A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

This paper employs the unified transform, also known as the Fokas method, to solve the advection-dispersion equation on the half-line. This method combines complex analysis with numerics. Compared to classical approaches used to solve linear partial differential equations (PDEs), the unified transform avoids the solution of ordinary differential equations and, more importantly, constructs an integral representation of the solution in the complex plane which is uniformly convergent at the boundaries. As a consequence, such solutions are well suited for numerical computations. Indeed, the numerical evaluation of the solution requires only the computation of a single contour integral involving an integrand which decays exponentially fast for large values of the integration variable. A novel contribution of this paper, with respect to the solution of linear evolution PDEs in general, and the implementation of the unified transform in particular, is the following: using the advection-dispersion equation as a generic example, it is shown that if the transforms of the given data can be computed analytically, then the unified transform yields a fast and accurate method that converges exponentially with the number of evaluations N yet only has complexity O(N). Furthermore, if the transforms are computed numerically using M evaluations, the unified transform gives rise to a method with complexity O(NM). Results are successfully compared to other existing solutions.F.P.J. de B. acknowledges the support from the National Science Foundation Grant No. EAR-1654009. M.J.C. is supported by EPSRC Grant No. EP/L016516/1. A.S.F. is supported by EPSRC, UK, via the senior fellowship

Coupled flow and contaminant transport modeling in large watersheds

A hybrid surface/subsurface flow and transport model is developed that blends distributed parameter models with simpler lumped parameter models. The hybrid model solves the channel flow and saturated groundwater flow domains in continuous time using fully distributed physically-based formulations. This system is supplemented with the overland flow and unsaturated groundwater flow that uses lumped parameter descriptions in discrete time. In the proposed model, a one-dimensional channel flow model is dynamically coupled with a two-dimensional vertically-averaged groundwater flow model along the river bed. As an alternative to the commonly applied iterative solution technique, a so-called simultaneous solution procedure is developed to provide a better understanding to the coupled flow problem. This new methodology is based on the principle of solving the two flow domains within a single matrix structure in a simultaneous manner. In addition to the flow model, a coupled contaminant transport model is also developed to simulate the migration of contaminants between surface and subsurface domains. The contaminant transport model dynamically couples a one-dimensional channel transport model with a two-dimensional vertically-averaged groundwater transport model. The coupling is performed at the river bed interface via advective and dispersive transport mechanisms. A modified extension of the proposed simultaneous solution procedure is also implemented to solve the coupled contaminant transport problem. The dynamic coupling provides the much needed understanding for the continuity of contaminants in strongly interacting surface/subsurface systems such as a river and an unconfined aquifer. The coupled flow and transport models are applied to the lower Altamaha watershed in southern Georgia. The flow model is used to perform simulations of hydrologic and hydraulic conditions along the river and in the dynamically linked surfacial aquifer. The model predicted the flood patterns including the magnitude of peaks and their arrival times with accuracy. Under the given flow conditions, the transport model is then implemented to test alternative contaminant transport patterns both in the river and within the aquifer. It has been found that the channel network would serve as a conduit for rapid transport of contaminants within the aquifer to large distances in small time frames.Ph.D.Committee Chair: Dr. Mustafa Aral; Committee Member: Dr. Paul Work; Committee Member: Dr. Philip Roberts; Committee Member: Dr. Terry Sturm; Committee Member: Dr. Turgay Uze

