Journal of Petroleum Science and Engineering 69 (2009) 247–254

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Journal of Petroleum Science and Engineering

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Research paper

Conductivity of carbonate formations with microfracture systems

Mikhail Markov 1, Aleksandr Mousatov ⁎, Elena Kazatchenko 1

Instituto Mexicano del Petróleo, Eje Central Norte, Lázaro Cárdenas 152, 07730 D. F., Mexico

⁎ Corresponding author. Fax: +52 55 9175 7089.E-mail addresses: mmarkov@imp.mx (M. Markov), a

(A. Mousatov), ekazatc@imp.mx (E. Kazatchenko).1 Fax: +52 55 9175 7089.

0920-4105/$ – see front matter © 2009 Elsevier B.V. Aldoi:10.1016/j.petrol.2009.08.019

a b s t r a c t

a r t i c l e i n f o

Article history:Received 27 August 2007Accepted 27 August 2009

Keywords:conductivity tensornon-interacting fractureselectrical anisotropy

This paper studies the influence of microfracture orientation, shape and conductivity on the effectiveelectrical properties of carbonate formations. The conductivity model of these formations can be presentedas a conductive porous matrix in which the oblate spheroids representing microfractures are embedded. Thespheroid aspect ratio varies in the range of 10−2–10−4 that corresponds to crack shapes in carbonates. Tosimulate the effective conductivity of formations with different aligned fracture systems we have used theapproach of non-interacting fractures (dilute inclusion concentration). Based on the comparison of themodeling results for the non-interactive approach and effective medium methods we have demonstratedthat the first-order equations can be applied for calculating the conductivity tensor for the fracture porosityup to 0.03. If the fracture conductivity is higher than the matrix one (fractures saturated with water of thesame or higher conductivity), the first-order expression is introduced for the effective conductivity tensor. Inthe case when the inclusion conductivity is lower than the matrix one (fractures filled with oil, gas, or sealedby solid phase), the first-order approximation should be applied to the resistivity tensor to provide thesimilar calculation accuracy in the required fracture-porosity range.We present the results of the effective conductivity simulation for formations with arbitrary orientedfractures, one system of aligned fractures, one system of fractures with normal distribution of orientationsalong one direction, and three systems of fractures (conductive and resistive) with non-orthogonalorientations.The modeling results demonstrate the feasibility of assessing the fracture parameters (conductivity,orientation, aspect ratio, and concentration) in carbonate formations using borehole measurements of theconductivity tensor.

mousat@imp.mx

l rights reserved.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Evaluation of reservoirs formed by carbonate formations frequentlyfaces a problem of characterization of the fracture systems. Informationabout the fracture orientation and porosity is required for the per-meability prediction, water control and optimization of oil production.

The micromechanics experiments on the crack generation showthat the thermal cracking produces a fairly isotropic distribution ofpredominantly intergranular cracks, while the stress-induced crack-ing leads to a strongly anisotropic distribution of intergranular andtransgranular cracks oriented parallel to the direction of maximalstress (David et al., 1999; Menendez et al., 1999). The carbonate oilfields exhibit both a random microfracture orientation and micro-fractures aligned along one or two–three preferential directionsmaking up a complex pattern of vertical, inclined and horizontal cracksystems (Bagrintseva, 1999). The system of aligned cracks saturated

with fluids (water or multiphase mixture containing water, gas andoil) affects significantly the formation resistivity and predeterminesvalues and orientations of the resistivity tensor components. Usingthe resistivity logs obtained by the electromagnetic tools speciallydesigned for determining the conductivity tensor (Kriegshäuser et al.,2000), the porosity and preferential direction of the borehole-scalemicrofractures can be estimated. This crack-structure reconstructionrequires the adequate formation resistivity model to predict theeffective conductivity tensor for complex distribution and orientationof microfracture systems.

The simple isotropic resistivity model of fractured formationswas considered by Pirson (1957) and Aguilera (1976). In this modelthe arbitrary oriented fractures approximated by thin plates areconnected only in parallel to rock blocks with a primary porosity.Itenberg and Schurman (1984) and Verzhbitskiy and Malinin (1986),using the same approximation of fracture systems, extended theresistivity model for electrically anisotropic formations with alignedcracks. The model of heterogeneous material where one componenttreated as spheroidal inclusions embedded into a conductive host wasproposed by Fricke (1924). Ovchinnikov (1950) extended theMaxwell–Garnete method for calculating the electromagnetic

248 M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

properties of the composite material with the ellipsoidal inclusions ofrandomorientation (isotropicmaterial) or aligned along one direction(anisotropic material). The conductivity tensor in carbonate double-porosity formations with several orthogonal fracture systems wasanalyzed in (Kazatchenko andMousatov, 2002) where microfractureswere approximated by flattened ellipsoids that asymptotically weresubstituted by thin plates for describing systems of aligned micro-fractures. Applying this combined approach Mousatov et al. (2003)analyzed the resistivity anisotropy in borehole vicinity related to theorientation of microfracture systems and their saturation withdifferent fluids.

The deferential effective medium (DEM) approach was developedby Veinberg (1966), Sen et al. (1981), and Giordano (2003) for thesimulation of the physical properties' of a heterogeneous isotropicmedium constituted by a homogeneous host and ellipsoidal inclusionsof a high concentration. Mendelson and Cohen (1982) formulatedthe DEM method to compute the conductivity tensor of anisotropicsedimentary rocks composed of aligned ellipsoidal grains. The self-consistent effective medium approximation (EMA) proposed byBruggeman (1935) was developed for the composite materials withellipsoidal components in severalworks (Berryman, 1995; Benveniste,1987). Based on this method Kazatchenko et al. (2004a,b, 2005)performed the resistivity modeling and inversion for double-porositycarbonate formations. The effective field method (EF) was applied byLevin and Markov (2004) for predicting the effective conductivity of amaterial with thin inclusions. The effective media and effective fieldmethods are sophisticated enough to predict correctly the conductiv-ity of rocks with high concentration (high porosity) of ellipsoidalinclusions arbitrary oriented (isotropic medium) or aligned along onedirection (transversely isotropic medium). However, these methodsface serious problems for calculating the conductivity tensor inarbitrary anisotropic formations where the anisotropy is producedby various microfracture systems with different orientation azimuths.

Taking into account that the microfracture concentration innaturally formed rocks does not exceed 2–3%, it is possible to obtainthe effective conductivity tensor of formations with complex crack-system structures applying a model of non-interactive inclusions(Shafiro and Kachanov, 2000).

In this paper we use the non-interactive approach (dilute inclusionconcentration) to study the influence of microfracture characteristicssuch as their orientation, shape and conductivity on the effective tensorof carbonate formations. The electrical model of these formations ispresented as a conductive porous matrix in which the oblate spheroidsrepresenting microfractures are embedded. The matrix conductivity isconsidered as a function of the matrix porosity (Kazatchenko et al.,2004a) andwater saturation (Kazatchenko et al., 2006). The aspect ratioof spheroids is selected in the range of 10−2–10−4 that corresponds tocrack shapes in carbonates (Cheng and Toksöz, 1979). To define therangeof fractureporositywhere themodel of non-interacting inclusionsis valid, we have compared the effective conductivity obtained by thedilute concentration approach with simulations performed by theeffectivemediummethods for the cases of randomly oriented (isotropicmedium) and aligned (transversely isotropic medium) fractures. If thefracture conductivity is higher than the matrix one (fractures saturatedwith water of the same or higher conductivity), the first-order expres-sion for the effective conductivity tensor can be used for the fractureporosities up to 0.03. In the case when the inclusion conductivity islower than the matrix one (fractures filled with oil, gas or solid phase),the first-order approximation should be applied to the resistivity tensorto provide similar accuracy of the effective conductivity calculation inthe required fracture-porosity range. We present the results of theeffective conductivity simulation for the following models: randomlyoriented fractures, one system of aligned fractures, system of fractureswith normal distribution of orientations along one direction, and threesystems of fractures (conductive and resistive) with non-orthogonalorientations.

