Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

In order to model non‐Fickian transport behaviour in groundwater aquifers, various forms of the time–space fractional advection–dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time–space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time–space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time–space fractional mass conservation equation and the time–space fractional water flux equation. Here, we apply these time–space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time–space. The numerical results demonstrate that the proposed time–space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field.     

Scaling and self‐similarity in one‐dimensional unsteady open channel flow

The conditions under which the Saint Venant equations system for unsteady open channel flow, as an initial–boundary value problem, becomes self‐similar are investigated by utilizing one‐parameter Lie group of point scaling transformations. One of the advantages of this methodology is that the self‐similarity conditions due to the initial and boundary conditions can also be investigated thoroughly in addition to the conditions due to the governing equation. The obtained self‐similarity conditions are compared with the scaling relationships that are derived through the Froude similitude. It is shown that the initial–boundary value problem of a one‐dimensional unsteady open channel flow process in a prototype domain can be self‐similar with that of several different scaled domains. However, the values of all the flow variables (at specified time and space) under different scaled domains can be upscaled to the same values in the prototype domain (at the corresponding time and space), as shown in this study. Distortion in scales of different space dimensions has been implemented extensively in physical hydraulic modelling, mainly because of cost, space and time limitations. Unlike the traditional approach, the distinction is made between the longitudinal–horizontal and transverse–horizontal length scales in this study. The scaled domain obtained by the proposed approach, when scaling ratios of channel width and water depth are equal, is particularly important for the similarity of flow characteristics in a cross‐section because the width‐to‐depth ratio and the inclination angles of the banks are conserved in a cross‐section. It is also shown that the scaling ratio of the roughness coefficient under distorted channel conditions depends on that of hydraulic radius and longitudinal length. The proposed scaling relations obtained by the Lie group scaling approach may provide additional spatial, temporal and economical flexibility in setting up physical hydraulic models. Copyright © 2013 John Wiley & Sons, Ltd.     

Modelling overland flow based on Saint‐Venant equations for a discretized hillslope system

Mathematical modelling of overland flow is a critical task in simulating transport of water, sediment and other pollutants from land surfaces to receiving waters. In this paper, an overland flow routing method is developed based on the Saint‐Venant equations using a discretized hillslope system for areas with high roughness and steep slope. Under these conditions, the momentum equation reduces to a unique relationship between the flow depth and discharge. A hillslope is treated as a system divided into several subplanes. A set of first‐order non‐linear differential equations for subsequent subplanes are solved analytically using Chezy's formula in lieu of the momentum equation. Comparison of the analytical solution of the first‐order non‐linear ordinary differential equations and a numerical solution using the Runge‐Kutta method shows a relative error of 0·3%. Using runoff data reported in the literature, comparison between the new approach and a numerical solution of the full Saint‐Venant equations showed a close agreement. Copyright © 2002 John Wiley & Sons, Ltd.     

黄河尾闾段一维耦合水沙数学模型研究及其应用

小浪底水库于1999年运用以后,该河道经历了长时间持续冲刷过程.为掌握小浪底水库运用后黄河尾闾段洪水演进特点及河床冲淤规律,采用一维水沙数学模型研究是一条重要的途径.本研究首先采用浑水控制方程,建立了一维耦合水沙数学模型,并利用2003年利津-西河口段汛期实测水沙及汛前断面地形资料对该模型进行率定,计算的流量、水位及含沙量等过程与实测值吻合较好;然后采用2015年利津—汊3段汛期实测资料对该模型进行验证,结果显示水位与冲淤量计算值与实测值较为符合;最后基于2015年实测洪水过程,计算了若干组不同断面间距下的洪水演进及冲淤过程,分析了不同断面间距对沿程水位及河段冲淤量等计算结果的影响,结果表明:采用不同断面间距对水位计算结果影响较小,而对冲淤量计算结果会产生一定影响;在河段水沙及冲淤特性复杂的情况下,采用一维数学水沙模型计算时应考虑断面间距的选择.     

