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Asimplegeometricmodelofsedimentaryrocktoconnecttransferandacousticproperties

ARTICLEinARABIANJOURNALOFGEOSCIENCES·MARCH2013

ImpactFactor:1.22·DOI:10.1007/s12517-013-0863-z

3AUTHORS,INCLUDING:

GaborKorvin

KingFahdUniversityofPetroleumandMin…

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AbdulazeezAbdulraheem

KingFahdUniversityofPetroleumandMin…

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Retrievedon:23August2015

ORIGINAL PAPER

A simple geometric model of sedimentary rock to connecttransfer and acoustic properties

Gabor Korvin & Klavdia Oleschko & Abdulazeez Abdulraheem

Received: 13 April 2012 /Accepted: 25 January 2013 /Published online: 12 February 2013# Saudi Society for Geosciences 2013

Abstract A simple rock model is presented which repro-duces the measured hydraulic and electric transport proper-ties of sedimentary rocks and connects these properties witheach other, as well as with the acoustic propagation veloc-ities and elastic moduli. The model has four geometricparameters (average coordination number Z of the pores,average pore radius r, average distance between nearestpores d, and average throat radius δ) which can be directlydetermined from the measured porosity Φ, hydraulic perme-ability k, and cementation exponent m of the rock via simpleanalytic expressions. Inversion examples are presented forpublished sandstone data, and for cores taken from SaudiArabian, Upper Jurassic and Permian carbonate reservoirs.For sandstone, the inversion works perfectly; for carbonates,the derived rock model shows order-of-magnitude agree-ment with the structure seen in thin sections. Inverting theequations, we express the transfer properties Φ, k, and m asfunctions of r, d, δ, and Z. Formulae are derived for the bulkdensity Db, formation factor F, and P-wave velocity in termsof the proposed geometrical parameters.

Keywords Digital rock physics . Petrophysics . Pore-spacemodel . Rock physics . Pore structure inversion

Introduction

The most exciting recent development in petrophysics hasbeen the emergence of digital rock physics (Andrä et al2012; Arns et al 2004; Boylan et al, 2002; Dong 2007;Dong et al. 2008; Dvorkin et al. 2009, 2011; Kalam et al.2011; Kayser and Ziauddin 2006; Keehm 2003; Knackstedtet al. 2007, 2009a, b, c; Peng et al. 2012; Rassenfoss 2011;Saenger et al. Sok et al. 2009; Sorbie and Skauge 2011;Touati et al. 2009; Zhang and Knackstedt 1995, etc.). Indigital rock physics (DPR), one reconstructs the pore spaceof a small (less than 1 cm3) piece of rock by computerizedX-ray tomography, digitizes the pore space, and then “nu-merically simulates various physical processes to obtainsuch macroscopic rock properties as permeability, electricconductivity and elastic moduli” (direct quote from Andrä etal. 2012). What is not always emphasized in DPR literature(see, however, Dvorkin et al. 2011) is that the tiny fragmentof rock selected for analysis must be a statistically repre-sentative realization of all possible random rock structuresthat could result from the same geologic processes of depo-sition, compaction, and diagenesis.

As the detailed pore geometries of any two cuttings fromthe same rock body are obviously different, there existinfinitely many “equivalent pore structures” which wouldlead, through computer simulation or laboratory measure-ment, to the same set of macroproperties. Among all differ-ent microstructures which would result in the samepetrophysical macroproperties, there must be a simple effec-tive pore-space model corresponding to that set of macro-properties. The search for this simple effective rock modelhad been the key idea of this study: we asked ourselves how

G. Korvin (*)Earth Sciences Department and Reservoir CharacterizationResearch Group, King Fahd University of Petroleum and Minerals,Dhahran, Saudi Arabiae-mail: gabor@kfupm.edu.sa

K. OleschkoCentro de Geociencias, Universidad Nacional Autónoma deMéxico (UNAM), Km. 15.5 Carretera Querétaro-San Luis Potosí,C. P. 76230, Juriquilla, Querétaro, Mexicoe-mail: olechko@servidor.unam.mx

A. AbdulraheemPetroleum Engineering Department, King Fahd University,Dhahran, Saudi Arabiae-mail: aazeez@kfupm.edu.sa

Arab J Geosci (2014) 7:1127–1138DOI 10.1007/s12517-013-0863-z

to find a simple pore-space model, with as few degrees offreedom as possible, that would correspond to a set ofcomputed or measured macroscopic rock properties. Thisidea is close to Biswal’s model (Biswal et al. 1999, 2009;Widjajakusuma et al. 1999) that embedded the microtomo-graphically derived pore space into three stochastic poremodels: two geostatistically constructed and one based onsimulated sedimentary processes.

The geometric rock model what we propose in order to fitmacroscopic properties is a simplified version of Doyen’smodel (1987). It consists of identical spherical poresconnected with cylindrical tubes of the same radius (as inAkbar 1993; Akbar et al. 1994). The model has four geo-metric parameters (average coordination number Z of thepores, average pore radius r, average distance betweennearest pores d, and average throat radius δ).

This rock physics to pore structure inversion is discussedin “Rock physics to rock texture inversion” section. In the“Inversion of hydraulic and electric transfer properties topore geometry” section, we review the direct inversion ofhydraulic and electric transport properties into pore geome-try, using the pore model of Doyen (1987) as example. Inthe “Inversion of hydraulic and electric transfer properties topore geometry” section, we introduce the equivalent poremodel and express the four geometric parameters from themeasured porosity Φ, hydraulic permeability k, and cemen-tation exponent m of the rock. The relevant Eqs. (21–23) or(24–26) are the most important results of this work.Numerical results, based on these equations, will be givenin the “Rock property to pore geometry inversion examples”section (Tables 1, 2, and 3 and Figs. 1, 2, and 3) for sixpublished (Jorgensen 1988) sandstone data and for ourmeasurements on 26 carbonate samples from the UpperJurassic Arab-D and the Permian Khuff formations (seeHalawani 2000; Meyer et al. 2000; Stenger et al. 2003, forthe geology).

In the “Use of the geometric model to connect poregeometry with transfer- and elastic properties” section,Eqs. (24–26) will be inverted, and we express the transferproperties Φ, k, and m as functions of r, d, δ, and Z. In the“Computation of the transfer properties and some other rock

properties from the equivalent rock model” section, the bulkdensity Db, formation factor F, and P-wave velocity areexpressed in terms of the proposed geometrical parameters,for the case of complete saturation. The case of incompletesaturation (“The case of incomplete saturation” section) issomewhat special because it needs a further independentparameter, the saturation exponent n. We do not attempt,in this paper, to relate n to the pore geometry (in this respect,see Knackstedt et al. 2007).

Finally, the “Discussion and concluding remarks” partgives a critical summary of the model and its applicationsand lists the several, still open problems.

Rock physics to rock texture inversion

Inversion of hydraulic and electric transfer propertiesto pore geometry

The prediction of the interior structure of sedimentary rocksfrom measured bulk physical properties is a nonlinear in-verse problem where the number of unknowns is muchlarger than the number of measurements. Such problemsallow an infinite number of solutions, out of which solutionswith small variance can be obtained by the Tikhonov regu-larization method (Lamm 1997); while those solutionswhich are most likely to occur in nature are given by themaximum entropy method (MEM, see Lifshitz andPitaevskii (1980) where the technique is explained throughthe derivation of Boltzman’s barometric equation orKorvin’s MEM treatment (1984) of shale compaction).The first application of MEM to rock physical inversionwas due to Doyen (1987) who determined the statisticalcrack geometry in igneous rocks from measured hydraulicconductivity and/or dc electric conductivity values at aseries of confining pressures. His rock model consisted ofa random distribution of spherical pores connected withnearby pores by tubes (“throats”). At a given referencepressure P=P0, the pore radii r are distributed according tosome probability density function (pdf) n(r); the throatlength l is distributed according to a pdf n(l); the throats

Table 1 Comparison of pub-lished (Jorgensen 1988)and computed (by Eq. 25) porethroat radii

Lithology Φ m k (mD) Hydraulic radius publishedin Jorgensen 1988 r (μm)

Geometric modelparameter δ (μm)

Berea sandstone 0.22 1.624 890 7 8.27

Pyrex 1 0.37 1.490 8.1 0.38 0.53

Pyrex 2 0.29 1.474 3,900 12 14.2

Nichola Buff sandstone 0.20 1.569 230 4.2 4.3

Eocene sandstone 0.22 1.694 340 4.4 5.26

Pennsylvania sandstone 1 0.16 1.635 180 4.1 4.47

Pennsylvania sandstone 2 0.21 1.644 390 5.5 5.65

1128 Arab J Geosci (2014) 7:1127–1138

have elliptic cross section with random semiaxes b and c.The pdf of the throat shape at the reference pressure is n(α,c) where α is the aspect ratio α=b/c. Using elasticity theoryfor the deformation of voids under pressure P (Bernabe et al.1982; Kuster and Toksöz 1974; Zimmerman 1991), thepressure-dependence of the pdf’s r(P), c(P), l(P), and α(P)can be determined. At every pressure step, the pdf n(α, c)satisfies the self-consistency equation (Kirkpatrick 1973).

Xa;c

n a; cð Þ g*hðPÞ � gh P; a; c; lð Þgh P; a; c; lð Þ þ Z

2 � 1� �

g*hðPÞ¼ 0; ð1aÞ

Xa;c

n a; cð Þ g*hðPÞ � gh P; a; c; lð Þgh P; a; c; lð Þ þ Z

2 � 1� �

g*hðPÞ¼ 0; ð1bÞ

for all P, where Z is average coordination number (number ofthroats incident to a pore),g*e ðPÞ; g*hðPÞ are themeasured bulkelectric and hydraulic conductances at pressure P, and gh=gh(P,α,c,l) and ge=ge (P,α,c,l) are theoretical conductances of anoriginally (α, c)-shaped throat of length l subjected to pressure

P. Doyen (1987) solved the system (1a, b) for the pdf’s of poresize n(r), of throat size n(l), and of throat ellipticity n(α, c)with the maximum entropy method.

In our study, we simplified Doyen’s model by assumingan equivalent pore space with as few parameters as possible.We had been encouraged by the success of the Cole–Coleequivalent circuit model of the induced potential effect inrocks (Cole and Cole 1941; Pelton et al. 1978; Telford et al.1990) or the three-parameter Thomeer model of mercurypressure injection (Thomeer 1960, 1983; Clerke 2003),which are classic examples that equivalent rock models withonly three degrees of freedom can realistically describe thebehavior of a complex geologic medium.

