impact of multi-set fracture pattern on the effective permeability of fractured porous media

$: Impact of multi-set fracture pattern on the effective permeability of fractured porous media$
Impact of multi-set fracture pattern on the effective permeabilityof fractured porous media

Amanzhol Kubeyev n

Imperial College London, South Kensington Campus, Exhibition Road, London SW7 2AZ, UK

a r t i c l e i n f o

Article history:Received 30 November 2012Accepted 14 October 2013

Keywords:permeabilitymulti-setsnumericalfractureanisotropy

a b s t r a c t

In fractured porous media, a significant contribution to flow is made through fracture–matrix networks.These networks create flow channels which play an important role in overall fluid transport. Previousfracture simulation studies were mostly done with stochastic patterns. In contrast, the analysis ofgeomechanically generated discrete fracture data sets that closely render naturally occurring systems,whilst account for flow through matrix and fractures simultaneously was presented in this study.The used propagation algorithm provides similar results to physical experiments. The impact of multiplesuperposed fracture sets within geological formations, created by multiple deformation events, on theeffective permeability has been analysed. A 2D fracture-matrix medium was simulated and effectivepermeability computed using a finite element based method. Specifically, the impact of certain detailedfracture characteristics, such as density, mean length, spacing, connectivity and matrix permeability on theflowwas measured. Results indicated the increase of effective permeability in multiple-sets of fracture sets.Fractures superimposed at different angles to the main set, parallel to the flow, increase connectivitybetween main flowlines and neighbouring fracture clusters. Thus, an increase in connectivity leads tohigher effective permeability of media and increased probability of percolation. Permeability anisotropywas also analysed in this study. An expression for the characterisation of anisotropic effective permeabilitywas proposed and simulated permeability was compared to analytical predictions.

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1. Introduction

Studying fractures can help us understand their influence on fluidtransport in differentially fractured geological formations (Shen et al.,2008). They have been shown to have a great impact on the porousmedia's overall effective permeability (Bogdanov et al., 2007).Numerical models are effective tools for the simulation and investi-gation of such structures. Utilizing robust numerical models for flowsimulation and realistic geometric representation of the fracturedmedia we aim to enhance flow prediction capability and rendernaturally occurring systems.

There are various modelling approaches used to simulate flowthrough fractured formations. The Equivalent Porous Medium, (EPM),(Barenblatt et al., 1960; Pruess et al., 1986) is a single continuumporous medium model, where permeability is a sum of fracture andporous medium permeability. Dual Continua Models, such as DCDM(Barenblatt et al., 1960; Warren and Root, 1963) contemplate con-nected fractures and disconnected matrix models. These have restric-tive assumptions such as no-flow in rock matrix, interconnected

fractures, and averaged grid block fracture and matrix properties.Another approach is the Discrete Fracture Network, (DFN),(Witherspoon et al., 1980; Dershowitz and Einstein, 1988), in whichthe porous medium is not represented, and all flow is transmittedthrough fractures. Network-based effective permeability models areefficient, but often less accurate (Bogdanov et al., 2007). These arebest suited for low permeability media (Long et al., 1985), such asintact crystalline rock, intact shales and halite, where permeabilitymay reach 10�8 mD (Ingebritsen and Sanford, 1998). In contrast, forflow where matrix has a significant contribution, both flow throughfractures and matrix must be accounted for. A model that capturesthis behaviour is the Discrete Fracture and Matrix model (DFM), inwhich the flow is simulated through the matrix and fracturessimultaneously (Matthai et al., 2007; Paluszny and Matthai, 2010).Effective permeability is a crucial conductivity parameter (Ingebritsenand Sanford, 1998), which measures the ability of medium to transmitfluids. It is an internal physical property not dependant on macro-scopic boundary conditions and is statistically homogeneous in largescales (Renard and Marsily, 1997). It is measured analytically, numeri-cally, and by correlating it with different physical properties, e.g.porosity. Heterogeneity may occur on many scales, leading to extremepermeability fluctuations, thus, in many cases the permeability is notaccurate (King, 1989). For numerical studies it is essential to include

