Structure-Preserving High Performance Computational Methods for Transport in Porous Media

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This dissertation aims at developing a novel and robust high performing paral- lel computational framework that can enhance the current predictive capabilities of numerical methods for large-scale subsurface transport applications. With increasing capacity and complexity of processors and memory systems, the need for improving the performance of subsurface flow and transport simulations has become an area of active research. Two of the most well known numerical deficiencies that the pop- ular formulations and existing simulation packages suffer from are the inability to meet the non-negative constraint under anisotropic diffusion (i.e., they violate dis- crete maximum principles) and the inability of the standard finite element formulation to ensure local mass balance. Moreover, there is no platform-agnostic performance model that can simultaneously document both the hardware/architectural and algo- rithmic efficiencies of any numerical method or software package that is sensitive to memory-bandwidth limitations. Several existing parallel scientific libraries, such as the Portable and Extensible Toolkit for Scientific Computations (PETSc) and the Massively Parallel Reactive Flow and Transport (PFLOTRAN) libraries, have been developed to help predict subsurface phenomena and are used by many of today’s leading hydrologists and geophysicists alike, but till date, the aforementioned con- cerns have not yet been resolved. Solutions to these numerical deficiencies have been proposed in literature but they do not address them concurrently in a high perfor- mance computing setting. Without a performance model, it is intractable to deter- mine whether proposed modifications to these subsurface software packages would be fast, scalable, or efficient. The objective of this dissertation is to present a computational framework that preserves important properties like local mass balance and positivity that can per- form at a high level. Performance tools and methodologies are presented to guide users on how to understand the performance of such frameworks. This dissertation comprises of three sections. First, we present a conceptual performance spectrum that covers time-to-solution, arithmetic intensity, and equations solved per second for any parallel computational framework that solves partial differential equations. As proof of concept, this spectrum is utilized on a wide variety of state-of-the-art sci- entific libraries and multi-grid solvers like the FEniCS/Firedrake Projects, HYPRE, and Trilinos. It is shown that this spectrum can augment one’s ability to understand both the hardware and algorithmic efficiencies of popular numerical techniques like the finite element method. Second, we propose an optimization-based computational framework that can ensure variationally consistent non-negative concentrations for large-scale anisotropic diffusion problems like Chromium remediation in the Sandia Canyon. The predicted computational performance of the proposed framework is based on a “perfect-cache” roofline model, loosely based on concepts originating from the performance spectrum, and it is shown that this roofline model can be used to predict how well the optimization-based framework can strong-scale. Third, we ex- tend the proposed computational framework to solve advection-diffusion equations by employing the variational inequality solver. We also enforce local/element-wise mass conservation by discretizing the advection-diffusion equation using the Discontinuous Galerkin finite element method. Our numerical experiments demonstrate that the proposed variational inequality approach conserves local mass balance, ensures non- negative concentration fields, and can accurately model large-scale and non-linear coupled flow and transport phenomena such as the miscible displacement of oil in a heterogeneous reservoir.

High performance computing, Parallel computing, Linear algebra, Finite element method, Optimization, Subsurface Flow and Transport, Advection-diffusion equations
Portions of this document appear in: Chang, Justin, Satish Karra, and Kalyana B. Nakshatrala. "Large-scale optimization-based non-negative computational framework for diffusion equations: Parallel implementation and performance studies." Journal of Scientific Computing 70, no. 1 (2017): 243-271. And in: Chang, J., and K. B. Nakshatrala. "Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations." Computer Methods in Applied Mechanics and Engineering 320 (2017): 287-334.