1. GEOPHYSICS, VOL. 64, NO. 5 (SEPTEMBER-OCTOBER 1999); P. 1447–1460, 8 FIGS., 1 TABLE.
Permeability prediction based on fractal pore-space geometry
Hansgeorg Pape∗, Christoph Clauser‡, and Joachim Iffland∗∗
ABSTRACT
Estimating permeability from grain-size distributions
or from well logs is attractive but difficult. In this paper
we present a new, generally applicable, and relatively
inexpensive approach which yields permeability infor-
mation on the scale of core samples and boreholes. The
approach is theoretically based on a fractal model for the
internal structure of a porous medium. It yields a general
and petrophysically justified relation linking porosity to
permeability, which may be calculated either from poros-
ity or from the pore-radius distribution. This general re-
lation can be tuned to the entire spectrum of sandstones,
ranging from clean to shaly. The resulting expressions for
the different rock types are calibrated to a comprehen-
sive data set of petrophysical and petrographical rock
properties measured on 640 sandstone core samples of
the Rotliegend Series (Lower Permian) in northeast-
ern Germany. With few modifications, this new straight-
forward and petrophysically motivated approach can
also be applied to metamorphic and igneous rocks. Per-
meability calculated with this procedure from industry
porosity logs compares very well with permeability mea-
sured on sedimentary and metamorphic rock samples.
INTRODUCTION
Permeability is one of the key petrophysical parameters for
managing hydrocarbon and geothermal reservoirs as well as
aquifers. Its magnitude may vary over several orders of mag-
nitude, even for a single rock type such as sandstone (Clauser,
1992; Nelson, 1994). Moreover, permeability is very sensitive
to changes in overburden pressure or to diagenetic alterations
(WalderandNur,1984;WalshandBrace,1984;LederandPark,
1986). For instance, drastic permeability reductions result from
the growth of minute amounts of secondary clay minerals on
quartz grains, since this changes the geometry of the hydraulic
Manuscript received by the Editor April 17, 1997; revised manuscript received October 20, 1998.
∗
Formerly GGA, Stilleweg 2, D-30655 Hannover, Germany; presently Geodynamik, Rheinische Friedrich-Wilhelms-Universit¨at, Nussallee 8,
D-53115 Bonn, Germany. E-mail: c.clauser@gga-hannover.de.
‡GGA, Stilleweg 2, D-30655 Hannover, Germany, E-mail: c.clauser@gga-hannover.de.
∗∗
LUNG Mecklenburg-Vorpommern, Pampower Str. 66/68, D-19061 Schwerin, Germany. E-mail: lung-sn@um.mv-regierung.de.
c 1999 Society of Exploration Geophysicists. All rights reserved.
capillaries. Generally speaking, permeability k is a function
of the properties of the pore space, such as porosity φ and
several structural parameters. In the past, different empirical
approaches were used to describe the observed highly non-
linear dependence of permeability on porosity by exponen-
tial or power-law relationships (e.g., Shenhav, 1971; Bourbie
and Zinszner, 1985; Jacquin and Adler, 1987). However, these
purely empirical approaches lack a petrophysical motivation.
Nelson (1994) reviews a number of these k–φ relationships. For
some homogeneous sandstone lithologies with a unique history
of sedimentation and diagenesis, the data show quasi-linear re-
lationships in φ–log(k) plots. The slopes of linear regressions
vary from moderate for sandy clay and silt (Bryant et al., 1974)
to very steep for channel sandstones (Luffel et al., 1991). This
suggests it would be very difficult to explain the φ–k relation-
ships of different lithologies with a single empirical expression.
As a consequence, a φ–log(k) plot of data from different sand-
stones with various lithologies is only very weakly correlated.
Nelson (1994) discusses several models ranging from purely
empirical to petrophysically based. Most of them express k as
the product of φ and a size parameter, taken to different pow-
ers. This size parameter may be grain diameter, pore radius,
or specific surface. In these power laws, the exponent of the
size parameter (or its reciprocal) is equal or close to two. The
exponent of φ in the numerator usually varies between 1 and 7,
and (1 − φ)2
is a factor in the denominator. Most nonempirical
models are based on the Kozeny–Carman equation (Carman,
1937, 1948, 1956), which links permeability to the effective pore
radius and the formation factor F [see equation (8)]. However,
none of these simple models can explain the sharp increase
of permeability with porosity observed for porosities greater
than 10%. For instance, Luffel et al. (1991) find for their chan-
nel sandstones that the pores are compressed to such a degree
that the hydraulic path is reduced to the free space between
adjacent grain asperities. Thus, the simple model of unconsol-
idated sand packing (a combination of ideal, smooth spheres
and large voids in between) no longer applies. This illustrates
that a real-world pore system is more complicated. It needs to
1447

2. 1448 Pape et al.
be viewed as a superposition of distinct structures at different
scales rather than as a simple arrangement of spheres or tubes.
In basin analysis it is common practice to use simple rela-
tionships of the modified Kozeny–Carman type (e.g., Ungerer
et al., 1990). Such an expression yields an incorrect permeabil-
ity prediction (e.g., Huang et al., 1996) if the size parameter (in
this case, the specific surface normalized to matrix volume) is
treated as a constant. In this expression the pore-size parame-
ter is much more important than the porosity term because it
is much more variable. Therefore, if the size parameter is elim-
inated by treating it as a constant, this must be compensated
for by a considerable increase in the exponent of φ, as seen in
equation (23).
Clean sandstones that experience a strong quartz cementa-
tion during their diagenesis show a different relationship be-
tween permeability and porosity compared to average sand-
stones. Based on percolation theory, Mavko and Nur (1997)
reinterpret a comprehensive data set for these rocks, originally
compiled and studied by Bourbie and Zinszner (1985). Details
and limitations of this approach are discussed at the end of
the section “Application of the General Fractal Permeability–
Porosity Relationship to Clean and Shaly Sandstones.”
The neutron porosity, gamma-ray, density, and sonic logs
used by Huang et al. (1996) in artificial neural network model-
ing contain in a complex way the information on the geometri-
cal pore structure, which is not accounted for in the approach
described above. However, in neural network modeling, there
is no requirement that the exact mathematical expression de-
scribing the dependence of the variables involved be specified,
in contrast to approaches based on explicit expressions such
as the Kozeny–Carman equation. Provided that the required
logs are available, the approach of Huang et al. (1996) yields
excellent permeability predictions.
Permeability is also estimated from acoustic measurements.
For a cylindrical borehole in a porous solid, Stoneley waves are
attenuated, and their velocity decreases with increasing perme-
ability. These effects are related to permeability as the waves
induce squirt flow in the porous medium. Winkler et al. (1989)
combine laboratory measurements and petrophysical models
based on Biot theory (Biot, 1956a,b) to study Stoneley wave
propagation in permeable media. Cheng and Cheng (1996) es-
timate permeabilities from low-frequency Stoneley wave ve-
locity and attenuation and find good agreement with measure-
ments on core samples.
In contrast to the methods discussed above and to the em-
pirically derived relations, fractal theory combined with petro-
physical relations provides alternative approaches (Adler,
1985; Thompson et al., 1987; Hansen and Skeltorp, 1988). The
number and definition of shape parameters required by these
different models vary according to the approach chosen. A
model particularly close to the natural appearance of pore
space in sedimentary rocks is the so-called pigeon-hole model
(Pape et al., 1987a). It is based on data measured on core sam-
ples from a variety of hydrocarbon reservoir rocks in northwest
Germany. Based on petrophysical principles, it yields relations
between various geometric, storage, and transport parameters
of these reservoir rocks. Like other fractal models, it starts
from the observation that the shape of the inner surface of
rock pores follows a self-similar rule. Thus, the theory of frac-
tals (Mandelbrot, 1977) can be applied. A key parameter in
this theory is the fractal dimension D.
Based on fractal relationships, we first derive permeability
from pore-radius distributions of a large petrophysical data set.
Next, a general relationship is derived which links permeability
directly to porosity. It is a three-term power law in which each
term dominates the relation in a different range of porosity.
This general relationship is then calibrated to different data
sets, from reservoir sandstones to basement rocks. These cal-
ibrated relationships are then used to calculate permeability
of reservoir sandstones from logs of porosity (and optionally
also from internal surface, shale factor, or irreducible water
saturation). Finally we demonstrate how this approach can
be extended to fracture porosity and show an application to
metamorphic basement rocks of the German Deep Continen-
tal Drillhole (KTB). This approach is attractive because it (1)
is petrophysically motivated, (2) requires in its simplest ver-
sion only information on porosity, (3) covers the entire range
of porosity, and (4) can incorporate additional information on
internal surface, shale factor, or irreducible water saturation,
where available.
PETROPHYSICAL DATA
Two different groups of petrophysical data are used in this
study. The first group is used for calibration and covers the
porosity–permeability spectrum for rocks ranging from clean
sandstones to shales. It consists of five data sets. The first one
was used by Pape et al. (1987a,b) to derive the general fractal
prorosity–permeability relationship. A much more compre-
hensive one, compiled over 15 years in East German hydrocar-
bon industry laboratories (Iffland and Voigt, 1996), comprises
measured porosity and permeability as well as distributions
of pore radii and grain sizes. The next three data sets con-
sist of porosity and permeability only: (3) shaly sandstone
(Kulenkampff, 1994; Debsch¨utz, 1995); (4) kaolinite (Michaels
and Lin, 1954); and (5) shales (Schl¨omer and Krooss, 1997). A
second group of data was used to demonstrate how the cali-
brated porosity–permeability relationships can be used to pre-
dict permeability from porosity logs. It consists of porosity and
permeability measured on different types of sandstone.
Among the calibration data, the set compiled by Iffland and
Voigt (1996) is by far the most comprehensive. It comprises
data for several petrophysical properties measured with var-
ious methods on sandstone core samples from the Permian
Rotliegend series in northeastern Germany. For this study the
following distributions of (1) porosity, (2) permeability, and (3)
pore radii were selected:
1) An average porosity φm was calculated for each rock sam-
ple as the arithmetic mean of three porosity values φIM ,
φ3MPa, and φCP which correspond to three different lab-
oratory techniques for porosity determination. Porosity
φIM is determined using the imbibition method with iso-
propanol as the saturant. Porosity φ3MPa is measured by
gas penetration with a confining pressure of 3 MPa in
a device also used for permeability. Porosity φCP is de-
termined during capillary pressure measurements which
yield pore radius distributions.
2) The gas-flow permeability k3MPa was measured during
high-pressure experiments at an initial confining pressure
of 3 MPa.
3) Capillary pressure measurements yield distributions of
pore radii weighted by volume. From the integrated

3. Fractal Permeability Prediction 1449
distribution functions, only the quartile values R25, R50,
and R75 are reported (Iffland and Voigt, 1996). Values
R25, R50, and R75 are defined as the smallest pore radii
of those capillaries which are completely filled when the
pore space is invaded with mercury up to 25%, 50%,
and 75%, respectively. Similarly, quartile values Q1 (first
quartile), MD (median), and Q3 (third quartile) are re-
ported for those grain-size distributions which had been
analyzed microscopically from thin sections by the chord-
section method.
FRACTAL MODEL FOR POROUS MEDIA
The classical petrophysical theory of hydraulic flow in granu-
lar beds was developed by Kozeny and Carman (Kozeny, 1927;
Carman, 1937, 1948, 1956), who describe the pore space by a
bundle of bent cylindrical capillaries and derive fundamental
relationships between various physical quantities. However, in
some cases these integer power-law expressions do not fit the
experimental data as well as some empirical relations do. This
is because the inner surface of a porous medium consists of
substructures of the pore walls that vary in size over many
orders of magnitude (Pape et al., 1982, 1984). Therefore, mea-
surements of the specific surface of sandstones with methods
of different resolving power (Pape et al., 1982) show scaling
properties characteristic for fractals (Mandelbrot, 1977).
The pigeon-hole model (Pape et al., 1987a) (Figure 1) yields
a particularly good description for grain packings such as those
found in sandstones (Korvin, 1992). The basic structure of this
model consists of two groups of spheres: the first represents the
grains with radius rgrain, and the second characterizes the pores
with radiusrsite (Figure 1). The grains as well as the pores form a
cascadeofhemisphericalsubstructures.Thepigeon-holemodel
refers to the shape of these structures when viewed from the
pore space. The pores are connected by hydraulic capillary
channels of radius reff . The radii rgrain, rsite, and reff are related
by an empirical relation (Pape et al., 1984), which is valid for a
FIG. 1. A sedimentary rock according to the hierarchical pi-
geon-hole model, showing geometrical pores with radiirsite and
hydraulic-model capillaries with effective radii reff (after Pape
et al., 1987a).
great variety of sandstones:
rgrain/rsite = (rgrain/reff)c1, with c1 = 0.39
if rgrain/reff > 30. (1)
Thesethreeradiiarethesizeparametersoftheporousmedium.
The self-similarity of the pigeon-hole model’s cascade of
hemispherical substructures is expressed by the self-similarity
ratio v =ri+1/ri < 1, whereri is the radius of the hemispheres of
the ith generation and each of these is again parent to N hemi-
spheres of radius ri+1. According to Mandelbrot (1977), fractal
behavior follows from self-similarity, and the fractal dimension
D is given by
D = log N/ log(1/v). (2)
This fractal dimension D is identical to the Hausdorff–
Besicovitch dimension, an important quantity in measure the-
ory. Therefore, many geometrical parameters of the porous
medium, such as internal surface and porosity, depend on the
resolution of the measuring method. The relations follow a
noninteger power law. For instance, the specific surface nor-
malized to total pore volume Spor is given by
Spor(λ1) = Spor(λ2)(λ1/λ2)(Dt −D)
. (3)
The values λ1 and λ2 are the minimum lengths that can be
resolved by the two methods of measuring, and Dt = 2 is the
topological dimension of a surface. Comparing corresponding
couples of surface areas for different resolutions, Pape et al.
(1987a) use equation (3) for the normalized specific surface
to determine that the average fractal dimension for a set of
sandstone samples is D = 2.36. Generally, D varies between
2.0 and 2.5 for most rocks; D = 2.36 is an average value repre-
sentative for the northwestern German hydrocarbon reservoir
rocks. Deviations from this value depend on the relative abun-
dance of diagenetic structures. In particular, a smaller value of
D is found for clean sandstones with large porosities.
A further property of fractals is that they involve size dis-
tributions. For instance, the pigeon-hole model is composed
of several classes of spherical bulges—the pigeon holes. These
are measured by the mercury intrusion method as pseudocap-
illaries. Typical distributions of sandstone pore radii exhibit a
long tail toward smaller radii. In a log–log plot, this part of
the curve has a linear asymptote with a slope equal to Dt − D,
where Dt = 3 is the topological dimension of a volume. This
can be also expressed similar to equation (3) for the normal-
ized specific surface:
1 − V (r1) = (1 − V (r2))(r1/r2)(3−D)
, (4)
where r1 and r2 are the smallest radii of pseudocapillaries that
are completely filled when the pore space is invaded by mer-
cury up to the relative volumes V (r1) and V (r2), respectively.
For the median and the third quartile radius of the pore radii,
equation (4) for the relative volumes reads
0.5 = 0.25(R50/R75)(3−D)
. (5)
RELATIONSHIPS BETWEEN PERMEABILITY
AND OTHER PETROPHYSICAL PROPERTIES
Three parameters required by the fractal model were deter-
mined from the data available from 640 Rotliegend sandstone

4. 1450 Pape et al.
samples (Iffland and Voigt, 1996): the fractal dimension D, the
fundamental parameter describing the structure of the pore
space, the formation factor F, and the tortuosity T . The latter
twogeometricalfactorsarerequirednotonlyintheexpressions
for permeability but also for electrical resistivity and porous-
medium diffusivity.
Fractal dimension
Equations (3)–(5) for normalized inner surface and for rel-
ative volumes were applied to the petrophysical data from the
Rotliegend series. The fractal dimension D was determined
from equation (5) for a normalized inner surface. However,
the scatter is large, and some values of the fractal dimension
were even less than two. This is obviously incorrect since the
fractal dimension of the pore-wall surface is always greater
than the Euclidian dimension two of a surface. To obtain bet-
ter agreement between data and model prediction, we need to
introduce several corrections to equations (4) and (5) for rel-
ative volumes. These account for the experimental conditions
during mercury imbibition (the method used to determine the
distribution of capillary radii), such as the shape of the contact
between mercury and pore wall and the bottleneck effect in
constricted pores. The fractal model always involves nonzero
residual volumes 1 − V (r1) and 1 − V (r2) in equation (4) for
relative volumes, which are defined by the product of surface
area and resolution length. For rough surfaces, this agrees well
with the physical situation. However, this is not the case for
a smooth surface, where a close contact between surface and
mercury is always established, irrespective of pressure. In this
case, the residual volume vanishes. To account for these effects,
a correction volume Vcorr needs to be added to the residual
volume (1 − V (r2)), corresponding to the higher resolution in
equations (4) and (5) for relative volumes:
Vcorr = α(R75/R50), (6)
where α < 1 is a factor to be determined.
Another difficulty stems from the assumption that the radii
calculated from the results of the capillary pressure experi-
ments are equal to the pore-site radii rsite. However, pore sites
of a given size are invaded through their smaller pore necks.
Similarly, even pigeon holes may be invaded by the mercury
through smaller inlets. Thus, a conversion of the radii R75 and
R50 to pigeon-hole radii is necessary in analogy to equation (1)
for the ratio of radii. This introduces an additional exponent
c2, which resembles the exponent c1 = 0.39 in equation (1) for
the ratio of radii. Thus, for the analysis of distributions of pore
radii, equation (5) for relative volumes is replaced by
0.5 = (0.25 + Vcorr)(R50/R75)(3−D)c2, (7)
where c2 and α are determined iteratively. While values for α
were increased from a starting value of 0.25 to the final value
of 0.275, values for c2 were varied around a value of 0.39. This
way, a value of c2 = 0.45 was determined for which the median
of D for all samples equals 2.36, which Pape et al. (1982, 1984,
1987a) find to be a typical value for sandstones. The value of
D = 2.36 takes into account bottleneck effects in the scaling re-
lation between the fractal dimension and the porosity of rocks
(Pape et al., 1987a). Thus, it is equivalent to higher values of
2.57–2.87 reported by Katz and Thompson (1985) where this
correction was not applied. As a result, our Rotliegend sam-
ples are characterized by a fractal dimension that varies within
a realistic range. By comparing the ensemble mean D = 2.36
with the fractal dimension of individual samples determined
from equation (7) for 50% and 75% saturation, it is possible
to distinguish between different types of sandstone. Larger in-
dividual D values indicate the pore system is more structured
as a result of higher clay content.
Tortuosity
A fundamental expression for the permeability k of a porous
medium is given by the modified Kozeny–Carman equation
(Kozeny, 1927; Carman, 1956):
k = r2
eff (8F), (8)
where F is the formation factor (Archie, 1942). It is defined as
the ratio of tortuosity T and porosity φ:
F = T/φ. (9)
Originally, the formation (resistivity) factor was defined as the
ratio of the electrical resistivity of a porous medium saturated
with an electrolyte and the resistivity of this electrolyte. It
is a purely geometric parameter, describing how the porous
medium obstructs transport processes. It is related to per-
meability via the Kozeny–Carman equation [equation (8)].
An empirical relationship between F and φ is expressed by
Archie’s first law:
F = a/φm
. (10)
Here, a is a factor depending on lithology and m is the cemen-
tation or tortuosity factor, which depends on rock structure.
The parameters a and m vary in the ranges 0.6 < a < 2 and
1 < m < 3 (Serra, 1984). Later, Archie’s law [equation (10)]
was also derived from a fractal model for porous rocks by
Pape and Schopper (1988) and Shashwati and Tarafdar (1997).
For a pigeon-hole model of common sandstones, this deriva-
tion starts with equation (9), expressing F by tortuosity T and
porosity φ and equations (15), (16), and (17) defined later in
the text. These equations express the fractal relationship be-
tween T and φ. Equations (17) and (9) combined yield a = 0.67
and m = 2.
Pape et al. (1987a) show that tortuosity behaves as a fractal
and depends on the ratio of reff and rgrain, with an exponent
involving the fractal dimension D:
T = 1.34(rgrain/reff)0.67(D−2)
. (11)
This implies that tortuosity increases with increasing fractal
dimension. However, this relation is valid only in the range
of 2 < D < 2.4. In contrast, strongly fractured rocks (such
as cataclastic rocks) or claystones show fractal dimensions in
the range of 2.4 < D < 3. These rocks are characterized by a
high degree of connectivity of the pore system, which in turn
reduces the tortuosity (Pape and Schopper, 1988).
Since we do not know reff at this stage, we replace it in equa-
tion (11) by the pore radius R50 and rgrain by the median grain
radius MD. Further, reservoir sandstones are limited in tor-
tuosity (e.g., Kulenkampff, 1994). Therefore equation (11) is
modified so the new expression is identical to equation (11)
for large pore radii reff; it yields Tmax = 10 in the limit of reff

5. Fractal Permeability Prediction 1451
approaching zero. This avoids extreme and unrealistic tortu-
osites. This is realized by equation (12) for the reciprocal of
tortuosity, in which the constant factor for the ratio R50/MD
is determined from a fit to the data measured on Rotliegend
sandstones in northwestern Germany (K¨ohler et al., 1993):
1/T = 0.1 + 0.5(R50/MD)0.67(D−2)
. (12)
This substitution makes sense as seen in Figure 2, a crossplot
of R50 and reff, calculated according to equations (8), (9), and
(12) from measured permeability, porosity, and R50. While
equation (11) is of the type of Richardson’s formula (1961)
for an ideal fractal, equation (12) for tortuosity represents a
so-called nonideal fractal (Rigaut, 1984). Now the formation
factor F can be determined from equation (9). Finally, the ef-
fective hydraulic pore radius reff can be obtained by inserting
the measured permeabilty k and the formation factor F into
equation (8).
