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Analysis of one-dimensional solute transport through porous media with spatially variable retardation factor
(1990) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Peter K. Kitanidis
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.
One-dimensional solute transport in porous media with partial well-to-well recirculation: Application to field experiments
(1990) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Paul V. Roberts
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.
A three-dimensional steady-state atmospheric dispersion-deposition model for emissions from a ground-level area source
(1992) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; P. V. Roberts; L . M. Hildemman
An analytical solution to the steady-state three-dimensional atmospheric dispersion equation has been developed for the transport of non-buoyant emissions from a continuous ground-level area source. The model incorporates power law profiles for the variation of wind speed and vertical eddy diffusivity with height, represents the lateral eddy diffusivity as a function of wind speed and the crosswind dispersion coefficient, and includes dry deposition as a removal mechanism. The model is well suited for accurate prediction of emission concentration levels in the vicinity of an area source, as well as farther downwind, under neutral or stable atmospheric conditions. The impact of the important model parameters on contaminant dispersion is examined. The results from several simulations, compared with point and line sources of equivalent source strength, indicate that at short downwind distances, predictions of contaminant concentrations emitted from area sources may be unacceptably inaccurate unless the structure of the source is properly taken into account.
Generalized TaylorAris moment analysis of the transport of sorbing solutes through porous media with spatiallyperiodic retardation factor
(Kluwer Academic Publishers, 1992) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Paul V. Roberts; Peter K. Kitanidis
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.
Macrodispersion of sorbing solutes in heterogeneous porous formations with spatially periodic retardation factor and velocity field
(1992) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Peter K. Kitanidis
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.
Artificial tracers for geothermal reservoir studies
(1993) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος
Safe disposal of thermally spent geothermal brines that contain environmentally hazardous constituents is commonly obtained by reinjection. The reinjection process also serves to maintain reservoir pressure, enhance thermal recovery, and eliminate possible compactional subsidence. To avoid premature thermal breakthrough of reinjected fluids, tracer tests are employed for detection and evaluation of preferential path networks. In this paper some promising tracers that have not received much attention in geothermal reservoir studies are discussed, and a comprehensive tabulation of field sites of artificial tracer utilization is presented. Chemical and transport processes responsible for tracer retention by the formation of reservoir solids, as well as available tracer detection techniques, are emphasized.
A study of crystal packing in a series of closely related square-planar palladium(II) and platinum(II) complexes
(Pergamon, 1994) Psyllaki Eleftheria; Ψυλλακη Ελευθερια; Bardwell David A.; Crossle John G.; Orpen, A. Guy; Jeffer John C.; Tilley Elizabeth E. M.
The crystal structures of the series of four-coordinate complexes [PdL1Cl) · CH2Cl2 (1·CH2Cl2), [PtL1Cl]·CH2Cl2 (2·CH2Cl2), [PdL3Cl][PF6 · CH2Cl2 (3 · CH2Cl2) and [PdL4Cl][PF6 (4) [HL1 = 6-(2-hydroxyphenyl)-2,2′-bipyridine; L3 = 6-(2-dimethylaminophenyl)-2,2′-bipyridine; L4 = 2-(2-dimethylaminophenyl)-1,10-phenanthroline] are compared and contrasted to examine the extent to which aromatic π-stacking interactions contribute to crystal packing. Complexes 1 and 2 have very similar planar molecular structures but stack in different ways; molecules of 1 form a linear stack in which overlap of the aromatic ligands is maximized between adjacent molecules and there is no PdPd interaction, whereas in 2 there are axial PtPt interactions within the linear stack but less overlap between adjacent aromatic ligands.
Analytical solutions for one-dimensional colloid transport in saturated fractures
(1994) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Assem Abdel-Salam
Closed-form analytical solutions for colloid transport in single rock fractures with and without colloid penetration into the rock matrix are derived for constant concentration as well as constant flux boundary conditions. A single fracture is idealized as two semi-infinite parallel plates. It is assumed that colloidal particles undergo irreversible deposition onto fracture surfaces and may penetrate into the rock matrix, and deposit irreversibly onto rock matrix solid surfaces. The solutions are obtained by taking Laplace transforms to the governing transport equations and boundary conditions with respect to time and space. For the case of no colloid penetration into the rock matrix, the solutions are expressed in terms of exponentials and complimentary error functions; whereas, for the case of colloid penetration into the rock matrix, the solutions are expressed in terms of convolution integrals and modified Bessel functions. The impact of the model parameters on colloid transport is examined. The results from several simulations indicate that liquid-phase as well as deposited colloid concentrations in the fracture are sensitive to the fracture surface deposition coefficient, the fracture aperture, and the Brownian diffusion coefficient for colloidal particles penetrating the rock matrix. Furthermore, it is shown that the differences between the two boundary conditions investigated are minimized at dominant advective transport conditions. The constant concentration condition overestimates liquid-phase colloid concentrations, whereas the constant flux condition leads to conservation of mass.
