How To Analyze The Flow Patterns Of A Fracture Network

Transcription

1 Flud Flow Complexty n Fracture Networks: Analyss wth Graph Theory and LBM Ghaffar, H.O. Department of Cvl Engneerng and Lassonde Insttute, Unversty of Toronto, Toronto, Canada Nasser, M.H.B. Department of Cvl Engneerng and Lassonde Insttute, Unversty of Toronto, Toronto, Canada Young, R.P. Department of Cvl Engneerng and Lassonde Insttute, Unversty of Toronto, Toronto, Canada s prohbted. Permsson to reproduce n prnt s restrcted to an abstract of not more than words; llustratons may not be coped. The abstract must contan conspcuous acknowledgement of where and by whom the paper was presented. ABSTRACT: Through ths research, embedded synthetc fracture networks n rock masses are studed. To analyss the flud flow complexty n fracture networks wth respect to the varaton of connectvty patterns, two dfferent approaches are employed, namely, the Lattce Boltzmann method and graph theory. The Lattce Boltzmann method s used to show the senstvty of the permeablty and flud velocty dstrbuton to synthetc fracture networks connectvty patterns. Furthermore, the fracture networks are mapped nto the graphs, and the characterstcs of these graphs are compared to the man spatal fracture networks. Among dfferent characterstcs of networks, we dstngush the modularty of networks and sub-graphs dstrbutons. We map the flow regmes nto the proper regons of the network s modularty space. Also, for each type of flud regme, correspondng motfs shapes are scaled. Implemented power law dstrbutons of fracture length n spatal fracture networks yelded the same node s degree dstrbuton n transformed networks. Two general spatal networks are consdered: random networks and networks wth hubness propertes mmckng a spatal damage zone (both wth power law dstrbuton of fracture length). In the frst case, the fractures are embedded n unformly dstrbuted fracture sets; the second case covers spatal fracture zones. We prove numercally that the abnormal change (transton) n permeablty s controlled by the hub growth rate. Also, comparng LBM results wth the characterstc mean length of transformed networks lnks shows a reverse relatonshp between the aforementoned parameters. In addton, the abnormaltes n advecton through nodes are presented.. INTRODUCTION Fracture networks embedded n dfferent structures are subject to the vared research areas. However, the man relevance of fracture networks s the propagaton of fractures (cracks) n brttle materals. The complex form of lnked cracks has nspred scentsts to develop several theores of fracture network formaton s general mechansms. One of these theoretcal branches s assocated wth sngle fracture mechancs. In complement wth ths approach, the nteracton of fractures yelds modfed stress feld. Then, the concentraton of stress or varaton of stran satsfes yeldng crtera []. Inducng dsorder nto the system (say ntact rock) gves the fracture s path of propagaton. Ths branch of fracture propagaton ncludes a wde range of methods: such as, lnear elastc fracture methods; effectve medum theory; fnte or boundary element methods to tackle the complex geometres; molecular dynamcs-based methods (.e., dstnct element methods) to gve detaled nsght nto the system; lattce-based methods assocated wth random fuse models or random beam methods; and, fnally, the methods based on statstcal feld theory lke nterface or strng growth methods [-5]. The second approach corresponds to the collectve observatons from controllable laboratory tests and feld data. The obtaned patterns thus are analyzed usng pattern recognton technques. Eventually, the am s to organze smple rules to satsfy the same characterstcs (features or attrbutes) for the fracture system. Such smplstc methods have orgnated from assocated wth statstcal physcs. To be specfc, the most well-known employed methods fall under the second approach, applyng statstcal technques to fracture networks. It satsfes general characterstcs of fracture networks lke dstrbuton law over densty, length, drectonalty, dstance, and fractal dmenson as well as

