Research Article A Model of Anisotropic Property of Seepage and …downloads.hindawi.com/journals/jam/2013/420536.pdf · for representing mechanical behavior of the fractured rock

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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 420536 19 pageshttpdxdoiorg1011552013420536

Research ArticleA Model of Anisotropic Property of Seepage and Stress forJointed Rock Mass

Pei-tao Wang12 Tian-hong Yang12 Tao Xu12 Qing-lei Yu12 and Hong-lei Liu12

1 Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines Northeastern University Shenyang 110819 China2 School of Resources amp Civil Engineering Northeastern University Shenyang 110819 China

Correspondence should be addressed to Tian-hong Yang yang tianhong126com

Received 30 March 2013 Revised 18 June 2013 Accepted 19 June 2013

Academic Editor Pengcheng Fu

Copyright copy 2013 Pei-tao Wang 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

Joints often have important effects on seepage and elastic properties of jointed rock mass and therefore on the rock slope stabilityIn the present paper a model for discrete jointed network is established using contact-freemeasurement technique and geometricalstatistic method A coupled mathematical model for characterizing anisotropic permeability tensor and stress tensor was presentedand finally introduced to a finite element model A case study of roadway stability at the Heishan Metal Mine in Hebei ProvinceChina was performed to investigate the influence of joints orientation on the anisotropic properties of seepage and elasticity ofthe surrounding rock mass around roadways in underground mining In this work the influence of the principal direction of themechanical properties of the rock mass on associated stress field seepage field and damage zone of the surrounding rock mass wasnumerically studiedThe numerical simulations indicate that flow velocity water pressure and stress field are greatly dependent onthe principal direction of joint planes It is found that the principal direction of joints is the most important factor controlling thefailure mode of the surrounding rock mass around roadways

1 Introduction

Underground mining has been considered a high-risk activ-ity worldwide Violent roof failure or rock burst inducedby mining has always been a serious threat to the safetyand efficiency of mines in China Accurate and detailedcharacterization for rockmasses can control stable excavationspans support requirements cavability and subsidence char-acteristics and thus influence the design of mining layoutsand safety of mines Rock mass is a geologic body composingof the discontinuities which have a critical influence ondeformational behavior of blocky rock systems [1] Themechanical behavior of this material depends principally onthe state of intact rock whose mechanical properties couldbe determined by laboratory tests and existing discontinuitiescontaining bedding planes faults joints and other structuralfeatures The distributions and strength of these discontinu-ities are both the key influencing factors for characterizing thediscontinuous and anisotropic materials Amadei [2] pointedout the importance of anisotropy of jointed rock mass anddiscussed the interaction existing between rock anisotropy

and rock stress Because of computational complexity and thedifficulty of determining the necessary elastic constants itis usual for only the simplest form of anisotropy transverseisotropy to be used in design analysis [3] The key work is tostudy the principal direction of elasticity or permeability andthen assume the jointed rock mass as transversely isotropicgeomaterial

Extensive efforts have been made to investigate themechanical response of transversely anisotropic rock mate-rial Zhang and Sanderson [4] used the fractal dimensionto describe the connectivity and compactness of fracturenetwork and found that the deformability and the overallpermeability of fractured rock masses increase greatly withincreasing fracture density By using the artificial transverselyisotropic rock blocks the mechanical properties with differ-ent dip angles were obtained by Tien and Tsao [5] Broschet al [6] evaluated the fabric-dependent anisotropy of aparticular gneiss by studying the strength values and elasticparameters along different directions Exadaktylos andKaklis[7] presented the explicit representations of stresses and

2 Journal of Applied Mathematics

strains at any point of the anisotropic circular disc com-pressed diametrically The stress concentration in uniaxialcompression for a plane strain model was investigated and aformulation of failure criteria for elastic-damage and elastic-plastic anisotropic geomaterials is formed by Exadaktylos [8]Amethodology to determine the equivalent elastic propertiesof fractured rockmasses was established by explicit represen-tations of stochastic fracture systems [9] and the conditionsfor the application of the equivalent continuum approachfor representing mechanical behavior of the fractured rockmasses were investigated Such anisotropy has an effect onthe interpretation of stress distributions Amadeirsquos resultsindicate that an anisotropy ratio (damaged elastic modulusto intact elastic modulus) of between 114 and 133 will havea definite effect on the interpreted in situ state of stress[10] Hakala et al [11] concluded that the anisotropy ratio ofFinland rock specimens is about 14 and suggested to be takeninto account in the interpretation of stress measurementresults

Moreover in rock engineering like mining and tunnelengineering interactions between in situ stress and seepagepressure of groundwater have an important role Groundwa-ter under pressure in the joints defining rock blocks reducesthe normal effective stress between the rock surfaces andtherefore reduces the potential shear resistance which can bemobilized by friction Since rock behaviormay be determinedby its geohydrological environment it may be essentialin some cases to maintain close control of groundwaterconditions in the mine area Therefore accurate descriptionof joints is an important topic for estimating and evaluatingthe deformability and seepage properties of rock masses

Many researchers have done a lot of studies on theseepage and anisotropic mechanical behaviors of the jointedrock mass Using homogeneous samples like granite orbasalt Witherspoon et al [12] investigated and definedthe permeability by fracture aperture in a closed fractureBased on geometrical statistics Oda [13] has studied anddetermined the crack tensor of moderately jointed graniteby treating statistically the crack orientation data via astereographic projection Using Odarsquos method Sun and Zhao[14] determined the anisotropy in permeability using thefracture orientation and the in situ stress information fromthe field survey For fractured rock system attempts havebeen made by Jing et al [15ndash18] and Bao et al [19] toinvestigate the permeability Using UDEC code Jing et alstudied the permeability of discrete fracture network suchas the existence of REV in [15] relations between fracturelength and aperture in [16] or stress effect on permeability in[17] or solute transport in [18] Bao et al [19] have discussedthe mesh effect on effective permeability for a fracturedsystem using the upscaled permeability field It would helpthe workers to spend less computational effort and memoryrequirement to investigate effective permeability The key fordetermining deformationmodulus and hydraulic parametersis to study representative elementary volume (REV) andscale effects of fractured rock mass [9 20ndash22] However thedifficulty of testing jointed rock specimens at scales sufficientto represent the equivalent continuum indicated that it isnecessary to postulate and verify methods of synthesizing

rock mass properties from those of the constituent elementslike intact rock and fractures

Although great progress has been made it is difficultto study the anisotropic deformation of large-scale rockmass ground water permeability change and interactionbetween stress and seepage due to the geological complexityof discontinuities Qiao et al [23] have pointed out that therock mass which is not highly fractured and has only few setsof joint system usually behaves anisotropically However themechanical properties directionality of highly jointed rockmass is usually ignored In particular since the limitationof mesh generation the numerical method can hardly dealwith the highly jointed specimensThe discontinuity like faultcould be treated as specific boundary An effective solutionshould be found to characterize the influence of joints on therock mass

In this paper based on equivalent continuum theory andtheoretical analysis a mathematical model for anisotropicproperty of seepage and elasticity of jointed rock mass isdescribed A permeability analysis code is developed toevaluate the anisotropic permeability for DFN model basedon VC++ 60 in this paper The DFN model could beanalyzed to evaluate the REV size and anisotropic propertyof permeability which would provide important evidencefor the finite element model The outline is as follows First3D images involving detailed geometrical properties of rockmass such as trace lengths outcrop areas joint orientationsand joint spacing were captured using ShapMetriX3D sys-tem According to the statistical parameters for each set ofdiscontinuities fracture network is generated using MonteCarlo method and jointed rock samples in different sectionscould be captured Next permeability tensors and elasticitytensors of rock mass of the sample are calculated by discretemedium seepage method and geometrical damage theoryrespectively Finally using the finite element method (FEM)an anisotropic mechanical model for rock mass is built Anengineering practice of roadway stability at theHeishanMetalMine in Hebei Iron and Steel Group Mining Company isdescribed Then the stress and seepage fields surroundingthe roadway were numerically simulated On the basis ofthe modeled results the influences of joint planes on stressseepage and damage zone were analyzed It is expectedthroughout this study to gain an insight into the influencesof discontinuities on the mechanical behavior of rock massand offer some scientific evidence for the design of mininglayouts or support requirements

2 Generation of a Fracture System Model byShapeMetriX3D

Traditional methods for rock mass structural parameters inmining engineering applications include scanline surveying[24] and drilling core method [25] The process generallyrequires physical contact with rock mass exposure and there-fore is hazardous Additionally takingmanual measurementsis time consuming and prone to errors due to sampling diffi-culties or instrument errors ShapeMetriX3D is a tool for thegeological and geotechnical data collection and assessment

Journal of Applied Mathematics 3

Imaging distance Baseline

Rock mass exposure

P2 (u )

P1 (u )

VL (XY Z)

VR (XY Z)

E

S

H

Figure 1 Imaging principle of the 3D surface measurement

for rock masses [26] The equipment is used for the metricacquisition of rock and terrain surfaces and for the contact-free measurement of geologicalgeotechnical parameters bymetric 3D images The images are captured using NikonD80 camera with 223 megapixel As shown in Figure 1 twoimages are acquired to reproduce the 3D rock mass faceDetails information related to the measurements is reportedin [27] by Gaich et al The measurements could be taken atany required number and extent even in regions that arenot accessible During measuring process two digital imagestaken by a calibrated camera serve for a 3D reconstructionof the rock face geometry which is represented on the com-puter by a photorealistic spatial characterization as shownin Figure 1 From it measurements are taken by markingvisible rock mass features such as spatial orientations of jointsurfaces and traces as well as areas lengths or positionsFinally the probability statistical models of discontinuitiesare established It is generally applied in the typical situationssuch as long rock faces at small height and rock slopeswith complex geometries [28] Almost any rock face canbe reconstructed at its optimum resolution by using thisequipment and its matching software By using this systemwe can increase working safety reduce mapping time andimprove data quality

