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A PROCESS-BASED DISTRIBUTED HYDROLOGIC MODEL AND
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Chaopeng Shen

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A PROCESS-BASED DISTRIBUTED HYDROLOGIC MODEL AND ITS
APPLICATION TO A MICHIGAN WATERSHED

By

Chaopeng Shen

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Environmental Engineering

2009

ABSTRACT

A PROCESS-BASED DISTRIBUTED HYDROLOGIC MODEL AND ITS
APPLICATION TO A MICHIGAN WATERSHED

By
Chaopeng Shen

The PAWS (Process-based Adaptive Watershed Simulator) model is a novel
distributed hydrologic model that is based on solving partial differential equations
(PDE) for physical conservation laws of the hydrologic cycle. The objective is to
create an efﬁcient physically-based modeling framework to describe the linkages
between processes at different scales and to improve the applicability of
physically-based models. The model simulates evapotranspiration, overland ﬂow,
channel flow, unsaturated soil moisture, groundwater ﬂow, depression storage,
vegetation growth and snowpack. PAWS focuses on the dynamic surface- subsurface
interactions and integrated responses by efficiently coupling runoff and groundwater
ﬂow to the vadose zone processes governed by the Richards equation. This novel
approach solves a long-standing bottleneck in PDE-based subsurface ﬂow modeling
by removing the computational limitations while maintaining physically consistent
solutions. Surface flow is solved by an efficient Runge-Kutta Finite Volume (RKFV)
scheme.

We follow the Freeze and Harlan (1969) blueprint in that we believe each component
of the model should be veriﬁable by itself. All ﬂow components have been

independently veriﬁed using analytical solutions and experimental data where

applicable. PAWS utilizes readily available data from national databases. The model is
applied to a medium-sized watershed in Michigan achieving high performance
metrics in terms of streamﬂow prediction at two gages during the calibration period
and the veriﬁcation period. The baseflow ﬂow periods are described particularly well.
Starting from a rough initial estimate of the groundwater heads, the model describes
the observed groundwater heads well (R2=0.98). The annual hydrologic ﬂuxes are
close to those estimated by a calibrated SWAT model. The model is considerably less
expensive than previous physically-based models of similar complexity.

The model is able to elucidate the complex interactions of processes in space and time.
Such detailed, quantitative and mechanistic descriptions cannot be produced by
conceptual models. The watershed is found to be a subsurface-dominated system with
saturation excess being the main runoff generation mechanism. Inﬁltration, recharge
and ET are also found to be strongly related to topography and groundwater flow. The
large seasonal variation of energy input drives the strong annual cycle and markedly

different responses in streamﬂow.

To my father, Hongxun Shen, and my mother; Jieying Li,
who have given me the best they have, and

all the love in the world

iv

Acknowledgements

I would like to express my sincere appreciation to my research and academic advisor,
Dr. Mantha S. Phanikumar, for his diligent, insightful and incredibly careful guidance
throughout my graduate studies, without which this dissertation would have been
impossible. I still have much to learn from his relentless enthusiasm, rigorous
professionalism and uncompromising pursuit of quality. I am grateful that I had the
opportunity to work with him on such a grand project.

I would like to thank my committee members: Dr. Joan B. Rose, Dr. Shu-Guang Li
and Dr. Ashton M. Shortridge for their invaluable comments, suggestions and advice
to improve this thesis to its ﬁnal form.

I would like to express my sincere appreciation for NOAA Center for excellence for
Great Lakes and Human Health which has provided the ﬁnancial support for my Ph.D.
study.

I would like to thank lab mate Niu Jie for his assistance with data processing and GUI
development. I also thank Mehmet Oztan, Hassan Abbas and Dr. Huasheng Liao in Dr.
Shu-Guang Li’s group for their assistance with groundwater data processing.

I would like to acknowledge the folks at MSU High Performance Computing Center
(HPCC) for the numerous help they offered. Special thanks go to Andrew Keen and
Dr. Dirk Colbry.

I want to take the chance to thank the many who have helped with experimental data

collection reported in this dissertation, including Dr. Aslam Irfan, Dr. Shawn
McElmurry and Heather Stevens, to name a few.

I would pay special thanks to Dr. Susan Masten who has initially given me the chance
to study at MSU. Otherwise this dissertation would have been non-existent.

Finally I want to thank Miss Wenqian Shan for her endless care and dedicated support,

which has given me much of the motivation at all stages of my Ph.D. study.

vi

ll

 

Table of Contents

List of Tables .................................................................................................................. x
List of Figures .............................................................................................................. xii
Chapter 1. Background and literature review ................................................................ 1
1.1. Motivation for a new hydrologic model .................................................... 1
1.2. Review of hydrologic models .................................................................... 4

Chapter 2. Development of the Hydrologic Model: Mathematical Bases and Test

Cases .....
2.1.
2.2.
2.3.

2.4.

2.5.

2.6.
2.7.
2.8.

....................................................................................................................... 12
Model Processes ....................................................................................... 13
Discretization and Representations .......................................................... 15
Hydrologic Processes and Solution to each ﬂow domain ........................ 20

2.3. 1. Evapotranspiration ............................................................................. 20

2.3.2. Vegetation .......................................................................................... 29

2.3.3. Overland ﬂow .................................................................................... 31

2.3.4. Channel ﬂow ...................................................................................... 43

2.3.5. Unsaturated vadose zone model ......................................................... 48

2.3.6. Saturated groundwater ﬂow model .................................................... 54
Interactions between domains .................................................................. 61

2.4.1. Coupling of the Unsaturated Richards equation and the groundwater

flow equation: .............................................................................................. 61

2.4.2. Coupling of the vadose zone and surface ﬂow: ................................. 75

2.4.3. Interaction between overland ﬂow and channel ﬂow ........................ 79

2.4.4. Interaction between groundwater and channel ﬂow .......................... 84
Data preparation steps general to all watersheds ..................................... 86

2.5.1. Digital elevation data ......................................................................... 86

2.5.3. Soils .................................................................................................... 91

2.5.4. Climatic data ...................................................................................... 91
Model Flow Diagram ............................................................................... 93
Test cases ................................................................................................. 95
Summary and Conclusions .................................................................... 112

Chapter 3. Development of the Hydrologic Model: Model Applications and

Comparisons .............................................................................................................. 114
3.1. The Red Cedar River watershed model ................................................. 114
3.1.1. Study site and input data .................................................................. 114
Groundwater data ....................................................................................... 128

vii

3.2.

3.3.

3.4.
3.5.
3.6.

3.1.2. Model Calibration ............................................................................ 130

SWAT Model for comparison ................................................................ 131
3.2.1. Brief summary of SWAT mathematical bases ................................. 132
3.2.2. SWAT Model Setup .......................................................................... 136

Results and Discussions ......................................................................... 139
3.3.1. Model Evaluation .......................................................................... 139
3.3.2. Model performance evaluations ....................................................... 139
3.3.3. Additional model results and hydrologic system of the RCR
watershed ................................................................................................... 151
3.3.4. Understanding the hydrology ........................................................ 153

Summary and Conclusions .................................................................... 167

Limitations and future research ............................................................. 168

Software package ................................................................................... 168

Chapter 4. Estimating Longitudinal Dispersion and Surface Storage in Streams Using

Acoustic Doppler Current Proﬁlers ........................................................................... 170
4.1. Introduction .............................................................................................. 170

4.2. Description of Sites .................................................................................. 179

4.3. Materials and Methods ............................................................................. 182
4.3.1. Transient Storage Modeling ............................................................. 188

4.3.2. Multi-resolution Wavelet Decomposition of ADCP Data ................ 191

4.3.3. Estimating the Longitudinal Dispersion Coeﬁicient ....................... 196

4.4. Results ......................................................................................................... 200
4.4.4. Evaluation of Channel Features and Potential for Hyporheic
Exchange in the RCR ................................................................................. 200

4.4.5. Evaluation of TS Model Parameters for RCR ................................. 202

4.4.6. Results from ADCP Surveys and Wavelet Analysis ........................ 207

4.4.7. Estimating Longitudinal Dispersion in Rivers ................................. 215

4.5. Conclusions ................................................................................................. 229
Appendix A. Supplemental Tables ............................................................................ 232
Appendix B. User’s Manual for the model Graphical User Interface ....................... 238
B. 1. Creating and Running the model in interactive mode ................................ 238

B. 1.1. Installing and starting the model ..................................................... 239

B. l .2. Loading the data .............................................................................. 240

B.1.3. Setting up the grid ........................................................................... 243

3.1.5. Setting up solution schemes and time steps .................................... 247

B. l .6. Final model set up ........................................................................... 249

B.1.7. Saving and Loading model .............................................................. 250

8.1.8. Running the model .......................................................................... 250

82. Running the model in non-interactive mode .............................................. 252

viii

References .................................................................................................................. 254

List of Tables

Table 2.1. Modeled Processes ...................................................................................... 13
Table 2.2. Major state variables and their symbols ...................................................... 15
Table 2.3. Supported land use types and their representations .................................... 18

Table 2.4. Simplied decision table for the atmospheric transmittance 1: from [Spokas

and F orcella, 2006] ...................................................................................................... 25
Table 2.5. Soil Parameters for the test case: Inﬁltration into very dry soils .............. 103
Table 2.6. Soil Parameters for the test case: Vauclin experiment .............................. 108

Table 3.1. Climatic Data sources and data availability. The dates in YYYYMMDD
format are the dates when a station starts to have records; Frac is the fraction of the
RCR watershed that is controlled by this weather station; PRCP PCT is the percentage
of valid precipitation records from 1998 to 2007 ...................................................... 119

Table 3.2. Land use percentages from the NLCD database and the MDNR databse 122
Table 3.3. SWAT Land use class percentages after apply thresholds. URLD: Low
Intensity Urban; URMD: Medium Intensity Urban; WETF: Forested Wetland; URHD:
High Intensity Urban; HAY: Hay/Forage Crops; AGRR: General Agriculture; FRSD:

Deciduous Forest ....................................................................................................... 138

Table 3.4. Performance metrics evaluating the model applied to the RCR watershed

.................................................................................................................................... 142
Table 3.5. Metric of Groundwater heads comparison to wellogic data ..................... 147
Table 3.6. Annual Average Fluxes compared to SWAT ............................................. 151

Table 4.1. Parameters in the Transient Storage model estimated for four different ﬂow
rates ............................................................................................................................ 205

Table 4.2. Comparison of the Relative Sizes of the Transient storage zones estimated
using tracer data and independently using the ADCP data for Reach A .................... 213

Table A. 1. List of input data to the watershed model and format .............................. 232

Table A2. An example transformation matrix from MDNR dataset to model classes
.................................................................................................................................... 233

