Permeability prediction

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 distributionsor from well logs is attractive but difficult. In this paperwe present a new, generally applicable, and relativelyinexpensive approach which yields permeability infor-mation on the scale of core samples and boreholes. Theapproach is theoretically based on a fractalmodel for theinternal structure of a porousmedium. It yields a generaland petrophysically justified relation linking porosity topermeability,whichmaybe calculated either fromporos-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 forthe different rock types are calibrated to a comprehen-sive data set of petrophysical and petrographical rockproperties measured on 640 sandstone core samples ofthe Rotliegend Series (Lower Permian) in northeast-ern Germany. With few modifications, this new straight-forward and petrophysically motivated approach canalso be applied to metamorphic and igneous rocks. Per-meability calculated with this procedure from industryporosity logs compares very well with permeability mea-sured on sedimentary and metamorphic rock samples.

INTRODUCTION

Permeability is one of the key petrophysical parameters formanaging hydrocarbon and geothermal reservoirs as well asaquifers. 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 sensitiveto changes in overburden pressure or to diagenetic alterations(Walder andNur, 1984;Walsh andBrace, 1984;Leder andPark,1986). For instance, drastic permeability reductions result fromthe growth of minute amounts of secondary clay minerals onquartz 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-Universitat, Nussallee 8,D-53115 Bonn, Germany. E-mail: [email protected].‡GGA, Stilleweg 2, D-30655 Hannover, Germany, E-mail: [email protected].∗∗LUNGMecklenburg-Vorpommern, Pampower Str. 66/68, D-19061 Schwerin, Germany. E-mail: [email protected]© 1999 Society of Exploration Geophysicists. All rights reserved.

capillaries. Generally speaking, permeability k is a functionof the properties of the pore space, such as porosity φ andseveral structural parameters. In the past, different empiricalapproaches 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; Bourbieand Zinszner, 1985; Jacquin and Adler, 1987). However, thesepurely empirical approaches lack a petrophysical motivation.Nelson (1994) reviews a number of these k–φ relationships. Forsomehomogeneous sandstone lithologieswith a uniquehistoryof sedimentation and diagenesis, the data show quasi-linear re-lationships in φ–log(k) plots. The slopes of linear regressionsvary frommoderate for sandy clay and silt (Bryant et al., 1974)to very steep for channel sandstones (Luffel et al., 1991). Thissuggests 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 asthe 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 thesize parameter (or its reciprocal) is equal or close to two. Theexponent of φ in the numerator usually varies between 1 and 7,and (1− φ)2 is a factor in the denominator. Most nonempiricalmodels are based on the Kozeny–Carman equation (Carman,1937, 1948, 1956),which links permeability to the effective poreradius and the formation factor F [see equation (8)]. However,none of these simple models can explain the sharp increaseof permeability with porosity observed for porosities greaterthan 10%. For instance, Luffel et al. (1991) find for their chan-nel sandstones that the pores are compressed to such a degreethat the hydraulic path is reduced to the free space betweenadjacent grain asperities. Thus, the simple model of unconsol-idated sand packing (a combination of ideal, smooth spheresand large voids in between) no longer applies. This illustratesthat a real-world pore system is more complicated. It needs to

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1448 Pape et al.

be viewed as a superposition of distinct structures at differentscales 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., Ungereret al., 1990). Such an expression yields an incorrect permeabil-ity prediction (e.g., Huang et al., 1996) if the size parameter (inthis case, the specific surface normalized to matrix volume) istreated as a constant. In this expression the pore-size parame-ter is much more important than the porosity term because itis much more variable. Therefore, if the size parameter is elim-inated by treating it as a constant, this must be compensatedfor by a considerable increase in the exponent of φ, as seen inequation (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, originallycompiled and studied by Bourbie and Zinszner (1985). Detailsand limitations of this approach are discussed at the end ofthe 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 approachdescribed above. However, in neural network modeling, thereis 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 suchas the Kozeny–Carman equation. Provided that the requiredlogs are available, the approach of Huang et al. (1996) yieldsexcellent permeability predictions.Permeability is also estimated from acoustic measurements.

For a cylindrical borehole in a porous solid, Stoneley waves areattenuated, and their velocity decreaseswith increasing perme-ability. These effects are related to permeability as the wavesinduce squirt flow in the porous medium. Winkler et al. (1989)combine laboratory measurements and petrophysical modelsbased on Biot theory (Biot, 1956a,b) to study Stoneley wavepropagation 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). Thenumber and definition of shape parameters required by thesedifferent models vary according to the approach chosen. Amodel particularly close to the natural appearance of porespace in sedimentary rocks is the so-called pigeon-hole model(Pape et al., 1987a). It is based on data measured on core sam-ples froma variety of hydrocarbon reservoir rocks in northwestGermany. Based on petrophysical principles, it yields relationsbetween various geometric, storage, and transport parametersof these reservoir rocks. Like other fractal models, it startsfrom the observation that the shape of the inner surface ofrock pores follows a self-similar rule. Thus, the theory of frac-tals (Mandelbrot, 1977) can be applied. A key parameter inthis theory is the fractal dimension D.

