Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

This article compares for the first time, local longitudinal and transverse dispersion coefficients obtained by homogenization with experimental data of dispersion coefficients in porous media, using the correct porosity dependence. It is shown that the longitudinal dispersion coefficient can be reasonably represented by a simple periodic unit cell (PUC), which consists of a single sphere in a cube. We present a slightly modified and simplified approach to derive the homogenized equations, which emphasizes physical aspects of homogenization. Subsequently, we give full dimensional expressions for the dispersion tensor based on a comparison with the convective dispersion equation used for contaminant transport, inclusive the correct dependence on porosity. For the PUC of choice, the dispersion relations are identical to the relations obtained for periodic media. We show that commercial finite element software can be readily used to compute longitudinal and transverse dispersion coefficients in 2D and 3D. The 3D results are for the first time obtained at relevant Peclet numbers. There is good agreement for longitudinal dispersion. The computed transverse dispersion coefficients for a single sphere in a cube are much too low. The effect of adsorption on the dispersion coefficient is also studied. Adsorption does not affect the transverse dispersion coefficient. However, adsorption enhances the longitudinal dispersion coefficient in agreement with an analysis of homogenization applied to Taylor dispersion discussed in the literature.     

Transverse dispersion at high Peclet numbers in short porous media

Transverse dispersion is examined in short ceramic foams for a variety of pore sizes and flow rates. Experiments were performed with a localized smoke source as a tracer and a coflow of air. Digital images were taken of the resulting dispersion patterns, and an approximate transverse dispersion coefficient was determined. A precise dispersion coefficient cannot be defined at these high Peclet numbers (10⁶–10⁸) because the dispersion becomes non-Fickian. Nevertheless, the approximate coefficients are shown to be consistent with the values obtained by other researchers for lower Peclet numbers and longer porous media.     

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     

Viscous Fingering Instability in Porous Media: Effect of Anisotropic Velocity-Dependent Dispersion Tensor

The viscous fingering of miscible flow displacements in a homogeneous porous media is examined to determine the effects of an anisotropic dispersion tensor on the development of the instability. In particular, the role of velocity-dependent transverse and longitudinal dispersions is investigated through linear stability analysis and nonlinear simulations. It is found that an isotropic velocity-dependent dispersion tensor does not affect substantially the development of the instability and effectively has the same effect as molecular diffusion. On the other hand, an anisotropic velocity-dependent dispersion tensor results in different instability characteristics and more intricate finger structures. It is shown that anisotropic dispersion has profound effects on the development of the fingers and on the mechanisms of interactions between neighboring fingers. The development of the new finger structures is explained by examining the velocity field and characterized qualitatively through a spectral analysis of the average concentration and an analysis of the variations of the sweep efficiency and relative contact area.     

Probabilistic approach for transport of contaminants through porous media

The channels formed between individual particles in porous media have variable dimensions and orientations. The porosity, permeability and its anisotropy exhibit random spatial distributions. The probabilistic approach can effectively describe the transport of contaminants through porous media and is analysed in this paper. Numerical results are obtained by considering (I) random dispersion coefficients without and with spatial structure, (II) random time distribution of concentration at the inlet boundary, (III) random velocity distribution in the flow field without and (IV) with variable dispersion coefficient, (V) non-linearity of the governing equation and (VI) anisotropy of the dispersion coefficient. Two methods are used for probabilistic predictions: (1) Gaussian field approach in conjunction with Monte Carlo method and (2) random walk method. The input random parameters are assumed to have normal and log-normal distributions according to available experimental data. The probability distribution functions of the contaminant concentration at different locations within the flow domain are calculated and compared with the input distributions as a function of the mean and fluctuation Peclet numbers. The one-dimensional case is analysed in detail and the illustrative numerical predictions are compared with analytical and experimental results. The extension to a two-dimensional domain is discussed in the last part of this paper.     

Heat dispersion in stationary mixed convection flow about horizontal surfaces in porous media

An analysis has been performed to study the influence of velocity dependent dispersion on transverse heat transfer in mixed convection flow above a horizontal wall of prescribed temperature in a saturated porous medium. The Boussinesq approximation and boundary layer analysis were used to numerically obtain gravity affected temperature and velocity distributions within the frames of Darcy's law and a total thermal diffusivity tensor comprising both of constant coefficient heat conduction and velocity proportional mechanical heat dispersion. Dependending on Pe_∞, the molecular Peclét number basing on the effective thermal diffusivity and the velocity of the oncoming flow, density coupling has distinct influences on heat transfer rates between the wall surface and the porous medium flow region. For small Peclét numbers, when heat conduction is the prevailing mechanism, wall heat fluxes are the higher the larger the density difference between the oncoming and the near wall fluid is. The opposite is true for larger Peclét numbers, when mechanical heat dispersion is the main cause of heat spreading. For Pe_∞ tending to infinity these wall heat fluxes approach finite maximum values in the total heat diffusivity model, they grow beyond any limit if only constant coefficient heat conduction is considered. Thus, the inclusion of mechanical heat dispersion effects yields physically more realistic predictions. Received on 18 September 1996     

