Parallel inexact constraint preconditioning for ill-conditioned consolidation problems

Constraint preconditioners have proved very efficient for the solution of ill-conditioned finite element (FE) coupled consolidation problems in a sequential computing environment. Their implementation on parallel computers, however, is not straightforward because of their inherent sequentiality. The present paper describes a novel parallel inexact constraint preconditioner (ParICP) for the efficient solution of linear algebraic systems arising from the FE discretization of the coupled poro-elasticity equations. The ParICP implementation is based on the use of the block factorized sparse approximate inverse incomplete Cholesky preconditioner, which is a very recent and effective development for the parallel preconditioning of symmetric positive definite matrices. The ParICP performance is experimented with in real 3D coupled consolidation problems, proving a scalable and efficient implementation of the constraint preconditioning for high-performance computing. ParICP appears to be a very robust algorithm for solving ill-conditioned large-size coupled models in a parallel computing environment.     

Preconditioners in computational geomechanics: A survey

The finite element (FE) solution of geomechanical problems in realistic settings raises a few numerical issues depending on the actual process addressed by the analysis. There are two basic problems where the linear solver efficiency may play a crucial role: 1. fully coupled consolidation and 2. faulted uncoupled consolidation. A class of general solvers becoming increasingly popular relies on the Krylov subspace (or Conjugate Gradient‐like) methods, provided that an efficient preconditioner is available. For both problems mentioned above, the possible preconditioners include the diagonal scaling (DS), the Incomplete LU decomposition (ILU), the mixed constraint preconditioning (MCP) and the multilevel incomplete factorization (MIF). The development and the performance of these algorithms have been the topic of several recent works. The present paper aims at providing a survey of the preconditioners available to date in computational geomechanics. In particular, a review and a critical discussion of DS, ILU, MCP and MIF are given along with some comparative numerical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

Finite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 John Wiley & Sons, Ltd.     

Cell to node projections: An assessment of error

Reservoir simulators typically use cell‐centered finite volume schemes and do not model directly the coupling of the flow processes with the geomechanics. Coupling of geomechanics with fluid flow can be important in many cases, but introducing fully coupled geomechanical effects in those simulators is not a trivial issue, because the geomechanics is better done by using the Galerkin vertex‐centered finite element methods by which the solid displacements are computed at the vertices of the cells. This creates difficulties in interfacing cell variables with nodal variables. Uncoupled or loosely coupled models are used by many researchers/practitioners by which a reservoir model is coupled to a geomechanical model by staggering in‐time flow and deformation via a sophisticated interface that repeatedly calls first flow and then mechanics. The method therefore requires projection of the reservoir cell variables onto the nodes of the geomechanics Galerkin finite element mesh. In this note, we attempt to quantify the errors associated with cell to node projection operations. For that purpose, we use a simple model of the pressure equation for a heterogeneous medium in one dimension. We are able to derive the exact analytical solution for this problem for both nodal and cell pressures. This allows us to compute the errors due to projection analytically, function of meshing refinement and permeability field variations. We compute upper and lower bounds for the errors, and analyze their magnitude for a variety of cases. We conclude that, in general, cell to node projection operations lead to substantial errors. Copyright © 2010 John Wiley & Sons, Ltd.     

A supercoarsening multigrid method for poroelasticity in 3D coupled flow and geomechanics modeling

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO₂ injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.     

A modified Jacobi preconditioner for solving ill‐conditioned Biot's consolidation equations using symmetric quasi‐minimal residual method

This paper identifies imbalanced columns (or rows) as a significant source of ill‐conditioning in the preconditioned coefficient matrix using the standard Jacobi preconditioner, for finite element solution of Biot's consolidation equations. A simple and heuristic preconditioner is proposed to reduce this source of ill‐conditioning. The proposed preconditioner modifies the standard Jacobi preconditioner by scaling the excess pore pressure degree‐of‐freedoms in the standard Jacobi preconditioner with appropriate factors. The performance of such preconditioner is examined using the symmetric quasi‐minimal residual method. To alleviate storage requirements, element‐by‐element iterative strategies are implemented. Numerical experiment results show that the proposed preconditioner reduces both the number of iteration and CPU execution time significantly as compared with the standard Jacobi preconditioner. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Accelerating strategies to the numerical simulation of large‐scale models for sequential excavation

