Hector Klie

A parallel, implicit, cell‐centered method for two‐phase flow with a preconditioned Newton–Krylov solver

A new parallel solution technique is developed for the fully implicit three‐dimensional two‐phase flow model. An expandedcell‐centered finite difference scheme which allows for a full permeability tensor is employed for the spatial discretization, and backwardEuler is used for the time discretization. The discrete systems are solved using a novel inexact Newton method that reuses the Krylov information generated by the GMRES linear iterative solver. Fast nonlinear convergence can be achieved by composing inexact Newton steps with quasi‐Newton steps restricted to the underlying Krylov subspace. Furthermore, robustness and efficiency are achieved with a line‐search backtracking globalization strategy for the nonlinear systems and a preconditioner for each coupled linear system to be solved. This inexact Newton method also makes use of forcing terms suggested by Eisenstat and Walker which prevent oversolving of the Jacobian systems. The preconditioner is a new two‐stage method which involves a decoupling strategy plus the separate solutions of both nonwetting‐phase pressure and saturation equations. Numerical results show that these nonlinear and linear solvers are very effective.

An Autonomic Reservoir Framework for the Stochastic Optimization of Well Placement

The adequate location of wells in oil and environmental applications has a significant economic impact on reservoir management. However, the determination of optimal well locations is both challenging and computationally expensive. The overall goal of this research is to use the emerging Grid infrastructure to realize an autonomic self-optimizing reservoir framework. In this paper, we present a policy-driven peer-to-peer Grid middleware substrate to enable the use of the Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm, coupled with the Integrated Parallel Accurate Reservoir Simulator (IPARS) and an economic model to find the optimal solution for the well placement problem.

Algebraic Multigrid Methods (AMG) for the Efficient Solution of Fully Implicit Formulations in Reservoir Simulation

A primary challenge for a new generation of reservoir simulators is the accurate description of multiphase flow in highly heterogeneous media and very complex geometries. However, many initiatives in this direction have encountered difficulties in that current solver technology is still insufficient to account for the increasing complexity of coupled linear systems arising in fully implicit formulations. In this respect, a few works have made particular progress in partially exploiting the physics of the problem in the form of two-stage preconditioners. Two-stage preconditioners are based on the idea that coupled system solutions are mainly determined by the solution of their elliptic components (i.e., pressure). Thus, the procedure consists of extracting and accurately solving pressure subsystems. Residuals associated with this solution are corrected with an additional preconditioning step that recovers part of the global information contained in the original system. Optimized and highly complex hierarchical methods such as algebraic multigrid (AMG) offer an efficient alternative for solving linear systems that show a discretely elliptic nature. When applicable, the major advantage of AMG is its numerical scalability; that is, the numerical work required to solve a given type of matrix problem grows only linearly with the number of variables. Consequently, interest in incorporating AMG methods as basic linear solvers in industrial oil reservoir simulation codes has been steadily increasing for the solution of pressure blocks. Generally, however, the preconditioner influences the properties of the pressure block to some extent by performing certain algebraic manipulations. Often, the modified pressure blocks are “less favorable” for an efficient treatment by AMG. In this work, we discuss strategies for solving the fully implicit systems that preserve (or generate) the desired ellipticity property required by AMG methods. Additionally, we introduce an iterative coupling scheme as an alternative to fully implicit formulations that is faster and also amenable for AMG implementations. Hence, we demonstrate that our AMG implementation can be applied to efficiently deal with the mixed elliptic-hyperbolic character of these problems. Numerical experiments reveal that the proposed methodology is promising for solving large-scale, complex reservoir problems.

Solving Sparse Linear Systems on NVIDIA Tesla GPUs

Current many-core GPUs have enormous processing power, and unlocking this power for general-purpose computing is very attractive due to their low cost and efficient power utilization. However, the fine-grained parallelism and the stream-programming model supported by these GPUs require a paradigm shift, especially for algorithm designers. In this paper we present the design of a GPU-based sparse linear solver using the Generalized Minimum RESidual (GMRES) algorithm in the CUDA programming environment. Our implementation achieved a speedup of over 20x on the Tesla T10P based GTX280 GPU card for benchmarks with from a few thousands to a few millions unknowns.

Application of Grid-enabled technologies for solving optimization problems in data-driven reservoir studies

This paper presents the use of numerical simulations coupled with optimization techniques in oil reservoir modeling and production optimization. We describe three main components of an autonomic oil production management framework. The framework implements a dynamic, data-driven approach and enables Grid-based large scale optimization formulations in reservoir modeling.