Based on fractal relationships, we first derive permeabilityfrompore-radius distributions of a large petrophysical data set.Next, a general relationship is derivedwhich links permeabilitydirectly to porosity. It is a three-term power law in which eachterm dominates the relation in a different range of porosity.This general relationship is then calibrated to different datasets, from reservoir sandstones to basement rocks. These cal-ibrated relationships are then used to calculate permeabilityof reservoir sandstones from logs of porosity (and optionallyalso from internal surface, shale factor, or irreducible watersaturation). Finally we demonstrate how this approach canbe extended to fracture porosity and show an application tometamorphic 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 rangeof porosity, and (4) can incorporate additional information oninternal surface, shale factor, or irreducible water saturation,where available.

PETROPHYSICAL DATA

Two different groups of petrophysical data are used in thisstudy. The first group is used for calibration and covers theporosity–permeability spectrum for rocks ranging from cleansandstones to shales. It consists of five data sets. The first onewas used by Pape et al. (1987a,b) to derive the general fractalprorosity–permeability relationship. A much more compre-hensive one, compiled over 15 years in East German hydrocar-bon industry laboratories (Iffland and Voigt, 1996), comprisesmeasured porosity and permeability as well as distributionsof pore radii and grain sizes. The next three data sets con-sist of porosity and permeability only: (3) shaly sandstone(Kulenkampff, 1994; Debschutz, 1995); (4) kaolinite (Michaelsand Lin, 1954); and (5) shales (Schlomer and Krooss, 1997). Asecond 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 andpermeability 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 comprisesdata for several petrophysical properties measured with var-ious methods on sandstone core samples from the PermianRotliegend series in northeastern Germany. For this study thefollowing distributions of (1) porosity, (2) permeability, and (3)pore radii were selected:

1) Anaverage porosityφm 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. PorosityφIM is determined using the imbibition method with iso-propanol as the saturant. Porosity φ3MPa is measured bygas penetration with a confining pressure of 3 MPa ina device also used for permeability. Porosity φCP is de-termined during capillary pressure measurements whichyield pore radius distributions.

2) The gas-flow permeability k3MPa was measured duringhigh-pressure experiments at an initial confining pressureof 3 MPa.

3) Capillary pressure measurements yield distributions ofpore radii weighted by volume. From the integrated

Fractal Permeability Prediction 1449

distribution functions, only the quartile values R25, R50,and R75 are reported (Iffland and Voigt, 1996). ValuesR25, R50, and R75 are defined as the smallest pore radiiof those capillaries which are completely filled when thepore space is invaded with mercury up to 25%, 50%,and 75%, respectively. Similarly, quartile values Q1 (firstquartile), MD (median), and Q3 (third quartile) are re-ported for those grain-size distributions which had beenanalyzedmicroscopically 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 abundle of bent cylindrical capillaries and derive fundamentalrelationships between various physical quantities. However, insome cases these integer power-law expressions do not fit theexperimental data as well as some empirical relations do. Thisis because the inner surface of a porous medium consists ofsubstructures of the pore walls that vary in size over manyorders of magnitude (Pape et al., 1982, 1984). Therefore, mea-surements of the specific surface of sandstones with methodsof different resolving power (Pape et al., 1982) show scalingproperties 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 thosefound in sandstones (Korvin, 1992). The basic structure of thismodel consists of two groups of spheres: the first represents thegrains with radius rgrain, and the second characterizes the poreswith radius rsite (Figure 1). The grains aswell as the pores formacascadeofhemispherical substructures.Thepigeon-holemodelrefers to the shape of these structures when viewed from thepore space. The pores are connected by hydraulic capillarychannels of radius reff . The radii rgrain, rsite, and reff are relatedby an empirical relation (Pape et al., 1984), which is valid for a

FIG. 1. A sedimentary rock according to the hierarchical pi-geon-holemodel, showing geometrical poreswith radii rsite andhydraulic-model capillaries with effective radii reff (after Papeet al., 1987a).

great variety of sandstones:

rgrain/rsite = (rgrain/reff)c1 , with c1 = 0.39

if rgrain/reff > 30. (1)

These three radii are the sizeparametersof theporousmedium.The self-similarity of the pigeon-hole model’s cascade of

hemispherical substructures is expressed by the self-similarityratio v = ri+1/ri < 1, where ri is the radius of the hemispheres ofthe ith generation and each of these is again parent to N hemi-spheres of radius ri+1. According toMandelbrot (1977), fractalbehavior follows from self-similarity, and the fractal dimensionD 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 porousmedium, such as internal surface and porosity, depend on theresolution of the measuring method. The relations follow anoninteger 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 beresolved by the two methods of measuring, and Dt = 2 is thetopological dimension of a surface. Comparing correspondingcouples of surface areas for different resolutions, Pape et al.(1987a) use equation (3) for the normalized specific surfaceto determine that the average fractal dimension for a set ofsandstone samples is D = 2.36. Generally, D varies between2.0 and 2.5 for most rocks; D = 2.36 is an average value repre-sentative for the northwestern German hydrocarbon reservoirrocks. Deviations from this value depend on the relative abun-dance of diagenetic structures. In particular, a smaller value ofD 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 composedof several classes of spherical bulges—the pigeon holes. Theseare measured by the mercury intrusion method as pseudocap-illaries. Typical distributions of sandstone pore radii exhibit along tail toward smaller radii. In a log–log plot, this part ofthe curve has a linear asymptote with a slope equal to Dt − D,where Dt = 3 is the topological dimension of a volume. Thiscan 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 thatare 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 PERMEABILITYAND OTHER PETROPHYSICAL PROPERTIES