Modeling of contaminant transport resulting from dissolution of nonaqueous phase liquid pools in saturated porous media
(1994) Evangelos A. Voudrias; Marios M. Fyrillas
A mathematical model for transient contaminant transport resulting from the dissolution of a single component nonaqueous phase liquid (NAPL) pool in two-dimensional, saturated, homogeneous porous media was developed. An analytical solution was derived for a semi-infinite medium under local equilibrium conditions accounting for solvent decay. The solution was obtained by taking Laplace transforms to the equations with respect to time and Fourier transforms with respect to the longitudinal spatial coordinate. The analytical solution is given in terms of a single integral which is easily determined by numerical integration techniques. The model is applicable to both denser and lighter than water NAPL pools. The model successfully simulated responses of a 1,1,2- trichloroethane (TCA) pool at the bottom of a two-dimensional porous medium under controlled laboratory conditions.
Three-dimensional analytical models of contaminant transport from nonaqueous phase liquid pool dissolution in saturated subsurface formations
(1995) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος
Closed form analytical solutions are derived for three-dimensional transient contaminant transport resulting from dissolution of single-component nonaqueous phase liquid pools in saturated porous media. The solutions are suitable for homogeneous porous media with unidirectional interstitial velocity. The dissolved solute may undergo first-order decay or may sorb under local equilibrium conditions. The solutions are obtained for rectangular and elliptic as well as circular source geometries, assuming that the dissolution process is mass transfer limited, by applying Laplace and Fourier transforms. Although the solutions contain integral expressions, these integrals are easily evaluated numerically. These solutions are useful for verifying the accuracy of numerical solutions to more comprehensive models and for design and interpretation of experiments in laboratory-packed beds and possibly some field studies. The results of several simulations indicate that for short downstream distances, predictions of contaminant concentrations are sensitive to the source structure and orientation with respect to the direction of interstitial flow .
Modeling of colloid and colloid-facilitated contaminant transport in a two-dimensional fracture with spatially variable aperture
(1995) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Assem Abdel-Salam
Mathematical models are developed for two-dimensional transient transport of colloids, and cotransport of contaminant/colloids in a fracture-rock matrix system with spatially variable fracture aperture. The aperture in the fracture plane is considered as a lognormally distributed random variable with spatial fluctuations described by an exponential autocovariance function. Colloids are envisioned to irreversibly deposit onto fracture surfaces without penetrating the rock matrix; whereas, the contaminant is assumed to decay, sorb onto fracture surfaces and onto colloidal particles, as well as to diffuse into the rock matrix. The governing stochastic transport equations are solved numerically for each realization of the aperture fluctuations by a fully implicit finite difference scheme. Emphasis is given on the effects of variable aperture on colloid and colloid-facilitated contaminant transport. Simulated breakthrough curves of ensemble averages of several realizations show enhanced colloid transport and more pronounced fingering when colloids are subject to size exclusion from regions of small aperture size. Moreover, it is shown that an increase in the fracture aperture fluctuations leads to faster transport and increases dispersion. For the case of contaminant/colloids cotransport it is shown, for the conditions considered in this work, that colloids enhance contaminant mobility and increase contaminant dispersion.
Analytical models for one-dimensional virus transport in saturated porous media
(1995) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Youn Sim
Analytical solutions to two mathematical models for virus transport in one-dimensional homogeneous, saturated porous media are presented, for constant flux as well as constant concentration boundary conditions, accounting for first-order inactivation of suspended and adsorbed (or filtered) viruses with different inactivation constants. Two processes for virus attachment onto the solid matrix are considered. The first process is the nonequilibrium reversible adsorption, which is applicable to viruses behaving as solutes; whereas, the second is the filtration process, which is suitable for viruses behaving as colloids. Since the governing transport equations corresponding to each physical process have identical mathematical forms, only one generalized closed-form analytical solution is developed by Laplace transform techniques. The impact of the model parameters on virus transport is examined. An empirical relation between inactivation rate and subsurface temperature is employed to investigate the effect of temperature on virus transport. It is shown that the differences between the two boundary conditions are minimized at advection-dominated transport conditions.
Numerical modeling of three-dimensional contaminant migration from dissolution of multicomponent NAPL pools in saturated porous media
(1995) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; K. Y. Lee
A three-dimensional model for contaminant transport resulting from the dissolution of multicomponent nonaqueous phase liquid (NAPL) pools in threedimensional saturated subsurface formations is developed. The solution is obtained numerically by a finite-difference scheme, and it is suitable for homogeneous porous media with unidirectional interstitial velocity. Each dissolved component may undergo first-order decay and may sorb under local equilibrium conditions. It is also assumed that the dissolution process is mass transfer limited. The nonaqueous phase activity coefficients of the NAPL pool components are evaluated at each time step. The model behavior is illustrated through a synthetic example with a NAPL pool consisting of a mixture of TCA (1,1,2- trichloroethane) and TCE (trichloroethylene). The numerical solution presented in this work is in good agreement with a recently developed analytical solution for the special case of a single component NAPL pool. The results indicate the importance of accounting for the necessary changes in the organic phase activity which significantly affects the equilibrium aqueous solubility
Analysis of a ,model for contaminant transport in fracture media in the presence of colloid
(1995) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Assem Abdel-Salam
A mathematical model has been developed to study the cotransport of contaminants with colloids in saturated rock fractures. The contaminant is assumed to decay, and sorb on to fracture surfaces and on to colloidal particles, as well as to diffuse into the rock matrix; whereas, colloids are envisioned to deposit irreversibly on to fracture surfaces without penetration into the rock matrix. The governing one-dimensional equations describing the contaminant and the colloid transport in the fracture, colloid deposition on to fracture surfaces, and contaminant diffusion into the rock matrix are coupled. This coupling is accomplished by assuming that the amount of contaminant mass captured by colloidal particles in solution and the amount captured by deposited colloids on fracture surfaces are described by modified Freundlich reversible equilibrium sorption relationships, and that mass transport by diffusion into the rock matrix is a first-order process. The contaminant sorption on to fracture surfaces is described by a linear equilibrium sorption isotherm, while the deposition of colloids is incorporated into the model as a first-order process. The resulting coupled contaminant transport non-linear equation is solved numerically with the fully implicit finite difference method. The constant concentration as well as the constant flux boundary conditions have been considered. The impact of the presence of colloids on contaminant transport is examined. According to model simulations the results show that, depending on the conditions of the physical system considered, colloids can increase or decrease the mobility of contaminants.
Unsaturated flow in a quasi-three-dimensional fractured medium with spatially variable aperture
(1996) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Assem Abdel-Salam
Transient moisture flow in a variably saturated quasi-three-dimensional fracture-rock matrix system is investigated. The fracture is assumed to possess a spatially variable aperture in its two-dimensional plane, whereas the rock matrix is treated as a two-dimensional homogeneous and tight porous medium. The aperture fluctuations in the fracture plane are described stochastically. Moisture exchange between the fracture and the rock matrix is accounted for via an advective coupling term that governs the transfer of moisture at the fracture-matrix interface and takes into account the effect of a fracturesurface coating material. Although the variable aperture fracture is two-dimensional, the coupling term between the fracture and the rock matrix accounts for the threedimensional nature of the physical system. The stochastic nonlinear set of partial differential equations is solved numerically by the Galerkin finite element method in conjunction with the Picard iterative scheme and an automatic time step marching. Simulations are performed to investigate phenomena which have been ignored in previous studies. It is demonstrated that, for the case of no moisture exchange with the rock matrix, the moisture follows preferential flow paths within the fracture plane and exhibits pronounced fingering effects. Furthermore, it is shown that the larger the fracture aperture fluctuations the more extended the moisture flow in the fracture. In addition, for the case where there exists moisture exchange with the rock matrix, the movement of the moisture front is considerably reduced, whereas fracture-surface coatings tend to slow down moisture absorption by the rock matrix.