Analytical Solutions for Macrodispersion in a 3D Heterogeneous Porous Medium with Random Hydraulic Conductivity and Dispersivity

A stochastic analysis of macrodispersion for conservative solute transport in three-dimension (3D) heterogeneous statistically isotropic and anisotropic porous media when both hydraulic conductivity and local dispersivity are random is presented. Analytical expressions of macrodispersivity are derived using Laplace and Fourier transforms. The effects of various parameters such as ratio of transverse to longitudinal local dispersivity, correlation length ratio, correlation coefficient and direction of flow on asymptotic macrodispersion are studied. The behaviour of growth of macrodispersivity in preasymptotic stage is also shown in this paper. The variation in local dispersion coefficient causes change in transverse macrodispersivity. The consideration of random dispersivity along with random hydraulic conductivity indicates that the total dispersion is affected and important in the case when the hydraulic conductivity and dispersivity are correlated. It is observed that the pre-asymptotic behavior of the macrodispersivity is not sensitive to the choice of spectral density functions.     

Prediction on dispersion in elastoplastic unsaturated granular media

This study focuses on the propagation of the plane wave in the elastoplastic unsaturated granular media, and the wave equations and dispersion equations are derived for the media under the framework of Cosserat theory. Due to symmetry, five different wave modes are considered and predicted for the elastoplastic unsaturated granular media based on the Cosserat theory, including two longitudinal waves, one rotational longitudinal wave and two coupled transverse–rotational transverse waves. The correspondence is discussed between these Cosserat wave modes and the classical wave modes. Based on the dispersion equations, the dispersion behaviors are obtained for the five Cosserat wave modes. The results indicated that the different stress-strain stages,including the elastic, hardening and softening stages, have obvious effect on the dispersion behaviors of the Cosserat wave modes.     

Miscible Displacements of Reactive and Anisotropic Dispersive Flows in Porous Media

The viscous fingering instability of miscible reactive–dispersive flows in a homogeneous porous media is investigated through nonlinear numerical simulations. In particular, the role of velocity-dependent transverse and longitudinal dispersions as well as the type and rate of auto-catalytic chemical reactions is analyzed. It is found that for a third-order auto-catalytic reaction, the higher the reaction rate, the more complex the finger structures. Furthermore, major differences between the flow development of third-order and second-order autocatalytic reactions are reported. In addition, the anisotropy and velocity dependence of the dispersion tensor are found to have a more profound effect on the fingering instability in the case of reactive flows than in the non-reactive ones. The qualitative characterization of the finger structures is explained by examining the flow velocity field and further quantified through an analysis of the average concentration and relative contact area.     

Moisture Removal from a Two-Layer Porous Media: A Conceptual Model and Experimental Results

Moisture removal from a two-layer porous media in which air is circulated through one layer and moisture is removed from the second has not been well studied due to the emphasis given to single-layer systems. This two-layer configuration is common in natural and engineered systems and can be used as a means to create a barrier to downward migrating fluids and to remove liquids and gases that may be present in the finer layer. However, there is little data on moisture removal from a two-layer porous media in which air is circulated through one layer parallel to the interface and moisture is removed from the finer second layer by evaporation. A conceptual model of the moisture removal from a two-layer porous media system was developed and compared to experimental moisture removal rates from laboratory scale dry barriers. The limited experimental data agrees well with the results predicted by the conceptual model, providing an initial validation.     

Upscaling of Nonwetting Phase Residual Transport in Porous Media: A Network Approach

Based on the volume averaging method, a macroscopic model is developed for the upscaling of NAPL transport in a porous medium idealised by a network model. Under the assumption of local mass non-equilibrium, a macroscopic equation involving a dispersion tensor, additional convective terms and a linear form for the interfacial mass flux is obtained. The resolution of the two local closure problems obtained allow the determination of the local properties without adjustable parmeters. These problems are solved in a semi-analytical, semi-numerical manner on the network. The originality of this work is the association of the upscaling by volume averaging method with the network approach. The local properties, including the dispersion tensor and the mass exchange coefficient, can therefore be calculated over a large number of pore-bodies and pore-throats in a computationaly tractable manner, thus leading to more significant results. Results are presented for 3D, spatially periodic models of porous media.     

