anisotropy of permeability and complex resistivity of tight sandstones subjected to hydrostatic...

Journal of Applied Geophysics 68 (2009) 356–370

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Journal of Applied Geophysics

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Anisotropy of permeability and complex resistivity of tight sandstones subjected tohydrostatic pressure

Norbert Zisser ⁎, Georg NoverUniversity of Bonn, Steinmann Institute, Poppelsdorfer Schloss, 53115 Bonn, Germany

⁎ Corresponding author. Tel.: +49 228/732766.E-mail address: [email protected] (N. Zisser).

0926-9851/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.jappgeo.2009.02.010

a b s t r a c t
a r t i c l e i n f o
Article history:
The dependence of perme Received 16 January 2008Received in revised form 3 February 2009Accepted 12 February 2009
Keywords:Confining pressurePermeabilityComplex resistivityAnisotropy

ability and complex electrical resistivity on direction was measured for low-permeable sandstone samples from a tight gas reservoir. Both properties were measured in three directionsat hydrostatic pressures up to 100 MPa.The decrease of permeability as a function of effective pressure (measured with a modified pressure-transient method) can be described by a power function. The pressure dependence is more controlled by theclosure of thin aspect ratio pores and cracks than by the minor reduction of porosity. The anisotropy ofpermeability is also a function of pressure. For some samples the preferred direction of flow changes withincreasing pressure.The Cole–Cole response-function can be fitted well to the complex resistivity spectra (kHz–MHz). Interfacialpolarization is the dominant polarization effect in this frequency range. The relaxation time of the Cole–Colemodel increases with increasing effective pressure, whereas the frequency exponent does not show anycontinuous behavior. According to the model of Lysne (1983) the geometrical distribution of pore shapes andtheir orientation can be derived from these quantities.The formation resistivity factor, taken from the real part of resistivity at 10 kHz, also increases with pressure.As the porosity does not change significantly, this increase means an increase of Archie's cementationexponent.Both, relaxation time and formation factor are also a function of the considered direction. But an overallrelationship between these quantities and permeability could not be observed; neither for absolute valuesnor for their anisotropy.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In view of the strong world gas prices the oil and gas industry isgiving more attention to non-conventional gas accumulations, liketight gas reservoirs. These reservoirs are characterized by lowmatrix–permeability, often combined with significant porosity and high porepressure, which yields to a large amount of gas.

Associated fractures can raise the permeability significantly.Distribution and orientation of the fractures and other rock fabricelements (like sedimentary layering or lineation) are important forthe anisotropy of the permeability, electrical resistivity and seismicvelocity (Dürrast et al., 2002).

An essential element of formation evaluation is the estimation ofpermeability. Therefore knowledge of the pressure dependenceof permeability of rocks from cores is important to deduce in-situpermeability and afterwards to calibrate permeability relationshipsderived from borehole logging tools. The effect of increasing

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hydrostatic pressure on changes in permeability is well-investigated(e.g. Bernabe et al., 2003; Bernabe, 1991; David et al., 1994; Fredrichet al., 1993; Walsh and Brace, 1984; Worthington, 2008; Zhu andWong, 1997). Also the permeability of tight gas sandstones wasinvestigated (e.g. Brower and Morrow, 1985; Ganghi, 1978; Jones andOwens, 1980; Kilmer et al., 1987). But there is a lack of knowledge onthe influence of hydrostatic pressure on the anisotropy of perme-ability, particularly with regard to micro-fractured, anisotropicsandstones. The variation of permeability and its anisotropy ateffective pressures up to 100 MPa are discussed in the first part ofthis paper.

The second part deals wit the pressure dependence of the complexresistivity in the frequency range of 1 kHz to 1 MHz. In this frequencyrange different types of polarization are effective: electronic, ionic(atomic), orientation (dipole) and interfacial (Maxwell–Wagner–Sillars) polarization. Electronic polarization occurs at frequencies upto 1016 Hz, being caused by the displacement of electron cloudssurrounding a nucleus by applying an electric field. Ionic polarization,caused by the relative movement of ions in a crystal lattice atfrequencies below 1013 Hz, is not of concern in wet rocks. Orientationpolarization occurs at frequencies below 1010 Hz in the presence of

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Table 1Mineral composition, porosity and matrix density of the tight sandstone samples.

Sample Mineral composition [vol.%] Porosity[%] (std.)

Matrixdensity[g/cm3]

Quartz Feldspar Mica Chlorite Dolomite

S1 62.24 12.39 24.05 1.31 0.00 2.62 (0.26) 2.68S2 37.65 5.65 51.09 5.62 0.00 1.26 (0.10) 2.72S3 73.01 2.03 16.11 2.91 5.93 2.60 (0.18) 2.70S4 84.42 2.30 11.42 1.04 0.82 7.96 (0.31) 2.65S5 79.88 3.47 14.10 0.69 1.86 9.60 (0.48) 2.68S6 81.35 4.73 11.57 0.78 1.58 9.22 (0.51) 2.68S7 77.96 8.20 12.59 1.24 0.00 8.20 (0.38) 2.67S8 78.30 9.42 9.66 0.73 1.89 4.46 (0.53) 2.66S9 80.43 5.06 12.92 0.58 1.01 5.38 (0.49) 2.72S10 74.85 7.88 14.97 0.39 1.91 8.45 (0.51) 2.66S11 71.89 5.93 12.99 0.81 8.38 5.99 (0.44) 2.70

The standard deviations (std.) of the porosity result from the measurements of thethree directional plugs (X, Y and Z) of each sample. The standard deviations for mineralcomposition and matrix density are less or close to the accuracy of measurements.

357N. Zisser, G. Nover / Journal of Applied Geophysics 68 (2009) 356–370

molecules with permanent dipole moment, like water. If an externalelectric field is applied the dipolar molecules tend to align in directionof the field. This polarization mechanism is used by various boreholelogging tools to estimate the porosity and water saturation by usingbasic mixing laws (cp. Ellis and Singer, 2007).

The most dominant polarization effect in the frequency rangebetween 1 kHz and 1MHz is the interfacial polarization, which acts upto frequencies of 108 Hz. Other polarization mechanisms with theirrelated low relaxation times have only a negligible and frequencyindependent contribution to the total polarization. The interfacialpolarization is also often called Maxwell–Wagner–Sillars (MWS)polarization, referring to the pioneer works of Maxwell (1892),Wagner (1914) and Sillars (1936). When an electric field is applied toa water saturated rock some of the initially free ions dissolved in thewater concentrate in the regions where electrical properties undergosudden changes, thus a build-up of charges at the interface betweeninsulation rock matrix and conducting pore fluid occur. If the electricfield is switched off, the ions return to their initial distribution. Thisprocess is controlled by a characteristic relaxation time, whichtypically takes values between 10−4 s and 10−8 s (Bona et al.,1998). The dispersion of the dielectrical and electrical quantities ofsuch a single relaxation process can be described by the Debye model(Debye, 1929). In natural rocks a large number of different relaxationprocesses occur, causing a distribution of relaxation times. This yieldsto a broadening of the dispersion which can often be described by theempirical Cole–Cole model (Cole and Cole, 1941). Although notheoretical model exists to explain experimental observations indetail, it is universally accepted that the interfacial polarization,characterized by relaxation time and dielectric dispersion, depends onthe complex micro-geometry of the rocks or rather on the geometricaldistribution and shape of the fluid-filled pores (Bona et al., 2002;Haslund, 1996; Hilfer, 1991; Lesmes and Morgan, 2001; Knight andNur, 1987; Lysne, 1983; Ruffet et al., 1991; Sen, 1981; Su et al., 2000;Trukhan, 1963).