Table A3. Summary of several watershed-scale hydrologic models ........................ 236

xi

List of Figures

(Images in this dissertation are presented in color)

Figure 2.1. Deﬁnition sketch of the model. T: transpiration, p: precipitation, EO
Evaporation from overland ﬂow/stream, EB: evaporation from bare soil; 1: inﬁltration;
R: Recharge .................................................................................................................. 14

Figure 2.2. Illustration of Arakawa-C grid ................................................................... 37

Figure 2.3. Relationship between K, 0 and h at different parameter values with the van
Genuchten formulation for a hypothetic soil type (a) at different a; (b): at different it;
(c) at different N. THE in the ﬁgure heading means 0. The base parameter for this

comparison is a= 2.49(1/m), N = 1.507, 05 = 0.43, 0, = 0.01; Ks = 0.175 (m/day); it =

-0. 14. Unit in the ﬁgure is the same as the base parameter set .................................... 49

Figure 2.4. sketch of the Vauclin 1979 test problem. HW(x) is the water table location

at x, z is measured 0 at the bottom. .............................................................................. 65
Figure 2.5. Illustration of Assumptions 1 and 2 ........................................................... 66
Figure 2.6. h(x,z) as a function of z and DR values ..................................................... 70

Figure 2.7. sketch of a river cell in the model. (a). cross-sectional view, (b) plane view
...................................................................................................................................... 82

Figure 2.8. Calculation of River/Land exchange ......................................................... 83

Figure 2.9. Discrepancies between NED and DEM when aggregated into the same
grid ............................................................................................................................... 87

Figure 2.10. the river bed elevations estimated based on the DEM and NED as

compared to the groundwater head .............................................................................. 90
Figure 2.11. Rainfall pattern of the Midwest from ...................................................... 92
Figure 2.12. Program Flow Chart for the proposed model .......................................... 94

Figure 2.13. Outﬂow hydrograph of the inclined plane test problem compared to the
analytical solution ........................................................................................................ 97

xii

Figure 2.14. Sketch of the V-Catchment test problem ................................................. 99

Figure 2.15. Solution to the V-Catchment test problem: Comparison of River outﬂow
hydrograph with analytical solution and the ﬁnite element (FE) solution reported in
[DiGiammarco]. (a) River outﬂow (b) Plane side outﬂow ........................................ 100

Figure 2.16. Simulated channel ﬂow vs measured data by [Irfan, 2002]. Red circle:
measured data; blue solid line: Simulated ................................................................. 102

Figure 2.17. Inﬁltration into very dry soil test problem at t = 6hr. circle: Analytical
solution; solid line: Ax= 0.075cm; dashed line: Ax= 0.25cm; dashed line with cross:
Ax: 0.6cm. ................................................................................................................. 104

Figure 2.18. Pumping near impervious wall test problem ......................................... 106

Figure 2.19. Solution to the pumping near impervious wall test problem: transient
comparison; ................................................................................................................ 107

Figure 2.20. Comparison of the proposed model with experimental results from
[Vauclin et al., 1979] .................................................................................................. 109

Figure 2.21. Comparison of soil moisture proﬁle obtained using the approach in the

proposed and traditional method, 03:0.3, EBC=Equation Boundary Condition

(Present approach); DBC=DirichIet Boundary Condition (Conventional approach) 111

Figure 3.1. Location of the Red Cedar River Watershed ........................................... 116
Figure 3.2. Elevation map of the Red Cedar River watershed ................................... 117
Figure 3.3. Locations of weather stations used for climatic input data ..................... 118

Figure 3.4. Basin-average Annual precipitation for the years from 1988 to 2008 (mm)

.................................................................................................................................... 119
Figure 3.5. NLCD Land use/Land cover map for the RCR watershed ...................... 121
Figure 3.6. River system of the RCR watershed with USGS ﬂow gages .................. 123

Figure 3.7. Geology maps of the RCR watershed. a. Land Systems; b. Bedrock
Geology ...................................................................................................................... 125

xiii

 

.. “4.5.!4 i

Figure 3.8. Soil property maps of the RCR watershed as reported in the STATSGO
database. a. Sand percent; b. Soil Available Water Capacity; c. Saturated Hydraulic

Conductivity (Ksat) .................................................................................................... 127

Figure 3.9. Watersheds delineated for the SWAT2005 model using the ArcSWAT GIS
interface ...................................................................................................................... 138

Figure 3.10. Comparison of the observed and simulated daily ﬂow at USGS gage
04112500 at East Lansing. Upper: Entire period from 1998/09/01 to 2005/12/31,
Middle and lower: Close-up look of the hydrograph ................................................. 141

Figure 3.1]. Comparison of the observed and simulated daily ﬂow at USGS gage
04111379 at Williamston ... ........................................................................................ 142

Figure 3.12. Comparison of the simulated streamﬂow from PAWS and SWAT and
observed USGS streamﬂow data. Upper: from 1999 to 2005, Lower: Close-up look of
water year 2001 .......................................................................................................... 144

Figure 3.13. Comparison of the simulated streamﬂow from PAWS and SWAT and
observed USGS streamﬂow data in log-scale ............................................................ 146

Figure 3.14. (a) 5 year averaged simulated groundwater, (b) observed heads
interpolated ﬁ'om wellogic database, (c) initial groundwater heads used to start the
model .......................................................................................................................... 148

Figure 3.15. Observed vs simulated groundwater heads. (a) Observed data vs 5 year
averaged value, (b) Observed data vs the rough initial guess. X-axis is the observed
data obtained from Kriging interpolation of the wellogic data .................................. 150

Figure 3.16. Pie chart showing the comparison of the average annual ﬂuxes as a
percentage of total precipitation between PAWS and SWAT ..................................... 152

Figure 3.17. Hydrograph partitioning into streamﬂow generation mechanisms:
Inﬁltration excess, saturation excess and groundwater contribution. X-axis are dates in
YY/MM/DD format. SatE: Saturation Excess, InfE: Inﬁltration Excess, Qgc:
Groundwater contribution, Qout: Red Cedar River outﬂow ..................................... 154

Figure 3.18. Temporal dynamics of ﬂuxes and state variables. Upper: Monthly mean
ﬂuxes (mm). Lower: Daily ﬂuxes (close up of 2004). Inf: Inﬁltration; ro: Overland
contribution to channel; Qgc: Groundwater contribution to channel; Rchrg: Recharge;
InfE: Inﬁltration Excess; SatE: Saturation Excess; Dperc: Deep Percolation ........... 155

xiv

Figure 3.19. Basin average state variables (a) Soil Water Content (change from initial
state), (b) Snow Water Equivalent. X-axis marks the beginning of each year (YY). 157

Figure 3.20. Daily ﬂuxes of recharge and groundwater baseﬂow. (a) Rising limb of
baseﬂow, (b) falling limb of baseﬂow. Again, dates are in YY/MM/DD format. Unit
of the ﬁgure is m ........................................................................................................ 159

Figure 3.21. Average annual runoff generation (mm) (a) Saturation Excess (b)

Topographic Index ..................................................................................................... 163
Figure 3.22. Average Annual Inﬁltration Excess ....................................................... 164
Figure 3.23. Annual Inﬁltration (a) and recharge map (b) ......................................... 165

Figure 3.24. Annual ET (a) and EvapG (b) map. EvapG is evaporation from the
ground. ....................................................................................................................... 166

Figure 4.1. Red Cedar River and the sampling locations .......................................... 180

Figure 4.2. Red Cedar River and the study region showing the sampling locations of
the Grand River tracer study ...................................................................................... 182

Figure 4.3. Comparison of the numerical solution with the analytical solution of De
Smedt et al [2005] ...................................................................................................... 191

Figure 4.4. Schematic diagram illustrating the concept of discrete wavelet
decomposition. S denotes the original signal (the 2D image of the velocity ﬁeld
measured by ADCP). L and H denote the low-frequency approximations and
high-frequency details, respectively. Sufﬁxes denote wavelet scale levels ............... 195

Figure 4.5. Comparison of observed and simulated tracer concentrations for four slug
releases conducted during summer 2002. (a) Q=2.49 m3/s. Sampling locations at
x=0.87km (Bogue Street Bridge) (b) O = 14.41 m3/s, sampling locations are, namely,
Farm Lane Bridge (x = 1.40km). Kellogg Bridge (x = 3.10km) and the Kalamazoo
Bridge (x=5.08km), respectively. (c) Q = 16.82km3/s, (d) Q= 19.06m3/s ................. 203

Figure 4.6. Observed mean velocity ﬁelds at two different stations in reach A in the
Red Cedar River obtained using a 1200 kHz ADCP (a and b) O = 5.49m3/s (8
November 2003). (c and (1) Q = 19.89 m3/s (19 March 2006). Note that during the
high event the adjacent low lying areas near the banks were ﬁlled with relatively
stagnant water which were not available during low discharge ................................. 209

XV

Figure 4.7. Multiresolution wavelet approximations for the images shown in Figure
4.6. After completing the wavelet analysis, the low-frequency content at wavelet
levels 1 and 2 (denoted by L1 and L2) was plotted in the physical space. (a and b)
Hagadorn Bridge. (c and (1) Farm Lane Bridge. ........................................................ 211

Figure 4.8. Sample ADCP transect data used for computing the longitudinal
dispersion coefﬁcients. .............................................................................................. 218

Figure 4.9. Vertical velocity proﬁles in the Ohio and St. Clair Rivers showing the
effect of ﬁtting a power law (black lines) and logarithmic proﬁles (red lines). The raw
data from the ADCP is shown using symbols ............................................................ 220

Figure 4.10. Effects of smoothing (logarithmic, power-law or no-smoothing) and
velocity projection methods (rotating velocities in the streamwise direction or
projecting them in a direction normal to the transect track) on the dispersion estimates
from ADCP ................................................................................................................ 225

Figure 4.11. Comparisons between ADCP and tracer estimates of the dispersion
coefﬁcient: (a) Box plots denote the variability in D estimated using the ADCP
method within a given river reach. Tracer estimates based on the ADE and the
transient storage modeling are shown using different symbols. (b) Comparisons
between ADCP and tracer estimates plotted on top of similar results reported in the
literature. .................................................................................................................... 227

Figure 4.12. Comparison of ADCP estimates of the dispersion coefﬁcient with tracer
estimates and results from empirical relations. (b) Box plots showing the deviation

between estimates of D based on the ADCP method and empirical relations ........... 228

