Eulerian-Lagrangian formulation of the equations for groundwater denitrification using bacterial activity

A mathematical model for groundwater denitrification using bacterial activity is presented. The model includes the momentum and mass balance equations for water and nitrogen, substrate and bacteria, and chemical reactions between them. The resulting multiphase, multicomponent, flow and transport governing equations, are coupled and nonlinear. A Eulerian-Lagrangian formulation of the equations is developed. The water and gas flow and transport equations are split into forward advection along characteristics, and a residual at a fixed frame of reference. Discontinuities, sharp fronts and steep gradients of the dependent variables are imposed on the advection mode and solved exactly. It is believed that this novel method will avoid numerical artifacts for the solution of the multiphase flow equations (e.g., upstream permeability) and numerical dispersion for the transport equation.     

THE EFFECTS OF INACCURATE INPUT PARAMETERS ON DEPOSITION OF SUSPENDED SEDIMENT

I.~OntjcnONEngineeringPredictionsofsedimenttransPOrtinopen-Channelsystems,especiallyifunsteadyflowconditionmustbeconsidered,isfraughtwithambigUityduetospatialandtemporalvariability,measurementerrors,lumtedsamplingoftheParameters,bounceandscallconditionS,andsink/sourcetenns.Duringrecentyearsmanysophisticatedmathematicalmodelshavebecomeavailabletocalculateunsteadyflowinone,tWoorthreedimensionS.ThesemodelswillbecoupledwithtranSportequationSforbedloadorsuSpendedloadtocalculateforexamplelongt…     

Non-isothermal soil water transport and evaporation

A detailed model was formulated to describe the non-isothermal transport of water in the unsaturated soil zone. The model consists of the coupled equations of mass conservation for the liquid phase, gas phase and water vapor and the energy conservation equation. The water transport mechanisms considered are convection in the liquid phase, and convection, diffusion and dispersion of vapor in the gas phase. The boundary conditions at the soil–atmosphere interface include dynamical mass flux and energy flux that accounts for radiation transport. Comparison of numerical simulations results with published experimental data demonstrated that the present model is able to describe water and energy transport dynamics, including situations of low and moderate soil moisture contents. Analysis of field studies on soil drying suggests that that dispersion flux of the water vapor near the soil surface, which is seldom considered in soil drying models, can make a significant contribution to the total water flux.     

A kinetic approach for simulating redox‐controlled fringe and core biodegradation processes in groundwater: model development and application to a landfill site in Piedmont,Italy

A three‐dimensional model for predicting redox controlled, multi‐species reactive transport processes in groundwater systems is presented. The model equations were fully integrated within a MODFLOW‐family reactive transport code, RT3D. The model can simulate organic compound biodegradation coupled to different terminal electron acceptor processes. A computational approach, which uses the spatial and temporal distribution of the rates of different redox reactions, is proposed to map redox zones. The method allows one to quantify and visualize the biological degradation reactions occurring in three distinct patterns involving fringe, pseudo‐core and core processes. The capabilities of the numerical model are demonstrated using two hypothetical examples: a batch problem and a simplified two‐dimensional reactive transport problem. The model is then applied to an unconfined aquifer underlying a leaking landfill located near the city of Turin, in Piedmont (Italy). At this site, high organic load from the landfill leachate activates different biogeochemical processes, including aerobic degradation, denitrification, manganese reduction, iron reduction, sulfate reduction and methanogenesis. The model was able to describe and quantify these complex biogeochemical processes. The proposed model offers a rational framework for simulating coupled reactive transport processes occurring beneath a landfill site. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytical Solutions for Water Flow and Solute Transport in the Unsaturated Zone

Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge and White nonlinear model to solve the Richards’ equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advection-dispersion equation. The method of characteristics is adopted to determine the location of a solute front in the unsaturated zone. The dispersion component is incorporated into the final solution using a singular perturbation method. The formulation of the analytical solutions is simple, and a complete solution is generated without resorting to computationally demanding numerical schemes. Indeed, the simple analytical solutions can be used as tools to verify the accuracy of numerical models of water flow and solute transport. Comparison with a finite-element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one-fourth the capillary length scale of the porous medium. However, when numerically solving the solute transport equation at this level of discretization, numerical dispersion and spatial oscillations were significant.     

