Influence of coordination number and percolation probability

on rock permeability estimation

U. FauziEarth Physics Laboratory, Department of Physics, Institute of Technology Bandung, Indonesia

A. Hoerdt and F. M. NeubauerInstitute of Geophysics and Meteorology, University of Cologne, Germany

Received 9 May 2001; revised 12 January 2002; accepted 12 January 2002; published 26 April 2002.

[1] Estimation of permeability from image sections can be

conducted by means of Local Porosity Theory (LPT) and the

effective medium approximation (EMA). Using these approaches,

the coordination number that is important for rock modeling is

included. In this paper, we extend the EMA approach by

considering the percolation probability as an additional

parameter. Both equations show that permeability will be higher

as coordination number as well as percolation probability

increases. The permeability increases most rapidly close to the

percolation threshold. Several thin sections of sandstone were

created by digital image analysis. Two Point Correlation Functions

were applied to estimate porosity and specific surface area.

Permeability distribution as an input to EMAwas created from the

local permeability. The effective permeability was then calculated

from the EMA equation iteratively. The result shows that

considering coordination number and percolation probability will

improve permeability estimation. INDEX TERMS: 5114

Physical Properties of Rocks: Permeability and porosity; 5112

Physical Properties of Rocks: Microstructure; 5194 Physical

Properties of Rocks: Instruments and techniques

1. Introduction

[2] The effective medium approximation (EMA) is widelyused to describe various physical properties of disordered media[Berryman, 1995; Koplik, 1981]. Kirkpatrick [1973] proposed aneffective medium approximation to describe conducting proper-ties of heterogeneous media. This approach was compared withdirect numerical calculation for several lattices [Koplik, 1981]and several probability distribution functions [David et al.,1990]. The method is excellent for non-critical random linearnetworks independent of coordination number, network size andshape [Koplik, 1981]. It works well for a uniform distributioneven when defined with wide ranges of random variables [Davidet al., 1990; King, 1989]. The EMA is relevant, with amaximum error of 7%, in the worst case of peak-like distribu-tions. The narrower the distribution is, the better the agreementbetween EMA and numerical calculation. The EMA is not suitedfor describing the conducting properties of heterogeneous mediawith decreasing exponential distributions [David et al., 1990]. Inthe worst case, the EMA shows an error of as much as 30%[King, 1989].[3] The EMA was applied to estimate transport properties of

rocks, i.e.: their electrical and hydraulic conductances, by Kopliket al. [1984] and Doyen [1988]. Koplik et al. [1984] found that theratio of estimated to measured permeability is about an order ofmagnitude. This overestimate may be due to non-percolation, and

shape of the channels, heterogeneity and anisotropy of the samples[Koplik et al., 1984]. Doyen [1988] used the same method asKoplik et al. [1984], however, he distinguished the pore and throatsizes using an erosion operation. He pointed out that the EMApredictions of electrical and hydraulic transport properties areaccurate to within a factor of 3 over a porosity range of about5%–22%.[4] In this paper we use specific surface area rather than pore or

throat size as a parameter to calculate hydraulic permeability. Weextend this method by considering percolation probability, whichhas significant influences on transport properties. The influence ofcoordination number and percolation probability on permeabilityestimation is also studied using LPT.

2. Theoretical Background

[5] The EMA equation derived for random resistor networks byKirkpatrick [1973] reads:

Zkeq � k

k þ z2� 1

� �keq

f kð Þdk ¼ 0 ð1Þ

where keq: the equivalent transport properties, f(k): the probabilityfunction of transport properties k, and z: the coordination number(number of throats meeting at a node). For a continuum model, z/2should be replaced by the dimension of the system [King, 1989;Sahimi, 1995]. Equation (1) can be extended by adding thepercolation probability (l). The equation then can be divided intopercolates (kp) and non-percolating blocks(kb) [Hilfer, 1991], i.e.:

Zkeq � kp kð Þ

kp kð Þ þ z2� 1

� �keq

lf kð Þdk

þZ

keq � kb kð Þkb kð Þ þ z

2� 1

� �keq

1� lð Þf kð Þdk ¼ 0 ð2Þ

Because kb = 0. Equation (2) leads to:

Zkeq � kp kð Þ

kp kð Þ þ z2� 1

� �keq

lf kð Þdk þZ

keqz2� 1

� �keq

1� lð Þf kð Þdk ¼ 0

ð3Þ

Simplifying withRf (k)dk = 1, we get:

Zf kð Þ

keq �z2l�1ð Þz2�1ð Þ kp kð Þ

kp kð Þ þ z2� 1

� �keq

0B@

1CAdk ¼ 0 ð4Þ

GEOPHYSICAL RESEARCH LETTERS, VOL. 29, NO. 8, 1237, 10.1029/2001GL013414, 2002

Copyright 2002 by the American Geophysical Union.0094-8276/02/2001GL013414

78 - 1

2.1. Several Limiting Cases

[6] If all cells percolate, i.e. l = 1, equation (4) will tend to theequation (1). For l = 1 and z = 2 this equation will give the lowerbound (equation 5)

keq ¼Z

f kð Þkp kð Þdk ð5Þ

and it gives the upper bound for l = 1 and z = 1 (6).

keq ¼1R f kð Þdkkp kð Þ

ð6Þ

If l = 2/z it reaches the percolation threshold, since keq will bezero. If f(k) is set to 1 then this equation reduces to the simpleKozeny-equation.

