Numerical Modelling in Geomechanics

NUMERICAL MODELLING AN EFFECTIVE TOOL FOR MINE PLANNING : 

NUMERICAL MODELLING AN EFFECTIVE TOOL FOR MINE PLANNING U.Siva Sankar, M.Tech Under Manager, Project Planning SCCL Email : uss_7@yahoo.com

Modelling : 

Modelling Proper understanding of complex behaviour of rock mass has always been difficult for reliable design and safe operation of mining excavations. Understanding the behaviour of rock in general and the jointed rock mass, in particular, has always been difficult for mining 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, reference and cost effectiveness ( Starfield & Beloch, 1986).

Modelling : 

Modelling Various models used in Mining:

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Physical Model: It is a miniature replica of some physical systems is of use. These are more commonly abstractions of reality. Models are used to simulate in the laboratory the behaviour of full scale prototype Physical Modelling Photo elasticity is an experimental method to determine stress distribution in a material. The method is mostly used in cases where mathematical methods become quite cumbersome. The photoelastic stress analysis technique depends upon the fact that certain optical properties of most transparent material change when these materials are 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 deformable rockmasses develop stress after being loaded at boundaries and gelatin – highly deformable rockmasses develop stress under own weight When a polarised light passes through a stress birefringent material patterns of coloured or black fringes are produced. Fringes gives trajectories of principle stresses and its direction.

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Photo Elastic Models Photoelastic pattern in a glass plate model containing a central circular hole from which vertical tensile cracks have propagated. Photoelastic pattern – Concentration of stresses in Lower part of a Slope

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Photo Elastic Models CSIR Polariscope for Photoelastic model analysis

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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.

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Equivalent Material Model Model in loading Frame ready for testing Model deformation w.r.t roof cracking

Mathematical Modelling : 

Mathematical Model: The representation of a physical system by mathematical expressions from which the behaviour of the system can be deduced with known accuracy. Analytical solutions Closed Form Solutions; These are mathematical relations between stresses and displacements for every point in the surrounding material. Analytical solution for stresses and displacements around a circular hole in a biaxillay loaded elastic plate (Kirsch in 1898) Analytical solution for stresses and displacements around a parallel sided slot in an infinite elastic medium (Salomon, 1968 & 1974). Analytical solution for stresses and displacements around a elliptical opening (Brady & Brown, 1985). Rock-support interaction analysis (Hoek & Brown, 1980) Mathematical Modelling

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2. Limit Equilibrium solutions; In this technique gravitational stresses acting on a rigid wedge or block separated from surrounding rockmass by discontinuities are calculated and are checked against shear resistance offered by the contact surfaces to determine whether the block can fall or slide. Surrounding stress field is ignored in this technique e.g. Slope analysis, Concept of dead weight design for designing bolting in galleries Slope Rock Load in a gallery

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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

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Numerical Modelling Approach adopted in all numerical methods is to “divide the problem into small physical and mathematical components and Then sum the influence of the components to approximate the behaviour of the whole system”. The series of complete mathematical equations formed in this process are then solved approximately. By definition, the computational solutions are always approximations of the 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.

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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.

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Numerical Modelling - Approaches Fig: (a) Continuous and (b) Discontinuous behaviour of Uniaxially Loaded Specimen Continuum methods Finite Difference Method (FDM) Finite Element Method (FEM) Boundary Element Method (BEM). 2. Discontinuum methods Discrete Element Method (DEM), 3. Hybrid Continuum / Discontinuum methods Hybrid FEM/BEM, Hybrid DEM/DEM, Hybrid FEM/DEM, and Other hybrid models.

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Numerical Modelling - Approaches Fig: Domain Method Fig: Boundary Method Boundary Element Method (BEM): This method derives its name from the fact that the user ‘discretizes’, or divides into elements, only boundaries of the problem geometry (i.e., excavation surfaces, the free surface for shallow problems, joint surfaces and material interfaces), thus reducing the problem dimensions by one and greatly simplifying the input requirements. In this method the conditions on a surface could be related to the state at all points throughout the remaining medium, even to infinity. The information required in the solution domain is separately calculated from the information on the boundary, which is obtained by solution of boundary integral equation.