Sediment yield modelling for small agricultural catchments: land‐cover parameterization based on remote sensing data analysis

Vegetation and soil properties and their associated changes through time and space affect the various stages of soil erosion. The island of Ishigaki in Okinawa Prefecture, Japan is of particular concern because of the propensity of the red‐soil‐dominated watersheds in the area to contribute substantial sediment discharge to adjacent coastal areas. This paper discusses the application of remote sensing techniques in the retrieval of vegetation and soil parameters necessary for the distributed soil‐loss modelling in small agricultural catchments and analyses the variation in erosional patterns and sediment distribution during rainfall events using numerical solutions of overland flow simulations and sediment continuity equations. To account for the spatial as well as temporal variability of selected parameters of the soil‐loss equations, a method is proposed to account for the variability of associated vegetation cover based on their spectral characteristics as captured by remotely sensed data. To allow for complete spatial integration, modelling the movement of sediment is accomplished under a loose‐coupled GIS computational framework. This study lends a theoretical support and empirical evidence to the role of vegetation as a potential agent for soil erosion control. Copyright © 2003 John Wiley & Sons, Ltd.     

Alternative split-operator approach for solving chemical reaction/groundwater transport models

Various schemes are available to solve coupled transport/reaction mathematical models, one of the most efficient and easy to apply being the two-step split-operator method in which the transport and reaction steps are performed separately. Operator splitting, however, does not solve exactly the fully coupled numerical model derived from the governing partial differential and algebraic equations describing the transport and reaction processes. An error, proportional to Δt (the time step used in the numerical solution) is introduced. Thus, small time steps must be used to ensure that accurate solutions result. An alternative scheme is presented, which iterates to the exact solution of the fully coupled numerical model. The new scheme enables accurate solutions to be calculated more efficiently than the two-step method, while maintaining separation of the transport and reaction steps in the calculations. As in the two-step method, the reaction calculations are performed node-wise throughout the computation grid. However, because the scheme relies on LU factorisation of the coefficient matrix in the transport equation solution, the reaction calculations must be performed in sequence, the sequence order being determined by the ordering of the nodes in the grid. Also, because LU factorisation is used, the scheme is limited to solute transport problems for which LU factorisation is a practical solution method.     

Influence of Boundary Condition Types on Unstable Density‐Dependent Flow

Boundary conditions are required to close the mathematical formulation of unstable density‐dependent flow systems. Proper implementation of boundary conditions, for both flow and transport equations, in numerical simulation are critical. In this paper, numerical simulations using the FEFLOW model are employed to study the influence of the different boundary conditions for unstable density‐dependent flow systems. A similar set up to the Elder problem is studied. It is well known that the numerical simulation results of the standard Elder problem are strongly dependent on spatial discretization. This work shows that for the cases where a solute mass flux boundary condition is employed instead of a specified concentration boundary condition at the solute source, the numerical simulation results do not vary between different convective solution modes (i.e., plume configurations) due to the spatial discretization. Also, the influence of various boundary condition types for nonsource boundaries was studied. It is shown that in addition to other factors such as spatial and temporal discretization, the forms of the solute transport equation such as divergent and convective forms as well as the type of boundary condition employed in the nonsource boundary conditions influence the convective solution mode in coarser meshes. On basis of the numerical experiments performed here, higher sensitivities regarding the numerical solution stability are observed for the Adams‐Bashford/Backward Trapezoidal time integration approach in comparison to the Euler‐Backward/Euler‐Forward time marching approach. The results of this study emphasize the significant consequences of boundary condition choice in the numerical modeling of unstable density‐dependent flow.     

