WO2024124069A2

WO2024124069A2 - Rapid sucker rod pump downhole dynacard estimation for deviated wells

Info

Publication number: WO2024124069A2
Application number: PCT/US2023/082998
Authority: WO
Inventors: Albert Hoefel; Jahir Pabon
Original assignee: Sensia LLC
Current assignee: Sensia LLC
Priority date: 2022-12-08
Filing date: 2023-12-07
Publication date: 2024-06-13
Anticipated expiration: 2025-06-08
Also published as: US20240191614A1; EP4630650A2; WO2024124069A3; CN120457265A

Abstract

Systems and methods provided herein relate to a pump system. The pump system includes a pump disposed within a well, an actuator operable to move a rod including a surface end coupled to the actuator and a downhole end coupled to the pump, and a controller. The controller is configured to identify a first impulse response and a second impulse response associated with the pump system based on a first model of the pump system. The controller is further configured to generate a second model of the pump system based on the first impulse response and the second impulse response. The controller is further configured to operate the pump system based on the second model.

Description

Atty. Dkt. No.: 123960-0813 P.C.T. PATENT APPLICATION FOR RAPID SUCKER ROD PUMP DOWNHOLE DYNACARD ESTIMATION FOR DEVIATED WELLS Inventors: ALBERT HOEFEL JAHIR PABON 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 RAPID SUCKER ROD PUMP DOWNHOLE DYNACARD ESTIMATION FOR DEVIATED WELLS CROSS REFERENCE TO RELATED APPLICATIONS This application claims the benefit of and priority to U.S. Provisional Patent Application No. 63/431,156, filed December 8, 2022, the entirety of which is incorporated by reference herein. BACKGROUND Various types of equipment can be utilized in a subterranean environment. As an example, a pump system can be utilized to move fluid in a well in a subterranean environment. U.S. Patent No. 8036829 describes analysis and control of a reciprocating pump system. SUMMARY One implementation of the present disclosure relates to a pump system. The pump system includes a pump disposed within a well, an actuator operable to move a rod including a surface end coupled to the actuator and a downhole end coupled to the pump, and a controller. The controller is configured to identify a first impulse response and a second impulse response associated with the pump system. The identification includes measuring a first set of position data associated with the surface end of the rod and generating, based on a first model of the pump system and the position data, a first set of data associated with simulated operation of the pump system with a load stimulus, and a second set of data associated with simulated operation of the pump system without the load stimulus. The first impulse response and the second impulse response are based on a comparison of the first set of data and the second set of data. The controller is further configured to generate a second model of the pump system. Generating the second model of the pump system includes measuring, during operation of the pump system, a second set of position data and a set of force data associated with the rod, estimating, based on the identified first impulse response, the force data, and the position data, one or more force values of a downhole condition of the rod, and estimating, based on the identified second impulse response and the one or more -2- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 force values, one or more position values of a downhole condition of the rod. The controller is further configured to operate the pump system based on the second model. Another embodiment of the present disclosure relates to a method of controlling a pump system. The pump system includes a pump disposed within a well and an actuator operable to move a rod including a surface end coupled to the actuator and a downhole end coupled to the pump. The method includes identifying a first impulse response and a second impulse response associated with the pump system. The identification includes measuring a first set of position data associated with the surface end of the rod and generating, based on a first model of the pump system and the position data, a first set of data associated with simulated operation of the pump system with a load stimulus, and a second set of data associated with simulated operation of the pump system without the load stimulus. The first impulse response and the second impulse response are based on a comparison of the first set of data and the second set of data. The method further includes generating a second model of the pump system. Generating the second model of the pump system includes measuring, during operation of the pump system, a second set of position data and a set of force data associated with the rod, estimating, based on the identified first impulse response, the force data, and the position data, one or more force values of a downhole condition of the rod, and estimating, based on the identified second impulse response and the one or more force values, one or more position values of a downhole condition of the rod. The method further includes operating the pump system based on the second model. Yet another embodiment of the present disclosure relates to a controller for controlling a pump system. The pump system includes a pump disposed within a well and an actuator operable to move a rod including a surface end coupled to the actuator and a downhole end coupled to the pump. The controller includes one or more processors and a memory. The one or more processors are configured to identify a first impulse response and a second impulse response associated with the pump system. The identification includes measuring a first set of position data associated with the surface end of the rod and generating, based on a first model of the pump system and the position data, a first set of data associated with simulated operation of the pump system with a load stimulus, and a second set of data associated with simulated operation of the pump system without the load stimulus. -3- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 The first impulse response and the second impulse response are based on a comparison of the first set of data and the second set of data. The one or more processors are further configured to generate a second model of the pump system. Generating the second model of the pump system includes measuring, during operation of the pump system, a second set of position data and a set of force data associated with the rod, estimating, based on the identified first impulse response, the force data, and the position data, one or more force values of a downhole condition of the rod, and estimating, based on the identified second impulse response and the one or more force values, one or more position values of a downhole condition of the rod. The one or more processors are further configured to operate the pump system based on the second model. Some embodiments relate to a controller for controlling a pump system including a pump disposed for use at a well. The controller includes one or more processors and a memory. The one or more processors are configured to provide a first impulse response using a first model in response to a surface surface position input associated with the pump and a surface load input associated with the pump. The first model is a neural network. The one or more processors are also configured to provide a downhole position associated with the pump and a downhole load associated with the pump in response to the surface surface position input associated with the pump and a surface load input associated with the pump and the first impulse response using a second model. The second model is a regression model or a neural network model. The one or more processors are also configured to operate the pump system using the second model. Some embodiments relate to a controller for controlling a pump system comprising a pump disposed for use at a well. The controller includes one or more processors and a memory. The one or more processors are configured to provide a downhole position associated with the pump and a downhole load associated with the pump in response to the surface surface position input associated with the pump and a surface load input associated with the pump using a model, the model being a regression model or a neural network model trained using will specific data. The one or more processors are also configured to operate the pump system using the second model. -4- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 This summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the devices or processes described herein will become apparent in the detailed description set forth herein, taken in conjunction with the accompanying figures, wherein like reference numerals refer to like elements. BRIEF DESCRIPTION OF THE DRAWINGS Various objects, aspects, features, and advantages of the disclosure will become more apparent and better understood by referring to the detailed description taken in conjunction with the accompanying drawings, in which like reference characters identify corresponding elements throughout. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. FIG. 1 is a schematic diagram of a system that includes a pump disposed in a subterranean environment, according to one embodiment. FIG. 2 is a schematic diagram of a method of operating the pump assembly of FIG. 1, according to one embodiment. FIG. 3 is a schematic diagram of an instrumented pump system and a dynacard plot regarding the operation of the pump system based on measurements of the instrumented pump system, according to one embodiment. FIG. 4 is a flow diagram of a method for generating the dynacard plot of FIG. 3, according to one embodiment. FIG. 5 is a flow diagram of a method for generating diagnostics, according to one embodiment. FIG. 6 is a flow diagram of a method for training a recurrent neural network model for use in the method illustrated in FIG. 5, according to one embodiment. FIG. 7 is a block diagram of a a recurrent neural model for use in the method illustrated in FIG. 5, according to one embodiment. -5- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 FIG. 8 is a block diagram of a method for training a regression model for use in the method illustrated in FIG. 5, according to one embodiment. DETAILED DESCRIPTION Before turning to the figures, which illustrate certain exemplary embodiments in detail, it should be understood that the present disclosure is not limited to the details or methodology set forth in the description or illustrated in the figures. It should also be understood that the terminology used herein is for the purpose of description only and should not be regarded as limiting. The present disclosure relates to pump systems, including, but not limited to, estimating one or more conditions associated with downhole pump systems and operating pump systems in accordance therewith. Reciprocating pump systems, such as sucker rod pump (SRP) systems, may extract fluids from a well and employ a downhole pump connected to a driving source (e.g., an actuator) at the surface. A rod string connects a surface driving force to the downhole pump in the well. When operated, the driving source cyclically raises and lowers the downhole pump, and with each stroke, the downhole pump lifts well fluids toward the surface. For example, on an upward motion of each stroke, a standing valve at the bottom is open and fluid is sucked into the bottom side of the below the piston, while the fluid on top of the piston is lifted up. On the downward motion of each stroke, a traveling valve opens, and the standing valve is closed, which allows a barrel on top of the piston to refill with fluid. If the pump is partly filled with gas, there is a delay before the traveling valve opens. In some embodiments, the pumping system is used in the petroleum industry, water industry, waste industry and general processing/manufacturing plants. In some embodiments the systems and methods provide for condition monitoring of equipment involved in the petroleum industry, water industry, waste industry and general processing/manufacturing plants. In some embodiments, the systems and methods are used in integrated well site automation products in the field, in integrated cloud products (for instance reservoir monitoring, modeling, validation, planning, optimization), and for -6- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 statistical data analytics for process and design improvements. In some embodiments, the systems and methods provide process estimates for SRP automation in deviated wells. Referring now to FIG. 1, a pump system 100 is shown, according to some embodiments. The system 100 includes a pump assembly 101 as driven by a pump drive system 104 that is operatively coupled to a controller 122. For example, the pump assembly 101 and drive system 104 may be arranged as a beam pump. In some embodiments, the system 100 further includes a walking beam 138 that reciprocates a rod string 144. The rod string 144 may include a polished rod portion 146 that can move in a bore of a stuffing box 150 of a well head assembly that includes a discharge port in fluid communication with a flowline 152. The rod string 144 may be suspended from the walking beam 138 via one or more cables 142 hung from a horse head 140 for actuating a downhole pump 110 of the pump assembly 101 where the downhole pump 110 is positioned in a well 102. For example, the well 102 may be in a subterranean environment, and the downhole pump 110 may be positioned near a bottom 112 of the well 102. In some embodiments, the well 102 may be a cased well or an open well. For example, a partially cased well may include an open well portion or portions. As shown in FIG. 1, the well 102 includes casing 106 that defines a cased bore where tubing 108 is disposed in the cased bore. An annular space may exist between an outer surface of the tubing 108 and an inner surface of the casing 106. In some embodiments, the walking beam 138 is actuated by a pitman arm (or pitman arms), which is reciprocated by a crank arm (or crank arms) 134 driven by a prime mover 130 (e.g., electric motor, etc.). For example, the prime mover 130 may be coupled to the crank arm 134 through a gear reduction mechanism, such as gears of a gearbox 132. In some cases, the prime mover 130 is a three-phase AC induction motor that can be controlled via circuitry of the controller 122, which may be connected to a power supply. The gearbox 132 of the pump drive system 104 may convert electric motor torque to a low speed, high torque output for driving the crank arm 134. The crank arm 134 may be operatively coupled to one or more counterweights 142 that serve to balance the rod string 144 and other equipment as suspended from the horse head 140 of the walking beam 138. A counterbalance may be provided by an air cylinder such as those found on air-balanced units. -7- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 In some embodiments, the downhole pump 110 is a reciprocating type of pump that includes a plunger 116 attached to an end of the rod string 144 and a pump barrel 114, which may be attached to an end of the tubing 108 in the well 102. The plunger 116 can include a traveling valve 118 and a standing valve 120 positioned at or near a bottom of the pump barrel 114. During operation, for an up stroke where the rod string 144 translates upwardly, the traveling valve 118 