Dispersion in Alluvial River

River pollution is the contamination of river water by pollutant being discharged directly or indirectly on it. Depending on the degree of pollutant concentration, subsequent negative environmental effects such as oxygen depletion and severe reductions in water quality may occur which affect the whole environment. River pollution can then cause a serious threat for fresh water and as well as the entire living creatures. Dispersion in natural stream is the ability of a stream to dilute soluble pollutants. Different types of pollution, such as accidental spill of toxic chemicals, industrial waste, intermittent discharge from combined sewer overflows and temperature variations produced by thermal outflows, may generate a cloud whose longitudinal spreading strongly affects the pollutant concentration dynamics. Pollutants discharging form a point source is easier to control where as pollutant discharging from non point sources arehardlycontrollable and may represent severe threat to the river ecosystem. The longitudinal dispersion coefficient is used to describe the change in characteristics of a solute cloud from an initial state of high concentration and low spatial variance to a downstream state of lower concentration and higher spatial variance. Therefore, in order to correctly estimate the degree of pollutionwithin a stream and ensure an efficient and informed management of riverine environments,a reliable estimationof the dispersion withinthe stream is a crucial concern. The objective of my research is to develop a mathematical model for determining the dispersion in alluvial river. In order to achieve the goal, a model has been developed which provides an analytical relation for the prediction of the dispersion coefficient in natural streams, given the planimetric configuration of the river and the relevant hydrodynamic and morphodynamic parameters (i.e., width to depth ratio, the sediment grain size, scaled with the flow depth, the Shields stress). One of the most striking features of alluvial rivers is their tendency to develop regular meandering plan forms. Their geometry is in fact characterized by a sequence of symmetrical curves which amplify over time due to erosion processes at the outer bank and deposition at the inner bank. This planimetric pattern affects both the hydrodynamics of the river and the distribution of bed elevations, as well as its hydraulic response, as the average bed slope is progressively reduced along with the flow cross sections. The flow filed that establishes in meandering rivers has clearly a great relevance on the behavior of the pollutant cloud and hence on the dispersion that drives its microscopic evolution. To develop a dispersion coefficient predicting model, the analytical models of flow field establishing in the cross section of a straightriver [TubinoansColombini, 1992] and of a meandering river [Frascati and Lanzoni, 2013] aredeveloped. The two dimensional mass balance equation governing the dynamics of a pollutant is then solved using asymptoticexpression and Morse and Feshbach[1953] formalism. Finally, using the two dimensional spatial distributions of the concentration, the flow depth and the velocity, the dispersion coefficient are obtained. For straight rivers the cross-sectional velocityand the theoretically predicted dispersion coefficients with the field datacollected by Godfrey and Frederick (1970)in two rivers (Clinch River, Copper Creek). The comparison is reasonably good. The performance of the model is also tested with reference to the predictions provided by the model proposed by Deng (2001). The resultant model is found to give prediction closer to 80% of the experimental data,a much better performance agreement with respect to the model of Deng (2001). The results of the model developed to estimate the dispersion coefficients in meandering river, have been compared with the experimental data available in experimental and referring to six different rivers. Also in this case the agreement between the dispersion coefficient predicted theoretically and those calculated on the basis of tracer tests is quite good and better than that ensured by the other theoretical and empirical predictors available in literatur

Environmental Impact Assessment of Liquid Waste Ponds in Uranium Milling Installations

A detailed environmental impact assessment is required in many countries when considering the disposal of wastes containing radioactive materials. In this paper we present hybrid numerical-analytical solutions for the subsurface transport of radioactive contaminant decay chains that may be used for such an assessment. The model involves the advective–dispersive transport of multiple radionuclide species within separate but coupled saturated and unsaturated soil domains. The resulting partial differential equations were solved using the Generalized Integral Transform Technique to yield analytical expressions for the concentration distributions versus distance, and analytical or numerical solutions as a function of time. The potential of the hybrid modeling approach is illustrated by means of an environmental impact assessment of an uranium milling liquid waste pond near Caetité, Brazil. Calculated radionuclide concentration distributions were for this purpose used in subsequent radiation dose calculations. Several scenarios were analyzed, including operational, human intrusion, farmer/road builder, and leaching scenarios. Numerical results show that for all scenarios analyzed the total dose is less than the Brazilian regulatory dose limit of 0.3 mSv/year above natural background for a period of 10,000 years.Indisponível

Development of watershed-based modeling approach to pollution source identification