Figure 4.13. Depth-averaged longitudinal (u) velocity proﬁles plotted as a function of

the transverse coordinate y ........................................................................................ 229
Figure B. 1. GUI after the model is started ................................................................. 240
Figure 8.2. Data GUI. (a) before loading data (b) after loading data ﬁles ................ 242
Figure 8.3. GUI after the model is started ................................................................. 244
Figure B.4. Discretization GUI .................................................................................. 246
Figure B.5. Solvers GUI. (a) after it is opened (b) after (it ﬁle is loaded .................. 248
Figure B.6. Loading solvers data into solvers functions GUI .................................... 249

xvi

Chapter 1. Background and literature review

1.1. Motivation for a new hydrologic model

In the twenty-ﬁrst century the management of water resources has gained
unprecedented scientiﬁc as well as strategic importance. On one hand, the growing
human population raises larger and larger demand for usable and available water
resources. Satisfying such a need can be a serious challenge in many parts of the
world. On the other hand, humans are equipped with the ability greater than ever to
harness water in nature for their own end, and they often opt to do so. These
interventions have led to some fundamental modiﬁcations to the hydrosphere. In the
past century we have essentially transformed the Earth’s ecosystem, even on the
grandest scale, into a human-dominated one [Vitozrsek et al., 1997]. While many of the
large-scale hydraulic projects proved essential for the economic development and
well-being of people and their positive impacts should not be downplayed or
understated, the hydrological, ecological and environmental consequences are usually
not fully understood. Today the Aral Sea in Central Asia, once the fourth largest
inland lake in the world, has been desiccated to 1/10 of its original size with its entire
ecological system destroyed. Studies reveal. that this overwhelmingly owns to the
diversions of water from the sea’s tributaries for irrigation expansion [Cretaux et al.,
2005; Mick/in, 1986; 1988]. The impacts of such interventions are so far reaching,

profound and irreversible that all aspects of the hydrologic system need to be

carefully assessed.

The surface-subsurface water interactions are often overlooked. Due to the slow
response of the subsurface water system, impacts of large—scale projects - benign,
malign or neutral - can take a long time to manifest themselves. The Aswan Dam in
Egypt and its reservoir, Lake Nasser, still witness newly discovered impacts that could
not be predicted 30 years ago when the dam was constructed. Some have observed
reduced carbon emission and more sustainable development [Prasad et al., 2001;
Strzepek et al., 2008], while others reported. erosion, salinization, and pollution that
induced decline in agricultural productivity and loss of land and coastal lagoons
[Stanley and Wame, 1993].

These challenges are further complicated by the more and more pronounced trend of
climate change. Climate change is expected to exacerbate current water stresses
[IPCC, 2007]. Semi-arid regions and drought-affected areas are projected with high
conﬁdence to suffer decreased water resources due to climate change. For example,
the Lakes Mead and Powell created by the Hoover Dam, the lifeblood of US.
southwest, are estimated to dry up by 2021 if no changes to the current water usage
are made [Barnett and Pierce, 2008; 2009]. The study has attributed this result to
global warming and current operating conditions.

Understanding water ﬂuxes is also important from a human health perspective. A
variety of pollutants including chemical and biological agents pose threats to human

and ecosystem health [U.S.EPA, 2000]. A well-known USGS study that involved 139

streams in 30 US states between 1999 and 2000 found pharmaceuticals, hormones and
a number of emerging contaminants in 80 % of the streams sampled [Kc/pin et al.,
2002]. Similarly a majority of rivers sampled in Michigan tested positive for the
presence of viable enteric viruses [Jenkins et al., 2005]. Therefore understanding
factors that inﬂuence the fate and transport of contaminants in rivers and streams is
extremely important from the point of predicting human health risks and protecting
the public. The ﬂow generation process governs the source and form of contamination.
Pollutants can reach streams via point source discharge, non-point source (overland
ﬂow), or subsurface seepage [Jamieson et al., 2004].

All these challenges call for resolute but well-informed decision making. Sound
decision making is best aided by good understanding of the complex and interrelated
hydrologic systems. Better understanding comes most elucidated with the ability to
explicitly describe the hydrologic processes in space and time. This is where a reliable
and veriﬁable process-based hydrologic model with good. predictive power could play
an important role.

The present study attempts to create a process-based hydrologic model that ﬁnds a
good balance between process modeling and applicability. This model should link
processes that occur at different scales and illustrate the interactions among the
hydrologic domains, including, surface water, soil water, groundwater, river, canopy
and atmosphere. It is hoped that this model will help address some of the challenges

posed by human intervention as well as climate change.

1.2. Review of hydrologic models

Many hydrologic models have been historically developed to study different
hydrologic systems. Several models that are recently cited are reviewed in Table A3.
Models are often developed with speciﬁc scientiﬁc objectives in mind. Thus each
model has its own strengths in some areas and may be inadequate in some other areas.
For example, SWAT is designed as a long term water balance and non-point source
pollution simulator [Arnold and Allen, 1996; Arnold and Fohrer, 2005]. Thus
simulating short term point source pollution with SWAT may not be advantageous.
Before developing our own model it is helpful to understand the current state of
research.

Broadly, there are two categories of models: conceptual and physically-based (or
mechanistic). Generally, in conceptual hydrologic models (CHM), the modeler forms
hypotheses, either from experience or his own perceptions, about the hydrologic
processes and proposes mathematical formulations to represent these processes with
sometimes strong simplifying assumptions. CHM are often based on empirical
relations and conceptual state variables that cannot be always measured.
Physically-based hydrologic models (PBHM), on the other hand, are derived
deductively from established physical principles with appropriate assumptions and
physically meaningful/measurable parameters [Beven, 2002]. Historically there have

been heated discussions about the relative advantages and disadvantages of the two

approaches. In 1966, the paper [Freeze and Harlan, 1969] laid out a blueprint for
physically based hydrologic modeling, writing out the equations for different ﬂow
processes and the linkages via common boundary conditions. The development of
such a physically based model in the years followed, however, has been limited by the
computational power, data availability, understanding of the complex hydrologic
system and, to a lesser extent, the accumulation of mathematical technique. [Beven,
2002] has challenged that blueprint and provided an alternative blueprint to
hydrologic modeling that is based on lumped conceptual models. These models do not
use the process theory to build a model structure a priori, but rely on observed data to
deﬁne an appropriate model structure (described by Beven as ‘hypothesis testing’).
He also stressed on the importance of quantifying the uncertainty of the models. With
the advancement of computer power, Geographic Information System (GIS) and
readily available databases, recently published models lean more and more toward the
physically-based approach [Karvonen et al., 1999]. However, to date there does not
seem to be a conclusion to that debate and the two schools of models continue to be
created and advanced.

In fact, both types of models have their own advantages and disadvantages. Some of
the notable conceptual models include TOPMODEL [Beven and Kirkby, 1979],
DLBRM [Croley and He, 2005], Sacramento Soil Moisture Accounting Model
(SAC-SMA) [Burnash, 1995], HEC-HMS [HEC, 2000], VIC-3L [Liang and Xie,

2001; Reed et al., 2004], etc (also see [Borah and Bera, 2003; Reed et al., 2004]).

Usually, conceptual models require less physical input as its components are idealized.
The models tend to be structurally simple, computationally inexpensive and more
easily operational. However, they need long term monitoring data to calibrate. The
parameters generally cannot be applied for ungauged watersheds. Moreover, the
conceptualization process blurs the underlying dynamics and extension of conceptual
models beyond the range of calibration is questionable [Beven, 1985]. It is not rare
that a CHM that describes completely different physics from the study region and still
get fair results after optimization, but the physics can be far from reality. In fact, a
large number of papers have been published to quantify the uncertainties associated
with the conceptual models (e.g. see [Beven, 2006; Beven and Binley, 1992; van
Griensven and Bauwens, 2003; Vrzrgt et al., 2005; Yang et al., 2008], etc). The large
uncertainties undermine the reliability of the model, especially since estimated
parameters cannot be interpreted.

A review of current physically-based models, on the other hand, reveals that there is
still much room for improvement, especially in achieving the right balance between
process descriptions and computational demands. PBHM generally tend to be data
intensive and computationally expensive and thus their applicability tends to be
limited. The reported results from PBHMs are often simulations for small areas
during short periods of time. The published comparisons with observations indicate
that these models, as commented by [Ivanov et al., 2004a], are yet to emerge as the

preferred tool for prediction and analysis. Some well-documented physically based

hydrologic models include the MIKESHE model [DH], 2001 ], Soil & Water
Assessment Tool (SWAT) [Arnold and Fohrer, 2005], Water and Energy Transfer
Process (WEP) model [Jia et al., 2001; Jia et al., 2006], CASC2D/GSSHA [Downer
and Ogden, 2004b] and tRIBS [Ivanov et al., 2004b], etc. We must mention here that
some regard SWAT as a semi-physically based model since considerable amount of
empiricism is included in the model, as highlighted in section 3.2. Although almost all
PBHMs still carry some level of empiricism, the models can be constrained much
better by real data since most parameters are physically based.

Here we review six watershed-scale and one macro-scale (VIC) PBHM. Table A3 in
the appendix lists, in alphabetical order, seven models that are currently being
published and cited in the literature. This is by no means a complete list of hydrologic
models that are of interest, but it does give a representative coverage. In its
completeness, watershed hydrologic models should incorporate several ﬂow domains
that cover various ﬂow paths possible after rain drops to the ground: overland ﬂow,
unsaturated subsurface ﬂow (the vadose zone model), saturated subsurface ﬂowv
(groundwater ﬂow) and channel ﬂow. Water also exists in canopy interception,
snowpack, biomass and depression storage. Besides the ﬂow domain, one of the
processes of utmost importance is the evapotranspiration, which on average is
estimated to take out 70% of the rainfall in North America [Jensen et al., 1990]. In
order to account for the seasonal dynamics of a watershed, a reasonable vegetation

growth module should also be included. A sufﬁcient PBHM should contain all of the

relevant processes. Table A3 also lists the spatial discretization method and solution

schemes to the ﬂow problems.

The InHM [Vandeeraak and Loague, 2001] is a processe-based model that
integrates overland ﬂow with the mixed form three-dimensional Richards equation for
the subsurface. InHM simulations in showed that both the Horton and Dunne overland
ﬂow mechanisms can be important streamﬂow generation processes. The InHM
simulations also suggested that accurate accounting of soil water storage can be as
important as exhaustive characterization of spatial variations in near-surface
permeability. However, the InHM is very computationally demanding such that it can

only applied at the plot-scale.