Our geometric rock model consists of spherical pores ofthe same size, connected with cylindrical tubes (as in Akbar1993; Akbar et al. 1994). In the next section, we shall definethe model and derive its parameters from transfer propertiesmeasured at a single pressure step rather than using a seriesof pressures as in Doyen (1987). Section “Rock property topore geometry inversion examples” will show numericalexamples for this new kind of “rock to texture” inversion.

Table 2 Input parameters for the rock physics to pore structure inversion, carbonate samples

Well Sample Lithology k Permeability (mD) Φ Porosity m Cementation exponent n Saturation exponent F Formation factor

A 10-A Dol Wkst/Pkst 0.547 0.1731 1.764 NA 22.06

A 15-A Dol Wkst 1.41 0.2258 1.755 NA 13.62

A 19-A Dol Wkst 1.3 0.2137 1.742 NA 14.71

A 28-A Dol Wkst 1.28 0.2227 1.870 NA 16.58

A 41-A Dol Mdst 0.071 0.1037 1.900 NA 74.05

A 45-A Dol Mdst 0.108 0.1203 1.919 NA 58.21

A 113-A Sucrosic Dol 56.120 0.2240 1.927 1.236 17.86

A 114-A Sucrosic Dol 125.0 0.2289 1.994 1.158 18.93

A 122-A Sucrosic Dol 3.23 0.1563 1.775 1.419 26.95

A 149-A Sucrosic Dol 122 0.3187 1.895 1.687 8.73

A 247-A Sucrosic Dol 94.8 0.1990 1.913 1.464 21.96

A 253-A Sucrosic Dol 0.046 0.0751 1.670 0.862 75.49

A 257-A Anhyd Dol 0.003 0.0315 1.400 NA 126.42

A 268-A Sucrosic Dol 247.0 0.2623 1.836 1.257 11.67

A 272-A Sucrosic Dol 2.73 0.2059 2.034 1.251 24.89

A 273-A Sucrosic Dol 117.0 0.2104 1.964 1.224 21.35

B 42-B Oomoldic Grst 0.007 0.1257 2.117 NA 80.59

B 59-B Oomoldic Grst 0.021 0.1126 2.092 NA 96.32

B 71-B Oomoldic Grst 0.002 0.0686 1.993 NA 177.50

B 274-B Sucrosic Dol 0.2796 1.972 1.566 12.34

B 278-B Sucrosic Dol 111.0 0.2457 1.982 1.452 16.15

B 282-B Sucrosic Dol 33.7 0.2157 1.940 1.426 19.61

B 283-B Sucrosic Dol 58.2 0.2965 1.947 1.245 10.67

B 287-B Oomoldic Grst 0.103 0.1720 2.324 2.195 59.77

B 294-B Oomoldic Grst 0.05 0.1600 2.420 2.004 84.33

B 355-B Dol Lst 93.0 0.1820 2.009 2.098 30.66

Arab J Geosci (2014) 7:1127–1138 1129

The equivalent rock model

Assume that the porosity Φ, permeability k, and cementationexponent m (figuring in Archie's law, 1942; modern treat-ment: Perez-Rozales 1982; Worthington 1993; Glover 2009)are measured for some sedimentary rock at a referencepressure P. We will construct an equivalent rock modelwhich has the same measured Φ, k, and m values. The modelconsists of a connected system of pores and throats distrib-uted within a homogeneous rock matrix. Pores are spheresof radius r, and they are distributed in the rock matrix insuch a way that the Euclidian distance between two nearestpores is d. A pore is connected on the average by Z neigh-boring pores with tortuous cylindrical pipes (throats) ofradius δ. The parameters of the equivalent rock model (r,

Table 3 Equivalent rock model computed from Table 2

Well Sample Z Coordinationnumber

r Meanporeradius (μ)

δ Mean throatradius (μ)

d Distancebetweenpores (μ)

A 10-A 4.62 1.9 0.25 6.08

A 15-A 4.65 1.99 0.34 6

A 19-A 4.70 2.04 0.34 6.2

A 28-A 4.30 2.34 0.34 6.92

A 41-A 4.22 2 0.12 7.22

A 45-A 4.18 2.04 0.14 7

A 113-A 4.16 16.85 2.28 49.22

A 114-A 4.01 27 3.41 77.5

A 122-A 4.58 5.51 0.63 18.05

A 149-A 4.23 12.81 2.76 35.06

A 247-A 4.19 26.14 3.13 78.71

A 253-A 4.99 1.48 0.1 6.05

A 257-A 7 0.49 0.04 2.75

A 268-A 4.39 23.48 4.30 67.47

A 272-A 3.93 5.13 0.54 15.06

A 273-A 4.07 28.79 3.42 84.85

B 42-B 3.79 0.73 0.03 2.44

B 59-B 3.83 1.46 0.06 5.04

B 71-B 4.01 0.82 0.02 3.32

B 274-B

B 278-B 4.04 21.98 3.09 62.33

B 282-B 4.13 14.23 1.80 41.84

B 283-B 4.11 10.80 2.01 29.65

B 287-B 3.51 2.31 0.12 6.92

B 294-B 3.41 2.22 0.09 6.76

B 355-B 3.98 35.66 3.31 108.33

r, δ, and d are given in microns

Fig. 1 Microscopic image from the sucrosic dolomite sample 268-Afrom the Permian Khuff formation. The model parameters (Z=4.39; r=23.48 μ, δ = 4.30 μ, d=67.47 μ) determined from m, k, and Φ usingEqs. (2) and (24–26) show order-of-magnitude agreement with thegeometry of the rock