In this paper, a novel combination of well‐established numerical procedures is explored in order to accelerate the simulation of sequential excavation. Usually, large‐scale models are used to represent these problems. Due to the high number of equations involved, the solver algorithm represents the critical aspect which makes the simulation very time consuming. The mutable nature of the excavation models makes this problem even more pronounced. To accomplish the representation of geometrical and mechanical aspects in an efficient and simple manner, the proposed solution employs the boundary element method with a multiple‐region strategy. Together with this representational system, a segmented storage scheme and a time‐ordered tracking of the changes form an adequate basis for the usage of fast updating methods instead of frontal solvers. The present development employs the Sherman–Morrison–Woodbury method to speed up the calculation due to sequential changes. The efficiency of the proposed framework is illustrated through the simulation of test examples of 2D and 3D models. Copyright © 2006 John Wiley & Sons, Ltd.     

基于粗细网格的有限元并行分析方法

并行计算己成为求解大规模岩土工程问题的一种强大趋势。探讨了粗细网格与预处理共轭梯度法结合的并行有限元算法。从多重网格刚度矩阵推得有效的预处理子。该算法在工作站机群上实现。用地基处理时土体强夯的数值模拟分析进行了数值测试,对其并行性能进行了详细分析。计算结果表明：该算法具有良好的并行加速比和效率,是一种有效的并行算法。     

Block-preconditioned Newton–Krylov solvers for fully coupled flow and geomechanics

The focus of this work is efficient solution methods for mixed finite element models of variably saturated fluid flow through deformable porous media. In particular, we examine preconditioning techniques to accelerate the convergence of implicit Newton–Krylov solvers. We highlight an approach in which preconditioners are built from block-factorizations of the coupled system. The key result of the work is the identification of effective preconditioners for the various sub-problems that appear within the block decomposition. We use numerical examples drawn from both linear and nonlinear hydromechanical models to test the robustness and scalability of the proposed methods. Results demonstrate that an algebraic multigrid variant of the block preconditioner leads to mesh-independent convergence, good parallel efficiency, and insensitivity to the material parameters of the medium.     

Dynamic stress analysis of a class of geomechanics problems by the boundary element method

This paper presents the application of an advanced BEM for periodic and transient dynamic stress analyses of a class of geomechanics problems. For transient dynamic analysis, the problem is first solved in the Laplace transform space, which happens to be similar to the periodic dynamic analysis, and then the time domain solution is obtained by numerical inversion of transform domain solutions. The numerical implementation of the BEM used to present the results in this paper is complete and most general available to date. It is capable of treating very large, multi-layered problems by substructuring and satisfying the equilibrium and compatibilities at the interfaces. With the help of this substructuring, capability problems related to layered media and soil–structure interaction have been analysed. A number of examples are presented and through comparisons with available analytical and numerical results, the applicability and usefulness of the present analysis to real geomechanical problems are established.     

Numerical performance of projection methods in finite element consolidation models

Projection, or conjugate gradient like, methods are becoming increasingly popular for the efficient solution of large sparse sets of unsymmetric indefinite equations arising from the numerical integration of (initial) boundary value problems. One such problem is soil consolidation coupling a flow and a structural model, typically solved by finite elements (FE) in space and a marching scheme in time (e.g. the Crank–Nicolson scheme). The attraction of a projection method stems from a number of factors, including the ease of implementation, the requirement of limited core memory and the low computational cost if a cheap and effective matrix preconditioner is available. In the present paper, biconjugate gradient stabilized (Bi‐ CGSTAB) is used to solve FE consolidation equations in 2‐D and 3‐D settings with variable time integration steps. Three different nodal orderings are selected along with the preconditioner ILUT based on incomplete triangular factorization and variable fill‐in. The overall cost of the solver is made up of the preconditioning cost plus the cost to converge which is in turn related to the number of iterations and the elementary operations required by each iteration. The results show that nodal ordering affects the perfor mance of Bi‐CGSTAB. For normally conditioned consolidation problems Bi‐CGSTAB with the best ILUT preconditioner may converge in a number of iterations up to two order of magnitude smaller than the size of the FE model and proves an accurate, cost‐effective and robust alternative to direct methods. Copyright © 2001 John Wiley & Sons, Ltd.     