can close and lift fluid (e.g., oil, water, etc.) above the plunger 116 to a top of the well 102 and the standing valve 120 can open to allow additional fluid from a reservoir to flow into the pump barrel 114. As to a down stroke where the rod string 144 translates downwardly, the traveling valve 118 can open and the standing valve 120 can close to prepare for a subsequent cycle. Operation of the downhole pump 110 may be controlled such that a fluid level is maintained in the pump barrel 114 where the fluid level can be sufficient to maintain the lower end of the rod string 144 in the fluid over its entire stroke. As an example, the system 100 can include a beam pump system. As explained, a prime mover can rotate a crank arm, whose movement is converted to reciprocal movement through a beam. The beam can include counterweights or a compressed air cylinder to help reduce load on the beam pump system during the upstroke. The beam can be attached to a polished rod by cables hung from a horsehead at the end of the beam. The polished rod can pass through a stuffing box and be operatively coupled to the rod string. As explained, the rod string can be lifted and lowered within the production tubing of a cased well by the reciprocal movement of the beam, enabling the downhole pump to capture and lift formation fluid(s) in a direction toward surface (e.g., with a flow vector component against gravity) in the tubing and through a pumping tee that directs the fluid into a flowline. As an example, the prime mover may be an internal combustion engine or an electric motor that provides power to the pumping unit. As an example, a prime mover can deliver highspeed, low-torque power to a gear reducer, which converts that energy into the low-speed, high-torque energy utilized by the surface pump. As shown in FIG. 1, a beam pumping unit, beam pump system or merely beam pump, converts the rotational motion of the prime mover into a reciprocating vertical motion that lifts and lowers a rod string connected to a subsurface pump. -8- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 Some aspects of a system can include prime mover type; pumping unit size, stroke length and speed setting; rod and tubing diameter; and downhole pump diameter, for example, based at least in part on reservoir fluid composition, wellbore fluid depth and reservoir productivity. As an example, a design framework may facilitate some decisions as to design, for example, to arrive at a desired pump speed to attain production targets without overloading the system or overwhelming the formation’s ability to deliver fluids to a wellbore. Beam pumps may be constructed in a variety of sizes and configurations. Some systems include design aspects that can aim to better manage torque, rod wear and/or footprint. For example, as to some design aspects, consider locating counterweights on the crank arm or on the beam and use of compressed air rather than weight to assist in load balancing. Further examples can involve changes to crank, gear reducer and motor position relative to the beam, as well as alternative beam designs, where such factors may change system loading. As an example, a system may place heavier rods, or sinker bars, in the lower section of the rod string to keep the rod string in tension, which reduces buckling and may help prevent contact with the tubing wall. Rod strings may also include stabilizer bars between sinker bars to centralize the rods, further reducing tubing wear. Rod guides, which may be made of reinforced plastics, may be molded onto steel rods at depths where engineers may predict the rods will experience side loading due to a deviated wellbore path. The guides can act like bearings between the tubing wall and the rod to prevent rod and tubing wear. Sliding guides may be able to move between molded guides during the pump cycle, aiding production by scraping paraffin from the tubing wall, which helps prevent well plugging. A rod rotator or tubing rotator may be used to rotate the rod a small fraction of a revolution on each stroke of the pumping unit to further extend rod string life. As an example, slow rotation of rod guides may help scrape paraffin from the tubing wall. -9- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 Sucker rods may be connected to the surface pumping unit by a polished rod. A polished rod, for example, made of standard alloy steel and hard-surface spray metal coating, can support loads created during a pump cycle and help to ensure a seal through a stuffing box at a top of a well. The stuffing box can be attached to a wellhead or pumping tee and can form a low-pressure tight seal against a polished rod. The seal can form a barrier between a well and atmosphere and may allow flow to be diverted into a flowline, for example, via a pumping tee. FIG. 2 shows cut-away view of the downhole pump 110, which shows a portion of a rod 144, the pump barrel 114, the plunger 116, the traveling valve 118, and the standing valve 120 positioned at or near the bottom of the pump barrel 114. Further shown in FIG. 2 are an opening 117 for inflow of fluid(s) and a chamber 119, which is shown to be in a space disposed at least in part between the traveling valve 118 and the standing valve 120. The downhole pump 110 is an example of a pump mechanism that can move fluid, where such fluid can differ with respect to time. As an example, fluid can be liquid and/or gas. As an example, fluid can include entrained solids, semi-solids, etc. FIG. 2 shows an example of a method 200 with actions or states 210, 220, 230 and 240, which can be portions of a cycle (e.g., cycle actions, cycle states, etc.). As to the action 210, the pump 110 has achieved a maximum downward reach of a cycle. In the action 220, a beam can begin its upward movement such that the rod 144 and plunger 116 are pulled upwardly, forcing the ball of the traveling valve 118 to be on to its seat. This upward movement reduces the pressure in the pump chamber 119 until it is less than the pressure at the pump intake 117. The ball in the standing valve 120 can then come off its seat, allowing formation fluid to enter via the intake 117 and flow to the pump chamber 119. As to the action 230, the standing valve 120 is closed as the plunger 116 is at the end of the upward stroke. As to the action 240, as the plunger travels down, the pump chamber 119 experiences a pressure increase, pushing the ball in the traveling valve 118 off its seat. The action 240 allows the formation fluid to flow from the pump chamber 119 into the tubing via the plunger 116 as the plunger 116 continues to move downwardly in the pump 110. A cycle can include the actions 210, 220, 230 and 240. Such a cycle can be repeated thousands of times per day. -10- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 The fluid displaced into the tubing may be carried toward surface on subsequent upward strokes of the plunger 116. FIG. 3 shows an example of a system 300 with a controller 322 and various sensors that include position sensors and load sensors. For example, as to position sensors consider an inclinometer 332 and proximity switches 333 (e.g., Hall Effect sensors); and, for example, as to load sensors, consider a load cell 334, current sensors 335 and a beam transducer 336. Such sensors can be operatively coupled to the controller 322 (e.g., via wire and/or wirelessly through wireless circuitry). As an example, the load cell 334 can be a load- capable dynamometer attached to the polished rod for acquiring dynamic data, which may be transmitted and/or otherwise accessed by one or more pieces of equipment. A controller can utilize sensor data to calculate rod loading (e.g., a surface condition) and, coupled with various models (e.g., algorithms), to estimate downhole pump fill (e.g., a downhole condition). A frequent challenge to downhole pump operation is the entry of gas into the pump, leading to fluid pound or gas interference. Fluid pound occurs when the plunger travels down quickly through low-pressure gas and then suddenly hits liquid fluid; the resulting compressive shock can damage rod strings and the prime mover gearbox. Gas interference is less damaging and occurs when the plunger travels down through high- pressure gas. Both conditions can reduce system efficiency. To combat gas interference, gas separators may be placed below the pump to redirect the gas into the wellbore annulus around the pump. Other modifications may be made to a completion to counter or reduce the effects of heavy oil and sand or other produced solids. Operators can diagnose gas interference, liquid fluid pound severity and various other operating conditions using a dynamometer, which plots rod tension versus displacement measurements at the surface and downhole at the pump. The shape of an ideal downhole graph, called a dynamometer card, is rectangular and indicative of a full pump. Deviations from the ideal shape can indicate performance issues, such as gas interference, -11- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 system leaks, stuck pumps, parted rods and various other anomalies that may be identified and accounted for automatically or through manual intervention. As rod pumping systems are relatively inexpensive to install and operate and have a relatively long life, rod pumping systems tend to be a quite common form of artificial lift. They tend to be “simple” machines that have a long and well-documented history in the industry, and they tend to be adjustable to meet changing well or field conditions. The use of rod pumps is likely to increase as the industry continues to expand its involvement in shale formations and other unconventional plays, which require operators to use high numbers of relatively low-flow-rate wells to exploit each field. Initial high pressures and high production volumes from these hydraulically fractured horizontal wells are quickly followed by low bottomhole pressures and steep production decline rates; production is possible through the use of artificial lift systems, of which rod pumps tend to be efficient at these low rates. Referring now to FIG. 3, a plot of a dynamometer card for a pump system, such as the pump system 100, is shown according to some embodiments. A dynamometer card is a record made by a dynamometer. A dynamometer is an instrument used in sucker-rod pumping to record the variation between a polished rod load and a polished rod displacement. Dynamometer cards may be used in the oilfield industry (among other settings) as force versus position to assess the integrity of a downhole displacement pump condition. The downhole force is estimated from a direct surface force and position measurement at the polished rod or related measurements through a mathematical model, generally referred to as the Gibbs wave equation. An analysis of dynamometer measurements may reveal a defective pump, leaky tubing, inadequate balance of the pumping unit, a partially plugged mud anchor, gas locking of the pump or an undersized pumping unit. A dynamometer card may be in the form of a graph, such as a dynagraph. Even if not the initial artificial lift system of choice, rod pumping systems tend to be installed on many types of wells as production rates decline and the economics of initial systems are undone by higher operating costs. As a consequence, rod pumping systems are likely to maintain their position as a frequently deployed artificial lift technique. -12- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 FIG. 3 also shows a surface condition plot 370 and a downhole condition plot 390, which are plots of load versus distance with respect to time, for example, with respect to one or more cycles that include the actions 210, 220, 230 and 240 of FIG. 2. As to the downhole condition plot 390, as mentioned, it can be based on a model. For example, the downhole force may be estimated from a direct surface force and surface position measurement at the polished rod (and/or or related measurements) through a mathematical model, generally referred to as the Gibbs wave equation (e.g., the “wave equation”). The wave equation describes the relation between surface and downhole force and position acting on the rod. Depending on the implementation, the wave equation may include various types of factors such as velocity of sound in a rod, modulus of elasticity of the material of rods, length of a rod string, number of increments in position, number of discretization in time, pump velocity (e.g., cycles per minute, strokes per minute, etc.), rod stroke length, rod diameter, specific weight of rod material, a factor of dimensionless damping, specific gravity of fluid, diameter of tubing, etc. Referring now to FIG. 4, a flow 400 for determining a dynacard model of the pump system 100 is shown, according to some embodiments. As suggested above with reference to FIG. 3, dynacards are force versus position plots used in the oilfield industry (or other applicable industries) to assess the integrity of a downhole displacement pump operation (e.g., the pump assembly 101 of the pump system 100). In some cases, the pump assembly 101 may be instrumented in order to determine the various defects mentioned above. However, instrumenting the pump assembly 101 may be expensive and unpractical in some cases, due to the subterranean environment defining and otherwise surrounding the well 102. Accordingly, a pump pressure and downhole pump position may be indirectly assessed from a downhole force acting on the pump plunger (e.g., a downhole force on the plunger 116). In some embodiments, the downhole force on the plunger 116 is estimated from a direct surface force measurement and a direct position measurement at the polished rod 146 (and/or or related measurements) through a mathematical model, generally referred to as the Gibbs wave equation. In some embodiments, the Gibbs wave equation describes a relationship between (1) a surface force measurement on the polished rod 146 and a surface position -13- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 measurement of the polished rod 146 and (2) a downhole force on the rod 144 and a downhole position of the rod 144 (e.g., a well trajectory problem). Depending on the implementation, the Gibbs wave equation may solve the well trajectory problem using factors involving the rod 144 such as a force or moment balance between a Newton inertial force, a distributed elastic force, a solid friction force, a viscous damping force, a gravity force, and a buoyant force. Accordingly, to solve the well trajectory problem, properties of the rod string 144 and fluid properties of the pump assembly 101 (e.g., fluid interactions of the plunger 116 within the downhole pump 110, etc.) may need to be known. In some embodiments, the Gibbs wave equation may be applied to a vertical well (e.g., a well that extends in a substantially one-dimensional vertical direction). With regard to vertical wells, the Gibbs wave equation may solve the well trajectory problem by way of a direct solution, either through a piecewise analytical solution based on a Fourier series of the acquired signals (e.g., the direct surface force measurement and the direct position measurement at the polished rod 146). Alternatively, a discretized solution as proposed by an Everitt-Jennings algorithm may be used. In regards to deviated wells (e.g., a well that extends in various directions and dimensions beyond those that characterize vertical wells), applying the Gibbs wave equation to solve the well trajectory problem may be more