A two-fluid hyperbolic model in a porous medium

Abstract

Abstract. The paper is devoted to the computation of two-phase flows in a porous medium when applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them corresponding to solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme, are shown to give wrong approximations in some cases involving sharp porous profiles. The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal. 43 (2006)

A fractional step method to compute a class of compressible gas–liquid flows

“…Nonetheless, due to the small variation of P g through λ = U g (see table 1), the rate of convergence seems to be a bit higher than 1/2; a better agreement is observed for the second test case, which has the same structure, but involves a stronger jump of P g through the gas contact wave. These results are in agreement with those of [13,30].…”

“…Setting ψ = m φ , the cell scheme (10) enables to rewrite : A drawback is that the scheme is less accurate than many approximate Riemann solvers. Nonetheless, as recalled in the last section, it enables to obtain convergent approximations when the mesh size diminishes (see [17,13,30] also for similar convergence studies).…”

“…The second test case is another Riemann problem taken from [13], where initial conditions are given in table 2. This Riemann problem also contains a contact wave associated with U g , and a right-going gas shock wave.…”

“…We emphasize that this approach was recently extended to the framework of two-phase flow in porous media [19,13] . The same procedure provides some way to tackle the modeling of three-phase flows [18] .…”

The Coupling of Multi-phase Flow Models

“…The father model involved in the coupling problems belonging to class (E4) corresponds to the porous model, when focusing on the homogeneous approach 23, 24 or on the two-fluid approach. 15,17,26 Details on reconstruction procedures can be found in the above-mentionned references. These reconstructions may become difficult when using complex equations of state ; in fact they usually require solving non-linear scalar equations in cells touching the coupling interface.…”