Exact solutions for one‐dimensional transient response of fluid‐saturated porous media
Exact solutions for one‐dimensional transient response of fluid‐saturated porous media
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DOI：
10.1002/nag.904
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发表时间：
2011-03
影响因子：
4
通讯作者：
Zhendong Shan;D. Ling;H. Ding
中科院分区：
文献类型：
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作者：
Zhendong Shan;D. Ling;H. Ding
基于毕奥理论，在考虑惯性、粘性和力学耦合的情况下，假定流体和固体颗粒均为可压缩，推导了单层饱和多孔介质和半无限介质一维瞬态响应的精确解.首先，以矩阵形式表示关于固体位移u和相对位移w的控制方程。对于均质边界条件下的单层问题，利用分离变量法求出特征值和特征函数，并利用搜索法求出位移向量u。在非齐次边界条件的情况下，首先将边界条件齐次化，然后基于特征函数构造位移场。利用本征函数的正交性，得到了一系列关于无量纲时间的常微分方程及其相应的初始条件。用状态空间法求解这些微分方程，得到了三种典型非齐次边界条件的级数解。对于半无限介质，通过对基本方程进行余弦和正弦变换，给出了两种非齐次边界条件的积分形式的精确解。最后，通过三个算例验证了所提方法的有效性，并分析了动态渗透系数和流体惯性对多孔介质瞬态响应的影响。版权所有© 2010约翰威利父子有限公司.
Based on the Biot theory, the exact solutions for one‐dimensional transient response of single layer of fluid‐saturated porous media and semi‐infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen‐values and the eigen‐functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen‐functions. Making use of the orthogonality of eigen‐functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state‐space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi‐infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media. Copyright © 2010 John Wiley & Sons, Ltd.