不可压缩多孔弹性介质中液体过滤的双孔隙度模型
Double Porosity Models for Liquid Filtration in Incompressible Poroelastic Media
AI总结：
本文基于均匀化理论，通过迭代均匀化方法推导了不可压缩多孔弹性介质中液体过滤的双孔隙度模型，得到了Biot-Terzaghi方程组及绝对刚性体的对应模型。
摘要：
本文通过均匀化理论推导了天然裂缝性油藏中液体过滤的双孔隙度模型。微观层面的控制方程包括：占据裂缝-孔隙空间（液体域）的不可压缩粘性流体的定常Stokes方程组，以及不可压缩弹性固体骨架的定常Lamé方程组，二者通过“固体骨架-液体域”公共边界上的对应边界条件耦合。假设液体域由裂缝（裂隙）和孔隙两个独立系统组成，且孔隙的无量纲尺寸δ依赖于裂缝的无量纲尺寸ε，满足δ=ε^r（r>1）。当裂缝的无量纲尺寸趋于零且固体为几何周期结构时，本文对均匀化过程进行了严格论证。最终推导得到了著名的多孔弹性介质中液体过滤的Biot-Terzaghi方程组，该方程组包含裂缝中液体的常规Darcy定律、固体骨架与孔隙中液体共同位移的各向异性Lamé方程，以及混合物速度的连续性方程。证明基于G. Allaire和M. Briane提出的迭代均匀化方法。作为主要结果的推论，本文还推导了绝对刚性体中不可压缩液体过滤的双孔隙度模型。
英文摘要：
Double porosity models for the liquid filtration in a naturally fractured reservoir is derived from the homogenization theory. The governing equations on the microscopic level consist of the stationary Stokes system for an incompressible viscous fluid, occupying a crack-pore space (liquid domain), and stationary Lame equations for an incompressible elastic solid skeleton, coupled with corresponding boundary conditions on the common boundary "solid skeleton-liquid domain". We suppose that the liquid domain is a union of two independent systems of cracks (fissures) and pores, and that the dimensionless size $δ$ of pores depends on the dimensionless size $\varepsilon$ of cracks: $δ=\varepsilon^{r}$ with $r>1$. The rigorous justification is fulfilled for homogenization procedure as the dimensionless size of the cracks tends to zero, while the solid body is geometrically periodic. As the result we derive the well-known Biot -- Terzaghi system of liquid filtration in poroelastic media, which consists of the usual Darcy law for the liquid in cracks coupled with anisotropic Lame's equation for the common displacements in the solid skeleton and in the liquid in pores and a continuity equation for the velocity of a mixture. The proofs are based on the method of reiterated homogenization, suggested by G. Allaire and M. Briane. As a consequence of the main result we derive the double porosity model for the filtration of the incompressible liquid in an absolutely rigid body.
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