complex. In some embodiments, the Gibbs wave equation may be estimated through a model. In other words, a model may be used to estimate (e.g., anticipate, model, predict, etc.) the calculations of the Gibbs wave equation as it would be used to solve the well trajectory problem. Depending on the implementation, the model may be one-dimensional, two- dimensional, or three-dimensional in nature. In this sense, a one-dimensional model may anticipate vertical (e.g., upwards and downwards in terms of the rod string 144 as depicted with reference to FIG. 1, as an example) forces and/or displacements of the rod string 144. A two-dimensional model may anticipate forces and/or displacements of the rod string 144 in terms of the one-dimensional model, with an added dimension for lateral horizontal forces and/or displacement (left and right in terms of the rod string 144 as depicted with reference to FIG. 1, as an example). Finally, a three-dimensional model may anticipate forces and/or displacements of the rod string 144 in terms of the two-dimensional model, with an added dimension for torsional forces and/or displacement, as well as abscissa forces and/or -14- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 displacement (twisting, as well as forward - e.g., out of the page - and backward - e.g., into the page - in terms of the rod string 144 as depicted with reference to FIG. 1, as an example). As suggested above, the Gibbs wave equation may be solved in a relatively simplistic manner for vertical wells using a one-dimensional model. However, deviated wells may often require a two-dimensional or three-dimensional model for accurately solving the well trajectory problem using the Gibbs wave equation. Alternatively, the Gibbs wave equation may be solved for deviated wells using a one-dimensional model. However, this may require a simplified estimation that necessitates ignoring forces relating to multi- dimensional aspects such as bending moments, buckling, and/or torsional stiffness of the rod string 144. Accordingly, it would be advantageous to provide a solution to the Gibbs wave equation for deviated wells using a two-dimensional or three-dimensional model that provides a more accurate incorporation of the actual conditions associated with deviated wells. For example, such solutions may not only allow for a proper accounting of bending moments and/or torsional stiffness properties of the rod string 144, but may further allow a more accurate determination of motion in all dimensions in order to model the effects of buckling in the rod string 144. In some embodiments, utilization of two-dimensional models and three- dimensional models of the Gibbs wave equation may each offer advantages relative to each other. On the one hand, two-dimensional models may require less computation and total bandwidth for a supervisory device such as the controller 122. On the other hand, three- dimensional models may require more computation and total bandwidth for the controller 122. For example, two-dimensional models may require three interrelated wave equations (structured to model the Gibbs wave equation) that identify vertical and horizontal forces and/or displacements regarding the rod-string 144, as suggested above. Three-dimensional models, conversely, may require six interrelated wave equations: the three wave equations mentioned above, further integrated with three additional wave equations for identifying abscissa forces and/or displacement, as well as torsional forces and/or displacement. In spite of the additional computational requirements mentioned above, three-dimensional models may of course provide a more accurate model of the Gibbs wave equation in terms of solving the well trajectory problem. Accordingly, either a two-dimensional or three-dimensional -15- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 model may be desirable, dependent upon the particular complexity of well deviation, available computing resources, and so on. Generally, however, both the two-dimensional and three-dimensional models may each provide a substantial increase in computational complexity relative to one-dimensional models. Regardless of a selection between two- dimensional and three-dimensional models, solving the Gibbs wave equation in accordance therewith may offer practical challenges associated with an amount of time required for computation, numeric stability (e.g., uncertainty) challenges, and so on. The systems and methods described herein may provide an advantageous solution for capitalizing on the increased accuracy of using two-dimensional and/or three-dimensional models of the Gibbs wave equation for solving the well trajectory problem, while also limiting (or otherwise eliminating) at least the challenges mentioned above (if not others) otherwise associated with utilization of such models as opposed to a one-dimensional model. As discussed above, the Gibbs wave equation may be solved in multiple ways in terms of a number of dimensions depicted by the model of the Gibbs wave equation. However, as suggested above, it would be advantageous to provide systems and methods that not only leverage the improved accuracy associated with two-dimensional and/or three- dimensional models, while also limiting side-effects associated with increased computational requirements. Accordingly, the systems and methods provide herein may relate to a multi- stage process (as defined by the flow 400 below). For example, at a first stage of implementation and/or operation of the pump system 100 (e.g., a “planning phase”), it may be advantageous to leverage the advantages of two-dimensional and/or three-dimensional models for determining a first model configured to solve the Gibbs wave equation. At this planning phase, relationships between surface conditions and downhole conditions for the pump system 100 may be determined. At a second stage, the pump system 100 may then leverage one or more aspects of the first model (as described in greater detail below) to identify a second model that is otherwise less complex, and therefore more efficient, relative to the first model. For example, the first stage may be a planning phase (e.g., a phase primarily directed toward providing the pump system 100 for a new well, such as the well 102). At this first stage, where various expenses with construction and implementation of the pump system in the well 102 are involved, accuracy in a model of the Gibbs wave equation may be paramount. The second stage may thus be a “diagnostic stage” associated with actual -16- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 operation of the pump system 100 and determining downhole conditions in real-time. In the second stage, computational advantages produced at the first stage with regard to the first model may be leveraged to a particular point, though at the second stage an emphasis may be shifted, somewhat, toward agile computation, therefore presenting greater advantages in a leaner model for solving the Gibbs wave equation, as described in greater detail below. Referring now with greater particularity to the flow 400, a “planning phase” for downhole dynamometer card estimation is initiated at process 401. As described in greater detail below with reference to processes 402-410, the “planning phase” may involve a forward model being used to determine one or more impulse responses that describe one or more relationships between surface conditions and downhole conditions in a generalized context. The forward model may apply assumed formation properties regarding various downhole force profiles to receive surface position values and a downhole force profile as input and provide a force distribution along the rod 144 as output. The force distribution may accordingly include the one or more impulse responses, which may be used to operate the pump system in a diagnostic phase of actual operation. At process 402, surface position values can be a priori simulated by the controller 122 or any other computing system configured to simulate conditions of the pump system 100. While “surface position values” as used herein may be simulated in regards to various moving components of the surface portion of the pump system 100 (e.g., the counterweights 142, the crank arm 134, the beam 138, the horsehead 140, the cables 142, and so on), in an exemplary embodiment of the present disclosure, the surface position of the polished rod 146 may be simulated in order to provide the systems and methods described herein. In some embodiments, measurements of surface conditions of the pump system 100 can be acquired from the actual surface position measurements and surface force measurements regarding the polished rod 146. For example, the surface position of the polished rod 146 may be detected by one or more position sensors of the pump system 100 (e.g., the inclinometer 322, the proximity switches 333, and/or other applicable sensors configured to detect a position of an object). The one or more position sensors of the pump system 100 may in turn provide the controller 122 with a steady transmission of the one or more position measurements of the polished rod 146. Therefore, the controller 122 may -17- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 compile at least one of a simulated or an acquired steady-state surface position signal X(t) with a known sampling rate - the number of values simulated by the controller 122 or measurements received by the controller 122 from the one or more position sensors within a given time frame. In some embodiments, the time frame may be a standard measure of time, such as one second. In other embodiments, the controller may further determine an amount of time required for one cycle of movement for the polished rod 146 (e.g., moving from an initial point and returning to the initial point through one complete cycle of the operation of the pump system 100), and base the known sampling rate upon this determined time frame. As described in greater detail below, X(t) may be used as an input stimulus for one or more simulations of the pump system 100 using the forward model. At each of processes 403 and 404, the forward model may be used to simulate the operation of the pump system 100. As discussed above, the forward model may be the two-dimensional or three-dimensional model configured to solve the Gibbs wave equation. In particular, the forward model may utilize parameter and observer techniques from control theory. For example, the forward model can include an input that stimulates the pump system 100 (as simulated via the forward model) and an output that can be measured. In general, the input stimuli of the forward model may be a surface position value of the polished rod 146 (e.g., the simulated or acquired surface position values X_sf(t)), and a reference downhole force fdh(t). As opposed to Xsf(t), fdh(t) may not be an acquired signal, in terms of practical measurement. Rather, a series of reference downhole force values may be applied as f_dh(t) in order to determine a relationship between an estimated consequential (e.g., actual) downhole force F_dh(t) and the other variables involved in operation of the pump system 100, as described in greater detail below. Accordingly, the model that is used for the simulation(s) involved in processes 403 and 404 may be “forward” in the sense that an output variable is first provided as input variable in the form of pre-selected reference values for determining one or more relationships for actually estimating the output variable. Of course, fdh(t) may be provided in the same state (e.g., over the same sequence of steady-state samples defined by the t values as simulated or acquired in process 401) as X_sf(t). In some embodiments, the simulations conducted at processes 403 and 404 may differ based on the value(s) provided as fdh(t). In the case of process 402, a first -18- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 simulation may be performed by the controller 122 via the forward model via input stimuli X_sf(t) and f_dh(t), where f_dh(t) = 0 (e.g., the reference downhole force across t is zero). The pump system 100 may then be simulated over the sequence of t values in order to determine an estimated series of surface force values Fsf(t, 0) (the first input variable being t and the second input variable being f_dh(t) = 0) and an estimated series of downhole position values Xdh(t, 0) at process 405. In the case of process 404, a second simulation may be performed by the controller 122 via the forward model via stimuli X(t) and fdh(t), where fdh(t) is a pulse load (f_dh(t) = F_pulse(t)). Depending on the implementation, F_pulse(t) may be provided in a number of variable formats, for example, F_pulse(t) can be an anticipated load amplitude. Thus, at process 406, the controller 122 may determine two estimated series of values Fsf(t, Fpulse(t)) and Xdh(t, F_pulse(t)) based on the second simulation. At process 407, the controller 122 may determine a series of values indicating differences between Fsf(t, Fpulse(t)) and Fsf(t, 0) over t. In other words, at each value of t for which the pump system 100 was simulated by the first and second models, Fsf(t, 0) may be subtracted from F_sf(t, F_pulse(t)) in order to determine the series of values ΔF_{sf_p}(t), where “p” is denoted as indication that the difference was generated based on system modeling that utilizes the simulated or acquired surface position measurements X(t). Likewise, at process 408, the controller 122 may determine a series of values similarly indicating differences between Xdh(t, Fpulse(t)) and Xdh(t, 0) over t in order to similarly determine ΔXdh_p(t). At processes 409 and 410, impulse responses are determined that relate simulated or measured surface conditions of the pump system 100 (e.g., surface position values and surface force values regarding the polished rod 146) and calculable downhole conditions of the pump system 100 (e.g., downhole position values and downhole force values regarding the rod 144). Generally, an impulse response is a reaction of any dynamic system (such as the calculated downhole conditions of the rod 144) in response to some external change (such as a change to a simulated or measured surface condition regarding the polished rod 146). As described in greater detail below, the impulse responses may be used to estimate downhole force and downhole position values regarding the rod 144 based on simulated or measured surface position and surface force values regarding the polished rod 146. -19- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 In some embodiments, the shape of the downhole pump load may be unknown. When the simulation of the pulse load f_dh(t) is generated, it provides a transfer behavior of the rod 144. The transfer behavior of the rod 144 can be used to determine a first impulse response HF(τ). At process 409, the first impulse response HF(τ) that correlates ΔF_{sf_p}(t) and F_pulse(t) can be determined. In some embodiments, the first impulse response HF(τ) is determined by way of expressing these functions relative to one another using a convolution function that incorporates the first impulse response HF(τ) in an inverted format as a first transfer function h_F(τ). For example, h_F(τ) and F_pulse(t) may be expressed as inputs of a first convolution function, where ΔF_{sf_p}(t) is the output of the first convolution function. In general, a convolution function is a mathematical operation on two input functions (hF(τ) and F_pulse(t-τ)), that produces an output function