Permeability estimates based on pore-radius distributions
In equation (8), permeability varies with the square of the
effective pore radius reff but only linearly with the reciprocal of
the formation factor. As outlined previously, the effective pore
radii reff for Rotliegend sandstones can be calculated from the
measured permeabilities and the distribution of pore radii. A
comparison of the computed effective pore radii and the quar-
tiles of the distribution of measured pore radii shows that, for
most samples, reff is close to R50 (Figure 2). However, for val-
ues of reff < 1 µm, the ratio reff/R50 > 1. In these cases the R25
value with an empirical weighting factor for porosity should be
used for estimating reff because R25 is larger than R50. There-
FIG. 2. Log–log plot of the median radius R50 (determined
from measured pore radius distribution) versus the effective
hydraulic pore radius reff according to equation (8). The solid
line is the linear regression log(R50) = −0.41 + 1.09 log(reff)
with R2
= 0.76, where R is the correlation coefficient. The
broken line is the identity function R50 = reff.
fore,thefollowingpermeabilitypredictionsarebasedonporos-
ity and both quartile radii R50 and R25. Tortuosity T can be
estimated from equation (12) using the default values D = 2.36
(Pape et al., 1982, 1984, and 1987a) and rgrain = 200 000 nm (an
average grain radius in our sandstone data sets) and replacing
reff by R50 or R25. Then kcalc is calculated from equations (13)
and (14), depending on the relative size of R50 and R25:
kcalc = (φ/8T )(R50)2
if R50 ≥ R25/2, (13)
kcalc = (0.25φ/8T )(R25)2
if R50 < R25/2, (14)
where R25and R50aregiveninnanometers.Equation(14)pre-
vents the calculated permeabilities from becoming too low as
they might be if they were estimated using only equation (13).
Figure 3 shows a crossplot of measured permeability versus
calculated permeability according to equations (13) and (14).
The scatter in Figure 3 is caused by the differences found
in the grain-size distributions of rocks from any certain forma-
tion. These primary differences are further enhanced during
the subsequent diagenesis. The linear trend with a slope close
to one indicates good agreement between measured perme-
ability data on the one hand and theoretical predictions on the
other hand, which are based on fractal theory and measured
pore-radius distributions.
Permeability–porosity relationship
Our derivation starts from the Kozeny–Carman equation (8)
and the expression for the formation factor [equation (9)].
Combining equations (8) and (9), permeability k is expressed
by the effective pore radius reff, tortuosity T , and porosity φ.
FIG. 3. Permeability calculated from two quartiles of
pore-radius distribution, R50 and R25, and porosity φ, plot-
ted against measured permeability. The solid line is the lin-
ear regression log(kcalculated) = −0.23 + 1.06 log(kmeasured) with
R2
= 0.83, where R is the correlation coefficient. The broken
line is the identity function.

6. 1452 Pape et al.
In general, reff and T vary with φ. For the moment we neglect
the limiting case of reff approaching zero, for which Tmax = 10.
Then, according to Pape et al. (1987a), T and reff are expressed
by φ:
T = 1.34(rgrain/reff)0.67(D−2)
= 1.34(rgrain/reff)0.24
(15)
and
φ = 0.534(rgrain/reff)0.39(D−3)
= 0.5(rgrain/reff)−0.25
,
(16)
with D = 2.36. Neglecting slight differences in the exponents,
equations (15) and (16) for tortuosity and porosity yield
T ≈ 0.67/φ (17)
and
r2
eff = r2
grain(2φ)8
. (18)
Inserting equations (9), (17), and (18) into equation (8), and
using a default value of rgrain = 200 000 nm (an average grain
radius in our sandstone data sets), k can now be written as
k = 191(10φ)10
(nm2
). (19)
Equation (19) defines a linear asymptote in a log–log plot of
k versus φ. It is valid for porosities larger than 0.1, whereas
for lower porosities the measured permeabilities exceed those
predicted by equation (19). An explanation is suggested by
the fact that the effective pore radii of the samples studied by
Pape et al. (1987a,b) do not decrease as rapidly with decreas-
ing porosity as suggested by equation (18). Therefore, we can
improve the permeability estimates for 0.01 < φ < 0.1 by as-
signing a fixed value to the effective hydraulic pore radius in
equation (8): reff = reff, fix = 200 nm. This is equal or near to
the mean of reff determined from measured permeabilities and
porosities and equations (8), (9), and (17) for 0.01 < φ < 0.1.
Inserting reff,fix = 200 nm into the combination of equations (8),
(9), and (17) then yields
k = φ2
r2
eff,fix (8 × 0.67) = 7463 φ2
(nm2
). (20)
However, for φ < 0.01 this expression still yields permeabili-
ties too small in comparison with the measured data. A limiting
value for reff in this range is reff,min = 50 nm. Inserting this value
for R50 into equation (12) yields a maximum value for tortuos-
ity, Tmax = 10. Using this value and inserting equation (17) into
equation (20) then yields
k = φr2
eff, min (8Tmax) = 31φ (nm2
). (21)
The sum of the expressions for permeability in equations (19),
(20), and (21) finally provides an average permeability–
porosity relationship for the entire porosity range:
k = 31φ + 7463φ2
+ 191(10φ)10
(nm2
). (22)
The linear combination of the expressions for the low-,
medium-, and high-porosity ranges is permissible since, for
a given porosity, the expressions for the other two porosity
ranges do not contribute significantly because of the difference
in powers of porosity.
The third term in equation (22) characterizes the fractal be-
havior of sandstones. The power of 10 corresponds to the frac-
tal dimension D = 2.36. A fractal dimension 2 < D < 2.36
corresponds to a power between 3 and 10. A fractal dimension
D > 2.36 yields a power greater than 10.
The factors of the three terms in equation (22) need to be cal-
ibrated when applied to a specific basin. Equation (22) tends to
underestimate the permeability measured on the Rotliegend
samples because the northeastern Germany Rotliegend sand-
stones are characterized by a relatively large permeability at
any given porosity. The coefficients of φ, φ2
, and φ10
in equa-
tion (22) are calibrated to the Rotliegend data by nonlinear
regression, which yields
k = 155φ + 37 315φ2
+ 630(10φ)10
(nm2
). (23)
Figure 4 shows this expression. The correlation is expressed by
R2
= 0.83, where R is the correlation coefficient.
If the fractal dimension of the investigated rocks deviates
significantly from the general value of D = 2.36 assumed in
our study, not only the coefficients but also the exponents of
the general permeability–porosity relationship need to be ad-
justed. A derivation similar to that for D = 2.36 [equations (15–
22)] yields a comprehensive expression valid for all types of
structures:
k = Aφ + Bφexp1
+ C(10φ)exp2
. (24)
It is valid for arbitrary values of D and the exponent m in
Archie’s law [equation (10)]. The exponents are given by
exp1 = m, exp2 = m + 2/(c1(3-D)) and 0.39 < c1 < 1. The
parameter c1 relates the effective pore radius reff to the pore ra-
dius rsite and the grain radius rgrain [equation (1)]. Equation (24)
with modified constants in the exponents will be used later to
FIG. 4. Log–log plot of permeability versus porosity. Open cir-
cles are measured permeability and porosity. The line shows
the fractal permeability prediction according to equation (23).
Correlation is given by R2
= 0.83, where R is the correlation
coefficient of the regression.

