Showing papers by "Mary F. Wheeler published in 2007"

A Multiscale Mortar Mixed Finite Element Method

Abstract: We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar space achieves approximation comparable to the fine scale on its coarse grid by using higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale finite element and variational multiscale methods. We derive a priori error estimates and show, with appropriate choice of the mortar space, optimal order convergence and some superconvergence on the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate subdomain and mortar grids. Numerical experi...

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case

Abstract: In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed finite element method and the displacements by a Galerkin method. Theoretical convergence error estimates are derived in a continuous in-time setting for a strictly positive constrained specific storage coefficient. Of particular interest is the case when the lowest-order Raviart–Thomas approximating space or cell-centered finite differences are used in the mixed formulation, and continuous piecewise linear approximations are used for displacements. This approach appears to be the one most frequently applied to existing reservoir engineering simulators.

Convergence of a symmetric MPFA method on quadrilateral grids

Abstract: This paper investigates different variants of the multipoint flux approximation (MPFA) O-method in 2D, which rely on a transformation to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different methods appear, depending on where the transformed permeability is evaluated. Midpoint and corner-point evaluations are considered. Relations to mixed finite element (MFE) methods with different velocity finite element spaces are further discussed. Convergence of the MPFA methods is investigated numerically. For corner-point evaluation of the reference permeability, the same convergence behavior as the O-method in the physical space is achieved when the grids are refined uniformly or when grid perturbations of order h 2 are allowed. For h 2-perturbed grids, the convergence of the normal velocities is slower for the midpoint evaluation than for the corner-point evaluation. However, for rough grids, i.e., grids with perturbations of order h, contrary to the physical space method, convergence cannot be claimed for any of the investigated reference space methods. The relations to the MFE methods are used to explain the loss of convergence.

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

Abstract: 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.

Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity

Abstract: We present a finite element formulation for coupled flow and geomechanics. We use mixed finite element spaces to approximate pressure and continuous Galerkin methods for displacements. In solving the coupled system, pressure and displacements can be solved either simultaneously in a fully coupled scheme or sequentially in a loosely coupled scheme. In this paper we formulate an iterative method where pressure and displacement solutions are staggered during a time step until a convergence tolerance is satisfied. A priori convergence results for the iterative coupling are also presented, along with a summary of the convergence results for the fully coupled scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 785-797, 2007

Stochastic collocation and mixed flnite elements for ∞ow in porous media

Abstract: The aim of this paper is to quantify uncertainty of ∞ow in porous media through stochastic modeling and computation of statistical moments. The governing equations are based on Darcy’s law with stochastic permeability. Starting from a specifled covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loµeve expansion. Mixed flnite element approximations are used in the spatial domain and collocation at the zeros of tensor product Hermite polynomials is used in the stochastic dimensions. Error analysis is performed and experimentally verifled with numerical simulations. Computational results include incompressible and slightly compressible single and two-phase ∞ow.

A multiscale mortar mixed finite element method for slightly compressible flows in porous media

Abstract: We consider multiscale mortar mixed finite element discretizations for slightly compressible Darcy flows in porous media. This paper is an extension of the formulation introduced by Arbogast et al. for the incompressible problem [2]. In this method, flux continuity is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. Optimal fine scale convergence is obtained by an appropriate choice of mortar grid and polynomial degree of approximation. Parallel numerical simulations on some multiscale benchmark problems are given to show the efficiency and effectiveness of the method.

Deflation AMG Solvers for Highly Ill-Conditioned Reservoir Simulation Problems

Abstract: 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.

Local problem-based a posteriori error estimators for discontinuous Galerkin approximations of reactive transport

Abstract: We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuska-Baumann version of DG. Numerical results are provided to illustrate the effectiveness of the proposed error estimators.

