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Macrodispersion of Sorbing Solutes in Heterogeneous Porous Formations With Spatially Periodic Retardation Factor and Velocity Field
Water Resources Research
Abstract
Expressions for the macroscopic velocity vector and dispersion tensor for sorbing solute transport in heterogeneous porous formations whose hydrogeologic properties are repeated at intervals were derived via Taylor-Aris-Brenner moment analysis. An idealized three-dimensional porous formation of infinite domain with spatially periodic retardation factor, velocity field, and microdispersion coefficients in all three directions was considered. Sorption was assumed to be governed by a linear equilibrium isotherm under local chemical equilibrium conditions. The analytical expressions presented are based on a perturbation method where all of the spatially periodic parameters employed were assumed to have ``small'' fluctuations. It was shown that the effective velocity vector is given by the volume-averaged interstitial velocity vector divided by the volume-averaged retardation factor, and the effective dispersion dyadic (second-order tensor) is given by the volume-averaged microdispersion dyadic divided by the volume-averaged dimensionless retardation factor plus a dyadic expressing the increase in solute spreading caused by the spatial variability of the parameters.
... Last, the zero-, first-and second-order global moments are obtained with the solution of the local moments since certain boundary conditions imposed at the surfaces of the unit element are satisfied by the local moments [185]. The zero-order moment (M 0 ) is a scalar that characterizes the total change in the hydraulic head or solute concentration. ...
... Kitanidis [188] derived general expressions for effective conductivity by solving the first spatial moment of the solute concentration in heterogeneous porous media with random time-invariant flow velocities. The expressions of macrodispersion of sorbing solutes are investigated, and it was found that a second term and a third term were integrated by the longitudinal macrodispersion, and these terms account for the effect of averaging the distribution coefficient and the first-order sorption rate in the equilibrium and kinetic sorption relation, separately [185]. Field formations exhibit recurrent geochemical characteristics which are not absolutely periodic, and as such, a porous medium that is assumed to be geochemically spatial and periodic may be criticized. ...
... Kitanidis [189] derived general expressions for effective conductivity by solving the first spatial moment of the solute concentration in heterogeneous porous media with random, timeinvariant flow velocities. The expressions of macrodispersion of sorbing solutes are investigated and found the longitudinal macrodispersion integrates a second and a third term that represents the effect of distribution coefficient averaging and the first-order sorption rate in the equilibrium and kinetic sorption relation, separately [185]. By modeling the periodic medium as a discrete graphical network, the method is applied to homogenize the resulting global equation to explicitly express the effective solute velocity, the effective first-order irreversible reaction rate constant, and the effective dispersivity dyadic [191]. ...
Physical and biogeochemical heterogeneity dramatically impacts fluid flow and reactive solute transport behaviors in geological formations across scales. From micro pores to regional reservoirs, upscaling has been proven to be a valid approach to estimate large-scale parameters by using data measured at small scales. Upscaling has considerable practical importance in oil and gas production, energy storage, carbon geologic sequestration, contamination remediation, and nuclear waste disposal. This review covers, in a comprehensive manner, the upscaling approaches available in the literature and their applications on various processes, such as advection, dispersion, matrix diffusion, sorption, and chemical reactions. We enclose newly developed approaches and distinguish two main categories of upscaling methodologies, deterministic and stochastic. Volume averaging, one of the deterministic methods, has the advantage of upscaling different kinds of parameters and wide applications by requiring only a few assumptions with improved formulations. Stochastic analytical methods have been extensively developed but have limited impacts in practice due to their requirement for global statistical assumptions. With rapid improvements in computing power, numerical solutions have become more popular for upscaling. In order to tackle complex fluid flow and transport problems, the working principles and limitations of these methods are emphasized. Still, a large gap exists between the approach algorithms and real-world applications. To bridge the gap, an integrated upscaling framework is needed to incorporate in the current upscaling algorithms, uncertainty quantification techniques, data sciences, and artificial intelligence to acquire laboratory and field-scale measurements and validate the upscaled models and parameters with multi-scale observations in future geo-energy research.
... Substantially different is the role of nano-scale heterogeneity, which has also been explored (Bradford et al., 2016;Pazmino et al., 2014). Several studies have explored the effect of spatially or temporally variable physicochemical parameters on contaminant and particle transport in water saturated porous media (Bekhit & Hassan, 2005;Chrysikopoulos & Sim, 1996;Chrysikopoulos et al., 1990aChrysikopoulos et al., , 1992aChrysikopoulos et al., , 1992bGao et al., 2013;Garabedian, 1987;Huysmans & Dassargues, 2009;Maghrebi et al., 2013;Sun et al., 2001;Valocchi, 1989) and investigated the influence of attachment rate distributions (Li et al., 2004;Tufenkji & Elimelech, 2005). However, the geochemical heterogeneity of subsurface formations has not been thoroughly examined in the literature. ...
... Suspended viruses remained over greater areas within the geochemically heterogeneous (see Figures 5m and 5n) than the geochemically homogeneous aquifers (see Figures 5f and 5g); whereas, for the case of irreversible attachment, the suspended virus concentrations progressively disappeared (see Figures 5t and 5u). This retarded biocolloid migration, which is caused by increased spreading in both upstream and downstream directions, is not intuitive; however, it is similar to the results observed in previous studies of contaminant transport with spatially variable retardation (Chrysikopoulos et al., 1990a(Chrysikopoulos et al., , 1992a(Chrysikopoulos et al., , 1992b. ...
... Clearly, the spatial variability of a enhanced the movement of the center of virus mass. These results are analogous to the enhanced migration and spreading of sorbing contaminants in porous media with spatially variable retardation (Chrysikopoulos et al., 1990a(Chrysikopoulos et al., , 1992a(Chrysikopoulos et al., , 1992b. ...
The effect of spatially variable attachment coefficient on biocolloid transport in geochemically heterogeneous porous formations was investigated numerically with a newly developed three–dimensional mathematical model. The biocolloid transport model accounts for horizontal uniform flow in water saturated porous media, and assumes that the biocolloid attachment varies spatially with a constant mean and random fluctuations. Biocolloid particles can either be suspended in the aqueous phase or attached (reversibly or irreversibly) onto the solid matrix. Multiple random realizations of geochemically heterogeneous porous media were employed in order to obtain appropriate ensemble mean concentration distributions, which subsequently were used for classical moment analysis. Emphasis was given in the proper selection of the number of realizations required for the correct ensemble mean estimation of a stochastic variable. The results showed that the existence of spatially variable biocolloid attachment efficiency, caused by geochemical heterogeneity, strongly contributes to an early time substantial increase in biocolloid spreading, an effect that asymptotically dissipated when the migrating biocolloid plume had sampled all of the geochemical heterogeneity within the porous formation. Furthermore, biocolloid plume spreading and enhanced transport were shown to increase with increasing variability of the attachment coefficient. Our findings suggested that neglecting to account for aquifer chemical heterogeneity may lead to erroneous predictions of biocolloid transport in porous media.
... [4] Although the method of volume averaging has been used extensively in chemical engineering, it has not been as widely employed in subsurface hydrology. Notable exceptions include the works of Cushman [1984], Gray et al. [1993], Kitanidis [1988Kitanidis [ , 1992, Plumb and Whitaker [1988], Chrysikopoulos et al. [1992], and Quintard and Whitaker [1994a, 1994b, 1994c, 1998a, 1998b. In this paper, we use volume averaging with closure to develop a series solution (which becomes an integral solution in the limit of increasing averaging volume) for the effective dispersion tensor. ...
... The use of spatially periodic model does not imply that the results apply only to periodic media or that the structure of the porous media is actually assumed to be periodic (see the discussion by Renard and de Marsily [1997, sect. 4.7], Pickup et al. [1994], Chrysikopoulos et al. [1992], Wang and Kitanidis [1999, Appendix A], and Eames and Bush [1999]). Quintard and Whitaker [1994a, 1994b, 1994c have shown that equation (4) represents the proper volume average for a disordered system and that periodic models are entirely suitable for the determination of effective transport coefficients associated with disordered systems. ...
In this work we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium; closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell. Our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions. Although this rectification is something that is generally believed to be true, there has been very little research that explores this issue explicitly. We show that under certain limiting (but mild) assumptions, the closure problem provides a Fourier series solution for the effective dispersion tensor. When second order spatial stationarity is imposed on the velocity field, the method yields a simple Fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity. This limiting result is identical to the quasi-Fickian forms that have been developed previously via ensemble average. As a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and it volume averaged statistics.
... It is also noteworthy that although the physical heterogeneity of the porous medium is essential, many studies have focused on the effects of spatially variable geochemical heterogeneities because the heterogeneous soil cannot be treated as uniform and homogeneous in the natural subsurface environment. The spatially variable geochemical heterogeneities affect pollutant transport significantly [12][13][14][15]. Additionally, many researchers have investigated the transport behavior of molecular size pollutants as well as suspended particle pollutants and found that the transport behavior between the solute type and suspended particle type of pollutants is very different [15,16]. ...
Modeling pollutant transport in heterogeneous media is an important task of hydrology. Pollutant transport in a non-homogeneous environment typically exhibits non-local transport dynamics, whose efficient characterization requires a parsimonious model with the non-local feature. This study encapsulates the non-local transport characteristic of pollutants into the peridynamic differential operator (PDDO) and develops a PDDO-based model for quantifying the observed pollutant non-local transport behavior. The simulation results show that the proposed model can describe pollutant non-local transport behavior in various heterogeneous media. The non-local nature of pollutant transport can be adjusted by pre-defined weight function w(|ξ|) and horizon Hx. Applications show that the PDDO-based model can better capture pollutant non-local transport behavior than the classical advection–diffusion equation (ADE) model, especially for quantifying the tail of the experimental data late. Analyses further reveal that the PDDO-based model can characterize both normal (Fickian) and anomalous (Lévy) diffusion regimes.
... For computational studies of permeability upscaling (Bøe, 1994;Durlofsky, 1991;Kitanidis, 1990;Renard & De Marsily, 1997) macrodispersion (Beaudoin & De Dreuzy, 2013;Chrysikopoulos et al., 1992;Hansen et al., 2018), subsurface mixing (Hidalgo et al., 2012;Wright et al., 2018), and geochemical reaction (Cirpka, 2002;de Anna, 2012;Hansen et al., 2014;Wright et al., 2017), it is often advantageous to employ a quasi-infinite domain so that the effect of heterogeneous flow and other physics on the process of interest may be studied without interference of simulation boundaries. As any numerical simulation is finite, a quasi-infinite domain must be simulated by "tiling" identical heterogeneous domains, which amounts to simulating a single tile with spatially periodic boundary conditions. ...
We present a finite volume approach for generating doubly‐periodic, 2D heterogeneous groundwater velocity fields, given an arbitrary hydraulic conductivity field discretized on a rectangular lattice. The method conserves mass, allows direct specification of any desired mean flow direction and computes the corresponding field with a single solve operation, and solves for inter‐cell fluxes that may be used directly for particle tracking with the Pollock method without further interpolation. We demonstrate an open‐source Python implementation of the approach that we created.
... This reduced the retention of nHAP and improved the transport of total Cd (II). Even though the retardation factor R f is often a space dependent parameter (Chrysikopoulos et al., 1990(Chrysikopoulos et al., , 1992, in this study, the R f values were compared in/near the plateau period of the penetration curves under various experimental conditions to better explain the influence of colloids on Cd(II) transport. Compared with nHAP-Cd binary component system, the coefficient K d of Cd(II) in nHAP-FA-Cd ternary component systems decreased by 12.70 and 13.33 times, R f decreased by 6.17 and 6.32 times, and the velocity of groundwater u increased by 6.17 and 6.32 times for 10 mg/L and 50 mg/L FA, respectively (Table S1). ...
Soil colloids can affect the cotransport of nanoparticles and pollutants. In this study, the influencing mechanisms of organic fulvic acid (FA) and inorganic montmorillonite colloid (MONT) on the cotransport of nHAP and Cd(II) were investigated. Column experiments combined with Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, attachment efficiency calculation and two-site kinetic retention model were applied to study the mechanisms. Results showed that the co-existence of FA or MONT made the transport of nHAP improve by 58%-75% and 33%-59%, respectively. Both of them could improve the stability of nHAP particles and enhance electrostatic repulsion between nHAP particles and sand. Retention of nHAP in the sand was mainly caused by secondary energy minimum and physical straining. The co-existence of FA or MONT changed the amount of sorbed species of Cd(II) and decreased the retardation effect of nHAP on Cd(II) transport. With increasing FA concentration, soluble FA•Cd and suspended nHAP•FA•Cd complexes in the system increased. Transport of soluble Cd(II) and total Cd(II) were strengthened due to the concentration effect of FA and the improved stability of nHAP particles. With increasing MONT concentration, the amount of soluble Cd(II) decreased and colloidal Cd(II) (nHAP•Cd and MONT•Cd) increased. Transport of soluble Cd(II) was weakened and colloidal Cd(II) was enhanced, and the transport of total Cd(II) improved by 34%-57% finally. The findings of this study can help to understand the fate of nanoparticles and Cd(II) in natural water and soil.
