# PHYSICS BASED, INTEGRATED MODELING OF HYDROLOGY AND HYDRAULICS AT WATERSHED SCALES

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PHYSICS BASED, INTEGRATED MODELING OF HYDROLOGY AND HYDRAULICS AT WATERSHED SCALES

# ABSTRACT

This thesis presents the major findings in the development of the hydrology and hydraulics modules of a first principle, physics-based watershed model (WASH123D Version 1.5).  The numerical model simulates water movement in watersheds with individual water flow components of one-dimensional stream/channel network flow, twodimensional overland flow and three-dimensional variably saturated subsurface flow and their interactions.

Firstly, the complete Saint Venant equations/2-D shallow water equations (dynamic wave equations) and the kinematic wave or diffusion wave approximations were implemented as three solution options for 1-D channel network and 2-D overland flow. Different solution techniques are considered for the governing equations based on physical reasoning and their mathematical property. A characteristic based finite element method is chosen for the hyperbolic-type dynamic wave model. And the Galerkin finite element method is used to solve the diffusion wave model.  Careful choice of numerical methods is needed even for the simple kinematic wave model. Since the kinematic wave equation is of pure advection, the backward method of characteristics is used for kinematic wave model. Diffusion wave and kinematic wave approximations are found in many surface runoff routing models. The error in these models has been characterized for some cases of overland flow over simple geometry. However, the nature and propagation of these approximation errors under more complex 2-D flow conditions are not well known. These issues are evaluated within WASH123D with comparison of simulation

results of several example problems. The accuracy of the three wave models for 1-D channel flow was evaluated with several non-trivial (trans-critical flow; varied bottom slopes with frictions and non-prismatic cross-section) benchmark problems (MacDonnell et al., 1997). The test examples for 2-D overland flow include: (1) a simple rainfallrunoff process on a single plane with constant rainfall excess that has a kinematic analytical solution under steep slope condition. A range of bottom slopes (mild, average and steep slope) are numerically solved with the three wave models and compared; (2) Iwagaki (1955) overland flow experiments on a cascade of three planes with shock waves; (3) overland flow in a hypothetical wetland. The applicability of dynamic-wave, diffusion-wave and kinematic-wave models to real watershed modeling is discussed with simulation results from these numerical experiments.  It was concluded that kinematic wave model could lead to significant errors in most applications. On the other hand, diffusion wave model is adequate for modeling overland flow in most natural watersheds. The complete dynamic wave equations are required in low-terrain areas such as flood plains or wetlands and many transient fast flow situations.

Secondly, issues about the coupling between surface water and subsurface flow are investigated. In the core of an integrated watershed model is the coupling among surface water and subsurface water flows. Recently, there is a tendency of claiming the fully coupled approach for surface water and groundwater interactions in the hydrology literature. One example is the assumption of a gradient type flux equation based on Darcy’s Law (linkage term) and the numerical solution of all governing equations in a single global matrix. We argue that this is only a special case of all possible coupling combinations and if not applied with caution, the non-physical interface parameter becomes a calibration tool.   Generally, there are two cases based on physical nature of the interface: continuous or discontinuous assumption, when a sediment layer exists at the interface, the discontinuous assumption may be justified. As for numerical schemes, there are three cases: time-lagged, iterative and simultaneous solutions.  Since modelers often resort to the simplest, fastest schemes in practical applications, it is desirable to quantify the potential error and performance of different coupling schemes. We evaluate these coupling schemes in a finite element watershed model, WASH123D. Numerical experiments are used to compare the performance of each coupling approach for different types of surface water and groundwater interactions. These are in terms of surface water and subsurface water solutions and exchange fluxes (e.g. infiltration/seepage rate). It is concluded that different coupling approaches are justified for flow problems of different spatial and temporal scales and the physical setting of the interface.

