1  

 

Multi-scale approach to invasion percolation of rock fracture networks

Ali N. Ebrahimi, Falk K. Wittel*, Nuno A.M. Araújo, Hans J. Herrmann

Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich,

Stefano-Franscini-Platz 3, CH-8093 Zurich, Switzerland *corresponding author: Tel. +41 44 633 2871. E-mail address: fwittel@ethz.ch

Abstract

A multi-scale scheme for the invasion percolation of rock fracture networks with heterogeneous fracture aperture fields is proposed. Inside fractures, fluid transport is calculated on the finest scale and found to be localized in channels as a consequence of the aperture field. The channel network is characterized and reduced to a vectorized artificial channel network (ACN). Different realizations of ACNs are used to systematically calculate efficient apertures for fluid transport inside differently sized fractures as well as fracture intersection and entry properties. Typical situations in fracture networks are parameterized by fracture inclination, flow path length along the fracture and intersection lengths in the entrance and outlet zones of fractures. Using these scaling relations obtained from the finer scales, we simulate the invasion process of immiscible fluids into saturated discrete fracture networks, which were studied in previous works.

Keywords: Fracture network, aperture field, permeability, invasion percolation, sub-surface flow, two-phase flow

1. Introduction

Transport of fluids in fractured geological media plays an important role in different applications such as subsurface hydrology, hydrocarbon recovery from natural reservoirs, safe storage facilities for captured CO2 or hazardous wastes (Sahimi, 1995; Adler and Thovert, 1999). The prediction of the penetration of an immiscible fluid into a fully saturated fracture network and its percolation is among the most challenging topics in sub-surface hydrology. Fractured rock, instead of behaving like an equivalent continuum, is only sparsely fractured, restricting fluid flow to a small part of the connected fractures (Reeves, et al., 2008). Fluids can take multiple pathways, can be trapped, and exhibit path instabilities and scale dependencies, what makes the prediction of effective properties rather challenging. Detailed descriptions however are rarely capable of capturing more than a handful of interconnected fractures and fail in representing the complex system dynamics emerging from the interaction of a huge number of locally interacting connected fractures. A partial solution to this dilemma was found in the Discrete Fracture Network (DFN) approach that assumes fluid flow to be entirely localized in the network of connected fractures (Smith and Schwartz, 1984; Huseby et al., 2001). DFN models have been widely applied to study fluid flow and transport characteristics of several fractured rocks with low permeability and fracture density (Reeves, et al., 2008; Sisavath, et al., 2004; Renshaw, 1999). As one expects, transport properties of the networks are strongly affected by the fracture density, size and interactions (Khamforoush and Shams, 2007; Koudina, et al., 1998; Dreuzy, et al., 2000). For a detailed description of the

  

2  

 

physics of fracture networks we refer to the reviews by Berkowitz (2002), Sahimi (1995), and Adler and Thovert (1999).

Since fluid flow in a single fracture is at the bottom of any DFN model, a detailed description of its geometry and hydraulic behavior is essential (Charmet, et al., 1990; Pompe, et al., 1990; Adler and Thovert, 1999). Realistic topological measurements of natural fracture planes revealed spatial correlations (Candela et al., 2012). From those, aperture field distributions can be obtained (Kumar, et al., 1997) and applied for studying the hydraulic behavior of the single fracture (Konzuk and Kueper, 2004). The most striking observation is that significant parts of the natural fractures have zero aperture since they are contact zones, while the fluid flow is naturally localized in areas with non-zero aperture which are also called open zones. Hence, the fluctuations in the aperture field lead to fluid flow in a network of channels (Katsumi, et al., 2009) and fracture interactions are governed by the characteristics of these two networks. To model the displacement of immiscible fluids inside DFN models, invasion percolation (IP) theory proved to be an efficient tool (Wilkinson and Willemsen, 1983). In fact, IP is a modified form of ordinary percolation (Broadbent and Hammersley, 1957), only with a well-defined sequence of invasion events. In IP theory both gravity and viscous effects are neglected and only capillary forces are considered. However, IP is a valid approximation to describe the slow immiscible displacement of two phase flow in fracture networks.

In this work, details of the single fracture with its inherent disorder are upscaled to the fracture network by a two-step coarse graining method. Vectorized artificial channel networks (ACN) are used, that are highly compressed representations of channelized flow on the single fracture scale obtained from Finite Element (FE) simulations with heterogeneous aperture fields. ACNs are used to calculate the scaling behavior of hydraulic transport properties of the single fractures in terms of size dependent equivalent apertures, as well as size and angle dependent fracture-fracture interactions for entry apertures. The numerically obtained rules are then incorporated into a fracture-network consisting of 4000 intersecting fractures for a modified invasion percolation (MIP) simulation with two-phase flow (Wettstein et al., 2012). This way we obtain a more realistic physical description of the invasion process with coarse grained information from the fracture scale.

2. Methodology

In this methodological section, first flow at the fracture scale is addressed by FE simulations and as prerequisite for data compression for the ACN. As a next step we describe fracture interaction parameters, effective properties and their consideration on the DFN scale in trapping MIP.

2.1. Flow at the fracture scale

Single fractures have fundamental meaning in fracture network modeling (Cappa, 2011) since from this, scale size effects emerge. The effect of fracture aperture and roughness on hydraulic properties was already addressed from the theoretical and numerical perspectives in the past (Drazer and Koplik, 2000; Drazer and Koplik, 2002; Auradou, et al., 2001; Berkowitz, 2002; Sahimi, 1995). In general, fracture surfaces exhibit spatial correlation (Méheust and Schmittbuhl, 2001) and are self-affine with roughness (Hurst) exponent close to 0.8 for diverse materials (Bouchaud, et al., 1990; Ansari-Rad, et al., 2012; Dyer, et al., 2012). However also roughness exponents close to 0.6 can be observed in the direction of slip at

  

 

 

laboratofaults (BfractureHagen-Pand Kuebetween2009). Dcan be f(2009). fractionsurface homogesurfaces(2005);

In this random geologicutilized apertureand xi bet al. 20system chosen sites, va

Fig. 1. C

To modflow for

ory scale saBistacchi, e

e opening hPoiseuille feper, 2004)n two facinDetailed exfound for exIsotropic se

nal Browniais vertical

eneous sheas while overDurham (1

study, uncm disordered

cal formatidisorder i

e hi of eachbeing a unifo002). In geand L the with respec

alues are int

Coarse grain(d) bina

del fluid flor small Rey

amples (Amet al., 2011have constaflow as well. However h

ng fracture xplanations xample in Aelf-affine fran motion (lly and horar displacerlapping an997); Katsu

correlated rd aperture fieon. Howevis given byh site i can ormly distrineral we hasize of the ct to the chaterpolated b

ing procedurarized flow a

ow inside oynolds num

mitrano and ; Candela,

ant aperturel as the cubhuge fluctusurfaces (Bof self-affi

Adler and Tacture surfa(Schrenk, erizontally dment. The d contacting

umi, et al. (2

random diselds are use

ver any typey a power-be generate

ibuted randoave sites waperture fi

aracteristic by bi-cubic i

re: (a) Apertand (e) skelet

of single fraber (Re<1)

