for example, via the one or more communication links  814 ). The one or more applications  858  may generate output data and store the output data in the memory  803 , hard drive  807 , in another local medium, or in one or more remote devices (for example, by sending the output data via the communication link  814 ). 
     Modifications, additions, or omissions may be made to  FIG.  8    without departing from the scope of the present disclosure. For example,  FIG.  8    shows a particular configuration of components of information handling system  800 . However, any suitable configurations of components may be used. For example, components of information handling system  800  may be implemented either as physical or logical components. Furthermore, in some embodiments, functionality associated with components of information handling system  800  may be implemented in special purpose circuits or components. In other embodiments, functionality associated with components of information handling system  800  may be implemented in configurable general-purpose circuit or components. For example, components of information handling system  800  may be implemented by configured computer program instructions. 
     Memory controller hub  802  may include a memory controller for directing information to or from various system memory components within the information handling system  800 , such as memory  803 , storage element  806 , and hard drive  807 . The memory controller hub  802  may be coupled to memory  803  and a graphics processing unit (GPU)  804 . Memory controller hub  802  may also be coupled to an I/O controller hub (ICH) or south bridge  805 . I/O controller hub  805  is coupled to storage elements of the information handling system  800 , including a storage element  806 , which may comprise a flash ROM that includes a basic input/output system (BIOS) of the computer system. I/O controller hub  805  is also coupled to the hard drive  807  of the information handling system  800 . I/O controller hub  805  may also be coupled to an I/O chip or interface, for example, a Super I/O chip  808 , which is itself coupled to several of the I/O ports of the computer system, including a keyboard  809 , a mouse  810 , a monitor  812  and one or more communications link  814 . Any one or more input/output devices receive and transmit data in analog or digital form over one or more communication links  814  such as a serial link, a wireless link (for example, infrared, radio frequency, or others), a parallel link, or another type of link. The one or more communication links  814  may comprise any type of communication channel, connector, data communication network, or other link. For example, the one or more communication links  814  may comprise a wireless or a wired network, a Local Area Network (LAN), a Wide Area Network (WAN), a private network, a public network (such as the Internet), a WiFi network, a network that includes a satellite link, or another type of data communication network. 
     A memory or storage device primarily stores one or more software applications or programs, which may also be described as program modules containing computer-executable instructions, which may be executed by the computing unit for implementing one or more embodiments of the present disclosure. The memory, therefore, may include one or more applications including, for example, a transmitter control application, a receiver control application, and one or more applications enabling one or more of the processes or sub-processes illustrated in  FIG.  2    and may produce outputs like those shown in  FIGS.  4 - 5   . These applications may integrate functionality from additional or third-party application programs or from system files stored in memory or on a storage device. An application may perform one or more of the steps in  FIG.  2   . System files, such as an ASCII text file may be used to store the instructions, data input, or both for the applications as may be required in, for example, one or more steps of  FIG.  2   . In certain embodiments, any one or more other applications may be used in combination. In certain embodiments, any one or more other applications may be used in combination may be used as stand-alone applications. 
     Although the computing device  800  is shown as having one or more generalized memories, the computing device  800  typically includes a variety of non-transitory computer readable media. By way of example, and not limitation, non-transitory computer readable media may comprise computer storage media and communication media. The memory may include computer storage media, such as a ROM and RAM in the form of volatile memory, nonvolatile memory, or both. A BIOS containing the basic routines that help to transfer information between elements within the computing unit, such as during start-up, is typically stored in the ROM. RAM typically contains data, program modules, other executable instructions, or any combination thereof that are immediately accessible to, presently being operated on, or both by the processing unit. By way of example, and not limitation, the computing device  800  may include an operating system, application programs, other program modules, and program data. 
