A test case for the simulation of three-dimensional variable-density flow and solute transport in discretely-fractured porous media

A test case for the simulation of three-dimensional variable-density flow and solute transport in discretely-fractured porous media
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  A test case for the simulation of three-dimensional variable-density flowand solute transport in discretely-fractured porous media Thomas Graf  a, * , René Therrien b a Center of Geosciences, Georg-August-University Göttingen, Goldschmidtstraße 3, 37077 Göttingen, Germany b Département de Géologie et Génie Géologique, Université Laval. Ste-Foy, Québec G1K 7P4, Canada a r t i c l e i n f o  Article history: Received 16 November 2007Received in revised form 16 June 2008Accepted 1 July 2008Available online 11 July 2008 Keywords: Test case3DDensityFractureNumerical model a b s t r a c t A test case has been developed for three-dimensional simulations of variable-density flow and solutetransport in discretely-fractured porous media. The simulation domain is a low-permeability porousmatrix cube containing a single non-planar fracture. The initial solute concentration is zero everywhere.A constant solute concentration is assigned to the top of the domain, which increases near-top fluid den-sity andinduces downward density-drivenflow. Thetest caseis therefore comparable todownwelling of a dense brine below a saline disposal basin or a waste repository. Numerous fingers and distinct convec-tion cells develop early in the fracture but the fingers later coalesce and convection becomes less appar-ent. To help test other variable-density flow and transport models, results of the test case are presentedboth qualitatively (concentration contours and velocity fields) and quantitatively (penetration depth,mass flux, total mass stored, maximum fracture and matrix velocity).   2008 Elsevier Ltd. All rights reserved. 1. Introduction Groundwater density varies as a function of fluid temperature,solute concentration and, to a lesser extent, fluid pressure. Spatialvariations of fluid density play an important role in contaminantmigration. When, for example, a high-density fluid overlies a lessdense fluid, unstable density-driven flow may occur, which damp-ens or eliminates the density stratification and eventually stabi-lizes flow. Transient flow can also induce temporal changes influid density [21]. Examples of density-driven flow and solute transport can be found in many areas of subsurface hydrology,oceanography,meteorology,geophysics,hazardouswastedisposal,and geothermal reservoirs [10,15,23,28].Numerical models used to simulate variable-density flow andsolutetransportingroundwatermustbetestedtoensurethattheyrepresent the required physical processes and are numerically rig-orous. Model testing relies in part on  test cases , which are well-de-fined analytical, numerical, laboratory or field results [35,47].A test case possesses a physical and mathematical character,such as nonlinearity, but has a simplifiedgeometry such that com-parable numerical solutions of accepted quality are available [1,7].Theevaluationofnewnumerical groundwatermodelstypicallyre-liesoninternalconsistencytestsincludingmassbalanceindicatorsand external tests such as comparison with other numerical mod-els. Clearly, the more test cases available for model testing, themore rigorous the testing process.Several test cases exist for variable-density flow in homoge-neous porous media [5,8,20,35,36,39,42,47]. In contrast, few test cases exist for variable-density flow in discretely-fractured porousmedia. Results presented by Shikaze et al. [41] can be used to testdense plume migration in a set of vertical fractures by comparingisohalinesfor givensimulationtimes. Variable-densityflowresultsin orthogonal fracture networks presented by Graf and Therrien[17] are equally useful. The inclined-fracture problem introducedby Graf and Therrien [15] can be used in both a qualitative (usingisohalines) and a quantitative (using detailed information onbreakthrough curves, mass fluxes, maximum velocities, etc.) man-ner to rigorously