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Please cite this article as: Kubeyev, A., Impact of multi-set fracture pattern on the effective permeability of fractured porous media. J.Petrol. Sci. Eng. (2013), http://dx.doi.org/10.1016/j.petrol.2013.10.022i

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heterogeneities in the computation of flow conductivity. Conventionalsimulators often neglect to incorporate fractures as discrete entities,leading to inadequate results (Philip et al., 2005). For instance, toobtain a reasonable production history match, gridblock permeabilityis often increased (Philip et al., 2005). In some cases, it was foundthat flow in simulated models was less than that found in a naturalfractured network, where naturally occurring fracture patterns arearranged in a way that improves connectivity and hence permeability(Odling and Webman, 1991). Some simulated models of fracturesets were developed with a degree of similarity to those obtainedexperimentally (Renshaw and Pollard, 1994). It was shown thatconnectivity is of a primary importance to flow (Bogdanov et al.,2007). Previously, simulation algorithms were extremely time-consuming and demanded much computational power (Wawrzynekand Ingraffea, 1989). However, with technological advancementgeomechanical modelling of complicated fracture patterns has nowbecome a viable tool (Ingraffea and Saouma, 1985). It is demanding tomeasure the effective permeability of a model with a high level ofgeological realism that incorporates geomechanical properties, suchas density and spatial fracture distribution. Paluszny and Matthai(2010) presented a 2D numerical model that measures effectivepermeability through a geomechanically generated network takinginto account flow through the matrix and fractures, and examines theeffect of the level of detail in fracture representation on the accuracyof effective permeability measurements.

Previous studies were done for single set fracture patterns only,which assume a sole episode of deformation (Paluszny andMatthai, 2010). However, in nature rock masses are often sub-jected to multiple stress and strain regimes throughout their burialhistory, resulting in the superposition of a multiple fracture sets.These patterns are common in geological formations (Rives et al.,1994; Cosgrove, 2005; Odling and Roden, 1997; Mandl, 1999;Grechka and Tsvankin, 2003). The study of the nature of fracturesand their geometry is critical as bulk properties of fractured media,such as permeability for example, are determined by fracturing,rather than by intrinsic rock properties (Cosgrove, 2005). Thus,multi-sets with superimposition angles should render naturallyoccurring structures more precisely than the randomly orienteddatasets.

The purpose of this paper is to measure the effective permeabilityof multi-sets, with respect to levels of geomechanical realism:curving, aperture distribution and connectivity. A comparative 2Dstudy between the effective permeability of rock masses containingsingle-set and multi-set fracture patterns is presented. Single-setfracture patterns are developed geomechanically and subsequently

superimposed. This is performed numerically, using finite elementbased method established from: a fracture criterion, a propagationcriterion, and propagation angle (Paluszny and Matthai, 2009).

This paper is organized as follows. Section 2 describes themethodology, computation of effective permeability, methods offracture growth, generation of multi-set fracture patterns andexperimental setup. Section 3 compares and analyses the resultsof single-set and multi-set fracture pattern model. Conclusion ofresults is provided in Section 4.

2. Material and methods

2.1. Flow model and equations

We measure effective permeability by applying a pressuregradient to the model and computing fluid pressure field. Weconsider steady-state single-phase fluid flow, which obeys volumeconservation law

∇q¼ 0 ð1Þwhere q (m/s) is the filtration velocity or Darcy flux:

q¼ � kμ∇P ð2Þ

where P (Pa) and μ (Pa s) are the fluid pressure and dynamicviscosity respectively, and k is the permeability of the rock. Theequation is discretized using finite element method, where eachelement of discretized domain has permeability related to it. Wedefine a constant matrix permeability km, for fractures we definean equivalent porous medium fracture permeability kf, by applyinga parallel plate approximation. We apply effective permeabilityboundary conditions to calculate total model throughput q. Forgiven fluid flow we approximate keff as

keff ¼qμL

AðPðuÞ�PðdÞÞ ð3Þ

where L (m) is length of model in the direction of flow, A (m2) is thearea of a cross section perpendicular to the flow, PðuÞ and PðdÞ areupstream and downstream pressures (Pa) respectively. Here, effectivepermeability, keff, is the permeability of the fractured porous mediumas it accounts for the flow through matrix and fractures simulta-neously. Fluid pressure is solved for using the finite element method,PðuÞ and PðdÞ are integrated over the boundaries of the model. Fig. 1ashows that we apply a fluid pressure gradient as boundary condition.The effective permeability is expressed in dimensionless term, Rk

Nomenclature

A area cross sectiona aperture sizeα anisotropy angled dimensionless fracture density in porous mediak permeabilitykeff effective permeabilitykf fracture permeabilitykm matrix permeabilityKIC material toughnessK hydraulic conductivityL length of the flow regionl mean fracture lengthli i fracture lengthlo side length of a square measuring regionLr total length of fractures i in region

P pressureP(u), P(d)fluid pressure gradients, upstream and downstreamq influxRk effective permeability ratio, dimensionlesss spacingμ viscosityθ superimposition angle of fracture sets

Subscripts

c fracture centrelinef fracture walleff effectivef fracturem matrixt total

A. Kubeyev / Journal of Petroleum Science and Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





(Philip et al., 2005), as

Rk ¼ keff =km ð4Þ

2.2. Fracture permeability

Fractures are discretized by line elements at the middle of thefracture, with locally defined fracture permeability. Permeabilityand porosity are weighed with local apertures. Fig. 1b illustratesschematically multi-set fracture mapping. Fracture permeability isapproximated using the parallel plate law (Nelson, 1976; Kranzet al., 1979; Witherspoon et al., 1979) in a piecewise fashion. Thevolume flow rate per local fracture segment width is defined as

q¼ kAdPμdL

¼ a3

12μdPdL

ð5Þ

kf ¼ a2=12 ð6Þwhere a is the plate opening or height of local fracture aperture. Itwas shown that using single aperture the entire fracture methodoverpredicts flow in fractures by a factor up to 2, as it neglectsfracture roughness impact on flow (Zimmerman et al., 1991;Mourzenko et al., 1995; Zimmerman and Bodvarsson, 1996). Tosimulate flow, fractures are discretized by line elements at themiddle of the fracture, with locally defined fracture permeability(Fig. 1b). Properties such as permeability and porosity are weightedat these lines with local apertures. This piecewise definition candefine kf as a function of local aperture. This method supportsfracture intersections, as at each junction there is a node and kf iscomputed at the center of each segment.

2.3. Multi-set fracture generation

Multi-set fracture patterns occur due to reorientation andstress displacement of geological formations. Fractured media is

characterized by one or more angles of superposition, measuredbetween the main orientation axes of the fracture sets. Models aregenerated by superimposing several single-set fracture patterns, atan angle θ. Each of these is grown independently using a finiteelement based geomechanical simulator (Paluszny and Matthai,2009). Local failure, propagation, and angle criteria control frac-ture tip extension at each step by associating the local strainenergy release rate with the maximum global energy of all tips,controlling the propagation by weighting it with an empiricalvelocity index (Renshaw and Pollard, 1994). The model propagatespre-existing cracks, or flaws, which have random sizes andlocations which follow Gaussian size distribution (Underwood,1970). Fig. 2 depicts four cases of superposition of straightfractures at different angles θ¼01, 151, 601, and 901.

2.4. Experimental setup

To simulate fractured porous media numerically, DiscreteFracture and Matrix (DFM) model was used. This method offracture modelling and computation of effective permeabilityand fracture–matrix flux ratio was developed at Imperial Collegeand ETH Zurich. Code was written on Cþþ programming lan-guage based on Complex Systems Modelling Platform (CSMPþþ)mechanics module, an object-oriented finite-element basedlibrary for multi-physics modelling (Matthai et al., 2007).

Following representative limestone geomechanical propertieswere assigned: Young's modulus of 20 GPa, Poisson's ratio of 0.25,material toughness KIC ¼ 15� 106 Pa m0:5. A model is 2D withdimensions of 1 m �4 m, where fractured area is 0.8 m �2 m,and an observational area is 0.46 m �0.46 m as shown in Fig. 3.Cracks develop from 100 initial flaws defined within the flawarea of the model. The dataset was then operated to extract onlythose fractures within a smaller observation area, from which wecompute the keff ratio. During analysis the keff was measuredat sequential growth stages with equal initial conditions before

Fig. 1. (a) Effective permeability boundary conditions of the observation area length L. The fluxes qin and qout are total in and out fluxes respectively. (b) Multi-set fracturemapping, fracture centerline and aperture approximation. Black dots illustrate fracture wall (fn, f nþ1, f n�1). White dotes are centerline nodes, ci and ciþ1. Fracture tip is fn.Aperture of fracture, ai, corresponds to centerline node ci. Here, ai is a measure of distance between f nþ1 and f n�1. The aperture assigned to the local line element; afi is theaverage between ai and aiþ1.