Lysne (1983) derives from the Maxwell–Wagner–Sillars theory amodel that relates the relaxation time τ to the geometric shape of afluid inclusion, which is assumed to have a spherical form:

τ =em n − 1ð Þ + ew

σwð1Þ

where εm is the permittivity of the insulation matrix; σw and εw arethe conductivity and permittivity of water; and n is a dimensionlessshape factor. The shape factor is a function of the aspect ratio of thepore. For spherical inclusions n takes a value of 3; for prolatespheroids the shape factor gets very high and increases rapidly withincreasing aspect ratios (Lysne, 1983). For a rock, consisting ofdifferent shaped and orientated pores, each of the related shapefactors or rather relaxation times contribute to the overall complexresistivity of the rock with a Debye-like function. Hence meanrelaxation time and dispersion of the rock are a function of thedistribution of pore shapes. Thusmeasurements of the relaxation timepromise to be a non-invasive tool for estimation of pore structure.Bona et al. (2002) pointed out that the shape factor n is no shapeindicator in the usual sense, as the shape factor is also controlled byother quantities, like the surface to volume ratio. Nevertheless apressure dependence and dependence on direction of the relaxationtime and dielectric dispersion could be expected for micro-fracturedsandstones.

A few datasets of frequency dependent complex resistivity andcomplex permittivity in dependence of hydrostatic pressure arepublished in literature (e.g. Freund and Nover, 1993; Garrouch andSharma, 1994; Glover et al., 1996; Heikamp and Nover, 2003; Locknerand Byerlee,1985; Nover, 2005). But no one of these deals withmicro-fractured sedimentary rocks particularly with regard to the change ofanisotropy.

In standard electrical borehole logging tools (as electrode orinduction devices) only the real part of complex resistivity at a singlelow frequency is used for calculation of the formation factor andcementation exponent according to Archie (1942). Thus the pressuredependence of these quantities will be discussed additionally.

In the last section of this paper the pressure dependence ofpermeability will be compared with that of the electrical quantities.Both, anisotropy of permeability and anisotropy of complex resistivityare affected by the same micro-structure, characterized by thefractures and their total volume and size distribution, aspect ratiodistribution, orientation distribution and connectivity (Glover et al.,1996). Hence a relationship between the permeability and electricalparameters could be expected, in particular for the correlationbetween permeability and formation factor, as there are numerousexamples in literature providing a site specific correlation of thesequantities (e.g. Doussal, 1989; Doyen, 1988).

2. Samples

Eleven sandstone samples of a core segment from a boreholelocated in the Northwest German Basin were selected for this study.The samples are part of a tight gas reservoir and come from depths ofabout 3000 m from a sandstone–shale–coal sequence. The elevensamples are numbered with increasing depth from S1 to S11. Theinvestigated core segment has a total length of about 20 m.

Three cylindrical plugs indicated by X, Yand Z (diameter=30mm,length=20–35 mm) were prepared for further investigations. Due totoo poor core recovery of the core segment for sample S2 only twoplugs (Y and Z) could be prepared. The Z-direction is parallel to thecore-axis. The X- and Y-direction are perpendicular to the core-axisand to each other. Thus the anisotropy of permeability and complexresistivity can be surveyed in the three planes perpendicular to eachother XY, XZ and YZ.

According to the classification of Pettijohn et al. (1987) thesandstones can be described as lithic graywackes, because they consistmainly of the detrital components monocrystalline quartz, rockfragments and feldspar (Dürrast et al., 2002). The main mineralogicalphases are quartz, feldspar (predominantly albit), mica (predomi-nantly sericite) and in minor parts chlorite and dolomite (Table 1).The mineral distribution was calculated from X-ray powder diffracto-metry data with Rietveld-analyses. This was done for all samples anddirections. The difference between the three directional plugs is for allphases of one sample less than 0.1%.

The porosity and matrix density of the samples were measuredusing the gas compression/expansion method (cp. Tiab and Donald-son, 2004). The matrix density varies from 2.65 to 2.72 g/cm3, which

Fig. 1. Illustration of the autoclave and equipment used for measurement of the permeability by the modified pressure-transient method. The upstream pressure p1 is held constant,while the increase of pressure p2 in the downstream reservoir V2 is recorded as a function of time. By pressure transducers (PTs) the pressures p1, p2 and the confining pressure pcwere logged by computer and related control unit. Deviations of p1 or pc from given values were automatically adjusted.

358 N. Zisser, G. Nover / Journal of Applied Geophysics 68 (2009) 356–370

reflects the similar mineral distributions (Table 1). Porosity rangesfrom 1.26 up to 9.60% (Table 1). The lowest porosity is associated withthe lowest quartz- and highest mica-content in sample S2. But there isno general trend between the mineral composition and porosity. Thestandard deviations of porosity result frommeasurements of the threedirectional plugs (X, Y and Z) of each sample. They identify the minordifferences between the directional plugs and point out the generalproblem of anisotropy-determination in natural heterogenic media atdifferent plugs (Rasolofosaon and Zinszner, 2002).

Two samples differ from the others. Sample S4 is characterized byan approximately 1.5 mm wide, preponderant mineralized fracture,which is located parallel to the Y-direction with an angle of about 45°between X- and Z-direction. Sample S2 shows the highest content ofmica and is characterized by a significant smaller mean grain size(derived from qualitative thin section analyses).

3. Permeability

3.1. Experimental setup

Before assembling the samples to the autoclave for measurementof permeability they were dried for 48 h at 105 °C to remove alladsorbedwater from the samples. After that the plugs were stored in avacuum vessel to avoid adsorbance of air humidity.

The permeability was measured in a gas-autoclave using amodified pressure-transient method. Argon gas was used as thepermeating medium. The top and bottom sides of the cylindricalsamples were put between carved steel-adapters, which are con-nected to the gas volume outside of the autoclave (Fig. 1). Thisassembly is held together by a shrink tubing, which also acts as aninsulator against the confining pressure medium.

Moreover, the deformation of the sample in direction of flow wasmeasured by displacement transducers that are located outside theautoclave at both sides. The displacements transducers are notmounted to the autoclave and can therefore measure the displace-ment of the tubes against the autoclave. Thus under the assumption ofisotropic deformation or by measuring of plugs of different directionsthe change of geometry of the plug can be specified. Obviously thereduction of porosity can be also estimated by this setup.

