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

Carbon capture and storage (CCS) is broadly recognised as having the potential to play a key role in meeting climate change targets, delivering low carbon heat and power, decarbonising industry and, more recently, its ability to facilitate the net removal of CO2 from the atmosphere. However, despite this broad consensus and its technical maturity, CCS has not yet been deployed on a scale commensurate with the ambitions articulated a decade ago. Thus, in this paper we review the current state-of-the-art of CO2 capture, transport, utilisation and storage from a multi-scale perspective, moving from the global to molecular scales. In light of the COP21 commitments to limit warming to less than 2 °C, we extend the remit of this study to include the key negative emissions technologies (NETs) of bioenergy with CCS (BECCS), and direct air capture (DAC). Cognisant of the non-technical barriers to deploying CCS, we reflect on recent experience from the UK's CCS commercialisation programme and consider the commercial and political barriers to the large-scale deployment of CCS. In all areas, we focus on identifying and clearly articulating the key research challenges that could usefully be addressed in the coming decade.

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Carbon capture and storage (CCS): the way forward

A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

Numerical simulations of particulate suspensions via a discretized boltzmann equation: part 1. theoretical foundation

https://www.math.utah.edu/~epshteyn/selectedpapers.html/paperHou1.pdf

A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

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Nonlinear Model Reduction via Discrete Empirical Interpolation

A numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented. The method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield, upon appropriate averaging of the fine scale velocity field, the coarse scale or equivalent grid block permeability. When the region over which this coarse scale permeability is computed constitutes a representative elementary volume (REV), the resulting equivalent permeability may be interpreted as the effective permeability of the region. Solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle-based finite element numerical procedure, which allows for the modeling of geometrically complex features. The specification of periodic boundary conditions is shown to yield symmetric, positive definite equivalent permeability tensors in all cases. The method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields. The applicability and limitations of the method for these more general heterogeneity fields are discussed.

Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media

An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators

A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.

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Dynamic simulation of hydrodynamically interacting particles

Determining the optimum type and location of new wells is an essential component in the efficient development of oil and gas fields. The optimization problem is, however, demanding due to the potentially high dimension of the search space and the computational requirements associated with function evaluations, which, in this case, entail full reservoir simulations. In this paper, the particle swarm optimization (PSO) algorithm is applied for the determination of optimal well type and location. The PSO algorithm is a stochastic procedure that uses a population of solutions, called particles, which move in the search space. Particle positions are updated iteratively according to particle fitness (objective function value) and position relative to other particles. The general PSO procedure is first discussed, and then the particular variant implemented for well optimization is described. Four example cases are considered. These involve vertical, deviated, and dual-lateral wells and optimization over single and multiple reservoir realizations. For each case, both the PSO algorithm and the widely used genetic algorithm (GA) are applied to maximize net present value. Multiple runs of both algorithms are performed and the results are averaged in order to achieve meaningful comparisons. It is shown that, on average, PSO outperforms GA in all cases considered, though the relative advantages of PSO vary from case to case. Taken in total, these findings are very promising and demonstrate the applicability of PSO for this challenging problem.

Louis J. Durlofsky

Papers

Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media

An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators

Dynamic simulation of hydrodynamically interacting particles

Application of a particle swarm optimization algorithm for determining optimum well location and type

A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations