Analysis, Pages 12 (2812 words)
Petroleum Engineer, Chevron Corp. & Masters Degree Candidate Advisor Dr. Jianhua Huang With help from PHD Candidate Sophia Chen Department of Statistics, Texas A&M, College Station MARCH 2011 ABSTRACT A common metric in Petroleum Engineering is “Mean Time Between Failures” or “Average Run Life”. It is used to characterize wells and artificial lift types, as a metric to compare production conditions, as well as a measure of the performance of a given surveillance & monitoring program.
Although survival curve analysis has been in existence for many years, the more rigorous analyses are relatively new in the area of Petroleum Engineering.
This paper describes the basic theory behind survival analysis and the application of those techniques to the particular problem of Electrical Submersible Pump (ESP) Run Life. In addition to the general application of these techniques to an ESP data set, this paper also attempts to answer: Is there a significant difference between the survival curves of an ESP system with and without emulsion present in the well?
Although survival curve analysis has been in existence for many years, the more rigorous analyses are relatively new in the area of Petroleum Engineering. As an example of the growth of these analysis techniques in the petroleum industry, Electrical Submersible Pump (ESP) survival analysis has been sparsely documented in technical journals for the last 20 years: ? ? ? First papers on the fitting of Weibull & Exponential curves to ESP run life data in 1990 (Upchurch) & 1993 (Patterson) Papers discussing the inclusion of censored data in 1996 (Brookbank) & 1999 (Sawaryn) Paper discussing the use of Cox Regression in 2005 (Bailey)
Unfortunately, the papers applying these techniques did little to transfer the knowledge to the practicing Petroleum Engineers.
They shared the technical concepts and equations, but not the practical knowledge of how to apply them to real life problems or why these analyses improved upon the “take the average of the run life of failed wells” technique most commonly used. THEORY OF SURVIVAL ANALYSIS Survival analysis models the time it takes for events to occur and focuses on the distribution of the survival times.
It can be used in many fields of study where survival time can indicate anything from time to death (medical studies) to time to equipment failure (reliability metrics). This paper will present three methodologies for estimating survival distributions as well as a technique for modeling the relationship between the survival distribution and one or more predictor variables (both covariates and factors). Appendix A has a list of important definitions relevant to survival analysis. KAPLAN MEIER (NON-PARAMETRIC) Non-parametric survival analysis characterizes survival functions without assuming an underlying distribution.
The analysis is limited to reliability estimates for the failure times included in the data set (not prediction outside the range of data values) and comparison of survival curves one factor at a time (not multiple explanatory variables). A common non-parametric analysis is Kaplan Meier (KM). KM is characterized by a decreasing step function with jumps at the observed event times. The size of the jump depends on the number of events at that time t and the number of survivors prior to time t. The KM estimator provides the ability to estimate survival functions for right censored data. ti is the time at which a “death” occurs. i is the number of deaths that occur at time ti. When there is no censoring, ni is the number of survivors just prior to time ti. With censoring, ni is the number of survivors minus the number of censored units. The resulting curve, as noted, is a decreasing step function with jumps at the times of “death” ti. The MTBF is the area under the resulting curve; the P50 (median) time to failure is (t) 0. 5. Upper and lower confidence intervals can be calculated for the KM curve using statistical software. A back-of-the-envelope calculation for the confidence interval is the KM estimator +/2 standard deviations.
Greenwood’s formula can be used to estimate the variance for nonparametric data (Cran. R-project): Figure 1: Example Kaplan Meier survival curve showing estimate, 95% confidence interval, and censored data points When comparing two survival curves differing by a factor, a visual inspection of the null hypothesis Ho: survival curves are equal, can be conducted by plotting two survival curves and their confidence intervals. If the confidence intervals do not overlap, there is significant evidence that the survival curves are different (with alpha < 0. 05%) COX PROPORTIONAL HAZARD (SEMI-PARAMETRIC)
Semi-Parametric analysis enables more insight than the Non-Parametric method. It can estimate the survival curve from a set of data as well as account for right censoring, but it also conducts regression based on multiple factors/covariates as well a judge the contribution of a given factor/covariate to a survival curve. CPH is not as efficient as a parametric model (Weibull or Exponential), but the proportional hazards assumption is less restrictive than the parametric assumptions (Fox). Instead of assuming a distribution, the proportional hazards model assumes that the failure rate (hazard rate) of a unit is the product of: ? a baseline failure rate (which doesn’t need to be specified and is only a function of time) and a positive function which incorporates the effects of factors & covariates xi1 – xik (independent of time) This model is called semi-parametric because while the baseline hazard can take any form, the covariates enter the model linearly. Given two observations i & i’ with the same baseline failure rate function, but that differ in their x values (ie two wells with different operating parameters xk), the hazard ratio for these two observations are independent of time:
The above ratio is why the Cox model is a proportional-hazards model; even though the baseline failure rate h0(t) is unspecified, the ? parameters in the Cox model can still be estimated by the method of partial likelihood. After fitting the Cox model, it is possible to get an estimate of the baseline failure rate and survival function (Fox). A result of the regression is an estimate for the various ? coefficients and an R-square value describing the amount of variability explained in the hazard function by fitting this model. Relative contributions of factors/covariates can be interpreted as: ? ? ? >0, covariate decreases the survival time as value increases, by factor of exp(? ) ? 0 scale; k>0 shape ?(ln(2))1/k The Weibull shape parameter, k, is also known as the Weibull slope. Values of k less than 1 indicate that the failure rate is decreasing with time (infant failures). Values of k equal to 1 indicate a failure rate that does not vary over time (random failures). Values of k greater than 1 indicate that the failure rate is increasing with time (mechanical wear out) (Weibull). A change in the scale parameter, ? , has the same effect on the distribution as a change of the X axis scale.