The GSSHA model, developed for the Army Corps of Engineers, evolved from
CASCZD [Downer and Ogden, 2004b]. CASC2D uses Green and Ampt inﬁltration
method [Green and Ampt, 1911] with redistribution (GAR) for moisture accounting.
However, it has been found that the processes modeled in the CASC2D model cannot
adequately describe watersheds where saturation excess is important [Downer et al.,
2002]. This research highlights the importance of incorporating the correct processes
in physically-based models and applying hydrologic models in the settings where they
are applicable. In GSSHA, the subsurface ﬂow component is improved by coupling

the Richards equation with the groundwater ﬂow equation. The height of the water

table is provided as the lower boundary condition to the Richards equation. A
constantly changing discretization is used to cope with the rise and fall of the
groundwater table. As will be shown later in section 2.4.1, this coupling approach will
cause soil moisture proﬁle to be inconsistent with the location of the groundwater
table. Moreover, it is difﬁcult to evaluate the potential of GSSHA or CASC2D as a
large-scale long term analysis tool as most of the published results are for small
catchment areas (20 km2 and 3.64 km2 in [Downer and Ogden, 2004a] and 3 km2 in
[Downer and Ogden, 2004b]) in a short time frame (<200 days). The comparison is

also limited to streamﬂow measurements.

The WEP was originally developed by [Jia et al., 2001] and applied in Japan to a
small size urban watershed. Later, it was expanded into a large scale version WEP-L
and applied to a very large basin — the Yellow River Basin in China [Jia et al., 2006].
The WEP model uses the mosaic approach which allows sub-grid heterogeneity to be
parameterized. Three layers of soils are deﬁned in the model and water is allowed to
move only vertically in the soil. The soil layers are connected to the unconﬁned
aquifer which is modeled using the Boussinesq equation. The proposed model shares
some similarities in the structure with the WEP model. The results reported for both
the Japanese watershed and the Yellow River basin are promising. Unfortunately, this
model is not available so direct comparison is not possible. It is felt that applications

to some medium-sized watersheds with different hydrologic settings and geologic

conﬁgurations may help further illustrate the general applicability of the WEP model.

[Panday and Huyakorn, 2004] details the processes included in the MODHMS model,
which is a commercial package. Overland ﬂow exchange with channel is modeled as
ﬂow over a rectangular weir. A similar approach is used by the proposed model. The
subsurface is modeled with a variably saturated 3D ﬂow model. 3D modeling is
generally considered to be computationally too expensive for watershed-scale
modeling. Since in this paper no application to real watershed is presented, we cannot

assess the applicability of this approach.

tRIBS uses a unique Triangular Irregular Network (TIN)-based discretization and thus
is thought to be capable of more properly capturing the spatial heterogeneity in
t0pography, land use or soils. However, the mapping of data into the triangular
element (‘Voronoi Regions’ in the original paper) requires special mesh generation
and processing procedures, which can be non-trivial for other modelers. The tRIBs
model has been applied to several medium-sized watersheds [Ivanov et al., 2004a]
and obtained reasonable results. More recently, a sophisticated weather generator
[Ivanov et al., 2004b] and a vegetation growth module have been added to the model.
The impact of topographic controls has been studied in detail, prompting new

directions of catchment hydrology [Ivanov et al., 2008a; b].

10

The MIKE-SHE model is a commercial hydrologic model that contains
comprehensive modules. The model was developed in Europe. It has a 1D Richards
equation for the unsaturated zone linked to the groundwater aquifers. 3D groundwater
ﬂow is simulated. Procedurally, 3D saturated ﬂow modeling is not very much
different from 2D modeling. The difficulty is mainly with the unsaturated vadose zone.

To be able to simulate transport process in a 8.7 km2 watershed, [Thompson et al.,

2004] created a MIKESHE model with a 30x 30m grid.

The PAWS model proposed in this dissertation attempts to improve the applicability
of PBHM by ﬁnding the right balance between processes, data and computation. The
model follows the Freeze and Harlan blueprint in that we believe each component of
the model should be veriﬁable by itself. PAWS uses accurate and computationally
efﬁcient schemes to solve the physically-based governing equations. The descriptions
of the unsaturated soil water domain by the Richards equation allow dynamic
interactions among different components. In the next chapter, we will develop the

mathematical details.

11

Chapter 2. Development of the Hydrologic

Model: Mathematical Bases and Test Cases

Mathematical models are, by deﬁnition, simpliﬁcations and abstractions of the real
world. The model designer chooses, using his own judgment, the processes that are
important to certain objectives and endpoints, and builds mathematical representations
of the processes. For the hydrologic modeling in our context, the processes involved
start at the moment precipitation reaches the ground (or canopy) and ends when water
exits the system via channel, overland ﬂow boundary, groundwater ﬂow boundary or
evapotranspiration.

In this chapter we discuss the mathematical basis of PAWS. We lay out our general
conceptual representation of the watershed and the modeled processes in the ﬁrst
section. We then describe the governing equations for each hydrologic unit and how
the 3D physical space of the watershed is discretized into computational grids in the
next section. In the third section the numerical solution schemes to solve the equations
or general calculation steps are described. In the last section, we compare our
numerical solution for each component with the analytical solution for one or more
test cases to ensure the accuracy of the numerical code. Where applicable,
experimental data are compared with numerical solutions to test the model

performance.

12

2.1. Model Processes

The processes modeled in PAWS are graphically presented in Figure 2.1 and
summarized in Table 2.1. The eight compartments where most calculations take place
are, respectively, surface ponding layer, canopy storage layer, impervious cover
storage layer, overland ﬂow layer, snowpack, soil moisture, groundwater aquifers and

channels. The major state variables are summarized in Table 2.2.

Table 2.1. Modeled Processes

 

Processes

Governing Equations

 

Snowfall Accumulation and melting

Mass and energy balance (U EB) [Luce
and Tarboton, 2004]

 

Canopy interception

Bucket model, storage capacity related to
Leaf Area Index (LAI)

 

Depression Storage

Speciﬁc depth

 

Runoff

Manning's formula + Kinematic wave
formulation+Coupled to Richards

equaﬁon

 

lnﬁltration/Exﬁltration

Coupled to Richards equation

 

Overland ﬂow

2D Diffusive wave equation

 

Overland/Channel Exchange

Weir formulation

 

Channel network

Dynamic wave or diffusive wave

 

Evapotranspiration

Penman Monteith + Root extraction

 

Soil Moisture

Richards equation

 

Lateral Groundwater Flow

quasi-3D

 

Recharge/Discharge

Non-iterative coupling inside Richards

 

 

(Vadose zone/Groundwater interaction) equation
Stream/Groundwater interaction Conductance/Leakance
Vegetation Growth Simpliﬁed Growth Cycle

 

 

 

 

Figure 2.1. Deﬁnition sketch of the model. T: transpiration, p: precipitation, E0
Evaporation from overland ﬂow/stream, EB: evaporation from bare soil; 1:
inﬁltration; R: Recharge

14

Table 2.2. Major state variables and their symbols

 

 

 

 

 

 

 

 

 

State Variable Symbol ’ Unit .
Soil Water Content, 0 (-),
Pressure Head“ h m
Surface Ponding Depth hl m
Overland Flow Depth" ‘ h m
Canopy Storage CS m
Snow Water Equivalent, SWE, m,
Snow Energy Content U kJ/m2
Groundwater Head H in
River Cross Sectional Area A m2
Ponding storage on impervious cover hi m

 

 

 

 

 

* There is no confusion between soil pressure head and overland ﬂow depth since
they are described in different sections, thus they share the use of symbol h.

It is clear that the vadose zone plays a central role in the model as it connects to
almost every other component. The vadose zone links surface water with groundwater
and is responsible for applying evapotranspiration. Having an efﬁcient, ﬂexible, and

robust vadose zone module is thus critical to the success of the model.

2.2. Discretization and Representations

2.2.1. Horizontal representation

The spatial domain of the watershed of interest is discretized into a structured grid as
illustrated in Figure 2.1a. Rivers are modeled as separate objects in a network of
channels. Each river may contain different number of cells with possibly variable

spatial step sizes. Within each river cell, the river characteristics such as river width,

15

bottom elevation, water depth, are assumed to be uniform. The intersecting length of
river cells with land cells are pre-computed and stored in sparse matrices. These
matrices are used to convert the exchange ﬂuxes from land grid to river cell grid as
will be explained in 2.4.3. This ﬂexible strategy allows river and land discretizations

to be independent and the coupling can be done regardless.

Within each land cell, sub-grid heterogeneity is modeled using a method similar to the
mosaic approach reported in [Jia et al., 2001]. Depending on the grid size, a cell may
possess a mixture of land use/land cover types. Since different land use/land cover
classes respond differently to hydrologic events, the treatment of sub-cell
heterogeneity is important. However, nationally-prepared, readily available datasets
contain too many plant types for a model to simulate. A cell may contain many classes
each taking up only a small fraction of the cell area. Moreover, some classes in the
database are a complex group of several plant species. For example, low intensity
urban commonly has non-negligible amount of soil-vegetation cover. Considering
these factors, land use data are re-classiﬁed into model classes which are represented
by certain plant types. Table 2.3 summarizes the representative plant types (RPT)
currently modeled. Table A2 provides an example transformation matrix from the
land use classiﬁcation provided by the Michigan Department of Natural Resources
[ll/HDNR, 2010b] to the model classes. For instance, the ﬁrst row of the table which

reads ‘low intensity urban’ is divided into 40% of impervious, 20% of Deciduous

l6

forest (modeled by Oak), and 40% of grass cover. The urban information is obtained
from [NRCS, 1986] Then, the fractional areas of each RPT inside a cell are summed
up. Then the number of land use classes that are going to be modeled (nRPT) should
be determined. Considerations include cell sizes, the complexity of the landscape, and
the computational cost. Smaller cell size would need correspondingly smaller nRPT.
Further, the ﬁrst nRPT largest RPTs are selected as the model classes in the cell. Other
types would be removed and have their areas assigned to the selected RPTs. Rather
than simply allocating the areas that belong to the removed types proportional to the
areas of the selected RPTs, they are assigned to the RPT in the same group (as in
Table 2.3). This processing logic attempts to represent land use classes as close to the

original land use as possible, while reducing the demand for computational resources.

Currently, ﬁve characteristics completely characterize a certain RPT: crop coefﬁcient

for evapotranspiration (Kc), canopy height (he), rooting depth (Root), Leaf Area Index

(LAI) and growth periods (Tg). These characteristics are found from literature [Breuer

et al., 2003; FA 0, 1998; Neitsch et al., 2005].

17

Table 2.3. Supported land use types and their representations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Land use Class Representation Group

Water Water Water

Urban or Industrial Impervious Impervious
Evergreen Forest Pine Tall Vegetation
Deciduous Forest Oak Tall Vegetation
Bush and shrub Shrub Short Vegetation
Herbaceous Grass Short Vegetation
Row Crops Com Agricutural
Bare Soil ' Bare Bare

Forage Crops Alfalfa Agricultural

The soils data may come from STATSGO or SSURGO in the United States. In each
cell, as we only model one soil column, the most dominant soil type is used. For the
elevation, either the mean or the median value within a cell is taken as the elevation
for this cell, if the grid size is larger than the resolution of the digital elevation dataset

provided.