2 characterstcs of flow propagaton lke permeablty [- 9]. The frst steps n analyzng the fractures structural confguratons ncluded the fractalty of the fracture systems. Havng a fractal dmenson wth a small varaton ndcates that the system has a unversalty property. However, there s not such unversal dmensonalty n fracture networks. Researchers have recognzed two fractal dmensons for the roughness (non-drectonal) of a sngle fracture []. Durng the past decade, developments n graph theory [-5] have opened a new chapter n the study of large nterwoven systems (complex systems). Ths new perspectve tackles another facet of the complextes embedded n fractured materals, called structural or topologcal complexty. In contrast, statstcal methods nvestgate uncertantes such as probablty or, n more advanced forms, other uncertantes approaches (lke fuzzy set theory, rough set theory or evdence theores). The relevant topcs n ths feld mght be summarzed as follows: ) transformaton of spatal networks n graph forms; ) nformaton propagaton (flud flow, energy or waves) through networks; ) deformaton of fractures due to mechancal or hydro-mechancal forces and, more generally, the reacton of a dsordered system (lke a fracture system) to external forces. Our focus n ths study s to generate dfferent confguratons of spatal fracture networks. In ths pursut, we try to use the smplest algorthms to fracture network generaton. Then, we transform spatal fracture networks nto graph forms where we gnore the spatal dstrbuton of fractures. The advantage of these transformatons s that t lnks the regular fracture networks to modern graph theory. The fracture zones as a hgh densty of fractures also as the abnormal emergence of fractures are another part of our research. We present a smple algorthm that gves the drectonalty and effectve radus of the hubs, whle takng nto account the growth of the hubs under a background growth of random jonts. Advecton of nformaton (here flud flow) through the generated spatal network and transformed networks s the man part of ths study. To analyze flud flow, three dfferent methods were employed. We use lattce Boltzmann (LBM) and fnte element methods (FEM) to obtan the flud flow patterns, velocty, pressure dstrbuton and permeablty for the spatal fracture networks. Addtonally, an advecton- based network equaton s used for transformed networks. The organzaton of the paper s as follows. The frst secton presents a smple algorthm to generate fracture networks and show how they are converted nto the graphs. In ths secton, we also ntroduce some basc characterstcs of graphs. The next secton summarzes three methods LBM, FEM and advecton-based networks to model flud flow (lamnar) n fracture networks. The man part of our study wll be covered n secton, where we present the results and dscuss the accuracy of the employed methods. Fnally, the summary and concluson of the present work s presented.. FRACTURE NETWORKS AND GRAPH THEORY In ths part, a smple algorthm s ntroduced to cover the man characterstcs of fracture networks. The transformaton of constructed fracture networks nto graph forms, and the dstngushng characterstcs of the graphs, are demonstrated next... Random Fracture Networks Several algorthms have been proposed to generate and cover the man statstcal propertes of natural fracture networks (based on the second approach presented n our ntroducton). The smplest algorthms n D consder the dstrbuton of fracture length, dp drecton of jont sets and jont spacng [7-9]. New generatons of fracture networks algorthms (or Dscrete Fracture Networks: DFN) consder other parameters that ncrease the accuracy of the estmatons, such as: the fracture densty parameter (number of fractures per length/area or volume) and fractal dmenson, where the fracture center s fractalty s mposed through a herarchal multplcatve process [9]. Most of the aforementoned algorthms (and codes) mpose a power law or modfed power law dstrbuton to fracture length. The cut-off value n power law dstrbuton shows a collapsng data set, ndcatng a smlar mechansm/sgnature n the modeled data set. The basc dea behnd the power law dstrbuton of fractures remans a challengng queston. However, recent developments n graph theory have tred to provde a reasonable answer. In three-dmensonal scenaros, each ndvdual fracture s assumed as trangle, crcle or any other polygon shapes where the consecutve generatons based on the pre-set parameters are accomplshed [7]. Another man attrbute of the fracture system s the densty or spatal densty of fractures. Ths property yelds the possble concentraton of fractures n space as well as fracture tps, around tunnels, the nterface between two layers and generally any sharp or dsturbed sub-felds wthn the system. The equvalent defnton of such fracture zones may be found n modularty and the hubness concept, n whch groups, communtes or dense clusters, through random lnks, communcate wth each other []. Our algorthm captures the power law dstrbuton, drectonalty of the jont sets and the hubness propertes of the networks, as descrbed below (Fgure ).