Pore size

0

200

400

600

800

1000

0 200 400 600 800 1000

Estimated r (microns)

Cal

cula

ted

r (

mic

ron

s)

Pore size

Fig. 2 Crossplot of the microscopically observed average pore radiiand the model pore radii r (computed by Eq. 24)

d

0

200

400

600

800

1000

0 200 400 600 800 1000

Estimated d (microns)

Cal

cula

ted

d (

mic

ron

s)

d

Fig. 3 Crossplot of the microscopically observed average throatlengths and the model throat lengths d (computed by Eq. 26)

1130 Arab J Geosci (2014) 7:1127–1138

d, Z, and δ) can be uniquely determined as functions of Φ, k,and m if we make the following seven assumptions:

(1) The equivalent rock model is only applicable for mac-roscopically homogeneous and isotropic parts of thereservoir. In case of heterogeneity or anisotropy, themodel parameters should be varied and then smoothlyfitted together. Optimally, the model is applied to plug-sized or cutting-sized rock fragments. In such a smallhomogeneous and isotropic rock volume, the theoreti-cal porosity and hydraulic permeability of the equiva-lent rock model are assumed to be the same as themeasured Φ and k values;

(2) The resistivity for 100 % saturation satisfies Archie’slaw ρbulk ¼ a

Φm ρw (resistivities are denoted by ρ,densities by D to avoid confusion);

(3) The hydraulic permeability of porous rocks obeys theKozeny–Carman equation (Kozeny 1927; Carman

1937; Walsh and Brace 1984) k ¼ 1b � Φ3 � 1

S2spec� 1t2 ;

(4) In the Kozeny–Carman equation, C is the hydraulictortuosity. We assume that the rock is not very tight, sothat the electric and hydraulic tortuosity values can betaken as equal. For low-permeability rocks, the hydraulicpermeability can be an order of magnitude higher thanthe electric one (Zhang and Knackstedt 1995; Duda et al.2011;Worthington 2011), or, in case of clayey rocks, it isdivergent at the percolation threshold (Korvin 1992a, b).

(5) The average coordination number and the cementationexponent m are related as Z ¼ 2m

m�1 (Yonezawa andCohen 1983);

(6) The hydraulic tortuosity C and formation factor F arerelated as C=F ·Φ (Perez-Rozales 1982; Glover 2009).

(7) During hydraulic flow, the total porosity consists of astagnant and a flowing part, Φ=Φs+Φf where, accord-ing to Perez-Rozales (1982), Φf=Φ

m.

We note that for the determination of (r, d, Z, and δ), themeasured resistivity value and Archie’s saturation exponent nare not needed.

To relate the transfer properties Φ, k, and m as functionsof (r, d, δ, and Z), we first use the cementation exponent m toexpress the average coordination number Z and the hydrau-lic tortuosity τ. According to the self-similar effective medi-um approximation for granular media (Yonezawa andCohen 1983), the average coordination number and thecementation exponent are related as

Z ¼ 2m

m� 1; ð2Þ

while (see Perez-Rozales 1982; Glover 2009) tortuosity isgiven by

t ¼ F � Φ: ð3Þ

Using Archie’s equation,

t ¼ F � Φ ¼ aΦΦm ¼ a

Φm�1 ð4Þ

where if a is not known, one usually takes its default values:i.e., a=0.62 for unconsolidated sand; a=0.81 for consoli-dated sand; a=1 for carbonate formations (Schlumberger1991). According to Glover (2009), a=1 is the only properchoice because otherwise lim

Φ!1ρbulk ¼ a ρw , which is

counterintuitive.The hydraulic permeability of porous rocks is, by the

Kozeny–Carman equation (Walsh and Brace 1984):

k ¼ 1

b� Φ3 � 1

S2spec� 1t2

ð5Þ

where b is a geometric factor equal to 2 for cylindricalthroats and 3 for cracks and Sspec is the specific surface. Ifwe measure distances in millimeter, surface areas in squaremillimeter, and permeability in millidarcy, Eq. (5) becomes(Korvin 1992a, b)

k md½ � ¼ 1

b� Φ3 � 1

Sspec mm�1½ �� �2 � 1t2 � 109 : ð6Þ

Taking b=2 and using Eq. (3), the specific surface Sspec is

Sspec ¼ Φ1:5 � 104:520:5 �t � k0:5 ¼ 104:5 � 1

F�

ffiffiffiffiffiffiΦ2k

rð7Þ

Suppose that there are altogether N uniformly distributedspherical pores of radius r within a 1 mm×1 mm×1-mm-sized cube V of the rock. Each pore occupies a separatedomain of average volume 1

N , that is, the mean distancebetween two nearby pores is

d � 13 ffiffiffiffiNp : ð8Þ

On the average, each pore is connected with Z neighbor-ing pores by means of tortuous throats with circular crosssection of radius δ. Denote the total length of these throatsby L. We write up two equations to determine N, L, r, and δ.The first equation expresses that the total pore space in rockvolume V is built up of N spherical pores and of a totallength L of cylindrical throats:

