Two approaches to modeling unstable flow and mixing of variable density fluids in porous media

Density variation of aqueous phase fluids flowing in a porous medium, resulting from spatial and temporal variation of solute concentration, often gives rise to unstable flow, and therefore has a significant effect on solute transport. Studies on simulating unstable flow and mixing of variable density fluids in seemingly homogeneous porous media are rare. In this study, a three-dimensional (3-D) and a one-dimensional (1-D) model were developed to simulate unstable flow and mixing in a vertical, nominally 1-D system. In the 3-D model, the fluid flow and solute transport equations were solved numerically with a very fine spatial discretization. The 1-D numerical model was derived from a theoretical model to simulate the flow and mixing of fluids with variable density and viscosity at the field scale. To evaluate the models, simulated results were compared with experimental data from displacement experiments in a vertical sand column. The results show that the 1-D model provides fairly good prediction of breakthrough curves and that the 3-D model is able to qualitatively simulate breakthrough curves for highly unstable flow and mixing.Contribution from the Alabama Agric. Exp. Sta. as AAES Journal No. 3-955037.     

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.     

Comparison of two- and three-dimensional modeling of invert trap for sewer solid management

In the present study, five different invert trap configurations （rectangular with and without lids on both sides; trapezoidal, trapezoidal with rectangular base and rectangular with trapezoidal base with lids on both sides） were simulated for both two-dimensional （2D） and three-dimensional （3D） flow conditions for three sediment types （sand, styrocell and plastic beads） at six flow rates （0.35, 0.70, 1.05, 1.35, 4.55 and 9.95 L/s） for each trap. Computational fluid dynamics （CFD）-based modeling using FLUENT software with Renormalization Group （RNG） k-e model along with discrete phase model （DPM） were used in the simulations. A hexagonal/tetrahedral and map-type non-uniform grid was chosen to discretize the entire computational domain and a control volume finite difference method was used to solve the governing equations. The flow rates selected in the present study cover the entire range of flow rate expected for dry weather and monsoon. The simulation is capable of differentiating between 2D and 3D modeling of particle trajectories, the effects of flow rate and trap geometry on flow patterns developed in the trap. The sediment retention ratio for 2D is higher than that for 3D modeling for all flow conditions, particle types and model geometry due to inclusion of lateral effects in 3D modeling. The invert trap having rectangular shape with trapezoidal base is found to be the most efficient configuration in both 2D and 3D modeling.     

Approximate analytical solutions for solute transport in two-layer porous media

Mathematical models for transport in layered media are important for investigating how restricting layers affect rates of solute migration in soil profiles; they may also improve the analysis of solute displacement experiments. This study reports an (approximate) analytical solution for solute transport during steady-state flow in a two-layer medium requiring continuity of solute fluxes and resident concentrations at the interface. The solutions were derived with Laplace transformations making use of the binomial theorem. Results based on this solution were found to be in relatively good agreement with those obtained using numerical inversion of the Laplace transform. An expression for the flux-averaged concentration in the second layer was also obtained. Zero- and first-order approximations for the solute distribution in the second layer were derived for a thin first layer representing a water film or crust on top of the medium. These thin-layer approximations did not perform as well as the binomial solution, except for the first-order approximation when the Peclet number,P, of the first layer, was low (P<5). Results of this study indicate that the ordering of two layers will affect the predicted breakthrough curves at the outlet of the medium. The two-layer solution was used to illustrate the effects of dispersion in the inlet or outlet reservoirs using previously published data on apparatus-induced dispersion.The U.S. Government right to retain a non-exclusive, royalty free licence in and to any copyright is acknowledged.     

Modeling of Solute Transport in a 3D Rough-Walled Fracture–Matrix System

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.     

Finite element modeling of transport in porous media

Modern computational techniques enable, in principle, the modeling of transport in porous media, involving convection, adsorption and dispersion. Implementation of the techniques for practical problems leads to various difficulties, however. One of these is the difference in horizontal and vertical scales in natural situations; other difficulties encountered are numerical dispersion and the flow near singularities. In order to overcome these difficulties a two-dimensional flow model has been adapted to incorporate three-dimensional velocity components. This procedure takes into account that in regional flow fields the horizontal flow components in aquifers are much larger than the vertical components, and yet it enables to observe transport in vertical direction. Numerical dispersion is suppressed by particle tracking.     

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.     

On the Influence of Pore-Scale Dispersion in Nonergodic Transport in Heterogeneous Formations

Flow of an inert solute in an heterogeneous aquifer is usually considered as dominated by large-scale advection. As a consequence, the pore-scale dispersion, i.e. the pore scale mechanism acting at scales lower than that characteristic of the heterogeneous field, is usually neglected in the computation of global quantities like the solute plume spatial moments. Here the effect of pore-scale dispersion is taken into account in order to find its influence on the longitudinal asymptotic dispersivity D₁₁we examine both the two-dimensional and the three-dimensional flow cases. In the calculations, we consider the finite size of the solute initial plume, i.e. we analyze both the ergodic and the nonergodic cases. With Pe the Péclat number, defined as Pe=U/D, where U, , D are the mean fluid velocity, the heterogeneity characteristic length and the pore-scale dispersion coefficient respectively, we show that the infinite Péclat approximation is in most cases quite adequate, at least in the range of Péclat number usually encountered in practice (Pe > 10²). A noteworthy exception is when the formation log-conductivity field is highly anisotropic. In this case, pore-scale may have a significant impact on D₁₁, especially when the solute plume initial dimensions are not much larger than the heterogeneities' lengthscale. In all cases, D₁₁ appears to be more sensitive to the pore-scale dispersive mechanisms under nonergodic conditions, i.e. for plume initial size less than about 10 log-conductivity integral scales.     