... Because the critical level of soil organic matter is low (0.1% for the case of benzene), the adsorption of organic contaminant onto the aquifer is mostly dominated by the hydrophobic reaction of soil organic matter. In field conditions, heterogeneous soil properties, such as organic matter, further complicate the fate of organic contaminants [15][16][17]. ...
Multi-dimensional transport studies are necessary in order to better explain the fate of contaminants in groundwater. In this study, a two-dimensional transport experiment with organic contaminants in saturated sand was conducted to investigate the migration of the organic contaminant plume in multi-dimensional flow conditions. The transport test was conducted using toluene as a model organic contaminant in a saturated sand box under steady flow conditions. The initial plume was generated via injection at a point source. After 24 h, the plume distribution was delineated by interpolating toluene concentrations in the porewater samples. The mass centers of the toluene and the conservative tracer were almost coincident, but the size of the toluene plume was significantly reduced in longitudinal as well as transversal directions. The appropriateness of several types of sorption models were compared to describe the toluene sorption in two-dimensional transport system using numerical modeling. Among the sorption models, the Langmuir model was found to be the most appropriate to describe the sorption of toluene during two-dimensional transport. The results showed that two-dimensional experiments are better than one-dimensional column experiments in identifying the adsorption characteristics that occur during transport in saturated aquifers.
... Along with the physical heterogeneity (variation of hydraulic conductivity), biochemical heterogeneities, such as spatially variable abiotic degradation rate, sorption coefficient, and microbial activity, have also been investigated through analytical analysis or numerical simulation for reactive solute transport (e.g. Chrysikopoulos et al., 1990Chrysikopoulos et al., , 1992aChrysikopoulos et al., , 1992bCvetkovic and Dagan, 1994;Dagan and Cvetkovic, 1993;Bellin et al., 1993;Hu et al., 1995Hu et al., , 1997Valocchi, 1989;Garabedian et al., 1988;Sposito, 1991, 1994;Robin et al., 1991;Burr et al., 1994;Miralles-Wilhelm and Gelhar, 1996a;Miralles-Wilhelm et al., 1997;Miralles-Wilhelm and Gelhar, 2000;Oya and Valocchi, 1998;Xin and Zhang, 1998;Kaluarachchi et al., 2000;Katzourakis and Chrysikopoulos, 2018). Up-scaled quantities that characterize the complex interplay between transport and sorption/ biodegradation processes at lower scale are obtained. ...
The correct characterization of macro-scale contaminant transport and transformation rates is an important issue for modeling reactive transport in heterogeneous aquifers. While previous studies have investigated field-scale heterogeneity of transport and biochemical properties, the effects of local transverse dispersion on macro-scale transport and transformation rates have not been well understood. In this paper, the process of oxygen-limited biodegradation in a stratified aquifer is analysed by spectral perturbation approach, and longitudinal macrodispersivity, effective biodegradation rate, effective retardation factor and effective velocity are derived for the coupled transport equations of a system consisting of a contaminant and an oxidizing agent (oxygen). The effects of local transverse dispersion on these macro-scale coefficients are studied. It is shown that local transverse dispersion can smooth the heterogeneity in biodegradation and sorption processes and enlarge effective biodegradation rate and retardation factor. The local transverse dispersion can also limit the effects of heterogeneity in biodegradation process on longitudinal macrodispersivities and effective velocities for the contaminant and dissolved oxygen. But the effects of heterogeneity in sorption process on the contaminant macrodispersivity is likely to be magnified by local transverse dispersion.
... Physically dominant mechanism of dispersion of contaminants in aquifers seems to be the spatial variability of groundwater flow velocities associated with primarily with the spatial distributions of hydraulic conductivity. This mechanism is commonly referred to as macrodispersion (Dagan, 1984;Chrysikopoulos et al., 1992;Gelhar et al., 1992) despite of the conservative and non-conservative solutes. Understanding the characteristic behind such transport pathways related to macrodispersion is a crucial step to ensuring the reliable prediction of a contaminant behavior (Uffink, 1985;Tompson, 1993). ...
In this study, intermediate scale dye tracer experiments were conducted in a 1m length, 1m height and 0.03m thickness sandbox to quantify macrodispersion phenomena in stratified porous formations and to elucidate the effect of the layering on the degree of macrodispersion. Spatial and temporal moment approaches based on a snapshot of tracer distribution and NaCl concentration at a point, respectively, were applied to identify the macrodispersivities in longitudinal and lateral directions. The experimental results indicated that the longitudinal macrodispersivity depends on the travel distance of solute and the degree of heterogeneity in stratified formations with two or four layers, while the transverse macrodispersivity decreases with the increase of travel distance and depends on the magnitude of the initial solute distribution. A difference between longitudinal macrodispersivity estimates using spatio-temporal moments was also clarified.
... In addition to the study of heterogeneity of physical properties (e.g., hydraulic conductivity), heterogeneity of biochemical properties (e.g., decay rate, retardation factor and attachment coefficient) has also been investigated (e.g., Chrysikopoulos et al. 1990Chrysikopoulos et al. , 1992aKatzourakis and Chrysikopoulos 2018). Dagan (1989) analyzed the center of mass displacement of a solute under spatially variable decay rate and retardation factor. ...
Estimating the values of dispersion and biochemical reaction rates of heterogeneous aquifers is critical to predicting the temporal evolution and fate of reactive solutes. While previous studies have investigated field-scale heterogeneity of transport and biochemical properties of porous media, effects of local dispersion have not been well understood. In this paper, longitudinal macro-dispersivity, effective decay rate, and effective solute velocity are derived for a stratified aquifer, and the effects of local dispersion, especially the local transverse dispersion, are studied. It is shown that the inclusion of local transverse dispersion leads to enlarged effective decay rate, and that ignoring it may significantly underestimate the effective rate. The Damkohler (Da) number and the coefficient of variation (CV) of decay rate have slight influence to macro-coefficients under very small Pe number (with large local transverse dispersion). However, Da number has growing effect on the asymptotic effective decay rate with the decrease in Pe number, and results in constant asymptotic values regardless of Da number under the condition with very large Pe number. Larger CV of decay rate leads to smaller effective decay rate and effective velocity, and longitudinal macro-coefficient. The longitudinal macro-dispersivity is found to depend on the correlation between the hydraulic conductivity and the decay rate if the local longitudinal dispersion is spatially variable.
... The sorption process occurring at the representative elemental volume (REV) scale is represented by a reversible linear equilibrium isotherm in which the amount of solutes sorbed onto the solid is proportional to the concentration of dissolved solutes (Chrysikopoulos et al., 1990(Chrysikopoulos et al., , 1992Robin et al., 1991). The constant of proportionality is known as the distribution coefficient or partition coefficient, ...
Even under the simple linear isotherm adsorption model, the parameters controlling adsorption under field conditions are frequently approximated by the values derived from batch experiments. First, the measurement scale and conditions are very different from those at the model scale. Second, the parameters are heterogeneous in space and, at most, there is some information about them at a few locations within an aquifer. For these two reasons, there is a need to consider how to treat the heterogeneity of the parameters that control adsorption, i.e. the retardation factor or distribution coefficient, and a need to establish upscaling rules to transfer the information about parameters at the measurement scale to those at the scale of model grid blocks. This paper presents some best upscaling rules for the distribution coefficient accounting for different heterogeneous structures for both the hydraulic conductivity and the distribution coefficient. Exhaustive numerical simulations are carried out by combining different heterogeneity patterns of the hydraulic conductivity and the distribution coefficient, the cross-correlation between them, overall degree of variability, and time dependence. It is demonstrated that under certain conditions, e.g. large variances and small correlation lengths of hydraulic conductivity lnK(x) and distribution coefficient lnK d(x), the geometric mean is a good approximation for the upscaled retardation factor.
... Macroscopic dispersion is traditionally determined by averaging pore-scale simulations of dispersion, which are based on detailed descriptions of the pore structure [van Milligen and Bons, 2012;Icardi et al., 2014]. For periodic porous media, dispersion coefficients may be determined with volume averaging and generalized Taylor-Aris-Brenner techniques, by defining a representative elementary volume (REV) [Whitaker, 1986;Chrysikopoulos et al., 1992aChrysikopoulos et al., , 1992b. For nonperiodic or heterogeneous porous media, where REV selection Previous studies have suggested that colloid transport in porous media is significantly affected by particle size [Fontes et al., 1991, Gannon et al., 1991. ...
Laboratory and field studies have demonstrated that dispersion coefficients evaluated by fitting advection-dispersion transport models to nonreactive tracer breakthrough curves do not adequately describe colloid transport under the same flow field conditions. Here an extensive laboratory study was undertaken to assess whether the dispersivity, which traditionally has been considered to be a property of the porous medium, is dependent on colloid particle size and interstitial velocity. A total of 48 colloid transport experiments were performed in columns packed with glass beads under chemically unfavorable colloid attachment conditions. Nine different colloid diameters and various flow velocities were examined. The breakthrough curves were successfully simulated with a mathematical model describing colloid transport in homogeneous, water-saturated porous media. The experimental data set collected in this study demonstrated that the dispersivity is positively correlated with colloid particle size, and increases with increasing velocity. The dispersivity values determined in this laboratory study were compared with 380 dispersivity values from earlier laboratory and field-scale solute, colloid and biocolloid transport studies published in the literature.
... Chrysikopoulos et al. (1990) se sont intéressés à l'effet sur le transport d'une variation spatiale du coefficient de retard dans un milieu physiquement homogène. Bellin et al. (1993Bellin et al. ( , 1995 ainsi que Chrysikopoulos et al. (1992) ont étudié le transport d'un soluté soumis à une sorption linéaire à l'équilibre dans le cas d'un milieu chimiquement et physiquement hétérogène. Bellin et al. (1993) ont étendu l'analyse stochastique de au cas du transport de soluté réactif, en considérant les facteurs de retard et les conductivités hydrauliques comme des variables aléatoires avec éventuellement des corrélations entre les deux types d'hétérogénéités, et ont dérivé des solutions analytiques de l'équation du transport. ...
Thèse publiée dans la collection des Mémoires du CAREN (ISSN 1761-2810) : Mémoire n° 5 (ISBN 2-914375-17-4)
... The transport of reactive solutes is attributed to chemical Dagan and Cvetkovic 1993) or physical (Cushey and Rubin 1997) sorption, or to a combination of both processes (Miralles-Wilhelm and Gelhar 1996). However, to simplify the prediction of transport by applying equilibrium-based solute-transport models, sorption reactions are assumed sufficiently fast (Kabala and Sposito 1991;Chrysikopoulos et al. 1992). But evidence from laboratory and field investigations indicates significant kinetic effects and much slower sorption rates at the field scale than generally assumed (Nkedi-Kizza et al. 1983;Ptacek and Gillham 1992). ...
This study, considering evidences of slower sorption rates of reactive solutes in the field than in laboratory, quantifies the velocity and retardation factor of a sodium fluorescein (uranin: C20H10Na 2O5) plume over its travel path in a heterogeneous aquifer. The transport process of uranin was evaluated by batch experiments and from breakthrough curves (BTCs) by using solute-transport models. Method of time moments analysed BTCs of uranin and bromide to derive the velocity and retardation factor. A constant velocity of the bromide plume, 0.64 m/day, implies a spatially and temporally uniform velocity field where groundwater flows at steady-state condition. A large dimensionless index (195) of chemical non-equilibrium model and equilibrium distribution coefficient (0.32) of uranin are indicative of chemical non-equilibrium transport process. The travel time of uranin plume increases asymptotically, following power law, with travel path of the plume. Good agreement of the exponent of power law with that of Freundlich isotherm is a result of nonlinear sorption, and provides an independent way of estimating the exponent of the isotherm. The local velocity of the plume decreases asymptotically in time and is predicted by the derivative of the relationship between travel path and travel time of the plume. The retardation factor, which increases in time following power law, when estimated from the local velocity, is considerably larger than that estimated from travel time of the plume.
... TheGelbar and Axness, 1983;Dagan, 1984]Tompson et al., 1996;Miralies-Wilhelm and Gelbar, 1996]Robin et al., 1991;Mackay et al., 1986;Barber et al., 1992;Davis et al., 1993][Chrysikopoulos et al., 1990;Hu et al., 1995;Miralies-Wilhelm and Gelhar, !996;Kabala and Sposito, 1991;Quinodoz and Valocchi, 1993;Bellin et al., 1993;Bosma et al., 1993;Dagan et al., 1992] ...
