Asymptotic Behavior of the Solutions of an Elliptic-Parabolic System Arising in Flow in Porous Media

Abstract

We study the asymptotic behavior, with respect to high Péclet numbers, of the solutions of the nonlinear elliptic-parabolic system governing the displacement of one incompressible fluid by another, completely miscible with the first, in a porous medium. Using compensated compactness techniques, we obtain the existence of a global weak solution to the nonlinear degenerate elliptic-parabolic system modelling the flow when the molecular diffusion effects are neglected.

An Arbitrary-Order Scheme on Generic Meshes for Miscible Displacements in Porous Media

“…In that case, the parabolic equation takes on a degenerate form, which makes the analysis of the model even more complex. [1] establishes the existence of a solution in the case of a vanishing molecular diffusion and regular source terms. Given the scale of the reservoir and the well bores, it is customary in simulations to consider wells concentrated on measures (Dirac measures in 2D, measures along lines in 3D).…”

“…where µ(0) is the viscosity of the oil and M is the mobility ratio between the oil and the injected solvent, given by M = µ(0) µ (1) . In order to maintain a balance of mass in the domain, the boundary of the reservoir ∂Ω is taken to be impermeable.…”

Convergence Analysis of a Mixed Finite Volume Scheme for an Elliptic-Parabolic System Modeling Miscible Fluid Flows in Porous Media

“…(5) Because of the homogeneous Neumann boundary conditions on U, the injection and production source terms have to satisfy the compatibility condition Ω q + (·, x) dx = Ω q − (·, x) dx in (0, T ), and since the pressure is defined only up to an arbitrary constant, we normalize p by the following condition: Ω p(·, x) dx = 0 in (0, T ). (6) The viscosity μ is usually determined by the following mixing rule in [0, 1], (7) where M = μ(0) μ (1) is the mobility ratio (μ can be extended to R by letting μ = μ(0) on (−∞, 0) and μ = μ(1) on (1, ∞)). The porosity Φ and the permeability K are in general assumed to be bounded from above and from below by positive constants (or positive multiples of I for the tensor K).…”

Asymptotic Analysis of a Nonlinear Parabolic Problem Modelling Miscible Compressible Displacement in Porous Media

“…Amirat, Hamdache and Ziani consider in [1] the parabolic-hyperbolic problem with D m = D p (u) = 0 when the viscosity μ is constant. The more realistic case D m = 0, D p (u) = 0 and μ = μ(c) is treated by Amirat and Ziani in [3] for an incompressible flow, that is when the divergence of the Darcy velocity u is fixed and then completely controlled. The one-dimensional case has been more studied.…”