The Geomechanics of CO2 Storage in Deep Sedimentary Formations

This paper provides a review of the geomechanics and modeling of geomechanics associated with geologic carbon storage (GCS), focusing on storage in deep sedimentary formations, in particular saline aquifers. The paper first introduces the concept of storage in deep sedimentary formations, the geomechanical processes and issues related with such an operation, and the relevant geomechanical modeling tools. This is followed by a more detailed review of geomechanical aspects, including reservoir stress-strain and microseismicity, well integrity, caprock sealing performance, and the potential for fault reactivation and notable (felt) seismic events. Geomechanical observations at current GCS field deployments, mainly at the In Salah CO₂ storage project in Algeria, are also integrated into the review. The In Salah project, with its injection into a relatively thin, low-permeability sandstone is an excellent analogue to the saline aquifers that might be used for large scale GCS in parts of Northwest Europe, the U.S. Midwest, and China. Some of the lessons learned at In Salah related to geomechanics are discussed, including how monitoring of geomechanical responses is used for detecting subsurface geomechanical changes and tracking fluid movements, and how such monitoring and geomechanical analyses have led to preventative changes in the injection parameters. Recently, the importance of geomechanics has become more widely recognized among GCS stakeholders, especially with respect to the potential for triggering notable (felt) seismic events and how such events could impact the long-term integrity of a CO₂ repository (as well as how it could impact the public perception of GCS). As described in the paper, to date, no notable seismic event has been reported from any of the current CO₂ storage projects, although some unfelt microseismic activities have been detected by geophones. However, potential future commercial GCS operations from large power plants will require injection at a much larger scale. For such large-scale injections, a staged, learn-as-you-go approach is recommended, involving a gradual increase of injection rates combined with continuous monitoring of geomechanical changes, as well as siting beneath a multiple layered overburden for multiple flow barrier protection, should an unexpected deep fault reactivation occur.     

Mixed precision parallel preconditioner and linear solver for compositional reservoir simulation

Reservoir simulation role in value creation and strategic management decisions cannot be over emphasized. Simulation of complex challenging reservoirs with millions of grid blocks especially in compositional mode is very time-consuming even with fast modern computers. On the other hand, high price of cluster supercomputers prevents them for being commonly used for fast simulation of such reservoirs. In recent years, the development of many-core processors like cell processors, DSPs, and graphical processing units (GPUs) has provided a very cost-effective hardware platform for fast computational operations. However, programming for such processors is much more difficult than conventional CPUs, and new parallel algorithm design and special parallel implementation methods are needed. Using the computational power of CPUs, GPUs, and/or any other processing unit, Open Computing Language (OpenCL) provides a framework for programming for heterogeneous platforms. In this paper, OpenCL is used to employ the computational power of a GPU to build a preconditioner and solve the linear system arising from compositional formulation of multiphase flow in porous media. The proposed parallel preconditioner is proved to be quite effective, even in heterogeneous porous media. Using data-parallel modules on GPU, the preconditioner/solver runtime reduced at least 1 order of magnitude compared to their serial implementation on CPU.     

ParCYCLIC: finite element modelling of earthquake liquefaction response on parallel computers

This paper presents the computational procedures and solution strategy employed in ParCYCLIC, a parallel non‐linear finite element program developed based on an existing serial code CYCLIC for the analysis of cyclic seismically‐induced liquefaction problems. In ParCYCLIC, finite elements are employed within an incremental plasticity, coupled solid–fluid formulation. A constitutive model developed for simulating liquefaction‐induced deformations is a main component of this analysis framework. The elements of the computational strategy, designed for distributed‐memory message‐passing parallel computer systems, include: (a) an automatic domain decomposer to partition the finite element mesh; (b) nodal ordering strategies to minimize storage space for the matrix coefficients; (c) an efficient scheme for the allocation of sparse matrix coefficients among the processors; and (d) a parallel sparse direct solver. Application of ParCYCLIC to simulate 3‐D geotechnical experimental models is demonstrated. The computational results show excellent parallel performance and scalability of ParCYCLIC on parallel computers with a large number of processors. Copyright © 2004 John Wiley & Sons, Ltd.     

