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## Piece-wise constant approximation for phase concentrations

The simplifying hypotheses accepted to advance the qualitative theory used in various papers do not concern only the mixing laws, but also various details of the solution struc-ture. The most significant assumption which is very frequently used, especially in the case of three-phase flow with high number of components, consists in the particular behavior of the independent phase concentrations.

The phase concentrations are frequently assumed to be piece-wise constant. The discon-tinuities in the behavior of phase concentrations, which are authorized by the conservation equations, are mobile.

This hypothesis is based on the fact that the transport of the phase concentrations is almost linear. Indeed, the non-linearity in the concentration transport can be caused by fluid compressibility or by non-linear diﬀusion. Within the framework of the ideal mixing model, the fluid is globally incompressible and the diﬀusion is totally neglected. So the transport of phase concentrations is purely convective (advective) and is not explicitly influenced by nonlinear eﬀects. Such a transport is expected to be described by a linear transport convection equation. The Riemann problem with constant initial and boundary conditions for a linear hyperbolic equation has a piece-wise constant solution with the shock propagating at the given velocity.

**Two-phase partially miscible displacement theory**

In the case of two-phase compositional system the Reimann problem has analytical solutions that have a suﬃciently simple structure.

[Buckley and Leverett 1941] developed the first analytical solutions for one-dimensional immiscible two-phase flow through porous media, as the model of oil displacement by water. The phases were also incompressible. The authors constructed the graphical-analytical solution.

[Welge 1952] simplified the water-oil displacement problem with a graphical method to construct the shock. He has shown that the shock conditions have a graphical interpreta-tion, being presented by a straight line tangent to the fractional flow function (F (s)) on the F − s diagram. This tangent line is known in the mathematical theory of quasi-linear diﬀerential equations as the Oleinik enveloped. In [ Helﬀerich 1979] the general analytical theory based on the method of characteristics for two-phase multi-component displacement was presented.

The analysis of the miscible gas and surfactant injection in oil was initially limited by ternary systems (three chemical components) : [Helﬀerich 1982 ; Hirasaki 1981 ; Larson 1979 ; Wachman 1964 ; Zanotti et al. 1983].

The model with three components, represents two diﬀerential equations, hyperbolic and non-linear, formulated for the total concentrations of two components. The solution is strongly based on the fact that the phase concentrations remain pieces-wise constant and can change only in a discontinuous way. Such a property of solution induced the introduction of the concept of tie-lines that are the lines of a constant phase composition on the phase diagram. Each rarefaction wave of the total concentration in the structure of the solution has been shown to correspond to an invariable phase concentration, or two a fixed tie-line. The shocks between the rarefaction waves are then the shocks between two diﬀerent tie lines.

The two-phase three-component system determines the main properties of the solution of the Riemann problem. It is entirely characterized by two external tie-lines : one of them corresponds to the composition of the injected fluid, while the second one represents the composition of the initial fluid in the reservoir. Both the external tie-lines are imposed by the Riemann conditions. The variation of the phase composition happens suddenly, in the form of a shock relating two tie lines. This shocks is double as it determines also the shock of the total composition (and the liquid saturation too). It was also shown that in the zone of invariable phase composition, the total composition (and saturation) can also have shocks, which is however single. These principles of constructing the solution were used further in multicomponent case.

The first solutions to quaternary systems were developed by [Monroe et al. 1990]. Through this solution he has shown that in a two-phase system with the number of compo-nents higher than three, the crossover tie-lines arise, which are intermediate between two external tie-lines and which are unknown. Monroe accepted that the phase concentrations can vary only in a discontinuous way while the passage from a tie-line to another one hap-pened through the shocks (only). This assumption has given him the possibility to develop the technique able to determine the crossover tie-lines.