3 Ths algorthm s based on power law dstrbuton of fracture length pl () l γ where l s the fracture length and gamma s the power that controls the connectvty and length. When gamma ncreases, the probablty of fndng fractures wth long length reduces. Other parameters are: the number of jont sets and fracture zones (hubs), hub growth, background rock jont growth (hereafter called the external lnks or back jonts), maxmum dameter of growth n x and/or y drectons, azmuths of jont sets and mean aperture of each fracture. The parameter related to hub and back growth s reported as recprocal values. Then large values of growth show slow propagaton of lnks. The spatal dstrbutons of hubs are accomplshed by employng a Gaussan dstrbuton. The growth n hub zones and complementary background fractures are based on a power law dstrbuton of fracture length and the unform dstrbuton of jont sets dps. We show that, for a wde range of gamma parameters, the dstrbuton of lnks n the transformed shape of fracture networks obeys the power law dstrbuton, ndcatng scale free networks. In other words, wth the descrbed method, we acheved modular networks wth power dstrbuton n transformed networks. n_g=; %Number of Generatons n=; %Number of grds NJS =; %Number of Jont set gama=.55; %parameter related to broadness of fracture length pl () l γ ; γ =.55 alpha=5*n./; %parameter to scale the power law hub_growth=;% recprocal of growth rate of fractures around hubs Back_growth=;%recprocal of growth rate of background fractures N.F.Z=: %number of fracture zones (hubs) r = round( +.*rand(,)); r = round( +.*rand(,)); %radus of hubs -effectve zone of fracture zones (homogeneous cores) Fg.. Part of the parameters employed n fracture network code-the code was compled n M-fle format/matlab... Modern Graph Theory A network conssts of the nodes and edges connectng them [-]. In order to analyze the network propertes of the generated fracture networks, each fracture generaton was mapped nto a correspondng node n a graph space. When two fractures are ntersected, two nodes are connected by a lnk. Ths procedure s llustrated n Fgure, whch enables us to nvestgate the networks usng the tools from modern network theory dscussed n the followng. Fg.. The transformaton procedure of a spatal fracture network n a graph shape wth nodes and edges (also see []). Let us ntroduce some propertes of the networks, such as: the clusterng coeffcent (C), degree dstrbuton ( P ( k ) ) and average path length ( L ). The clusterng coeffcent descrbes the degree to whch k neghbors of a partcular node are connected to each other. Neghbors are defned as the connected nodes to a partcular node. The clusterng coeffcent shows the collaboraton between the connected nodes,.e., the local structures. Assumng that the th node has k neghborng nodes, there can exst at most k( k )/ edges between the neghbors. c s defned as the rato: c= th number of edges between the neghbors of the node k(k-)/ () Then, the clusterng coeffcent s gven by the average of c over all the nodes n the network []: N C = c. N = For k we defnec. The closer C s to one, the larger the nterconnectedness of the network. The connectvty dstrbuton (or degree dstrbuton), Pk ( ), s the probablty of fndng nodes wth k edges n a network. In large networks, there wll always be some fluctuaton n the degree of dstrbuton. Large fluctuatons from the average value ( < k > ) refer to the hghly heterogeneous networks, whle homogeneous networks dsplay low fluctuatons [-]. The average (characterstc) path length, L, s the mean length of the shortest paths connectng any two nodes on the graph. The shortest path between a par (, j) of nodes n a network can be assumed as ther geodesc dstance, g j, wth a mean geodesc dstance L gven as below []: L= gj, NN ( ) () < j ()

4 where g j s the geodesc dstance (shortest dstance) between node and j, and N s the number of nodes. Based on the aforementoned network characterstcs, two lower and upper bounds of networks can be recognzed: regular networks and random networks (or Erdős Rény networks []). Regular networks have a hgh clusterng coeffcent (C /) and a long average path length. Random networks (whose constructon s based on the random connecton of nodes) have a low clusterng coeffcent and the shortest possble average path length. However, Watts and Strogatz [-] ntroduced a new type of network wth hgh clusterng coeffcent and small (much smaller than the regular ones) average path length. Analyss of the nternal network structures, usng the obtaned networks, s presented by sub-graphs and motfs. The subgraphs are the nodes wthn the network wth the specal shape(s) of connectvty. The relatve abundance of subgraphs has been shown to be an ndex to the