There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

/pdf/unified-analysis-of-discontinuous-galerkin-methods-for-9numgtph62.pdf

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

/pdf/the-local-discontinuous-galerkin-method-for-time-dependent-23o5v9uy8q.pdf

The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

A Phase-Field Based Approach for Modeling the Cementation and Shear Slip of Fracture Networks


 Discretization methods have been developed to accompany a novel cut-cell gridding technique for reservoir simulation that preserves the orthogonality characteristic in the lateral direction. A major drawback of the cut-cell gridding method is that polyhedral cells emerge near faults that have relatively small volumes. Pragmatic but non-rigorous approximation methods have been developed in the past to merge these cells with their neighbors so that the grid representation fits the two-point flux approximation (TPFA) framework. In this work, we take a different approach and investigate the global and local applications of select consistent discretization methods in the vicinity of fault representations on cut-cell grids.
 We develop and test consistent discretization methods that are of low computational cost and do not require major intrusive changes to the solver structure of commercial reservoir simulators. Cell-centered methods such as multi-point flux approximation (MPFA), average multi-point flux approximation (AvgMPFA), and nonlinear two-point flux approximation (NTPFA) methods fit naturally into the framework of existing industrial-grade simulators. Therefore, we develop and test variants of the AvgMPFA and NTPFA methods that are specifically designed to operate on cut-cell grids. An implementation of the well-established but computationally expensive MPFA method is also made for cut-cell grids to serve as a reference to computations with AvgMPFA and NTPFA. All investigated methods are implemented within the framework of a full-physics 3D research simulator with a general compositional formulation, which encompasses black-oil models.
 We use a set of synthetic cut-cell grid models of varying complexity including conceptual models and a field-scale model. We compare the novel cut-cell adapted AvgMPFA and NTPFA simulation results in terms of accuracy and computational performance against the ones computed with reference MPFA and TPFA methods. We observe that AvgMPFA consistently yields more accurate and computationally efficient simulations than NTPFA on cut-cell grids. Moreover, AvgMPFA hybrids run faster than NTPFA hybrids when compared on the same problem for the same hybridization strategy. On the other hand, the computational performance of AvgMPFA degrades more rapidly compared to NTPFA with increasing "rings" of orthogonal blocks around cut-cells owing to its relatively wider stencil. Auspiciously, only one or two "rings" of orthogonal blocks around cut cells are sufficient with AvgMPFA to deliver high accuracy.

Consistent Discretization Methods for Reservoir Simulation on Cut-Cell Grids

The main objective of the present work is to propose and evaluate a learning computational engine for history matching, which is based on a hybrid multilevel search methodology. According to this methodology, the parameter space is globally explored and sampled by the simultaneous perturbation stochastic approximation (SPSA) algorithm at a given resolution level. This estimation is followed by further analysis by using a neural learning engine for evaluating the sensitiveness of the objective function with respect to variations of each individual model parameter in the vicinity of the promising optimal solution explored by the SPSA algorithm. The proposed methodology is used to numerically determine how additional sources of information may aid in reducing the ill-posedness associated with permeability estimation via conventional history matching procedures. The additional sources of information considered in this work are related to pressures, concentrations and fluid velocities at given locations in a reliable fashion, which in practical scenarios might be estimated from high resolution seismic surveys, or directly obtained as in situ measurements provided by sensors. This additional information is incorporated, along with production data, into a multi-objective function that is mismatched between the observed and the predicted data. The preliminary results presented in this work shed light on future research avenues for optimizing the use of additional sources of information such as seismic or sensor data in history matching procedures.

A learning computational engine for

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)

Stochastic Krylov subspace methods for flow in random porous media

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)

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.200700929

Mary F. Wheeler

Papers

A Phase-Field Based Approach for Modeling the Cementation and Shear Slip of Fracture Networks

Consistent Discretization Methods for Reservoir Simulation on Cut-Cell Grids

A learning computational engine for

Stochastic Krylov subspace methods for flow in random porous media

SPSA for oil‐parameter estimation and reservoir management