On 2D approximations for dissolution problems in Hele-Shaw cells

In this paper, we study the dissolution problems occurring in laterally large 3D systems with very small dimensions along the third coordinate, such as fractures or Hele-Shaw cells. On the basis of the scale separation assumption, we apply upscaling to the 3D pore-scale model using the volume averaging method to develop 2D averaged equations. The influence of the choice of momentum equations on the accuracy of the 2D Hele-Shaw model is discussed, and we show that the results obtained using Darcy-Brinkman equations are better than those obtained using Darcy's law, because of the consideration of the viscous boundary layer. The validity and accuracy of the resulting 2D model are assessed based on comparisons with full 3D solutions for problems corresponding to the existence of geometrical 3D features to which a simple averaging procedure along a line (i.e., the height of the gap) perpendicular to the 2D plane cannot be applied, such as the dissolution of pillars. The results show that when Peclet and Reynolds numbers exceed certain limits, 3D effects must be considered. Moreover, natural convection effects are important when the Rayleigh number is large.

Automated summary

This paper focuses on the dissolution problems occurring in laterally large 3D systems with very small dimensions along the third coordinate, such as fractures or Hele-Shaw cells. The study applies upscaling to the 3D pore-scale model using volume averaging method to develop 2D averaged equations. The influence of momentum equations on the accuracy of the 2D Hele-Shaw model is discussed, and it is shown that the results obtained using Darcy-Brinkman equations are better due to the consideration of the viscous boundary layer. The validity and accuracy of the resulting 2D model are assessed based on comparisons with full 3D solutions.