Research ArticleDetermining the REV for Fracture Rock MassBased on Seepage Theory
Lili Zhang12 Lu Xia2 and Qingchun Yu2
1Department of Earthquake Science Institute of Disaster Prevention Sanhe 065201 China2School of Water Resources and Environment China University of Geosciences Beijing 100083 China
Correspondence should be addressed to Lili Zhang zhanglili168126com
Received 9 January 2017 Revised 8 March 2017 Accepted 20 April 2017 Published 14 May 2017
Academic Editor Shuyu Sun
Copyright copy 2017 Lili Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Seepage problems of the fractured rock mass have always been a heated topic within hydrogeology and engineering geology Theequivalent porous medium model method is the main method in the study of the seepage of the fractured rock mass and itsengineering application The key to the method is to determine a representative elementary volume (REV) The FractureToKarstsoftware that is discrete element software is a main analysis tool in this paper and developed by a number of authors Accordingto the standard of rock classification established by ISRM this paper aims to discuss the existence and the size of REV of fracturedrock masses with medium tractility and provide a general method to determine the existence of REV It can be gleaned from thestudy that the existence condition of fractured rock mass with medium tractility features average fracture spacing smaller than06m If average fracture spacing is larger than 06m there is no existence of REVThe rationality of the model is verified by a casestudy The present research provides a method for the simulation of seepage field in fissured rocks
1 Introduction
Seepage problems of the fractured rock mass have alwaysbeen a heated topic within hydrogeology and engineeringgeology Many problems that are closely related to theresearch of fracture seepages including the stability of damfoundation and seepage the stability of bedrock side slopesunder the influence of groundwater fissure deposit the pre-vention of mine water inrush and the leakage and diffusionof nuclear waste However the study of the seepage problemsof the fractured rock mass is still in its preliminary stageat present The general method regards fractured media asporous media and uses the permeability tensor of porousmedia to describe the seepage characteristics of the fracturedmedia There are different types of structural plane in frac-tured rock mass which lead to the complexity of rock masscharacteristics Therefore the study on rock mass model isalways one of the important problems in rock mechanicsThe representative elementary volume (REV) is the basis todetermine a rock mass mechanics model and it is necessaryto research the REV of fractured rock mass effectively so thatthe REV size of the fractured rock mass can be determined
The concept of the REV was the first introduced incontinuum mechanics by Bear [1] and it is to be used todescribe the flow in the porous media The parameter ofinterest is both homogeneous and statistically stationarywhich will ensure consistency in flow simulation studiesTheREV is defined in two situations on (1) unit cell in a periodicmicrostructure and (2) volume containing a very large setof microscale elements possessing statistically homogeneousproperties The REV has been discussed by many authors[2ndash17] The REV of a fractured rock mass is the smallestvolume in during the study of parameter when the hydraulicconductivity is a constant value One special concern is theevaluation of the REV of the fractured rock masses due tothe fact that fluid flow in fractured rockmasses is of high scaleeffect [2 18ndash23] Previous studies assumed that the anisotropywas achieved bymaking use of different correlation lengths inthe horizontal and vertical directions and flow barriers weremodeled stochastically [24ndash36]
Snow [37] concluded the math expressions of singleand infinite fracture permeability tensors assuming that thefracture seepages did not interfere with each other whilethe overall permeability tensor of fracture network was the
HindawiGeofluidsVolume 2017 Article ID 4129240 8 pageshttpsdoiorg10115520174129240
2 Geofluids
linear superposition of all fractures As the fracture networkwas the same as the porous media the permeability can beexpressed by a permeability tensor A fractured network canbe approximately viewed as a porous medium which was ifthe equivalent porousmedia of fractures are existent then thepermeability coefficients can be expressed by a symmetricsecond-order tensor Li and Zhang [38] also conductedresearch on the REV of fractures Bear [1] proved that if119903 = 11987011990212 or 119903 = 119870119895minus12 in the polar coordinate systemmapping can form an ellipse (under the condition of three-dimensional ellipsoid) where 119903 represents radius vector 119870119902represents permeability coefficient in the direction of the flowline and 119870119895 represents permeability coefficient in the direc-tion of the hydraulic gradient Long and Witherspoon [39]proposed that if the representative elementary volume ofthe fractured rock mass (REV) existed the following twoconditions must be met
(1) The media must be uniform in the area of study thatis the average permeability coefficient of the area ofstudy changes along with the scope of the study withbig change then it can be determined that this area ofstudy is uniform
(2) In the polar coordinate system the equivalent perme-ability coefficient k in each directionwithin the area ofstudy can be described using an ellipse approximatelyat this time 119903 = radic119870
2 Introduction to FractureToKarst Software
This paper uses discrete fracture network (DFN) softwareaccording to the standard of rock classification established byISRM and discusses the existence of REV of fractured rockmasses with medium tractility The discrete fracture network(DFN) software FractureToKarst for seepage in fracture rockmass is a kind of software using the Monte Carlo method Itcan generate a two-dimensional fracture network of arbitraryshapes set the common statistical parameters of fracturefilter the fractures within the area of study and proceedwith automatic discrete can be set The head value andequivalent permeability coefficient of each node can becalculated by using the water balance principle In the studyset an aspect ratio for the area of study 2 1 [40] calculate anequivalent permeability coefficient every 10∘ rotation of thearea of study every area of study can get 36 equivalent perme-ability coefficients and discuss whether the direction of theequivalent permeability coefficient in polar coordinates canbe described as a permeability coefficient ellipse or not thusdetermining the existence of REV
21 Mathematical Model Mathematical model is the waterdynamic model of the system Fracture network is composedof a single flat smooth fracture in a state of the laminar flowin a single fracture and can be described by the cubic law
Type 119876 is the boundary flux 120588 is the density of water 119892is the gravitational acceleration 120583 is flow dynamic viscositycoefficient 119861 is fracture width Δ119867 is two-head difference ofthe fracture 119871 is the length of the fracture section
In the fracture network each node can establish ahydraulic link equation by the water balance principle
Type 119876119894119895 is the node flux 119876119887 is the boundary flux Thehydrodynamic equations can be constructed by combiningformula (1) and formula (2) and the head distribution ofthe fracture node within the system can be derived by theequations
22 Model Algorithm Model algorithm is a numerical meth-od The detailed steps are as follows
(1) Grid discretization all the fractures can be dividedinto the smallest fracture section by the nodes Each node andconnected fractures make a unit balance zone as shown inFigure 1
(2) For the hydraulic head assign an initial value andboundary treatment the node head is constant in the bound-ary of fixed water levelThe initial water head of the rest of thenodes is 099 times the maximum elevation value The nodehead of the impermeable boundary is equal to the adjacentnodes water head
(3) For the hydraulic head do iteration calculationcalculate the hydraulic head values of all nodes using theiterative methods based on formulas (1) and (2)
(4) Calculate the permeability coefficient completing thehydraulic head calculation follows evaluating the equivalentpermeability coefficient K in the flow direction using Darcyrsquoslaw
(5) Calculate the permeability coefficient in any directionkeep the shape of the area of study unchanged make it rotatearound its center and calculate an equivalent permeabilitycoefficient every 10∘simrotation then we can get 36 equivalentpermeability coefficients119870
(6) Draw in the polar coordinate system it is appropriateto use the equivalent permeability coefficients of 36 directionsin Step (5) to draw the diagram to see whether it can form anellipse so we know whether the REV is existent or not
Geofluids 3
1 2
3 4
5
6
7
8
Figure 2 Sketch of fracture network
3 Analytical Solution for Validation of theSoftware FractureToKarst
Assume that there is a fracture network as shown in Figure2 the right and left boundaries being the given head borderand the upper and lower two boundaries being the imperme-able boundary the start-point and end-point coordinatesof the fractures are [(1 0) (1 2)] [(05 15) (3 0)] [(0 08)(4 08)]
Assume the hydraulic head of two kinds of in and out ofboundary values1198671 = 2051198672 = 05 respectively Using theprinciple of water balance and the cubic flow law there is
If 119860 = 120588119892119861312120583 there are1198671 = 2051198672 = 051198601198676 minus 119867311987163 = 01198601198678 minus 119867411987184 = 01198601198677 minus 119867511987175 = 0119860(1198671 minus 119867611987116 + 1198673 minus 119867611987136 + 1198678 minus 119867611987186 + 1198677 minus 119867611987176 ) = 0119860(1198676 minus 119867711987167 + 1198678 minus 119867711987187 + 1198675 minus 119867711987157 ) = 0119860(1198676 minus 119867811987168 + 1198674 minus 119867811987148 + 1198672 minus 119867811987128 + 1198677 minus 119867811987178 )= 0
(4)
Type 1198671sim1198678 is the water head 119871 119894119895 is the length of fracturesection between the two nodes 120588 is the density of water 119892is the gravitational acceleration 120583 is flow dynamic viscositycoefficient
Through (4) we can obtain that the values of 119871 119894119895 and thehydraulic head values for each node are
The equivalent permeability coefficient of the flow direc-tion 119896 is obtained through Darcyrsquos law
119896 = 01173ms (6)
Use FractureToKarst to build fracture network and inputparameters the value of the water head and the equivalentpermeability coefficient are shown in Figure 3
The calculation result in that the program equals theman-ual computation result proving that the program is correct
4 Simulation of the Fracture Network
According to the standard of rock classification establishedby ISRM fractured rock mass with medium tractility refersto rock mass whose trace length is more than 3m and lessthan 10m Because the permeability of two sets of orthog-onal fracture rock masses is closest to being isotropic thedescribed ellipse is closest to being a circle
According to the fracture spacing classification of ISRMthe spacing within 20sim60mm is very dense spacing Within10m times 10m two sets of orthogonal fractures were generatedthe average spacing is 006m and the aperture is 00001mThe two sets of fracture parameters are shown in Table 1 (twosets of fracture identification for I and II in Table 1) Here putsome fractures in the same direction as a set of fractures Thedistribution types of the trace length and the direction arethe normal distribution and the gap width is the logarithmicnormal distribution The right and left boundaries are the
4 Geofluids
Table 1 Input data of the simulation fractures
Geometric parameters Average Standard deviation Minimum Maximum
Figure 3 Calculation results of the FractureToKarst
given head border and the upper and lower two boundariesare the impermeable boundary
FromTable 1 two sets of orthogonal fractures can be builtThe first set has 276 fractures and the second set has 208 frac-tures with a total of 484 generating fractures The diagramof generated fractures is shown in Figure 4(a) The equiva-lent permeability coefficient of the study area changes withthe scope of the study without changes so the study area isuniform Within 10m times 10m select five study areas of 10mtimes 05m 20m times 10m 30m times 15m 40m times 20m and60m times 30m and the diagram of generated fractures ofFractureToKarst is shown in Figures 4(b)ndash4(f)
Acquire the permeability coefficients of all the directionsin the five regions and make a comprehensive comparisonchart of permeability coefficients (shown in Figure 5) Ana-lyzing Figure 5 it can be concluded that every equivalent per-meability coefficient is basically stable when the area of studyis larger than 10m times 05mThere is no dramatical change forthe permeability coefficients in the four regions 20mtimes 10m30m times 15m 40m times 20m and 60m times 30m so more than10mtimes 05m in the area of study can be approximately viewedas a homogeneous medium region
In Figure 5 the meaning of all the symbols as followsldquordquo is the area of study of 10m times 05m ldquoerdquo is the area of
study of 20mtimes 10m ldquo998810rdquo is the area of study of 30mtimes 15mldquoIrdquo is the area of study of 60m times 30m ldquordquo is the area ofstudy of 70m times 35m
5 The Fitting Calculation ofthe Uniform Basin
51 Calculation of Regional Rotation As the uniform basin10mtimes 05mwas determined above taking the initial angle as
the angle between the horizontal direction and the flowdirection rotate thewhole area of study clockwise to calculatean equivalent permeability coefficient every 10∘ then eacharea of study has 36 equivalent permeability coefficients asshown in Table 2
Table 2 shows that themaximumvalue of the permeabilitycoefficient is 0127ms when the angle is 0∘ and 180∘ betweenthe flow direction and the horizontal orientation (because theangle between the two direction is 180∘ so they are the sameone flow field) the minimum value of the permeability coef-ficient is 0076ms when the angle is 40∘ and 220∘ betweenthe flow direction and the horizontal orientation Thereason for this phenomenon is that the calculation in Table 2is derived from the data of the fracture network in Table 1where 0∘ (180∘) and 90∘ (270∘) are two groups of orthogonalfractures In Table 1 at four points at angles 0∘ (180∘) and 90∘(270∘) there will be a local maximum valueThe permeabilitycoefficient at 0∘ (180∘) is the local maximum value of fracturesin the 0∘ (180∘) direction and the permeability coefficient at40∘ (220∘) is the minimum value between the two local max-imum values
52 The Polar Coordinate Fitting of the Permeability Coeffi-cient Assuming that point P is any one point on the ellipsein the polar coordinate system A B are respectively the twoendpoints and C is the focus of the ellipse The semisimmajoraxis and the semisimminor axis of the ellipse are assumed as 119886and 119887 respectively This results in
