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Keywords: stiffness matrix

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 52nd U.S. Rock Mechanics/Geomechanics Symposium, June 17–20, 2018

Paper Number: ARMA-2018-686

... stress tensor

**stiffness****matrix**equation anisotropy 1. INTRODUCTION Calculation of the distribution of stress around a borehole subjected to far field in situ stress is essential in drilling wellbore stability analysis [1,2], borehole breakout analysis [3,4], sonic log interpretation [5-7] as well...
Abstract

ABSTRACT: An explicit analytical workflow for calculating the distribution of stress around a circular borehole in an arbitrary tilted transversely isotropic (TTI) formation is presented. This workflow is developed based on the formalism of Lekhnitskii and Amadei who gave a closed-form stress solution for a borehole in an arbitrary anisotropic medium. However, their solution is numerically singular when a formation is isotropic or transversely isotropic with the isotropy plane perpendicular to the borehole axis. It does not naturally reduce to Kirsch solution in the case of isotropy. Also, the borehole stress solution of Amadei is given as a function of the roots of a characteristic equation that is a sixth order polynomial function, which needs to be solved numerically for TTI media. In order to make Amadei solution easy to implement in practices, I derive the analytical roots of the characteristic equation and resolve the numerical singularity issue by introducing a modification to the stress solution. The proposed workflow for borehole stress calculation in TTI media is given in a completely explicit form and is stable for arbitrary TTI anisotropy (including isotropy).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 52nd U.S. Rock Mechanics/Geomechanics Symposium, June 17–20, 2018

Paper Number: ARMA-2018-052

... & Gas input parameter indentation moduli

**stiffness****matrix**matrix multiscale model kerogen random variable elastic property consolidated clay fraction stiffness tensor 1. INTRODUCTION Organic-rich shales are multiphase geo-materials exhibiting multi-scale microstructure, highly...
Abstract

ABSTRACT: A probabilistic multiscale modeling for predicting poromechanical properties of shales is presented. To this end, a framework of experimental characterization, physically-based multiscale modeling and uncertainty quantification that spans from nanoscale to macroscale is utilized. To account for the uncertainty in the model input parameters, they are modeled as random variables. To this end, input parameters are divided into two classes of random variables: tensor-valued and scalar random variables and their corresponding statistical description is constructed by employing Maximum Entropy principle (MaxEnt) based on available information. Then, to propagate uncertainty across different length scales the Monte Carlo simulation is carried out and consequently probabilistic descriptions of macro-scale properties are constructed. Furthermore, a global sensitivity analysis is carried out to characterize the contribution of each source of uncertainty on the overall response. Finally, methodological developments are validated against experimental test database. The integration of experimental characterization, multiscale modeling and uncertainty quantification utilized in this work improves the robustness and reliability of predictive models for poromechanical behavior of shales.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 51st U.S. Rock Mechanics/Geomechanics Symposium, June 25–28, 2017

Paper Number: ARMA-2017-0382

... of state strength hydraulic fracturing hydraulic fracture propagation fracture trace fracture network Reservoir Characterization Upstream Oil & Gas fracture process mechanical anisotropy fracture

**stiffness****matrix**Rutqvist reservoir geomechanics node Simulation plane injection...
Abstract

ABSTRACT: This study investigates hydraulic fracture propagation using a coupled hydro-mechanical simulation code, TOUGH-RBSN. The modeling tool combines a multiphase fluid flow and heat transport simulator, TOUGH2, with a geomechanical and fracture-damage modeling approach, called the rigid-body-spring network (RBSN). The model provides a discrete representation of material failure and fracture processes, and hydrological properties (e.g., permeability, porosity) of fractured elements are evaluated by transient fracture aperture changes. Modeling capabilities for coupled hydro-mechanical processes are presented through hydraulic fracturing simulations of laboratory test samples with multiple pre-existing natural fractures. Sensitivity studies are conducted by changing the reservoir configurations, such as the viscosity of injected fluid and the confining stress condition. More realistic reservoir modeling at a larger scale is conducted with configurations of mechanical anisotropy (e.g., elastic moduli, strength parameters), which demonstrates the capability of modeling directional fracturing paths aligned with anisotropic rock properties. 1. INTRODUCTION

Proceedings Papers

K. C. Das, S. S. Sandha, I. Carol, P. E. Vargas, N. A. González, E. Rodrigues, J. M. Segura, M. R. Lakshmikantha, U. Mello

Publisher: American Rock Mechanics Association

Paper presented at the 50th U.S. Rock Mechanics/Geomechanics Symposium, June 26–29, 2016

Paper Number: ARMA-2016-553

... mass cohesive discontinuity fault plane Computation shear stress enriched element displacement profile reservoir geomechanics displacement discontinuity

**stiffness****matrix**enrichment function Reservoir Characterization subdivision XFEM representation matrix displacement field 1...
Abstract