7. Fractal Permeability Prediction 1453
derive an expression for permeability of magmatic and meta-
morphic rocks. Pape et al. (1984) derive values of m = 1.53 and
c1 = 0.47 from laboratory experiments on unconsolidated sand
for small ratios of grain to effective radius, rgrain/reff ≈ 2.9.
Application of general fractal permeability–porosity
relationship to clean and shaly sandstones
The discussion of the permeability–porosity relationship
presented above is not restricted to the special case of sand-
stones from the Rotliegend series. We demonstrate this us-
ing several data sets of permeability and porosity measured
on samples from different types of sandstones. Figure 5 shows
these data as well as the curves corresponding to equations (22)
and (23) for comparison. For permeabilities greater than
107
nm2
, these curves reach the field of unconsolidated sands
represented by two samples of Schopper (1967). Data from
a set of unconsolidated kaolinite samples from Michaels and
Lin (1954) provide further orientation in the high-permeability
range. These data give an excellent example for a power law
with an exponent of 10. Unconsolidated rocks represent the
high-porosity, high-permeability limit within the total data
field. Curves within this permeability–porosity field can be in-
terpreted as corresponding to diagenetic paths. Different dia-
genetic processes may superpose each other, such as mechan-
ical compaction, mineral solution, and cementation. As long
as the fractal dimension of the pore space equals the standard
value D = 2.36, the diagenetic path is characterized by a power-
law curve with an exponent of 10 in the porosity–permeability
plot (Figure 5).
However, conditions may be different in the low-permea-
bility range. Usually the decrease of the effective pore radius
reff and the increase of the tortuosity T is less pronounced for
low permeabilities, i.e., at later stages of diagenesis. This is
reflected by the smaller increase in permeability with poros-
ity in the low-porosity range corresponding to the linear and
quadratic terms in equation (22). In addition to these curves,
other curves of the same type can be modeled but with different
coefficients, reflecting the rock parameters rgrain; reff,min; reff,fix;
and Tmax. One example is equation (23) for the northeastern
Germany Rotliegend sandstones with larger coefficients in the
three term of equation (22). To characterize samples with lower
permeability than predicted by equation (22) (see Figure 5), a
new expression was derived by reducing the three coefficients
in equation (22). This was done in such a way that the new co-
efficients A, B, and C in equation (24) are smaller than those
in equation (22) by the same proportion as the coefficients of
equation (22) are smaller than those in equation (23), for in-
stance, Aeq.22/Aeq.23 = Aeq.24/Aeq.22. This yields
k = 6.2φ + 1493φ2
+ 58(10φ)10
(nm2
). (25)
Finally, a fourth curve is constructed as a low-permeability
limit at any given porosity in Figure 5 in such a way that it
satisfies the data measured on high-porosity kaolinite samples
and on low-porosity shales. To this end, all coefficients in equa-
tion (25) are reduced by a constant factor of 58, yielding
k = 0.1φ + 26φ2
+ (10φ)10
(nm2
). (26)
With the aid of these four k–φ curves, different types of
sandstones can be characterized (symbols, Figure 5). The data
shown in Figure 5 were measured on samples of unconsolidated
sand and Fontainebleau sandstone as well as on samples from
the Dogger, Keuper, Rotliegend, Bunter, and Carboniferous
formations (Kulenkampff, 1994; Debsch¨utz, 1995; Petrophys-
ical Research Group, Institute of Geophysics, University of
Clausthal, personal communication, 1998). The Dogger and
northwestern Germany Rotliegend sandstones plot mainly
between the curves defined by equations (23) and (25). In
comparison,thenortheasternGermanyRotliegendsandstones
(see also Figure 4) in general show higher permeabilities at
FIG. 5. Log–log plot of permeability versus porosity for dif-
ferent consolidated and unconsolidated clean and shaly sand-
stones. Symbols represent measured permeability and porosity
data discussed in the text. The lines defined by equations (26)
and (27) limit the data field toward large and small porosities,
respectively. The line defined by equation (22) separates clean
sandstones (left and above) from shaly sandstones (right and
below).

8. 1454 Pape et al.
any given porosity. Bunter sandstones as well as Carbonif-
erous feldspar-rich sandstones and greywackes plot upon the
curve defined by equation (25). Additional data on shales from
the Jurassic, Rotliegend, and Carboniferous are provided by
Schl¨omer and Krooss (1997). Most of these data fall between
the curves of equations (25) and (26). They are characterized
by a wide grain-size distribution and a high content in clay
minerals. The field between the two limiting curves defined by
equations (25) and (26) characterizes the so-called shaly sand-
stones. Very clean sandstones, made up mainly of quartz grains
of equal size, are found on the high-permeability end of the
curve defined by equation (23). The Oligocene Fontainebleau
Sandstone is an extreme example of this very pure type of sand-
stone. Bourbie and Zinszner (1985) found the following good
relation between permeability and porosity:
k = 303(100φ)3.05
(nm)2
for φ > 0.08 (27a)
k = 0.0275(100φ)7.33
(nm2
) for φ ≤ 0.08. (27b)
At first glance, equation (27a) seems to contradict equations
such as equation (22). However, it can be explained by the frac-
talconcept,assumingthefractaldimension D equalsabouttwo.
In this case, both tortuosity T and the specific surface Spor are
constant, and Spor is inversely proportional to both the poros-
ity and the effective pore radius reff. Thus reff is proportional to
porosity. Inserting equation (9) for the formation factor F into
equation (8) for the Kozeny–Carman and substituting φ for reff
yields an exponent equal to three. This is much smaller than
the exponent in equation (19). Similarly, a modified Kozeny–
Carman equation of this type can be derived from equation (8)
if a nonfractal spherical grain packing model is assumed (yield-
ing a constant tortuosity) and the effective pore radius is sub-
stituted by a term involving the specific surface expressed by
the grain radius and the porosity. The resulting equation
k = 0.5(rgrain)2
φ3
(1 − φ)2
(28)
is frequently used in basin analysis (e.g., Ungerer, 1990). Fig-
ure 5 illustrates very clearly that this approximation (assuming
rgrain = 0.1 mm) can be valid only in a very limited domain of
the total k–φ field.
The value D = 2 means the quartz grains are smooth. This
is the case for well-rounded grains and for an intense quartz
cementation with well-developed crystal faces. In addition, the
grains must be of equal size. The fractal dimension can also
be estimated from the distribution of pore radii. Bourbie and
Zinszner (1985) present pore-radius distributions for differ-
ent porosities. These curves are symmetrical and narrow for
porosities >8%, corresponding to equation (27a). This corre-
sponds to the limiting case, when the fractal model consists of
only one generation of pigeon holes such as in a simple cap-
illary model. However, for porosities <8% [equation (27b)],
the distribution curves of pore radii are broader and skewed
toward smaller values. From this shape a fractal dimension
D > 2 can be derived, which corresponds to an exponent
much greater than three. This is expressed by the exponent
of 7.33 in equation (27b). This, in turn, corresponds to a fractal
dimension 2 < D < 2.36. The curve corresponding to equa-
tions (27a,b) can thus be regarded as a diagenetic path, start-
ing with a slightly inclined branch. But for porosities <8%,
the slope of the curve increases, indicating that the diagenetic
processes have generated a truly fractal pore space similar to
average sandstones [equation (22)]. In Figure 5, the curves de-
finedbyequations(27a,b)aresupplementedbydatafromsome
samples of Fontainebleau sandstone (J. R. Schopper, personal
communication, 1998). Obviously, equation (22) does not ap-
ply to these very clean sandstones. This is because, during di-
agenesis, a strong porosity reduction is first caused by quartz
cementation, indicated by small exponents of the porosity in
the k–φ relation. As pressure increases as a result of burial,
the pore volume partially collapses, which is indicated by large
exponents of the porosity in the k–φ relation. In contrast, me-
chanical and chemical diagenesis are in reverse order for rocks
containing clay minerals because even minute amounts of clay
minerals may prevent the precipitation of dissolved silica. For
these rocks, porosity is first reduced by compaction (yielding
large exponents of φ) and then by cementation (with small
exponents of φ). Thus, the area in Figure 5 between the two
curves defined by equations (22) and (27) specifies the do-
main of clean sandstones. Most of the data of clean Rhaet
Sandstones from the Triassic Keuper epoch fall in this do-
main. They are generally good aquifers and good hydrocarbon
reservoirs.