Contribution to the Modelling of Coupled Heat and Mass Transfers on 3D Real Structure of Heterogeneous Building Materials: Application to Hemp Concrete

Hydraulic flow, electrical flow and the passage of elastic waves through porous media are all linked by electrokinetic processes. In its simplest form, the passage of elastic waves through the porous medium causes fluid to flow through that medium and that flow gives rise to an electrical streaming potential and electrical counter-current. These processes are frequency-dependent and governed by coupling coefficients which are themselves frequency-dependent. The link between fluid pressure and fluid flow is described by dynamic permeability, which is characterised by the hydraulic coupling coefficient (C_hp). The link between fluid pressure and electrical streaming potential is characterised by the streaming potential coefficient (C_sp). While the steady-state values of such coefficients are well studied and understood, their frequency dependence is not. Previous work has been confined to unconsolidated and disaggregated materials such as sands, gravels and soils. In this work, we present an apparatus for measuring the hydraulic and streaming potential coefficients of high porosity, high permeability consolidated porous media as a function of frequency. The apparatus operates in the range 1 Hz to 2 kHz with a sample of 10 mm diameter and 5–30 mm in length. The full design and validation of the apparatus are described together with the experimental protocol it uses. Initial data are presented for three samples of Boise sandstone, which present as dispersive media with the critical transition frequency of 918.3?±?99.4 Hz. The in-phase and in-quadrature components of the measured hydraulic and streaming potential coefficients have been compared to the Debye-type dispersion model as well as theoretical models based on bundles of capillary tubes and porous media. Initial results indicate that the dynamic permeability data present an extremely good fit to the capillary bundle and Debye-type dispersion models, while the streaming potential coefficient presents an extremely good fit to all of the models up to the critical transition frequency, but diverges at higher frequencies. The streaming potential coefficient data are best fitted by the Pride model and its Walker and Glover simplification. Characteristic pore size values calculated from the measured critical transition frequency fell within 1.73% of independent measures of this parameter, while the values calculated directly from the Packard model showed an underestimation by about 12%.

     

Impact of local diffusion on macroscopic dispersion in three-dimensional porous media

While macroscopic longitudinal and transverse dispersion in three-dimensional porous media has been simulated previously mostly under purely advective conditions, the impact of diffusion on macroscopic dispersion in 3D remains an open question. Furthermore, both in 2D and 3D, recurring difficulties have been encountered due to computer limitation or analytical approximation. In this work, we use the Lagrangian velocity covariance function and the temporal derivative of second-order moments to study the influence of diffusion on dispersion in highly heterogeneous 2D and 3D porous media. The first approach characterizes the correlation between the values of Eulerian velocity components sampled by particles undergoing diffusion at two times. The second approach allows the estimation of dispersion coefficients and the analysis of their behaviours as functions of diffusion. These two approaches allowed us to reach new results. The influence of diffusion on dispersion seems to be globally similar between highly heterogeneous 2D and 3D porous media. Diffusion induces a decrease in the dispersion in the direction parallel to the flow direction and an increase in the dispersion in the direction perpendicular to the flow direction. However, the amplification of these two effects with the permeability variance is clearly different between 2D and 3D. For the direction parallel to the flow direction, the amplification is more important in 3D than in 2D. It is reversed in the direction perpendicular to the flow direction.     

Dispersion in consolidated sandstone with radial flow

This paper presents some experimental and theoretical results for dispersion processes occurring in consolidated Berea sandstone with radial flow geometry. A comprehensive review of the derivation and application of several analytical solutions is also presented. The Galerkin finite element method is applied to solve the advection-dispersion equation for unidimensional radial flow.Individual and combined effects of mechanical dispersion and molecular diffusion are examined using velocity-dependent dispersion models. Comparison of simulated results with experimental data is made. The effect of flow rates is examined. The results suggest that a linear dispersion model,D=u, whereD is the dispersion coefficient,u the velocity and a constant, is not a good approximation despite its wide acceptance in the literature. The most suitable mathematical formulation is given by an empirical form of , whereD _ois the molecular diffusion coefficient. For the range of Péclet number (Pe=vd/D _m,wherev is the characteristic velocity,d the characteristic length andD _mthe molecular diffusion coefficient in porous media) examined (Pe=0.5 to 285), a power constant ofm=1.2 is obtained which agrees with the value reported by some other workers for the same regime.     