Abstract: Reservoir Geomechanics is playing an increasingly important role in developing and producing hydrocarbon reserves. One of the main challenges in reservoir modeling is accurate and efficient simulation of arbitrary intersecting faults. In this paper, we propose a new formulation to model multiple intersecting cohesive discontinuities (faults) in reservoirs using the XFEM framework.This formulation involves construction of enrichment functions and computation of stiffness sub-matrices for bulk (rock mass), faults and their interaction DOFs. A sub-divisional scheme has been developed, to perform volume and surface integrations. The sub-divisional scheme can efficiently handle most possible configurations of multiple intersecting/nonintersecting faults in 3D finite elements. In order to be able to model cohesive behaviour of intersecting faults, we have performed additional surface integrations which, to the knowedge of the authors, are not reported in the literature. For example, if an element is intersected by two cohesive faults, we compute 16 volumes an d 8 surface in tegrations for stiffness matrices and 4 surface integrations for force vectors, to form elemental equations. The current distributed implementation using C++/MPI can simulate very large meshes. Three-dimensional benchmark cases are presented to validate the accuracy of the approach, and the potential benefits in applications. The present study shows that interaction DOFs and their associated surface stiffness matrices play a significant role in accurate modeling of cohesive faults via XFEM. Introduction Faults are geological entities with thickness several orders of magnitude smaller than the grid blocks typically used to discretize the domain. Due to frictional forces between the surfaces, the natural occurring faults in rock masses are cohesive. Cohesive faults may result in relative displacements in the plane of the fracture. Due to arbitrary and multiple occurrence of faults, many of them may intersect and result in complex 3D geometries. Intersecting faults affect the strength and deformability of rock mass and thus their accurate modelling is required to understand the geomechanical behaviour of reservoirs. Numerical modelling techniques can be classified on the basis of the procedure used to represent faults. Faults can be represented implicitly by using constitutive equation for the continuum and additional deformations due to opening/slip. The explicitly representation of faults may be done in various ways, the more classical one is via zero-thickness (or sometimes thin layer) elements inserted in-between element faces/edges while all nodal variables maintain the traditional meaning as regula displacements (Goodman et al., 1968; Gens et al., 1988). We do implicit representation of the faults using level set functions (Das et al., 2015). The discontinuity can cut the elements in arbitrary ways, which is captured using zero values of level set function. This approach can define the discontinuities with complex geometries. The jump in the displacement field near the fault plane is captured using appropriate enrichment function. For example, for rock faults modeling, ‘Heaviside function’ is used to enrich nodes (Wells and Sluys, 2001).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 50th U.S. Rock Mechanics/Geomechanics Symposium, June 26–29, 2016

Paper Number: ARMA-2016-046

... compression test

**stiffness****matrix**perpendicular 1 1. INTRODUCTION Geomechanical anisotropy of reservoir rocks plays an important role in wellbore stability analysis (Cui and Abousleiman, 1998) as well as the design of hydraulic fracture treatments (Sesetty and Ghassemi, 2016). Laboratory characterization...
Abstract

Abstract: Rock anisotropy is important in many reservoirs geomechanics problems, but is it often challenging to characterize rock anisotropy In this work, a testing program is developed to determine the orientation of material bedding planes through a hydrostatic compression test and to reveal the nature of the rock anisotropy. Also, static and dynamic measurements are carried out on a whole core to measure the coefficients of elastic anisotropy. The core used is from Utica Shale, a Middle Ordovician formation in the Appalachian Basin, underlying much of the northeastern United States and adjacent parts of Canada. The geomechanical properties of this shale are determined using multiple methods at multiple scales. The measurements are first made on a 4” diameter core using three strain rosettes to completely characterize a 3D hydrostatic strain tensor, which then is used to identify the type of anisotropy (transversely isotropic vs orthotropic), as well as the strike and the dip of the core bedding planes with respect to the locations of the strain rosettes. Dynamic measurements on the whole core are also made using multiple transducers placed on the sample surface. The sample is then sub-cored and plugs are extracted for deviatoric compression tests, which allow for complete determination of all the anisotropic elastic constants. The results from static and dynamic measurements are presented and compared, for whole core as well as the sub-cored plugs. Scale effects are observed to play a role in the measurement of mechanical properties. Introduction Geomechanical anisotropy of reservoir rocks plays an important role in wellbore stability analysis (Cui and Abousleiman, 1998) as well as the design of hydraulic fracture treatments (Sesetty and Ghassemi, 2016). Laboratory characterization of anisotropy is typically performed, by extracting core plugs from at least three orientations with respect to the bedding planes (Aoki et al 1993) – this can be done only if the orientation of the bedding planes in the core sample is completely known. Gonzaga et al. (2008) proposed a technique to determine the elastic coefficients for transversely isotropic rock from a single core sample. However, the technique requires a sample with obliquely dipping bedding planes with respect to the cylindrical sample axis (i.e., a standard parallel or perpendicular plug cannot be used since only one material axis is loaded under a deviatoric compression test for either of these orientations).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 48th U.S. Rock Mechanics/Geomechanics Symposium, June 1–4, 2014

Paper Number: ARMA-2014-7094

... in hydraulics, rock mass stability, and in coastal and offshore engineering. rock-fluid incompatibility oilfield chemistry Production Chemistry chemical treatment Upstream Oil & Gas experiment algorithm rigid block rock/fluid interaction tsunami wave

**stiffness****matrix**matrix formation...
Abstract

Abstract We present a three dimensional fluid-structure coupling between SPH and 3D-DDA for modelling rock-fluid interactions. The Navier-Stokes equation is simulated using the SPH method and the motions of the blocks are tracked by a Lagrangian algorithm based on a newly developed, explicit, 3D-DDA formulation. The coupled model is employed to investigate the water entry of a sliding block and the resulting wave(s). The coupled SPH-DDA algorithm provides a promising computational tool to for modelling a variety of solid-fluid interaction problems in many potential applications in hydraulics, rock mass stability, and in coastal and offshore engineering.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 47th U.S. Rock Mechanics/Geomechanics Symposium, June 23–26, 2013

Paper Number: ARMA-2013-416

... Abstract: We present a 3D-DDA formulation that uses an explicit time integration procedure and an efficient contact detection algorithm optimized to minimize the computational effort. The advantages of the explicit formulation are that the global

**stiffness****matrix**does not need...
Abstract

Abstract: We present a 3D-DDA formulation that uses an explicit time integration procedure and an efficient contact detection algorithm optimized to minimize the computational effort. The advantages of the explicit formulation are that the global stiffness matrix does not need to be assembled and the linear equations do not need to be solved by matrix inversion. Consequently, the computational effort and memory requirement can be reduced considerably, which is important for efficient solution of large 3D problems. In addition, the computational efficiency is increased by eliminating unnecessary contact computations using a grid based nearest neighbor search. The grid divides space into a number of cells of equal size and each object is then associated with the cells it overlaps. As only objects overlapping a common cell can possibly be in contact, in-depth tests are only performed on objects found sharing cells with the block tested for collision. The contacts between the blocks are detected by using Fast Common- Plane (FCP) approach. The halfedge (HE) data structure approach is used to handle the navigation into the topological information associated with polyherdral objects (vertices, edges, faces). The halfedge data structure allows for quick traversal between faces, edges, and vertices due to the explicitly linked structure of the network. Examples are provided which demonstrate the capabilities of new algorithm and the size of problem that can be analyzed.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 45th U.S. Rock Mechanics / Geomechanics Symposium, June 26–29, 2011

