Numerical simulation of the sucker-rod pumping system

Abstract

The sucker rod pump is an artificial lift method frequently applied in onshore petroleum wells. This system can be described using a numerical simulation based on the behavior of a rod string. In the past, the elastic behavior of the rod string made it difficult to model the system. However, since the 1960s and with the advent of digital computers, it has been modeled numerically. The rod string behaves like a slender bar, and thus, the propagation of elastic waves along the bar can be represented by a one-dimensional equation. Gibbs (1963) presented a mathematical model based on the wave equation, which is described on the basis of the analysis of forces on the rod string and is incorporated into a boundary value problem involving partial differential equations. The use of the finite difference method allows for a numerical solution by the discretization of the wave equation developed in the mathematical formulation with appropriate boundary and initial conditions. This work presents a methodology for implementing an academic computer code that allows simulation of the upstroke and downstroke motion of the rod string described by the wave equation under ideal operating conditions, assuming a harmonic motion of the rod at one end and downhole pump at the other end. The goal of this study is to generate the downhole dynamometer card, an important and consolidated tool that controls the pump system by diagnosing operational conditions of the downhole pump.