1. MechaRock International Consultants www.mecharock.com Modeling Fractured Rocks with the Finite Element code DISROC About Disroc DISROC is a Finite Element code specially conceived for modeling geotechnical projects in fractured rocks. It is based on more than 20 years of shared experience between an association of researchers, engineering consultants and experts in numerical modeling of civil engineering structures, geotechnical and mining projects in fractured rock formations.

2. 2 Finite Element Method for modelling engineering structures Finite Element Method is the most powerful numerical method for modelling mechanical, hydraulic and thermal behaviour of engineering structures. Most of geotechnical projects are designed by Finite Element Method, and softwares using this method are highly appreciated by engineers. However, in presence of fractures and discontinuities, softwares based on Finite Difference or Distinct Element methods seems to be needed, even if these methods are less efficient or pleasant to use (time duration, geometry limitations, outputs…).

3. 3 Joint Elements for fractures in Finite Element Method Rockjoint, Masonry mortar Fracture Contact interface Joint Element (Goodman 1976) 1 2 3 4 With appropriate parameters, joint elements can reproduce the behavior of fractures, rockjoints, interfaces and contact surfaces. However, their use in presence of a great number of discontinuities or fractures poses the difficulty of Conform Finite Element mesh creation. Disroc has solved this problem. Zero thickness Joint Element was proposed by (Goodman 1976) for modeling discontinuities in the Finite Element Method. With appropriate parameters, joint elements can reproduce the behavior of fractures, rockjoints, interfaces and contact surfaces.

4. 4 Conform Finite Element mesh generation for fractured medium DISROC ® is the first Finite Element code especially conceived for fractured rocks. Its powerful meshing tool DISCRAC ® allows easily creating a conform mesh and special Joint Elements for fractured media.  DIScontinuous ROCk  DIScretization of CRACked media     Joint : K n, K t  c, 

5. 5 Modelling fractured rocks with DISROC With DISROC it becomes easy to model geotechnical projects like dams, tunnels, bridges and rock cuttings in fractured rocks. Rock Slope Stability Tunnel in fractured rock

6. 6 6 Bolting fractured rock Homogenization of fractured rock properties Bolts are very often used to reinforce and stabilize fractured rocks, but are difficult to model when they cross fractures: DISROC® is the only Finite Element software capable to model properly rock bolts crossing fractures. Modelling fractured rocks with DISROC Effective elastic properties of fractured rock masses are very often needed for projects design: DISROC® has a “Large scale Homogenization” module for determination of effective parameters of fracture rock masses (deformation modulus, cohesion, angle of internal friction).

7. 7 Modelling fractures and bolts Modeling fractures and bolts with DISROC is very easy. The following tunnel/road project includes: - a rock mass with two sets of fractures (possibly non persistent) - non persistent fractures (cracks) on the tunnel’s wall, - rock bolts to stabilize the rock slope and the rock cut over the road. All these elements are easily introduced in the Finite Element model created by DISROC.

8. 8 Meshing with Discrac® The Finite Element mesh created by the software GID (www.gidhome.com) is transformed by the module Discrac® to generate specific elements for fractures, bolts and cables.www.gidhome.com The meshing tool integrates: - Intersecting fractures (a) - Non persistent fractures (b) - Rockbolts passing through fractures (c ) (c) (a) (b)

9. 9 Tunnels 1: Example of a project with rock cutting in a fractured rockmass The project includes a tunnel and a rock cutting for a road in a fractured sedimentary formation. The formation is constituted of alternate layers of two limestones varieties. The interfaces between layers are modeled as fractures (Fracture1). Two faults are present in the formation (Fracture2). Modeling passes through the following stages. I) The fractures are generated stochastically (Fracture1)and faults are placed in the model with their known position (Fracture2). II) Other lines defining the soil profile, the tunnel contour, the cutting contour and the rock bolts are introduced in the model. road