     The components shown in the memory may also be included in other removable/non-removable, volatile/nonvolatile non-transitory computer storage media or the components may be implemented in the computing device  800  through an application program interface (“API”) or cloud computing, which may reside on a separate computing device coupled through a computer system or network (not shown). For example and without limitation, a hard disk drive may read from or write to non-removable, nonvolatile magnetic media, a magnetic disk drive may read from or write to a removable, nonvolatile magnetic disk, and an optical disk drive may read from or write to a removable, nonvolatile optical disk such as a CD-ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that may be used in the exemplary operating environment may include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, or the like. The drives and their associated computer storage media discussed above provide storage of computer readable instructions, data structures, program modules, and other data for the computing unit. 
     The computing device  800  may receive commands or information from a user through one or more input devices such as the keyboard  809  and the mouse  810 . Additional input devices may comprise a microphone, joystick, touchscreen, scanner, voice or gesture recognition, one or more sensors including one or more seismic sensors, and the like (not shown). These and other input devices may be coupled to the processing unit through the Super I/O chip  808  that is coupled to the ICH  805 , but may be coupled by other interface and bus structures, such as a parallel port or a universal serial bus (USB) (not shown). 
     A monitor or other type of display device (not shown) may be coupled to the MCH  802  via an interface, such as the GPU  804  or via Super I/O chip  808 . A graphical user interface (“GUI”) may also be used with the video interface  804  to receive instructions from a user and transmit instructions to the central processing unit  801 . A GUI may be used to display the outputs of the processes described in in  FIGS.  2  and  7   , including, without limitation, as shown in  FIGS.  4 - 5   , and may be used to prompt or display modification of subsurface operations or production activities. The computing device  800  may comprise peripheral output devices such as speakers, printer, external memory, any other device, or any combination thereof, which may be coupled through any output peripheral interface. 
     Any one or more input/output devices may receive and transmit data in analog or digital form over one or more communication links  814  such as a serial link, a wireless link (for example, infrared, radio frequency, or others), a parallel link, or another type of link. The one or more communication links  814  may comprise any type of communication channel, connector, data communication network, or other link. For example, the one or more communication links  814  may comprise a wireless or a wired network, a Local Area Network (LAN), a Wide Area Network (WAN), a private network, a public network (such as the Internet), a wireless fidelity or WiFi network, a network that includes a satellite link, or another type of data communication network. 
     Although many other internal components of the computing device  800  are not shown, those of ordinary skill in the art will appreciate that such components and their interconnection are well known. 
     Any one or more embodiments of the present disclosure may be implemented through a computer-executable program of instructions, such as program modules, generally referred to as software applications or application programs executed by a computer. A software application may include, for example, routines, programs, objects, components, data structures, any other executable instructions, or any combination thereof, that perform particular tasks or implement particular abstract data types. The software application forms an interface to allow a computer to react according to a source of input. For example, an interface application may be used to implement any one or more embodiments of the present disclosure. The software application may also cooperate with other applications or code segments to initiate a variety of tasks based, at least in part, on data received, a source of data, or any combination thereof. Other applications or code segments may provide optimization components including, but not limited to, neural networks, earth modeling, history-matching, optimization, visualization, data management, and economics. The software application may be stored, carried, or both on any variety of memory such as CD-ROM, magnetic disk, optical disk, bubble memory, and semiconductor memory (for example, various types of RAM or ROM). Furthermore, the software application and one or more inputs or outputs may be transmitted over a variety of carrier media including, but not limited to wireless, wired, optical fiber, metallic wire, telemetry, any one or more networks (such as the Internet), or any combination thereof. 
     Moreover, those skilled in the art will appreciate that one or more of the embodiments may comprise a variety of computer-system configurations, including hand-held devices, multiprocessor systems, microprocessor-based or programmable consumer electronics, minicomputers, mainframe computers, and any combination thereof. Any number of computer-systems and computer networks are acceptable for use with the present disclosure. The disclosure may be practiced in distributed-computing environments where tasks are performed by remote-processing devices that are linked through a communications network. In a distributed-computing environment, program modules may be located in both local and remote computer-storage media including memory storage devices. The present disclosure may, therefore, be implemented in connection with various hardware, software, or any combination thereof, in a computer system, information handling system, or other processing system. 