Bubble deformation and flow structure measured by double shadow images and PIV/LIF

An experiment on bubble motion in a simple shear layer was performed in order to obtain fundamental knowledge of the force on the bubble and its lateral motion induced by the surrounding flow field. We explored the flow structure in the vicinity of the bubble in one plane and its deformation in two planes by particle image velocimetry (PIV)–laser-induced fluorescence (LIF) and a projection technique for two perpendicular planes, respectively. For our experiment, we chose a single air bubble with an equivalent bubble diameter D _eq of 2~6 mm in a vertical shear flow. Velocity measurements were made using a digital high-speed CCD camera for PIV with fluorescent tracer particles. The second and third CCD cameras were used to detect the bubbles shape and motion via backlighting from an array of infrared LEDs. We quantitatively studied the three-dimensional wake structure from measurements of the two-dimensional vortex structure and approximated three-dimensional shape deformation arranged from two perpendicular bubble images.     

Solute Transport Analysis Through Heterogeneous Media in Nonuniform in the Average Flow by a Stochastic Approach

The stochastic approach has been shown to be an excellent tool for the characterisation and analysis of velocity fields and transport processes through heterogeneous porous formations. The main results (linear theory) have been obtained for problems with simplified flow conditions, usually in the assumption of uniform in the average flow, but a great effort is spent to reach theoretical results for more complex situations.This paper deals with 2D heterogeneous aquifers subject to uniform recharge; the stochastic approach is adopted to characterise, as ensemble behaviour, the velocity field and transport processes of a nonreactive solute. The impact of transmissivity conditioning on solute particles trajectories is analysed and an application is carried out. The analytical formulations, obtained by a first order analysis, are compared to the one resulting from constant in the average hydraulic gradient, and their reliability is investigated with numerical tests performed by a Monte Carlo method.The result of this study is that strong non-stationarities are present in the flow and transport process. A detailed analysis shows that the theoretical results cannot be extended to cases with high heterogeneity level, unlike the uniform in the average flow fields.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. 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 a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

Experimental investigation of solute transport in large,homogeneous and heterogeneous,saturated soil columns

Laboratory tracer experiments were conducted to investigate solute transport in 12.5-m long, horizontally placed soil columns during steady saturated water flow. Two columns having cross-sectional areas of 10×10cm² were used: a uniformly packed homogeneous sandy column and a heterogeneous column containing layered, mixed, and lenticular formations of various shapes and sizes. The heterogeneous soil column gradually changed, on average, from coarse-textured at one end to fine-textured at the other end. NaCl breakthrough curves (BTC's) in the columns were measured with electrical conductivity probes inserted at 50- or 100-cm intervals. Observed BTC's in the homogeneous sandy column were relatively smooth and sigmoidal (S-shaped), while those in the heterogeneous column were very irregular, nonsigmoidal, and exhibited extensive tailing. Effective average pore-water velocities (v _eff) and dispersion coefficients (D _eff) were estimated simultaneously by fitting an analytical solution of the convection-dispersion equation to the observed BTC's. Velocity variations in the heterogeneous medium were found to be much larger than those in the homogeneous sand. Values of the dispersivity,=D _eff/v _eff, for the homogeneous sandy column ranged from 0.1 to 5.0 cm, while those for the heterogeneous column were as high as 200cm. The dispersivity for transport in both columns increased with travel distance or travel time, thus exhibiting scale-dependency. The heterogeneous soil column also showed the effects of preferential flow, i.e., some locations in the column showed earlier solute breakthrough than several locations closer to the inlet boundary. Spatial fluctuations in the dispersivity could be explained qualitatively by the particular makeup of the heterogeneities in the column.     