Improvement of surface run‐off in the hydrological model ParFlow by a scale‐consistent river parameterization

We propose an improvement of the overland‐flow parameterization in a distributed hydrological model, which uses a constant horizontal grid resolution and employs the kinematic wave approximation for both hillslope and river channel flow. The standard parameterization lacks any channel flow characteristics for rivers, which results in reduced river flow velocities for streams narrower than the horizontal grid resolution. Moreover, the surface areas, through which these wider model rivers may exchange water with the subsurface, are larger than the real river channels potentially leading to unrealistic vertical flows. We propose an approximation of the subscale channel flow by scaling Manning's roughness in the kinematic wave formulation via a relationship between river width and grid cell size, following a simplified version of the Barré de Saint‐Venant equations (Manning–Strickler equations). The too large exchange areas between model rivers and the subsurface are compensated by a grid resolution‐dependent scaling of the infiltration/exfiltration rate across river beds. We test both scaling approaches in the integrated hydrological model ParFlow. An empirical relation is used for estimating the true river width from the mean annual discharge. Our simulations show that the scaling of the roughness coefficient and the hydraulic conductivity effectively corrects overland flow velocities calculated on the coarse grid leading to a better representation of flood waves in the river channels.     

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

Fractals are famous for their self‐similar nature at different spatial scales. Similar to fractals, solutions of scale invariant processes are self‐similar at different space–time scales. This unique property of scale‐invariant processes can be utilized to translate the solution of the processes at a much larger or smaller space–time scale (domain) based on the solution calculated on the original space–time scale. This study investigates scale invariance conditions of kinematic wave overland flow process in one‐parameter Lie group of point transformations framework. Scaling (stretching) transformation is one of the one‐parameter Lie group of point transformations and it has a unique importance among the other transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process yields a self‐similar solution at different space–time scales. However, the conditions for the process to be scale invariant usually dictate various relationships between the scaling coefficients of the dependent and independent variables of the process. Therefore, the scale invariance of a process does not assure a self‐similar solution at any arbitrary space and time scale. The kinematic wave overland flow process is modelled mathematically as initial‐boundary value problem. The conditions to be satisfied by the system of governing equations as well as the initial and boundary conditions of the kinematic wave overland flow process are established in order for the process to be scale invariant. Also, self‐similarity of the solution of the kinematic wave overland flow under the established invariance conditions is demonstrated by various numerical example problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Two‐dimensional continuous simulation of spatiotemporally varied hydrological processes using the time‐area method

Distributed, continuous hydrologic models promote better understanding of hydrology and enable integrated hydrologic analyses by providing a more detailed picture of water transport processes across the varying landscape. However, such models are not widely used in routine modelling practices, due in part to the extensive data input requirements, computational demands, and complexity of routing algorithms. We developed a two‐dimensional continuous hydrologic model, HYSTAR, using a time‐area method within a grid‐based spatial data model with the goal of providing an alternative way to simulate spatiotemporally varied watershed‐scale hydrologic processes. The model calculates the direct runoff hydrograph by coupling a time‐area routing scheme with a dynamic rainfall excess sub‐model implemented here using a modified curve number method with an hourly time step, explicitly considering downstream ‘reinfiltration’ of routed surface runoff. Soil moisture content is determined at each time interval based on a water balance equation, and overland and channel runoff is routed on time‐area maps, representing spatial variation in hydraulic characteristics for each time interval in a storm event. Simulating runoff hydrographs does not depend on unit hydrograph theory or on solution of the Saint Venant equation, yet retains the simplicity of a unit hydrograph approach and the capability of explicitly simulating two‐dimensional flow routing. The model provided acceptable performance in predicting daily and monthly runoff for a 6‐year period for a watershed in Virginia (USA) using readily available geographic information about the watershed landscape. Spatial and temporal variability in simulated effective runoff depth and time area maps dynamically show the areas of the watershed contributing to the direct runoff hydrograph at the outlet over time, consistent with the variable source area overland flow generation mechanism. The model offers a way to simulate watershed processes and runoff hydrographs using the time‐area method, providing a simple, efficient, and sound framework that explicitly represents mechanisms of spatially and temporally varied hydrologic processes. Copyright © 2015 John Wiley & Sons, Ltd.     

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 unifying framework for watershed thermodynamics: constitutive relationships

The balance equations for mass and momentum, averaged over the scale of a watershed entity, need to be supplemented with constitutive equations relating flow velocities, pressure potential differences, as well as mass and force exchanges within and across the boundaries of a watershed. In this paper, the procedure for the derivation of such constitutive relationships is described in detail. This procedure is based on the method pioneered by Coleman and Noll through exploitation of the second law of thermodynamics acting as a constraint-type relationship. The method is illustrated by its application to some common situations occurring in real world watersheds. Thermodynamically admissible and physically consistent constitutive relationships for mass exchange terms among the subregions constituting the watershed (subsurface zones, overland flow regions, channel) are proposed. These constitutive equations are subsequently combined with equations of mass balance for the subregions. In addition, constitutive relationships for forces exchanged amongst the subregions are also derived within the same thermodynamic framework. It is shown that, after linearisation of the latter constitutive relations in terms of the velocity, a watershed-scale Darcy's law governing flow in the unsaturated and saturated zones can be obtained. For the overland flow, a second order constitutive relationship with respect to velocity is proposed for the momentum exchange terms, leading to a watershed-scale Chezy formula. For the channel network REW-scale Saint–Venant equations are derived. Thus, within the framework of this approach new relationships governing exchange terms for mass and momentum are obtained and, moreover, some well-known experimental results are derived in a rigorous manner.     

