Stratified Porous Medium Research Articles

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF LNAPL MOVEMENT IN STRATIFIED MEDIA DURING SOIL REMEDIATION

The saturation distribution and clean up efficiency of light non-aqueous phase liquid (LNAPL) in the strata beneath the earth has been the subject of many studies. Better understanding of LNAPL infiltration into layered soil is important for the effective design of remediation strategies. The objective of this study was to simulate LNAPL movement in homogenous and stratified porous media using gravity assisted inert gas injection (GAIGI) process as a cleaning technique. We used homogeneous and layered sandpacked transparent models that allows for visual observation of LNAPL movement in order to study LNAPL redistribution in a layered porous medium. Pore volume, porosity, absolute permeability, connate water saturation, and oil saturation of the models were determined experimentally. Seasonal water table movement and contaminated zone were established and then, under GAIGI process, clean up efficiency was determined. The downward displacement of LNAPL by gas drive resulted in very high LNAPL clean up efficiency. Using the contaminant production history in the homogeneous model, the LNAPL relative permeability was calculated and the results were extended to layered media. The numerical multi-phase flow model in porous media was validated with regard to the experimental results. This model is able to adequately reproduce the experimental LNAPL saturation profile and clean up efficiency.

Effective dispersion in a chemically heterogeneous medium under temporally fluctuating flow conditions

We investigate effective solute transport in a chemically heterogeneous medium subject to temporal fluctuations of the flow conditions. Focusing on spatial variations in the equilibrium adsorption properties, the corresponding fluctuating retardation factor is modeled as a stationary random space function. The temporal variability of the flow is represented by a stationary temporal random process. Solute spreading is quantified by effective dispersion coefficients, which are derived from the ensemble average of the second centered moments of the normalized solute distribution in a single disorder realization. Using first-order expansions in the variances of the respective random fields, we derive explicit compact expressions for the time behavior of the disorder induced contributions to the effective dispersion coefficients. Focusing on the contributions due to chemical heterogeneity and temporal fluctuations, we find enhanced transverse spreading characterized by a transverse effective dispersion coefficient that, in contrast to transport in steady flow fields, evolves to a disorder-induced macroscopic value (i.e., independent of local dispersion). At the same time, the asymptotic longitudinal dispersion coefficient can decrease. Under certain conditions the contribution to the longitudinal effective dispersion coefficient shows superdiffusive behavior, similar to that observed for transport in s stratified porous medium, before it decreases to its asymptotic value. The presented compact and easy to use expressions for the longitudinal and transverse effective dispersion coefficients can be used for the quantification of effective spreading and mixing in the context of the groundwater remediation based on hydraulic manipulation and for the effective modeling of reactive transport in heterogeneous media in general.

Upscaling, relaxation and reversibility of dispersive flow in stratified porous media

Dispersive tracer released in a unidirectional velocity field belonging to a stratified porous of finite height describes a transition, called relaxation, from a convective dominated behaviour for short times to Fickian behaviour for asymptotic long times. The temporal relaxation state of the tracer is controlled by the transverse mixing term. In most practical applications, the orders of the time and length scales of the relaxation mechanism are such that in an upscaled model of a stratified medium the dispersive flux is in a pre-asymptotic state. Explicit modelling of the relaxation of the dispersive flux in the pre-asymptotic region is required to improve the accuracy. This paper derives a pre-asymptotic one-dimensional upscaled model for the transverse averaged tracer concentration. The model generalises Taylor dispersion (Proc. R. Soc. London 219, 186–203 (1953)) and extends the method of Camacho (Phys. Rev. E 47(2), 1049–1053 (1993a); Phys. Rev. E 48 (1993b)) to dispersion tensors that may vary as function of the transverse direction. In the averaging step, the governing two-dimensional equation is first spectrally decomposed in terms of the eigenfunctions of the transverse mixing term. Next, the resulting modal relaxation equations are combined into an effective relaxation equation for the extended dispersive Taylor flux. Contrary to the one-dimensional Fickian approach, the upscaled model approximates the multi-scale relaxation behaviour as a single scale relaxation process and accounts for the partial reversibility of convective dispersion upon reversal of the flow direction. The upscaled model is evaluated against the original two-dimensional model by means of moment analysis. The longitudinal tracer variance predicted by our model is quantitatively correct in the short and long time limits and is qualitatively correct for intermediate times.

Blowing/Suction Effects on Hydromagnetic Simultaneous Heat and Mass Transfer by Natural Convection from a Vertical Cylinder Embedded in a Thermally Stratified Porous Medium

The problem of steady, laminar, hydromagnetic, combined heat and mass transfer in non-Darcy natural convection flow along a permeable vertical cylinder embedded in a thermally stratified, fluid-saturated porous medium with a stratified free stream in the presence of thermal and solutal dispersion is investigated. The surface of the cylinder is assumed to be permeable to allow for possible fluid wall blowing or suction. A generalized flow model is used to include the effects of macroscopic viscous term and microscopic inertial force. The nonsimilar form of the transformed equations is solved numerically. A parametric study illustrating the influence of the magnetic field, the local thermal stratification parameter ξ, the thermal dispersion parameter Dsγ, the solutal dispersion parameter Dsζ, and suction/blowing parameter fω is made on the profiles of velocity, temperature, and concentration. Numerical data for the skin friction factor, the local Nusselt number, and the local Sherwood number have been presented in tabular form for various parametric conditions.