**Three-phase immiscible displacement theory**

The immiscible three-phase displacement was extensively studied in a number of pu-blications. The MC for three-phase case is applied in the similar way as in two-phase case. [Falls and Schulte 1992 and Guzman 1995] suggest the detailed discussion on the construc-tion of some analogues to compositional paths for three-phase immiscible flow. At the same time, the three-phase case described by the system of two coupled nonlinear hyperbolic equations, has revealed two new research problems which had no analogues in two-phase case.

First of all, there is the problem of non-uniqueness of the entropy conditions. A consi-derable number of mathematical researches were devoted to this problem. First of all it was shown that the classical Lax and Liu entropy conditions at the shock often fail or are insuﬃcient : [Azevedo and Marchesin 1995 ; Azevedo et al. 1996, 1999, and 2002]. Some more general conditions can be obtained by the classical method of including dissipative eﬀects (diﬀusion-dispersion) in the flow model and by tending next these terms to zero. The resulting conditions determine the unique solution [Isaacson et al. 1990 ; Marchesin et al. 1997 and 1999]. However it was shown that various dispersion tensors lead to diﬀerent zero-dispersion limits. In particular, this implies that a numerical solution influenced by the artificial numerical dispersion can converge to a false solution of equation with true physical dispersion [Marchesin et al. 1997]. It was also shown that in three-phase case a new kind of transitional shocks arises : [Isaacson et al. 1990]. [Schecter et al. 1996] discusses the origin and the construction of transitional shocks in detail.

The second new problem was the demonstration of a probable flow instability in three-phase case. It was discovered in [Bell, Trangenstein, and Shubin, 1986] that two quasi-linear equations describing three-phase flow without dissipative terms can loss the hyperbolicity and become elliptic. As the initial problem for an elliptic system is unstable, this means the physical instability arising in the system behavior within a region of saturation va-riation. [Shearer and Trangenstein 1989] showed that the probable loss of hyperbolicity is influenced by the form of the relative permeability curves and can be reached for many physically meaningful relative permeability models at least at one saturation value. Thus, in contrast to the two-phase flow model, the three-phase flow without dissipative terms may not be necessarily hyperbolic, but also non-strictly hyperbolic, or mixed hyperbolic and even elliptic, depending on the relative permeability model used [Guzman, 1995]. At the next step it was shown that only the relative permeability model in which each phase permeability depends only on its own phase saturation (the model of Stone) are guaran-teed to not have elliptic regions [Trangenstein 1989]. Even in models where non strictly hyperbolic behavior is guaranteed, the hyperbolicity may fail at more than one interior saturation values [Guzman and Fayers 1994].

**HT-splitting in the ideal mixing model**

First of all, [Entov 1997] showed that the Riemann problem for the ideal mixing model in two-phase multicomponent case can be formally split into a physico-chemical and a hydrodynamic parts, if the piece-wise constant approximation for phase concentrations is assumed (section 1.1.5). Such an approximation really means automatically that the H and T are split, but the introduction of this hypothesis is simply postulated. Moreover, the splitting is reached within the ideal mixing model, which does not improve the modeling of real systems.

[Pires, Bedrikovetsky and Shapiro 2005] suggested another way to split H and T , by introducing the specific replacement of variables into the two-phase compositional model. They reduced the system of equations to auxiliary system containing only N − 1 ther-modynamical variables and to one flow equation containing one hydrodynamic parameter (saturation). Unfortunately this method was developed for the ideal mixing model for the cases which can be treated by traditional technique.

**Asymptotic HT-split model for real systems**

The asymptotic splitting between thermodynamic and hydrodynamics in gas-liquid flow was proposed in [Oladyshkin and Panfilov 2006]. In this work the full compositional model is used and none of the simplifying assumption introduced in the ideal mixing model (section 1.1.2) is used. The true properties of the fluids and the true laws of mixing are considered.