Practical implementation of the fractional flow approach to multi-phase flow simulation

Fractional flow formulations of the multi-phase flow equations exhibit several attractive attributes for numerical simulations. The governing equations are a saturation equation having an advection diffusion form, for which characteristic methods are suited, and a global pressure equation whose form is elliptic. The fractional flow approach to the governing equations is compared with other approaches and the implication of equation form for numerical methods discussed. The fractional flow equations are solved with a modified method of characteristics for the saturation equation and a finite element method for the pressure equation. An iterative algorithm for determination of the general boundary conditions is implemented. Comparisons are made with a numerical method based on the two-pressure formulation of the governing equations. While the fractional flow approach is attractive for model problems, the performance of numerical methods based on these equations is relatively poor when the method is applied to general boundary conditions. We expect similar difficulties with the fractional flow approach for more general problems involving heterogenous material properties and multiple spatial dimensions.     

Combined hillslope diffusion and sediment transport simulation applied to landscape dynamics modelling

We present a new numerical approach for simulating geomorphic and stratigraphic processes that combines open‐channel flow with non‐uniform sediment transport law and semi‐empirical diffusive mass wasting. It is designed to facilitate modelling of surface processes across multiple space‐ and time‐scales, and under a variety of environmental and tectonic conditions. The physics of open‐channel flow is primarily based on an adapted Lagrangian formulation of shallow‐water equations. The interaction between flow and surface geology is performed by a non‐uniform total‐load sediment transport law. Additional hillslope processes are simulated using a semi‐empirical method based on a diffusion approach. In the implementation, the resolution of flow dynamics is made on a triangulated grid automatically mapped and adaptively remeshed over a regular orthogonal stratigraphic mesh. These new methods reduce computational time while preserving stability and accuracy of the physical solutions. In order to illustrate the potential of this method for landscape and sedimentary system modelling, we present a set of three generic experiments focusing on assessing the influence of contrasting erodibilities on the evolution of an active bedrock landscape. The modelled ridges morphometrics satisfy established relationships for drainage network geometry and slope distribution, and provide quantitative information on the relative impact of hillslope and channel processes, sediment discharge and alluviation. Our results suggest that contrasting erodibility can stimulate autogenic changes in erosion rate and influence the landscape morphology and preservation. This approach offers new opportunities to investigate joint landscape and sedimentary systems response to external perturbations. The possibility to define and track a large number of materials makes the implementation highly suited to model source‐to‐sink problems where material dispersion is the key question that needs to be addressed, such as natural resources exploration and basin analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

NUMERICAL SIMULATIONS OF SCOUR AND DEPOSITION IN A CHANNEL NETWORK

1 INTRODUCTION Evolution of the river bed in alluvial channels has been studied by many researchers using analytical and numerical approaches. The use of analytical approach alone is insufficient for solving natural river engineering problems. With rapid growth in computer technology, numerical models have become a popular means for the study of mobile bed hydraulics. During the past decade, several numerical models have been developed. Most of the computer codes, such as HEC2SR (Si…     

MODFLOW‐Based Coupled Surface Water Routing and Groundwater‐Flow Simulation

In this paper, we present a flexible approach for simulating one‐ and two‐dimensional routing of surface water using a numerical surface water routing (SWR) code implicitly coupled to the groundwater‐flow process in MODFLOW. Surface water routing in SWR can be simulated using a diffusive‐wave approximation of the Saint‐Venant equations and/or a simplified level‐pool approach. SWR can account for surface water flow controlled by backwater conditions caused by small water‐surface gradients or surface water control structures. A number of typical surface water control structures, such as culverts, weirs, and gates, can be represented, and it is possible to implement operational rules to manage surface water stages and streamflow. The nonlinear system of surface water flow equations formulated in SWR is solved by using Newton methods and direct or iterative solvers. SWR was tested by simulating the (1) Lal axisymmetric overland flow, (2) V‐catchment, and (3) modified Pinder‐Sauer problems. Simulated results for these problems compare well with other published results and indicate that SWR provides accurate results for surface water‐only and coupled surface water/groundwater problems. Results for an application of SWR and MODFLOW to the Snapper Creek area of Miami‐Dade County, Florida, USA are also presented and demonstrate the value of coupled surface water and groundwater simulation in managed, low‐relief coastal settings.     