Thirdly, The Method of Characteristics (MOC) in the context of finite element method was applied to the complete 2-D shallow water equations for 2-D overland flow. For two-dimensional overland flow, finite element or finite volume methods are more flexible in dealing with complex boundary. Recently, finite volume methods have been very popular in numerical solution of the shallow water equations. Some have pointed out that finite volume methods for 2-D flow are fundamentally one-dimensional (normal to the cell interface). The results may rely on the grid orientation.  The search for genuinely multidimensional numerical schemes for 2-D flow is an active topic.  We consider the Method of Characteristics (MOC) in the context of finite element method as a good alternative. Many researchers have pointed out the advantage of MOC in solving 2-D shallow water equations that are of the hyperbolic type that has wave-like solutions and at the same time, considered MOC for 2-D overland flow being non-tractable on complex topography. The intrinsic difficulty in implementing MOC for 2-D overland flow is that there are infinite numbers of wave characteristics in the 2-D context, although only three independent wave directions are needed for a well-posed solution to the characteristic equations. We have implemented a numerical scheme that attempts to diagonalize the characteristic equations based on pressure and velocity gradient relationship. This new scheme was evaluated by comparison with other choice of wave characteristic directions in the literature.  Example problems of mixed sub-critical flow/super-critical flow in a channel with approximate analytical solution was used to verify the numerical algorithm. Then experiments of overland flow on a cascade of three planes (Iwagaki 1955) were solved by the new method. The circular dam break problem was solved with different selections of wave characteristic directions and the performance of each selection was evaluated based on accuracy and numerical stability. Finally, 2-D overland flow over complex topography in a wetland setting with very mild slope was solved by the new numerical method to demonstrate its applicability.

Finally, the physics-based, integrated watershed model was tested and validated with the hydrologic simulation of a pilot constructed wetland in South Florida. For this field problem strong surface water and groundwater interactions are a key component of the hydrologic processes. The site has extensive field measurement and monitoring that provide point scale and distributed data on surface water levels, groundwater levels and physical range of hydraulic parameters and hydrologic fluxes.  The uniqueness of this modeling study includes (1) the point scale and distributed comparison of model results with observed data, for example, the spatial distribution of measured vertical flux in the wetland is available; (2) model parameters are based on available field test data; and (3) water flows in the study area consist of 2-D overland flow, hydraulic structures/levees, 3D subsurface flow and 1-D canal flow and their interactions.  This study demonstrates the need and the utility of a physics-based modeling approach for strong surface water and groundwater interactions.

LIST OF FIGURES…………………………………………………………………………………………xi

LIST OF TABLES………………………………………………………………………………..xv

ACKNOWLEDGEMENTS……………………………………………………………………………..xvi

Chapter 1  Introduction……………………………………………………………………………………1

1.1 Overview of the Mathematical Modeling of Watersheds………………………….1

1.2 Physics-based, integrated watershed models…………………………………………..4

1.3 Motivation and Objectives……………………………………………………………………5

1.4 Format……………………………………………………………………………………………….8

References……………………………………………………………………………………………….9

Chapter 2  On simulating surface water flows with dynamic, diffusion and

kinematic waves……………………………………………………………………………………….14

Abstract…………………………………………………………………………………………………..14 2.1 Introduction………………………………………………………………………………………..15

2.2 Governing Equations …………………………………………………………………………..17

2.2.1 Dynamic wave equations…………………………………………………………….18

2.2.2 Diffusion wave equation……………………………………………………………..20

2.2.3 Kinematic wave equation ……………………………………………………………21

2.3 Numerical Methods …………………………………………………………………………….22

2.3.1 Dynamic wave model…………………………………………………………………22

2.3.2 Diffusion wave model ………………………………………………………………..27

2.3.3 Kinematic wave model……………………………………………………………….28

2.4 Comparative examples…………………………………………………………………………29

2.4.1 Verification and comparison of steady flow in one-dimensional

channels……………………………………………………………………………………..30

2.4.2 Rainfall-runoff over a plane ………………………………………………………..37 2.4.3 Two-dimensional partial dam break with friction…………………………..40

2.4.4 Two-dimensional unsteady flow in a treatment wetland………………….44

2.4.5 Two-dimensional river flow at a river bend…………………………………..51

2.5 Discussions………………………………………………………………………………………..54

2.6 Summary and Conclusions …………………………………………………………………..55

Acknowledgements…………………………………………………………………………………..55

References……………………………………………………………………………………………….56

Chapter 3  Dynamic Wave Modeling of Two-dimensional Overland Flow using

Characteristics-based Finite Element Method………………………………………………60 Abstract…………………………………………………………………………………………………..60

