Cubic equations of state

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Table of contents

1 Introduction 
1.1 Compositional model and structure of solutions
1.1.1 About the mathematical model of compositional flow
1.1.2 Reduced model of ideal mixing
1.1.3 Shocks in the solution structure and related problems
1.1.4 General structure of solutions
1.1.5 Piece-wise constant approximation for phase concentrations
1.2 Ideal mixing theory for two- and three-phase compositional problems : State of the art
1.2.1 Two-phase partially miscible displacement theory
1.2.2 Three-phase immiscible displacement theory
1.2.3 Three-phase partially miscible displacement theory
1.2.4 Numerical modeling of multiphase compositional flow
1.3 HT-split compositional models
1.3.1 HT-splitting in the ideal mixing model
1.3.2 Asymptotic HT-split model for real systems
1.4 Outline of this thesis
1.4.1 New results obtained
1.4.2 Structure of the first part-Two phase compositional flow
1.4.3 Structure of the second part-Phase transition
1.4.4 Structure of the third part-Three phase compositional flow
2 Two-phase compositional model 
2.1 Hydrodynamic equations
2.1.1 Mass and momentum conservation
2.1.2 Dimensionless form of the conservation equations
2.1.3 Main parameters
2.2 Thermodynamic subsystem
2.2.1 General thermodynamic closure relationships
2.2.2 General relation for chemical potential
2.2.3 Cubic equations of state
2.2.4 Ideal mixing law
2.3 Approximative dissolution laws
2.3.1 Raoult’s Law
2.3.2 Henry’s Law
2.3.3 Approximation of constant K-values
3 Asymptotic compositional model 95
3.1 Asymptotic model and HT-splitting
3.1.1 Formulation of the general flow problem
3.1.2 « Canonical form » of the compositional model
3.1.3 Asymptotic expansion of the compositional model
3.1.4 Zero-order asymptotic model
3.1.5 Mnemonic rule for deriving the asymptotic model
3.1.6 Separation of the thermodynamics
3.1.7 Asymptotic thermodynamic model
3.1.8 Physical meaning of the differential thermodynamic equations
3.1.9 Improved asymptotic hydrodynamic compositional model
3.2 1D asymptotic compositional model
3.2.1 Streamline technique of flow modelling
3.2.2 1D asymptotic model along streamlines
3.2.3 First integral. Advanced equation for saturation
3.2.4 1D asymptotic model in cartesian frame
3.2.5 Relation for q
4 Riemann problem for two-phase case 
4.1 Formulation of the problem in terms of the asymptotic model
4.1.1 Hydrodynamic equations
4.1.2 Closure thermodynamic equations
4.1.3 Displacement problem
4.1.4 Twice Cauchy problem for the thermodynamic differential subsystem
4.2 Discontinuities in the asymptotic model
4.2.1 Coefficient discontinuity of the hydrodynamic equations
4.2.2 Discontinuities in the solution structure
4.2.3 Hugoniot conditions at S-chocks
4.2.4 Entropy condition
4.2.5 Admissible discontinuities
4.2.6 Lack of Hugoniot conditions
4.2.7 Additional Hugoniot conditions for coefficient discontinuities
4.2.8 Asymptotic form of the Hugoniot conditions at SC-shocks
4.2.9 S-shock : Continuity of the total velocity
5 Solution to the Riemann problem : semi-analytical method 
5.1 Diagrammatical representation of the thermodynamic part
5.1.1 Phase diagrams and tie lines
5.1.2 Phase diagram for variable pressure
5.1.3 Concept of a P-Surface
5.2 Development of the analytical front tracker
5.2.1 HT-splitting in Hugoniot conditions
5.2.2 Number of SC-chocks and crossover P-surfaces
5.2.3 Front tracker for concentrations
5.2.4 Front tracker for saturations
5.2.5 Algorithm of solution of the Riemann problem
5.3 Continuous solutions of the flow equations
5.3.1 Method of characteristics
5.3.2 Example of solution
5.4 Results of solution for three- and four- components problem
5.4.1 Description of the analyzed examples
5.4.2 Calculation of densities and viscosities
5.4.3 Three-components two-phase problem
5.4.4 Four-components two-phase problem
5.4.5 Determination of an intermediate P-surface
5.5 Comparison with numerical and analytical solutions
5.5.1 Construction of the numerical solution
5.5.2 Comparison with numerical solution
5.5.3 Construction of a particular analytical solution
5.5.4 Comparison with the analytical solution
6 Modeling two-phase compositional flow with over/under-saturated single-phase zones
6.1 Description of the problematic
6.1.1 Phase transitions and over/under-saturated zones
6.1.2 Compositional model for a binary mixture
6.1.3 Local phase equilibrium and diffusion. Curie principle
6.1.4 Equations of single-phase compositional flow in over-saturated zones
6.2 Formulation of the problem of two-phase flow with over-saturated zones
6.2.1 Physical formulation
6.2.2 Boundary and initial conditions in terms of the saturation
6.2.3 Boundary and initial conditions in terms of the total concentration
6.2.4 Conditions at the interfaces of phase transition
6.3 Problems in modeling
6.3.1 Inconsistence of the equations and variables in various zones
6.3.2 Classic method : replacement of saturation by total concentration .
6.4 Method of negative (extended) saturations
6.4.1 Equivalence principle and consistence conditions
6.4.2 Proof to the equivalence principle
6.4.3 Extended concept of saturation
6.4.4 Boundary, initial and Hugoniot conditions in terms of negative saturation
6.5 Application of the method to solve compositional flow problems
6.5.1 Example 1 : ideal mixing. Comparison to the analytical solution .
6.5.2 Example 2 : non-ideal mixing, variable pressure
6.5.3 Example 3 : non-ideal solution, variable pressure, diffusion
6.5.4 On the continuity of the fronts of phase transition induced by diffusion
6.6 Method of negative saturation in the asymptotic compositional model
6.6.1 Three-components two-phase problem
6.6.2 Four-components two-phase problem
7 Asymptotic HT-split model for three-phase compositional flow 
7.1 Mathematical model of three-phase multicomponent flow
7.1.1 Hydrodynamics equations
7.1.2 Thermodynamic closure relations
7.1.3 Particularity of a three-phase system
7.1.4 Dimensionless form of the conservation equations
7.2 Asymptotic three-phase model and HT-splitting
7.2.1 « Canonical » form of the compositional model
7.2.2 Derivation of the canonical form
7.2.3 Zero-order asymptotic model
7.2.4 Separation of the thermodynamics
7.2.5 Asymptotic thermodynamic model
7.2.6 Improved asymptotic hydrodynamic model
7.2.7 1D asymptotic three-phase model
7.3 Diagrammatical representation of the split thermodynamics
7.3.1 Phase diagrams, tie-surfaces, and triangles
7.3.2 P-surfaces and P-volumes
7.4 Riemann problem in terms of the asymptotic model
7.4.1 Problem formulation
7.4.2 Structure of the solution
7.4.3 SC-shocks
7.5 Solution to the asymptotic three-phase Riemann problem
7.5.1 HT-splitting in Hugoniot conditions
7.5.2 Concentration tracker. Number of SC-shocks
7.6 Results for three-component problem
7.6.1 Description of the cases
7.6.2 Water injection in immobile oil
7.6.3 Water injection in mobile oil
7.6.4 Carbon dioxide injection : water is absent
7.6.5 Carbon dioxide injection : immobile water
7.6.6 Carbon dioxide injection : mobile water
Conclusions
A Hougoniot conditions for inhomogeneous differential equations
B Entropy condition for inhomogeneous differential equations
Bibliographie