Numerical modelling in Geo mechanics


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Fundamntals of rock behaviour modelling and numerical modelling methods and applications in the field of rock mechanics

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Numerical modelling in Geo mechanics

  1. 1. NUMERICAL MODELLING AN EFFECTIVE TOOL FOR MINE PLANNING U.Siva Sankar, M.Tech Under Manager, Project Planning SCCL Email : Modelling Proper understanding of complex behaviour of rock mass has always always been difficult for reliable design and safe operation of mining excavations. Understanding the behaviour of rock in general and the jointed rock rock mass, in particular, has always been difficult for mining engineers engineers involved in reliable planning and design, and safe operation of mining projects under complex and difficult conditions.Model: It is any representation or abstraction of a system or process. A model is an intellectual abstraction that includes purpose, process. reference and cost effectiveness ( Starfield & Beloch, 1986). 1
  2. 2. Modelling Various models used in Mining: Photo-Elastic Models Physical Models Equivalent Material Models Models Closed Form Solutions Analytical Models Mathematical Limit Equilibrium Models Solutions Numerical Models Physical ModellingPhysical Model: It is a miniature replica of some physical systems is of use.These are more commonly abstractions of reality. Models are used to simulatein the laboratory the behaviour of full scale prototypePhoto elasticity is an experimental method to determine stress distribution in amaterial. The method is mostly used in cases where mathematical methodsbecome quite cumbersome. The photoelastic stress analysis technique depends upon the fact that certainoptical properties of most transparent material change when these materialsare subject to stress. The model is machined from a stress birefringent material like glass or plastic,for, e.g., tunnel represented as a circular hole in a plate Glass, PE rubber or epoxy resin – for hard and moderate deformablerockmasses develop stress after being loaded at boundaries and gelatin – highlydeformable rockmasses develop stress under own weight When a polarised light passes through a stress birefringent material patternsof coloured or black fringes are produced. Fringes gives trajectories of principle stresses and its direction. 2
  3. 3. Photo Elastic ModelsPhotoelastic pattern in a glass platemodel containing a central circular hole Photoelastic pattern – Concentration offrom which vertical tensile cracks have stresses in Lower part of a Slopepropagated. Photo Elastic Models CSIR Polariscope for Photoelastic model analysis 3
  4. 4. Equivalent Material Model Equivalent Material Model: the purpose of this model or realistic model is to simulate in the laboratory the behaviour of full scale prototype Elastic, plastic behaviour, viscous flow, fracture of the modeled structure can be simulated Selection of Model materials and loading conditions to be carefully done Models are built on principles of dimensional Similitude Model Materials : generally weak fabricated materials, materials are blended to simulate stratification, jointing and other realistic geological features. Plaster of Paris, lead oxide saw dust oil , gypsum plaster Disadvantages are time taking, involves labor , for every study different models are to be built. Equivalent Material ModelModel in loading Frame ready for testing Model deformation w.r.t roof cracking 4
  5. 5. Mathematical ModellingMathematical Model: The representation of a physical system by mathematical Model: expressions from which the behaviour of the system can be deduced with known deduced accuracy.Analytical solutions1. Closed Form Solutions;These are mathematical relations between stresses and displacements for every displacements point in the surrounding material. Analytical solution for stresses and displacements around a circular hole in a circular biaxillay loaded elastic plate (Kirsch in 1898) (Kirsch Analytical solution for stresses and displacements around a parallel sided slot in parallel an infinite elastic medium (Salomon, 1968 & 1974). 1974) . Analytical solution for stresses and displacements around a elliptical opening elliptical (Brady & Brown, 1985). 1985). Rock-support interaction analysis (Hoek & Brown, 1980) Rock-2. Limit Equilibrium solutions;In this technique gravitational stresses acting on a rigid wedge or block separatedfrom surrounding rockmass by discontinuities are calculated and are checked againstshear resistance offered by the contact surfaces to determine whether the block canfall or slide.Surrounding stress field is ignored in this techniquee.g. Slope analysis, Concept of dead weight design for designing bolting ingalleries Slope Rock Load in a gallery 5
  6. 6. Numerical Modelling In general, the numerical, or analytical, design methods are derived from the fundamental laws of force, stress, and elasticity. Numerical modelling techniques require far more computational power than analytical techniques, but they are well suited to address complex geometries and material behaviour. Most of the Numerical modelling undertaken in the process of mine planning and design involves using linear elastic, static, and boundary element programs. The speed, memory efficiency and ease of use of these codes renders them well suited to quick design analysis. Numerical models can represent complex geometries with a high degree of accuracy. Numerical Modelling• Approach adopted in all numerical methods is to “divide the probleminto small physical and mathematical components andThen sum the influence of the components to approximate the behaviourof the whole system”.• The series of complete mathematical equations formed in this processare then solved approximately.• By definition, the computational solutions are always approximations ofthe exact solution.A numerical model code is simply capable of: Solving the equations of equilibrium, Satisfying the strain compatibility equations, and Following certain constitutive equations - when prescribed boundary conditions are set forth. 6
