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Modern geochemistry aims to provide an accurate description of geological processes, and a set of models and quantitative rules that help predict the evolution of geological systems. This work is an introduction to the mathematical methods of geochemical modeling, largely based on examples presented with full solutions. It shows how geochemical problems, dealing with mass balance, equilibrium, fractionation and dynamics and transport in the igneous, sedimentary and oceanic environments, can be reformulated in terms of equations. Its practical approach then leads to simple but efficient methods of solution. This book should help the motivated reader to overcome the formal difficulties of geochemical modeling, and bring state-of-the-art methods within reach of advanced students in geochemistry and geophysics, as well as in physics and chemistry.
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This book provides a quantitative treatment for a variety of geochemical problems involving mass balance, equilibrium, dynamics, and transport. Numerous applications from igneous and sedimentary environments are presented in the form of problems and their explicit solutions. It will particularly appeal to geochemists who need a proper grounding in the essential modeling methods brought to the field from physics and chemistry. Applications to natural environment make these methods also of interest to the geophysics, physics and chemistry community. INTRODUCTION TO GEOCHEMICAL MODELING INTRODUCTION TO GEOCHEMICAL MODELING FRANCIS ALBAREDE Ecole Normale Superieure de Lyon CAMBRIDGE UNIVERSITY PRESS Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211. USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1995 First published 1995 First paperback edition (with corrections) 1996 A catalogue record for this book is available from the British Library Library of congress cataloguing in publication data Albarede, Francis. Introduction to geochemical modeling/Francis Albarede. p. cm. ISBN 0-521-45451-4 1. Geochemistry - Mathematical models. I. Title. QE515.A53 1995 553.9'Ol'5118-dc2O 93-49747 CIP ISBN 0 521 45451 4 ISBN 0 521 57804 3 hardback paperback Transferred to digital printing 2002 PR To Benjamin, Olivier, and Isabelle as a sign of deep affection. Contents Foreword Preface xv xvii 1 Mass balance, mixing, and fractionation 1.1 Concentrations as mixing variables 1.1.1 Basic concepts 1.1.2 Special case: binary mixing 1.1.3 Ternary mixing and removal 1.1.4 The inverse approach 1.2 Reactional assemblage 1.3 Working with ratios 1.3.1 Introduction 1.3.2 Ratio-concentration relationships in binary mixing 1.3.3 Ratio-ratio relationships in binary mixing 1.3.4 Mixing hyperbola: the inverse problem 1.3.5 Ratio-ratio relationships in ternary mixing 1.4 Normalized variables 1.5 Incremental processes (distillation) 1.5.1 Introduction 1.5.2 Concentration changes upon closed-system crystallization 1.5.3 Changes in element and isotope ratios upon closed-system crystallization 1.5.4 FeO-MgO fractionation during olivine crystallization in basalts 1.5.5 Elemental fractionation during basalt differentiation 1.5.6 Fractional melting 1.5.7 Fractional condensation 1.5.8 Open-system isotopic exchanges 1 1 1 3 6 9 9 11 11 15 18 26 28 31 34 34 35 2 Linear algebra 2.1 A matrix refresher 2.1.1 Definitions 2.1.2 A few rules for matrix manipulation 52 52 52 53 IX 36 39 41 43 46 47 x 2.2 2.3 2.4 2.5 2.6 Contents 2.1.3 The common-dimension expansion of the matrix product 2.1.4 The subspaces of a matrix Square matrices 2.2.1 The determinant