Three-dimensional reconstruction of cardiac flows based on multi-planar velocity fields

Measurement of the three-dimensional flow field inside the cardiac chambers has proven to be a challenging task. This is mainly due to the fact that generalized full-volume velocimetry techniques cannot be easily implemented to the heart chambers. In addition, the rapid pace of the events in the heart does not allow for accurate real-time flow measurements in 3D using imaging modalities such as magnetic resonance imaging, which neglects the transient variations of the flow due to averaging of the flow over multiple heartbeats. In order to overcome these current limitations, we introduce a multi-planar velocity reconstruction approach that can characterize 3D incompressible flows based on the reconstruction of 2D velocity fields. Here, two-dimensional, two-component velocity fields acquired on multiple perpendicular planes are reconstructed into a 3D velocity field through Kriging interpolation and by imposing the incompressibility constraint. Subsequently, the scattered experimental data are projected into a divergence-free vector field space using a fractional step approach. We validate the method in exemplary 3D flows, including the Hill’s spherical vortex and a numerically simulated flow downstream of a 3D orifice. During the process of validation, different signal-to-noise ratios are introduced to the flow field, and the method’s performance is assessed accordingly. The results show that as the signal-to-noise ratio decreases, the corrected velocity field significantly improves. The method is also applied to the experimental flow inside a mock model of the heart’s right ventricle. Taking advantage of the periodicity of the flow, multiple 2D velocity fields in multiple perpendicular planes at different locations of the mock model are measured while being phase-locked for the 3D reconstruction. The results suggest the metamorphosis of the original transvalvular vortex, which forms downstream of the inlet valve during the early filling phase of the right ventricular model, into a streamline single-leg vortex extending toward the outlet.     

Diffusion–Convection in a Porous Medium with Impervious Inclusions at Low Flow Rates

The aim of this paper is to develop a macroscopic model for the transport of a passive solute, by diffusion and convection, in a heterogeneous medium consisting of impervious solids periodically distributed in a porous matrix. In the porous part, the flow is described by Darcy's law. Attempt is made to derive the macroscopic equation governing the average concentration field in the equivalent macroscopic medium and the macroscopic transport parameters. The analysis is conducted in the case when convection and diffusion are of the same order of magnitude at the macroscopic level, that is, when the Péclet number is of order 1. The proposed macroscopic model is obtained using the homogenization method for periodic structures with a double scale asymptotic expansion, in which the small parameter is the ratio between the two characteristic lengths l (the period scale of the impervious bodies distribution) and L (the scale of the macroscopic sample). The macroscopic parameters, which characterize the multiporous medium, depend solely on the transport parameters in the porous matrix and on the geometry of the impervious inclusions without any phenomenological assumption. Numerical computations are performed using a finite element method for several geometries of the solid inclusions, in two- and three-dimensional cases.     

Numerical study on the shear-flow behavior and transport process in rough rock fractures

This paper presents a systematic research for understanding mechanical shearing effects on the fluid flow and the solute transport behavior of rough fractures through a numerical simulation approach. The aperture fields were modeled based on a real rock fracture geometry and the normal displacement obtained from the shear-flow test. The fluid flow through the rough fracture under shear was simulated using a finite element code that solves the Reynolds equation, and the transport behavior through the rough fracture under shear was simulated calculating the advection–dispersion equation. The results show that the fracture apertures increase as the shear displacement increases, with a few major flow channels detected through the fracture. The shear-induced flow channels increase both flow connectivity and transport connectivity, which accelerate the movement of solutes in a particular direction and lead to early breakthrough of the contaminants. Adsorption, acting as a retardation term, has a decisive influence on the transport process. These results can give a basic knowledge of the hydromechanical and solute transport progress through fracture, and will be helpful to safety assessment for high-level radioactive waste disposal facilities.     

Three-dimensional particle image velocimetry for use in three-phase fluidization systems

A three-dimensional particle image velocimetry (3-DPIV) system is developed to measure the three-dimensional local flow properties of gas-liquid and gas-liquid-solid fluidization systems. The 3-DPIV system requires one camera to simultaneous record orthogonal views of the flow field created by a special optical arrangement. The 3-DPIV has been successfully calibrated and is capable of providing qualitative and quantitative flow information including three-dimensional, full-field, instantaneous velocities, accelerations and holdups of each phase. In this study, sample results of the application of the 3-DPIV technique to a three-dimensional gas-liquid-solid fluidization system operating in the dispersed bubble flow regime demonstrate that the 3-DPIV technique is an effective instrument in studying the local, transient flow phenomena in multiphase systems.Notation D focal length of camera - d _a horizontal offset - l particle displacement - t time interval between consecutive video frames - U particle velocity - Ul superficial liquid velocity - Ug superficial gas velocity - V virtual viewpoint of orthogonal projection - x, y, z spatial coordinates - z _a vertical offset This work was supported by the NSF Grant CTS-9200793.     

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.     

An Analytical Solution to the One-Dimensional Solute Advection-Dispersion Equation in Multi-Layer Porous Media

An analytical solution to the one-dimensional solute advection-dispersion equation in multi-layer porous media is derived using a generalized integral transform method. The solution was derived under conditions of steady-state flow and arbitrary initial and inlet boundary conditions. The results obtained by this solution agree well with the results obtained by numerically inverting Laplace transform-generated solutions previously published in the literature. The analytical solution presented in this paper provides more flexibility with regard to the inlet conditions. The numerical evaluation of eigenvalues and matrix exponentials required in this solution technique can be accurately and efficiently computed using the sign-count method and eigenvalue evaluation methods commonly available. The illustrative calculations presented herein have shown how an analytical solution can provide insight into contaminant distribution and breakthrough in transport through well defined layered column systems. We also note that the method described here is readily adaptable to two and three-dimensional transport problems.