Simulation of pore-scale flow using finite element-methods

I present a new finite element (FE) simulation method to simulate pore-scale flow. Within the pore-space, I solve a simplified form of the incompressible Navier-Stoke’s equation, yielding the velocity field in a two-step solution approach. First, Poisson’s equation is solved with homogeneous boundary conditions, and then the pore pressure is computed and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field I estimate the effective permeability of porous media samples characterized by thin section micrographs, micro-CT scans and synthetically generated grain packings. This two-step process is much simpler than solving the full Navier Stokes equation and therefore provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier-Stoke’s equation. My numerical model is verified with an analytical solution and validated on samples whose permeabilities and porosities had been measured in laboratory experiments (Akanji and Matthai, 2010). Comparisons were also made with Stokes solver, published experimental, approximate and exact permeability data. Starting with a numerically constructed synthetic grain packings, I also investigated the extent to which the details of pore micro-structure affect the hydraulic permeability (Garcia et al., 2009). I then estimate the hydraulic anisotropy of unconsolidated granular packings. With the future aim to simulate multiphase flow within the pore-space, I also compute the radii and derive capillary pressure from the Young-Laplace equation (Akanji and Matthai,2010

Synthetic modelling study of marine controlled-source electromagnetic data for hydrocarbon exploration

The marine controlled-source electromagnetic method (CSEM) is a geophysical technique for mapping subsurface electrical resistivity structure in the offshore environment. It has gained ground in recent years as a tool for remote detection and mapping of hydrocarbon reservoirs as it serves as an independent yet complementary method to seismic acquisition. While CSEM data contains useful information about the subsurface, modelling and inversion are required to convert data into interpretable resistivity images. Improvement of modelling tools will assist in closing the gap between acquisition and interpretation of CSEM data. The primary focus of this study was to explore the limits of our present modelling capabilities in the context of marine electromagnetic scenarios. Software based on the three-dimensional CSEM finite-element forward code CSEM3DFWD (Ansari and Farquharson, 2014; Ansari et al., 2015) was employed in this study. While testing of this software had been expanded to models of relevance to mineral exploration, its performance for models which are representative of marine geologic environments, in particular those which are encountered in offshore oil and gas exploration, had not yet been investigated. In this study, marine models of increasing complexity were built and tested, with the ultimate goal of synthesizing marine CSEM data for three-dimensional earth models which were complete in their description of the subsurface. Computed responses were compared to results existing in the literature, when available. To investigate the capability of the code in modelling realistic scenarios, forward solutions were computed for a marine reservoir model based on the real-life North Amethyst oil field, located in the Jeanne d’Arc Basin, offshore Newfoundland. When the capability of modelling realistic earth models is fully realized, forward modelling may be used to assess the utility of the marine CSEM method as a tool for hydrocarbon detection and delineation in specific offshore scenarios

Failure processes in soft and quasi-brittle materials with nonhomogeneous microstructures