Identification of unknown pollution sources is essential to environmental protection and emergency response. A review of recent publications in source identification revealed that there are very limited numbers of research in modeling methods for rivers. What’s more, the majority of these attempts were to find the source strength and release time, while only a few of them discussed how to identify source locations. Comparisons of these works indicated that a combination of biological, mathematical and geographical method could effectively identify unknown source area(s), which was a more practical trial in a watershed. This thesis presents a watershed-based modeling approach to identification of critical source area. The new approach involves (1) identification of pollution source in rivers using a moment-based method and (2) identification of critical source area in a watershed using a hydrograph-based method and high-resolution radar rainfall data. In terms of the moment-based method, the first two moment equations are derived through the Laplace transform of the Variable Residence Time (VART) model. The first moment is used to determine the source location, while the second moment can be employed to estimate the total mass of released pollutant. The two moment equations are tested using conservative tracer injection data collected from 23 reaches of five rivers in Louisiana, USA, ranging from about 3km to 300 km. Results showed that the first moment equation is able to predict the pollution source location with a percent error of less than 18% in general. The predicted total mass has a larger percent error, but a correction could be added to reduce the error significantly. Additionally, the moment-based method can be applied to identify the source location of reactive pollutants, provided that the special and temporal concentrations are recorded in downstream stations. In terms of the hydrograph-based method, observed hydrographs corresponding to pollution events can be utilized to identify the critical source area in a watershed. The time of concentration could provide a unique fingerprint for each subbasin in the watershed. The observation of abnormally high bacterial levels along with high resolution radar rainfall data can be used to match the most possible storm events and thus the critical source area

Integrated Environmental Modelling Framework for Cumulative Effects Assessment

Global warming and population growth have resulted in an increase in the intensity of natural and anthropogenic stressors. Investigating the complex nature of environmental problems requires the integration of different environmental processes across major components of the environment, including water, climate, ecology, air, and land. Cumulative effects assessment (CEA) not only includes analyzing and modeling environmental changes, but also supports planning alternatives that promote environmental monitoring and management. Disjointed and narrowly focused environmental management approaches have proved dissatisfactory. The adoption of integrated modelling approaches has sparked interests in the development of frameworks which may be used to investigate the processes of individual environmental component and the ways they interact with each other. Integrated modelling systems and frameworks are often the only way to take into account the important environmental processes and interactions, relevant spatial and temporal scales, and feedback mechanisms of complex systems for CEA. This book examines the ways in which interactions and relationships between environmental components are understood, paying special attention to climate, land, water quantity and quality, and both anthropogenic and natural stressors. It reviews modelling approaches for each component and reviews existing integrated modelling systems for CEA. Finally, it proposes an integrated modelling framework and provides perspectives on future research avenues for cumulative effects assessment

WATER QUALITY IMPACT OF NON-POINT SOURCE CONTAMINANTS IN SMALL TIDAL RIVERS

An integrated numerical modeling methodology was developed for analysis of non-point contaminant production and water quality impacts in estuarine receiving waters. The methodology was calibrated against contaminant source observations and estuarine data. Contaminant source loading is based on the runoff curve number technique combined with a partial contributing area approach. Sediment washoff is used as an indicator of contaminant quantities. Hydrographs and contaminant load time series are introduced into a set of one-dimensional estuary models. Freshwater flow and tidal height are forcing functions for dynamic computation of velocity and water surface profiles. The effect of longitudinal salinity distribution is included through a closed form solution of governing differential equations. Sensitivity of the models to key parameters was investigated through numerical experiments. Runoff volume and contaminant weight were greatest on impervious areas of the watershed. Hydrologic analysis could be calibrated by a single parameter. Contaminant load was primarily affected by sediment transport coefficients. Hydrodynamic model results demonstrated that the key components of the dynamic force balance are surface slope and acceleration of the water mass. Friction was an important but smaller component. The hierarchy of physical processes affecting contaminant dispersion was dominated by tidal currents. Density driven circulation is most significant in deep sections with a large longitudinal density gradient. Calibration runs and application to analysis of non-point source water quality impacts in the Oyster River, New Hampshire, were performed. Both salinity and dye distributions were adequately simulated. Investigation of receiving water impacts in the Oyster River Estuary revealed that following major storm events, BOD concentration and dissolved oxygen deficit can be dominated by the effect of non-point source contaminants