     In one or more embodiments, a method for stochastically projecting a well trajectory of a bottom hole assembly in a subsurface formation comprises receiving a first one or more system model parameters from a system model parameter probability distribution, receiving a first one or more steering inputs, receiving a first one or more values corresponding to the bottom hole assembly initial conditions at a first position within the subsurface formation, and stochastically projecting a first one or more trajectories of the bottom hole assembly from the first position within the subsurface formation to a second position within the subsurface formation based at least in part on one or more of the first one or more system model parameters, the first one or more steering inputs, and the first one or more values corresponding to the bottom hole assembly initial conditions. 
     In one or more embodiments, the method of further comprises stochastically projecting the first confidence region between the first position and the second position based at least in part one or more of the first one or more stochastically projected trajectories, the first one or more system model parameters, the received one or more steering inputs, and the received one or more values corresponding to the bottom hole assembly initial conditions. In one or more embodiments, the method further comprises providing one or more of the first one or more stochastically projected trajectories and the first confidence region to one or more of a display and a trajectory controller. In one or more embodiments, the method further comprises discarding one or more outliers in the first one or more stochastically projected trajectories of the bottom hole assembly before stochastically projecting the first confidence region. In one or more embodiments, the method further comprises advancing the bottom hole assembly from the first position to the second position. In one or more embodiments, the method further comprises stochastically projecting a second one or more trajectories of the bottom hole assembly from the second position to a third position based at least in part on one or more of the first one or more system model parameters, the received one or more steering inputs, and the received one or more values corresponding to the bottom hole assembly initial conditions. In one or more embodiments, the method further comprises receiving a second one or more system model parameters from the system model parameter probability distribution, stochastically projecting a second one or more trajectories of the bottom hole assembly from the second position to a third position based at least in part on one or more of the second one or more system model parameters, the received one or more steering inputs, and the received one or more values corresponding to the bottom hole assembly initial conditions. 
     In one or more embodiments, the first one or more system model parameters may be randomly selected from the system model parameter probability distribution. In one or more embodiments, the method further comprises generating a second one or more one or more steering inputs and stochastically projecting a second one or more trajectories of the bottom hole assembly from the second position to a third position based at least in part on one or more of the selected one or more system model parameters, the second one or more steering inputs, and the received one or more values corresponding to the bottom hole assembly initial conditions. In one or more embodiments, stochastically projecting the first one or more trajectories of the bottom hole assembly occurs in real-time. In one or more embodiments, selecting a second one or more steering inputs may be based at least in part on one or more of the first one or more stochastically projected trajectories and the first confidence region to one or more of a display and a trajectory controller. In one or more embodiments, the method further comprises receiving a second one or more system model parameters from the system model parameter probability distribution, receiving a second one or more steering inputs, receiving a second one or more values corresponding to the bottom hole assembly initial conditions at a second position within the subsurface formation, and stochastically projecting a second one or more trajectories of the bottom hole assembly from the second position in the subsurface formation to a third position in the subsurface formation based at least in part on one or more of the second one or more system model parameters, the second one or more steering inputs, and the second one or more values corresponding to the bottom hole assembly initial conditions. 
     In one or more embodiments, a system for stochastically projecting a well trajectory of a bottom hole assembly comprises a bottom hole assembly comprising one or more transducers, a trajectory controller coupled to the bottom hole assembly, and an information handling system coupled to the transducers, where the information system comprises a processor, and a non-transitory computer readable medium for storing one or more instructions that, when executed, causes the processor to receive a first one or more system model parameters from a system model parameter probability distribution, receive a first one or more steering inputs, receive a first one or more values corresponding to the bottom hole assembly initial conditions from the one or more transducers at a first position within a subsurface formation, and stochastically project a first one or more trajectories of the bottom hole assembly from the first position within the subsurface formation to a second position within the subsurface formation based at least in part on one or more of the first one or more system model parameters, the first one or more steering inputs, and the first one or more values corresponding to the bottom hole assembly initial conditions. 