functonalty of networks wth respect to nformaton processng. They also correlate wth the global characterstcs of the networks [9-]. The network motfs ntroduced by Mlo et al. [9] are the partcular sub-graphs representng patterns of local nterconnectons between the nodes n the network. A motf s a sub-graph that appears more than a certan amount (other crtera can be found n the lterature). A motf of sze k (contanng k nodes) s called a k-motf (or, more generally, a sub-graph)... Advecton Based Network Equaton For a gven network wth N nodes, the degree of the node and the Laplacan of the connectvty matrx are defned by [5]: N k = A ; L = A kδ () j j j j j= where k, A j, Lj are the degree of th node, elements of a symmetrc adjacency matrx, and the network Laplacan matrx, respectvely. The flow of nformaton ( u ) s expressed by the network s mode of dffuson or advecton []: N d u () t = f ( u ) + ε L j u j dt (5) j= where f ( u ) and ε are the local dynamcs of each profle (node) and dffuson constant, respectvely (n our smulaton f( u ) = ; ε = ). Ths equaton s a dscrete form of the classcal dffuson (advecton) equaton. In a steady state case, we have N ε Lu j j = whch s equvalent wth u = j= whle each node s connected to k of other nodes nstead of or nodes. The applcaton of Eq. as coupled oscllators also has been nvestgated n synchronzaton (or the reachng of a steady tme state) of oscllators mounted on each node.. LATTICE BOLTZMANN METHOD (LBM) Hstorcally, the LBM results from efforts to mprove the Lattce Gas Cellular Automata (LGCA) []. The LBM predcts the dstrbuton functon of fcttous flud partcles on fxed lattce stes for dscrete tme steps n dscrete drectons. Two man steps n the LBM algorthm are the streamng and collson. In the streamng step, partcles move to the nearest neghbor along ther drecton-wse dscretzed veloctes. The streamng process s followed by the collson step, where the partcles relax towards a local equlbrum dstrbuton functon. In the sngle-relaxaton tme LBM, the collson operator s smplfed to nclude only one relaxaton tme for all the modes. In the DQ9 LBM, the flud partcles at each node are allowed to move to ther eght nearest neghbors wth eght dfferent veloctes, e. The nnth partcle s at rest and does not move. The flud densty and macroscopc veloctes are calculated by properly ntegratng the partcle dstrbuton functons on each node. The evoluton of dstrbuton functon n sngle-relaxaton tme LBM [- ] s gven by: Δt eq f( x+ eδ tt, +Δ t) = f(,) x t [ f(,) x t f (,)] x t () τ where for any lattce node, x + e Δ t s ts nearest node along the drecton ; τ s the relaxaton tme; and f s the equlbrum dstrbuton functon n drecton. The macroscopc flud varables, densty ρ and velocty can be obtaned from the moments of the dstrbuton functons as follows: ρ = f; ρv= fe (7) whle the flud pressure feld p s determned by the equaton of the state of an deal gas: p= Cs ρ () where C s the flud speed of sound and, for DQ9, the s lattce s equal to. The relaxaton tme characterzes the tme-scale behavor of flud partcle collsons and determnes the lattce flud vscosty: ν = ( τ ) (9) Once the macroscopc velocty feld s determned, the permeablty of the medum under study can be predcted usng Darcy s law: eq

5 aperture(arbtary unt) K v =. p () μ where v, K, p and μ are the volume averaged flow velocty, permeablty tensor, pressure gradent vector and the dynamc vscosty of the flud, respectvely. The flud flow s drven by pressure (or by some external force) on the boundares. To extract the correct permeablty values, the system should be n the steady state. The computaton s n the steady state when the relatve change of the average velocty s less than a tolerance varable (here -9 ). More detals about the dfferent numercal and techncal aspects of the LBM can be found elsewhere [-].. RESULTS AND DISCUSSION In ths secton, the results of our smulaton, based on the aforementoned methods, are presented. Twodmensonal fracture networks were generated at the percolaton threshold, where there s a contnuous path from one sde of the system to the other. All of the cases had 7 *7 grds except one case whch ncludes 5*5 grds. The frst case (upper row n Fgure a) was a complex one that ncluded a unform dstrbuton of apertures for the fractures examned. However, n all of the other cases, smlar values of openng (aspect rato) were chosen for the generated fractures. As t has been depcted n Fgures and, ncreasng the gamma parameter, fracture length and ncreases the number of fractures wth small length. Ths changes the total permeablty and velocty dtrbuton of flud. Furthermore, the ncrement of gamma parameter nduces a less complex flud pattern (Fgure ). A broad range of drectonalty n fracture generatons also