Multilevel spectral coarsening for graph Laplacian problems with application to reservoir simulation

We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in FAS (full approximation scheme) for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. We illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation

Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and solved without ever re-visiting the fine level, an essential element for multigrid algorithms to achieve optimal scalability. Numerical examples comparing relative performance of the proposed nonlinear multigrid solvers with standard single-level approaches -- Picard's and Newton's methods -- are presented. Results show that the proposed solver consistently outperforms the single-level methods, both in efficiency and robustness

Model Calibration, Drainage Volume Calculation and Optimization in Heterogeneous Fractured Reservoirs

We propose a rigorous approach for well drainage volume calculations in gas reservoirs based on the flux field derived from dual porosity finite-difference simulation and demonstrate its application to optimize well placement. Our approach relies on a high frequency asymptotic solution of the diffusivity equation and emulates the propagation of a 'pressure front' in the reservoir along gas streamlines. The proposed approach is a generalization of the radius of drainage concept in well test analysis (Lee 1982), which allows us not only to compute rigorously the well drainage volumes as a function of time but also to examine the potential impact of infill wells on the drainage volumes of existing producers. Using these results, we present a systematic approach to optimize well placement to maximize the Estimated Ultimate Recovery. A history matching algorithm is proposed that sequentially calibrates reservoir parameters from the global-to-local scale considering parameter uncertainty and the resolution of the data. Parameter updates are constrained to the prior geologic heterogeneity and performed parsimoniously to the smallest spatial scales at which they can be resolved by the available data. In the first step of the workflow, Genetic Algorithm is used to assess the uncertainty in global parameters that influence field-scale flow behavior, specifically reservoir energy. To identify the reservoir volume over which each regional multiplier is applied, we have developed a novel approach to heterogeneity segmentation from spectral clustering theory. The proposed clustering can capture main feature of prior model by using second eigenvector of graph affinity matrix. In the second stage of the workflow, we parameterize the high-resolution heterogeneity in the spectral domain using the Grid Connectivity based Transform to severely compress the dimension of the calibration parameter set. The GCT implicitly imposes geological continuity and promotes minimal changes to each prior model in the ensemble during the calibration process. The field scale utility of the workflow is then demonstrated with the calibration of a model characterizing a structurally complex and highly fractured reservoir

Effective Modeling Approaches for CO2 EOR Developments

We present a simulation study of a mature reservoir for COv2 Enhanced Oil Recovery (EOR) development. This project is currently recognized as the world’s largest project utilizing post-combustion COv2 from power generation flue gases. With a fluvial formation geology and sharp hydraulic conductivity contrasts, this is a challenging and novel application of COv2 EOR. The objective of this study is to obtain a reliable predictive reservoir model by integrating multi-decadal production data at different temporal resolutions into the available geologic model. This will be useful for understanding flow units, heterogeneity features and their impact on subsurface flow mechanisms to guide the optimization of the injection scheme and maximize COv2 sweep and oil recovery from the reservoir. Our strategy consists of a hierarchical approach for geologic model calibration incorporating available pressure and multiphase production data. The model calibration is carried out using regional multipliers whereby the regions are defined using a novel Adjacency Based Transform (ABT) accounting for the underlying geologic heterogeneity. The Genetic Algorithm (GA) is used to match 70-year pressure and cumulative production by adjusting pore volume and aquifer strength. This leads to an efficient and robust workflow for field scale history matching. The history matched model provided important information about reservoir volumes, flow zones and aquifer support that led to additional insight to the prior geological and simulation studies. The history matched field-scale model is used to define and initialize a detailed fine-scale model for a COv2 pilot area which will be utilized for studying the impact of fine-scale heterogeneity on COv2 sweep and oil recovery. The uniqueness of this work is the application of a novel geologic model parameterization and history matching workflow for modeling of a mature oil field with decades of production history and which is currently being developed with COv2 EOR. In addition to the history matching studies, we developed an embedded discrete fracture model (EDFM) which is currently recognized as a promising alternative to conventional fracture modeling approaches including multiple continuum models and unstructured discrete fracture models because of its accuracy and computational efficiency. We tested the developed model with several examples including water flood and COv2 flood scenarios and confirmed applicability of the EDFM

Generalization of Mixed Multiscale Finite Element Methods with Applications

Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms. For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces. Two enrichment approaches are proposed, one is based on generalized multiscale finite element methods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (ρAMGe). The former one, which is called mixed GMs- FEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor. The latter enrichment approach is based on ρAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces. Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. Our numerical results show that the proposed methods are more efficient than the conventional methods while still being able to produce reliable solution for our targeted applications such as reservoir simulation. Moreover, the robustness of the mixed GMsFEM for linear elasticity with respect to the high contrast heterogeneity in Poisson ratio is evident from our numerical experiments. Lastly, our empirical results show good speedup and approximation by the proposed multilevel coarsening method

Multiscale Spectral-Domain Parameterization for History Matching in Structured and Unstructured Grid Geometries

Reservoir model calibration to production data, also known as history matching, is an essential tool for the prediction of fluid displacement patterns and related decisions concerning reservoir management and field development. The history matching of high resolution geologic models is, however, known to define an ill-posed inverse problem such that the solution of geologic heterogeneity is always non-unique and potentially unstable. A common approach to improving ill-posedness is to parameterize the estimable geologic model components, imposing a type of regularization that exploits geologic continuity by explicitly or implicitly grouping similar properties while retaining at least the minimum heterogeneity resolution required to reproduce the data. This dissertation develops novel methods of model parameterization within the class of techniques based on a linear transformation. Three principal research contributions are made in this dissertation. First is the development of an adaptive multiscale history matching formulation in the frequency domain using the discrete cosine parameterization. Geologic model calibration is performed by its sequential refinement to a spatial scale sufficient to match the data. The approach enables improvement in solution non-uniqueness and stability, and further balances model and data resolution as determined by a parameter identifiability metric. Second, a model-independent parameterization based on grid connectivity information is developed as a generalization of the cosine parameterization for applicability to generic grid geometries. The parameterization relates the spatial reservoir parameters to the modal shapes or harmonics of the grid on which they are defined, merging with a Fourier analysis in special cases (i.e., for rectangular grid cells of constant dimensions), and enabling a multiscale calibration of the reservoir model in the spectral domain. Third, a model-dependent parameterization is developed to combine grid connectivity with prior geologic information within a spectral domain representation. The resulting parameterization is capable of reducing geologic models while imposing prior heterogeneity on the calibrated model using the adaptive multiscale workflow. In addition to methodological developments of the parameterization methods, an important consideration in this dissertation is their applicability to field scale reservoir models with varying levels of prior geologic complexity on par with current industry standards

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds