Survival Analysis: The Statistically Rigorous Method for Analyzing Electrical Submersible Pump System Performance

Abstract

Summary A rigorous statistical methodology using survival analysis (SA) was developed and applied to electrical submersible pump (ESP) system performance data. The approach extracts unbiased information from performance data and permits lifetime modeling, with parameter combinations employing all available data. The analysis explicitly accounts for ESPs that are still operational at the time of the study, thus removing a historical source of statistical bias. The analysis uses Kaplan-Meier (KM) (Kaplan and Meier 1958) and Cox proportional hazards (CPHs) (Cox 1972) modeling to determine statistical significance of explanatory variables (EVs). Methods developed to facilitate EV factor collapsing are also discussed (the partitioning of levels of each factor into nonempty subsets of statistically similar response), so that an acceptable degree of parsimony is achieved. Essential definitions necessary for preliminary data structure are also covered. We demonstrate the practical utility of this methodology on a comprehensive data set to enable unbiased and conclusive appraisal of ESP performance, thereby resolving a common concern about comparative-system reckoning. The paper concludes that SA, suitably applied to properly censored data, is essentially the only reliable method of evaluating ESP system performance (and other types of time-to-event data). Introduction The critical importance of ESP system performance to field economics and deliverability has been well documented (Allis and Capps 1984; Upchurch 1990; Brookbank 1996). What has been missing, however, is agreement as to what constitutes the most appropriate methodology for analyzing the wealth of performance data that are collected. The manner in which the available data are scrutinized, analyzed, qualified, and presented is influential to timely, economic, and accurate well and field design. This paper shows how we can extract the full profundity of useful information that inhabits even a moderately sized data set through application of, what we consider to be, the only truly appropriate technique for analyzing time-to-event data: SA. The objective of any SA is to identify variables that influence survival and to predict survival probabilities. This is achieved by finding a suitable statistical model that fits the data closely. We then examine variables included in the model and, finally, make predictions about ESP system performance for well- and field-planning purposes. The structure of this article is as follows:First, we outline the problems and inconsistencies inherent with existing (non-SA) analysis approaches to ESP performance analysis.Background on SA is then provided, along with an outline of the three main classes of SA methodology.We define terminology; in particular, the terms "system" and "components."We then present a summary of the extensive data set employed in the analysis.The remainder of the paper presents a detailed and sequential SA for a rich data set to demonstrate how parameter interaction, factor collapsing, and appropriate goodness-of-fit measures can be employed to achieve a parsimonious model of the given data. Note that a parsimonious model, in this context, refers to one containing the minimum number of significant parameters that adequately represents the data.Finally, we conclude that SA is the only viable and appropriate unbiased time-to-event methodology for evaluating ESP system performance. The analytical process we propose represents the given data parsimoniously and provides performance indicators (with confidence bounds) in response to specific questions. At the risk of premature presentation of results, Fig. 1 shows just one such SA plot that can be generated. This plot provides unbiased answers to a specific question: "What is the performance between Cable Manufacturers A, B, and C when required to last 500 days, when installed with Motor Series "D" in a well that has sand (abrasion) present" Also, "What degree of confidence can we have in these results" We see from Fig. 1 that 57% of pumps installed with Cable Manufacturer "A" are expected to survive to 500 days, while only approximately 33% of pumps with Cable Manufacturer "B and C" survived to this time (note that B and C are grouped together, as are Motor Series Types "D, L, and I," because there is no statistical difference between them). For Cable Manufacturer A, we are 95% confident that the interval between 40 and 80% covers the true (unknown) proportion surviving to 500 days. The equivalent range for Cable Manufacturers B and C is between 25 and 48%. We conclude, therefore, that for the specific conditions stated in the question (and everything else being equal), Manufacturer A performs better than B and C by the margins stated. Associated validation statistics also conclude that we may draw reliable statistical inference from these results. We now have a solid platform upon which to base performance contracts. By the end of this article, we hope to have demonstrated a rigorous and unbiased methodology for analyzing ESP system performance data and essentially any time-to-event data.