Accurate geological modelling of features such as faults, fractures or erosion requires grids that are flexible with respect to geometry. Such grids generally contain polyhedral cells and complex grid-cell connectivities. The grid representation for polyhedral grids in turn affects the efficient implementation of numerical methods for subsurface flow simulations. It is well known that conventional two-point flux-approximation methods are only consistent for K-orthogonal grids and will, therefore, not converge in the general case. In recent years, there has been significant research into consistent and convergent methods, including mixed, multipoint and mimetic discretisation methods. Likewise, the so-called multiscale methods based upon hierarchically coarsened grids have received a lot of attention. The paper does not propose novel mathematical methods but instead presents an open-source Matlab® toolkit that can be used as an efficient test platform for (new) discretisation and solution methods in reservoir simulation. The aim of the toolkit is to support reproducible research and simplify the development, verification and validation and testing and comparison of new discretisation and solution methods on general unstructured grids, including in particular corner point and 2.5D PEBI grids. The toolkit consists of a set of data structures and routines for creating, manipulating and visualising petrophysical data, fluid models and (unstructured) grids, including support for industry standard input formats, as well as routines for computing single and multiphase (incompressible) flow. We review key features of the toolkit and discuss a generic mimetic formulation that includes many known discretisation methods, including both the standard two-point method as well as consistent and convergent multipoint and mimetic methods. Apart from the core routines and data structures, the toolkit contains addon modules that implement more advanced solvers and functionality. Herein, we show examples of multiscale methods and adjoint methods for use in optimisation of rates and placement of wells.

No abstract

We analyse and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an upscaling method by solving the pressure equation on a (moderate-sized) coarse grid, while at the same time produce a detailed and conservative velocity field on the underlying fine grid. Earlier work has shown that the multiscale method performs excellently on highly heterogeneous cases using uniform coarse grids. In this paper, we extend the methodology to nonuniform and unstructured coarse grids and discuss various formulations for generating the coarse-grid basis functions. Moreover, we focus on the impact of large-scale features such as barriers or high-permeable channels and discuss potentially problematic flow cases. To improve the accuracy of the multiscale solution, we introduce adaptive strategies for the coarse grids, based on either local hierarchical refinement or on adapting the coarse grid more directly to large-scale permeability structures of arbitrary shape. The resulting method is very flexible with respect to the size and the geometry of coarse-grid cells, meaning that grid refinement/adaptation can be performed in a straightforward manner. The suggested strategies are illustrated in several numerical experiments.

Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid models for highly heterogeneous petroleum reservoirs. Unlike traditional simulation, approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating subscale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately sized) coarse grid, one can retain the efficiency of an upscaling method and, at the same time, produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finite-element method (MsMFEM) has been shown to be a particularly versatile approach. In this paper, we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. To implement MsMFEM, one needs a discretisation method for solving local flow problems on the underlying fine grids. In principle, any stable and conservative method can be used. Here, we use a mimetic discretisation, which is a generalisation of mixed finite elements that gives a discrete inner product, allows for polyhedral elements, and can (easily) be extended to curved grid faces. The coarse grid can, in principle, be any partition of the subgrid, where each coarse block is a connected collection of subgrid cells. However, we argue that, when generating coarse grids, one should follow certain simple guidelines to achieve improved accuracy. We discuss partitioning in both index space and physical space and suggest simple processing techniques. The versatility and accuracy of the new multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. A variety of coarse grids, both violating and obeying the above mentioned guidelines, are employed. The MsMFEM solutions are compared with a reference solution obtained by direct simulation on the subgrid.

Flow diagnostics, as referred to herein, are computational tools derived from controlled numerical flow experiments that yield quantitative information regarding the flow behavior of a reservoir model in settings much simpler than would be encountered in the actual field. In contrast to output from traditional reservoir simulators, flow-diagnostic measures can be obtained within seconds. The methodology can be used to evaluate, rank, and/or compare realizations or strategies, and the computational speed makes it ideal for interactive visualization output. We also consider application of flow diagnostics as proxies in optimization of reservoirmanagement work flows. In particular, by use of finite-volume discretizations for pressure, time of flight (TOF), and stationary tracers, we efficiently compute general Lorenz coefficients (and variants) that are shown to correlate well with simulated recovery. For efficient optimization, we develop an adjoint code for gradient computations of the considered flow-diagnostic measures. We present several numerical examples, including optimization of rates, well placements, and drilling sequences for two-and threephase synthetic and real field models. Overall, optimizing the diagnostic measures implies substantial improvement in simulation-based objectives. IntroductionComputational tools for reservoir modeling play a critical role in the development of strategies for optimal recovery of hydrocarbon resources. These tools can be simply viewed as a means of forecasting recovery given a set of data, assumptions, and operating constraints; (e.g., to validate alternative hypotheses about the reservoir or systematically explore different strategies for optimal recovery).By nature, reservoir modeling is an interdisciplinary exercise, and reservoir-modeling tools must be well-integrated to promote collaboration between scientists and engineers with different backgrounds. New work flows are emerging because of recent advances in static reservoir characterization and dynamic flow simulation. Modern numerical flow simulators have evolved to include more-general grids, complex fluid descriptions, flow physics, well controls, and couplings to surface facilities. These generalizations have helped to more realistically describe fluid flow in the reservoir on the time scales associated with reservoir management. Meanwhile, modern reservoir-characterization techniques have shifted away from traditional variogram-based models toward object-and feature-based models that more accurately describe real geologic structures. To quantify uncertainty in the characterization, it is necessary to generate an ensemble of reservoir models that may include one thousand or more individual realizations. Reservoir simulation is computationally demanding, and a single simulation on a full reservoir model may require from some minutes to hours or even days. Direct evaluation of multiple production scenarios on large ensembles of Earth models is therefore impractical with full-featured flow simulators, and the comp...