Stein Krogstad

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.

A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids

We analyze 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 ...

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.

Nonlinear output constraints handling for production optimization of oil reservoirs

Adjoint-based gradient computations for oil reservoirs have been increasingly used in closed-loop reservoir management optimizations. Most constraints in the optimizations are for the control input, which may either be bound constraints or equality constraints. This paper addresses output constraints for both state and control variables. We propose to use a (interior) barrier function approach, where the output constraints are added as a barrier term to the objective function. As we assume there always exist feasible initial control inputs, the method maintains the feasibility of the constraints. Three case examples are presented. The results show that the proposed method is able to preserve the computational efficiency of the adjoint methods.

The Application of Flow Diagnostics for Reservoir Management

Flow-diagnostics, as referred to herein, are computational tools based on 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 reservoir management workflows. In particular, based on finite volume discretizations for pressure, time-offlight (TOF) and stationary tracer, we efficiently compute general Lorenz coefficients (and variants) which 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 three phase synthetic and real field models. Overall, optimizing the diagnostic measures imply substantial improvement in simulation-based objectives.

Adjoint Multiscale Mixed Finite Elements

The topic of this thesis is fast and accurate simulation techniques used for simulations of flow and transport in porous media, in particular petroleum reservoirs. Fast and accurate simulation techniques are becoming increasingly important for reservoir management and development, as the geological models increase in size and level of detail and require more computational resources to be utilized. The multiscale framework is a promising approach to facilitate simulation of detailed geological models. In contrast to traditional upscaling approaches, the multiscale methods have the detailed geological models present at all times. The work in this thesis includes development of a multiscale-multiphysics method for naturally fractured reservoirs and a new coarsening strategy for geological models to facilitate fast and accurate transport simulations in a multiscale framework. In addition, the work comprises an application of the multiscale framework for flow and transport simulation for rate optimization loops. The coarsening strategy generates flow-based transport grids and is based on amalgamating cells from a fine model, typically the geological model, according to an indicator function. The research indicates a great potential for flexibility and scalability suitable for multi-fidelity simulators

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

Multiscale Mixed Finite Element Modeling of Coupled Wellbore/Near-Well Flow

An accurate well modeling capability is important for both production and reservoir engineering calculations. Ideally, the models used for these two applications should be related in a logical manner. Multiscale methods allow varying degrees of resolution and can therefore provide a natural linkage between production and reservoir models. In this work, we present a coupled wellbore-reservoir flow model that is based on a multiscale mixed finite element formulation for reservoir flow linked to a drift-flux wellbore flow representation. The model is able to capture efficiently the effects of near-well heterogeneity in the reservoir and phase holdup and pressure variation in the wellbore. The formulation presented here is for oil-water systems. The basic reservoir-wellbore linkage is described and validated through comparison to results from an existing simulator. The multiscale methodology is then applied to a heterogeneous reservoir model. Both vertical and deviated wells are considered. Comparisons of the multiscale solution to the fully resolved (fine-scale) solution demonstrate the high degree of accuracy of the method, for both reservoir and wellbore quantities, as well as its efficiency.

A Low Complexity Lie Group Method on the Stiefel Manifold

A low complexity Lie group method for numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is presented. Based on the quotient space representation of the Stiefel manifold we provide a representation of the tangent space suitable for Lie group methods. According to this representation a special type of generalized polar coordinates (GPC) is defined and used as a coordinate map. The GPC maps prove to adapt well to the Stiefel manifold. For the n×k matrix representation of the Stiefel manifold the arithmetic complexity of the method presented is of order nk2, and for n≫k this leads to huge savings in computation time compared to ordinary Lie group methods. Numerical experiments compare the method to a standard Lie group method using the matrix exponential, and conclude that on the examples presented, the methods perform equally on both accuracy and maintaining orthogonality.

On enumeration problems in Lie-Butcher theory

The algebraic structure underlying non-commutative Lie-Butcher series is the free Lie algebra over ordered trees. In this paper we present a characterization of this algebra in terms of balanced Lyndon words over a binary alphabet. This yields a systematic manner of enumerating terms in non-commutative Lie-Butcher series.