A Local Model for the Energy Transfer in a Saturated Static Mixture

In the present work the transient energy transfer in a nonsaturated porous medium is studied, using a mixture theory viewpoint. The porous matrix is assumed homogeneous, rigid and isotropic, while the fluid is a Newtonian incompressible one and both are assumed static. Since the homogeneous matrix is not saturated, gradients of concentration are present. The porous medium and the fluid (a liquid) will be regarded as continuous constituents of a mixture that will have also a third constituent, an inert gas, assumed with zero mass density and thermal conductivity. The problem is described by a set of two partial differential equations which represent the energy balances for the fluid and the solid constituents. Isovalues for these two constituents are plotted, considering representative time instants and selected values for the energy equations coefficients and for the saturation.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

Saturated Elastic Porous Solids: Incompressible,Compressible and Hybrid Binary Models

Constitutive models for a general binary elastic-porous media are investigated by two complementary approaches. These models include both constituents treated as compressible/incompressible, a compressible solid phase with an incompressible fluid phase (hybrid model of first type), and an incompressible solid phase with a compressible fluid phase (hybrid model of second type). The macroscopic continuum mechanical approach uses evaluation of entropy inequality with the saturation condition always considered as a constraint. This constraint leads to an interface pressure acting in both constituents. Two constitutive equations for the interface pressure, one for each phase, are identified, thus closing the set of field equations. The micromechanical approach shows that the results of Didwania and de Boer can be easily extended to general binary porous media.     

GURTIN-TYPE VARIATIONAL PRINCIPLES FOR DYNAMICS OF A NON-LOCAL THERMAL EQUILIBRIUM SATURATED POROUS MEDIUM

Based on the porous media theory and by taking into account the effects of the pore fluid viscidity, energy exchanges due to the additional thermal conduction and convection between solid and fluid phases, a mathematical model for the dynamic-thermo-hydro-mechanical coupling of a non-local thermal equilibrium fluid-saturated porous medium, in which the two constituents are assumed to be incompressible and immiscible, is established under the assumption of small deformation of the solid phase, small velocity of the fluid phase and small temperature changes of the two constituents. The mathematical model of a local thermal equilibrium fluid-saturated porous medium can be obtained directly from the above one. Several Gurtin-type variational principles,especially Hu-Washizu type variational principles, for the initial boundary value problems of dynamic and quasi-static responses are presented. It should be pointed out that these variational principles can be degenerated easily into the case of isothermal incompressible fluid-saturated elastic porous media, which have been discussed previously.     

Analytical Solution of Compression,Free Swelling and Electrical Loading of Saturated Charged Porous Media

Analytical solutions are derived for one-dimensional consolidation, free swelling and electrical loading of a saturated charged porous medium. The governing equations describe infinitesimal deformations of linear elastic isotropic charged porous media saturated with a mono-valent ionic solution. From the governing equations a coupled diffusion equation in state space notation is derived for the electro-chemical potentials, which is decoupled introducing a set of normal parameters, being a linear combination of the eigenvectors of the diffusivity matrix. The magnitude of the eigenvalues of the diffusivity matrix correspond to the time scales for Darcy flow, diffusion of ionic constituents and diffusion of electrical potential.     

Macroscopic modelling of transport phenomena in porous media. 2: Applications to mass,momentum and energy transport

In this second paper, the averaging rules presented in Part 1 are employed in order to develop a general macroscopic balance equation and particular equations for mass, mass of a component, momentum and energy, all of a phase in a porous medium domain. These balance equations involve averaged fluxes. Then macroscopic equations are developed for advective, dispersive and diffusive fluxes, all in terms of averaged state variables of the system. These are combined with the macroscopic balance equations to yield field equations that serve as the core of the mathematical models that describe the transport of extensive quantities in a porous medium domain. It is shown that the methodology of averaging leads to a better understanding of the effective stress concept employed in dealing with transport phenomena in deformable porous media.     

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.     

Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: II. Constitutive Theory

In Part I macroscopic field equations of mass, linear and angular momentum, energy, and the quasistatic form of Maxwell's equations for a multiphase, multicomponent medium were derived. Here we exploit the entropy inequality to obtain restrictions on constitutive relations at the macroscale for a 2-phase, multiple-constituent, polarizable mixture of fluids and solids. Specific emphasis is placed on charged porous media in the presence of electrolytes. The governing equations for the stress tensors of each phase, flow of the fluid through a deforming medium, and diffusion of constituents through such a medium are derived. The results have applications in swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, chromotography, drug delivery, and other swelling systems.     

层状横观各向同性饱和土的非轴对称动力响应

通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的Biot非轴对称波动方 程转化为一组一阶常微分方程组. 然后基于径向Hankel变换,建立问题的状态方程;求解状态方程后,得到传递矩阵. 进而利用传递矩阵,结合饱和层状地基的边界条件、排水条件及层间接触和连续条件,求解 了任意震源力作用下层状横观各向同性饱和地基频域动力响应问题. 时域解可通过频率的Fourier积分得到.     

Equations of the mechanics of porous media saturated with liquid and gas

The approach proposed by Podil'chuk [1] is used to derive a system of equations of motion for saturated porous media, allowance being made for the mutual influence of the solid, liquid, and gas phases. The permeabilities of the anisotropic porous medium are assumed to depend on the direction. It is shown that when there are no gas phases and the liquid is incompressible the system of equations reduces to the general equations of the theory of elasticity of an anisotropic body with fictitious stress components. For a porous medium saturated with liquid, the relationships between the permeabilities and the anisotropy constants are obtained. The motion of liquid in an elastic porous medium in the form of an orthotropic cylindrical region with a cavity in the form of a circular cylinder is considered as an example.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 82–87, July–August, 1981.     

The effective homogeneous behavior of heterogeneous porous media

The macroscopic equations that govern the processes of one- and two-phase flow through heterogeneous porous media are derived by using the method of multiple scales. The resulting equations are mathematically similar to the point equations, with the fundamental difference that the local permeabilities are replaced by effective parameters. The method allows the determination of these parameters from a knowledge of the geometrical structure of the medium and its heterogeneities. The technique is applied to determine the effective parameters for one- and two-phase flows through heterogeneous porous media made up of two homogeneous porous media.     

Thermo-Chemo-Electro-Mechanical Formulation of Saturated Charged Porous Solids

A thermo-chemo-electro-mechanical formulation of quasi-static finite deformation of swelling incompressible porous media is derived from a mixture theory including the volume fraction concept. The model consists of an electrically charged porous solid saturated with an ionic solution. Incompressible deformation is assumed. The mixture as a whole is assumed locally electroneutral. Different constituents following different kinematic paths are defined: solid, fluid, anions, cations and neutral solutes. Balance laws are derived for each constituent and for the mixture as a whole. A Lagrangian form of the second law of thermodynamics for incompressible porous media is used to derive the constitutive restrictions of the medium. The material properties are shown to be contained in one strain energy function and a matrix of frictional tensors. A principle of reversibility results from the constitutive restrictions. Existing theories of swelling media should be evaluated with respect to this principle.     

On the motion of a fluid-saturated porous solid

Two-parameter structure model of a porous solid is proposed as an approximation of a real porous structure and the macroscopic mass and momentum balance equations are derived for such a medium filled with liquid. The approach presented leads to the equations of motion for a fluid-saturated porous medium with coupling terms via cross-mass couplings. The linear form of these equations is equivalent to the well-known Biot equations.     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.     

Motion equations of the dynamic theory of consolidation from the point of view of the theory of transient pipe flows

An elastic fluid-saturated porous medium is modeled as a bundle of parallel cylindrical tubes aligned in a direction parallel to the fluid movement. The pore space is filled with viscous compressible liquid. A cell model and the theory of transient pipe flow are used to derive one-dimensional governing equations in such media. All macroscopic constants in these equations are defined by the individual material constants of the fluid and solid. The interaction force includes an additional term not found in Biot's theory.     

