Exponentiation Bias in Production and Reserves Estimates Made by Use of Exponential-Decline Models

Abstract

Summary The standard method to evaluate an oil- or gas-production-decline curve estimated with an exponential function—taking the logarithms of both sides of the equation, estimating the parameters of the transformed function through linear regression, and exponentiating—leads to biased estimates of future production. The bias arises in the process of exponentiation. The direction and magnitude of exponentiation bias depend on three driver variables: the variance of the post-peak-production history; the number of post-peak observations on production; and the estimated rates of production during the forecast period. A correction factor, dependent on the confluent-hypergeometric-limit function, applied to the biased estimators produces unbiased estimates of future production. The correction factor can be quickly evaluated and introduced into the work flow for use in evaluating exponential-decline curves. The net bias in estimates of future production is more likely to be negative than positive. Negative bias understates remaining resources and reserves. The probability of negative, rather than positive net bias, is an increasing function of the maturity of production at the point of evaluation. The absolute magnitude of the bias is a direct function of both the variance of the empirical post-peak-production history and the forecasted rates of future production. It is an inverse function of the length of the post-peak-production history. A data set of 54,254 completion-level monthly production histories from the Gulf of Mexico (GOM) was used to quantify the bias and show the characteristics of production that determine its direction and magnitude. In this data set, exponentiation bias in estimates of remaining resources usually results in small absolute errors. Holding out varying fractions of the production histories of the completions analyzed, the interquartile range for errors in the estimated remaining resources (relative to unbiased estimates) extends from an underestimate of 886 to an overestimate of 2,105 BOE. However, at the extreme ends of the distribution of errors, maximum underestimates of 8.3 million BOE and overestimates as large as 22.5 million BOE were found. More than 14% of the completions analyzed had forecast errors of more than 30%. Extreme biases are predictably associated with specific ranges and combinations of values of the three driver variables. Therefore, exponentiation bias can have very large and predictable effects on the economic value of estimated remaining resources, but they can be reliably corrected.