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

Introduction to linear and nonlinear programming

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

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Inverse problems: A Bayesian perspective

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

History matching is a type of inverse problem in which observed reservoir behavior is used to estimate reservoir model variables that caused the behavior. Obtaining even a single history-matched reservoir model requires a substantial amount of effort, but the past decade has seen remarkable progress in the ability to generate reservoir simulation models that match large amounts of production data. Progress can be partially attributed to an increase in computational power, but the widespread adoption of geostatistics and Monte Carlo methods has also contributed indirectly. In this review paper, we will summarize key developments in history matching and then review many of the accomplishments of the past decade, including developments in reparameterization of the model variables, methods for computation of the sensitivity coefficients, and methods for quantifying uncertainty. An attempt has been made to compare representative procedures and to identify possible limitations of each.

Recent progress on reservoir history matching: a review

Handling Conflicting Multiple Objectives Using Pareto-Based Evolutionary Algorithm for History Matching of Reservoir Performance

Full-Field Streamline Tracing in Complex Faulted Systems With Non-Neighbor Connections

A Hierarchical Streamline-Assisted History Matching Approach With Global and Local Parameter Updates

Multiphase Streamline Modeling in Three Dimensions: Further Generalizations and a Field Application

\n Multistage hydraulically fractured horizontal wells provide an effective means to exploit unconventional reservoirs. The current industry practice in the interpretation of field response often uses empirical decline-curve analysis or pressure-transient analysis/rate-transient analysis (PTA/RTA) for characterization of these reservoirs and fractures. These analytical tools depend on simplifying assumptions and do not provide a detailed description of the evolving reservoir-drainage volume accessed from a well. Understanding of the transient-drainage volume is essential for unconventional-reservoir and fracture assessment and optimization.\n In our previous study (Yang et al. 2015), we developed a “data-driven” methodology for the production rate and pressure analysis of shale-gas and shale-oil reservoirs. There are no underlying assumptions of fracture geometry, reservoir homogeneity, and flow regimes in the method proposed in our previous study. This approach depends on the high-frequency asymptotic solution of the diffusivity equation in heterogeneous reservoirs. It allows us to determine the well-drainage volume and the instantaneous recovery ratio (IRR), which is the ratio of the produced volume to the drainage volume, directly from the production data. In addition, a new w(τ) plot has been proposed to provide better insight into the depletion mechanisms and the fracture geometry. w(τ) is the derivative of pore volume with respect to τ.\n In this paper, we build upon our previous approach to propose a novel diagnostic tool for the interpretation of the characteristics of (potentially) complex fracture systems and drainage volume. We have used the w(τ) and IRR plots for the identification of characteristic signatures that imply complex fracture geometry, formation linear flow, partial reservoir completions, and fracture-interference/compaction effects during production. The w(τ) analysis gives us the fracture surface area and formation diffusivity, while the IRR analysis provides additional information on fracture conductivity. In addition, quantitative analysis is conducted using the novel w(τ) plot to interpret fracture-interference time, formation permeability, total fracture surface area, and stimulated reservoir volume (SRV).\n The major advantages of this current approach are the model-free analysis without assuming planar fractures, homogeneous formation properties, and specific flow regimes. In addition, the w(τ) plot captures high-resolution flow patterns not observed in traditional PTA/RTA analysis. The analysis leads to a simple and intuitive understanding of the transient-drainage volume and fracture conductivity. The results of the analysis are useful for hydraulic-fracturing-design optimization and matrix- and fracture-parameter estimation.

Akhil Datta-Gupta

Papers

Handling Conflicting Multiple Objectives Using Pareto-Based Evolutionary Algorithm for History Matching of Reservoir Performance

Full-Field Streamline Tracing in Complex Faulted Systems With Non-Neighbor Connections

A Hierarchical Streamline-Assisted History Matching Approach With Global and Local Parameter Updates

Multiphase Streamline Modeling in Three Dimensions: Further Generalizations and a Field Application

Reservoir and Fracture-Flow Characterization Using Novel Diagnostic Plots