A small perturbation approach is used to analyze the impact of chemical heterogeneity on the one-dimensional transport of a pollutant that undergoes linear kinetic adsorption. We make an important simplifying assumption that the aquifer is physically homogeneous but chemically heterogeneous. The aquifer is assumed to be comprised of two distinct zones: reactive and nonreactive; a Bernoulli random process is used to characterize the spatial distribution of reactive zones along the aquifer. We develop analytical solutions to study the distribution of the ensemble mean, standard deviation, and coefficient of variation of the dissolved concentration after an instantaneous injection of contaminant. In addition, numerical solutions based on a Monte Carlo approach are used to determine the validity of the analytical solutions. Finally, an analysis involving temporal and spatial moments is used to derive expressions for the large-scale effective parameters (velocity and dispersion) that capture the impact of chemical heterogeneity. Temporal moment analysis provides closed-form analytical expressions for the asymptotic effective velocity and dispersion, while spatial moment analysis explains the effect of chemical heterogeneity on the preasymptotic value of these effective parameters. A key result from our analysis shows that chemical heterogeneity creates ``pseudokinetic'' or ``macrokinetic'' conditions characterized by a time-dependent effective retardation coefficient even when the local equilibrium assumption is invoked.
... al . 1991;Ball & Roberts 1991). The effect of sorption heterogeneity combined with flow heterogeneity of solute transport in soils and aquifers has been the topic of several numerical and/or analytical studies over the past decade (van der Zee & van Riemsdijk 1987;Dagan 1989;Cvetkovic & Shapiro 1990;Kabala & Sposito 1991;Destouni & Cvetkovic 1991;Chrysikopoulos et al . 1992;Bellin et al . 1993;Tompson 1993;Burr et al . 1994;Berglund 1995;Simmons et al . 1995;Ginn et al . 1995;Hu et al . 1995;Tompson et al . 1996). ...
We consider migration of contaminants in groundwater and wish to characterize transport globally using spatial and tempora moments. The specific problem addressed in this work is how to simultaneously account for the spatial variability of the hydrauli conductivity, K, and of one or several sorption parameters, P. The Lagrangian framework for reactive transport in aquifers of Cvetkovic and Dagan (1994) and Dagan and Cvetkovic (1996 is extended to incorporate the spatial variability in sorption parameters. For arbitrary sorption reactions, the general resul can be used for simplified Monte Carlo simulations, where a three–dimensional advection–sorption problem is reduced to a three–dimensiona advection and one–dimensional advection–sorption problem. The first two spatial moments characterize the spatial extent o a contaminant plume and are derived for ergodic transport, for cases of continuous and pulse injection. Expressions for th first three temporal moments which characterize field–scale contaminant discharge, are derived for linear sorption reactions.
All the derived expressions for the global transport quantities are given in terms of Lagrangian statistics of the fluid velocit and the sorption parameter(s) random fields. Analytical solutions are provided for a few sorption models which are most frequen in applications: nonlinear equilibrium sorption and linear nonequilibrium sorption. Analytical results are given in terms of Lagrangian statistics of the ‘reaction flow path’, μ, which integrates the sorption parameter along an advection flow path with time as the integration variable. Lagrangian statistic of μ are related to the Eulerian statistics of the hydraulic conductivity, K, and the sorption parameter, P, analytically and using Monte Carlo, particle–tracking simulations. The derived analytical expressions are robust for th considered range of variabilities, when compared to simulation results. For extraction of a contaminant subject to Langmui sorption, the effect of spatial variability in the sorption capacity on the first two moments of the displacament front i suppressed by the effect of nonlinearity. For linear nonequilibrium sorption, spatial variability in the forward rate coefficien has a more significant influence than in the backward rate, on the first three temporal moments.
... Although realistic aquifers exhibit spatial variability on a hierarchy of scales, periodic or quasi-periodic models similar to the one outlined above (given by equations (5.7)–(5.9)) have often been used due to their simplicity [80, 81]. In this paper we shall take (Figure 13 shows the regions in the domain that have violated the lower and upper bounds on the concentrations of the invariants and the product under the Galerkin formulation. ...
We present a novel computational framework for diffusive-reactive systems
that satisfies the non-negative constraint and maximum principles on general
computational grids. The governing equations for the concentration of reactants
and product are written in terms of tensorial diffusion-reaction equations. %
We restrict our studies to fast irreversible bimolecular reactions. If one
assumes that the reaction is diffusion-limited and all chemical species have
the same diffusion coefficient, one can employ a linear transformation to
rewrite the governing equations in terms of invariants, which are unaffected by
the reaction. This results in two uncoupled tensorial diffusion equations in
terms of these invariants, which are solved using a novel non-negative solver
for tensorial diffusion-type equations. The concentrations of the reactants and
the product are then calculated from invariants using algebraic manipulations.
The novel aspect of the proposed computational framework is that it will always
produce physically meaningful non-negative values for the concentrations of all
chemical species. Several representative numerical examples are presented to
illustrate the robustness, convergence, and the numerical performance of the
proposed computational framework. We will also compare the proposed framework
with other popular formulations. In particular, we will show that the Galerkin
formulation (which is the standard single-field formulation) does not produce
reliable solutions, and the reason can be attributed to the fact that the
single-field formulation does not guarantee non-negative solutions. We will
also show that the clipping procedure (which produces non-negative solutions
but is considered as a variational crime) does not give accurate results when
compared with the proposed computational framework.
... Spatial moment analysis was introduced by Aris [35] and since then it has been applied to numerous solute transport studies [36][37][38][39][40]. In this study, the distributions of suspended colloids within the fracture (snapshots) are analyzed by the absolute spatial moments, which are defined as: ...
... Relatively few studies have been carried out about other equivalent flow and transport parameters than hydraulic conductivity. Gelhar and Axness (1983), Dagan (1982) and Neuman et al. (1987) Chrysikopoulos et al. 1992; Hu et al. 1995; Ptak and Schmid 1996; Metzger et al. 1996; Reichle et al. 1998). The discussion of the effective diffusion coefficient has been limited to the steady-state case (Özisik 1993), in which the same expressions hold as for the mathematically similar processes of steady-state groundwater or heat flow. ...
Diffusion is an important transport process in low permeability media, which play an important role in contamination and remediation
of natural environments. The calculation of equivalent diffusion parameters has however not been extensively explored. In
this paper, expressions of the equivalent diffusion coefficient and the equivalent diffusion accessible porosity normal to
the layering in a layered porous medium are derived based on analytical solutions of the diffusion equation. The expressions
show that the equivalent diffusion coefficient changes with time. It is equal to the power average with p = −0.5 for small times and converges to the harmonic average for large times. The equivalent diffusion accessible porosity
is the harmonic average of the porosities of the individual layers for all times. The expressions are verified numerically
for several test cases.
... Here we have placed all the terms containing the dependent variable on the left-hand side, while the source terms involving the temporal and spatial derivatives of {(p (3)(3} have been placed on the right-hand side. On the basis of eqn (29) we can see that the right-hand side of eqn (32) (32), we need to estimate both the time and space derivatives and also the pressure difference. We begin with the right-hand side of eqn (32) and represent the derivatives according to the following equation (see Ref. 10 Here one must think of ..::l {(p (3)(3} as some identifiable change in the large-scale average pressure, with the characteristic time t* and the characteristic length L p being defined by the estimates represented by eqns (34) and (35). ...
In this paper we develop the regional form of Darcy's law when the condition of large-scale mechanical equilibrium is valid. This condition occurs for many steady or quasi-steady flows of practical importance, and analytical constraints are presented which define the domain of validity for large-scale mechanical equilibrium. Given a two-region model of a heterogeneous porous medium, the intrinsic region-averaged velocities can be expressed in terms of the large-scale average velocity according to ϕη{〈vβ〉gh}η = Mβη∗ · {〈vβ〉}, in the η - region ϕω{〈vβ〉ω}ω = Mβω∗ · {〈Vβ〉}, in the ω - region. Here the mapping tensors Mβη∗ and Mβω∗, are specified by the same closure problem used to determine the effective thermal conductivity tensor for a two-phase system, and they are directly related to the regional permeability tensors Kβν∗ and Kβω∗ Regional permeability tensors are determined for stratified systems, for a nodular model of heterogeneous porous media, and for fractured porous media. These large-scale permeabilities are sensitive to the structure of the mechanical heterogeneities and in general they are not equal to the local permeability that is used to characterize the mechanical heterogeneities.
... Also, in risk assessment as it is applied in nuclear repository safety studies, the probability distribution of times of arrival in a certain place may be sufficient, even if it is not an ergodic property of the medium. Real physical diffusion/dispersion and its modification by heterogeneity of the retardation factor is investigated by Chrysikopoulos et al. (1992) who looked at a single medium with a periodically varying retardation coefficient. For a sinusoidal distribution they find that the effective dispersion coefficient is ...
In order to be able to extrapolate reactive pollutant transport behaviour from the laboratory experiment in the aquifer field scale it is necessary to combine the deterministic laboratory results with the stochastics of the field situation to produce effective macroscopic properties. For simple linear adsorption standard methods can be applied to arrive at effective retardation and dispersion parameters. Effective time-scales for chemical reactions such as adsorption and degradation may no longer be determined by the time-scales of processes on the molecular scale only but also by rate-limiting steps arising from the variable accessibility of reactive parts of the medium. For a bacterial degradation reaction it is suggested that an exchange coefficient between mobile water and reactive phase can parametrize subscale processes. The coefficient depends on the degree of heterogeneity of the medium.
... Worthy to note is that for the estimation of the hydrodynamic dispersion coefficients for solute and contaminant transport in porous media without the use of correlations, which are typically derived from experimental observations, several theoretical and numerical procedures are available in the literature. For periodic media, dispersion coefficient estimation may be determined with volume averaging and generalized Taylor-Aris-Brenner techniques, by defining a representative elementary volume (REV) [46][47][48]. For non-periodic or heterogeneous porous media, where REV selection may not be a trivial task, random-walk particle tracking methods are often employed [49]. ...
This study presents a non-invasive imaging method for in situ concentration determination of conservative tracers and pollutants in a two-dimensional glass pore network model. The method presented is an extension to the work by Huang et al., and Thomas and Chysikopoulos. The method consists of fabricating the glass pore network model using a photolithography technique, conducting flowthrough contaminant transport experiments, taking digital photographs at various times of the two-dimensional pore network under ultraviolet or visible light source, and determining the spatially-distributed pollutant concentrations by measuring the color intensity in the photographs with comparative image analysis. Therefore, the method is limited to fluorescent or colored pollutants and tracers. The method was successfully employed to in situ concentration determination of uranine and red color tracers.
Modeling transport process at large scale requires proper scale-up of subsurface heterogeneity and an understanding of its interaction with the underlying transport mechanisms. A technique based on volume averaging is applied to quantitatively assess the scaling characteristics of effective mass transfer coefficient in heterogeneous reservoir models. The effective mass transfer coefficient represents the combined contribution from diffusion and dispersion to the transport of non-reactive solute particles within a fluid phase. Although treatment of transport problems with the volume averaging technique has been published in the past, application to geological systems exhibiting realistic spatial variability remains a challenge. Previously, the authors developed a new procedure where results from a fine-scale numerical flow simulation reflecting the full physics of the transport process albeit over a sub-volume of the reservoir are integrated with the volume averaging technique to provide effective description of transport properties. The procedure is extended such that spatial averaging is performed at the local-heterogeneity scale.
In this paper, the transport of a passive (non-reactive) solute is simulated on multiple reservoir models exhibiting different patterns of heterogeneities, and the scaling behavior of effective mass transfer coefficient (Keff) is examined and compared. One such set of models exhibit power-law (fractal) characteristics, and the variability of dispersion and Keff with scale is in good agreement with analytical expressions described in the literature.
This work offers an insight into the impacts of heterogeneity on the scaling of effective transport parameters. A key finding is that spatial heterogeneity models with similar univariate and bivariate statistics may exhibit different scaling characteristics because of the influence of higher order statistics. More mixing is observed in the channelized models with higher-order continuity. It reinforces the notion that the flow response is influenced by the higher-order statistical description of heterogeneity. An important implication is that when scaling-up transport response from lab-scale results to the field scale, it is necessary to account for the scale-up of heterogeneity. Since the characteristics of higher-order multivariate distributions and large-scale heterogeneity are typically not captured in small-scale experiments, a reservoir modeling framework that captures the uncertainty in heterogeneity description should be adopted.
It is now widely recognized that groundwater aquifers exhibit significant three-dimensional, small-scale variability in their hydraulic properties and that this variability controls the migration and dispersion of contaminants at the field scale. Quantitative study of the impact of small-scale variability upon field-scale transport has been a central theme of groundwater research in recent years; this research has been motivated by a host of important questions. How can properties measured on small samples in the laboratory be extrapolated to larger scales? Are fundamental constitutive relations derived from studies at the laboratory scale valid at field scales? How can we quantify the inherent uncertainty in our information on spatially varying soil properties? What is the effect of this uncertainty upon the reliability of model predictions?