Mavko and Nur (1997) expand the range of validity of the
k–φ relationship [equation (28)] for a smooth-grain packing
model by introducing a percolation threshold porosity φc. This
yields
k = B(2rgrain)2
(φ − φc)3
(1 + φc − φ)2
, (29)
where B is a constant that must be calibrated for each basin.
Equation (29) yields a good fit when calibrated to the data for
the clean Fontainebleau Sandstone of Bourbie and Zinszner
(1985), i.e., setting B = 5, 2rgrain = 250 µm, and φc = 0.025. This
replaces the two linear approximations [equation (27a,b)] by a
single expression. While this is a perfectly valid empirical ap-
proach, it is still under dispute whether there really is a percola-
tion threshold for flow through porous rocks (e.g., Diedericks
and Du Plessis, 1996; Shashwati and Tarafdar, 1997). Intro-
ducing a threshold porosity φc into equation (28) means that
the formation factor F in the Kozeny–Carman equation (8) is
replaced by T/(φ − φc), rather than by T/φ, as in our fractal
approach. Both approaches are based on the concept of an
effective porosity. While porosity is restricted by tortuosity in
the fractal approach, it is reduced by the threshold porosity in
the percolation approach. However, in contrast to Archie’s law
[equation (10)], the new expression F = T/(φ − φc) for the for-
mation factor tends to infinity as φ approaches φc. However,
this is generally not observed in measurements on sedimen-
tary rocks. As a result, it does not appear that there is a general
percolation threshold porosity for permeability (see Figure 5).
Measured permeabilities are reported even for porosities <1%
(Schl¨omer and Krooss, 1997).
PERMEABILITY DERIVED FROM LOGGING DATA
For practical applications the fractal approach allows one to
obtain permeability logs from conventional industry porosity
logs or from laboratory-derived pore-radius distributions.

9. Fractal Permeability Prediction 1455
However, the prediction of permeability based on borehole
measurements can be much further improved if, in addition
to a porosity log, additional logs can be interpreted to allow
estimation of specific surface, clay content and bound water,
irreducible water saturation, and cation exchange capacity. For
instance, modern commercial log interpretation systems yield
porosity, quartz as well as shale content, and gas saturation
based on input from sonic, density, neutron, gamma-ray, and
electrical resistivity logs. A pore-size parameter can then be
derived from porosity and shale content. Finally, new devel-
opments which promise a more direct access to values of the
specific surface are the nuclear magnetic resonance method
(Kenyon et al., 1995) and the interpretation of complete de-
cay curves of induced polarization in the time domain (Pape
Table 1. Principles of measurement of some wireline logs and their use for the prediction of permeability.
Information in
Dependence on rock Main direct combination
Log type Physical response or fluid properties information with other logs
Resistivity Conduction of electro- Formation water salinity, Formation factor F Tortuosity T ,
lytes in the pore fluid water saturation, specific surface Spor
and of cations at the formation factor,
surface of the specific surface
pore wall
Gamma ray Total radiation of Content of thorium Shale content Specific surface Spor
natural potassium, adsorbed to pore walls,
thorium, and minerals containing
uranium potassium (e.g.,
feldspars and clays),
thorium, or uranium
Spectral Individual radiation The same radioactive Shale content, Specific surface Spor
gamma ray of natural potassium, sources as above, but specific surface Spor
thorium, and with respect to content
uranium in potassium,
thorium, and
uranium
Sonic Interval transit time of Elastic properties of P-wave velocity Porosity, shale
acoustic waves minerals and fluids, content (specific
water saturation, grain surface Spor)
contacts, rock
structure, porosity
Density Attenuation of Number of electrons Electron density Porosity, matrix density
collimated gamma per volume (correlated to
rays emitted rock density)
from a source
Neutron Velocity loss by elastic Number of hydrogen Hydrogen index Porosity, water
scattering of high- nuclei per volume saturation
energy neutrons
emitted from a
source
Induced Transient electro- Polarization of minerals Frequency-dependent Specific surface Spor,
polarization magnetic response with metallic conduction AC resistivity formation water
of a rock to an (sulfides, graphite), amplitude and phase salinity
electric current like polarization of the cation- angle, or voltage
a resistor combined rich water layer at the decay after DC
with capacitors pore wall surface, shut-off
effects depend on
rock structure and
salinity of the pore fluid
Nuclear Proton spins are first Build-up or decay of spin; Content of free Specific surface Spor
magnetic polarized by a DC polarization obeys an and bound water,
resonance magnetic field; exponential law; the time viscosity
following its shut-off, constant is largest for the
spins precess in the pore fluid’s free water and
earth’s magnetic field smaller for water close
to pore walls
et al., 1996). Table 1 summarizes borehole logs that can be com-
bined to obtain petrophysical information relevant for calcu-
lating permeability. A joint log interpretation yields porosity φ,
tortuosity T , and specific surface Spor . From these values, per-
meability can be calculated from a modified Kozeny–Carman
equation.
Reservoir permeability from a porosity log
We apply the average k–φ relationship, equation (22), to
porosity values obtained from acoustic, density, and neutron
logs (Figure 6a). The logs were recorded in a borehole in the
Rotliegend Formation in northwestern Germany. In this basin
the formation consists of an upper shaly siltstone and a lower

10. 1456 Pape et al.
sandstone, the main gas reservoir. Most of the section shown
was cored. Porosity and permeability were determined in the
laboratory and are available for calibration. Core permeability
is well correlated with permeability calculated from core poros-
ity and equation (22). This is expressed by R2
= 0.81, where R
is the correlation coefficient. Therefore, equation (22) appears
to be adequate for calculating a continuous permeability log
in this basin from log porosity without further calibration. The
result is in good agreement with the individual permeability
values measured on core (Figure 6b).
Reservoir permeability based on log data of porosity
and internal surface
High-resolution data of specific surface Spor obtained from
borehole logs correspond to laboratory data obtained with
the Brunauer–Emmett–Teller (BET) method (Brunauer et al.,
1938). However, what is really required is a specific surface
estimate that corresponds to the much smoother surface of
a hydraulic capillary. Its specific surface area is much smaller
than the one obtained from borehole logs. Therefore, when
FIG. 6. Permeability prediction for reservoir rocks. (a) Porosity
measured on cores (circles) and derived from sonic, density,
and neutron logs (line); (b) permeability measured on cores
(circles) and derived from the average permeability–porosity
relationship, equation (22), applied to the porosity log shown
in (a).
using equation (8), the effective pore radius reff is substituted
by reff = 2/Spor(λ1) (Pape et al., 1987a). The specific surface
Spor(λ1) corresponds to the resolution length λ1 =reff of the
permeability measurement, which is larger than the resolution
length of the BET method. The value Spor(λ1) can be calcu-
lated from equation (3) using the high-resolution specific sur-
face Spor(λ2) of the BET measurement and its resolution length
λ2, which is on the order of the dimension of gas molecules. In
sedimentary rocks with a high content of clay minerals, the
specific surface measured with the BET method is larger by
a factor q than the high-resolution surface which would result
for the pigeon-hole model and the effective pore radius reff of
these samples. The factor q > 1 is called lamella factor (Pape
et al., 1987a,b). Replacing the effective pore radius reff in equa-
tion (8) with the BET specific surface divided by the lamella
factor q, as described above, yields the so-called Paris equation
(Papeetal.,1982).Itrelatespermeabilitytospecificsurfaceand
accounts for the different resolutions:
k = 0.332φ2
(Spor(λ2)/q)−3.11
Sporin nm−1
, k in nm2
.
(30)
Log interpretation systems frequently provide the so-called
shale indicator Vsh, which is related to the specific surface. The
shale indicator is part of the solid content (1 − φ) of the rock
and corresponds to the rock’s clay content. Calibrating with
laboratory data from the same basin of the Rotliegend Forma-
tion in northwestern Germany, K¨ohler et al. (1993) had estab-
lished the following empirical relationship between the shale
factor and the specific surface:
Spor(λ2) = 0.313Vsh (nm−1
). (31)
If both porosity and shale factor are available from borehole
logs, the expression for specific surface [equation (31)] can be
inserted into equation (30). In the following example, porosity
and shale factor were determined by combined log interpreta-
tion using acoustic, density, neutron, and resistivity logs from
a borehole close to that of the first example (Figure 6). The
section of the borehole shown is a small part of the Rotliegend
Formation where reservoir sandstones are overlain by shaly
siltstones. Figure 7a shows porosity values measured on core
and the porosity log as well as the shale factor log. Figure 7b
shows the permeability values measured on core and perme-
ability logs calculated on the base of different equations. First,
permeability was calculated from the average permeability–
porosity relationship [equation (22)]. The other three logs use
the shale indicator as additional information. With Spor from
equation (31) for specific surface and φ, a permeability log was
calculated from equation (30). For the reservoir sandstone,
these permeabilities agree well with those of equation (22).
However, for the shaly siltstone the permeabilities calculated
with Spor are distinctly lower. As the core permeabilities scatter
widely in this part, it is impossible to decide which of the two
permeability logs is better.
For comparison, two more logs are calculated and plotted in
Figure 7b using as additional information the irreducible water
saturation Swi . The first of these two logs is calculated according
to a relation suggested by Zawisza (1993):
k = 45 584φ3.15
(1 − Swi )2
(nm2
). (32)

11. Fractal Permeability Prediction 1457
This expression is similar to equation (28) and corresponds to
a grain-packing model. The factor (1 − Swi ), a reduction of the
effective pore radius, accounts for deviations from this model
because of the clay content of natural sandstones. According
to Zawisza (1993), the irreducible water saturation Swi may be
determined from Vsh based on the empirical relationship
Swi = V 0.61
sh (1 − 2.5φ)3.18
. (33)
The permeability log obtained from equation (33) into equa-
tion (32) agrees well with the other logs and the core perme-
abilities for the reservoir sandstone. However, permeabilities
in the shaly siltstone section seem too high compared with core
permeability. Additionally, the permeability contrast between
sandstone and shaly siltstone seems not sharp enough.
The second permeability log based on porosity and the
irreducible water saturation Swi [equation (33)] was calcu-
lated according to an expression proposed by Timur (1968,
1969):
k = 0.33 φ4.4
S2
wi
0.83
(nm2
). (34)
This permeability log, shown in Figure 7b, is nearly identi-
cal with the log calculated according to equation (22), which,
however, does not require additional information on the shale
factor.
FIG. 7. Permeability prediction for reservoir rocks. (a) Poros-
ity measured on cores (circles) and porosity (blue line) and
shale factor Vsh (green line) derived from sonic, density, and
neutron logs; (b) permeability measured on cores (circles) and
derived from the porosity shown in (a) and the average perme-
ability–porosity relationship, equation (22) (blue line). Three
permeability logs are calculated from logs of porosity and
shale factor, shown in (a): the Paris equation (30) (red line);
the expression suggested by Zawisza (1993) [equation (32)]
(brown line); and the expression suggested by Timur (1968,
1969) [equation (34)] (green line).
Permeability of magmatic and metamorphic rocks
A fractal geometry relates permeability to porosity. A sim-
ilar fractal model can be established in magmatic and meta-
morphic rocks. The effective pore radius reff of the cylindrical
capillaries in a porous rock, however, is now replaced by the
hydraulically effective fissure aperture. The Kozeny–Carman
equation (8) can be applied not only to systems of capillary
cylindrical tubes but also to systems of narrow fissures bounded
by parallel plates. Therefore, the derivations discussed above
remain valid. Smooth fractures have a fractal dimension of
D = 2. In this case equation (28) yields permeability from frac-
ture porosity, and the grain radius rgrain is identified with the
linear dimension of the unfractured domains between the frac-
tures. In contrast, hierarchical substructures lead to a fractal di-
mension 2 < D < 3, as in porous rocks, and the effective pore
radius reff is identified with the effective fissure width. Pape and
Schopper (1987) discussed an application of this approach to
granite. In this study the pore space geometry of the fissured
rock is described by a model in which the pigeon-hole-type in-
ternal surface is additionally overgrown by two generations of
secondary minerals forming lamellar structures. The internal
surface of this composite system is accordingly larger than in
the original pigeon-hole model. This is accounted for by the
lamella factor q in the equation for the conversion of the spe-
cific surface area normalized to pore volume, Spor, to the res-
olution lengths λ1 and λ2 of different measurement methods
[equation (3)]:
Spor(λ1) = Spor(λ2)q(λ1/λ2)(Dt −D)
. (35)
The lamella factor q > 1 accounts for the increase in spe-
cific surface area resulting from the overgrowth of secondary
minerals, which is detected by measurement methods with a
resolution length λ1 but is not resolved by methods with a res-
olution length λ2. As in equation (3), Dt = 2 is the topological
dimension of a surface, and D is the fractal dimension. An av-
erage lamella factor q = 5.8 and an average fractal dimension
D = 2.24 were determined directly from equation (35) by Pape
and Schopper (1987) using low-resolution optical and high-
resolution hydraulic measurements on 14 granite samples from
the Falkenberg borehole in southeastern Germany.