Paper Number: ARMA-11-203

... deformation analysis

**stiffness****matrix**Deformation analysis illustrative example contact constraint equation 1. INTRODUCTION The discontinuous deformation analysis (DDA), which is an energy-based method, is an alternative to the distinct element method (DEM) for discontinuity-based problems [1...
Abstract

ABSTRACT In the present high-order 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, but may lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violates the physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in this research. In this paper, contact constraints are enforced in high-order 3-D DDA using the augmented Lagrangian method and the formulations are presented. Moreover, a code has been programmed by Visual C++ and an illustrative example is used to validate the new formulations and the code. Using the augmented Lagrangian method to enforce contact restraints retains the simplicity of the Penalty method and reduces the disadvantages of it. 1. INTRODUCTION The discontinuous deformation analysis (DDA), which is an energy-based method, is an alternative to the distinct element method (DEM) for discontinuity-based problems [1]. This method can be used to solve problems involving discontinuous media. Original DDA formulation utilizes first order displacement functions to describe the block movement and deformation. Therefore, stress or strain is assumed constant through the block and the capability of block deformation is limited. This may yield unreasonable results when the block deformation is large and geometry of the block is irregular. There are some published papers on deformable blocks in 3-D DDA. Beyabanaki et al. [2-4] implemented Trilinear and Serendipity hexahedron FEM Meshes into 3-D DDA. Beyabanaki et al. [5-7] presented 3-D DDA with second-and third-order displacement functions. Beyabanaki et al. [8] presented 3-D DDA with n th -order displacement functions. Recently, contact theory of n th -order 3-D DDA is presented by Beyabanaki et al. [9]. The penalty method was originally used by the abovementioned 3-D DDA researchers to enforce contact constraints at the block interface. The accuracy of the contact solution depends highly on the choice of the penalty number and the optimal number cannot be explicitly found beforehand. Obviously, the penalty number should be very large to achieve zero interpenetration distance. However, a very high penalty number leads to progressive ill-conditioning of the resulting system and thus one cannot hope to achieve high-accuracy solutions with this approach. A well-known method to overcome these problems for equality constrained problems is the augmented Lagrangian method [10]. The augmented Lagrangian method has been advocated by Lin et al. [11] in two dimensional discontinuous deformation analysis. In this research, the same method has been implemented in high-order three-dimensional discontinuous deformation analysis and an illustrative example is presented for demonstrating this new approach. 2. HIGH-ORDER 3-D DDA In the original 3-D DDA, the block displacements function is equivalent to the complete first-order displacement approximation; constant strains and constant stresses are assumed within each block. When displacement functions are taken as n th -order functions: The high-order function is necessary in most engineering analyses since it can represent stress concentrations within one block. In two dimensions, the contact types between blocks include corner-to-corner, corner-to-edge and edge-toedge;

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 45th U.S. Rock Mechanics / Geomechanics Symposium, June 26–29, 2011

Paper Number: ARMA-11-492

.... Artificial Intelligence maximum axial force decoupled rock bolt Reservoir Characterization Upstream Oil & Gas

**stiffness****matrix**matrix axial force boundary rock mass displacement metals & mining elasto-plastic rock mass Rock mechanics rock bolt elastic rock mass neutral point...
Abstract

ABSTRACT Rock bolts have been widely used as a primary support system to stabilize rock masses around tunnels, underground mine galleries, slopes and others structures. To model the interaction between fully grouted bolts and the rock mass a numerical procedure is developed called „enriched finite element method (EFEM)?. Conceptually if a solid finite element is intersected by a grouted rock bolt, it becomes an „enriched? element. Nodes of enriched elements have additional degrees of freedom which are used to determine displacements and stresses in the bolt. Stiffness of enriched elements is formulated based on properties of the rock mass, bolt rod and grout, orientation of the bolt and borehole diameter. This paper quantitatively evaluates bolt performance in different shapes of underground openings viz circular, rectangular and D-shaped, using the proposed enriched finite element method (EFEM) combined with elasto-plastic behaviour of rock mass and grout material. In addition, a comparative study of bolt performance is also presented considering both coupled and decoupled behaviour of rock bolts. INTRODUCTION Rock bolts have been widely used as a primary support system to stabilize slopes, hydro dams and underground structures such as tunnels and mine workings and others structure made in rock masses. The term “rock bolt” is defined in geomechanics as a form of mechanical support that is inserted into the rock mass with the primary objective of increasing its stiffness and/or strength with respect to tensile or shear loads. In general, rock bolts reinforce rock masses through restraining the deformation within rock masses and reduces the yield region around the excavation boundary. During the last four decades, different types of rock bolts have been practiced, out of which fully grouted active/passive bolts were the most common types. For a fully grouted passive rock bolt installed in deformable rock masses, a neutral point exists on the bolt, where shear stress at the interface between the bolt and grout material vanishes. Based on neutral point concepts, shear stresses and axial loads developed along a bolt rod are analytically formulated by many researchers. Bolt grout interactions around a circular tunnel in Hoek-Brown medium have been formulated analytically considering a bolt density factor. Considering different approaches to bolt performance Stille presented a closed form elasto-plastic analytical solutions of grouted bolts. Based on the shear lag model (SLM), Cai et al. derived an analytical solution of rock bolts for describing the interaction behaviours of rock bolt, grout material and rock mass. Brady and Lorig. numerically analyzed the interactions of bolt grout in Mohr Coulomb media using the finite difference method (FDM) technique. In addition, numerous studies have been published on the analytical solution of stresses and displacements around a circular tunnel considering elasto-plastic rock mass with Mohr-Coulomb yield criterion. Elwi and Hrudey and d?Avila et al., proposed embedded finite element method for reinforcing curved layers and concrete structures respectively. Finite Element Method (FEM) and/or FDM based procedures are also developed for the analysis of the said problem and have been presented in many references for solving geotechnical problems.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the Alaska Rocks 2005, The 40th U.S. Symposium on Rock Mechanics (USRMS), June 25–29, 2005