An explicitly coupled hydro‐geomechanical model for simulating hydraulic fracturing in arbitrary discrete fracture networks

Modeling hydraulic fracturing in the presence of a natural fracture network is a challenging task, owing to the complex interactions between fluid, rock matrix, and rock interfaces, as well as the interactions between propagating fractures and existing natural interfaces. Understanding these complex interactions through numerical modeling is critical to the design of optimum stimulation strategies. In this paper, we present an explicitly integrated, fully coupled discrete‐finite element approach for the simulation of hydraulic fracturing in arbitrary fracture networks. The individual physical processes involved in hydraulic fracturing are identified and addressed as separate modules: a finite element approach for geomechanics in the rock matrix, a finite volume approach for resolving hydrodynamics, a geomechanical joint model for interfacial resolution, and an adaptive remeshing module. The model is verified against the Khristianovich–Geertsma–DeKlerk closed‐form solution for the propagation of a single hydraulic fracture and validated against laboratory testing results on the interaction between a propagating hydraulic fracture and an existing fracture. Preliminary results of simulating hydraulic fracturing in a natural fracture system consisting of multiple fractures are also presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Template elastic‐plastic computations in geomechanics

In this paper we present a new approach to computations in elasto‐plastic geomechanics. The approach is based on the object oriented design philosophy and observations on similarity of most incremental elastic–plastic material models. This new approach to elastic–plastic computations in geomechanics allows for creation of template material models. The analysis of template material models will in turn allow for an easy implementation of other elastic–plastic material models based on the object oriented design principles. Furthermore we present some illustrative implementation details. Finally we present analysis results that emphasize features of template elastic–plastic computations in geomechanics. Copyright © 2002 John Wiley & Sons, Ltd.     

Joint optimization of oil well placement and controls

Well placement and control optimization in oil field development are commonly performed in a sequential manner. In this work, we propose a joint approach that embeds well control optimization within the search for optimum well placement configurations. We solve for well placement using derivative-free methods based on pattern search. Control optimization is solved by sequential quadratic programming using gradients efficiently computed through adjoints. Joint optimization yields a significant increase, of up to 20% in net present value, when compared to reasonable sequential approaches. The joint approach does, however, require about an order of magnitude increase in the number of objective function evaluations compared to sequential procedures. This increase is somewhat mitigated by the parallel implementation of some of the pattern-search algorithms used in this work. Two pattern-search algorithms using eight and 20 computing cores yield speedup factors of 4.1 and 6.4, respectively. A third pattern-search procedure based on a serial evaluation of the objective function is less efficient in terms of clock time, but the optimized cost function value obtained with this scheme is marginally better.     

大区域地下水模拟的预优并行GMRES(m)算法研究

大区域研究区由于涉及范围大、水文地质参数复杂多变,一直是进行地下水数值模拟的热点和难点。针对大区域地下水模拟的特点,在MPI环境中对Krylov子空间GMRES(m)算法的并行性进行分析,提出基于区域分解法的并行实现策略,并对不同的预条件子的加速效果进行比较。数值实验结果表明:并行GMRES(m)算法在求解大区域三维地下水模型时可以显著的加快求解速度,且具有较好的可扩展性。另外,Jacobi预条件子与GMRES算法的组合具有更优的加速比和执行效率,是一种求解大型化、复杂化地下水水流问题的可行方案。     

A Review of Some Rock Mechanics Issues in Geothermal Reservoir Development

Rock mechanics and geomechanical studies can provide crucial information for economic geothermal reservoir development. Although significant progress has been made in reservoir geomechanics, technical challenges specific to the geothermal area (high temps, data collection, experimentation issues) have prevented widespread use of geomechanics in geothermal reservoir development. However, as the geothermal industry moves to develop more challenging resources using the concept of enhanced geothermal systems (EGS), and to maximize productivity from conventional resources, the need for improved understanding of geomechanical issues and developing specific technologies for geothermal reservoirs has become critical. Rock mechanics research and improved technologies can impact areas related to in-situ stress characterization, initiation and propagation of artificial and natural fractures, and the effects of coupled hydro-thermo-chemo-mechanical processes on fracture permeability and induced seismicity. Rock mechanics/geomechanics research, including experimental and theoretical investigations as well as numerical and analytical solutions, has an important role in optimizing reservoir design and heat extraction strategies for sustainable geothermal energy development. A number of major areas where rock mechanics research can facilitate geothermal systems development are reviewed in this paper with particular emphasis on EGS design and management.