benchmark variable-density flow in an inclinedfracture embedded in a porous matrix. Caltagirone [5] has pre-sented an analytical solution for the onset of convection in homo-geneousmedia.Thesolutioncanbeappliedtoverticalandinclinedfractures by assuming a homogeneous fracture of constant aper-ture and by introducing a cosine weight to account for fracture in-cline [14]. Although the solution of Caltagirone [5] is the only available analytical solution for variable-density flow applied to abox type domain of finite dimensions, it is not truly an analyticalsolution,butratherusesanalyticalprocedurestoderivethecriticalRayleigh number for different aspect ratios in order to determineconditions that govern stable and unstable states of convectionfor box type domains. The solution presented by Caltagirone [5]also forms the basis for newly proposed benchmark problems pre-sented by Weatherill et al. [47]. 0309-1708/$ - see front matter    2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.advwatres.2008.07.003 *  Corresponding author. E-mail addresses:, (R.Therrien). Advances in Water Resources 31 (2008) 1352–1363 Contents lists available at ScienceDirect Advances in Water Resources journal homepage:  Existingtestcasesforvariable-densityflowinfracturedrockare2D test problems [5,15,17,41,47]. Test cases using results from either physical or laboratory experiments for variable-density flowin fractured rock do not exist. Solute transport through fracturedrock has previously been investigated in two laboratory studies,where fractures were embedded either in an impermeable [31]or in a porous [22] matrix. The physical model used in the twostudies was built from a horizontally oriented sheet of plexiglas,into which a series of fractures was cut. A tracer solution contain-ing dye and salt was injected and the tracer was transported by animposed background flow. Conducting a similar laboratory exper-iment in 2D(and possibly3D) fracturedmedia where density vari-ations are significant remains a major challenge.To date, a robust and well accepted (numerical) test case forvariable-density flow in fractures oriented in 3D and embeddedin a 3D porous matrix does not exist. We therefore propose herea new test problem for variable-density flow in a non-planar frac-ture embedded in a 3D porous matrix. The test case is developedwith the HydroGeoSphere model [15,44,45], which solves 3D var- iable-density flowandsolutetransport indiscretely-fractured por-ousmedia.Withthemodel, variable-densityflowis describedforarock matrix containing a single non-planar fracture [16,19]. The fracture is represented by nine planar facets and discretized withtriangular fracture elements. The transient test case describesadvection, mechanical dispersion, molecular diffusion in both thefracture and the porous matrix, as well as fracture–matrix diffu-sion.Weprovidedetaileddataonmassfluxes,solutebreakthroughcurves,concentrationcontoursandsolutepenetrationdepthtoen-surerigorousobjectivetestingofanewmodel.Wealsogivedetailson the numerical methods used to ensure further rigorousness of thenewtestcase. Thenewtestproblemallowsnumerical compar-ison for other variable-density flow models. 2. Numerical model  2.1. The HydroGeoSphere model HydroGeoSphere is a numerical 3D variable-density, variably-saturatedgroundwaterflowandmulti-componentsolutetransportmodel for fractured porous media, and is based on the FRAC3DVSmodel. The HydroGeoSphere model applies the control volume fi-nite element (CVFE) method to the flow equation, the Galerkin fi-nite element method with full upstream weighting to the solutetransport equation, and a finite difference scheme to time deriva-tives [15,44]. It is assumed that 2D fracture elements and 3D ma- trix elements share common nodes in the 3D grid. Thus,hydraulic heads and concentrations are assumed identical alongthe fracture–matrix interface.The HydroGeoSphere model applies the first level of the Ober-beck–Boussinesq (OB) approximation to discretize the flow andtransport equations [4,21,27,34]. The OB assumption reflects the degree to which density variations are accounted for. Level 1 of the OB approach