Minimum environmental impact discharging

Many contaminants exhibit decay. Decay mechanisms include consumption by bacteria or radioactive decay (temporal decay uniform across the flow), heat loss or evaporation through the surface (decay decreasing with depth), and break up by turbulence (decay proportional to the product of velocity and depth). This thesis investigates how the decay of pollutants in a river effects the dilution process and the selection of discharge siting to achieve minimum environmental impact. For a non-symmetric river with non-reversing flow, exact solutions are presented that illustrate the effect on the optimal position for a steady discharge of cross-channel variation in the decay (uniform, decreasing or increasing with depth). The optimal position is shifted to deeper or to shallower water accordingly as the temporal decay divided by flow speed decreases or increases with water depth. When advection dominates diffusion, there are special directions (rays) along which information is carried. For steady, unstratified, plane parallel flow, the effects of decay are allowed for in specifying these special directions. Two special cases are considered. Firstly, for a smoothly varying depth, a general result has been derived for the curvature of the rays as effected by spatial non-uniformity in decay, mixing, flow speed and flow direction. Secondly, for discontinuous variations in depth, diffusivity, velocity and decay, approximate concentration formulae are derived. Ray bending indicates that the downstream propagation of pollutant is principally in the low-decay region. Computational results are used to give pictorial illustration of the concentration distributions and of the difference between discharging at non-optimal and optimal sites.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

