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.
Recently, a great deal of attention has been focused on the construction of exponential integrators for semi-linear problems. In this paper we describe a matlab package which aims to facilitate the quick deployment and testing of exponential integrators, of Runge-Kutta, multistep and general linear type. A large number of integrators are included in this package along with several well known examples. The so-called ϕ-functions and their evaluation is crucial for stability and speed of exponential integrators, and the approach taken here is through a modification of the scaling and squaring technique; the most common approach used for computing the matrix exponential.
We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of B-series and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [Math. Comp., 46 (1986), pp. 119-133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.