considers density effects only in the buoyancyterm of the momentum equations (Darcy equations) and neglectsdensity in the fluid and solute mass conservation equations. Thisassumptioniscorrectif spatialdensityvariationsaresmallrelative Nomenclature Latin letters (2 b ) [L] fracture aperture  A  [L  2 ] surface area b  [L  2 T  1 ] boundary flux vector c   [–] relative solute concentration d  [–] microscopic length D d  [L  2 T  1 ] free-solution diffusion coefficient D ij  [L  2 T  1 ] hydrodynamic dispersion f   [L  3 T  1 ] groundwater flux vector  g   [L T  2 ] acceleration due to gravity  g  r , h  [–] grid at discretization levels  r  and  h g   [L  2 T  1 ] buoyancy vector h 0  [L] equivalent freshwater head H  [L T  1 ] conductance or stiffness matrixI + , I  [–] fracture–matrix interface K  0 ij  [L T  1 ] freshwater hydraulic conductivity of porous matrix K  fr0  [L T  1 ] freshwater hydraulic conductivity of fracture L v  [L] geometry of porous matrix element  v =  x ,  y ,  z Pe g  [–] grid Peclet number for triangular fracture element q  [L T  1 ] Darcy flux Q   [T  1 ] groundwater flux Re  [–] Reynolds number S  [L] fluid mass matrix S  S  [L   1 ] specific storage t   [T] time v  [L T  1 ] linear flow velocity V   [L  3 volume w  [–] approximation function  x  [L] space Greek letters a  [M  1 L T 2 ] matrix compressibility a l  [L] longitudinal dispersivity a t  [L] transverse dispersivity b  [M  1 L T 2 ] water compressibility c  [–] maximum relative density d ij  [–] Kronecker delta function g  j  [–] indicator for flow direction h  [–] temporal discretization level j ij  [L  2 ] permeability of porous matrix l  [M L   1 T  1 ] water viscosity q  [M L   3 ] water density q max  [M L   3 ] maximum water density r  [–] spatial discretization level s  [–] factor of tortuosity /  [–] matrix porosity u  [1  ] fracture incline X  [M M  1 T  1 ] advective–dispersive–diffusive solute flux Sub- and superscripts 0 [–] reference fluide [–] porous matrix elementfe [–] fracture elementfr [–] fracture i ,  j  [–] spatial indicesI [–] nodal indexn [–] normal direction Special symbols o  [–] partial differential operator D  [–] difference T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363  1353  to density [21,33]. Although the OB assumption ‘‘has not been completely justified” [26], it has been adapted in a number of var- iable-density flow studies (e.g. [9,15,27,41]). Referring to the OB assumption,level 1, Joseph[26] alsostates‘‘that thereisnospecialreason besides our lack of proofs to doubt the validity of the non-linear OB-equations”.Density variations cause weak nonlinearities in the flow equa-tion. Thus, groundwater flow and solute transport are physicallycoupled. In the numerical model, the coupled system of equationsis solved by Picard iteration. The spatiotemporally discretized ma-trixequationsaresolvedusingtheWATSITiterativesolverpackagefor general sparse matrices [6] and a conjugate gradient stabilized(CGSTAB) acceleration technique [38]. Governing equations of groundwater flow and solute transport are presented below.  2.2. Governing equations 2.2.1. Equations for the 3D porous matrix The following equations describe variable-density flow and sol-ute transport in 3D porous media [3,21]:   o q i o  x i ¼  S  S o h 0 o t  i  ¼  1 ; 2 ; 3  ð 1 Þ oo  x i / D ij o c  o  x  j   q i c    ¼  o ð / c  Þ o t  i ;  j  ¼  1 ; 2 ; 3  ð 2 Þ where q i  [L T  1 ] Darcy flux S  S  [L   1 ] specific storage h 0  [L] equivalent freshwater head /  [–] matrix porosity D ij  [L  2 T  1 ] hydrodynamic dispersion c   [–] solute concentration, expressed as relative concentration The specific storage  S  S  is defined as [12]: S  S  ¼  q 0  g  ð a þ  /b Þ ð 3 Þ where q 0  [M L   3 ] freshwater density  g   [L T  2 ] gravitational acceleration a  [M  1 L T 2 ] matrix compressibility b  [M  1 L T 2 ] fluid compressibility The coefficients of the hydrodynamic dispersion tensor are gi-ven by [3]: / D ij  ¼ ð a l   a t Þ q i q  j j q j þ a t j q j d ij  þ  / s D d d ij  i ;  j  ¼  1 ; 2 ; 3  ð 4 Þ where a l  [L] longitudinal dispersivity a t  [L] transverse dispersivity d ij  [–] Kronecker