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# Comparison of permeability estimation models

Many approaches to estimating permeability exist. Recognizing the importance of rock type, various petrophysical (grain size, surface area, and pore size) models have been developed. (See links to these in Single phase permeability). Additional techniques have been developed for applying well logs and other data to the problem of predicting permeability [*k* or log(*k*)] in uncored wells. The variation in models and approaches can be confusing. This page explores the differences among models, the reasons for those differences, and answers some common questions about how to decide what approach to use.

## Importance of pore throat size

The best petrophysical models show that pore throat size *r* is a prime control on *k*. Yet, *r* is missing from most example data sets (Figs. 5 through 11 in Rock type influence on permeability) and from the diagram summarizing diagenetic processes (Fig. 2 in Rock type influence on permeability). With the models, it is possible to consider what the range of pore sizes might have been when sediments were deposited and what the range is in the present state of consolidation. As an example, consider the samples from the Hosston formation (Fig. 5). The uppermost vertical bar in Fig. 3 from Rock type influence on permeability shows the range of grain size reported for the Hosston formation. The range of initial pore throat sizes (vertical hachured bar in Fig. 3) was computed from the present-day grain size using data from Beard and Weyl^{[1]} (Fig. 4) to transform grain size to initial permeability, followed by application of the Katz-Thompson equation (Eq. 4 from Estimating permeability from well log data) to obtain pore throat size from *k* and *Φ*. Present-day pore throat sizes (lowermost solid bar in Fig 3 in Rock type influence on permeability) were computed from present-day *k* and *Φ* with the Katz-Thompson^{[2]} equation (Eq. 4 from Estimating permeability from well log data). From this exercise, one can see that the largest pore throat size has diminished from 250 (initial) to 45 (present day) μm, that the smallest present-day sizes are <2 μm, and that the range of pore throat size has broadened considerably on the logarithmic phi scale.

## Expressions relating permeability to porosity

Expressions relating permeability *k* to porosity *Φ* are summarized in **Table 1**. There, permeability is in millidarcies, and grain sizes (*d*) and pore sizes (*r*) are expressed in micrometers, so the coefficients may differ from the originating equation in the text. From these representative equations, it can be seen that:

- Predictive equations are simple in form
*k*is related to a power of*Φ*(except for the Krumbein and Monk^{[3]}equation)*k*is related to the square of either a characteristic length or a measure of surface area

We have seen that models relying on estimates of surface area, whether that estimate comes from irreducible water saturation, NMR, gas adsorption, or cation exchange data, require porosity raised to a power of ≈4. How can surface-area models requiring a porosity power of 4 (Figs. 4 and 8 from Estimating permeability based on surface area and water saturation) be reconciled with pore dimension models requiring a power of 2 (Fig 2 and 3 from Estimating permeability based on pore dimension)?

The answer lies in the pore-size distribution. Because most of the surface area is contributed by the smallest grains (pores), measures of surface area emphasize the small end of the pore-size spectrum. Yet, the small pores contribute least to permeability. The high (≈4) power of porosity serves to unweight the contribution of the small pores. In the surface-area models, porosity serves a dual role, first as a measure of tortuosity and second as a measure of the pore-size distribution function.

A similar question arises with the grain size models. Both models by Berg^{[4]} (Fig. 2 in Estimating permeability based on grain size) and Van Baaren^{[5]} require a porosity power of ≈5, multiplied by the square of a dominant grain size. Why is the porosity power so high?

The probable reason is that the dominant grain size becomes a progressively poorer measure of dominant pore size as the spread in grain size increases and small grains (pores) become more abundant. Again, porosity serves as both a measure of tortuosity and a weighting factor to compensate for the presence of small pores at lower porosities. Moreover, the retention of a sorting term in Eqs. 2a and 2b of Estimating permeability based on grain size is inadequate compensation for small pores, even though a sorting term is all that is needed in sized samples (Eq. 1 in Estimating permeability based on grain size).

## Models incorporating an estimator of pore size

Models incorporating an estimator of pore size (those labeled 5d, 19, and 20 in **Table 1**) include porosity raised to a power of m (≈2). Estimates of dominant or characteristic pore size are more effective at predicting *k* than estimates of grain size or surface area, so the higher exponent of porosity to compensate for the low end of the porosity spectrum is not required. Given a measure of *r* and *Φ*, the more information that *r* contains regarding the large through-going pores, the lower the dependence on *Φ* is. Indeed, the findings of Beard and Weyl,^{[1]} Swanson,^{[6]} Pittman,^{[7]} and Katz and Thompson^{[2]} all show that *Φ* is not so important as a predictor of *k* as long as the dominant *r* is well specified. Conversely, using Pittman’s findings of Table 1 in Estimating permeability based on pore dimension, as *r* decreases below *r*_{apex}, *r* becomes a progressively poorer estimator of the dominant *r*, and a higher exponent of *Φ* is required to compensate for the inclusion of pore throats that do not contribute to flow.

The preceding considerations hold for predicting *k* on individual samples from a wide range of rock formations, whereas Rock type influence on permeability shows that *Φ* can be a good predictor of *k* for samples from a given rock formation. Why is this?

The pore-size models produce curves of constant pore size that transgress the steeper log(*k*)-*Φ* data trends. The cutting of the log(*k*)-*Φ* trends by the curves of constant pore size shows that porosity reduction is always accompanied by a reduction in characteristic pore size. As rocks from a common source are compacted and undergo diagenesis, pore space is reduced, and permeable pathways are progressively blocked in a systematic way that maintains a consistent relationship between *Φ* and *r*. Samples from different formations that have undergone different diagenetic processes follow different evolutionary paths in log(*k*)-*Φ*-*r* space and thus produce different trends on a log(*k*)-*Φ* plot.