10. 10 Tunnels : Modeling stages III) A conform Finite Element mesh is created by DISCRAC ® +GID. Specific joint elements for fractures and bolt elements for rockbolts are created automatically. The material properties are assigned to limestone layers, fractures and rock bolts. In this example, the limestone varieties 1a, 1b, 2a, 2b are identical to Limestone1 and Limestone2 and are introduced for determination of the initial in situ stresses before tunnel excavation and rock cutting. Tunnels 1: Example of a project with rock cutting in a fractured rockmass

11. 11 Tunnels In situ stress (  yy ) before excavation Tunnels 1: Example of a project with rock cutting in a fractured rockmass Vertical displacement U y due to tunnel excavation Rock bolts are placed (activated) in the model at this stage with a pre-stress S L = 0.1 T Vertical stress (  yy ) after tunnel excavation IV) The next steps are achieved like in classical Finite Element codes: prescribing loads and boundary conditions, modeling excavation stages, displaying results… SLSL

12. 12 Tunnels Tunnels 1: Example of a project with rock cutting in a fractured rockmass Vertical stress (  yy ) after rock cutting Vertical displacement showing uplift after rock cutting Vertical displacement details showing fractures opening Bolts stresses change when crossing fractures and attain a maximum value of 2 T.

13. 13 A double line tunnel in a sedimentary rock mass Tunnels 2: Case Study

14. 14 Tunnels 3: Example of bolting effects on the Safety Factor

15. Tunnels Tunnels 4: Bloc fall in a tunnel in blocky rockmass Tunnel in a blocky rockmass Displacement at the roof of the tunnel versus the excavation ratio Calculations diverge before total excavation and can not go beyond the excavation ratio of 0.9. The displacement field at this stage shows the existence of instable blocks at the roof of the tunnel. Instable blocks at the roof of the tunnel Non convergence

16. 16  Fractures can be introduced in the model by stochastic distribution laws or in a deterministic way.  Gravity load can be applied step by step to determine the safety factor of the slope.  Horizontal and vertical accelerations can be applied in order to analyze the stability against seismic loads. Rock slope with two types of fractures Finite Element mesh created by DISCRAC ® and GID Displacement under prescribed load Shear stress on fractures Analysis and stabilization of natural rock slopes, rock cuttings and open pit mines Rock slope stability analysis

17. 17 Rock slope stability analysis  Fractures can be introduced in the model by stochastic distribution laws or in a deterministic way.  Gravity load can be applied step by step to determine the safety factor of the slope.  Horizontal and vertical accelerations can be applied in order to analyze the stability against seismic loads. Rock slope with two types of fractures Finite Element mesh created by DISCRAC ® and GID Displacement under prescribed load Shear stress on fractures The stability of natural rock slopes, rock cuttings and open pit mines is easily analyzed with DISROC.

18. 18 Slope stability under seismic load Application of gravity forces to define the initial state of stress Rock cut in a blocky rockmass Displacement of the point A versus seismic load ratio. The calculations can not go beyond 0.7 g horizontal acceleration and diverge at this stage. Addition of 1 g horizontal acceleration to represent seismic load The displacement field at 0.7g horizontal acceleration reveals an instable block (blue in the figure) A (A)

19. 19 Slope design optimization Design modification Modified model in DISROC Meshing facilities of DISROC for fractured rocks allow easy optimization of rock cutting design. If the projected slope reveals instable, it is easy to change quickly the design in DISROC and analyze the modified project. Initial slope design revealed to be instable

20. 20 Cross section of an Earth Dam lying on a rock mass foundation with two sets of discontinuities (DISROC  ) Rock foundation along with the dam and the dam-foundation interaction are analyzed in a unique model enclosing all the fractures’ sets Stability of dams on fractured rock

21. Effective model for fractured rockmass Fractures and faults modeled individually as discontinuities Fractures replaced by a continuous effective material Far-field fractures act only by their global effects, and only in elastic phase. A preliminary homogenization allows replacing the fractured rock mass by a continuous media with adequate effective properties. Great discontinuities like faults can be introduced in the final model as individual lines. Combination of fractures modeled individually (near-field) and replaced by an effective material (far-field). Fractures and faults modeled individually as discontinuities Fractures replaced by a continuous effective material Combination of fractures modeled individually (near-field) and replaced by an effective material (far-field). Fractures and faults modeled individually as discontinuities ?