delta function s  [–] matrix tortuosity D d  [L  2 T  1 ] free-solution diffusion coefficient  2.2.2. Equations for the 2D fracture The variable-density flowand solutetransport equations for 2Ddiscrete fractures are [21,41,44]: ð 2 b Þ  o q fr i o  x i  S  frS o h fr0 o t  ( ) þ Q  n j I  þ   Q  n j I    ¼ 0  i ;  j ¼ 1 ; 2  ð 5 Þð 2 b Þ  oo  x i D fr ij o c  fr o  x  j  q fr i  c  fr    o c  fr o t    þ X n j I þ   X n j I   ¼ 0  i ;  j ¼ 1 ; 2  ð 6 Þ where (2 b ) [L] is fracture aperture, and the last two terms in eachequation denote components of fluid flux and solute mass flux nor-mal to fracture–matrix interfaces I + and  I   . Specific storage in thefracture can be derived from Eq. (3) by assuming that the fractureis incompressible, such that  a =0, and by setting its porosity to 1[15]: S  frS  ¼  q 0  g  b  ð 7 Þ Coefficients of the hydrodynamic dispersion tensor of the frac-ture are defined by [44]: D fr ij  ¼ ð a frl   a frt  Þ q fr i  q fr  j j q fr j þ a frt  j q fr j d ij  þ  D d d ij  i ;  j  ¼  1 ; 2  ð 8 Þ  2.2.3. Darcy flux The variable-density Darcy flux,  q i , is a function of freshwaterhead,  h 0 , and solute concentration,  c  . Darcy fluxes for matrix andfracture are given by [3]: q i  ¼  K  0 ij o h 0 o  x  j þ c c  g  j    i ;  j  ¼  1 ; 2 ; 3  ð 9 Þ q fr i  ¼  K  fr0 o h fr0 o  x  j þ c c  fr g  j  cos u  !  i ;  j  ¼  1 ; 2  ð 10 Þ where K  0 ij  [LT  1 ] freshwater hydraulic conductivity of rock matrix K  fr0  [LT  1 ] freshwater hydraulic conductivity of fracture g  j  [–] indicator for flow direction:  g  j  = 0 in horizontal directions; g  j  = 1 otherwise c  [–] maximum relative density,  c = ( q max / q 0 )  1 q max  [M L   3 ] maximum fluid density u  [  ] fracture incline:  u  = 0   for a vertical fracture;  u  = 90   for ahorizontal fracture and where the assumption is made that solute concentrations of afluid with densities  q = q max  and  q = q 0  are  c   =1 and  c   =0, respec-tively. It is also assumed that fluid viscosity is constant [15,41].Freshwater hydraulic conductivities are given by [3]: K  0 ij  ¼  j ij q 0  g  l 0 i ;  j  ¼  1 ; 2 ; 3  ð 11 Þ K  fr0  ¼ ð 2 b Þ 2 q 0  g  12 l 0 ð 12 Þ where j ij  [L  2 ] permeability of the rock matrix l 0  [M L   1 T  1 ] freshwater viscosity  2.3. Numerical formulation of buoyancy In the present study, variable-density flow is simulated in a 3Dporous rock matrix (represented by hexahedral 3D elements) con-taininganon-planar2Dfracture(representedbytriangular2Dele-ments). Variable-density flow in a 3D rock matrix and inrectangular 2D fracture elements has previously been simulated[15]. However,rectangular2Delementsareinadequatetodescribea non-planar inclined fracture in 3D [19]. Therefore, the HydroGe- oSphere model has been modified here to incorporate triangularfracture elements (Fig. 1) in the study of variable-density flowand transport in 3D fractured rock. The enhanced model is thenused to define the new 3D test case.Triangular elements are generated in the model such thattheir nodes must correspond to nodes of 3D hexahedral ele-ments. With that constraint, a total of 56 different triangularelements, of different incline, can be generated for the 8-nodehexahedral element shown in Fig. 1a. For example, the triangu-lar element shown in Fig. 1a, whose nodes are 2–7–5, is oneof 56 possible triangular elements contained in a hexahedralelement. 1354  T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363  ApplicationoftheCVFEmethodtovariable-densityflowEqs.(1)and (5) gives semi-discrete global matrix-equations for the porousmedium and fractures, respectively: H  h 0  þ S   o h 0 o t   þ g   ¼  b  ð 13 Þ H fr  h fr 0  þ S fr   o h fr 0 o t   þ g  fr ¼  b fr ð 14 Þ where H ,  H fr [L T  1 ] conductance or stiffness matrices S ,  S fr [L] fluid mass matrices g  ,  g  fr [L  2 T  1 ] buoyancy vectors b ,  b fr [L  2 T  1 ] boundary flux vectors Vector  g   and  g  fr can be written as g   ¼ X e g  e ð 15 Þ g  fr ¼ X  fe g  fe ð 16 Þ where g  e [L  2 T  1 ] buoyancy vector of matrix element e g  fe [L  2 T  1 ] buoyancy vector of fracture element fe  2.3.1. Numerical formulation of buoyancy in 3D hexahedral matrixelements According to Frind [12], nodal entries of   g  e are calculated as  g  eI  ¼ Z  V  e K  0 ij c  c  e   o w eI o  z   d V  e I  ¼  1 ; 2 ;  . . .  ; 8  ð 17 Þ where V  e [L  3 ] volume of matrix element e  c  e [–] average solute concentration in e w eI  [–] value of the 3D approximation function in e at node I Using usual approximation functions for 3D hexahedral ele-ments [24] gives the following buoyancy vector: g  e ¼  K  0 ij c  c  e  L  x L  y 4  f 1   1   1   1 1 1 1 1 g T ð 18 Þ where  L  x  and  L  y  [L] are sizes of element e in  x - and  y -direction,respectively.  2.3.2. Numerical formulation of buoyancy in 2D triangular fractureelements According to Frind [12] and Graf and Therrien [15], nodal en- tries of   g  fe are calculated as  g  feI  ¼ Z   A fe K  fr0 c  c  fe   cos u o w feI o   z   d  A fe I   ¼  1 ; 2 ; 3  ð 19 Þ where  A fe [L  2 ] surface area of fracture element fe  c  fe [–] average solute concentration in fe w feI  [–] value of the 2D approximation function in fe at node I   z   [L] local  z  -axis of fe Usingusualapproximationfunctionsfor2Dtriangularelements[24], we have derived the following buoyancy vector: g  fe ¼  K  fr0 c  c  fe  cos u 12   x 3      x 2   x 1      x 3   x 2      x 1 8><>:9>=>; ð 20 Þ where    x I  [L] is the value of the local  x -coordinate of node I.  2.4. Verification of numerical buoyancy formulation Wesimulatedvariable-densityflowinvariablyinclinedfracturesconsistingoftriangularfractureelementsandcomparedtheresultswiththosepresentedbyGrafandTherrien[15] forrectangular ele-ments.Theporousmatrixisdiscretizedbyhexahedral3Delements.Themodeldomainisaverticalsliceofunitthicknessanddimensions12 m  10 m. Model domain, initial and boundary conditions andflow/transporttimeweightingschemesusedinthesimulationscon-ductedhereareidenticaltothoseusedbyGrafandTherrien[15].Fig. 2 shows results from three different inclines and demonstratesagreementofresultswhendiscretizinganinclinedfracturebytrian-gular(thisstudy)andrectangular[15] fractureelements. 3. The 3D test case  3.1. Problem design The model domain for the test case is a cubic box of a sidelengthequal to 10 m. The box represents a porous matrix and con-tains a single non-planar fracture. The fracture is represented bynine planar facets as shown in Fig. 3. Fracture geometry and facetvertex locations are given in Tables A.1 and A.2.Lateral boundaries are assumed to be impermeable and the topand bottom boundaries are assigned a constant hydraulic head h 0  =0. A constant concentration  c   =1 is assigned to the top bound-ary, and all other boundaries are assigned a zero-dispersive fluxboundary condition [41]. The initial condition for flow is  h 0  =0,andfortransport c   =1onthedomaintopand c   =0everywhereelse[14,17,41].Thetotalsimulationtimeis10years.Fullyimplicittime a b Fig. 1.  Geometry and node numbering conventions for (a) a 3D porous matrix element containing eight nodes (1  8) and (b) a 2D fracture element containing three nodes(  1   3). The local  x -axis (   x ) is horizontal, points to the right and fracture nodes are numbered counterclockwise. T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363  1355  weighting is used for both flow and transport [13,15]. Model parameters for the test case are summarized in Table A.3.ThePicarditerationisterminatedwhenoneoftwoconvergencecriteria is met. The two convergence criteria are the maximumnumber of iterations, set to 100, and the maximumchange in con-centration and hydraulic head between iterations, set to 0.05. InSection 3.4 we examine the sensitivity of the results to Picard con-vergence criteria.  3.2. Spatial discretization As pointed out by [25], investigating grid convergence (pre- sented in Section 3.4) requires a hierarchy of uniformly refinedgrids. Therefore, we did not use locally refined grids.Here, the 3D grid at spatial discretization level  r ( r = 1,2,  . . . ,11) consists of (10 r +1) 3 grid nodes and (10 r ) 3 iden-tical hexahedral porous matrix elements, which have a uniformsize  D  x  = D  y  = D  z   =1/ r  m. Increasing grid levels correspond tofiner grids. The locations of fracture facet vertexes (Table A.2) havebeenchosensuchthat thelocationof vertexesandsegmentsofthenine fracture facets is preserved for all grid levels. For each level r ,triangularfractureelementsthatrepresentthenon-planarfracturehave been selected prior to conducting the simulation. Therefore,changing the grid does not modify the location, shape and orienta-tion of the non-planar fracture.  