Evaluating Proportions of Undetected Geological Events in the Case of Erroneous Identifications
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Evaluating Proportions of Undetected Geological Events in the Case of Erroneous Identifications1 Jerry L. Jensen,2 Jeffrey D. Hart,3 and Brian J. Willis4 Some geological events occur infrequently but still have a significant impact upon reservoir characteristics. By their very nature, however, it can be difficult to properly estimate the proportions of uncommon events because they may not appear during limited sampling. For example, even with 40 observations and an event proportion of 0.05, there is a 0.13 chance that no events will be observed. We provide some results and guidance concerning methods to estimate proportions when such events are not detected. Two cases are discussed, estimating proportions without errors in identification and estimating proportions when errors may arise. It is well-known that the distribution of possible proportions in the error-free case can be calculated using Bayesian analysis. If one assumes a standard uniform distribution as the prior for the proportion, Bayesian analysis gives a Beta distribution for the posterior. The situation becomes more complicated, however, when detection errors are included; the true proportion has a distribution consisting of several Beta distributions. The difference in results between the error-free and witherror situations can be considerable. For example, when 10 error-free observations are made and no uncommon events are detected, there is a 0.50 chance that the true proportion exceeds 0.06 and a 0.10 chance that it exceeds 0.19. Including the effects of erroneous identifications, however, increases the median proportion to 0.09 and the upper decile to 0.27. We also examine the case where there may be prior geological information, which can be incorporated by amending the prior distribution of the proportion. We find that the use of such a prior makes little difference unless there are very few observations or there are major differences between the anticipated and the observed proportions. KEY WORDS: rare events; Binomial distribution; Beta distribution; erroneous detection; Bayes theorem.
INTRODUCTION The ability of fluids to move through a hydrocarbon reservoir may reflect not only the mean rock permeability but also can depend critically on the abundance 1 Received
6 December 2004; accepted 9 June 2005; Published online: 16 May 2006. of Petroleum Engineering and Geology & Geophysics, Texas A&M University, 3116 TAMU, College Station, Texas, 77843-3116; e-mail:[email protected]. 3 Department of Statistics, Texas A&M University, College Station, Texas, 77843-3143. 4 Department of Geology and Geophysics, Texas A&M University, College Station, Texas, 77843-3115. 2 Departments
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and lateral continuity of high- and low-permeability constituents within the rock volume. Where the rock is very heterogeneous, showing pronounced local variability in permeability, extreme-value permeability constituents may fundamentally control fluid
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