11990921198862 + 1199102
1198872 = 1119909 = 120588 cos 120579119910 = 120588 sin 120579
(7)
Assuming that = 119879 + 120579 119879 is a parameter that is theangle between the principal axis of the ellipse and the polarcoordinate 0∘ axis
Solve (7) where the ellipse equation of the permeabilitycoefficient in the polar coordinate system is
120588 = 119886119887radic1198862 minus (1198862 minus 1198872) cos2 (120579 + 119879) (8)
Draw the ellipse according to Table 2 and (8) in the polarcoordinate system and fit it as shown in Figure 6 Fittingparameters and fitting values are shown in Table 3
Geofluids 5
(a) All the fractures (b) 10m times 05m
(c) 20m times 10m (d) 30m times 15m
(e) 60m times 30m (f) 70m times 35m
Figure 4 Sketch of fracture of research area
0
45
90
135
180
225
270
315
0 01 02 03 04 05
Figure 5 Comprehensive comparison chart of permeability coeffi-cients of the five research areas
The semisimmajor axis and the semisimminor axis of thefitting ellipse are 119886 = 03121083582 119887 = 0298487332respectively where 119879 = 2359065441 radians and the fittingequation is
120588= 0093160391radic0097411627 minus 000831694cos2 (120579 + 2359065441)
(9)
0
45
90
135
180
225
270
315
0 01 02 0403
Figure 6 Curve fitting
6 Determination of the REV
Ohman et al have conducted a considerable amount of usefulresearch [41ndash43] on the equivalent medium in the fracturedrocks to compare the similarity between the equivalent per-meability coefficient of numerical simulation and the ellipsein different area of studies These two scholars have obtained
6 Geofluids
Table 2 Geometry parameters of the simulation fractures
RMS is the fitting correlation coefficient of the ellipse119870119878(120579) is the simulation value and 119870119891(120579) is the fitting valueDepending on its similarity to the elliptic curve RMS can
be divided into three categories
(1) When RMS le 02 the size of the area of study can beused as the REV of the fractured rock mass
(2) When 04gtRMSgt 02 the size of the area of study canbe used as the REV of the fractured rock mass undercertain conditions
(3) When RMS ge 04 the size of the area of study cannotbe used as the REV of the fractured rock mass
Therefore after comparing Tables 2 and 3 it can begleaned that the fitting result is quite ideal with the correla-tion coefficient RMS = 0132064722TheREV of the fracturedrock mass of this kind exists whose size is 10m times 05m
Further research shows that the REV of very densespacing (lt002m) of the fractured rock mass exists and itis less than 10m times 05m The REV of the dense spacing(006sim02m) of the fractured rock mass is 20m times 10m TheREV of medium spacing (02sim06m) of the fractured rock
Geofluids 7
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
linear superposition of all fractures As the fracture networkwas the same as the porous media the permeability can beexpressed by a permeability tensor A fractured network canbe approximately viewed as a porous medium which was ifthe equivalent porousmedia of fractures are existent then thepermeability coefficients can be expressed by a symmetricsecond-order tensor Li and Zhang [38] also conductedresearch on the REV of fractures Bear [1] proved that if119903 = 11987011990212 or 119903 = 119870119895minus12 in the polar coordinate systemmapping can form an ellipse (under the condition of three-dimensional ellipsoid) where 119903 represents radius vector 119870119902represents permeability coefficient in the direction of the flowline and 119870119895 represents permeability coefficient in the direc-tion of the hydraulic gradient Long and Witherspoon [39]proposed that if the representative elementary volume ofthe fractured rock mass (REV) existed the following twoconditions must be met
(1) The media must be uniform in the area of study thatis the average permeability coefficient of the area ofstudy changes along with the scope of the study withbig change then it can be determined that this area ofstudy is uniform
(2) In the polar coordinate system the equivalent perme-ability coefficient k in each directionwithin the area ofstudy can be described using an ellipse approximatelyat this time 119903 = radic119870
2 Introduction to FractureToKarst Software
This paper uses discrete fracture network (DFN) softwareaccording to the standard of rock classification established byISRM and discusses the existence of REV of fractured rockmasses with medium tractility The discrete fracture network(DFN) software FractureToKarst for seepage in fracture rockmass is a kind of software using the Monte Carlo method Itcan generate a two-dimensional fracture network of arbitraryshapes set the common statistical parameters of fracturefilter the fractures within the area of study and proceedwith automatic discrete can be set The head value andequivalent permeability coefficient of each node can becalculated by using the water balance principle In the studyset an aspect ratio for the area of study 2 1 [40] calculate anequivalent permeability coefficient every 10∘ rotation of thearea of study every area of study can get 36 equivalent perme-ability coefficients and discuss whether the direction of theequivalent permeability coefficient in polar coordinates canbe described as a permeability coefficient ellipse or not thusdetermining the existence of REV
21 Mathematical Model Mathematical model is the waterdynamic model of the system Fracture network is composedof a single flat smooth fracture in a state of the laminar flowin a single fracture and can be described by the cubic law
Type 119876 is the boundary flux 120588 is the density of water 119892is the gravitational acceleration 120583 is flow dynamic viscositycoefficient 119861 is fracture width Δ119867 is two-head difference ofthe fracture 119871 is the length of the fracture section
In the fracture network each node can establish ahydraulic link equation by the water balance principle
Type 119876119894119895 is the node flux 119876119887 is the boundary flux Thehydrodynamic equations can be constructed by combiningformula (1) and formula (2) and the head distribution ofthe fracture node within the system can be derived by theequations
22 Model Algorithm Model algorithm is a numerical meth-od The detailed steps are as follows
(1) Grid discretization all the fractures can be dividedinto the smallest fracture section by the nodes Each node andconnected fractures make a unit balance zone as shown inFigure 1
(2) For the hydraulic head assign an initial value andboundary treatment the node head is constant in the bound-ary of fixed water levelThe initial water head of the rest of thenodes is 099 times the maximum elevation value The nodehead of the impermeable boundary is equal to the adjacentnodes water head
(3) For the hydraulic head do iteration calculationcalculate the hydraulic head values of all nodes using theiterative methods based on formulas (1) and (2)
(4) Calculate the permeability coefficient completing thehydraulic head calculation follows evaluating the equivalentpermeability coefficient K in the flow direction using Darcyrsquoslaw
(5) Calculate the permeability coefficient in any directionkeep the shape of the area of study unchanged make it rotatearound its center and calculate an equivalent permeabilitycoefficient every 10∘simrotation then we can get 36 equivalentpermeability coefficients119870