and thus expresses how the shape of one function (ΔF_{sf_p}(t)) is modified by the other (F_pulse(t)). In other words, the first convolution function may indicate how the difference between surface force on the polished rod 146 (as varied between the first simulation without the downhole force input stimulus, and the second simulation with the downhole force input stimulus) changes based on a change to the downhole force input stimulus regarding the rod 144. In such convolution functions, t is a constant and τ is a variable of integration for determining the output of the convolution function. The first convolution function is provided below as an illustrative example. ΔFsf_p(t) = conv(hF(τ), Fpulse(t-τ)) In some embodiments, the first impulse response H_F(τ) may then be determined by way of a system identification process that inverts h_F(τ). For example, the transfer function hF(τ) may be expressed in matrix form by expressing the associated functions ΔF_{sf_p}(t) and F_pulse(t-τ) in vector forms, where F_pulse(t-τ) is the input vector and ΔF_{sf_p}(t) is the output vector. The matrix may then be inverted (e.g., by way of inverting the associated vectors) to obtain HF(τ). In some embodiments, the first impulse response HF(τ) can be a Hankel matrix that can include the impulse responses as lines of the relation between the downhole force F_dh(t) and the surface differential force ΔF_{sf_p}(t). The following convolution function is provided below as an illustrative example. ΔFsf_p(t) = conv(hF(τ), Fdh(t)) = HF(τ) Fdh(t) -20- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 In some embodiments, deconvolution is implemented to determine the downhole force F_dh(t) =