Paper Number: ARMA-05-808

... geomechanics displacement Upstream Oil & Gas calibration MPa vector metals & mining block-spring model active failure area analytical solution Failure area Reservoir Characterization orientation contour

**stiffness****matrix**three-dimensional block-spring model excavation contact force...
Abstract

ABSTRACT: A three-dimensional block-spring model and a computer code BSM3D have been developed for simulating the behavior of jointed rocks. Joints and discontinuities in rock masses are interpreted as interfaces between blocks. Large displacements are modeled by using an iterative procedure, in which the contact stiffness and forces vary according to contact modes governed by the Mohr-Coulomb failure criterion. Calibration of the discrete model is carried out following a benchmarking procedure: verify the formulation and program using analytical solutions; compare the results with those from referred commercial software; and, examine the displacements using in situ measurements. The calibration results have shown that BSM3D is correct and reliable for most of the problems involved in jointed rocks, though some limitations may apply. It can be used as a practical tool in the planning of mine excavations and slope stability analyses. INTRODUCTION The three-dimensional block-spring model and the computer software BSM3D have been developed in CANMET-MMSL, Natural Resources Canada, for analyzing large displacement and stability of jointed rock masses. BSM3D simulates the discontinuous rocks by assembling blocks divided by joints and faults. The blocks are allowed to slide and detach along joint faces and soft layers. Contact forces are computed from the relative displacements between blocks and then examined by the failure conditions. An iterative procedure is adopted to simulate the large-scale displacements of the rock blocks. In order to simulate more complex geometrical and geological conditions of the jointed rocks, different shapes of the blocks can be formed, including 8- node, 6-node, 5-node, and 4-node 3D elements. Boundary displacements and tractions, in situ stress, underground water flow, excavation and backfill procedures can be considered. The software is equipped with fast solvers (in-core and out-of-core direct and iterative sparse solvers) and a multifunctional visual processor to help users easily set up models and view analysis results. The program has been verified through analytical solutions [1] and applied in several mine projects. In order to further demonstrate the validation and reliability as well as the limitations of the software, more examples have been analyzed and evaluated against theoretical solutions and other software codes, such as FLAC3D and UDEC.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 35th U.S. Symposium on Rock Mechanics (USRMS), June 5–7, 1995

Paper Number: ARMA-95-0045

... of the results were compared with those of the finite element method. The method of DDA with improvement has been proven to be very effective to analyze rock stability problems in discontinuous deformable rock masses. Upstream Oil & Gas lining rock mass Wellbore Design rockbolt

**stiffness****matrix**...
Abstract

INTRODUCTION ABSTRACT: Discontinuous deformation analysis (DDA) was created by Shi & Goodman. It solves a finite element type of mesh where all the elements are isolated blocks and bounded by pre--exisfing discontinuities under kinematic conditions of dynamic and quasi-static motion. The authors introduced an elasto-plasfic yield criterion in the analysis, and added several new elements to handle with practical rock mechanics problems. The results show validity of the method for practical use. A new numerical methoddiscontinuous deformation analysis (DDA) was invented by Shi & Goodman (1984, 1985) and further developed by Shi & Goodman (1988, 1989)since then. This method uses the displacements and strains as unknown variables in an element block, and solves the equilibrium equations in the same manner as the matrix analysis of structures in the Finite Element Method. The original DDA only uses an elastic material property for a block and the friction is activated along a block-to-block interface. In this paper improvements of the original DDA are described and the new code is applied to solve rock stability problems. The block element in the new code can deform as an elasto-plastic material following the DruckerPrager associated constitutive law. The main purpose to take into account the block yielding is to analyze soft rock mass behavior subjected to various loading conditions such as excavation and embankment. The interface between blocks behaves according to the Mohr-Coulomb's criterion including cohesive force. Damping coefficient was implemented to take into account the block collision. The rockbolt element was introduced to represent the effect of confinement for rock masses. The bonding element which fuses two blocks was also invented to represent shotcrete in a tunnel structure. The new DDA code was calibrated in comparison with laboratory model tests. Stability of rock slope and tunnel in a discontinuous rock mass is analyzed, and the effect of lining and rock bolts is discussed. Rockfall on a very steep slope was also calculated. Some of the results were compared with those of the finite element method. The method of DDA with improvement has been proven to be very effective to analyze rock stability problems in discontinuous deformable rock masses.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 33rd U.S. Symposium on Rock Mechanics (USRMS), June 3–5, 1992

Paper Number: ARMA-92-0775

... ABSTRACT: The fragment size distribution was calculated for caved materials; naturally caved from the caving methods of mining. The precalculation of fragment sizes was made based on the probabilistic model of rock joint systems surveyed in the field and the

**stiffness****matrix**of continuum...
Abstract

ABSTRACT: The fragment size distribution was calculated for caved materials; naturally caved from the caving methods of mining. The precalculation of fragment sizes was made based on the probabilistic model of rock joint systems surveyed in the field and the stiffness matrix of continuum mechanics to search the joints around a nodal point and to compute the block volumes formed by the joints. The actual computation of fragment sizes was done in two steps. First, the joint systems were simulated on the computer to duplicate the statistics of joint parameters. The joint simulation was localized by using geostatistics for localized fragmentation. Then, based on the simulated joint systems, the stiffness matrix of the finite element method was applied to test the continuity between two adjacent nodal points. By using the fine grid nodal point system developed within a local area, rock blocks are searched and located to calculate their volumes. Finally, the statistics of block volumes was cumulated to build the fragment size distribution. INTRODUCTION The precalculation of fragment size distribution has been desirable for a long time for caved materials from the caving method of mining. It is a very important design criteria for drawpoints, and loading and hauling systems of caving methods such as the block caving method or sublevel caving method. Also, precalculated fragment size distribu- tion has a broad application in general geotechnical engineering where large scale excavation is involved, as in large underground space construction, surface mine production, or any large-scale blasting for civil and mining projects. In this paper, a technical method was developed for searching and locating rock blocks, as well as computing their volumes that could be formed by joint systems in the rock mass. Also, a systematic algorithm was introduced to obtain the fragment size distribution including the local stochastic modeling of joint systems by non- parametric geostatistical techniques. The actual computation of fragment sizes was done in two steps. First, the joint systems were simulated on the computer to duplicate the statistics of joint parameters observed in the field. The joint simulation was localized by using geostatistics for the localized fragmentations, a local area where caving was designed. Then, based on the simulated joint systems, the stiffness matrix of the finite element method was applied to test the continuity between two adjacent nodal points for searching rock blocks and computing their volumes. The local area of rock mass was divided into a discrete cell-block model. The rectangular element used in this cell-block model consisted of 8 nodes at the corners, and there were 28 bars that could connect two adjacent nodal points within an element. Therefore, 2B bars were tested if they were cut by joints, and the test results were used to develop the final global stiffness matrix system of the entire rock mass modeled. By using a finer elemental model, rock blocks are searched and located on the global stiffness matrix to calculate their volumes, which are formed by the simulated joint systems. Finally, the statistics of block volumes are cumulated to construct the fragment size distributions or so-called S-curves.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 26th U.S. Symposium on Rock Mechanics (USRMS), June 26–28, 1985

Paper Number: ARMA-85-0131-1

... and solving, for each load step, the incremental equilibrium equations [K]{au} = {at} where [K] is the

**stiffness****matrix**of the finite element assemblage, {An} and {AF} the incremental nodal displacement and force vector. Many different schemes for achieving this end exist; some do not require reformulation...
Abstract

ABSTRACT 1 INTRODUCTION Development of computer technology and numerical modeling techniques in the last decade have made feasible large-scale computations of geotechnical problems. General purpose finite element programs have been developed that provide a capability for static and dynamic analyses of rock masses with material inhomogeneity, complex non- linear behavior, and structural discontinuities. While detailed calculations incorporating these attributes of rock masses are theoretically possible, there are practical constraints that create a limit of resolution beneath which the rock must be considered as homogeneous. In other words, a representative element of some average dimension must be treated as "continuum" and be assigned homogeneous stress-strain relations (Fig. 1). The length of resolution, denoted here as k is dependent upon the size and requirements of the problems modeled and in practice may correspond to the size of the smallest region represented in a numerical idealization; such as the smallest element in a finite element model. For large scale analysis, this resolution length k may be one, two, or more orders of magnitude greater than the dimension of the rock samples tested in the laboratory. This is not a problem if the rock is genuinely continuous and isotropic. In practice, because the laboratory sample and the "representative" element contain discontinuities that may influence the material behavior differently at the two levels of resolution, the average stress-strain relation can differ substantially. Several empirical procedures for scaling the laboratory values of the elastic moduli and strength of the rock to in-situ values on the basis of quality indices, such as RQD, Q, or RHR have been proposed (e.g., Bieniawski 1979). However, these procedures are generally deficient because they fail to account for the resolution at which numerical modeling is performed. Accordingly, several investigators have developed semiempirical procedures in which the material properties of a given volume are related to the corresponding proper- ties of a laboratory sample via relationships involving sample dimensions as well as quality (e.g., Hardy and Hocking 1978). These have found most application in development of the properties of rock pillars, but are conceptually suited for numerical modeling, providing that the material properties of each zone in the model are adjusted in accordance with its dimensions. The alternatives to simple empirical procedures involve developing models that directly account for the presence of the fractures. Along those lines, equivalent anisotropic models and ubiquitous joint models have received much attention. As discussed in the following section, these approaches have their limitations. A better, but often impractical approach, is to explicitly model each fracture. Here we propose a more practical alternative; one that involves two levels of definition of the rock mass. The first, or global level corresponds to the continuum representation at the scale of resolution we designated k. The second, is a macroscopic level, at which the discontinuities and inhomogeneties of the rock can be treated. FIGURE 1. EQUIVALENT CONTINUUM FOR LARGE SCALE MODELING(available in full paper) 2 BACKGROUND Equivalent continuum models for fractured rock mass are based on the assumption that single or multiple set of parallel joints exist in the rock mass, and that the precise location of the joints is unimportant.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 24th U.S. Symposium on Rock Mechanics (USRMS), June 20–23, 1983

Paper Number: ARMA-83-0053

... rock mass principal stress interface element method block assembly contour plot traction Upstream Oil & Gas equation in-situ state computational model

**stiffness****matrix**block domain distinct element \ 26ch U.S. Sympostum on Rock Hechantca June 1983 STRESS DISTRIBUTION IN A JOINTED...
Abstract

ABSTRACT ABSTRACT The in-situ state of stress is a basic factor in many rock mechanics problems. This state of stress, resulting from the rock mass geologic history, is generally complex and heterogeneous. In-situ measurement techniques sample only small volumes of rock, and both a properly formulated sampling strategy and a statistical treatment of the results are required to obtain representative, volume averaged stresses. A computational model has been developed, on which to conduct studies of a jointed and fractured rock mass. It involves coupling a deformable block version of the Distinct Element Method and the Boundary Element Method, in order to analyze the mechanical response of such a medium. It exploits the assumption that the far-field, modeled with the boundary elements, may be represented by linear elastic behavior. Nonlinear response occurs essentially on the discontinuities, and is confined to the near-field, modeled with the distinct elements. The model has been applied to several simple block assemblies, in some introductory analyses. Stress distribution in these systems resulting from different load paths and loading conditions are presented. INTRODUCTION Knowledge of the in-situ state of stress is an essential requirement for the design of underground excavations, as well as in many other rock mechanics problems. The complex processes occurring during the formation of a rock mass, and throughout its geologic history, are likely to produce rather heterogeneous stress distributions. The mechanical behavior of geologic materials, the heterogeneity of rock masses, diverse patterns of jointing and the existence of penetrative structural features, may affect the ambient state of stress directly. They also determine the response of the rock mass to any subsequent loading episodes, of tectonic origins, for example and further increase the complexity of the in-situ state of stress. Current techniques of field measurement of stresses sample only small volumes of rock, and the considerable scatter in results obtained in practice confirms the above considerations. As a consequence, isolated observations are not meaningful, and a large number of measurements is needed to make possible the determination of the representative, macroscopically- averaged stresses that are required for engineering design purposes. A statistical treatment of these field results, which tests their mutual consistency and reflects their spatial distribution, must then be undertaken to confirm the validity of the solution. The effectiveness of the proposed process will be improved if the sampling strategy is defined according to the specific site conditions, especially the basic jointing patterns in the rock mass and the existence and characteristics of any major, transgressive features. NUMERICAL MODELING The definition of guidelines for a sampling strategy for in-situ stress measurements requires an investigation on how this state of stress is related to the rock mass structure and properties, and to various episodes of loading that occurred in the past. Numerical modeling is an appropriate tool for such an investigation. Different types of jointing patterns or state of fracturing, and various constitutive behaviors can be modeled in an easy, cost-effective manner. A series of numerical experiments designed to determine the response of the rock mass to different load paths will then produce a data set of stress distributions, against which some statistical treatment may be evaluated.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 23rd U.S Symposium on Rock Mechanics (USRMS), August 25–27, 1982

