Convective cooling/heating induced thermal stresses in a fluid saturated porous medium undergoing local thermal non-equilibrium

Local thermal non-equilibrium (LTNE) may have profound effects on the pore pressure and thermal stresses in fluid saturated porous media under transient thermal loads. This work investigates the temperature, pore pressure, and thermal stress distributions in a porous medium subjected to convective cooling/heating on its boundary. The LTNE thermo-poroelasticity equations are solved by means of Laplace transform for two fundamental problems in petroleum engineering and nuclear waste storage applications, i.e., an infinite porous medium containing a cylindrical hole or a spherical cavity subjected to symmetrical thermo-mechanical loads on the cavity boundary. Numerical examples are presented to examine the effects of LTNE under convective cooling/heating conditions on the temperature, pore pressure and thermal stresses around the cavities. The results show that the LTNE effects become more pronounced when the convective heat transfer boundary conditions are employed. For the cylindrical hole problem of a sandstone formation, the thermally induced pore pressure and the magnitude of thermal stresses are significantly higher than the corresponding values in the classical poroelasticity, which is particularly true under convective cooling with moderate Biot numbers. For the spherical cavity problem of a clay medium, the LTNE effect may become significant depending on the boundary conditions employed in the classical theory.     

饱和多孔介质化学-热-弹性模型及应用

本文基于热局部非平衡(LTNE)条件和加权平均温度概念,并假设孔隙流体由溶质和溶剂两组元组成,对页岩（饱和多孔介质）,推导给出了一种LTNE条件下的化学-热-弹性模型,同时讨论了耦合方程组的解耦求解问题．作为模型的应用,考虑无限大平面含一圆形孔的情况,研究了冷/热对流以及溶质摩尔分数突变边界条件下圆孔附近的孔隙压力和化-热应力问题,用Laplace变换得到了平面轴对称情况下有关力学变量的表达式．数值分析了圆孔边界上冷/热对流的Biot数和溶质摩尔分数改变量对圆孔附近孔隙压力和化-热应力的影响．结果表明：在Biot数为中等值(1~5)范围内,LTNE效应是非常明显的;化学作用对孔隙压力和固相应力的影响不可忽视．     

THERMAL CONSOLIDATION OF LAYERED POROUS HALF-SPACE TO VARIABLE THERMAL LOADING

An analytical method was derived for the thermal consolidation of layered, saturated porous half-space to variable thermal loading with time. In the coupled governing equations of linear thermoelastic media, the influences of thermo-osmosis effect and thermal filtration effect were introduced. Solutions in Laplace transform space were first obtained and then numerically inverted. The responses of a double-layered porous space subjected to exponential decaying thermal loading were studied. The influences of the differences between the properties of the two layers (e.g., the coefficient of thermal consolidation, elastic modulus) on thermal consolidation were discussed. The studies show that the coupling effects of displacement and stress fields on temperature field can be completely neglected, however, the thermo-osmosis effect has an obvious influence on thermal responses.     

Stationary Fingering Instability in a Non-equilibrium Porous Medium with Coupled Molecular Diffusion

Finger type double diffusive convective instability in a fluid-saturated porous medium is studied in the presence of coupled heat-solute diffusion. A local thermal non-equilibrium (LTNE) condition is invoked to model the Darcian porous medium which takes into account the energy transfer between the fluid and solid phases. Linear stability theory is implemented to compute the critical thermal Rayleigh number and the corresponding wavenumber exactly for the onset of stationary convection. The effects of Soret and Dufour cross-diffusion parameters, inter-phase heat transfer coefficient and porosity modified conductivity ratio on the instability of the system are investigated. The analysis shows that positive Soret mass flux triggers instability and positive Dufour energy flux enhances stability whereas their combined influence depends on the product of solutal Rayleigh number and Lewis number. It also reveals that cell width at the convection threshold gets affected only in the presence of both the cross-diffusion fluxes. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient and porosity modified conductivity ratio are found. An excellent agreement is found between the exact and asymptotic solutions.     

Response of saturated porous media subjected to local thermal loading on the surface of semi-infinite space

Heat source function method is adopted in the present paper to derive elementary solutions of coupled thermo-hydro-mechanical consolidation for saturated porous media under conjunct actions of instantaneous point heat source, instantaneous point fluid source and constant volume force. By using the so-called fictitious heat source method and images method, the solutions of a semi-infinite saturated porous medium subjected to a local heat source with time-varied intensity on its free surface are developed from elementary solutions. The numerical integral methods for calculating the unsteady temperature, pore pressure and displacement fields are given. The thermomechanical response are analyzed for the case of a circular planar heat source. Besides, the thermal consolidation characteristics of a saturated porous medium subjected to a harmonic thermal loading are also given, and the fluctuation processes of the field variables located below the center of heat source are analyzed. The project supported by the National Natural Science Foundation of China (50578008) The English text was polished byYunming Chen.     