Increasing the value of the scale parameter, while holding the shape parameter constant, has the effect of stretching out the PDF and survival curve (Weibull). Figure 2: Example Weibull curves with varying shape & scale parameters The Weibull regression model is the same as the Cox regression model with the Weibull distribution as the baseline hazard. The proportional hazards assumption used by the CPH method, when applied to a survival curve with a Weibull function baseline hazard, only holds if two survival curves vary by a difference in the scale parameter (? ) not by a difference in the shape parameter (k).
If goodness of fit to the Weibull distribution can be achieved, a confidence interval can be calculated for the curve, the median value and its confidence interval can be calculated, and a comparison of the differences in two survival curves can be conducted. Goodness of fit can be tested in R using an Anderson Darling calculation and verified with a Weibull probability plot. Poor fit in the tails of the Weibull distribution is a common occurrence for reliability data due to infant mortality and longer than expected wear out time. STEPWISE COX & W EIBULL REGRESSION
Given a large number of explanatory variables and the larger number of potential interactions, not all of those variables may be necessary to develop a model that characterizes the survival curve. One way of determining a model is by using Stepwise model selection through minimization of AIC (Akaike Information Criterion). This model selection technique allows variables to enter/exit the model using their impact on the AIC calculated at that step. AIC is an improvement over maximizing the R-Square in that it’s a criterion that rewards goodness of fit while penalizing for model complexity.
APPLICATION TO AN ESP DATA SET As stated previously, these survival analysis techniques can be applied to many types of data in many industries ranging from survival data for people in a medical study to survival data for equipment in a reliability study. These methodologies have many uses in the petroleum industry; from surface equipment system and component reliability used by facility and reliability engineers, to well and downhole system and component reliability used by petroleum and production engineers.
As an example, this paper illustrates the use of these techniques on the run life of Electrical Submersible Pumps (ESP). ESPs are a type of artificial lift for bringing produced liquids to the surface from within a wellbore. Appendix B includes a diagram of an ESP. For this paper, the run life will refer to the run life of an ESP system, not the individual components within the ESP system. While this paper focuses on ESP systems, these same techniques could be applied to other areas of Petroleum Engineer interests including run life of individual ESP components, other types of artificial lift, entire well systems, etc.
DATA DESCRIPTION ESP-RIFTS JIP (Electrical Submersible Pump Reliability Information and Failure Tracking System Joint Industry Project) is a group of 14 international oilfield operators who have joined efforts to gain a better understanding of circumstances that lead to a success or failure in a specific ESP application. The JIP includes access to a data set of 566 oil fields, 27861 wells, 89232 ESP installations, and 182 explanatory factors/covariates related to either the description of the ESP application or the description of the ESP failure.
For the analysis described in this paper, a subset of the data has been used, restricted to: ? ? ? ? ? ? Observations related to Chevron operated fields observations with no conflicting information (as defined by the JIP’s data validation techniques) factors that were related to the description of the ESP application (excluded 27) factors not confounded with or multiples of other factors (excluded 30) factors with a large number (>90%) of non-missing data points (excluded 78) factors that were not free-form comment fields (excluded 27)
Appendix C has a list of the original 182 variables with comments on why they were removed from the analyzed data set, below is a table of the 20 remaining explanatory variables included in this analysis. SUMMARY TABLE OF DATA INCLUDED IN THE CPH/REGRESSION ANALYSIS: OBSERVATIONS: 1588 DESCRIPTION RunLife Censor Country Offshore Oil Water Gas Scale CO2 Emulsion CtrlPanelType NoPumpHouse PumpVendor NoPumpStage NoSealHouse
NoMotorHouse MotorPowerRating NoIntakes NoCableSys CableSize DHMonitorInstalled DeployMethod COVARIATE/FACTOR & # OF LEVELS Response Censor Flag (0, 1) Factor (7 levels) Factor (2 levels) Covariate Covariate Covariate Factor (5 levels) Covariate Factor (3 levels) Factor (2 levels) Covariate Factor (2levels) Covariate Covariate Covariate Covariate Covariate Covariate Covariate Factor (2 levels) Factor (2 levels) DESCRIPTION Time between date put on production and date stopped or censored 1 if ESP failure 0 if still running or stopped for a different reason Country & Field in which the ESP is operated Indication of whether the ESP was an onshore or offshore installation Estimated average surface oil rate (m3/day) Estimated average surface water rate (m3/day) Estimated average surface gas rate (1000m3/day) Qualitative level of scaling present in the well % of CO2 present in the well Qualitative level of emulsion present in the well Type of surface control panel used Number of pump housings Pump Vendor Number of pump stages Number of seal housings Number of motor housings Motor rated power at 60Hz (HP) Number of intakes Number of cable systems Size of cable Flag for installation of a downhole monitor Method of ESP deployment into the well FINDING THE P50 TIME TO FAILURE FOR A DATASET Example 1: Using the entire data set, what is the P50 estimate for the runtime of a Chevron ESP? The answers differ considerably for the 4 calculation types: METHODOLOGY Mean or Median Kaplan Meier Median CPH Median INCLUDES CENSORED?