3.1 Introduction………………………………………………………………………………………..61

3.2 Governing Equations …………………………………………………………………………..65

3.3 Numerical Methods …………………………………………………………………………….71

3.3.1 Backward Tracking approach………………………………………………………73

3.3.2 Characteristic wave directions……………………………………………………..73

3.4 Verification and Validation Examples……………………………………………………76

3.4.1 Example 1: non-uniform steady flow in a straight channel………………76

3.4.2 Example 2: Rainfall-runoff on a sloping plane ………………………………79

3.4.3 Example 3:  Two-dimensional Circular dam break problem ……………81

3.4.4 Example 4: rainfall-runoff process over a three-plane cascade

surface ……………………………………………………………………………………….86

3.4.5 Example 5: Overland flow in a hypothetical wetland ……………………..89

3.5 Discussions………………………………………………………………………………………..97

3.6 Summary and Conclusions …………………………………………………………………..98

Acknowledgements…………………………………………………………………………………..99

References……………………………………………………………………………………………….99

Chapter 4  A Comparative Study of coupling Approaches for Surface water and

groundwater interactions……………………………………………………………………………104

Abstract…………………………………………………………………………………………………..104

4.1 Introduction………………………………………………………………………………………..103

4.2 Governing equations and Interface Conditions………………………………………..107

4.2.1 Channel Network Flow……………………………………………………………….107

4.2.2 Overland Flow…………………………………………………………………………..109 4.2.3 Subsurface Flow………………………………………………………………………..111

4.2.4 Interface Conditions …………………………………………………………………..113

4.3 Numerical Schemes for Surface and Subsurface Flow Coupling……………….115

4.3.1 Coupling between overland flow and subsurface flow…………………….120 4.3.2 Coupling between channel flow and subsurface flow……………………..121

4.3.3 Coupling Procedure……………………………………………………………………122

4.4 Numerical Examples……………………………………………………………………………125

4.4.1 Coupled overland and subsurface flow: rainfall-runoff process………..126

4.4.2 Surface water-groundwater interaction in a constructed wetland………129

4.4.3 Stream-aquifer interaction…………………………………………………………..133

4.5 Discussion and Conclusions…………………………………………………………………138

Acknowledgements…………………………………………………………………………………..139

References……………………………………………………………………………………………….139

Chapter 5   Integrated Modeling of Groundwater and Surface Water interactions

in a constructed wetland ……………………………………………………………………………143 Abstract…………………………………………………………………………………………………………143

5.1 Introduction………………………………………………………………………………………..144

5.2 Site Description ………………………………………………………………………………….147

5.3 Model Tool: WASH123D…………………………………………………………………….152

5.4 Input Data ………………………………………………………………………………………….152

5.5 Model Setup……………………………………………………………………………………….153

5.6 Model Calibration and Validation …………………………………………………………158

5.7 Model Simulation Results ……………………………………………………………………159

5.7.1 Surface water flow……………………………………………………………………..160 5.7.2 Subsurface Flow………………………………………………………………………..165

5.7.3 Exchange Fluxes………………………………………………………………………..173

5.8 Discussions………………………………………………………………………………………..174

5.9 Summary and Conclusions …………………………………………………………………..175

Acknowledgements…………………………………………………………………………………..176

References……………………………………………………………………………………………….176

Chapter 6  Summary and Conclusions……………………………………………………………….181

Chapter 1

Introduction

# 1.1 Overview of the Mathematical Modeling of Watersheds

A watershed model is an integrated representation of nearly any hydrological process of the hydrologic cycle; major processes within a watershed include precipitation, interception, evapotransiration, infiltration, overland flow, channel/stream flow, and subsurface flow.

Watershed models are essential tools used to address various water resources and environmental protection problems (Singh and Frevert, 2006). The mathematical modeling of watersheds has been extensive since the creation of modern digital computers in the 1950s (Singh and Woolhiser, 2002). There is a plethora of all kinds of watershed models, ranging from purely empirical, black box type models to physics based, integrated models.

Many lumped or semi-distributed watershed models are similar in model structure and principle. The well-known conceptual Stanford watershed model (Crawford and Linsley, 1966) and its successor, HSPF (Donigian and Imhoff, 2006), are representative of these watershed model types. The model structure is based on water balance in predefined conceptual storage zones, and model parameters are often without clear physical meaning.  The limitation of lumped watershed models is well known – model parameters must be calibrated with historic data; estimated parameters are not transferable and distributed information on flow velocity and water depth cannot be provided for water quality modeling.