Concepts of reactive contaminant transport in porous media that involve mass transfer processes, such as sorption, and chemical and microbial transformation are introduced. These reactive transport concepts are quantitatively described within the context of hydrogeologic heterogeneity. In particular, the degree of fate and transport of a single reactive contaminant in the subsurface subjected to certain spatially heterogeneous physical or chemical processes is addressed. This paper presents some results from recent research in the area of reactive contaminant transport that has attempted to create models that integrate these previously disparate viewpoints by including both physical and chemical heterogeneity.
Sorption is a well-known phenomenon that may cause the retardation effect of zinc in the subsurface environment. In this study, the governing process for zinc sorption during transport was investigated by conducting 2-D plume tests in a laboratory scale sand tank model using the time domain reflectometry (TDR) method. Tracer solutions of NaNO3 and ZnSO4 were applied at a constant flow rate as a pulse type to capture the plumes of both solutes based on TDR-measured resistance. It was revealed that the observed zinc sulfate plume showed no retardation relative to sodium nitrate with a retardation factor of R ≈ 1. Instead of retardation, a prominent reduction of zinc sulfate mass occurred during transport through the tank model due to the irreversible sorption as well as longitudinal dispersion. This indicates that the controlling factor for the sorption process of zinc sulfate in the sand tank model is kinetic rather than equilibrium. These hydrogeological parameters would provide valuable information on the prediction of the fate of zinc in sandy aquifer materials.
Adsorbing solute transport in two-dimensional heterogeneous unsaturated soil was studied by means of stochastic numerical simulations. Heterogeneities in the soil's hydraulic properties and in the adsorption isotherm were simulated using random fields having specified statistical structures. Macrodispersion was analyzed using the spatial moments of numerically generated solute plumes. Among different realizations of the heterogeneous soil, the discrepancies between second-order moments and macrodispersion coefficients were large. Macrodispersivities of unsaturated soils increased with decreasing water content. Also, heterogeneous adsorption of solute enhanced the solute spreading. When the adsorption coefficient was negatively correlated with the saturated hydraulic conductivity, solute spreading was greater than when adsorption was uncorrelated or positively correlated with the conductivity.
A water-saturated fracture, partially clogged with porous material coating the fracture surfaces, is considered. Fluid flow and contaminant transport in this fracture are significantly altered relative to an unclogged fracture. Analytical expressions are developed for the water velocities in the clogged and the unclogged regions in the fracture and the asymptotic longitudinal dispersion coefficient for the system. For highly adsorbing dissolved contaminants or large colloids, the slow diffusion within the porous region causes enhanced dispersion. In a standard tracer test, colloidal contaminants will arrive earlier than dissolved tracers for either one of two reasons: (1) colloids confined to the unclogged portion of the fracture will have a larger average velocity or (2) colloids that can diffuse into the porous region have very low Brownian dififusivities, resulting in a large longitudinal dispersion due to the inverse relationship between Taylor dispersion and Brownian diffusion. The average velocity or asymptotic longitudinal dispersion coefficient can be orders of magnitude greater than that for a molecular tracer.
The plume is characterized by the expected values of its spatial moments: mass, centroid coordinates, and tensor of second spatial moment. Under ergodic conditions, assumed to prevail, the one-realization spatial moments are approximately equal to their ensemble means. By adopting a first-order approximation in the log conductivity variance and with neglect of pore-scale dispersion, the spatial moments of a plume of a conservative solute were evaluated in the past as functions of the transport time or distance from the input zone (Dagan, 1988) by a Lagrangian approach. The moments were expressed in terms of a quadrature that had to be evaluated numerically. The present study generalizes these results to a reactive solute which undergoes kinetically controlled sorption, according to a linear relationship. The reaction depends on two coefficients: a reaction time tr and an equilibrium partition coefficient Kd which are assumed to be constant. The solution of the transport problem is obtained by the Lagrangian approach. -from Authors
Effective parameters for flow in saturated porous media are obtained via Taylor-Aris-Brenner moment analysis considering both periodic as well as stationary porous medium properties. It is assumed that a slug is instantaneously introduced into an unbounded, anisotropic porous medium having a compressible matrix, and that the correlation length of the local hydraulic conductivity and specific storage fluctuations is smaller than the correlation length of hydraulic head fluctuations (gradually varying flow). It is shown that the effective specific storage is equal to its volume average. The effective hydraulic conductivity is derived by a small-perturbation analysis and it is shown to consist of its volume average and of a second term which accounts for the ‘small’ local conductivity fluctuations.
Steady flow of incompressible fluids takes place in geological formations of spatially variable permeability. The permeability is regarded as a stationary random space function (RSF) of given statistical moments. The fluid carries reactive solutes and we consider, for illustration purposes, two types of reactions: nonlinear equilibrium sorption of a single species and mineral dissolution (linear kinetics). In addition, we analyse the nonlinear problem of horizontal flow of two immiscible fluids (the Buckley-Leverett flow). We consider injection at constant concentration in a semi-infinite domain at constant initial concentration and we neglect the effect of pore scale dispersion. The field-scale transport problem consists of characterizing an erratic plume, or displacement front, emanating from a given source area along distinct random flow paths. Reactive transport along three-dimensional flow paths is transformed to a one-dimensional Lagrangian-Eulerian domain (tau , t), where tau is the fluid residence time and t is the real time. Due to nonlinearity, discontinuities (shock waves) along a flow path may develop. Close form solutions are obtained for the expected values of the spatial and temporal moments of a nonlinearly reacting solute plume, or of two immiscible fluids. These results generalize the previous results for linearly reacting solute (Cvetkovic & Dagan 1994). The general results are illustrated and discussed in part II.
The conceptual framework developed in Dagan & Cvetkovic (1996) is applied to specific cases of flow and reactive transport in geological media. Two types of reactions are considered: Langmuir sorption and mineral dissolution. Two-phase immiscible displacement of oil by water (Buckley-Leverett flow) is also analysed. Both reactions and heterogeneity enhance the field-scale dispersive effects. The relative influence of the heterogeneity and reactions on the field-scale dispersion depends on the scale of the transport problem relative to the heterogeneity integral scale on the one hand, and the reaction parameters on the other. The dispersive effect due to sorption nonlinearity may dominate over the effects of heterogeneity. Similarly, for heavier oil, i.e. for small water to oil viscosity ratio, the effect of nonlinearity in the two-phase immiscible dynamics will generally dominate over the heterogeneity for large transport times. If the characteristic transport time, defined as the ratio between the reservoir scale and the flow velocity, is comparable to the heterogeneity characteristic time I/U, heterogeneous fingering and nonlinear two-phase dynamics yield comparable rates of dispersion.
A stochastic-convective reactive (SCR) transport method is developed for one-dimensional steady transport in physically heterogeneous media with nonlinear degradation. The method is free of perturbation amplitude limitations and circumvents the difficulty of scale dependence of phenomenological parameters by avoiding volume-averaged specifications of diffusive/dispersive fluxes. The transport system is conceptualized as an ensemble of independent convective-reactive streamlines, each characterized by a randomized convective velocity (or travel time). Dispersive effects are treated as a component of the randomness in the streamline velocity ensemble, so no explicit expression for hydrodynamic dispersive flux is written in the streamline transport equation. The expected value of the transport over the stream tube ensemble is obtained as an average of solutions to the reactive convection equation according to the stream tube (travel time) probability distribution function. In this way, transport with reaction can be expressed in terms of global-scale random variables, such as solute travel time and travel distance, which are integrals of the stochastic variables such as velocity. Derivations support the hypothesis that via the SCR the decay process can be factored out of the mechanical transport behavior (as reflected by movement of a passive tracer) and scaled independently. Solution strategies are presented for general linear and nonlinear kinetic reactions. Demonstration simulations show that for Fickian transport with nonlinear reactions the SCR and convention dispersion equation can give different results. 75 refs., 6 figs.
Temporal moments analysis is used to study the impact of chemical heterogeneity on the one-dimensional transport of a pollutant that undergoes linear kinetic adsorption. We make an important simplifying assumption by considering that the aquifer is physically homogeneous. but chemically heterogeneous. The aquifer is assumed to be comprised of distinct reactive and nonreactive zones; in the reactive zones the pollutant undergoes linear kinetic adsorption. The distribution of reactive sites along the porous medium is characterized using two different approaches. stochastic and deterministic. A Bernoulli random process is used in the first approach. whereas a Fourier series is used in the second. In the stochastic approach the adsorption rate constant is also allowed to vary randomly. An approximate analytical solution for the temporal moments of the stochastic problem and an exact solution for the periodic problem are obtained. Temporal moments derived from this analysis are used to compute effective parameters (velocity and dispersion) as a function of the distance from the initial contaminant injection. Our results show that the overall effective dispersion coefficient is a sum of three terms accounting for the processes of equilibrium adsorption. kinetic adsorption. and chemical heterogeneity. respectively. We examine the impact of the adsorption rate parameter upon overall spreading and show that the magnitude of this rate parameter governs the relative contribution of chemical heterogeneity. A comparison between the effective parameters for the stochastic and periodic problem and the sensitivity of the results to physical and chemical parameters are also addressed.
Bacterial infiltration through the subsurface has been studied experimentally under different conditions of interest and is dependent on a variety of physical, chemical and biological factors. However, most bacterial transport studies fail to adequately represent the complex processes occurring in natural systems. Bacteria are frequently detected in stormwater runoff, and may present risk of microbial contamination during stormwater recharge into groundwater. Mixing of stormwater runoff with groundwater during infiltration results in changes in local solution chemistry, which may lead to changes in both bacterial and collector surface properties and subsequent bacterial attachment rates. This study focuses on quantifying changes in bacterial transport behavior under variable solution chemistry, and on comparing the influences of chemical variability and physical variability on bacterial attachment rates. Bacterial attachment rate at the soil-water interface was predicted analytically using a combined rate equation, which varies temporally and spatially with respect to changes in solution chemistry. Two-phase Monte Carlo analysis was conducted and an overall input-output correlation coefficient was calculated to quantitatively describe the importance of physiochemical variation on the estimates of attachment rate. Among physical variables, soil particle size has the highest correlation coefficient, followed by porosity of the soil media, bacterial size and flow velocity. Among chemical variables, ionic strength has the highest correlation coefficient. A semi-reactive microbial transport model was developed within HP1 (HYDRUS1D-PHREEQC) and applied to column transport experiments with constant and variable solution chemistries. Bacterial attachment rates varied from 9.10×10(-3)min(-1) to 3.71×10(-3)min(-1) due to mixing of synthetic stormwater (SSW) with artificial groundwater (AGW), while bacterial attachment remained constant at 9.10×10(-3)min(-1) in a constant solution chemistry (AGW only). The model matched observed bacterial breakthrough curves well. Although limitations exist in the application of a semi-reactive microbial transport model, this method represents one step towards a more realistic model of bacterial transport in complex microbial-water-soil systems.
Numerical experiments are conducted to examine the effect of gravity on monodisperse and polydisperse colloid transport in water-saturated fractures with uniform aperture. Dense colloids travel in water-saturated fractures by advection and diffusion while subject to the influence of gravity. Colloids are assumed to neither attach onto the fracture walls nor penetrate the rock matrix based on the assumption that they are inert and their size is larger than the pore size of the surrounding solid matrix. Both the size distribution of a colloid plume and colloid density are shown to be significant factors impacting their transport when gravitational forces are important. A constant- spatial-step particle-tracking code simulates colloid plumes with increasing densities transporting in water- saturated fractures while accounting for three forces acting on each particle: a deterministic advective force due to the Poiseuille flow field within the fracture, a random force caused by Brownian diffusion, and gravitational force. Integer angles of fracture orientation with respect to the horizontal ranging from -90 to +90 degrees are considered, and three log-normally distributed colloid plumes with mean particle size of 1 μm and standard deviation of 0.6, 1.2, and 1.8 μm are examined. Colloid plumes are assigned densities of 1.25, 1.5, 1.75, and 2.0 g/cm3. The first four spatial moments and the first two temporal moments are estimated as functions of fracture orientation angle and colloid density. Several snapshots of colloid plumes in fractures of different orientations are presented. Results are strongly dependent upon fracture orientation angle. In all cases, larger particles tend to spread over wider sections of the fracture in the flow direction, but smaller particles can travel faster or slower than larger particles depending on fracture orientation angle. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
A Lagrangian framework is used for analysing reactive solute transport by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The reaction considered is kinetically controlled sorption–desorption. Transport is quantified by the expected values of spatial and temporal moments that are derived as functions of the non-reactive moments and a distribution function which characterizes sorption kinetics. Thus the results of this study generalize the previously obtained results for transport of non-reactive solutes in heterogeneous formations (Dagan 1984; Dagan et
al. 1992). The results are illustrated for first-order linear sorption reactions. The general effect of sorption is to retard the solute movement. For short time, the transport process coincides with a non-reactive case, whereas for large time sorption is in equilibrium and solute is simply retarded by a factor R = 1+Kd, where Kd is the partitioning coefficient. Within these limits, the interaction between the heterogeniety and kinetics yields characteristic nonlinearities in the first three spatial moments. Asymmetry in the spatial solute distribution is a typical kinetic effect. Critical parameters that control sorptive transport asymptotically are the ratio εr between a typical reaction length and the longitudinal effective (non-reactive) dispersivity, and Kd. The asymptotic effective dispersivity for equilibrium conditions is derived as a function of parameters εr and Kd. A qualitative agreement with field data is illustrated for the zero- and first-order spatial moments.
