On the basis of conservation laws and basic principles of thermodynamics, a mathematical model is developed for flows of a two-phase granular fluid. The phases consist of a viscoplastic granular Bingham fluid and a viscous Newtonian fluid. As an application, one dimensional flows are studied in a channel to address the stability of the proppant pack which fills a hydro-fracture. We find correlations between the phase flow rates and the pressure gradient. Such correlations are similar to a Darcy law. We determine a criterion for the initiation of motion of a granular phase in a porous medium. Given a yield stress of the granular phase, it is proved that this phase does not flow if either the pressure gradient or the channel thickness is small. The phase flow rates are studied numerically at various input parameters such as the phase viscosities, yield stresses and etc. The factors slowing down the penetration of the solid phase into the porous medium are revealed.
|Journal||Journal of Physics: Conference Series|
|Publication status||Published - 20 Nov 2020|
|Event||9th International Conference on Lavrentyev Readings on Mathematics, Mechanics and Physics - Novosibirsk, Russian Federation|
Duration: 7 Sep 2020 → 11 Sep 2020