A permeability model for naturally fractured carbonate reservoirs

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Marine and Petroleum Geology 40 (2013) 115e134

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Marine and Petroleum Geology

journal homepage: www.elsevier .com/locate/marpetgeo

A permeability model for naturally fractured carbonate reservoirs

Vincenzo Guerriero a,*, Stefano Mazzoli a, Alessandro Iannace a, Stefano Vitale a, Armando Carravetta b,Christoph Strauss c

aDipartimento di Scienze della Terra, Università degli studi di Napoli ‘Federico II’, Largo San Marcellino 10, 80138 Napoli, ItalybDipartimento di Ingegneria Idraulica, Geotecnica e Ambientale, Università degli studi di Napoli ‘Federico II’, Via Claudio 21, 80125 Napoli, Italyc Shell Italia E&P, Piazza Indipendenza 11/B, 00153 Rome, Italy

a r t i c l e i n f o

Article history:Received 27 April 2012Received in revised form27 October 2012Accepted 2 November 2012Available online 13 November 2012

Keywords:Petroleum geologyStatistical fracture analysisStructural modelingNumerical modelingReservoir simulation

* Corresponding author. Tel.: þ39 0812538124; faxE-mail address: [email protected] (V. G

0264-8172/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.marpetgeo.2012.11.002

a b s t r a c t

Based on a detailed structural study performed on a reservoir surface analogue, a fracture permeabilitymodel for carbonate reservoirs is proposed. This involves four hierarchical systems, each one assumed toconvey fluids exclusively to that of immediately higher order. New basic equations are then provided,simulating hydrodynamic reservoir response. The large scale, first-order structures consist of faults, towhich high-permeability structures (damage zones) are associated. At the meter scale, stratabound jointsystems, together with bedding joints bounding the mechanical layers, form a connected networktransporting fluids to the fault system. At a lower scale, down to crystal size, non-stratabound jointsconstitute a pervasive and capillary fracture network which conveys fluids within the rock mass. Thenon-fractured host rock constitutes the lower permeability system (at the crystal scale). As no publishedworks include an effective integration of detailed structural and numerical models, the present studyaims at covering a significant gap in the literature, also providing appropriate theoretical basis forsubsequent studies on fracture network modeling and reservoir simulation. Besides its application in thefield of reservoir development and management, the illustrated model may also be useful in environ-mental studies involving ground fluids such as e.g. in the field of special/toxic/nuclear wastemanagement.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Reservoir simulation is the main tool in modern reservoirmanagement. The possibility to predict reservoir response toseveral techniques of oil and gas recovery enables the selection ofthe economically most attractive strategy to optimize oil produc-tion (e.g. Gharbi, 2004; Adamson et al., 1996). A correct character-ization of the fracture network (FN) plays a central role in reservoirsimulation. In fact, physical models describing the hydraulicbehavior of fractured reservoirs ascribe to fracture systems a majorrole in controlling rock permeability (Warren and Root, 1963),particularly in low-porosity reservoirs (Schmoker et al., 1985;Nelson, 2001). Furthermore, reservoir characterization approachesbased on the use of decline-curve analysis require the knowledge ofappropriate probability models for fracture attribute statisticaldistribution (Camacho-Velazquez et al., 2008).

Most numerical simulation criteria use relatively simple struc-tural models to characterize the 3D space. In the present paper wepropose a novel approach intended to provide a more realistic

: þ39 0815525611.uerriero).

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description of permeability structures within a naturally fracturedcarbonate reservoir. The key issue of our approach lies in theidentification of fracture sets showing specific statistical behaviors.For each fracture set, the correct probability distribution model isprovided for several fracture attributes (with particular reference toopening displacement and fracture density) over a range of scales.As maximum fracture density (MFD, detectable at the micro-scale)can substantially affect the dynamic response of the fracturedrock, this parameter e and its dependence on grain size e is alsoanalyzed.

A further main research topic of this study deals with theanalysis of hydrodynamic behavior of the proposed structuralmodel for reservoir rocks. The related flow equations are discussedand new basic finite difference equations derived. These latter mayallow one to obtain more effective and reliable numerical simula-tion algorithms. We point out that this work provides the first steptoward the implementation of a full numerical model. Additionalwork would be required in order to provide a technique that can beapplied to reservoir simulation, an issue that is beyond the scope ofthis paper, being object of a future specific research project.

The present model is based on the results of fracture analysiscarried out on a series of carbonate outcrops in the Sorrento

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Peninsula (Italy), which have been selected as a surface analogue ofthe buried reservoir of the large Basilicata oil fields in the SouthernApennines (Shiner et al., 2004; Guerriero et al., 2010). The analyzedsuccession displays structural features that are very common incarbonate rocks. Therefore the proposed model, effectively inte-grating well established concepts concerning the occurrence ofdifferent joint typologies in bedded rocks (Odling et al., 1999) andthe hierarchical organization of flow in carbonate rocks (Massonnatand Viszkok, 2002), may be of general applicability to carbonatereservoirs.

2. Why is a new fracture network model needed?

Most recent theories on fracture network characterization andhydrodynamic simulation appear to show serious limitations:

(1) A common statistical behavior is often assumed for all fracturescontained within a given rock volume. However, carbonaterocks commonly display various typologies of fractures, char-acterized by different statistical behaviors. For instance, somefracture sets show a self-similar behavior, whilst some othersare size restricted. Fractures do also behave statisticallysignificantly different with respect to spatial fracture distri-bution (e.g. random, clustered, regular) and probability distri-bution of fracture aperture, length etc.

(2) Many studies commonly assume that fracture parameters suchas attitude, aperture and density, can be readily ‘translated’ interms of permeability coefficients to be applied to the entire 3Dspace of a reservoir. Accordingly, single porosity models assigna permeability tensor to each specific point/cell of the reservoir.On the other hand, dual-porosity models (Warren and Root,1963) cannot operate at different scales (as it is proposed inthis study). In such models, the FN is usually assumed as con-sisting of faults and a matrix including all further minor frac-tures. The behavior of each grid-point of the permeable matrix(made up of fractured rock) is described by a permeabilitytensor. However, as illustrated below, it can be shown thatelementary volumes of fractured porous rock characterized bythe same permeability in steady-state conditions can exhibitsubstantially different dynamic behavior in the case of non-steady-state flow. Therefore, a permeability tensor cannot beused to describe the dynamic behavior of a rock elementaryvolume in a non-steady-state flow, without introducingsignificant approximation errors.

These issues are dealt with in the following two sections.