In order to accurately represent rock discontinuitiesShapeMetriX3D (3GSM) in this paper is used for the metricacquisition of rock mass exposure and for the contact-freemeasurement of geological parameters by metric 3D imagesStereoscopic photogrammetry deals with themeasurement ofthree-dimensional information from two images showing thesame object or surface but taken from two different anglesjust as shown in Figure 1

From the determined orientation between the two imagesand a pair of corresponding image points 119875

1(119906 V) and

1198752(119906 V) imaging rays (colored in red) are reconstructed

ES

HProfile I-I

Figure 2 Geological mapping and geometric measurements ofstereoscopic restructuring model

whose intersection leads to a 3D surface point 119875 (119883 119884 119885)By automatic identification of corresponding points withinthe image pair the result of the acquisition is a metric 3Dimage that covers the geometry of the rock exposure Oncethe image of a rock wall is ready geometric measurementscan be taken as shown in Figure 2 There are a total of threegroups of discontinuities in this bench face

Figure 3 is facilitated to show the 3-dimensional distri-bution of the trace along the section of Profile I-I from thestereoscopic model shown in Figure 2 The height of thisrock face is approximately 50m with a distance of 20mperpendicular to the trace The measured orientation in ahemispherical plot could also be captured as in Figure 4(a)Figure 4(b) shows the results of the distribution of joints

The generated 3D rock face is about 5m times 5mwith a highresolution enough to distinguish the fractures According to


02minus18

2

1

0

minus1

minus2

minus3

minus4

Distance (m)

Hei

ght (

m)

Figure 3 Length and sketch of Profile I-I by stereoscopic model

the survey results the geologic data like joint density dipangle trace length and spacing can be acquired Baghbananand Jing [16] generated the DFN models whose orientationsof fracture sets followed the Fisher distribution In thispaper four different probability statistical models are usedto generate the DFN models The dip angle dip directiontrace length and spacing all follow one particular probabilitystatistical model Table 1 shows the basic information aboutthe fracture system parameters Type I of the probabilitystatistical model in Table 1 stands for negative exponentialdistribution Type II for normal distribution Type III forlogarithmic normal distribution and Type IV for uniformdistribution Based on the probability models the particularfracture network could be generated using Monte Carlomethod [29 30]

3 Constitutive Relation ofAnisotropic Rock Mass

Due to the existence of joints and cracks the mechanicalproperties (Youngrsquos modulus Poissonrsquos ratio strength etc)of rock masses are generally heterogeneous and anisotropicThree preconditions should be confirmed in this section(i) anisotropy of rock mass is mainly caused by IV or V-class structure or rock masses containing a large number ofdiscontinuities (ii) rock masses according to (i) could betreated as homogeneous and anisotropic elastic material (iii)seepage tensor and damage tensor of rock mass with multisetof joints could be captured by the scale of representativeelementary volume (REV)

31 Stress Analysis Elasticity represents the most commonconstitutive behavior of engineering materials includingmany rocks and it forms a useful basis for the description of

30∘

60∘

330∘

300∘

120∘

150∘210∘

240∘

EW

N

S

(a)

(b)

Figure 4 Stereographic plot and the distribution of total three setsof joints

more complex behaviorThemost general statement of linearelastic constitutive behavior is a generalized form of HookersquoLaw in which any strain component is a linear function of allthe stress components that is

120576119894119895= [S] 120590119894119895 (1)

where [S] is the flexibility matrix 120576119894119895is the strain and 120590

119894119895is

the stressMany underground excavation design analyses involving

openings where the length to cross-section dimension ratiois high are facilitated considerably by the relative simplicityof the excavation geometry In this section the roadway is


Table 1 Characteristics of 3 sets of discontinuities for the slope exposure

Set Density(mminus1)

Fracture characteristicsDip direction (∘) Dip angle (∘) Trace length (m) Spacing (m)

Type Meanvalue

Standarddeviation Type Mean

valueStandarddeviation Type Mean


valueStandarddeviation

1 12 II 9073 1388 III 7124 1505 II 086 023 IV 083 0752 13 II 28996 2289 IV 5985 84 II 089 028 IV 076 0733 33 III 18905 1455 III 5529 1335 II 085 025 IV 03 031

Z

X

Y

O

Figure 5 A diagrammatic sketch for underground excavation

uniform cross section along the length and could be properlyanalyzed by assuming that the stress distribution is the samein all planes perpendicular to the long axis of the excavation(Figure 5) Thus this problem could be analyzed in terms ofplane geometry

The state of stress at any point can be defined in termsof the plane components of stress (120590

11 12059022 and 120590

12) and

the components (12059033 12059023 and 120590

31) In this research the 119885

direction is assumed to be a principal axis and the antiplane

shear stress components would vanish The plane geometricproblem could then be analyzed in terms of the planecomponents of stress since the 120590

33component is frequently

neglected Equation (1) in this case may be recast in the form

[

[

12057611

12057622

12057612

]

]

= [S] [120590]

=

[[[[[[[[[

[

1

1198641

minus]231

1198643

minus(]12

1198641

+]231

1198643

) 0

minus(]12

1198641

+]231

1198643

)1

1198641

minus]231

1198643

0

0 02 (1 + V

12)

1198641

]]]]]]]]]

]

times [

[

12059011

12059022

12059012

]

]

(2)where [S] is the flexibility matrix of the material under planestrain conditions The inverse matrix [S]minus1 (or [E]) could beexpressed in the form

[

[

12059011

12059022

12059012

]

]

= [E] [120576] =

[[[[[[[[[[

[

1198642

1(1198643minus 1198641V231)

Δ

minus1198642

1(1198643V21

+ 1198641V231)

Δ0

minus1198642

1(1198643V21

+ 1198641V231)

Δ

1198642

1(1198641V231

minus 1198643)

Δ0

0 01198641

2 (1 + V12)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(3)

where Δ is expressed as follows

Δ = minus11986411198643+ 11986411198643V221

+ 1198642

1V231

+ 1198642

1V231

+ 21198642

1V21V231 (4)

The moduli 1198641and 119864

3and Poissonrsquos ratios V

31and

V12

could be provided by uniaxial strength compression ortension in 1 (or 2) and 3 directions

The mechanical tests including laboratory and in situtests for rock masses in large scale can hardly capture

the elastic properties directly Hoek-Brown criterion [31ndash33] could calculate the mechanical properties of weak rocksmasses by introducing the Geological Strength Index (GSI)Nevertheless the anisotropic properties cannot be capturedusing this criterion and thus the method for analyzinganisotropy of jointed rock mass needs a further study

In this paper the original joint damage in rock massis considered as macro damage field In elastic damagemechanics the elastic modulus of the jointed material may


H = hb1

H = hb2

V = 0 V = 0

M

M998400

N

N998400

Figure 6 Fracture network schematic diagram in seepage areaMN andM1015840N1015840 are the constant head boundaries with the hydraulichead of ℎ

1198871and ℎ

1198872respectively MM1015840 and NN1015840 are the impervious

boundaries with flow 119881 of zero

degrade and the Youngrsquos modulus of the damaged element isdefined as follows [34]

119864 = 1198640(1 minus 119863) (5)

where D represents the damage variable and E and 1198640are

the elastic moduli of the damaged and intact rock samplesrespectively In this equation all parameters are scalar

With the geometric information of the fracture samplethe damage tensor [35] could be defined as

119863119894119895=

119897

119881

119873

sum

119896=1

120572(119896)

(119899(119896)

otimes 119899(119896)

) (119894 119895 = 1 2 3) (6)

where N is the number of joints l is the minimum spacingbetween joints V is the volume of rock mass n(119896) is thenormal vector of the kth joint and a(119896) is the trace length ofthe kth joint (for 2 dimensions)

According to the principal of energy equivalence [36] theflexibility matrix for the jointed rock sample can be obtainedas

1198781015840

119894119895= (1 minus 119863

119894)minus1

119878119894119895(1 minus 119863

119895)minus1

(7)

where 119878119894119895is the flexibilitymatrix for intact rock119863

119894and119863

119895are

the principal damage values in 119894 and 119895 directions respectivelyFor plane strain geometric problem the constitutive relationwhere the coordinates and principal damage have the samedirection could be expressed as follows

[

[

12059011

12059022

12059012

]

]

=

[[[[[[[[[[

[

1198640(V0minus 1) (119863

1minus 1)2

2V20+ V0minus 1

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

0

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

1198640(V0minus 1) (119863

2minus 1)2

2V20+ V0minus 1

0

0 01198640(1198631minus 1) (119863

2minus 1)

2 (1 + V0)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(8)

where 1198640is the Youngrsquos modulus of intact rock and V

0is

Poissonrsquos ratio for intact rockThe parameters could be easily captured by laboratory

tests Equation (7) gives the principal damage values 1198631and

1198632 Based on geometrical damage mechanics all elements

in this matrix could be obtained by the method mentionedpreviously and the anisotropic constitutive relation of jointedrock sample could finally be confirmed

32 Seepage Analysis The seepage parameters of rock massare quantized form of permeability and also are the basisto solve seepage field of equivalent continuous mediumBased on the attribute of fractured rock mass and by takingengineering design into account fractured rock mass is oftenconsidered to be anisotropic continuous medium In thefracture network shown in Figure 6 a total number of119873 cross

points or water heads and 119872 line elements are containedParameters can be obtained with the model such as waterhead the related line elements equivalent mechanical fissurewidth seepage coefficient and so on The correspondingcoordinates of each point could be acquired For a fluid flowanalysis based on the law of mass conservation the fluidequations on a certain water head take the form [37]