Three parameters required by the fractal model were deter-mined from the data available from 640 Rotliegend sandstone

1450 Pape et al.

samples (Iffland and Voigt, 1996): the fractal dimension D, thefundamental parameter describing the structure of the porespace, the formation factor F , and the tortuosity T . The lattertwogeometrical factors are requirednotonly in theexpressionsfor 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 theRotliegend series. The fractal dimension D was determinedfrom equation (5) for a normalized inner surface. However,the scatter is large, and some values of the fractal dimensionwere even less than two. This is obviously incorrect since thefractal dimension of the pore-wall surface is always greaterthan the Euclidian dimension two of a surface. To obtain bet-ter agreement between data and model prediction, we need tointroduce several corrections to equations (4) and (5) for rel-ative volumes. These account for the experimental conditionsduring mercury imbibition (the method used to determine thedistribution of capillary radii), such as the shape of the contactbetween mercury and pore wall and the bottleneck effect inconstricted pores. The fractal model always involves nonzeroresidual volumes 1− V (r1) and 1− V (r2) in equation (4) forrelative volumes, which are defined by the product of surfacearea and resolution length. For rough surfaces, this agrees wellwith the physical situation. However, this is not the case fora smooth surface, where a close contact between surface andmercury is always established, irrespective of pressure. In thiscase, the residual volume vanishes. To account for these effects,a correction volume Vcorr needs to be added to the residualvolume (1− V (r2)), corresponding to the higher resolution inequations (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 sitesof a given size are invaded through their smaller pore necks.Similarly, even pigeon holes may be invaded by the mercurythrough smaller inlets. Thus, a conversion of the radii R75 andR50 to pigeon-hole radii is necessary in analogy to equation (1)for the ratio of radii. This introduces an additional exponentc2, which resembles the exponent c1 = 0.39 in equation (1) forthe ratio of radii. Thus, for the analysis of distributions of poreradii, 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 valueof 0.275, values for c2 were varied around a value of 0.39. Thisway, a value of c2 = 0.45 was determined for which the medianof D for all samples equals 2.36, which Pape et al. (1982, 1984,1987a) find to be a typical value for sandstones. The value ofD = 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 of2.57–2.87 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 withina realistic range. By comparing the ensemble mean D = 2.36with the fractal dimension of individual samples determinedfrom equation (7) for 50% and 75% saturation, it is possibleto distinguish between different types of sandstone. Larger in-dividual D values indicate the pore system is more structuredas a result of higher clay content.

Tortuosity

A fundamental expression for the permeability k of a porousmedium is given by the modified Kozeny–Carman equation(Kozeny, 1927; Carman, 1956):

k = r2eff/(8F), (8)

where F is the formation factor (Archie, 1942). It is defined asthe ratio of tortuosity T and porosity φ:

F = T/φ. (9)

Originally, the formation (resistivity) factor was defined as theratio of the electrical resistivity of a porous medium saturatedwith an electrolyte and the resistivity of this electrolyte. Itis a purely geometric parameter, describing how the porousmedium 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 byArchie’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 and1 < m < 3 (Serra, 1984). Later, Archie’s law [equation (10)]was also derived from a fractal model for porous rocks byPape 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 andporosity φ and equations (15), (16), and (17) defined later inthe text. These equations express the fractal relationship be-tween T and φ. Equations (17) and (9) combined yield a = 0.67and m = 2.Pape et al. (1987a) show that tortuosity behaves as a fractal

and depends on the ratio of reff and rgrain, with an exponentinvolving the fractal dimension D:

T = 1.34(rgrain/reff)0.67(D−2). (11)

This implies that tortuosity increases with increasing fractaldimension. However, this relation is valid only in the rangeof 2 < D < 2.4. In contrast, strongly fractured rocks (suchas cataclastic rocks) or claystones show fractal dimensions inthe range of 2.4 < D < 3. These rocks are characterized by ahigh degree of connectivity of the pore system, which in turnreduces 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 grainradius MD. Further, reservoir sandstones are limited in tor-tuosity (e.g., Kulenkampff, 1994). Therefore equation (11) ismodified so the new expression is identical to equation (11)for large pore radii reff; it yields Tmax = 10 in the limit of reff


approaching zero. This avoids extreme and unrealistic tortu-osites. This is realized by equation (12) for the reciprocal oftortuosity, in which the constant factor for the ratio R50/MDis determined from a fit to the data measured on Rotliegendsandstones in northwestern Germany (Kohler 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 crossplotof R50 and reff, calculated according to equations (8), (9), and(12) from measured permeability, porosity, and R50. Whileequation (11) is of the type of Richardson’s formula (1961)for an ideal fractal, equation (12) for tortuosity represents aso-called nonideal fractal (Rigaut, 1984). Now the formationfactor F can be determined from equation (9). Finally, the ef-fective hydraulic pore radius reff can be obtained by insertingthe measured permeabilty k and the formation factor F intoequation (8).