Mixture modeling of jointed fluidsaturated porous media

The quasi-static equations of motion are studied for bi-laminated fluid-saturated porous media within the framework of non-phenomenological mixture theories. The flow-deformation coupled behavior of the media is governed by Biot's theory for which all constituents are considered compressible. The asymptotic analysis for a periodic microstructure with multiple scales, developed by Hegemier and Murakami, is adopted to obtain the equations of equilibrium and mass conservation in a binary saturated porous medium. The multiscale analysis appears to be advantageous for dealing with consolidation phenomena because it is capable of transforming a coupled, transient problem into two decoupled, steady-state ones. Various models with different degrees of approximation are generated, and among them a theory for saturated rocks with a single joint system is described. Mixture properties are expressed explicitly in terms of characteristics of intact and joint material. The most distinctive feature of this model comes from the fact that some cross terms, that have not been included in previous models, appear in the constitutive equations for fluid mass change and fluid flux. These cross terms are physically understood because they simply take into account effects occurring on the local level: the deformation-flow coupled phenomenon, the stress continuity and displacement compatibility conditions. These novel results may have far-reaching consequences for future theoretical modeling and experimental programs in two-phase fluid-filled porous media.     

TEMPERATURE PROFILES OF LOCAL THERMAL NONEQUILIBRIUM FOR THERMAL DEVELOPING FORCED CONVECTION IN POROUS MEDIUM PARALLEL PLATE CHANNEL

Based on the two-energy equation model, taking into account viscous dissipation due to the interaction between solid skeleton and pore fluid flow, temperature expressions of the solid skeleton and pore fluid flow are obtained analytically for the thermally developing forced convection in a saturated porous medium parallel plate channel, with walls being at constant temperature. It is proved that the temperatures of the two phases for the local thermal nonequilibrium approach to the temperature derived from the one-energy equation model for the local thermal equilibrium when the heat exchange coefficient goes to infinite. The temperature profiles are shown in figures for different dimensionless parameters and the effects of the parameters on the local thermal nonequilibrium are revealed by parameter study.     

Effect of Local Thermal Non-equilibrium on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is analytically studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Darcy model is employed. The effect of local thermal non-equilibrium among the particle, fluid, and solid-matrix phases is investigated using a three-temperature model. The analysis reveals that in some circumstances the effect of LTNE can be significant, but for a typical dilute nanofluid (with large Lewis number and with small particle-to-fluid heat capacity ratio) the effect is small.     

Thermal Dispersion–Radiation Effects on Non-Darcy Natural Convection in a Fluid Saturated Porous Medium

The effects of thermal dispersion and thermal radiation on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium are studied. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. Rosseland approximation is used to describe the radiative heat flux in the energy equation. Similarity solution for the transformed governing equations is obtained. Numerical results for the details of the velocity and temperature profiles which are shown on graphs have been presented. The combined effect of thermal dispersion and thermal radiation, for the two cases Darcy and non-Darcy porous medium, on the heat transfer rate which are entered in tables is discussed.     

一维流体饱和粘弹性多孔介质层的动力响应

本文研究了不可压流体饱和粘弹性多孔介质层的一维动力响应问题。基于粘弹性理论和多孔介质理论,在流相和固相微观不可压、固相骨架服从粘弹性积分型本构关系和小变形的假定下,建立了不可压流体饱和粘弹性多孔介质层一维动力响应的数学模型,利用Laplace变换,求得了原初边值问题在变换空间中的解析解,并利用Laplace逆变换的Crump数值反演方法,得到原动力响应问题的数值解。数值研究了饱和标准线性粘弹性多孔介质层的动力响应,分析了固相位移、渗流速度、孔隙压力及固相有效应力等的响应特征。结果表明,与不可压流体饱和弹性多孔介质相同,不可压流体饱和粘弹性多孔介质中亦只存在一个纵波,并且固相骨架的粘性对动力行为有显著的影响。     