流固边界耦合介质高阶有限差分地震正演模拟方法

本文针对流固边界耦合介质提出了一种高效、稳定的正演数值模拟方法. 首先,从一阶位移-应力弹性波方程出发,基于海底流固边界的位移和应力的连续性条件,采用三次样条海底界面定量表征方法,推导出不规则海底界面下流固边界耦合介质中的地震波波动方程;其次,通过空间微分的高阶差分格式提高数值模拟的空间精度,并结合已推导的地震波波动方程,将四阶时间微分转换至高阶空间微分,进一步提高了数值模拟的时间精度;最后,在与标量波波动方程数值模拟结果对比分析的基础上,分别利用简单的水平层状模型和复杂海底模型,验证和讨论了本文提出的流固边界耦合介质高阶有限差分地震波正演模拟方法的有效性和准确性.      

Numerical mass balance for soil-moisture transport problems

The Galerkin finite element method coupled with the Crank-Nicolson time advance procedure is often used as a numerical analog for unsaturated soil-moisture transport problems. The Crank-Nicolson procedure leads to numerical mass balance problems which results in instability. A new temporal and spatial integration procedure is proposed that exactly satisfies mass balance for the approximating function used. This is accomplished by fitting polynomials continuously throughout the time and space domain and integrating the governing differential equations. To reduce computational effort, the resulting higher order polynomials are reduced to quadratic and linear piece-wise continuous polynomial approximation functions analogous to the finite element approach. Results indicate a substantial improvement in accuracy over the combined Galerkin and Crank-Nicolson methods when comparing to simplified problems where analytical solutions are available.     

A stochastic integral equation analog of rainfall-runoff processes for evaluating modeling uncertainty

In this paper a very general rainfall-runoff model structure (described below) is shown to reduce to a unit hydrograph model structure. For the general model, a multi-linear unit hydrograph approach is used to develop subarea runoff, and is coupled to a multi-linear channel flow routing method to develop a link-node rainfall-runoff model network. The spatial and temporal rainfall distribution over the catchment is probabilistically related to a known rainfall data source located in the catchment in order to account for the stochastic nature of rainfall with respect to the rain gauge measured data. The resulting link node model structure is a series of stochastic integral equations, one equation for each subarea. A cumulative stochastic integral equation is developed as a sum of the above series, and includes the complete spatial and temporal variabilities of the rainfall over the catchment. The resulting stochastic integral equation is seen to be an extension of the well-known single area unit hydrograph method, except that the model output of a runoff hydrograph is a distribution of outcomes (or realizations) when applied to problems involving prediction of storm runoff; that is, the model output is a set of probable runoff hydrographs, each outcome being the results of calibration to a known storm event.     

A stochastic integral equation analog of rainfall-runoff processes for evaluating modeling uncertainty

In this paper a very general rainfall-runoff model structure (described below) is shown to reduce to a unit hydrograph model structure. For the general model, a multi-linear unit hydrograph approach is used to develop subarea runoff, and is coupled to a multi-linear channel flow routing method to develop a link-node rainfall-runoff model network. The spatial and temporal rainfall distribution over the catchment is probabilistically related to a known rainfall data source located in the catchment in order to account for the stochastic nature of rainfall with respect to the rain gauge measured data. The resulting link node model structure is a series of stochastic integral equations, one equation for each subarea. A cumulative stochastic integral equation is developed as a sum of the above series, and includes the complete spatial and temporal variabilities of the rainfall over the catchment. The resulting stochastic integral equation is seen to be an extension of the well-known single area unit hydrograph method, except that the model output of a runoff hydrograph is a distribution of outcomes (or realizations) when applied to problems involving prediction of storm runoff; that is, the model output is a set of probable runoff hydrographs, each outcome being the results of calibration to a known storm event.     