2.2. Local Permeability

[7] The local permeability distribution ( f(k)) and the localpermeability (kp) as input parameters for equation (4) are createdwith the help of local porosity theory. An image section is dividedinto several cells having cell size L*. This cell size L* is obtainedfrom the extremum of the information functions which are afunction of the local porosity distribution [Hilfer, 1992]. Anintermediate length scale is also expected to exist for the localspecific surface area. The characteristic length may also beestimated from the first minimum of the two-point correlationfunctions.[8] The local permeability is calculated from the Kozeny

equation, where the local porosity and the local specific surfacearea are estimated from the two-point correlation functions (TPCF)[Debye et al., 1957; Berryman and Blair, 1987]. The TPCF isdefined as [Berryman and Blair, 1987]:

T rð Þ ¼ 1

V

ZV

f xð Þf xþ rð Þd3r ð7Þ

f xð Þ ¼ 0 if x in solid

1 if x in pore space

�ð8Þ

T 0 0ð Þ ¼ � S

4ð9Þ

The local specific surface area (S) then may be estimated from theimage section from the slope of the TPCF at its origin [Berrymanand Blair, 1987] or by polynomial fitting. The discretized versionof equation (7) reads:

T m; nð Þ ¼ 1

N

X1�i�imax

1� j� j max

fij fiþm; jþn ð10Þ

where:

fi j ¼0 if i; jð Þ in solid

1 if i; jð Þ in pore space

�ð11Þ

with imax = L* � m, jmax = L* � n, N = imax jmax, for 0 � m,n �(L* � 1). The isotropic average of the TPCF is then expressed as:

T Vð Þ ¼ 1

2Vþ 1

X2Vl¼0

T V; qð Þ ð12Þ

where T V; qð Þ T V cos pl4V ; V sin pl

4V

�is the bilinear interpolation

given in Press et al. [1986]. We used z as the integer value of the(L*/2). The local hydraulic conductance is then expressed as:

hk �f3k

th f S2k

L*ð Þ2

Lð13Þ

where fk: local porosity of the k-th cell, tk: local tortuosity of thek-th cell., Sk: local specific surface area of the k-th cell, f: shapefactor, L*: cell size, and L: length of the cell.

2.3. Coordination Number

[9] The coordination number (z) is defined as the number ofbranches meeting at a node. This parameter is not directlymeasurable from a single planar cross section, since it needs a3-dimensional model. Doyen [1988], however, defined a pseudotwo-dimensional coordination number as the total number ofthroats divided by the total number of pore elements in a thinsection. For the largest porosity, however, he assumed that z = 6.Yuan [1981] pointed out that 3-dimensional packing has anaverage value of coordination number of 6. An empirical correla-tion between coordination number and porosity can expressed as[Dullien, 1992]:

1� f ¼ 1:072� 0:11937zþ 0:0043z 2 ð14Þ

This equation shows that coordination number increases asporosity increases as also shown by Doyen [1988] for pseudo-coordination number. Figure 1 shows this tendency.

Figure 1. Coordination number versus porosity.

Figure 2. Digital image processing equipments for rock perme-ability estimation.

78 - 2 FAUZI ET AL.: INFLUENCE OF COORDINATION NUMBER AND PERCOLATION PROBABILITY

[10] It is shown that coordination number increases as porositybecomes larger.

2.4. Percolation Probability

[11] Percolation probability indicates whether a cell percolatesor blocks. It equals 1 for percolating cells and 0 otherwise. Theaverage percolation probability is an important parameter, which isnot easy to measure. A three-dimensional model is needed toestimate this percolation probability. Percolation probability for asample may be anisotropic (Hilfer, pers. comm.). Hilfer [1991]studied several special models to estimate percolation probability.The percolation probability is constant for a simple uniformcapillary model. It depends on the percolation threshold for aconsolidation model and on porosity for centered pore model. Asan adjustable parameter the percolation probability may beexpressed as a function of porosity [Hansen et al., 1993; Haslundet al., 1994]:

l ¼ afb ð15Þ

where a and b are positive constant parameters, and f is porosity. Itis clear that percolation probability increases as porosity goes up.Based on the permeability-porosity data, Mavko and Nur [1997]

found a critical porosity. They considered this critical porosity tomodify the Kozeny-Carman relation.