Transient analysis of ionic replacements in elastic–plastic expansive clays

Chemically active saturated clays are considered in a two-phase framework. The solid phase contains clay particles, absorbed water and dissolved ions, Na⁺, K⁺ and Cl⁻. The fluid phase, or pore water, contains free water and the same ionic species. Water and ions can transfer between the two phases. In addition, they diffuse through the porous medium. A global understanding of all phenomena, mass transfer, diffusion/advection and deformation is provided. The coupled constitutive equations associated to these phenomena are developed. Emphasis is laid on the electro-chemo-mechanical constitutive equations in an elastic–plastic setting.A finite element formulation embodying all the above aspects is proposed and simulations of oedometer tests are presented and commented. Of particular interest are the consolidation and swelling that occur during salinization and desalinization of an external reservoir in contact with the specimen, and the more subtle, but important effects of replacing an NaCl pore solution by a KCl pore solution, and conversely.     

Effect of rotation on ferromagnetic porous convection with a thermal non-equilibrium model

The effect of rotation on the onset of thermal convection in a horizontal layer of ferrofluid saturated Brinkman porous medium is investigated in the presence of a uniform vertical magnetic field using a local thermal non-equilibrium (LTNE) model. A two-field model for temperature representing the solid and fluid phases separately is used for energy equation. The condition for the occurrence of stationary and oscillatory convection is obtained analytically. The stability of the system has been analyzed when the magnetic and buoyancy forces are acting together as well as in isolation and the similarities as well as differences between the two are highlighted. In contrast to the non-rotating case, it is shown that decrease in the Darcy number Da and an increase in the ratio of effective viscosity to fluid viscosity Λ is to hasten the onset of stationary convection at high rotation rates and a coupling between these two parameters is identified in destabilizing the system. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. Besides, the influence of magnetic parameters and also parameters representing LTNE on the stability of the system is discussed and the veracity of LTNE model over the LTE model is also analyzed.     

Thermal radiation effect on non-Darcy natural convection with lateral mass transfer

 A boundary layer analysis has been presented to study the influence of thermal radiation and lateral mass flux on non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. Similarity solution for the transformed governing equations is obtained and the combined effect of thermal radiation and fluid suction/injection on the heat transfer rate is discussed. Numerical results for the details of the velocity and temperature profiles as well as Nusselt number have been presented. Received on 7 July 1999     

Modeling and Simulation of Local Thermal Non-equilibrium Effects in Porous Media with Small Thermal Conductivity

The two-equation model in porous media can describe the local thermal non-equilibrium (LTNE) effects between fluid and solid at REV scale, with the temperature differences in a solid particle neglected. A multi-scale model has been proposed in this study. In the model, the temperature differences in a solid particle are considered by the coupling of the fluid energy equation at REV scale with the heat conduction equation of a solid particle at pore scale. The experiments were conducted to verify the model and numerical strategy. The multi-scale model is more suitable than the two-equation model to predict the LTNE effects in porous media with small thermal conductivity. The effects of particle diameter, mass flow rate, and solid material on the LTNE effects have been investigated numerically when cryogenic nitrogen flows through the porous bed with small thermal conductivity. The results indicate that the temperature difference between solid center and fluid has the same trend at different particle diameters and mass flow rates, while the time to reach the local thermal equilibrium is affected by solid diameter dramatically. The results also show that the temperature difference between solid center and surface is much greater than that between solid surface and fluid. The values of \( \rho {\text{c}} \) for different materials have important influence on the time to reach the local thermal equilibrium between solid and fluid.     

Ferromagnetic Convection in a Heterogeneous Darcy Porous Medium Using a Local Thermal Non-equilibrium (LTNE) Model

The combined effects of vertical heterogeneity of permeability and local thermal non-equilibrium (LTNE) on the onset of ferromagnetic convection in a ferrofluid saturated Darcy porous medium in the presence of a uniform vertical magnetic field are investigated. A two-field model for temperature representing the solid and fluid phases separately is used. The eigenvalue problem is solved numerically using the Galerkin method for different forms of permeability heterogeneity function Γ(z) and their effect on the stability characteristics of the system has been analyzed in detail. It is observed that the general quadratic variation of Γ(z) with depth has more destabilizing effect on the system when compared to the homogeneous porous medium case. Besides, the influence of LTNE and magnetic parameters on the criterion for the onset of ferromagnetic convection is also assessed.     