The transient method can be derived from the fundamental Darcy-equation:

K = − η � ujp

ð2Þ

where K is the permeability of the rock, u the volume flow density, ηthe dynamic viscosity of the fluid and p the fluid pressure. A directmeasurement of the permeability according to Eq. (2) is notpracticable for low-permeable samples, because the flow density istoo low for standard flow meters. Under specific conditions (lowporosity, constant temperature and approximately constant compres-sibility β) the permeability can bemeasured by the pressure-transientmethod and be calculated by applying Eq. (3) (Zoback and Byerlee,1975b).

p1 tð Þ− p2 tð Þ = p1 0ð Þ− p2 0ð Þð Þ � exp −KGηβ

� 1V1

+1V2

� �� t

� �ð3Þ

The geometry-factor G is given by the quotient of the crosssectional area A and length l of the sample. The dynamic viscosity η isa function of temperature. The parameter t is the time after startingthe measurement and px(t) denotes the related pressure in VolumeVx. For the setup shown in Fig. 1 with constant p1 and related infinityVolume V1 Eq. (3) can be simplified to

p2 tð Þ = p1 + p2 0ð Þ− p1ð Þ � exp −KGηβV2

t� �

ð4Þ

So the permeability K can be calculated from the measured timeseries of p2(t) by linear regression. Eq. (4) makes no account to the“Klinkenberg-Effect", which characterizes the effect of increasingapparent (gas) permeability Kmeas by decreasing average absolute gaspressure pav in the sample (Klinkenberg, 1941). The absolutepermeability Kabs (equivalent to the liquid permeability) can becalculated from

Kabs = Kmeas pavð Þ � 1 +Bpav

� �−1ð5Þ

Fig. 2. Permeability variation with increasing effective pressure. a) Measured pressure dependence of sample S4 (filled symbols), sample S9 (open symbols) and fitted powerfunction (solid lines). Circles, squares and triangles indicate plugs in X, Y and Z-direction. b) Variation of normalized permeability (K/K0) with pressure calculated from pressuresensitivity exponent AK (Table 2). The symbols mark samples S4 and S9 in X, Y and Z-direction. Dashed lines indicate the pressure dependence of all measured plugs.

Table 2Normal-pressure permeability K0 and pressure sensitivity exponent AK calculated bylinear regression for each individual plug (Eq. (6)).

Sample d K0 [10−18 m2] AK R2

S1 X 1.82 1.14 0.95Y 1.73 1.11 0.98Z 0.36 0.77 0.94

S3 X 0.34 0.89 0.94Y 0.88 0.84 0.93Z 0.16 0.61 0.91

S4 X 12.8 0.45 0.98Y 7.02 0.36 0.96Z 19.6 0.64 0.95

S5 X 11.4 0.45 0.95Y 6.98 0.56 0.97Z 35.4 0.38 0.98

S6 X 44.7 0.44 0.93Y 11.8 0.71 0.96Z 6.49 0.64 0.97

S7 X 11.8 0.58 0.93Y 16.6 0.74 0.98Z 6.66 0.57 0.98

S8 X 9.36 0.96 0.98Y 36.7 1.23 0.98Z 4.00 0.89 0.96

S9 X 7.34 1.15 0.97Y 5.97 1.18 0.97Z 3.18 0.83 0.97

S10 X 9.70 0.42 0.99Y 9.12 0.41 0.99Z 14.7 0.49 0.97

S11 X 11.5 1.17 0.91Y 16.5 1.19 0.94Z 6.36 1.01 0.93

Regression coefficients R2 are general greater than 0.91, which indicate theapplicableness of the power function for describing the permeability variation withincreasing effective pressure.


where B denotes the Klinkenberg-factor. To estimate the absolutepermeability from the pressure-transient method, the time series wassplit into several subintervals. For each interval the apparentpermeability Kmeas(pav) was calculated by means of Eq. (4) and therelated mean gas pressure pav was calculated by the average gaspressure in the interval. The absolute permeability was estimated bylinear regression from Eq. (5). This procedure was used for thepermeability estimation of the eleven investigated samples and yieldssatisfactory results (realistic Klinkenberg-factors). The absolutepermeability is for the samples up to a factor of about three smallerthan the gas-permeability. Permeability data in this paper thus are“Klinkenberg”-corrected.

The measurement of permeability was performed only up toeffective pressure of 100 MPa to avoid irreversible compaction of thesample. At higher effective pressure the elastic domain ends andcataclastic deformation can occur, i.e. by grain crushing and porecollapse. By rerunning the experiment after a few weeks it wasensured that no irreversible deformation or hysteresis effect wasinduced by the application of confining pressure. The minimum of theconfining pressure is 7.5 MPa, because p1 is set to 5 MPa to get anapproximately constant compressibility during the measurement.That is why the minimal effective pressure is limited to 2.5 MPa,because confining pressure has to be significant greater than porepressure to avoid gas leaking. Assuming the poroelastic coefficient forpermeability to be equal to one, the effective pressure peff wascalculated from the difference between confining pressure andinternal pore pressure. Although the coefficient may differ signifi-cantly from one (cp. Zoback and Byerlee, 1975a; Berryman, 1992) thisassumption has no significant influence on the estimated effectivepressure for the presented experiments, because of the constant andrelatively low internal pore pressure compared with confiningpressure.

3.2. Pressure dependence

The samples exhibited in all three directions a decrease ofpermeability with increasing pressure (Fig. 2a). The continuousdecrease is an indicator that no fracturing in the samples occurs atincreasing pressure. The samples, except sample S2, show in theconsidered pressure range values of permeability between 10−20 and10−16 m2. The permeability of low-porosity sample S2 is even atlowest confining pressure beneath the detection limit of thepermeameter for this setup (b10−21 m2).

The permeability decreases rapidly at low effective pressure andstabilizes with increasing pressure (Fig. 2a). The high decrease of

permeability at relatively low pressure is caused by the closure ofopen cracks and therefore of the preferred flow paths of the fluid. Aspointed out by various authors by theoretical or experimentalinvestigations (e.g. Brower and Morrow, 1985; Ganghi, 1978; Jonesand Owens, 1980; Kilmer et al., 1987) the pressure dependence of lowpermeability rocks cannot be interpreted by basic network modelslike Kozeny–Carman equations (see e.g. Dullien, 1992), where thedecrease of permeability is predominantly related to decreasingporosity and increasing tortuosity of the flow paths. Of course thesetwo quantities also contribute to the decrease of permeability of tightsandstones, but the pressure dependence is more controlled by theclosure of thin aspect ratio pores and cracks (e.g. Ganghi, 1978; Kilmer

Fig. 3. Relationship between pressure sensitivity exponentAK,K0 and other petrophysical parameter (circles, squares and triangles indicate plugs in X, Yand Z-direction). a) No correlationbetween AK and K0 is obvious for the 30 measured plugs. b) An approximate linear reduction of AK with increasing porosity can be detected (a=−0.077; b=1.255; R2=0.45). c) Theincrease of zero-pressure permeabilityK0 with increasing porosity can be described by a power function (Ka=1.04E−19;A=2.30;R2=0.64). d) A reduction ofK0with increasing contentof chlorite is observed (Ka=2.87E−17, A=−1.48; R2=0.56). Note that low permeability and high content of chlorite sample S2 would also match to this trend.


et al., 1987). Correspondingly the decrease of porosity is for allmeasured plugs below the detection limit, which means that thedisplacement measured at the pipes against autoclave are less than1 µm. From this follows that the decrease of porosity is for all samplesless than 0.03%. Thus the porosity is nearly independent of confiningpressure.

