Sucker rod string dynamics in deviated wells

Abstract

Beam pumping is the most common artificial lift system. The down-hole dynamometer card, calculated using surface measurements, is an important tool to determine the pump operating condition and optimize the well production strategy. The description of the dynamics of a vertical sucker rod string has been well understood since S. G. Gibbs paper in the sixties [1]. The vertical well problem can be easily solved because the rod string is accurately approximated by a 1D linear elastic model, where the fluid drag is also linear with velocity. This can be solved in closed form and the parameters inferred from it, such as the down-hole card or pump fillage factor, have been widely used as the basis of most automatic well controllers.The number of deviated wells in the world has been steadily increasing since new technologies such as directional drilling became available and unlocked the potential of unconventional resources, where the possibility of drilling horizontal wells or multipad directional wells is critical for field development. These directional wells many times are produced using beam pumps after the primary depletion stage. Thus the description of the dynamics of a sucker rod string in non-vertical cases has become an important problem in recent years. These wells need to be analyzed including the effects of the non-vertical well trajectory and its consequent rod-tubing Coulomb friction losses. Coulomb friction is a highly non-linear phenomena that introduces very important dynamical effects. The increase in the complexity of this system is probably the cause of a lack of a consensus on how to deal with the problem. This fact has a clear impact on the industry, most well controllers do not include the possibility of calculating the down-hole card for a deviated well and many patents with different numerical schemes trying to solve the system have been disclosed recently.In this article a new model for the system that includes all the relevant physical mechanism that need to be incorporated in the description is proposed. The model is fully self contained and can be solved by numerical means, giving an accurate representation of the sucker rod string dynamics for directional wells. In contrast to other approaches to the problem, this model solves a closed system of equations without any external estimation of the variables in the system. Using this framework both the forward, or predictive, scheme, useful in the design of the rod string, and the down-hole card problem, needed to monitor down-hole conditions (diagnostic mode) are solved. This theory is used to move one step ahead and predict new effects not previously discussed in the literature, such as multiple micro sticking events of the rod string that are generated by static Coulomb friction.Field cases are analyzed and agree with the predictions of the model. These examples demonstrate that it is critical to take into account Coulomb friction to properly describe qualitative and quantitative features, that cannot be approximated by any means using a one dimensional vertical well approximation.