ΔF_{sf_p}(t). For example, in some embodiments, a regularized implementation of a pseudoinverse matrix H^-1F(τ) can be utilized. As one example, a direct inversion process may be used. As another example, a direct inversion process with regularization may be used. As another example, a Wiener filter may be applied in order to invert the matrix. As yet another example, a Tikhonov regularization may be applied in order to invert the matrix. As another example still, the inversion may be solved with a direct solver in real time. In one example, selection of a regularization parameter may be applied, e.g., selecting regularization as a relatively small fraction of the relatively large Eigenvalue. For example, the matrix can be firstly transformed into a diagonal form, regularization values are added where Eigenvalues are below a threshold and then backwards transformation from the diagonal form can be performed. In one example, the inversion problem can be solved as a minimum search optimization problem. For example, the least square solution can be applied, and minimization problem to minimize the quadratic error can also be addressed through a gradient descent method or a conjugate gradient method. At process 410, a second impulse response HX(τ) that correlates ΔXdh(t) and Fpulse(t) is determined. Similar to process 409, HX(τ) is determined by way of expressing these functions relative to one another using a convolution function that incorporates the first impulse response HX(τ) in an inverted format as a first transfer function hX(τ). The second convolution function may indicate how the downhole force input stimulus changes based on the difference between estimated downhole position (as varied between the first simulation without the downhole force input stimulus, and the second simulation with the downhole force input stimulus). The second convolution function is provided below in an illustrative example, and H_X(τ) may be determined in a manner similar to H_F(τ) as discussed above (e.g., inversion of a matrix where ΔX_{dh_p}(t-τ) is the input vector and F_pulse(t-τ) is the output vector). Fpulse(t) = conv(hX(τ), ΔXdh_p(t-τ)) At process 411, a “diagnostic phase” may be initiated that applies hF(τ) and h_X(τ) in order to determine downhole force and position estimations regarding the rod 144 (e.g., downhole conditions) based on measured force and position values regarding the polished rod 146 (e.g., surface conditions). The diagnostic phase may be characterized as -21- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 described below with reference to processes 412-417. Depending on the implementation, the planning phase may be used in surveillance and control situations during practical operation of the pump system 100. At process 412, surface position measurements and surface force measurements are acquired by the controller 122. As compared to the surface position measurements simulated (or acquired) by the controller 122 with reference to process 402, the surface position measurements regarding the polished rod 146 acquired at process 412 may be used in real-time to determine downhole position and force values regarding the rod 144 as described in greater detail below. As described above with reference to process 401, the surface position measurements may be simulated (or acquired by position sensors of the pump system 100) and the controller 122 may compile a simulated (or acquired) steady-state surface position signal X₂(t) with a known sampling rate. The surface force values may be indicative of a load on the polished rod 146 (e.g., a load on the surface portion of the rod 144). When the surface force measurements are acquired from the actual surface conditions of the pump system 100, the surface force measurements may be detected by one or more load sensors of the pump system 100 (e.g., the load cell 334, the current sensors 335, the beam transducer 336, and/or other applicable sensors configured to detect a load on an object). The one or more load sensors of the pump system 100 may in turn provide the controller 122 with a steady transmission of the one or more force measurements on the polished rod 146. Therefore, the controller 122 may compile a simulated or acquired steady- state surface force signal F(t) with a known sampling rate. At process 413, the surface position signal X₂(t) and the surface force signal F(t) are synchronized with the surface position signal X1(t) (e.g., the surface position signal simulated or acquired at process 401 in the planning phase). For example, for each of X₂(t) and F(t), the signal phases associated therewith may be adjusted (e.g., shifted) to scale and match the signal phase associated with X1(t). At process 414, the controller 122 determines a difference ΔFSF(t, FDH) between the synchronized surface force values F(t) and the surface force values F_sf(t, 0) (the estimated force values determined at process 405 based on the simulation of the pump system 100 with a reference downhole force of zero at process 403) across time (t). ΔFSF is detailed -22- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 here to be a function of F_DH (as opposed to 0 or F_pulse(t) from the simulations described above with reference to processes 403-406) because ΔF_SF is considered to be a function of the “actual” downhole force FDH that is presently unknown and to be calculated as described below, rather than assumed via reference downhole forces (0, and Fpulse(t)). At process 415, the actual downhole force values F_DH(t) mentioned above are now estimated by applying the impulse response HF(τ) (e.g., calculated based on a correlation between ΔFsf_p(t) and Fpulse(t) as described above with reference to process 409) to ΔFSF(t, F_DH). At process 416, actual downhole position values X_DH(t) are similarly estimated by applying the impulse response