Steady and Unsteady Heat Transfer in a Channel Partially Filled with Porous Media Under Thermal Non-Equilibrium Condition

Steady and pulsatile flow and heat transfer in a channel lined with two porous layers subject to constant wall heat flux under local thermal non-equilibrium (LTNE) condition is numerically investigated. To do this, a physical boundary condition in the interface of porous media and clear region of the channel is derived. The objective of this work is, first, to assess the effects of local solid-to-fluid heat transfer (a criterion indicating on departure from local thermal equilibrium (LTE) condition), solid-to-fluid thermal conductivity ratio and porous layer thickness on convective heat transfer in steady condition inside a channel partially filled with porous media; second, to examine the impact of pulsatile flow on heat transfer in the same channel. The effects of LTNE condition and thermal conductivity ratio in pulsatile flow are also briefly discussed. It is observed that Nusselt number inside the channel increases when the problem is tending to LTE condition. Therefore, careless consideration of LTE may lead to overestimation of heat transfer. Solid-to-fluid thermal conductivity ratio is also shown to enhance heat transfer in constant porous media thickness. It is also revealed that an increase in the amplitude of pulsation may result in enhancement of Nusselt number, while Nusselt number has a minimum in a certain frequency for each value of amplitude.     

Macroscopic constitutive equations of thermo-poroviscoelasticity derived using eigenstrains

Macroscopic constitutive equations for thermo-viscoelastic processes in a fully saturated porous medium are re-derived from basic principles of micromechanics applicable to solid multi-phase materials such as composites. Simple derivations of the constitutive relations and the void occupancy relationship are presented. The derivations use the notion of eigenstrain or, equivalently, eigenstress applied to the separate phases of a porous medium. Governing coupled equations for the displacement components and the fluid pressure are also obtained.     

衰变热源作用下饱和多孔介质热固结问题的扩展精细积分法

热源作用下饱和多孔介质热固结效应是土木及能源工程领域的一个重要课题.由于问题的复杂性,已有的研究大多将介质假定为均匀各向同性,且将热源假定为恒定强度.实际工程中,天然饱和多孔介质常表现出明显的分层特性,热源强度也存在衰变性,为此本工作采用扩展精细积分法对衰变热源作用下层状饱和多孔介质的热固结问题进行研究.借助于积分变换,将饱和多孔介质热固结问题的偏微分方程转化为变换域内的常微分方程;然后对饱和多孔介质微层元进行合并消元,并结合边界条件,推导出衰变热源作用下层状饱和多孔介质热固结问题在积分变换域内的扩展精细积分解;对所得解答进行相应的数值积分逆变换,可获得所求温度、超静孔压及竖向位移在物理域内的解答.基于上述求解过程,编制相应的计算程序进行数值计算,通过与已有文献对比,验证本文扩展精细积分法在求解层状饱和多孔介质热固结问题中的适应性和正确性;最后通过几组算例,分析热源衰变周期、热源埋深及介质的成层性对热固结效应的影响.结果表明:热源衰变周期对温度和超静孔压的峰值、以及达到峰值的时间均有明显影响,衰变周期越长,二者峰值均越大,且达到峰值所需时间越长;热源埋深对超静孔压及竖向位移变化影响显著,深埋热源作用时热源两侧竖向位移呈对称分布,而浅埋热源两侧则无此现象;饱和多孔介质的分层特性对热固结效应影响明显.     

非饱和土-隧道系统动力响应计算的波函数法

针对非饱和地基土中埋置隧道的三维动力响应计算问题, 提出了波函数法.采用无限长的Flügge薄壁圆柱壳模拟圆形隧道衬砌,采用流、固、气组成的三相介质模拟非饱和地基土体.分别采用分离变量法以及Helmholtz矢量分解定理求解薄壁圆柱壳的振动控制方程与非饱和土的波动方程.根据隧-土交界面与地表面处的应力、位移以及孔隙流体压力等边界条件,利用平面波与柱面波的转换性质,实现了隧道内作用单位简谐载荷时隧道衬砌与土体系统动力响应的耦合求解.通过与既有单相弹性介质2.5维有限元-边界元法、两相饱和多孔介质2.5维有限元-边界元法以及三相非饱和介质Pip in Pip半解析法的计算结果进行对比, 验证了本文计算方法的可靠性. 最后,基于该方法, 通过算例分析了不同饱和度下非饱和土-隧道系统的动力响应特征.结果表明, 饱和度对土体动位移与超孔隙水压力的幅值响应有较大影响.该方法的非饱和地基土参数退化后,也可用来计算和分析饱和地基土或单相弹性地基土与隧道系统的动力响应.      

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.