2. Model description

The electrical conductivity of fractured carbonate formations canbe defined as an effective conductivity of a two-component materialcomposed of the conductive porous matrix and inclusions represent-ing fractures. The matrix conductivity depends on the concentrationof small-scale connected matrix pores (matrix porosity ϕm) saturatedpartially or completely with mineralized conductive water. The frac-ture porosity ϕf in carbonates does not exceed the values of 0.02–0.03even in strongly fractured formations (Bagrintseva, 1999). We treatthe porousmatrix as a homogeneous isotropic mediumwith the givenconductivity σm. The matrix conductivity of carbonate formations canbe introduced based on the statistical analysis of experimental data(Kazatchenko and Mousatov, 2002) or theoretically calculatedapplying the self-consistent symmetrical approach for a mixture oflow conductive solid grains and high conductive pores filled withwater (Kazatchenko et al., 2004a).

The secondary pores in carbonates, including fractures, are char-acterized by large-scale sizes and specific shapes. We approximatefractures by similar oblate spheroids which have the same aspect ratioα and different sizes. The fracture orientation is given by normals η tothe spheroid maximal cross section. Fractures can be arbitraryoriented (isotropic medium) or have non-random distributions oforientations with respect to some preferential directions formingdifferently oriented fracture systems (arbitrary anisotropic medium).We assume that the fractures' centers are randomly distributed.

The aspect ratios of fractures in carbonates lie in the range of3⁎10−3–3⁎10−4 as it was demonstrated by Cheng and Toksöz (1979)and Bagrintseva (1999). The fracture conductivity σ f depends on theelectrical properties of a filling material and it can vary in a wide range.Generally, the presence of microfractures is considered as a factor thatincreases the conductivity of carbonate rocks. In fact, when micro-fractures are filled with conductive fluids such as a borehole mudfiltrate or connate water, their conductivity exceeds significantly thematrix one. In this case the ratio between the fracture and matrixconductivities S can achieve the values of 102–103 for low matrixporosity. However, in oil-saturated carbonates (especially oil-wet for-mations) the fracture-matrix conductivity ratio can go down to 10−1.The lowest values of this ratio (S=10−2–10−3) correspond tocemented fractures filled with non-conductive solids of extremelylow microporosity. The conductive and resistive fracture systems ofdifferent orientations can be simultaneously presented in carbonateformations and can lead to additional electrical anisotropy increase.

3. Approximations for non-interacting fractures

To calculate the effective conductivity tensor of media with acomplex non-random distribution of fracture orientations we applyan approach that does not account for the interaction of fractures. Inthis case each fracture embedded in the matrix is affected by the samegiven electric field that is assumed unperturbed by other fractures.The non-interacting approximation is correct for small inclusionconcentrations and widely used in micromechanics.

In a medium with inclusions the following basic equations foraverage vectors of the electric field ⟨Ei⟩ and current density ⟨Ji⟩ can bewritten

⟨Ei⟩ = ð1−ϕf Þ⟨Emi ⟩ + ϕf ⟨E

fi ⟩; ð1Þ

⟨ Ji⟩ = ð1−ϕf Þ⟨Jmi ⟩ + ϕf ⟨Jfi ⟩; ð2Þ

Jmi = σmEmi ; ð3Þ

J fi = σ fE fi ; ð4Þ

249M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

⟨Ji⟩ = σ⁎ij⟨Ej⟩; ð5Þ

where

Eim and Ei

f are the electric fields in matrix and fracture,correspondingly,

J im and J if are the current densities in the matrix and fracture,correspondingly,

σm and σ f are the scalar matrix and fracture conductivities,correspondingly,

σij⁎ is the effective conductivity tensor of a heterogeneousmedium,

ϕf is the porosity of fractures,⟨⟩ is averaging symbol.