  7. 7. Numerical Modelling The main sources of the input data for the numerical model are, site investigations, and laboratory and field tests. Numerical methods will give approximate solution, but not the exact solution of the problem. Numerical Approaches: The methods are categorized as Continuum, Discontinuum and Hybrid Continuum/Discontinuum. The Continuum assumption implies that at all points in a problem region; the materials cannot be torn open or broken into pieces. All material points originally in the neighbourhood of a certain point in the problem region remain in the same neighbourhood throughout the deformation or transport process. Numerical Modelling - Approaches1. Continuum methods Finite Difference Method (FDM) Finite Element Method (FEM) Boundary Element Method (BEM).2. Discontinuum methods Fig: (a) Continuous and (b) Discontinuous behaviour Discrete Element Method (DEM), of Uniaxially Loaded Specimen3. Hybrid Continuum / Discontinuum methods Hybrid FEM/BEM, Hybrid DEM/DEM, Hybrid FEM/DEM, and Other hybrid models. 7
  8. 8. Numerical Modelling - Approaches Free Surface Excavation Zone or Element Excavation Boundary Element Finite Boundary or Zone of influence Fig: Boundary Method Fig: Domain MethodBoundary Element Method (BEM):This method derives its name from the fact that the user ‘discretizes’, or dividesinto elements, only boundaries of the problem geometry (i.e., excavationsurfaces, the free surface for shallow problems, joint surfaces andmaterial interfaces), thus reducing the problem dimensions by one andgreatly simplifying the input requirements.In this method the conditions on a surface could be related to the state at allpoints throughout the remaining medium, even to infinity. The informationrequired in the solution domain is separately calculated from the information onthe boundary, which is obtained by solution of boundary integral equation. Numerical Modelling - ApproachesBEMs are simpler and faster, but usually not powerful enough toaccommodate complex geometry and excessive variations in rock massproperties.Suitable for large scale mine modellingE.g. BESOL, MUSLIM/NLFinite Element Method (FEM):The continuum is approximated as a series of discrete elements connected toadjacent elements only at specific shared points called nodes. The behaviourof each element is then described individually using exact differentialequations. The global behaviour of the material is modeled by combining allindividual elements.Fig: Finite Element method 8
  9. 9. Numerical Modelling - ApproachesFEM is perhaps the most versatile of all methods and capable of yielding themost realistic results even in complex geo-mining conditions. Complexity inproblem formulation and requirements of long computer time and large memoryspace seem to be its major shortcomings. e.g. ANSYS, ABAQUS, NASTRAN, COSFLOW, NISAFinite Difference Method (FDM):The continuum is represented by a series of discrete gridpoint at which displacements, velocities andaccelerations are calculated. The displacement field iscomputed by approximating the differentialequations for the system as a set of differenceequations (central, Forward or backward) that Fig: Finite Difference Methodare solved discretely at each grid point. Thedifferential equations are approximated through the useof difference equations. Numerical Modelling - ApproachesFDM results into conditionally stable solution. That is, the convergence of thesolution at different stages of iteration to a true solution depends on the size ofelements and size of the load steps. It has also got the advantage of time-stepping which allows a better understanding of the trend and mode of afailure”.e.g. FLAC (Fast Langrangian Analysis of Continua)Discrete Element Method (DEM) :The DEM for modeling a discontinuum is relatively different compared withBEM, FEM and FDM, and focuses mainly on applications in the fields offractured or particulate geological media. The essence of DEM is torepresent the fractured medium as assemblages of blocks formed by connectedfractures in the problem region, and solve the equations of motion of theseblocks through continuous detection and treatment of contacts between theblocks. The blocks can be rigid or be deformable with FDM or FEMdiscretizations.The distinct element method is ideally suited to modelling of both large scalegeological discontinuities such as faults, dykes and highly fracturedassemblages of rock blocks.e.g. UDEC, 3 DEC 9
