可变形双重多孔介质中流动的Biot-Darcy-Brinkman模型;均匀化和数值模拟
The Biot-Darcy-Brinkman model of flow in deformable double porous media; homogenization and numerical modelling
摘要：
在本文中，我们提出了两个层次的均匀化的流动在一个可变形的双多孔结构描述在两个特征尺度。与介观结构相关联的较高水平的孔隙率由基质中的通道构成，所述基质由微孔材料制成，所述微孔材料由弹性骨架和被粘性流体饱和的孔组成。宏观模型是由均匀化的非均质结构中的流动，其特征在于由两个小参数参与的两级渐近分析，从而采用缩放系数来尊重孔径的差异。第一级放大的流体-结构相互作用的问题产生的Biot连续描述的介观矩阵耦合的斯托克斯流在通道中。第二步的均匀化导致一个宏观模型，包括三个方程的位移，介观流速和压力。由于两个孔隙之间的相互作用，宏观流动由Darcy-Brinkman模型控制，该模型包括两个方程，该方程与关于两相介质的分级结构的总体平衡方程相耦合。根据介观结构的特征响应，导出了均匀化双孔隙连续介质的有效宏观参数的表达式。一些对称性和互易性关系示出和边界条件的问题进行了讨论。该模型已在有限元代码SfePy中实现，该代码非常适合计算均匀化。最后给出了一个用混合有限元法求解非定常问题的算例。
英文摘要：
In this paper we present the two-level homogenization of the flow in a deformable double-porous structure described at two characteristic scales. The higher level porosity associated with the mesoscopic structure is constituted by channels in a matrix made of a microporous material consisting of elastic skeleton and pores saturated by a viscous fluid. The macroscopic model is derived by the homogenization of the flow in the heterogeneous structure characterized by two small parameters involved in the two-level asymptotic analysis, whereby a scaling ansatz is adopted to respect the pore size differences. The first level upscaling of the fluid-structure interaction problem yields a Biot continuum describing the mesoscopic matrix coupled with the Stokes flow in the channels. The second step of the homogenization leads to a macroscopic model involving three equations for displacements, the mesoscopic flow velocity and the micropore pressure. Due to interactions between the two porosities, the macroscopic flow is governed by a Darcy-Brinkman model comprising two equations which are coupled with the overall equilibrium equation respecting the hierarchical structure of the two-phase medium. Expressions of the effective macroscopic parameters of the homogenized double-porosity continuum are derived, depending on the characteristic responses of the mesoscopic structure. Some symmetry and reciprocity relationships are shown and issues of boundary conditions are discussed. The model has been implemented in the finite element code SfePy which is well-suited for computational homogenization. A numerical example of solving a nonstationary problem using mixed finite element method is included.
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