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.

根据守恒定律和热力学基本原理，建立了两相颗粒流体流动的数学模型。这些相由粘性粒状宾汉流体和粘性牛顿流体组成。作为一种应用，研究了通道中的一维流动，以解决填充水力裂缝的支撑剂填充物的稳定性。我们发现相流速和压力梯度之间的相关性。这种相关性类似于达西定律。我们确定了多孔介质中颗粒相运动开始的标准。给定颗粒相的屈服应力，证明如果压力梯度或通道厚度小，该相不会流动。在各种输入参数下对相流速进行数值研究，例如相粘度，