(

1198731015840

sum

119895=1

119902119895)

119894

+ 119876119894= 0 (119894 = 1 2 119873) (9)

where 119902119895is the quantity of flow from line element 119895 to water

head 119894 1198731015840 is the total number of line elements intersect at i

and119876119894is the fluid source term In the joint network each line

element would be assigned a length 119897119895and fissure width 119887

119895to

investigate the permeability


According to the hydraulic theory [38] for a singlejoint seepage the flow quantity 119902

119895of line element 119895 can be

expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)

whereΔℎ119895is the hydraulic gradient 120583 is the coefficient of flow

viscosity 120588 is the density of water and 119887119895is the fissure width

of joint On the basis of (9) and (10) the governing equationscan be represented as (11) for seepage in fracture network

(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)

Based on the discrete fracture network method [39] anequivalent continuummodel for seepage has been established[28 40 41]The hydraulic conductivity can be acquired basedon Darcyrsquos Law on the basis of water quantity in the network(see (12))

Boundary conditions in the model shown in Figure 6 areas follows

(a) MN andM1015840N1015840 as the constant head boundary(b) MM1015840 and NN1015840 as the impervious boundary with flow

119881 of 0

Then the hydraulic conductivity can be defined as

119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)

where Δ119902 is the total quantity of water in the model region(m2sdotsminus1) Δ119867 is the water pressure difference between inflowand outflow boundaries (m) K is the equivalent hydraulicconductivity coefficient in the MM1015840 direction (msdotsminus1) andMN andM1015840M are the side lengths of the region (m)

Based on Biot equations [42] the steady flow model isgiven by

119870119894119895nabla2119901 = 0 (13)

where119870119894119895is the hydraulic conductivity and p is the hydraulic

pressure For plane problems the dominating equation ofseepage flow is as follows in (14) The direction of jointplanes is considered to be the principal direction of hydraulicconductivity

119870119894119895= [

11987011

11987012

11987021

11987022

] (14)

33 Coupling Mechanism of Seepage and Stress

331 Seepage Inducted by Stress The coupling actionbetween seepage and stress makes the failure mechanismof rock complex The investigations on this problem havepervasive theoretical meaning and practical value Theprincipal directions associated with the symmetric cracktensor are coaxial with those of the permeability tensor

Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)

Figure 7The relationship between planes of joints and the principalstress direction

The first invariant of the crack tensor is proportional to themean permeability while the deviatoric part is related tothe anisotropic permeability [13] Generally the change ofstress which is perpendicular to the joints plane is the mainfactor leading to the increase or decrease of the ground waterpermeability In the numerical model seepage is coupledto stress describing the permeability change induced bythe change of the stress field The coupling function can bedescribed as follows as given by Louis [43]

119870119891= 1198700119890minus120573120590

(15)

where119870119891is the current groundwater hydraulic conductivity

1198700is the initial hydraulic conductivity 120590 is the stress perpen-

dicular to the joints plane and 120573 is the coupling parameter(stress sensitive factor to be measured by experiment) thatreflects the influence of stress The larger 120573 is the greater therange of stress induced the permeability [44]

332 Stress Inducted by Seepage On the basis of generalizedTerzaghirsquos effective stress principle [45] the stress equilibriumequation could be expressed as follows for the water-bearingjointed specimen

120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)

where 120590119894119895is the total stress tensor 119864

119894119895119896119897is the elastic tensor of

the solid phase 120576119896119897is the strain tensor120572

119894119895is a positive constant

which is equal to 1 when individual grains are much moreincompressible than the grain skeleton P is the hydraulicpressure and 120575

119894119895is the Kronecker delta function

34 Coordinate Transformation Generally planes of jointsare inclined at an angle to the major principal stress directionas shown in Figure 7 In establishing these equations the XY and 1 3 axes are taken to have the same Z (2) axis and theangel 120579 is measured from the 119909 to the 1 axis


Roadway Rock mass exposure(northern slope)

Roadway

S

W

N

E

10m

10m

Figure 8 The rock mass exposure of northern slope in Heishan open pit mine

Considering the hydraulic pressure needs no coordinatetransformation and (16) will be expressed as

[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)

where [T120590]minus1 is the reverse matrix for stress coordinates

transformation and [T120576] is the strain coordinates transforma-

tion matrix and they can be expressed as follows

[T120590]minus1

= [

[

cos2120579 sin2120579 minus2 cos 120579 sin 120579

sin2120579 cos2120579 2 sin 120579 cos 120579sin 120579 cos 120579 minus sin 120579 cos 120579 cos2120579 minus sin2120579

]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579

sin2120579 cos2120579 minus sin 120579 cos 120579minus2 sin 120579 cos 120579 2 sin 120579 cos 120579 cos2120579 minus sin2120579

]

]

(18)

Similarly the coordinate transformation form can beexpressed as

11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840

11is the hydraulic conductivity coefficient along the

distribution direction of joint planes 119870101584022

is perpendicularto the distribution direction and 120579 is measured from thecoordinate system 119909 to the optimal direction of permeability

4 A Case Study

41 Description of the Area under Study Themodel describedabove is applied to evaluate the seepage field and stress field ofjointed rock roadway in the Heishan Metal Mine (Figure 8)Themine is located inChengde cityHebei province in north-ern China HeishanMetalMine has transferred fromopen pitmining to underground mining since 2009 The deformationand stability of the roadways for mining the hanging-wall orebecome the key technique issue The elevation of the crest ofthe slope is 920m The roadway in the research is in 674mlevel of the high northern slope The rock mass is mainlycomposed of anorthosite and norite in the northern slopeareaThe anisotropic properties of seepage and stress fieldwillbe discussed in detail

42 Capture of Joint Network Depending on the 3D contact-free measuring system discontinuities on the exposure couldbe easily captured The collection process for the geologyinformation of northern slope in Heishan Metal Mine hasbeen discussed previously A 3D fracture network of rockmass could be obtained using the Monte Carlo method byanalyzing the statistical parameters (Figure 9(a)) Finally thefracture network is generated and expanded to the roadwayand the fracture network perpendicular to the center line ofroadway could be easily captured (Figure 9(b))

43 Permeability Investigation According to the symmetry ofgeometry the method for solving the hydraulic conductivitycoefficient in different directions of fracture network ispresented to save computational time [41]

Rotated every 15∘ clockwise and exerted certain waterpressure on the boundary shown in Figure 10 the hydraulicconductivity coefficient of verticaldirection 119870

[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]

30∘ direc-tion 119870

[30] and 15∘ direction 119870

[15]could be acquired respec-

tively Then the permeability tensors by fracture parameters


(a) (b)

Figure 9 Joint network based on the statistical data usingMonte Carlo method (a) 3D joint network in rock mass (b) profile with 10m edgelength perpendicular to the roadway

0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘

Figure 10 Generated fracture network by Monte Carlo method (side length of internal squares is 1m 4m 7m 8m and 9m resp)

can be determined By enlarging the geometric size ofthe fracture network the permeability scale effect can beinvestigated finally

Based on the algorithm in Section 32 the size effect ofrock mass in different sizes of statistics window is studiedThe region of fracture network is enlarged from 1m to 9mwith different step lengths until the equivalent parametersin different directions achieve a stable value With a fixedcenter of this region the equivalent permeability is calculatedrotating every 15∘ clockwise as shown in Figure 11

The variation of hydraulic conductivity coefficients withthe increase of direction angles and sample size is depictedin Figure 12 The main direction angle is approximated to15∘ It can be seen that the anisotropy of seepage propertyis apparent with the distribution of joints Seven hydraulicconductivity coefficients of different sizes (1m 3m 4m 5m6m 8m and 9m) are chosen to be compared with thoseof the sample whose size is 7m The coefficient deviationbetween one particular size and 7m would be found and

presented in Figure 13 Values of permeability decrease withthe increase of sample size and tend to stabilize when thesample size comes to 7m

The principal values of permeability in maximum andminimum in a rock samplewith joint plane angle being 15∘ arelisted in Table 2 The principal permeability values decreasefrom 423 (times10minus6ms) to 266 (times10minus6ms) as the samplesize increases from 1m to 9m along the main directionAccording to the results discussed previously the REV is7m times 7m for this fracture sample and the related hydraulicconductivity is 277 (times10minus6ms) for maximum and 087(times10minus6ms) for minimum The ratio for the maximum tominimum hydraulic conductivity value is 318

44 Damage Tensor Similar to the algorithm in Section 31the size effect of sample damage in different sizes of statisticswindow is studiedThe region of fracture network is enlargedfrom 3m to 12m with different step lengths until the damage


ang0∘ ang15∘ ang30∘


Figure 11 Fracture network in the study area with edge length of 7m

values in different directions stabilize With a fixed centerof this region the principal damage values are calculatedrotating every 15∘ clockwise Figure 14 shows the sample dam-age in different sizes and directions Similar to the deviationanalysis of permeability the damage values also fluctuatinglytend to stability according to the deviation of the damagetensor in different sizes and directions shown in Figure 15In this study the size of representative elementary volumeis also 7m in length and the initial damage tensor of REVof jointed rock mass could also be obtained The principaldamage values 119863

1and 119863

2for fracture sample are 017 for

minimumand 050 formaximum and the principal directionangle 120579 of damage tensor is approximately 105∘ which isperpendicular to the direction of principal permeabilityThe related rock parameters including undamaged Youngrsquosmodulus 119864

0 Poissonrsquos ratio V

0 and uniaxial tensile strength

et al are listed in Table 3 It should be noted that the elasticityand strength of rock could be determined in the laboratorytests Finally the constitutive relation can be found accordingto (8) Equations (17) and (19) can be introduced into the FEMsimulations

45 Geometry and Boundary Conditions A numerical modelof Heishan Metal Mine is established in order to simulatethe mechanism of stress and seepage in jointed rock mass

Table 2 Principal values 119870[15]

of fracture hydraulic conductivity atdifferent sizes

Sample size (m) 1 4 7 8 9Principal value ofhydraulic conductivity(times10minus6 ms)

Maximum 423 328 277 269 266Minimum 080 097 087 088 085Ratio 529 338 318 306 313

Table 3 Parameters used in the model to validate the model insimulating the anisotropic properties of stress and seepage

Material parameters ValuesUndamaged Youngrsquos modulus 119864

0(GPa) 65

Undamaged Poissonrsquos ratio V0

023Density (kgmminus3) 2700Principal direction angle of joint plane 120579 (∘) 15Coupling parameter 120573 (Paminus1) 05

taking into account the anisotropic property as shown inFigure 16 Table 3 shows the basic mechanical propertiesParameters listed in Table 4 are the hydraulic conductivitycoefficients 119870

119894119895 damage tensor 119863

119894119894and Youngrsquos modulus 119864

119894119895

and shear modulus 119866119894119895of the fracture sample along joint


015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m

Figure 12 Permeability tensor of the rock mass (unit ms)

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘

Figure 13 Deviation of the permeability values to that of 7msample under different directions of samples in different sizes

Table 4 The hydraulic conductivity coefficients119870119894119895 damage tensor


119894119895 and shear modulus 119866

119894119895of the fracture

sample along joint plane direction

Subscripts 119894119895 119870119894119895(times10minus6ms) 119863

119894119894119864119894119895(GPa) 119866

119894119895(GPa)

11 277 050 1883 mdash22 087 017 5191 mdash12 mdash mdash 934 1097

000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m

Figure 14 Damage tensor (times10minus1)

0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘

Figure 15 Deviation of the damage tensor to 7m sample underdifferent directions of samples in different sizes

plane direction used in the finite element code The modelcontains two roadways with a 35mtimes 3m three-centered archsection within a 50m times 50m domain The bottom boundaryof the domain is fixed in all directions and the left andthe right boundaries are fixed in the horizontal directionIn this regard a pressure (120590

119904) of 597MPa is applied on

the top boundary of the model to represent the 221m deepoverburden strata Under the steady-state groundwater flowcondition a hydrostatic pressure 119901

119908 of 221MPa is applied


120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)

Figure 16 Plane strain roadway model (a) boundary conditions(b) the finite element mesh

upon top boundary No-flow conditions are imposed on thethree boundaries of the rectangular domainThe initial waterpressure on the roadways boundaries is 0MPa All the gov-erning equations described previously are implemented intoCOMSOL Multiphysics a powerful PDE-based multiphysicsmodeling environmentThemodel is assumed to be in a stateof plane strain (with no change in elastic strain in the verticaldirection) and static mechanical equilibrium

46 Results and Discussion

461 Stress Distribution Adverse performance of the rockmass in the postexcavation stress field may be caused byeither failure of the anisotropic medium or slip on the

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104

Figure 17 Contours of the first principal stress for 120579 = 15∘ case

0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)

Coupled resultDecoupled result

Displacement from left side of A-A998400 section (m)

Figure 18 Contrast of first principal stress between coupled anddecoupled model

weakness planes [3] The elastic stress distribution aroundthe roadways directly influences the deformation of rock andthus determines the design process

Figure 17 shows the contour of first principal stress cou-pledwith the seepage processThe orientation andmagnitudeofmaximumprincipal stress controlled the distribution of thestress concentration in the heterogeneous media The sim-ulation result shows that the principal stress concentrationzones appearmainly in rock surrounding the roadwaysThereexistmaximumstress concentration areas in the arch foot andfloor Measures should be taken to control the deformationand assure the construction safety

To characterize the response of the stress to the hydraulicmechanics a comparison of two scenarios is also presentedas shown in Figure 18 The first principal stress in the stress-seepage coupled model along the horizontal section A-A1015840where 119910 = 27 is compared with a decoupled modelsThe result shows that the first principal stress increaseswhen the seepage process is considered Figure 19 shows


Fluid pressure in coupled modelFluid pressure in decoupled modelPermeability in coupled modelPermeability in decoupled model

Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0Displacement from left side of A-A998400 section (m)

0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06

Figure 19 Observed changes of normal stress in joint plane direction and hydraulic conductivity between coupled and decoupled models

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2

Figure 20 Darcyrsquos velocity magnitude (ms)

a plot of normal stress in joint plane direction and hydraulicconductivity along the horizontal section A-A1015840 where 119910 = 27

as is shown in Figure 16 The coupled and decoupled modelcould be analyzed using Comsol Multiphysics code In thecoupled case the governing equations for solid and fluidphase are solved in weakly coupled sense For the sake ofconvenient contrast the normal stress distribution as well ashydraulic conductivity is plotted when no seepage-coupledprocess is considered The result shows that the normalstress will be underestimated when no coupling of seepageprocess is included Moreover the compressive stress leads tothe decrease of permeability and the hydraulic conductivityincreases when seepage process is included

462 Seepage Distribution Flow velocity with the angle ofjoint plane being 15∘ is shown in Figure 20 All the processis not considered time dependent Therefore the velocity

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106

Figure 21 The seepage pressure distribution (Pa)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16

Figure 22 Damage zone of the simulating model (the directionof damage zone is approximately perpendicular to that of the jointplanes whose angle is 15∘)


50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)

Figure 23 Contour of the fluid pressure and flow velocity vectors in different joint plane directions

Journal of Applied Mathematics 15Y

flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05

Figure 24 The flow velocity in 119910 direction along the horizontalsection A-A1015840 for different numerical models at 120579 = 0∘ 30∘ 60∘ and90∘

change is induced by the permeability variation caused by thecompressive stress upon the joint plane In this case studywater flows along the principal direction of joint plane wherethe permeability is largest in the model An asymmetricseepage field is observed and maximum of flow velocity isdistributed on right top of the roadways roof Moreover theseepage pressure is also asymmetrically distributed as shownin Figure 21

463 Damage Zone For underground mining two prin-cipal engineering properties of the joint planes should beconsidered They are low tensile strength in the directionperpendicular to the joint plane and the relatively low shearstrength of the surfaces The anisotropic strength parametersas tensile and compressive strength parameters for jointedrock mass are discussed by Chen et al [46] Claesson andBohloli [47] Nasseri et al [48] Gonzaga et al [49] andCho et al [50] As discussed previously the fluid pressureand velocity are sensitive to the joint plane angles 120579 In thissection the Hoffman anisotropic strength criterion is used toassess the damage zone in this numerical model as shown in

1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)

The properties 119883119905and 119884

119905represent the tensile strength

along joint plane direction and perpendicular to joint planedirection respectively 119883

119888and 119884

119888represent the compressive

strength along joint plane direction and perpendicular tojoint plane direction respectively S is the shear strengthof the material along the joint direction 120590

1represents the

normal stress along the principle direction of elasticity and1205902represents the normal stress perpendicular to the principle

direction of elasticity 12059112represents the shear stress

The shear strength of the joints can be described by thesimple Coulomb law

119878 = 119888 + 1205902tan120601 (21)

where 119888 is cohesive strength and 120601 is the effective angle offriction of the joint surfaces

Table 5Mechanical properties of rockmass in the direction of jointplane

Tensile strength (MPa) 119883119905

160119884119905

40

Compressive strength (MPa) 119883119888

120119884119888

96

Shear strength (MPa) 119888 15120601 50

It should be noted that the mechanical parameters beacquired from laboratory or in situ tests However it isdifficult to directly employ the strength parameters forjointed rock mass due to the inaccessibility of the tests forhuge rock mass Based on the in situ and laboratory testsof Heishan Metal Mine the tensile compressive and shearstrength parameters are listed in Table 5

The direction of joint plane is 15∘ and thus the tensilestrength in the direction perpendicular to joint plane is rel-atively low compared with that in other directions Figure 22shows the damage zone in this case study and the failurearea mainly distributes in the direction perpendicular to theweakness plane which gives an illustration of the roadwayfailure mode in tabular orebodies Moreover the failure ofcovered rock mass and the rock pillar in and between the tworoadways does not influence each other in this case study andthus the choice of roadwayrsquos interval is proper from the aspectof the mechanics analysis

47 Further Discussion These simulations were performedto develop an understanding of the mechanics of joints andinfluence on stress and seepage fields and to gauge the abilityof the proposed transversely anisotropic model to capturethe response of jointed rock mass For this purpose a totalof six scenarios with the joint plane angles 120579 ranging from0∘ to 150∘ with an interval of 30∘ are simulated in order toexamine the effect of joint plane directions see Figure 7 forthe definition of 120579 And the anisotropic properties of seepagefield and damage zones are examined in these simulations

471 Seepage Field Figure 23 presents the fluid pressuredistribution and flow field arrows with different joint planedirections It can be seen that the maximum pressure whichis located on the top boundary equals the initial fluid pressure(221MPa) The fluid pressure distribution on top of theroadways differs with the increase of directions

When the joint plane is parallel or perpendicular tothe floors of roadways the fluid pressure distributions andflow arrows are all axially symmetric which agrees withexpectations The fluid pressure in the 30∘ or 60∘ case isdistributed unsymmetrically as shown in Figures 23(b) and23(c)

The flow velocity in 119910 direction along the horizontalsection A-A1015840 where 119910 = 27m as shown in Figure 16(a)curves are plotted in Figure 24 for 120579 = 0∘ 30∘ 60∘ and90∘ respectively In all cases the absolute value of velocityincreases approximately exponentially above the roadways


