Permeability Modeling

Abstract

Whereas porosity is a measure of the pore space relative to the bulk volume in a rock formation, permeability is the ability of a porous material or membrane to allow gas or liquid to pass through it. In heterogeneous rocks, permeability can vary in several orders of magnitude and generally has a highly skewed distribution with many small values and much fewer large values. Measured data for permeability are generally very limited. The combination of high variability and limited data makes permeability modeling difficult. Using the relationship between porosity and permeability is the most practical way to build 3D permeability models. However, several pitfalls exist in calibrating permeability to porosity, including the effect of lithofacies, the effect of scale, and discrepancies between core and well-log permeabilities. This chapter presents methods and related pitfalls in modeling 3D permeability.

The greatest learning disability of all may be pattern blindness—the inability to see relationships or detect meaning.

Marilyn Ferguson

Appendix 20.1: A Short Tale of Long Tails of Skewed Histograms

One expression of subsurface heterogeneities is a skewed histogram of some petrophysical properties (Roislien and Omre 2006), in which there are abundant occurrences of small values and a long tail of lower occurrences of high values. This is often true for permeability, fracture length, and some rare metal grades. This type of histogram often shows a shoulder-shape on one side and an initial rapid falloff, followed by gradual decreasing frequencies on the other side (Fig. 20.1a). This is often termed heavy-tailed distribution. There are discrepancies in the literature regarding the definition of heavy-tailed distributions; sometimes the terms long-tailed, heavy-tailed, and fat-tailed distributions are used differently and sometimes similarly. For geoscience applications, we can simply consider a long-tailed distribution as a histogram that has one-sided skewed distribution with a relatively small number of extreme values. In short, a long-tailed distribution has high-frequencies of small values followed by lower frequencies of larger values that gradually tail off asymptotically. The two-common long-tailed histograms are lognormal and power law distributions.

A random variable is lognormally distributed if its logarithm is normally distributed. The lognormal probability density function is defined by Eq. 2.4 in Chap. 2. A lognormal distribution is skewed, and thus its mean is greater than median, and its median is greater than its mode.

The power law model has been increasingly used in economics, engineering, and information theory (Easley and Kleinberg 2010). Its probability density function satisfies

$$ f(x)={ax}^{-b} $$

(20.18)

where x is a random variable, a and b are constants.

Figure 20.13 shows one lognormal and two power law histograms. They all have a long tail. In a linear scale, it is seen that the smallest values represent either nearly 60% or even 80% of data; many large values have much lower frequencies and they are hardly observable. In the logarithmic scale, some of them become observable, but not all of them. Table 20.1 compares the parameters of these histograms. Although the first power law distribution has a similar arithmetic mean as the lognormal histogram, its maximal value and standard deviation are dramatically higher. The second power law distribution has similar frequencies as the lognormal distribution for the first few smallest value bins, but it falls off much more gradually afterwards, which can be better seen in the logarithmic scale (Fig. 20.13b).

Fig. 20.13

(a) Comparing three histograms in linear scale: black is a lognormal distribution; white and gray are two power law distributions (see Table 20.1; White is Power law 1). Only small values of the property are shown; values greater than 80 have very small frequencies that would not be shown up. (b) Same as (a), but in logarithmic scale with a much wider range; values greater than 8000 are not shown (see Table 20.1 for other statistics)

Table 20.1 Comparing two frequency distributions: lognormal versus power law

Some applied geoscientists might wonder whether a normal distribution can be said to have a long tail. Although a normal distribution is theoretically defined from negative infinity to positive infinity, it is generally not considered to have a long tail because 99.7% data are within three standard deviations. A long-tailed distribution falls off much more gradually and the standard deviations are much larger, especially true for power law distributions.