Permeability estimates based on pore-radius distributions

In equation (8), permeability varies with the square of theeffective pore radius reff but only linearly with the reciprocal ofthe formation factor. As outlined previously, the effective poreradii reff for Rotliegend sandstones can be calculated from themeasured permeabilities and the distribution of pore radii. Acomparison of the computed effective pore radii and the quar-tiles of the distribution of measured pore radii shows that, formost samples, reff is close to R50 (Figure 2). However, for val-ues of reff < 1µm, the ratio reff/R50 > 1. In these cases the R25value with an empirical weighting factor for porosity should beused for estimating reff because R25 is larger than R50. There-

FIG. 2. Log–log plot of the median radius R50 (determinedfrom measured pore radius distribution) versus the effectivehydraulic pore radius reff according to equation (8). The solidline is the linear regression log(R50) = −0.41 + 1.09 log(reff)with R2 = 0.76, where R is the correlation coefficient. Thebroken line is the identity function R50 = reff.

fore, the followingpermeabilitypredictionsarebasedonporos-ity and both quartile radii R50 and R25. Tortuosity T can beestimated from equation (12) using the default values D = 2.36(Pape et al., 1982, 1984, and 1987a) and rgrain = 200 000 nm (anaverage grain radius in our sandstone data sets) and replacingreff 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 R50aregiven innanometers.Equation (14)pre-vents the calculated permeabilities from becoming too low asthey might be if they were estimated using only equation (13).Figure 3 shows a crossplot of measured permeability versuscalculated permeability according to equations (13) and (14).The scatter in Figure 3 is caused by the differences foundin the grain-size distributions of rocks from any certain forma-tion. These primary differences are further enhanced duringthe subsequent diagenesis. The linear trend with a slope closeto one indicates good agreement between measured perme-ability data on the one hand and theoretical predictions on theother hand, which are based on fractal theory and measuredpore-radius distributions.

Permeability–porosity relationship

Our derivation starts from theKozeny–Carman equation (8)and the expression for the formation factor [equation (9)].Combining equations (8) and (9), permeability k is expressedby the effective pore radius reff, tortuosity T , and porosity φ.

FIG. 3. Permeability calculated from two quartiles ofpore-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) withR2 = 0.83, where R is the correlation coefficient. The brokenline is the identity function.

1452 Pape et al.

In general, reff and T vary with φ. For the moment we neglectthe limiting case of reff approaching zero, for which Tmax = 10.Then, according to Pape et al. (1987a), T and reff are expressedby 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

r2eff = r2grain(2φ)8. (18)

Inserting equations (9), (17), and (18) into equation (8), andusing a default value of rgrain = 200 000 nm (an average grainradius 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 ofk versus φ. It is valid for porosities larger than 0.1, whereasfor lower porosities the measured permeabilities exceed thosepredicted by equation (19). An explanation is suggested bythe fact that the effective pore radii of the samples studied byPape et al. (1987a,b) do not decrease as rapidly with decreas-ing porosity as suggested by equation (18). Therefore, we canimprove the permeability estimates for 0.01 < φ < 0.1 by as-signing a fixed value to the effective hydraulic pore radius inequation (8): reff = reff, fix = 200 nm. This is equal or near tothe mean of reff determined frommeasured permeabilities andporosities 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 = φ2r2eff,fix/(8× 0.67) = 7463φ2 (nm2). (20)

However, for φ < 0.01 this expression still yields permeabili-ties too small in comparisonwith themeasured data.A limitingvalue for reff in this range is reff,min = 50 nm. Inserting this valuefor R50 into equation (12) yields amaximum value for tortuos-ity, Tmax = 10. Using this value and inserting equation (17) intoequation (20) then yields

k = φr2eff,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, fora given porosity, the expressions for the other two porosityranges do not contribute significantly because of the differencein 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 < 2.36corresponds to a power between 3 and 10. A fractal dimensionD > 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 tounderestimate the permeability measured on the Rotliegendsamples because the northeastern Germany Rotliegend sand-stones are characterized by a relatively large permeability atany given porosity. The coefficients of φ, φ2, and φ10 in equa-tion (22) are calibrated to the Rotliegend data by nonlinearregression, which yields

k = 155φ + 37 315φ2 + 630(10φ)10 (nm2). (23)

Figure 4 shows this expression. The correlation is expressed byR2 = 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 inour study, not only the coefficients but also the exponents ofthe general permeability–porosity relationship need to be ad-justed.Aderivation similar to that for D = 2.36 [equations (15–22)] yields a comprehensive expression valid for all types ofstructures:

k = Aφ + Bφexp1 + C(10φ)exp2. (24)

It is valid for arbitrary values of D and the exponent m inArchie’s law [equation (10)]. The exponents are given byexp1= m, exp2= m + 2/(c1(3-D)) and 0.39 < c1 < 1. Theparameter 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 showsthe fractal permeability prediction according to equation (23).Correlation is given by R2 = 0.83, where R is the correlationcoefficient of the regression.


derive an expression for permeability of magmatic and meta-morphic rocks. Pape et al. (1984) derive values ofm = 1.53 andc1 = 0.47 from laboratory experiments on unconsolidated sandfor small ratios of grain to effective radius, rgrain/reff ≈ 2.9.