22. 22 Example : sedimentary bedded rock E = 10 GPa, = 0.25, K n = 10 GPa.m, Kt= 2.5 GPa.m, D = 1m Goodman formulae:

23. 23 Homogenization in DISROC The geometry and mechanical properties of fractures are introduced in DISROC which determines equivalent elastic properties of the rock mass by a numerical homogenization method. The following slides show different stages of this process. DISROC contains a specific module for determining a continuum equivalent model for a fractured rock mass. Fractures geometry in a sedimentary rock mass Equivalent elastic modulus in different directions determined by homogenization

24. 24 Homogenization in DISROC : Fracturing model data acquisition I) For each family of fractures, the fractures’ orientation, length, spacing and mechanical parameters are specified. II) Fractures sets are generated stochastically according to specified parameters. III) A conform Finite Element mesh is created by Discrac® + GID.

25. 25 Homogenization in DISROC : Load application on the REV U y displacement under uniaxial compression  yy U x displacement under shear stress  xy V) The average stresses and strains in the REV, taking into account the fractures opening, are computed for each loading case and the homogenized elastic properties of the fractured rock mass are determined from the average values. IV) 3 different basic loads; uniaxial compression in x and y directions and pure xy shear, are applied on the REV’s contour. Anisotropic elastic coefficients for the homogenized behavior

26. 26 Homogenization : Anisotropic stiffness and compliance tensor calculation The stiffness and compliance tensors lines are computed automatically by imposing boundary conditions corresponding to macroscopic strain or stress in different directions.

27. 27 Homogenization : Anisotropic stiffness and compliance tensor calculation The homogenized stiffness and compliance tensors lines are given as a direct result of calculation.

28. 28 Rockmass with general configuration of fractures The effective elastic coefficients C ij are directly calculated by DISROC Homogenization module, and can be introduced as material parameters for modeling the rock mass by its effective properties. ? 31200

29. 29 Case study - Granitic rockmass of La Vienne, France Homogenization of Strength Properties with Disroc Disroc allows the calculation of stress-strain curve of fractured media with elastic-plastic behavior. This makes possible determination of the effective cohesion and friction angle for fractured rockmasses.

30. 30 Special model for rockmass with one fracture family For a fractured rockmass containing a family of parallel and infinite fractures, a special material model is implemented in Disroc which provides the corresponding effective behavior. ? 31400 E K t K n K tn D  Numerical homogenization is not needed for this case: the parameters Cij are computed automatically based on theoretical relations.

31. 31 Case Study 1: Stability assessment for retrofitting purposes with bolts Masonry Structures Evolution of the damage state in the bridge Opening of the active fractures Vertical stress maps Concentration of stress near the fractures zone Analysis of masonry structures needs combining discontinuous modeling with continuous modeling. For instance the vault of masonry bridges are modeled as assemblage of blocs and walls or fill materials as continuous materials. Disroc allows easily combining theses two types of models the same process.

32. 32 Case study 2 – Temple: Assessment of the temple’s stability for retrofitting purposes Masonry Structures Yanouh Roman temple, Lebanon Mesh generation in presence of fractures and Stress maps

33. 33 Modeling bolts 41110 :Elastic-plastic bolt + elastic-plastic bolt/roc contact Nb = 8 Param1 = E (bolt elastic modulus) Param2 = K t (bolt/rock contact shear stifness ) Param3 = K n (bolt/rock contact normal stifness ) Param4 = K nt = K tn (bolt/rock contact ns stifness ) Param5 = Y s (bolt elastic limit) Param6 = c (bolt/rock contact cohesion) Param7 =  (bolt/rock contact friction angle) Param8 = s 0 (bolt pres-stress) Complete models for bolts, anchors and bars are available in Disroc with full integration of the grout behavior by an elastic-plastic interface model. Bolts can cross fractures. The model of intersection allows discontinuity of rock displacement at the two side of the fracture with continuity of the bolt rod. Disroc is the only Finite Element software allowing this modeling.

