Halvor Møll Nilsen

Open-source MATLAB implementation of consistent discretisations on complex grids

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 add-on 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.

MRST-AD - an Open-Source Framework for Rapid Prototyping and Evaluation of Reservoir Simulation Problems

We present MRST-AD, a free, open-source framework written as part of the Matlab Reservoir Simulation Toolbox and designed to provide researchers with the means for rapid prototyping and experimentation for problems in reservoir simulation. The article outlines the design principles and programming techniques used and explains in detail the implementation of a full-featured, industry-standard black-oil model on unstructured grids. The resulting simulator has been thoroughly validated against a leading commercial simulator on benchmarks from the SPE Comparative Solution Projects, as well as on a real-field model (Voador, Brazil). We also show in detail how practitioners can easily extend the black-oil model with new constitutive relationships, or additional features such as polymer flooding, thermal and reactive effects, and immediately benefit from existing functionality such as constrained-pressure-residual (CPR) type preconditioning, sensitivities and adjoint-based gradients. Technically, MRST-AD combines three key features: (i) a highly vectorized scripting language that enables the user to work with high-level mathematical objects and continue to develop a program while it runs; (ii) a flexible grid structure that enables simple construction of discrete differential operators; and (iii) automatic differentiation that ensures that no analytical derivatives have to be programmed explicitly as long as the discrete flow equations and constitutive relationships are implemented as a sequence of algebraic operations. We have implemented a modular, efficient framework for implementing and comparing different physical models, discretizations, and solution strategies by combining imperative and object-oriented paradigms with functional programming. The toolbox also offers additional features such as upscaling and grid coarsening, consistent discretizations, multiscale solvers, flow diagnostics and interactive visualization.

Fully-implicit simulation of vertical-equilibrium models with hysteresis and capillary fringe

Geological carbon storage represents a new and substantial challenge for the subsurface geosciences. To increase understanding and make good engineering decisions, containment processes and large-scale storage operations must be simulated in a thousand year perspective. A hierarchy of models of increasing computational complexity for analysis and simulation of large-scale CO2 storage has been implemented as a separate module of the open-source Matlab Reservoir Simulation Toolbox (MRST). This paper describes a general family of two-scale models available in this module. The models consist of two-dimensional flow equations formulated in terms of effective quantities obtained from hydrostatic reconstructions of vertical pressure and saturation distributions. The corresponding formulation is fully implicit and is the first to give a mass-conservative treatment and include general (non-linearized) CO2 properties. In particular, the models account for compressibility, dissolution, and hysteresis effects in the fine-scale capillary and relative permeability functions and can be used to accurately and efficiently study the combined large-scale and long-term effects of structural, residual, and solubility trapping.

A Multiscale Mixed Finite Element Solver for Three Phase Black Oil Flow

Previous research has shown that multiscale methods are robust and capable of providing more accurate solutions than traditional upscaling methods. Multiscale methods solve the pressure equation on a coarse grid, but capture the effects from fine-scale heterogeneities through basis functions computed numerically from local single-phase problems on the underlying geocellular grid. Published results have so far been limited to simple Cartesian grids and/or incompressible flow. Here, we present a multiscale mixed finite-element method for three-phase black-oil flow on geomodels with industry-standard complexity. In particular, we discuss which effects can be incorporated in the multiscale basis functions and which effects should be modeled only on the coarsened simulation grid. Moreover, we describe how to handle degenerate hexahedral cells and non-matching interfaces that occur across faults. Finally, we present results of flow simulations on models of industry-standard complexity and demonstrate how multiscale methods can be used to simulate three-phase black-oil flow directly on high-resolution geomodels. The multiscale methods presented herein enable varying resolution and provide a systematic procedure for coarsening or refining the simulation model. Introduction For the oil industry to succeed in increasing oil recovery there is a growing trend for model-based decisions. New and exciting developments are seen in a variety of areas such as real-time reservoir management, uncertainty quantification, integrated operations, closed-loop management, and production optimization. Common to all these fields of endeavor is the requirement for fast flow simulation in which the simulation model is tightly coupled to the geology and dynamic data sources. However, there is a significant, and increasing, gap between the level of detail seen in geological models and the capabilities of contemporary reservoir simulators. Mature fields have a large amount of geological and geophysical data that can be used to create static models, and sizes of high-resolution geological models range from a few million and up to a billion cells. Contemporary reservoir simulators typically operate on model sizes from tens of thousands to a few million cells. Similarly, mature fields usually have a lot of dynamic data (pressure tests, production data, 4-D seismics, etc) that could be used to calibrate and history match the high-resolution geological models. Unfortunately, instead of focusing on understanding the physical characteristics of reservoirs and the economic consequences of different developments, a lot of valuable human resources is diverted to upscaling (and downscaling) and its negative consequences for the representation of heterogeneities and fluid flow. Upscaling is a costly process which additionally wastes much of the information inherent in high-resolution geological models since local flow structures are only preserved in an average sense on the upscaled grid. Enabling the oil industry to make a step-change in its work processes therefore calls for a radical speedup of flow simulation and for simulators that are equipped to utilize both static data and the vast amount of dynamic data that becomes available. As an example, it would be highly attractive if reservoir simulation could be performed at seismic resolution in order to use 4-D seismics to history-match simulation models. There are several technological developments that can contribute to a radical speedup of flow simulation: advances in hardware, parallel algorithms, improved (non)linear solvers, and alternative formulations (streamlines, operator splitting), to name a few. Another important contribution may come from multiscale methods, as will be discussed herein. Generally speaking, multiscale methods are numerical methods and strategies that aim to describe physical phenomena on coarse grids while accounting for the influence of fine-scale structures in the porous media. However, unlike traditional upscaling techniques, multiscale methods often provide a mechanism to recover an approximate fine-scale solutions. Multiscale modeling of flow and transport in porous media has become a hot research topic in recent years. A quite comprehensive overview of current developments is found in a recent issue of the Computational Geosciences journal (Juanes and Tchelepi 2008). Common for all these methods is that they seek efficient solutions of elliptic (or parabolic) equations with rough coefficients in the absence of scale separation, which is often assumed in many other multiscale methods. In the race for making a

A simulation workflow for large-scale CO2 storage in the Norwegian North Sea

Large-scale CO2 injection problems have revived the interest in simple models, like percolation and vertically-averaged models, for simulating fluid flow in reservoirs and aquifers. A series of such models have been collected and implemented together with standard reservoir simulation capabilities in a high-level scripting language as part of the open-source MATLAB Reservoir Simulation Toolbox (MRST) to give a set of simulation methods of increasing computational complexity. Herein, we outline the methods and discuss how they can be combined to create a flexible tool-chain for investigating CO2 storage on a scale that would have significant impact on European CO2 emissions. In particular, we discuss geometrical methods for identifying structural traps, percolation-type methods for identifying potential spill paths, and vertical-equilibrium methods that can efficiently simulate structural, residual, and solubility trapping in a thousand-year perspective. The utility of the overall workflow is demonstrated using real-life depth and thickness maps of two geological formations from the recent CO2 Storage Atlas of the Norwegian North Sea.

Vertically Averaged Equations with Variable Density for \hbox {CO}_2 Flow in Porous Media

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Spill-point analysis and structural trapping capacity in saline aquifers using MRST-co2lab

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Robust Simulation of Sharp-Interface Models for Fast Estimation of CO2 Trapping Capacity in Large-Scale Aquifer Systems

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