(6) Draw in the polar coordinate system it is appropriateto use the equivalent permeability coefficients of 36 directionsin Step (5) to draw the diagram to see whether it can form anellipse so we know whether the REV is existent or not
Geofluids 3
1 2
3 4
5
6
7
8
Figure 2 Sketch of fracture network
3 Analytical Solution for Validation of theSoftware FractureToKarst
Assume that there is a fracture network as shown in Figure2 the right and left boundaries being the given head borderand the upper and lower two boundaries being the imperme-able boundary the start-point and end-point coordinatesof the fractures are [(1 0) (1 2)] [(05 15) (3 0)] [(0 08)(4 08)]
Assume the hydraulic head of two kinds of in and out ofboundary values1198671 = 2051198672 = 05 respectively Using theprinciple of water balance and the cubic flow law there is
If 119860 = 120588119892119861312120583 there are1198671 = 2051198672 = 051198601198676 minus 119867311987163 = 01198601198678 minus 119867411987184 = 01198601198677 minus 119867511987175 = 0119860(1198671 minus 119867611987116 + 1198673 minus 119867611987136 + 1198678 minus 119867611987186 + 1198677 minus 119867611987176 ) = 0119860(1198676 minus 119867711987167 + 1198678 minus 119867711987187 + 1198675 minus 119867711987157 ) = 0119860(1198676 minus 119867811987168 + 1198674 minus 119867811987148 + 1198672 minus 119867811987128 + 1198677 minus 119867811987178 )= 0
(4)
Type 1198671sim1198678 is the water head 119871 119894119895 is the length of fracturesection between the two nodes 120588 is the density of water 119892is the gravitational acceleration 120583 is flow dynamic viscositycoefficient
Through (4) we can obtain that the values of 119871 119894119895 and thehydraulic head values for each node are
The equivalent permeability coefficient of the flow direc-tion 119896 is obtained through Darcyrsquos law
119896 = 01173ms (6)
Use FractureToKarst to build fracture network and inputparameters the value of the water head and the equivalentpermeability coefficient are shown in Figure 3
The calculation result in that the program equals theman-ual computation result proving that the program is correct
4 Simulation of the Fracture Network
According to the standard of rock classification establishedby ISRM fractured rock mass with medium tractility refersto rock mass whose trace length is more than 3m and lessthan 10m Because the permeability of two sets of orthog-onal fracture rock masses is closest to being isotropic thedescribed ellipse is closest to being a circle
According to the fracture spacing classification of ISRMthe spacing within 20sim60mm is very dense spacing Within10m times 10m two sets of orthogonal fractures were generatedthe average spacing is 006m and the aperture is 00001mThe two sets of fracture parameters are shown in Table 1 (twosets of fracture identification for I and II in Table 1) Here putsome fractures in the same direction as a set of fractures Thedistribution types of the trace length and the direction arethe normal distribution and the gap width is the logarithmicnormal distribution The right and left boundaries are the
4 Geofluids
Table 1 Input data of the simulation fractures
Geometric parameters Average Standard deviation Minimum Maximum
Figure 3 Calculation results of the FractureToKarst
given head border and the upper and lower two boundariesare the impermeable boundary
FromTable 1 two sets of orthogonal fractures can be builtThe first set has 276 fractures and the second set has 208 frac-tures with a total of 484 generating fractures The diagramof generated fractures is shown in Figure 4(a) The equiva-lent permeability coefficient of the study area changes withthe scope of the study without changes so the study area isuniform Within 10m times 10m select five study areas of 10mtimes 05m 20m times 10m 30m times 15m 40m times 20m and60m times 30m and the diagram of generated fractures ofFractureToKarst is shown in Figures 4(b)ndash4(f)
Acquire the permeability coefficients of all the directionsin the five regions and make a comprehensive comparisonchart of permeability coefficients (shown in Figure 5) Ana-lyzing Figure 5 it can be concluded that every equivalent per-meability coefficient is basically stable when the area of studyis larger than 10m times 05mThere is no dramatical change forthe permeability coefficients in the four regions 20mtimes 10m30m times 15m 40m times 20m and 60m times 30m so more than10mtimes 05m in the area of study can be approximately viewedas a homogeneous medium region
In Figure 5 the meaning of all the symbols as followsldquordquo is the area of study of 10m times 05m ldquoerdquo is the area of
study of 20mtimes 10m ldquo998810rdquo is the area of study of 30mtimes 15mldquoIrdquo is the area of study of 60m times 30m ldquordquo is the area ofstudy of 70m times 35m
5 The Fitting Calculation ofthe Uniform Basin
51 Calculation of Regional Rotation As the uniform basin10mtimes 05mwas determined above taking the initial angle as
the angle between the horizontal direction and the flowdirection rotate thewhole area of study clockwise to calculatean equivalent permeability coefficient every 10∘ then eacharea of study has 36 equivalent permeability coefficients asshown in Table 2
Table 2 shows that themaximumvalue of the permeabilitycoefficient is 0127ms when the angle is 0∘ and 180∘ betweenthe flow direction and the horizontal orientation (because theangle between the two direction is 180∘ so they are the sameone flow field) the minimum value of the permeability coef-ficient is 0076ms when the angle is 40∘ and 220∘ betweenthe flow direction and the horizontal orientation Thereason for this phenomenon is that the calculation in Table 2is derived from the data of the fracture network in Table 1where 0∘ (180∘) and 90∘ (270∘) are two groups of orthogonalfractures In Table 1 at four points at angles 0∘ (180∘) and 90∘(270∘) there will be a local maximum valueThe permeabilitycoefficient at 0∘ (180∘) is the local maximum value of fracturesin the 0∘ (180∘) direction and the permeability coefficient at40∘ (220∘) is the minimum value between the two local max-imum values
52 The Polar Coordinate Fitting of the Permeability Coeffi-cient Assuming that point P is any one point on the ellipsein the polar coordinate system A B are respectively the twoendpoints and C is the focus of the ellipse The semisimmajoraxis and the semisimminor axis of the ellipse are assumed as 119886and 119887 respectively This results in
11990921198862 + 1199102
1198872 = 1119909 = 120588 cos 120579119910 = 120588 sin 120579
(7)
Assuming that = 119879 + 120579 119879 is a parameter that is theangle between the principal axis of the ellipse and the polarcoordinate 0∘ axis
Solve (7) where the ellipse equation of the permeabilitycoefficient in the polar coordinate system is
120588 = 119886119887radic1198862 minus (1198862 minus 1198872) cos2 (120579 + 119879) (8)
Draw the ellipse according to Table 2 and (8) in the polarcoordinate system and fit it as shown in Figure 6 Fittingparameters and fitting values are shown in Table 3
Geofluids 5
(a) All the fractures (b) 10m times 05m
(c) 20m times 10m (d) 30m times 15m
(e) 60m times 30m (f) 70m times 35m
Figure 4 Sketch of fracture of research area
0
45
90
135
180
225
270
315
0 01 02 03 04 05
Figure 5 Comprehensive comparison chart of permeability coeffi-cients of the five research areas
The semisimmajor axis and the semisimminor axis of thefitting ellipse are 119886 = 03121083582 119887 = 0298487332respectively where 119879 = 2359065441 radians and the fittingequation is
