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.     

Measurement of the Coefficient of Transverse Dispersion in Flow Through Packed Beds for a Wide Range of Values of the Schmidt Number

Experimental values of the coefficient of transverse dispersion (D _T) were measured with the system 2-naphthol/water, over a range of temperatures between 293K and 373K, which corresponds to a range of values of viscosity () between 2.83×10^–4 Ns/m² and 1.01×10^–3 Ns/m² and of molecular diffusion coefficient (D _m) between 1.03×10^–9 m²/s and 5.49×10^–9 m²/s. Since the density () of water is close to 10³ kg/m³, the corresponding variation of the Schmidt number (Sc=/D _m) was in the range 1000 – 50. More than 200 experimental values of the transverse dispersion coefficient were obtained using beds of silica sand with average particle sizes (d) of 0.297 and 0.496mm, operated over a range of interstitial liquid velocities (u) between 0.1mm/s and 14mm/s and this gave a variation of the Reynolds number (Re=du/) between 0.01 and 3.5.Plots of the dimensionless coefficient of transverse dispersion (D _T/D _m) vs. the Peclet number (Pe_m=ud/D _m) based on molecular diffusion bring into evidence the influence of Sc on transverse dispersion. As the temperature is increased, the value of Sc decreases and the values of D _T/D _m gradually approach the line corresponding to gas behaviour (i.e. Sc 1), which is known to be well approximated by the equation D _T/D _m=1/+ud/12D _m, where is the tortuosity with regard to diffusion.     

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     

Full-field measurements of flow through a scaled metal foam replica

Open-celled foam geometries show great promise in heat/mass transfer, chemical treatment, and enhanced mixing applications. Flow measurements on these geometries have consisted primarily of observations of the upstream and downstream effects the foam has on the velocity field. Unfortunately, these observations give little insight into the flow inside the foam. We have performed quantitative flow measurements inside a scaled replica of a metal foam, ϕ = 0.921, D _Cell = 2.5 mm, by Magnetic Resonance Velocimetry to better understand the fluid motion inside the foam and give an alternative method to determine the foam cell and pore sizes. Through these 3-D, spatially resolved measurements of the flow field for a cell Reynolds number of 840, we have shown that the transverse motion of the fluid has velocities 20–30% of the superficial velocity passing through the foam. This strong transverse motion creates and dissipates streamwise jets with peak velocities 2–3 times the superficial velocity and whose coherence length is strongly correlated to the cell size of the foam. This complex fluid motion is described as “mechanical mixing” and is attributed to the geometry of the foam. A mechanical dispersion coefficient, D _M, was formed which demonstrates the transverse dispersion of this geometry to be 14 times the kinematic viscosity and 10 times the thermal diffusivity of air at 20°C and 1 atm.     

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.     

Heat transfer and hydrodynamic studies on two- and three-phase fluidized beds of floating bubble breakers

Heat transfer characteristics in three-phase fluidized beds of floating bubble breakers have been studied in a 0.142 m I.D. x 2.0 m high Plexiglas column fitted with an axially mounted cylindrical heater.Effects of the liquid and gas velocities, the particle size, the volume ratio of floating bubble breaker to particles on phase holdup, the vertical bubble length, and the heat transfer coefficient have been determined.In the bubble-coalescing regime, the heat transfer coefficient in three-phase fluidized beds having the volume ratio V_f/V_s of 10–15% produced a maximum increase in heat transfer coefficient of about 20% in comparison to that in the bed without floating bubble breakers. Also, bubble length and gas-phase holdups exhibited their maximum and minimum values at a volume ratio of 10–15%. The heat transfer coefficient in three-phase fluidized beds of floating bubble breakers can be estimated from the surface renewal model with isotropic turbulence theory.Heat transfer coefficients expressed in terms of the Nusselt number have been correlated with the particle Reynolds number and the volume ratio of floating bubble breakers to particles.     

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.     

An equation of motion for an incompressible Newtonian fluid in a packed bed

The application of a volume average Navier-Stokes equation for the prediction of pressure drop in packed beds consisting of uniform spherical particles is presented. The development of the bed permeability from an assumed porous microstructure model is given. The final model is quasi-empirical in nature, and is able to correlate a wide variety of literature data over a large Reynolds number range. In beds with wall effects present the model correlates experimental data with an error of less than 10%. Numerical solutions of the volume averaged equation are obtained using a penalty finite element method.Nomenclatures d length of a representative unit cell - d _e flow length in Representative Unit Cell - d _p characteristic pore size - D _T column diameter - D _P equivalent particle diameter - e _v energy loss coefficient for elbow - f _app apparent friction factor - f _v packed bed friction factor, defined by Equation (30) - F term representing impermeability of the porous medium - I integral defined by Equation (3) - L length of packed column - N Number of RUC in model microstructure - P pressure - P interstitial pressure - P pressure deviation - Re_p Reynolds number,v _p d _p/ - Re_s Reynolds number,v _s d/gm - Re_b Reynolds number,v _s D _p/ - S _fs fluid solid contact area - T tortuosity - v fluid velocity - v velocity deviation - v _p velocity in a pore - v _s superficial velocity in the medium - v interstitial velocity - V _o total volume of representative unit cell - V pore volume of representative unit cell - change in indicated property - u normal vector onS _fs - porosity - viscosity - density - coefficient in unconsolidated permeability model     

Research on particle dispersion in a plane mixing layer with coherent structures

The numerical simulation with two-way coupling was performed in a liquid -particle mixing layer and the corresponding experiment study was made. In the process of vortex rolling up and vortices pairing, the particles with different St number have a very different pattern of dispersion. The mean velocity of particle with St = 1 is higher than that of the fluid phase on the low-speed side, and lower than that of the fluid phase on the high-speed side. The RMS of particle approaches that of the fluid phase with decreasing particle St number. The RMS in the transverse direction is smaller than that in the streamwise direction. The velocity fluctuation correlation of particle is smaller than the Reynolds shear stress, the “overshoot“ phenomenon that the velocity fluctuation correlation of particle is larger than the Reynolds shear stress does not appear. The larger the St number of particle is, the wider the range of the particle dispersion will be. The computed results are in agreement with the experimental ones.     

Measurements of mass transfer between particles and gas in packed tubes at very low tube to particle diameter ratios

Experimental results for the mass transfer between spherical naphthalene particles and air in packed tubes of very low tube to particle diameter ratio (D/d=1.41, 1.98, and 3.77) are presented. During the experiments the Reynolds number (2.4Re ₀<1500), the=" bed=" length,=" and=" the=" test=" temperature=" have=" been=" varied.=" data=" reduction=" has=" been=" carried=" out=" with=" and=" without=" accounting=" for=" axial=" dispersion=" in=" the=" model.=" the=" measured=" sherwood=" numbers=" are=" compared=" with=" the=" predictions=" according=" to=" the=" correlation=" of=" gnielinski=" and=" of=" wakao/funazkri,=" originally=" developed=" for=" large=" packed=" beds.=" the=" porosity=" of=" packings=" at=">D/d-ratios is discussed.Es werden Versuchsergebnisse über die Stoffübertragung zwischen kugelförmigen Naphthalinpartikeln und Luft in Festbetten mit sehr kleinem Verhältnis zwischen dem Rohr- und dem Partikeldurchmesser (D/d=1,41, 1,98 und 3,77) mitgeteilt. Während der Experimente wurden die Reynoldszahl (2,4Re ₀<1500), die=" bettlänge=" und=" die=" temperatur=" variiert.=" die=" versuchsauswertung=" erfolgte=" sowohl=" mit=" als=" auch=" ohne=" berücksichtigung=" der=" axialen=" dispersion.=" die=" experimentell=" ermittelten=" sherwoodzahlen=" werden=" mit=" den=" voraussagen=" der=" korrelationen=" von=" gnielinski=" und=" von=" wakao/funazkri=" verglichen;=" beide=" korrelationen=" wurden=" für=" betten=" mit=" einem=">D/d-Verhältnis entwickelt. Außerdem wird die Porosität von Betten im Bereich kleinerD/d-Quotienten behandelt.Dedicated to Prof. Dr.-Ing. K. Stephan on the occasion of his 60th birthday     

Combined Effect of Stress, Pore Pressure and Temperature on Methane Permeability in Anthracite Coal: An Experimental Study

The effects of particle-size distribution on the longitudinal dispersion coefficient ( $D_{\mathrm{L}})$ D L ) in packed beds of spherical particles are studied by simulating a tracer column experiment. The packed-bed models consist of uniform and different-sized spherical particles with a ratio of maximum to minimum particle diameter in the range of 1–4. The modified version of Euclidian Voronoi diagrams is used to discretize the system of particles into cells that each contains one sphere. The local flow distribution is derived with the use of Laurent series. The flow pattern at low particle Reynolds number is then obtained by minimization of dissipation rate of energy for the dual stream function. The value of $D_{\mathrm{L}}$ D L is obtained by comparing the effluent curve from large discrete systems of spherical particles to the solution of the one-dimensional advection–dispersion equation. Main results are that at Peclet numbers above 1, increasing the width of the particle-size distribution increases the values of $D_{\mathrm{L}}$ D L in the packed bed. At Peclet numbers below 1, increasing the width of the particle-size distribution slightly lowers $D_{\mathrm{L}}$ D L .     

Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media

We review and discuss diffusion and hydrodynamic dispersion in a heterogeneous porous medium. Two types of heterogeneities are considered. One is percolation disorder in which a fraction of the pores do not allow transport to take place at all. In the other type, the permeabilities of various regions of the pore space are fractally distributed with long-range correlations. Both systems give rise to unusual transport in which the mean square displacement <r ²(t)> of a particle grows nonlinearly with time. Depending on the heterogeneities and the mechanism of diffusion and disperison, we may havefractal transport in which <r ²> growsslower than linearly with time, orsuperdiffusive transport in which <r ²> growsfaster than linearly with time. We show that percolation models can give rise to both types of transport with scale-dependent transport coefficients such as diffusivity and dispersion coefficients, which are consistent with many experimental observations.     

On the modeling of miscible flow in multi-component porous media

An analytical model of miscible flow in multi-component porous media is presented to demonstrate the influence of pore capacitance in extending diffusive tailing. Solute attenuation is represented naturally by accommodating diffusive and convective flux components in macropores amd micropores as elicited by the local solute concentration and velocity fields. A set of twin, coupled differential equations result from the Laplace transform and are solved simultaneously using a differential operator for one-dimensional flow geometry. The solutions in real space are achieved using numeric inversion. In addition, to represent more faithfully the dominant physical processes, this approach enables efficient and stable semi-analytical solution procedure of the coupled system that is significantly more complex than current capacitance type models. Parametric studies are completed to illustrate the ability of the model to represent sharp breakthrough and lengthy tailing, as well as investigating the form of the nested heterogeneity as a result of solute exchange between macropores and micropores. Data from a laboratory column experiment is examined using the present model and satisfactory agreement results.Roman Letters a rate coefficient of internal flow - b velocity ratio (v ₁/v ₂) - h dispersion ratio (D ₂/D ₁) - c ₁ macropore concentration - c ₂ micropore concentration - ¯c ₁ macropore concentration in Laplace space - ¯c ₂ micropore concentration in Laplace space - c ₁ ⁰ macropore concentration at source location - c ₂ ⁰ micropore concentration at source location - D ₁ macropore dispersion coefficient - D ₂ micropore dispersion coefficient - f fraction of pore space occupied by fluid in primary channel - L length of laboratory sample column - K mass exchange rate - t time from initial stage - v ₁ primary flow channel velocity - v ₂ micropore interstitial velocity - x distance from source - y dimensionless distance Greek Letters equivalent Péclet number - dimensionless time, or injected pore volume     

TANDEM CYLINDERS IN IMPULSIVELY STARTED FLOW

The impulsively started flow field for circular cylinders of equal diameter arranged in tandem was investigated using flow visualization and particle image velocimetry (PIV), over a longitudinal pitch ratio range ofL /D=1·0–3·0, and for Reynolds numbers from Re=1200–3800. The PIV technique was used to obtain a time history of the instantaneous in-plane vorticity field from the moment of impulsive start, from which the spatial and temporal development of the flow was studied. Measurements of vortex strength and vortex position relative to the cylinders were obtained from these data. Three types of fluid behaviour were identified based on L/D: single bluff-body behaviour when the cylinders are in contact, constrained streamwise growth and lateral expansion of the gap recirculation zones at small and intermediate L/D, and independent formation of recirculation zones similar to a single impulsively started circular cylinder at larger L/D.     

Transient behavior of heat removal from a cylindrical heat storage vessel packed with spherical porous particles

The transient heat transfer behavior in the case of heat removal from a cylindrical heat storage vessel packed with spherical particles was investigated experimentally for various factors (flow rate, diameter of spherical particles packed, temperature difference between flowing cold air and spherical particles accumulating heat, and physical properties of spherical particles). The experiments were covered in ranges of Reynolds number based on the mean diameter of spherical particles packed Re_d = 10.3–2200, porosity?=0.310 to 0.475, ratio of spherical particle diameter to cylinder diameterd/D = 0.0075–0.177 and ratio of length of the cylinder to cylinder diameterL/D=2.5–10. It was found that especially the flow rate and the dimension of spherical particles played an important role in estimating the transient local heat transfer characteristics near the wall of the cylindrical vessel in the present heat storage system. As flow rate and diameter of spherical particles were increased under a given diameter of the cylinder heat storage vessel, the mean heat transfer coefficient between the flow cold air and the hot spherical particles increased and the time period to finish removing heat from the vessel reduced. In addition, the useful experimental correlation equations of mean heat transfer coefficient between both phases and the time period to finish removing heat from the vessel were derived with the functional relationship of Nusselt numberNu _d=f [modified Prandtl numberPr ^* (d/D), Re_d) and Fourier numberFo = f(d/D, L/D, Pr^*, Re_d).     

The equations of miscible flow with negligible molecular diffusion

A set of equations with generalized permeability functions has been proposed by de la Cruz and Spanos, Whitaker, and Kalaydjian to describe three-dimensional immiscible two-phase flow. We have employed the zero interfacial tension limit of these equations to model two phase miscible flow with negligible molecular diffusion. A solution to these equations is found; we find the generalized permeabilities to depend upon two empirically determined functions of saturation which we denote asA andB. This solution is also used to analyze how dispersion arises in miscible flow; in particular we show that the dispersion evolves at a constant rate. In turn this permits us to predict and understand the asymmetry and long tailing in breakthrough curves, and the scale and fluid velocity dependence of the longitudinal dispersion coefficient. Finally, we illustrate how an experimental breakthrough curve can be used to infer the saturation dependence of the underlying functionsA andB.Roman Letters A a surface area; cross-sectional area of a slim tube or core - A _1s pore scale area of interface between solid and fluid 1 - A ₁₂ pore scale area of interface between fluid 1 and fluid 2 - A(S ₁) fluid flow weighting function defined by Equation (3.21) - a _i ,b _a ,c _a ,d _i macro scale parameters,i=1...2 (Section 3); polynomial coefficients,i=1...N (Section 7) - B(S ₁) fluid flow weighting function defined by Equation (3.16) - c _e effluent concentration - c _i mass concentration fluidi=1...2 - c _fi fractional mass concentration of fluidi=1...2 - D dispersion tensor - D _m mechanical dispersion tensor - D ₀ molecular dispersion tensor - D _L longitudinal dispersion coefficient - D _T transverse dispersion coefficient - D _L ⁰ defined by Equation (6.21) - F(c _f2) defined by Equation (5.17) - f ₁(S ₁) fractional flow - g acceleration of gravity - j ₂ deviation mass flux of fluid 2 - K permeability of porous medium - K _ij generalized relative permeability function,i=1...2,j=1...2 - K _ri relative permeability functions,i=1...2 - L length of a slim tube or core - M _i total mass of fluidi=1...2 in volumeV - N number of points used to generate numerical curves - n unit normal to a surface - P pressure - P _i pressure in fluidi=1...2 - P _c capillary pressure - P ₁₂ macroscopic capillary pressure parameter - P(x) normal distribution function - q Darcy velocity of total fluid - q _i Darcy velocity of fluidi=1...2 - S _i saturation of fluidi=1...2 - S _L a low saturation value forS ₁ - S _H a high saturation value forS ₁ - u average intersitial fluid velocity - u _S isosaturation velocity - V volume used for volume averaging - V(c _f2) function defined by Equation (6.28) - V _e effluent volume - V _f fluid volume - V _i volume of fluidi=1...2 (Section 2); injected fluid volume - V _p pore volume of a slim tube or core - v macro scale fluid velocity - v _i macro scale velocity of fluidi=1...2 - _q (S ₁) isosaturation speed - _g (S ₁) component of isosaturation velocity due to gravity - w(S _L,S _H,t) width of a displacement front - w(t) overall width of a displacement front Greek Letters static interfacial tension - _ME macroscopic dispersivity - divergence operator - porosity - _i fraction of pore space occupied by fluidi=1...2 - (S ₁) effective viscosity of the fluid - _i viscosity of fluidi=1...2 - ₁₂ macroscopic fluid viscosity coupling parameter - macro scale fluid density - _i density of fluid i=1...2 - _q effective gravitational fluid density     