3. Experimental Methods

[12] Digital image processing is used to study the influence ofcoordination number and percolation probability for permeabilityestimation. Figure 2 shows the procedure of estimating perme-ability using the above approaches.[13] First, a thin section was cut and placed on an optical

microscope. The scanning procedure was done using a digitalcamera mounted on an optical microscope that is connecteddirectly to a PC. The size of each pixel is approximately 10 mm.A binary image is created after selecting a threshold. After anappropriate threshold range is chosen, each pixel is set to one(pore) if it is within the range or to zero (matrix) if it is outside therange. The LPT and EMA are then applied to estimate permeabilityas a function of coordination number and percolation probability.[14] We applied the TPCF to calculate porosity and specific

surface area. The parameters obtained are then used as inputparameters to LPT. The LPT formula applied to estimate perme-ability reads as follows [Hilfer, 1992]:

k �f; �S� �

��f3

f th �S2 l �f; �S� � l �f; �S

� �� pc

� �t ð16Þ

where k is permeability in, f is a shape factor which depends on thecross-sectional shape of the capillaries, th is hydraulic tortuosity, �Sis the average local specific surface area, �f is the average localporosity, pc is percolation threshold, l is the percolationprobability, and t = 1 in the effective medium approximation[Hilfer, 1992].[15] The shape factor varies between 2 (for circular shape) and 3

(for an infinite rectangular disk) [Georgi and Menger, 1994].Hydraulic tortuosity th is defined as the square of the ratio ofthe effective average path length to the shortest distance along thedirection of flow [Dullien, 1992]. The percolation thresholddepends on the choice of Bravais lattice. In the effective mediumapproximation pc = 2/z [Hilfer, 1992; Sahimi, 1995], where z is thecoordination number of the lattice of measurement cells.

Figure 3. Rock image and local permeability distribution.

Figure 4. Permeability as a function of coordination number. Themeasured permeability is taken from Fauzi [1997].

Figure 5. Permeability as a function of percolation probability.The measured permeability is taken from Fauzi [1997].

FAUZI ET AL.: INFLUENCE OF COORDINATION NUMBER AND PERCOLATION PROBABILITY 78 - 3

[16] Estimation of permeability as a function coordinationnumber and percolation probability by means of EMA is conductedas follows. The binary image was divided into several cells withthe size L* obtained from local porosity theory or two pointcorrelation functions [Fauzi et al., 1996]. Figure 3 show the binaryimages and local permeability distribution for sample used in thisstudy.[17] The black color refers to pores and white indicates matrix.

The corresponding permeability distribution shows that the highporosity areas are related to high permeability. The estimatedpermeability as a function of coordination number is shown inFigure 4.[18] Permeability goes up as coordination number rises. The

increase of permeability as a function of coordination numbercalculated from EMA is slower than using LPT. The calculatedpermeability is close to the measured air permeability for very lowcoordination number. It does not mean that the low coordinationnumber necessarily belongs to the above sample, since the meas-ured permeability is due to three-dimensional properties and theimage considers only two-dimensional case.[19] Influence of the percolation probability on permeability

estimation is shown in Figure 5:[20] Percolation probability significantly influences permeabil-

ity calculation using LPT as well as with EMA. Permeability willbe larger as percolation probability increases. The permeabilitycalculated by the LPT approach gives a rapid increase for lowpercolation probability and increases more slowly at higher perco-lation probability.

4. Conclusion and Discussion

[21] The influences of coordination number and percolationprobability can be studied by means of LPT and EMA with thehelp of image analysis. The results show the permeability increasesrapidly at small coordination number as well as at low percolationprobability. We can see clearly that the influences of coordinationnumber and percolation probability are not linear. It may representthe complexity of transport mechanism.[22] Three-dimensional rock modeling and reconstruction are

very useful in determining coordination number and percolationprobability. In turn they will improve the permeability estimationof real rocks.

[23] Acknowledgments. This work was partly supported by URGEproject and RUT VI from the Republic of Indonesia, and also IGMUniversity of Cologne-Germany. We are grateful to reviewers and Prof.K. Vozoff for their constructive comments and suggestions.

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�����������U. Fauzi, Earth Physics Laboratory, Department of Physics, Institute of

Technology Bandung, Indonesia.A. Hoerdt and F. M. Neubauer, Institute of Geophysics and Meteorology,

University of Cologne, Germany.

78 - 4 FAUZI ET AL.: INFLUENCE OF COORDINATION NUMBER AND PERCOLATION PROBABILITY