Studies of Robust Two Stage Preconditioners for the Solution of Fully Implicit Multiphase Flow Problems

The solution of the linear system of equations for a large scale reservoir simulation has several challenges. Preconditioners are used to speed up the convergence rate of the solution of such systems. In theory, a preconditioner defines a matrix M that can be inexpensively inverted and represents a good approximation of a given matrix A. In this work, two-stage preconditioners consisting of the approximated inverses M1 and M2 are investigated for multiphase flow in porous media. The first-stage preconditioner, M1, is approximated from A using four different solution methods: (1) constrained pressure residuals (CPR), (2) lower block Gauss-Seidel, (3) upper block Gauss-Seidel, and (4) one iteration of block Gauss-Seidel. The pressure block solution in each of these different schemes is calculated using the Algebraic Multi Grid (AMG) method. The inverse of the saturation (or more generally, the nonpressure) blocks are approximated using Line Successive Over Relaxation (LSOR). The second stage preconditioner, M2, is a global preconditioner based on LSOR iterations for the matrix A that captures part of the original interaction of different coefficient blocks. Several techniques are also employed to weaken the coupling between the pressure block and the nonpressure blocks. Effective decoupling is achieved by: (1) an IMPES-like approach designed to preserve the integrity of pressure coefficients, (2) Householder transformations, (3) the alternate block factorization (ABF), and (4) the balanced decoupling strategy (BDS) based on least squares. The fourth method is a new technique developed in this work. The aforementioned preconditioning techniques were implemented in a parallel reservoir simulation environment, and tested for large-scale two-phase and three-phase black oil simulation models. This study demonstrates that a two-stage preconditioner based on balanced decoupling strategy (BDS) or ABF combined with Gauss-Seidel sweeps, that also incorporate nonpressure solutions for M, delivers both the fastest convergence rate and the most robust option overall without compromising parallel scalability.

Towards dynamic data-driven management of the ruby gulch waste repository

Previous work in the Instrumented Oil-Field DDDAS project has enabled a new generation of data-driven, interactive and dynamically adaptive strategies for subsurface characterization and oil reservoir management. This work has led to the implementation of advanced multi-physics, multi-scale, and multi-block numerical models and an autonomic software stack for DDDAS applications. The stack implements a Grid-based adaptive execution engine, distributed data management services for real-time data access, exploration, and coupling, and self-managing middleware services for seamless discovery and composition of components, services, and data on the Grid. This paper investigates how these solutions can be leveraged and applied to address another DDDAS application of strategic importance – the data-driven management of Ruby Gulch Waste Repository.

Models, methods and middleware for grid-enabled multiphysics oil reservoir management

Meeting the demands for energy entails a better understanding and characterization of the fundamental processes of reservoirs and of how human made objects affect these systems. The need to perform extensive reservoir studies for either uncertainty assessment or optimal exploitation plans brings up demands of computing power and data management in a more pervasive way. This work focuses on high performance numerical methods, tools and grid-enabled middleware systems for scalable and data-driven computations for multiphysics simulation and decision-making processes in integrated multiphase flow applications. The proposed suite of tools and systems consists of (1) a scalable reservoir simulator, (2) novel stochastic optimization algorithms, (3) decentralized autonomic grid middleware tools, and (4) middleware systems for large-scale data storage, querying, and retrieval. The aforementioned components offer enormous potential for performing data-driven studies and efficient execution of complex, large-scale reservoir models in a collaborative environment.

Deflation AMG Solvers for Highly Ill-Conditioned Reservoir Simulation Problems

In recent years, deflation methods have received increasingly particular attention as a means to improving the convergence of linear iterative solvers. This is due to the fact that deflation operators provide a way to remove the negative effect that extreme (usually small) eigenvalues have on the convergence of Krylov iterative methods for solving general symmetric and non-symmetric systems. In this work, we use deflation methods to extend the capabilities of algebraic multigrid (AMG) for handling highly non-symmetric and indefinite problems, such as those arising in fully implicit formulations of multiphase flow in porous media. The idea is to ensure that components of the solution that remain unresolved by AMG (due to the coupling of roughness and indefiniteness introduced by different block coefficients) are removed from the problem. This translates to a constraint to the AMG iteration matrix spectrum within the unit circle to achieve convergence. This approach interweaves AMG (V, W or V-W) cycles with deflation steps that are computable either from the underlying Krylov basis produced by the GMRES accelerator (Krylov-based deflation) or from the reservoir decomposition given by high property contrasts (domain-based deflation). This work represents an efficient extension to the Generalized Global Basis (GGB) method that was recently proposed for the solution of the elastic wave equation with geometric multigrid and an out-of-core computation of eigenvalues. Hence, the present approach offers the possibility of applying AMG to more general large-scale reservoir settings without further modifications to the AMG implementation or algebraic manipulation of the linear system (as suggested by two-stage preconditioning methods). Promising results are supported by a suite of numerical experiments with extreme permeability contrasts.

A Timestepping Scheme for Coupled Reservoir Flow and Geomechanics on Nonmatching Grids

This paper presents numerical techniques for coupled simulations with different time scales and space discretizations for reservoir flow and geomechanics. We use an explicitly coupled approach together with an iterative coupling to increase stability and reduce time discretization error. An error indicator is proposed to determine when displacement must be updated and whether the explicit or iterative coupling technique is required. Under this setting, one geomechanics calculation is performed for several reservoir flow steps. For time steps without geomechanics updates linear extrapolated pore volume is used for porous flow calculations. The resulting algorithm is computationally more efficient than the iterative coupling, and it is more stable and accurate than the loosely coupled technique. In the event that different meshes are used for the reservoir flow and geomechanics models, special treatments are required for the integration of the coupling terms over each element. To avoid complex 3D grid intersection calculations we propose to divide an element into a number of subelements and apply the midpoint integration rule over each subelement. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method for coupled simulations with different time and space discretizations.