Fig. 2. Superimposition of different fracture sets at different angles. Single-set is θ(01), multiple sets are angles θ(151), θ(601), θ(901).

A. Kubeyev / Journal of Petroleum Science and Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





simulation of each single and multi fracture sets. The fracturedensity, spacing and length was then measured to characterizegrowth. The observation area was selected arbitrarily within flawarea as permeability is compared versus dimensionless fracturedensity. The latter is defined by Underwood (1970) as proportionalto the fracture set development stage, also increasing with fractureset superposition

d¼ 1A

∑n

i ¼ 1

li4

2

ð7Þ

where, A is a flow area, li is the i fracture length, n is the amountof fractures in set. Fracture spacing is a perpendicular distancebetween two near parallel cracks, defined by Wu and Pollard(1995):

s¼ Aloþ∑n

i ¼ 1li¼ AloþL

ð8Þ

where lo is a side length of a square measuring region, Lr is a totallength of fractures i in that region, n is a number of fractures and Ais area. Permeability of multi-sets, obtained by merging two ormore single-sets, was analysed and the relationship betweenthe geometric anisotropy of the patterns and the anisotropy of keffratio for straight and curved fracture patterns was systematicallystudied. A workflow of the experiments is depicted in Fig. 4

3. Results and analysis

At each simulation stage, density gradually increased and initiallyseparate fractures started to connect. An increase in fracture con-nectivity and effective permeability was observed, as shown inFig. 5a. Under the percolation threshold, flow is mostly controlledby the matrix and fracture network. As fractures have higherconductivity than matrix, main flowing regions parallel to thefracture's orientation occur. These flow regions in single-set modelsshowed relatively low connectivity, and thus flow depended both on

fractures and the matrix's ability to transmit fluid. In contrast, amulti-set model relied on connectivity to feed fast flowing channels,and exhibited higher Rk for the same densities, when compared totheir single-set counterparts.

3.1. Impact of fracture set angle on the effective permeability

Rk of the observation area was measured with a straightfractures model, a low permeability matrix km ¼ 10�15 m2, and afracture aperture of a¼0.01 m. The impact of various intersectionangles on effective permeability is summarized in Fig. 5a and b,where a single-set pattern is θ¼01. Multi-set fracture patternsexhibit greater impact on the permeability ratio than single-sets.There is an increase from 1 to 4.3 in permeability ratio at θ¼151and the model reached maximum increment when do0:65. Thedifference between multi-sets and single-sets is significant for anysuperposition angles. For do0:7 orthogonal and single set havelowest permeability increment, whereas θ¼151–601 angles exhibitsimilar high Rk trend. For d >0.7 permeability of orthogonal trendincreased sharply due to connection of fractures into network andthis trend for all multi-sets becomes similar. Displacement angleθ¼901 corresponds in nature to tensile fractures, which occur as aresult of high internal fluid pressure or during the contraction oflayers and sediment desiccation. Results are consistent for θ¼301and 901 with those of a study done by Nakashima et al. (2000).However, the latter research was done by averaging fracture-onlyvelocity fields thus neglecting flow through matrix and substan-tially underestimating the effective permeability (Bogdanov et al.,2003). Impact of superposition angle θ on effective permeability isshown in Fig. 5b.

3.2. Impact of fracture spacing and length on permeability ratio

Fig. 6a illustrates the dependence of Rk on the mean fracturelength in three scenarios: single set θ¼01, double set θ¼301andorthogonal double set θ¼901. An exponential increase in effective

Fig. 3. Model with fracture and observation areas (not in scale).

No. Experiment name

Input parameters Output

Km, m2 0 a, m Iterations

no.