dramatcally changes the complexty of the flow. Ths ssue s revealed n the curvature of the flow paths,.e. the tortuosty. It nduces more consumpton of energy n order to reach steady state or the flow to drve. The fracture systems wth regular rock jont sets exhbt relatvely smpler, and therefore more predctable, flud flow paths (Fgure 5). Partcularly, the relatve drecton of flow to the dp of fracture sets changes the spread of permeablty (the range of varaton of permeablty [Fgure ]). The dstrbuton of velocty drectly affects the total permeablty of the system as well as synchronzaton tme (tme to reach steady state). The results of the modelng wth LBM and FEM (Fgures, 5 and 7) show dstngushed abnormalty and dfferentated spatal area n velocty dstrbuton. As t has been vsualzed n fgures and 5, the dense fractures resulted from low values of gamma parameters havng more complcated dscretzed areas (blocks) than Hereafter, regular rock jonts stands for jonts wth sets (two or three). For example see: Permeablty of Three-D Random Fber Webs by Koponen et al n PRL-VOL.-99 regular rock jont sets. We detect -edged blocks (as loops) and other non-loops -pont nodes n transformed graphs. It has been shown that the general confguratons of the generated systems vary, but that the system ensures a nearly dentcal dstrbuton of - pont loops/non-loops. Ths s correct for a broad range of parameter varatons n terms of gamma. In the fgure above, gamma s spelled as 'gama'. Should t be gamma there too? p p gama= gama=.5 gama=.75 ; two jont set q y 5 gama=.75; two jont set gama=.75; Modular Network gama=.75; Modular Network Multple hubs Fg..Varaton of fracture networks depends on gamma parameter jont sets and the densty of fractures. The angular dstrbuton of jonts and fracture length has been shown as the nsets. -5 γ = -5 γ = p() l l γ - γ =.5 Fg.. Complex fracture networks wth dfferent dstrbuton n fracture length and correspondng flud flow patterns, obtaned by LBM. The velocty feld s n logarthmc scale. To model the heterogenety n fracture densty, we set a unform core for each module (fracture zone) wth a maxmum or unts (arbtrary) radus, and wth starlke drectonalty (wth unform dstrbuton). Background random lnks have a unform dstrbuton of dp n the range of - (half star). Modelng wth LBM (and also wth FEM) shows how fracture zones trap the flow and change pressure dstrbutons (Fgure and 7). Generally, the fracture zones act as the relaxaton It can be nvestgated further that transformed networks produce sub-graphs wth dfferent shapes. p -5

6 of velocty and rapd reducton of velocty (and then the gradent of pressure). More fracture zones generally reduce the permeablty of the total system; however, ts reducton magntude depends on qualty of lnks among modules (background rock jont). p Tme step s γ =.75 γ =.75 γ = Fg. 5. Other confguratons of complex fracture networks wth dfferent dstrbutons n fracture length, the drectonalty of jont sets wth two sets and correspondng flud flow patterns, obtaned by LBM. The velocty feld s n logarthmc scale. Fg. 7. Flud flow modelng wth FEM n a) Sngle hub pattern wth gamma =.75, hub growth=, back growth =; b) Multple hubs wth 5 grds and c) two jont sets (, 9 degree azmuths). The rght hand pctures are the pressure dstrbuton wth the correspondng networks. a) b) c) - - P(k) frequency Estmated Probablty Densty Functon - The fttness coeffecnet s The correlaton coeffecnets s ndex# k.5 e) f) Two general ponts should be consdered n the analyss of permeablty: the nfluence of nternal lnks n the fracture zones and the effects of external lnks. Each of these parameters effectvely changes the nformaton (flud) of the flow. Weak external lnks yeld concentraton of partcles n especal modules whle frequent external lnks reduce concentraton of partcles and the flow from modules. The nternal fractures of the fracture zone mpose local complexty, and can be assumed to be sub-fracture networks wthn the system. The combnaton of the aforementoned factors gves rse to a complcated stuaton d) - log(clusterng coeffcent) Fg.. Left) Sngle hub pattern wth gamma =.75, hub growth=, back growth =, ts velocty dstrbuton usng LBM and pressure dstrbuton; Rght) Sngle hub pattern wth gamma =.75, hub growth=, back growth =, ts velocty dstrbuton and pressure dstrbuton, from up to down log(node degree) Nodes# Fg.. a) Network wth generaton per each hub wth hubs. Hub growth and back-growth parameters respectvely are and wth gamma=.5; b) sub-graphs dstrbuton of ponts sub-graphs over transformed fracture networks n graphs; c) dstrbuton of node s degree; d) scalng of clusterng coeffcent wth number of lnks; e) dstrbuton of velocty varaton n advecton network system; and f) vsualzaton of node s dstance.