Flow between parallel walls containing the lines of neutrally buoyant circular cylinders

The motions of a single and two lines of neutrally buoyant circular cylinders in fluid between flat parallel walls are numerically investigated over the range of the Reynolds number of 12 < Re < 96, the ratio of the diameter of the cylinder D_s to the channel width D of 0.25≤D_s/D≤0.5, and the ratio of the streamwise spacing of the cylinders L to the channel width of 0.75≤L/D≤2. The lattice Boltzmann method is used for computations of the fluid phase and the cylinders are moved according to Newton’s law of motion. The Segré–Silberberg effect is found for both a single and two lines of cylinders. It is also found that for two lines of cylinders with D_s/D=0.25 and L/D=1, the equilibrium positions of the two lines are arranged to be staggered with respect to each other in the flow direction. The effects of the Reynolds number Re, D_s/D, and L/D on the equilibrium position of the lines of cylinders and on the friction factor of the cylinder–fluid mixture are presented and discussed.     

Experimental Study on the Effective Particle Diameter of a Packed Bed with Non-Spherical Particles

An experimental study is conducted to determine the characteristics of frictional pressure drops of fluid flow in porous beds packed with non-spherical particles. The objective is to examine the applicability of the Ergun equation to flow resistance assessment for packed beds with non-spherical particles. The experiments are carried out on the POMECOFL facility at KTH. Hollow spheres and cylinders are used to pack the beds. Either water or air is chosen as the working fluid. The experimental data show that the Ergun equation is applicable to all the test beds if the effective particle diameter used in the equation is chosen as the equivalent diameter of the particles, which is the product of Sauter mean diameter and shape factor of the particles in each bed.     

Flow regimes in wet conical spouted beds using glass bead mixtures

Using a high-viscosity Newtonian fluid, glycerol, an experimental investigation was carried out to evaluate the stable spouting regime in conical spouted beds using four particle mixtures： a reference （monoparticles）, a binary mixture, two ternary mixtures with flat and Gaussian distributions respectively. The mixtures were selected for particle diameters （dp） ranging from 1.09 to 4.98 mm and particle diameter ratios （dpL/dps） ranging from 1.98 to 4.0. Experimental data show that pressure fluctuation signals of the bed, as indicated by changes in their standard deviations, provide suitable information to identify the range of operational conditions for stable spouting. However, the analysis of skewness of curves of pressure fluctuation as a function of air velocity appears not sufficient to identify a particular flow regime. For glycerol in the spouting regime, the standard deviation is noted to increase with increasing glycerol concentration due to the growth of interparticle forces. The implications of these research findings on the drying of suspensions in conical spouted beds using glass bead mixtures are also discussed.     

Estimation of Macrodispersion by Different Approximation Methods for Flow and Transport in Randomly Heterogeneous Media

We present two- and three-dimensional calculations for the longitudinal and transverse macrodispersion coefficient for conservative solutes derived by particle tracking in a velocity field which is based on the linearized flow equation. The simulations were performed upto 5000 correlation lengths in order to reach the asymptotic regime. We used a simulation method which does not need any grid and therefore allows simulations of very large transport times and distances.Our findings are compared with results obtained from linearized transport, from Corrsin's Conjecture and from renormalization group methods. All calculations are performed with and without local dispersion. The variance of the logarithm of the hydraulic conductivity field was chosen to be one to investigate realistic model cases.While in two dimensions the linear transport approximation seems to be very good even for this high variance of the logarithmic hydraulic conductivity, in three dimensions renormalization group results are closer to the numerical calculations. Here Dagan's theory and the theory of Gelhar and Axness underestimate the transverse macrodispersion by far. Corrsin's Conjecture always overestimates the transverse dispersion. Local dispersion does not significantly influence the asymptotic behavior of the various approximations examined for two-dimensional and three-dimensional calculations.