3 

Schmittbuhet al., 2012

e and couldbic law (Zimuations of thBrown, 198ine fractureThovert (19aces are conet al. (2013displaced w

local aperg regions ar2009)).

sordered fred for genere of apertu-law probabed as hi=exom number

with i=1,…,Lield, e.g. oflength l for

interpolatio

ture field, (btonized field

actures, the , as well as

hl, 2002) an2). In princd be reducemmerman ahe aperture f87; Dreuzy,e surfaces a99), Brown

nstructed us3)). In a cowith respecrture is there assigned

racture surfrality, since ure field canbility densxp(B(xi-1)) in [0,1] (O

Ld, with d df a square lr the apertun (see Fig.1

b) flow rate fifor a single

lubrications negligible

nd even at tiple, fractured to two pnd Bodvarsfield arise d

et al., 201and aperturen (1995) anding a spectronsecutive ct to the oren taken aszero apertu

faces and hthere is no

n be treatedity distribuwith the diliveira, et aldenoting thlattice. The ure field. Be1(a)).

ield, (c) equadisordered f

n approximagravity and

the scale oures with a parallel plasson, 1996; due to conta12; Katsume field distrd Hamzehoral method bstep a copy

original, ress distance ures (Aurado

hence uncofocus on a

d analogouution such isorder paraal., 2011; Brhe dimensio

number ofetween neig

alized flow rfracture.

ation, henced inertial fo

f natural constant

ates with Konzuk

act zones mi, et al.,

ributions ur et al., based on y of the sembling between ou, et al.

orrelated peculiar

usly. The that the

ameter B raunstein on of the f cells is ghboring

ate field,

e Stokes orces are

  

4  

 

assumed. The Stokes equation between two parallel plates relates the pressure field p to the velocity field by

p (1) with the dynamic viscosity of fluid. For a parallel plate the velocity only depends on plate distance (z-direction) while the pressure p is only dependent on the x-direction. Hence Eq. 1 can be rewritten as

2

2,

dp dC

dx dz

(2)

where C is a constant (Méheust and Schmittbuhl, 2001). By solving Eq. 2, one gets the well-known “cubic law”, which gives a good estimate of the volumetric flow rate q through a single fracture as a function of the pressure gradient p in the flow direction. A cubic law equation is given by

3

,12

hq W p

L (3)

where h is the plate distance or fracture aperture and W denotes the width of the fracture perpendicular to the flow direction (see Fig. 1(a)).

To simulate fluid flow through a rough fracture, an aperture field has to be used. Using the lubrication approximation, pressure p only depends on the x- and y-direction, leading to the following form of the Stokes equation:

2

2

( , , )( , ) .

d x y zp x y

dz

(4)

In order to simplify Eq. 4, the local direction u(x,y) and the local coordinate u along this direction for the pressure gradient are introduced by

2

2

( , , ).

dp d x y z

du dz

(5)

In order to reduce the problem to a two dimensional system, the local flow rate direction ˆ( , )q x y can be obtained by integration along the z direction

3ˆ( , )

ˆ( , ) ( , , )12

u

l

z

z

h x yq x y x y z dz p

, (6)

with zl and zu being the lower and upper surface heights respectively, hence the local cubic

law with the aperture field ˆ , h x y . For this two-dimensional approach, the main flow

direction is assumed to be the x -direction with constant pressure at the inlet pin and outlet pout. Also no-flow conditions are defined on boundaries parallel to the flow direction. Note that all quantities are kept in a dimensionless form by using the characteristic units of the numerical simulation, hence the characteristic length l of the aperture field has length unity.

The FE method (FEM) is chosen to discretize Eq. 6 using the COMSOL Multiphysics® software, with the Matlab interface. The aperture field is discretized using structured quadrilateral elements of mesh size d. The weak form of the cubic law is obtained by the linear test functions and by applying Green’s theorem (integration by parts) over the domain

  

 

 

of the erepresenMUltifrmentionassumedmaximu

2.2. I

Since Fand numwhich ianalysisnormalirate fielsuccessithe follo

F

In orderintensitythe MAsignificachannelMorphothe netwunconnerepetitivskeletontopolog

element. Sint extrapolrontal Massned boundard to be zeroum aperture

Identificati

FEM simulamber of realis based ons. First chaized flow ralds along theive construcowing.

Fig. 2. Algor

r to identifyy-level of th

ATLAB Imaant improvels are segological opework to a ected brancve eliminatnized netwical proper

nce Gauss lated valuesively Parary conditiono and the ape.

on and gen

ations are clizations forn the stron

annel netwoate fields. Ie identified ction of the

rithm to iden

y the channehe flow rateage Processiement in th

gmented byerations on system of iches and sption of thork is shorties of th

quadrature s from the

allel Sparsens, aperturepertures of i

neration of t

computationr one value ng channeliorks are exn a second network sk

e artificial ch

ntify the chan

els from thee matrix is iing Toolbox

he visibility y binarizatthe binary identical topurs with e pixels w

own in Fighe network

5 

is used foe integratioe Direct Soe values at ainlet and ou

the artificia

nally expenof disorder

ization of fxtracted by

step, severkeletons. Thhannel netw

nnels and to

e normalizedincreased byx (Gonzalezof channel

tion using images, lik

opology butlow impac

with coordg. 1(e). It

such as

or the numeon points. olver (MUMany boundautlet are trea

al channel n

nsive, consir, a coarse gfluid flow using imag

ral propertiehe channel iwork is sket

create the ar

d flow rate y a histograz, et al., 200s. To extrac

the Otsuke the medit of only o

ct on the todination nu

is the starchannels l

erical integrThe system

MPS). In ory with no

ated like a b

network (AC

dering the graining strainside frac

ge analysis es are extradentificatiotched in Fig

rtificial chan

field, the dyam equalizat04). Fig. 1(ct the skele

u’s methodial axis tranone pixel wotal flow rmber one.

