A subcell-enriched Galerkin method for advection problems

In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard convergence analysis for both meshes. By allowing an arbitrary degree of enrichment on both, the coarse and the fine mesh (also including the case of no enrichment), our analysis technique is very general in the sense that our results cover the range from the standard continuous finite element method to the standard discontinuous Galerkin (DG) method with (or without) local subcell enrichment. Numerical experiments confirm our analytical results and indicate good robustness of the proposed method

The parallel finite element system M++ with integrated multilevel preconditioning and multilevel Monte Carlo methods

We present a parallel data structure for the discretization of partial differential equations which is based on distributed point objects and which enables the flexible, transparent, and efficient realization of conforming, nonconforming, and mixed finite elements. This concepts is realized for elliptic, parabolic and hyperbolic model problems, and sample applications are provided by a tutorial complementing a lecture on scientific computing. The corresponding open-source software is based on this parallel data structure, and it supports multilevel methods on nested meshes and 2D and 3D as well as in space-time. Here, we present generic results on porous media applications including multilevel preconditioning and multilevel Monte Carlo methods for uncertainty quantification

Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media

A conservative flux postprocessing algorithm is presented for both steady-state and dynamic flow models. The postprocessed flux is shown to have the same convergence order as the original flux. An arbitrary flux approximation is projected into a conservative subspace by adding a piecewise constant correction that is minimized in a weighted $L^2$ norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard $L^2$ norm which has been considered in earlier works. We also study the effect of different flux calculations on the domain boundary. In particular we consider the continuous Galerkin finite element method for solving Darcy flow and couple it with a discontinuous Galerkin finite element method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table

Vers des méthodes immerées generalisées:une approche Shifted Boundary P1 avec des flux d'ordre pour les équations de Darcy

In this paper, we propose to extend the recent embedded boundary method known as "shifted boundary method" to the Darcy flow problems. The aim is to provide an improved formulation that would give, using linear approximation, at least second order accuracy on bothflux and pressure variables, for any kind of boundary condition, considering embedded simulations.The strategy adopted here is to enrich the approximation of the pressure using Taylor expansionsalong the edges. The objective of this enrichment is to give a quadratic shape to the pressure. The resulted scheme provides high order accuracy on both variables for embedded simulations with an overall second order accuracy, that is bumped to third order for the pressure when only Dirichletboundaries are embedded

Enriched mixed finite element models for dynamic analysis of continuous and fractured porous media

The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.cma.2018.08.011 © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/Enriched Finite Element Models are presented to more accurately investigate the transient and wave propagation responses of continuous and fractured porous media based on mixture theory. Firstly, the Generalized Finite Element Method (GFEM) trigonometric enrichments are introduced to suppress the spurious oscillations that may appear in dynamic analysis with the regular Finite Element Method (FEM) due to numerical dispersion/Gibbs phenomenon. Secondly, the Phantom Node Method (PNM) is employed to model multiple arbitrary fractures independently of the mesh topology. Thirdly, frictional contact behavior is simulated using an Augmented Lagrange Multiplier technique. Mixed Lagrangian interpolants, bi-quadratic for displacements and bi-linear for pore pressure, are used for the underlying FEM basis functions. Transient (non-wave propagation) response of fractured porous media is effectively modeled using the PNM. Wave propagation in continuous porous media is effectively modeled using the mixed GFEM. Wave propagation in fractured porous media is simulated using a mixed GFEM-enriched Phantom Node Method (PNM-GFEM-M). The developed mixed GFEM portion of the model is verified through a transient consolidation problem. Subsequently, the ability of the enriched FEM models to capture the dynamic response of fractured fully-saturated porous media under mechanical and hydraulic stimulations is illustrated. The superior ability of the PNM-GFEM-M in inhibiting spurious oscillations is shown in comparison against the regular finite element solutions of some impact problems. It is demonstrated that by embedding appropriate enrichment basis functions in both displacement and pore pressure fields the results obtained are more accurate than those obtained using standard finite element approximations or approximations in which only the displacement is enriched.Natural Sciences and Engineering Research Council of Canad

Simulation of Hydraulic Stimulation: Acoustic Wave Emission in Fractured Porous Media Using Local and Global Partition-of-Unity Finite Element