## Practical applications

The problem of predicting permeability has been reviewed by compiling data and predictive algorithms from the literature. Which approach should be used to estimate permeability from core and well log data? As a practical matter, it depends on what data are available from a given well or field:

1. In cases in which no core data are available, one can proceed by analogy using data developed in formations with geological properties similar to the one under study. Figs. 5-8 in Rock type influence on permeability are examples of the types of analog data that one might use.

2. When porosity and grain size estimates are available, refer to Fig. 2 in Estimating permeability based on grain size. This chart appears to give good estimates for many consolidated rocks. Exceptions will exist, such as rocks containing illite in pore space and low-permeability formations such as those shown in Fig. 8 in Rock type influence on permeability.

3. In situations in which porosity and water saturation can be estimated, permeability can be estimated from Timur’s^{[8]} relationship (see Estimating permeability based on surface area and water saturation). In clay-bearing rocks, the dual-water relationship for permeability is an interesting enhancement, but the user is required to provide estimates of both interstitial and bound water.

4. When NMR logs are available, one can make use of the permeability transforms developed for such logs. Laboratory determination of a *t*_{2} cutoff is advised.

5. Permeability is controlled by a pore dimension of a selected subset of the pore population and can be determined from capillary pressure by mercury injection (see Estimating permeability based on pore dimension). Mercury injection can be applied to determine the permeability of small or fragmented samples.

6. In field developments in which core data are abundant and a relatively simple (linear) log(*k*)-*Φ* relation is the result of a fairly uniform lithology and uncomplicated diagnetic history, then one can turn to regression methods to predict *k* from well log estimates of *Φ*.

7. In heterogeneous reservoirs, a high degree of scatter on a log(*k*)-*Φ* plot requires that the reservoir be zoned before k can be estimated. One must choose a petrophysical parameter with which to zone the reservoir rocks. Various practitioners have used *r*_{35}, the flow zone indicator (*I*), the square root of *k*/*Φ*, and even *k* itself (note that each of these four parameters has the dimension of length or length squared). One must also choose a method of zoning or clustering; among the candidates are:

- Linear regression
- Neural networks
- Data binning
- Fuzzy clustering

A good set of core data is required to establish the zones or clusters. After the method is tested, then well logs are used to compute a value of the zonation parameter. The more geological information that can be incorporated into the zonation procedure, the better. In fact, the breadth of petrophysical, well test, and geological data is probably more important than the particular zonation parameter or clustering methodology chosen. Complex reservoirs require complex methods.

8. Fractured reservoirs are a special and difficult case. Fracture permeability cannot be measured with core samples, so it is difficult to establish ground truth. Methods of estimating fracture permeability from fracture aperture and fracture density are tenuous because aperture varies throughout the fracture plane, some fractures are sealed with mineral deposits, and some are open. Combinations of techniques seem to work well. Examination of core can provide:

- Orientation and number of fractures
- Facies descriptions
- Mineralization on fracture surfaces

Borehole images provide:

- Fracture number
- Orientation
- Aperture

Flow (spinner) logs reveal zones of fluid flow into the wellbore. Sonic waveform logs show fracture location, and if conditions permit, permeability estimates can be extracted from Stoneley waves. Other well logs provide porosity estimates. Well tests provide estimates of permeability over isolated intervals. Analysis of the state of stress can provide insight on fracture location and the probability of being open or closed. Analysis of complementary data sets can provide insights that cannot be obtained from isolated data sets.

In all cases, one must bear in mind that a permeability predictor will be unique to the field or formation for which it is developed. This unfortunate fact is a result of the multiple pathways that can be followed during burial and diagenesis.

## References

- ↑
^{1.0}^{1.1}Beard, D.C. and Weyl, P.K. 1973. Influence of Texture on Porosity and Permeability of Unconsolidated Sand. American Association of Petroleum Geologists Bull. 57 (2): 349-369. - ↑
^{2.0}^{2.1}Katz, A.J. and Thompson, A.H. 1986. Quantitative Prediction of Permeability in Porous Rock. Physical Review B 34 (11): 8179. - ↑ Krumbein, W.C. and Monk, G.D. 1943. Permeability as a Function of Size parameters of Unconsolidated Sand. Trans., AIME 151 (1): 153. http://dx.doi.org/ 10.2118/943153-G
- ↑ Berg, R.R. 1970. Method for Determining Permeability From Reservoir Rock Properties. Trans., Gulf Coast Association of Geological Societies 20: 303-335.
- ↑ Ahmed, U., Crary, S.F., and Coates, G.R. 1991. Permeability Estimation: The Various Sources and Their Interrelationships. J Pet Technol 43 (5): 578-587. SPE-19604-PA. http://dx.doi.org/10.2118/19604-PA
- ↑ Swanson, B.F. 1981. A Simple Correlation Between Permeabilities and Mercury Capillary Pressures. J Pet Technol 33 (12): 2498-2504. SPE-8234-PA. http://dx.doi.org/10.2118/8234-PA
- ↑ Pittman, E.D. 1992. Relationship of Porosity and Permeability to Various parameters Derived From Mercury Injection—Capillary Pressure Curves for Sandstone. American Association of Petroleum Geologists Bull. 76 (2): 191-198.
- ↑ Timur, A. 1968. An Investigation Of Permeability, Porosity, & Residual Water Saturation Relationships For Sandstone Reservoirs. The Log Analyst IX (4). SPWLA-1968-vIXn4a2.

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## See also

Relative permeability and capillary pressure