Stochastic Analysis of Reactive Solute Transport in Heterogeneous,Fractured Porous Media: A Dual-Permeability Approach

A nonlocal, first-order, Eulerian stochastic theory is developed for reactive chemical transport in a heterogeneous, fractured porous medium. A dual-permeability model is adopted to describe the flow and transport in the medium, where the solute convection and dispersion in the matrix are considered. The chemical is under linear nonequilibrium sorption and first-order degradation. The hydraulic conductivities, sorption coefficients, degradation rates in both fracture and matrix regions, and interregional mass transfer coefficient are all assumed to be random variables. The resultant theory for mean concentrations in both regions is nonlocal in space and time. Under spatial Fourier and temporal Laplace transforms, the mean concentrations are explicitly expressed. The transformed results are then numerically inverted to the real space via Fast Fourier Transform method. The theory developed in this study generalizes the stochastic studies for a reactive chemical transport in a one-domain flow field (Hu et al., 1997a) and in a mobile/immobile flow field (Huang and Hu, 2001). In comparison with one-domain transport, the dual-permeability model predicts a larger second moment in the longitudinal direction, but smaller one in the transverse direction. In addition, various simplification assumptions have been made based on the general solution. The validity of these assumptions has been tested via the spatial moments of the mean concentration in both fracture and matrix regions.     

A Numerical Method of Moments for Solute Transport in Nonstationary Flow Fields

A Lagrangian perturbation approach has been applied to develop the method of moments for predicting mean and variance of solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity may stem from medium nonstationarity, finite domain boundaries, and/or fluid pumping and injecting. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at the control plane. The analytically derived moment equations for solute transport in a nonstationary flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. This method is also compared with the numerical Monte Carlo method. The calculation results indicate the two methods match very well when the variance of log-conductivity is small, but the method of moment is more efficient in computation.     

Large Time Profiles in Reactive Solute Transport

In this paper we consider the large time structure of reactive solute plumes in two dimensional, macroscopically homogeneous, flow domains. The reactions between the dissolved chemicals and the porous matrix are equilibrium adsorption reactions, given by an isotherm of Freundlich type. We also incorporate the effect of partial and full decay. We use the method of asymptotic balancing to obtain, to leading order, the large time behaviour of the solute concentration and the relevant moments (mass, centre of mass,variance). The method of balancing is based on certain conjectures about the form of the temporal decay and partial spreading of the solute. These conjectures are verified numerically.     

Asymptotic Analysis of Solute Transport with Linear Nonequilibrium Sorption in Porous Media

We analysed the asymptotic behaviour of breakthrough curves (BTCs) obtained after a single pulse injection in a 1D flow domain. Five different types of solute transport with nonequilibrium sorption were considered. The properties of the porous medium were assumed to be spatially constant. For long times, the concentration at a fixed position in time was found to decay like exp(–t) where depends on both the transport parameters and the parameters describing the nonequilibrium process. The results from the asymptotic analysis were compared with 1D numerical transport calculations. For all cases examined a good agreement between numerical calculations and the asymptotic analysis was found. The results from the asymptotic analysis provide an alternative way to determine transport and sorption related parameters from BTCs. The derived relationships between and the model parameters are however only valid for large times. This requires that the very low concentrations need to be measured and not only the bulk mass, too. By either increasing or decreasing the velocity during BTC experiments additional asymptotic equations are obtained which can be used to determine the value of the model parameters. The results from the asymptotic analysis can also be used in standard inverse modelling techniques to either obtain good initial guesses or to reduce the parameter space. The fact that linear nonequilibrium processes decay like exp(–t) can be used to qualitatively evaluate observed BTCs. The asymptotic analysis can also be easily extended to a larger class of transport problems (e.g. transport of solutes with microbial decay) provided that the overall transport problem remains linear in the concentration.     