Material failure pervades the fields of materials science and engineering; it occurs at various scales and in various contexts. Understanding the mechanisms by which a material fails can lead to advancements in the way we design and build the world around us. For example, in structural engineering, understanding the fracture of concrete and steel can lead to improved structural systems and safer designs; in geological engineering, understanding the fracture of rock can lead to increased efficiency in oil and gas extraction; and in biological engineering, understanding the fracture of bone can lead to improvements in the design of bio-composites and medical implants. In this thesis, we numerically investigate a wide spectrum of failure behavior; in soft and quasi-brittle materials with nonhomogeneous microstructures considering a statistical distribution of material properties. The first topic we investigate considers the influence of interfacial interactions on the macroscopic constitutive response of particle reinforced elastomers. When a particle is embedded into an elastomer, the polymer chains in the elastomer tend to adsorb (or anchor) onto the surface of the particle; creating a region in the vicinity of each particle (often referred to as an interphase) with distinct properties from those in the bulk elastomer. This interphasial region has been known to exist for many decades, but is primarily omitted in computational investigations of such composites. In this thesis, we present an investigation into the influence of interphases on the macroscopic constitutive response of particle filled elastomers undergoing large deformations. In addition, at large deformations, a localized region of failure tends to accumulate around inclusions. To capture this localized region of failure (often referred to as interfacial debonding), we use cohesive zone elements which follow the Park-Paulino-Roesler traction-separation relation. To account for friction, we present a new, coupled cohesive-friction relation and detail its formulation and implementation. In the process of this investigation, we developed a small library of cohesive elements for use with a commercially available finite element analysis software package. Additionally, in this thesis, we present a series of methods for reducing mesh dependency in two-dimensional dynamic cohesive fracture simulations of quasi-brittle materials. In this setting, cracks are only permitted to propagate along element facets, thus a poorly designed discretization of the problem domain can introduce artifacts into the fracture behavior. To reduce mesh induced artifacts, we consider unstructured polygonal finite elements. A randomly-seeded polygonal mesh leads to an isotropic discretization of the problem domain, which does not bias the direction of crack propagation. However, polygonal meshes tend to limit the possible directions a crack may travel at each node, making this discretization a poor candidate for dynamic cohesive fracture simulations. To alleviate this problem, we propose two new topological operators. The first operator we propose is adaptive element-splitting, and the second is adaptive mesh refinement. Both operators are designed to improve the ability of unstructured polygonal meshes to capture crack patterns in dynamic cohesive fracture simulations. However, we demonstrate that element-splitting is more suited to pervasive fracture problems, whereas, adaptive refinement is more suited to problems exhibiting a dominant crack. Finally, we investigate the use of geometric and constitutive design features to regularize pervasive fragmentation behavior in three-dimensions. Throughout pervasive fracture simulations, many cracks initiate, propagate, branch and coalesce simultaneously. Because of the cohesive element method's unique framework, this behavior can be captured in a regularized manner. In this investigation, unstructuring techniques are used to introduce randomness into a numerical model. The behavior of quasi-brittle materials undergoing pervasive fracture and fragmentation is then examined using three examples. The examples are selected to investigate some of the significant factors influencing pervasive fracture and fragmentation behavior; including, geometric features, loading conditions, and material gradation

Numerical simulation of fracture pattern development and implications for fuid flow

Simulations are instrumental to understanding flow through discrete fracture geometric representations that capture the large-scale permeability structure of fractured porous media. The contribution of this thesis is threefold: an efficient finite-element finite-volume discretisation of the advection/diffusion flow equations, a geomechanical fracture propagation algorithm to create fractured rock analogues, and a study of the effect of growth on hydraulic conductivity. We describe an iterative geomechanics-based finite-element model to simulate quasi-static crack propagation in a linear elastic matrix from an initial set of random flaws. The cornerstones are a failure and propagation criterion as well as a geometric kernel for dynamic shape housekeeping and automatic remeshing. Two-dimensional patterns exhibit connectivity, spacing, and density distributions reproducing en echelon crack linkage, tip hooking, and polygonal shrinkage forms. Differential stresses at the boundaries yield fracture curving. A stress field study shows that curvature can be suppressed by layer interaction effects. Our method is appropriate to model layered media where interaction with neighbouring layers does not dominate deformation. Geomechanically generated fracture patterns are the input to single-phase flow simulations through fractures and matrix. Thus, results are applicable to fractured porous media in addition to crystalline rocks. Stress state and deformation history control emergent local fracture apertures. Results depend on the number of initial flaws, their initial random distribution, and the permeability of the matrix. Straightpath fracture pattern simplifications yield a lower effective permeability in comparison to their curved counterparts. Fixed apertures overestimate the conductivity of the rock by up to six orders of magnitude. Local sample percolation effects are representative of the entire model flow behaviour for geomechanical apertures. Effective permeability in fracture dataset subregions are higher than the overall conductivity of the system. The presented methodology captures emerging patterns due to evolving geometric and flow properties essential to the realistic simulation of subsurface processes

Nondifferentiable energy minimization for cohesive fracture in a discontinuous Galerkin finite element framework