120588= 0093160391radic0097411627 minus 000831694cos2 (120579 + 2359065441)
(9)
0
45
90
135
180
225
270
315
0 01 02 0403
Figure 6 Curve fitting
6 Determination of the REV
Ohman et al have conducted a considerable amount of usefulresearch [41ndash43] on the equivalent medium in the fracturedrocks to compare the similarity between the equivalent per-meability coefficient of numerical simulation and the ellipsein different area of studies These two scholars have obtained
6 Geofluids
Table 2 Geometry parameters of the simulation fractures
RMS is the fitting correlation coefficient of the ellipse119870119878(120579) is the simulation value and 119870119891(120579) is the fitting valueDepending on its similarity to the elliptic curve RMS can
be divided into three categories
(1) When RMS le 02 the size of the area of study can beused as the REV of the fractured rock mass
(2) When 04gtRMSgt 02 the size of the area of study canbe used as the REV of the fractured rock mass undercertain conditions
(3) When RMS ge 04 the size of the area of study cannotbe used as the REV of the fractured rock mass
Therefore after comparing Tables 2 and 3 it can begleaned that the fitting result is quite ideal with the correla-tion coefficient RMS = 0132064722TheREV of the fracturedrock mass of this kind exists whose size is 10m times 05m
Further research shows that the REV of very densespacing (lt002m) of the fractured rock mass exists and itis less than 10m times 05m The REV of the dense spacing(006sim02m) of the fractured rock mass is 20m times 10m TheREV of medium spacing (02sim06m) of the fractured rock
Geofluids 7
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
3 Analytical Solution for Validation of theSoftware FractureToKarst
Assume that there is a fracture network as shown in Figure2 the right and left boundaries being the given head borderand the upper and lower two boundaries being the imperme-able boundary the start-point and end-point coordinatesof the fractures are [(1 0) (1 2)] [(05 15) (3 0)] [(0 08)(4 08)]
Assume the hydraulic head of two kinds of in and out ofboundary values1198671 = 2051198672 = 05 respectively Using theprinciple of water balance and the cubic flow law there is
If 119860 = 120588119892119861312120583 there are1198671 = 2051198672 = 051198601198676 minus 119867311987163 = 01198601198678 minus 119867411987184 = 01198601198677 minus 119867511987175 = 0119860(1198671 minus 119867611987116 + 1198673 minus 119867611987136 + 1198678 minus 119867611987186 + 1198677 minus 119867611987176 ) = 0119860(1198676 minus 119867711987167 + 1198678 minus 119867711987187 + 1198675 minus 119867711987157 ) = 0119860(1198676 minus 119867811987168 + 1198674 minus 119867811987148 + 1198672 minus 119867811987128 + 1198677 minus 119867811987178 )= 0
(4)
Type 1198671sim1198678 is the water head 119871 119894119895 is the length of fracturesection between the two nodes 120588 is the density of water 119892is the gravitational acceleration 120583 is flow dynamic viscositycoefficient
Through (4) we can obtain that the values of 119871 119894119895 and thehydraulic head values for each node are
The equivalent permeability coefficient of the flow direc-tion 119896 is obtained through Darcyrsquos law
119896 = 01173ms (6)
Use FractureToKarst to build fracture network and inputparameters the value of the water head and the equivalentpermeability coefficient are shown in Figure 3
The calculation result in that the program equals theman-ual computation result proving that the program is correct
4 Simulation of the Fracture Network
According to the standard of rock classification establishedby ISRM fractured rock mass with medium tractility refersto rock mass whose trace length is more than 3m and lessthan 10m Because the permeability of two sets of orthog-onal fracture rock masses is closest to being isotropic thedescribed ellipse is closest to being a circle
According to the fracture spacing classification of ISRMthe spacing within 20sim60mm is very dense spacing Within10m times 10m two sets of orthogonal fractures were generatedthe average spacing is 006m and the aperture is 00001mThe two sets of fracture parameters are shown in Table 1 (twosets of fracture identification for I and II in Table 1) Here putsome fractures in the same direction as a set of fractures Thedistribution types of the trace length and the direction arethe normal distribution and the gap width is the logarithmicnormal distribution The right and left boundaries are the
4 Geofluids
Table 1 Input data of the simulation fractures
Geometric parameters Average Standard deviation Minimum Maximum
Figure 3 Calculation results of the FractureToKarst
given head border and the upper and lower two boundariesare the impermeable boundary
FromTable 1 two sets of orthogonal fractures can be builtThe first set has 276 fractures and the second set has 208 frac-tures with a total of 484 generating fractures The diagramof generated fractures is shown in Figure 4(a) The equiva-lent permeability coefficient of the study area changes withthe scope of the study without changes so the study area isuniform Within 10m times 10m select five study areas of 10mtimes 05m 20m times 10m 30m times 15m 40m times 20m and60m times 30m and the diagram of generated fractures ofFractureToKarst is shown in Figures 4(b)ndash4(f)
Acquire the permeability coefficients of all the directionsin the five regions and make a comprehensive comparisonchart of permeability coefficients (shown in Figure 5) Ana-lyzing Figure 5 it can be concluded that every equivalent per-meability coefficient is basically stable when the area of studyis larger than 10m times 05mThere is no dramatical change forthe permeability coefficients in the four regions 20mtimes 10m30m times 15m 40m times 20m and 60m times 30m so more than10mtimes 05m in the area of study can be approximately viewedas a homogeneous medium region
In Figure 5 the meaning of all the symbols as followsldquordquo is the area of study of 10m times 05m ldquoerdquo is the area of
study of 20mtimes 10m ldquo998810rdquo is the area of study of 30mtimes 15mldquoIrdquo is the area of study of 60m times 30m ldquordquo is the area ofstudy of 70m times 35m
5 The Fitting Calculation ofthe Uniform Basin
51 Calculation of Regional Rotation As the uniform basin10mtimes 05mwas determined above taking the initial angle as
the angle between the horizontal direction and the flowdirection rotate thewhole area of study clockwise to calculatean equivalent permeability coefficient every 10∘ then eacharea of study has 36 equivalent permeability coefficients asshown in Table 2
Table 2 shows that themaximumvalue of the permeabilitycoefficient is 0127ms when the angle is 0∘ and 180∘ betweenthe flow direction and the horizontal orientation (because theangle between the two direction is 180∘ so they are the sameone flow field) the minimum value of the permeability coef-ficient is 0076ms when the angle is 40∘ and 220∘ betweenthe flow direction and the horizontal orientation Thereason for this phenomenon is that the calculation in Table 2is derived from the data of the fracture network in Table 1where 0∘ (180∘) and 90∘ (270∘) are two groups of orthogonalfractures In Table 1 at four points at angles 0∘ (180∘) and 90∘(270∘) there will be a local maximum valueThe permeabilitycoefficient at 0∘ (180∘) is the local maximum value of fracturesin the 0∘ (180∘) direction and the permeability coefficient at40∘ (220∘) is the minimum value between the two local max-imum values
