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Estimating permeability based on grain size
Rock type influence on permeability discusses how permeability can be significantly affected by rock type, grain size, and extent of compaction or cementation. This page discusses several models that have been developed for estimating permeability based on grain size.
Krumbein and Monk's equation
Using experimental procedures that were later adopted by Beard and Weyl,^{[1]} Krumbein and Monk^{[2]} measured permeability in sandpacks of constant 40% porosity for specified size and sorting ranges. Analysis of their data, coupled with dimensional analysis of the definition of permeability, led to
where:
- k is given in darcies
- d_{g} is the geometric mean grain diameter (in mm)
- σ_{D} is the standard deviation of grain diameter in phi units, where phi=-log_{2}(d) and d is expressed in millimeters
Although the Krumbein and Monk equation is based on sandpacks of 40% porosity and does not include porosity as a parameter, Beard and Weyl showed that Eq. 1 fits their own data fairly well even though porosity of the Beard and Weyl samples ranges from 23% to 43%. In fact, because of difficulties in obtaining homogeneous sandpacks, Beard and Weyl chose to use computed k values from Eq. 1 rather than their measured data in tabulating values for fine and very fine samples with poor or very poor sorting. If Eq. 1 can predict k for a varying Φ in unconsolidated sandpacks, then the exponential dependence on sorting must be adequate to describe all the effects associated with porosity reduction. In other words, both k and Φ reduction pictured in Fig. 1 are due primarily to a decline in degree of sorting.
The laboratory studies of Krumbein and Monk^{[2]} and Beard and Weyl^{[1]} dealt with sieved sands from a common source, so such grain properties as angularity, sphericity, and surface texture did not vary much. Moreover, sorting was purposely c=ontrolled to be log normal. In situations where these ideal conditions are not met, other techniques must be invoked to predict permeability in unconsolidated sands. A disproportionate amount of fines can drastically reduce k in unconsolidated sands. Morrow et al.,^{[3]} using statistical techniques on data from Gulf Coast sands, found that permeability correlated best with the logarithm of grain size times sorting if the fines fraction, taken to be <44 μm, was accounted for.
Berg’s model
An interesting model linking petrological variables—grain size, shape, and sorting—to permeability is that of Berg.^{[4]} Berg considers "rectilinear pores," defined as those pores that penetrate the solid without change in shape or direction, in various packings of spheres. Simple expressions for k are derived from each packing, which form a linear trend when log(k) is plotted against log(Φ). From these geometrical considerations comes an expression relating k to Φ raised to a power and to the square of the grain diameter,
where:
k is given in darcies d (in mm) is the median grain diameter Φ is porosity in percent p, a sorting term
If permeability is expressed in millidarcies, d in micrometers, and Φ as fractional porosity, this expression becomes
To account for a range of grain sizes, Berg considered two mixtures of spheres and assumed that k will be controlled primarily by the smaller grains. This introduces a sorting term p=P_{90}-P_{10}, called the percentile deviation, to account for the spread in grain size. The p term is expressed in phi units, where phi=-log_{2}d (in mm). For a sample with a median grain diameter of 0.177 mm, a value of 1.0 for p implies that 10% of the grains are >0.25 mm and 10% are <0.125 mm.
Berg’s expression (Eq. 2b) is illustrated in Fig. 2 for p=1 and varying d. Permeability increases rapidly with increasing porosity, depending on Φ to the fifth power, and the curves migrate downward and to the right with decreasing grain size. Nelson^{[5]} finds that Fig. 2 ^{[4]} is remarkably concordant with several published data sets. Berg’s model appears to be a usable means of estimating permeability in unconsolidated sands and in relatively clean consolidated quartzose rocks. This is true even though Berg did not expect his model to be applicable for porosity values <30%.
Van Baaren’s model
Proceeding along more empirical lines, Van Baaren^{[6]} obtains a result nearly identical to that of Berg. Van Baaren begins with Kozeny-Carman's equation where surface area is based on the ratio of pore surface area to rock volume and makes a series of substitutions (see summary by Nelson^{[5]}) that result in
where d_{d} (in μm) is the dominant grain size from petrological observation, m is the cementation exponent, and C is a sorting index that ranges from 0.7 for very well sorted to 1.0 for poorly sorted sandstones. Consequently, Eq. 3a can be used to estimate k from petrological observations on dominant grain diameter d_{d} and degree of sorting, along with a porosity estimate obtained from either core or logs.
Assuming that the dominant grain size d_{d} is equivalent to Berg’s median grain diameter d, then Eq. 3a is very similar in form to Eq. 2a. For example, a sorting parameter p=1 in Berg’s Eq. 2b results in
where k is given in millidarcies, whereas for a well-sorted sandstone, C=0.84 and Eq. 3a becomes
Van Baaren’s Eq. 3b is so close to Berg’s Eq. 2c that a separate log(k)-Φ plot is not warranted here. Van Baaren’s expression is probably easier to use because the parameters are directly related to practical petrological variables. Both models display a porosity exponent >5, and both are compatible with the data of Beard and Weyl on unconsolidated sands in that k increases with the square of grain size.
Nomenclature
d | = | grain diameter |
k | = | permeability |
m | = | Archie cementation exponent |
p | = | pressure |
Φ | = | porosity |
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} 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
- ↑ Morrow, N.R., Huppler, J.D., and Simmons, A.B. III. 1969. Porosity and Permeability of Unconsolidated, Upper Miocene Sands From Grain-Size Analysis. J. of Sedimentary Petrology 39 (1): 312-321. http://dx.doi.org/ 10.1306/74D71C3F-2B21-11D7-8648000102C1865D
- ↑ ^{4.0} ^{4.1} Berg, R.R. 1970. Method for Determining Permeability From Reservoir Rock Properties. Trans., Gulf Coast Association of Geological Societies 20: 303-335.
- ↑ ^{5.0} ^{5.1} Nelson, P. 1994. Permeability-porosity relationships in sedimentary rocks. The Log Analyst 35 (3): 38–62.
- ↑ Van Baaren, J.P. 1979. Quick-Look Permeability Estimates Using Sidewall Samples and Porosity Logs. Trans., 6th Annual European Logging Symposium, Society of Professional Well Log Analysts.
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