Evaluation of Krylov Subspaces Iterative Methods for Calculating Fluid Flow in Three-Dimensional Discrete Fracture Networks

Document Type : Research Article

Authors

Dept. of Mining, Petroleum & Geophysics, Shahrood University of Technology, Shahrood, Iran

10.29252/anm.2019.8651.1301

Abstract

Summary
Computation of fluid flow in fractured rocks is very important. The rock-mass is consisted of intact rock and fractures. Number and connectivity pattern of the fractures are two key factors controlling the fluid flow in the rock-masses. One of the most accurate methods to model geometrical structure of the rock-masses is discrete fracture network (DFN). Anisotropy and heterogeneity of the rock masses often affects the computations of the flow, therefore, three-dimensional DFN has been more desirable in literatures. Numerical calculation of the fluid flow requires solving a large system of equations which are generated by discretization schemes. Solving these systems are not usually straightforward and it needs more special and complex methods to converge the result. One of the best methods in this regard are Krylov subspaces methods. Evaluation of different Krylov subspaces methods which have been validated in comparison with a direct method and 3DEC modeling, has been considered in this research and the most optimized methods have been determined using a series of sensitivity analyses. Therefore, CG, CR and IOM have been characterized as the most accurate and fastest Krylov subspaces methods. The provided results in this research can be a sufficient guideline for the researchers who want to study the fluid flow in fractured rocks.
 
Introduction
In this research, a numerical model is developed to calculate flow field in fractured rocks. The rock-mass is modeled geometrically from crude mapping data and an optimized algorithm is chosen to triangulate the geometrical framework of the model. The flow filed is discretize using a numerical scheme and the generated system of equations are solved using different iterative methods. The application of different iterative solving methods has been validated in comparison with direct one and a series of sensitivity analysis is performed to determine the most optimal iterative methods.
 
Methodology and Approaches
3D-DFN forms the geometrical framework of the present geometrical model as one of the most accurate and used methods to simulate fractured rocks. The algorithm provided by Erhel et. al. is the foundation of meshing process and FEM method is used to discretize the geometrical structure. The large system of equation generated by FEM is solved using different Krylov subspaces methods and the results are validated by LQ factorization. The sensitivity analysis is performed on two key parameters, precise and CPU run-time of the model to determine the most optimal method.
 
Results and Conclusions
The results of sensitivity analysis show that the methods CG, CR and IOM are the most optimal methods of Krylov subspaces which can be well compatible to calculation of fluid flow in 3D-DFN models.

Keywords

Main Subjects

Full Text

توده‌سنگ ترکیبی از ماده‌سنگ و ناپیوستگی است. ناپیوستگی‌ها شامل تخلخل، شکستگی‌، درزه‌، گسل‌ و صفحات لایه‌بندی هستند و نقش مهمی در حوزه مهندسی سنگ به‌ویژه در برآورد جریان سیال به‌ منظور طراحی و اجرای مخازن زیرزمینی برای ذخیره‌سازی ایمن مواد رادیواکتیو و دفن زباله‌های خطرناک، انتقال آب زیرزمینی در سفره‌ها، حرکت نفت و گاز در مخازن هیدروکربوری و تعیین میزان آبگذری در پی سدها و جداره‌های تونل‌ها دارند. همچنین شکستگی‌های سنگ عامل تعیین‌کننده‌ای در ناپایداری شیب‌های سنگی و معدن‌کاری روباز و زیرزمینی در شرایط وجود آب زیرزمینی به‌حساب می‌آیند [1-5].

نحوه اتصال شکستگی‌ها به یکدیگر (اتصال‌پذیری) در توده‌سنگ، الگو‌ی جریان سیال را تعیین می‌نماید. هنگامی‌که نفوذ‌پذیری سنگ بکر در مقایسه با نفوذ‌پذیری شکستگی‌ها بسیار کمتر باشد (به ‌ویژه برای سنگ‌های با تخلخل پایین) سیال در امتداد مسیر‌های ایجاد شده به‌ وسیله شکستگی‌های متصل ‌به ‌هم جریان می‌یابد. هنگامی‌که جریان سیال در مدل فیزیکی توده‌سنگ به حد تراوش نزدیک می‌شود، میدان جریان از الگوی اتصال‌پذیری شکستگی‌ها در مقایسه با تعداد آنها، تبعیت بیشتری خواهد کرد. در چنین حالتی، حتی یک تغییر کوچک در نحوه اتصال‌پذیری شکستگی‌ها (به‌ عنوان‌ مثال اضافه شدن یک شکستگی کوچک به مدل) می‌تواند موجب تغییر قابل‌توجهی در میدان جریان شود ]6[.

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Journal of Analytical and Numerical Methods in Mining Engineering
Volume 10, Issue 22
May 2020
Pages 15-31

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History

  • Receive Date: 29 November 2017
  • Revise Date: 23 September 2019
  • Accept Date: 12 June 2018

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