The permeability–pressure relationship for all plugs can bedescribed by a power function:

K peffð Þ = K0p−AKeff ð6Þ

The exponent AK is a measure of the pressure sensitivity ofpermeability. AK adopts values between 0.36 and 1.23 (Table 2), whichis similar to the values detected by Kilmer et al. (1987) for low-permeable sandstones from several tight gas reservoirs. The normal-pressure permeability K0 and the pressure sensitivity exponent AK

show a slight trend to be approximately independent of the directionand therefore a characteristic of the sample (Table 2). Neverthelessslight differences of AK cause important differences of permeability athigher effective pressure (Fig. 2). From the pressure dependence ofnormalized permeability (K(peff)/K0) it can be seen, that for someplugs the permeability decreases with increasing pressure by morethan two orders of magnitude whereas in other plugs just a decreaseof a factor of 2 was detected (Fig. 2b). But there is no general trend inthe magnitude and pressure dependence of permeability for thedifferent directions (Table 2).

3.3. Correlations

A general trend described in various publications is that in lowpermeability and low porosity rocks the relative reduction ofpermeability (represented by AK) is significantly stronger than inhigh permeability and high porosity rocks (e.g. Jones and Owens,1980; Kilmer et al., 1987). At least for the dependency on permeability

the presented data does not confirm with this general or any othertrend, as can be seen by the plot of AK versus K0 (Fig. 3a). This could beexplained either by the high anisotropy of the samples or by theassumption that the investigated range of permeability is too small toevolve the described general trend, as the pressure dependence of allsamples is controlled by microcracks. However the pressure sensitiv-ity exponent follows the general trend and shows a weak linearcorrelation to porosity (Fig. 3b).

Additional regression analyses were made to detect other possiblecorrelations between K0, AK and any other measured petrophysicalparameter. The pressure sensitivity exponent AK shows no additionalcorrelations. However K0 is weakly related to the porosity by a powerfunction (Fig. 3c). Such correlation between porosity and permeabilityare typical for samples from one formation (e.g. Nelson, 1994). Moreunexpected is the trend to low permeability with increasing content ofchlorite (Fig. 3d). This trend is more obvious with regard to the lowestpermeability sample S2, which has the highest content of chlorite. Togive an explanation of this trend it would be necessary to analyze thedistribution of chlorite in the pore structure (e.g. pore-lining or pore-filling texture). A relationship between pressure sensitivity exponentAK and content of sodium feldspar, as pointed out by Kilmer et al.(1987), is unverifiable for this dataset.

3.4. Anisotropy

As mentioned above, the three directions X, Y and Z areperpendicular to each other and therefore the anisotropy of perme-ability can be considered in the three planes XY, XZ and YZ.

The anisotropy of the permeability αK for these planes is defined as

αK d1; d2; peffð Þ = lgKd1

peffð ÞKd2

peffð Þ

!ð7Þ

Fig. 4. Variation of the anisotropy of permeability αK with increasing effective pressure. a) Pressure dependence of anisotropy of sample S4 (filled symbols) and S9 (open symbols).The anisotropy is calculated by Eq. (8) (values of αK0 and AαK are given in Table 3). Circles, squares and triangles indicate planes XY, XZ and YZ. Note that anisotropy of zero stands forisotropic permeability in the plane and a change of sign of αK indicates a change of the predominant direction of permeability. b) Normalized pressure dependence of anisotropy. Thesymbols mark samples S4 and S9 in the XY, XZ and YZ planes. Dashed lines indicate the pressure dependence of all measured plugs. The wide differences in pressure dependence ofanisotropy for different samples and planes are obvious.

Table 3Normal-pressure anisotropy of permeability αK0 and pressure sensitivity exponent AαK

describing the anisotropy of the considered planes (given by direction d1/d2) of eachsample.

Sample d1/d2 αK0 AαK piso [MPa]

S1 X/Y 1.05 −0.021 11.2X/Z 5.09 −0.367 84.5Y/Z 4.84 −0.346 95.5

S3 X/Y 0.39 −0.053 b1X/Z 2.09 −0.277 14.4Y/Z 5.39 −0.224 N200

S4 X/Y 1.82 −0.092 N200X/Z 0.65 0.194 9.1Y/Z 0.36 0.286 35.8

S5 X/Y 0.69 0.140 b1X/Z 0.32 −0.036 b1Y/Z 0.20 −0.175 b1

S6 X/Y 3.78 0.262 b1X/Z 6.89 0.193 b1Y/Z 1.82 −0.069 N200

S7 X/Y 0.71 0.168 7.5X/Z 1.77 −0.009 N200Y/Z 2.49 −0.177 171.5

S8 X/Y 0.26 0.275 143.8X/Z 2.34 −0.062 N200Y/Z 9.18 −0.337 N200

S9 X/Y 1.23 0.029 b1X/Z 2.31 −0.321 13.6Y/Z 1.88 −0.349 6.1

S10 X/Y 1.06 −0.017 39.4X/Z 0.66 0.071 N200Y/Z 0.62 0.088 N200

S11 X/Y 0.70 0.020 N200X/Z 1.81 −0.169 33.2Y/Z 2.60 −0.190 154.6

Anisotropy of permeability is calculated by applying Eq. (8). Isotropic pressure-pointspiso indicate the effective pressure at which the preferred flow paths respectively thesign of the anisotropy change. Isotropic pressure-points that are smaller than 1 MPa areindicated with b1, as this range of pressure is not covered by the experimental setup.Also piso greater than 200 MPa are given by N200, as higher effective pressures aredefinitely outside of the elastic domain of the samples.


where d denotes the direction (X, Y or Z) and Kd the associatedpermeability at given effective pressure. As the pressure dependenceof permeability for all measured plugs can be described adequately bya power function (Table 2), the anisotropy of permeability can becalculated by the power functions of the respective directional plugs:

αK d1; d2; peffð Þ = lg αK0ð Þ + AαK � lg peffð Þ ð8aÞ

αK0 =K0;d1K0;d2

ð8bÞ

AαK = AK;d2− AK;d1

ð8cÞ

The logarithmical formulation of anisotropy causes negative valuesfor values of Kd2

greater than Kd1, so that a change of sign with

increasing pressure means a change of the preferred direction of flow(Fig. 4a). The pressure sensitivity exponent for anisotropy ofpermeability AαK ranges from −0.367 to 0.286 (Table 3). Thusanisotropy of permeability is less sensitive to increasing pressurethan permeability. The different signs of AαK indicate that bothdecrease and increase of anisotropy with increasing effective pressureoccur. By consideration of the normalized anisotropy the differenttrends of anisotropy with increasing pressure become apparent(Fig. 4b). In some planes just slight changes of anisotropy withincreasing pressure occur, while the anisotropy in others changes upto a factor of 0.7 in the considered range of pressure, which compliesto a change of the Kd1

/Kd2ratio of about 5.