Paper Number: ARMA-82-628

... conditions. Upstream Oil & Gas discrete element

**stiffness****matrix**element domain underground excavation displacement procedure hybrid discrete element-boundary element method formulation Reservoir Characterization element formulation requirement rock mass discrete element method...
Abstract

ABSTRACT The Discrete Element Method is a numerical technique suitable for use in modeling the discontinuum behavior of jointed rock. The disadvantage of this method, in its application to analysis of underground excavations, is the necessity to discretize the complete problem domain. Since the far-field rock responds to the excavation as an elastic continuum, it may be appropriately represented by the Boundary Element Method. This exploits the inherent advantage of the Boundary Element Method in representing a quasi-infinite domain in terms of its internal surface geometry and associated boundary conditions. The procedures employed in developing a first-generation coupled Discrete Element-Boundary Element algorithm are described and the solution to a simple problem verifying the performance of the coupled code is presented. A procedure for numerical treatment of installed support systems is also presented. INTRODUCTION The design of underground excavations requires the capacity to determine support requirements and to assess the adequacy and timeliness of various types of installed support and reinforcement. The interactive nature of support mechanics is often explained in terms of a reaction curve for the rock medium, and a force-displacement curve for the support system. Thus, the mechanical behavior of both the local rock mass and the support is used to determine the point at which load applied to the peripheral rock by the support system re-establishes equilibrium in the rock mass. For the case of a circular excavation made in a homogeneous isotropic mass in a hydrostatic pre-mining stress field, the load-deformation curve for the rock mass can be determined and used in calculating the support required to achieve control of boundary displacements. Two implicit assumptions in the application of the ground reaction curve--radial symmetry and rock mass continuum behavior--limit the ability of the model to analyze many real situations. In particular, the assumption of radial symmetry cannot take into account the flexural properties of the support. The assumption of rock mass continuum behavior does not permit consideration of cases where slim or separation occurs on discontinuity surfaces. The hybrid computational schemes described here are being developed to eliminate these difficulties. They will provide a numerical model of rock-support structure interaction suitable for use in designing support for arbitrarily shaped excavations in a jointed rock mass. A HYBRID DISCRETE ELEMENT-BOUNDARY ELEMENT METHOD Discontinuities in the rock mass frequently exercise a significant role in the local response of the rock to excavation development. For hard rock, or low field stresses, the main modes of local rock mass response involve separation and slip on discontinuity surfaces. Numerical models of such media must account for displacements which are orders of magnitude greater than block elastic deformations. The computational scheme must also model the progressive loosening behavior of the jointed rock mass. The only procedure which fulfills these requirements is the Discrete Element Method (also known as the Rigid Block or Distinct Element Method). Use of this method in the analysis of underground excavations requires the introduction of arbitrary external boundaries and boundary conditions.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 23rd U.S Symposium on Rock Mechanics (USRMS), August 25–27, 1982

Paper Number: ARMA-82-604

... storage requirements may be at least partially avoided by the use of "skyline" solution algorithms (Bathe and Wilson, 1976; Zienkiewicz, 1977). The effect on solution time of several large "skyscrapers" in the column skyline of the global

**stiffness****matrix**has apparently not been investigated. Another...
Abstract

INTRODUCTION Methods of linking boundary integral (BI) solutions with finite element (FE) solutions have been well described theoretically in other publications (e.g., Zienkiewicz, et al., 1977). The purpose of this paper is to point out some practical aspects of implementing these linkage algorithms for rock mechanics applications and to describe an alternative linkage algorithm. A not so obvious motivation for using FE-BI linkages is illustrated in Figure 1. The cost of modeling the excavation of the crusher station by the BI method can equal or exceed the cost of an equivalent FE model. This is due to the large surface area which must be modeled by the BI method. In such cases, a more economical approach might be to model the region immediately adjacent to the crusher station by finite elements and link the solution at the periphery of the FE model with a solution for the exterior region. Further economies can be realized if some flexibility in the type of exterior solution is possible. For example, it would be desirable to use a coarse discretization of the FE-BI interface for a BI solution, which would be less demanding of computer storage and time, or to use an analytical solution for the exterior region. The alternative linkage algorithm to be discussed allows such flexibility. BOUNDARY INTEGRAL METHODS There are basically two formulations of BI methods: the direct and the indirect. FE-BI linkage algorithms depend on which formulation is used. The associated increase in computer storage requirements may be at least partially avoided by the use of "skyline" solution algorithms (Bathe and Wilson, 1976; Zienkiewicz, 1977). The effect on solution time of several large "skyscrapers" in the column skyline of the global stiffness matrix has apparently not been investigated. Another aspect of the above FE-BI linkage algorithm is that the discretization of the interface S is the same for both the FE and BI solutions. In the interests of economy and practicality, it may be desirable to use a coarser discretization for the BI solution or, even better, an analytical solution, if possible. The alternating algorithm described in the next section allows such flexibility in the solution for the region H of Figure 2. ALTERNATING ALGORITHM In the alternating algorithm, the FE and BI solutions are treated separately. Displacements and tractions produced by the two solutions at the interface S between the FE region and the region H are matched in an iterative manner. A test example of the application of this algorithm to a two dimensional problem in plane strain is shown in Figure 3. The FE model is intended to approximate the problem of a hole of radius a(=l) in an infinite medium with a compressive stress at infinity and zero stress at r = a. The solution for the region exterior to the FE region is the classical Lame solution of a hole of radius b(=3) in an infinite medium with a displacement boundary condition at r = b and a stress boundary condition at infinity.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 18th U.S. Symposium on Rock Mechanics (USRMS), June 22–24, 1977