Application of general fractal permeability–porosityrelationship to clean and shaly sandstones

The discussion of the permeability–porosity relationshippresented 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 measuredon samples from different types of sandstones. Figure 5 showsthese data aswell as the curves corresponding to equations (22)and (23) for comparison. For permeabilities greater than107 nm2, these curves reach the field of unconsolidated sandsrepresented by two samples of Schopper (1967). Data froma set of unconsolidated kaolinite samples from Michaels andLin (1954) provide further orientation in the high-permeabilityrange. These data give an excellent example for a power lawwith an exponent of 10. Unconsolidated rocks represent thehigh-porosity, high-permeability limit within the total datafield. 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 longas the fractal dimension of the pore space equals the standardvalue D = 2.36, the diagenetic path is characterized by a power-law curve with an exponent of 10 in the porosity–permeabilityplot (Figure 5).However, conditions may be different in the low-permea-

bility range. Usually the decrease of the effective pore radiusreff and the increase of the tortuosity T is less pronounced forlow permeabilities, i.e., at later stages of diagenesis. This isreflected by the smaller increase in permeability with poros-ity in the low-porosity range corresponding to the linear andquadratic terms in equation (22). In addition to these curves,other curves of the same type canbemodeledbutwith differentcoefficients, reflecting the rock parameters rgrain; reff,min; reff,fix;and Tmax. One example is equation (23) for the northeasternGermany Rotliegend sandstones with larger coefficients in thethree termof equation (22). To characterize sampleswith lowerpermeability than predicted by equation (22) (see Figure 5), anew expression was derived by reducing the three coefficientsin equation (22). This was done in such a way that the new co-efficients A, B, and C in equation (24) are smaller than thosein equation (22) by the same proportion as the coefficients ofequation (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-permeabilitylimit at any given porosity in Figure 5 in such a way that itsatisfies the data measured on high-porosity kaolinite samplesand 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 ofsandstones can be characterized (symbols, Figure 5). The datashown inFigure 5weremeasuredon samples of unconsolidatedsand and Fontainebleau sandstone as well as on samples fromthe Dogger, Keuper, Rotliegend, Bunter, and Carboniferousformations (Kulenkampff, 1994; Debschutz, 1995; Petrophys-ical Research Group, Institute of Geophysics, University ofClausthal, personal communication, 1998). The Dogger andnorthwestern Germany Rotliegend sandstones plot mainlybetween the curves defined by equations (23) and (25). Incomparison, thenortheasternGermanyRotliegend sandstones(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 representmeasured permeability and porositydata 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 cleansandstones (left and above) from shaly sandstones (right andbelow).

1454 Pape et al.

any given porosity. Bunter sandstones as well as Carbonif-erous feldspar-rich sandstones and greywackes plot upon thecurve defined by equation (25). Additional data on shales fromthe Jurassic, Rotliegend, and Carboniferous are provided bySchlomer and Krooss (1997). Most of these data fall betweenthe curves of equations (25) and (26). They are characterizedby a wide grain-size distribution and a high content in clayminerals. The field between the two limiting curves defined byequations (25) and (26) characterizes the so-called shaly sand-stones. Very clean sandstones, made upmainly of quartz grainsof equal size, are found on the high-permeability end of thecurve defined by equation (23). The Oligocene FontainebleauSandstone is an extreme example of this very pure type of sand-stone. Bourbie and Zinszner (1985) found the following goodrelation 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 equationssuch as equation (22).However, it can be explained by the frac-tal concept, assuming the fractal dimension D equals about two.In this case, both tortuosity T and the specific surface Spor areconstant, and Spor is inversely proportional to both the poros-ity and the effective pore radius reff. Thus reff is proportional toporosity. Inserting equation (9) for the formation factor F intoequation (8) for the Kozeny–Carman and substituting φ for reffyields an exponent equal to three. This is much smaller thanthe exponent in equation (19). Similarly, a modified Kozeny–Carman equation of this type can be derived from equation (8)if a nonfractal spherical grain packingmodel is assumed (yield-ing a constant tortuosity) and the effective pore radius is sub-stituted by a term involving the specific surface expressed bythe grain radius and the porosity. The resulting equation

k = 0.5(rgrain)2φ3/(1− φ)2 (28)