2.1. Previous studies on reservoir structural characterization

Most of recently adopted criteria for FN characterization withinfractured reservoirs consider fracture attributes (e.g., aperture,fracture density, etc.) as random variables (RV) distributed withinthe reservoir volume according to certain e experimentallyestablished e statistical models. A common approach in thesestudies (Belfield, 1994; Delay and Lamotte, 2000; Sahimi, 2000;Tran and Rahman, 2006; Tran et al., 2006; Etminan and Seifi, 2008;Mata-Lima, 2008) consists in producing some determination ofthose RV affecting reservoir behavior (e.g. fracture density, porosity,permeability), whose statistical distribution is in agreement withobservational data. In addition, when such determinations are usedas input parameters for reservoir simulators, it is required that theanalysis yields results matching with real production data. InEtminan and Seifi (2008), as well in Mata-Lima (2008), the simu-lation of parameters describing the FN is carried out by directsequential simulation (Soares, 2001), while preserving the

variogram (spatial variability; e.g. Diggle and Ribeiro, 2007) ofporosity, permeability and other RV of interest. The permeabilityfield defined in this way provides the input for reservoir simula-tions. The reservoir dynamic response is then compared with realproduction data (history matching), and this procedure is repeateduntil the disagreement between simulation and real productiondata reaches a desirable minimum. In Delay and Lamotte (2000), inaddition to variogram preservation of FN attributes, the entropy isintroduced as spatial disorder descriptor of the RV under consid-eration. Further authors make the assumption that the FN showsfractal and/or multi-fractal scaling (Belfield, 1994; Sahimi, 2000;Tran and Rahman, 2006; Tran et al., 2006; Camacho-Velazquezet al., 2008). However, as it will be shown below, not all fracturesets observed in carbonate rocks show fractal geometry. Althoughprevious studies provide fundamental theoretical tools for FNcharacterization and simulation, they are funded on simplifiedstructural models not always appropriate for a fully correct inter-pretation of fracture characteristics. The major limitations of thesesimplified models are related with the assumption of a commonstatistical behavior for all fractures contained within the rock mass.The main goal of our study approach is to provide, by means of anappropriate statistical analysis, correct probability models of frac-ture attribute distribution for different fracture types (as well asbedding joints) that can be encountered in a fractured carbonatereservoir.

2.2. Steady-state and dynamic behavior of fractured rock

In order to illustrate how elementary volumes of a fracturedporous rock showing similar permeability values in steady-stateconditions can exhibit substantially different dynamic behavior,we have carried out numerical simulations involving two two-dimensional fractured porous bodies B1 and B2, constituted by thesame porous and permeable medium (Fig. 1a). Each of these bodies,having a square section of 10 cm each side, contains two orthogonalfracture sets with constant spacing and aperture values. B1 showshigher values of fracture spacing and aperture than B2, but bothexhibit the same permeability value for steady-state flow. Thesimulation has been performed by difference equations, with thefollowing boundary conditions:

h ¼ 0:01; at each grid point; for t ¼ 0;

h ¼ 0; along the upper side; andh ¼ 0:01 along the lower side; for t > 0:

where h (here expressed in meters) is the hydraulic head, t is timeand the fluid flow is supposed to be parallel to the lateral sides. Theresults have been compared with the behavior of an elementaryvolume of non-fractured porous medium, having the samepermeability of B1 and B2.

The simulation reveals that B1 and B2 show very differentdynamic response and release fluid slower than the equivalentporous medium. System B1, characterized by higher values offracture spacing, releases fluid much slower than B2 (Fig. 1b and c).The diagram in Figure 1c shows the discharged volume of fluid e

calculated by time integration of the difference between ingoingand outgoing flow across the boundary surface e as a function oftime. Note that, in order to release 50% of the total fluid volume(2� 10�8 m3), the body B1 needs a time that is more than one orderof magnitude larger than that needed by body B2.

The different behavior of bodies B1 and B2 occurs because theaverage time spent by a fluid particle contained within the matrixto reach a fracture is proportional to the average path, i.e. one half

Figure 1. Model of fracture-parallel flow in a fissured porous body. (a) Two fractured porous bodies showing similar permeability values in case of steady-state flow. (b) Diagramshowing the flow rate (m3/s) as function of time for the two fractured bodies and the equivalent homogeneous permeable system. (c) Diagram showing the cumulative releasedfluid as function of time. Should be noted that to release the 50% of the total discharged fluid volume (2 � 10�8 m3), the body B1 employs a time of more than one order of magnitudegreater than that spent by the body B2.

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of mean fracture spacing. Therefore, for a given porous matrixpermeability, the system characterized by a higher fracture densityallows a more efficient fluid discharge (although this is an illus-trative and therefore simplified example, it should be noted thatdifferent fracture density values are usually associated withdifferentmatrix properties; this issue is dealt with in next sections).As hydrocarbon recovery involves non-steady-state flowwithin thereservoir, translating fracture parameters e such as spatialarrangement and density e into permeability values (e.g. Etminanand Seifi, 2008) may not be appropriate to achieve a reliablereservoir simulation. Describing the hydraulic behavior of anelementary rock volume by a permeability tensor in numericalmodels is equivalent to substituting it with a porous non-fractured

medium exhibiting the same permeability value, neglecting itspotentially different hydrodynamic behavior. By doing so, largeapproximation errors may be introduced in the calculations, asillustrated by the previous example.

3. The study area

The studied carbonates are part of the thick Mesozoic shallow-water succession cropping out in the Sorrento Peninsula(southern Italy; Fig. 2) These platform carbonates tectonicallyoverly a tectonic pile comprising Mesozoic-Tertiary pelagic basin(Lagonegro) successions, Neogene flysch-type deposits andshallow-water and slope facies carbonates of the Apulian Platform

Figure 2. Geological framework of field study area. (a) Geological sketch map of the Naples e Sorrento Peninsula area of the southern Apennines (from Guerriero et al., 2010). (b)Geological sketch map of field analogue area. (c) Sketch (after Mazzoli et al., 2008) showing depth structure of the southern Apennines. Pl: Pliocene siliciclastic beds overlying theApulian Platform carbonates; M: Melange zone occurring at the base of the allochthonous units (Mazzoli et al., 2001). P-Tr: Permo-Triassic siliciclastic beds underlying the ApulianPlatform carbonates. (d) Field photograph showing one of the studied outcrops. (e) General lithostratigraphy of the Apennine carbonate platform (studied part of the Cretaceousstratigraphy arrowed).

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(e.g. Mazzoli et al., 2001; Butler et al., 2004; Ascione et al., 2012;Iannace et al., 2012, and references therein). Significant oil reservesare trapped within the tectonically buried Apulian Platformcarbonates, characterized by broad, long-wavelength antiformsproduced by thick-skinned inversion (Shiner et al., 2004; Mazzoliet al., 2008, 2012). The studied carbonates are very similar, interms of age, lithology, facies, and rock texture, to the buried

Cretaceous reservoirs of the major oil fields (Val d’Agri, TempaRossa) of the Basilicata region (Fig. 2; Iannace et al., 2008; Vitaleet al., 2012), an area in which the Apennine fold and thrust belt ischaracterized by active, post-orogenic, upper crustal extension (e.g.Macchiavelli et al., 2012). The outcrop is located along a road cut onthe southern slope of Monte Faito, approximately 30 km SE ofNaples (Fig. 2) and consists of a 50 m-thick alternation of

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limestones and dolomite of Lower Cretaceous (Albian) age. Lime-stones are generally micrite-rich facies (mudstone, wackestone andmud-rich packstone) deposited in lagoonal marine and palustrineenvironments. Dolomite beds include medium- and coarse-grainedvarieties, both formed by early diagenetic replacement (Iannaceet al., 2008). Bed thicknesses range between 5 and 210 cm, witha mean value of 54 cm. Limestone, medium- and coarse-graineddolomite beds account for 24%, 46% and 30% of the exposedcarbonate succession, respectively. Minor marly and silicifiedlayers, also occurring within the studied succession, are not takeninto account in the present study.

4. Fracture data collection

Detailed studies based on high-quality datasets from fracturesystems from several reservoir analogues (Odling et al., 1999) haveshown that in layered rocks two different types of tensionaljoint sets may be identified, which exhibit different statisticalbehaviors:

Figure 3. Examples of hierarchical fracture systems

(i) ‘stratabound’ systems, in which fractures cut across mechan-ical layers, having terminations on the bedding joints;

(ii) ‘non-stratabound’ systems (Fig. 3), in which fractures do notterminate against bedding joints.

