WSEAS TRANSACTIONS on SYSTEMS Luiz H S Torres Leizer Schnitman. method Section 3 presents a mathematical model of, a real sucker rod pumping system Section 4 is de. voted to a brief description of the algorithm for the. adaptive controller used The results obtained from. the adaptive controller are presented and its perfor. mance demonstrated in section 5 Section 6 presents. the conclusions, 2 Sucker Rod Pumping System, In this artificial lift method the rotary movement of. a prime mover either an electric motor or a combus. tion engine localized at the surface of the pumping. unit is converted to the reciprocating movement of a. rod string This column transmits the reciprocating. movement to the pump components located at the bot. tom of the well which are responsible to elevating the Figure 2 Bottomhole schematic with sucker rod. fluid from the reservoir to the surface The sucker pumping system. rod pumping system can be divided into downhole and. surface elements see Fig 1, velocity which in turn varies the pumping speed mea. sured in cycles per minute CPM In this control strat. egy the variable speed drive VSD technique is used. This allows the pumping speed to be adjusted through. a frequency inverter device 21 17 It is important, to note that the production performance is affected by. the annular fluid level and operating at the minimum. possible annular level i e the minimum bottomhole, pressure maximizes the reservoir oil outflow 16 In. the design of a control system to increase oil produc. tion a dynamic model of the sucker rod pumping sys. tem may relate the pumping speed of the unit to the. fluid level in the annular well The dynamic model, may also reveal the relationship between the model. parameters and the real process According to the lit. erature 22 16 these parameters are usually related. to sources such as the fluid characteristics in the well. the environmental properties at the bottom of the hole. the electrical devices and the mechanical assembly. Figure 1 Components of a sucker rod pumping sys, 3 System Modeling. The rod string is the link between the pumping, unit located on the surface and the bottomhole pump In systems that use sucker rod pumps it is often desir. The bottomhole pump is a simple type of positive able for an operating range on the downhole fluid level. displacement reciprocating pump in which the fluid to be very close to the pump inlet level as showed. is displaced in one direction of the alternating move in Fig 2 This operating range is characterized by. ment The function of the bottomhole pump is to pro complete pump filling with the minimum possible bot. vide energy to the fluid in the reservoir 1 In Fig 2 a tomhole pressure This produces the minimum back. schematic of the bottomhole is presented The annular pressure in the production zone of the reservoir and in. well and pump inlet level are also shown turn increases oil production 16 In the Laborato rio. The pumping cycle due to the relative motion of de Elevac a o Artificial LEA in English the Artifi. the valves affects the bottomhole pressure The oil cial Lift Lab at the Universidade Federal da Bahia. production is controlled by varying the prime mover UFBA there is a real plant of a sucker rod pump. E ISSN 2224 2678 54 Volume 17 2018, WSEAS TRANSACTIONS on SYSTEMS Luiz H S Torres Leizer Schnitman. with an artificial well of 32 m height fully instru is considered as the process output and the pumping. mented with full access and a visible bottomhole All speed N t measured in CPM as the process input. components of this equipment are industrial grade and It is possible to obtain the volumetric balance of. the plant includes a supervisory system for data ac the liquid drainage from the annular well It can be. quisition and control A schematic of this well is pre written as follows. sented in Fig 3 and Fig 4, qa t qr t qp t 1, where qa t t is the flow in cubic meter per day. m3 d from anullar well to the production tubing, where is located the bottomhole pump qr t is flow. in m3 d from the reservoir to the production tubing. and qp t is the pump flow in m3 d The flow from, anullar well to the production tubing is defined as. qa t Aa h t 2, where dt h t is the level rate in the anullar well. and Aa is the anullar cross sectional area in square. inches in2 This area is calculated as follow, Figure 3 Infrastructure and layout. Aa Dic 2 Det 2 3, where Dic is the internal diameter in in of the casing. tubing and Det is the external diameter in in of the. production tubing The flow from the