An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators

Control-volume discretizations using multipoint flux approximations (MPFA) were developed in the last decade. This paper gives an introduction to these methods for quadrilateral grids in two and three dimensions. The introduction is kept on a basic level, and a brief summary to more advanced results is given. Only the O-method with surface midpoints as continuity points is discussed. Flux expressions are derived both in physical and in curvilinear space. Equations for calculation of the transmissibility coefficients are given, and an explicit solution is shown for constant coefficients. K-orthogonality, stability and monotonicity are discussed, and an iterative solution technique is presented. Two numerical examples close the paper.

An Introduction to Multipoint Flux Approximations for Quadrilateral Grids

http://www.math.pitt.edu/~yotov/teaching/16-2/math2602/MFD-JCP-survey.pdf

Mimetic finite difference method

This paper describes hierarchical approach to modeling flow in a naturally fractured formation. Our model is based on calculating the effective permeability of a fractured formation, as a function of grid block size, and using the results in a conventional finite difference flow simulator. On the basis of their length (lf) relative to the finite difference grid size (lg), fractures are classified as belonging to one of three groups: (1) short fractures (lf≪ lg), (2) medium-length fractures (lf ∼ lg), and (3) long fractures (lf≫ lg). The effects of the fractures belonging to each class are computed in a hierarchical manner. The permeability contribution from short fractures is derived in an analytical expression and used as an enhanced matrix permeability for the next-scale (medium-length) calculation. The effective matrix permeability associated with medium-length fractures is numerically solved using a boundary element method. The long fractures are modeled explicitly as major fluid conduits. As numerical examples, tracer transport in fractured formations was illustrated. The numerical results clearly indicated that effective tensor permeability well represented directional, enhanced permeability in fractured formations. The fluid-conduit formulation captured the efficient fluid transport by long fractures.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000WR900340

Hierarchical modeling of flow in naturally fractured formations with multiple length scales

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.

/pdf/open-source-matlab-implementation-of-consistent-4xy2cqn5o8.pdf

Open-source MATLAB implementation of consistent discretisations on complex grids

Discretization methods are proposed for control-volume formulations on polygonal and triangular grid cells in two space dimensions. The methods are applicable for any system of conservation laws where the flow density is defined by a gradient law, like Darcy's law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy. This paper gives a derivation of the methods, and the relation to previously published methods is also discussed. A further discussion of the methods, including numerical examples, is given in the companion paper, Part II [SIAM J. Sci. Comput., pp. 1717--1736].

Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods

Discretization on Non-Orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media

Curvilinear grids applied in reservoir simulation and elsewhere, are usually orthogonal. Enhanced flexibility is the main reason why non-orthogonal, curvilinear grids are considered here. The model equations are transformed from Cartesian to curvilinear space, and then discretized with a CVFD-method. In two space dimensions, the molecule used to approximate a flux through a cell face contains six grid points. The proposed methods for calculating transmissibilities fully account for anisotropy, inhomogeneities, and non-orthogonality of the grid. In the vicinity of inactive cell faces or fluid contacts, the flux molecule may have to be redefined. Close to a fluid contact, different flux molecules will be used for each phase. Since fluid contacts generally are moving, this requires dynamic flux molecules.

Discretization on Non-Orthogonal, Curvilinear Grids for Multi-Phase Flow

A class of flux-conservative diseretization schemes are presented for alternative grids in two space dimensions. The family of methods represents a unified approach to diseretization in anisotropic, inhomogeneous media. Both triangular, quadrilateral and polygonal grids are eonsidered. A common feature for all diseretizations is the ability to handle media inhomogeneities in combination with full tensor anisotropy.

A Class of Discretization Methods for Structured and Unstructured Grids in Anisotropic, Inhomogeneous Media

Discretization methods are presented for control-volume formulations on quadrilateral, triangular and polygonal grid cells in two space dimensions. The methods are applicable for any system of conservation laws where the flow density is defined by a gradient law, like Darcy’s law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy.

Discretization on general grids for general media

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Ø. Bøe

Author Tools

Chat about Author