An important result of the pressure dependence of anisotropy is,that many samples exhibit a change of sign of the anisotropy for thegiven pressure range, with the consequence that isotropic pressure-points occur piso (Fig. 4a; Table 3).

That implies a pressure dependence of the preferred flow paths. Asthe reduction of porosity with increasing pressure is unattached of theconsidered direction, the change of preferred flow paths can beexclusively related to the closure of thin aspect ratio pores and cracks.

The strong pressure dependence of anisotropy and permeabilitypoint out, that it seems to be necessary for reservoir characterizationtomeasure these quantities at related in-situ pressure conditions. Alsothe variation of permeability and its anisotropy with increasingpressure must be known for the simulation of gas production, as

during production a considerable drop in pore pressure occurs, whichmeans an increasing effective pressure. In addition, for estimation ofpermeability at real simulated in-situ conditions, the measurementshave to be done in a triaxial experimental setup in respect to the

Fig. 5. Illustration of the piston-cylinder autoclave and equipment used for the measurement of complex resistivity (a) and top view of the carved electrode (b).


anisotropic stress condition in the reservoir (cp. Khan and Teufel,2000).

4. Resistivity

4.1. Experimental setup

The complex resistivity measurements are carried out with a two-electrode configuration (Fig. 5a). The material of the electrodes isstainless steel located on plexiglass-discs. The electrodes are radiallycarved, so that by increasing confining pressure and related decreaseof the porosity of the sample the pore fluid can move to these chinks(Fig. 5b). This movement ensures a constant inner-pore pressure ofthe sample. Test measurements have shown that the water-filledchinks have no influence on the measured impedance. To accomplishan adequate contact between electrode and sample, high adsorbentnon-polarizable paper-membranes (saturated with the pore fluid) arelocated between electrode and sample. The sample-electrode-systemis held together by an elastic shrink tubing which also acts to insulatethe sample from the pressure medium glycerin. Special account had tobe made to the conceptual design of the electrical leadthrough in theautoclave (cp. Glover et al., 1994). Leadthroughs were designed withcapacities less than 6 pF.

Before assembling the samples to the autoclave for measurementof complex resistivity they were dried for 48 h at 105 °C to remove alladsorbedwater from the samples. The dried plugswere fully saturatedwith a 0.1 M NaCl solution in a vacuum vessel for at least 48 h toensure complete saturation (cp. Taherian et al., 1990). By comparisonof the weight of the saturated plug and the calculated weight fromthe known values of porosity, matrix density and fluid density thecompleteness of saturation was verified. The plugs were left inthe NaCl solution until the conductivity of the solution was for morethan 24 h constant. Thus it can be assumed that the sample is in achemical equilibrium with the solution. Due to evaporation andsolving processes the resistivity of the pore fluid decreases somewhatafter the saturation. The resistivity of the solution varies between 0.96and 1.01 Ωm. But it was ensured that all plugs of each sample weresaturated with solution of identical resistivity.

The electrical impedance Z* (* denotes complex values) wasmeasured with a HP4192A impedance analyzer in the series mode at amaximum voltage of 1 V (for details see e.g. Knight and Nur, 1987).From the impedance the complex resistivity ρ* is calculated bymultiplying a geometry-factor (A/l), where A denotes the crosssectional area and l the length of the cylindrical plugs. The complexresistivity can be expressed either in rectangular or polar form. Thuscomplex resistivity ρ* is given by real ρ′ and imaginary ρ″ part or bymagnitude |ρ| and phase Φ of the resistivity:

ρ4 ωð Þ = ρ0 ωð Þ + iρ00 ωð Þ = jρ j ωð ÞeiΦ ωð Þ ð9Þ

with i2=−1 and the angular frequency ω=2πf. Instead of complexresistivity the conductive and capacitive properties of rocks can berepresented by complex conductivity σ*, complex permittivity ε* andrelative complex permittivity κ*:

σ4 ωð Þ = 1ρ4 ωð Þ = iωe4 ωð Þ = iωe0κ4 ωð Þ ð10Þ

The permittivity of vacuum ε0 is given by 8.85E−12 F/m. Inliterature the relative complex permittivity at high frequencies is oftencalled ‘dielectric constant,’ which does not make the frequencydependence clear. In the following complex electrical and dielectricalquantities are only discussed in terms of complex resistivity, as mostlogging tools, that operate in this frequency range, are used to measureresistivity, even though they do not consider polarization effects.

With the HP4192A impedance analyzer the electrical impedance canbemeasured in the frequency range of 5Hz to 13MHz. But only a limitedportion of the response in this frequency range can be related to thedirect response of the sample. At low frequencies the two-electrode-technique results in polarization at the sample-electrode interface. Toavoid this effect theminimum frequency is set to 1 kHz. Nevertheless forsome spectra electrode polarization (characterized by large relaxationtimes) is observed. A quantification of the electrode impedance bymeasuring with various sample length (e.g. Lesmes and Frye, 2001)could not be applied to the samples, as they had to stay intact. Thus for

Fig. 6. Dependence of the formation factor on effective pressure for all measured samples and directions. The formation factor is calculated from the real part of complex resistivity ata frequency of 10 kHz (Eq. (11)). The variation of formation factor with increasing pressure can be described either by a power or a linear function (Eq. (12), Table 4). As no significantreduction of porosity occurs in the regarded pressure range, the increasing formation factor indicates the increase of Archie's cementation exponent with increasing pressure. Notethe damage of sample S2 in Y-direction at effective pressures between 40 and 50 MPa, which causes a small decrease of the formation factor.


this spectra the lower frequencies were removed from the dataset forfurther consideration. The significant influence of electrode polarizationto the measured spectra for all measurements starts at frequencies lessthan 10 kHz. The upper limit of frequency is set to 1 MHz. At higherfrequencies the capacitive effect of the leadthroughs gets significant andparasitic capacities have to be taken into account.

Ten logarithmically spaced datasets were collected per decade offrequency. The complex resistivity was measured for confiningpressures up to 100 MPa at a constant inner-pore pressure of about0.1 MPa. Thus the confining pressure approximately conforms to theeffective pressure.

Five samples were selected for the measurement of complexresistivity: the low-permeable sample S2, the by a mineralizedfracture dominated sample S4 and samples S3, S8 and S9 that showa representative pressure dependence of the permeability. Themeasurements were carried out at the very same plugs as themeasurements of permeability.

4.2. Formation resistivity factor

TheArchie formula is often used to relate the resistivity of brinefilledand partially saturated reservoir rocks to porosity and saturation.Measurements on core plugs can be used to calibrate electrical loggingdata and to identify hydrocarbon. The formation resistivity factor F isdefined as the ratio of the resistivity of the saturated plug ρ0 to theresistivity of the saturating fluid ρw. The formation factor is linked to theporosity and saturation by Archie's formula (Archie, 1942), where ρtdenotes the resistivity of the sample with the brine saturation Sw.