1,2,4-Trichlorobenzene Flow Characteristics in Saturated Homogeneous and Stratified Porous Media

Subsurface contamination by nonaqueous phase liquids, such as 1,2,4-trichlorobenzene, is a widespread problem at numerous sites. In this work, the behaviour of 1,2,4-trichlorobenzene in the saturated zone was investigated using two natural porous media. To this end, cylindrical columns were packed with different materials and the initial water saturation was displaced by loading a constant level of trichlorobenzene on top of the porous bed. The experiments were carried out with different configurations to simulate natural media with several stratigraphic units. Also, the effect of pressure gradient on nonaqueous phase liquid flow was investigated. Both the pressure gradient and the configuration of the layers of porous media have shown a strong effect on water recoveries and fluxes through the column.

Effects of variable viscosity on non-Darcy MHD free convection along a non-isothermal vertical surface in a thermally stratified porous medium

An analysis is performed for non-Darcy free convection flow of an electrically conducting fluid over an impermeable vertical plate embedded in a thermally stratified, fluid saturated porous medium for the case of power-law surface temperature. The present work examines the effects of non-Darcian flow phenomena, variable viscosity, Hartmann–Darcy number and thermal stratification on free convective transport and demonstrates the variation in heat transfer prediction based on three different flow models. The wall effect on porosity variation is approximated by an exponential function. The effects of thermal dispersion and variable stagnant thermal conductivity are taken into consideration in the energy equation. The resulting non-similar system of equations is solved using a finite difference method. Results are presented for velocity, temperature profiles and local Nusselt number for representative values of different controlling parameters.

Unsteady Free Convection along an Infinite Vertical Flat Plate Embedded in a Stably Stratified Fluid-Saturated Porous Medium

The problem of unsteady free convection heat transfer from a one-dimensional (parallel) flow along an infinite vertical flat plate embedded in a thermally stratified fluid-saturated porous medium is considered. Flows are induced by a sudden change in the arbitrary temporal plate temperature. By a formal reduction of the corresponding boundary value problems to well-known Fourier heat conduction problems, analytical solutions of the Darcy and energy equations are obtained. Several special cases are discussed in detail.

A Numerical Method for Wave Propagation in Viscoelastic Stratified Porous Media

This paper presents a numerical method, a transmission matrix method, for the wave propagation in viscoelastic stratified saturated porous media. The wave propagation in saturated media, based on Biot theory, is a coupled problem. In this stratified three-dimensional model we do the Laplace transform for the time variable and the Fourier transform for the horizontal space coordinate. The original problem is transformed into ordinary differential equations with six independent unknown variables, which are only the function of the coordinate of depth. Thus, we get a transmission matrix of the wave problem for each layer. In the process of solution we use numerical method to calculate the eigenvalues and the eigenvectors of the transmission matrices. In the first step of the solution process we can obtain the wave field in the transformed space. The fast Fourier transform (FFT) method is used to do the inverse Laplace and the inverse Fourier transforms to get the solution in the time space. The detailed formulae are derived and some numerical examples are given.

Modelagem numérica para o processo da evaporação da água do solo

Foram realizados vários ensaios laboratoriais para avaliar o desempenho de um modelo numérico em simular o processo unidimensional da evaporação da água do solo. Este modelo foi desenvolvido a partir da linearização da equação de Richards no espaço e do uso da técnica iterativa de Newton-Raphson, para resolver essa equação não-linear no tempo, e leva em conta o fluxo de vapor dentro do perfil de solo e a taxa de evaporação na sua superfície. Os resultados mostraram que o modelo foi capaz de simular satisfatoriamente o processo, tanto em meios porosos homogêneos quanto estratificados

Upscaling and reversibility of Taylor dispersion in heterogeneous porous media

Tracer flow in stratified porous media is dominated by the interaction between convective transport and transverse diffusive mixing. By averaging the tracer concentration in the transverse direction, a one-dimensional non-Fickian dispersion model is derived. The model accounts for the relaxation process that reduces the convective transport to dispersive mixing. This process is (short-) time correlated and partially reversible upon reversal of flow direction. For multiscale velocity fields, the relaxation is a multiscale process. To date only single scale processes have been successfully upscaled. Our procedure extends this to multiscale processes, using scale separation. The model parameters can be calculated a priori based on the velocity profile. For periodic flow reversal, the results are essentially the same. Despite the non-Fickian behavior during a cycle, the net contribution of each cycle to the spreading relaxes to a Fickian process in a similar way as for unidirectional flow. The cycle time averaged dispersion coefficient is a monotonically increasing function of the reversal time. It asymptotically converges towards the effective dispersion coefficient in the absence of flow reversal.