7 The fttness coeffecnet s The correlaton coeffecnets s log(node degree) The fttness coeffecnet s The correlaton coeffecnets s -..9 log(node degree) Estmated Probablty Densty Functon k log(clusterng coeffcent) a) d) b) e) c) Fg. 9. a) Network wth generatons per each hub wth hubs. Hub growth and back-growth parameters respectvely are and wth gamma=.5; b) sub-graphs dstrbuton of - ponts sub-graphs over transformed fracture networks n graphs; c) dstrbuton of node s degree; d) scalng of clusterng coeffcent wth number of lnks; e) dstrbuton of velocty varaton n advecton network system; and f) vsualzaton of node s dstance. a) b) c) 5 - f) the stablty of the soluton wth respect to Δt and Δ x (ε s constant per each fracture whch s equvalent wth permeablty of each jont). In fgures to, fve man statstcal characterstcs of graphs have been shown. Part b n the aforementoned fgures shows the dstrbuton of -pont sub graphs, ndcated by ndexes to. Despte dfferent confguratons n hubness mplementaton (fgures and 9), the frequency of sub-graphs follows a smlar trend. Decrease n the frequency of the ndex, as a rectangular loop, and ncrease n the ndex 5 are common features of the frequences. Ths s not observed n regular jonts wth perpendcular ntersecton (Fgure ) n whch the ndex shows a slght ncrease. Referrng to smulaton wth LBM (or FEM), t reveals that the regular fracture networks wth pre-set jont sets present much more unform dstrbuton n the velocty than n fracture zones (all see Fgure wth advectonnetwork equaton). log(clusterng coeffcent) d) e) f) P(k) Fg.. a) Network wth generaton per each hub wth hubs And gamma=.5 wth two rock jont sets; b) sub-graphs dstrbuton of -ponts sub-graphs over transformed fracture networks n graphs; c) dstrbuton of node s degree; d) scalng of clusterng coeffcent wth number of lnks; e) dstrbuton of velocty dstrbuton n advecton network system; and f) vsualzaton of node s dstance. It s notced that, due to the random nature of the fractures webs, Monte Carlo smulaton s necessary to get a precse answer. For LBM, only 5- fractures for each case (each constant parameter n the model) have been analyzed snce the procedure s tme-consumng. For modelng n FEM, t has been assumed that each fracture has the dmenson n z drecton as the aperture /openng of each jont. The results of smulaton wth LBM and FEM (5, and 7) show approxmately the same patterns n flud flow path. It s expected that the results for permeablty or twstablty wll show the same answer usng both methods. Now, we map the spatal jont networks nto graphs, usng the method descrbed n secton.. It has been dstngushed how dfferent hubness parameters affect the herarchcal patterns n fractures. Usng the method descrbed n secton., the dstrbuton of velocty over nodes n steady state s presented and compared wth the results obtaned wth the FEM method. To model Eq., the Euler method has been used n ntegraton, consderng P(u) HG=,BG= HG=,BG= HG=,BG= HG=,BG= HG=5,BG= HG=,BG= HG=,BG= HG=,BG=5 HG=,BG= u Fgure. Dstrbuton of velocty n transformed networks usng the steady state soluton of the advecton-network system over dfferent confguratons of hub growth (HG) and back fracture growth (BG). The values of growths are recprocal to the propagatons of fracture. Total number (max.) of hubs could be. The smulated system had snks and sources nodes wth and values of velocty (or concentraton of partcles). The frst and last fracture networks were consdered as sources and snks respectvely. It suggests that the ndex (and somehow 5 ) mples the most channelzed flow, possbly wth a low value of twstablty. Indexes and 5 ndcate such abnormalty n flud flow. Index possbly shows the most The results presented here are the mean results for 5 tmes realzatons. The njecton of partcles from not-smlar ponts lke fgures -7 also mght be questonable, but t can be proved that the general dstrbuton would not be affected. 5 We analyzed ths fact n a sngle shear fracture; see H.Ghaffar et al., Flud flow analyss n a rough fracture (type II) usng complex networks and lattce Boltzmann method, PanAm-CGS (), Toronto, Canada.