rting point ength, wid

ration, nodm is solveorder to apflow condit

big channel

CN)

required reategy is woctures and algorithms

acted from on algorithmg.2 and desc

nnel network

ynamic rangtion techniq

(c) demonsteton of the nd (see Fignsformation

with. In a nrate are pru The prunfor extrac

dth and de

e values ed by a pply the tions are with the

esolution orked out

network s on the the flow

m and the cribed in

k.

ge of the que from rates the network,

g. 1(d)). n, reduce next step uned by ned and cting the egree of

  

6  

 

connectivity. For this purpose the networks need to be separated into interconnected channels, what can be obtained by simply removing pixels with more than two connections from the skeleton. For further evaluation, removed pixels are stored as connection nodes. The degree of connectivity for each node is obtained by counting the number of the connections of each node to other channels. Note that channels shorter than the characteristic length of the aperture are merged. Since channels are now separated objects, they are labeled and further analyzed. Each channel gets evaluated by measuring its equivalent aperture, width and length, as well as the degree of their connectivity. These properties are measured along and in the vicinity of the channel directly from the field output of the FE simulation. Details on the procedure for calculating the equivalent apertures and widths of channels are given in Appendix A.

Based on extracted channel and network characteristics, an algorithm is developed to generate a reasonable, coarse grained ACN. This highly compressed representation of the fracture flow field will be used to numerically quantify the effect of system size on effective fracture apertures, effective entry apertures for fracture intersections and disorder averaged over many realizations. In principle many different construction principles for the ACN can be used alike. We base our algorithm on vectorizable random lattices (Mourkarzel and Herrmann, 1992). The construction of the ACN starts on a square lattice with Na nodes and characteristic size a that is in principle obtained from the previously extracted node density ρ. The tortuosity of channels is irrelevant, and only the channel length is used for the construction. Due to the merging condition mentioned above, the connected nodes in the ACN model cannot be closer than the minimum distance constraint between two connected channels dmin obtained from the distribution of distance between two nodes. This condition is imposed by shrinking each grid cell to the size b=a-dmin with fixed center. The positions of nodes are set inside these cells. In the next step, the degree of connectivity for each node is randomly distributed from the previously measured degree of node connectivity of the channels. Each node can only be connected to the nearest neighborhood nodes. Finally, the equivalent aperture and channel width is assigned to each connection, sampling from the previously numerically evaluated distribution functions.

2.3. Properties of fracture intersections

Experimental results point at the critical role of the correct treatment of fracture intersections on the dynamics and transport pathways in invasion and transport processes (Ji, et al., 2006, Glass and LaViolette, 2004; Smith and Schwartz, 1984; Wood et al., 2005). Intersections between fractures behave as capillary barriers that control the flow or invasion rate into the fracture network. Intersections of fractures can be regarded as ACNs interacting along intersection lines. Important properties are the equivalent aperture between two intersections, as well as the entry aperture e.g. from fracture i to fracture j. They depend on the characteristics of the underlying channel networks with disorder B, the intersection length Lij and distance dik, as well as their relative spatial orientation (see Fig. 3). In this work, we refrained from trying to find relations of general validity and focused on a numerical approach that calculates intersections of different ACNs instead.

In Fig. 3 flow through fracture j from fracture i to fracture k is sketched. The original equivalent aperture hj

eq of reference fracture j must be modified by considering the path length along the fracture dik

j, as well as the two lengths of the fracture intersection lines at the inlet Lij and outlet Ljk as well as the contact angles between the fractures ( and ). Considering the geometrical configuration of fracture j, hj

eq and wj are modified at the intersection, based on

  

 

 

the rotat

The totall invol

Fig. 3.

In ordechannelchannelassume Euler aconside<10°, in>10°, thbetweendetaileda Gaussfrom the

Once alaperturemust berealizati

2.4. D

The aim(Wettsteand Khregular For the inscribethe inter

tion angles

ˆ sieq eqj jh h

tal flow ratelved channe

Schematic rc

er to obtainls of one frls geometrithat channe

angles anred by the nstantaneouhe entry apen two channd fracture simsian profile.e height of t

ll entry apere for fracture calculated ions, before

Discrete Fr

m of this wein et al., 2

hamforoush or irregularsake of sim

ed in a disk rval [rmin, r

by the follo

ˆin( ); w

e from fractuels in the in

representaticross section

n effective racture onlyically intersel i is in thend aroundlength of th

us invasion erture of thenels in the mulations, t Therefore the intersec

rtures for ovre j in the infor a wide

e they are em

racture Netw

work is to i012). The Dand Shams

r, polygonalmplicity, eac

with radiusrmax], and th

owing equat

ˆ sin(j jww

ure i to fractersection li

ion of the in of the inte

entry apery contributsect. To ste reference d the localhe overlap bof geometr

e single chareference pthe equivalefor the inter

ct point (see

verlapping cntersection orange of po

mployed in

work Mode

improve thDFN constrs (2007). Bl fractures. Tch fracture is RD. The ra

he number o

7 

tions:

( ). (

cture k is obine of fractu

nteraction beersection betw

rtures of frte to fluid ttudy possibplane, connl y and z abetween chrically overannel hij

E caplane. As went aperturersecting cha

e inset in Fig

channels arof fracture iossible fractthe large fr

el

he trapping ruction is exBriefly, the The fracturis modeled adii of the m

of vertices N

(7)

btained by sure j and k.

etween threeween two cha

fracture intetransport in

ble intersecnected to a axes. The annels i andrlapping chaan be obtain

will be explae size hj

eq anannels, hij

E ig. 3).

re calculatedi, is obtaineture intersecracture netw

MIP of Dxplained in fracture nees are orienby an indivmentioned cNv is chosen

umming up

e fractures (annels (inset

ersections, nto an interts betweenchannel j throtation arod j. For critannels is asned from thained in Secnd width wj

in the inters

d from the ted. The effections and a

work simulat

FNs previodetail by H

etwork is conted around vidual polygcircles are rn in the inte

p the flow ra

(i,j,k) and sct).

it is assumrsecting fra

n two chanhat is rotateound the xtical rotatiossumed. Fo

he intersectic. 3.1, basej of channelsection is ca

total flow, tective entry averaged ovtion.

ously develHuseby et alomposed ofthe mean d

gon with Nv

randomly cerval [3, Nm

ates over

chematic

med that acture, if nnels we ed by the x axis is n angles

or angles ion point d on the ls follow alculated

the entry aperture

ver many

oped by l. (2001) f planar,

direction. vertices

hosen in max], both

  

8  

 

from uniform distributions. The vertices are uniformly directed on the perimeter of the circle with uniformly distributed angles from [0,2π]. All fractures have centers that are uniformly distributed in the simulation box of size LN. The orientation of each fracture is adjusted by the unit normal vector of its plane. In order to generate anisotropic fracture networks, the fracture normal vectors are distributed according to the Fisher distribution. In the standard cylindrical

coordinates ' [0, ] and ' [0,2 ] , the Fisher distribution of a random set of normal vectors is defined with a probability density function

' ' '( , ) sin exp( cos ),4 sinh

kf k k

k

(8)

where k>0 is the dispersion parameter about the mean direction which controls the anisotropy in the DFN model. Note that Eq. 8 represents the particular case where initial coordinates '

0and '

0 are equal to zero. The distribution is rotationally symmetric around the initial mean

direction, which coincides with the z-axis. Increasing the value of k concentrates the distribution around this axis. The distribution is unimodal for k>0 and is uniform on the sphere for k=0. In order to build a spanning cluster over the entire domain of the DFN, the fracture density of the system should be high enough. The detailed analysis for the effects of anisotropy and fracture density for the DFN model on the number of intersections of fractures is presented by Huseby et al. (2001) and Khamforoush et al. (2008). For the invaded DFN model, the global system matrix is composed of invaded fractures as elements and the connections as nodes using a simple channel model following Cacas et al. (1990) and Wettstein et al. (2012). After the DFN is constructed, the invasion of immiscible fluids can be simulated. Note that our algorithm should more correctly be named trapping MIP, since it considers entirely encircled non-invaded regions using a burning algorithm (Herrmann et al., 1984). After pruning the mentioned traps and dead ends from the percolating cluster, the invasion percolation backbone is used to calculate the flow through the remaining hydraulic network. The entry zone is chosen in the xz-plane on the left side at y=0, and the exit zone correspondingly as the xz -plane on the right side at y=LN. Periodic boundaries are applied on the other four sides. A detailed description of the algorithm can be found in Wettstein et al. (2012).

3. Coarse graining calculations

Up-scaling in general goes along with loss or compression of information. Hence it is important to quantify the accuracy of the procedures. First a fixed degree of disorder is chosen for demonstration purposes. The distribution of channel network characteristics is studied in detail, before they are used to construct artificial ACNs. After verifying their behavior, parametric studies of intersecting networks are performed to obtain quantitative scaling relations for improving DFNs.

3.1. Quantification of channel properties

The main assumption of the proposed approach is that flow inside fractures is localized within a channel network with distinct topological characteristics. The channels are created due to the dispersion of apertures and can been identified as flow pathways. In the past, numerical and experimental studies have explained the impact of these channels on the permeability of the single fracture at both experimental and field scales (Ishibashi, et al., 2009; Watanabe, et

  

 

 

al., 200obtain pdimensidistribua characviscosit0. We dquality, convergsmaller.numerichave a R

The emthe aperwell. Hrate comincreasito strontheir owdisorder

Fig. 4. E

This stuIn fact, entirely could bethe flownetworkpropertias well networkfunction

8; Talon, etpressure fielional apertu

ution by paracteristic lenty of 10-4 discretize th

a converggence. Reas. The paracal simulatiReynolds nu

mergence andrture field. Cence, first w

mpared to toing heterogengly localizewn, due to thr system can

Effect of diso

udy is limitethis high im

y localized we considere

w for the dik topology ies. The two

as their eqk (Sec.2.2) ns f (see Fig

t al., 2010; lds that are ure fields wameter B (sngth l of unenters at a

he system bence study

sonable accuameters areion. In ordeumber lowe

d impact ofConsequentwe monitorotal flow in eneity of theed flow in he increase n be identif

rder on the f

ed to the campact of thewithin the nd as a paralfferent realto identify

o main propquivalent apwe obtain

g. 5):

Talon and Aused to calcwith disord

see Sec.2.1.)nit length anpressure pin

by a squareof decreas

uracy is obe rendered er to use theer than one.

f channelizatly the charar the effect the entire fre aperture fichannels thof contact z

fied with res

fraction of c

ase of B=4, we channels o

network of cllel plate floizations of

y underlyinperties for cperture heq. information

9 

Auradou, 20culate flow der, defined). The undend a size oin = 10-6 ande grid of elesing l/d revbtained for m

dimensione cubic law

ation strongacteristic siof the diso

fracture, as wfield B fromhat form a zones. In thspect to the

channelized f

when approon the total

channels. Now parallel aperture fie

ng length scchannels arUsing the d

n on all cha

010). Thererate fields fd by the perlying two-of L x W ofd leaves theement edgevealed an exmeshes wit

nless by thw, all dimen

ly depends izes and toporder on thewell as the o

m zero, one snetwork w

his way, the system size

flow as well a

oximately 9l flow rate aote that a loto the channelds allowscales for sce distributiodescribed idannels and

efore we solfor specific power-law -dimensionaf 80 x 80. Fe system at length d. Txcellent expth values ofhe charactensionless pa

on the degrpologies of ne ratio of thopen zone rshifts from aith charactetransition f

e (Andrade,

as the ratio o

4% of the fallows us toower concennel networkus to chara

caling arguon functiondentificationhence norm

lve the cubirealizationsprobability al aperture fFluid of a dzero pressuTo assess thponent of 2f about d=0

eristic unitsarameters a

ree of disornetworks ch

he channelizratio (see Fia parallel pleristic propfrom weak t Jr, et al., 2

of the open zo

flow is chano assume flontration of ck. Channelizacterize the uments of tns for their wn procedure

malized dist

ic law to s of two-

density field has dynamic

ure pout = he mesh 2.025 of 0.25 and s of the are set to

rder B of hange as zed flow ig.4). By late flow erties of to strong 009).

one area.

nnelized. ow to be channels zation of channel

transport width w, e for the tribution

  

 

 

( )eqf h

Fig. 5. Ts

3.2. C

To be connectnode deits valuenodes aThe resconnect6).

Fig. 6

A muchresults. node i is

0.014exp

The distribusystem (B=4)

Characteriz

able to contivity and censity is defe for the dis

are distributsulting distrted nodes, c

6. Distributio

h better valiTo do so, ts represente

0p

2 0.0

eqh

ution of prop). Solid lines

zation and v

nstruct an channel lengfined as a nsordered syted betweenribution of can be comp

on of the dist

idation is ththe cubic laed by

2

2

.0091,

301

erties of chashow the Ga

validation o

ACN, topogth need to

number of nstem with B

n 3, 4 and 5channel le

pared to the

tances betweorigin

he comparisw is applied

10 

( ) 0.f w

annels, (a) wiaussian fit (E

of ACNs

ological feao be measurnodes Na peB=4 is ρ=0.75 connectioengths, hence original fr

een interconnnal fracture

son of pressd to each ch

.014exp

idth, and (b) Eq. 9) of the m

atures, suchred from th

er area of th70±0.01and

ons with 81%ce the distaacture, show

nected nodes(line).

sure and flohannel. Hen

2

1.3902

2 0.1988

w

equivalent smeasured va

h as node he skeletonihe fracture Nd the degree%, 16% anance distribwing good a

s for the ACN

ow rates betnce, mass b

2

2

2.