Anisotropy

angleDensity,dim.-less

Rk, dim.-less

Fracture length, m

Spacing, m

Permeability

anisotropy

1

Impact of fracture set angle on the effectivepermeability 1*10-15

0o, 15o, 30o, 60o, 90o 0.01 60

- + + - - -

2 Impact of fracture length and spacing 1*10-13

0o, 30o, 90o 0.001 60

- + + + + -

3 Curved versus straight multi-sets 1*10-15 0o, 30o 0.001 110

- + + - - -

4

Effective permeability anisotropy

1*10-13-1*10-15 0o, 17o

0.1-0.001 60

+ + + - - +

5

Multi sets above percolation threshold

1*10-13-1*10-16 30o, 90o 0.01 160

- + + - + -

Fig. 4. Experiment setup workflow chart. Here, "þ" - analysed, "�" - not analysed.






permeability occurs in both models, caused by increase in fracturedevelopment and thus fracture length.

Both double sets show a gradual exponential increase in Rk atlow fracture lengths of l¼0.02–0.085 m, and sharp Rk increaseat l¼0.085–0.12 m mean fracture length from 3 to 4.25 in θ¼301and from 1.8 to 4.3 in orthogonal model. The θ¼901 and single setmodels have lower permeability increment compared to θ¼301multi-set.

Connectivity of main flowing regions is lower in single-setthan in multi-set θ¼301 and similar to the orthogonal model tilll¼0.12 m. In single and orthogonal models where l¼0.02–0.12 m,flow is mostly controlled by matrix intrinsic properties andremains similar, after which the Rk in orthogonal model increased

sharply by three order of magnitudes if length is increased. This isdue to connection of separated fractures into a main flowingregions that increase connectivity during increase of fracturelength sufficient enough to conduct fluid from one domain toanother through highly conductive fracture network. Once thedomains are connected the Rk increases significantly and thedifference between that models becomes apparent.

Dependence of permeability on spacing, shown in Fig. 6b showsa similar conclusion with a typical higher permeability ratioincrement seen in multi-sets. Results are consistent with theprevious study (Nakashima et al., 2000), where stochastic discretemodels showed increase in keff with increase in mean and totalfracture length. Effect of superposition of fracture sets that analysed

Fig. 5. (a) Permeability ratio as a function of density at different superposition angles, straight fracture pattern. (b) Rk as a function superposition angles for various fracturedensity. For both km ¼ 10�15 m2.

Fig. 6. Effective permeability ratio as a function of (a) mean length and (b) spacing during fracture propagation, θ¼01, θ¼301 and θ¼901. Here, km ¼ 10�13 m2 anda¼0.001 m.

Fig. 7. Curved (a) versus straight (b) fracture propagation models. Dimensionless permeability ratio as a function of dimensionless fracture density. Here, a¼0.001 m.






in consequent study (Nakashima et al., 2001) uses fracture velocityfields averaging technique that showed decrease in keff value forθ¼301–901. However, this method neglects flow through matrixand considerably underestimates the keff (Bogdanov et al., 2003).

3.3. Curved versus straight multi-sets

The fracture's growth under remote boundary conditions atarbitrary angles is controlled by local stress states at the fracturetips. By restricting the modelling of the curvature straight fracturegeometries was obtained. The main analysis is conducted withstraight fixed aperture model, but a comparison with curvedgeometries is provided in this section. Results depicted in Fig. 7show that the curved model has higher Rk¼4.2 than in straightRk¼3.5 multi-sets (θ¼301) for the same d¼1. In single-sets weobserve a stepwise Rk increase, lower than in multi-sets. Whend¼1 in curved single-set models the permeability ratio Rk¼3,whereas in straight models Rk¼2.9. Differences between doubleand single set models become significant at d¼0.5 in bothgeometries. Both models show typically more Rk in multi-sets,and higher permeability increments when fractures have a curvedgeometry.