Taylor-Aris dispersion theory, as generalized by Brenner, is employed to investigate the macroscopic behavior of sorbing solute transport in a three-dimensional, hydraulically homogeneous porous medium under steady, unidirectional flow. The porous medium is considered to possess spatially periodic geochemical characteristics in all three directions, where the spatial periods define a rectangular parallelepiped or a unit-element. The spatially-variable geochemical parameters of the solid matrix are incorporated into the transport equation by a spatially-periodic distribution coefficient and consequently a spatially-periodic retardation factor. Expressions for the effective or large-time coefficients governing the macroscopic solute transport are derived for solute sorbing according to a linear equilibrium isotherm as well as for the case of a first-order kinetic sorption relationship. The results indicate that for the case of a chemical equilibrium sorption isotherm the longitudinal macrodispersion incorporates a second term that accounts for the eflect of averaging the distribution coefficient over the volume of a unit element. Furthermore, for the case of a kinetic sorption relation, the longitudinal macrodispersion expression includes a third term that accounts for the effect of the first-order sorption rate. Therefore, increased solute spreading is expected if the local chemical equilibrium assumption is not valid. The derived expressions of the apparent parameters governing the macroscopic solute transport under local equilibrium conditions agreed reasonably with the results of numerical computations using particle tracking techniques.
In order to investigate the effects of reactive transport processes within a heterogeneous porous aquifer, two small-scale forced gradient tracer tests were conducted at the ‘Horkheimer Insel’ field site. During the experiments, two fluorescent tracers were injected simultaneously in the same fully penetrating groundwater monitoring well, located approximately 10 m from the pumping well. Fluoresceine and Rhodamine WT were used to represent the classes of practically non-sorbing and sorbing solutes, respectively. Multilevel breakthrough curves with a temporal resolution of 1 min were measured for both tracers at different depths within the pumping well using fibre-optic fluorimeters. This paper presents the tracer test design, the fibre-optic fluorimetry instrumentation, the experimental results and the interpretation of the measured multilevel breakthrough curves in terms of temporal moments and effective transport parameters. Significant sorption of Rhodamine WT is apparent from the effective retardation factors. Furthermore, an enhanced tailing of Rhodamine WT breakthrough curves is observed, which is possibly caused by a variability of aquifer sorption properties. The determined effective parameters are spatially variable, suggesting that a complex numerical flow and transport modelling approach within a stochastic framework will be needed to adequately describe the transport behaviour observed in the two experiments. Therefore, the tracer test results will serve in future work for the validation of numerical stochastic transport simulations taking into account the spatial variability of hydraulic conductivity and sorption-related aquifer properties.
The time and scale dependence of the retardation factor is examined for transport of a linearly adsorbing solute in an aquifer with spatially variable distribution coefficient (Kd). The variability of 1n Kd and the logarithm of the hydraulic conductivity (ln K) are modeled as stationary random fields. Two definitions of the effective retardation factor are introduced: (i) the ratio of the ensemble average centroid velocities of a passive and retarded solute and (ii) the ratio of the ensemble average arrival time of a passive solute to that of a retarded solute. The effective retardation factor based on the first definition is shown to be time-dependent and that based on the second definition is shown to be distance (scale) dependent. Theoretical expressions are developed for computing the effective retardation factors based on a knowledge of statistical parameters of the ln K and ln Kd fields. Illustrative computations are presented for one-dimensional and three-dimensional heterogeneous media.
A stochastic model for one-dimensional virus transport in homogeneous, saturated, semi-infinite porous media is developed. The model accounts for first-order inactivation of liquid-phase and adsorbed viruses with different inactivation rate constants, and time-dependent distribution coefficient. It is hypothesized that the virus adsorption process is described by a local equilibrium expression with a stochastic time-dependent distribution coefficient. A closed form analytical solution is obtained by the method of small perturbation or first-order approximation for a semi-infinite porous medium with a flux-type inlet boundary condition. The results from several simulations indicate that a time-dependent distribution coefficient results in an enhanced spreading of the liquid-phase virus concentration.
The objectives of this research were 2-fold: (1) to test the hypothesis that the rate of desorption of a halogenated alkene from a water-saturated aquifer material equals the rate of sorption in that system and (2) to develop a technique for measuring desorption rates that would be useful in characterizing a large-scale, heterogeneous Subsurface environment. A batch desorption methodology (intermittent purging) was developed as an extension of a documented, long-term equilibration technique (flame-sealed ampules). A batch model incorporating radial pore diffusion with internal retardation captured the dynamics of the observed desorption behavior. However, the model consistently underestimated desorption rates at early times and overestimated rates at later times. The best-fitting effective pore diffusion coefficient values (D-p) for the Borden sand-fractions ranged over nearly 2 orders of magnitude (7 X 10(-10) to 5 x 10(-8) cm(2)/s) and were, in most cases, two to four times lower than previous sorption rate estimates for the Borden sand. Possible reasons for the discrepancy are presented and discussed.
The problem of transport of a passive solute in a porous medium by convection and dispersion is analysed by the method of homogenization. Assuming that the geometry is periodic, the expressions for the macroscopic dispersion coefficients are derived. A few possible scalings are compared and we find that the most interesting one provides a local balance between drift and diffusion.
The long-term behavior of five organic solutes during transport over a period of 2 years in ground water under natural gradient conditions was characterized quantitatively by means of moment estimates. Total mass was conserved for two of the organic compounds, carbon tetrachloride and tetrachloroethylene, while the total mass declined for three other compounds, bromoform, 1,2-dichlorobenzene, and hexachloroethane. The declines in mass for the latter three compounds are interpreted as evidence of transformation of the compounds. Retardation factors for the organic solutes, relative to chloride, ranged from 1.5 to 9.0, being generally greater for the more strongly hydrophobic compounds. The retardation is attributed to sorption. The apparent retardation factor increased markedly for all compounds over the duration of the experiment, by as much as 150%. Results from temporal and spatial sampling were in good agreement when compared at the same scale of time and distance.
The cumulant expansion method, used previously by Sposito and Barry (1987) to derive an ensemble average transport equation for a tracer moving in a heterogeneous aquifer, is generalized to the case of a reactive solute that can adsorb linearly and undergo first-order decay. In the process we also generalize the Van Kampen (1987) result for the cumulant expansion of a multiplicative stochastic differential equation containing a time-dependent sure matrix. The resulting partial differential equation exhibits terms with field-scale coefficients that are analogous to those in the corresponding nonstochastic local-scale transport equation. There are also new terms in the third- and fourth-order spatial derivatives of the ensemble average concentration. It is demonstrated that the effective solute velocity for the aqueous concentration, not that for the total concentration (aqueous plus sorbed), is relevant for a field-scale description of solute transport. The field-scale effective solute velocity, dispersion coefficient, retardation factor, and first-order decay parameters, unlike their local-scale counterparts, are time-dependent because of autocorrelations and cross correlations among the random local solute velocity, retardation factor, and first-order decay constant. It is shown also that negative cross correlations between the random tracer solute velocity and the inverse of the local retardation factor may produce both enhanced dispersion and a temporal growth in the field-scale retardation factor. These effects are possible in any heterogeneous aquifer for which a stochastic description of aquifer spatial variability is appropriate.
A large-scale field experiment on natural gradient transport of solutes in groundwater has been conducted at a site in Borden, Ontario. Well-defined initial conditions were achieved by the pulse injection of 12 m3 of a uniform solution containing known masses of two inorganic tracers (chloride and bromide) and five halogenated organic chemicals (bromoform, carbon tetrachloride, tetrachloroethylene, 1,2-dichlorobenzene, and hexachloroethane). A dense, three-dimensional array of over 5000 sampling points was installed throughout the zone traversed by the solutes. Over 19,900 samples have been collected over a 3-year period. The tracers followed a linear horizontal trajectory at an approximately constant velocity, both of which compare well with expectations based on water table contours and estimates of hydraulic head gradient, porosity, and hydraulic conductivity. The vertical displacement over the duration of the experiment was small. Spreading was much more pronounced in the horizontal longitudinal than in the horizontal transverse direction; vertical spreading was very small. The organic solutes were retarded in mobility, as expected.
The problem of solute transport in a heterogeneous formation whose transmissivity or hydraulic conductivity are subject to uncertainty is studied for two- and three-dimensional flows. Approximate closed form solutions are derived for the case of a solute pulse in an average uniform flow through a formation of unconditional stationary random transmissivity. The solute concentration, regarded as a random variable, is determined in terms of its expectation and variance and is found to be subject to a high degree of uncertainty. The uncertainty is greatly reduced if the solute input zone is large compared to the transmissivity integral scale. In any case the concentration expectation does not obey a diffusion type equation in the case of two-dimensional flows, unless the solute body has traveled a distance larger than a few tens transmissivity integral scales. This distance may be exceedingly large in many conceivable applications.
This paper presents several exact and approximate analytical solutions of the equations describing convective-dispersive solute transport through large cylindrical macropores with simultaneous radial diffusion from the larger pores into the surrounding soil matrix. Adsorption effects were included through the introduction of linear isotherms for both the macropore region and the soil bulk matrix. In one formulation the macropores are surrounded by cylindrical soil mantles of finite thickness. Another formulation considers diffusion from a single cylindrical macropore into a radially infinite soil system. A relatively simple but very accurate approximate solution that ignores dispersion in the macropore region is also derived. The various analytical solutions in this paper can be used to calculate temporal and spatial concentration distributions in the macropore system. In addition, approximate solutions are presented for the radial concentration distribution within the adjacent soil matrix. By means of an example, it is demonstrated that at early times, little accuracy is lost when the radially finite soil mantle is replaced by an infinite system.
Solute transport in porous formations is governed by the large-scale heterogeneity of hydraulic conductivity. The two typical lengthscales are the local one (of the order of metres) and the regional one (of the order of kilometres). The formation is modelled as a random fixed structure, to reflect the uncertainty of the space distribution of conductivity, which has a lognormal probability distribution function. A first-order perturbation approximation, valid for small log-conductivity variance, is used in order to derive closed-form expressions of the Eulerian velocity covariances for uniform average flow. The concentration expectation value is determined by using a similar approximation, and it satisfies a diffusion equation with time-dependent apparent dispersion coefficients. The longitudinal coefficients tend to constant values in both two- and three-dimensional flows only after the solute body has travelled a few tens of conductivity integral scales. This may be an exceedingly large distance in many applications for which the transient stage prevails. Comparison of theoretical results with recent field experimental data is quite satisfactory.
A general theory of Taylor dispersion phenomena is presented based upon an analysis of the stochastic trajectory of a single Brownian particle undergoing convection and diffusion and subject to conservative external forces. The key to the analysis devolves upon classifying the independent variables specifying the instantaneous position of the Brownian particle into two distinct groups - local coordinates x, which are of bounded variation, and global coordinates X, which are unbounded. Circumstances are considered in which the phenomenological functions appearing in the constitutive equations for the respective particle flux-density vectors j and J in the x and X subspaces depend only upon the local coordinates x, but not the global coordinates X. The results are asymptotically valid for time intervals sufficiently long for the Brownian particle to have effectively achieved equilibrium with regard to its x-space transport, but yet of sufficiently short duration such that equilibrium does not obtain with respect to the X-space transport.
Generalized Taylor dispersion theory was employed to provide the context of a theory of time-periodic transport phenomena within spatially periodic porous media and related composite (multiphase) media. The novel time-periodic feature, distinguishing the present analysis from its predecessors in this series, permits modelling of rhythmical and cyclical flow and dispersion processes in porous and composite media, such as might be encountered in various biological, physiological and geophysical systems. It is pointed out that the Taylor dispersivity for situations wherein the time-average convective transport is identically zero may exceed, by many orders of magnitude, the purely molecular diffusion which occurs in situations where the convective transport is zero for all time. In this context, it is demonstrated that, at low oscillation frequencies, the dispersion may be treated as a quasistatic process, thereby permitting the use of available steady-state experimental dispersivity data in porous media to estimate the comparable dispersivity occurring during time-periodic processes.
