| Radial dispersion refers to the dispersive transport process of a solute under a radial flow field, and it is a common phenomenon occurring in many areas. A typical example is the injection of a tracer in an aquifer via an injection well for the parameter estimation. Some chemicals or microbes which are injected into the aquifer to remedy the polluted groundwater, heat exchanges in pumping/injecting wells in geothermal exploration, or CO2injection may also be described as radial dispersion. By carefully reviewing the literature, we found the there were many studies on the radial dispersion; however, many problems still exist. For example, previous studies of the radial dispersion mostly concentrated on the single aquifer or simplified the effect of the adjacent aquitards, ignored the influence of the skin effect of the injection well, assumed that the sorption coefficients were independent with the groundwater flow field, ignored the effect of the aquifer dip on the solute transport. After considering these factors, the movement of the solute may be totally different from the previous case, and we name such solute transport as abnormal radial dispersion. In this study, we focused on the radial dispersion from three aspects, including new analytical solutions, new numerical methods, and new inverse parameter estimation method.The first part mainly concentrated on the newly developed analytical solutions. Generally, analytical solution of the solute transport problems entailed some mathematical approximations which were often more restrictive than those required for the numerical simulation, and therefore many scientists and engineers may preferred to the numerical simulation. However, we found the numerical solutions of advection-dispersion equation may suffer numerical oscillations and artificial dispersion for dominating advection problems which always occurred near the well. Furthermore, another type of numerical errors cannot be reduced easily when modelling solute transport near interfaces of abruptly changing media parameters, such as aquifer-aquitard interface. In this study, we presented several new analytical solutions of radial dispersion, including the new analytical solutions of solute transport in the aquifer bounded by the infinite aquitards and finite aquitards, new solutions considering the skin effect in the aquifer-aquitard system, and new solutions with velocity dependent sorption effect. These new analytical solutions could render parameter estimation easier, more stable, and computationally efficient.Due to the low permeability and effective porosity of the aquitard, the aquitard was generally treated as the nature barrier of the aquifer, and therefore it is reasonable to take it as the aquiclude. Recent studies showed that the total porosity of the aquitard was very high and the absorbing ability to the contaminants was very strong. From the long term, treating the aquitard as the aquiclude may cause unacceptable errors. Chen (1985) presented an analytical solution of the radial dispersion in the aquifer-aquitard system, and subsequently this solute was widely used to simulate the radial dispersion problems. However, this solution was based on some assumptions, for example, assuming the mass flux across the aquifer-aquitard interface as a thickness averaged source/sink term in the governing equation of transport in the aquifer, the advection and dispersion in the aquitard can be ignored, and the vertical dispersion in the aquifer could be ignored. Zhan et al.(2009) pointed out that such assumptions only fit for the problems related to the solute transport in the fracture-matrix system, and named this solution as AA solution. However, some fundamental differences between an aquifer-aquitard system and a fracture-matrix system could make above assumptions invalid for solute transport in an aquifer-aquitard system. Firstly, groundwater leakage across an aquitard is likely to occur if a considerable hydraulic gradient exists across the aquitard, so the advective transport in the aquitard cannot be neglected. Secondly, the thickness of an aquifer is much greater than the fracture aperture and solute across the aquifer thickness is rarely quickly mixed. Therefore, the variation of solute concentration along the aquifer thickness should be considered specifically. In this study, we derived the analytical solutions of solute transport in the aquifer-aquitard system, considering aquitard advection, aquitard dispersion including mechanical dispersion and molecular diffusion, aquifer radial advection, aquifer radial dispersion, aquifer vertical dispersion, the first-order biodegradation or radioactive decay, and retardation in both aquifer and aquitard. We employed Laplace-Fourier method to solve the coupled governing equations of transport in the aquifer and aquitard, supplemented by the continuity of concentration and mass flux across the aquifer-aquitard interface. The new solutions were the extension of the AA