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Numerical Modelling - Approaches BEMs are simpler and faster, but usually not powerful enough to accommodate complex geometry and excessive variations in rock mass properties. Suitable for large scale mine modelling E.g. BESOL, MUSLIM/NL Finite Element Method (FEM): The continuum is approximated as a series of discrete elements connected to adjacent elements only at specific shared points called nodes. The behaviour of each element is then described individually using exact differential equations. The global behaviour of the material is modeled by combining all individual elements. Fig: Finite Element method

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FEM is perhaps the most versatile of all methods and capable of yielding the most realistic results even in complex geo-mining conditions. Complexity in problem formulation and requirements of long computer time and large memory space seem to be its major shortcomings. e.g. ANSYS, ABAQUS, NASTRAN, COSFLOW, NISA Numerical Modelling - Approaches Finite Difference Method (FDM): The continuum is represented by a series of discrete grid point at which displacements, velocities and accelerations are calculated. The displacement field is computed by approximating the differential equations for the system as a set of difference equations (central, Forward or backward) that are solved discretely at each grid point. The differential equations are approximated through the use of difference equations. Fig: Finite Difference Method

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Numerical Modelling - Approaches FDM results into conditionally stable solution. That is, the convergence of the solution at different stages of iteration to a true solution depends on the size of elements 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 a failure”. e.g. FLAC (Fast Langrangian Analysis of Continua) Discrete Element Method (DEM) : The DEM for modeling a discontinuum is relatively different compared with BEM, FEM and FDM, and focuses mainly on applications in the fields of fractured or particulate geological media. The essence of DEM is to represent the fractured medium as assemblages of blocks formed by connected fractures in the problem region, and solve the equations of motion of these blocks through continuous detection and treatment of contacts between the blocks. The blocks can be rigid or be deformable with FDM or FEM discretizations. The distinct element method is ideally suited to modelling of both large scale geological discontinuities such as faults, dykes and highly fractured assemblages of rock blocks. e.g. UDEC, 3 DEC

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Numerical Modelling - Approaches Fig: Various Numerical Approaches IMPLICIT and EXPLICIT SOLUTION TECHNIQUES Once the model has been descritized, material properties are assigned and loads have been prescribed, some technique must be used to redistribute the any unbalanced loads and thus determine the solution to a new state of equilibrium. The techniques used are implicit and explicit – with respect to time. The response of a non-linear system generally depends on the sequence of loading, and thus it is necessary that the load path modeled be representative of the actual load path experienced by the body. This is achieved by breaking the total applied load into increments, each increment being sufficiently small to ensure solution convergence for the increment after only a few iterations.

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Implicit techniques use principle of Potential energy and assemble systems of linear equations, which are then solved by standard techniques of matrix formulations and reduction. Dynamic relaxation scheme described by Otter et al. (1966), and first applied in modelling by Cundall (1971). In this technique no matrices are formed, solution proceeds explicitly inn the time domain – unbalanced forces acting at a material integration point result in acceleration of the mass that is associated with the point; The application of Newton’s law of motion expressed as a difference equation yields incremental displacements; applying the appropriate constitutive 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 implicit solutions tend to perform faster than explicit solution. However, as the degree of non-linearity of the system increases imposed loads must be applied in smaller increments, which implies a greater number of matrix formulations and reductions and, therefore, increased computational expense. Hence highly non-linear problems are best handled by packages that employ an explicit solution technique. Numerical Modelling - Approaches

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Comparison of Numerical methods

Applications of numerical Modelling : 

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.

Usage of Numerical Models : 

Usage of Numerical Models Interpretation: use of models to help us interpret field or laboratory data. Design: use models to compare the relative performance of various design alternatives, with less emphasis on the final predicted performance. Prediction: use a model to provide a final, quantifiable prediction of actual field behaviour. Majority of model application to the categories of Interpretation and Design say 90 to 95%, i.e., unfortunately 5 to 10% of modelling effort to prediction

Numerical Model Calibration : 

Numerical Model Calibration Fig: Information required for calibration of the Model

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Comparison of various Numerical Modeling Softwares NIOSH

Conclusions : 

Conclusions Numerical modeling is a very promising and effective tool in understanding the rock mass response subjected to complex loading conditions. Efficient use of this tool for reliable design and fixing of strata management problems requires a thorough knowledge of the modeling theory, scope and limitations. Using numerical models, shield, rock strata, coal seam and goaf interactions can be modeled effectively for different insitu loading conditions. Proper analysis of model response is very important which requires the basic understanding of the mechanisms involved in the physical 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 simulations and other analytical methods for precise numerical simulation.