No Yes Yes P50 ESTIMATE (DAYS) Mean: 563 Median: 439 1044 1043 ASSUMPTION None None None (as no comparison of levels/covariates, essentially same results as KM) Anderson Darling GOF for Weibull Distribution N/A N/A N/A ASSUMPTIONS MET ? Weibull Median Yes 1067 NO (rejected the null hypothesis of good fit, due to poor fit in the tails) In this example, the biggest impact on the difference between the methods is the inclusion of censored data. A large number of the ESPs in this data set have been running for >3000 days without a failure and were excluded in the often used calculation of the average run life of all failed ESPs. Given that the Weibull distribution did not pass the Anderson Darling goodness of fit test, the most appropriate calculation would have been the KM or CMH.
Appendix E has the output from the various methodologies. The interpretation of these results is that the P50 estimate of run life for an ESP installation in Chevron is ~ 1044 days. Additional, output from the KM analysis sets the 95% confidence interval at 952 to 1113 days. Figure 3: Comparison of estimation methods for full data survival curve. Note the deviation of the Weibull in the tails of the data. COMPARING TWO SURVIVAL CURVES DIFFERING BY A FACTOR Example 2: Using the 2 level factor emulsion, does the presence of emulsion in the well make a significant difference in the P50 run life of an ESP system? METHODOLOGY Mean or Median Kaplan Meier Median CPH Median INCLUDE CENSOR?
No Yes Yes EMULSION P50 (DAYS) Mean 600 Median 458 606 533 NO EMULSION P50 (DAYS) Mean 536 Median 424 1508 1408 SIGNIFICANT DIFFERENCE? Don’t know Yes (visual Inspection of CI) Yes, with a Likelihood ratio test and a pvalue of 0, reject that B’s are the same. Yes, with a z test statistic and a pvalue of 0, reject that the scale values are the same. INTERPRETATION Well performance is about the same Wells without emulsion perform much better Wells without emulsion survive longer. Exp(B) indicates 2. 5 times increased survival time for no emulsion. Wells without emulsion survive longer. Scale parameter value indicates 2. 75 times increased survival time for no emulsion. ASSUMPTIONS MET ? No. (Reject null hypothesis of prop. hazards with a p value of 0. 01. ) No. Reject null hypothesis of good of fit due to poor fit in the tails) Weibull Median Yes 531 1463 The more complex the methodology used, the more information is available to interpret the results. Again, the addition of censored data resulted in a very different interpretation of the data than just using the mean/median value of all failed ESPs; not just in the order of magnitude of the results, but also determination of which condition resulted in a longer run life. The results of both the CPH & Weibull methodologies are suspect due to their failure to meet the prerequisite assumptions. Looking at the plots, it is apparent that the fit is poor in the tails.
Appendix F has the output from the various methodologies The interpretation of these results is that wells without emulsion have > a 2x increase in their P50 run life than wells with emulsion. It should be noted that given the other factors that differ in the operation of these ESPs, this difference may not be fully attributed only to the difference in emulsion, but this interpretation should lead to further investigation. Figure 4: KM estimated survival curves for ESPs with and without emulsion with confidence interval Figure 5: Comparison of estimation methods (KM, CPH, Weibull) for ESPs with and without emulsion CHOOSING THE VARIABLES THAT CHARACTERIZE A SURVIVAL CURVE
Example 3: Of the variables collected by the JIP, which most describe the survival function? Do the variables collected in the dataset capture the variation in the survival function? As stated previously, both Weibull & Cox regression fit a model using explanatory variables. The introduction of Stepwise variable selection to that regression allows the preferential fitting of the model by minimizing the AIC. As Weibull regression is a special case of Cox regression with a Weibull baseline hazard function, and as Cox regression has less restrictive assumptions than parametric regression, this example will focus solely on Cox regression using Stepwise