  10. 10. Numerical Modelling - Approaches Fig: Various Numerical ApproachesIMPLICIT and EXPLICIT SOLUTION TECHNIQUESOnce the model has been descritized, material properties are assigned andloads have been prescribed, some technique must be used to redistribute theany unbalanced loads and thus determine the solution to a new state ofequilibrium. The techniques used are implicit and explicit – with respect to time.The response of a non-linear system generally depends on the sequence ofloading, and thus it is necessary that the load path modeled be representativeof the actual load path experienced by the body. This is achieved by breakingthe total applied load into increments, each increment being sufficiently small toensure solution convergence for the increment after only a few iterations. Numerical Modelling - Approaches Implicit techniques use principle of Potential energy and assemblesystems of linear equations, which are then solved by standard techniques ofmatrix formulations and reduction. Dynamic relaxation scheme described by Otter et al. (1966), and firstapplied in modelling by Cundall (1971). In this technique no matrices are formed, solution proceeds explicitly innthe time domain – unbalanced forces acting at a material integration pointresult in acceleration of the mass that is associated with the point; The application of Newton’s law of motion expressed as a differenceequation yields incremental displacements; applying the appropriateconstitutive relation produces new set of forces, and so on marching in time,for each material integration point in the model. For Linear problems and problems of moderate non-linearity implicitsolutions tend to perform faster than explicit solution. However, as the degree of non-linearity of the system increases imposedloads must be applied in smaller increments, which implies a greater number ofmatrix formulations and reductions and, therefore, increased computationalexpense. Hence highly non-linear problems are best handled by packages that employan explicit solution technique. 10
  11. 11. Comparison of Numerical methodsMetho Advantages Disadvantages dBEM •Far-field Far- condition inherently represented •Coefficient Matrix fully populated •Only boundaries require discretizations, •Solution time increases with exponentially with result in early solution than any other number of elements used method •Limited potential for handling heterogeneous and non-linear materials non-FEM & •Potential for easily handling material •Entire volume must be descretized, results in descretized,FDM heterogeneity longer solution time •Material & geometric non-linearity handled non- •Far-field Far- boundary conditions must be efficiently, especially when explicit solution approximated is used •For linear problems explicit solutions are •When explicit solution is used skill is relatively slow required for user in assessing numerical •Solution time increases with exponentially with convergence increase in number of elements in implicit solution •When implicit solution is used matrix are technique bandedDEM •Solutiontime increases with linearly with •Solution time much slower than for linear number of elements used problems •Very general constitutive relations may be •Results can be sensitive to assumed values of used with little penalty in terms of modelling parameters computational expense Applications of numerical Modelling Design of Openings, and Pillars. Design of Supports for mine workings. Design of pit slopes and spoil dumps and estimating their stability. Prediction of Main and periodic weightings in Bord & Pillar and Longwall workings. Analysis of support interaction vis a vis strata. Analysis of long term stability of permanent mine excavations. Prediction of surface subsidence over mine excavations., and Simulating effects of blasting on stability of mine workings in Underground as well as in opencast mines. 11
  12. 12. Usage of Numerical ModelsInterpretation: use of models to help us interpret field orlaboratory data.Design: use models to compare the relative performanceof various design alternatives, with less emphasis on thefinal predicted performance.Prediction: use a model to provide a final, quantifiableprediction of actual field behaviour.Majority of model application to the categories ofInterpretation and Design say 90 to 95%, i.e.,unfortunately 5 to 10% of modelling effort to prediction Numerical Model Calibration Fig: Information required for calibration of the Model 12
  13. 13. Comparison of various Numerical Modeling SoftwaresCode Source Type Use ComplexityBESOL Mining Stress Systems 2D/3D BEM Common SimpleEXAMINE Roc Science Inc 2D/3D BEM Rare MediocreMAP 3D Mine Modelling Ltd 3D BEM Moderate MediocreLaMODEL NIOSH -- 3D BEM Moderate SimpleMUSLIM/NL USBM 3D BEM Moderate MediocreFLAC Itasca Consultancy Ltd 2D/3D FDM Common Advanced/ComplexCOSFLOW CSIRO 3D FEM Rare AdvancedPhase2 Roc Science Inc 2D FEM Moderate SimpleANSYS ANSYS, Inc 2D/3D FEM Moderate AdvancedABAQUS Dassault Systems FEM Moderate Advanced Simulia CorpPFC Itasca Consultancy Ltd 2D/3D DEM Rare Complex3DEC Itasca Consultancy Ltd 3D DEM Rare ComplexUDEC Itasca Consultancy Ltd 2D DEM Moderate AdvancedBEFE -- 2D/3D FE Rare Complex &BEMELFEN Rockfield Software Ltd 2D/3D FE Rare Complex &DEM Conclusions Numerical modeling is a very promising and effective tool in understanding the rock mass response subjected to complex loading loading conditions. Efficient use of this tool for reliable design and fixing of strata fixing management problems requires a thorough knowledge of the modeling modeling theory, scope and limitations. Using numerical models, shield, rock strata, coal seam and goaf interactions can be modeled effectively for different insitu loading loading conditions. Proper analysis of model response is very important which requires the requires basic understanding of the mechanisms involved in the physical process process being modeled and the requirement for its numerical simulation. Results from numerical simulation should be compared with field measurements for back calculations and improved input data. More experiences are needed in comparative study between numerical numerical simulations and other analytical methods for precise numerical simulation. simulation. 13