solution. Our new analytical solution is almost the same with the numerical solution by COMSOL Multiphysics when the difference of the parameters between the aquifer and aquitard is small. By comparing the new solutions with the AA solutions, one could draw the following conclusions:a) AA solutions had considerably overestimated the mass in the upper aquitard when an upward advection existed in the upper aquitard. b) The rates of dimensionless mass change in the upper aquitard calculated from the new solution and the AA solution increased with time following sub-linear fashions, while the one from the AA solution was universally greater than that calculated from the new solution at any given time. c) The times corresponding to the peak values of RTDs for the AA solutions and the new solution were almost the same. From a groundwater remediation perspective, utilization of the AA solution or the expanded AA solution may lead to much longer design time and higher cost for cleaning up the aquifer-aquitard system.Many studies demonstrated that the aquitard thickness is much high than the aquifer, however, Bradbury et al. found that the aquitard is very thin in South Carolina in USA. Rezaei et a finite-difference solution. To test the VDS effect, we derived a new benchmark solution with spatially averaged velocity-independent sorption (VIS) rates. Results showed that the difference between VDS and VIS solutions were obvious, and VDS solutions were greater than their VIS counterparts at the early stage. A larger detachment rate could shorten the time required for reaching the equilibrium whereas a larger attachment rate had the opposite effect.Laplace transform method has widely applied to solve the problems related the radial dispersion, however, the inverse Laplace transform was difficult to be analytically carried out since the solution in Laplace domain were very complex, and then a large number of different numerical methods were called in. By reviewing the literature, we found that there were many methods for numerical inverse Laplace transform, such as the Stehfest method, the de Hong method, the Honig-Hirdes method, the Schapery method, the Talbot method, the Weeks method, the Simon method and the Zakian method. However, many studies showed that the numerical inverse Laplace transform was generally an ill-posed problem, and there was no universal method which worked well for all problems. Meanwhile, there were several the optimal free parameters for each method, in which the values were given by empirical method. Presently, many scientists and engineers wanted to know which method is accurate and effiective for the radial dispersion? Did the empirical parameters still work for the radial dispersion? To answer these question, we summarized eight numerical inverse Laplace transform methods, of which eight were selected to be evaluated in the numerical accuracy and the computational efficiency for the radial dispersion problems, including the Stehfest method, the de Hong method, the Honig-Hirdes method, the Schapery method, the Talbot method, the Weeks method, the Simon method and the Zakian method. The finite-element solutions by the COMSOL Multiphysics package were introduced as the benchmarks. Meanwhile, we investigated the optimal free parameters of each method, including the number of terms used in the summation and the numerical tolerance of approaching pole, and so on. We found some previous recommended values of the free parameters did not work very well, especially for the advection-dominating problems. Using trial-and-error method, we renewed the recommended value table. We concluded that the Schapery method might be not fit for the radial dispersion problems, while the de Hong method, the Talbot method, and the Simon method worked very well, regardless of the dispersion-dominating or advection-dominating problems. The Weeks method can be used to solve the dispersion-dominating problems, not the advection-dominating problems. The Stehfest method, the Honig-Hirdes method, and the Zakian method were recommended to be applied for the inverse Laplace transform only for the dispersion-dominating problems.The second part mainly focused on the new numerical solutions for groundwater flow and solute transport sloping aquifer, and developed MODFLOW-SP and MT3DMS-SP packages numerical simulation based on the MODFLOW and MT3DMS packages. For the radial dispersion, rectangular grid system of the finite-difference method was less effective than the triangular grid system. Since the MODFLOW and MT3DMS packages were very powerful, which based on the finite-difference method, they were widely used to solve the problems of groundwater and solute transport around the well in the aquiferUsing numerical simulation, non-horizontal-model-layer (NHML) grid system was more al.