Generalized Taylor dispersion theory is extended, so as to incorporate time-periodic global transport phenomena. The analysis expands upon a prior work, which was limited to transport with steady, time-independent convection. Special emphasis is placed upon periodic convection processes, such as might be encountered in rhythmical physiological or biological systems, or in geophysical phenomena involving periodic flows, like the ebb and flow of tides. Low frequency processes (where the period is long compared with the local-space diffusional relaxation time) are analysed. In accord with intuition, it is formally demonstrated that the effective Taylor dispersivity for this case can be calculated by supposing the dispersion process to be quasistatic.
Consider the problem of flow in a porous medium with hydraulic
conductivity which fluctuates locally about a mean value. The flow is
unsteady but gradually or slowly varying, i.e., the correlation length
of head fluctuations is considerably larger than the correlation length
of hydraulic-conductivity fluctuations. The equations which must be
satisfied by the effective conductivity tensor are derived under general
conditions using a method of volume averaging and spatial moments. The
generality of the derived equations is shown by replicating some known
results.
An analytical solution for describing the transport of dissolved
substances in heterogeneous porous media with a distance-dependent
dispersion relationship has been developed. The analytical solution can
be used to characterize differences in the transport process relative to
the classical convection-dispersion equation which assumes that the
hydrodynamic dispersion in the porous medium remains constant. The form
of the hydrodynamic dispersion function used in the analytical solution
is D(x) = α(x(v¯ + diffusion,
where α(x) is the distance-dependent dispersivity and v¯ is the
average pore water velocity. It is shown that for models which differ only in
how the dispersion function is expressed, erroneous model parameters may
result from parameter estimation techniques which assume a constant
hydrodynamic dispersion coefficient if the porous medium is more accurately
characterized by a distance-dependent dispersion relationship. For such
situations, the proposed model could be used to provide an alternate means
for obtaining these parameters. The overall features of the solution are
illustrated by several examples for constant concentration and flux boundary
conditions.
Relating the variability of permeability to the variability of head is a
central part of linear estimation techniques such as cokriging. Only a
few analytic relationships between log permeability covariances and head
covariances presently exist. This paper describes a general numerical
procedure which computes head covariances (ordinary or generalized) and
cross covariances for any proper log permeability covariance. The
numerical spectral method, a discrete analog of Fourier-Stieltjes
analysis, employs the pertinent linearized (small-perturbation
approximation) equations describing the physics of flow. The domain is
taken as finite, with boundary effects considered negligible. The
numerical spectral method can reproduce all pertinent analytic results
with excellent agreement. Furthermore, we demonstrate the method's
generality by finding the covariance relations for a case where no
analytical results presently exist.
Langevin techniques are employed to establish the stochastic trajectory of a Brownian tracer particle moving within the global/local hyperspace whose coordinates geometrically characterize the space of events of generalized Taylor dispersion theory. This dynamical approach contrasts with prior kinematical (i.e., Fokker-Planck) analyses employed in this series, corresponding to the use of a convective-diffusion equation describing the temporal evolution of a probability density. Expressions derived for the mean velocity vector Ū∗, and dispersivity dyadic D̄∗ of the Brownian particle are shown to be identical to those derived by the kinematical arguments utilized in prior contributions. Eigenfunction techniques are introduced, and subsequently employed to further confirm, through detailed analyses of several illustrative examples, equality of the alternative kinematical and dynamical schemes. Reconciliation of the different-appearing dynamical and kinematical formulas obtained for the Taylor dispersion dyadic results in a new integral representation of the B-vector field. A simple physical interpretation of this kinematical field is thereby suggested, which sheds new light on its fundamental role in dispersion phenomena.
The sampled data reveal that a number of outliers (low ln (K) values) are present in the data base. These low values cause difficulties in both variogram estimation and determining population statistics. The analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate. The classical, Cressie/Hawkins and squared median of the absolute deviations estimators are used to compute experimental variograms. None of these estimators provides completely satisfactory variograms for the Borden data with the exception of the classical estimator with outliers removed from the data set. Theoretical exponential variogram parameters are determined from nonlinear (NL) estimation. Differences are obtained between NL fits and those of Sudicky (1986). For the classical-screened estimated variogram, NL fits produce in ln (K) variance of 0.24, nugget of 0.07, and integral scales of 5.1 m horizontal and 0.21 m vertical along A- A′. For B-B′ these values are 0.37, 0.11, 8.3 and 0.34. The fitted parameter set for B-B′ data (horizontal and vertical) is statistically different than the parameter set determined for A- A′. We also evaluate a probabilistic form of Dagan's (1982, 1987) equations relating geostatistical parameters to a tracer cloud's spreading moments. -from Authors
The three-dimensional movement of a tracer plume containing bromide and chloride is investigated using the data base from a large-scale natural gradient field experiment on groundwater solute transport. The analysis focuses on the zeroth-, first-, and second-order spatial moments of the concentration distribution. These moments define integrated measures of the dissolved mass, mean solute velocity, and dispersion of the plume. Moments are estimated from the point observations using quadrature approximations tailored to the density of the sampling network. The estimators appear to be robust, with acceptable sampling variability. Estimates of the mass in solution for both bromide and chloride demonstrate that the tracers behaved conservatively, as expected. Analysis of the first-order moment estimates indicates that the experimental tracer plumes traveled along identical trajectories. The horizontal trajectory is linear and aligned with the hydraulic gradient. The vertical trajectory is curvilinear, concave upward. The total vertical displacement is small, however, so that the vertical component of the mean solute velocity vector is negligible. The estimated mean solute velocity is identical for both tracers (0. 091 m/day) and is spatially and temporally uniform for the first 647 days of travel time.
The spatial variability of hydraulic conductivity at the site of a long-term tracer test performed in the Borden aquifer was examined in great detail by conducting permeability measurements on a series of cores taken along two cross sections, one along and the other transverse to the mean flow direction. Along the two cross sections, a regular-spaced grid of hydraulic conductivity data with 0.05 m vertical and 1.0 m horizontal spatial discretization revealed that the aquifer is comprised of numerous thin, discontinuous lenses of contrasting hydraulic conductivity. Estimation of the three-dimensional covariance structure of the aquifer from the log-transformed data indicates that an exponential covariance model with a variance equal to 0.29, an isotropic horizontal correlation length equal to about 2.8 m, and a vertical correlation length equal to 0.12 m is representative. A value for the longitudinal macrodispersivity calculated from these statistical parameters using three-dimensional stochastic transport theory developed by L. W. Gelhar and C. L. Axness (1983) is about 0.6 m. For the vertically averaged case, the two-dimensional theory developed by G. Dagan (1982, 1984) yields a longitudinal djspersivity equal to 0.45 m. Use of the estimated statistical parameters describing the ln (K) variability in Dagan's transient equations closely predicted the observed longitudinal and horizontal transverse spread of the tracer with time. Weak vertical and horizontal dispersion that is controlled essentially by local-scale dispersion was obtained from the analysis. Because the dispersion predicted independently from the statistical description of the Borden aquifer is consistent with the spread of the injected tracer, it is felt that the theory holds promise for providing meaningful estimates of effective transport parameters in other complex-structured aquifers.
The magnitude of longitudinal dispersivity in a sandy stratified aquifer was investigated using laboratory column and field tracer tests. The field investigations included two-single-well injection-withdrawal tracer tests using 131I and a two-well recirculating withdrawal-injection tracer test using 51Cr-EDTA. The tracer movement within the aquifer was monitored in great detail with multilevel point-sampling instrumentation. A constant value for dispersivity of 0.7 cm was found to be representative (and independent of travel distance) at the scale of an individual level within the aquifer. A dispersivity of 0.035 cm was determined from laboratory column tracer tests as a representative laboratory-scale value for sand from the field site. The scale effect observed between the laboratory dispersivity and the dispersivity from individual levels in the aquifer is caused by the greater inhomogeneity of the aquifer (e.g., laminations within individual layers) and the averaging caused by the groundwater sampling system. Full-aquifer dispersivities of 3 and 9 cm obtained from the single-well tests indicate a scale effect with the value obtained being dependent mainly on the effect of transverse migration of tracer between the layers and the total injection volume. The full-aquifer dispersivity of 50 cm from the two-well test is scale-dependent, controlled by the distance between the injection and withdrawal wells (8 m) and hydraulic conductivity distribution in the aquifer. Scale-dependent full-aquifer dispersivity expressions were derived relating dispersivity to the statistical properties of a stratified geologic system where the hydraulic conductivity distribution is normal, log normal or arbitrary. In the developed expressions, dispersivity is a linear function of the mean travel distance. Proportionality constants ranged from 0.041 to 0.256 for the hydraulic conductivity distributions obtained from the field tracer tests.
Conventional modeling of mass transport in groundwater systems usually involves use of the dispersion-convection equation with large values of porous medium dispersivity to account for macroscopic dispersion. This work describes a modeling concept which accounts for macroscopic dispersion not as a large-scale diffusion process but as mixing caused by spatial heterogeneities in hydraulic conductivity. The two-dimensional spatially autocorrelated hydraulic conductivity field is generated as a first-order nearest-neighbor stochastic process. Analysis of a variety of hypothetical media shows that over finite domains a population of tracer particles convected through this statistically homogeneous conductivity field does not have the normal distribution and does not yield the constant dispersivity that classic theory would predict. This problem occurs because of insufficient spatial averaging in the macroscopic velocity field by the moving tracer particles. Our analyses suggest that the diffusion model for macroscopic dispersion may be inadequate to describe mass transport in geologic units. Sensitivity analysis with the model has shown that features of transport, such as first arrival of a tracer, are dependent on porous medium structure and that even when the statistical features of porous media are known, considerable uncertainty in the model result can be expected.
A theory is presented which accounts for nonlinearity caused by the deviation of plume “particles” from their mean trajectory in three-dimensional, statistically homogeneous but anisotropic porous media under an exponential covariance of log hydraulic conductivities. Existing linear theories predict that, in the absence of local dispersion, transverse dispersivities tend asymptotically to zero as Fickian conditions are reached. According to our new quasi-linear theory these dispersivities ascend to peak values and then diminish gradually toward nonzero Fickian asymptotes which are proportional to σY4 when the log hydraulic conductivity variance σY2 is much less than 1. All existing theories agree that in isotropic media the asymptotic longitudinal dispersivity is proportional to σY2 when σY2 < 1, and all are nominally restricted to mildly heterogeneous media in which this inequality is satisfied. However, the quasi-linear theory appears to be less prone to error than linear theories when extended to strongly heterogeneous media because it deals with the above nonlinearity without formally limiting σY2. It predicts that when σY ≫ 1 in isotropic media, both the longitudinal and transverse dispersivities ascend monotonically toward Fickian asymptotes proportional to σY.
An interpretation is offered for the observation that dispersivities increase with scale. Apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments, each field having its own exponential autocovariance, associated integral scale, and variance that increases as a power of scale. Such a hierarchy is shown theoretically to form a self-similar random field with homogeneous increments. Regardless of whether or not the underlying assumption is valid, one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi-linear theory of non-Fickian and Fickian dispersion in homogeneous media which supports the notion of a self-similar hierarchy a posteriori. The hierarchy is revealed to possess a semivariogram γ(s;) ≊ cs½, where c is a constant, and a fractal dimension D ≊ E + 0.75, where E is the topological dimension of interest. This can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous. As such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum, one must question the utility of associating medium properties with representative elementary volumes and relying on Fickian models of dispersion over more than relatively narrow scale intervals. Porous and fractured media appear to follow the same idealized scaling rule for both flow and transport, raising a question about the validity of many distinctions commonly drawn between such media. Finally, the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large-scale modes from the hierarchy.
A solute transport model incorporating well-to-well recirculation was developed to facilitate the interpretation of pilot-scale field experiments conducted for the evaluation of a test zone chosen for in situ restoration studies of contaminated aquifers, where flow was induced by recirculation of the extracted fluid. A semianalytical and an approximate analytical solution were derived to the one-dimensional advection-dispersion equation for a semi-infinite medium under local equilibrium conditions, with a flux-type inlet boundary condition accounting for solute recirculation between the extraction-injection well pair. Solutions were obtained by taking Laplace transforms to the equations with respect to time and space. The semianalytical solution is presented in Laplace domain and requires numerical inversion, while the approximate analytical solution is given in terms of a series of simple nested convolution integrals which are easily determined by numerical integration techniques. The applicability of the well-to-well recirculation model is limited to field situations where the actual flow field is one dimensional or where an induced flow field is obtained such that the streamlines in the neighborhood of the monitoring wells are nearly parallel. However, the model is fully applicable to studies of solute transport through packed columns with recirculation under controlled laboratory conditions. The model successfully simulated tracer breakthrough responses at a field solute transport study, where an induced flow field superimposed on the natural gradient within the confined aquifer was created by a well pair with extraction to injection rates of 10: 1.4.