Flow through isotropic granular porous media

A generalization of the Navier-Stokes equation is developed to include laminar flow through a rigid isotropic granular porous medium of spatially varying permeability. The model is based on a theory of interspersed continua and the mean geometrical properties of an idealized granular porous microstructure. The derived momentum transport equations are applicable to granular porous media over the entire porosity range from zero through unity. No restriction with respect to flow velocity is imposed, except for the assumption of laminar flow within the pores. The results provide useful and versatile equations and substantiate many of the empirical equations currently in use. One of the major advantages of the generalized momentum equation is its adaptability to numerical simulation.     

On balance equations for heterogeneous continua

Summary The paper presents, from the viewpoint of classical continuum physics, a rigorous derivation of the general balance equations for heterogeneous continua-substances consisting of many distinct, possibly interacting, constituents. The treatment is a direct generalization of the well known theory for a simple (one-substance) continuum and covers multicomponent media in solid, liquid and gaseous forms. The work rests on axioms in integral form stated for the constituents of the heterogeneous continuum, using well defined material volumes for the medium as a whole. An explicit hypothesis is introduced asserting how a constituent of a heterogeneous medium is influenced by the totality of the others. This hypothesis allows for general interactions among the constituents and, in principle, covers the special results for liquid helium mixtures given by Prigogine and Mazur. All properties of the heterogeneous continuum as a whole are consistently deduced from the corresponding properties of the constituents. In particular, the overall integral balance equations are derived by summing over all components the corresponding balance equations for the constituents. The differential balance equations for the heterogeneous continuum are consequences of the corresponding integral balance equations and a requirement that specifies the mathematical form of the resultant equations. Herein, this requirement is a general structural hypothesis that, in certain special cases of gaseous mixtures, agrees with known results of kinetic theory. While our development differs from that of Truesdell, it may be considered as an integral equivalent to his and a completion of the program he started.This work was in part supported by the National Aeronautics and Space Administration research grant NsG-381. It was completed while the author served as consultant for the Army Missile Command. The support of both agencies is gratefully acknowledged.     

Infiltration of liquid droplets into porous media: Effects of dynamic contact angle and contact angle hysteresis

We present a detailed theory for infiltration, which accounts for a general model for the dynamic contact angle between the droplet and the porous medium as well as contact angle hysteresis, and analyze the resulting equations of motion. The theory shows that infiltration of droplets into dry porous media involves three phases due to contact angle hysteresis: (1) An increasing drawing area (IDA) phase during which the interface between the droplet and the porous medium increases, (2) a constant drawing area (CDA) phase during which the contact line of the droplet remains pinned, and (3) a decreasing drawing area (DDA) phase. The theory is based on the following assumptions: (1) The droplet has the shape of a spherical cap, (2) the porous medium consists of a bundle of vertical tubes of same size, and (3) the pressure within the droplet is uniform. We find that infiltration always consists of a cascade process formed by the IDA, CDA, and DDA phases, where the entire process may begin or end in any of the three phases. The entire process is formulated with four nondimensional parameters: Three contact angles (initial, advancing, and receding) and a porous permeability parameter. A comparison of our theory to experimental data suggests that one should use different parameterizations for the dynamic contact angle models of the IDA and DDA phases. In general, the IDA and DDA phases are described by integro-differential equations. A numerical-solution approach is presented for solving the dynamic equations for infiltration. The total time of infiltration and the time dependence of drawing area are critically affected by the occurrence of the IDA, CDA, and DDA phases as well as by the permeability. With ordinary differential equations (ODEs), we are able to approximate the IDA phase and to describe exactly infiltration processes that start out with the CDA or DDA phase.     

A solution of steady-state fluid flow in multiply fractured isotropic porous media

Herein a plane, steady-state fluid flow solution for fractured porous media is first presented. The solution is based on the theory of complex potentials, the theory of Cauchy integrals, and of singular integral equations. Subsequently, a numerical method is illustrated that may be used for the accurate estimation of the pore pressure and pore pressure gradient fields due to specified hydraulic pressure or pore pressure gradient acting on the lips of one or multiple non-intersecting curvilinear cracks in a homogeneous and isotropic porous medium. It is shown that the numerical integration algorithm of the singular integral equations is fast and converges rapidly. After the successful validation of the numerical scheme several cases of multiple curvilinear cracks are illustrated.