Φ ¼ N � 4 r3 p3

þ L � d2 p: ð9Þ

The second equation states that the total pore surface withinthe microscopic cube (i.e., the specific surface) is the sum ofthe spherical pore surface areas minus the holes on this surfacecut out by the throats, plus the areas of the throat walls:

Sspec ¼ N � 4r2p � NZ d2 p þ 2pd � L ð10Þ

Arab J Geosci (2014) 7:1127–1138 1131

As δ<<r, we neglect the term which is quadratic in δ:

Sspec ¼ N � 4r2p þ 2pd � L: ð11Þ

To eliminate L from Eqs. (9) and (11), note that

L ¼ N � Z2� d � t: ð12Þ

The meaning of the factor 1/2 is that every throat belongsto two pores, the C factor expresses the tortuosity of the pipeconnecting two pores a distance d apart. By Eqs. (2), (3),and (8),

L ¼ N � Z2� d � t ¼ N 2m

m�1 t

2 � 3 ffiffiffiffiNp ¼ N23 � m

m� 1� Φ �F ð13Þ

Using Eq. (13) in Eqs. (9) and (10):

Φ ¼ N � 4r3p3

þ d2p N23 � m

m� 1� Φ F ð14Þ

Sspec ¼ N � 4r2p þ 2d p N23 � m

m� 1� Φ F: ð15Þ

The physical meaning of Eq. (14) is that the totalporosity is the sum of a hydraulically inactive “stagnant”part (consisting of the pores) and of a part where thefluid is actually moving (the “flowing porosity” consti-tuted by the throats):

Φ ¼ N � 4 r3 p3

þ d2 p � N 23 � m

m� 1� Φ � F ¼ Φs þΦf :

ð16Þ

According to Perez-Rozales (1982),

Φf ¼ Φm ð17aÞ

Φs ¼ Φ�Φm : ð17bÞ

Matching the two terms in Eq. (14) with Eqs. (17a) and(17b), respectively, and solving for r and δ,

r ¼ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 Φ�Φmð Þ

4p

r� 1

N13

ð18Þ

d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim� 1

m� 1

pF� Φm�1

r� 1

N13

: ð19Þ

To find 1

N13, substitute Eqs. (18) and (19) to Eq. (15) in

place of r and δ, and use Eq. (7) to obtain

1

N13

¼ 10�4:5 � 2p � ffiffiffiffiffi2k

p � FffiffiffiffiΦ

p

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φmþ1 �F

r" #:

ð20ÞFinally, from Eqs. (18) and (19),

r ¼ 10�4:5 � 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 Φ�Φmð Þ

4p

r� 2p � ffiffiffiffiffi

2kp � F� ffiffiffiffi

Φp

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φmþ1 �F

r" #ð21Þ

d ¼ 10�4:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim� 1

m� Φ

m�1

pF

s� 2p � ffiffiffiffiffi

2kp � F

Φm � ffiffiffiffiΦ

p

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φmþ1 �F

r" #;

ð22Þand from Eqs. (8) and (20),

d ¼ 10�4:5 � 2p � ffiffiffiffiffi2k

p � FffiffiffiffiΦ

p

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φmþ1 F

r" #: ð23Þ

Equations (21) to (23), combined with Eq. (2) completelysolve the problem of expressing (r, d, Z, and δ) as functionsof Φ, k, and m.

If we accept Glover’s argument (2009, his Eq. 7) that onemust have a=1 in Archie’s second equation, then F ¼ 1