(e.g., calculated based on a correlation between ΔX_{dh_p}(t) and F_pulse(t) as described above with reference to process 410) to ΔX_SF(t, X_DH). At process 417, the acquired surface position measurements X₂(t) and acquired surface force measurements F(t) (see process 412) are mapped with reference to the estimated actual downhole position values XDH(t) (see process 415) and the estimated actual downhole force values F_DH(t) (see process 416) in order to generate a downhole dynacard that correlates X2(t) and F(t) with XDH(t) and FDH(t) across the timescale t. As described herein, a pump system 100 can implement one or more offline techniques and one or more online or live techniques to generate a digital twin, according to some embodiments. The digital twin may be an instantiation of one or more reduced order models (ROMs) that digitally encapsulates necessary model attributes across an expected operating space as a system, and may include design, installation, and model variables. The digital twin can be an instantiation of the ROMs at a particular point in time and may operate in real-time based on measurements and/or real-time information, for example, real-time inputs. The digital twin can output real-time outputs of any of the ROMs that are included in the digital twin. The digital twin may be implemented as one of the live techniques of the pump system 100, using real-time inputs (e.g., sensor data, measurements, etc.) and outputting real-time outputs (e.g., predicted values of one or more variables of a system, calculated values of one or more variables of the system, values of calibration variables of the system, etc.). The digital twin may be configured to estimate or predict values of variables -23- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 that are relatively more difficult to measure such as gas content, intake pressure, damping, and the like. It should be understood that these particular variables that are more difficult to measure are presented as an example and should not be understood as limiting. According to one embodiment, a ROM can be developed and database created for impulse responses over a parameter range for the method of FIG. 4. In some embodiments, the ROM can be an interpolatory tensorial reduced order model (tROM). In some embodiments, machine learning models may be utilized by method 400. For example, the planning or learning phase of method 400 can be implemented in two major steps: (i) calculating impulse responses H_F(τ), H_X(τ) and (ii) utilizing a regression model of the impulse responses

, HX(τ) based on input parameters (e.g., at least one or more of density, damping or viscosity, or the like). In some embodiments, the pump system 100 and/or method 400 can execute these two steps of the planning or learning phase in a single step through the recurrent neural networks, that can be, for example, recursive neural networks. From excitation simulation of a pulse force downhole fdh(t) = Fpulse(t) over the operation parameter and speed range, calculation of individual impulse responses H_F(τ),

correlated to this range can be obtained as, for example, a learning sample for a regression. In some embodiments, a recursive form of the dynamic equations can be obtained that can be further used for deconvolution in the subsequent run-time step(s) of the diagnostic phase of method 400. For example, the learning sample of impulse responses HF(τ), HX(τ) can be used to process a regression model that, in real time at the well site, facilitates calculation of the impulse responses H_F(τ),

at the estimated and measured operational parameters. In some embodiments, during a diagnostic phase for workflow of a hydrocarbon, oil, or petroleum system, or any other device of the pump system 100, initially, the surface values such as surface values of force F_sf(t) and position X_sf(t) can be used. These surface values can be utilized to determine estimated values for the damping, density Rho and friction coefficients, also initial estimated values for other relevant parameters of the pump system 100 may be determined. In some embodiments, a dynacard prediction model can be based on simulated learning samples. In some embodiments, the dynacard prediction model can be executed by interpolation with, for example, a look up table. Further, in some embodiments, Gaussian or neural network regression can be utilized. -24- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 In some embodiments, a prediction of parameters based on linearization at an operational point and autotuning for the method of FIG. 4 can be utilized. A regression model can be trained with preexisting well data and simulation data for the well. The output of the first regression model can then be used for a regression model input to predict the first impulse response H_F(τ) between the downhole force F_dh(t) and the surface differential force ΔFsf_p(t) (as to simulated no-load condition) and the impulse response