A Solute Transport in Stratified Media

Two-dimensional and steady solute transport in a stratified porous formation is analysed under assumption that the effect of pore-scale dispersion is negligible. The longitudinal dispersion produced as a result of the vertical variation of hydraulic conductivity is analysed by averaging the variability of a solute flux concentration and conductivity. The evolution of the solute flux concentration is expressed with respect to the correlated variable, that is the travel (arrival) time at a fixed location and the averaging procedure is constructed to satisfy the boundary condition where the inlet concentration is a known function of time. In such a statement, a velocity-averaged solute flux concentration is described by a conventional dispersion model (CDM) with a dispersion coefficient which is a function of the arrival time. It is demonstrated that such CDM satisfies the assumption that hydraulic conductivity of the layers is gamma distributed with the parameter of distribution which is chosen to represent a reasonable value of the field scale solute dispersion. The overall behaviour of the model is illustrated by several examples of two-dimensional mass transport.     

Experiments on Solute Transport in Aggregated Porous Media: Are Diffusions Within Aggregates and Hydrodynamic Dispersion Independent?

Two region models for solute transport in porous media assume that hydrodynamic dispersion in mobile water and solute diffusion within immobile water regions are independent. Experimental and theoretical results for transport through a macropore indicate that hydrodynamic dispersion and solute exchange are interdependent. Experiments were carried out to investigate this problem for a column packed with spherical porous aggregates. The effective diffusion coefficient of a tracer within the agreggates was determined from specific experiments. The dispersivity of the bed was determined from experiments carried out with a column filled with nonporous beads. We took advantage of the dependence of hydrodynamic dispersion on density ratios between the invading and displaced solutions to obtain a set of breakthrough curves corresponding to situations where the diffusion coefficient remains constant, whereas the dispersivity varies. Simulations reproduce correctly the experiments. Small discrepancies are noted that can be corrected either by increasing the dispersion coefficient or by fitting the external mass transfer coefficient. Increased dispersion coefficients probably reveal a modification of Taylor dispersion due to solute exchange. The fitted external mass transfer coefficients are close to the values obtained with classical correlations of the chemical engineering literature.     

Modeling Variable Density Flow and Solute Transport in Porous Medium: 1. Numerical Model and Verification

A new numerical model for the resolution of density coupled flow and transport in porous media is presented. The model is based on the mixed hybrid finite elements (MHFE) and discontinuous finite elements (DFE) methods. MHFE is used to solve the flow equation and the dispersive part of the transport equation. This method is more accurate in the calculation of velocities and ensures continuity of fluxes from one element to the adjacent one. DFE is used to solve the convective part of the transport equation. Combined with a slope limiting procedure, it avoids numerical instabilities and creates a very limited numerical dispersion, even for high grid Peclet number.Flow and transport equations are coupled by a standard iterative scheme. Residual based criterion is used to stop the iterations. Simulations of an unstable equilibrium show the effects of the criteria used to stop the iterations and the stopping criterion in the solver. The effects are more important for finer grids than for coarser grids.The numerical model is verified by the simulation of standard benchmarks: the Henry and the Elder test cases. A good agreement is found between the revised semianalytical Henry solution and the numerical solution. The Elder test case was also studied. The simulations were similar to those presented in previous works but with significantly less unknowns (i.e. coarser grids). These results show the efficiency of the used numerical schemes.     

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.     

Conjugate Phase Change Heat Transfer in an Inclined Compound Cavity Partially Filled with a Porous Medium: A Deformed Mesh Approach

