A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes

AbstractThis paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented

Large scale ab-initio simulations of dislocations

We present a novel methodology to compute relaxed dislocations core configurations, and their energies in crystalline metallic materials using large-scale ab-intio simulations. The approach is based on MacroDFT, a coarse-grained density functional theory method that accurately computes the electronic structure with sub-linear scaling resulting in a tremendous reduction in cost. Due to its implementation in real-space, MacroDFT has the ability to harness petascale resources to study materials and alloys through accurate ab-initio calculations. Thus, the proposed methodology can be used to investigate dislocation cores and other defects where long range elastic effects play an important role, such as in dislocation cores, grain boundaries and near precipitates in crystalline materials. We demonstrate the method by computing the relaxed dislocation cores in prismatic dislocation loops and dislocation segments in magnesium (Mg). We also study the interaction energy with a line of Aluminum (Al) solutes. Our simulations elucidate the essential coupling between the quantum mechanical aspects of the dislocation core and the long range elastic fields that they generate. In particular, our quantum mechanical simulations are able to describe the logarithmic divergence of the energy in the far field as is known from classical elastic theory. In order to reach such scaling, the number of atoms in the simulation cell has to be exceedingly large, and cannot be achieved with the state-of-the-art density functional theory implementations

The 1999 Center for Simulation of Dynamic Response in Materials Annual Technical Report

Introduction: This annual report describes research accomplishments for FY 99 of the Center for Simulation of Dynamic Response of Materials. The Center is constructing a virtual shock physics facility in which the full three dimensional response of a variety of target materials can be computed for a wide range of compressive, ten- sional, and shear loadings, including those produced by detonation of energetic materials. The goals are to facilitate computation of a variety of experiments in which strong shock and detonation waves are made to impinge on targets consisting of various combinations of materials, compute the subsequent dy- namic response of the target materials, and validate these computations against experimental data

Block preconditioning for fault/fracture mechanics saddle-point problems

The efficient simulation of fault and fracture mechanics is a key issue in several applications and is attracting a growing interest by the scientific community. Using a formulation based on Lagrange multipliers, the Jacobian matrix resulting from the Finite Element discretization of the governing equations has a non-symmetric generalized saddlepoint structure. In this work, we propose a family of block preconditioners to accelerate the convergence of Krylov methods for such problems. We critically review possible advantages and difficulties of using various Schur complement approximations, based on both physical and algebraic considerations. The proposed approaches are tested in a number of real-world applications, showing their robustness and efficiency also in large-size and ill-conditioned problems

Mixed Formulations in Space and Time Discretizations for the Fixed-Stress Split Method in Poromechanics

Coupled flow and geomechanics become one of the important research topics in oil and gas industry for development of unconventional petroleum reservoirs such as gas shale, tight gas, and gas hydrates. In particular, these reservoirs are naturally born with its complex behavior, exhibiting strong non-linearity, anisotropy, and heterogeneity effects within each geomaterial and fluid by itself. In addition, the coupling between flow and geomechanics is more complicated for unconsolidated reservoirs or shale formations. Thus, it is critical to assess these complex coupled processes properly through poromechanics with forward numerical simulation and to provide more accurate solutions in order to predict the reservoir performance more precisely. The main objective of this study is to address several numerical issues that are accompanied with simulation in poromechanics. We perform in-depth analysis on mathematical conditions to satisfy for numerically stable and accurate solution, employing various mixed formulations in space and time discretization. Specifically, in space discretization, we deal with the spatial instability that occurs at early times in poromechanics simulation, such as a consolidation problem. We identify two types of spatial instabilities caused by violation of two different conditions: the condition due to discontinuity in pressure and the inf-sup condition related to incompressible fluid, which both occur at early times. We find that the fixed-stress split with the finite volume method for flow and finite element method for geomechanics can provide stability in space, allowing discontinuity of pressure and circumventing violation of the inf-sup condition. In time discretization, we investigate the order of accuracy in time integration with the fixed-stress sequential method. In the study, two-pass and deferred correction methods are to be considered for studying the high-order methods in time integration. We find that there are two different inherent constraint structures that still cause order reductions against high-order accuracy while applying the two methods. As an additional in-depth analysis, we study a large deformation system, considering anisotropic properties for geomechanical and fluid flow parameters, the traverse isotropy and permeability anisotropy ratio. Seeking more accurate solutions, we adopt the total Lagrangian method in geomechanics and multi-point flux approximation in fluid flow. By comparing it to the infinitesimal transformation with two-point flux approximation, we find that substantial differences between the two approaches can exist. For a field application, we study large-scale geomechanics simulation that can honor measured well data, which leads to a constrained geomechanics problem. We employ the Uzawa’s algorithm to solve the saddle point problem from the constrained poromechanics. From numerical parallel simulations, we estimate initial stress distribution in the shale gas reservoir, which will be used for the field development plan. From this study, we find several mathematical conditions for numerically stable and accurate solution of poromechanics problems, when we take the various mixed formulations. By considering the conditions, we can overcome the numerical issues. Then, reliable and precise prediction of reservoir behavior can be obtained for coupled flow-geomechanics problems