Φm ,and Eqs. (21–23) simplify to

r ¼ 10�4 �3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 Φ�Φmð Þ

4p

r� 2p � ffiffiffiffiffi

2kp

Φm � ffiffiffiffiffiffiffiffi10Φ

p

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φ

r" #ð24Þ

d ¼ 10�4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim� 1

m� Φ

2m�1

p

s� 2p � ffiffiffiffiffi

2kp

Φm � ffiffiffiffiffiffiffiffi10Φ

p

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φ

r" #ð25Þ

1132 Arab J Geosci (2014) 7:1127–1138

d ¼ 10�4 � 2p � ffiffiffiffiffi2k

p

Φm � ffiffiffiffiffiffiffiffi10Φ

p

� 2 � 3 Φ�Φmð Þ4p

� �23

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p� m

m� 1� Φ

r" #: ð26Þ

Rock property to pore geometry inversion examples

Numerical results, based on Eqs. (5) and (24–26) are givenin Tables 1, 2, and 3 for seven published sandstone samples(taken from Jorgensen 1988) and for our laboratory meas-urements and microscopy of 26 carbonate samples from theUpper Jurassic Arab-D and the Permian Khuff formations(Halawani 2000; Meyer et al. 2000; Stenger et al. 2003).Note that if k is in millidarcy in these equations, r, δ, and dwill be obtained in millimeter. For convenience, however, inTables 1 and 3, these values are shown in microns. For theseven sandstone samples (Table 1) only one geometricproperty, the hydraulic radius had been published, and thisis indeed in near-perfect agreement with the throat radius δcomputed by Eq. 25. For carbonate rocks, we cannot expectsuch a close agreement because most carbonates have dou-ble or triple porosity (Warren and Root 1963; Abdassah andErshaghi 1986; Lucia 1998; Ahr 2008; Pal 2012; Pulido etal. 2007, 2011; etc.), as carbonate pore space consists ofinter-matrix pores (for which our model is applicable), vugs,and fractures. Also, for triple-porosity rocks, the value ofthe transfer factors (also called shape factors) betweendifferent kinds of pores (first introduced by Warren andRoot 1963; more recent developments: Lim and Aziz1995; Hassanzadeh and Pooladi-Darvish 2006) are stillpoorly understood. The geometric model should be modi-fied for triple-porosity rocks; this will be discussed—as afuture research problem—in “Discussion and concludingremarks” section. To check whether for carbonate rocksthe computed model parameters are not physically mean-ingless, that is, they have the same order of magnitude as themicroscopically determined pore radius r, average distancebetween nearest pores d, and average throat radius δ, weanalyzed the micrographs of 26 carbonate samples. Figure 1is a typical example; it is the microscopic image of a thinsection from sample 268-A (a sucrosic dolomite from thePermian Khuff formation). Visual inspection shows that thecomputed values (Z=4.39; r=23.48 μ, δ = 4.30 μ, d=67.47 μ) are reasonably close (in an order-of-magnitudesense) to the geometry of the rock. Further results are givenas in Tables 2 and 3 and as crossplots in Figs. 2 and 3. Inspite of the large scatter, there is an order-of-magnitudeagreement between computed and observed results, andwe are optimistic that the model can be extended and fine-tuned for carbonates. We note that Tables 2 and 3 and thecrossplots do not contain the throat radius because, in most

of the cases, it could not be determined from the micro-graphs by visual inspection.

Use of the geometric model to connect pore geometrywith transfer and elastic properties

Computation of the transfer properties and some other rockproperties from the equivalent rock model

In what follows, the physical units will always be noted bysquare brackets, […]. The geometric model parameters r (inmillimeter), d (in millimeter), δ (in millimeter), and Z (pos-itive real number) can be used in a straightforward way tocompute m (cementation exponent), Φ (porosity, 0≤Φ≤1), k(permeability), and D (density). We also need a which is thelithology-dependent dimensionless constant in Archie’s lawρbulk ¼ a

Φm ρw . Accepting the empirical laws of Archie(1942) and Wyllie et al. (1956, 1958), the formation factorF, rock resistivity ρ, density D, and P-wave velocity VP canbe estimated as follows.

The cementation factor is obtained from Eq. 2:

m ¼ Z

Z � 2: ð27Þ

From Eq. (8), the number of spherical pores in a smallvolume V of 1 mm3 is

N ¼ 1

d3; ð28Þ

that is, from Eq. (19),

Φ ¼ m

m� 1� pa � d

d

� �2" # 1

ð2m�1Þ

: ð29Þ

This immediately yields the dimensionless formation fac-tor from Archie’s law:

F ¼ a

Φm ¼ m

m� 1� pa � d

d

� �2" #� 1

ð2m�1Þ0@

1A; ð30aÞ

and the tortuosity, by Eq. (3), (based on Perez-Rozales 1982;Glover 2009):

t :¼ FΦ ¼ a

Φm�1 ¼ am

m� 1� pa � d

d

� �2" #� ðm�1Þ

2m�1ð Þ0@

1A:

ð30bÞ

In case of fully saturated rocks, Eq. (29) immediatelygives the bulk density, bulk resistivity, and (by

Arab J Geosci (2014) 7:1127–1138 1133

Wyllie’s empirical equation) the approximate P-wavevelocity as

Dbulk ¼ ΦDfluid þ 1� Φð ÞDmatrix ð31Þ

ρbulk ¼a

Φm � ρfluid ð32Þ

V P;Wyllie ¼ 1Φ

V fluidþ 1�Φ

Vmatrix

: ð33Þ

In Eqs. (31–33), all densities are in kilogram per cubicmeter, resistivities in ohm meter, and velocities in meter persecond. As well known (Mavko et al. 1998), Wyllie’s Eq. (33)is a heuristic, approximate equation for Φ<0.2 porosities. Atheoretically exact expression for P-wave velocity, based onBiot’s theory (1956) and also valid in case of partial satura-tion, will be derived in “The case of incomplete saturation”section (Eq. 44).

The hydraulic permeability k is first expressed in milli-darcy units:

k millidarcy½ � ¼ 5�108a2 � Φ2mþ1 � 1

S2spec

� �¼ 5�108

a2 � mm�1 � pa � d

d

� �2h ið2mþ1Þð2m�1Þ

� 4p r2

d3þ 2pma

m�1dd2

mm�1 � pa � d

d

� �2h i 1�m2m�1

� ��2

ð34aÞFor the further calculations, we convert the permeability

to SI units (in square meter, see Turcotte and Schubert 1982,p 382):

k m2 ¼ 9:8697� 10�16 k millidarcy½ �: ð34bÞ

The case of incomplete saturation

By the laws of Archie (1942), the resistivity of a rock,whose pore space is only partially saturated with a conduct-ing fluid of resistivity ρw, is

ρS ¼ a

Φm S�n ρw ð35aÞ

where n is the saturation exponent. The saturation S isnormalized to lie between 0 and 1. In particular,

ρS¼1 ¼a

Φm ρw ð35bÞ

How could partial saturation be incorporated to theequivalent rock model? There are two possible approaches:The first is to assume that the same rock model describeselectric conduction in case of partial saturation as well, butinstead of ρw, the fluid has a saturation-dependent resistivity

