Flow in deformable porous media. Part 1 Simple analysis
Flow in deformable porous media. Part 1 Simple analysis
复制标题
可变形多孔介质中的流动。
DOI：
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发表时间：
1993
影响因子：
3.7
通讯作者：
M. Spiegelman
中科院分区：
文献类型：
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作者：
M. Spiegelman
地球上的许多过程，如岩浆迁移，可以用低粘度流体在粘性可变形、可渗透的基质中的流动来描述。这篇文章和一篇配套论文的目的是为了更好地理解控制这些两相流的方程。本文给出了控制方程的一系列解析近似解，表明这些方程描述了两种不同的矩阵变形模式。基质的剪切变形受Stokes方程控制，可导致孔隙率驱动的对流。基质的体积变化受孔隙度的非线性色散波动方程控制。孔隙度波的存在是因为流体流量是孔隙度的递增函数，并且基质可以随着流体流量的变化而膨胀或紧凑。波的速度和行为取决于渗透率和孔隙度之间的函数关系。如果渗透率相对于孔隙度的偏导数∂kϕ/∂ϕ也是孔隙度的递增函数，则波的传播速度比孔隙中的流体更快，并可能陡峭地形成孔隙度冲击。然而，孔隙率波的传播被基质对体积变化的粘性阻力所阻挡。线性分析表明，粘性应力导致平面波分散，并提供额外的压力梯度，使流体绕障碍物的流动偏转。当在非线性方程中忽略粘性阻力时，流体通量中的障碍物会形成孔隙率激波。使用特征方法，我们量化了冲击在一维和二维发展的具体标准。另一篇论文使用数值格式证明，在完整的方程中，对体积变化的粘性阻力导致简单的激波弥散成一连串的非线性孤立波。
Many processes in the Earth, such as magma migration, can be described by the flow of a low-viscosity fluid in a viscously deformable, permeable matrix. The purpose of this and a companion paper is to develop a better physical understanding of the equations governing these two-phase flows. This paper presents a series of analytic approximate solutions to the governing equations to show that the equations describe two different modes of matrix deformation. Shear deformation of the matrix is governed by Stokes equation and can lead to porosity-driven convection. Volume changes of the matrix are governed by a nonlinear dispersive wave equation for porosity. Porosity waves exist because the fluid flux is an increasing function of porosity and the matrix can expand or compact in response to variations in the fluid flux. The speed and behaviour of the waves depend on the functional relationship between permeability and porosity. If the partial derivative of the permeability with respect to porosity, ∂kϕ/∂ϕ, is also an increasing function of porosity, then the waves travel faster than the fluid in the pores and can steepen into porosity shocks. The propagation of porosity waves, however, is resisted by the viscous resistance of the matrix to volume changes. Linear analysis shows that viscous stresses cause plane waves to disperse and provide additional pressure gradients that deflect the flow of fluid around obstacles. When viscous resistance is neglected in the nonlinear equations, porosity shock waves form from obstructions in the fluid flux. Using the method of characteristics, we quantify the specific criteria for shocks to develop in one and two dimensions. A companion paper uses numerical schemes to show that in the full equations, viscous resistance to volume changes causes simple shocks to disperse into trains of nonlinear solitary waves.