개수로에서 2차원 분산계수 추적법 및 원격관측 추적자 실험에의 적용

학위논문(박사)--서울대학교 대학원 :공과대학 건설환경공학부,2020. 2. 서일원.The depth-averaged two-dimensional (2D) advection-dispersion equation (ADE) has been widely used to analyze the mixing phenomenon of the various dissolved and suspended matters in river systems. In depth-averaged 2D ADE, dispersion coefficients are essential parameters to explain the spreading of pollutant clouds, caused by complexities of flow structures and river environments. The dispersion coefficients in 2D ADE can be calculated by the routing based-observation methods using the tracer test data. The 2D stream-tube routing procedure (2D STRP) has been the only method to calculate both longitudinal and transverse dispersion coefficients of 2D ADE simultaneously for the transient concentration conditions. In this study, the limitations of 2D STRP were quantitatively analyzed using the hypothetically generated data. Besides, the new routing-based observation method (2D STRP-i) and the remote sensing-based experimental framework for tracer tests were developed to overcome the limitations of existing determination methods for dispersion coefficients. The performance of existing 2D STRP was evaluated in terms of the variation of Peclet number, and the spatially varied velocity distributions. The results of the evaluations showed that the existing 2D STRP well provided the temporal distribution of tracer concentration in the high Peclet number, but it could not reproduce the reliable results when the tracer clouds reached wall boundaries and the Peclet number decreased. The new routing-based observation method (2D STRP-i) was developed to improve the drawback of the existing method. The 2D STRP-i derived from the 2D ADE in the orthogonal curvilinear coordinate system, assuming the steady-state flow condition. 2D STRP-i could adequately reproduce the reliable temporal distribution even if the effect of wall boundaries was significant. A remote sensing based-experimental framework for tracer tests was developed to acquire the spatio-temporal concentration distribution of tracer clouds in open channel flows. Tracer tests using Rhodamine WT were conducted in the large-scaled experimental channels to collect the RGB images using a commercial digital camera mounted on a UAV and the concentration of Rhodamine WT using in-situ fluorometric probes. The empirical relationship between the image data and the Rhodamine WT concentration data was estimated using an artificial neural network (ANN) model. The acquired spatio-temporal concentration distributions by the proposed method gave general as well as detailed views of the tracer cloud moving dynamically in open channel flows that cannot be easily observed using conventional in-situ measurements. The 2D STRP-i was applied to the remotely measured tracer data to calculate the dispersion coefficients. The Latin Hypercube Simulation was adopted to determine the optimum values of dispersion coefficients. The results of simulations showed that the RMSE distributions were non-convex with many local minima. In addition, the optimal dispersion coefficients were differently found according to the evaluation functions. In this study, the multiple evaluation indices were selected to determine the dispersion coefficients more robustly. The results showed that the longitudinal dispersion coefficients were similarly determined by both 2D STRP and STRP-i, while the existing 2D STRP generally underestimated the values of transverse dispersion coefficients compared to the results of 2D STRP-i.2차원 수심평균 된 이송-분산 방정식은 하천환경에서 다양한 용존성 오염물질의 혼합현상을 모의하기 위해 널리 활용되어 왔다. 수심평균 된 2차원 이송-분산 방정식의 분산계수는 3차원 이송-확산 방정식에 대해 수심 평균을 취하는 과정에서 연직방향상 질량이송의 편차를 Fick의 법칙에 의해 모형화함에 따라 생성된다. 2차원 혼합모형에서 분산계수는 하천의 전단흐름에 의해 야기되는 오염물질의 퍼짐 현상을 표현하는 중요한 인자로서 작용하므로 정교한 오염물질 혼합거동을 모의하기 위해서는 적절한 분산계수를 산정하는 것이 필수적이다. 