A Statistical Study of Reservoir Permeability: Distributions, Correlations, and Averages

Abstract

Summary. We propose that reservoir permeability may be statistically distributed in a variety of ways. Two hypothetical cases of reservoir layering are statistically analyzed. This analysis suggests that a restricted family of functions all related to the normal distributioncan be used to represent permeability distributions. The log-normal distribution is one member of the family. Several sets of field data are analyzed. The analyses show thatpermeability data are not necessarily log-normally distributed,all the permeability distributions considered are closely approximated by members permeability distributions considered are closely approximated by members of the proposed family of functions, andimproved porosity/permeability relationships result when the permeability distribution is known. Introduction For many years, reservoir permeability has been recognized as a random-valued property of the formation. Law was among the first to analyze reservoir permeability statistically. He studied three horizons from a sandstone reservoir and concluded that permeability has a log-normal probability density function (PDF). Law also showed that, by knowing the mean and variance of the PDF for Core plug data from a new well, he could successfully predict the well's productivity index. Since Law's study, many investigators have analyzed permeability distributions. Bennion analyzed 60,000 sandstone, and 24,000 limestone permeabilities from Canadian reservoirs. He concluded that the logarithm of permeability gave a PDF that is skewed to the right and leptokurtic in comparison to a normal PDF. Lambert studied the data from 689 wells in 22 fields and compared the permeability distribution of each well with three possible PDF's: normal, log-normal, and exponential. She concluded possible PDF's: normal, log-normal, and exponential. She concluded that 285 wells have approximately normal PDF's, 297 have distributions best described as log-normal, and 102 wells have approximately exponential PDF's. Freeze, by considering such empirical studies as Bennion's and Law's and indirect evidence, concludes that permeability is log-normally distributed. These analyses have lacked a statistical framework within which the studies could be conducted. No definitive assurance can be given that another study will not yield substantially different conclusions. Furthermore, no appropriate alternatives to the log-normal distribution have been proposed to accommodate inconclusive results such as Bennion's. This paper presents analyses of six data sets for Permeability distributions in light of a theoretical framework. The theory indicates that a continuous range of PDF's is possible and that the log-normal and normal distributions are two members in the range. How to select the most appropriate PDF for the data set being analyzed is discussed. Correlations involving permeability improve when the PDF is chosen appropriately. Also, the behavior of estimators of the average reservoir permeability vary as the PDF is changed. PDF is changed. Theoretical Analysis of Permeability Distributions The central limit theorem (CLT) is first reviewed because of its essential role in the analysis. Then, two cases of reservoir heterogeneity are studied to obtain the permeability PDF for each. From these cases, we develop a general proposition for the distribution of permeability. CLT. Consider a set of n + 1 random variables, xj, i=1,2.....n, and s n, where ............................................(1) The CLT states that, under certain conditions, s n becomes normally distributed as n increases without bound. The conditions require that moments up through the third exist, and that the xi be mutually independent. Not all versions of the CLT require that the variables xi be independent, but it is unlikely that the conditions required by these versions apply to our situations. In the limit, however, sn will be normally distributed regardless of the distribution of the xi, provided that the moment and independence conditions are satisfied. It has been observed that the sum sn may obtain near normality for quite small n. In practice, the requirement of independence is difficult to verify, but we assume that it does hold in the following analysis. Analysis of Layered Models. The two-layer models to be investigated are shown in Figs. 1 and 2. Each block of permeable material has n layers of equal size and the permeabilities of all the layers, ki, i = 1,2...n, are assumed independent and identically distributed. The total permeability of each model to flow as indicated in the figures is denoted ki. For the case of layering parallel to the flow direction (Fig. 1), .......................................(2) Layering in series with the flow path (Fig. 2) yields .................................(3) Comparisons of Eq. 1 with Eqs. 2 and 3 suggest that either kt is approximately normally distributed (AND) or that (kt) -1 is AND, depending on the layering configuration. We say "approximately" because neither kt nor (kt)-1 can be precisely normally distributed because both quantities are precisely normally distributed because both quantities are nonnegative by definition. A strictly normally distributed variable always has a nonzero probability of obtaining a negative value. If the ratio of the standard deviation to the mean of the distribution is small (i.e., less than 0.43), however, then the approximation is quite good. SPEFE P. 461