Paper Number: ARMA-77-0009

... axis crystalline rock plane reduction deformation borehole

**stiffness****matrix**initial stress AN ANALYSIS OF DEEP BOREHOLES IN HOT, DRY, CRYSTALLINE ROCK By C. M. St. John, Assistant Professor Department of Civil and Mineral Engineering University of Minnesota Mi nneapol i s, Minnesota ABSTRACT...
Abstract

ABSTRACT ABSTRACT The geothermal energy program has resulted in an increased interest in the deformation and stability of relatively large diameter holes drilled deep into crystalline rocks in regions where the geothermal gradient is high. Knowledge of the response of the rock surrounding such holes can be gained providing that an adequate model can be constructed. For complete generality the model should be capable of analysis of a hole which is arbitrarily orientated with respect to the principal axes of the initial stress field and also any planes of elastic symmetry of the rock. Under these conditions the deformation of the rock around the hole will not, in general, be in plane strain. Taking the ¿ axis as the axis of the hole then, providing the hole is long, the axial strain will be constant and the axial displacement will be a linear function of ¿. The other components of displacement, and therefore all the components of the stress and strain tensors, are independent of ¿. This kind of deformation is a particular case of a kind referred to as antiplane-strain. In this paper, it is shown how this condition can be modeled very simply without resorting to a full three-dimensional analysis. The formulation of a finite element code making use of this approach is presented very briefly and shown to give results in good agreement with those obtained analytically. Finally, some calculations of stresses and displacements around chilled holes deep in hot, dry crystalline rock are introduced. INTRODUCTION Drilling a borehole in any rock will result in disturbance of the initial stress and temperature states. When the hole is deep and the rock hot, these disturbances may be of practical significance as far as borehole closure and stability are concerned. Reviewing the literature, it is clear that this problem has not been extensively studied for the conditions that are likely to be encountered when drilling deep into hot, dry rock of the crystalline basement. Analysis of stresses around borehole has been mainly related to drilling in soft rock. In some cases the influence of the bottom of the hole is taken into account, but more commonly the hole is considered to be long and. an analysis conducted on the basis that the deformation will be in plane-strain. Often it is further assumed that the axial stress will be the intermediate principal stress and that it has no effect upon the strength of the rock. Whether these simplifications are reasonable depends-upon the nature of the rock and also the initial state of stress. To be completely general, however, full account should be taken of: The magnitude and orientation of the principal initial stresses Material anisotropy The orientation of the hole If all these factors are taken into account a conventional two-dimensional analysis assuming the deformation to be in plane-strain will not be adequate. In particular, such an analysis would not represent axial, or out of plane, displacements and associated stresses.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 18th U.S. Symposium on Rock Mechanics (USRMS), June 22–24, 1977

Paper Number: ARMA-77-0187

... problems. Clearly the method has many advantages for use in rock mechanics problems. The elements can be chosen in anyway and each element has its own

**stiffness****matrix**, so that the material need not be homogeneous or isotropic. This study focuses on processes of fracture propagation and faulting in rock...
Abstract

ABSTRACT INTRODUCTION Most rocks consist of aggregates of crystals and amorphous particles joined by varying amounts of cementitious materials. The properties of these structural particles may differ widely from one particle to the others. Moreover, most rocks are granular and porous, indicating the existence of cracks with the dimensions of small fractions of an inch. The overall mechanical properties of rock depend upon every one of its structural features and appearances. In a specimen subjected to unconfined compression, the first structural damage appears as elongated microcracks. As the load increases toward the ultimate strength, there is a pronounced increase in microcracking. At the point of ultimate strength, a macroscopic fracture plane develops. Finally the fracture plane extends toward the ends of the specimen and a rapid drop in the resistance of the specimen to the applied load occurs. The stiff testing machines make it possible to control the failure of rock specimens. The obtained complete stress-strain curve shows the gradual decrease of loading capacity after a point of maximum stress (1). In underground rock systems, the ability of partially failed rock to withstand load is of the greatest importance. The linear fracture mechanics has been applied to the failure of rocks by several authors (3). However, the explanation of this phenomenon is still not altogether clear. Recently, a new method known as the Finite Element Method (FEM) was developed for engineering problems. Clearly the method has many advantages for use in rock mechanics problems. The elements can be chosen in anyway and each element has its own stiffness matrix, so that the material need not be homogeneous or isotropic. This study focuses on processes of fracture propagation and faulting in rock specimens under uniaxial compression. The FEM is applied to simulate failure of rock which is heterogeneous and consists of various materials and/or pores. The calculation is carried out well into the post-failure region. 2. SIMULATION 2.1 Model of A Rock Specimen Rock can be seen as a kind of composite material which contains a sufficient number of structural particles. Accordingly the rock specimen is divided into 252 triangular elements. Each element of model rock corresponds to each mineral particle or pore with its elastic constants and strength randomly distributed. For theoretical solution, the FEM is applied to simulate the progressive failure of rock assuming the stress and strain within each element are uniform. Fig. 1 The Model Rock(available in full paper) 2.2 CRITERIA FOR FAILURE OF EACH ELEMENT An incremental uniform displacement is applied at the upper end of the model rock. At each increment stress distribution is obtained for each element and tested by Mohr's failure criterion for failure. The Mohr?s failure criterion can be written as (mathematical equation)(available in full paper) where Co is the uniaxial compression strength and To is the unlaxial tensile strength. It must be noted that there are two modes of failure. One is compression failure shown by Mohr's stress circle A in Fig. 2. The other is tensile failure shown by Mohr's stress circle B. Stress severity S is used to discuss the mode of failure precisely (2).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 11th U.S. Symposium on Rock Mechanics (USRMS), June 16–19, 1969

