We present here a multilayer model for shallow grain-fluid mixtures with dilatancy effects. It can be seen as a generalization of the depth-averaged model presented in Bouchut et al. (2016) [6], that includes dilatancy effects by considering a two-layer model, a mixture grain-fluid layer and an upper fluid layer, to allow the exchange of fluid between them. In the present work the approximation of the mixture layer is improved including normal variations of the velocities and concentrations of the two phases thanks to the multilayer approach. In the model presented here dilatancy effects induce in particular a non-hydrostatic pressure for both phases related to the excess pore fluid pressure. Contrary to the single-layer model, the computation of this excess pore pressure entrains a serious difficulty due to the multilayer approach. We identified here one of the main numerical difficulty of solving two-phase shallow debris flows models: the strongly non-linear behaviour and abrupt changes of the excess pore fluid pressure when starting from non-equilibrium conditions. We propose a simplified approach to approximate the excess pore fluid pressure in the simple case of uniform flows in the downslope direction and quantify the error made. Our method makes it possible to introduce two or three layers in the normal directions with a reasonable approximation. Analytical solutions for uniform grain-fluid flows over inclined planes, with and without side wall friction, are calculated and compared to the proposed model. The presented model preserves the total solid granular mass as in [6]. In the numerical results, we observe that the proposed model with a two layer description of the mixture accurately represents the velocity measured at the surface of the mixture in the laboratory experiments. This is obviously poorly represented by the depth-averaged velocity in single-layer models while the other quantities (solid volume fraction, basal excess pore fluid pressure) are similar to those obtained with single-layer models. Our numerical results show a significant impact of the parameters involved in dilatancy law, in particular on the calculation of the time evolution of the excess pore fluid pressure.

我们在这里提出了具有膨胀效应的浅颗粒-流体混合物的多层模型。可以看作是Bouchut等人提出的深度平均模型的概括。（2016）[6]，其中包括通过考虑两层模型，混合颗粒流体层和上部流体层来允许它们之间交换流体的膨胀效应。在目前的工作中，由于采用了多层方法，因此改善了混合层的近似性，包括速度和两相浓度的正常变化。在此处介绍的模型中，膨胀效应特别引起了与孔隙流体压力过高相关的两个阶段的非静水压力。与单层模型相反，由于多层方法，这种多余的孔隙压力的计算带来了严重的困难。我们在这里确定了解决两相浅层泥石流模型的主要数值困难之一：从非平衡条件开始时，强烈的非线性行为和过量孔隙流体压力的突然变化。我们提出了一种简化的方法，在下坡方向均匀流动的简单情况下，可以估算多余的孔隙流体压力，并量化误差。我们的方法可以在法线方向上合理地引入两层或三层。计算了在有和没有侧壁摩擦的情况下，在倾斜平面上均匀的流体流动的解析解，并与提出的模型进行了比较。提出的模型保留了[6]中的总固体颗粒质量。在数值结果中 我们观察到所提出的带有混合物两层描述的模型准确地代表了实验室实验中在混合物表面测得的速度。在单层模型中，这显然不能用深度平均速度来表示，而其他量（固体体积分数，基础过量孔隙流体压力）与单层模型中的相似。我们的数值结果显示了膨胀定律所涉及的参数的重大影响，特别是对多余孔隙流体压力随时间变化的计算。基本的多余孔隙流体压力）类似于单层模型获得的值。我们的数值结果显示了膨胀定律所涉及的参数的重大影响，特别是对多余孔隙流体压力随时间变化的计算。基本的多余孔隙流体压力）类似于单层模型获得的值。我们的数值结果表明，膨胀率定律所涉及的参数有重大影响，特别是对多余孔隙流体压力随时间变化的计算。