2.2.2. Vertical representation

The model structure in the vertical direction is given in Figure 2.1b. Water on the
ground is separated into two domains, the ponding water layer and the ﬂow domain.
The ponding domain will be subject to inﬁltration and evaporation whereas only
water in the ﬂow domain can move from one cell to another (including river cells).
Runoff from ponding to ﬂow domain can be calculated using several different
methods described in section 2.4.2. Depending on the climatic conditions, a layer of

snow, quantiﬁed by the Snow Water Equivalent (SWE) and snow cover fraction

18

(Afrac), may also exist on the ground. After taking out the canopy interceptions,
rainfall is joined by snowmelt to add to the ponding layer. Except for the nRPT land
use classes on the surface as described earlier, we model one soil column for each cell.
The soil column is responsible for computing inﬁltration from surface ground (the
ponding domain), soil evaporation, root extraction and percolation into unconﬁned
aquifer. Water is assumed to move only vertically in the soil column but once in the
aquifers it can move laterally. The unconﬁned aquifer has a thickness equal to its
water depth. It can exchange water and energy with the soil column. Water can
percolate ﬁirther down to deeper aquifers via a layer of aquitard and contribute to or
gain from river water through river bed materials. Pumping activities can also directly

extract water from the aquifer layers.

19

2.3. Hydrologic Processes and Solution to each ﬂow domain

The four ﬂow domains that the model considers are overland ﬂow, channel ﬂow, soil
moisture and saturated groundwater ﬂow. They will be described one after another in
this section. The other important component that will be discussed in detail is

evapotranspiration.

The mass balance equation for the ponding layer on the ground is written as:

3’71
'07 = P — 05W + SNOM — Eg — Inf — Fg (2.1)

Where It] is the water depth in the surface ponding layer (m), P—CSnew is the
precipitation (m/day) reaching the ground after subtracting canopy storage, SNOM is
the rate of snowmelt (m/day), Eg is the rate of evaporation on the ground (m/day), Inf

denotes inﬁltration (m/day), and F g is the runoff to the overland ﬂow domain.

2.3. l. Evapotranspiration

Evapotranspiration is the collective term of evaporation and transpiration and is a
major component of the hydrologic cycle. In the proposed model, we apply the
reference evapotranspiration (RED/adjustment approach to calculate ET. First the
reference ET is computed using climate input data for a type of reference plant, and
then it is adjusted for different plants and soil moisture conditions. Multiple
deﬁnitions of potential evapotranspiration and reference evapotranspiration exist in

the literature. In particular, the term 'potential evapotranspiration‘ has caused a great

20

deal of confusion. In this dissertation, we follow the deﬁnitions of ‘reference ET’
given in [FAQ 1998]: The evapotranspiration rate from a reference surface, not short

of water.

The reference Evapotranspiration is computed using the Penman-Monteith (PM)

equation:

AU?" —G)+pcp(e;J —ez)/ra

(2.2)
A+'y(1+r‘c/ra)

AET =

 

where ET is the reference evapotranspiration on a day (mm/day) for a given reference

plant, it is the latent heat of vaporization (Ml/kg), Rn is the net radiation (MJ/day), G
is the ground heat ﬂux (MJ/day), A is slope of the saturation vapor
pressure-temperature curve (kPa/°C), p is the air density (kg/m3), cp is the speciﬁc
heat of moisture at constant pressure (1.013><103 MJ/kg/°C), e? is the saturation
vapor pressure at height 2 (kPa), eZ is the actual vapor pressure at height 2 (kPa), ra is
the aerodynamic resistance (s/m), rC is the canopy resistance (s/m), and y is the

psychrometric constant (kPa/°C). it, y, A and e? are calculated as

A = 2.501— 2.361 x 10‘3Ta

 

 

 

7 : cpPa
0.622/\
16.78T — 116.9
e0 = exp “ (2-3)
0 Ta + 237.3
4098e0
A = a

 

(Ta + 237.3)2

where Ta is air temperature in Celsius (°C), Pa is air pressure (kPa) calculated for a

21

given elevation Elev (m):
Pa = 101.3 — 0.01152Elev + 0.544 x10-6Elev (2.4)
The actual vapor pressure is calculated from relative humidity, which is often

provided in the form of dew point temperature from climatic data sources:

 

 

 

 

16.7.811de _ 116.9
exp T (2.5)
RH : dfflli0+ 237-3

ea

where RH is relative humidity and Tdew is dew point temperature (°C).Given the

maximum and minimum temperatures me (°C) and Tmn (°C) in a day, the

sub-daily temperature is computed [Campbell, 1985], assuming highest temperature at

3PM local time:

T = :r + Mcos(0.2618(t —15)) (2.6)
2

av

in which hr is the hour of the day (local time), Tav is average temperature of the day
(°C).

Solar radiation data are normally scarce but can be adequately estimated from
sun-earth relationships and other available climatic data. Thus a solar radiation
calculation scheme was developed to ﬁll the data gaps. The net radiation, Rn, is
computed at the weather station sites and applied to the model domain using nearest
neighbor interpolation. The net radiation is the sum of absorbed incoming shortwave

radiation, incoming and outgoing long-wave radiation:

22

Rn = (1 — (1)123 + H17: + H16 (2.7)
Where Rn is the net radiation (Ml/day), a is the albedo, Rs is the incoming shortwave
radiation, H1; is the incoming long-wave radiation (Ml/day), Hle is the outgoing

long-wave radiation ('MJ/day). All terms are non-negative except for H16.

The shortwave solar radiation at a given time is calculated by summing up the direct
radiation (or beam radiation) and the diffuse radiation:

R5 2 Sb + Sd (2.8)
where Sb is the beam radiation and Sd is the diffuse radiation. Sb is calculated from
the following formula:

too 3

. . 2 . . . .
in which, ISC=1367 w/m IS the solar constant, IS IS the atmospheric correction factor,

Itoa=ISC00392 is the top of atmosphere solar irradiance corrected for incident angle, 02

is the zenith angle between the sun and a given surface on earth:

i—l

cos 62 = sinqbsiné + coscbcosdcos (2.10)

 

where (p is the latitude of the site, 5 is the solar declination angle, t is the time of the

day and tsn is the local solar noon time of the day. The solar declination angle can be
calculated from[Campbell and Norman, 1998]:

siné = 0.39793in[4.869 + 00172.] + 0.033455in(6.2238 + 0.0172J)]
(2.11)

where J is the Julian day of the year. The term 0.033455in(6.2238+0.01721) accounts

23

for the eccentricity correction factor for the Earth's orbit.

Much research has been done to estimate the atmospheric correction factor 1:5. The
methods range from complex models that need comprehensive climate input to simple
empirical relations that use only temperature. To create an efﬁcient method for
estimating hourly incoming solar insolation based on limited climatic input (namely,
daily precipitation, maximum and minimum temperature), [Spokas and Forcella,
2006] proposed the following simpliﬁed formula for Is:

7' = Tm (2.12)
where t is the atmospheric transmittance given in Table 2.4. m is the optical mass

number:

P
= —-“-—- 2.1
m 101300562 ( 3)

in which, P21 is the atmospheric pressure (kPa) at the site.
The incoming long-wave radiation is emitted from particles and gases in the air. It is
estimated as:

HI = 500T: (2.14)

i

where ea is the air emittance, o the Stephen-Boltzman constant (4.903X10-9

-2 -4 -l . . . . . . . . .
MJm K d ), and TK rs air temperature 1n Kelvrn. The air emrssrvrty 18 evaluated

considering cloud cover and using the Satterlund parameterization [Luce and

Tarboton, 2004; Satterlund, 1979]:

24

T

a.

5a 2 Cf +(1— Cf)1.08 1 —exp —(10ea)201 (2.15)

 

In which, ea is the vapor pressure in the air (kPa), Cf is the Bristow and Campbell
transmission factor, which measures how close the air transmittance is to clear sky
conditions:

T
Cf =1— T (2.16)

712.77

 

In which 1: and me are actual and maximum air transmittance. They are provided by

the solar radiation algorithm described above, using Table 2.4 proposed by [Spokas

and F orcella, 2006].

Table 2.4. Simplied decision table for the atmospheric transmittance r from [Spokas

 

 

 

and F orcella, 2006]
Conditions Value of t
No precipitation and A 'I>10C T =0.70
No precipitation on present day, 1 =0.60

but precipitation fell the previous day

 

Precipitation occurring on present day I =0.40

 

Precipitation today and also the previous day I =0.30

 

If it is needed, the topographic effect may also be considered as in [Piedallu and
Gegozrt, 2007].
The diffuse radiation in Eq. (2.8) can then be evaluated as [Campbell and Norman,
1998f

Sd = 0.3(1— 7771)]

(2.17)

too

which assumes 30% of scattered/absorbed solar radiation by the atmosphere is then

25

reﬂected to the ground. The above model is easy to use due to its simplicity and

limited data demands .

The outgoing long-wave radiation is calculated as:
4
in which, fc is a factor describing the effect of cloud, and as is the emittance of ground

surface, their calculations are from [Neitsch et al., 2005],[Jensen et al., 1990]:

 

m (2.19)
e = —(0.34 — 0.139 ea)

This completes the estimation of net radiation. So far, the only missing terms from Eq.

(2.2) are ra, the aerodynamic resistance and re, the canopy resistance.

The reference ET calculated using alfalfa as the reference crop is used as the total ET
demand. It is ﬁrst used to evaporate any water in the canopy storage and on the
ground. The ET demand on the impervious fraction of the cell is used to evaporate
depression storage on the impervious fraction. This ET demand, if not depleted, is not
used in any other way. The ET demand on the snow-covered fraction is also discarded
as the energy is assumed to be used for the melt of snow, which is computed by other

methods (reference for these equations):

(if? T = TETnfarfa (1 _ fsnow)(1 — Amp) (220)
dET = dET - EC — Es

26

where dET is the remaining ET demand, I'ETalfalfa is the reference ET computed for
alfalfa. fimp is the fraction of impervious cover, fsnow is the fraction of snow-covered

area, Ec is the evaporation from canopy storage and BS is the evaporation of water on
the ground surface:

EC = n1a.x(dET, CS)
E3 = max(dET — Ec,h)

(2.21)
where h is the water on the ground surface (m). The transpiration demand for each
plant type is calculated by multiplying dET by the crop coefﬁcient:

dTPz. 2 K01? .- dET (2.22)
In which, dTPi is the transpiration demand for the i-th plant type, Kci is the crop
coefﬁcient for the i-th plant type (alfalfa-based) and is a function of plant species and
growth stages. Common Kc values can be found in literature studies, e.g. [FAO,
1998]
Then we distribute transpiration demand to the soil layers considering root

distribution and soil moisture constraints. A uniform root zone distribution function is

assumed for all plant types:
(2.23)

where giJ is the root zone distribution function for the i-th plant type in the j-th layer,

rti is the root density in the j-th layer. Vegetation roots experience difficulty in

extracting water when the soil moisture is low. The amount of water that roots can

extract depends on the soil moisture and this relationship is called the root efﬁciency.

