Gaussian Quadrature Accurately Approximates the Relative Weights of Each Reserves Category of the PRMS Matrix Through a Cumulative Distribution Function
Authors: Moridis Nefeli , Lee W. John , Sim Wayne , Blasingame Thomas 
Published: April 2019

Abstract
The objective of this work is to develop a methodology to estimate the fraction of Reserves assigned to each Reserves category (1P, 2P, and 3P) of the PRMS resources classification matrix using a cumulative distribution function (CDF). Previous published work has often used Swanson's Mean (SM) as the basis for allocating Reserves to individual categories, but we found that this method, which relates the Reserves categories through a CDF for a normal distribution, is an inaccurate means to determine the relationship of the Reserves categories with asymmetric distributions, and our work identified a better method, Gaussian Quadrature (GQ).
Production data are lognormally distributed, regardless of basin type, and thus are not compatible with the SM concept. The GQ algorithm provides a methodology to estimate the fraction of Reserves that lie within the 1P, 2P, and 3P categories — known as their weights. GQ is a numerical integration method that uses discrete random variables and a distribution that matches the original data. For this work, we associate the lognormal CDF with a set of discrete random variables that replace the production data, and determine the associated probabilities. The production data for both conventional and unconventional fields are lognormally distributed, thus we expect that this methodology can be implemented in any field.
We selected 38 wells from a Permian Basin dataset available to us, and we performed probabilistic decline curve analysis (DCA) using the Arps Hyperbolic model and Monte Carlo simulation (MCS) to obtain a probability distribution of the 1P, 2P, and 3P volumes. We considered this information to be our "truth case," to which we compared relative weights of different Reserves categories from the GQ and SM methodologies. We also performed probabilistic rate transient analysis (RTA) using the IHS Harmony software to obtain the 1P, 2P, and 3P volumes, and calculated the relative weights of each Reserves category. Once we completed these first two steps, we implemented a 3-point GQ to obtain the weights and percentiles for each well. We analyzed the GQ results by calculating the percentage differences between the probabilistic DCA, RTA, and GQ results.
The probabilistic DCA results indicated that the SM method is an inaccurate method for estimating the relative weights of each Reserves category. Our results show that the GQ method was able to capture an accurate representation of the Reserves weights, we note that we assumed an expected lognormal CDF for Reserves. We believe that 1C, 2C, 3C, and 1U, 2U, and 3U Contingent and Prospective Resources are distributed in a similar way (i.e., as a lognormal CDF) but have greater variance.
Based on our results, we conclude that the GQ method is accurate and can be used to approximate the relationship between the relative weights of resources in PRMS categories. This relationship will aid entities in reporting Reserves of different categories to regulatory agencies because it can be recreated for any field, play, or region. These distributions of Reserves and Resources Other than Reserves (ROTR) are important for planning and for resource inventorying. The GQ method provides a measure of confidence in our prediction of the Reserves weights because of the relatively smaller percentage differences between the probabilistic DCA, RTA, and GQ weights than those implied by the SM method. For reference, our proposed methodology can be implemented in both conventional and unconventional reservoirs.