ρ(S). By Archie’s laws (35a, b), we would have then ρS ¼ aΦm

ρðSÞ ¼ aΦm S�n ρw , from where

ρðSÞ ¼ S�n ρw ð36Þ(special cases are ρ(1)=ρw, ρ(0)=∞). The second possibilitywould be to assume that, in case of partial saturation, it is thecoordination number Z which depends on S, that is, only afraction of throats contains conducting brine, the rest containsnonconducting hydrocarbon. By Eq. (2), this would force thecementation exponent m to also depend on S and to satisfy

ρðSÞ ¼ a

Φm S�n ρw ¼ a

ΦmðSÞ ρw; ð37Þ

which would imply the functional equation

ΦmðSÞ ¼ Φm Sn : ð38ÞThis model, however, is unphysical because for S→0, it

implies ΦmðSÞ ! 0 ; i:e: ; mðSÞ ! 1 ; because 0 < Φ < 1,which contradicts the observed fact (Mavko et al. 1998) thatmis usually close to 2 and rarely exceeds 3.

Thus, only Eq. (36) is feasible, and for a partially satu-rated rock, the resistivity is given by Eqs. (30a) and (36):

ρðSÞ :¼ am

m� 1pa

dd

� �2" #� m

ð2m�1Þ

S�n ρw : ð39Þ

Consider now the acoustics of the fully or partially,saturated rock model. According to Biot’s theory (1956),the saturated bulk modulus differs from the dry bulk mod-ulus in a term ΔK (Φ, S):

KS ¼ Kdry þK Φ; Sð Þ:We estimate first the effective bulk modulus of the fluid,

through the Reuss (1929) average (Mavko et al. 1998):

Kporefiller :¼ S

K fluidþ 1� S

Kair

� ��1

; ð40Þ

then we compute

ΔK Φ; Sð Þ ¼ Ksolid �Kdry

� �2Ksolid 1� Φ� Kdry

Ksolidþ Φ Ksolid

Kporefiller

h i ; ð41Þ

and finally get

KS ¼ Kdry þΔK Φ; Sð Þ: ð42ÞAll compressibilities in Eqs. (43–45) are in Pascal=kilo-

gram per meter per square second, the values of the moduliKdry, Ksolid, Kfluid, Kair must be specified. The density, in caseof partial saturation, is

D Φ; Sð Þ ¼ ΦS Dfluid þΦ 1� Sð ÞDair þ 1� Φð ÞDmatrix

ð43Þ

1134 Arab J Geosci (2014) 7:1127–1138

The shear modulus in micropascal kilogram per meterper square second of the rock is independent of saturation orof the fluid properties. Recalling that

VP ¼ffiffiffiffiffiffiffiffiffiKþ4μ

3D

q; V Shear ¼

ffiffiffiμD

p ð44Þ

and using quantities already calculated in Eqs. (42) and (43),

VP :¼ffiffiffiffiffiffiffiffiffiffiffiKS þ4μ

3D Φ;Sð Þ

r; V shear :¼

ffiffiffiffiffiffiffiffiffiffiffiμ

D Φ;Sð Þq

: ð45Þ

Discussion and concluding remarks

The proposed equivalent geometric model of sedimentaryrocks belongs to the family of effective medium models(modern examples: Kachanov 1994; Sayers and Kachanov1995; Schubnel and Guéguen 2003; Fortin et al. 2005; etc.),its parameters (Z, r, δ, and d) can be easily derived from afew measured rock properties (k, Φ, and m). The converse isalso true: from the values (Z, r, δ, and d), one can calculatethe bulk rock properties (k, Φ, and m). If the specific matrixand fluid properties are also known, the elastic constants andthe P- and S-velocities can be calculated both for the fullysaturated case and for partial saturation. The dc resistivitycan be computed in case of complete saturation by Archie’slaw, but for partial saturation, we need a further rock prop-erty, the saturation exponent n. We note that, while in digitalrock physics, n can be estimated by simulation (Knackstedtet al. 2007), in our model, we could not derive it in terms of(Z, r, δ, and d) or (k, Φ, and m) using physical arguments.Also, while the parameters (Z, r, δ, and d) are sufficient tofind first the coordination number m (by Eq. 2) and then tocalculate single-phase permeability (using Eq. 34a), wehave not succeeded to describe relative permeabilities fortwo-phase flow. This failure is possibly explained by thecomputer simulation study of Sok et al. (2002), whoreported that, for a proper description of two-phase flow ina pore network, one needs to specify network topology athigher order than the coordination number, by using coor-dination number sequences (Grosse-Kunstleve et al. 1996).The geometrical model implies that mechanical properties(density, P- and S-velocities, elastic moduli) depend on thepore coordination number Z, because all of them depend onporosity Φ, by Eq. (29) Φ depends on m, and by Eq. (2), m isrelated to Z. This is in accord with granular medium theories(e.g., Brandt 1955; Walton 1987) where porosity, grain-to-grain coordination number, and elastic properties are linked,though the relation between grain-to-grain and pore-to-porecoordination numbers still needs a separate careful study.Through the entire study, we have heavily relied on the

