Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - a _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - A _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - V velocity vector in the-phase, m/s - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment

In this paper we examine the closure problem associated with the volume averaged form of the Stokes equations presented in Part II. For both ordered and disordered porous media, we make use of a spatially periodic model of a porous medium. Under these circumstances the closure problem, in terms of theclosure variables, is independent of the weighting functions used in the spatial smoothing process. Comparison between theory and experiment suggests that the geometrical characteristics of the unit cell dominate the calculated value of the Darcy's law permeability tensor, whereas the periodic conditions required for thelocal form of the closure problem play only a minor role.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface associated with the local closure problem, m² - A _p surface area of a particle, m² - b vector used to represent the pressure deviation, m^–1 - B ⁰ B+I, a second order tensor that maps v _m ontov - B second-order tensor used to represent the velocity deviation - d _p 6V _p/A_p, effective particle diameter, m - d a vector related to the pressure, m - D a second-order tensor related to the velocity, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor calculated on the basis of a spatially periodic model, m² - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p characteristic length for the volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - characteristic length (pore scale) for the-phase - _i i=1, 2, 3 lattice vectors, m - weighting function - m(-y) , convolution product weighting function - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - n unit normal vector pointing from the-phase toward the -phase - p pressure in the-phase, N/m² - p _m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p – p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function - r position vector, m - r position vector locating points in the-phase, m. - V averaging volume, m³ - B volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v _m superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - v traditional superficial volume averaged velocity, m/s - v – v _m , spatial deviation velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the -phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - _m m * , weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v traditional superficial volume averaged velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V/V, volume average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

A study of two-dimensional dispersion in unconsolidated porous media

An experimental and numerical investigation into the magnitude of longitudinal and transverse dispersion in a two-dimensional flow field over a particle Peclet number range of 50–8500 is reported. Numerical modelling using a Galerkin finite element method is used to test various models, notably those of Fried and combarnous and Koch and Brady. Dispersion at low Peclet numbers (< 200) is found to be described adequately by either model, which at large Peclet, the degree of dispersion is significantly underestimated. An improved dispersion model for Peclet numbers greater than 200 is proposed. The transverse dispersion term and the choice of inlet boundary condition are found to have a negligible effect on the shape of the breakthrough curve.Nomenclature A (z) Polynomial in the z plane - B (z) Polynomial in the z plane - C Concentration - C _f Feed concentration - C _o Concentration at the entrance - D Dispersion tensor - D _f Molecular diffusion coefficient - D ₁ Longitudinal dispersion coefficient - D _p Particle diameter - D _t Transverse dispersion coefficient - k Permeability/viscosity - k Dimensionless permiability in the Koch and Brady model - P Pressure - Pe _k Modified Peclet number, Pe _p k - Pe _p Particle Peclet number vD _p /D _f - v Velocity - z Axial coordinate or complex variable Greek letters Solution domain - Boundary - Porosity     

多孔介质中二维对流传热的分叉

本文利用分叉理论研究了流体饱和的二维多孔介质从底部加热所引起的自然对流,用有限差分方法确定对流的分叉进程;揭示其模式转换机理及分叉对非正常流动图象形成的影响;同时确定了矩形截面宽高比与临界端利数的关系。还提出了一个判别分支稳定笥的简明方法。     

Dispersion of inert solutes in spatially periodic,two-dimensional model porous media

Taylor dispersion of a passive solute within a fluid flowing through a porous medium is characterized by an effective or Darcy scale, transversely isotropic dispersitivity , which depends upon the geometrical microstructure, mean fluid velocity, and physicochemical properties of the system. The longitudinal, and lateral, dispersivity components for two-dimensional, spatially periodic arrays of circular cylinders are here calculated by finite element techniques. The effects of bed voidage, packing arrangement, and microscale Péclet and Reynolds numbers upon these dispersivities are systematically investigated.The longitudinal dispersivity component is found to increase with the microscale Péclet number at a rate less than Pe². This accords with previous calculations by Eidsath et al. (1983), although the latter calculations were found to yield significantly lower longitudinal dispersivities than those obtained with the present numerical scheme. With increasing Péclet number, a Pe² dependence is, however, approached asymptotically, particularly for square cylindrical arrays - owing to the creation of a linear streamline zone between cylinders.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Measurement of two-dimensional velocity fields in porous media by particle image displacement velocimetry

Local measurements of the phase function and of two components of the velocity can be performed in transparent porous media by means of particle image displacement velocimetry (P.I.D.V.). Some preliminary results are presented and discussed.     

