Abstract
A classical model for three-phase capillary immiscible flows in a porous medium is considered. Capillarity pressure functions are found, with a corresponding diffusion-capillarity tensor being triangular. The model is reduced to a degenerate quasilin-ear parabolic system. A global existence theorem is proved under some hypotheses on the model data.
Highlights
We study the question of global solvability for the 2 × 2 quasi-linear parabolic system ut + f (u)x = (B(u)ux )x, x ∈ Ω = {x ∈ R : |x| < 1}, 0 < t < T, (1.1)
As is well known by reservoir engineers, mobilities and capillary pressures can be plotted as functions of the saturations only in the case of flows where just two phases are present [9, 6], that is, when u ∈ ∂4
Equations (1.13) can be used to recover the capillary pressures pij in the entire phase triangle ∆ by the formulas p13 (u1, u2 ) = φ13 (ξ) −
Summary
As is well known by reservoir engineers, mobilities and capillary pressures can be plotted as functions of the saturations only in the case of flows where just two phases are present [9, 6], that is, when u ∈ ∂4. The constraints (1.13) amount to a linear hyperbolic system of partial differential equations for the capillary pressures p13 and p23 , whose coefficients involve the mobility functions, for which we set the boundary conditions. We study in detail the case when the mobilities are linear functions of the corresponding phase saturation λi = ki ui , ki = const > 0. Equations (1.13) can be used to recover the capillary pressures pij in the entire phase triangle ∆ by the formulas p13 (u1 , u2 ) = φ13 (ξ) −. ≡ 0 and under the assumption that some solution’s norms are finite a priori
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