     In one or more embodiments, the one or more instructions, when executed, further causes the processor to stochastically project a confidence region for the projected trajectory of the bottom hole assembly between the first position within the subsurface formation to the second position within the subsurface formation. In one or more embodiments, the system further comprises a display and the one or more instructions, when executed, further causes the processor to provide one or more of the first one or more stochastically projected trajectories and the first confidence region to one or more of the display and the trajectory controller. In one or more embodiments, the one or more instructions, when executed, further causes the processor to randomly select the first one or more system model parameters from the system model parameter probability distribution. In one or more embodiments, the one or more instructions, when executed, further causes the processor to one or more of stochastically project the trajectory of the bottom hole assembly or stochastically project the confidence region for the projected trajectory of the bottom hole assembly in real time. In one or more embodiments, the one or more instructions, when executed, further causes the processor to receive a second one or more system model parameters from the system model parameter probability distribution, receive a second one or more steering inputs; receive a second one or more values corresponding to the bottom hole assembly initial conditions at the second position within the subsurface formation, and stochastically project a second one or more trajectories of the bottom hole assembly from the second position in the subsurface formation to a third position in the subsurface formation based at least in part on one or more of the second one or more system model parameters, the second one or more steering inputs, and the second one or more values corresponding to the bottom hole assembly initial conditions. 
     In one or more embodiments, a method for stochastically projecting a well trajectory of a bottom hole assembly in a subsurface formation in real time comprises receiving a first one or more system model parameters from a system model parameter probability distribution, receiving a first one or more steering inputs, receiving a first one or more values corresponding to the bottom hole assembly initial conditions at a first position within the subsurface formation, stochastically projecting a first one or more trajectories of the bottom hole assembly from the first position within the subsurface formation to a second position within the subsurface formation, advancing the bottom hole assembly from the first position to the second position, receiving a second one or more system model parameters from the system model parameter probability distribution, receiving a second one or more steering inputs, receiving a second one or more values corresponding to the bottom hole assembly initial conditions at a second position within the subsurface formation, and stochastically projecting a second one or more trajectories of the bottom hole assembly from the second position within the subsurface formation to a third position within the subsurface formation. In one or more embodiments, the method further comprises stochastically projecting the first one or more confidence regions based on the stochastically projected first one or more trajectories of the bottom hole assembly between the first position within the subsurface formation and the second position within the subsurface formation and further comprising stochastically projecting the second one or more confidence regions based on the stochastically projected second one or more trajectories of the bottom hole assembly between the second position within the subsurface formation and the third position within the subsurface formation. 
     While the present disclosure has been described in connection with presently preferred embodiments, it will be understood by those skilled in the art that it is not intended to limit the disclosure to those embodiments. It is therefore, contemplated that various alternative embodiments and modifications may be made to the disclosed embodiments without departing from the spirit and scope of the disclosure defined by the appended claims and equivalents thereof. In particular, with regards to the methods disclosed, one or more steps may not be required in all embodiments of the methods and the steps disclosed in the methods may be performed in a different order than was described. Furthermore, no limitations are intended to the details of construction or design herein shown, other than as described in the claims below. It is therefore evident that the particular illustrative embodiments disclosed above may be altered or modified and all such variations are considered within the scope and spirit of the present disclosure. In particular, every range of values (for example, “from about a to about b,” or, equivalently, “from approximately a to b,” or, equivalently, “from approximately a-b”) disclosed herein is to be understood as referring to the power set (the set of all subsets) of the respective range of values. The terms in the claims have their plain, ordinary meaning unless otherwise explicitly and clearly defined by the patentee.