분산계수를 실험적으로 산정하는 방법으로는 크게 모멘트법과 추적법으로 나뉘며, 비정상상태의 혼합거동에 대해 종방향 및 횡방향 분산계수를 모두 산정할 수 있는 방법은 추적법 계열의 2차원 유관추적법(2D STRP)이 유일하다. 본 연구에서는 해석해 및 수치해 기반의 자료를 바탕으로 2D STRP의 적용범위 및 정확도를 정량적으로 분석하였다. 분석된 정보를 바탕으로 2D STRP 및 기존의 접촉식 계측 기반 추적자실험법의 한계를 극복하기 위해 개선된 2차원 유관추적법(2D STRP-i) 및 원격탐사 기반의 추적자 실험법을 개발하였다. 기존 2D STRP의 성능은 다양한 Peclet 수의 범위 및 비균등 유속분포에 대해서 평가되었다. 평가 결과, 2D STRP는 Peclet 수가 높은 조건일수록 농도분포의 예측 정확도가 상승하였으나, 추적자의 농도분포가 하안 경계에 도달하는 경우에는 부정확한 결과를 초래하는 것으로 나타났다. 본 연구에서는 기존 2D STRP의 한계를 보완하여 더욱 정확한 분산계수를 산정하고자, 하안 경계면 조건을 고려한 2차원 유관추적법(2D STRP-i)을 개발하였다. 2D STRP-i는 직교-곡선좌표계 기반의 2차원 이송-분산 방정식을 바탕으로 횡방향 유속분포 및 하안 경계조건을 고려하여 유도되었다. 제안된 2D STRP-i는 공간적으로 상이한 이송효과 및 하안경계 조건을 적절히 반영함으로써 기존의 2D STRP에 비해 농도분포의 예측 정확도를 월등히 개선시키는 것으로 평가되었다. 또한, 본 연구에서는 시공간적으로 높은 해상도의 추적자 농도 자료를 취득하고자 원격탐사기반의 추적자실험법을 개발하였다. 원격탐사기반의 추적자실험법은 소형 무인항공기에 장착된 상용 디지털카메라의 각 밴드별 화소강도의 변화를 이용하여 하천에 주입된 형광색소(Rhodamine WT)의 농도를 측정하는 방법이다. 추적자실험을 통해 취득된 형광색소의 농도 및 디지털 영상자료를 이용하여 인공신경망 기반의 경험적 회귀모형을 구축하였으며, 학습된 회귀모형은 결정계수 0.9이상의 우수한 정확도를 보이는 것으로 나타났다. 원격탐사기반 추적자실험법에 의해 측정된 농도자료에 대해 기존의 2D STRP와 본 연구에서 제안된 2D STRP-i를 적용하여 분산계수를 산정 및 비교하였다. 최적 분산계수의 결정은 Latin Hypercube 모의법을 이용하였으며, 최적의 분산계수는 실측된 농도분포와 예측된 농도분포 간 오차를 평가하는 기준에 따라 상이한 값으로 산정되었다. 따라서, 본 연구에서는 다양한 통계적 특성을 만족시키는 강건한 분산계수를 산정하기 위해 정규화된 다수의 평가지표를 고려하여 분산계수의 최적값을 결정하였다. 기존 2D STRP 및 제안된 2D STRP-i에 의해 산정된 분산계수값을 비교한 결과, 기존 2D STRP 및 2D STRP-i에 의해 산정된 종방향 분산계수는 유사한 경향을 보이는 것으로 나타났으나, 횡방향 분산계수는 하안 경계면의 효과에 의해 기존 2D STRP의 결과가 과소산정되는 것으로 나타냈다.1. Introduction 1 1.1 Background and necessities of study 1 1.2 Objectives and methodology 7 2. Theoretical backgrounds 10 2.1. Pollutant transport model in the natural river 10 2.1.1 Necessity of 2D pollutant transport models 10 2.1.2 2D ADE in the cartesian coordinate system 15 2.1.3 2D ADE in the natural coordinate system 22 2.2 Determination of the dispersion coefficient from tracer test data 26 2.2.1 Experimental methods for conventional tracer tests 26 2.2.2 The routing procedure for 1D mixing models. 29 2.2.3 Routing procedure for 2D dispersion coefficients 34 2.3 Remote sensing techniques for water environments 41 3. Implementation of a numerical tool for 2D ADE 46 3.1 Outline of the numerical tool 46 3.2 Discretization of governing equation 47 3.2.1 Interpolation schemes for advection term 51 3.2.2 Interpolation schemes for dispersion term 60 3.2.3 Discretization for the time derivative 63 3.2.4 Conditions for numerical stability 66 3.2.5 Boundary conditions 70 3.2.6 Initial conditions 75 3.3 Evaluation of numerical model 76 4. Evaluation of 2D stream-tube routing procedure 91 4.1 Overview of the evaluation procedure for 2D STRP 91 4.2 Evaluation of 2D STRP in the condition of uniform flow 97 4.3 Evaluation of 2D STRP in the condition of spatially varied flows 112 5. Development of the improved 2D STRP 121 5.1 Development of the improved 2D STRP 121 5.2 Evaluation of the improved 2D STRP 131 6. Development of remote sensing-based tracer tests 134 6.1 Outline of the experimental procedure 134 6.2 Acquisition of aerial imagery 137 6.3 Preprocessing of aerial imagery 139 6.4 ANN model of retrieving concentration from image data 141 6.5 Application of remote sensing-based tracer test in REC channels 146 6.5.1 Tracer tests based on in-situ measurements in REC 146 6.5.2 Acquisition of aerial imagery in REC 151 6.5.3 Retrieval of tracer cloud distribution from image data 154 6.5.4 Spatio-temporal distribution of tracer clouds 166 6.5.5 Assessment of cross-applicability 170 7. Application of the improved 2D STRP 174 7.1 Pre-processing of remotely sensed data 174 7.2 Calculation of 2D dispersion coefficients 181 8. Conclusions 199 8.1 Conclusions of this study 199 8.2 Future study 203 References 205 Appendix 217 Appendix A. Program code of numerical model for 2D ADE 217 Appendix B. Tracer test results in EXP-A315 223 Appendix C. Tracer test results in EXP-A317 258 국문초록 300Docto