Stochastic analysis of two-phase immiscible flow in stratified porous media

This paper investigates the physics of two-phase, immiscible flow in stratified porous media. A review of previous studies reveals the major role played by the interaction of heterogeneity and viscous forces in the development of large-scale flow regimes. The stabilizing effects of gravity for flow in vertically layered porous media is introduced in a first order theoretical model and illustrated with the results of Monte Carlo simulations.

Stochastic analysis of two-phase immiscible flow in stratified porous media

This paper investigates the physics of two-phase, immiscible flow in stratified porous media. A review of previous studies reveals the major role played by the interaction of heterogeneity and viscous forces in the development of large-scale flow regimes. The stabilizing effects of gravity for flow in vertically layered porous media is introduced in a first order theoretical model and illustrated with the results of Monte Carlo simulations.

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media. First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions. In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.

Effect of Double Stratification on Free Convection in a Darcian Porous Medium

The problem of free convection heat and mass transfer from vertical surface embedded in a doubly stratified porous medium has been studied. The similarity solution is presented for the case of uniform heat and mass flux conditions when the thermal and solutal stratification of the medium are assumed to have the power function form x1/3. The flow, temperature and concentration fields are effected by the complex interactions among the diffusion ratio parameter Le and buoyancy ratio parameter N in addition to the flow driving Darcy-Rayleigh number Rax. The temperature and concentration profiles are effected due to double stratification of the medium and its effect on the Nusselt and Sherwood numbers is discussed.

Double-Diffusive Natural Convection Induced by a Wavy Surface in a Stratified Porous Medium

Double-Diffusive Natural Convection Induced by a Wavy Surface in a Stratified Porous Medium

Fluid Intake Cavities in Stratified Porous Media

Fluid Intake Cavities in Stratified Porous Media

Non-Darcy free convection induced by a vertical wavy surface in a thermally stratified porous medium

The partial differential equations governing the natural convection heat transfer from a vertical wavy surface in a thermally stratified fluid saturated porous medium are analysed under Forchheimer based non-Darcian assumptions. Based on non-similarity transformation deduced by scale analysis the governing equations are reduced to boundary layer equations. These simplified partial differential equations are solved numerically by a finite difference scheme following the Keller Box approach. Extensive numerical simulations are carried out for various values of wavelength-to-amplitude ratio of wavy vertical surface at different thermal stratification levels of porous medium both under Darcian and non-Darcian assumptions. Results from the current study are compared with those available in literature. In Darcian case local heat fluxes along the wavy vertical surface are periodic with an oscillatory pattern of period, which is exactly half of the period of vertical wavy surface. In the non-Darcian case local heat fluxes continue to be periodic but with a complex oscillatory pattern of period exactly same as that of the vertical wavy surface. Increasing S or Gr ∗ or a leads to a fall in local Nusselt number.

Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid‐saturated stratified porous media

We present a full‐waveform modeling technique of the coupled seismoelectromagnetic wave propagation in fluid‐saturated stratified porous media. Our simulation code uses the macroscopic governing equations derived by Pride [1994], which couple Biot's theory and Maxwell equations via flux/force transport equations. In this theory the coupling mechanism is explained by electrokinetic effects taking place at the pore level. The synthetic seismoelectrograms and seismomagnetrograms are computed by extending the generalized reflection and transmission matrix method and by using a discrete wave number integration of the global reflectivity obtained in the frequency wave number domain. Synthetic time sections and snapshots of the wave propagation are used to study the seismic, electromagnetic, and seismoelectromagnetic waves properties in fluid‐saturated layered porous media. Two wave phenomena are investigated: (1) the electric and magnetic fields induced by the propagation of a seismic perturbation in a homogeneous porous medium and (2) the electromagnetic waves generated at depth when seismic waves propagate through a vertically heterogeneous porous medium. Concentrating on the second effect, we show that the zone which effectively contributes to the generation of EM disturbances along a plane interface coincides with the first Fresnel zone associated with a seismic‐to‐electromagnetic wave conversion. A numerical sensitivity study shows that the EM waves generated at depth by the passage of seismic waves through an interface are particularly sensitive to contrasts in porosity, permeability, fluid salinity, and fluid viscosity. Our numerical simulations highlight the potential of artificially generated seismoelectromagnetic converted waves for the characterization of the subsurface and its fluid content.

Gas flow in a stratified porous medium with crossflow

A new model called semi-permeable wall model is presented for multilayer gas reservoir. The model is used to study the influence of crossflow on pressure transient well tests and other single-phase flow problems. It is suggested here to use this model to approximate the actual multilayer gas reservoir, so that the problem is greatly simplified mathematically. Its differential equation is established here for multilayer gas reservoirs, and is linearized by normalized pseudo pressure and pseudo time. Simulation program is developed by finite-difference method when all layers are perforated. The feature of wellbore pressure and rate is clarified by analyzing the results of numerical simulation.