In stratabound systems spacing is regular and increases withmechanical layer thickness. According to several authors, averagespacing is a linear function of layer thickness (Price, 1966; Huangand Angelier, 1989; Narr and Suppe, 1991; Gross, 1993; Mandalet al., 1994; Gross and Engelder, 1995; Wu and Pollard, 1995;Narr, 1996; Pascal et al., 1997; Odling et al., 1999; Bai and Pollard,2000). In non-stratabound systems spacing values exhibita rather irregular spatial trend. Fracture aperture, denoted by b,shows a cumulative frequency e here defined as the number offractures per meter (fpm) having aperture greater than b (Guer-riero, 2012) e following a power law distribution at several scalesof observation (Ortega et al., 2006).

Our structural analysis has been carried out taking into accountthe distinction of the two mentioned types of fracture, in order to

of this study, at various scales of observation.

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characterize the specific statistical behavior of each set. The basictechnique of statistical data sampling adopted in this study con-sisted of measuring fractures along a line lying over the outcropsurface (scan line).

For stratabound fracture spacing, a scan line of about 3 m lengthhas been carried out for each mechanical layer over the wholeoutcropping succession (ca. 50 m thick). For each detected fracture,(i) attitude, (ii) distance from the scan line origin, and (iii)mechanical layer thickness have been recorded. Based on theacquired data set, statistical correlations between lithology, layerthickness and fracturing characteristics have been investigated.

The analysis of non-stratabound fracture systems has beencarried out over three scales of observation (details about multi-scale scan line analysis are provided in Guerriero, 2012; Guerrieroet al., 2011) by: (i) traditional scan lines (3e4 m long) on theoutcrop (Fig. 4a), (ii) micro-scan lines using a digital micro-camera(Proscope) with a magnification of 50� (Fig. 4b), and (iii) micro-scan lines by optical microscopy on oriented thin sections, ata magnification of 50� and 200� (Fig. 4c). A furthermicro-scan linehas been carried out on a drill core sample from the Tempa Rossa oilfield (Fig. 4d).

The analyzed dataset includes both filled and non-filled joints,viewed as a unique fracture population as they exhibit a statisticallyhomogeneous behavior.Microcracks, intended as fractures involvinga single or few neighboring crystals and showing low aspect ratio

Figure 4. Examples of fracture sampling techniques at different scales. (a) Traditional scan50�) images. (c) Micro-scan line on thin section (magnification 200�). (d) Micro-scan line

values, have been excluded from the analysis, as they do notcontribute to fluid flow in reservoir rocks, being usually sealed bymineralization and non-connected (Hooker et al., 2009; Dati et al.,2011). It is worth noting that, according to the definition above,microcracks are different from small fractures (or small joints). Thelatter, in fact, although showing aperture values of few microns,exhibit high aspect ratios (102 at least) and involve a larger number ofcrystals. From this point of view, those microcracks that are notsealede and therefore contribute to oil stocking e behave as matrixpores, being not directly connected to a fracture system.

The traditional scan lines have been carried out on mudstone(beds N. 1, 56, 107), medium- (beds N. 66, 71) and coarse-graineddolomite (beds N. 102, 118, 120). The following characteristicshave been recorded for each detected fracture: type of feature (vein,joint, shear fracture), distance from scan line origin, attitude,length, opening displacement, morphology, crosscutting relation-ships, composition and texture of fracture fill together withmechanical layer thickness of the hosting bed. Opening displace-ment (or kinematic aperture; hereafter simply termed aperture)has been recorded using the logarithmically graduated comparatorproposed by Ortega et al. (2006).

Digital micro-camera (Proscope) scan lines have been per-formed on mudstone, either directly on outcrop (bed 56) or onoriented polished rock samples of medium- (bed 66) and coarse-grained (bed 118) dolomite. Micro-structural analysis has also

line on outcrop. (b) Micro-scan line on digital micro-camera (Proscope; magnificationon peel from drill core sample from the Tempa Rossa oil field (magnification 50�).

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been performed by observation of three oriented thin sections onmudstone (beds 56, 107) and medium-grained dolomite (bed 66).Further micro-scan lines have been performed by optical micros-copy (at a magnification of 50�) on a drill core sample of medium-grained dolomite from the Tempa Rossa oil field (depth 5083 m,grain size 20e80 mm) and on an oriented sample from bed 66, usingpeels obtained from a polished surface of the rock (Fig. 4d). Sucha technique has provided higher quality images with respect to thedigital micro-camera (Proscope), at the same magnification level.This has allowed us to compare fracture data from a buried reser-voir rock and from its outcropping geological analogue. Micro-structural analysis was mainly aimed to reveal aperture andspacing distribution of joints. Therefore, only the following char-acteristics have been recorded for each detected structure: fracturetype, attitude (only by specifying the main set, previously detectedon outcrop, to which it belongs), distance from scan line origin, andaperture. It should be noted that field measurement of fractureaperture in reservoir analogues may be biased/affected by severalcauses (e.g. surface weathering, release of pre-existing confiningstress related with overburden, etc.). These issues are not tackled inthis work, as here we are mainly interested in deducing relationsbetween fracture aperture and host rock properties (e.g. grain size,bed thickness) and inferring statistical properties of aperturedistribution (type of distribution, scale dependence/independenceetc.), rather than quantifying the parameters of fracture attributestatistical distribution.

Finally, in order to study fault damage zone characteristics, an18 m long scan line has been performed, perpendicular to a verticalfault and across several mechanical layers.

5. Statistics and scaling analysis

The structural analysis was mainly aimed to study tensionalfractures (mode I fractures, i.e. veins and joints). In the study area,two main homogeneously distributed joint sets have been detec-ted. They are at a high angle to each other and orthogonal tobedding (Fig. 5).

5.1. Stratabound joint sets

The structural analysis carried out on stratabound fracturesystems has unraveled a regular spacing, mainly controlled bymechanical layer thickness (t), and aperture values in the order ofthe millimeters. The correlation analysis shows that mean spacing(s) is proportional to bed thickness (Fig. 6a). Lithology does notaffect significantly fracture spacing, as the ratio s/t (i.e. the pro-portionality coefficient) attains similar values for the three differentlithologies encountered in outcrop (Fig. 6b). Nevertheless, lithologyappears to be related to mechanical layer thickness. Thicknesscumulative distribution F(t) e calculated irrespective of lithology e

is well approximated by a lognormal law (Fig. 7a; the best-fitlognormal laws have been calculated using the methodsdescribed in Mazzoli et al., 2009, for joint spacing analysis). On theother hand, this differs significantly from the distributions calcu-lated for each lithology (Fig. 7b). In order to verify whether eachlithology exhibits a different distribution, a classical statisticalcompatibility test may be applied assuming that thickness showsa theoretical distribution F(t) (i.e., that calculated irrespective oflithology). For each thickness value ti belonging to a sample of sizen, the sampling distribution of the number y of determinationslesser or equal to ti follows a binomial probability law Fbin(y)(Dekking et al., 2005) characterized by a probability p ¼ F(ti). The tivalue is chosen, for each lithology, according to the KolmogoroveSmirnov criterion (i.e., that yielding the maximum disagreementbetween observed and theoretical cumulative distribution;

Dekking et al., 2005). For a given data set, the compatibilityhypothesis is discarded if y attains too large or small values for thechosen ti value, namely if it is: Fbin(y) < 0.05, or Fbin(y) > 0.95. Intwo cases e coarse dolomite and mudstone e the observeddistribution results to be incompatible (in probabilistic terms) withthat calculated based on all lithologies, thus confirming thatthickness distribution follows different lognormal laws for differentlithologies. The three distributions of Figure 7b exhibit substantialdifferences in standard deviation values, whilst the mean andgeometric mean ewhich is the most appropriate parameter whendealing with lognormal AVs e show small differences.