Coupling water flow and solute transport into a physically-based surface-subsurface hydrological model

A distributed-parameter physically-based solute transport model using a novel approach to describe surface-subsurface interactions is coupled to an existing flow model. In the integrated model the same surface routing and mass transport equations are used for both hillslope and channel processes, but with different parametrizations for these two cases. For the subsurface an advanced time-splitting procedure is used to solve the advection-dispersion equation for transport and a standard finite element scheme is used to solve Richards equation for flow. The surface-subsurface interactions are resolved using a mass balance-based surface boundary condition switching algorithm that partitions water and solute into actual fluxes across the land surface and changes in water and mass storage. The time stepping strategy allows the different time scales that characterize surface and subsurface water and solute dynamics to be efficiently and accurately captured. The model features and performance are demonstrated in a series of numerical experiments of hillslope drainage and runoff generation.     

Numerical model of local scour considering the unsteady sediment inflow and sediment sorting: Application to scour upstream of slit weir

The current study proposes a novel framework for the numerical model for estimating the temporal scour considering unsteady sediment inflow and the sediment sorting process. The framework was applied to local scour upstream of a slit weir. The scour model is based on an ordinary nonlinear differential equation derived from sediment continuity and scour rate equations. A one-dimensional(1-D)Boussinesq-type model coupled with nonequilibrium sediment transport was incorporated in the scour model to...     

Two‐dimensional physically based finite element runoff model for small agricultural watersheds: I. Model development

Soil erosion by water is the root cause of ecological degradation in the Shiwalik foothills of Northern India. Simulation of runoff and its component processes is a pre‐requisite to develop the management strategies to tackle the problem, successfully. A two‐dimensional physically based distributed numerical model, ROMO2D has been developed to simulate runoff from small agricultural watersheds on an event basis. The model employs the 2‐D Richards equation with sink term to simulate infiltration and soil moisture dynamics in the vadoze zone under variable rainfall conditions, and 2‐D Saint‐Venant equations under the kinematic wave approximation along with Manning's equation as the stage‐discharge equation for runoff routing. The various flow‐governing equations have been solved numerically by employing a Galerkin finite element method for spatial discretization using quadrilateral elements and finite difference techniques for temporal solutions. The ROMO2D computer program has been developed as a class‐based program, coded in C + + in such a way that with minor modifications, the model can be used to simulate runoff on a continuous basis. The model writes output for a runoff hydrograph of each storm. Model development is described in this paper and the results of model testing and field application are to be presented in a subsequent paper. Copyright © 2008 John Wiley & Sons, Ltd.     

A MATHEMATICAL QUASI-2D OVERLAND FLOW MODEL

1 INTRODUCTIONThe prediction of future impacts on terrestrial ecosystems by atmospheric, climatic and land-usechanges is the aim of watershed management. Meeting these requirements scientists, managers and policymakers try to achieve the sustainable management of the vitally important resources of watersheds due toan integrated ecosystem approach at the catchment scale. As composite landscapes often have a highdegree of contingency between its elements, the transport over these landscape s…     

Analysis of tracer tomography using temporal moments of tracer breakthrough curves

Hydraulic/partitioning tracer tomography (HPTT) was recently developed by Yeh and Zhu [Yeh T-CJ, Zhu J. Hydraulic/partitioning tracer tomography for characterization of dense nonaqueous phase liquid source zones, Water Resour Res 2007;43:W06435. doi:10.1029/2006WR004877.] for estimating spatial distribution of dense nonaqueous phase liquids (DNAPLs) in the subsurface. Since discrete tracer concentration data are directly utilized for the estimation of DNAPLs, this approach solves the hyperbolic convection–dispersion equation. Solution to the convection–dispersion equation however demands fine temporal and spatial discretization, resulting in high computational cost for an HPTT analysis. In this work, we use temporal moments of tracer breakthrough curves instead of discrete concentration data to estimate DNAPL distribution. This approach solves time independent partial differential equations of the temporal moments, and therefore avoids solving the convection–dispersion equation using a time marching scheme, resulting in a dramatic reduction of computational cost. To reduce numerical oscillations associated with convection dominated transport problems such as in inter-well tracer tests, the approach uses a finite element solver adopting the streamline upwind Petrov–Galerkin method to calculate moments and sensitivities. We test the temporal moment approach through numerical simulations. Comparing the computational costs between utilizing moments and discrete concentrations, we find that temporal moments significantly reduce the computation time. We also find that tracer moment data collected through a tomographic survey alone are able to yield reasonable estimates of hydraulic conductivity, as indicated by a correlation of 0.588 between estimated and true hydraulic conductivity fields in the synthetic case study.     