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)

Figure 25 The damaged zone under different principal elastic directions (red arrows represent the flow velocity vector)

As shown in Figure 24 the flow velocity in scenario where120579 = 0∘ is in the lowest level The reason is that the jointplane in this scenario is horizontally distributed and thecompressive stress leads to the decrease of permeabilityWhen angle of joint plane increases to 30∘ flow velocitiesabove both roadways increase and Roadway I has a morehigher velocity than Roadway II The scenario where 120579 = 60∘has similar performance However when the joint plane isvertically distributed flow velocity above Roadway I is lower

than that where 120579 = 60∘ The flow velocity in the case wherejoints are vertically or horizontally distributed is symmetric

472 Damage Zone The effect of joint plane angle on thedamage zone is illustrated by using the Hoffman anisotropicstrength criterion as shown in (23) The damage zones indifferent angles of joint planes are shown in Figure 25 andcould be an index to visualize the potential failure mode ofthe roadway


It is clear from Figures 25(a) to 25(f) that an increase ofthe joint plane angle has significantly influenced the shapeand size of damage zone When the joints are horizontallydistributed (120579 = 0∘) the tensile strength in the directionperpendicular to joint planes is much lower and thus thedamage zone mainly concentrates within roof and bottomof roadways This failure mainly manifests as roof fallingand floor heave Similarly when the angle increases to 30∘the damage zones mainly concentrate in the rock mass ofleft roof and right floor within the roadways which is alsoin the direction perpendicular to joint planes Comparedwith the result in Figure 25(b) the direction of damage zonewith 120579 = 60∘ shown in Figure 25(c) rotated significantlyand the area also increases When the joints are verticallydistributed (120579 = 90∘) the principal direction of damagezone is nearly horizontal Lateral rock mass surrounding theroadways stabilizes with the increase of joint plane angle inthis study and the pillar between two roadways is also stableThe scenarios where 120579 = 120∘ or 120579 = 150∘ have the similarresponse to the joint plane direction with that of 60∘ or 30∘Qualitatively the simulation results are in good agreementwith the results by Jia et al [51] and Huang et al [52] Themodel in this study could to some extent be effectively usedto analyze the anisotropic property for jointed rock mass

5 Conclusion

The main purpose is to introduce the anisotropic model ofseepage and stress and apply the model to the jointed rockmass The coupled finite element analysis was performedand the effects of joint planes direction on stress field flowvelocity and Darcyrsquos velocity were verified A more reliablemodel for the stability analysis of rock engineering and riskevaluation of water inrush was provided Based on the resultsof a series of numerical simulations under different scenariosthe following conclusions are drawn

(1) A linkage of digital information of fractures andmechanical analysis is realized Relations betweendigital images involving detailed geometrical proper-ties of rock mass and the quantitative determinationof hydraulic parameters as well as elastic propertieswere realized The results show that the scale for bothdamage tensor and permeability tensor of the rockmassrsquos REV in northern slope of Heishan Metal Mineis 7m times 7m

(2) We examine the model for seepage-stress coupledanalysis on anisotropic properties The numericalsimulations in this study have indicated that theexistence of joint planes greatly affects the seepageproperties and stress field In anisotropic rock waterflows mainly along the joint planes and water pres-sure is asymmetrically distributed where the angle ofjoint plane is 15∘ in the northern slope of HeishanMetal Mine

(3) The influence of fractures cannot be neglected instability analysis of rock mass The numerical resultsvisualized the damage zones in different directions of

joint planesThe direction of damage zoneswas foundperpendicular to that of joint planes which agrees wellwith the field observations and theoretical analysis

The work reported in this paper is an initial effort on theinfluence of joint orientation on the anisotropic property ofseepage and stress in jointed rock masses A model for morecomplex jointed rock mass remains to be quantified

Nomenclature

119863 Damage variable (dimensionless)119864 Damaged Youngrsquos modulus (Pa)1198640 Undamaged Youngrsquos modulus (Pa)

119866 Shear modulus (Pa)119870119894119895 Hydraulic conductivity coefficient (ms)

1198701015840

119894119894 Hydraulic conductivity coefficient along

joint plane direction (ms)119901 Fluid pressure (Pa)119878119894119895 Flexibility coefficient (Paminus1)

119873 The number of joints119897 Minimum spacing between joints (m)119881 Volume of rock mass (m3)Δℎ119895 The hydraulic gradient

119899(119896) The normal vector of the kth joint

(dimensionless)119886(119896) The trace length of the kth joint (m)

119876119894 The fluid source term (m3)

[T120590]minus1 The reverse matrix for stress coordinates

transformation[T120576] The strain coordinates transformation

matrix119883119905 The tensile strength in joint direction (Pa)

119884119905 The tensile strength perpendicular to

direction (Pa)119883119888 The compressive strength in joint

direction (Pa)119884119888 The compressive strength perpendicular

to direction (Pa)119878 The shear strength of the material (Pa)

Greek Symbols

120572119894119895 Positive constant

120573 Coupling coefficient (Paminus1)120590119894119895 Stress tensor (Pa)

1205901 1205902 1205903 The first second and third principalstresses (Pa)

120576119894119895 Total strain tensor (dimensionless)

120579 Angle of joint plane (∘)V Damaged Poissonrsquos ratio (dimensionless)V0 Undamaged Poissonrsquos ratio

(dimensionless)120588 Density of water (kgm3)120583 Coefficient of flow viscosity (Pasdots)120601 Effective angle of friction of the joint

surfaces


Conflict of Interests

The authors declare that there is no conflict of interests

Acknowledgments

This study was performed at Center for Rock Instabilityamp Seismicity Research (CRISR) Northeastern UniversityShenyang ChinaThework presented in this paperwas finan-cially supported by the National Basic Research Program ofChina (No 2013CB227900)The Natural Science FoundationofChina (GrantNo 51174045 50904013 51034001 5093400641172265) the basic scientific research funds of Ministry ofEducation of China (No N090101001 N120601002) and theproject supported by the Specialized Research Fund for theDoctoral Program of Higher Education of China (Grant No20120042120053) also give financial support to this research

References

[1] W G Pariseau S Puri and S C Schmelter ldquoA new modelfor effects of impersistent joint sets on rock slope stabilityrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 45 no 2 pp 122ndash131 2008

[2] B Amadei ldquoImportance of anisotropy when estimating andmeasuring in situ stresses in rockrdquo International Journal of RockMechanics and Mining Sciences and Geomechanics vol 33 no3 pp 293ndash325 1996

[3] B H G Brady and E T Brown Rock Mechanics For Under-ground Mining Springer Amsterdam The Netherlands 2004

[4] X Zhang and D J Sanderson ldquoNumerical study of criticalbehaviour of deformation and permeability of fractured rockmassesrdquoMarine and Petroleum Geology vol 15 no 6 pp 535ndash548 1998

[5] Y M Tien and P F Tsao ldquoPreparation and mechanicalproperties of artificial transversely isotropic rockrdquo InternationalJournal of RockMechanics andMining Sciences vol 37 no 6 pp1001ndash1012 2000

[6] F J Brosch K Schachner M Blumel A Fasching and H FritzldquoPreliminary investigation results on fabrics and related phys-ical properties of an anisotropic gneissrdquo Journal of StructuralGeology vol 22 no 11-12 pp 1773ndash1787 2000

[7] G E Exadaktylos and K N Kaklis ldquoApplications of an explicitsolution for the transversely isotropic circular disc compresseddiametricallyrdquo International Journal of Rock Mechanics andMining Sciences vol 38 no 2 pp 227ndash243 2001

[8] G E Exadaktylos ldquoOn the constraints and relations of elasticconstants of transversely isotropic geomaterialsrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 38 no 7pp 941ndash956 2001

[9] K-B Min and L Jing ldquoNumerical determination of the equiva-lent elastic compliance tensor for fractured rock masses usingthe distinct element methodrdquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 795ndash816 2003

[10] B Amadei Rock Anisotropy and the Theory of Stress Measure-ments Springer 1983

[11] M Hakala H Kuula and J A Hudson ldquoEstimating thetransversely isotropic elastic intact rock properties for in situstressmeasurement data reduction a case study of theOlkiluotomica gneiss Finlandrdquo International Journal of Rock Mechanicsand Mining Sciences vol 44 no 1 pp 14ndash46 2007

[12] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

[13] M Oda ldquoPermeability tensor for discontinuous rock massesrdquoGeotechnique vol 35 no 4 pp 483ndash495 1985

[14] J Sun and Z Zhao ldquoEffects of anisotropic permeability offractured rock masses on underground oil storage cavernsrdquoTunnelling andUnderground Space Technology vol 25 no 5 pp629ndash637 2010

[15] 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

[16] 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

[17] A Baghbanan and L Jing ldquoStress effects on permeability ina fractured rock mass with correlated fracture length andaperturerdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 8 pp 1320ndash1334 2008

[18] Z Zhao L Jing I Neretnieks and L Moreno ldquoNumericalmodeling of stress effects on solute transport in fractured rocksrdquoComputers and Geotechnics vol 38 no 2 pp 113ndash126 2011

[19] K Bao A Salama and S Sun ldquoUpscaling of permeabilityfield of fractured rock system numerical examplesrdquo Journal ofApplied Mathematics Article ID 546203 20 pages 2012

[20] W Z Chen J P Yang J L Yang X B Qiu and C C CaoldquoHydromechanical coupled model of jointed rock mass andits application to pressure tunnelsrdquo Chinese Journal of RockMechanics and Engineering vol 27 no 9 pp 1569ndash1571 2006

[21] W Chen J Yang X Zou and C Zhou ldquoResearch onmacrome-chanical parameters of fractured rock massesrdquo Chinese Journalof Rock Mechanics and Engineering vol 27 no 8 pp 1569ndash15752008