The development of watershed models is often guided by an intentional application and modeling objective. When information on the total quantity and timing of surface runoff at the watershed outlet (such as in flood control) is enough for the modeling objective, the traditional lumped watershed models can be used if historic data is available, in order to calibrate the model. But in other cases, such as non-point source pollution, soil erosion, land use effect, climate change, etc., users need information about the flow field (water velocity and depth distribution) or the timing of any extensive alterations to the watershed system from land use or deforestation; from here, a physics based, distributed hydrologic model is necessary. There are also great needs in real-world water management for such integrated models. For example, the importance of surface water and groundwater interaction has led to integrated models developed by local government agencies in South Florida (Lal et al., 2005) and California (LaBolle et al., 2003).

Since Freeze and Harlan published their blueprint for a three-dimensional model of watersheds (Freeze and Harlan, 1969) more than three decades ago, there has been much progress in this field.   The SHE and its derivatives MIKE SHE are very popular in Europe (Abbott et al., 1986).  In the United States, the CASC2D and its newest version GSSHA (Downer, 2004) is a finite difference code that can do integrated modeling in a less rigorous approach. There are some model codes developed around the popular MODFLOW groundwater model [for example, MODBRANCH (Swain and Wexler

(1996), MOD-HMS (Panday and Huyakorn (2004)].

There has been much debate and controversy on physics based, distributed watershed models (e.g., Grayson et al., 1992; Woolhiser, 1996; Beven, 2002 and Loague and VanderKwaak, 2004). Beven (2002) concluded that a radical change in paradigm is needed for watershed models.  The Freeze and Harlan (1969) blueprint is however, flawed, and will eventually be abandoned. For instance, a major flaw is related to both scale (the point-scale mechanistic partial differential equations may not be valid at model grid scale) and equifinality (model over-parameterization).

Reggiani et al. (1998, 1999, 2000 and 2005) offer an alternative watershed model structure based on the discretization of a watershed into spatial units, termed ‘representative elementary watersheds’ (REWs). The point-scale conservation equations for mass, momentum, and energy is integrated over a sub-watershed. This flux-based formulation approach can be traced back to earlier research by Duffy (1996).  The difficulties in determining hydrological fluxes are a major problem and, if the size of REWs is very small, it will run into the same scale problem as the point-scale, physics based models. On the other hand, if the size of REWs is large, the physical meaning of state variables, such as flow velocity, pressure head, and water depth, are only nominal at

best.

Young (2003) offers another alternative, termed the ‘data based mechanistic approach’; its use is intended for hydrologic models. It is said to be a stochastic model with a top-down approach that can incorporate physical reasoning into the statistical model. This approach is better suited for practical application (e.g., real-time flood forecasting) and not for what-if type explanation-oriented modeling.

In the proceeding chapters, the Freeze and Harlan (1969) blueprint of a physics based, three-dimensional watershed model (as a viable modeling approach for integrated watershed models) will be explored. The increasing availability of spatially distributed hydrological data through remote sensing, radar rainfall, GIS, and new measurement techniques also support such physics-based distributed models.

# 1.2 Physics-based, integrated watershed models

The mechanistic, process-oriented modeling of fluid flow in watersheds can be conveniently divided into flow component models and coupling mechanism. There have been extensive studies on component models (for example, overland flow models, channel network flow models, and subsurface flow models) and among them; the flow component coupling is another important issue.

Water flow components in a watershed are essentially comprised of surface water and subsurface flow, where ‘surface water flows’ include overland and channel flow. There are at least three distinct runoff generation mechanisms. The first one is the

Hortonian overland flow mechanism’, usually called ‘infiltration excess mechanism’.  This has been the dominant concept in the current generation of surface hydrologic models.  Under this paradigm, overland flow was extensively modeled, but the infiltration and subsurface flow components are treated empirically, as a water sink/loss. Field observation and further research have demonstrated that Hortonian overland flow is rare and of little impact in humid, forested regions, while the subsurface storm flow mechanism is the major contribution to runoff.  In this case, most rainfall is to be absorbed by the soil, and subsurface flow becomes an important contribution to stream flow. Another runoff generation mechanism is called saturated excess, which represents the runoff process in place such as wetlands where the soil is saturated most of the time. Only a physics-based, integrated watershed models can simulate all these runoff generation mechanisms in a single model.

