Asymptotic Dykstra–Parsons Distribution, Estimates and Confidence Intervals

Abstract

The sample Dykstra–Parsons (DP) coefficient, the most popular heterogeneity static measure among petroleum engineers, may exhibit significant sampling errors. Moreover, approximations of its probability distributions (uncertainty estimates) are only available for specific families of permeability models (e.g., log-normal). The cited probability distributions allow for the specification of confidence intervals and other inferences for the theoretical DP, which will be useful for reservoir screening purposes, or to establish if a more detailed study is justified. This paper presents the development of an asymptotic approximation of the distribution of the sample Dykstra–Parsons coefficient, which is independent of the permeability probability distribution. The effectiveness (bias and confidence intervals) of the proposed approach is demonstrated using analytical and field case studies and by comparing the results gleaned with those obtained using a well-known parametric approximation, under different scenarios of reservoir maturity levels (i.e., number of wells) and different degrees of deviation from the log-normal probability density function assumption. The results show that, in the vast majority of the case studies, the proposed approach outperformed the parametric approximation; in particular, our approach resulted in a significant reduction of the bias and the confidence intervals always including the theoretical DP coefficient. In addition, an excellent agreement was observed between the asymptotic cumulative distribution of the DP coefficient and the corresponding empirical distribution for sample sizes as small as one hundred, which suggests that high success rates can be obtained when reservoirs are classified according to the asymptotic DP coefficient.