Page 38 - Physical Principles of Sedimentary Basin Analysis

20 Properties of porous media

one where b =−23 and another where b =−20. We notice that the log–linear relationship
is useful both for sandstones, clays and shales. It is often the best porosity–permeability
description available unless detailed measurements are made. The sandstone permeabilities
from Figure 2.12 are also plotted in Figure 2.13, and they are three orders of magnitude
above the upper limit for shale permeability. Figure 2.13b shows how the permeability
might decrease with depth for shale. The porosity–permeability area in Figure 2.13ais
mapped to permeability depth using Athy’s depth–porosity function (2.4) with Sclater and
Christie (1980) parameters in Table 2.1 for shale. We see that we may expect shale perme-
2
ability as low as 10 −20 m in the depth interval below 1.5 km. It has been claimed that the
permeability of shale and clay are dependent on scale because of heterogeneities like for
instance fractures. Fractures exist on all length scales and can be important for the average
permeability with increasing block size. Neuzil (1994) also compiled permeability esti-
mates from inverse hydrological analyses that cover a much larger length scale than what
is possible to measure in the laboratory and found that these permeabilities are consistent
with the low permeabilities from the compilation of laboratory measurements.
A third group of rocks that should be mentioned are carbonates, which are also less
studied with respect to permeability than the sandstones. The carbonates have a pore space
that often contains vugs or small cavities. To what degree the vugs are connected and
therefore important for the permeability may be field or site dependent. Nelson (2004b)
discusses two data sets for carbonates in addition to several sandstones, and it appears
that the permeability of the two carbonates are in a similar porosity–permeability range as
the sandstones. Mallon and Swarbrick (2008) compare two chalk data sets from the North
Sea and conclude that routine core analyses may overestimate the permeability by orders
of magnitude for low permeability cores. Nelson (2004a) and Nelson and Kibler (2003)
present a collection of porosity and permeability data sets for core plugs in silicilastic
rocks (which may be downloaded via the internet).

Exercise 2.11 Let the permeability be k 1 at a porosity φ 1 and k 2 at a different poros-
ity φ 2 . Show that a and b in the linear fit (2.50)are b = log (k 2 /k 1 )/(φ 2 − φ 1 ) and
10
a = (φ 2 log (k 1 ) − φ 1 log (k 2 ))/(φ 2 − φ 1 ).
10
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2.8 The rotation matrix

We will need to rotate vectors and matrices in the work with anisotropic permeability.
Rotations are done with a matrix R(θ) that transforms a vector from one coordinate system
to a coordinate system rotated an angle θ. It is straightforward to derive in 2D. Figure 2.14
shows the transformation of a vector from the unprimed coordinate system to the primed
coordinate system, and we have

x = a cos(α − θ) = a cos α cos θ + a sin α sin θ = x cos θ + z sin θ

z = a sin(α − θ) = a sin α cos θ − a cos α sin θ =−x sin θ + z cos θ