34. 34 Representing bolt stresses F FEM mesh for the sample, bolt and fracture Axial force S L in the bolt represented in two different ways. S L passes by a local maximum when crossing the fracture. S L (MN) SLSL  Deformation at the roof of a bolted tunnel Weight S L (MN) Axial force S L in the bolt represented in two different ways. S L passes by a maximum when crossing the fracture.  Pull out test on a bolt crossing a fracture FEM mesh for the rock, Bolt and fracture SLSL

35. 35 Hydraulic module The hydraulic module of Disroc allows: - Modeling flow in a fracture network under pressure gradient and gravity forces - Modeling flow in a porous/fractured rock mass - Determination of the effective permeability of fractures rock masses The flow in fractures is modeled by the Poiseuille law, and in the rock matrix by the Darcy’s law, and fracture/matrix mass exchange are fully taken into account. The pressure field calculated in fractures can be injected as a pressure load in the mechanical module in order to take into account its effects on the mechanical stability. The two types of calculations, on a discrete fracture network and on fractures in a porous matrix, can be performed on the same geometry and mesh. This makes very easy to estimate the effect of a matrix permeability. An example for effective permeability calculation is given in the following page.

36. 36 Hydraulic module : Effective permeability of fractured rockmass Effective permeability can be calculated for a discrete fracture network (impervious matrix) or with taking into account a matrix permeability. The necessary boundary conditions are prescribed automatically and the effective permeability given as a direct output of the calculation. Pressure field The average fluid velocity in the domain calculated automatically  Effective permeability Rockmass with two fracture families Unit pressure gradient on the boundary P1P1 P0P0 Flow in fractures Effective permeability

37. 37 Materials models A great variety of classical constitutive models are available in DISROC for rocks, fractures, joints and rockbolts. Solid materials: Elastic-plastic behavior: - Linear isotropic or anisotropic elasticity - Mohr-Coulomb, Drucker-Prager, Hoek & Brown plastic failure criteria - Anisotropic Darcy’s law for hydraulic diffusion Discontinuities: fractures, faults, rock joints and interfaces - Linear or non linear Barton-Bandis elasticity - Mohr-Coulomb (Cohesion, friction angle) yield criterion - Poiseuille’s law for flow in fractures Rockbolts and cables - Elastic and plastic limit for steel rod, - Elastic stiffness, cohesion and friction angle for rock– grout interface Joint : K n, K t  C, 

38. 38 Displaying results in DISROC Stress vectors on rock joints Normal stress on rock joints A variety of different representations of the results are possible, specially those concerning rock joints and fractures. Example: Deformation of the fractured REV under shear stress  xy : U x displacement

39. 39 Architecture GID is a powerful pre and post processor developed by Cimne: www.gidhome.com www.gidhome.com WinDisroc manages data acquisition and generates fractured rockmasses Discrac allows joint elements creation Disroc is the calculation module Geometry Boundary conditions Mesh GID Post Process GID Discrac WinDisroc Disroc input file Fracture generation Parameters output file

40. 40 DISROC functionalities DISROC has the following main functionalities: Elastic-plastic modeling of rocks, rock-joints and rockbolts with incremental loading Incremental multistage excavation of underground openings and rock cuttings Stability of rock slopes under seismic loads (horizontal and vertical acceleration) Analysis of block fall down risk in tunnels in blocky rockmasses Modeling rock bolts, bars and cables in fractured rock Modeling fluid flow in fractured porous rocks or in discrete fracture networks Taking into account fluid pressure effects in the mechanical stability analysis Homogenization of fractured rock mass mechanical and hydraulic properties: - determination of the effective elastic parameters - simulation of effective stress-strain curve to determine effective strength properties - determination of the effective permeability of fractured rock masses DISROC is interfaced with the powerful pre and post-processor GID (www.gidhome.com) that allows easily defining the geometry and materials model, generating mesh, and displaying the calculation results in the form of contours and curves, etc.www.gidhome.com

41. 41 MechaRock International Consultants www.mecharock.com For more information, please send an email to: info@mecharock.com