In the non-interacting approximation it is generally assumed thatthe average electric field is equal to the matrix field unaltered by thepresence of inclusions

⟨Ei⟩ = Emi : ð6Þ

Taking into account this condition the electric field inside aninclusion can be expressed as

⟨E fi ⟩ = AijE

mj ; ð7Þ

where Aij is a second rank tensor depending on the inclusion shape.After algebraic transformations of Eqs. (1)–(7) the first-order

expression for the effective conductivity tensor is found (Landau andLifshitz, 1960; Shvidler, 1985)

σ⁎ij = σmδij + ϕf ðσ f−σmÞAij; ð8Þ

where δij is Kronecker's symbol.Tensor Aij is obtained from the solution of the one-particle

problem for the ellipsoidal inclusion (Frank, 1963; Mendelson andCohen, 1982)

Aij = δij + nijðσmÞ−1ðσ f−σmÞh i−1

; ð9Þ

where nij is the depolarizing factor of ellipsoid (Stratton, 1941;Landau and Lifshitz, 1960).

When the axes of a spheroidal inclusion coincidewith the Cartesiancoordinate system the tensor A has only three non-zero principaldiagonal components

Aii =σm

σm + niðσ f−σmÞ ; ð10Þ

and the components of depolarizing factor ni are expressed explicitlyby algebraic functions

n1 = n2 =12ð1−n3Þ;

n3 =1−m2

2m3 ln1 + m1−m

−2m� �

;

ð11Þ

where; m =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−α2;

pα=a1/a3 is the aspect ratio,a1 and a3 are the spheroid semi-axes (a1=a2>a3).

To determine the limits of fracture porosity ϕf or fracture densityd=ϕf/α valid for applying the non-interacting approach, we havecompared the effective conductivity obtained by the first-order ap-

proximations with the results computing by the Differential EffectiveMedium method (DEM).

The DEM method is a well-known homogenization method inmechanics of composite materials and geophysics that was firstsuggested by Bruggeman (1935) for describing properties of dielectriccomposites. This method consists of iteratively adding inclusions of adilute concentration into the host with the effective parametersdetermined in the previous step. It is convenient to use this methodfor calculation of effective properties of composite materials with highconcentration of inclusions arbitrary oriented (isotropic medium) oraligned along one azimuth (transversely isotropic medium). Unfor-tunately, the application of this method for an arbitrary anisotropicmedium with non-random orientations of inclusions faces seriousmathematical problems that impede the DEM use in the case offormations with different fracture systems. The DEM theoreticalbackground and calculation procedure for predicting the effectiveconductivity tensor of porous rocks were completely described in theworks of Sen et al. (1981) (isotropic formations, spheroidal poreshapes) and Mendelson and Cohen (1982) (transversely isotropicmedium with spheroidal inclusions).

The comparison between the conductivity curves calculated by thenon-interacting approximation and DEM for randomly and alignedfractures shows that the relative difference between the simulationmethods is less than 10% for the range of fracture porosities up to0.03 in both cases (isotropic and anisotropic media). The conductivitycurves for the high fracture-matrix conductivity ratio S=3⁎103 andspheroid aspect ratio α=10−2–10−3 are presented in Fig. 1 A, B.

For randomly oriented fractures with the conductivity lower thanthe matrix one the first-order approximation for conductivity (8) de-scribes the effective conductivity with accuracy of 10% just for the verylow fracture concentrations (ϕf<0.005). Then error increases rapidlyand achieves the value about 100% for the fracture concentration of 0.01(Fig. 2 A).

To extend the non-interacting approach for the heterogeneousmedia where the resistive inclusions are embedded into the homo-geneous conductive matrix, we have formulated the first-orderapproximation for the resistivity tensor. Taking into account that inthese media the electric current avoids the low conductive inclusions,we can assume that the average current density in Eq. (2) is mostlydefined by the average current density in the matrix. In this case wehave to substitute the expressions (6) and (7) by the correspondingconditions for the current densities

⟨J⟩ = Jm; ð12Þ

⟨J f ⟩ = BijJm; ð13Þ

where B is a second rank tensor depending on the inclusion shape.The tensor B relates the current density inside an inclusion to the

current density far from inclusion and can be rewritten as

Bij =σ f

σm Aij: ð14Þ

From Eqs. (1)–(5) and taking into account the expressions ((12)–(14))we find the first-order approximation for the effective resistivitytensor ρ⁎=1/σ⁎

ρij⁎ = ρmδij + ϕf ðρ f−ρmÞρm

ρ fAij; ð15Þ

where ρm and ρf are the matrix and fracture resistivities, respectively.

Fig. 1. A, B. Effective conductivities for the non-interaction (NI) and DEM methods forrandomly oriented (A) and aligned (B) fractures when the fracture conductivity ishigher than the matrix one (α=10−3). σm=0.01 Ω−1m−1, σ f=3.0 Ω−1m−1.

Fig. 2. A, B. Effective conductivities for the non-interaction (NI) and DEM methods forrandomly oriented (A) and aligned (B) fractures when the fracture conductivity islower than the matrix one (α=10−3). σm=0.01 Ω−1m−1, σ f=3.3⁎10−3Ω−1m−1.

250 M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

The compliance of Eq. (15) corresponds to the effective conduc-tivity tensor for non-interactive low conductive fractures

σij⁎ = σm δij−ϕf

σ f−σm

σm

!Aij

" #−1

: ð16Þ

This approximation gives better accuracy than the linear first-order expression (8) for the conductivity tensor when the ratio S<1(fracture conductivity is lower than the matrix one). The first-orderapproximation (16) can be used for the wider range of the fractureporosity than the expression (8) even for the very low ratio S=1/300(Fig. 2 A, B). Using this approximation the effective conductivity withaccuracy of 10% can be calculated up to the ϕf=0.03 for randomlyoriented and aligned fractures with aspect ratio α=10−3.

The results of comparison demonstrate that the equationsobtained with the assumption of non-interacting conductive andresistive fractures (with respect to the matrix conductivity) can beapplied for the fractured formationmodeling because the real fractureporosity does not exceed 0.02–0.03.

4. Conductivity tensor of non-randomly oriented fractures

Frequently, the orientations of stress-formed fractures are notarbitrary and fluctuate nearby some preferential directions predeter-

mined by geological and mechanical conditions. While the fractureporosity (fracture density) remains in the range of the non-interactingfractures' approximation, the effective conductivity tensor of suchcomplex structures can be treated as a superposition of each fractureeffects and calculated by applying the first-order linear Eqs. (8) or(16).

In these equations the tensor Aij (or Bij) for an inclusion (oblatespheroid) randomly oriented with respect to the coordinate systemcan be presented as

Aij⁎ = Annlinljn; ð17Þ

where l is the rotation matrix

l = j cosφ −cos θ sinφ sin θ sinφsinφ cos θ cosφ −sin θ sinφ0 sin θ cosθ

j: ð18Þ

The variables θ and φ are the colatitudinal (angle with the axis z)and azimuthal angles of the fracture normal η in the spherical coor-dinate system.

A variation of fractures' orientations can be described by thedistribution function f (θ, φ)

f ðθ;φÞ = 1ϕf

dϕf ðθ;φÞdω

; ð19Þ

Fig. 3. A, B. Effective conductivity components (A) and anisotropy coefficient (B) asfunctions of the angle θ for the Hudson model. σm=0.01 Ω−1m−1, σ f=1.0 Ω−1m−1,α=3⁎10−3.

251M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

where dϕf is the porosity value of fractures with the normal orienta-tion corresponding to the solid angle dω=sinθdθdφ.

Taking into account the distribution of fractures' orientations, fromEq. (8) we can obtain the effective conductivity tensor for a mediumwith non-randomly oriented fractures of the same conductivities andshapes (similar spheroids)