is frequently used in basin analysis (e.g., Ungerer, 1990). Fig-ure 5 illustrates very clearly that this approximation (assumingrgrain = 0.1 mm) can be valid only in a very limited domain ofthe 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 quartzcementation with well-developed crystal faces. In addition, thegrains must be of equal size. The fractal dimension can alsobe estimated from the distribution of pore radii. Bourbie andZinszner (1985) present pore-radius distributions for differ-ent porosities. These curves are symmetrical and narrow forporosities >8%, corresponding to equation (27a). This corre-sponds to the limiting case, when the fractal model consists ofonly 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 skewedtoward smaller values. From this shape a fractal dimensionD > 2 can be derived, which corresponds to an exponentmuch greater than three. This is expressed by the exponentof 7.33 in equation (27b). This, in turn, corresponds to a fractaldimension 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 diageneticprocesses have generated a truly fractal pore space similar toaverage sandstones [equation (22)]. In Figure 5, the curves de-finedbyequations (27a,b) are supplementedbydata fromsomesamples of Fontainebleau sandstone (J. R. Schopper, personalcommunication, 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 quartzcementation, indicated by small exponents of the porosity inthe k–φ relation. As pressure increases as a result of burial,the pore volume partially collapses, which is indicated by largeexponents of the porosity in the k–φ relation. In contrast, me-chanical and chemical diagenesis are in reverse order for rockscontaining clay minerals because even minute amounts of clayminerals may prevent the precipitation of dissolved silica. Forthese rocks, porosity is first reduced by compaction (yieldinglarge exponents of φ) and then by cementation (with smallexponents of φ). Thus, the area in Figure 5 between the twocurves defined by equations (22) and (27) specifies the do-main of clean sandstones. Most of the data of clean RhaetSandstones from the Triassic Keuper epoch fall in this do-main. They are generally good aquifers and good hydrocarbonreservoirs.Mavko and Nur (1997) expand the range of validity of the

k–φ relationship [equation (28)] for a smooth-grain packingmodel by introducing a percolation threshold porosity φc. Thisyields

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 forthe clean Fontainebleau Sandstone of Bourbie and Zinszner(1985), i.e., setting B = 5, 2rgrain = 250 µm, and φc = 0.025. Thisreplaces the two linear approximations [equation (27a,b)] by asingle expression. While this is a perfectly valid empirical ap-proach, it is still under disputewhether there really is a percola-tion threshold for flow through porous rocks (e.g., Diedericksand Du Plessis, 1996; Shashwati and Tarafdar, 1997). Intro-ducing a threshold porosity φc into equation (28) means thatthe formation factor F in the Kozeny–Carman equation (8) isreplaced by T/(φ − φc), rather than by T/φ, as in our fractalapproach. Both approaches are based on the concept of aneffective porosity. While porosity is restricted by tortuosity inthe fractal approach, it is reduced by the threshold porosity inthe percolation approach.However, in contrast toArchie’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 generalpercolation threshold porosity for permeability (see Figure 5).Measured permeabilities are reported even for porosities<1%(Schlomer and Krooss, 1997).

PERMEABILITY DERIVED FROM LOGGING DATA

For practical applications the fractal approach allows one toobtain permeability logs from conventional industry porositylogs or from laboratory-derived pore-radius distributions.


However, the prediction of permeability based on boreholemeasurements can be much further improved if, in additionto a porosity log, additional logs can be interpreted to allowestimation of specific surface, clay content and bound water,irreducible water saturation, and cation exchange capacity. Forinstance, modern commercial log interpretation systems yieldporosity, quartz as well as shale content, and gas saturationbased on input from sonic, density, neutron, gamma-ray, andelectrical resistivity logs. A pore-size parameter can then bederived from porosity and shale content. Finally, new devel-opments which promise a more direct access to values of thespecific 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 inDependence 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 Sporand of cations at the formation factor,surface of the specific surfacepore wall

Gamma ray Total radiation of Content of thorium Shale content Specific surface Spornatural potassium, adsorbed to pore walls,thorium, and minerals containinguranium potassium (e.g.,

feldspars and clays),thorium, or uranium

Spectral Individual radiation The same radioactive Shale content, Specific surface Sporgamma ray of natural potassium, sources as above, but specific surface Spor

thorium, and with respect to contenturanium in potassium,

thorium, anduranium

Sonic Interval transit time of Elastic properties of P-wave velocity Porosity, shaleacoustic waves minerals and fluids, content (specific

water saturation, grain surface Spor)contacts, rockstructure, porosity

Density Attenuation of Number of electrons Electron density Porosity, matrix densitycollimated gamma per volume (correlated torays emitted rock density)from a source

Neutron Velocity loss by elastic Number of hydrogen Hydrogen index Porosity, waterscattering of high- nuclei per volume saturationenergy neutronsemitted from asource

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 salinityelectric current like polarization of the cation- angle, or voltagea resistor combined rich water layer at the decay after DCwith capacitors pore wall surface, shut-off

effects depend onrock structure andsalinity of the pore fluid

Nuclear Proton spins are first Build-up or decay of spin; Content of free Specific surface Spormagnetic 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 thespins precess in the pore fluid’s free water andearth’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 porosityφ,tortuosity T , and specific surface Spor . From these values, per-meability can be calculated from a modified Kozeny–Carmanequation.

Reservoir permeability from a porosity log

We apply the average k–φ relationship, equation (22), toporosity values obtained from acoustic, density, and neutronlogs (Figure 6a). The logs were recorded in a borehole in theRotliegend Formation in northwestern Germany. In this basinthe formation consists of an upper shaly siltstone and a lower

1456 Pape et al.

sandstone, the main gas reservoir. Most of the section shownwas cored. Porosity and permeability were determined in thelaboratory and are available for calibration. Core permeabilityiswell correlatedwithpermeability calculated fromcoreporos-ity and equation (22). This is expressed by R2 = 0.81, where Ris the correlation coefficient. Therefore, equation (22) appearsto be adequate for calculating a continuous permeability login this basin from log porosity without further calibration. Theresult is in good agreement with the individual permeabilityvalues measured on core (Figure 6b).