F =ρ0

ρw= Φ−m ; I =

ρt

ρ0= S−n

w ð11Þ

If the cementation exponent m and the saturation exponent n aredetermined by experimental core analysis, the saturation can becalculated from the resistivity indexes I, derived fromwell logging. Asthe presented measurements were all carried out on fully brinesaturated plugs (Sw=1), the saturation exponent could not beevaluated. Nevertheless, knowledge of the pressure dependence offormation factor and cementation exponent is essentially for estima-tion of hydrocarbon saturation of reservoirs.

For calculation of the formation factor the real part of complexresistivity at a frequency of 10 kHz were chosen. This is the lowestfrequency, where the measured spectra are not affected byelectrode polarization.

The formation factor of all plugs increase with increasingpressure (Fig. 6). Due to insignificant pressure dependence ofporosity the increase of formation factor is mainly caused by anincreasing tortuosity of the transport path of the free chargecarriers and by the closure of thin aspect ratio pores and cracks,which can create new dead-end pores.

The formation factor takes values between 50 and 250, except ofsample S4 in X- and Z-direction, where F reaches values up to 4000.This behavior can be related to the highly resistive mineralizedfracture in sample S4, which is located approximately perpendicularto the direction of the electric field for the plugs in X- and Z-direction. These plugs are also characterized by an increase of F by upto one order of magnitude in the regarded range of pressure, which isa stronger pressure dependence as detected for the other plugs(including sample S4 in Y-direction). Thus the formation factor isvery sensitive for such dominant rock fabric elements.

Sample S2 in Y-direction differs from the continuous behavior ofthe other samples. This plug is characterized by an abrupt change toa lower formation factor at effective pressure between 40 and

Table 4Pressure dependence of formation factor (calculated from Eq. (11) at a frequency of10 kHz) and relaxation time (from Cole–Cole model, Eq. (13)).

Sample D F0 AF R2 τ0 [10−8s] Aτ R2

S2 Y 63.64 0.064 0.76 2.51 0.055 0.7961.86⁎ 0.703⁎ 0.97⁎

Z 145.57 0.018 0.62 8.27⁎ 1.33E−10⁎ 0.87⁎

146.43⁎ 0.184⁎ 0.98⁎

S3 X 123.68 0.029 0.94 2.67 0.116 0.95Y 189.80 0.054 0.93 9.23 0.520 0.87Z 113.28 0.026 0.78 7.11⁎ (0)⁎ (–)⁎

115.19⁎ 0.190⁎ 0.96⁎

S4 X 236.55 0.617 0.81 42.8 1.109 0.87Y 62.38 0.196 0.75 1.01# 0.046# 0.99#

61.73⁎ 1.735⁎ 0.99⁎

Z 230.30 0.600 0.95 3.51 0.979 0.97S8 X 83.01 0.117 0.85 1.72 0.207 0.87

Y 60.99 0.238 0.98 2.01 0.290 0.94Z 149.47 0.166 0.95 0.45 1.308 0.93

S9 X 129.77 0.049 0.99 6.80 0.108 0.96Y 58.78 0.047 0.98 2.51 0.057 0.79Z 100.71 0.071 0.98 3.97 0.192 0.97

The pressure dependence of these quantities can be described either by a powerfunction (Eq. (12a) and (14a), linear function (Eq. (12b) and (14b); marked by ⁎) orexponential function (Eq. (14c); marked by #).


50 MPa. This change can be related to a damage of the plug, causedby the development of a crack, which can be seen after removing thesample from the autoclave. The crack was located perpendicular tothe abutting face of the plug. After cracking of the plug the formationfactor stays constant with increasing pressure. As the plug S2 in Z-direction stays intact up to a pressure of 100 MPa, the instability ofthe plug in Y-direction must be caused by local heterogeneity or bymechanical stress before running of the experiment. Neverthelessthe reaction of the formation factor on cracking demonstrates thepossibility to detect such damage by electrical measurements.

The pressure dependence of the formation factor for most of theplugs follows a power function:

F peffð Þ = F0pAFeff ð12aÞ

By linear regression derived parameter F0 and the pressuresensitivity exponent AF are listed in Table 4. Some pressure

Fig. 7. Variation of the anisotropy of formation resistivity factor αF with increasing effectivTable 4. a) Pressure dependence of anisotropy of sample S4 (filled symbols) and S9 (open symand S8 aremarked by asterisks and crosses. b) Normalized pressure dependence of the anisotsample S4 in the XY and YZ planes.

dependencies (S2-Y, S2-Z, S3-Z, S4-Y) can be better described by alinear function:

F peffð Þ = F0 + AFpeff ð12bÞ

Sample S4 and sample S8 are indicated by the highest pressuresensitivity exponents, while the exponents of the other samples takevalues down to 0.01. Thus the pressure sensitivity seems to be morerelated to the samples than to a specific direction. But the number ofinvestigated samples is too small to verify this presumed trend.

By Eq. (11) the formation factor is related by a power function to theporosity. As mentioned above the pressure-deduced decrease of porosityis insignificantly small. Hence the increase of the formation factor withincreasing pressure implies an increase of the cementation exponent inthe regarded pressure range of up to a factor of two. For a correct estima-tion of the cementation exponent measurements at different salinities ofthe NaCl solution are required to determine the contribution of surfaceconductivity on total rock conductivity. Due to the fact that the mea-surementswere only carried outwith a constant resistivity of the solutionno quantitative discussion of the pressure dependence ofm can be given.

The anisotropy αF of formation resistivity factor was calculated inanalogy to the anisotropy of permeability (Eq. (7)). The highest valuesof anisotropy occur in samples S4 in the regarded planes XY and YZ,which are made up of a direction perpendicular to the low resistivefracture (X, Z) and parallel to it (Fig. 7a). These planes also show thestrongest pressure dependence of anisotropy (Fig. 7b). All otherplanes take absolute values up to 0.4 and they are weakly affected byhydrostatic pressure (Fig. 7). A change of sign of the anisotropy for thegiven pressure range, which means the occurrence of isotropicpressure-points, appears only for sample S8 in the XY plane.

4.3. Complex resistivity spectra

The spectra of complex resistivity are characterized by a strongdispersion. This can be seen from the Argand-diagram (Fig. 8, upperrow) or the Bode-diagram (Fig. 8, middle and bottom row). In theArgand-diagram the imaginary part of complex resistivity is plottedversus the real part for all measured frequencies (kHz–MHz), where thehigh frequencies values are located on the left side and the lowfrequencies values on the right side of the graph. The graph conforms toa section of a single (depressed) semicircle, centered on or below thereal part axis. This indicates complex resistivity spectra, which are

e pressure. The anisotropy is calculated according to Eq. (7) using the values given inbols). Circles, squares and triangles indicate considered planes XY, XZ and YZ. Sample S3ropy of formation factor. Note the high values and pressure dependence of anisotropy for

Fig. 8. Pressure dependence of the complex resistivity spectra of sample S4 for the three directions X (first column), Y (second column) and Z (third column), plotted in Argand- (top)and Bode-(middle and bottom) diagrams. The arrows indicate the direction of increasing effective pressure. Note the differences between Y-direction and the two other directions.The electrical dispersion of the Y-direction is in qualitative sense representative for the not shown spectra of samples S2, S3, S8 and S9.


dominated by a relaxation process. This process is also obvious in theBode-diagramfor thedispersionof themagnitude andphase of complexresistivity. The magnitude decreases with increasing frequency with aninflection point at the frequency corresponding to the characteristicrelaxation time (Fig. 8, middle row). The absolute value of phase takestheir maximum value at this frequency (Fig. 8, bottom row).