σij⁎ = σmδij + ϕf ðσm−σ f Þ⟨Aij

⁎f ðθ;φÞ⟩: ð20Þ

Here the symbol ⟨⟩ means averaging operator over a solid angle

⟨Aij⁎f ðθ;φÞ⟩ = Akk

12π

∫2π

0

dφ ∫π=2

0

lkilkjf ðθ;φÞ sinθdθ; ð21Þ

and the distribution f (θ, φ) satisfies the condition

12π

∫2π

0

dφ ∫π=2

0

f ðθ;φÞ sinθdθ = 1: ð22Þ

In the case of fractures aligned along a single direction (all fracturenormals have the same angles θ= θ0 and φ=φ0) with thedistribution

f ðθ;φÞ = 2πδðθ−θ0Þsinθ

δðφ−φ0Þ; ð23Þ

the tensor ⟨Aij⁎ f (θ, φ)⟩ is obtained explicitly and the effective

conductivity components are

σ11⁎ = σm + ϕf ðσ f−σmÞðA11 cos

2 φ0 + A11cos2 θ0 sin

2 φ0 + A33 sin2 θ0 sin

2 φ0Þ;σ12⁎ = ϕf ðσ f−σmÞðA11sinφ0 cosφ0−A11cos

2 θ0 sinφ0 cosφ0−A33 sin2θ0 sinφ0 cosφ0Þ;

σ13⁎ = ϕf ðσ f−σmÞðA33 cos θ0 sin θ0 sinφ0−A11 sin θ0 cos θ0 sinφ0Þ;

σ22⁎ = σm + ϕf ðσ f−σmÞðA11 sin

2 φ0 + A11 cos2 θ0 cos

2 φ0 + A33 sin2 θ0 cos

2 φ0Þ;σ23⁎ = ϕf ðσ f−σmÞðA11 sin θ0 cos θ0 cosφ0−A33 cos θ0 sin θ0 sin

2 φ0Þσ33⁎ = σm + φf ðσ f−σmÞðA11 sin

2 θ0 + A33 cos2 θ0Þ;

σ21⁎ = σ12

⁎ ; σ31⁎ = σ13

⁎ ; σ32⁎ = σ23

⁎ ; ð24Þ

The conductivity components for the particular case of fracturesaligned in the x–y plane (θ=0° and φ is undefined) were consideredin the previous section (Figs. 1 B and 2B).

When the normals of all fractures have the same given angle withthe z-axis (anisotropy axis) and their projections onto the x–y planeare randomly oriented, the distribution function depends only on theangle θ0

f ðθ;φÞ = f1ðθÞf2ðφÞ =δðθ−θ0Þsinθ

; ð25Þ

where f1ðθÞ = δðθ−θ0Þsinθ and f2(φ)=1.

This model was proposed by Hudson (1990) for studying thecracked transversely isotropic medium.With such a crack orientation,the anisotropy axis (symmetry axis) of the cracked medium is alwaysconserved parallel to the vertical axis while the angle θ varies from 0°up to 90°.

The averaging procedure of the expression (24) over the angle φgives the following tensor components for this model

σ11⁎ = σm +

12ϕf ðσ f−σmÞðA11 + A11 cos

2 θ0 + A33 sin2 θoÞ;

σ11⁎ = σ22

⁎ ;

σ33⁎ = σm + ϕf ðσ f−σmÞðA11 sin

2θ0 + A33 cos

2θ0Þ:

ð26Þ

The graphs of the conductivity components as functions of theangle θ and fracture concentration are shown in the Fig. 3A. Here the

conductive fractures (S=102) is approximated by spheroids withaspect ratio α=3⁎10−3. It should be noted that the verticalcomponent σz is more sensitive to the orientation angle and variesin the wider range than the horizontal components σx and σy. Theanisotropy coefficient defined as λ=(σx/σz)1/2 is high enough forsmall orientation angles and achieves the values of λ=1.7 at thefractures concentration ϕf=0.025 (Fig. 3B). Starting from the anglesθ=50° the anisotropy coefficient becomes less than 1 and goes to theasymptotic value 2−1/2 with increasing the fracture aspect ratio forhigh conductive and vertically oriented fractures.

If fractures are not strongly aligned along one direction andtheir normals vary about the polar angle θ0with a distribution functionf (θ, φ)= f1(θ) (f2(φ)=1), this effective medium is transverselyisotropic and the non-zero components of the conductivity tensor are

σ11⁎ = σm +

12ϕf ðσ f−σmÞ⟨A11 + A11 cos

2ðθ−θ0Þ + A33 sin2ðθ−θ0Þ⟩;

σ11⁎ = σ22

⁎ ;

σ33⁎ = σm + ϕf ðσ f−σmÞ⟨A11 sin

2ðθ−θ0Þ + A33 cos2ðθ−θ0Þ⟩: ð27Þ

To evaluate the influence of fracture disorientations on the con-ductivity tensor we have performed calculations with the tangentialdistribution functions (Fedorishin, 1976)

f1ðθÞ = Q exp½−tg2ðθ−θ0Þ= tg2θd�; ð28Þ

252 M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

where θd is the disorientation angle between the fracture normal andaxis z, the coefficient Q is the normalization factor to satisfy thecondition (22).

The function (28) is able to describe all kinds of transverselyisotropic media related to a non-random orientation of fractures. Thedisorientation angle θd determines dispersion of distribution. Forexample, when θd=π/2, the fractures are arbitrary oriented and sucha model corresponds to the isotropic media. For θd=0 the expression(28) gives the distribution for the medium with strictly orientedfractures.

Fig. 4 A shows the graphs of the effective conductivity componentsas a function of the fracture porosity simulated for the models wherethe orientations of fractures vary about the horizontal plane (θ0=0)with the disorientation angles θd=10°, 20°, 30°. The vertical compo-nentσz ismore sensitive to the fracture orientation than the horizontalcomponents. The anisotropy coefficient λ drops with increasing thedisorientation angle but it remains significant even for the anglesθd=20°–30° (Fig. 4B).

5. Conductivity tensor of fracture systems withdifferent orientations

The approach of non-interactive fractures allows us to determinethe effective conductivity of formations with an arbitrary anisotropyrelated to the fracture systems which have different preferentialdirections. In this case, for each fracture system p from Eqs. (10) and(11) we can firstly calculate the tensors Aii

p in the local coordinates

Fig. 4. A, B. Effective conductivity components (A) and anisotropy coefficient (B) asfunctions of the fracture porosity ϕf and disorientation angle θ. σm=0.01 Ω−1m−1,σ f=1.0 Ω−1m−1, θ0=0, α=3⁎10−3.

coinciding with the given fracture-system orientation. Then, thesetensors should be transformed in the global coordinate system

Ap⁎ij = Ap

kllpikl

pjl; ð29Þ

where the lp is defined by Eq. (18) with the polar angles θ=θp andφ=φp corresponding to the p-th fracture system.

The effective conductivity tensor can be calculated as a sum of theeffects produced by each system of fractures with correspondingporosity ϕf

p

σij⁎ = σmδij + ∑

p

1ϕpf ðσm−σ f ÞAp⁎

ij : ð30Þ

The graphs of the conductivity components for two conductivefracture systems are demonstrated in Fig. 5 A. The first system ofporosity ϕf1=0.01 is composed of vertical fractures which are orthog-onal to the y-axis and the second one of the variable porosity ϕf2 isalso vertical and forms the angle φ12 with the first system. The angleφ12 and the porosity of the second fracture system affect significantlythe horizontal conductivity component and coefficients of anisotropy(Fig. 5 B).

Applying the non-interacting approach we have simulated theeffective conductivity tensor for formations which contain simulta-neously as the fractures sealed by solid materials or filled with highresistive fluids (oil or gas) as well the open fractures saturated with

Fig. 5. A, B. Effective conductivity components (A) and anisotropy coefficient (B) fortwo conductive fracture systems vs. fracture porosity ϕf2 and orientation angle φ12.σm=0.01 Ω−1m−1, σ f=1.0 Ω−1m−1.