To derive a k–φ relationship for magmatic and metamorphic
rocks, we start with equation (24), which is valid for all types
of structures. The first and second exponents of porosity in this
three-term expression contain as unknowns the fractal dimen-
sion D, the exponent m in Archie’s first law [equation (10)], and
the parameter c1 relating the different radii in equation (1). The
fractal dimension D can be determined, for instance, by mea-
suring the specific surface using optical and hydraulic measure-
ments as well as from electrical measurements of the induced
polarization (IP). The parameter c1 is set to c1 = 1 because the
relation between the different radii is now expressed using the
lamella factor q. Pape and Schopper (1988) suggest the fol-
lowing expressions for tortuosity T , porosity φ, and formation
factor F:
T = 0.1469(rgrain/reff)0.19
, (36)
φ = 31.62q0.76
(rgrain/reff)−0.76
, (37)

12. 1458 Pape et al.
and
F = 1.67q0.19
φ−1.25
. (38)
Comparing the exponents in equations (36), (37), and (38)
with those in equations (15), (16), and (10), we find the fractal
dimension D and the Archie exponent m were determined as
D = 2.24 and m = 1.25. Using these values and c1 = 1 for calcu-
latingthetwoexponentsinequation(24)yieldsexp1 = 1.25and
exp2 = 3.88.Withthisthethreecoefficientsofequation(24)are
calibrated with permeability and porosity measured on cores
from the Falkenberg borehole, yielding A = 0, B = 234, and
C = 2094. Thus, the relation between permeability and porosity
of microfissured granite is
k = 234φ1.25
+ 2094(10φ)3.88
(nm2
). (39)
The correlation of this expression in relation to the mea-
sured data is expressed by R2
= 0.66, where R is the corre-
lation coefficient. This comparatively low correlation is at-
tributable to the large variation found for the lamella factor,
i.e., qmin = 1 ≤ q ≤ 19 = qmax.
While the fractal dimension D = 2.24 determined for the
Falkenberg granite with its microfissure system is small com-
pared to the average value of D = 2.36 of sedimentary rocks, a
larger value of D = 2.5 was found for a dioritic breccia from the
Hardangerregion,Norway,(Pape,1997)usingopticalmethods.
In the framework of the KTB program, two boreholes of
4000 and 9100 m depth were drilled in the immediate vicin-
ity of the Falkenberg borehole (Emmermann and Lauterjung,
1997). They provide an exellent opportunity to study the physi-
calpropertiesofmetamorphicrocks,mostlygneissandmetaba-
site, in the continental crust using laboratory and borehole
measurements. The pilot borehole was completely cored, and
a great number of porosity and permeability measurements
were performed on drill cores (Berckhemer et al., 1997). A
porosity log was calculated for the pilot borehole from a multi-
variate statistical approach based on resistivity, sonic, density,
and neutron logs (Zimmermann, 1991; Zimmermann et al.,
1992; Pechnig et al., 1997).
Measurements of the induced polarization in the KTB bore-
holes showed that most of the KTB rocks are characterized by
a fractal dimension close to two and low porosities, while my-
lonitic zones have larger fractal dimension and porosity (Pape
and Vogelsang, 1996). It turns out that the second term of the
k–φ relationshipfortheFalkenberggranite[equation(39)]pro-
vides a good fit for the measured k–φ data in the mylonitic high-
porosity range. Neglecting the first term and calibrating with
known permeabilities and porosities from core samples yields
a coefficient C = 311 for the remaining term. The low-porosity
range with D = 2, corresponding to smooth fracture walls, can
also be fitted by the second term of equation (39) but with an
exponent exp 2 = m + 2/(c1(3-D)) modified for D = 2. Setting
c1 = 1 as before and m = 1 yields an exponent exp 2 = 3. The
coefficient of this term was also calibrated with known perme-
abilities and porosities from core samples, yielding C = 45. The
sum of both terms yields a k–φ relationship for the entire range
of porosity encountered in the KTB boreholes:
k = 45(10φ)3
+ 311(10φ)3.88
(nm2
). (40)
Based on this k–φ relationship for metamorphic rocks from
the KTB, a permeability log can be calculated from a given
porosity log. Figure 8 shows a comparison between permeabil-
ity calculated from equation (40) and the porosity log for the
KTB pilot borehole (Pechnig et al., 1997) on the one hand, and
permeability measured on core samples on the other hand.
CONCLUSIONS
The Kozeny–Carman equation (8) expresses a strong depen-
dence of permeability on the effective hydraulic pore radius reff
and additionally on the formation factor F, the ratio of tortu-
osity T and porosity φ. In the classical capillary-bundle model
of Kozeny and Carman, reff, T , and φ are independent param-
eters. In fact, this is true for general porous media. However,
in rocks which have experienced the same sedimentation and
diagenesis history, these parameters become correlated as in
the case of the analyzed Rotliegend Sandstone samples.
Based on a fractal model for porous rocks, several new ge-
ometrical relations were established in which effective radius,
tortuosity, and porosity are connected through the fractal di-
mension D. This is the fundamental geometric parameter for
the description of the pore-space structure. A default value of
FIG. 8. Permeability prediction for basement rocks. (a) Poros-
ity measured on cores (circles) and derived from resistivity
and sonic logs (line); (b) permeability measured on cores
(circles) and permeability log derived from the permeabil-
ity–porosity relationship calibrated for KTB basement rocks
[equation (40)] applied to the porosity log shown in (a).

13. Fractal Permeability Prediction 1459
D = 2.36 was shown to be useful for interpreting data from an
ensemble of Rotliegend Sandstone samples.
Permeability can be estimated from laboratory capillary
pressure measurements using a modified Kozeny–Carman
equation: The effective pore radius is expressed by the first
quartile or the median of the distribution of pore radii, and
the tortuosity is expressed by the ratio of median pore radius
and median grain radius. The predictions agree very well with
measured permeability (Figure 3).
Finally, based on fractal theory, a power-law relation was
established between permeability and porosity. This fractal
model is flexible and applicable over a wide range of porosities.
Moreover, it can be adjusted to different rock types. As an ex-
ample, the fractal permeability–porosity relationship was suc-
cessfully calibrated to a comprehensive data set measured on
Rotliegend Sandstone samples. A group of variants of the aver-
age fractal porosity–permeability relationship [equation (22)]
corresponds to different types of sandstones, ranging from
clean to shaly. These curves define a permeability–porosity
domain for sandstones which is validated by measured data
reported in the literature. Thus, they provide some general
guidance into the porosity–permeability relationship for sand-
stones. When applying this relationship to a specific basin, how-
ever, the coefficients A, B, andC in the general k–φ relationship
k = Aφ + Bφ2
+ C(10φ)10
need to be determined anew, based
on core data from this specific basin.
This novel approach is not restricted to sedimentary rocks.
Although magmatic and metamorphic rocks differ from sed-
imentary rocks with respect to pore geometry, an analogous
fractal model can be established. The main theoretical con-
siderations still remain valid in spite of the fact that the void
space in crystalline rocks is formed mainly by fissures and mi-
crocracks (in contrast to the porous networks of grain pack-
ings). In this case the effective pore radius reff of the cylindrical
capillaries is replaced by the effective fissure aperture.
In conclusion, the fractal approach provides a simple and
versatile technique that can be used easily to derive perme-
ability from porosity determined from conventional industry
porosity logs or from laboratory-derived pore radius distribu-
tions for a wide variety of different sedimentary and basement
rocks.
ACKNOWLEDGMENTS
The research reported in this paper was supported by the
German Federal Ministry for Education, Science, Research,
and Technology (BMBF) under grant 032 69 95. G¨unter Zim-
mermann (Tech. Univ. Berlin) kindly made available for this
study a porosity log from the KTB pilot borehole. Zhijing
(Zee) Wang (Chevron, La Habra), Zehui Huang (GSC At-
lantic, Dartmouth), Bin Wang (Mobil Technology, Dallas), and
one anonymous reviewer are gratefully acknowledged for their
constructive comments. Various versions of this manuscript
were critically read and improved by John Sass (USGS,
Flagstaff), Kurt Bram, Dietmar Grubert, Georg Kosakowski,
and Daniel Pribnow (all GGA, Hannover).
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