From Fig. 8 the strong pressure dependence of complex resistivityspectra is obvious. With increasing effective pressure magnitude,phase, real and imaginary part of complex resistivity increase for allfrequencies. The spectra of different directions of one sample showpartly strong variations. For example sample S4 in Y-direction hassignificant lower values of the magnitude and phase than the spectraof the two other directions (Fig. 8). Moreover the graphs in theArgand-diagram cover for the Y-direction the lower frequency regionof the semicircle while for X- and Z-direction in the same frequencyrange the higher region is covered (Fig. 8 top). Thus the anisotropy ofthe complex resistivity spectra can be described qualitatively verywell for such a high anisotropic sample as sample S4.

For the other measured samples, which are not characterized bysuch a dominant rock fabric element as sample S4, qualitative

differences between different directions are less significant, as theyall show a dispersion of the complex resistivity complex comparableto that of sample S4 in Y-direction.

To quantify the pressure-induced changes of dispersion thecomplex resistivity spectra have to be described in terms of relaxationmodels. There are many empirical models to characterize complexresistivity spectra. The most conventional models are the Cole–Cole(Cole and Cole, 1941), Davidson–Cole (Davidson and Cole, 1950),Havriliak–Negami (Havriliak and Negami, 1966) and Double–Cole–Cole models (Pelton et al., 1978). An overview and comparison of themost common models is given by Dias (2000). To identify the bestmodel for describing the presented data a combination of a geneticalgorithm and a standard nonlinear iterative least-squares methodwere used (cp. Zisser, 2005). This pseudo-hybrid algorithm enables toinvert the data with different models without any assumptions aboutthe value of the parameters of the respective models. By comparisonof the lowest root-mean-squares of the different models the bestmodel for the data can be identified.

Themostpracticalmodel,whichmeans themodel that can representthe data with the smallest number of independent parameters, for the

Fig. 9. Pressure dependence of the relaxation time of the Cole–Cole model for all measured samples and directions. The variation of relaxation time with increasing pressure can bedescribed either by a power, linear or exponential function (Eq. (14), Table 4). According to the model of Lysne (1983) the increase of relaxation time is caused by an increase of theaspect ratio of the pores (Eq. (1)). Note that the damage of sample S2 in Y-direction at effective pressures between 40 and 50 MPa cannot be observed fromvalues of relaxation time.


measured spectra is the empirical Cole–Colemodel (Cole andCole,1941)for resistivity after Pelton et al. (1978):

ρ4 ωð Þ = ρ0 1− m 1− 11 + iωτð Þc

� �� ð13Þ

where ρ0 is the direct-current resistivity (DC-resistivity), m thechargeability, τ the main relaxation time and c the frequencyexponent. As expected the DC-resistivity is only slightly higher thanthe real part of complex resistivity at 10 kHz. Thus the pressuredependence of ρ0 is similar to that of the formation resistivity factor,which is discussed in the previous section. The pressure dependenceof chargeability is also of minor interest. The chargeability is definedas the normalized difference between DC-resistivity and resistivity atinfinity frequency. In contrast to low frequencies (approx. forfb105 Hz for the measured spectra of this study), where conductioncurrent is the predominant contribution to total electrical current(ωε/σ≪1), at high frequencies (approx. for fN106 Hz) conductioncurrent is negligible with respect to displacement current (ωε/σ≫1).Thus the resistivity converge to zero with increasing frequencies andtherefore the chargeability takes values equal or close to one.Conduction and displacement current can be treated as physicallydistinct quantities, but the measured quantities are effective quan-tities, which are a combination of the true quantities (Fuller andWard,1970). This is the reason why Lockner and Byerlee (1985) pointed outthat a separation of the total current in conduction and displacementcurrent is inappropriate for the description and interpretation of thecomplex resistivity or permittivity data.

Because of the high values of chargeability the values of relaxationtime and frequency exponent can be determinedwith high resolution,even if the peak of the phase is not recorded.

The relaxation time shows a dependence on pressure that can bedescribed either by a power, linear or exponential function (Fig. 9;Table 4):

τ peffð Þ = τ0pAτeff ð14aÞ

τ peffð Þ = τ0pAτeff ð14bÞ

τ peffð Þ = τ0exp Aτpeffð Þ ð14cÞ

Except of samples S3 in Z-direction the relaxation time takeshigher values with increasing pressure. This increase conforms to theexpected behavior of τ predicted by the model of Lysne (1983), as theincrease of the relaxation timemeans an increase of the aspect ratio ofthe pores with increasing hydrostatic pressure (Eq. (1)). The fastclosing process of the microcracks at low pressure is related to astrong increase of relaxation time. High pressure sensitivity andabsolute values of τ are observed for sample S4 in X- and Z-direction,caused by the dominant mineralized fracture that also dominates thepressure dependence of the formation resistivity factor.

A special case is the behavior of sample S3 in Z-direction, as therelaxation time in the considered pressure range is nearly constant.This means in terms of Lysne's model that the pore aspect ratio in Z-direction does not change. Minor variationwith increasing pressure isalso observed for the low-porosity sample S2.

From the anisotropy of relaxation time it can be seen that τ is notonly a function of the geometry but also a function of the orientationof the pores or rather microcracks (Fig. 10). The anisotropy ofrelaxation time ατ is calculated according to Eq. (7). The anisotropytakes very high values up to 2.8. As expected the highest values arerelated to sample S4 (Fig. 10a). But samples S3 and S9 show for some

Fig. 10. Variation of the anisotropy of relaxation time ατ with increasing effective pressure. The anisotropy is calculated according to Eq. (7) using the values given in Table 4.a) Pressure dependence of anisotropy of samples S4 (filled symbols) and S9 (open symbols). Circles, squares and triangles indicate considered planes XY, XZ and YZ. Samples S3 andS8 are marked by asterisks and crosses. b) Normalized pressure dependence of the anisotropy of relaxation time.


planes stronger pressure sensitivity of anisotropy as sample S4. Interms of Lysne's model the pressure dependence of anisotropy impliesthat the microcracks are not distributed uniformly in the sample, asthe aspect ratios of the pores represented by the shape factors do notdecrease in the same extent for the different directions. A special caseis the variation of anisotropy of relaxation time in sample S4 (Fig. 10).The absolute values of anisotropy increase in the planes XY and YZ upto a hydrostatic pressure of about 30 MPa and decreases thereaftercontinuously. An explanation for this behavior could be a decrease ofthe aspect ratio of the mineralized fracture up to a pressure of 30 MPa.After that the high aspect ratio fracture is partially closed and several

Fig. 11. Dependence of the frequency exponent (Cole–Cole model) fro

smaller sized pores were formed, which causes a lower influence ofthe dominant fracture on the distribution of relaxation times.