Reservoir permeability based on log data of porosityand internal surface

High-resolution data of specific surface Spor obtained fromborehole logs correspond to laboratory data obtained withthe Brunauer–Emmett–Teller (BET)method (Brunauer et al.,1938). However, what is really required is a specific surfaceestimate that corresponds to the much smoother surface ofa hydraulic capillary. Its specific surface area is much smallerthan the one obtained from borehole logs. Therefore, when

FIG. 6. Permeability prediction for reservoir rocks. (a) Porositymeasured on cores (circles) and derived from sonic, density,and neutron logs (line); (b) permeability measured on cores(circles) and derived from the average permeability–porosityrelationship, equation (22), applied to the porosity log shownin (a).

using equation (8), the effective pore radius reff is substitutedby reff = 2/Spor(λ1) (Pape et al., 1987a). The specific surfaceSpor(λ1) corresponds to the resolution length λ1 = reff of thepermeability measurement, which is larger than the resolutionlength 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 theBETmeasurement and its resolution lengthλ2, which is on the order of the dimension of gas molecules. Insedimentary rocks with a high content of clay minerals, thespecific surface measured with the BET method is larger bya factor q than the high-resolution surface which would resultfor the pigeon-hole model and the effective pore radius reff ofthese samples. The factor q > 1 is called lamella factor (Papeet al., 1987a,b). Replacing the effective pore radius reff in equa-tion (8) with the BET specific surface divided by the lamellafactor q , as described above, yields the so-called Paris equation(Papeet al., 1982). It relates permeability to specific surface andaccounts 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-calledshale indicator Vsh , which is related to the specific surface. Theshale indicator is part of the solid content (1− φ) of the rockand corresponds to the rock’s clay content. Calibrating withlaboratory data from the same basin of the Rotliegend Forma-tion in northwestern Germany, Kohler et al. (1993) had estab-lished the following empirical relationship between the shalefactor and the specific surface:

Spor(λ2) = 0.313Vsh (nm−1). (31)

If both porosity and shale factor are available from boreholelogs, the expression for specific surface [equation (31)] can beinserted into equation (30). In the following example, porosityand shale factor were determined by combined log interpreta-tion using acoustic, density, neutron, and resistivity logs froma borehole close to that of the first example (Figure 6). Thesection of the borehole shown is a small part of the RotliegendFormation where reservoir sandstones are overlain by shalysiltstones. Figure 7a shows porosity values measured on coreand the porosity log as well as the shale factor log. Figure 7bshows 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 usethe shale indicator as additional information. With Spor fromequation (31) for specific surface and φ, a permeability log wascalculated from equation (30). For the reservoir sandstone,these permeabilities agree well with those of equation (22).However, for the shaly siltstone the permeabilities calculatedwith Spor are distinctly lower.As the core permeabilities scatterwidely in this part, it is impossible to decide which of the twopermeability logs is better.For comparison, two more logs are calculated and plotted in

Figure 7b using as additional information the irreducible watersaturation Swi . Thefirst of these two logs is calculated accordingto a relation suggested by Zawisza (1993):

k = 45 584φ3.15(1− Swi )2 (nm2). (32)


This expression is similar to equation (28) and corresponds toa grain-packing model. The factor (1− Swi ), a reduction of theeffective pore radius, accounts for deviations from this modelbecause of the clay content of natural sandstones. Accordingto Zawisza (1993), the irreducible water saturation Swi may bedetermined from Vsh based on the empirical relationship

Swi = V 0.61sh (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, permeabilitiesin the shaly siltstone section seem too high compared with corepermeability. Additionally, the permeability contrast betweensandstone 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/S2wi

)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 shalefactor.

FIG. 7. Permeability prediction for reservoir rocks. (a) Poros-ity measured on cores (circles) and porosity (blue line) andshale factor Vsh (green line) derived from sonic, density, andneutron logs; (b) permeability measured on cores (circles) andderived from the porosity shown in (a) and the average perme-ability–porosity relationship, equation (22) (blue line). Threepermeability logs are calculated from logs of porosity andshale 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 cylindricalcapillaries in a porous rock, however, is now replaced by thehydraulically effective fissure aperture. The Kozeny–Carmanequation (8) can be applied not only to systems of capillarycylindrical tubes but also to systems of narrowfissures boundedby parallel plates. Therefore, the derivations discussed aboveremain valid. Smooth fractures have a fractal dimension ofD = 2. In this case equation (28) yields permeability from frac-ture porosity, and the grain radius rgrain is identified with thelinear 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 poreradius reff is identifiedwith the effective fissurewidth. Pape andSchopper (1987) discussed an application of this approach togranite. In this study the pore space geometry of the fissuredrock is described by a model in which the pigeon-hole-type in-ternal surface is additionally overgrown by two generations ofsecondary minerals forming lamellar structures. The internalsurface of this composite system is accordingly larger than inthe original pigeon-hole model. This is accounted for by thelamella 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 secondaryminerals, which is detected by measurement methods with aresolution length λ1 but is not resolved by methods with a res-olution length λ2. As in equation (3), Dt = 2 is the topologicaldimension of a surface, and D is the fractal dimension. An av-erage lamella factor q = 5.8 and an average fractal dimensionD = 2.24 were determined directly from equation (35) by Papeand Schopper (1987) using low-resolution optical and high-resolution hydraulicmeasurements on 14 granite samples fromthe Falkenberg borehole in southeastern Germany.To derive a k–φ relationship for magmatic andmetamorphic