[22] J P Yang Study of macro mechanical parameters of fracturedrock mass [PhD thesis] Wuhan institute of the Chineseacademy of sciences geotechnical engineering Wuhan China2009

[23] C Qiao Z Zhang and X Li ldquoAnisotropic strength charac-teristics of jointed rock massrdquo in Proceedings of the ISRM-Sponsored International Symposium on Rock Mechanics RockCharacterisation Modelling and Engineering Design Methods p140 2009

[24] A Robertson ldquoThe interpretation of geological factors foruse in slope theoryrdquo in Proceedings of the Symposium onTheoretical Background to Planning of Open PitMines pp 55ndash71Johannesburg South Africa 1970

[25] C-YWang and K T Law ldquoReview of borehole camera technol-ogyrdquoChinese Journal of RockMechanics and Engineering vol 24no 19 pp 3440ndash3448 2005

[26] Austrian Startup Company ShapeMetriX3DModel Merger UserManual Earth Products China Shenyang China 2008

[27] A Gaich W Schubert and M Poetsch ldquoThree-dimensionalrock mass documentation in conventional tunnelling usingJoint-MetriX3Drdquo in Proceedings of the 31st ITA-AITES WorldTunnel Congress E Yucel and T Solak Eds pp 59ndash64AA Balkema Publishers Istanbul Turkey 2005 UndergroundSpace Use Analysis of the Past and Lessons for the Future


[28] T Yang Q Yu S Chen H Liu T Yu and L Yang ldquoRock massstructure digital recognition and hydro-mechanical parameterscharacterization of sandstone in Fangezhuang coal minerdquo Chi-nese Journal of Rock Mechanics and Engineering vol 28 no 12pp 2482ndash2489 2009

[29] P T Wang T H Yang Q L Yu H L Liu and P H ZhangldquoCharacterization on jointed rock masses based on PFC2DrdquoFrontiers of Structural and Civil Engineering vol 7 no 1 pp 32ndash38 2013

[30] Y Y Jiao X L Zhang and T C Li DDARF Method ForSimulating theWhole Process of Rock Failure Science PublishingHouse Beijing China 2010

[31] E Hoek and E T Brown ldquoPractical estimates of rock massstrengthrdquo International Journal of Rock Mechanics and MiningSciences vol 34 no 8 pp 1165ndash1186 1997

[32] P Marinos and E Hoek ldquoEstimating the geotechnical proper-ties of heterogeneous rock masses such as flyschrdquo Bulletin ofEngineering Geology and the Environment vol 60 no 2 pp 85ndash92 2001

[33] E Hoek C Carranza-Torres and B Corkum ldquoHoek-Brownfailure criterion-2002 editionrdquo in Proceedings of the 5th NorthAmerican Rock Mechanics Symposium and 17th Tunneling Asso-ciation of Canada Conference (NARMS-TAC rsquo02) pp 267ndash2712002

[34] J Lemaitre and R Desmorat Engineering Damage MechanicsSpringer Berlin Germany 2005

[35] T Kawamoto Y Ichikawa and T Kyoya ldquoDeformation andfracturing behaviour of discontinuous rock mass and damagemechanics theoryrdquo International Journal for Numerical andAnalyticalMethods inGeomechanics vol 12 no 1 pp 1ndash30 1988

[36] F Sidoroff ldquoDescription of anisotropic damage application toelasticityrdquo in Proceedings of the IUTAM Colloquium on PhysicalNonlinearities in structural analysis pp 237ndash244 1981

[37] Y Wu and Z Zhang An Introduction to Rock Mass HydraulicsSouthwest Jiaotong University Press Chengdu China 2005

[38] D T Snow ldquoAnisotropie Permeability of Fractured MediardquoWater Resources Research vol 5 no 6 pp 1273ndash1289 1969

[39] K Hestir and J C S Long ldquoAnalytical expressions for thepermeability of random two-dimensional Poisson fracture net-works based on regular lattice percolation and equivalentmediatheoriesrdquo Journal of Geophysical Research vol 95 no 13 pp21565ndash21581 1990

[40] T Yang and Y Xiao ldquoStructural plane survey and analysis ofpermeability characteristics of open pit mine slope rock massrdquoSite Investigation Science and Technology vol 16 no 3 pp 27ndash30 1998

[41] Y X Xiao C F Lee and S JWang ldquoAssessment of an equivalentporous medium for coupled stress and fluid flow in fracturedrockrdquo International Journal of Rock Mechanics and MiningSciences vol 36 no 7 pp 871ndash881 1999

[42] M A Biot ldquoGeneral theory of three-dimensional consolida-tionrdquo Journal of Applied Physics vol 12 no 2 pp 155ndash164 1941

[43] C Louis ldquoRock hydraulicsrdquo in Rock Mechanics L Muller Edpp 287ndash299 Springer New York NY USA 1974

[44] T H Yang T Xu H Y Liu C A Tang B M Shi and Q XYu ldquoStress-damage-flow coupling model and its application topressure relief coal bed methane in deep coal seamrdquo Interna-tional Journal of Coal Geology vol 86 no 4 pp 357ndash366 2011

[45] K Terzaghi R B Peck and G Mesri Soil Mechanics inEngineering Practice Wiley-Interscience New York NY USA1996

[46] C-S Chen E Pan and B Amadei ldquoDetermination of deforma-bility and tensile strength of anisotropic rock using Braziliantestsrdquo International Journal of Rock Mechanics and MiningSciences vol 35 no 1 pp 43ndash61 1998

[47] J Claesson and B Bohloli ldquoBrazilian test stress field and tensilestrength of anisotropic rocks using an analytical solutionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 39 no 8 pp 991ndash1004 2002

[48] M H B Nasseri K S Rao and T Ramamurthy ldquoAnisotropicstrength and deformation behavior of Himalayan schistsrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 40 no 1 pp 3ndash23 2003

[49] G G Gonzaga M H Leite and R Corthesy ldquoDeterminationof anisotropic deformability parameters from a single standardrock specimenrdquo International Journal of Rock Mechanics andMining Sciences vol 45 no 8 pp 1420ndash1438 2008

[50] J-W Cho H Kim S Jeon and K-B Min ldquoDeformationand strength anisotropy of Asan gneiss Boryeong shale andYeoncheon schistrdquo International Journal of Rock Mechanics andMining Sciences vol 50 pp 158ndash169 2012

[51] P Jia C-A Tang T-H Yang and S-H Wang ldquoNumeri-cal stability analysis of surrounding rock mass layered bystructural planes with different obliquitiesrdquo Dongbei DaxueXuebaoJournal of Northeastern University vol 27 no 11 pp1275ndash1278 2006

[52] S Huang J Xu X Ding and A Wu ldquoStudy of layered rockmass composite model based on characteristics of structuralplane and its applicationrdquo Chinese Journal of Rock Mechanicsand Engineering vol 29 no 4 pp 743ndash756 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


strains at any point of the anisotropic circular disc com-pressed diametrically The stress concentration in uniaxialcompression for a plane strain model was investigated and aformulation of failure criteria for elastic-damage and elastic-plastic anisotropic geomaterials is formed by Exadaktylos [8]Amethodology to determine the equivalent elastic propertiesof fractured rockmasses was established by explicit represen-tations of stochastic fracture systems [9] and the conditionsfor the application of the equivalent continuum approachfor representing mechanical behavior of the fractured rockmasses were investigated Such anisotropy has an effect onthe interpretation of stress distributions Amadeirsquos resultsindicate that an anisotropy ratio (damaged elastic modulusto intact elastic modulus) of between 114 and 133 will havea definite effect on the interpreted in situ state of stress[10] Hakala et al [11] concluded that the anisotropy ratio ofFinland rock specimens is about 14 and suggested to be takeninto account in the interpretation of stress measurementresults

Moreover in rock engineering like mining and tunnelengineering interactions between in situ stress and seepagepressure of groundwater have an important role Groundwa-ter under pressure in the joints defining rock blocks reducesthe normal effective stress between the rock surfaces andtherefore reduces the potential shear resistance which can bemobilized by friction Since rock behaviormay be determinedby its geohydrological environment it may be essentialin some cases to maintain close control of groundwaterconditions in the mine area Therefore accurate descriptionof joints is an important topic for estimating and evaluatingthe deformability and seepage properties of rock masses

Many researchers have done a lot of studies on theseepage and anisotropic mechanical behaviors of the jointedrock mass Using homogeneous samples like granite orbasalt Witherspoon et al [12] investigated and definedthe permeability by fracture aperture in a closed fractureBased on geometrical statistics Oda [13] has studied anddetermined the crack tensor of moderately jointed graniteby treating statistically the crack orientation data via astereographic projection Using Odarsquos method Sun and Zhao[14] determined the anisotropy in permeability using thefracture orientation and the in situ stress information fromthe field survey For fractured rock system attempts havebeen made by Jing et al [15ndash18] and Bao et al [19] toinvestigate the permeability Using UDEC code Jing et alstudied the permeability of discrete fracture network suchas the existence of REV in [15] relations between fracturelength and aperture in [16] or stress effect on permeability in[17] or solute transport in [18] Bao et al [19] have discussedthe mesh effect on effective permeability for a fracturedsystem using the upscaled permeability field It would helpthe workers to spend less computational effort and memoryrequirement to investigate effective permeability The key fordetermining deformationmodulus and hydraulic parametersis to study representative elementary volume (REV) andscale effects of fractured rock mass [9 20ndash22] However thedifficulty of testing jointed rock specimens at scales sufficientto represent the equivalent continuum indicated that it isnecessary to postulate and verify methods of synthesizing

rock mass properties from those of the constituent elementslike intact rock and fractures