(2013) established a model in which the aquifer was bounded the adjacent thin aquitards, assuming the flow was uniform in the aquifer. In this study, we established a new model of radial dispersion in the aquifer-aquitard system, and derived a new analytical solution. This model considered the aquitard advection, aquitard dispersion including mechanical dispersion and molecular diffusion, aquifer radial advection, aquifer radial dispersion, aquifer vertical dispersion, the first-order biodegradation or radioactive decay, and retardation in both aquifer and aquitard. By comparing the new solution with the pervious solution with infinite aquitard, one could see:a) The aquitard thickness could affect the solute transport in the aquifer when the aquitard/aquifer thickness ratio was smaller than2.0. b) The difference between the new solution and the pervious solution became more obvious with smaller aquitard-aquifer thickness ratio. Such difference between the solutions in the upper aquitard was more obvious than the one in the lower aquitard. c) Previous solutions overestimated the concentration at the aquifer-aquitard interface for the smaller aquitard-aquifer thickness ratio. d) The BTC value was smaller when the values of aquitard-aquifer thickness ratio were smaller. The time, that solute transport became steady state, was smaller for the smaller aquitard-aquifer thickness ratio. Although the new solution with thin aquitard was an extension of the solution with infinite aquitard, the format of latter was much simple, which was recommended when the difference between two solution can be ignored.Regardless of the injection well or the observed well, the aquifer might be disturbed around a well, which was produced by an artificial filter-pack of the well, or by disturbing the aquifer fonnation surrounding the well screen during the well drilling and installation procedure. Such zone was named skin zone. The effect of the well skin could be either positive or negative. The positive skin zone refers to the permeability of the skin zone smaller than that of the aquifer. The negative skin effect is that the permeability of such a skin zone is greater than that of the aquifer. In the previous studies, the skin effect was ignored. In this studies, we investigated radial reactive transport in an aquifer-aquitard system considering the skin effect, where the thickness of the skin zone was assumed to be a finite distance. The results showed:a) The values of the BTCs increased with the increasing of the ratio of radial dispersion in the skin zone and the formation zone (Ω), while the values of the RTDs decreased with the increasing Ω. b) The time RTD arriving the peak value was almost the same for different Ω. c) When the flow direction was upward, the concentration in the aquifer and the upper aquitard increased with the increasing Ω and the thickness of the skin zone, while the values of the concentration in the lower aquitard were not sensitive to Ω and the thickness of the skin zone. The positive skin effect is more important to the solute transport, and cannot be ignored.Different from the surface water, the contaminants transport in the aquifer may adsorb in porous media. Previous studies on solute transport assumed the sorption rates to be constant, while numerous experimental evidence demonstrated that sorption rates might linearly increase with the groundwater flow velocity for some microbes. In this study, we established a mathematical model of reactive solute transport under the radial flow field, where the sorption effect was described by a hybrid equilibrium-kinetic model with velocity-dependent sorption (VDS) rates. A new semi-analytical solution was obtained by combining the piecewise linear fitness and the Laplace transform methods, and the accuracy of this new solution was checked by accurate than the horizontal-model-layer (HML) grid system to describe the fluid flow in a sloping aquifer on the basis of MODFLOW-2000and MT3DMS. However, the finite-difference scheme of NHML was based on the Dupuit-Forchheimer assumption that the streamlines were horizontal for MODFLOW-2000, which was acceptable for slope less than0.10. MT3DMS also contained the similar assumptions which may result in great errors for the large slope angle. In chapter VII, We investigated the mechanics of the groundwater and solute transport in the sloping aquifer, and developed new packages, MODFLOW-SP and MT3DMS-SP, for solving such problems.In this study, the assumptions involved in existing analytical and numerical solutions of groundwater flow in an unconfined sloping aquifer were carefully analyzed. The problems and potential numerical errors associated with MODFLOW-2000were also discussed in great details. We proposed a new numerical scheme based the non-horizontal-model-layer grid system considering the characteristics of the sloping base. This new numerical scheme was specifically programed in a SLOPE package that was integrated into the MODFLOW-2000program to create MODFLOW-SP which could handle the problems related to sloping aquifers. The solutions by MODFLOW-SP agreed with a widely used numerical benchmark solution