单层非饱和多孔介质一维瞬态响应半解析解

基于Zienkiewicz提出的非饱和多孔介质波动理论,考虑两相流体和固体颗粒的压缩性以及惯性、黏滞和机械耦合作用,采用半解析的方法获得了一类典型边界条件下单层非饱和多孔介质一维瞬态响应解。首先推导出无量纲化后以位移表示的控制方程,并将其写成矩阵形式;然后,将边界条件齐次化,求解控制方程所对应的特征值问题,得到了满足齐次边界条件的特征值和相对应的特征函数。根据变异系数法并利用特征函数的正交性,得到了一系列仅黏滞耦合的关于时间的二阶常微分方程及相应的初始条件。在此基础上,运用精细时程积分法给出了常微分方程组的数值解。最后,通过若干算例验证了结果的正确性并探讨了单层非饱和多孔介质一维瞬态动力响应的特点。该方法可推广应用于其他典型的边界条件。     

单层不可压缩饱和多孔介质一维瞬态响应精确解

基于Biot理论,考虑惯性、黏滞和机械耦合作用,假定固体颗粒和流体均不可压缩,得到了表面任意竖向荷载作用下单层饱和多孔介质一维瞬态响应的精确解。导出了以固体骨架位移表示的无量纲控制方程,并将边界条件齐次化。求解对应无黏滞耦合作用的特征值问题,得到一组满足齐次边界条件、关于空间坐标的正交函数基。利用变异系数法和基函数的正交性,得到一系列相互解耦的、关于时间的二阶常微分方程及相应的初始条件,并采用状态空间法求解常微分方程,得到位移分量。对整体平衡方程关于空间坐标积分,根据边界条件可确定总应力,并进而求得孔隙压力。通过算例验证所得解法的正确性     

A semi-analytical solution for the transient response of one-dimensional unsaturated single-layer poroviscoelastic media

This paper develops a semi-analytical solution for the transient response of an unsaturated single-layer poroviscoelastic medium with two immiscible fluids by using the Laplace transformation and the state-space method. Using the elastic–viscoelastic correspondence principle, we first introduce the Kelvin–Voigt model into Zienkiewicz’s unsaturated poroelastic model. The vibrational response for unsaturated porous material can be obtained by combining these two models and assuming that the wetting and non-wetting fluids are compressible, the solid skeleton and solid particles are viscoelastic, and the inertial and mechanical couplings are taken into account. The Laplace transformation and state-space method are used to solve the basic equations with the associated initial and boundary conditions, and the analytical solution in the Laplace domain is developed. To evaluate the responses in the time domain, Durbin’s numerical inverse Laplace transform method is used to obtain the semi-analytical solution. There are three compressional waves in porous media with two immiscible fluids. Moreover, to observe the three compressional waves clearly, we assume the two immiscible fluids are water and oil. Finally, several examples are provided to show the validity of the semi-analytical solution and to assess the influences of the viscosity coefficients and dynamic permeability coefficients on the behavior of the three compressional waves.     

Exact solution for the 1D transient response of saturated single-layer poroviscoelastic media

The basic equations for fluid-saturated porous media proposed by Biot are modified by replacing the classical linear elastic model of the solid skeleton with the Kelvin–Voigt model. Thus, the new theory can take into account the viscoelastic effect of the solid skeleton. After the establishment of appropriate boundary and initial conditions, a time-domain series solution for the transient response of a fluid-saturated single-layer poroviscoelastic medium is obtained by using the finite Fourier transform and the corresponding analytical inverse transform. Several numerical examples are provided to illustrate the validity of the exact solution and to investigate the influence of the viscosity coefficient, permeability coefficient, and load frequency on the transient response of a fluid-saturated single-layer poroviscoelastic medium.     

Exact solutions for one‐dimensional transient response of fluid‐saturated porous media

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.     

An analytical solution for the transient response of saturated linear elastic porous media

The motions of fluid and solid phases in saturated porous media are coupled by inertial, viscous and mechanical interactions as described by Biot's equations. A one-dimensional exact analytical solution of the Biot's equations for the completely general solution of the transient problem in saturated, linear, elastic, porous media is presented. The problem is solved by using the Fourier series. The transient response of porous media is shown for typical material properties of a natural granular deposit and for different degrees of viscous coupling. The analytical results show the mechanics of dispersive wave propagation in saturated porous media and they should provide a useful comparison term for the existing numerical solution methods.     

A semianalytical solution for one-dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer

One-dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer is studied in this paper. An analytical solution for a special case with a dynamic permeability coefficient k_f → ∞ and a semianalytical solution for a general case with an arbitrary dynamic permeability coefficient are presented. The eigenfunction expansion and precise time step integration methods are employed. The solution is presented in series form, and thus, the long-term dynamic responses of saturated porous media with small permeability coefficients can be easily computed. We first transform the nonhomogeneous boundary conditions into homogeneous boundary conditions, and then we obtain the eigenvalues and orthogonal eigenfunctions of the fluid–solid system. Finally, the solutions in the time domain are developed. As the model is one dimensional, geometric attenuation is absent, and only the attenuation in the saturated porous medium is considered. We can apply this model to analyse the influences of different seabed types on the propagation of acoustic waves in the fluid layer, which is very important in ocean acoustics and ocean seismic. This solution can also be employed to validate the accuracies of various numerical methods.     