27

The root efﬁciency function is taken from [Lai and Katul, 2000]:

6 _ 0w r/(o—ow)

77(6)=[ 0

S

 

(2.24)

 

in which n is the root efﬁciency function, 0 is the soil moisture content, 05 is the
saturated water content, 0w is the wilting point and T is an empirical parameter.
[Brand et al., 2005] has examined this root efﬁciency function and found it performed
well compared to a ﬁeld-measured soybean dataset.

At this point, we can calculate the transpiration for the j-th soil layer:

nRPT
TPJ. = Z1 giejnjdz. -dET (2.25)
2::

where j is the vertical soil layer number, TP; is the transpiration in soil layer j, nRPT

is the number of RPT in the current land cell and 8; is the leaf-cover fraction of the

i-th RPT. The leaf-cover fraction of the RPT is calculated using Beer Lambert law:

62'. = j; -(1 — exp(—0.5LAIi)) (2.26)

in which, fi is the land use fraction of the i-th RPT, LAIi is the leaf area index of this

RPT.

The evaporation, on the other hand, is applied to the area that is not covered by

 

leaves:
nRPT

6 9

E]. :7, g]. 1— E 6i dET (2.27)
121
where j is the vertical soil layer number
6 1

9]: = "2‘ (2.28)

28

and the water constraint function is:

(229)

 

 

2.3.2. Vegetation
At the current stage of model development, a simple vegetation growth module is
included. The LAI, rooting depth (RMX), crop ET coefficient (Kc) and canopy height

(be) are updated daily according to a piecewise linear function described in [Dingman,

 

 

20021
ranlin JD < JP(1)
.n0-—Jr(1)
Vmin + JP(2) _ JP(1)( max _ min) JP(1) S JD < JP(2)
V 2 ‘Vmax JP(2) _<_ JD < JP(3) (2.30)
JI)-Jr(2)
nm—W®_Wmhmpnm)wwgm<nm

1’ J7(1)5;JL>

l min

 

where V is either LAI, RMX, Kc or he. Vmax is the maximum value of the variable
and Vmin its minimum value. JD is the Julian day, and JP is the Julian days of the
control points, JP(1), JP(2) , JP(3) and JP(4) corresponding, respectively, to the
starting day of growth, the day on which the plant reaches its maximum canopy, the
day on which the leaves begin to wilt, and the day on which canopy comes back to its
minimum. Maximum and minimum LAI, RMX and hC have been obtained from
literature values [Brezrer et al., 2003; Neitsch et al., 2005]. Kc for the alfalfa based

reference ET is found from [FA 0, 1998].

29

Vegetation Canopy Storage
The maximum Canopy storage of a cell is updated each day using its LAI [Noilhan

and Planton, 1989]:

nRPT
W7 =2><10‘4 Z 6iLAIz. (2.31)

7711
121

Where Wrmx is the maximum canopy storage (m), and 5 and nRPT is deﬁned above.
The canopy interception is then calculated using a bucket model:

C5 = min(Wr

new "ILL'

— CS, P) (2.32)

in which, CS is the current canopy storage (m), CSnew is the new interception during

the time step (m), and P is the precipitation during the time step (m).

30

2.3.3. Overland ﬂow

Overland ﬂow is an important contributor to channel ﬂow. The magnitude of overland
ﬂow is an extremely important factor that determines erosion and the amount of solids
carried into the river system. It is of great signiﬁcance to the transport of land applied

chemicals to surface waters and thus to water quality and human health issues.

Rainfall, snowmelt, or occasionally subsurface exﬁltration are the main causes of
overland ﬂow. Based on the source of the excess water, overland ﬂow is
conventionally categorized into inﬁltration excess (Hortonian) [Horton, 1933; Leach
et al., 1933] and saturation excess [Dunne and Black, 1970]. In inﬁltration excess, the
rate of rainfall exceeds the inﬁltrating capacity of the receiving surface. Hortonian
runoff is often more signiﬁcant in arid and semi-arid regions due to the absence of
well developed soil and vegetation covers [Lange et al., 2003]. The inﬁltrating
capacity can be further reduced by surface sealing that is a result of sudden wetting,
heavy rainfall on bare soils or other reasons [Assouline, 2004]. The saturation excess,
on the other hand, results from the underlying soil being saturated and unable to
accept any more inﬁltration. However, it is becoming more and more widely accepted
that subsurface ﬂow plays an important role in runoff generation even in cases where

Hortonian runoff is traditionally thought to be dominant.

Overland ﬂow is characterized by fast, ephemeral ﬂows over rough, uneven surfaces

31

and over temporally discontinuous ﬂow domains. In most hydrologic models,
overland ﬂow is considered as a thin ﬁlm of water which evenly covers the entire
surface of the ﬂow area (e. g. [Downer and Ogden, 2004b; Jia et al., 2001]). However,
water tends to rapidly concentrate into multiple rivulets and the ideal thin ﬁlm ﬂow
generally does not exist except for the ﬁrst couple of hundred meters. The National
Resources Conservation Services (NRCS) describes the three stages of overland ﬂow
[Division, 1986] as: (1) sheet ﬂow, the ﬂow over plane which occurs in the headwater
of streams, normally less than 300 feet of travel length; (2) shallow concentrated ﬂow;
and (3) open channel, well developed ﬂow paths such as gullies, pipes, rills and
ditches. Although many models do attempt to explicitly model major rivers, the
existence of these well-deﬁned ﬂow paths is so ubiquitous that it is practically
impossible to model all of them. Figure 3.6 shows well-developed channels that have
been registered in the National Hydrography Dataset (NHD) in a 1000 km2 region in
Michigan. We observe that the river network forms a dense web. Explicitly modeling
these ﬂow webs has far surpassed current available computational resources. Thus the

overland ﬂow models are inevitably describing the bulk effects.

After inﬁltration is computed in the vadose zone model, the water left on the ground
may become surface runoff. PAWS contains two compartments: the runoff
compartment (or the overland ﬂow layer) and the surface storage compartment (or the

surface ponding layer). The runoff compartment describes sheet ﬂow, shallow

32

concentrated ﬂow, ﬂows in pipes, furrows, ditches and other pathways. The surface
storage compartment models the layer of water that interacts with soil. Once water
moves into a ﬂow pathway, it no longer inﬁltrates over the entire surface like the
surface storage does. As a result, we treat water in the runoff compartment as

non-inﬁltrating unless the water ﬂows back into the storage.

Although the thickness of the overland ﬂow is generally within the order of a couple
of centimeters, which qualiﬁes for the laminar flow regime description, the highly
uneven, rough and variable surface favors a turbulent treatment [Gunduz and Aral,
2005; Singh, 1996]. Due to this reason, the depth-integrated Saint Venant Equations
(SVE) for shallow, gradually varied unsteady ﬂow, sometimes called the shallow

water equations or the Dynamic Wave Equations, can be used:

Qﬂ + (90m) 8011)) _

 

— ——3
8t 82 By

011 Bu Bu (9h

61 ”(‘91: ”(9y gee 9“” f) ( )

(91' 0'1) (0 8h

—+ —+ ———- + S —S

0t 0:1: 8y ya “051 1‘)

Where, h is the overland ﬂow water depth [L], u and v are the x- and y-direction

water velocrtres (m/s), g 15 the gravrtatronal acceleration [L/T ], 5 IS the source term as

precipitation, exﬁltration from subsurface, or sink term as inﬁltration or evaporation,

33

S0 is the slope ([L/L]), Sfis the frictional slope ([L/L]).
The full non-linear Saint Venant equations govern the conservation of volume (lSt

. . . . . . d (1
equation, this IS also called the contrnurty equation) and momentum (2n and 3r

equations) of shallow water ﬂow. This set of equations includes gravity (i.e. the slope
of the underlying surface), pressure gradient and local and convective accelerations as
the sources of momentum, and frictional loss as the resistance to ﬂow. In overland
ﬂow routing, we regard the effect of wind as negligible.

Elegant mathematical theories of these equations have been developed and details are
available in [Sing/2, 1996] . By recognizing the dominant role of gravity in overland
ﬂow and the complex nature of surface roughness characteristics, Equations (2.33)
can be simpliﬁed by dropping terms. The Diffusive Wave Equation is obtained by

ignoring local and convective acceleration (that is inertia) terms:

Q + BUM) + 3(hri) = 3

at 8.7: By

 

8h

0=——+ S S
981: g((

f) (2.34)

)2 —
(9h

The Diffusive Wave (DW) Equation retains gravity and pressure gradient and thus can

describe backwater and ﬂooding.

The simplest form of the Saint Venant equation is the Kinematic Wave (KW)

34

Equation, which further ignores pressure gradient from the Equation (2.34):

_c'?_h + 8(hu) + 802.11)

at 8:!) 8y

2.35
50.7: : Sf ( )
50y : Sf

This equation assumes steady state ﬂow conditions (i.e, surface slope is equal to
friction slope). For a given surface slope, the ﬂow direction is always pre-deﬁned. As
a result, the KW Equation cannot describe backwater [Borah and Bera, 2003] and

ﬂooding effects.

For the closure of the DW and KW equations, an additional equation is needed. This
equation must relate the state of ﬂow to the ﬂow resistance forces. The formulation

that has been most widely used is the Manning’s formula,

2
(2.36)

 

nu
Sf " [ h2/3

 

where n is the manning’s roughness coefﬁcient [Bl/3T], h is the ﬂow depth [L] and u
is the mean ﬂow velocity [LT-l]

In the proposed model, three schemes for solving the overland ﬂow have been
included to meet the different needs, namely, a Runge-Kutta Finite Volume scheme
(hereby abbreviated as RKF V), a Semi-Implicit Semi-Lagrangian (SISL) scheme, and

a shock-capturing scheme based on the Weighted Essentially Non-Oscillatory

35

(WENO) method [Shu and Osher, 1989; Shu, 1997]. The RKFV can solve the KWE
and DWE. It is sufficient for modeling overland ﬂow in most cases. The SISL scheme
is a powerful scheme for the SWE and is most useful for coastal areas, estuaries, .
shallow lakes and wetlands where surfaces are mostly submerged, water systems are
complex and the effect of topography is less predominant. The WENO scheme which
solves the DWE and SWE is reserved for future research. Its primary applications are
modeling extreme hydrological events, including ﬂashﬂood and dam-break scenarios.
As one of the most important requirements for water routing, all three schemes are

mass-conservative.