3.3. Time discretization The 3D test case is simulated for increasing levels of time dis-cretization h ( h =1,2, 3).Toguaranteenumericalstability,thesolu-tion process is completed using a variable time-step proceduresimilar to the one outlined by [11,44,48]. After obtaining the solu- tionattimelevel L ,thefollowingtime-stepsizeiscomputedusing: ð D t  Þ L þ 1 ¼  c   ð h Þ max I j c  L þ 1I    c  L I jð D t  Þ L ð 21 Þ where ( D t  ) [T] is time-step size and  c  * ( h ) [–] is the maximumchange in relative solute concentration during a single time-step.To ensure further numerical stability, the maximum multiplier of time-step sizes is 2.0, and the maximum time-step size is 1 month.The valueof   c  * ( h ) is a functionof timediscretization level andgivenby: c   ð h Þ ¼  0 : 2   12   ð h  1 Þ ð 22 Þ Increasing time discretization levels give smaller values of   c  * ( h )and smaller time-step sizes. Variable time-stepping allows the useof increasingly larger time-step sizes when there are only smallchanges in concentration. Thus, CPU time is reduced withoutdecreasing numerical accuracy. Similarly, drastic concentrationchanges increase time step sizes. It can be shown that the numer-ical stability of a simulation using adaptive time-stepping isidentical to the stability of a simulation using very small constanttime-step sizes. The Courant-criterion is satisfied in simulationswith adaptive time-stepping if the initial time-step size chosen isvery small.  3.4. Numerical accuracy A grid convergence study is carried out to determine the neces-sary discretization level in space and time. The study involves per-forming a simulation on successively finer grids and smaller timestep sizes. As discretization levels  r  and  h  increase, the discretiza-tion errors should exclude computer round-off errors and asymp-totically approach zero. In addition to the grid convergencestudy, the Picard convergence criterion is examined to guaranteefurther numerical accuracy.We define  g  r , h  here as the grid of spatial discretization level  r and time discretization level  h . Simulations were carried out atincreasing grid levels, and time-variable solute mass flux  f  ([L  3 T  1 ] when using dimensionless relative concentration)through the domain top was observed. In the grid convergencestudy,wechosemassfluxesastherepresentativequantitybecausemass flux is an integral parameter that does not yield informationat a specific point in the grid. We denote the mass fluxes using  g  r , h at 1 yr, . . . ,10 yr by  f  1 r ; h , . . . ,  f  10 r ; h  and define: f  r ; h  : ¼ f  f  1 r ; h ;  . . .  ;  f  10 r ; h g T 2 R 10 ð 23 Þ We use the  L 2 -norm to estimate the error  e r +1/2, h  [L  3 T  1 ] betweensolutions corresponding to increasing spatial discretization levels: e r þ 1 = 2 ; h  ¼ k f  r þ 1 ; h   f  r ; h k N   ð 24 Þ Likewise, theerrore r , h +1/2  [L  3 T  1 ]betweensolutionscorrespondingto increasing time discretization levels is 5 10 15 20 time [yr]   s  o   l  u   t  e  c  o  n  c  e  n   t  r  a   t   i  o  n   [  -   ] 45°60°30° trianglesrectangles Fig. 2.  Simulated breakthrough curve at  z   =6 mto verify variable-density flow in asingle fracture of variable incline using triangular (this study) and rectangular [15]fracture elements. x   -   d   i   r  e  c  t   i   o  n    [   m   ]    0246810  y  - d i r ec t io n  [  m ] 0 2 4 6 8 10   z  -   d   i  r  e  c   t   i  o  n   [  m   ] 0246810  5 1  4 987 6  3 21 23456789 10 Fig. 3.  Geometry of the triangulated non-planar fracture for the test case. Circlednumbers are facet IDs and numbers in italic are facet vertex IDs.1356  T. Graf, R. Therrien/Advances in Water Resources 31 (2008) 1352–1363
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