Kozeny–Carman equation (Walsh and Brace 1984) k ¼ 1b

�Φ3 � 1S2spec

� 1t2 (Eq. 5). This is an empirical law which has

been improved lately using percolation theory (Guéguenand Dienes 1989; Korvin 1992a; Benson et al. 2006a, b).Even though empirical, the Kozeny–Carman Law is ex-tremely useful and versatile. Its Sspec factor makes it possibleto incorporate pore size distribution in the model, while thehydraulic tortuosity τ can describe the permeability behaviorat the percolation threshold (Korvin 1992a, b). Throughoutthe paper, we have mixed empirical laws (Archie’s,Wyllie’s, and that of Kozeny–Carman) with “theoretical”ones (such as Kirkpatrick’s Eq. 1a, 1b or Biot’s theory),and even the theoretical equations have empirical elements(as the Reuss average, Eq. 40, hidden in Biot’s theory) or adhoc assumptions (such as the self-consistency assumption inKirkpatrick 1973; the maximum entropy assumption inDoyen 1987; or the effective media approximation inYonezawa and Cohen 1983). Such a mixture of theoreticaland empirical equations naturally results in empirical equa-tions of limited validity, and our main equations, Eqs. (21)to (23), or their special cases (24) to (26) for a=1) should beconsidered empirical, as the majority of equations of petro-physics and well log analysis are.

As seen in section “Rock property to pore geometryinversion examples,” the geometric model works well forsandstones; for carbonates, the resulting pore parameters arenot physically impossible, but they show only order-of-magnitude agreement and very poor correlation with themicroscopic rock structure. We see no difficulty to similarlymodel fracture porosities or vug porosities which are moreappropriate for carbonates, but the integration of the threemodels into a single triple-porosity model—which will bethe next step of this research—is a great challenge.Throughout the paper, we assumed the statistical homoge-neity and isotropy of the rock volume, which might be truefor a small cutting analyzed in DPR, less true for a plug-sized sample, and is an obviously invalid assumption onreservoir scale. Issues of upscaling the model to reservoirscale, making it heterogeneous and anisotropic, are amongthe further tasks to be solved.

Acknowledgments The authors gratefully acknowledge the excel-lent conditions for research provided by their home institutions, theKing Fahd University of Petroleum and Minerals, Saudi Arabia, andUniversidad Nacional Autónoma de México. Thanks are due toDr. Nabil Akbar who, many years ago, suggested the topic andacquainted us with his 1993 Stanford Ph.D. thesis. Parts of the workhave been previously presented at two workshops: the SymposiumArreglos de Fractura (UNAM, Mexico City, 25 February, 2002) andat the Theoretical Physics Day (KFUPM, Dhahran, Saudi Arabia, 13May, 2007). The work of Dr. G.K. and Dr. A.A. has been partlysupported by the King Abdulaziz City for Science and Technologythrough project no. 08-OIL82-4, National Science Technology Inno-vation Plan. Dr. G.K. is grateful for the financial support from theproject no. 168638 SENERCONACYT-Hidrocarburos Yacimiento Pet-rolero como un Reactor Fractal (project leader Professor Klavdia

Arab J Geosci (2014) 7:1127–1138 1135

Oleshko) which enabled him to visit the Research Group of ProfessorOleshko in Juriquilla, Querétaro, Mexico, and to work with her onsome critical aspects of this research.

Appendix: Notations

a Dimensionless constant in Archie’s lawρbulk ¼ a

Φm ρwα=b/c Aspect ratio of the throat’s cross

section’s in the model of Doyen (1987)b Geometrical constant in Eq. (5). For

circular tubes b=2b, c Semiaxes of the elliptical throat’s cross

section in the model of Doyen (1987)d Model parameter, average geometric

distance between two nearest pores(millimeter)

Dfluid, Dmatrix

etc.Specific densities (in gram per cubiccentimeter)

δ Model parameter, mean throat radius(millimeter)

F Formation factor in Archie’s law (1942)Φ Total porosity, fraction (0≤Φ≤1)Φs, Φf Stagnant, resp. flowing (hydraulically

effective) parts of the porosity (seeEq. 16, and Perez-Rozales 1982; Glover2009)

ge, gh Electric, resp. hydraulic conductancesin the Kirkpatrick equation (Eq. 1)

g*e ; g*h Bulk electric, resp. hydraulic

conductances in the Kirkpatrickequation (Eq. 1)

k Permeability (in square meter)Kdry, Ksolid, Kfluid,Kair etc.,

Specific bulk moduli (in Pascal)

l Throat length in the model of Doyen(1987)

l Average distance followed by the fluidpath (in millimeter), l

d

� � ¼ tL Total length of the throats in a unit

volume of rock (in millimeter)m Cementation exponent in Archie’s lawm(S) Saturation-dependent cementation

exponent, Eqs. (37) and (38)μ Shear modulus in Pascaln Saturation exponent (in Archie’s law)n(r) Pore radius probability distribution

function in the model of Doyen (1987)n(α, c) Throat-shape probability distribution

function in the model of Doyen (1987)N Number of pores in a unit volume

(1 mm×1 mm×1 mm) of rock

P, P0 Pressure in the model of Doyen (1987)r Model parameter, mean pore radius

(in millimeter)ρ100%, ρbulk, ρs,ρw

Specific resistivities in ohm meter

ρ(S) Saturation-dependent resistivity in Eqs.(36) and (37)

S Saturation, between [0,1]Sspec [1/mm] Specific surface (i.e., total surface area

of a unit volume of rock)C Hydraulic tortuosityV Rock volume for which the geometric

model holdsVfluid, Vmatrix etc. Specific P-wave velocities in meter per

secondZ Coordination number (average number

of throats emerging from one pore)

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