of Mac Cormack (2003) very well. The difference between MODFLOW-2000and MODFLOW-SP was small but maybe non-negligible when the aquifer slope was0.27, while the difference was obvious when the aquifer slope was0.50. Under the steady-state flow condition, the groundwater flow direction was almost parallel to the aquifer base except in narrow regions near the left and right boundaries. MODFLOW-SP can be used to predict the hydraulic head along the E’-V-F’profile very well (see Fig.7-2B). The errors associated with above two constrains used in MODFLOW-SP were negligible when the slope was smaller than0.50. Such errors were small but noticeable when the slope was0.75, and they became significant when the slope was1.0.Meanwhile, we proposed a new finite-difference scheme of solute transport related to the sloping aquifer. This new numerical scheme was specifically programed in a SLOPE package that was integrated into the MT3DMS program to create MT3DMS-SP which could handle the problems related to sloping aquifers. The difference between MT3DMS and MT3DMS-SP was obvious when the aquifer slope was larger than0.50. MT3DMS-SP could be used to predict the hydraulic head along the E’-V-F’profile very well, and the errors associated were negligible when the slope was smaller than0.75.When the slope of the aquifer was zero, the MODFLOW-SP and MT3DMS-SP packages reduced to MODFLOW and MT3DMS packages, respectively. These two packages can be used to solve the problems in either unconfined aquifer or confined aquifer.The third part mainly concentrated on the new method for inverse parameter estimation. To study contaminant transport in groundwater, an essential requirement was a robust and accurate estimation of the transport parameters such as dispersion coefficient. The commonly used methods in dispersion coefficient estimation using the breakthrough curves (BTCs) data included inverse error function methods (IEFM), genetic algorithm, simulated annealing method, and so on. Many studies showed that optimized methods were very sensitive to the initial values, time consuming, or entered into local optimum, and therefore IEFM was widely applied. In this study, we proved that IEFM may cause unacceptable errors, and the main reason was that the random error in the measured concentrations, which might be described by a normal distribution, would no longer follow the normal distribution after the IEFM transformation.In this study, we proposed a new method using the weighted least squares method (WLSM) to estimate the dispersion coefficient and seepage velocity. The weights were calculated based on the slope of the observed BTCs. We tested the new method against other methods such as genetic algorithm and CXTFIT program and found great agreement. This new method acknowledged different characteristics of solute transport at early, intermediate, and late time stages and divided BTCs into three sections for analysis. The developed method was applied to interpret three column tracer experiments by introducing continuous, constant-concentration of sodium chloride (NaCl) into columns filled with sand, gravel, and sand-gravel media. This study showed that IEFM performed well only when the observed data points were located in the linear (intermediate time) section of BTCs; it performed poorly when data points were in the early and late time stages. The new WLSM method, however, performed well for data points scattering over the entire BTCs and appeared promising in parameter estimation for solute transport in a column.In summary, we conducted our research of radial dispersion from the following three aspects, new analytical solutions, new numerical solutions and new method for inverse parameter estimation. As for the analytical solutions, we developed some analytical solutions for different conceptual models using Laplace-Fourier transform methods. Based on the experiment data, we established a new model describing the sorption processes, and we proposed a new analytical method for the radial dispersion with new scorpion model. We summarized eight inverse numerical Laplace transform methods, and investigated the free parameter values in each method for radial dispersion. In the numerical simulation, we presented a new numerical scheme for groundwater flow and solute transport in the sloping aquifer. We developed two new packages, MODFLOW-SP and MT3DMS-SP, based on the original codes of MODFLO-2000and MT3DMS. Therefore, MODFLOW-SP and MT3DMS-SP could also be used to solve the problems related to the radial dispersion. For the inverse parameter estimation, we developed a new method. This study showed that IEFM performed well only when the observed data points were located in the linear section of BTCs; it performed poorly when data points were in the early and late time stages. The new method performed well for data points scattering over the entire BTCs and appeared promising in parameter estimation for solute transport in a column. |
PDF Full Text Request