Content of

Introduction to Geochemical Modeling

by Francis Albarede

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

2 Linear algebra

2.1 A matrix refresher
2.1.1 Definitions 2.1.2 A few rules for matrix manipulation 2.1.3 The common-dimension expansion of the matrix product 2.1.4 The subspaces of a matrix
2.2 Square matrices
2.2.1 The determinant of a matrix 2.2.2 The inverse of a matrix 2.2.3 Orthogonal matrices 2.2.4 The trace of a matrix 2.2.5 The fundamental geometric transformations 2.2.6 The metric tensor and oblique projections 2.2.7 Gram-Schmidt orthogonalization
2.3 Eigencomponents
2.3.1 General 2.3.2 Computation of eigencomponents 2.3.3 Eigencomponents of symmetric matrices
2.4 Quadratic forms and associated quadrics
2.4.1 Quadrics associated with symmetric matrices 2.4.2 Gerschgorin's circles theorem
2.5 Systems of linear differential equations
2.5.1 First-order linear homogeneous equations 2.5.2 Linear equations of order higher than one 2.5.3 Stability of solutions to linear systems of differential equations
2.6 Linear function spaces
2.6.1 General 2.6.2 Fourier series 2.6.3 Legendre polynomials 2.6.4 Associated Legendre polynomials 2.6.5 Spherical harmonics

3 Useful numerical analysis

3.1 Functions of a single variable
3.1.1 Derivatives 3.1.2 Equation of the tangent to a curve 3.1.3 Leibniz's rule for the derivative of a definite integral 3.1.4 Taylor series 3.1.5 Roots of implicit equations and extrema of functions: the Newton's method 3.1.6 Ordinary differential equations: the Euler method 3.1.7 Ordinary differential equations: the Runge-Kutta method 3.1.8 Interpolation with spline functions
3.2 Functions of several variables
3.2.1 Introduction 3.2.2 System of implicit non-linear equations: the Newton-Raphson method 3.2.3 Extrema: the steepest-descent method 3.2.4 Constrained minimization 3.2.5 The Runge-Kutta method for a system of differential equations 3.2.6 Interpolation with spline functions
3.3 Partial differential equations: the finite differences method
3.3.1 One-dimensional diffusion problems: general 3.3.2 More boundary conditions 3.3.3 A word about advection 3.3.4 Two space coordinates: The ADI method

4 Probability and statistics

4.1 A single random variable
4.1.1 General 4.1.2 Expectation and moments 4.1.3 A compendium of some common probability density functions 4.1.4 Some relationships between fundamental distributions 4.1.5 Estimators 4.1.6 Change of variable 4.1.7 Confidence intervals 4.1.8 Random deviates
4.2 Several random variables
4.2.1 Estimators 4.2.2 Useful multivariate distributions 4.2.3 Change of variables 4.2.4 Confidence region of a sample from a normal population
4.3 Error propagation and error calculation
4.3.1 General concepts 4.3.2 Linear error propagation 4.3.3 Linearized error propagation for non-linear relationships 4.3.4 Monte-Carlo simulations
4.4 Principal component analysis

5 Inverse methods

5.1 Linear estimates
5.1.1 General 5.1.2 The least-square straight line and least-square plane 5.1.3 Least-square polynomials 5.1.4 Least-square hyperbola 5.1.5 The periodogram 5.1.6 Fitting global data with spherical harmonics
5.2 Non-linear least-squares
5.3 Constrained least-squares
5.3.1 Linear constraints: the closure condition 5.3.2 Quadratic constraints: mineral reactions
5.4 Handling errors in least-square problems
5.4.1 A simple illustration: the weighted mean 5.4.2 Linear least-square systems 5.4.3 Non-linear least-square systems. Isochrons
5.5 Gradient projection and the total inverse
5.6 The continuous inverse model