Permeability coefficient tensor in the Kozeny-Carman capillary model

The representation of the permeability coefficient tensor for capillary models of porous media displaying isotropic and anisotropic flow properties is considered. The representation proposed is compared with the Kozeny-Carman formula. It is shown that in general, as distinct from the widely accepted representation of the Carman constant in the form of a product of the form factor and tortuosity, this constant is equal to a combination of three coefficients, namely the form factor, the tortuosity, and the structure coefficient. The presence of the latter is due to the fact that in periodic capillary models the porosity is not equal to the surface porosity. It is shown that the Carman constant, the form factor, and the structure coefficient are not universal and their intervals of variation are calculated. The results obtained make it possible to explain and interpret numerous experimental data on the determination of the Carman constant in various porous media.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 96–104, July–August, 1996.     

The effect of leaching on mixing in porous media with a single heterogeneity

This paper presents an exploratory study of the effect of leaching on mixing in a porous medium containing a single heterogeneity to investigate the effect of the heterogeneity and time-dependent pore structure on dispersion. A percolation-convection simulation (PCS) model is used along with laboratory model experiments to study the mixing. The results show that mixing changes when the pores of the models are leached and that there is a change in regime influence during leaching. The simulation represents the mixing through a first leach for homogeneous media and for heterogeneous media with significant changes in permeability. If the pore structure is changing with time, prediction of mixing must include effects of heterogeneity and regime influence. Although the experimental results are representative of idealized laboratory sized systems they provide insight into the effects of leaching in heterogeneous media. Further the simulation may be useful on a field scale.Nomenclature b molecular weight, gm/mol - C concentration, mol/cm³ - C ₀ initial concentration, mol/cm³ - d _rms root-mean-squared distance, cm - d ₅₀ 50% grain size, cm - D channel depth, cm - f _n fraction of input tracer in effluent at time t _n - K ₁ permeability of flow field outside of heterogeneity, cm² - k ₂ permeability of heterogeneity, cm² - k _S reaction rate constant, cm/min - K _L microscopic dispersion coefficient in the longitudinal direction, cm²/sec - K _O overall dispersion coefficient in the longitudinal direction, cm²/sec - K _T microscopic dispersion coefficient in the transverse direction, cm²/sec - L length of channel, cm - n exponent for velocity - P pressure, N/M² - Pe Peclet number, Lv/K _O - P _ext local pressure outside heterogeneity, N/M² - P _int local pressure inside heterogeneity, N/M² - Q volumetric flow rate, cm³/sec - R channel half width, cm - t time, sec - W _c channel width, cm - W _c0 initial channel width, cm - v interstitial fluid velocity, cm/sec - v _k macroscopic velocity in transverse direction, cm/sec - v _y macroscopic velocity in longitudinal direction, cm/sec - v fluid velocity entering the medium, cm/sec - x _i transverse location of parcel at time t _i, cm - y _i longitudinal location of parcel at time t _i, cm - x microscopic movement in transverse direction, cm - y microscopic movement in longitudinal direction, cm Greek Letters t time increment, sec - ₀ overall dispersivity, cm - ² longitudinal variance of the distribution, cm² - porosity - _B bulk density, gm/cm³ - fractional grade of leachable material Currently with Center for Naval Analysis.     

Flow stability in a channel with porous walls

We study the stability of the flow which forms in a plane channel with influx of an incompressible viscous fluid through its porous parallel walls. Under certain assumptions the study of the stability reduces to the solution of modified Orr-Sommerfeld equation accounting for the transverse component of the main-flow velocity. As a result of numerical integration of this equation we find the dependence of the local critical Reynolds number on the blowing Reynolds number R₀, which may be defined by two factors: the variation of the longitudinal velocity profile with R₀ and the presence of the transverse velocity component. A qualitative comparison is made of the computational results with experimental data on transition from laminar to turbulent flow regimes in channels with porous walls, which confirms that it is necessary to take into account the effect of the transverse component of the main-flow velocity on the main-flow stability in the problem in question.Flows in channels with porous walls are of interest for hydrodynamic stability theory in view of the fact that they can be described by the exact solutions of the Navier-Stokes equations by analogy with the known Poiseuille and Couette flows. However, in contrast with the latter, the flows in channels with porous walls (studies in [1], for example) will be nonparallel.The theory of hydrodynamic stability of parallel flows has frequently been applied to nonparallel flows (in the boundary layer, for example). In so doing the nonparallel nature of the flow has been taken into account only by varying the longitudinal velocity component profiles. A study was made in [2, 3] of the effect of the transverse component of the main flow on its stability. In the case of the boundary layer in a compressible gas, a considerable influence of the transverse velocity component on the critical Reynolds number was found in [2] and confirmed experimentally. A strong influence of the transverse velocity component on the instability region was also found in [3] in a study of the flow stability in a boundary layer with suction for an incompressible fluid.     

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.