3.4. Effective permeability anisotropy

In fractured media, effective permeability is often anisotropicas various fracture patterns affect the total flux of the systemaround the formation. An anisotropic directional study is pre-sented here according to a range (01–3601) of angles around theobservational area. Rotational boundary conditions are appliedand permeability is measured with respect to rotation angle, andplot as a function of direction (Fig. 8). Input parameters wereaperture size, high km ¼ 10�13 m2 and low km ¼ 10�15 m2. Fig. 8aillustrates a behaviour for single-set θ¼01 and double-set θ¼171for km¼10�15 m2. Theoretical effective permeability according toDarcy's law gives an ellipse with the maximum on the fracture'sset direction axis and minimum on the perpendicular axes of1=

ffiffiffiffiK

p(Long et al., 1982). There is an increase in keff in maximum

directions from 1�10�15 up to 5�10�14 m2, compared to single-sets that show increase until 2–2.5�10�14 m2 on average.Furthermore, the dispersion of keff in rotation angles 3301–301, inmulti-sets is 4�10�14 m2 on average, whereas in single-setsat 301–3501, it is 2�10�14 m2, for a constant density d¼1.24.A similar trend is seen in Fig. 8b in high km ¼ 10�13 m2 matrix,where the average single-set keff is 2:5–3� 10�12 m2 in themaximum direction. Results for the same initial conditions but

Fig. 8. Single-set θ¼01, vs. Multi-sets θ¼171, for density¼1.24. Effective permeability as a function of direction, for a¼0.001 m. (a) Low km ¼ 10�15 m2 matrix. (b) Highkm ¼ 10�13 m2 matrix; (c) a¼0.1 m, km ¼ 10�13 m2, and (d) Log plot of multi-sets keff as a function of direction. Here θ¼901, a¼0.1 m, km ¼ 10�15 m2.






using an aperture size a¼0.1 m and km ¼ 10�13 m2 are shown inFig. 8c. Here, conductivity of a single fracture is much higher,and matrix permeability 100 times higher, km ¼ 10�13 m2, hencevalues and dispersion of keff in single-set patterns are typicallylarger than in early models. In orthogonal geometries, θ¼901, thefracture network reaches percolation, shown in Fig. 8d, and keff ata maximum of 10�1 m2 within 401–3501, around the model. Whenrotation angle is 901, there is no percolation, whereas at theopposite angle 2801, we observe percolation of up to 1�10�2

m2 in keff. In nature, second data-set fractures tend to abut intoearlier sets, thus we observe smaller keff in these directions. Thebehaviour of effective permeability anisotropy in geomechanicallygenerated simulation model in multi-set fractured media isexpressed by

RkðαÞ ¼ keff ðαÞ=km ¼ 10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið18:734d2�5:1688d�0:42Þn cos ð17αþ13Þ

q

ð9Þ

and effective permeability anisotropy in single-set fractured mediais given by

RkðαÞ ¼ keff ðαÞ=km¼ 1:1616e3:46dn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos ð17αþ12:8Þ� sin ð cos ð17αþ12:8ÞÞ

p

ð10Þwhere α is the anisotropy angle, d the density. Figs. 9 and 10aillustrate an application of expressions in Eqs. (7) and (8) andcomparison with numerical values.

3.5. Analytical vs. numerical effective permeabilities

The comparison of geomechanically generated fracture networkwith analytical solution is provided in this subsection. Deterministicand stochastic approaches are two ways of calculating effectivepermeability analytically in fractured media. An analytical approachof stochastic fractures network calculates permeability in a medium

Fig. 9. Eq. (7) for effective permeability anisotropy in multi-set fracture patters compared with simulation results. Here, a¼0.001 m, α¼ 171 in (a) km ¼ 10�15 m2, and (b)km ¼ 10�13 m2.

Fig. 10. (a) Eq. (8) for single set fracture patters anisotropy permeability ratio; (b) comparison between analytical defined by Zimmerman (1996) and geomechanicallygenerated numerical results of effective permeability ratio as a function of density.






with two dimensional elliptical inclusions of aspect ratio as definedby Zimmerman (1996), in systems below the percolation threshold.These inclusions are randomly oriented and much more permeablethan the matrix representing the fracture set. However, it is valid forfractures of infinite extent which is not realistic in nature:

Rk ¼ ð1þðπ=4Þ dÞ=ð1�ðπ=4Þ dÞ ð11ÞFig. 10b depicts a good match between analytical and orthogonalmulti-sets for all density ranges. In matrix dominated flow, low d,results are in the same range of magnitude, analytical and numer-ical approximations match within 20% on average for all models.There is a good match for do0:5 between analytical, single andorthogonal double sets, but analytical curve exceeds single set atd40:5 where Rk reaches 2.7 and 4.7 respectively. For do0:7, themulti-sets θ¼151 have higher permeability increments comparedto others, up to two orders of magnitude. For high fracture densityd40:7, we observe similar values of Rk around 4.4 in double setsand analytical solutions.

