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Myron B. Allen - The Mathematics of Fluid Flow Through Porous Media

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Master the techniques necessary to build and use computational models of porous media fluid flow

In The Mathematics of Fluid Flow Through Porous Media, distinguished professor and mathematician Dr. Myron B. Allen delivers a one-stop and mathematically rigorous source of the foundational principles of porous medium flow modeling. The book shows readers how to design intelligent computation models for groundwater flow, contaminant transport, and petroleum reservoir simulation.

Discussions of the mathematical fundamentals allow readers to prepare to work on computational problems at the frontiers of the field. Introducing several advanced techniques, including the method of characteristics, fundamental solutions, similarity methods, and dimensional analysis, The Mathematics of Fluid Flow Through Porous Media is an indispensable resource for students who have not previously encountered these concepts and need to master them to conduct computer simulations.

Teaching mastery of a subject that has increasingly become a standard tool for engineers and applied mathematicians, and containing 75 exercises suitable for self-study or as part of a formal course, the book also includes:

  • A thorough introduction to the mechanics of fluid flow in porous media, including the kinematics of simple continua, single-continuum balance laws, and constitutive relationships
  • An exploration of single-fluid flows in porous media, including Darcys Law, non-Darcy flows, the single-phase flow equation, areal flows, and flows with wells
  • Practical discussions of solute transport, including the transport equation, hydrodynamic dispersion, one-dimensional transport, and transport with adsorption
  • A treatment of multiphase flows, including capillarity at the micro- and macroscale

Perfect for graduate students in mathematics, civil engineering, petroleum engineering, soil science, and geophysics, The Mathematics of Fluid Flow Through Porous Media also belongs on the bookshelves of any researcher who wishes to extend their research into areas involving flows in porous media.

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Table of Contents List of Tables Chapter 3 Chapter 7 Appendix A List of - photo 1
Table of Contents
List of Tables
  1. Chapter 3
  2. Chapter 7
  3. Appendix A
List of Illustrations
  1. Chapter 2
  2. Chapter 3
  3. Chapter 4
  4. Chapter 5
  5. Chapter 6
  6. Chapter 7
  7. Appendix B
  8. Appendix D
Guide
Pages
The Mathematics of Fluid Flow Through Porous Media

Myron B. Allen

University of Wyoming

The Mathematics of Fluid Flow Through Porous Media - image 2

This edition first published 2021

2021 John Wiley & Sons, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Myron B. Allen to be identified as the author of this work has been asserted in accordance with law.

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The contents of this work are intended to further general scientific research, understanding, and discussion only and are not intended and should not be relied upon as recommending or promoting scientific method, diagnosis, or treatment by physicians for any particular patient. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of medicines, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each medicine, equipment, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress CataloginginPublication Data Applied for:

ISBN: 9781119663843

Cover Design: Wiley

Cover Image: Myron B. Allen

To Professor George F. Pinder, who has lit the path for so many.


Preface

Seldom turns out the way it does in the song.

Robert Hunter

This book provides a semesterlength course in the mathematics of fluid flows in porous media. Over a 20year span, I taught such a course every few years to doctoral students in engineering, mathematics, and geophysics. Most of these students' research involved flow and transport in groundwater aquifers, soils, and petroleum reservoirs. The students' mathematical backgrounds ranged from standard undergraduate engineering requirements to more advanced, graduatelevel training.

The book emphasizes analytic aspects of flows in porous media. This focus may seem odd: Most mathematically oriented scholarship in the area is computational in nature, owing both to the heterogeneity of natural porous media and to the inherent nonlinearity of many underground flow models. Nevertheless, while many superb books cover computational methods for flows in porous media, intelligent design of numerical approximations also requires a grasp of certain analytic questions:

  • Where do the governing equations come from?
  • What physics do they model, and what physics do they neglect?
  • What qualitative properties do their solutions exhibit?

Where appropriate, the book discusses numerical implications of these questions.

The exposition should be accessible to anyone who has completed a baccalaureate program in engineering, mathematics, or physics at a US university. The book makes extensive use of multivariable calculus, including the integral theorems of vector field theory, and ordinary differential equations. Several sections exploit concepts from firstsemester linear algebra. No prior study of partial differential equations is necessary, but some exposure to them is helpful.

After a brief introduction in introduces the mass and momentum balance laws from which the governing partial differential equations arise. This chapter sets the stage for a pattern that appears throughout the book: We derive governing equations, then analyze representative or generic solutions to infer important attributes of the flows.

focuses on model formulation.

Two features of the book deserve comment.

  • Over 100 exercises, most of them straightforward, appear throughout the text. Their main purpose is to engage the reader in some of the steps required to develop the theory.
  • There are four appendices. The first simply lists symbols that have dedicated physical meanings. The remaining appendices cover three common curvilinear coordinate systems, the Buckingham Pi theorem of dimensional analysis, and some aspects of surface integrals. While needed at certain junctures in the text, these topics seem ancillary to the book's main focus.

I owe thanks to dozens of students at the University of Wyoming who endured early versions of the notes for this book. These men and women convinced me of its utility and offered many corrections and suggestions for improvement. Professor Frederico Furtado kindly offered additional corrections, generous encouragement, and insights deeper than he will admit. I also owe sincerest thanks to my colleagues in the University of Wyoming's Department of Mathematics and Statistics, from whom I have learned a lot. I cannot have asked for a better academic home. Finally, my wife, Adele Aldrich, deserves more gratitude than I know how to express, for her support through the entire process.

Myron B. Allen

Laramie, Wyoming

December, 2020


Introduction
1.1 Historical Setting

The mathematical theory of fluid flows in porous media has a distinguished history. Most of this theory ultimately rests on Henry Darcy's 1856 engineering study .

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