Our analysis has also pointed out a random spatial distributionfor the three lithologies (i.e., the composition of each layer isindependent from that of neighboring ones; Tables 1 and 2).Different lithologies occur according to the following proportions:medium dolomite 46%, coarse dolomite 30%, mudstone 24%.Comparing observed (Table 1) and theoretical random (Table 2)lithology distributions suggests that different lithologies arerandomly distributed within the analyzed succession.

5.2. Non-stratabound joint sets

Non-stratabound fractures detected in the studied outcropshow a uniform random spatial distribution, to which an expo-nential distribution of spacing values is associated (Guerriero,2012; Guerriero et al., 2009, 2010, 2011).

Concerning fracture aperture, recent studies (Ortega et al., 2006)suggest that the cumulative frequency (F) of joint apertures (b) iswell described by a power law for a wide range of scales ofobservation:

F�b� ¼ c$b�m

where c and m are experimental constants. Such a distribution forjoint aperture implies that, as the scale of observation is reduced,observed fracture density increases (e.g., fracture density detectedin thin section can be much greater than that detected by nakedeye). Thus measured fracture density is meaningful only if thefracture size lower threshold is specified (e.g., by considering thedensity of fractures whose aperture is greater than 0.2 mm).Actually, the power law is a theoretical distribution model thatcannot extend its validity infinitely at small scales. In fact, accordingto a power law distribution, fracture density should increasetoward infinity as aperture values decrease but, quite obviously, inthin section areas of intact rock, devoid of fractures, are clearlyvisible (Fig. 4d).

Our multi-scale analysis has confirmed that, in fine grainedrocks, the cumulative frequency of joint aperture follows a powerlaw over several orders of magnitude (in this study, aperturedistribution has been evaluated taking into account either miner-alized and non-filled joints; Fig. 8). Furthermore, our data revealMFD to be mainly controlled by grain size (Table 3). In mudstone,characterized by grain size below 10 mm, a power law can beapplied down to aperture values of a few mm. The maximumregistered fracture density reaches values up to the range of 1000e2000 (Fig. 9 and Table 3). In medium dolomite (grain size 20e80 mm) MFD values ranging from 250 to 350 fpm have beendetected by peel examination (magnification 50�) (Fig. 9 andTable 3). In this rock type, micro-scan line performed on thinsection (magnification level 200�) has not provided enough frac-tures to derive a meaningful statistical sample. In coarse dolomite,even at the relatively low magnification of 50�, micro-scan linesdid not provide a statistically meaningful sample. Anyway, a frac-ture density in the order of a few tens of fpm can be estimated from

Figure 5. Orientation data for all fractures measured from eight scan lines carried out in the studied carbonate succession for different outcrop surface exposures (contours of polesto fracture planes are shown on lower hemisphere, equal-area projections).

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outcrop (Fig. 8c). A diagram of fracture density vs. grain size (Fig. 9)based on experimental data (Table 3), shows a close approximationto an inverse proportionality.

The micro-scan line obtained by means of the peels techniqueon medium-grained dolomite (bed 66) has provided high-qualityimages and a sufficiently long sample line (ca. 8 cm), allowing usto investigate the upper limit for fracture density. In such imagescrystals and non-fractured host rock were clearly visible. Theaperture cumulative frequency (Fig. 10a) has been calculated bytaking into account fractures whose aperture is greater than0.2 mm for outcrop data, within the range 0.02e0.2 mm forfractures detected by Proscope, and below 0.02 mm for those

observed on peels. The cumulative frequency of apertures followsa power law up to a density value F* of about 180 fpm(corresponding to an aperture value b* z 0.02 mm; Fig. 10a).Beyond this value the distribution deviates from a power lawapproximating a horizontal trend, in correspondence to themaximum observable fracture density. It may be envisaged that, inthis case, such a horizontal trend is not related to truncation arti-facts (e.g. Guerriero, 2012; Ortega et al., 2006) but rather to a realdeviation of distribution from the power law.

In order to compare micro-scan line data from the studiedreservoir analogue with those from the previously mentioned drillcore, we have plotted on the same diagram (Fig. 10b) the aperture

Figure 7. Statistical distribution of mechanical layer thickness. (a) Cumulative distribution calculated regardless of lithology. (b) Cumulative distributions calculated for eachlithotype. Distributions are well described by a lognormal RV, although different lithologies show different bed thickness distributions.

Figure 6. Analysis of correlation between fracture spacing and mechanical layer thickness. (a) Least squares line calculated regardless of lithology. (b) Best-fit line calculated foreach lithotype. The correlation coefficient attains similar values for the four diagrams, thus suggesting that spacing is mainly controlled by bed thickness and is independent oflithology.

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Table 1Lithology proportions of adjoining strata. The value into each cell is given by thenumber of strata with lithology denoted by the column to which it belongs,preceded by a bed with lithology denoted by the row, divided by the overall numberof strata (for example the value 0.12 in the column mudstone and row mediumdolomite means that 12% of detected strata are mudstone preceded by mediumdolomite).

Coarse dolomite Medium dolomite Mudstone

Coarse dolomite 0.12 0.11 0.07Medium dolomite 0.08 0.26 0.12Mudstone 0.1 0.09 0.05

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cumulative frequency for the drill core sample and that obtainedfor bed 66, which shows a similar grain size. The two data sets areclearly consistent, pointing out how fracture density values aremainly controlled by grain size at micro-scale. The presentedresults provide useful insights into the role of grain size incontrolling the lower validity limit of the power law describing theaperture cumulative frequency, as well as themaximum observablefracture density. The power law distribution is related to the self-similar geometry of joints. In a statistically homogeneousmedium subjected to a statically homogeneous stress field, self-similarity can be interpreted as a reproduction of the fracturingprocesses in a similar fashion at different scales. On the other hand,at the crystal scale themedium is heterogeneous and so is the stressdistribution. Thus, at this scale of observation, fractures are notcharacterized by self-similarity; a maximum fracture density (MFD)can be defined, as well as a minimum mean spacing (which is theinverse of the MFD). Our results also reveal an MFD increase asgrain size decreases.

5.3. Fault zones

The fault analyzed in this study is vertical and characterized bydip-slip kinematics (extensional with respect to moderately Ndipping bedding). An approximately 4 m thick damage zone isassociated with the fault (Fig. 11a and b). Fracture orientationanalysis (Fig. 11c) illustrated the occurrence of two main fracturesets: (i) a vertical one, roughly parallel to the main fault, and (ii)a set orthogonal to bedding. The latter includes regional joints (i.e.,‘background’ fractures characterizing the whole succession) thathave been widely analyzed in the scan lines described in theprevious sections. In this section we focus on the vertical fractureset, which is clearly associated with the fault. This assumption isbased on the fact that fracture density approaches 0 fpm in the hostrock a few meters away from the fault (Fig. 11b. The analysis hasbeen carried out exclusively within the outcrop, on fracturesdetectable by naked eye and characterized by angles of dip >80�.Local fracture density has been estimated according to thefollowing method: for every five sampled fractures along the scanline, local fracture density is calculated as the ratio 5/D, where D isthe distance between the first and the 5th fracture. This allows toobtain estimates based on sample of the same size (in terms offracture number). Apart from a peak value of 120 fpm, which is

Table 2Theoretical distribution of lithology proportions for consecutive strata. The theo-retical percentages are obtained by the product of the percentages associated withindividual lithologies (for example, the value 0.11 in the cell mudstone-mediumdolomite is given by the product between 0.24 and 0.46, which are the proportions ofmudstone and medium dolomite, respectively).