Stochastic analysis of two-phase flow in porous media: II. Comparison between perturbation and Monte-Carlo results

The first paper (Changet al., 1995) described stochastic analysis of two-phase flow in a fully liquid-saturated system. In this paper, the results of Monte-Carlo simulations are compared with the closed-form expressions obtained using the perturbation approach. We present analytical solutions to the one-dimensional, steady-state oil-and-water flow equations. These solutions are subsequently used in the Monte-Carlo analysis to estimate the statistical properties of the key output processes. The comparison between the results of perturbation and Monte-Carlo approaches shows a good agreement between the two methods over a wide range lnk (k is the intrinsic permeability) variability with three different combinations of input stochastic processes of lnk and soil parameter . In particular, a good agreement was obtained for capillary and individual pressure variances and effective phase conductivities. The results showed that as the mean capillary pressure in the profile increased, the variance of capillary pressure increased while the variances of individual pressures decreased. Overall comparison between the two methods showed that the first-order perturbation theory can be successfully used to describe the effective behavior of large-scale, two-phase systems.     

Models and problems for saturated porous media

A multiphase model of a liquid- and gas-saturated porous medium is proposed. The model takes into account the finite deformations of a skeleton, arbitrary flows of liquids and gases, and the phase mass transfer between the skeleton and the liquids. Some basic relations are given for the corresponding boundary value problems in the cases of arbitrary and small motions. The constitutive relations describing the properties of the skeleton resistance to deformation (skeleton stresses) and the mutual resistance of components (internal interaction) are considered in detail. The internal actions exerted on the skeleton by a moving liquid (gas) are discussed when these actions may take the form of drag forces, lift (displacing) forces, overturning moments, and rotational (screw) moments.     

On tortuosity and areosity tensors for porous media

An average streamwise channel velocity is proposed as a more accurate representation of the actual intrapore velocity than the intrinsic phase average velocity. A relationship is derived between the average streamwise channel velocity and the interstitial velocity and superficial velocity. New definitions of tortuosity and areosity as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. The definitions produce results which conform with several other published results and are applicable to anisotropic media in general.     

Natural convection of non-Newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media

Numerical analysis of the free convection coupled heat and mass transfer is presented for non-Newtonian power-law fluids with the yield stress flowing over a two-dimensional or axisymmetric body of an arbitrary shape in a fluid-saturated porous medium. The governing boundary layer equations and boundary conditions are cast into a dimensionless form by the similarity transformation. The resulting system of equations is solved by a finite difference method. The parameters studied are the rheological constants, the buoyancy ratio, and the Lewis number. Representative velocity, temperature, and concentration profiles are presented and discussed. It is found that the results depend strongly on the values of the yield stress parameter and the power-law index of the non-Newtonian fluid.     

Natural convection of non-Newtonian power-law fluid over axisymmetric and two-dimensional bodies of arbitrary shape in fluid-saturated porous media

Numerical analysis of the free convection coupled heat and mass transfer is presented for non-Newtonian power-law fluids with the yield stress flowing over a two-dimensional or axisymmetric body of an arbitrary shape in a fluid-saturated porous medium. The governing boundary layer equations and boundary conditions are cast into a dimensionless form by the similarity transformation. The resulting system of equations is solved by a finite difference method. The parameters studied are the rheological constants, the buoyancy ratio, and the Lewis number. Representative velocity, temperature, and concentration profiles are presented and discussed. It is found that the results depend strongly on the values of the yield stress parameter and the power-law index of the non-Newtonian fluid.     

Transport equations for elastic and other waves in random media

We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.     

Thermal stretching in two-phase porous media: Physical basis for Maxwell model

An alternate yet general form of the classical effective thermal conductivity model (Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physical basis. It is demonstrated that the reduced effective thermal conductivity of the porous media due to non-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combination of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).     

A LGA model for dispersion in heterogeneous porous media

The lattice gas automaton (LGA) model proposed in the previous paper is applied to the problem of simulating dispersion and mixing in heterogeneous porous media. We demonstrate here that tracer breakthrough profiles and longitudinal dispersion coefficients can be computed for heterogeneous porous media.     

A scale-dependent equation for solute transport in porous media

The governing equation describing solute transport in porous media is reformulated using standard volume averaging techniques. The alternative formulation is based on a modified definition of the deviation, which allows for variation of macroscopic velocity across the REV. The new equation contains additional scale-dependent terms which are functions of the size of the averaging volume (REV). This result indicates that the scale-dependent nature of the dispersion phenomenon is inherent even at the scale of the REV.     

New conductivity probes for dispersivity measurements in porous media

The new electric probes described in this note are fabricated using printed circuit techniques. They offer various advantages over the existing models: high miniaturization, small disturbance of the porous medium, precise location, low cost and easy fabricatio     

Capillary ascension in porous media: A scaling law

A relationship between the microstructure of a porous medium and the height of ascension of a fluid in the material is proposed. The porous medium is modelled by means of a square lattice randomly filled with matter. A scaling law is derived from a numerical simulation. An analytical solution is also obtained and compared with the numerical results.