3.6. Multi-sets above the percolation threshold

In this case, we normalize keff by dividing it by a2 to present resultsin dimensionless terms. Fig. 11 shows that below percolation forboth high and low matrix permeabilities, i.e. km ¼ 10�13 m2 andkm ¼ 10�15 m2 respectively, effective permeability increases graduallyup to over an order of magnitude, during which flow is dominated bymatrix and the area between fractures. A percolation is observed, anda sharp increase in keff =a2 from 10�11 up to 10�1 at θ¼901 super-position angle and d¼0.9. The same is observed for the higher densityd¼1.25 at θ¼301. Fluid flow at that stage is dominated by the fracturenetwork.

The percolation appears at much lower density d¼0.9 inmultiple sets, whereas normally in stochastic systems percolationdensity is d¼1.2–1.4 (Berkowitz, 1995; Zimmerman, 1996). Thismay be due to increase in connectivity in multi-sets and ability ofgeomechanically generated patterns to render natural formations.However, the numerical study of crossing fractures was done bysuperimposition technique, but in nature the late fractures tend toabut into the earlier set (Rives et al., 1994; Cosgrove, 2005). It addsuncertainty and discrepancy may occur. Results indicate higherpercolation probability in the case of multi-sets compared tosingle set models, where percolation was not achieved.

4. Conclusions

A numerical study analysing the effect of several fracturesets superposition cases, that represent a multiple fractured

formations, on the effective permeability was presented in this paperand results were analysed quantitatively and qualitatively. Geomecha-nically grown models account for the flow through matrix andfractures simultaneously, using finite element method and localpropagation failure criteria. Results may be summarised as follows:

(1) Superimposition of fracture sets results in enhanced interac-tion between main fluid flowing regions and increases con-nectivity of the fractures. This leads to increasing effectivepermeability of the system. keff increases in all directionsaround the model that represents a permeability anisotropy.

(2) Effective permeability increases at all superposition angles.There is a sinusoidal trend to increase in keff and it reachesmaximum values when the angle θ is in range 151–301 andat 901.

(3) Permeability increment of multi-set fractured media com-pared to single set of fractures tends to increase in high andlow permeability matrices, high and low aperture sizes, and incurved and straight crack geometries. Within same fracturedensity we observe typically bigger keff in curved fracturesgeometry than in a model with straight fracture tips in multi-ple fracture sets.

(4) The percolation in multi-sets is reached at a smaller densityd¼0.9 compared to stochastic systems where percolationthreshold reached at d¼1.2–1.4 that may be due to higherconnectivity of the main flow regions and higher fractureinterconnection. Thus, multi-set fracture patterns are bettersuited to study flow behaviour above the percolation threshold.

(5) An expression that calculates the effective permeability aniso-tropy of the fractured porous media, containing a multiple-setand single-set of fractures was presented in this study.

The applications of this work may be considered in subsurface andnumerical sciences, including a simulation of the fractured forma-tions, reservoir engineering and subsurface contaminant storage.E.g. subsurface contaminant transport studies address the difficul-ties in predicting its flow behaviour (Black, 1993). Solutes, radio-nuclides and colloids in fractured media may travel for a longdistances and enter the ground and surface water. This study mayimprove a predictions and enhance the efficiency of rendering anaturally occurring systems.

Acknowledgments

Author would like to thanks Dr. Adriana Paluszny and Prof.Robert W. Zimmerman who greatly assisted this research.

Fig. 11. Multi-sets above percolation threshold, keff/a2 dimensionless as a function of (a) dimensionless density and (b) spacing for high 10�13 m2 and low 10�15 m2 matrixpermeability.






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impact of multi-set fracture pattern on the effective permeability of fractured porous media

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