between downhole force Fdh(t) and downhole position Xdh(t) (relative to the simulated no-load condition at the same operational point). An additional regression model may be used to estimate the no-load curves (e.g., the curves without load) as a function of the operational point. In some embodiments, the impulse response H_F(τ) can be used together with the surface difference force ΔF_{sf_p}(t) in a deconvolution to identify a downhole force estimated value Fdh(t) and with the estimated value Fdh(t) through a convolution to determine estimated value Xdh(t) for downhole position. With these estimated values Fdh(t) and Xdh(t), a reconstruction of the downhole dynacard can be created. In one embodiment, the dynacard can be used in a multicolor image (such as for example, including two colors) for a regression model to predict the gas content and intake pressure. Optionally or alternatively, a third color can be also used to include the measured surface dynacard. The same or substantially the same image can be also used in a classification model. In some embodiments, convolutional neural networks (CNNs) for regression, autotuning, and estimation of parameters for the method of FIG. 4, can be utilized. Each learning sample database for each model can also be complemented with general dynacard data from other wells to increase the regression and classification learning sample database. This labeled data can be used for supervised learning; unlabeled data can be used for semi- supervised learning. To obtain autotuning, the forward model is run multiple times with varying well parameters so as to find the closest match to the downhole dynacard as expected from the pump parameters. -25- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 In some embodiments, deconvolution can be performed by at least one of two methods: (i) tested conjugate gradient method and (ii) pre-calculated Tikhonov regularization. In some embodiments, an initial regression model can be tested with the CNN, that can be retrained with latest data and/or complemented with unscaled and/or normalized data (of, e.g., surface force values). In some embodiments, a subsequent regression model can be a regression-based tROM; for example, resolution obtained according to this method can have higher quality for the relatively deep wells. In some embodiments, the final regression model and/or a classification model can be tested with the CNN, retrained with the latest data, and/or complemented with the unscaled and/or normalized data. In some embodiments, classification model can be a semi-supervised learning CNN. In some embodiments, the surface dynacard can also be complemented in a multicolor image with a surface pressure dynacard, e.g., surface tubing pressure p(xsf) over the surface position value or measurement. Although the convolutional neural networks are described herein for dynacard models, any other regression and classification model can be used. In some embodiments, a fast SRP downhole dynacard estimation for deviated well can be achieved by using machine learning aspects including but not limited to those described below. Dynacards can be embodied as force versus position plots used in the oilfield industry to assess the integrity of a downhole displacement pump operation. The primary interest is in the pump gas content related to the pump pressure of certain system. Instrumenting the pump for direct position and pressure measurements is generally expensive and unpractical, therefore, the pump pressure and pump position are indirectly assessed from the downhole force acting on the pump plunger in a SRP. The downhole force is estimated from a direct surface force and position measurement at the polished rod or related measurements through a mathematical model, generally referred to as the Gibbs wave equation in some embodiments. The wave equation describes the relation between surface and downhole force and position acting on the rod. It can be solved in multiple ways. In the planning phase, a forward model can be used. Based on assumed formation properties various downhole force profiles can be considered. The downhole force profile and the surface -26- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 motion are the inputs for the forward model. The output of the forward model is the force distribution along the rod string. It is primarily used to properly size the rod string. For surveillance and control in operation a different solution of the wave equation is used, also referred as the diagnostic solution. In this case the input for the solution of the wave equations is surface position and force and the output is downhole position and force. In some embodiments, a fast and robust solution that is based on initial simulation solutions of the forward model in the operational point of the surveilled well derived from the surface position measurement can be provided. The relation between surface signal and downhole signal simulation results are then approximated with a simpler dynamic model and its inverse. As long as the operation point does not change drastically, the inverse solution of the dynamic model approximation allows a calculation of downhole force and position that is fast enough for assessment of the pump health as well as for dynamic control in some embodiments. With reference to FIGS 5-9, embodiments of models are discussed which can be used as an alternative or modification to flow 400 (FIG. 4) and which can be used with SRPs. With reference to FIG.5, a recurrent neural network (RNN) implements a relation between downhole and surface predictions, measurements, or characteristics. An SRP flow 500 includes a model 502, a model 504, a model 506 and a model 508. Model 502 is a predictive surface dynacard regression model that receives periodic input of a surface force represented by function F_sf(x_sf) and provides a prediction of Rho Eta and Fr based on simulated learning samples. Model 504 is a predictive tROM model that receives periodic input of a surface force represented by function F_sf(t) and prediction of Rho Eta and Fr from model 502 and provides impulse results according to function h(t,Eta, Rho,fr). Model 504 uses interpolation with a look up table in some embodiments. In some embodiments, Gaussian or neural network (NN) regression can be used in model 504. Model 506 receives surface force represented by function F_sf(t) and position represented by the function x_sf(t) and impulse response or results from model 504 and provides a downhole force result and downhole position result represented by respective functions f_dh(t) and x_dh(t). Model 506 is a deconvolution model. In some embodiments, models 506 can be eliminated and replaced by RNN prediction in some embodiments. Model -27- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 508 receives surface force represented by function F_sf(x) and the downhole force represented by function f_dh(t) and provides alarms or classifications and gas content, Rho and intake pressure. Model 508 is a dynacard model and uses simulated learnings in some embodiments. A convolutional neural network (CNN) model 512 can replace models 502 and 404, and a recurrent neural network (RNN) model 514 can replace model 506 in some embodiments. Model 512 receives surface force represented by function F_sf(t) and position represented by the function x_sf(t) and provides impulse response or results. Model 514 receives surface force represented by function F_sf(t) and position represented by the function x_sf(t) and impulse response or results from model 504 and provides a downhole force result and downhole position result represented by respective functions f_dh(t) and x_dh(t). With reference to FIG. 6, recurrent NN training can utilize a flow 600 that accesses a simulation database of values including but not limited to: surface force (F_sf), surface position (X_sf), downhole force F_dh), gas content, Rho, Eta, fr, intake pressure (pi), etc. An RNN training operation 602 is used to provide RNN model 604 which can be used as model 514 (FIG. 5). Training operation 602 uses physical model parameters specific to the particular well in some embodiments. The parameters can include fixed physical parameters and parameters that change with time. With reference to FIG. 7, a regression for direct prediction with design parameters for a dynacard can be used. A NN model 702 receives surface force, surface position, and SRP model parameters (e.g., trajectory, rod sections, stiffness diameters, etc.) and provides Rfo, Eta, fr, gas fillage, intake pressure. Model 702 does not require dynamic models, has lower complexity, and requires less insight into underlying physics. Model 702 can be any regression model. With reference to FIG. 8, training can utilize a flow 800 that accesses a simulation database of values including but not limited to: surface force (F_sf), surface position (X_sf), downhole force F_dh), gas content, Rho, Eta, fr, intake pressure (pi), etc. Training operation 802 is used to provide regression model 804 which can be used as model 504 (FIG. 5) in some embodiments. Model 802 can be model 702 formed using training operation 802. Training operation 802 uses physical model parameters specific to the -28- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 particular well in some embodiments. The parameters can include fixed physical parameters and parameters that change with time. In some embodiments, model 704 uses parameters that change with time after operation 802 in some embodiments. Configuration of Exemplary Embodiments As utilized herein, the terms “approximately,” “about,” “substantially”, and similar terms are intended to have a broad meaning in harmony with the common and accepted usage by those of ordinary skill in the art to which the subject matter of this disclosure pertains. It should be understood by those of skill in the art who review this disclosure that these terms are intended to allow a description of certain features described and claimed without restricting the scope of these features to the precise numerical ranges provided. Accordingly, these terms should be interpreted as indicating that insubstantial or inconsequential modifications or alterations of the subject matter described and claimed are considered to be within the scope of the disclosure as recited in the appended claims. It should be noted that the term “exemplary” and variations thereof, as used herein to describe various embodiments, are intended to indicate that such embodiments are possible examples, representations, or illustrations of possible embodiments (and such terms are not intended to connote that such embodiments are necessarily extraordinary or superlative examples). The term “coupled” and variations thereof, as used herein, means the joining of two members directly or indirectly to one another. Such joining may be stationary (i.e., permanent or fixed) or moveable (i.e., removable or releasable). Such joining may be achieved with the two members coupled directly to each other, with the two members coupled to each other using a separate intervening member and any additional intermediate members coupled with one another, or with the two members coupled to each other using an intervening member that is integrally formed as a single unitary body with one of the two members. If “coupled” or variations thereof are modified by an additional term (i.e., directly coupled), the generic definition of “coupled” provided above is modified by the plain language meaning of the additional term (i.e., “directly coupled” means the joining of two members without any separate intervening member), resulting in a narrower definition than -29- 4885-4682-7157.1 Atty. Dkt. No.: 123960-0813 the generic definition of “coupled” provided above. Such coupling may be mechanical, electrical, or fluidic. The term “or,” as used herein, is used in its inclusive sense (and not in its exclusive sense) so that when used to connect a list of elements, the term “or” means one, some, or all of the elements in the list. Conjunctive language such as the phrase “at least one of X, Y, and Z,” unless specifically stated otherwise, is understood to convey that an element may be either X, Y, Z; X and Y; X and Z; Y and Z; or X, Y, and Z (i.e., any combination of X, Y, and Z). Thus, such conjunctive language is not generally intended to imply that certain embodiments require at least one of X, at least one of Y, and at least one of Z to each be present, unless otherwise indicated. References herein to the positions of elements (i.e., “top,” “bottom,” “above,” “below”) are merely used to describe the orientation of various elements in the FIGURES. It should be noted that the orientation of various elements may differ according to other exemplary embodiments, and that such variations are intended to be encompassed by the present disclosure. Although the figures and description may illustrate a specific order of method steps, the order of such steps may differ from what is depicted and described, unless specified differently above. Also, two or more steps may be performed concurrently or with partial concurrence, unless specified differently above. Such variation may depend, for example, on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. It is important to note that the construction and arrangement of the apparatus as shown in the various exemplary embodiments is illustrative only. Additionally, any element disclosed in one embodiment may be incorporated or utilized with any other embodiment disclosed herein. Although only one example of an element from one embodiment that can be incorporated or utilized in another embodiment has been described above, it should be appreciated that other elements of the various embodiments may be incorporated or utilized with any of the other embodiments disclosed herein. -30- 4885-4682-7157.1