(9)

size for a disalues (circles

density, deized netwoNa/L

2. In the of connectnd 3%, respbution betwagreement (

N (circles) an

tween ACNbalance for a

ordered

s).

egree of rks. The is study, tivity for ectively.

ween two (see Fig.

nd the

N and FE a typical

  

 

 

where zequivalefluid pronodes tothan theflow in one for that shoequivaleconsideperform

3.3. P

The twoequivaleintersecrespect example

Fig.rota

To fit thconside

The par

where b

,1

z

i jj

pG

z denotes thent apertureoperties areo obtain thee one of thechannels wB=4. The

ould be accuent aperturered as sm

mance gain.

Parameteri

o importantent aperture

ctions. Usingto the leng

e in Fig. 7 f

. 7. Equivaleation angle

he simulatired by the e

1eqeq

ik a ah

rameters a

1eq eqn na b b

bnieq, i=1,2,3

i jpp

he degree oe of the chae taken as foe pressure ae detailed frwas consideequivalent urately prede of the detaall with re

ization of fr

t effective e for flow g the procedgth of the ifor specific

ent aperture and path le

ion results, expression

2 3( )eq eija L a

and dikj are i

2 sin( )eqn ijb

3 and n=1,2

Tab

0 with G

of connectivnnel connec

or the FE caat the nodesacture. This

ered that wafracture ape

dicted from ailed fracturespect to t

racture inte

properties tbetween tw

dures descrintersectionsvalues).

size over thength (≈60°

the quadra

3 4( )eqjk

eqL a

involved in

3 ,eq jn ikb d

2,…,5 are th

b. 1. Fitting p

11 

(

12

eqij

ij ij

hG w

vity for nodcting node alculation. Ts. The total s differenceas quantifieerture is onthe ACN m

re and ACNthe huge d

eractions

to considerwo fractureribed in Secs (Lij, Ljk) p

e length of thand dik≈35) (Lij=Ljk≈20).

atic depende

25( ) eqq

jiL a

the fitting p

he fitting pa

parameters f

3),

ijl (10)

de i, wij, lij,i and j. Iden

The set of liflow inside

e is largely ed before (Fne of the phmodel. HereN was calcudata reduct

r in the dise intersectio. 2.3., we obpath length

he fracture inand (b) with.

ency on the

2( ) .qjkL

parameters a

arameters su

for Eq. 12 (B

and hij arentical boundnear equatio

e the ACN iexplained b

Fig. 4) to behysically ime the relativulated to be ion by the

crete fractuons and thebtain equivadik

j and ro

ntersections h constant int

e length of

(11)

aneq through

(12)

ummarized i

B=4).

e width, lendary conditons is solveis about 8%by the fact te ~94% of

mportant parve error betw

~11%, whae ACN an

ure networke entry apealent apertu

otation angl

(a) with contersection len

the interse

h

in Tab. 1.

ngth and ions and ed for all

% smaller that only the total rameters ween the at can be d hence

k are the erture of ures with le (see

stant ngth

ctions is

  

 

 

For largand pathin Figurdirectly connect

To obtafracture

The entrthe leng

where aa lineargiven in

The depthe twoimpact.

ge angles , h lengths. Tre 7b for

y the connecting fracture

ain the values Lij, averag

Figure 8.

ry aperture gth of the in

1E E

ij ah

aiE, i=1,2,3,

r dependencn Tab. 2:

1E En na b b

pendency oo channels a

n

1 

2 

3 

4 

5 

fracture i aThis causes =75°. Howction betwees is exceed

ue of the eges over 100

. Dependenc

can be fittentersection, e

2 ( )EijLa

are the fittice on the ro

2 sin( ).Enb

f the rotatioand ignored

Tab

1eqnb

‐9.997e‐04 

‐1.884e‐04 

1.416e‐04 

3.344e‐05 

‐2.90e‐06 

and k can dira sudden in

wever this cen fracture

ded.

ntry apertu00 realizatio

e of the entr

ed with reasexpressed b

23 ( ) ,E

ija L

ing parametotation angl

on angle arod in the cal

b.2. Fitting p

n

1  2.

12 

2eqnb

‐4.225e‐0

7.102e‐04

7.73e‐04

‐6.91e‐05

‐1.66e‐05

rectly intersncrease in thcase is mori and k onc

ure hijE as f

ons of the A

ry aperture o

sonable accuby

(13)

ters for the le around y

(14)

ound the z-lculation of

parameters f

1Enb

. 22e‐2

3eqnb

03  2.371e

4  ‐1.123

4  ‐3.973

5  ‐1.48e

5  8.702e

sect due to he equivalenre straightfoce the criter

function ofACNs are m

on intersectio

uracy by us

entry apertuy-axis with

-direction isf the entry

for Eq. 14 (B

2Enb

‐3.65e‐3

3q

e‐03 

e‐04 

e‐04 

e‐06 

e‐06 

different rotnt aperture orwardly sorion ½Lij≥ d

the intersemade (see Fi

on length and

ing a quadr

ure which afitting para

s included iaperture du

=4).

otations, intevalue that i

olved by addik

j·cos()-1

ection lengtig. 8).

d angle .

ratic depend

are assumedameters bi

E,

in the interaue to its ne

ersection is visible ddressing 1 for two

h of the

dency on

d to have i=1,2,3

action of egligible

  

 

 

4. Inv

At the transporexplainesize disquantifithe singanalysisfracturethe previmmiscicontrastfracturethe num

Fig. 9. Dprocess

The netconstruc

vasion per

scale of thert is causeed. At the nstribution aied in Sec. 3gle fractures is essentiaed rocks (Haviously stuible displact to the pree orientationmerically fou

DFN model f. Colors indi

twork consction lower

rcolation

e single fraed by the network scaand density3.3. Only fe and the fral due to thamzehpour,udied problcement throuevious modn, intersectiound scaling

for full invasicate the inva

sists of Nfr=r and uppe

2  33  ‐1

of DFNs

acture and hheterogen

ale howevery, as well ew researcheracture netwhe strong e, et al., 2009lems of twugh isotrop

difications, on length, arelations fo

ion with trapasion sequen

fra

=4000 fracter bounds

13 

.68e‐41.60e‐6

hence chaneous geomr, the compas size ef

es have attework scale, effects of b9; Dreuzy,

wo-phase inpic and anis

fracture inas well as thor B=4.

