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.

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Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models. Herein, we consider the problem of computing the effective permeability of rock samples based on high-resolution 3D CT scans containing millions of voxels. We use the Stokes-Brinkman equations in the entire domain, covering regions of free flow governed by the Stokes equations, porous Darcy flow, and transitions between them. The presence of different length scales and large (ten orders of magnitude) contrasts in permeability leads to highly ill-conditioned linear systems of equations, which are difficult to solve. To obtain a problem that is computationally tractable, we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models. We find that, in terms of effective permeability, the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability. All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows, depending on if there do or do not exist percolating free-flow regions. The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.

Storage of CO2 in deep saline aquifers is considered an important means to reduce anthropogenic CO2 in the atmosphere. Assessing the risk of storage operations requires accurate modeling of migration of injected CO2. However, since potential injection sites typically are very large and timescales long, flow simulation with traditional methods from the petroleum industry is often not feasible. Also, CO2 is very mobile and the flow is usually confined to thin layers, which put severe requirements on the vertical grid resolution. Using a vertical equilibrium assumption, the flow of a layer of CO2 can be approximated in terms of its thickness to obtain a 2D simulation model. Although this approach reduces the dimension of the model, important information of the heterogeneities in the underlying 3D medium is preserved. In this paper, we consider the Johansen formation, a candidate for CO2 sequestration, to compare the use of 3D simulations to simulations with a vertical equilibrium 2D model. We discuss numerical aspects of using the different methods, and demonstrate that the vertical equilibrium model provides more accurate results when the vertical grid resolution is low. Moreover, we investigate how averaging of parameters influences the accuracy of the vertically equilibrium solution. Recently, there has been a renewed interest in VE methods as a means to simulate large-scale CO 2 migration, for which a sharp-interface assumption with vertical equilibrium may be reasonable due to

Cavities and fractures can significantly affect the flow paths of carbonate reservoirs and should be accurately accounted for during flow simulation. Herein, our goal is to compute the effective permeability of rock samples based on high-resolution 3D CT-scans containing millions of voxels. To this end, we need a flow model that properly accounts for the effects of Darcy flow in the porous material and Stokes flow in the void volumes on all relevant scales. The presence of different length scales and large contrasts in the petrophysical parameters leads to highly ill-conditioned linear systems that make such a flow model very difficult to solve, even on large-scale parallel computers. To identify simplifications that render the problem computationally tractable, we analyze the relative importance of the Stokes and Darcy terms for a wide variety of parameter ranges on an idealized 2D model. We find that a system with a through-going free flow region surrounded by a low permeable matrix can be accurately modeled by ignoring the Darcy matrix and simulating only the Stokes flow. Using the obtained insight, we are able to compute the effective permeability of a specific model from a CT-scan that contains more than eight million voxels.