On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains

Abstract

We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in [Guo et al., 2014] or the asymptotic approach in [Yortsos, 1995]. The latter approach uses a geometrical scaling. We show the efficiency of the approach in asymptotically flat domains for Darcy regimes. Moreover, we prove that it is vastly equivalent to the multiscale model approach.We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws [Hassanizadeh and Gray, 1993].spatially three-dimensional settings [20]. Therefore it is an important issue to perform model reductions if possible, in particular for any kinds of limit regimes.

On the limit of a two‐phase flow problem in thin porous media domains of Brinkman type

“…An example is the dimensional reduction approach by vertical integration in the field of petroleum studies, 2 hydrogeology, 1,3 and carbon dioxide sequestration 4,5 . Another example is the asymptotic approach for two‐phase flows in Darcy 6 and Brinkman regimes 7 and for single‐phase flow in deformable porous materials 8 . We also refer to the multiscale model approach in Guo et al 9 A comparative study on the accuracy and efficiency of the asymptotic approach over the others is preformed in Armiti‐Juber and Rohde, 7 where also an equivalence result with the multiscale approach is proved.…”

“…Another example is the asymptotic approach for two‐phase flows in Darcy 6 and Brinkman regimes 7 and for single‐phase flow in deformable porous materials 8 . We also refer to the multiscale model approach in Guo et al 9 A comparative study on the accuracy and efficiency of the asymptotic approach over the others is preformed in Armiti‐Juber and Rohde, 7 where also an equivalence result with the multiscale approach is proved. A recent approach suggests an adaptive algorithm that couples the dimensional reduction approach with the full model 10 .…”

“…In this paper, we provide a mathematically rigorous derivation of the reduced model from the asymptotic approach in Armiti‐Juber and Rohde 7 by exploring the limit of the full two‐phase flow model in porous media domains of Brinkman type as the width–length ratio of the domain tends to zero. Precisely, we prove the analytical convergence of weak solutions for the full model to weak solutions for the reduced model.…”

“…We consider the homogenized flow of two incompressible immiscible fluids in the rectangular domain such that H ≪ L (Figure 1), where is the geometrical parameter. Using dimensionless variables, governing equations for such flows are given by the so‐called Brinkman two‐phase flow model (BTP model), in the dimensionless domain Ω × (0, T ), where and T > 0 7 . We refer to Appendix for details on the derivation of this model.…”

“…An illustration of the displacement process of a wetting phase into a thin domain Ω γ 7 [Colour figure can be viewed at wileyonlinelibrary.com]…”