In this paper, the melting process of a PCM inside an inclined compound enclosure partially filled with a porous medium is theoretically addressed using a novel deformed mesh method. The sub-domain area of the compound enclosure is made of a porous layer and clear region. The right wall of the enclosure is adjacent to the clear region and is subject to a constant temperature of T_c. The left wall, which is connected to the porous layer, is thick and thermally conductive. The thick wall is partially subject to the hot temperature of T_h. The remaining borders of the enclosure are well insulated. The governing equations for flow and heat transfer, including the phase change effects and conjugate heat transfer at the thick wall, are introduced and transformed into a non-dimensional form. A deformed grid method is utilized to track the phase change front in the solid and liquid regions. The melting front movement is controlled by the Stefan condition. The finite element method, along with Arbitrary Eulerian–Lagrangian (ALE) moving grid technique, is employed to solve the non-dimensional governing equations. The modeling approach and the accuracy of the utilized numerical approach are verified by comparison of the results with several experimental and numerical studies, available in the literature. The effect of conjugate wall thickness, inclination angle, and the porous layer thickness on the phase change heat transfer of PCM is investigated. The outcomes show that the rates of melting and heat transfer are enhanced as the thickness of the porous layer increases. The melting rate is the highest when the inclination angle of the enclosure is 45°. An increase in the wall thickness improves the melting rate.

     

A Two-Velocity Two-Temperature Model for a Bi-Dispersed Porous Medium: Forced Convection in a Channel

A two-velocity two-temperature model for bi-dispersed porous media is formulated. Using the model, an analytic solution is obtained for the problem of forced convection in a channel between parallel plane walls that are held either at uniform temperature or uniform heat flux. In each case, Nusselt number values are given as functions of a conductivity ratio, a velocity ratio, a volume fraction, and an internal heat exchange parameter.     

Transport with a Very Low Density Contrast in Hele–Shaw Cell and Porous Medium: Evolution of the Mixing Zone

The optimal concentration of a blue dye solution with 'tracer' properties, enabling a pollutant to be marked was determined by the use of numerical, theoretical and experimental approaches. Experimental investigations were performed on a transparent Hele–Shaw cell and the concentration distribution was analyzed using an optical technique based on dye light absorption properties. The injected optimal concentration was established thanks to a theoretical and experimental study carried out on the output signal dynamics. Using the same experimental conditions, numerical simulations were performed. The very good agreement between the data (experimental and numerical) clarified that: (i) the choice of the blue dye optimal concentration was valid and (ii) the concentration-dependent density should not be neglected in flow and transport equations even if it concerns a so-called 'tracer'. Following this remark, a theoretical aspect was developed in order to determine the analogous conditions between a Hele–Shaw cell and a porous medium for the variable density transport phenomenon. The structure of the concentration-dependent dispersion tensor used in the numerical code was obtained by homogenizing the Stokes flow of a bi-component mixture. The numerical results show that, as long as the tracer density does not exceed a certain value, it is not necessary to take into account a density contrast in terms of the dispersion tensor. The classical form of the Taylor dispersion tensor can be used successfully.     

Darcian Dynamics: A New Approach to the Mobilization of a Trapped Phase in Porous Media

We develop a new approach, which we term Darcian Dynamics, to simulate two-phase (liquid-gas) flow in porous media, when the gas phase is disconnected in the form of ganglia. The method is based on the assumption of homogeneous fluid flow for the liquid, although it does allow for heterogeneous capillary thresholds due to the pore microstructure. Using techniques from potential theory, the hydrodynamic interaction between liquid and gas is expressed through an integral representation over the ganglia interfaces. We use a numerical method to solve the resulting integral equation, and explore conditions for the onset of ganglia mobilization as well as for subsequent events, such as break-up, coalescence and stranding. The interaction between the ganglia and the flowing phase is influenced by the capillary and gravity (Bond) numbers, and by geometric factors, such as size, orientation, and ganglia density. The latter effect depends on the hydrodynamic interaction in addition to the intuitively expected crowding effect.     