pping MIP. (nce. (b) and (acture netwo

tures insideof radius o

‐5.70e‐51.71e‐7

nnel networkmetry and plex behavioffects of frempted to m

simultaneoboth scales

et al., 2012nvasion persotropic DFntersections he path leng

(a) and (c) in(d) show the ork.

e a cube ofof polygon

k, the compchannelizator originateracture inte

model the hyously. In faon the tran

2). In the forcolation (IPNs (Wettsteare now in

gth along the

nvaded fractupressure dis

f length LN

ns are set t

plexity of ttion as pres from the ersections tydraulic behact, this munsport propollowing, wIP) in the ein et al., 2ncluding the fractures b

ures during stribution in

N=10. For tto rmin=0.5

the fluid reviously

fracture that was havior of ulti-scale erties of

we revisit form of

2012). In he effect by using

invasion the flow

he DNF , rmax=1

  

 

 

respectiwith reschanged11(a),(bstarts atbreakthrproductpressureanalogo

Fig. 10. Ithe isotrimprov

The droinvasionthe inclMIP is isotropican be eand henreach thidentica

The caprelation the watanisotro

ively, whilespect to thed by settinb)) and =5t the entryrough, the tion rate at e distributious to the A

Invasion proropic DFN (

ved MIP. The

op of the apn process. Tlination effepresented ic and anisoexplained bnce more sthe percolat

al. Note that

pillary presn to the wateter saturatioopic cases a

e the numbee local fractg the Fish

50 for the hy zone and

invasion pthe inlet u

ons for thCN model o

ocess with ap=1) and anie breakthrou

dott

perture threThe comparect of interin Figs. 10 otropic caseby a higher teps to invation breaktt decreasing

ssure is caler saturationons in the are smaller

er of verticeture porositer dispersio

highly anisoprogresses

process is until no fu

he DFN moon the hydra

perture thresisotropic DFugh step is mted line for t

eshold for erison betweesections seeand 11. Thes is signifidisorder for

ade all nonhrough, as

g aperture th

lculated usn in Fig. 11new MIP athan in the

14 

es is limitety and fracon parametotropic ones in the dir

continued urther fractuodel, a onaulic netwo

shold and invN (=50). (a

marked by a dthe aperture

each invasioen trappinge (Wettstei

he number oicantly incrr equivalen-trapped fra well as t

hreshold me

sing the Yo, also referrat the full

e previous M

d to Nmax=8ture area, oter =1 for(Figs. 9-11

rection of to mimic

ures can bene phase flork.

vasion rate f) and (c) trapdashed line fthreshold va

on step charg MIP (CI =n et al. 20

of steps neereased by thnt apertures,actures. Thhe aperture

eans increas

oung-Laplacred to as wainvasion p

MIP model.

8. These pronly the frar the isotro1(c),(d)). Ththe outlet for examp

e invaded. low calcula

for each step pping MIP (for the invasialue.

racterizes th= 0.5; adjust12)) and theded for a fuhe multi-sca leading to e number oe thresholdsing the of e

ce equationater retentioprocess for This again

roperties areacture orienopic case (he invasion(x-direction

ple an infiIn order to

ation is pe

for trapping(CI = 0.5). (b)ion steps and

he dynamictment paramhe newly defull invasionale procedusmaller av

of steps reqd are more entry pressu

n and displon curve. As

both isotron is due to

e similar ntation is (Figs. 9-n process n). After nite gas o obtain erformed

g MIP on ) and (d) d with a

cs of the meter for eveloped n for the ure. This alanches

quired to or less

ure.

layed in s visible, opic and a higher

  

 

 

degree apertureincreasi

Fig. 11.isotrop

new detrappe

5. Sum

We intrcalculatnetworkand frcharactethese digrained,accuracscaling performand outljust a unintersecIt is shimmisci

of disorderes as well ing number

Capillary ppic DFN (=

eveloped MIPed water of p

mmary an

roduced a mtions on coks (ACNs) tracture inteerization aristributions , vectorizedy, but withbehavior o

med. These alet of the frniform aper

ctions, as weown for a ible fluids i

, leading toas the lackof invasion

ressure-wate=1) and anisoP. The total a

pruned “dead

nd conclu

multi-scale omputationthat are useeractions. re based onfor the cha

d ACNs caph highly reof transport are the pathacture. Discrture for a fell as entry previously

into fully sa

o the particik of the con steps for th

er saturationotropic DFN amount of trd end” fractu

usions

framework nally efficied to calculaThe utiliz

n robust imannel properpture the hyeduced info

properties h length, inccrete fracturfracture, buapertures thstudied ex

aturated, lar

15 

ipation of connectivity he multiscal

n relationship(=50). (a) a

rapped waterures that are

that maps ent, hydrauate system szed algori

mage procesrties ACNs

ydraulic propormation. T

of fractureclination as re networks

ut by assignhat considerxample, howrge fracture

channels wof the anisle approach

p after full inand (c)Trappr is marked be not invadab

flow in hetulically corspecific scaithms for sing routine

s are generaperties of si

To quantifyes, parametwell as inte

s can now bning equivalr intersectiow these rele networks

ith small ensotropic DFh.

nvasion for tping MIP (CI

by a dashed ble is indicat

terogeneousrrespondingling relation

channel es and netw

ated. It was ingle fractu

y the intra-er variation

ersection lene improved

lent apertureon lengths aations chanby trapping

ntry and eqFN, resultin

trapping MICI = 0.5). (b) a

line. The amted by the sol

s fractures fg artificial ns for fractuidentificati

work theoryshown, tha

ures with rea- and interns on the Angth of the d by considees between

and spatial rnge the invg MIP (Wet

quivalent ng in an

P on the and (d)

mount of lid line.

from FE channel

ure sizes on and y. Using at coarse asonable -fracture

ACN are entrance ering not fracture

relations. vasion of ttstein et

  

16  

 

al. 2012).

The entire procedure is based on the assumption that the origin of scale effects in fracture networks stems from the fracture flow that is mainly localized inside a channel network with characteristic, scale dependent network properties like a characteristic length scale among others. The study was limited to spatially uncorrelated fracture surfaces with power-law disorder. Also a degree of disorder was chosen that leads to clear flow localization in channels for Stokes flow. However different aperture fields, higher Reynolds numbers or parallel plate flow for non-channelized portions can be considered with small additional effort to combine the effects of heterogeneity in the single fracture scale on the network scale.

