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

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

Control-volume discretizations using multipoint flux approximations (MPFA) were developed in the last decade. This paper gives an introduction to these methods for quadrilateral grids in two and three dimensions. The introduction is kept on a basic level, and a brief summary to more advanced results is given. Only the O-method with surface midpoints as continuity points is discussed. Flux expressions are derived both in physical and in curvilinear space. Equations for calculation of the transmissibility coefficients are given, and an explicit solution is shown for constant coefficients. K-orthogonality, stability and monotonicity are discussed, and an iterative solution technique is presented. Two numerical examples close the paper.

An Introduction to Multipoint Flux Approximations for Quadrilateral Grids

http://www.math.pitt.edu/~yotov/teaching/16-2/math2602/MFD-JCP-survey.pdf

Mimetic finite difference method

This paper describes hierarchical approach to modeling flow in a naturally fractured formation. Our model is based on calculating the effective permeability of a fractured formation, as a function of grid block size, and using the results in a conventional finite difference flow simulator. On the basis of their length (lf) relative to the finite difference grid size (lg), fractures are classified as belonging to one of three groups: (1) short fractures (lf≪ lg), (2) medium-length fractures (lf ∼ lg), and (3) long fractures (lf≫ lg). The effects of the fractures belonging to each class are computed in a hierarchical manner. The permeability contribution from short fractures is derived in an analytical expression and used as an enhanced matrix permeability for the next-scale (medium-length) calculation. The effective matrix permeability associated with medium-length fractures is numerically solved using a boundary element method. The long fractures are modeled explicitly as major fluid conduits. As numerical examples, tracer transport in fractured formations was illustrated. The numerical results clearly indicated that effective tensor permeability well represented directional, enhanced permeability in fractured formations. The fluid-conduit formulation captured the efficient fluid transport by long fractures.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000WR900340

Hierarchical modeling of flow in naturally fractured formations with multiple length scales

Discretization methods are proposed for control-volume formulations on polygonal and triangular grid cells in two space dimensions. The methods are applicable for any system of conservation laws where the flow density is defined by a gradient law, like Darcy's law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy. This paper gives a derivation of the methods, and the relation to previously published methods is also discussed. A further discussion of the methods, including numerical examples, is given in the companion paper, Part II [SIAM J. Sci. Comput., pp. 1717--1736].

Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods

Discretization on Non-Orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media

Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property. It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied. Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in terms of monotonicity.

Monotonicity of control volume methods

Control-Volume Discretization Methods for 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs

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