Coarse dolomite Medium dolomite Mudstone

Coarse dolomite 0.09 0.14 0.07Medium dolomite 0.14 0.21 0.11Mudstone 0.7 0.11 0.06

associated with the cataclasite occurring within a ca. 10 cm thickfault core, local fracture densities of more than 50 fpm have beendetected (Fig. 11b).

The spatial distribution and degree of fracture clustering may beevaluated by means of the variation coefficient cv ¼ s/S (where s isthe standard deviation and S is the mean spacing value; Gillespieet al., 1999, 2001; Gillespie, 2003). In our instance, a value ofcv ¼ 2.28 underlines the substantially clustered spatial distributionof fault damage zone-related fractures.

6. Characteristics of the observed permeable structures

The statistical results illustrated in the previous sections, inte-grated with the results obtained by previous studies (e.g. Odlinget al., 1999; Massonnat and Viszkok, 2002), allow the distinctionof four hierarchical permeable systems, each characterized bya specific statistical behavior (Fig. 3). They are:

(i) fault zones and associated fracture sets,(ii) stratabound joint sets(iii) non-stratabound joint sets,(iv) non-fractured host rock.

In the following sections, the main features of each system willbe outlined, emphasizing their relevance to the hydraulic proper-ties of the host rock. The significance of this discrimination for theconstruction of numerical models will be subsequently discussed.

6.1. Fault zones

Fault zones may represent conduit or barrier at large scale. Thedominant behavior as conduit or barrier is controlled by thethickness and lateral continuity of two main fault zone compo-nents: the high-permeability damage zone, and the low-permeability fault core (e.g. Caine et al., 1996). Commonly faultattributes (such as length or displacement) as well as parameters(e.g., spacing or aperture) characterizing associated fracture setsshow self-similarity over several scales of observation (e.g.,Mandelbrot, 1983; Gudmundsson, 1987; Heffer and Bevan, 1990;Barton and Zoback, 1992; Gillespie et al., 2001; Barton, 1995; Celloet al., 1998; Odling et al., 1999; Cello et al., 2000, 2001; and manyothers). Fractures related to fault damage zone are characterized bya clustered spatial distribution (in our instance cv ¼ 2.28). Conse-quently, regions of several meters of tens of meters width occur atoutcrop scale, in which these fractures show very low frequenciesor are absent. The usually limited thickness and low continuity offault core-related cataclasites compared to well-developed, highfracture density damage zones characterize clustered fracture setsassociatedwith the latter as the first-order permeable system in thestudied carbonates.

6.2. Stratabound joints

At the meter scale, stratabound joints, together with beddingjoints, form a hydraulic, well-connected network conveying fluidstoward the fault system. Such a fracture network subdivides therock into roughly parallelepiped (Odling et al., 1999), centimeter- tometer-sized elementary blocks. The stratabound joint system ischaracterized by a significantly lower permeability with respect tofault systems, being characterized by lower fracture density values.As opposed to damage zone-related fracture sets, the strataboundjoint system is ‘size restricted’ (i.e., scale dependent), and itsgeometry depends on the distribution of mechanical layer thick-ness as well as lithology.

Figure 8. Cumulative frequencies of fracture apertures, obtained by the integration of outcrop- and micro-scale scan line data on: (a) mudstones (bed 56), (b) medium-graineddolomite (bed 66), and (c) coarse-grained dolomite (bed 118). The curves (a) and (b) follow a power law over several orders of magnitude.

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134 125

6.3. Non-stratabound joint system

At the scale of the elementary blocks (defined by strataboundfractures) and down to the crystal scale, non-stratabound jointsconstitute a pervasive and capillary network allowing fluids con-tained within the rock mass (inside pores and mircrofractures) toflow toward the more permeable fracture systems. In fact, fluidscontained within pores and/or microcracks reach this fracturesystem through short paths across non-fractured matrix, whoselength depends on fracture spacing. Since the cumulative frequencyof aperture values is well described by a power law over severalscales of observation, and the spatial distribution is uniformlyrandom, it follows that the non-stratabound joint system is

characterized by self-similarity (Ortega et al., 2006; Guerriero et al.,2011). According to Odling et al. (1999), the permeability associatedwith the fracture system within each elementary block of rockdepends on the connectivity of non-stratabound fractures. Such anaspect has been widely analyzed within the framework of thestatistical theory of percolation (Odling et al., 1999, and referencestherein). Synthetically, permeability is mainly controlled by thecumulative frequency of fracture lengths, and by the reciprocalorientation of fracture sets. Nevertheless, taking into account thatjoint apertures are distributed according to a power law andconsidering all small fractures, fracture density always reaches highvalues and connectivity may also attain high values. Therefore,permeability of the fractured rock is mainly controlled by the

Table 3Detected MFD values for several lithologies.

Lithology Method Grainsize (mm)

Max. fracturedensity(fractures/m)

Bed 56 set 1 Mudstone Thin section (200�) <10 mm 1800Bed 56 set 2 Mudstone Thin section (200�) <10 mm 1350Bed 107 set 1 Mudstone Thin section (200�) <10 mm 1300Bed 107 set 2 Mudstone Thin section (200�) <10 mm 1000Bed 66 set 1 Medium

dolomitePeels (50�) 25 mm 250

Bed 66 set 2 Mediumdolomite

Peels (50�) 25 mm 360

Bed 118 set 2 Coarsedolomite

Outcrop 120 mm 30

Bed 120 set 2 Coarsedolomite

Outcrop 140 mm 30

Figure 10. (a) ‘True’ cumulative frequency of fracture apertures on medium-graineddolomite (bed 66) at several scales. (b) Integrated data from drill core sample andcopping out reservoir analogue (Bed 66) multi-scale scan line, on rocks showing equalgrain size. Beyond the density value F* e to which corresponds the aperture value b* ethe power law is no longer valid and the distribution tends to assume an horizontaltrend up to the maximum fracture density (MFD) value.

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134126

statistical distribution of fracture apertures. It should be noted that,in terms of statistical analysis, filled and non-filled joints are dealtwith as a unique fracture population as they exhibit a statisticallyhomogeneous behavior. Nevertheless, they clearly play differentroles from the hydraulic point of view. Thus, once the statisticalproperties of the non-stratabound fracture population (includingfilled and non-filled joints) have been defined, the occurrence offilled joints needs to be quantified in order to assess the proportionof non-sealed joints. This evaluation can be carried out by applyingthe criteria illustrated by Laubach (2003).

A further issue needs to be addressed concerning the hydraulicbehavior of the non-stratabound fracture system: in carbonaterocks characterized by low primary porosity values, fractureporosity can provide a relevant fraction of voids volume. In theseconditions, the oil stocked within the fractures can be released ina significantly shorter time than that contained in the pores,because it does not flow through the porous matrix, whosepermeability is several orders of magnitude lower than that offractured rock.

6.4. Non-fractured porous host rock

The non-fractured portions of the host rock represent thelowest-scale (i.e., crystal-sized) permeable system. Such a systemdoes not show self-similarity because its properties dependmainly,in the case of interparticle porosity, on the size and shape of grains.The mean path length for a fluid to reach the fracture network from

Figure 9. Scatter diagram showing MFD vs grain size, for different carbonate rocks.The experimental data shows a close approximation to an inverse proportionalityrelationship. Note that the coefficient has been calculated according to length unitsutilized within the diagram (mm; m�1).

rock pores is equal to half of the mean (minimum) spacing.Therefore, mean path length increases with grain size. Neverthelessthe time tm spent by fluid particles within the porous matrix duringflow depends on such path length and on matrix permeability. Thislatter varies as a function of grain size and, although the relation-ship between (primary) permeability and grain size is still not fullyunderstood, it is reasonably well established that permeabilitydecreases significantly as grain size decreases (e.g. Nelson, 2001). Inorder to evaluate the time tm (the quantification of which goesbeyond the scope of this paper), the dependence of both perme-ability and MFD on grain size distribution should be determined byan appropriate correlation analysis. What is important in thiscontext is that the non-fractured host rock system is generallycharacterized by significantly lower permeability values withrespect to the previously described fracture systems.

7. On the behavior of naturally fractured carbonate rocks

7.1. Fluid paths within the joint network

Stratabound and non-stratabound joint networks display ratherdifferent geometries. Therefore, they play different roles incontrolling fluid flow within a rock volume. The considerationsillustrated in this section are funded on the basic assumption thatthe fracture system is significantly more permeable than the non-fractured matrix. Therefore, these concepts cannot be applied tothe limit case in which matrix permeability compares with thatassociated to the fracture system, as well as to the case of vuggyreservoirs, which are not dealt with in this paper. Stratabound

Figure 12. (a) Path of fluid particles within stratabound fracture network. The pathlength, between two generic points A and B, has the same order of magnitude of theEuclidean distance AB. (b) Path of fluid particles within non-stratabound fracturenetwork. Occurrence of filled joints can confer notable tortuosity to the path of fluidswithin this network and, therefore, the path length between two generic points A andB may become much greater than the Euclidean distance AB.

Figure 11. (a) Multi-layer scan line across a fault zone. (b) Diagram of fracture density vs. distance referred to fault damage zone-related fractures (showing angles of dip > 80�). (c)Orientation data for vertical fractures (contours of poles to fracture planes are shown on lower hemisphere, equal-area projections).

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134 127

fractures, together with bedding joints bounding mechanicallayers, form a well-connected network exhibiting a regular geom-etry. The path of fluid particles within these fractures also showsa regular geometry and its length, between two generic points Aand B, has the same order of magnitude of the Euclidean distanceAB (Fig. 12a). In our study area, the geometry of the strataboundjoint network and the related fluid path is mainly controlled by bedthickness and its spatial distribution. On the other hand, non-stratabound joints form an irregular network showing a markedspatial variability in terms of fracture spacing and aperture.Furthermore, the fluid path within this network is strongly affectedby the occurrence of joints sealed by mineralization (filled joints),which may result in highly tortuous flow patterns and/or nonegligible flow. Therefore, the fluid path length between twogeneric points A and B may become much greater than theEuclidean distance AB (Fig. 12b). Grain size significantly affects thehydraulic behavior of non-stratabound joint network due to thefollowing reasons: (i) MFD grows as grain size decreases, (ii) theproportion of filled joints reaches higher values for the smallestfractures, and (iii) the non-fractured host rock (matrix) perme-ability decreases as grain size decreases. For reasons (i) and (ii), infine grained rocks fluid paths may become markedly tortuous (orcompletely interrupted) and its irregularity propagates itself to thesmallest scales. Thus the path length between two points A and Bmay exceed the distance AB by several orders of magnitude. It isworth noting that, for a given difference between the hydraulichead relative to points A and B, the average gradient (and hence themagnitude of fluid velocity) is inversely proportional to the pathlength AB. Therefore, as grain size decreases, the time spent by fluidparticles within the non-stratabound joint network increases dueto both increasing path length and decreasing average gradient.

Although the average path length of fluids within the matrix isinversely proportional to MFD, the time spent by fluid particlesthrough the matrix depends also on its permeability. The latter, inturn e for the reason (iii) may attain very low values in fine grained

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134128

rocks as it decreases with grain size. Therefore, the time spent bya fluid particle tomove from a pore to the nearest fracturemay evenincrease as fracture spacing (and grain size) decreases. In this latterinstance, fluids move slower than they do within a coarse-grainedrock, within either matrix or micro-fracture networks. This mayexplain the different performance of (fine grained) limestone and(medium/coarse-grained) dolostone reservoirs in releasing oil,even when these rocks show similar values of porosity.

7.2. Simple model of porous fractured medium characterized byhierarchical permeable structures

In order to analyze the hydraulic behavior of fractured stratifiedrocks, the results of numerical dynamic simulations carried out ona simple two-dimensional model including a hierarchical system ofpermeable structureswill bedescribed (Fig.13). For simplicity reasons,in this and following sections we assume a single-phase Darcy flowwithin fractures. Nevertheless the model and related equations canalso be modified in order to consider a multi-phase non-Darcy flow(e.g. Altinos and Onder, 2008). The studied body has a square sectionwith a 10 cm side. It includes: (i) a highly permeable system ewhichwould emulate the stratabound FN e composed of two orthogonalfracture sets, characterized by aperture values of 1 mm and constantspacing of 24 mm; (ii) a lower permeability system e emulating thenon-stratabound FNe constituted by two orthogonal sets of fractureswith aperture and spacing equal to 0.1 mm and 4 mm, respectively;and (iii) the non-fractured matrix, with a permeability value of3�10�4m/s. The simulation has beenperformed byfinite differences,according to the same boundary conditions adopted in Section 2.2.

Figure 14 illustrates the trend of the hydraulic head related topore pressurewithin the studied body (FN andmatrix) as a functionof time. In Figure 15 h1, h2 and h3 are the hydraulic heads related topore pressure within stratabound, non-stratabound fractures andmatrix, respectively. h1, h2 and h3 tend to show a linear trend, alongthe flow axis, after a sufficiently long time, i.e. when the relatedpermeable system reaches steady-state flow conditions. The firstsystem reaching steady-state conditions is that characterized bythe highest permeability value (fractures with aperture of 1 mm),as it shows the fastest decrease of hydraulic head. The medium-sized FN characterized by a fracture aperture of 0.1 mm showsa delay in reaching steady-state flow conditions, while the non-fractured host rock needs an even longer. Therefore, within the

Figure 13. Two-dimensional system, including hierarchical permeable structures,utilized for numerical simulations.

transient stage, the function h2 is characterized by “domes” boun-ded by fractures with largest aperture (1 mm) and, consequently,the fluid moves toward this latter permeable system. At a smallerscale, the function h3 exhibits convexities bounded by minor frac-tures (i.e. those with aperture of 0.1 mm). Consequently, the fluidmoves from the matrix toward the minor FN and from the lattertoward the major FN (i.e. the fluid flow occurs from the less to themore permeable system, following the hierarchical order ofpermeable structures).