<Emphasis Type="Italic">A Priori</Emphasis> Parameter Estimation for the Thermodynamically Constrained Averaging Theory: Species Transport in a Saturated Porous Medium

The thermodynamically constrained averaging theory (TCAT) has been used to develop a simplified entropy inequality (SEI) for several major classes of macroscale porous medium models in previous works. These expressions can be used to formulate hierarchies of models of varying sophistication and fidelity. A limitation of the TCAT approach is that the determination of model parameters has not been addressed other than the guidance that an inverse problem must be solved. In this work we show how a previously derived SEI for single-fluid-phase flow and transport in a porous medium system can be reduced for the specific instance of diffusion in a dilute system to guide model closure. We further show how the parameter in this closure relation can be reliably predicted, adapting a Green’s function approach used in the method of volume averaging. Parameters are estimated for a variety of both isotropic and anisotropic media based upon a specified microscale structure. The direct parameter evaluation method is verified by comparing to direct numerical simulation over a unit cell at the microscale. This extension of TCAT constitutes a useful advancement for certain classes of problems amenable to this estimation approach.     

Driving Potentials of Heat and Mass Transport in Porous Building Materials: A Comparison Between General Linear,Thermodynamic and Micromechanical Derivation Schemes

In systems of coupled transport processes the question of the appropriate driving potentials is a point of discussion. In this article, three different approaches to derive models for transport currents are systematically compared. According to a general linear approach, an arbitrary full set of independent state variables and material properties is sufficient to describe any transport current. This approach is derived here from a symmetry principle. Thermodynamic and micromechanical approaches are more complex and even less general, but they allow additional statements about the transport coefficients and they reduce the number of transport processes. In the thermodynamic approach the additional information stems from the calculation of the entropy production rate; the micromechanical approach involves a microphysical model of the considered porous system. As a practical example, the three derivation schemes are applied to the often-encountered case of non-hysteretic heat and moisture transport in homogeneous building materials. It is shown, how the general state variables of a porous system are reduced to only two. Then from the general linear approach it can be seen, that all equations for the moisture transport current using a main driving potential (e.g. moisture content, vapour pressure, chemical potential) and an independent secondary driving potential (e.g. temperature, liquid pressure) are equivalent, without recurrence to the thermodynamic or micromechanical approach. However, the transport coefficients are arbitrary phenomenological functions depending on the two state variables. Based on a literature survey it is shown, which additional statements can be made in the thermodynamic and in the micromechanical approach. The latter yields the pressure-driven model (vapour and liquid pressure as the two driving potentials). Finally it is shown, what is to be expected, if in more complex systems the number of state variables increases.     

Thermally Developing Forced Convection in a Porous Medium: Circular Duct with Walls at Constant Temperature,with Longitudinal Conduction and Viscous Dissipation Effects

A modified Graetz methodology is applied to investigate the thermal development of forced convection in a circular duct filled by a saturated porous medium, with walls held at constant temperature, and with the effects of longitudinal conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Péclet number, Brinkman number).     

Response to Comment on “Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work” by A. Barletta and D. A. Nield,Transport in Porous Media,DOI 10.1007/s11242-008-9320-y, 2009

This short note is a response to the comment on our recently published paper cited in the title. All points raised by the author of the comment are discussed. It is shown that one of the remarks, concerning eigenflow solutions in the limiting case of forced convection, does not have a sound physical basis. In fact, it refers to a circumstance, a fluid with a thermal expansion coefficient greater than that of a perfect gas, of marginal or no interest in the framework of convection in porous media.     

Notes on Convection in a Porous Medium: (i) an Effctive Rayleigh Number for an Anisotropic Layer, (ii) the Malkus Hypothesis and Wavenumber Selection

Studies of convection in a layer of a saturated anisotropic porous medium uniformly heated from below are reviewed. Emphasis is placed on the usefulness of an effective Rayleigh number, defined (for certain boundary conditions) as the square harmonic-mean square root of horizontally-based and vertically-based Rayleigh numbers. The status of the Malkus hypothesis, together with its relationship with observed cell size, is also discussed. A possible explanation is provided for the observed phenomenon that in a porous medium the cell size decreases as the amplitude of convection increases, whereas in a clear fluid the cell size increases.