When additional knowledge about the finer scale is incorporated on larger scales, in general more physical results are obtained. We demonstrated how an efficient two-scale approach for invasion and flow in fracture networks can work with a model of only one input parameter that controls the heterogeneity of the aperture size distribution. We compare results from the two-scale MIP model with the previously developed MIP model for artificial fracture networks, since absolute data for verification is abundant. Since the presented model considers scale effects originating form disordered surface roughness such as intersections and flow path lengths of underlying channel networks, it can be seen as the improved and more accurate model to predict transport properties of the fractured rocks.

Acknowledgment

Financial support from nagra, ETH Zurich, and the European Research Council (ERC) under Advanced Grant No. 319968-FlowCCS is gratefully acknowledged. The authors are thankful to Paul Marshall (nagra) and Bill Lanyon (Fracture Systems Ltd.), for fruitful discussions and ongoing support.

Appendix A. Width and Equivalent Aperture Size of Channels

To measure the width of the channels, circles with different radii are drawn to find the intersection between channel wall and circles. Their centers are placed on the skeleton (see Fig. A1). The mean of the flow rate profile for different cross sections inside one typical channel is normal distributed, hence characterized by variance and amplitude. It is assumed that the flow rates in the boundaries of the channels are less that 5% of the maximum flow rate inside the channels. Circles with increasing radius are drawn and simultaneously the corresponding values of the flow rates for each radius are checked until the criterion is satisfied. The final diameter of the circle is taken as the width of the channel.

  

 

 

Fig. A(inset).

Using thcompute

where qnode i a

6. Ref

Adler, Univ

Amitranshear

AndradePaths

Ansari-Rquasi

Auradouin rou

Auradoushear

Berkowrevie

BistacchroughGeop

Bouchau

A1. Flow ratThe dashed

he cubic lawed by

312

(eqijh

w p

qij is the flowand j; l and w

ferences

P.M., Thoversity Pressno, D., Schmr band. J. Ge Jr, J. S., Os. Phys. RevRad, M.S., istatic fractuu, H., Hulinugh self-affiu, H., Drazer displacem

witz, B., 20ew. Adv. Whi, A., Griffihness at sephys, 168, 2aud, E., Lap

te profile of line indicat

w, the equi

2,

)ij ij

i j

q l

p p

w rate insidw are length

vert, J.-F., s, Oxford, Umittbuhl, J.

Geophys. ReOliveira, E.Av. Lett., 103Allaei, M.Vure surfacesn, J.P., Roufine fractureer, G., Huli

ment of rough02. Charac

Water Resourffith, W.A., ismogenic

2345-2363. passet, G.,

channels wites the skele

ivalent aper

de the channh and width

MourzenkUK. ., 2002. Fras., 107, s23

A., Moreira3, 225503. V., Sahimi, s. Phys. Revx, S., 2001.

es. Phys. Ren, J.P., Koph fracture w

cterizing flor., 25, 861–8Smith, S.Adepths from

Planès, J.,

17 

ith fitted Gaeton while th

the channel.

rture size hij

(

nel betweenh of the chan

ko, V.V., 2

acture rough375. a, A.A., Herr

M., 2012. Nv. E, 85, 02. Experimenev. E, 63, 06plik, J., 200walls. Waterow and tran884.

A.F., di Torom LiDAR a

, 1990. Fra

ussian curvee double sid.

ijeq for a ch

B1)

n node i andnnel, respec

2013. Frac

hness and g

rmann, H.J.

Nonunivers1121. ntal study o66306. 5. Permeabr Res. Res., nsport in f

o, G., Jonesand photogr

actal dimen

. Representaed arrow po

annel from

d j; pi and pj

ctively.

tured Poro

gouge distri

., 2009. Fra

ality of rou

of miscible d

ility anisotr41, W0942

fractured ge

, R.S., Nielrammetric a

nsion of fra

ation of a choint at the w

node i to n

pj are the pre

ous Media.

ibution of a

acturing the

ughness exp

displaceme

ropy induce23. eological m

lsen, S., 201analysis. Pu

actured sur

hannel width of

node j is

essure at

Oxford

a granite

Optimal

ponent of

nt fronts

ed by the

media: A

11. Fault ure appl.

rfaces: a

  

18  

 

universal value? Europhys. Lett., 13, 73-79. Braunstein, L.A., Buldyrev, S.V., Havlin, S., Stanley, H.E., 2002. Universality classes for

self-avoiding walks in a strongly disordered system. Phys. Rev. E., 65, 056128. Broadbent, S.R., Hammersley, J.M., 1957. Percolation processes I. Crystals and mazes. Math.

Proc. Camb. Phil. Soc., 53, 629-641. Brown, S.R., Kranz, R.L., Bonner, B.P., 1986. Correlation between the surfaces of natural

rock joints. Geophys. Res. Lett., 13, 1430-1433. Brown, S.R., 1987. Fluid flow through rock joints: The effect of surface roughness. J.

Geophys. Res., 1978-2012. Brown, S.R., 1995. Simple mathematical model of a rough fracture. J. Geophys. Res., 100,

5941-5952. Cacas, M.C., Ledoux, E., de Marsily, G., Tillie, B., Barbreau, A., Durand, E., Feug, B.,

Peaudecerf, P., 1990. Modeling fracture flow with a stochastic discrete fracture network: calibration and validation. I. The flow model. Water Res. Res., 26, 479-489.

Candela, T., Renard, F., Klinger, Y., Mair, K., Schmittbuhl, J., Brodsky, E.E., 2012. Roughness of fault surfaces over nine decades of length scales. J. Geophys. Res., 117, B08409.

Cappa, F., 2011. Influence of hydromechanical heterogeneities of fault zones on earthquake ruptures. Geophys. J. Int., 185, 1049-1058.

Charmet, J.C., Roux, S., Guyon, E., 1990. Disorder and Fracture. Springer. Drazer, G., Koplik, J., 2000. Permeability of self-affine rough fractures. Phys. Rev. E, 62,

8076-8085. Drazer, G., Koplik, J., 2002. Transport in rough self-affine fractures. Phys. Rev. E, 66,

026303. Dreuzy, J.-R. d., Méheust, Y., Pichot, G., 2012. Influence of fracture scale heterogeneity on

the flow properties of three-dimensional discrete fracture networks (DFN). J. Geophys. Res., 117, B11207.

Dreuzy, J.-R. d., Davy, P., Bour, O., 2000. Percolation parameter and percolation-threshold estimates for three-dimensional random ellipses with widely scattered distributions of eccentricity and size. Phys. Rev. E, 62, 5948–5952.