253M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

conductive water (Fig. 6 A, B). In this model two vertical fracturessystems of the same porosities and aspect ratios have the conductiv-ity (σ f12=1.0 Ω−1m−1) higher than the matrix one (σm=0.01 Ω−1

m−1) and the third horizontal system is composed of resistive frac-tures (σ f3=10−3Ω−1m−1). Because the porosity of resistive frac-tures was selected low enoughwe could use the first-order expression(8) for calculations of conductivity. When the porosity of resistivefractures is higher than 0.005–0.006 the effective conductivity shouldbe estimated using the first-order approximation for resistivity(Eq. (16)). In this case the conductivity tensor can be presented inthe following form

σij⁎ = σm δij−∑

p

1ϕpf

σ f−σm

σm

!Apij

" #−1

: ð31Þ

The components of the effective conductivity tensor for the modelwith two vertical and mutually orthogonal systems of resistive frac-tures (σ f12=10−3Ω−1m−1 and σm=0.01 Ω−1m−1) and one hori-zontal conductive system (σ f3=1.0Ω−1m−1) are presented in Fig. 7 A.The vertical resistive fracture systems affect strongly the horizontalcomponents which are measured by different types of conventionalelectrical and induction tools in vertical wells. The anisotropycoefficient of the formation with systems of the resistive fracturescan achieve large values and it is not limited by any asymptote incontrast to the case of conductive fractures.

Fig. 6. A, B. Effective conductivity tensor (A) and anisotropy coefficient (B) for the forma-tion (matrix conductivity σm=10−2Ω−1m−1) with one resistive (σ f3=10−3Ω−1m−1)and two conductive (σ f12=1.0Ω−1m−1) fracture systems.

Fig. 7. A, B. Effective conductivity components (A) and anisotropy coefficient (B) for theformation (matrix conductivity σm=10−2Ω−1m−1) with two vertical and mutuallyorthogonal systems of sealed fractures (σ f12=10−3Ω−1m−1) and one horizontalsystem of fractures saturated with water (σ f3=1.0 Ω−1mm−1).

6. Conclusions

We have presented an approach for calculating the effectiveconductivity tensor of formations containing the randomly orientedand aligned fractures. The approach is based on the model with non-interacting fractures. By comparison the effective conductivity calcu-lated by using the first-order equations and the differential effectivemedium method we have demonstrated that the non-interactiveapproximation can be applied for the fracture porosity up to 0.03 forboth isotropic and anisotropic media.

The simulation results have shown that the selection of the linearfirst-order equations depends on the fracture conductivity. The first-order expression for the conductivity tensor is more appropriate forpredicting the effective formation parameters when the fractureconductivity is higher than the matrix one (fractures saturated withwater of the same or higher conductivity). If the fracture conductivityis lower than the matrix one (fractures filled with oil, gas, or sealed bysolid phase), the first-order linear approximation should be applied tothe resistivity tensor to provide the similar calculation accuracy in therequired fracture-porosity range.

The results of modeling demonstrate the feasibility of conductivitytensor measurements in a borehole to detect the orientation offracture system in real carbonate formations. The assessment of thefracture orientation, aspect ratios, and concentration can be per-formed solving the inverse problem for borehole measurements withthree-axial induction tools and using additional well log or core data.

254 M. Markov et al. / Journal of Petroleum Science and Engineering 69 (2009) 247–254

Nomenclatureϕm matrix porosityϕf fracture porosityσm matrix conductivityσ f fracture conductivityα aspect ratio of spheroidal inclusionsη normal to the spheroid maximal cross sectionS ratio between the fracture and matrix conductivities⟨Ei⟩ average vector of the electric field⟨Ji⟩ average vector of the current densityEim electric field in the matrixEif electric field in the fracture

Jim current density in the matrixJ if current density in the fractureσij⁎ effective conductivity tensor of a heterogeneous mediumnij depolarizing factor of ellipsoida1, a3 spheroid semi-axes (a1=a2>a3)ρ⁎ij effective resistivity tensorθ colatitudinal angle of the fracture normal η in the spherical

coordinate systemφ azimuthal angle of the fracture normal η in the spherical

coordinate systemω solid angle in the spherical coordinate systemλ anisotropy coefficient

Acknowledgments

The authors are grateful to theMexican Petroleum Institute, wherein the frame work of the scientific “Hydrocarbon Recuperation”program this study was fulfilled. We would also like to thank Dr. V.Levine for useful discussions and comments.

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