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

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

Compositional model

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

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

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

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

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

Diagrammatical Representation of the Split Thermodynamics

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

Concept of a P-surface

Reimann problem in terms of Ht-split model

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

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.

Degenerating Hugoniot conditions Compact form of the Compositional model Where New Hugoniot conditions

Pure Saturation shocks Eliminating liquid velocity Eliminating gas velocity Adding up together

Total System of Hugoniot Conditions Where:

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.

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

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

Unknown phase concentrations and equations at sc-shock

Shock numberUnknown concentrations First ( from injection P-surface )2N Second up to (N-3)th4N Last (t o initial P-surface )2N Total4N(N-3) Type of equationNumber of utilization Total number (N-3) Hugoniot Conditions(N-2)(N-3)*(N-2) N chemical potential equilibrium2(N-3)2N(N-3) 2 normalizing equations2(N-3)4(N-3) N-2 differential thermodynamics equations (N-3)(N-2)(N-3) Total4N(N-3) Unknown phase concentrations: Equations at sc-shocks :

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.

Results for three and four components problem

Three Components Problem

Four Components Problem

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:

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