reservoir to the. production tubing is stated as, qr t P I Ps Pw t 4. where P I is called productivity index in kgf cm2, Ps is the static pressure in kgf cm2 and Pw t is the. well flowing pressure in kgf cm2 also called down, hole pressure Equation 4 also express a relation. ship between the oil production rate from the reservoir. qr t and the downhole pressure Pw t This relation, ship is called inflow performance relationship IPR. In order to obtain initially a simple and representative. reference model of the actual system in LEA the re. lationship presented in Eq 4 assumes a single phase. Figure 4 Well schematic of the rod pump in the LEA flow and the pressure in the porous media reservoir. is above the bubble point pressure saturation pres. In modeling procedures variables i e input and, sure Thus the flow qr t varies linearly with the. output signals are often used that measure a deviation. downhole pressure Pw t Fig 5 shows the linear IPR, from a reference or a nominal operating point In this. work a reference is chosen by considering a desired. The static pressure Ps is given by, operating range and the pump inlet level A schematic. of the well with the chosen reference and operating. Ps Pcs f AB 5, range is shown in Fig 4 It is important to note that the. production performance is associated with the annular where Pcs is the casing pressure in kgf cm2 in static. fluid level Therefore a dynamic model of sucker rod conditions i e the pumping system is turned off and. pumping may be developed by relating the pumping thus there is no production from the well f is the. speed of the unit with the fluid level in the annular well specific weight in Newton per cubic meter N m3. in order for example to assist in system monitoring or of the fluid and AB is the length in m between the. for control purposes Consequently in this model the static level hs and the casing as shown in Fig 2 In. level in the annular well h t measured in meters this work the pressure of the gas column on the fluid. E ISSN 2224 2678 55 Volume 17 2018, WSEAS TRANSACTIONS on SYSTEMS Luiz H S Torres Leizer Schnitman. in in2 Sp is the plunger stroke length in in and, N t is the pumping speed in cycles per minute The. value of the volumetric efficiency of the pump is, normally in the range of 70 and 80 Finally the level. rate into the anullar can be calculated as a function of. the pumping speed N t as follow, dt h t Aa h t, 2 36 10 2 Ap Sp 11. As shown in the Fig 2 the level in the anullar is, measured by using a pressure sensor The pump speed. can be varied by using a frequency inverter on the sur. face facilities, Figure 5 The inflow performance relationship. 4 Robust Adaptive Control Tech, in the annular well is not considered The well flowing In systems that use sucker rod pumps it is often de. pressure Pw t is given by sirable for the operating range on the downhole fluid. level to be very close to the pump inlet level In, Pw t Pcd f AB h t 6 this work a robust adaptive controller called indi. where Pcd is the casing pressure in kgf cm2 in dy rect variable structure model reference adaptive con. namics conditions i e the pumping system is turned trol IVS MRAC is applied to a sucker rod pump. on and thus there is production from the well and It is expected that the controller operate inside the. h t is the level as indicated in Fig 2 It is assumed desired range on the downhole fluid level More. here Pcs Pcd The flow from the reservoir to the pro over the controller must be able to adapt its parame. duction tubing in Eq 4 can be rewritten considering ters and demonstrate robustness in the case of process. the Eq 5 and Eq 6 changes uncertainties variations unmodeled dynam. ics and perturbations in the system It is essential to. qr t P I f h t 7 remark that such conditions are physical quantities of. a sucker rod pumping Variations for example can be. Thus the Eq 1 can be rewritten as follow the result of a change in the composition of the fluid. flow produced water oil and or gas or in some pa, d rameters of the system well reservoir such as Basic. Aa h t P I f h t qp t 8 Sediment and Water BS W Gas Oil Ratio GLR. dt Gas Oil Ratio GOR specific weight of the fluid and. the Productivity Index PI 2 There are also op, P I f erational problems that disrupt the system and may. h t h t qp t 9 compromise the productivity of the method such as. dt Aa Aa fluid pound phenomenon 16 leaks fluid in the travel. It could be observed that the dynamic model in ing and or standing valves or in the production tubing. Eq 9 is a linear