Although great progress has been made it is difficultto study the anisotropic deformation of large-scale rockmass ground water permeability change and interactionbetween stress and seepage due to the geological complexityof discontinuities Qiao et al [23] have pointed out that therock mass which is not highly fractured and has only few setsof joint system usually behaves anisotropically However themechanical properties directionality of highly jointed rockmass is usually ignored In particular since the limitationof mesh generation the numerical method can hardly dealwith the highly jointed specimensThe discontinuity like faultcould be treated as specific boundary An effective solutionshould be found to characterize the influence of joints on therock mass

In this paper based on equivalent continuum theory andtheoretical analysis a mathematical model for anisotropicproperty of seepage and elasticity of jointed rock mass isdescribed A permeability analysis code is developed toevaluate the anisotropic permeability for DFN model basedon VC++ 60 in this paper The DFN model could beanalyzed to evaluate the REV size and anisotropic propertyof permeability which would provide important evidencefor the finite element model The outline is as follows First3D images involving detailed geometrical properties of rockmass such as trace lengths outcrop areas joint orientationsand joint spacing were captured using ShapMetriX3D sys-tem According to the statistical parameters for each set ofdiscontinuities fracture network is generated using MonteCarlo method and jointed rock samples in different sectionscould be captured Next permeability tensors and elasticitytensors of rock mass of the sample are calculated by discretemedium seepage method and geometrical damage theoryrespectively Finally using the finite element method (FEM)an anisotropic mechanical model for rock mass is built Anengineering practice of roadway stability at theHeishanMetalMine in Hebei Iron and Steel Group Mining Company isdescribed Then the stress and seepage fields surroundingthe roadway were numerically simulated On the basis ofthe modeled results the influences of joint planes on stressseepage and damage zone were analyzed It is expectedthroughout this study to gain an insight into the influencesof discontinuities on the mechanical behavior of rock massand offer some scientific evidence for the design of mininglayouts or support requirements

2 Generation of a Fracture System Model byShapeMetriX3D

Traditional methods for rock mass structural parameters inmining engineering applications include scanline surveying[24] and drilling core method [25] The process generallyrequires physical contact with rock mass exposure and there-fore is hazardous Additionally takingmanual measurementsis time consuming and prone to errors due to sampling diffi-culties or instrument errors ShapeMetriX3D is a tool for thegeological and geotechnical data collection and assessment



Rock mass exposure

P2 (u )

P1 (u )

VL (XY Z)

VR (XY Z)

E

S

H





1(119906 V) and


ES

HProfile I-I






02minus18

2

1

0

minus1

minus2

minus3

minus4

Distance (m)

Hei

ght (

m)






30∘

60∘

330∘

300∘

120∘

150∘210∘

240∘

EW

N

S

(a)

(b)



120576119894119895= [S] 120590119894119895 (1)


119894119895is







Type Meanvalue






Z

X

Y

O




11 12059022 and 120590

12) and







[

[

12057611

12057622

12057612

]

]

= [S] [120590]

=

[[[[[[[[[

[

1

1198641

minus]231

1198643

minus(]12

1198641

+]231

1198643

) 0

minus(]12

1198641

+]231

1198643

)1

1198641

minus]231

1198643

0

0 02 (1 + V

12)

1198641

]]]]]]]]]

]

times [

[

12059011

12059022

12059012

]

]


[

[

12059011

12059022

12059012

]

]

= [E] [120576] =

[[[[[[[[[[

[

1198642

1(1198643minus 1198641V231)

Δ

minus1198642

1(1198643V21

+ 1198641V231)

Δ0

minus1198642

1(1198643V21

+ 1198641V231)

Δ

1198642

1(1198641V231

minus 1198643)

Δ0

0 01198641

2 (1 + V12)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(3)


Δ = minus11986411198643+ 11986411198643V221

+ 1198642

1V231

+ 1198642

1V231

+ 21198642

1V21V231 (4)



31and

V12






H = hb1

H = hb2

V = 0 V = 0

M

M998400

N

N998400


1198871and ℎ




119864 = 1198640(1 minus 119863) (5)




119863119894119895=

119897

119881

119873

sum

119896=1

120572(119896)

(119899(119896)

otimes 119899(119896)

) (119894 119895 = 1 2 3) (6)



1198781015840

119894119895= (1 minus 119863

119894)minus1

119878119894119895(1 minus 119863

119895)minus1

(7)


119894and119863

119895are


[

[

12059011

12059022

12059012

]

]

=

[[[[[[[[[[

[

1198640(V0minus 1) (119863

1minus 1)2

2V20+ V0minus 1

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

0

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

1198640(V0minus 1) (119863

2minus 1)2

2V20+ V0minus 1

0

0 01198640(1198631minus 1) (119863

2minus 1)

2 (1 + V0)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(8)


0is







(

1198731015840

sum

119895=1

119902119895)

119894

+ 119876119894= 0 (119894 = 1 2 119873) (9)





119895to





expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)




(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)




119881 of 0


119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)



119870119894119895nabla2119901 = 0 (13)



119870119894119895= [

11987011

11987012

11987021

11987022

] (14)



Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)



119870119891= 1198700119890minus120573120590

(15)





120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)










Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Rock mass exposure

P2 (u )

P1 (u )

VL (XY Z)

VR (XY Z)

E

S

H





1(119906 V) and


ES

HProfile I-I






02minus18

2

1

0

minus1

minus2

minus3

minus4

Distance (m)

Hei

ght (

m)






30∘

60∘

330∘

300∘

120∘

150∘210∘

240∘

EW

N

S

(a)

(b)



120576119894119895= [S] 120590119894119895 (1)


119894119895is







Type Meanvalue






Z

X

Y

O




11 12059022 and 120590

12) and







[

[

12057611

12057622

12057612

]

]

= [S] [120590]

=

[[[[[[[[[

[

1

1198641

minus]231

1198643

minus(]12

1198641

+]231

1198643

) 0

minus(]12

1198641

+]231

1198643

)1

1198641

minus]231

1198643

0

0 02 (1 + V

12)

1198641

]]]]]]]]]

]

times [

[

12059011

12059022

12059012

]

]


[

[

12059011

12059022

12059012

]

]

= [E] [120576] =

[[[[[[[[[[

[

1198642

1(1198643minus 1198641V231)

Δ

minus1198642

1(1198643V21

+ 1198641V231)

Δ0

minus1198642

1(1198643V21

+ 1198641V231)

Δ

1198642

1(1198641V231

minus 1198643)

Δ0

0 01198641

2 (1 + V12)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(3)


Δ = minus11986411198643+ 11986411198643V221

+ 1198642

1V231

+ 1198642

1V231

+ 21198642

1V21V231 (4)



31and

V12






H = hb1

H = hb2

V = 0 V = 0

M

M998400

N

N998400


1198871and ℎ




119864 = 1198640(1 minus 119863) (5)




119863119894119895=

119897

119881

119873

sum

119896=1

120572(119896)

(119899(119896)

otimes 119899(119896)

) (119894 119895 = 1 2 3) (6)



1198781015840

119894119895= (1 minus 119863

119894)minus1

119878119894119895(1 minus 119863

119895)minus1

(7)


119894and119863

119895are


[

[

12059011

12059022

12059012

]

]

=

[[[[[[[[[[

[

1198640(V0minus 1) (119863

1minus 1)2

2V20+ V0minus 1

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

0

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

1198640(V0minus 1) (119863

2minus 1)2

2V20+ V0minus 1

0

0 01198640(1198631minus 1) (119863

2minus 1)

2 (1 + V0)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(8)


0is







(

1198731015840

sum

119895=1

119902119895)

119894

+ 119876119894= 0 (119894 = 1 2 119873) (9)





119895to





expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)




(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)




119881 of 0


119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)



119870119894119895nabla2119901 = 0 (13)



119870119894119895= [

11987011

11987012

11987021

11987022

] (14)



Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)



119870119891= 1198700119890minus120573120590

(15)





120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)










Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02minus18

2

1

0

minus1

minus2

minus3

minus4

Distance (m)

Hei

ght (

m)






30∘

60∘

330∘

300∘

120∘

150∘210∘

240∘

EW

N

S

(a)

(b)



120576119894119895= [S] 120590119894119895 (1)


119894119895is







Type Meanvalue






Z

X

Y

O




11 12059022 and 120590

12) and







[

[

12057611

12057622

12057612

]

]

= [S] [120590]

=

[[[[[[[[[

[

1

1198641

minus]231

1198643

minus(]12

1198641

+]231

1198643

) 0

minus(]12

1198641

+]231

1198643

)1

1198641

minus]231

1198643

0

0 02 (1 + V

12)

1198641

]]]]]]]]]

]

times [

[

12059011

12059022

12059012

]

]


[

[

12059011

12059022

12059012

]

]

= [E] [120576] =

[[[[[[[[[[

[

1198642

1(1198643minus 1198641V231)

Δ

minus1198642

1(1198643V21

+ 1198641V231)

Δ0

minus1198642

1(1198643V21

+ 1198641V231)

Δ

1198642

1(1198641V231

minus 1198643)

Δ0

0 01198641

2 (1 + V12)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(3)


Δ = minus11986411198643+ 11986411198643V221

+ 1198642

1V231

+ 1198642

1V231

+ 21198642

1V21V231 (4)



31and

V12






H = hb1

H = hb2

V = 0 V = 0

M

M998400

N

N998400


1198871and ℎ




119864 = 1198640(1 minus 119863) (5)




119863119894119895=

119897

119881

119873

sum

119896=1

120572(119896)

(119899(119896)

otimes 119899(119896)

) (119894 119895 = 1 2 3) (6)