Mixed Formulations in Space and Time Discretizations for the Fixed-Stress Split Method in Poromechanics

Coupled flow and geomechanics become one of the important research topics in oil and gas industry for development of unconventional petroleum reservoirs such as gas shale, tight gas, and gas hydrates. In particular, these reservoirs are naturally born with its complex behavior, exhibiting strong non-linearity, anisotropy, and heterogeneity effects within each geomaterial and fluid by itself. In addition, the coupling between flow and geomechanics is more complicated for unconsolidated reservoirs or shale formations. Thus, it is critical to assess these complex coupled processes properly through poromechanics with forward numerical simulation and to provide more accurate solutions in order to predict the reservoir performance more precisely. The main objective of this study is to address several numerical issues that are accompanied with simulation in poromechanics. We perform in-depth analysis on mathematical conditions to satisfy for numerically stable and accurate solution, employing various mixed formulations in space and time discretization. Specifically, in space discretization, we deal with the spatial instability that occurs at early times in poromechanics simulation, such as a consolidation problem. We identify two types of spatial instabilities caused by violation of two different conditions: the condition due to discontinuity in pressure and the inf-sup condition related to incompressible fluid, which both occur at early times. We find that the fixed-stress split with the finite volume method for flow and finite element method for geomechanics can provide stability in space, allowing discontinuity of pressure and circumventing violation of the inf-sup condition. In time discretization, we investigate the order of accuracy in time integration with the fixed-stress sequential method. In the study, two-pass and deferred correction methods are to be considered for studying the high-order methods in time integration. We find that there are two different inherent constraint structures that still cause order reductions against high-order accuracy while applying the two methods. As an additional in-depth analysis, we study a large deformation system, considering anisotropic properties for geomechanical and fluid flow parameters, the traverse isotropy and permeability anisotropy ratio. Seeking more accurate solutions, we adopt the total Lagrangian method in geomechanics and multi-point flux approximation in fluid flow. By comparing it to the infinitesimal transformation with two-point flux approximation, we find that substantial differences between the two approaches can exist. For a field application, we study large-scale geomechanics simulation that can honor measured well data, which leads to a constrained geomechanics problem. We employ the Uzawa’s algorithm to solve the saddle point problem from the constrained poromechanics. From numerical parallel simulations, we estimate initial stress distribution in the shale gas reservoir, which will be used for the field development plan. From this study, we find several mathematical conditions for numerically stable and accurate solution of poromechanics problems, when we take the various mixed formulations. By considering the conditions, we can overcome the numerical issues. Then, reliable and precise prediction of reservoir behavior can be obtained for coupled flow-geomechanics problems

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above

Summary of Research 2000, Department of Mechanical Engineering

The views expressed in this report are those of the authors and do not reflect the official policy or position of the Department of Defense or U.S. Government.This report contains project summaries of the research projects in the Department of Mechanical Engineering. A list of recent publications is also included, which consists of conference presentations and publications, books, contributions to books, published journal papers, and technical reports. Thesis abstracts of students advised by faculty in the Department are also included

Coupled deformation, fluid flow and fracture propagation in porous media

Polygonal faults are non-tectonic fault systems which are layer-bound (at some vertical scale) and are widely developed in fine-grained sedimentary basins. Although several qualitative mechanisms have been hypothesised to explain the formation of these faults, there is a weak general consensus that they are formed by the coupled deformation and fluid expulsion of the host sediments (consolidation). This thesis presents a numerical framework that can be extended to investigate the role consolidation plays in the development and evolution of these faults. The method is also applicable to reservoir engineering and CO2 storage. An understanding of the coupled mechanical response and fluid flow is critical in determining compaction and subsidence in oil reservoirs and fault-seal integrity during CO2 disposal and storage. The technique uses a fracture mapping approach (FM) and the extended finite element method (XFEM) to modify the single phase FEM consolidation formulation. A key feature of FM-XFEM is its ability to include discontinuities into a model independently of the computational mesh. The fracture mapping approach is used to simulate the flow interaction between the matrix and existing fractures via a transfer function. Since fractures are represented using level set data, the need for complex meshing to describe fractures is not required. The XFEM component of the method simulates the influence of the pore fluid on the mechanical behaviour of the fractured medium. In XFEM, enrichment functions are added to the standard finite element approximation to ensure an accurate approximation of discontinuous fields within the simulation domain. FM-XFEM produces results comparative to the discrete fracture method on relatively coarse meshes. FM-XFEM has also been extended to model the propagation of existing fractures using a mixed-mode criterion applicable to geological media. Stress concentrations at the tips of existing fractures show good agreement with an analytical solution found in literature