非饱和介质中热能传输及水分迁移的数值积分解

在给出非饱和介质热能-水分传输的耦合质量控制方程和基于Fourier热传导定律的热能平衡方程的基础上,对热能传输及水分迁移的基本特征和机理进行了分析。其中,考虑了温度势、吸力势和重力势的耦合作用影响。给出有热源时控制方程的简化形式,并对半无限体自由表面作用平面热源条件下介质内非稳态温度场、体积含水率分布场进行数值积分求解。利用这些解答给出常热源强度和变热源强度两种情况下,温度场随时间的变化特征以及水分迁移的演化过程,并分析了重力项对计算结果的影响。     

Micromechanics of saturated and unsaturated porous media

The homogenization method is used to determine the formulation of the behaviour of both saturated and unsaturated porous media. This approach makes it possible to assess the validity of the effective stress concept as a function of the properties of the porous media at the microscopic scale. Furthermore, the influence of the morphologies of the solid and fluid phases on the macroscopic behaviour is studied. The strain induced by drying is examined as a function of the morphological properties. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.     

Equations of frost propagation in unsaturated porous media

The equations of soil freezing are established where the soil is partially water-saturated, i.e., when it contains air. We choose a macroscopic viewpoint, using leveled parameters (averages on a “small” volume surrounding the considered point). We assume that water can exist at a temperature below 0°C. Mechanical energy (kinetic energy and power of inner forces) is neglected with respect to thermal energy; radiation is also neglected. The establishment of the equations is based upon the expression (1) of the mass and energy conservation laws; and (2) of constitutive laws such as Fourier's law, Darcy's law, the curve of capillary pressure in terms of the saturation degree, etc.

We obtain a system of nonlinear partial differential equations with a free surface; the unknowns are the temperature, the saturation degree and the water pressure at each point and at each time; these unknowns are coupled in the differential equations.     

Ozonation of diesel fuel in unsaturated porous media

The objectives of this study were to determine the feasibility of ozonation in unsaturated porous media, and consequently to observe its features and to identify possible limiting factors. Diesel fuel was chosen to represent a complex organic contaminant that is widespread in the environment. In this experiment, the effects of several ozonation features were investigated. Sand was spiked with commercially available diesel fuel (17.024 g diesel/kg dry sand), and packed into a column. Ozone was supplied into the column in a downward direction. When the sand was treated for 7 h at 20 mg ozone/l of air, 40% of the diesel was removed. As the ozone concentration increased from 5 to 20 mg ozone/l, the removal efficiency increased. The removal rate varied significantly depending on the ozone concentration and the treatment duration. At higher ozone concentration, significant quantities of ozone were consumed by the intermediates produced by the ozonation process, and therefore, the removal efficiency and the apparent removal rate of diesel became lowered. The low removal efficiency of diesel results from the high concentrations of normal alkanes. Total hydrocarbon concentration (THC) in the effluent gas was measured using a total hydrocarbon analyzer. The THC decreased with the period of exposure and increasing ozone concentration. Ozonation decreased the gas-extractable fraction and accordingly, the THC decreased. Water-extractable fractions formed by the action of ozone were further oxidized by ozone. Due to the reduction of WEOC (water-extractable organic C) caused by ozone treatment, the potential spread of contamination can be reduced.     

多孔介质非饱和导水率预测的分形模型

多孔介质非饱和导水率是地下水污染预测与评价的重要参数。根据分形几何的基本原理和方法,推导出了与Campbell经验公式在形式上完全一致的多孔介质非饱和导水率的预测公式。公式中的幂指数为介质孔隙分维和随机行走分维的函数,分别体现了多孔介质的静态性质与动态性质对其中水分运动的影响,但静态性质的影响是主要的,即导水率主要受多孔介质的结构控制。根据文献中报道的大量数据,利用笔者推导的预测公式计算得到的幂指数的统计值与试验测定的幂指数的统计值基本一致,说明推导的理论公式预测多孔介质非饱和导水率是较为可靠的。     

成层多孔粘弹性地基在轴对称荷载作用下的解

给出一种有效的解析方法求解成层多孔粘弹性地基在轴对称荷载作用下的固结问题，通过关于时间t的拉普拉斯变换和关于半径r的亨克尔变换，将控制偏微分方程转化为常微分方程，从而得到用常微分短阵方程形式表示的控制方程，基于矩阵常微分方程的解以及拉普拉斯，亨克尔逆变换，可得到该固结问题的解，该方法理论及计算简捷，通过算例比较了该方法的精度，并研究了地基的粘弹性特性对固结过程的影响。     