The RKF V scheme

The Runge-Kutta Finite Volume (RKFV) scheme is designed speciﬁcally to solve the
Diffusive Wave equation for long term surface ﬂow routing. This scheme has been
found to be accurate, very efﬁcient and stable. Combining Manning’s formula Eq.

(2.36), the Diffusive Wave equation (2.34) is re-arranged as:

 

_3_l_z_ _ _ (90m) __ (9021!) + .s

 

at 8.7: 83/
1/2 ‘ , 1/2
= 1112/3 SO __0£ : —sgn ﬂ —1-h2/3 9-2 (237)
n ‘5 (9:1: 8:1: 77. (9:1:
1/2 ‘ f 1/2
021/12/3 S0 __8_h_ =—sgn ﬂ lh2/3ﬂ
n 3’ 8y By 77. 8g

 

 

where n is the free surface elevation [L], n = E+h, E being the ground surface

elevation. Due to the transient nature of overland ﬂow, wetting and drying are

36

frequent phenomena that must be described naturally. The Arakawa-C grid is used for
the discretization of Equation (2.34), Illustrated in Figure 2.2, the Arakawa-C grid
deﬁnes velocities on the cell boundaries, whereas water depth is deﬁned at the cell
center. The water depth at the interface is explicitly calculated, depending on the
water depth on two sides of the boundary.

V

 

i,j+1/2

W29 '1.) 9': |+1/2,j

”‘10

 

 

 

 

 

 

 

 

Top View
__ ___7r____
\ h IL
Surface
E
Datum
l 1
Side View

Figure 2.2. Illustration of Arakawa-C grid.

With the above sketch, Eq. (2.37) can discretized in space as:

hi—l/2.jui—1/2,j_ i+1/2,jui+1/2,j
6h _ h A2: h +3
6t 1 i,j—1/2ui,j—1/2 _ i,j+1/2ui,j+1/2

I Ay

 

 

37

(2.38)
Given the water depth on two sides of the boundary as hL and hR, and the free surface
elevation as ‘1L and 11R, we use the following logic to determine the ﬂow depth at the

boundary, h 1 )2:

 

 

 

 

r
Y,hL >hR'
Y,/1L >071 N,h]/2 :h’L
Nh =0
. , 1/2
7 >7 ?< 2.39
7” 7R nil/2 =0.5*(hL +113) ( )
Y,hR >hL?
N,,l]/2 =0

The interface ﬂow depth in the y direction can be calculated using the same approach.

The ﬁrst condition, 77 L > 77R ?, is to decide the upwind direction. For the DWE, the
free surface elevation always determines the direction of ﬂow. The second conditional

statement is to decide if the upper stream cell is dry. The third selection, hL > hR?, is
to ensure the interface ﬂux does not exceed what is available for outﬂow during the
explicit updating.

Once bug is obtained, we use h1/2 in Eq. (2.37) to calculate interface velocities u and

v, and ﬂuxes.

 

 

 

1/2
u.. _—_—Sgn(n,, _7' H)_h. I’li.j+1—ni.j|
z.]+1/2 1,]+1 22,] n z.J+l/2| Ami j+1/2 I
(2.40)
1/2
’U- . : —Sgn (77. __ 77 -~)lh 'l'li+1.j _ ”Ill
1+1/‘2.] 2+1.] 1,] n 1+1/2.}| Ayi+1/2j

Then equation (2.3 8) is used to update the solution. The procedure is computationally

38

very efﬁcient.

To improve accuracy and stability, Equation (2.38) is marched in time with the

explicit Runge-Kutta method. For any given partial differential equation:

8U
E- = f(U) (2.41)

The standard Runge Kutta method can be written as:

1’
12a%

3:0

r—l
W+zaa
320

U: = U" + At

 

 

(2.42)
F, = f

 

 

Where U is the unknown variable and F is the time derivative computed at a certain

time level. Coefficients cs and brs can be looked up from the Butcher’s table. Here we

employ a third order version of the TVD RK method. More speciﬁcally:

F0 = f(U")
F1 = f(U" +AtF0)

(2.43)
F1 = f

 

1
U" +ZAt(F0 +Fl)

 

-.+1_ . At _
U" —U”+?(F0+4F2+F1)

A second order accurate in time TVD RK is given as:

F0 = f(U")
F1 2 f(U"’ + AtFO) (244)
At

Un+1 = U” + ?(F0 + F1)

This is also a form of predictor/corrector scheme as described in the literature. Second

39

order accuracy in space is achieved with the above scheme when the

h.1 /2 = 0.5 * (hL + hR) branch in Procedure (2.39) is invoked.

The modiﬁed Semi-Implicit Semi-Lagrangian (SISL) scheme

The SISL scheme was originially developed by [Casulli, 1990; 1999] and then
advanced by [Martin and Gorelick, 2005]. Our development of the scheme largely
follows that of [Martin], but some changes were made to the scheme. In order to use
this scheme, the Chezy’s formulation is employed for the friction loss term. This
scheme solves the fully nonlinear Saint Venant equations with the inclusion of wind,
Coriolis and eddy viscosity effects [Martin and Gore/ick, 2005]:

_8_h + 8(hu) + 8(lw) : 8

8t 8m 8y

 

 

 

 

 

l 1
811 811 811 an 0911 6211 7T (U, — u) J13 + v2
—+u-,—-+v7—=—g—+€ + +————+g———u+fv
8t 8:1: 8y 8:7: [(932 33/21 H 022

l ) r
8U 811 8'0 87) 82v 821) 7T (Va — 1)) \ltr2 + v2
—+u—+v—=—g—+€——+ , +——+g—v+fu
8t 83 8y 8y [82:2 33/2) H C22

(2.45)

For the above hyperbolic PDE, solving the advective part of the equation, namely, the

 

highlighted portion:
8U 8U 8a 8271 8211 877 7T (U0 — U) \/u2 + 112
—+u—+v——=e— —+fv—g—+————+g———u
8t 82: 83/ 35172 8y2 8m H 022

 

 

 

 

 

 

 

 

40

_8_v+uﬂ+v_8£_e8_gv_+§2_v +f’u— 271 ———7T(Va_v)+ ———U2+v2v
8t 8:1: 8y 31:2 ﬁg? gay H g 022

 

 

normally poses the greatest challenge to the stability and efﬁciency of the numerical
scheme. Whereas the treatment of other terms generally admits much larger time steps.
This provides the motivation for the usage of operator splitting and the
semi-Lagrangian scheme to solve the advective components. Let us follow [Martin
and Gorelick, 2005] and denote the advective scheme as F and the solution to other

terms as S. Then Equation (2.45) is solved in two fractional steps:

11 F u
* _
v v”
rhn+1‘ r'hll.‘\ (246)
*
“n+1 =5 u
:1:
,U'n+1 L” J

 

 

 

 

 

 

The F operator is a semi-Lagrangian scheme involves two steps. In the ﬁrst step, the

current grid points (X ,Y) = :1: 0:0 y are traced back through At of the travel time
N N

along the travel paths that are deﬁned by the ﬂow ﬁeld of (u ,v ), and we thus ﬁnd

the departing location of (X,Y) as (XSLeYSL) [Martin and Gorelick, 2005] used a

semi-analytical approach. Although this is also implemented in the code, after some
initial comparisons, we found the classic 4-stage Runge Kutta method [Zheng, 2002]

to be more stable.

In the second step, we calculate the velocities (215,115?) at locations (XSLiYSLI

41

. . . . . . . N N . . .
This can be done usrng a srmple bilinear interpolation from (u ,v ). For a pornt (x1,y1)
such that X] éxi<x2 and y] éyi<y2 where x1,x2,y1,y2 are coordinates of the closest

cell centers in our grid, the bilinear interpolation of velocity u can be written as (2.47).

uN(:z:1,y1)
(332 — $0012 - 311)
N($2,y1)
(5’52 “ $0012 ’ 311) i
UNCEpy-Z)
($2 — $11012 “ 191)
“N($22y2)
(1'2 _ 9300/2 “ 91)

 

“31(1‘1'13/2') R1 ($2 '— 53,)(3/2 - 31))

u

 

(2.47)
+

 

(1172 _ 9700/, '— 91)

 

((Ei — ‘51)(3/1 ‘ 311):

Essentially, we trace back the points along the ﬂow lines and ﬁnd out their values at
the previous time step. We must note here that this is in fact not the most correct
implementation of semi-Lagrangian method, as a classic SL method traces back along

the characteristics, which are not necessarily the ﬂow lines.

We notice that the SISL scheme is considerably more complex in theory and much
more computationally expensive than the RKFV scheme. Although the time step can
be larger, it is not obvious whether the gain with larger time step would offset the
much more computations involved. Also, the semi-Lagrangian scheme may crash at
large discontinuities. The semi-Lagrangian step is useful only at places where ﬂow is
continuous (so that ﬂow path tracking is meaningful). In overland ﬂow, land is

constantly drying and wetting, and ﬂow is transient and discontinuous. This makes the

42

RKFV scheme more favorable for long term simulations, whereas the SISL scheme
may be of more interest for areas with shallow lakes and estuarine systems. We will

compare the performance of the two schemes in section 2.7.