rocks, we start with equation (24), which is valid for all typesof structures. The first and second exponents of porosity in thisthree-term expression contain as unknowns the fractal dimen-sion D, the exponentm inArchie’s first law [equation (10)], andtheparameter c1 relating thedifferent radii in equation (1). Thefractal dimension D can be determined, for instance, by mea-suring the specific surface using optical and hydraulicmeasure-ments as well as from electrical measurements of the inducedpolarization (IP). The parameter c1 is set to c1 = 1 because therelation between the different radii is now expressed using thelamella factor q . Pape and Schopper (1988) suggest the fol-lowing expressions for tortuosity T , porosity φ, and formationfactor F :

T = 0.1469(rgrain/reff)0.19, (36)

φ = 31.62q0.76(rgrain/reff)−0.76, (37)

1458 Pape et al.

and

F = 1.67q0.19φ−1.25. (38)

Comparing the exponents in equations (36), (37), and (38)with those in equations (15), (16), and (10), we find the fractaldimension D and the Archie exponent m were determined asD = 2.24 and m = 1.25. Using these values and c1 = 1 for calcu-lating the twoexponents inequation (24) yields exp1= 1.25andexp2= 3.88.With this the three coefficients of equation (24) arecalibrated with permeability and porosity measured on coresfrom the Falkenberg borehole, yielding A = 0, B = 234, andC = 2094. Thus, the relation betweenpermeability andporosityof 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, alarger value of D = 2.5was found for a dioritic breccia from theHardanger region,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-cal propertiesofmetamorphic rocks,mostly gneiss andmetaba-site, in the continental crust using laboratory and boreholemeasurements. The pilot borehole was completely cored, anda great number of porosity and permeability measurementswere performed on drill cores (Berckhemer et al., 1997). Aporosity log was calculated for the pilot borehole from amulti-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 bya fractal dimension close to two and low porosities, while my-lonitic zones have larger fractal dimension and porosity (Papeand Vogelsang, 1996). It turns out that the second term of thek–φ relationship for theFalkenberggranite [equation (39)] pro-vides a goodfit for themeasured k–φ data in themylonitic high-porosity range. Neglecting the first term and calibrating withknown permeabilities and porosities from core samples yieldsa coefficient C = 311 for the remaining term. The low-porosityrange with D = 2, corresponding to smooth fracture walls, canalso be fitted by the second term of equation (39) but with anexponent exp 2= m + 2/(c1(3-D)) modified for D = 2. Settingc1 = 1 as before and m = 1 yields an exponent exp 2= 3. Thecoefficient of this term was also calibrated with known perme-abilities and porosities from core samples, yieldingC = 45. Thesum of both terms yields a k–φ relationship for the entire rangeof 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 fromthe 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 theKTB pilot borehole (Pechnig et al., 1997) on the one hand, andpermeability measured on core samples on the other hand.

CONCLUSIONS

TheKozeny–Carman equation (8) expresses a strong depen-dence of permeability on the effective hydraulic pore radius reffand additionally on the formation factor F , the ratio of tortu-osity T and porosity φ. In the classical capillary-bundle modelof 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 anddiagenesis history, these parameters become correlated as inthe 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 forthe 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 resistivityand 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).


D = 2.36 was shown to be useful for interpreting data from anensemble of Rotliegend Sandstone samples.Permeability can be estimated from laboratory capillary

pressure measurements using a modified Kozeny–Carmanequation: The effective pore radius is expressed by the firstquartile or the median of the distribution of pore radii, andthe tortuosity is expressed by the ratio of median pore radiusand median grain radius. The predictions agree very well withmeasured permeability (Figure 3).Finally, based on fractal theory, a power-law relation was

established between permeability and porosity. This fractalmodel 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 onRotliegend Sandstone samples.A group of variants of the aver-age fractal porosity–permeability relationship [equation (22)]corresponds to different types of sandstones, ranging fromclean to shaly. These curves define a permeability–porositydomain for sandstones which is validated by measured datareported in the literature. Thus, they provide some generalguidance 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 relationshipk = Aφ + Bφ2 + C(10φ)10 need to be determined anew, basedon 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 analogousfractal model can be established. The main theoretical con-siderations still remain valid in spite of the fact that the voidspace 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 cylindricalcapillaries 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 industryporosity logs or from laboratory-derived pore radius distribu-tions for a wide variety of different sedimentary and basementrocks.

ACKNOWLEDGMENTS

The research reported in this paper was supported by theGerman Federal Ministry for Education, Science, Research,and Technology (BMBF) under grant 032 69 95. Gunter Zim-mermann (Tech. Univ. Berlin) kindly made available for thisstudy a porosity log from the KTB pilot borehole. Zhijing(Zee) Wang (Chevron, La Habra), Zehui Huang (GSC At-lantic, Dartmouth), BinWang (Mobil Technology, Dallas), andone anonymous reviewer are gratefully acknowledged for theirconstructive comments. Various versions of this manuscriptwere 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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