1198781015840

119894119895= (1 minus 119863

119894)minus1

119878119894119895(1 minus 119863

119895)minus1

(7)


119894and119863

119895are


[

[

12059011

12059022

12059012

]

]

=

[[[[[[[[[[

[

1198640(V0minus 1) (119863

1minus 1)2

2V20+ V0minus 1

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

0

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

1198640(V0minus 1) (119863

2minus 1)2

2V20+ V0minus 1

0

0 01198640(1198631minus 1) (119863

2minus 1)

2 (1 + V0)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(8)


0is







(

1198731015840

sum

119895=1

119902119895)

119894

+ 119876119894= 0 (119894 = 1 2 119873) (9)





119895to





expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)




(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)




119881 of 0


119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)



119870119894119895nabla2119901 = 0 (13)



119870119894119895= [

11987011

11987012

11987021

11987022

] (14)



Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)



119870119891= 1198700119890minus120573120590

(15)





120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)










Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Type Meanvalue






Z

X

Y

O




11 12059022 and 120590

12) and







[

[

12057611

12057622

12057612

]

]

= [S] [120590]

=

[[[[[[[[[

[

1

1198641

minus]231

1198643

minus(]12

1198641

+]231

1198643

) 0

minus(]12

1198641

+]231

1198643

)1

1198641

minus]231

1198643

0

0 02 (1 + V

12)

1198641

]]]]]]]]]

]

times [

[

12059011

12059022

12059012

]

]


[

[

12059011

12059022

12059012

]

]

= [E] [120576] =

[[[[[[[[[[

[

1198642

1(1198643minus 1198641V231)

Δ

minus1198642

1(1198643V21

+ 1198641V231)

Δ0

minus1198642

1(1198643V21

+ 1198641V231)

Δ

1198642

1(1198641V231

minus 1198643)

Δ0

0 01198641

2 (1 + V12)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(3)


Δ = minus11986411198643+ 11986411198643V221

+ 1198642

1V231

+ 1198642

1V231

+ 21198642

1V21V231 (4)



31and

V12






H = hb1

H = hb2

V = 0 V = 0

M

M998400

N

N998400


1198871and ℎ




119864 = 1198640(1 minus 119863) (5)




119863119894119895=

119897

119881

119873

sum

119896=1

120572(119896)

(119899(119896)

otimes 119899(119896)

) (119894 119895 = 1 2 3) (6)



1198781015840

119894119895= (1 minus 119863

119894)minus1

119878119894119895(1 minus 119863

119895)minus1

(7)


119894and119863

119895are


[

[

12059011

12059022

12059012

]

]

=

[[[[[[[[[[

[

1198640(V0minus 1) (119863

1minus 1)2

2V20+ V0minus 1

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

0

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

1198640(V0minus 1) (119863

2minus 1)2

2V20+ V0minus 1

0

0 01198640(1198631minus 1) (119863

2minus 1)

2 (1 + V0)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(8)


0is







(

1198731015840

sum

119895=1

119902119895)

119894

+ 119876119894= 0 (119894 = 1 2 119873) (9)





119895to





expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)




(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)




119881 of 0


119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)



119870119894119895nabla2119901 = 0 (13)



119870119894119895= [

11987011

11987012

11987021

11987022

] (14)



Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)



119870119891= 1198700119890minus120573120590

(15)





120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)










Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H = hb1

H = hb2

V = 0 V = 0

M

M998400

N

N998400


1198871and ℎ




119864 = 1198640(1 minus 119863) (5)




119863119894119895=

119897

119881

119873

sum

119896=1

120572(119896)

(119899(119896)

otimes 119899(119896)

) (119894 119895 = 1 2 3) (6)



1198781015840

119894119895= (1 minus 119863

119894)minus1

119878119894119895(1 minus 119863

119895)minus1

(7)


119894and119863

119895are


[

[

12059011

12059022

12059012

]

]

=

[[[[[[[[[[

[

1198640(V0minus 1) (119863

1minus 1)2

2V20+ V0minus 1

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

0

minus1198640V0(1198631minus 1) (119863

2minus 1)

2V20+ V0minus 1

1198640(V0minus 1) (119863

2minus 1)2

2V20+ V0minus 1

0

0 01198640(1198631minus 1) (119863

2minus 1)

2 (1 + V0)

]]]]]]]]]]

]

[

[

12057611

12057622

12057612

]

]

(8)


0is







(

1198731015840

sum

119895=1

119902119895)

119894

+ 119876119894= 0 (119894 = 1 2 119873) (9)





119895to





expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)




(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)




119881 of 0


119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)



119870119894119895nabla2119901 = 0 (13)



119870119894119895= [

11987011

11987012

11987021

11987022

] (14)



Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)



119870119891= 1198700119890minus120573120590

(15)





120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)










Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












expressed as

119902119895=

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

(10)




(

1198731015840

sum

119895=1

1205881198873

119895

12 120583sdot

Δℎ119895

119897119895

)

119894

+ 119876119894= 0 (11)




119881 of 0


119870 =Δ119902 sdot 119872

1015840119872

Δ119867 sdot 119872119873 (12)



119870119894119895nabla2119901 = 0 (13)



119870119894119895= [

11987011

11987012

11987021

11987022

] (14)



Joint plane

1205901

1205901

1205903 1205903

13

Y

XO

120579K22

K11

Z (2)



119870119891= 1198700119890minus120573120590

(15)





120590119894119895= 119864119894119895119896119897

120576119896119897minus 120572119894119895119875120575119894119895 (16)










Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Roadway

S

W

N

E

10m

10m



[

[

12059011

12059022

12059112

]

]

= [T120590]minus1

[E] [T120576] [[

12057611

12057622

12057412

]

]

minus 120572119894119895[

[

119875

119875

0

]

]

120575119894119895 (17)




[T120590]minus1

= [

[



]

]

[T120576] = [

[

cos2120579 sin2120579 cos 120579 sin 120579


]

]

(18)


11987011

=1198701015840

11+ 1198701015840

22

2+

1198701015840

11minus 1198701015840

22

2cos 2120579

11987012

= 11987021

= minus1198701015840

11minus 1198701015840

22

2sin 2120579

11987022

=1198701015840

11+ 1198701015840

22

2minus

1198701015840

11minus 1198701015840

22

2cos 2120579

(19)

where 1198701015840




4 A Case Study





[90] 75∘ direc-

tion 119870[75]

60∘ direction 119870[60]

45∘ direction 119870[45]






(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)


0∘15∘

30∘

45∘

60∘

75∘

90∘

115∘

130∘

145∘

160∘

175∘180∘













1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1and 119863













0(GPa) 65




119894119895 damage tensor 119863


119894119895



015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










015

30

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330

345

000E + 00

100E minus 06

200E minus 06

300E minus 06

400E minus 06

500E minus 06

1m4m7m

8m9m


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10Sample size (m)

Dev

iatio

n of

the h

ydra

ulic

cond

uctiv

ity co

effici

ents

()

0∘

15∘

30∘








119894119894119864119894119895(GPa) 119866

119894119895(GPa)


000

100

200

300

400

500

6000

1530

45

60

75

90

105

120

135

150165180

195210

225

240

255

270

285

300

315

330345

3m4m

7m

5m10m12m


0

10

20

30

40

50

60

2 3 4 5 6 7 8 9 10 11 12 13Sample size (m)

Dev

iatio

n of

the d

amag

e val

ues (

)

90∘

105∘

120∘







120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120597p120597x = 0120597p120597x = 0

Plane strain

50

m

35m

3m

10m

50m

pw = 221MPa

120590s = 597MPa

I II

A-A998400

(a)

(b)





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

3

25

2

15

1

05

31558 times106

times106

38969 times104


0010203040506070809

111

0 5 10 15 20 25 30 35 40 45 50

Firs

t prin

cipa

l stre

ss (M

Pa)









Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Nor

mal

stre

ss in

join

t pla

ne d

irect

ion

(MPa

)

Hyd

raul

ic co

nduc

tivity

coeffi

cien

t (m

s)

0 5 10 15 20 25 30 35 40 45 50

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05


0E + 00

5E minus 07

1E minus 06

15E minus 06

2E minus 06

25E minus 06


50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

10205 times10minus5

times10minus6

15581 times10minus9

10

8

6

4

2





50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

00

221 times106

times106


1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

16



50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

00

221 times106

times106

(a)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0 0

221 times106

times106

(b)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(c)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(d)

50

45

40

35

30

25

20

15

10

5

0

50454035302520151050

2

15

1

05

0

221 times106

times106

0

(e)

50

45

40

35

30

25

20

15

10

5

050454035302520151050

2

15

1

05

0

221 times106

times106

0

(f)



flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










flow

velo

city

(ms

)

X coordinate (m)

0 5 10 15 20 25 30 35 40 45 50

0∘

30∘60∘

90∘

0E + 00

minus2E minus 06

minus4E minus 06

minus6E minus 06

minus8E minus 06

minus1E minus 05

minus12E minus 05




1205902

1

119883119905119883119888

minus12059011205902

119883119905119883119888

+1205902

2

119884119905119884119888

+119883119888minus 119883119905

119883119905119883119888

1205901+

119884119888minus 119884119905

119884119905119884119888

1205902+

1205912

12

1198782= 1

(20)




119888and 119884



1represents the




119878 = 119888 + 1205902tan120601 (21)




160119884119905

40


120119884119888

96









1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(a)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(b)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(c)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(d)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(e)

1

0

10908070605040302010

20 25 30

34

32

30

28

26

24

22

20

18

(f)







5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusion







Nomenclature



1198701015840














Greek Symbols







surfaces




Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Model of Anisotropic Property of Seepage and …downloads.hindawi.com/journals/jam/2013/420536.pdf · for representing mechanical behavior of the fractured rock