A closed-form analytical small-perturbation (or first-order) solution to the one-dimensional advection-dispersion equation with spatially variable retardation factor is derived to investigate the transport of sorbing but otherwise nonreacting solutes in hydraulically homogeneous but geochemically heterogeneous porous formations. The solution is developed for a third- or flux-type inlet boundary condition, which is applicable when considering resident (volume-averaged) solute concentrations, and a semi-infinite porous medium. For mathematical simplicity it is hypothesized that the sorption processes are based on linear equilibrium isotherms and that the local chemical equilibrium assumption is valid. The results from several simulations, compared with predictions based on the classical advection-dispersion equation with constant coefficients, indicate that at early times, spatially variable retardation affects the transport behavior of sorbing solutes. The zeroth moments corresponding to constant and variable retardation are not necessarily equal. The impact of spatially variable retardation increases with increasing Péclet number. The center of mass appears to move more slowly, and solute spreading is enhanced in the variable retardation case. At late times, when the travel distance is much larger than the correlation scale of the retardation factor, the zeroth moment for the variable retardation case is identical to the case of invariant retardation. The small-perturbation solution agrees closely with a finite difference numerical approximation.
Spatial moment analysis is used in this paper to study the asymptotic,
long-time behavior of the depth-averaged solute plume for transport in a
perfectly stratified aquifer. The solute is assumed to adsorb onto the
aquifer solids according to a first-order reversible kinetic rate law;
steady, unidirectional, horizontal flow is assumed with arbitrary
vertical variation in pore water velocity, dispersion coefficients, and
adsorption reaction parameters. We derive general formulas to calculate
the effective dispersion coefficient governing the transport of the
depth-averaged plume. The results demonstrate that overall longitudinal
spreading of the plume results from three distinct factors: local Darcy
scale longitudinal dispersion, vertical variations in the pore water
velocity and retardation factor, and adsorption kinetics. For the
example of a two-layer aquifer, a simple nonequilibrium index is derived
which shows that deviations from local equilibrium diminish as the
degree of heterogeneity of the retarded pore water velocity increases.
It is also demonstrated that enhanced plume spreading can be caused by
negative correlation between the vertically varying pore water velocity
and retardation factor in addition to slow adsorption kinetics. Thus
great caution is warranted in interpreting the results of field scale
reactive tracer experiments.
In layered permeable deposits with flow predominately parallel to the
bedding, advection causes rapid solute transport in the more permeable
layers. As the solute advances more rapidly in these layers, solute mass
is continually transferred to the less permeable layers as a result of
molecular diffusion due to the concentration gradient between the
layers. The interlayer solute transfer causes the concentration to
decline along the permeable layers at the expense of increasing the
concentration in the less permeable layers, which produces strongly
dispersed concentration profiles in the direction of flow. The key
parameters affecting the dispersive capability of the layered system are
the diffusion coefficients for the less permeable layers, the
thicknesses of the layers, and the hydraulic conductivity contrasts
between the layers. Because interlayer solute transfer by transverse
molecular diffusion is a time-dependent process, the advection-diffusion
concept predicts a rate of longitudinal spreading during the development
of the dispersion process that is inconsistent with the classical
Fickian dispersion model. A second consequence of the solute-storage
effect offered by transverse diffusion into low-permeability layers is a
rate of migration of the frontal portion of a contaminant in the
permeable layers that is less than the groundwater velocity. Although
various lines of evidence are presented in support of the
advection-diffusion concept, more work is required to determine the
range of geological materials for which it is applicable and to develop
mathematical expressions that will make it useful as a predictive tool
for application to field cases of contaminant migration.
The dispersion of a conservative solute produced as a result of vertical
variations of hydraulic conductivity in a horizontal stratified aquifer
of finite thickness is analyzed by applying the moment method of Aris to
solve the governing advection-dispersion equation describing mass
transport. In the analysis it is assumed that the aquifer is of constant
thickness and of infinite lateral extent, the hydraulic conductivity is
a known function of the vertical coordinate only, and the flow is
unidirectional, parallel to the stratification. The applicable Aris
moment equations are developed in a suitable nondimensional form.
Analytical solutions are obtained for the zeroth and first moments and
for the time derivative of the second moment of the longitudinal
concentration distribution for the case of an instantaneous plane source
for several idealized hydraulic conductivity profiles (parabolic,
linear, step function, and cosine profiles and their even periodic
extensions). The analysis gives the time-dependent variation of the
longitudinal macrodispersivity for these idealized cases throughout the
transient development of the dispersion process. The results of the
analysis are applied to a field-measured hydraulic conductivity profile,
and predicted values of the longitudinal macrodispersivity are compared
with field results. An important conclusion from the analyses is that
nonuniformities in the hydraulic conductivity profile which persist over
long distances may produce rather large values of longitudinal
macrodispersivity which are comparable to those observed in some
aquifers and which are much larger than those predicted by some previous
stochastic analyses. Implications of the analytical results for field
dispersion problems are discussed.
The first paper in this series presented a description of a stochastic modeling concept for mass transport. In this paper we extend that analysis to consider a more realistic set of transport conditions in a groundwater basin with geologic layering, hydraulic anisotropy, spatial variations in porosity, and geochemical retardation. Uncertainties in transport predictions can be characterized by frequency distributions formed on the time of arrival of mass at the water table, on the exit location, and on the quantity of mass arriving at the water table as a function of time. Results show that transport is highly sensitive to porous medium heterogeneities. Considerable uncertainty is possible in predicting the spatial and temporal distribution of mass. Porous medium parameters capable of changing both the magnitude and direction of advective transport are of primary importance in influencing predictive ability. Most important in this respect are the arrangement of units with different mean hydraulic conductivities, the standard deviation and spatial continuity of hydraulic conductivity, and the hydraulic anisotropy. In certain geologic systems, patterns of contaminant migration can be predicted with more certainty. Features such as layering and hydraulic anisotropism can constrain the flow of mass to specific directions, thus limiting the size of the region through which mass is likely to spread. Depending upon the relative time scales of the release function and the transport process, lack of information about the timing and concentrations added at the source may cause greater uncertainty than the heterogeneity.
Laboratory investigations were conducted to determine whether the observed field retardation of bromoform, carbon tetrachloride, tetrachloroethylene, 1,2-dichlorobenzene, and hexachloroethane at the Borden field site could be explained by the linear, reversible, equilibrium sorption model. The five halogenated organic solutes, which have octanol-water partition coefficients ranging from 200 to 4000, were the same as those used in the field study. The sorbent, a medium sand containing 0.02% organic carbon, was excavated 11.5 m from the experimental well field at the Borden site. Sorption isotherms were linear in the aqueous concentration range from 1 to 50 μg/L and could be described by a single distribution coefficient Kd. The experimentally determined Kd exceed those predicted by the hydrophobic sorption model that accounts only for partitioning into organic matter, by factors ranging from 1.7 for hexachloroethane to 10 for tetrachloroethylene. Retardation factors inferred from the laboratory determined distribution coefficients fell within the range estimated from spatial sampling data in the field experiment.
Tailing of breakthrough responses, which has been experimentally
observed during flow through porous media, can be modeled by dividing
the porous medium into regions of mobile and immobile water, and
coupling the advective-dispersive solute transport equation with
expressions to describe diffusional transfer between the two regions.
Three-dimensional solutions to this coupled set of partial differential
equations with infinite boundary conditions are derived by applying the
Laplace transform to the equations with respect to time, and the Fourier
transform with respect to space. The solutions presented herein may be
useful in applying the two-region models to field settings.
The dispersion of soluble matter introduced into a slow stream of solvent in a capillary tube can be described by means of a virtual coefficient of diffusion (Taylor 1953a) which represents the combined action of variation of velocity over the cross-section of the tube and molecluar diffusion in a radial direction. The analogous problem of dispersion in turbulent flow can be solved in the same way. In that case the virtual coefficient of diffusion K is found to be 10\cdot 1av* or K = 7\cdot 14aU surd gamma . Here a is the radius of the pipe, U is the mean flow velocity, gamma is the resistance coefficient and v* 'friction velocity'. Experiments are described in which brine was injected into a straight 3/8 in. pipe and the conductivity recorded at a point downstream. The theoretical prediction was verified with both smooth and very rough pipes. A small amount of curvature was found to increase the dispersion greatly. When a fluid is forced into a pipe already full of another fluid with which it can mix, the interface spreads through a length S as it passes down the pipe. When the interface has moved through a distance X, theory leads to the formula S2 = 437aX(v*/U). Good agreement is found when this prediction is compared with experiments made in long pipe lines in America.
Sir Geoffrey Taylor has recently discussed the dispersion of a solute under the simultaneous action of molecular diffusion and variation of the velocity of the solvent. A new basis for his analysis is presented here which removes the restrictions imposed on some of the parameters at the expense of describing the distribution of solute in terms of its moments in the direction of flow. It is shown that the rate of growth of the variance is proportional to the sum of the molecular diffusion coefficient, D, and the Taylor diffusion coefficient kappa a2U2/D, where U is the mean velocity and a is a dimension characteristic of the cross-section of the tube. An expression for kappa is given in the most general case, and it is shown that a finite distribution of solute tends to become normally distributed.
In the mid-seventies, a new area of research has emerged in subsurface hydrology, namely sto chastic modeling of flow and transport. This development has been motivated by the recognition of the ubiquitous presence of heterogeneities in natural formations and of their effect upon transport and flow, on the one hand, and by the vast expansion of computational capability provided by elec tronic machines, on the other. Apart from this, one of the areas in which spatial variability of for mation properties plays a cardinal role is of contaminant transport, a subject of growing interest and concern. I have been quite fortunate to be engaged in research in this area from its inception and to wit ness the rapid growth of the community and of the literature on spatial variability and its impact upon subsurface hydrology. In view of this increasing interest, I decided a few years ago that it would be useful to present the subject in a systematic and comprehensive manner in order to help those who wish to engage themselves in research or application of this new field. I viewed as my primary task to analyze the large scale heterogeneity of aquifers and its effect, presuming that the reader already possesses a background in traditional hydrology. This is achieved in Parts 3, 4 and 5 of the text which incorporate the pertinent material."
The expected values of the spatial second-order moments of a solute body transported by groundwater are derived for flow through heterogeneous formations of a stationary random anisotropic structure. They are based on a general formulation, which reduces to most existing results in the literature as particular cases. Detailed results are given for the spatial variance as a function of time in the case of axisymmetric anisotropy, average flow parallel to the plane of isotropy, first-order approximation in the log conductivity variance sigma/sub Y/², and high Peclet numbers. These results fill the gap existing between the studies of Dagan (1982, 1984), on one hand, and those of Gelhar and Axness (1983) and Neuman et al. (1987), on the other. A preliminary investigation of the higher-order effects in sigma/sub Y/² suggests that the use of the first-order approximations is warranted, at present, only for sigma/sub Y/²
A rigorous theory of Brownian particle flow and dispersion phenomena in spatially periodic structures is presented within the context of generalized Taylor dispersion theory. The analysis expands upon a prior work, which was limited to transport within the continuous phase, to include convective and diffusive transport of the tracer particle within the interior of the discontinuous phase, as well as surface adsorption and transport along the phase boundary separating the discontinuous and continuous phases. Incorporated within the generalization are considerations of tracer particles of non-zero size, and situations wherein external forces act upon the tracer, the novel effect of each being to cause the tracer to move with a different velocity from that of the fluid in which it is suspended. Applications to various chromatographic separation phenomena are cited. Extensions of the analysis to heat-transfer problems and to situations involving homogeneous, first-order chemical reactions are also made. Both Eulerian and Lagrangian interpretations of the tracer transport phenomena are given.
A rigorous theory of dispersion in both granular and sintered spatially-periodic porous media is presented, utilizing concepts originating from Brownian motion theory. A precise prescription is derived for calculating both the Darcy-scale interstitial velocity vector {v}* and dispersivity dyadic {D}* of a tracer particle. These are expressed in terms of the local fluid velocity vector field v at each point within the interstices of a unit cell of the spatially periodic array and, for the dispersivity, the molecular diffusivity D of the tracer particle through the fluid. Though the theory is complete, numerical results are not yet available owing to the complex structure of the local interstitial velocity field v. However, as an illustrative exercise, the theory is shown to correctly reduce in an appropriate limiting case to the well-known Taylor-Aris results for dispersion in circular capillaries.