In the following, reverse boundary conditions for the studiedelementary volume are considered, in which its lower side expe-riences a pore pressure increase (Fig. 16):

h ¼ 0; at each grid point; for t ¼ 0;

h ¼ 0; along the upper side; andh ¼ 0:01 along the lower side; for t > 0:

Again, the most permeable system exhibits the fastest variationfor the related hydraulic head. Thus, h1 increases more rapidly thanh2 which, in turn, shows a faster increase than h3. As a consequence,opposite to the case of pressure depletion, the fluid flows from themore to the less permeable system. The dynamic response of anelementary volume of fractured porous medium under theseboundary conditions (pore pressure increase) is of relevance for theapplication of oil recovery techniques based on the injection of fluid/gas in the subsurface. As a consequence of the pore pressure increaserelated to injection,fluids are expected tomove fromthestrataboundtoward the non-stratabound FN (i.e. inwards each block; Fig. 16),thereby counteracting the recovery of oil stored within the non-stratabound FN and non-fractured host rock. A further interestingobservation concerns the flow rate across a generic surface withinthe illustrated two-dimensional model. It includes contributionsfrom both the stratabound and the non-stratabound FN, and acrossmatrix. In the illustrated simulations, the flow rate has been calcu-lated for different surface attitudes. In all cases, its value is equal tothat through the major fracture system (i.e. that with aperture of1 mm), while that along minor fractures and across the matrix isnegligible. Therefore, with a good approximation, we can assumethat neighboring elementary volumes of fractured rock exchangemutually fluids exclusively through the more permeable fracturesystem. This observation will be useful in building up a numericalmodel for fluid flow, as illustrated in the following section.

In this section we have considered a rather simple model con-sisting of a two-dimensional porous body containing two perfectlyparallel fracture sets, each one exhibiting constant spacing andaperture. The foregoing discussion has only illustrative purposes,aimed at explaining some peculiar characteristics of the dynamicalbehavior of porous bodies containing hierarchical fracture systems.In the following sections we will refer to more realistic three-dimensional models, including irregular/complex hierarchicalfracture systems.

7.3. Approach to flow modeling

A further main purpose of this work consists in providinga model of discrete calculus which takes into account the severalfracture typologies detectable at different scales of observation(from kilometer to sub-millimeter scale), utilizing a gridding widthwhich allows one performing reservoir simulations in a reasonabletime. To this purpose we will follow a similar approach to thatleading to the formulation of the dual-porosity model (Warren andRoot, 1963), adapted to the structure typologies commonly occur-ring in carbonate rocks.

Figure 14. Diagram showing the trend of the hydraulic head h, within the analyzed system, for several time values, where T0 ¼ 2 � 10�10 s. The diagrams point out how the functionh shows, for several time values, some convexities or domes bounded by larger fractures, whilst the detail image of a single cell (bounded by mayor fractures) allows to recognize, ata smaller scale, similar convexities bounded by smaller fractures. As fluid particles move they self parallel to the gradient of h, it follows that fluid flow occurs first from matrix tosmaller fissures and then from these latter ones to larger fractures, following the structure hierarchy.

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134 129

Let us imagine we want to simulate the behavior of a carbonatereservoir with a parallelepiped shape. Suppose that the parallele-piped has sides of ca. 500 m and is partitioned into elementaryvolumes of rock having a few meters long side (e.g. 5 m). Eachelementary volume will include several rock layers and fracturetypes. Let us consider, for the time being, blocks which are notintercepted by faults. Therefore, three permeable systems occurwithin the block: stratabound joints, non-stratabound fracturesand the non-fractured host rock (matrix). Let us consider the flowrate across the boundary surface of the generic elementary volume.This is provided by the sum of: (i) the flow rate across the strata-bound fracture network, (ii) the flow rate across the non-stratabound joints and (iii) the flow rate within matrix. Byassuming that the first of these quantities is much greater than the

other two, i.e. that neighboring elementary volumes transfer fluidsto each other exclusively by means of the stratabound jointnetwork (this being characterized by the highest permeabilityvalue; see Section 7.2.) the continuity equation attains thefollowing form:

F1 þ F12 ¼ �C1vP1vt

; (1)

where P1 is the mean pore pressure within stratabound joints(calculated according toWarren and Root, 1963); F1 is the flow rateper unit volume across stratabound joints; F12 is the flow rate perunit volume exchanged between stratabound and non-strataboundfractures within the considered volume (which is positive when

Figure 15. Diagram showing the trend of the heights h1, h2 and h3 separately and at several scales of observation, within the analyzed system, along the time. Diagrams illustratehow larger fracture system exhibit a more rapid pressure depletion than less permeable systems, constituted by small fissures and matrix. When T/N, fluid flow reaches a steady-state condition and functions h1, h2 and h3 show a linear trend along the flow axis, belonging to the same plane surface (note that the diagrams of these functions have differentscales).

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134130

outgoing from the former to the latter system); C1 is a compress-ibility coefficient depending on the compressibility of rock and fluid(Warren and Root, 1963). In Eq. (1) we have assumed that the flowrate of exchanged fluid between stratabound joints and matrix isnegligible in comparison with F1 and F12. Denoting as h1 thehydraulic head related to P1 and under the hypothesis that flowacross the stratabound fracture network follows the Darcy law, theexpression of F1 for a small volume is given by: F1 ¼ �V$KVh1,where V is the well-known differential vector nabla operator and Kis the permeability tensor associated with this network. ThereforeEq. (1) becomes:

�V$KVh1 þ F12 ¼ �C1vP1vt

; (1a)

The flow rate per unit volume F12 ewhich, for congruence withEq. (1), is positive when ingoing into the non-stratabound fracturesystem e produces an increase (or decrease if negative) of averagepore pressure within the non-stratabound joint network, P2. Thetotal flow rate per unit volume ingoing into this fracture network isprovided by the sum of F12 and that exchanged with the non-fractured rock, F23. Therefore it results:

F12 þ F23 ¼ C2vP2vt

; (2)

Analogously, the flow rate per unit volume F23 e which ispositive if outgoing from the matrix e causes a pore pressuredepletion within the matrix denoted by P3:

Figure 16. Diagram showing the trend of the hydraulic head for reverse boundary conditions. Note how excessive increase of stratabound fracture pore pressure (e.g. asa consequence of enhanced oil recovery by fluid injection) may cause fluid injection within each block, thereby counteracting the recovery of oil stored within the non-strataboundFN and non-fractured host rock.

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134 131

F23 ¼ �C3vP3vt

; (3)

Eq. (1) is valid also for those blocks that are intersected by faults,in which P1 denotes the average pore pressure within strataboundjoint and faults. It should be noted that, in this case, the blockpermeability tensor may be markedly anisotropic.

Although the illustrated model shows some conceptual analogywith the dual-porosity model proposed byWarren and Root (1963),it also exhibits relevant differences. In Eq. (1), the termF12 does notdenote the fluid flow exchanged between fractures and (non-frac-tured) matrix, as it does instead in the dual-porosity model. Rather,it expresses the fluid exchanged between the stratabound jointnetwork and the hierarchically low-grade system (i.e. the non-stratabound fracture network). Furthermore Eq. (2), involving theexchanged fluid between non-stratabound fractures and the non-fractured porous rock, is not included in the dual-porosity model.