Hydraulic Fracturing (HF) is an effective stimulation process for extracting oil and gas from unconventional low-permeable reservoirs. The process is conducted by injecting high-pressure fluids into the ground to generate fracture networks in rock masses and stimulate natural fractures to increase the permeability of formation and extract oil and gas. Due to the multiple and coupled-physics involved, hydraulic fracturing is a complex engineering process. The extent of the induced fractures and stimulated volume and reactivation of natural faults and fractures are some of the practical issues associated with hydraulic fracturing. Acoustic Emission (AE) monitoring and analysis are used to probe the behaviour of solid materials in such applications. The process of elastic wave propagation induced by an abrupt local release of stored strain energy is known as acoustic, microseismic, and seismic emission (depending on the context and the magnitude of the event). These emissions can be triggered by material bifurcation-instabilities like slope slipping, fault-reactivation, pore collapsing, and cracking - processes that are all categorized as localization phenomena. The microseismic monitoring industry attempts to relate acoustic emissions measured by geophones to the nature of the stimulated volume created during hydraulic fracturing. This process is full of uncertainties and researchers have not yet focused on both explicitly modeling the process of fracture reactivation and the accurate simulation of acoustic wave propagations resulting from the localization. The biggest gap in the modeling literature is that most of the previous works fail to accurately simulate the process of transient acoustic wave propagation through the fractured porous media following the elastic energy release. Instead of explicitly modeling fracturing and acoustic emission, most previous studies have aimed to relate energy release to seismic moment. To overcome some of the existing shortcomings in the numerical modeling of the coupled problem of interface localization-acoustic emission, this thesis is focused on developing new computational methods and programs for the simulation of microseismic wave emissions induced by interface slip instability in fractured porous media. As a coupled nonlinear mixed multi-physics problem, simulation of hydraulic stimulation involves several mathematical and computational complexities and difficulties in terms of modeling, stability, and convergence, such as the inf-sup stability problems that arise from mixed formulations due to the hydro-mechanical couplings and contact conditions. In AE modeling, due to the high-frequency transient nature of the problem, additional numerical problems emerging from the Gibbs phenomenon and artificial period elongation and amplitude decay are also involved. The thesis has three main objectives. The first objective is to develop a numerical model for simulation of wave propagation in discontinuous media, which is fulfilled in Chapter 2 of the thesis. In this chapter a new enriched finite element method is developed for simulation of wave propagation in fractured media. The method combines the advantages of the global Partition-of-Unity Method (PUM) with harmonic enrichment functions via the Generalized Finite Element Method (GFEM) with the local PUM via the Phantom Node Method (PNM). The GFEM enrichments suppress the spurious oscillations that can appear in regular Finite Element Method (FEM) analysis of dynamic/wave propagations due to numerical dispersions and Gibbs phenomenon. The PNM models arbitrary fractures independently of the original mesh. Through several numerical examples it has been demonstrated that the spurious oscillations that appear in propagation pattern of high-frequency waves in PNM simulations can be effectively suppressed by employing the enriched model. This is observed to be especially important in fractured media where both primary waves and the secondary reflected waves are present. The second objective of the thesis is to develop a mixed numerical model for simulation of wave propagation in discontinuous porous media and interface modeling. This objective is realized in Chapter 3 of the thesis. In this chapter, a new enriched mixed finite element model is introduced for simulation of wave propagation in fractured porous media, based on an extension of the developed numerical method in Chapter 2. Moreover, frictional contact at interfaces is modeled and realized using an augmented Lagrange multiplier scheme. Through various numerical examples, the effectiveness of the developed enriched FE model over conventional approaches is demonstrated. Moreover, it is shown that the most accurate wave results with the least amount of spurious oscillations are achieved when both the displacement and pore pressure fields are enriched with appropriate trigonometric functions. The third objective of the thesis is to develop computational models for the simulation of acoustic emissions induced by fracture reactivation and shear slip. This objective is realized in Chapter 4 of the thesis. In this chapter, an enriched mixed finite element model (introduced in Chapter 3) is developed to simulate the interface slip instability and the associated induced acoustic wave propagation processes, concurrently. Acoustic events are triggered through a sudden release of strain energy at the fracture interfaces due to shear slip instability. The shear slip is induced via hydraulic stimulation that switches the interface behaviour from a stick to slip condition. The superior capability of the proposed enriched mixed finite element model (i.e., PNM-GFEM-M) in comparison with regular finite element models in inhibiting the spurious oscillations and numerical dispersions of acoustic signals in both velocity and pore pressure fields is demonstrated through several numerical studies. Moreover, the effects of different characteristics of the system, such as permeability, viscous damping, and friction coefficient at the interface are investigated in various examples