6 Modeling chemical equilibrium

6.1 Introduction
6.2 The Newton-Raphson method applied to solutions
6.2.1 Homogeneous equilibrium in solutions 6.2.2 Heterogeneous equilibrium in solutions 6.2.3 More about scaling
6.3 Gibbs energy minimization
6.3.1.Mixtures of ideal gases 6.3.2 Pure coexisting phases

7 Dynamic systems

7.1 Introduction
7.2 Single-variable residence time analysis
7.2.1 Non-reactive species 7.2.2 Reactive species 7.2.3 Radioactive decay and first-order kinetics 7.2.4 Isotope and trace-element ratios 7.2.5 Heterogeneities, mixing time, and residence time 7.2.6 Stability of single-variable systems 7.2.7 Random geochemical variables 7.2.8 Population dynamics
7.3 One element in several interacting reservoirs
7.3.1 A closed-system 3-box model with concentrations as the variables 7.3.2 The general box model: an empirical model 7.3.3 The general box model with forcing terms
7.4 Several elements in several interacting reservoirs
7.4.1 Multiple reservoir isotopic systems 7.4.2 Non-linear coupling of geochemical reservoirs

8 Transport, advection, and diffusion

8.1 Fluxes
8.1.1 Basic definitions
8.2 The divergence theorem and the conservation equations
8.2.1.The continuity equation 8.2.2 The general transport equation
8.3 Advection and percolation
8.3.1 Effect of bioturbation on concentration profiles in sediments 8.3.2 Exposure ages and the assessment of erosion rates 8.3.3 Dispersal of a conservative tracer in a velocity field. 8.3.4 Percolation and infiltration metasomatism
8.4 Diffusion basics
8.4.1 The diffusion equation 8.4.2 The diffusion coefficient 8.4.3 The Matano interface
8.5 Solutions of the diffusion equation: parallel flux
8.5.1 Parallel flux: the instantaneous point source in the infinite medium 8.5.2 Two half-spaces with uniform initial concentrations 8.5.3 The infinite medium with a layer of uniform initial concentration 8.5.4 The infinite medium: an arbitrary anitial distribution 8.5.5 The infinite medium with C0(x) being a periodic function of x 8.5.6 The semi-infinite medium with constant surface concentration 8.5.7 The slab with uniform initial concentration 8.5.8 The slab with accumulation of a radiogenic isotope 8.5.9 Disequilibrium fractionation during solidification
8.6 Radial flux and spherical coordinates
8.6.1 Introduction 8.6.2 Radial diffusion in the sphere 8.6.3 Desorption from a sphere into a well-stirred solution 8.6.4 The sphere with accumulation of a radiogenic isotope
8.7 The diffusion coefficient varies with time
8.7.1 General 8.7.2 Cooling ages
8.8 Two useful steady-state solutions
8.8.1 Early diagenesis: sulfate reduction 8.8.2 The advection-diffusion model in the water column
8.9 Simultaneous precipitation and diffusion
Appendix 8A: The error function
Appendix 8B: The theta functions
Appendix 8C: Duhamel's principle

9 Trace-elements in magmatic processes

9.1 Introduction
9.2 Batch melting and crystallization
9.2.1 Introduction and forward problem 9.2.2 Inverse problem: the source composition is known 9.2.3 Inverse problem: when the source composition is unknown 9.2.4 Shaw's formulation
9.3 Incremental processes
9.3.1 Fractional crystallization: forward problem 9.3.2 Fractional crystallization: inverse problem 9.3.3 Fractional melting 9.3.4 Continuous melting
9.4 Open magmatic systems
9.4.1 The steady-state magma chamber 9.4.2 A periodically erupting, periodically refilled magma chamber 9.4.3 Assimilation-fractional crystallization (AFC) 9.4.4 Zone-refining 9.4.5 Percolation and magma segregation
9.5 Which element, which process?
9.5.1 The good use of compatible and incompatible elements 9.5.2 Elements and processes
9.6 Disequilibrium fractionation during crystal growth