Modeling flows in permeable pressure systems and channels

The study is focused on the velocity and pressure distributions in flows in pipeline systems, as well as hydrological parameters of open flow examined on the basis of Saint Venant equations with allowance made for lateral inflow and the equations of motion of incompressible fluid in perforated or porous pipes. Some analytical solutions are given for the formulated boundary problems, which can be applied to the calculation of operation regimes of sprinkling machines of the “Fregat” or “Valey” type.     

NUMERICAL SIMULATION OF SEDIMENT TRANSPORT IN DYNAMIC FLOW CONDITIONS

1 INTRODUCTION Rivers as a source of life can at the same time impose devastating conditions on the environment. It is , therefore, imperative to analyse and predict river behaviour for different given conditions and engineering activities. Therefore, the use of simulation tools in this field has become a necessity. Many computational tools for simulation of sediment transport in rivers are now available that can be used for prediction and design under different flow conditions. However, …     

MODELING SEDIMENT PROCESSES ON SMALL WATERSHEDS:A CONCEPTUAL FRAMEWORK Ⅰ. BROAD SHALLOW FLOW PROCESSE

A conceptual modeling framework for developing process-based mathematical models of sediment generation, transport, and deposition on broad shallow flow areas is presented. The governing equations relevant to process-based modeling of broad shallow flow sediment processes on small watersheds are presented and the effects of space and time averaging on the predictive equations are described. Starting from the most general one-dimensional, unsteady model of sediment processes, simpler model structures are obtained and the successive simplifications made on the governing equations in order to obtain simpler and less detailed formulations are described. Specific model formulations are given for illustrative purposes and applications of these models to erosion and sediment yield prediction from broad shallow flow areas are shown using sediment data from rainfall simulator plots. In spite of some progress made in the development of process-based erosion and sediment yield models from broad shallow flow areas, further developmental modeling efforts must be based on a clear separation between hydrologic and hydraulic processes, and the soil properties which are significant for each.     

Viscosity-excluded formulations of 2-D river flow with a new wetting and drying algorithm

An efficient method for simulating 2-D river flow is developed in which horizontal turbulent shears are omitted from the 2-D depth-averaged momentum equations. It is shown that a pseudo-viscosity can be reproduced to take into account the lost shear action, by incorporating the vertically integrated continuity equation to the momentum equations and transforming the latter into a discrete integral form. To simulate river flows with wet and dry areas, negative water depths are allowed when solving the continuity equation. The concept of negative water depth enables us to track flow boundaries with about the same accuracy but much less effort as compared with traditional numerical methods. An optimal threshold value defining dry areas is first obtained by one-dimensional theoretical analysis and then sought by trial-and-error for two-dimensional flow simulation with tolerable node-to-node spurious oscillations, while mass is best conserved. Numerical solutions using the new procedure are compared with the one-dimensional benchmark solution of the Saint Venant equations and the experimental data from a two-stage channel. Robustness of the present approach is also tested through the study of water flow in a natural river and a hypothetical channel with several bumps.     

MODELING SEDIMENT PROCESSES ON SMALL WATERSHEDS: A CONCEPTUAL FRAMEWORK Ⅱ. CONCENTRATED FLOW PROCESSES

L INTRODUCTIONThe purpose of this paper is to present a general modeling framework that can serveas a conceptual basis for developing sediment process models by concentrated flow systems on small watersheds. A survey of fundamental principles for developing sedimentprocess models is made with particular emphasis on the effects of space and time averaging on the governing equations. Starting from the most general one--dimensional,unsteady model of sediment processes, simpler model structur…     

Spatially non‐local model of inelastic deformations: applications for rock failure problem

A spatially non‐local model for inelastic deformation of solids is proposed and studied. The non‐locality of deformation is taken into account by the additional parameter of state beyond the classical parameters such as stress and strain tensors. This additional parameter is the curvature tensor expressed in terms of the metric strain tensor, and it is called the failure parameter. In the case of small deformation, it is equivalent to the Saint‐Venant incompatibility tensor. Thermodynamic properties of the model are studied, and governing differential equations for spatially non‐local model are formulated, which are composed by the elasticity equations and parabolic equation for the failure parameter. The model can be applied to the study of the rock failure problem, and as an example, the one‐dimensional problem for the deformation of half‐plane loaded by the normal surface stress is studied. Stationary and non‐stationary formulations of the problem are considered, and qualitative agreement with available experimental data is observed.