Durham, W.B., 1997. Laboratory observations of the hydraulic behavior of a permeable fracture from 3800 m depth in the KTB pilot hole. J. Geophys. Res., 102, 18405–18416.

Dyer, H., Amitrano, D., Boullier, A.-M., 2012. Scaling properties of fault rocks. Journal of Structural Geology., 45, 125-136.

Glass, J.R., LaViolette, R.A., 2004. Self-organized spatial-temporal structure within the fractured Vadose Zone: Influence of fracture intersections. Geophys. Res. Lett., 31, L15501.

Hamzehpour, H., Mourzenko, V.V., Thovert, J.-F., Adler, P.M., 2009. Percolation and permeability of networks of heterogeneous fractures. Phys. Rev. E,79, 036302.

Herrmann, H.J., Hong, D.C., Stanley, H.E., 1984. Backbone and Elastic Backbone of Percolation Clusters Obtained by the New Method of Burning. J. Phys. A Lett., 17, L261-L265.

Huseby, O., Thovert, J.-F., Adler, P.M., 2001. Dispersion in three-dimensional fracture networks. Phys. Fluids, 13, 594-615.

Ishibashi, T., Watanabe, N., Hirano, N., Okamoto, A., 2009. GeoFlow: A novel model simulator for prediction of the 3-D channeling flow in a rock fracture network. J. Geophys. Res., 114, B03205.

Ji, S.-H., Nicholl, M.J., Glass, R.J., Lee, K.-K., 2006. Influence of simple fracture

  

19  

 

intersections with differing aperture on density-driven immiscible flow: Wetting versus nonwetting flows. Water Res. Res., 42, W10416.

Katsumi, N., Watanabe, N., Hirano, N., Tsuchiya, N., 2009. Direct measurement of contact area and stress dependence of anisotropic flow through rock fracture with heterogeneous aperture distribution. Earth Planet. Sci. Lett., 281. 81–87.

Khamforoush, M., Shams, K., 2007. Percolation thresholds of a group of anisotropic three-dimensional fracture networks. Physica A: Statistical Mechanics and its Applications, 385, 407–420.

Khamforoush, M., Shams, K., Thovert, J.-F., Adler, P.M., 2008. Permeability and percolation of anisotropic three-dimensional fracture networks. Phys. Rev. E, 77, 056307

Konzuk, J.S., Kueper, B.H., 2004. Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture. Water Res. Res., 40, W02402.

Koudina, N., Gonzalez Garcia, R. Thovert, J.-F., Adler, P.M., 1998. Permeability of three-dimensional fracture networks. Phys. Rev. E, 57, 4466–4479.

Kumar, A.T.A, Majors, P.D., Rossen, W.R., 1997. Measurement of Aperture and Multiphase Flow in Fractures using NMR Imaging. Soc. Pet. Eng., 12, 101–108.

Méheust, Y., Schmittbuhl, J., 2001. Geometrical heterogeneities and permeability anisotropy of rough fractures. J. Geophys. Res., 106, 2089.

Moukarzel, C., Herrmann, H.J., 1992. A vectorizable random lattice. J. Stat. Phys. 68 (5-6), 911-923.

Oliveira, E.A., Schrenk, K.J., Araujo, N.A.M., Herrmann, H.J., Andrade Jr, J.S., 2011. Optimal path cracks in correlated and uncorrelated lattices. Phys. Rev. E, 83, 046113.

Pompe, W., Herrmann, H.J., Roux, S., 1990. Statistical models for the fracture of disordered media. North-Holland.

Reeves, D.M., Benson, D.A., Meerschaert, M.M., 2008. Transport of conservative solutes in simulated fracture networks: 1. Synthetic data generation. Water Res. Res., 44, W05404.

Renshaw, C.E., 1999. Connectivity of joint networks with power law length distributions. Water Res. Res., 35, 2661– 2670.

Sahimi, M., 1995. Flow and Transport in Porous Media and Fractured Rock. Wiley-VCH, New York.

Schrenk, K.J., Posé, N., Kranz, J.J., van Kessenich, L.V.M., Araújo, N.A.M., Herrmann, H.J., 2013. Percolation with long-range correlated disorder. Phys. Rev. E, 88, 052102.

Sisavath, S., Mourzenko, V.V., Genthon, P., Thovert, J.-F., Adler, P.M., 2004. Geometry, percolation and transport properties of fracture networks derived from line data. Geophy. J. Int., 157, 917–934.

Smith, L., Schwartz, F.W., 1984. An analysis on the influence of fracture geometry on mass transport in fractured media. Water Res. Res., 20, 1241– 1252.

Talon, L., Auradou, H., 2010. Permeability of self-affine aperture fields. Phys. Rev. E, 82, 046108.

Talon, L., Auradou, H., Hansen, A., 2010. Permeability estimates of self‐affine fracture faults based on generalization of the bottleneck concept. Water Res. Res., 46, W07601.

Wood, T.R., Nicholl, M.J., Glass, R.J., 2005. Influence of fracture intersections under unsaturated, low-flow conditions. Water Res. Res., 41, W04017.

Watanabe, N., Hirano, N., Tsuchiya, N., 2008. Determination of aperture structure and fluid flow in a rock fracture by high-resolution numerical modeling on the basis of a flow-through experiment under confining pressure. Water Res. Res., 44, W06412.

Wettstein, S.J., Wittel, F.K., Araujo, N.A.M., Lanyon, B., Herrmann, H.J., 2012. From invasion percolation to flow in rock fracture networks. Physica A, 391, 264–277.

  

 

 

Wilkinstheor

ZimmerPoro

Highlig

Ao

I E

i M

i Graphi

son, D., Wiry. J. Phys. rman, R.W.us Media, 2

ghts: A modifiedof the indivIndividual fEquivalent intersectionModificatioisotropic an

ical abstrac

illemsen, JA, 16, 3365, Bodvarsso

23, 1-30.

d percolationvidual fractufractures areand entry a

n length. ons are applnd anisotrop

ct:

.F., 1983. 5–3376. on, G.S., 19

n model is ures to incore coarse graapertures de

lied to simupic discrete

20 

Invasion p

996. Hydrau

developed brporate sizeained using epend on fr

ulate invasiofracture net

percolation:

ulic conduct

by considere effects. an artificial

racture inter

on percolatiotworks.

A new fo

tivity of roc

ring heterog

l channel nersection ang

on processe

orm of per

ck fractures.

geneity on t

etwork gle, path len

es through tw

rcolation

. Transp.

the scale

ngth and

wo large