52 The Polar Coordinate Fitting of the Permeability Coeffi-cient Assuming that point P is any one point on the ellipsein the polar coordinate system A B are respectively the twoendpoints and C is the focus of the ellipse The semisimmajoraxis and the semisimminor axis of the ellipse are assumed as 119886and 119887 respectively This results in
11990921198862 + 1199102
1198872 = 1119909 = 120588 cos 120579119910 = 120588 sin 120579
(7)
Assuming that = 119879 + 120579 119879 is a parameter that is theangle between the principal axis of the ellipse and the polarcoordinate 0∘ axis
Solve (7) where the ellipse equation of the permeabilitycoefficient in the polar coordinate system is
120588 = 119886119887radic1198862 minus (1198862 minus 1198872) cos2 (120579 + 119879) (8)
Draw the ellipse according to Table 2 and (8) in the polarcoordinate system and fit it as shown in Figure 6 Fittingparameters and fitting values are shown in Table 3
Geofluids 5
(a) All the fractures (b) 10m times 05m
(c) 20m times 10m (d) 30m times 15m
(e) 60m times 30m (f) 70m times 35m
Figure 4 Sketch of fracture of research area
0
45
90
135
180
225
270
315
0 01 02 03 04 05
Figure 5 Comprehensive comparison chart of permeability coeffi-cients of the five research areas
The semisimmajor axis and the semisimminor axis of thefitting ellipse are 119886 = 03121083582 119887 = 0298487332respectively where 119879 = 2359065441 radians and the fittingequation is
120588= 0093160391radic0097411627 minus 000831694cos2 (120579 + 2359065441)
(9)
0
45
90
135
180
225
270
315
0 01 02 0403
Figure 6 Curve fitting
6 Determination of the REV
Ohman et al have conducted a considerable amount of usefulresearch [41ndash43] on the equivalent medium in the fracturedrocks to compare the similarity between the equivalent per-meability coefficient of numerical simulation and the ellipsein different area of studies These two scholars have obtained
6 Geofluids
Table 2 Geometry parameters of the simulation fractures
RMS is the fitting correlation coefficient of the ellipse119870119878(120579) is the simulation value and 119870119891(120579) is the fitting valueDepending on its similarity to the elliptic curve RMS can
be divided into three categories
(1) When RMS le 02 the size of the area of study can beused as the REV of the fractured rock mass
(2) When 04gtRMSgt 02 the size of the area of study canbe used as the REV of the fractured rock mass undercertain conditions
(3) When RMS ge 04 the size of the area of study cannotbe used as the REV of the fractured rock mass
Therefore after comparing Tables 2 and 3 it can begleaned that the fitting result is quite ideal with the correla-tion coefficient RMS = 0132064722TheREV of the fracturedrock mass of this kind exists whose size is 10m times 05m
Further research shows that the REV of very densespacing (lt002m) of the fractured rock mass exists and itis less than 10m times 05m The REV of the dense spacing(006sim02m) of the fractured rock mass is 20m times 10m TheREV of medium spacing (02sim06m) of the fractured rock
Geofluids 7
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
Figure 3 Calculation results of the FractureToKarst
given head border and the upper and lower two boundariesare the impermeable boundary
FromTable 1 two sets of orthogonal fractures can be builtThe first set has 276 fractures and the second set has 208 frac-tures with a total of 484 generating fractures The diagramof generated fractures is shown in Figure 4(a) The equiva-lent permeability coefficient of the study area changes withthe scope of the study without changes so the study area isuniform Within 10m times 10m select five study areas of 10mtimes 05m 20m times 10m 30m times 15m 40m times 20m and60m times 30m and the diagram of generated fractures ofFractureToKarst is shown in Figures 4(b)ndash4(f)
Acquire the permeability coefficients of all the directionsin the five regions and make a comprehensive comparisonchart of permeability coefficients (shown in Figure 5) Ana-lyzing Figure 5 it can be concluded that every equivalent per-meability coefficient is basically stable when the area of studyis larger than 10m times 05mThere is no dramatical change forthe permeability coefficients in the four regions 20mtimes 10m30m times 15m 40m times 20m and 60m times 30m so more than10mtimes 05m in the area of study can be approximately viewedas a homogeneous medium region
In Figure 5 the meaning of all the symbols as followsldquordquo is the area of study of 10m times 05m ldquoerdquo is the area of
study of 20mtimes 10m ldquo998810rdquo is the area of study of 30mtimes 15mldquoIrdquo is the area of study of 60m times 30m ldquordquo is the area ofstudy of 70m times 35m
5 The Fitting Calculation ofthe Uniform Basin
51 Calculation of Regional Rotation As the uniform basin10mtimes 05mwas determined above taking the initial angle as
the angle between the horizontal direction and the flowdirection rotate thewhole area of study clockwise to calculatean equivalent permeability coefficient every 10∘ then eacharea of study has 36 equivalent permeability coefficients asshown in Table 2
Table 2 shows that themaximumvalue of the permeabilitycoefficient is 0127ms when the angle is 0∘ and 180∘ betweenthe flow direction and the horizontal orientation (because theangle between the two direction is 180∘ so they are the sameone flow field) the minimum value of the permeability coef-ficient is 0076ms when the angle is 40∘ and 220∘ betweenthe flow direction and the horizontal orientation Thereason for this phenomenon is that the calculation in Table 2is derived from the data of the fracture network in Table 1where 0∘ (180∘) and 90∘ (270∘) are two groups of orthogonalfractures In Table 1 at four points at angles 0∘ (180∘) and 90∘(270∘) there will be a local maximum valueThe permeabilitycoefficient at 0∘ (180∘) is the local maximum value of fracturesin the 0∘ (180∘) direction and the permeability coefficient at40∘ (220∘) is the minimum value between the two local max-imum values
52 The Polar Coordinate Fitting of the Permeability Coeffi-cient Assuming that point P is any one point on the ellipsein the polar coordinate system A B are respectively the twoendpoints and C is the focus of the ellipse The semisimmajoraxis and the semisimminor axis of the ellipse are assumed as 119886and 119887 respectively This results in
11990921198862 + 1199102
1198872 = 1119909 = 120588 cos 120579119910 = 120588 sin 120579
(7)
Assuming that = 119879 + 120579 119879 is a parameter that is theangle between the principal axis of the ellipse and the polarcoordinate 0∘ axis
Solve (7) where the ellipse equation of the permeabilitycoefficient in the polar coordinate system is
120588 = 119886119887radic1198862 minus (1198862 minus 1198872) cos2 (120579 + 119879) (8)
Draw the ellipse according to Table 2 and (8) in the polarcoordinate system and fit it as shown in Figure 6 Fittingparameters and fitting values are shown in Table 3
Geofluids 5
(a) All the fractures (b) 10m times 05m
(c) 20m times 10m (d) 30m times 15m
(e) 60m times 30m (f) 70m times 35m
Figure 4 Sketch of fracture of research area
0
45
90
135
180
225
270
315
0 01 02 03 04 05
Figure 5 Comprehensive comparison chart of permeability coeffi-cients of the five research areas
The semisimmajor axis and the semisimminor axis of thefitting ellipse are 119886 = 03121083582 119887 = 0298487332respectively where 119879 = 2359065441 radians and the fittingequation is
120588= 0093160391radic0097411627 minus 000831694cos2 (120579 + 2359065441)
(9)
0
45
90
135
180
225
270
315
0 01 02 0403
Figure 6 Curve fitting
6 Determination of the REV