A boundary integral method for steady flow in unsaturated porous media

The governing equation for steady flow in a homogeneous, partially saturated, porous medium can be written in a linear form if one adopts a hydraulic conductivity function which varies exponentially with capillary-pressure head. The resulting linear field equation is a steady Fokker–Planck equation and is well-suited to numerical solution by the boundary integral equation method (BIEM). The exponential conductivity function is often used in soil physics and is known to be a reasonable approximation over limited ranges of pressure head. A computer code based on the BIEM for obtaining numerical solutions is described and tested. The BIEM is found to exhibit quadratic convergence with element size reduction on smooth solutions and on singular problems, if mesh grading is used. Agreement between results from the BIEM code and a finite element code that solves the fully non-linear problem is excellent, and is achieved at a substantial advantage in computer processing time. As an illustrative example, the code is applied to determine the distribution of moisture in the vicinity of a tunnel.     

Humidity transfer in unsaturated heterogeneous porous media by homogenization

In the paper the one-equation model of humidity transfer in unsaturated macroscopically heterogeneous porous media is presented. The homogenization method by two-scale asymptotic expansions is used to derive the upscaled form of the Richard equation, which is commonly used when the medium is considered as macroscopically homogeneous. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics of the porous medium. The domain of validity of the model is explicitly given, namely: the length-scales separation, the characteristic time scale condition and the ratio of the hydrodynamic characteristics being of the same orders of magnitude. The effective capillary capacity and the effective hydraulic conductivity for an equivalent continuum are defined in terms of geometry and local hydrodynamic characteristics of the porous medium. A procedure of determination of the effective suction curve and the effective hydraulic conductivity curve as functions of the average water content for any type of the macroscopic heteregeneity for which the method can be applied, is provided. Since the problem is non-linear this procedure involves the resolution of a local boundary value problem formulated over a period for each value of suction. In two or three-dimensional cases, this problem can be solved using the numerical methods for any geometry of the medium. In a one-dimensional case it was shown that the analytical solution gives the well-known results of harmonic and arithmetic mean.     

Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with symmetric semi-permeable drainage boundary

This paper presents semi-analytical solutions to Fredlund and Hasan’s one-dimensional consolidation for unsaturated soils under symmetric semi-permeable drainage boundary conditions. Two variables are introduced to transform two coupled governing equations of pore-air and pore-water pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. Then, the pore-air and pore-water pressures, and soil settlement are obtained in the Laplace domain. Crump’s method is adopted to perform the inverse Laplace transform in order to obtain semi-analytical solutions in time domain. It is shown that the present solution is more applicable to various types of drainage boundary conditions, and in a good agreement with existing solutions from the literature. Furthermore, several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with traditional drainage boundary (single or double), and single-sided and double-sided semi-permeable drainage boundaries. Finally, it illustrates the changes in pore-air and pore-water pressures and soil settlement with time at different values of symmetric semi-permeable drainage boundary conditions parameters. In addition, parametric studies are conducted by the variations of pore-air and pore-water pressures at different ratios of air-water permeability coefficient and the depth.     

A model for moisture and heat flow in fractured unsaturated porous media

The present study investigates propagation of a cohesive crack in non‐isothermal unsaturated porous medium under mode I conditions. Basic points of skeleton deformation, moisture, and heat transfer for unsaturated porous medium are presented. Boundary conditions on the crack surface that consist of mechanical interaction of the crack and the porous medium, water, and heat flows through the crack are taken into consideration. For spatial discretization, the extended finite element method is used. This method uses enriched shape functions in addition to ordinary shape functions for approximation of displacement, pressure, and temperature fields. The Heaviside step function and the distance function are exploited as enrichment functions for representing the crack surfaces displacement and the discontinuous vertical gradients of the pressure and temperature fields along the crack, respectively. For temporal discretization, backward finite difference scheme is applied. Problems solved from the literature show the validity of the model as well as the dependency of structural response on the material properties and loading. Copyright © 2016 John Wiley & Sons, Ltd.     

MI-ECku: A novel methodology for estimating unsaturated hydraulic conductivity of porous media

Acta Geotechnica - Numerous applications of hydraulic conductivity of porous media (e.g., soils, clay liners, rocks, concrete, ceramic filters, etc.) in their unsaturated state are well established...     

横观各向同性含液饱和多孔介质黏弹性瞬态反应

在考虑横观各向同性含液饱和多孔介质固体骨架和流体可压缩性以及固体骨架的黏弹性特征下,基于横观各向同性含液饱和多孔介质u-w形式的三维动力控制方程,以固相位移u、液相相对位移w为基本未知量,综合运用Laplace变换、双重Fourier变换等方法,在直角坐标系下通过引入中间变量,将六元2阶动力控制方程组化为两组各含4个未知变量的常微分方程组,给出了直角坐标系下横观各向同性含液饱和多孔介质三维黏弹性动力反应的积分形式一般解;作为理论推导的验证,通过引入初始条件和边界条件,对横观各向同性含液饱和多孔介质半空间黏弹性瞬态反应问题进行了求解。解答的退化验证表明,所推导的理论解是正确的。