Physical and Computational Domain Decompositions for Modeling Subsurface Flows

Abstract: Modeling of multiphase flow in permeable media plays a central role in subsurface environmental remediation as well as in problems associated with production of hydrocarbon energy from existing oil and gas fields. Numerical simulation is essential for risk assessment, cost reduction, and rational and efficient use of resources. The contamination of groundwater is one of the most serious environmental problems facing the world. For example, more than 50% of drinking water in the Unites States comes from groundwater. More than 10,000 active military installations and over 6,200 closed installations in the United States require subsurface remediation. The process is difficult and extremely expensive and only now is technology emerging to cope with this severe and widespread problem. Hydrocarbons contribute almost two-thirds of the nation’s energy supply. Moreover, recoverable reserves are being increased twice as fast by enhanced oil recovery techniques as by exploration. Features that make the above problems difficult for numerical simulation include: multiple phases and chemical components, multi-scale heterogeneities, stiff gradients, irregular geometries with internal boundaries such as faults and layers, and multi-physics. Because of the uncertainty in the data, one frequently assumes stochastic coefficients and thus is forced to multiple realizations; therefore both computational efficiency and accuracy are crucial in the simulations. For efficiency, the future lies in developing parallel simulators which utilize domain decomposition algorithms. One may ask what are the important aspects of parallel computation for these complex physical models. First, in all cases, one must be able to partition dynamically the geological domain based upon the physics of the model. Second, efficient distribution of the computations must be performed. Critical issues here are load balancing and minimal communication overhead. It is important to note that the two decompositions may be different.

Superconvergence for Control-Volume Mixed Finite Element Methods on Rectangular Grids

Abstract: We consider control-volume mixed finite element methods for the approximation of second-order elliptic problems on rectangular grids. These methods associate control volumes (covolumes) with the vector variable as well as the scalar, obtaining local algebraic representation of the vector equation (e.g., Darcy’s law) as well as the scalar equation (e.g., conservation of mass). We establish $O(h^2)$ superconvergence for both the scalar variable in a discrete $L^2$-norm and the vector variable in a discrete $H({\rm div})$-norm. The analysis exploits a relationship between control-volume mixed finite element methods and the lowest order Raviart-Thomas mixed finite element methods.

A neural stochastic multiscale optimization framework for sensor-based parameter estimation

Abstract: This work presents a novel neural stochastic optimization framework for reservoir parameter estimation that combines two independent sources of spatial and temporal data: oil production data and dynamic sensor data of flow pressures and concentrations. A parameter estimation procedure is realized by minimizing a multi-objective mismatch function between observed and predicted data. In order to be able to efficiently perform large-scale parameter estimations, the parameter space is decomposed in different resolution levels by means of the singular value decomposition (SVD) and a wavelet upscaling process. The estimation is carried out incrementally from low to higher resolution levels by means of a neural stochastic multilevel optimization approach. At a given resolution level, the parameter space is globally explored and sampled by the simultaneous perturbation stochastic approximation (SPSA) algorithm. The sampling yielded by SPSA serves as training points for an artificial neural network that allows for evaluating the sensitivity of different multi-objective function components with respect to the model parameters. The proposed approach may be suitable for different engineering and scientific applications wherever the parameter space results from discretizing a set of partial differential equations on a given spatial domain.

A hybrid optimization approach for automated parameter estimation problems

Abstract: We present a hybrid optimization approach for solving automated parameter estimation models. The hybrid approach is based on the coupling of the Simultaneous Perturbation Stochastic Approximation (SPSA) [1] and a Newton-Krylov Interior-Point method (NKIP) [2] via a surrogate model. The global method SPSA performs a stochastic search to find target regions with low function values. Next, we generate a surrogate model based on the points of regions on which the local method NKIP algorithm is applied for finding an optimal solution. We illustrate the behavior of the hybrid optimization algorithm on one testcase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic Data-Driven Systems Approach for Simulation Based Optimizations

Abstract: This paper reviews recent developments in our project that are focused on dynamic data-driven methods for efficient and reliable simulation based optimization, which may be suitable for a wide range of different application problems. The emphasis in this paper is on the coupling of parallel multiblock predictive models with optimization, the development of autonomic execution engines for distributing the associated computations, and deployment of systems capable of handling large datasets. The integration of these components results in a powerful framework for developing large-scale and complex decision-making systems for dynamic data-driven applications.