7.4. Example of difference equation scheme for the fluid flow

Let us consider a mesh grid whose nodes have coordinatesindividuated by indexes i, j and k along the x, y and z axes,respectively. Let the spacing between the nodes be equal to D alongthe three directions. Denoting by B the considered elementaryblock, whose volume is equal to VB, andwith A its boundary surface,the flow rate per unit volume F1 is provided by:

F1 ¼ 1VB

$

Z

A

v.$dA

.¼ 1

VB$

Z

A

KVh1$dA.; (4)

where h1 is the average hydraulic head within the strataboundfracture system, v

. is the velocity vector associated to h1 gradientaccording to Darcy law and K is the permeability tensor associatedto this fracture system. Let us assume that the principal directions

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134132

of this latter tensor are parallel to the coordinate axes. Then theapproximated expression for the flow rate Fx across the boundarysurfaces orthogonal to the x axis is provided by:

Fxz

0B@�Kxiþ1jk þ Kxijk

�2

$

�hiþ1jk � hijk

�D

��Kxi�1jk þ Kxijk

�2

$

�hijk � hi�1jk

�D

1CA$D2;

(5)

The term ðhiþ1jk � hijkÞ=D in Eq. (5) expresses the approximatedvalue of the derivate at the middle point between those of indexes iand i þ 1. Analogously, the term ðKxiþ1jk þ KxijkÞ=2 represents theapproximated expression for the x component of the permeabilitytensor, defined as the mean of its value at the above mentionedpoints. In a similar way, the expressions for the flow rates across theother boundary surfaces Fy and Fz, are given by:

FyzD2

��Kyijþ1kþKyijk

�$�hijþ1k�hijk

���Kyij�1kþKyijk

�

$�hijk�hij�1k

��;

(6)

FzzD2

��Kzijkþ1þKzijk

�$�hijkþ1�hijk

���Kzijk�1þKzijk

�

$�hijk�hijk�1

��;

(6a)

Let us assume now that F12 ¼ F12(h2 � h1) and, in particular,that this function is (with a good approximation) linear: F12 ¼ l12(h2 � h1) (Warren and Root, 1963), where l12 is a constantdepending on system geometry, and h2 and h1 are the mean valuesof the hydraulic head (where the mean is obtained over theconsidered elementary volume) corresponding to pore pressurevalues P2 and P1 respectively. In a similar way, let us assume to be:F23 ¼ l23 $ (h3 � h2). Under these hypotheses, Eqs. (1)e(3) attainthe following form:

1

2D2

��Kxiþ1jkþKxijk

�$�h1iþ1jk�h1ijk

���Kxi�1jkþKxijk

�

$�h1ijk�h1i�1jk

�þ�Kyijþ1kþKyijk

�$�h1ijþ1k�h1ijk

�

��Kyij�1kþKyijk

�$�h1ijk�h1ij�1k

�þ�Kzijkþ1þKzijk

�

$�h1ijkþ1�h1ijk

���Kzijk�1þKzijk

�$�h1ijk�h1ijk�1

��

þl12$�h2ijk�h1ijk

�¼�C1g

vh1ijkvt

;

(7)

l12$�h2ijk � h1ijk

�þ l23$

�h3ijk � h2ijk

�¼ C2g

vh2ijkvt

; (8)

l23$�h3ijk � h2ijk

�¼ �C3g

vh3ijkvt

; (9)

where g denotes the weight per unit volume of the consideredfluid. The first term in Eq. (7) expresses the flow rate per unitvolume (Fx þ Fy þ Fz)/D3. The time derivates in Eqs. (7)e(9) can bedeveloped according to an appropriate difference scheme (e.g.forward derivate, backward, central, etc.). Therefore, these equa-tions provide a linear system of 3N equations over a grid netincluding N nodes. Note that, before solving this system, theconstants Kijk and lmn need to be preliminarily calculated, for eachblock, by appropriate criteria or algorithms. Providing such

algorithms is beyond the scope of this contribution, which isintended to provide a general framework for subsequent specificapplications. The illustrated difference scheme would only repre-sent an example of the approach that may be used to obtainnumerical solution for Eqs. (1)e(3). However, different approachesand discretization criteria are available in the literature (e.g. Jing,2003).

8. Discussion and concluding remarks

Structural analysis unraveled a hierarchical organization ofpermeable structures within layered carbonate rocks. Thesestructures can be ordered, according to both decreasing scale ofobservation and permeability, as follows: (i) fault network, (ii)stratabound FN, (iii) non-stratabound FN, and (iv) non-fracturedhost rock (matrix). The hydraulic behavior of hierarchical struc-tures cannot be modeled in terms of parallel or in series permeablemedia. Fluid flow follows the hierarchy of structures as (with a goodapproximation) the fluid is yielded/received by one of thesesystems exclusively to/from that of the next higher order. Thematrix exchanges fluids with non-stratabound FN, whereas theflow amount exchanged with the stratabound FN or faults isnegligible. Therefore, the dynamic behavior of thematrix is coupledwith that of the non-stratabound FN. The dynamic behavior of thislatter, accordingly, is coupled with that of stratabound fractures,which, in turn, interact with the fault network.

The proposed model represents a step ahead within the contextof carbonate reservoir characterization as it provides the correctapproach to structural analysis carried out by the integration ofdifferent information fromvarious field data sources. Fault networkcharacterization can be reasonably carried out on the basis ofcriteria available in the literature and discussed in Section 2.1. Thecharacteristics of the stratabound FN can be defined by an inte-grated study of (i) well data (mainly BHI data), from which infor-mation about layer thickness and fracture intensity can be acquired,and (ii) geological analogues, which may be used to characterizethe relations between joint spacing/aperture and host rock featuressuch as bed thickness, lithology, petrophysics, etc. In a similar way,the non-stratabound FN can be characterized bymeans of well coreanalysis, integrated with the study of reservoir analogues.

The analytical model presented in this paper provides the firststep toward the construction of a full numerical model allowingone to calculate the pore pressure within fractures, at several scalesof observation, in a reasonable time. The next step consists inderiving appropriate algorithms aimed at calculating the coeffi-cients described in Section 7.4, based on statistically-treatedstructural data, and will be object of a future research project.The present model also allows one obtaining a better under-standing of the hydraulic behavior of fractured porous rocks. First,the suggested approach to numerical modeling removes the errorassociated with substituting elementary volumes of porous frac-tured rock with homogeneous ones, showing the same perme-ability value. Such an error is not quantifiable a priori and mayattain relevant values. Furthermore, the numerical modelingapproach proposed here may allow to calculate the dynamicresponse and the amount of released/stocked fluid by each of thepermeable sub-systems occurring within each elementary block,along with time, as well as the ‘true’ path of fluids within the rock.Indeed, once the values of variables h1, h2 and h3 are calculated foreach block, the fluid path may be reconstructed bymeans of a moreclosely spaced gridding, applied only to appropriately selectedelementary volumes. This information is of fundamental interest inthe field of reservoir management, as it allows an appropriateplanning of extraction activities and enhanced oil recovery. Theresults obtained in this study may find application also in further

V. Guerriero et al. / Marine and Petroleum Geology 40 (2013) 115e134 133

research fields, such as (e.g.) the study of geothermal reservoirs, ofcontaminant migration of nuclear and/or toxic industrial wastesfrom underground storage repositories, as well as various types ofenvironmental studies involving ground fluids.

Acknowledgments

Thoughtful and constructive reviews by JMPG anonymousreviewers, together with the comments by Editor John Tipper,substantially improved the paper. Reservoir engineer at Shell ItaliaE&P, Rini Verbruggen, is also thanked for critical reading of themanuscript. Financial support by Shell Italia E&P for carrying outresearch on fractured carbonate reservoirs is gratefully acknowl-edged. This is publication N. 1 of the Italian Ministry of Universityand Research Project PRIN2009 N73SR4 (Responsible: A. Iannace).

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