Ohman et al have conducted a considerable amount of usefulresearch [41ndash43] on the equivalent medium in the fracturedrocks to compare the similarity between the equivalent per-meability coefficient of numerical simulation and the ellipsein different area of studies These two scholars have obtained
6 Geofluids
Table 2 Geometry parameters of the simulation fractures
RMS is the fitting correlation coefficient of the ellipse119870119878(120579) is the simulation value and 119870119891(120579) is the fitting valueDepending on its similarity to the elliptic curve RMS can
be divided into three categories
(1) When RMS le 02 the size of the area of study can beused as the REV of the fractured rock mass
(2) When 04gtRMSgt 02 the size of the area of study canbe used as the REV of the fractured rock mass undercertain conditions
(3) When RMS ge 04 the size of the area of study cannotbe used as the REV of the fractured rock mass
Therefore after comparing Tables 2 and 3 it can begleaned that the fitting result is quite ideal with the correla-tion coefficient RMS = 0132064722TheREV of the fracturedrock mass of this kind exists whose size is 10m times 05m
Further research shows that the REV of very densespacing (lt002m) of the fractured rock mass exists and itis less than 10m times 05m The REV of the dense spacing(006sim02m) of the fractured rock mass is 20m times 10m TheREV of medium spacing (02sim06m) of the fractured rock
Geofluids 7
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
Figure 5 Comprehensive comparison chart of permeability coeffi-cients of the five research areas
The semisimmajor axis and the semisimminor axis of thefitting ellipse are 119886 = 03121083582 119887 = 0298487332respectively where 119879 = 2359065441 radians and the fittingequation is
120588= 0093160391radic0097411627 minus 000831694cos2 (120579 + 2359065441)
(9)
0
45
90
135
180
225
270
315
0 01 02 0403
Figure 6 Curve fitting
6 Determination of the REV
Ohman et al have conducted a considerable amount of usefulresearch [41ndash43] on the equivalent medium in the fracturedrocks to compare the similarity between the equivalent per-meability coefficient of numerical simulation and the ellipsein different area of studies These two scholars have obtained
6 Geofluids
Table 2 Geometry parameters of the simulation fractures
RMS is the fitting correlation coefficient of the ellipse119870119878(120579) is the simulation value and 119870119891(120579) is the fitting valueDepending on its similarity to the elliptic curve RMS can
be divided into three categories
(1) When RMS le 02 the size of the area of study can beused as the REV of the fractured rock mass
(2) When 04gtRMSgt 02 the size of the area of study canbe used as the REV of the fractured rock mass undercertain conditions
(3) When RMS ge 04 the size of the area of study cannotbe used as the REV of the fractured rock mass
Therefore after comparing Tables 2 and 3 it can begleaned that the fitting result is quite ideal with the correla-tion coefficient RMS = 0132064722TheREV of the fracturedrock mass of this kind exists whose size is 10m times 05m
Further research shows that the REV of very densespacing (lt002m) of the fractured rock mass exists and itis less than 10m times 05m The REV of the dense spacing(006sim02m) of the fractured rock mass is 20m times 10m TheREV of medium spacing (02sim06m) of the fractured rock
Geofluids 7
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
RMS is the fitting correlation coefficient of the ellipse119870119878(120579) is the simulation value and 119870119891(120579) is the fitting valueDepending on its similarity to the elliptic curve RMS can
be divided into three categories
(1) When RMS le 02 the size of the area of study can beused as the REV of the fractured rock mass
(2) When 04gtRMSgt 02 the size of the area of study canbe used as the REV of the fractured rock mass undercertain conditions
(3) When RMS ge 04 the size of the area of study cannotbe used as the REV of the fractured rock mass
Therefore after comparing Tables 2 and 3 it can begleaned that the fitting result is quite ideal with the correla-tion coefficient RMS = 0132064722TheREV of the fracturedrock mass of this kind exists whose size is 10m times 05m
Further research shows that the REV of very densespacing (lt002m) of the fractured rock mass exists and itis less than 10m times 05m The REV of the dense spacing(006sim02m) of the fractured rock mass is 20m times 10m TheREV of medium spacing (02sim06m) of the fractured rock
Geofluids 7
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
mass is 100m times 50m The REV is not the existence of thefractured rock mass of wide spacing (06sim20m) with verywide spacing (20sim60m) and utmost spacing (gt06m)because their fractures have no connection and have nohydraulic conductivity and thus the REV does not exist
7 Conclusions and Suggestions
In conclusion the existence of REV is closely related to thefracture conditions Not all types of fractured rock mass haveREV The more intensive the fractures are the better thepenetration is the better the permeability of the rocks iswhich means the easier it is to become equivalent to porousmedia Thus the following conclusions can be made
(1) The existence condition of the REV for fracturedrock mass with medium tractility is that the averagespacing of the fractures should be less than or equal to06m that is to say the size of the REV of extremelydense spacing of the fractured rock mass is less than10m times 05m the size of the REV of very densespacing of the fractured rockmass is 10mtimes 05m thesize of the REV of dense spacing of the fractured rockmass is 20m times 10m the size of the REV of mediumspacing of the fractured rock mass is 100m times 50m
(2) When the average spacing of the fractures is morethan 06m the REV of wide spacing and very widespacing and extremely wide spacing of the fracturedrock mass does not exist
(3) Although this study has obtained certain achievementin the scale effects of the REV of the discrete fracturemedium there are still some shortcomings includingthe influence of the different distribution of the geo-metric elements and the influence of the distributionof the different gap lengths which all require furtherdiscussion
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was financially supported by the Foundation forTeachers from China Earthquake Administration (Grant no20150102) by the Foundation for Hebei Province Science andTechnology Planning Project (Grant no 162776436) and bythe National Natural Science Foundation of China (Grant no41272387)
References
[1] J BearDynamics of Fluids in Porous Media Elsevier New YorkUSA 1972
[2] J C S Long and P A Witherspoon ldquoPorous media equiva-lents for networks of discontinuous fracturesrdquoWater ResourcesResearch vol 18 no 3 pp 645ndash658 1982
[3] M Hassanizadeh and W G Gray ldquoGeneral conservation equa-tions for multi-phase systems 1 averaging procedurerdquo in Flow
through porous media recent developments a computationalmechanics publications G F Pinder Ed pp 1ndash16 1983
[4] H H Haldorsen ldquoSimulator parameter assignment and theproblem of scale in reservoir engineeringrdquo in Reservoir char-acterization L W Lake and H B Caroll Eds pp 293ndash340Academic Press Orlando USA 1986
[5] G H Shi ldquoManifold method of material analysisrdquo in Transac-tion of the 9th Army Conference on Applied Mathematics andComputing pp 57ndash76 USA 1991