Integrated time-lapse seismic inversion for reservoir petrophysics and fluid-flow imaging

Abstract: SUMMARY Seismic history matching has been used to reduce uncertainty and increase the accuracy in reservoir characterization. In this paper we propose a joint inversion scheme for quantitative reservoir petrophysics characterization and inside fluid flow imaging, which integrates as many data sources as available, such as timelapse seismic data, production data and sensor information. We demonstrate that this integrated framework leads to accurate seismic history matching and reservoir parameter estimation and that the incorporation of time-lapse seismic data does help speed up the convergence process. In addition, the imaging of the inside fluid flow is also obtained. Considering the unique feature of Bayesian inference in data integration and uncertainty analysis, we also propose a formulation to simultaneously integrate all data sources in a Bayesian framework and solve the problem by stochastically constructing the posterior probability distribution (PPD) using Markov Chain Monte Carlo (MCMC) methods. The coefficients of rock fluid physics models are also incorporated. This means that the coefficients are determined stochastically based on data rather from lab measurements or empirical relationships. Based on MCMC samples, uncertainty can be correctly quantified.

Multiscale deflation solvers for flow in porous media

Abstract: This work describes a novel physics-based deflation preconditioner approach for solving porous media flow problems characterized by highly heterogeneous media. The approach relies on high-conductivity block solutions after rearranging the linear system coefficients into high-conductive and low conductive blocks from a given physically driven threshold value. This rearranging relies on the Hoshen-Kopelman (H-K) algorithm that is commonly used to determine percolation clusters. The resulting preconditioner may alternatively be combined with a deflation preconditioning stage. The proposed approach is coined as a physics-based 2-stage deflation preconditioner (P2SDP). Numerical experiments on different permeability distributions reveal that P2SDP is a powerful means to solve pressure systems when compared to more conventional algebraic approaches such as the incomplete Cholesky factorization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Stochastic Krylov subspace methods for flow in random porous media

Abstract: The present work establishes a comparative analysis of two different Krylov-based methods for assessing uncertainty in porous media flow applications. They are: (1) the stochastic reduced basis method (SRBM) and the (2) Krylov-Karhunen-Loeve moment equation (KKLME). The former relies on the construction of an orthogonal basis from which projectors may be realized to perform the uncertainty analysis on a lower-dimensional space. The second approach relies on the idea of expressing input and corresponding outputs in terms of stochastic and perturbative polynomial expansions. By grouping terms and moments of the same order, the original stochastic equation is replaced by successive (and parallel) solutions of a set of deterministic equations. We provide a set of numerical experiments illustrating the capabilities of SRBM and KKLME against Monte Carlo simulations (MCS) on stationary permeability cases. We show that KKLME is superior to SRBM. Furthermore, KKLME appears to be also a more efficient alternative than MCS for non-stationary permeability field distributions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

SPSA for oil‐parameter estimation and reservoir management

Abstract: In this paper we implement and test the Simultaneous Perturbation Stochastic Approximation method (SPSA) in order to perform parameter estimation or history matching in reservoir simulation. The SPSA algorithm allows the determination of descent directions with very few forward model evaluations no matter the number of parameters. We test this approach for the estimation of transmissibilities for a single-phase flow problem in very heterogeneous environments. The global properties of SPSA allow the determination of reasonable good inverse solutions which is demonstrated through numerical experiments. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A multiscale method for parameter estimation in reservoir simulation

Abstract: We consider a multiscale approach to the problem of parameter estimation or history matching in reservoir simulation. To do this, we look at the problem upscaling as the most important component of the proposed method. For upscaling we suggest a global approach which seeks to minimize the misfit between fine scale and coarse scale simulations and then used the Simultaneous Perturbation Stochastic Approximation (SPSA) method to deal with the minimization process. The essential idea of multiscale history matching comsists of constructing a coarsening sequence of models, so that each one of them minimizes the misfit between preditctions and observations. Then, the cosarsest model is downscaled one step into the sequence and used as an initial guess for the next step of refinement. The process is repeated until no improvement is observed in the solution. Numerical experiment indicate that this approach produces consistent results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)