When a soluble substance is introduced into a fluid flowing slowly through a small-bore tube it spreads out under the combined action of molecular diffusion and the variation of velocity over the cross-section. It is shown analytically that the distribution of concentration produced in this way is centred on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow. The dispersion along the tube is governed by a virtual coefficient of diffusivity which can be calculated from observed distributions of concentration. Since the analysis relates the longitudinal diffusivity to the coefficient of molecular diffusion, observations of concentration along a tube provide a new method for measuring diffusion coefficients. The coefficient so obtained was found, with potassium permanganate, to agree with that measured in other ways. The results may be useful to physiologists who may wish to know how a soluble salt is dispersed in blood streams.
The dispersive mixing resulting from complex flow in three-dimensionally
heterogeneous porous media is analyzed using stochastic continuum
theory. Stochastic solutions of the perturbed steady flow and solute
transport equations are used to construct the macroscopic dispersive
flux and evaluate the resulting macrodispersivity tensor in terms of a
three-dimensional, statistically anisotropic input covariance describing
the hydraulic conductivity. With a statistically isotropic input
covariance, the longitudinal macrodispersivity is convectively
controlled, but the transverse macrodispersivity is proportional to the
local dispersivity and is several orders of magnitude smaller than the
longitudinal term. With an arbitrarily oriented anisotropic conductivity
covariance, all components of the macrodispersivity tensor are
convectively controlled, and the ratio of transverse to longitudinal
dispersivity is of the order of 10-1. In this case the
off-diagonal components of the dispersivity tensor are significant,
being numerically larger than the diagonal transverse terms, and the
transverse dispersion process can be highly anisotropic. Dispersivities
predicted by the stochastic theory are shown to be consistent with
controlled field experiments and Monte Carlo simulations. The theory,
which treats the asymptotic condition of large displacement, indicates
that a classical gradient transport (Fickian) relationship is valid for
large-scale displacements.
A three-dimensional theory is described for field-scale Fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities. The study relies partly on earlier work by the authors the attributes of which are briefly reviewed. It leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media. We express the dispersion tensor D as the sum of a local component d and a field-scale component DELTA . The local component is assumed to be independent of velocity (which is most appropriate if it represents molecular diffusion) and its principal terms are taken to act parallel and normal to the mean velocity vector mu . The field-scale component is written as alpha mu , where alpha is a dispersivity tensor and mu equals vertical mu vertical . We show that at large Peclet numbers P, the dispersivity tensor reduces to a single principal component parallel to the mean velocity, regardless of how mu is oriented. This result, valid for arbitrary covariance functions of log-hydraulic conductivity, differs from that of L. F. Gelhar and C. L. Axness (1983), according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value.
The direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions. Under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space, closed form expressions are derived for the effective conductivity or storativity and for the head variogram. Two types of unsteady flow are considered: transients and periodic flows. In the case of transients, it is assumed that initially the head is constant (no flow) and that ultimately the flow becomes uniform. It is found that the effective conductivity and transmissivity are time-dependent and drop from the arithmetic mean (initially) to the steady state value during a relaxation time. The latter is much larger for one-dimensional flows than for two- or three-dimensional ones. The head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag. While in three-dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions, the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale. Hence steady state analysis for transients might not be warranted for regional transient flows. In the case of periodic flows of seasonal nature, the head gradient is assumed to be made up from a steady component and a periodic one. If the amplitude of the latter is sufficiently small compared to the steady one, steady state value of effective transmissivity and storativity can be adopted. The head variogram has also its steady state form for large lags and quasisteady for small lags. Hence the variogram lies between the steady one (steady component of head gradient) and the quasisteady one (instantaneous head gradient).
One of the most important limitations to the application of modeling techniques for the analysis of mass transfer in groundwater systems is the difficulty in characterizing the dispersive character of natural systems. Although large quantities of experimental data exist for laboratory scale experiments, the handful of measurements which have been obtained from regional systems suggest that dispersion produced by large-scale porous medium nonidealities is considerably more important. In this paper the effects of macroscopic dispersion are simulated for uniform heterogeneous porous media under conditions of one- dimensional flow. The idealized media considered for detailed analysis consist of low permeability inclusions within a higher-permeability medium. When the inclusions are not arranged rather homogeneously within the region, a unique dispersivity value for the medium cannot be defined, and dispersivity changes as a function of space. The magnitude of dispersion is controlled by the contrast in hydraulic conductivity between the inclusions and the remainder of the medium, the number of inclusions, and the mode of aggregation. Generally, dispersivity is found to decrease as the conductivity contrast decreases and the structure of the medium is regularized. It will be possible to estimate the dispersivity of a medium by using stochastic analysis. Hypothetical porous media with characteristics similar to those of some actual medium will yield a range of dispersivity values. The basic data for this technique will be detailed statistical analysis of the mode of porous medium aggregation and the conductivity contrasts within the medium.
When the quasi-linear theory developed in paper 1 is applied to anisotropic media it shows, in contrast to the isotropic case, that longitudinal and transverse dispersivities may become asymptotically proportional to σY when the log hydraulic conductivity variance σY2 is much smaller than 1. It further implies, among other phenomena, that when the mean seepage velocity vector μ is at an angle to the principal axes of statistical anisotropy, the long axis of a plume is generally offset toward the direction of the largest log hydraulic conductivity correlation scale; when μ is at 45° to the bedding in strongly stratified media, the longitudinal axis is nearly parallel to the bedding under non-Fickian conditions. As Fickian conditions are approached, the plume rotates toward μ and stabilizes asymptotically at a relatively small angle of deflection depending on σY2. Application of the quasi-linear theory to depth-averaged concentration data from a tracer experiment at Borden, Ontario, yields a consistent and improved fit to a two-dimensional model without any need for parameter adjustment. Three-dimensional models are shown to be in fundamental conflict with observed behavior at Borden and in other stratified formations; we show that, in principle, this conflict is easy to resolve by accounting for local hydraulic anisotropy.
Suppose in a convection-dispersion equation, governing solute movement
in a saturated porous medium of infinite extent, the convection velocity
components are periodic functions of spatial coordinates. Then it
follows from a general mathematical result that the solute concentration
can be asymptotically approximated by a Gaussian density. Two
theoretical examples, with and without a constant vertical velocity, are
given to illustrate an application of this mathematical result to solute
dispersion in a parallel-bedded, three-dimensional aquifer of infinite
extent. Analytical expressions are obtained for dispersion coefficients
in the asymptotic Gaussian approximations in the two examples. The
dispersion coefficients functionally depend on a velocity scaling
parameter U0 and a spatial scaling parameter (period) a via
their product aU0, and this functional dependence on
aU0 is different in the two examples. The expressions for
dispersion coefficients are formally contrasted and compared with those
obtained previously by other authors in a parallel-bedded infinite
aquifer in which the convection velocity is a random field. Some
physical implications of dependence of dispersion coefficients on a
spatial scaling parameter (i.e., the scale effect) are discussed in the
context of modeling in an aquifer with evolving heterogeneities. This
interpretation of "scale effect" in dispersion coefficients is
contrasted with the preasymptotic interpretation in the current
literature.
For the special case of a stratified porous medium with flow parallel to
the bedding it is shown that the transport of solute cannot, in general,
be represented by the usual convection-diffusion equation, even for
large time. The necessary conditions for the appearance of a Fickian
diffusive process are discussed and compared with previous work done by
Gelhar et al. (1979) and Marie et al. (1967). It is shown, however, that
when the flow is not exactly parallel to the stratification, diffusive
behavior is much more likely to appear. The need for further work on the
mechanism of transport in porous media is then emphasized.
This work is concerned with the nonreactive transport of solute
materials in groundwater, or hydro-dispersive transfer. Several types of
flow fields are considered: linear (or uniform) flow with one- and
two-dimensional dispersion and radial flow under diverging and
converging conditions. The analysis includes the two main possibilities
for introduction of solutes into an aquifer: continuous and
instantaneous (or slug) injection. Different solutions from the
literature plus some original solutions for dispersion in a linear flow
field have been unified by transposing the solutions into dimensionless
variables of concentration (CR), time (tR), and
the Peclet number (P). This permits an analysis of the errors committed
in some commonly used approximations for dispersion as a function of P.
In the case of radial flow, a numerical method using finite differences
has been developed that can be applied to either diverging or converging
flow problems. Results in dimensionless form when compared to the only
analytical approximations that could be found (for continuous injection
in a diverging flow field) indicate that the approximate solutions are
in error when P ≤ 10. The radial flow results are also compared to
those for linear flow fields to demonstrate that in most cases either
approach can be used as long as P > 3. A series of dimensionless type
curves has been developed showing CR versus tR for
practical ranges of P. A simple method of interpreting tracer tests is
proposed using these type curves. One is able to directly determine
dispersivity and kinematic porosity using curve matching techniques.
Results from some recent field tests in France are analyzed using this
approach. There is definite confirmation from these investigations that
the apparent (macroscopic) dispersivity can vary depending on the
distances used in the field.
The longitudinal dispersion produced as a result of vertical variations
of hydraulic conductivity in a stratified aquifer is analyzed by
treating the variability of conductivity and concentration as
homogeneous stochastic processes. The mass transport process is
described using a first-order approximation which is analogous to that
of G. I. Taylor for flow in tubes. The resulting stochastic differential
equation describing the concentration field is solved using spectral
representations. The results of the analysis demonstrate that for large
time the longitudinal dispersivity approaches a constant value which is
dependent on statistical properties of the medium. The analysis also
describes the transient development of the dispersive process and some
non-Fickian effects which occur early in the displacement process.
A solute transport model incorporating well-to-well recirculation was developed to facilitate the interpretation of pilot-scale field experiments conducted for the evaluation of a test zone chosen for in situ restoration studies of contaminated aquifers, where flow was induced by recirculation of the extracted fluid. A semianalytical and an approximate analytical solution were derived to the one-dimensional advection-dispersion equation for a semi-infinite medium under local equilibrium conditions, with a flux-type inlet boundary condition accounting for solute recirculation between the extraction-injection well pair. Solutions were obtained by taking Laplace transforms to the equations with respect to time and space. The semianalytical solution is presented in Laplace domain and requires numerical inversion, while the approximate analytical solution is given in terms of a series of simple nested convolution integrals which are easily determined by numerical integration techniques. The applicability of the well-to-well recirculation model is limited to field situations where the actual flow field is one dimensional or where an induced flow field is obtained such that the streamlines in the neighborhood of the monitoring wells are nearly parallel. However, the model is fully applicable to studies of solute transport through packed columns with recirculation under controlled laboratory conditions. The model successfully simulated tracer breakthrough responses at a field solute transport study, where an induced flow field superimposed on the natural gradient within the confined aquifer was created by a well pair with extraction to injection rates of 10:1.4.
Natural geologic formations are highly heterogeneous. It is impossible and often unnecessary to describe in deterministic terms the spatial variability of their properties. However, the hydrogeologic parameters may be represented in probabilistic terms. Prediction of solute transport may then be defined as the derivation of the probabilistic properties of concentration. This work deals with the first two integral (or spatial) moments of solute concentration in a heterogeneous formation of infinite extent. The first moment is the vector of the mean position of the centroid of the plume and, in a generalized sense, represents advection. The second moment is the matrix of dyadics of the mean squared displacement about the average position of the centroid of the plume and, again in a generalized sense, represents dispersion. Assuming that the mixing at the laboratory scale is Fickian, with random but time-invariant velocities and lab-scale dispersion matrices, the differential equations satisfied by the first two moments are derived. An analytical first-order (or small perturbation) solution is obtained for stationary velocity, and compared with a numerical solution.
A previous generalized theory of Taylor dispersion phenomena is here extended so as to include “coupling” phenomena. Such direct coupling arises from the fact that—in certain circumstances—global transport processes may be driven by local gradients, and conversely. An obvious example of such direct coupling effects arises in problems pertaining to the gravitational settling and concomitant “sedimentation dispersion” of asymmetric Brownian particles lacking a center of symmetry. In the presence of such asymmetry the parlicle's translational and rotational motions are inseparably coupled to one another. This specific problem will be ueated in a companion paper as an example of the general coupling theory developed herein.
Generalized Taylor dispersion theory extends the basic long-time, asymptotic scheme of Taylor and Aris greatly beyond the class of rectilinear duct and channel flow dispersion problems originally addressed by them. This feature has rendered it indispensable for studying flow and dispersion phenomena in porous media, chromatographic separation processes, heat transfer in cellular media, sedimentation of non-spherical Brownian particles, and transport of flexible clusters of interacting Brownian particles, to mention just a few examples of the broad class of non-unidirectional transport phenomena encompassed by this scheme. Moreover, generalized Taylor dispersion theory enjoys the attractive feature of conferring a unified paradigmatic structure upon the analysis of such apparently disparate physical problems. For each of the problems thus treated it provides an asymptotic, macroscale description of the original microscale transport process, being based upon a convective-diffusive ‘model’ problem characterized by a set of constant (position- and time-independent) phenomenological coefficients.