An Improved Finite-Difference Calculation of Downhole Dynamometer Cards for Sucker-Rod Pumps

Abstract

Summary This paper presents a finite-difference representation of the wave equationdeveloped for diagnostic analyses of sucker-rod pumping systems. A consistentmethod of computing the viscous damping term associated with the damped-waveequation is also presented. Introduction Sucker-rod pumping is the most widely used means of artificial lift. About85% to 90% of all producing wells in the U.S. are rod-pumped. Thus, a reliablemethod of analyzing these pumping systems is a necessity. For many years, the surface dynamometer has been used to analyze sucker-rodsystems. Interpretation of actual pump conditions from surface dynamometercards is often difficult, if not impossible. Results obtained from surfacecards are strictly qualitative and are dependent on the analyzer'sexpertise. The ideal analysis procedure would be to measure the actual pump conditionswith a downhole dynamometer. However, this situation is not economicallyfeasible. Therefore, an accurate method of calculating downhole pump cards frommeasured surface cards is needed. This paper presents a method for calculatingthese downhole cards that uses a presents a method for calculating thesedownhole cards that uses a finite-difference representation of the waveequation. First, a brief description of previous calculation techniques isgiven. Previous Methods. Past work involving the analysis of sucker-rod pumping Previous Methods. Past work involving the analysis of sucker-rod pumpingsystems can be divided into two categories. One category involves predictingthe performance of new sucker-rod installations by calculating predicting theperformance of new sucker-rod installations by calculating surface load fromknown surface position and pump load. The other category deals with thediagnosis of existing pumping installations by determining actual pumpconditions from measured surface conditions. This paper focuses on the lattercategory. Snyder was the first to develop a method for calculating down-hole forcesand displacements. His technique incorporates the method of characteristics tosolve the undamped-wave equation. Snyder assumed that the tension in the rod isthe result of two force waves, f (downward wave) and g (upward wave). Thevalues of f and g, calculated from the surface dynamometer card, would beconstant over the entire rod string for the undamped solution. Snyder correctedfor damping using a concentrated damping force to advance the values of f and gdown the rod string. These two force waves are then used to compute thedownhole pump card. Snyder's method is rigorously valid only for a uniformsucker-rod string. Gibbs and Neely developed an analytical technique in 1966 for obtaining subsurfaceconditions. The method uses truncated Fourier series approximations of the 1D, damped-wave equation to determine load and displacement. The relativesmoothness of the load/time and displacement/time curves is important in a Fourier analysis; however, the load function approaches a square wave at thepump. The Fourier series solution oscillates at the discontinuities of thissquare wave, restricting the number of terms that can be taken in the seriessolution and still preserve accuracy. In turn, the smaller number of terms inthe series preserve accuracy. In turn, the smaller number of terms in theseries causes the solution to be less accurate. Gibbs and Neely's analyticalmethod has become the primary means of calculating downhole dynamometercards. Knapp was the first to present a method for computing down-hole dynamometercards using finite differences. His formulation does not account for variablerod diameter or rod material. Knapp's theory was used in the development of themodel presented here. Model Development The behavior of the sucker-rod pumping system is complex. This study entailsmodeling a portion of this system, namely the sucker-rod string from thesurface to the pump. The wave equation is ideal for this purpose because theproblem at hand involves the propagation of waves in a continuous medium. Wave Equation. The 1D wave equation is a linear hyperbolic differentialequation that describes the longitudinal vibrations of a long, slender rod. Using this equation with viscous damping, we can approximate the motion of thesucker-rod string. In its simplified form, the wave equation is given by (1) where v = . Eq. 1 is for the simplified case of a constant rod diameter. Multiplyingthrough by (/) modifies Eq. 1 to account for variable rod diameter: (2) which is the form of the wave equation used to develop the finite-differencemodel. Several researchers gave a detailed derivation of the wave equation, beginning with a force balance on an element of the sucker-rod string. Generally, solving the wave equation would require two boundary conditionsand two initial conditions because the equation contains second-orderderivatives in both time and space. However, the problem solved here does notrequire initial conditions because only periodic (steady-state) solutions aredesired. Because the effects of the initial conditions have faded in periodicsolutions, only two boundary conditions are required. The two required boundaryconditions are time histories of polished-rod load and displacement. Theseconditions can be obtained polished-rod load and displacement. These conditionscan be obtained directly from a surface dynamometer point plot, a graph ofpolished-rod load vs. displacement recorded at evenly spaced increments oftime. Surface cards are typically recorded as continuous plots, however, andnot as point plots at equal time increments. In this case, pumping-unitkinematics must be used to attain a relationship between time and polished-roddisplacement. Svinos developed a versatile method for polished-roddisplacement. Svinos developed a versatile method for performing the kinematicanalysis of pumping units. His method was used in performing the kinematicanalysis of pumping units. His method was used in this study to obtain surfaceposition at evenly spaced time increments. Constant speed was assumed for theprime mover, and inertia effects were neglected. SPEPE P. 121