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  • Research ArticleAn Improved Sucker Rod Pumping System Model and SwabbingParameters Optimized Design

    Weicheng Li ,1,2 Shimin Dong ,1 and Xiurong Sun 1

    1School of Mechanical Engineering, Yanshan University, Qinhuangdao, 066004, China2School of Engineering, Kings College, University of Aberdeen, Aberdeen, AB24 3UE Scotland, UK

    Correspondence should be addressed to Shimin Dong;

    Received 28 December 2017; Accepted 15 October 2018; Published 30 October 2018

    Academic Editor: Gen Q. Xu

    Copyright 2018 Weicheng Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    Considering the impact of fluid flowing into pump on sucker rod pumping system (SRPS) dynamic behaviors, an improvedSRPS model with new boundary model is presented, which is a fluid-solid coupled model with the interactions among surfacetransmission, rod string longitudinal vibration, plunger motion, and fluid flow. A uniform algorithm is adopted instead of themixed iteration algorithm for the surface transmission and downhole rod string vibration submodels, to reduce the difficulties ofsolving the entire SRPS model.The dynamic response comparison is executed between the improvedmodel and the currentmodel,and the results show that it will bring a calculation error on pump load and pump fullness if the progress of fluid flowing into thepump (PFFP) is ignored. Based on this improved model, a multitarget optimization model is proposed and the dynamic behaviorof SRPS is improved with the optimized swabbing parameters.

    1. Introduction

    The SRPS is widely used in oil fields. It comprises three parts:surface transmission unit converting rotational motion intolinear motion, sucker rod string as a joint between surfaceand downhole, and reciprocating pump exploiting the oil(see Figure 1). The importance of predicting the dynamicresponses of SRPS is to determine the operating situation andoil production [1]. For this equipment is usually set up in anopen-air environment, and the test data device mounted onit is usually broken, especially the main working subsystemwhich is located nearly one kilometer or more downhole,making it difficult to be tested. Therefore, a more accurateSRPS simulation model should be established although theresearch in this field of study has been carried out widely.Due to the slender rod string moving upwards and down-wards all time, an intense longitudinal vibration is produced.According to Figure 1, the SRPS model can be divided intorod string longitudinal vibration model, surface transmissionmodel, and downhole pumping model. Commonly, thismodel is solved using the rod string longitudinal vibrationequation as the foundation, surface and downhole model

    as boundary conditions. The most successful model of rodstring longitudinal vibration is the Gibbss wave equation,and based on that, the models are studied specifically on theenhancement of the surface and downhole boundaries withdifferent operating conditions [24]. The surface boundarycondition that includes the motor speed variations has beenextensively used and is more applicable in practice [5, 6]whereas the downhole boundary condition has been contin-uously improved. However, further study is still required dueto the inconsistent alteration and complication encounteredin downhole operation.

    The downhole boundary condition actually is a modeldescribing the pump operation. The pump operating modecan be divided into upstroke and downstroke. Duringupstroke, as the plunger moves upwards, the pump pressurewill decrease and the fluid will not be sucked into the pumpuntil this pressure drop down to pump inlet pressure. Forthe downstroke, the pressure will increase with the plungergoes down and the fluid will begin to be drained out whenthe pressure equals the pump outlet pressure. Based on theSRPSs operating state, the first universal downhole boundarymodel was divided into four phases with a vague formulation

    HindawiMathematical Problems in EngineeringVolume 2018, Article ID 4746210, 15 pages
  • 2 Mathematical Problems in Engineering

    Dynamic liquid level

    Pump inlet

    Pump outlet


    Pump depth

    Rod string

    Suspension point

    Horse headBeam



    Reduction gear box


    Link rod

    Figure 1: Sucker rod pumping system.

    [2]. In an ideal condition, during the fully loaded upstrokemovement, the pump load was set equivalent to the fluidload; pump load was set to zero for the unloaded downstrokemovement [7, 8]. As the pump operation phases are highlyrelated to pump pressure, it was revised with an explicitformation deduced by the interaction between pump outletpressure and the pressure in the pump barrel [914], asfollows:

    () = ( ) (1)where FPL is the pump load, N;Ap andAr are the cross sectionarea of plunger and rod string respectively, m2; pd is the pumpoutlet pressure, pa; p is the pump pressure, pa.