Claims

Atty. Dkt. No.: 123960-0765 WHAT IS CLAIMED IS: 1. A pump system comprising a pump disposed within a well, an actuator operable to move a rod comprising a surface end coupled to the actuator and a downhole end coupled to the pump, and a controller configured to: identify a first impulse response and a second impulse response associated with the pump system, wherein the identification comprises: simulating a first set of position data associated with the surface end of the rod; generate, based on a first model of the pump system and the position data, a first set of data associated with simulated operation of the pump system with a load stimulus, and a second set of data associated with simulated operation of the pump system without the load stimulus, wherein the first impulse response and the second impulse response are based on a comparison of the first set of data and the second set of data; generate a second model of the pump system, wherein generating the second model of the pump system comprises: measuring, during operation of the pump system, a second set of position data and a set of force data associated with the rod; estimating, based on the identified first impulse response, the force data, and the position data, one or more force values of a downhole condition of the rod; estimating, based on the identified second impulse response and the one or more force values, one or more position values of a downhole condition of the rod; and operate the pump system based on the second model. 2. The system of claim 1, wherein the second model is a dynacard. 3. The system of claim 1, wherein the first model is a two-dimensional model of the rod. 4. The system of claim 1, wherein the first model is a three-dimensional model of the rod. -31--4682-7157.1 Atty. Dkt. No.: 123960-0765 5. The system of claim 1, wherein the comparison of the first set of data and the second set of data identifies a difference in surface force values and a difference in downhole position values between the first set of data and the second set of data. 6. The system of claim 5, wherein identifying the first impulse response and second impulse response further comprises: identifying a first transfer function that correlates the difference in surface force values with the load stimulus; determining the first impulse response based on the first transfer function; identifying a second transfer function that correlates the difference in downhole position values with the load stimulus; and determining the second impulse response based on the second transfer function. 7. The system of claim 6, wherein determining the first impulse response based on the first transfer function comprises expressing the first transfer function as a first matrix based on a first vector of the difference in surface force values and a second vector of the load stimulus values, and wherein determining the second impulse response based on the second transfer function comprises expressing the second transfer function as a second matrix based on a third vector of the difference in downhole position values and the second vector of the load stimulus values. 8. A method of controlling a pump system, the pump system comprising a pump disposed within a well and an actuator operable to move a rod comprising a surface end coupled to the actuator and a downhole end coupled to the pump, the method comprising: identifying a first impulse response and a second impulse response associated with the pump system, wherein the identification comprises: simulating a first set of position data associated with the surface end of the rod; generate, based on a first model of the pump system and the position data, a first set of data associated with simulated operation of the pump system with a load stimulus, and a second set of data associated with simulated operation of the pump system -32--4682-7157.1 Atty. Dkt. No.: 123960-0765 without the load stimulus, wherein the first impulse response and the second impulse response are based on a comparison of the first set of data and the second set of data; generating a second model of the pump system, wherein generating the second model of the pump system comprises: measuring, during operation of the pump system, a second set of position data and a set of force data associated with the rod; estimating, based on the identified first impulse response, the force data, and the position data, one or more force values of a downhole condition of the rod; estimating, based on the identified second impulse response and the one or more force values, one or more position values of a downhole condition of the rod; and operating the pump system based on the second model. 9. The method of claim 8, wherein the second model is, comprises or is part of a dynacard, a regression, or neural network model. 10. The method of claim 8, wherein the first model is a two-dimensional model of the rod. 11. The method of claim 8, wherein the first model is a three-dimensional model of the rod. 12. The method of claim 8, wherein the comparison of the first set of data and the second set of data identifies a difference in surface force values and a difference in downhole position values between the first set of data and the second set of data. 13. The method of claim 12, wherein identifying the first impulse response and second impulse response further comprises: identifying a first transfer function that correlates the difference in surface force values with the load stimulus; determining the first impulse response based on the first transfer function; identifying a second transfer function that correlates the difference in downhole position values with the load stimulus; and -33--4682-7157.1 Atty. Dkt. No.: 123960-0765 determining the second impulse response based on the second transfer function. 14. The method of claim 13, wherein determining the first impulse response based on the first transfer function comprises expressing the first transfer function as a first matrix based on a first vector of the difference in surface force values and a second vector of the load stimulus values, and wherein determining the second impulse response based on the second transfer function comprises expressing the second transfer function as a second matrix based on a third vector of the difference in downhole position values and the second vector of the load stimulus values. 15. A controller for controlling a pump system comprising, wherein the pump system comprises a pump disposed within a well and an actuator operable to move a rod comprising a surface end coupled to the actuator and a downhole end coupled to the pump, wherein the controller comprises one or more processors and a memory, the one or more processors configured to: identify a first impulse response and a second impulse response associated with the pump system, wherein the identification comprises: simulating a first set of position data associated with the surface end of the rod; generate, based on a first model of the pump system and the position data, a first set of data associated with simulated operation of the pump system with a load stimulus, and a second set of data associated with simulated operation of the pump system without the load stimulus, wherein the first impulse response and the second impulse response are based on a comparison of the first set of data and the second set of data; generate a second model of the pump system, wherein generating the second model of the pump system comprises: measuring, during operation of the pump system, a second set of position data and a set of force data associated with the rod; estimating, based on the identified first impulse response, the force data, and the position data, one or more force values of a downhole condition of the rod; -34--4682-7157.1 Atty. Dkt. No.: 123960-0765 estimating, based on the identified second impulse response and the one or more force values, one or more position values of a downhole condition of the rod; and operate the pump system based on the second model. 16. The controller of claim 15, wherein the first model is a two-dimensional model of the rod. 17. The controller of claim 15, wherein the first model is a three-dimensional model of the rod. 18. The controller of claim 15, wherein the comparison of the first set of data and the second set of data identifies a difference in surface force values and a difference in downhole position values between the first set of data and the second set of data. 19. The controller of claim 18, wherein identifying the first impulse response and second impulse response further comprises: identifying a first transfer function that correlates the difference in surface force values with the load stimulus; determining the first impulse response based on the first transfer function; identifying a second transfer function that correlates the difference in downhole position values with the load stimulus; and determining the second impulse response based on the second transfer function. 20. The controller of claim 19, wherein determining the first impulse response based on the first transfer function comprises expressing the first transfer function as a first matrix based on a first vector of the difference in surface force values and a second vector of the load stimulus values, and wherein determining the second impulse response based on the second transfer function comprises expressing the second transfer function as a second matrix based on a third vector of the difference in downhole position values and the second vector of the load stimulus values. -35--4682-7157.1 Atty. Dkt. No.: 123960-0765 21. A controller for controlling a pump system comprising a pump disposed for use at a well, wherein the controller comprises one or more processors and a memory, the one or more processors configured to: provide a first impulse response using a first model in response to a surface surface position input associated with the pump and a surface load input associated with the pump, the first model being a neural network; provide a downhole position associated with the pump and a downhole load associated with the pump in response to the surface surface position input associated with the pump and a surface load input associated with the pump and the first impulse response using a second model , the second model being a regression model or a neural network model operate the pump system using the second model. 22. A controller for controlling a pump system comprising a pump disposed for use at a well, wherein the controller comprises one or more processors and a memory, the one or more processors configured to: provide a downhole position associated with the pump and a downhole load associated with the pump in response to the surface surface position input associated with the pump and a surface load input associated with the pump using a model, the model being a regression model or a neural network model trained using will specific data; and operate the pump system using the second model. -36--4682-7157.1