The procedure is based on two physical properties of the natural multiphase system :

i) the liquid-gas mobility ratio is small ; this is typical for any natural gas-liquid systems due to the high diﬀerence in phase viscosities. This assumption means that the cha-racteristic parameter responsible for the liquid-gas mobility ratio ω is small.

i) the time of stabilization of the pressure field is much smaller than the macroscopic time of total displacement in the reservoir. This property is also satisfied for all the natural reservoirs, which was sown in [ Panfilov 1986], [ Panfilov and Shilovich 2000]. Indeed the time of pressure stabilization is determined by fluid expansion or rock compaction, which is very fast processes having the characteristic time of several minutes, while the total time of displacement has the order of several years. Mathematically this means that the compositional model contains the parameter responsible for this time ratio ε which is also small.

Using these properties and assuming ε ω → 0 the authors have succeeded in transfor- ming N − 2 mass conservation equations into diﬀerential thermodynamic equations that did not include time and space variables. These new thermodynamic equations expressed the variation of composition in gas phase due to its exchange with low mobile liquid in an open thermodynamic system (the system in which gas flows through an immobile liquid is thermodynamically open).

Due to this the compositional model was split into the thermodynamic and hydrody-namic independent systems. Only the two-phase model was analyzed. For the two-phase case, the thermodynamic system includes 2N − 2 equations : N classic equations of phase equilibrium and N − 2 new diﬀerential thermodynamic equations. This systems contains 2N − 1 variables : 2(N − 1) phase concentrations and pressure. So this system determines the phase concentrations as the functions of pressure. The remaining hydrodynamic closed sub-system contains only two mass conservation equations with respect to two variables : pressure and saturation. The number of hydrodynamic equations does not depend of the number of chemical components.

The solution to two hydrodynamic equations was obtained for the problems of reser-voir depletion characterized by continuous behavior of all the functions. In this case the solution does not impose any technical problems. However for the Riemann problem with discontinuous solutions this model was never applied before.

### Dimensionless form of the conservation equations

In the next sections we will analyze the flow problem characterized by some geometry of the domain and some initial and boundary conditions which introduce characteristic scales of all the variables functions : the reservoir size L, the characteristic time of fluid displacement t∗, the maximal pressure diﬀerence over the domain P 0, the mean permea-bility and porosity K , Φ , the initial gas and liquid densities and viscosities ρ0g, ρ0l, µ0l and µ0g. In order to avoid the appearance of non-significant parameters, for phase densities we will use the same characteristic value ρ0g while assuming that the diﬀerence in gas and liquid densities is not too high (in contrast to the diﬀerence in viscosities). Then we can introduce the dimensionless variables : ϕg = ρg ϕl = ρl ψg = ρg µg0K ψl = ρlµl0K ρ0 ρ0 ρ0 µg K ρ0 µl K.

#### Cubic equations of state

The equation of state describes the behavior of the entire phase and determines the dependence between phase pressure, temperature, and volume for a fixed mass of the considered phase under stable equilibrium conditions.

In practice the most applications are related to the « cubic » equations of state. They has origin in the Van-Der-Waals equation (VDW EOS) which was suggested in 1983 and which is cubic with respect to the volume : P = RT − a (2.14).

where parameters a and b were introduced in order to take into account the forces of attrac-tion and repulsion between molecules. They are evaluated through the critical parameters of the fluid, by taking into account the fact that the first and the second derivatives of function P (V ) are zero at critical point.

**Formulation of the general flow problem**

Let us consider Eqs. (2.5) in the domain x ∈ Ω ⊂ R2, 0 < t ≤ T∗, where Ω is bounded and has « holes » or excluded points that represents wells (the sources). The boundary of each source is assumed to be circular. The characteristic distance between the sources L is much larger than the internal radius of a source rw , so that in several situations it is possible to consider only punctual sources. We will examine R2 only to simplify the developments.

We will analyze a suﬃciently general boundary-value problem that determines a wide spectrum of practical processes : – at the external boundary ∂Ω : p = p∗ (3.1a).

**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 **