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

Finite element discretizations for variable-order fractional diffusion problems

We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded domains. A number of challenges are encountered when discretizing these equations. The first comes from the heterogeneous kernel singularity in the fractional integral operator. The second comes from the dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. An additional challenge comes from the need to handle volume conditions-the generalization of classical local boundary conditions to the nonlocal setting. Satisfying these conditions requires that the effect of the whole domain, including both the interior and exterior regions, can be computed on every interior point in the discretization. Performed directly, this would result in quadratic complexity. To address these challenges, we propose a strategy that decomposes the stiffness matrix into three components. The first is a sparse matrix that handles the singular near-field separately and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far field and is approximated by a hierarchical $\mathcal{H}^{2}$ matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. Analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.Comment: 33 pages, 11 figure

Emerging Developments in Interfaces and Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a well established and flourishing area of research. This workshop focused on recent developments and emerging new themes. By bringing together experts in these fields we achieved progress in open questions and developed novel research directions in mathematics related to interfaces and free boundaries. This interdisciplinary workshop brought together researchers from distinct mathematical fields such as analysis, computation, optimisation and modelling to discuss emerging challenges

Innovative Approaches to the Numerical Approximation of PDEs

This workshop was about the numerical solution of PDEs for which classical approaches, such as the finite element method, are not well suited or need further (theoretical) underpinnings. A prominent example of PDEs for which classical methods are not well suited are PDEs posed in high space dimensions. New results on low rank tensor approximation for those problems were presented. Other presentations dealt with regularity of PDEs, the numerical solution of PDEs on surfaces, PDEs of fractional order, numerical solvers for PDEs that converge with exponential rates, and the application of deep neural networks for solving PDEs

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

Modeling and Simulation in Engineering

The general aim of this book is to present selected chapters of the following types: chapters with more focus on modeling with some necessary simulation details and chapters with less focus on modeling but with more simulation details. This book contains eleven chapters divided into two sections: Modeling in Continuum Mechanics and Modeling in Electronics and Engineering. We hope our book entitled "Modeling and Simulation in Engineering - Selected Problems" will serve as a useful reference to students, scientists, and engineers