relationship given by a first order low efficiency of the system due to incomplete filling. ordinary differential equation According to 23 the of the pump and a failure of the electric power system. pump flow in m3 d of a sucker rod pump can be frequency inverter. given by The IVS MRAC controller was initially proposed. in 24 as an alternative to the direct approach called. VS MRAC 25 Its mathematical description and the, qp t 2 36 10 2 Ap Sp N t 10 stability analysis and its proof in the presence of un. modeled dynamics and perturbations for the case of. where is the volumetric efficiency of the pump in relative degree one can be found in 26 A simpli. Ap is the cross sectional area of the pump plunger fied version including the practical application of the. E ISSN 2224 2678 56 Volume 17 2018, WSEAS TRANSACTIONS on SYSTEMS Luiz H S Torres Leizer Schnitman. IVS MRAC controller to the speed control of a three troller design is treated as the true model of the plant. phase induction motor can be found in 27 at the instant t The latter is used to calculate the. This technique was developed based on the indi controller parameters c t from the algebraic equa. rect MRAC approach called I MRAC The classic tion c t f p t The control law C and the. I MRAC method is one of the primary techniques in equation c f p are calculated to meet the perfor. adaptive control 28 29 and it is still generating new mance requirements for the model P p. research and publications 30 The desired response, is provided by a reference model and the control ob 4 1 Mathematical Description. jective is to minimize the error between the system un. der control and a model reference output In the IVS The mathematical development of the IVS MRAC. MRAC scheme there are new features added to the scheme related to adaptation of the controller algo. I MRAC approach By using the sliding mode con rithm to this specific problem will be presented here. trol based on variable structure control VSC 31 The transfer function of the plant can be described as. the I MRAC method is associated with fast transient follows. and good robustness to parameter uncertainties varia. tions and perturbations The IVS MRAC scheme es np s. timates the plant parameters instead of using the con dp s. troller parameters where kp is the high frequency gain For the case of. The IVS MRAC controller provides a straight relative degree one n 1 np and dp are monic. forward design for the relay amplitudes used in the polynomials set to be. switching laws of the control algorithm 27 because. the relays are directly associated with the plant pa n 1. rameters These parameters in turn represent the re np s sn 1 i sn 1 i 13. lationships among the physical parameters of the sys i 1. tem such as the resistances capacitances moments of n 1. inertia and friction coefficients that have uncertain dp s sn 1 sn 1 i 1 sn 1 i 14. ties that are relatively easily determined The block i 1. diagram shown in Fig 6 illustrates the general concept The desired response given by the transfer func. of the IVS MRAC scheme tion of the model reference signal can be written as. where ym is the model reference output It is assumed. the reference input r is a piecewise continuous signal. and uniformly bounded The polynomials nm and dm, are similarly defined as the following polynomials of. nm s sn 1 m i sn 1 i 16, dm s sn m 1 sn 1 m i 1 sn 1 i. Figure 6 The general concept of the IVS MRAC 17, scheme The vector of plant parameters which are as. sumed to be known is set to be, The model plant P p is parameterized in re. lation to the vector p An estimator generates p t p kp T 1 T T 18. by processing the input signal u and the output sig where Rn 1 contains the elements of i i n. nal y The estimate p t specifies a model charac 1 1 i R is element 1 in Eq 14 Rn 1. terized by P p t that for the purposes of the con contains the elements of i 1 i n 1 1 in. E ISSN 2224 2678 57 Volume 17 2018, WSEAS TRANSACTIONS on SYSTEMS Luiz H S Torres Leizer Schnitman. Eq 14 and in the same way one sets m m 1 and The values of kpnom k p and 1 are constants The. m in Eq 16 and Eq 17 When the plant param values k p and 1 are associated with the relay sizing. ies of the modeling and design of control systems for sucker rod pumping systems and other arti cial lift methods 15 16 17 However most of these sim ulations and models have been limited to theoretical studies that do not take into account the practical sit uations encountered in real production elds Addi

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