HT-splitting method for the Riemann problem of multicomponent two phase flow in porous media

1. Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée (LEMTA) HT-splitting method for the Riemann problem of multicomponent two phase flow in porous media Anahita ABADPOUR Mikhail PANFILOV

2. Introduction • Compositional model • Diagrammatical representation of the split thermodynamics • Reimann problem in terms of Ht-split model • Results for three and four components problem

3. Compositional model

4. Hydrodynamic Equations • Mass conservation of each component • Momentum balance for each phase Where

5. Thermodynamics Closure relations • Chemical potential equilibrium equations • Equations of phase state • Normalizing equations • Rheological equations of state

6. Main Parameters • Relative phase mobility • Perturbation ratio , Splitting of Hydrodynamics & Thermodynamics (S. Oladyshkin, M. Panfilov -2006 ) = + Thermodynamic independent model (2N+1 equations) Hydrodynamic independent model (2 equations) Compositional flow model (2N+3 equations)

7. Ht-split form of compositional model • Differential thermodynamic equations (DTE) • Hydrodynamic equations where

8. First integral of the HT-split model • Steady-state pressure • Equation of saturation transport Where Fractional mass flow function of gas phase

9. Diagrammatical Representation of the Split Thermodynamics

10. Phase diagrams and tie-lines P UndersaturatedLiquid Oversaturated Gas Equilibrium Liquid Equilibrium Liquid + Gas Equilibrium Gas C – total concentration of the light component

11. Concept of a P-surface

12. Reimann problem in terms of Ht-split model

13. Problem formulation • Initial state of gas saturation • Pressure boundary condition • Initial condition of phase composition

14. Lack of discontinuity conditions • Hugoniot condition for transport equation • Entropy condition ( Lax inequality ) • These two conditions are unfortunately largely insufficient, as a simultaneous shock of saturation and concentrations is determined by N + 1 parameters from one side of the shock: the shock velocity, N-1 concentrations and 1 saturation at the shock, where N is the number of components.

15. Degenerating Hugoniot conditions • Compact form of the Compositional model Where • New Hugoniot conditions

16. Pure Saturation shocks • Eliminating liquid velocity • Eliminating gas velocity • Adding up together

17. Total System of Hugoniot Conditions Where:

18. Intermediate P-surfaces One of the significant qualitative results of the classic theory of the Riemann problem announces that in an N-component two-phase system that does not change the number of phases, the phase concentrations should follow (N−1) different tie lines including two ”external” tie lines that correspond to the initial and the injection states and (N−3) ”intermediate” or ”crossover” tie-lines. In the case of variable pressure we assume the same result to be valid, so in a N-component two-phase system that does not change the number of phases the phase concentrations should follow (N−1) different P-surfaces including two ”external” P-surfaces that correspond to two boundary pressures and (N−3) ”intermediate” or ”crossover” P-surfaces. This means that the concentrations can have (N−2) internal shock.

19. Algorithm of solving the Riemann problem • Front tracing : • Determination of the backward and forward concentrations at all the CS-shocks: solution to the transcendent system of : • Degenerating Hugoniot relations at the shocks • Differential thermodynamic equations between the shocks • Chemical potential equilibrium equations • Determination of the shock saturations and velocities : solution to the transcendent system of : • Remaining Hugoniot Relations • Entropy Condition • Solution to the differential transport equation : • Solution to the saturation transport equation while taking into account the priori determined parameters and placing of all the shocks

20. Determining intermediate concentrations From: With some arithmetic calculations: Eliminating saturations and densities:

21. Unknown phase concentrations and equations at sc-shock

22. Unknown phase concentrations: • Equations at sc-shocks :

23. Determination of the sc-shocks saturations After finding the concentrations on each P-surface, we are able to use just 2 of these N-1 equations to find the saturations before and after each concentrations shock: Here the total velocity was assumed to be constant. It is significant that this assumption is not exact, but it is much weaker than the assumption of constant total velocity all over the problem in the classic theory.

24. Results for three and four components problem

25. Three Components Problem

26. Four Components Problem

27. Advantage of the developed method • Concentrations at the shocks can be determined explicitly as the solution to an algebraic system of equations coupled with the thermodynamic block. • For a sufficiently low number of components, this system can be solved in the analytical way. • Saturations at the discontinuity then could be determined using the intermediate concentrations. parameters of the shocks can be determined before constructing the solution to the Riemann problem:

28. Thank you for your attention!