Paper Number: ARMA-69-0237

...)(Available in full paper), and in the plastic range 12 Reservoir Characterization displacement plane-strain condition Upstream Oil & Gas Rock mechanics reservoir geomechanics material property iteration loading

**stiffness****matrix**procedure Artificial Intelligence wedge application...
Abstract

ABSTRACT The finite element method of analysis was introduced by Clough. 1 Within a short period, it has developed into a powerful solution technique in structural and continuum mechanics. It has been used to solve boundary value problems in two and three-dimensional elasticity. 2-5 Anisotropy and heterogeneity of materials were allowed for at the earliest Stages. 1,4 Its application has been extended to allow for cracks. 5 , creep and viscoelasticity, . 6,7 incremental construction, . 6-8 nonlinear elasticity, . 3,9 effect of joints in rock, . 10 and to materials with different behavior in tension and compression. 7 . Because of its versatility, the finite element method has found increasing application in the field of rock mechanics. In fact, several of the more significant developments have been motivated by problems in this field. Elasto-plastic material behavior based on the incremental or rate-type theory of plasticity rather than the deformation theory was considered by Reyes and Deere. . 11 A material law based on a generalization. 12 of the Mohr-Coulomb criterion was used. In this chapter, the formulation of material laws developed by Reyes and Deere. 11 is used. An efficient analysis procedure, incorporating recent developments in finite element analysis, is developed for plane-strain conditions. To demonstrate its effectiveness, the procedure is applied to three cases, one for which an exact solution is available, another for which field observations had been made, and a third one for which a parallel study was made, using an accepted limit-equilibrium method. PLANE-STRAIN PLASTICITY A yield function which is a proper three-dimensional generalization of the Mohr-Coulomb hypothesis was suggested by Drucker and Prager 12 in 1952: (Mathematical Equation)(Available in full paper), where a and k are material constants, J 1 is the first invariant of stress and J 2 is the second in variant of stress deviation (see Notation, Table 1). Eq. I plots as a circular cone in principal stress space (Fig. 1) with its axis equally inclined to the coordinate axes. When a=O, the yield surface reduces to the yon Mises criterion and plots as a circular cylinder. In the case of plane strain, Drucker and Prager 12 have shown that J 1 and J 2 become: (Mathematical Equation)(Available in full paper), in which F is angle of internal friction and c is cohesion of the material. Fig. 1--Mohr-Coulomb failure cone. (Available in full paper) (Mathematical Equation)(Available in full paper) The incremental stress-strain relations for plane strain can be conveniently Represented in matrix form,(Mathematical Equation)(Available in full paper), where E is the elastic modulus and v the Poisson's ratio for the isotropic material. In the plastic range, the elements of the stress-strain matrix, D , are functions of the state of stress. Reyes 13 presented a plastic stress-strain matrix which was based on the concept of plastic potential consistent with the incremental theory of plasticity: (Mathematical Equation)(Available in full paper) An interesting characteristic of the Mohr-Coulomb yield criterion( Eq. 1) becomes apparent in considering the axial restraint, s 3 ,in the elastic and plastic range. For plane-strain loading, the equation for s 3 in the elastic range is (mathematical Equation)(Available in full paper), and in the plastic range 12

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 11th U.S. Symposium on Rock Mechanics (USRMS), June 16–19, 1969

Paper Number: ARMA-69-0193

... by finite difference solution of the Laplace equation. The normalized maximum shear stresses 4 max /¿H, where ¿ is the density of the material and H is the height of the vertical slope, are plotted in Fig. 2. material property stress point

**stiffness****matrix**Reservoir Characterization Rock...
Abstract

ABSTRACT In many civil or mining engineering works, the stresses and deformations of natural or excavated slopes due to the combined effect of self-weight and initial stresses are of interest. Because of the mathematical complexities of this problem, however, no complete exact solution exists for the simplest case, that of assuming the stress-strain relation of the rock to be perfectly elastic. Efforts have therefore been directed toward numerical and experimental technique. The stresses within an elastics lope have been obtained by LaRochelie, 1 using the photoelastic and finite difference method to determine the difference and sum of principal stresses. Subsequently, Finn, 2 Hoyaux and Ladanyi, 3 and Duncan and Dunlop 4 attempted to solve the same problem by using the finite element method. The problem was also investigated by Brown and Goodman 5 and Brown and King 6 who emphasize the importance of the effect of loading or unloading procedure on the stress distribution within the slope. Although the geometry of the slope and the boundary conditions involved are identical or similar, the results obtained by these various authors by no means agree. It is therefore necessary to review briefly the results of elastic analysis in order to lend credence to the methods used in the solution. Although the stress-strain behavior of some rocks may be reasonably approximated by the elastic model, other rocks generally exhibit an elasto-plastic behavior with strain hardening or strain softening as shown in Fig. 1. Experiments show that for most rocks, the stress-strain relationship is approximately linear from the origin to the point P (neglecting the small concave region under low stresses). At P, the curve deviates (Fig. 1--Stress-strain relations of rook (Available in full paper)), from linearity and yielding starts to occur. Beyond P, the stress may either increase with strain as shown by curve PA (strain hardening) or decrease with strain (strain softening) after a peak stress is reached as shown by curve PB. The actual behavior depends on the type of rock and the confining pressure. For a given type of rock, an increase in confining pressure gradually transforms strain-softening behavior into strain hardening behavior. As a first step toward this more realistic solution,the stress-strain relationship is assumed to be elastic, perfectly plastic for the determination of the stresses and deformations in a vertical slope of soft rock. Two methods of solution were used in this paper, the lumped parameter Model 7 and the finite element method , so that the results and their effectiveness may be compared and large errors arising from computational techniques may be detected. ELASTIC SOLUTIONS Complete solutions of stresses and displacements within elastic slopes of several inclinations have been obtained by LaRochelie. 1 The principal stress difference was determined by using a gelatin model with Young's modulus E-2.5 psi and Poisson's ratio µ-0.5. The sum of the principal stresses was computed by finite difference solution of the Laplace equation. The normalized maximum shear stresses 4 max /¿H, where ¿ is the density of the material and H is the height of the vertical slope, are plotted in Fig. 2.