One-dimensional virus transport in porous media with time-dependent inactivation rate coefficients
(1996) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Youn Sim
A model for virus transport in one-dimensional, homogeneous, saturated porous media is developed, accounting for virus sorption and inactivation of liquid phase and adsorbed viruses with different time dependent rate coefficients. The virus inactivation process is represented by a pseudo first-order rate expression. The pseudo first-order approximation is shown to simulate available experimental data from three virus inactivation batch studies better than the frequently employed constant rate inactivation model. Model simulations indicated that the pseudo first-order approximation, compared to the constant inactivation, leads to extended survival of viruses and, consequently, more distant migration. Results from a parameter sensitivity analysis demonstrated that estimation of pseudo first-order inactivation rate coefficients from field observations requires data collection near the source of virus contamination during initial stages of virus transport.
One-dimensional virus transport in homogeneous porous media with time-dependent distribution coefficient
(1996) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Youn Sim
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.
Modeling colloid transport and deposition in saturated fractures
(1997) Chrysikopoulos Constantinos; Χρυσικοπουλος Κωνσταντινος; Assem Abdel-Salam
A model is developed to describe the transport of colloids in a saturated fracture with a spatially variable aperture, accounting for colloid deposition onto fracture surfaces under various physicochemical conditions. The fracture plane is partitioned into unit elements with different apertures generated stochastically from a log-normal distribution. The model also accounts for colloid size exclusion from fracture elements with small apertures. Both equilibrium and kinetic colloid deposition onto fracture surfaces are investigated. Colloid surface exclusion is incorporated in the dynamics of kinetic deposition. The impact of deposited colloids on further colloid deposition is described by either a linear or a non-linear blocking function. The resulting system of governing partial differential equations is solved numerically using the fully implicit finite difference method. Model simulations illustrate the presence of preferential colloid transport in the fracture plane. It is shown that size exclusion increases the dispersion of colloids and leads to earlier breakthrough, especially for large-size particles. Furthermore, it is demonstrated that surface exclusion enhances colloid transport, and the assumption of clean-bed media may underestimate liquid-phase colloid concentrations.
Square-prismatic vs. square-antiprismatic coordination in complexes of lead(II) with a simple bidentate chelating ligand; effects of intermolecular hydrogen bonding
(Royal Society of Chemistry, 1997) Elefteria Psillakis; JohnáC Jeffery
Whereas the eight-coordinate complex cation of [PbL4][(MeO)2PO2]·2H2O [L = 3-(2-pyridyl)pyrazole] has square prismatic geometry stabilised by hydrogen bonding between the pyrazolyl NH groups and the dimethylphosphate anion, the complex cation of [PbL4][PF6]2, in which there are no significant hydrogen-bonding interactions, has a structure distorted towards square antiprismatic with a 22° twist between the planes.
Lanthanide complexes of the hexadentate n-donor podand tris[3-(2-pyridyl)pyrazolyl]hydroborate: Solid-State and solution properties
(1997) Elefteria Psillakis; Peter L Jones; Angelo J Amoroso; Jon A McCleverty; John C Jeffery; Michael D Ward; Leigh H Rees
The hexadentate N6-donor podand tris[3-(2-pyridyl)pyrazolyl]hydroborate (TpPy) contains 2-pyridyl fragments attached to the pyrazolyl C3-positions such that each arm is a bidentate chelate. Three series of lanthanide(III) complexes were prepared: [M(TpPy)(MeOH)2F][PF6] (series A), [M(TpPy)(NO3)2] (series B), and [M(TpPy)2][BPh4] (series C). Crystallographic studies showed that series A and B have a 1:1 metal:TpPy ratio, with the metal ion lying within the podand cavity and the remaining coordination sites occupied by solvent molecules and/or counterions to give 9-coordination (A, with one fluoride and two methanol ligands) or 10-coordination (B, with two bidentate nitrate ligands). The C complexes were prepared in the absence of any coordinating anions and have a 1:2 metal:TpPy ratio with an unusual icosahedral geometry arising from coordination of the 12 nitrogen donors from two interleaved podands. Conductivity studies on the B complexes show that in water the nitrates dissociate to give [M(TpPy)(H2O)q](NO3)2; the relaxivity of [Gd(TpPy)(NO3)2] in water is 4.4 s-1 mM-1, a value comparable to those of clinically useful MRI contrast enhancement agents. Comparison of emission lifetimes of [M(TpPy)(NO3)2] (M = Eu, Tb) in H2O/D2O and CH3OH/CD3OD give values for q, the number of coordinated solvent molecules, of 3.6 (water) and 2.6 (methanol). The C complex [Tb(TpPy)2][BPh4] also has q = 2.6 in methanol, suggesting that partial ligand dissociation allows access of solvent molecules to the metal coordination sphere.

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