8 homogenety n nformaton flow. Such abnormalty n the dstrbuton of velocty (or partcle concentraton) through nodes (fractures) has been modeled wth the advecton based network equaton descrbed n secton., and llustrated n part e of the fgures to. In our model, the source and snk terms were kept n the frst and last generatons of fractures (wth values wth the amount and -, respectvely). Increasng external lnks (background jonts) nduces nformalty and homogenety n the pressure and velocty of spatal spread. In other words, t somehow suppresses the hubness and modularty attrbute of the system. Increasng the hubness aspect and decreasng external lnks yelds non-unform flow and concentraton of partcles n specal nodes. J= J= Sngle hub dstrbuton. In other words, our fracture algorthm (wth or wthout modularty), for a wde range of parameter varatons, gves scale-free networks. Part d demonstrates a space called the spectrum of graphs, whch s assocated wth the clusterng coeffcent (c) - node s degree (k) space. It has been shown that networks wth power law dstrbuton wth a lnear scalng of c-k present wth a herarchcal property [7]. Strong herarchcal patterns n modular networks and weak herarchcal patterns n regular fracture systems have been observed (Fgure ). K x x α K x x γ =.75 f(u) 5 multple hubs u γ θ Fg.. Left) Varatons of permeablty wth gamma parameter through complex fracture network confguraton; Rght) varatons of permeablty wth the drectonalty (azmuth) of jont sets under constant gamma parameter (no hubness property s nduced n facture patterns). f(u) u Fg.. Dstrbuton of velocty n spatal fracture networks usng FEM over confguratons of spatal fracture networks. The smulated systems have been shown n Fgure 7. Fgure shows the dstrbuton of velocty felds n the steady state case for dfferent confguratons of fracture zones parameters. The general trend of dstrbuton s Gaussan. For the system wth mnmum effect of modularty, we observed more fluctuaton n dstrbuton wth an evdent peak around zero velocty. Ths ndcates that decreasng modularty nduces a transton n velocty (and then total permeablty) dstrbuton. Deletng or cuttng external lnkages forces the system to have power law dstrbuton.e. abnormalty n pressure and velocty abundances wth less fluctuaton n velocty or pressure frequences. Plottng the dstrbuton of velocty feld -modeled wth FEM- shows a dstrbuton wth the long tal characterstc. (Fgure ). In the transformaton of spatal networks, the spatalty has been lost and the same propertes of the flow wth such a mappng nto the graphs have been recovered. Part c n the above fgures shows the degree dstrbuton, whch roughly obeys power law Recent analyss n network theory has hghlghted the abnormalty of dffuson wth random walk theory and stochastc partcle smulaton, for example see PRL 9,7(5) and PRE E,55(). As the last part of our research, we focus on the possble relaton between the mean path length of transformed networks and varaton of permeablty. Part f n fgures - shows the dstance of nodes, where cold colors ndcate near neghborhoods or mmedate ntersected jonts and hot colors ndcate far neghborhoods or non-connected nodes. Comparng the calculated mean path length or the graph s dameter and the smulated permeablty shows a reverse correlaton between the two parameters. Rch neghborhood attrbute exposes a hghly permeable fractured system, whle poor nodes reveal low permeablty from a connectvty pont of vew (Fgure ). K x <l> Fg.. An approxmate relaton between the mean path lengths of transformed fracture networks and smulated permeablty usng LBM. Each pont results from 5 tme realzatons.