In contrast to the relaxation time the frequency exponent c of theCole–Cole model does not show any continuous behavior withincreasing pressure (Fig. 11). According to Lysne's model c should bea measure of the distribution of pores, which are characterized bytheir shape factor and aspect ratio. Thus a decrease of the frequencyexponent is related to a wider distribution of shape factors. Anexplanation for this discontinuous behavior can be the formation of‘new’ pores. Such ‘new’ pores could be created by the splitting ofexisting pores in the zones of pore throats. In this case the two

m the effective pressure for all measured samples and directions.

Fig.12. Relationship between permeability and formation resistivity factor (a) as well as relaxation time (b) in the pressure range from 2MPa (open circle) to 100MPa (closed circle).A general increase of formation factor and relaxation time with decreasing permeability is observed for each individual sample and direction. But an overall relationship betweenthese quantities does not exist.


resulting ‘new’ pores could have aspect ratios that differ strongly fromthat of the original pore. As Lysne's shape factor does not depend onthe volume of the pore, the closure of pores at different hydrostaticpressures could explain the discontinuous behavior of the frequencyexponent and therefore of the distribution of relaxation times. Toprove this assumption more investigation into characterization of thepore structure is necessary (e.g. thin section analyses, tomography).

In various publications a relationship between the frequencyexponent and the specific surface area to volume ratio (e.g. Knight andNur, 1987) or the fractal dimension of the specific surface area ispointed out (e.g. Ruffet et al., 1991). The presented data disprove sucha relationship. The discontinuous behavior of c is physically notrealistic for the progression of the specific surface areawith increasingpressure, particularly with regard to the minor decrease of porosityand significant dependence on direction. Also the fractal dimension ofthe specific surface should not be a function of hydrostatic pressureand direction. Accordingly the estimated values of c varying between0.3 and 0.9 cause unrealistic fractal dimensions between 2 and 3 by

Fig. 13. Relationship between the anisotropy of permeability and formation resistivity facto100 MPa (closed circle). Both occur, negative and positive correlation between these qudependence of αF or ατ out.

applying, e.g., the relationships of Le Mehaute and Crepy (1983) orWong et al. (1989).

Thus the frequency exponent of the investigated tight sandstonesseems to be more related to the geometrical properties (such as poregeometry, pore orientation and pore distribution) than to surfaceproperties.

5. Relationship between permeability and resistivity

Both, permeability and complex resistivity, are a function ofeffective pressure. The pressure dependence of permeability, forma-tion resistivity factor and relaxation time are all induced by changes ofthe pore microgeometrical parameters. Thus a relationship betweenthese quantities could be anticipated. But only a relationship betweenthese properties for each single direction of a sample can be observed(Fig. 12). As the increase of the formation factor with increasingpressure is mainly caused by an increase of the cementation exponentand not by a reduction of porosity (unlike as in rocks dominated by

r (a) as well as relaxation time (b) in the pressure range from 2 MPa (open circle) toantities, which point the absence of any possibility to deduce αK from the pressure


intergranular porosity), it is obvious that a site specific F–Φ–Krelationship is not applicable to the investigated rocks.

As mentioned above the relaxation time is mainly a function of thepore shape and is not related to the pore volume and the connectivityof the pores. Thus it is evident that the relaxation time is not ameasure of the permeability.

Nevertheless there is a negative correlation between permeabilityand formation factor as well as relaxation time for each individualplug, as permeability generally decreases with increasing pressurewhile formation factor and relaxation time increases.

From comparison of the pressure dependence of the anisotropy ofpermeability with that of formation factor or relaxation time itbecomes clear that these quantities are sensitive to different propertiesof the pore structure (Fig. 13). Thus a general relationship betweenanisotropy of permeability and that of formation factor or relaxationtime does not exist. In contrast to the pressure dependence ofpermeability isotropic pressure-points of formation factor and relaxa-tion time do not occur in the considered range of pressure (except forαF of sample S8 in the XYplane). Hence there is neither a possibility toestimate absolute values of permeability nor the anisotropy ofpermeability from formation resistivity factor or relaxation time.

6. Conclusion

The permeabilities of the investigated fractured sandstonesdecreases continuously with increasing effective pressure (up to100 MPa) up to two orders of magnitude, while the porosities staynearby constant. The pressure dependence can be represented wellusing a power function. The anisotropy of permeability is also afunction of pressure, because increasing pressure causes a change ofthe preferred flow paths. As the reduction of porosity with increasingpressure is independent of the considered direction, the change ofpreferred flow paths can be exclusively related to the closure of thinaspect ratio pores and cracks. Thus measurements of samples underin-situ conditions are required to characterize the anisotropicbehavior of reservoir rocks.

The formation resistivity factor increases with increasing effectivepressure. According toArchie's law the increase of the formation factor iscaused by an increasing cementation exponent, as the porosity does notchange significantly with pressure. In addition the formation factorseems to be an indicator of the orientation of dominant rock fabricelements, e.g. for the mineralized fracture in sample S4.

The spectra of complex resistivity also depend on effectivepressure. All spectra can be described using the Cole–Cole model(Cole and Cole, 1941). The increase of the relaxation time can beconstrued by the model of Lysne (1983). Thus the increase of therelaxation time means an increase of the aspect ratios of the pores inrespect to the direction of the electric field. Hence the relaxation timeis a measure of the shape and orientation of the thin aspect ratio poresand cracks. The frequency exponent of the Cole–Cole model shows adiscontinuous behavior with increasing pressure. This behavior seemsto be related to the origin of ‘new’ shaped pores that could be createdby the splitting of pores at specific effective pressure. But thisassumption has to be proved by further investigations into the natureof the pore structure. Nevertheless the measurement of complexresistivity in the frequency range, where Maxwell–Wagner–Sillarspolarization dominates, has the potential to be a useful tool forcharacterization of geometrical structure of micro-fractured tightsandstones.

However an estimation of permeability from relaxation time orformation factor is not possible. Also the anisotropy of permeability isnot related to that of formation factor or relaxation time.

Therefore in the next step of our investigations we are going tocarry out measurements of complex resistivity in the low frequencyregion (mHz–kHz), where the polarization is dominated by electro-chemical polarization instead of Maxwell–Wagner–Sillars effects

(e.g. Leroy et al., 2008). Correlations between complex resistivity(b100 Hz) and permeability are delineated in various publications(e.g. Binley et al., 2005; Börner et al., 1996; Sturrock et al., 1999; Slater,2007; Tong et al., 2006). Measurements in the low frequency regionshould show if such a relationship is also detectable for micro-fractured samples subjected to hydrostatic pressure.

Acknowledgment

We thank former Preussag Energie GmbH (Lingen, Germany) andS. Siegesmund (University Göttingen) for providing the samples andthe admission to publish this data. Thanks also to the German Sciencefoundation (DFG) for sponsoring this work (project: NO294/13-2)and to Editor A. Hördt and the two anonymous reviewers for theirconstructive comments.

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anisotropy of permeability and complex resistivity of tight sandstones subjected to hydrostatic...

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