LISEM: A SINGLE-EVENT,PHYSICALLY BASED HYDROLOGICAL AND SOIL EROSION MODEL FOR DRAINAGE BASINS. II: SENSITIVITY ANALYSIS,VALIDATION AND APPLICATION

A new hydrological and soil erosion model has been developed and tested: LISEM, the Limburg soil erosion model. The model uses physically based equations to describe interception, infiltration and soil water transport, storage in surface depressions, splash and flow detachment, transport capacity and overland and channel flow. From the validation results it is clear that, although the model has several advantages over other models, the results of LISEM 1.0 are far from perfect. Based on the sensitivity analysis and field observations, the main reasons for these differences seems to be the spatial and temporal variability of the soil hydraulic conductivity and the initial pressure head at the basin scale. Another reason for the differences between measured and simulated results is our lack or understanding of the theory of hydrological and soil erosion processes.     

Evaluation of model complexity and space–time resolution on the prediction of long‐term soil salinity dynamics,western San Joaquin Valley,California

The numerical simulation of long‐term large‐scale (field to regional) variably saturated subsurface flow and transport remains a computational challenge, even with today's computing power. Therefore, it is appropriate to develop and use simplified models that focus on the main processes operating at the pertinent time and space scales, as long as the error introduced by the simpler model is small relative to the uncertainties associated with the spatial and temporal variation of boundary conditions and parameter values. This study investigates the effects of various model simplifications on the prediction of long‐term soil salinity and salt transport in irrigated soils. Average root‐zone salinity and cumulative annual drainage salt load were predicted for a 10‐year period using a one‐dimensional numerical flow and transport model (i.e. UNSATCHEM) that accounts for solute advection, dispersion and diffusion, and complex salt chemistry. The model uses daily values for rainfall, irrigation, and potential evapotranspiration rates. Model simulations consist of benchmark scenarios for different hypothetical cases that include shallow and deep water tables, different leaching fractions and soil gypsum content, and shallow groundwater salinity, with and without soil chemical reactions. These hypothetical benchmark simulations are compared with the results of various model simplifications that considered (i) annual average boundary conditions, (ii) coarser spatial discretization, and (iii) reducing the complexity of the salt‐soil reaction system. Based on the 10‐year simulation results, we conclude that salt transport modelling does not require daily boundary conditions, a fine spatial resolution, or complex salt chemistry. Instead, if the focus is on long‐term salinity, then a simplified modelling approach can be used, using annually averaged boundary conditions, a coarse spatial discretization, and inclusion of soil chemistry that only accounts for cation exchange and gypsum dissolution–precipitation. We also demonstrate that prediction errors due to these model simplifications may be small, when compared with effects of parameter uncertainty on model predictions. The proposed model simplifications lead to larger time steps and reduced computer simulation times by a factor of 1000. Copyright © 2006 John Wiley & Sons, Ltd.     

A general approach to advective–dispersive transport with multirate mass transfer

A numerical solution that is significantly more general than other semi-analytical solutions is presented for governing equations describing advective–dispersive transport with multirate mass transfer between mobile and immobile domains. The new solution approach is general in the sense that it does not impose any restrictive assumption on the spatial or temporal variability of advective and dispersive processes in the mobile domain. A single integro-differential equation (IDE) is developed for the concentration in the mobile domain by separating the concentration in the immobile domain from the set of two partial differential equations. The solution to the IDE requires the evaluation of a temporal integral of the concentration in the mobile domain, which is a function of the Laplace transform of the distribution of the mass transfer rate coefficient. The Laplace transform is not limited to flow fields with known constant velocities. The solutions for one- and two-dimensional examples obtained using the new approach agree with those obtained by existing semi-analytical and numerical approaches.