Asymptotics for the Muskat problem

作者：Booth R J S^*

来源：Journal of Engineering Mathematics, 2011, 69(2-3): 155-168.

DOI：10.1007/s10665-010-9370-3

• Asymptotic analysis
• Free-boundary problem
• Muskat problem
• Polymer flooding
• Smoothing

The Muskat problem is a simple model for the displacement of one fluid by another in a porous medium or Hele-Shaw cell. Solutions of the problem are unstable to small-wavelength disturbances, and there is evidence that the problem is ill-posed. A well-posed problem may be developed by including additional regularising physical effects, primarily surface tension. In practical situations involving oil recovery, one can engineer for there to be a gradual change in the viscosity of the fluids, either by the gradual introduction of a polymer capable of altering the viscosity of water, or by gradual diffusion between two miscible fluids. The regularising effect of smoothing the initial data is investigated. Thus rather than a sharp interface between the two fluids, and therefore a sharp jump in viscosity, there is a smooth transition from one viscosity to another. It is shown how the standard Muskat problem may be recovered as an asymptotic limit and the effective free-surface model that may be recovered is discussed. The stability of solutions to the smoothed Muskat problem is investigated, and the important effect of smoothing for disturbances with a large wavenumber is demonstrated.

• 出版日期2011-3