[6] G H Shi ldquoManifold methodrdquo in Manifold method Proceedingof first International Forum on Discontinuous DeformationAnalysis (DDA) and Simulations of Discontinuous Media TSIPress pp 52ndash204 Albuquerque NM Mexico USA 1996
[7] G Shi ldquoProducing joint polygons cutting joint blocks andfinding key blocks for general free surfacesrdquo Chinese Journal ofRock Mechanics and Engineering vol 25 no 11 pp 2161ndash21702006
[8] A Hurst ldquoSedimentary flow units in hydrocarbon reservoirssome shortcomings and a case for highresolution permeabilitydatardquo in The geological modelling of hydrocarbon reservoirs andoutcrop analogues S Flint and I D Bryant Eds pp 191ndash2041993 Special publication No 15 of the international associationof sedimentologists
[9] B Noetinger ldquoThe effective permeability of a heterogeneousporous mediumrdquo Transport in Porous Media vol 15 no 2 pp99ndash127 1994
[10] P S Ringrose Pickup G E J Jensen L and M Forrester ldquoTheArdross reservoir gridblock analogue sedimentology statisticalrepresentative and flow upsclaingrdquo inReservoir characterizationrecent advances Am assoc petrol geol memoir R Schatzingerand J Jordan Eds vol 71 pp 265ndash276 The Ardross reservoirgridblock analogue sedimentology 1999
[11] P S Ringrose E Skjetne and C Elfenbein ldquoPermeability Esti-mation Functions Based on Forward Modeling of SedimentaryHeterogeneityrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colo USA October 2003
[12] P Ringrose K Nordahl and R Wen ldquoVertical permeabilityestimation in heterolithic tidal deltaic sandstonesrdquo PetroleumGeoscience vol 11 no 1 pp 29ndash36 2005
[13] K Nordahl and P Ringrose Identifying the representativeelementary volume for permeability in heterolithic deposits usingnumerical rock models Math Geosci vol 40 2008
[14] R Al-Raoush and A Papadopoulos ldquoRepresentative elemen-tary volume analysis of porous media using X-ray computedtomographyrdquo Powder Technology vol 200 no 1-2 pp 69ndash772010
[15] L Xia M Li Y Chen Y Zheng and Q Yu ldquoBlockiness level ofrock mass around underground powerhouse of Three GorgesProjectrdquo Tunnelling and Underground Space Technology vol 48pp 67ndash76 2015
[16] L Xia Y Zheng and Q Yu ldquoEstimation of the REV size forblockiness of fractured rock massesrdquo Computers and Geotech-nics vol 76 pp 83ndash92 2016
[17] K Esmaieli J Hadjigeorgiou and M Grenon ldquoEstimatinggeometrical and mechanical REV based on synthetic rockmass models at Brunswick Minerdquo International Journal of RockMechanics andMining Sciences vol 47 no 6 pp 915ndash926 2010
[18] A M Shapiro and J Andersson ldquoSteady state fluid responsein fractured rock A boundary element solution for a coupleddiscrete fracture continuum modelrdquoWater Resources Researchvol 19 no 4 pp 959ndash969 1983
8 Geofluids
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
[19] C-H Lee B-WDeng and J-L Chang ldquoA continuumapproachfor estimating permeability in naturally fractured rocksrdquo Engi-neering Geology vol 39 no 1-2 pp 71ndash85 1995
[20] K-B Min L Jing and O Stephansson ldquoDetermining theequivalent permeability tensor for fractured rock masses usinga stochastic REV approach Method and application to the fielddata from Sellafield UKrdquo Hydrogeology Journal vol 12 no 5pp 497ndash510 2004
[21] P Blum R Mackay M S Riley and J L Knight ldquoPerformanceassessment of a nuclear waste repository upscaling coupledhydro-mechanical properties for far-field transport analysisrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 42 no 5-6 pp 781ndash792 2005
[22] P Blum R Mackay M Riley and J Knight ldquoHydraulischeModellierung und die Ermittlung des reprasentativen Elemen-tarvolumens (REV) im Kluftgesteinrdquo Grundwasser vol 12 no1 pp 48ndash65 2007
[23] A Baghbanan and L Jing ldquoHydraulic properties of fracturedrock masses with correlated fracture length and aperturerdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 44 no 5 pp 704ndash719 2007
[24] A Desbarats ldquoSupport effects and the spatial averaging oftransport propertiesrdquo Mathematical Geology vol 21 no 3 pp383ndash389 1989
[25] S H Begg R R Carter and P Dranfield ldquoAssigning effectivevalues to simulator gridblock parameters for heterogeneousreservoirsrdquo SPE Reservoir Engineering (Society of PetroleumEngineers) pp 455ndash463 1989
[27] A Henriette C G Jacquin and P M Adler ldquoThe effective per-meability of heterogeneous porous mediardquo Phys Chem Hydro-dyn vol 11 no 1 pp 63ndash80 1989
[28] R Norris and J Lewis ldquoThe Geological Modeling of EffectivePermeability in Complex Heterolithic Faciesrdquo in Proceedings ofthe 66th annual technical conference and exhibition Dallas TXUSA 1991
[29] B Noetinger and C Jacquin ldquoExperimental Tests of a SimplePermeability Composition Formulardquo Society of petroleum engi-neers preprint SPE 22841 1991
[30] P Corbett S Anggraeni and D Bowen ldquoThe use of the probepermeameter in carbonatesaddressing the problems of per-meability support and stationarityrdquo Log Analyst vol 40 no 5pp 316ndash326 1999
[31] V C Tidwell and J L Wilson ldquoPermeability upscaling mea-sured on a block of Berea Sandstone Results and interpreta-tionrdquoMathematical Geology vol 31 no 7 pp 749ndash769 1999
[32] V C Tidwell and J L Wilson ldquoHeterogeneity permeabilitypatterns and permeability upscaling Physical characterizationof a block of Massillon sandstone exhibiting nested scales ofheterogeneityrdquo SPE Reservoir Evaluation and Engineering vol3 no 4 pp 283ndash291 2000
[33] Y Bernabe U Mok B Evans and F J Herrmann ldquoPerme-ability and storativity of binary mixtures of high- And low-permeability materialsrdquo Journal of Geophysical Research B SolidEarth vol 109 no 12 pp 1ndash12 2004
[34] J M McKinley C D Lloyd and A H Ruffell ldquoUse of variogra-phy in permeability characterization of visually homogeneoussandstone reservoirs with examples from outcrop studiesrdquoMathematical Geology vol 36 no 7 pp 761ndash779 2004
[35] M D Jackson A H Muggeridge S Yoshida and H D John-son ldquoUpscaling permeability measurements within complexheterolithic tidal sandstonesrdquoMath Geol vol 35 no 5 pp 446ndash454 2003
[36] M D Jackson S Yoshida A H Muggeridge and H D John-son ldquoThree-dimensional reservoir characterization and flowsimulation of heterolithic tidal sandstonesrdquo AAPG Bulletin vol89 no 4 pp 507ndash528 2005
[37] D T Snow A parallel plate model of fractured permeable me-dia [dissertation] [dissertation thesis] Berkeley Univ of Calif[dissertation] 1965
[38] J H Li and L M Zhang ldquoGeometric parameters and REV of acrack network in soilrdquo Computers and Geotechnics vol 37 no4 pp 466ndash475 2010
[39] J C Long and P A Witherspoon ldquoThe relationship of thedegree of interconnection to permeability in fracture networksrdquoJournal of Geophysical Research vol 90 no B4 pp 3087ndash30971985
[40] Q Yu D Chen and G Xue ldquoHydrodynamics of discontinuousfracture networkrdquo Earth Science-journal of China University ofGeosciences vol 20 no 4 pp 474ndash478 1995
[41] J Ohman and A Niemi ldquoUpscaling of fracture hydraulics bymeans of an oriented correlated stochastic continuum modelrdquoWater Resources Research vol 39 no 10 pp 1277ndash1289 2003
[42] J Ohman A Niemi and C-F Tsang ldquoA regional-scale particle-tracking method for nonstationary fractured mediardquo WaterResources Research vol 41 no 3 pp 1ndash16 2005
[43] J Ohman A Niemi and C-F Tsang ldquoProbabilistic estima-tion of fracture transmissivity from Wellbore hydraulic dataaccounting for depth-dependent anisotropic rock stressrdquo Inter-national Journal of RockMechanics andMining Sciences vol 42no 5-6 pp 793ndash804 2005