Until recently, most works on the computational modelling of fracture relied on a Newtonian mechanics approach, i.e., momentum balance equations describing the motion of the body along with fracture criteria describing the evolution of fractures. Robustness issues associated with this approach have been identified in the previous literature, several of which, as this thesis shows, due to the discontinuous dependence of stress field on the deformation field at the time of insertion of displacement discontinuities. Lack of continuity limits applicability of the models and undermines reliability of the numerical solutions. In particular, solutions often show non-convergent behaviour with time step refinement and exhibit nonphysical velocity fields and crack activation patterns. In addition, implicit time-stepping schemes, which are favoured in quasi-static and low-velocity problems, are challenging in such models. This is not a coincidence but a manifestation of algorithmic pitfalls of such methods. Continuity of stresses is in general hard to achieve in a computational model that employs a crack initiation criterion. Energy (variational) approaches to fracture have gained increased popularity in recent years. An energy approach has been shown to avoid introduction of a crack initiation criterion. The central idea of this model is the minimization of a mechanical energy functional, whose term representing the energy due to the cracks is a nondifferentiable function of the interface openings at zero opening displacement. A consequence of this formulation is that crack initiation happens automatically as a by-product of energy minimization. This avoids the complexities arising from the introduction of an extrinsic activation criterion but entails minimization of a nondifferentiable potential. The aim of this research is to develop robust and efficient computational algorithms for numerical implementation of the energy approach to cohesive fracture. Two computational algorithms have been proposed in a discontinuous Galerkin finite element framework, including a continuation algorithm which entails successive smooth approximations of the nondifferentiable functional and a block coordinate descent algorithm which uses generalized differential calculus for the treatment of nondifferentiability. These methods allow for a seamless transition from the uncracked to the cracked state, making possible the use of iterative solvers with implicit time-stepping, and completely sidestepping robustness issues of previous computational frameworks. A critical component of this work is validation of the robustness of the proposed numerical methods. Various numerical simulations are presented including time step and mesh size convergence studies and qualitative and quantitative comparison of simulations with experimental observations and theoretical findings. In addition, an energy-based hydro-mechanical model and computational algorithm is presented for hydraulic fracturing in impermeable media, which shows the crucial importance of continuity in multi-physics modelling. A search algorithm is developed on the basis of graph theory to identify the set of fluid-pressurized cracks among cracks in naturally fractured domains

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

Micromechanical Study of Rock Fracture and Fragmentation under Dynamic Loads using Discrete Element Method

The study presented in this thesis aims to numerically explore the micro-mechanisms underlying rock fracture and fragmentation under dynamic loading. The approach adopted is based on the Discrete Element Method (DEM) coupled to the Cohesive Process Zone (CPZ) theory. It assumes rock material as assemblage of irregular-sized deformable fragments joining together at their cohesive boundaries. The simulation, which is referred to as Cohesive Fragment Model (CFM), takes advantage of DEM particle/contact logic to handle the fragments and boundaries in between. In this idealization, mechanical properties of particle and more dominantly those of contact control macroscopic response of the particle assemblage. A rate-dependent orthotropic cohesive law is developed for DEM contacts to capture rock material specific features, e.g. brittleness, anisotropy and rate-dependency. Rock experimental behavior is then modeled in order to assess individually the sensitivity of results to grain size, confining pressure, micromechanical parameters, stored strain energy, loading rate etc. The thesis is organized to approach the problem systematically. First, CFM application for static analysis is examined. It is shown that CFM quantitatively and qualitatively predicts compressive and tensile failure of hard and soft rocks as well as shear strength, dilatancy and degradation of rough rock joints. CFM micro-parameters, i.e., stiffness of particle and strength, stiffness, and friction of contact are calibrated using a combination of statistical disciplines and original closed-form expressions. The calibration process provides useful physical interpretation for each micro-parameter in terms of standard rock mechanical properties. These interpretations enable to understand how macroscopic behavior of rock material originates from its mineral microstructure. Energy needed to fully open a contact, the contact energy numerically represents material fracture energy in CFM. Experimental investigations suggest that fracture energy is independent of loading rate in quasi-static circumstances. Thus, contact energy is simply assumed as constant in static analysis. However, simulation on fast fracturing by CFM warns that this assumption causes serious deviations in fracture dynamic analysis. Laboratory observations reveal that fast-moving fracture consumes more energy than slow-moving one does. This inspires to consider contact energy as variable and rate-dependent to provide the model with the appropriate prediction of the fracture energy release process. Applying this new approach, fracture behavior of PMMA plates is investigated under different levels of stored strain energy. As the final stage, dynamic fracture toughness of rock samples, measured by the split-Hopkinson pressure bar test, is simulated and promising results are obtained. They demonstrate how numerical modeling can practically aid experimental methods in terms of measurement verification, error estimation, and performing appropriate corrections. The studies suggest that DEM is an effective and convenient tool to investigate fracture and fragmentation problems. While predictions by continuum models are restricted only to crack initiation, simulation by DEM made it possible to track both the initiation and progression of fracture over time by following consecutive damage of contacts. Moreover, the research specifically demonstrates that the proposed contact model properly predicts the experimental behavior of rock fracture under static and dynamic loading. This result verifies the model validity and adequacy for rock fracture analysis

Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method

A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions. Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results