9 5. CONCLUSIONS Fracture networks n rock masses have been studed n ths paper. The flud flow n fracture networks, as a functon of varaton of connectvty patterns, was analyzed. The Lattce Boltzmann method and FEM were used n modelng permeablty and flud velocty dstrbuton. Furthermore, fracture networks were mapped nto graphs, and the characterstcs of the obtaned graphs were compared wth the man spatal fracture networks. The hubness character of fractures as fracture zones has been dstngushed. The results showed that fracture zones, the nternal and external lnks of the damaged zones, dramatcally change the transport propertes of a rock mass. REFERENCES [] Adler M.P., Thovert J.F., Fractures and Fracture Networks, Kluwer Academc, 999. [] Alava M. J., Nukala P.K. V. V., Zapper S., Statstcal models of fracture, Advances n Physcs, 55,,, 9 7. [] Kachanov M., Elastc solds wth many cracks: A smple method of analyss, I nt. J. Solds Struct, 97, -. [] Bonamya D., Bouchaud E., Falure of heterogeneous materals: A dynamc phase transton?, Physcs Reports, 9,,. [5]. Renshaw C.E, and Pollard D.D., Numercal smulaton of fracture set formaton: A fracture mechancs model consstent wth expermental observatons, Journal of Geophyscal Research-Sold Earth, 99(B5), 99, [] Berkowtz and Adler, Stereologcal analyss of fracture network structure n geologcal formatons, Journal of Geophyscal Research, Vol., No. B7, Pages 5,9-5,, July, 99. [7] Bour et al., A statstcal scalng model for fracture network geometry, wth valdaton on a multscale mappng of a jont network (Hornelen Basn, Norway), Journal Of Geophyscal Research, Vol. 7, No. B,,.9/jb7,. [] Sornette A. Davy P.,, and Sornette D., Fault growth n brttle-ductle experments and the mechancs of contnental collsons, J. Geophys. Res.,9,,,9, 99. [9] Berkowtz B., Bour O., Davy P., and Odlng N., Scalng of fracture connectvty n geologcal formatons, Geophys. Res. Lett., 7(),,. [] Davy P., Sornette A., and Sornette D., Some consequences of a proposed fractal nature of contnental faultng, Nature,, 5 5, 99. [] Newman M.E.J., The structure and functon of complex networks, SIAM Revew 5 () 7-5. [] Albert R., Barabás A.-L., Statstcal mechancs of complex networks, Revew of Modern Physcs 7 () [] Barabás A.L., Albert R., Jeong H., Mean feld theory for scale-free random networks, 999.arxv:condmat/997v. [] K. Sanyal. Bomer, A. Vespganan, Network s scence, In: Annual Revew of Informaton Scence & Technology (7) [5] Watts D.J., Strogatz S.H., Collectve dynamcs of `smallworld' networks, Nature 9 (99) -. [] Andresen C.A., Propertes of fracture networks and other network systems, Norwegan Unversty of Scence and Technology Faculty of Natural Scences and Technology Department of Physcs, PhD thess,. [7] Gertsch L.S.. Three-dmensonal fracture network models from laboratory scale rock samples. Int. J. Rock Mech. Mn. Sc. & Geomech. Abstr., ():5 9, 995. [] Boccalett S.. Complex networks: Structure and dynamcs. Physcs Reports, :75,. [9] Mlo R., et al.. Network motfs: smple buldng: blocks of complex networks, Scence 9, 7. [] Mlo R., Itzkovtz S., Kashtan N., Levtt R., Shen-Orr S., Ayzenshtat I., et al.. Superfamles of evolved and desgned networks, Scence, 5 5. [] Berkowtz B.. Characterzng flow and transport n fractured geologcal meda: A revew. Advances n Water Resources, 5:,. [] Madad M., and Sahm M.,. Lattce Boltzmann smulaton of flud flow n fracture networks wth rough, selfaffne fractures, Phys. Rev. E 7,,9. [] Succ S.,. The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, Oxford Unversty Press, pp. [] Eker E., and Akn S.,. Lattce-Boltzmann smulaton of flud flows n synthetc fractures. Trans. Por. Meda, 5:. [5] McGraw P.N., & Menznger,M. Laplacan spectra as a dagnostc tool for network structure and dynamcs. Phys. Rev. E 77, (). [] Nakao H., & Mkhalov A. S.,. Turng patterns n network-organzed actvator nhbtor systems, Nature Phys., [7] Ravasz E. and Barabás A.-L., Phys. Rev. E 7, ().