    For the pd is always considered as a constant pressure, thekey of this research is to establish an accurate pump pressuremodel which consists of four phases as shown in Figure 2.With considering of the gas, this downhole boundary modelis improved as shown in (2).When the pump is at phases 1 and3, the pressure variation obeys the rules of gas state equation.As for in phases 2 and 4, the pump pressure is taken as the psand pd separately.

    = ( + 0 (/)) phase 1

    = phase 2 = ( + (/))

    phase 3 = phase 4


    phase 1 phase 2 phase 3 phase 4

    gas e







    gas c






    e flui


    Figure 2: Four phases of pump operation.

    where and are the gas column length when plungeris arriving at bottom dead center and top dead center,respectively, m; up is the plunger displacement, m; Ls is thepump stroke displacement, m; Lp is the plunger length, m; is the fluid dynamic viscosity, pas; is the clearance betweenplunger and pump barrel, m; Dd is the pump diameter, m;ts and tt are the open time of standing valve and travellingvalve, respectively, s; tu is the upstroke time; q is the liquidinstantaneous leakage volume.

    In this formula, the principle that the gas/oil ratio ofclearance volume (the space volume when plunger arrives atbottom dead center) equals the gas/oil ratio of pump inletis applied. However, at this time, the pump pressure shouldbe the same as the pump outlet pressure. Thus, it is revisedwith the pump outlet pressure and this improved model hasbeen widely used until now. [1518]. However, the currentdownhole boundary model still exists some shortcomingsneed to be improved. Due to ignoring the progress of PFFP,it is established with the hypothesis of regarding the pumpis filled with the fluid, whose gas/liquid ratio always equalsthat at pump intake, as well as the one in the clearancevolume. This assumption is not applicable for the oil wellwith insufficient oil well deliverability (OWD) which will

  • Mathematical Problems in Engineering 3



    Rod string longitudinal


    Plunger motionPump pressure variationFluid motion

    Figure 3: SRPS coupled model sketch.

    result in the pumping fluid being unable to keep pace withthe plunger and cause an incomplete fullness. Meanwhile, apump load calculation error will be produced with keepingthe pump pressure as constant when fluid flows into thepump.

    The SRPS model lays the foundation for optimizingthe swabbing parameters to improve the system operationstatus, except for predicting and evaluating its dynamicresponse. Miska et al., 1997 [19], propose a computer-aidedoptimization method relying on a simple linear algebraicsystem model, so as to minimize the energy consumption.Firu et al., 2003 [20], present an improve optimization cri-terion including eight operational parameters to achieve themaximum system efficiency. Liu and Qi, 2011 [21], apply fluidflow characteristics in coalbed methane reservoirs to estimatethe production capacity and build the system efficiencyoptimization model combined with SRPS performance. Theabove optimization models are built with a simplified pumpload; then an improved system efficiency optimization modelis built jointing with formula (2) [22, 23]. However, thecurrent optimization models ignore the effect of swabbingparameters on pump fullness, in that the SRPS modelsrestriction. Besides that, single target optimization cannotmake an accurate and comprehensive presentation for SRPSwhose pumping progress is complex, multicomponent andinteractive.

    In this paper, firstly, an improved SRPSmodel is presentedwith the new downhole boundary model, considering themovement of fluid flowing into pump with gas instantaneousdissolution and evolution. Secondly, for the new downholeboundary model, a nonlinear fluid-solid coupled model willincrease the complexity of this improved SRPS model. Aunified numerical algorithm is applied on the whole modelto decrease the calculation time, instead of traditional mixediteration algorithm. Thirdly, a multiobjective optimizationmodel is proposed based on the improved SRPS model.Fourthly, the surface dynamometer card is collected to verifythe improvedmodels accuracy. Fifthly, the dynamic responsecomparison on the current SRPS model and improved SRPSmodel is executed. Finally, the optimization program isapplied on a test well, and the results are


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