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Analytical solutions for one-dimensional two-phase multicomponent flows in porous media describe processes of enhanced oil recovery, environmental flows of waste disposal, and contaminant propagation in subterranean reservoirs and water management in aquifers. We derive the exact solution for

Exact self-similar solutions of Riemann problems for hyperbolic systems of conservation laws and non-self-similar solutions of hyperbolic wave interactions have been derived for various flows in gas dynamics, shallow waters, and chromatography (see monographs [

The scalar hyperbolic equations (

Riemann problem (

The system (

The particular case of so-called multicomponent polymer flooding is the dependency of the component sorption concentration of its own concentration only

The distinguished invariant feature of

The non-self-similar solution of system (

The splitting technique reduces number of equations in (

The structure of the text is as follows. The particular case of the general system (

Let us discuss the displacement of oil by aqueous chemical solution with water drive accounting for different salinities of formation and injected waters. In the following text, the component

The system of governing equations consists of mass balance equations for aqueous phase, for dissolved and adsorbed chemical, and for dissolved salt [

The fractional flow function (water flux) depends on the water saturation

Fractional flow curves and Riemann problem solution, where

The system (

The displacement of oil by chemical slug corresponds to the following initial-boundary problem:

The initial condition is denoted by

Generally

The solution of the Riemann problem is self-similar:

System of (

In the present section we briefly explain the splitting method for the solution of hyperbolic system of conservation laws equations (

As it follows from divergent (conservation law) form of equation for mass balance for water (

Introduction of potential function (Lagrangian coordinate) and mapping between independent variables.

Derivation of mass balance equation in Eulerian and Lagrangian coordinate systems.

From fluid mechanics point of view,

Let us derive the relationship between the elementary wave speeds of the system in

Speeds of a particle in Eulerian and Lagrangian coordinates.

Let us derive (

Applying the definition of the potential function equation (

The auxiliary system contains only equilibrium thermodynamic variables, while the initial system contains both hydrodynamic functions (phase’s relative permeabilities and viscosities) and equilibrium thermodynamic variables.

The above splitting procedure is applied to the solution of displacement of oil by polymer slug with alternated salinity in the next section.

Introducing new variables “density”

Projection of the space of the large system into that of auxiliary system and the lifting procedure using the solution of auxiliary system.

The boundary conditions for slug problem equation (

The initial conditions for slug problem equation (

Let us discuss the solution of the problem equations (

The mass balance conditions on shocks which follow from the conservation law (Hugoniot-Rankine condition) form of the system (

As it follows from equality (

The shock waves must obey the Lax evolutionary conditions [

The solution of auxiliary system is presented in Figure

Solution of the auxiliary problem. (a) Adsorption isotherm for chemical for different water salinities and the Riemann problem solution; (b) Riemann problem solution on the plane of chemical concentration

Figure

The image of the solution in (

The solution of lifting equation with known concentrations (

Solution of the auxiliary and lifting system for slug problem in

Time

Non-self-similar solution of the problem for wave interactions in

Solution of the Riemann problem: (a) trajectories of shock fronts and characteristic lines in

Now let us solve the slug problem equations (

So, zone I in Figure

Now let us solve the lifting equation (

The Hugoniot-Rankine condition for the rear slug front is

The solution of the slug injection problem: (a) trajectories of shock fronts and characteristic lines in

Here the trajectory of the rear slug front

Solution of the lifting equation in

In reality, there is no chemical initially in the reservoir during the majority of chemical enhanced oil recovery applications; that is,

Solution of the lifting system in (

Non-self-similar solution of the problem for wave interactions in

Following exact solution equations (

During continuous injection

The trajectories of shocks 2–>3, 3–>4, and 4–>

The corresponding profiles of saturation and concentrations are shown in Figure

Injection of water without chemical and with salinity

The propagation of the rear slug front from the beginning of water drive injection in the reservoir is shown in Figure

The profiles are shown in Figures

Application of the splitting method to

The method of splitting between hydrodynamics and thermodynamics in system of two-phase multicomponent flow in porous media allows obtaining an exact solution for non-self-similar problem of displacement of oil by chemical slug with different water salinity for the case of linear polymer adsorption affected by water salinity.

The solution consists of explicit formulae for water saturation and polymer and salt concentrations in the continuity domains and of implicit formulae for front trajectories.

First integrals for front trajectories allow for graphical interpretation at the hodograph plane, yielding a graphical method for finding the front trajectories.

For linear sorption isotherms, the solution depends on three fractional flow curves that correspond to initial reservoir state

For linear sorption isotherms, the only continuous wave is

Introduction of salinity dependency for sorption of the chemical introduces the intermediate

The exact solution shows that the injected chemical slug dissolves in the connate reservoir water rather than in the chemical-free water injected after the slug.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Long-term cooperation in hyperbolic systems and fruitful discussions with Professors A. Shapiro (Technical University of Denmark), Y. Yortsos (University of Southern California), A. Roberts (University of Adelaide), A. Polyanin (Russian Academy of Sciences), M. Lurie, and V. Maron (Moscow Oil and Gas Gubkin University) are gratefully acknowledged. The reviewers are gratefully acknowledged for their critical comments yielding to significant improvement of the text.