# 1.3 Motivation and Objectives

The watershed modeling in the past 30 years has been focused on individual flow components, e.g., overland flow, channel flow, groundwater flow, etc. Most overland flow modeling studies did not couple with subsurface flow and, in subsurface hydrology, the surface water processes were often only considered as a simple source/sink. When coupling is considered, weak coupling or artificially created linkage terms are often used for flux-exchange calculations.

Freeze and Harlan (1969) were the first to develop a blueprint of the physically based watershed model. Smith and Woolhiser (1972) were one of the first to apply the one-dimensional Richards’ equation for unsaturated flow in their overland model with a 1-D kinematic wave equation. Akan and Yen (1981) coupled a two-dimensional subsurface flow component with overland flow. One of the first attempts to build a comprehensive numerical model of a watershed based on the Freeze and Harlan blueprint is the SHE model in Europe (Abbott et al., 1986). In the SHE watershed model, the unsaturated zone is represented with one-dimensional Richards’ equation. In USA, the focus has been toward a practical engineering application, so most of the watershed models are lumped.  Among the few physically based models developed are CASC2D by Julien et al. (1995) and KINEROS by Woolhiser et al. (1990). CASC2D included the diffusion wave approximation of overland flow and channel flow; infiltration is based on the Green and Ampt model and the explicit finite difference method is used. KINEROS is based on the kinematic wave model for overland flow routing with empirical infiltration equations.

This thesis research concerns with the physics based, integrated mathematical modeling of watershed hydrology and hydraulics. The computational aspect of physicsbased, integrated watershed models is investigated in term of proper selection of governing equations, coupling approaches, and numerical methods, etc. The hydrology and hydraulics modules of the watershed model are particularly designed as the basis for modeling transport of sediments and pollutant at watershed scales.

Some recently developed physics-based, integrated watershed models are briefly compared in Table 1-1. All of these models apply Richards’ equation for subsurface flow; however, the governing equations for surface flow, numerical methods and coupling approach are quite different. Therefore, several critical issues are still in need of further study, even though the Freeze and Harlan (1969) blueprint was proposed more than three decades ago.

Table 1-1: Comparison of watershed models

 Code Channel flow Overland flow Coupling between Surface and Subsurface Flows Reference GSSHA DIW (FDM) DIW (FDM) Time-lagged Downer and Orgden (2004) INHM DIW ( CVFEM) Discontinuous with linear Linkage-terms VanderKwaak(1999) MODHMS DIW (FDM) DIW (FDM) Discontinuous with linear Linkage-terms Panday and Huyakorn (2004) Morita and Yen DIW(FDM) Continuous Morita and Yen (2002) WASH123D DYW (MOC) DIW (FEM, ELM) KIW (ELM) DYW (MOC) DIW (FEM, ELM) KIW (ELM) Continuous or Discontinuous with general linkage terms (linear or nonlinear) Yeh et al. (2006)

Note: DIW: diffusion wave; DYW: dynamic wave; KIW: kinematic wave; ELM: Eulerian-Lagragian method; CVFEM: control volume finite element; FDM: finite difference method;  MOC: method of characteristics.

# 1.4 Format

Chapter 1 is an introduction and review of state-of-the art physics based, integrated watershed modeling.  The major findings from this thesis research are presented in the form of four journal articles, self titled as Chapters 2 through 5. Chapter 2 regards the accuracy and applicability of dynamic, diffusion, and kinematic wave models for surface runoff; it is based on a paper prepared for Journal of Hydrologic Engineering, ASCE. Chapter 3 regards dynamic wave modeling of overland flow, with a characteristics-based finite element method; it is based a paper prepared for the International Journal of Numerical Methods in Fluids. Chapter 4 is a comparative study of coupling approaches for surface water and groundwater interactions based on a paper intended for submission to the Journal of Hydrology.  Chapter 5 is an application on surface water and groundwater interactions in a constructed wetland, based on a paper prepared for possible publication on Journal of Hydrologic Engineering, ASCE.   The last chapter, Chapter 6, is summary of work presented and a presentation of suggested research and future applications.

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PHYSICS BASED, INTEGRATED MODELING OF HYDROLOGY AND HYDRAULICS AT WATERSHED SCALES