2.3.4. Channel ﬂow

The channel ﬂow model is based on the one dimensional Saint Venant equations.
Similar to the overland ﬂow, the channel ﬂow can also be described by, from full
complexity to simpliﬁed forms, the Dynamic Wave, Diffusive Wave and Kinematic
Wave equations. To account for the variation of ﬂow area along channels, the
conservation of mass and momentum takes a slightly different form. The full

nonlinear Saint Venant equation for channel routing, or the dynamic wave equation,

 

 

 

 

 

 

 

 

reads:
8A 8(«uA)
E+W2TMQIHHC+QI 248)
Or, one may write it in a matrix form:
A 11.4 —(7~b + ql + (196 + qt)
"AtJrAgngug -gA(SO—Sf)+-ij:0 (2‘49)

 

 

where A is the cross-sectional area of the channel (m2), u is the ﬂow velocity (m/s),

7; is the river stage (also called free surface elevation, m), r is precipitation (m/s), ql is
. 3 2 . . .
lateral inﬂow from overland ﬂow (m /m /s), qgc IS the groundwater contribution

(m3/m2/s), S0 is the slope and Sf is the friction slope. The computation of Qgc will be

43

described in section 2.4.2. As with the overland ﬂow, several solvers have been
implemented in PAWS, including a Runge-Kutta Finite Volume scheme that is similar
to the overland RKF V scheme and a second order MacCormack scheme. The RKFV
scheme solves the simpliﬁed Diffusive Wave Equation, in which the momentum

conservation equation is replaced with:

1/2
11 = lh1:[501: — 9—1-24] 2 —sgn[QQ]-1-h

71 8:1: 8:1: '17. c

2711/2

2.50
8:1: ( )

 

 

Where n is the free surface elevation (m), n is the manning’s roughness coefficient
and hc is the channel ﬂow depth (m). Kinematic wave Equation is found to be
unusable for channel routing, as it cannot describe backwater effects and its
topography-driven nature is unsuitable for the stream networks where the elevation
variation is mild. The RKFV scheme serves as the long-term channel routing scheme
and its details have been given in previous section. The difference is that the temporal
derivative of area is calculated using:

8A 1
8t— : ——A—;; Ai—1/2u1-1/2 _ Ai+1/2”z‘+1/2 + 32- (2.51)

Where, for a rectangular channel

A1+1/2 Z wi+1/2hct+1/2 (2-52)

Wi+1/2 is the channel width at cell interfaces. The channel width is an input value to

the model and may be obtained from ﬁeld or aerial observations or regression

techniques. hci+1/2 is determined from he; and hci+1 and the free surface elevations on

both sides again using the procedure (2.39). The formulation in (2.51) also admits

’44

other channel geometries, in which case different functional relationships between A
and h must be provided. However for the current research, rectangular channels are
deemed sufﬁcient. Other widely used models such as SWAT [Neitsch et al., 2005] use

rectangular channels only. We employ an operator splitting approach such that the

groundwater contribution qgc is solved implicitly as described in section 2.4.2.

Boundary conditions

Internal Boundary Conditions

The river’s physical domain is discretized into M one-dimensional cells whose
mass/stage is deﬁned on each cell’s center. A ghost cell is added to both ends of the
grid. A ghost cell is a. frictional cell whose existence is only for the purpose of
implementing the boundary conditions. When the equation is solved, only cells inside
the physical domain are updated. The boundary conditions are implemented by setting
up the values of the ghost cells to reﬂect the correct physical inﬂow/outﬂow
conditions. When a river drains into a downstream river, we call the receiving one the
main river and the contributing one a tributary. The main river’s cell at which the
tributary meets is identiﬁed as the conﬂuence cell. The stage of the conﬂuence cell is

taken as a Dirichlet boundary condition for the tributaryz.

EM+1,t=Ec,m
(2.53)
N~N+1 __ N
hM+1,t _ ham

Where EN+1,t is the bed elevation of the N+1-th cell (a ghost cell) of the tributary

45

river, EC,m is the bed elevation of the main river at the place the tributary conﬂuences
the main river l1N~N+1 is the SH e of the head of the host cell of the tributa
° 11+1¢ ‘g g ry

. . . N N+1 . .
river during the time step t to t , whrle hit” rs the head of the conﬂuence cell on

. . N
the main rrver at the end of t .

The river models are run in an upstream-downstream cascade sequence, so the
contributions from tributaries are always computed before the model is run for the

main river. The tributary inﬂows are converted into a source term to the main river:

tN+1

N~N+1 N~N+1 1 nt
. = . —— , .dt .

In which, 3 is the source term throughout the new time step of the conﬂuence cell on

the main river [Lz/T], At and Ax are the temporal and spatial steps of the main river

N +1
1
and ft N qjdt is the accumulative inﬂow of its j-th tributary during this time step.

The above boundary condition implicitly assumes that the downstream river stage
does not change much during the time step, which is generally valid since the main
river is a larger river. But its limitation is that during high ﬂow periods the time step
of the river network may need to be reduced to maintain stability. Each river is
checked against its own Courant number restriction and the smallest time is used for
the entire river network. This ensures that the rivers communicate frequently enough
and the Dirichlet downstream boundary is properly updated. At the highest ﬂows, this
setting has constrained the time step of the river network to be around 1 to 2 minutes.

However, this is not unreasonable as the durations of high ﬂows are often not long.

46

For regular ﬂows, the river network uses a At of 10 minutes. It is possible to have
negative inﬂow, i.e., backwater from the main river into tributaries. This situation is
handled without the need for any further modiﬁcation by the above internal boundary

conditions.

At the ﬁnal outlet of the river network, if downstream river stage record is provided, it
can be used in PAWS. Otherwise, a simple free outﬂow condition can be used. Since
the physical condition is not available, this method extrapolates the head of the ghost
cell from its internal cells:
hCM+1 = 2th - th_1 (2.55)

If the upstream end of the river is a headwater, a Dirichlet value of ho = O is simply
given to the ghost cell. It is possible to read in an inﬂow record ﬁle if measured data
are available. Moreover, sometimes it is desirable to split a river into several domains.
In this case, the upstream river inﬂow can be treated as a tributary and use the BC

detailed above.

As described in the previous section, some researchers [Gzrnduz and Aral, 2005;
Ivanov et al., 2004a], have argued for ﬁnite element methods for overland ﬂow
modeling. Their main motivation is the ﬂexibility of the ﬁnite element mesh to
resolve the topography, hydrologic response units and rivers. Although the concern is

legitimate, we follow a structured grid approach for its simplicity and generality.

47

2.3.5. Unsaturated vadose zone model
Vertical water moisture movement in the soil compartment is described by the mixed
form of the Richards’s Equation [Celia et al., 1990; van Dam and F eddes, 2000]

8h

8h 8
Cl—z—Kh— 1+Wl .
(flat 32 ”8.2+ (I) (256)

 

 

 

 

In which, K(h) is the unsaturated hydraulic conductivity (L/T), h is the soil water
pressure head (L), W(h) is the volumetric source and sink term, including evaporation,
plant root extraction and bypass ﬂow and z is the vertical coordinate (positive upward)
(L). The two state variables, pressure head h and water content 0 are linked via C(h)=
86 / 8h , the differential water capacity (L-l). This equation assumes that for a given
type of soil, there is a one-to-one relationship between the pressure head and the water
content (water retention curve) and thus we can solve only for one of the variables.

The solution to the Richards’ Equation is of central importance to the overall
performance of the watershed model, because this compartment links inﬁltration and
percolation, and serves as the vital linkage between surface water and groundwater,
and governs the actual evapotranspiration. The unsaturated hydraulic conductivity and
pressure head are both functions of the soil moisture content. We choose the

Mualem-van Genuchten (VG) formulation:

48

_ n —<n—1)/n
Szam azb 14]
93 — 0,,
( 1)/ 2 (2.57)
n— n
K(h) = K35A 1 — (1 — 9/0"”)

 

 

Where S is the relative saturation, 0 is the soil moisture content, 03 is the saturated

water content, (E)r is the residual water content, 11 is a measure of the pore-size

distribution, a is a parameter related to the inverse of the air entry suction and k is a
pre tortuosity/ connectivity parameter [van Genuchten, 1980].

Eq. (2.57) is widely used in literature. However, with this formulation, the equation is
highly nonlinear and is known to produce results of a very transient nature. Figure 2.3

shows relationship between K, 0 and h at different 11, it and a values.

49

Figure 2.3. Relationship between K, 0 and h at different parameter values with the van
Genuchten formulation for a hypothetic soil type (a) at different a; (b): at different it;
(c) at different N. THE in the ﬁgure heading means 0. The base parameter for this

comparison is a= 2.49(l/m), N = 1.507, 03 = 0.43, 0, = 0.01; Ks = 0.175 (m/day); )1 =

-0. 14. Unit in the ﬁgure is the same as the base parameter set

Kvs 0 at different a

 

 

 

 

 

 

 

 

0 I I I e-
g ...... O1
2 —10 — ' ' ‘05 +-
3 ------ 2
_20 I I 1 g4
0 0.1 0.2 9 0.3 0.4 0.5

 

 

 

 

 

 

8
‘9'
a)
2
_ 1 1 I
-§00 —150 -100 -50 0
0
(a)

50

Figure 2.3 (cont’d.)

K vs 0 at different N

 

 

 

 

 

 

 

   

 

1091 0( 0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-§OO —1 50 —1 00 —SO 0
h
(b)
Kvs 0 at different it
0 I I g A _ - W
9 .‘1‘1&?‘*aﬁﬂ‘.’: ‘ """ -O.I
§_10_${' "'-0.5.
- ------ -1
3 1 1 I I _-5
—20
O 0.1 0.2 9 0.3 0.4 0.5
0 vs h at different it
0 I l I
8
§ —1 —
.8’
_ I I I
-%00 -150 —100 —50 0
h
(C)

In this ﬁgure, a base parameter set is used (see ﬁgure caption) while one of the

51

parameters, either 11, it or 01, is replaced by the values indicated by the ﬁgure legend.
Then the K as a function of 0 and 0 as a function of h are plotted for the altered
parameter sets. We observe that K can vary by many orders of magnitude at different
0. All of n, it and or greatly inﬂuence the relationships, in which, n is the most

sensitive.

A naive ﬁnite difference discretization in time for the LHS of equation (2.56) would
read:

. . j+1/2 [+1 '
6]+1_ a] : Ci (6,} — 12.3) (2.58)
At At

 

Where 0j+1 is the water content at the next time level. (To avoid confusion with the
connectivity parameter n, in this section we used j to denote the time levels and use i
for vertical cell index, i is positive downward). However, due to the high non-linearity
of the differential water capacity term, temporal discretizations similar as above are
known to produce serious mass balance errors [Lehmann and Ackerer, 1998; van Dam
and Feddes, 2000]. Here we employ the widely used implicit iteration scheme by
[Celia et al., 1990].

92341 _ 62/ : 034L142 (ht-141.1) _ hzj'p) + 9541.154 _ 613' (2.59)

In which p is the iteration number and C is evaluated as:

52

Tl

ahj’p (n — 1)

 

 

071-11) : % ___ _ (93 ‘97-) ,
,

8h 271—1 hip (2.60)

. 71. n
a h] ‘ p

 

1+

 

 

 

 

In other words, the differential water capacity C(0) is updated at every iteration using
the newly obtained hj’p. The iteration repeats until both h and 0 converge. This stable
implicit scheme can almost guarantee mass balance error to be close to rounding off
errors [van Dam and F eddes, 2000]. When we apply Equation (2.59) in to (2.56) we
obtain the full discretization formula:

_1_

j+1,p j+1.p_ j+1.p—1 j+1.p—1_ j
N cl. (hi hi +02. 9,.

 

j '+1, '+1, 1' j+1,p j+1.p
A23. Az._1/2 A22. Az.+1/2 ’

2 'l

 

 

(2.61)

We can write this into a tri-diagonal form:

7' j+l-.p 71p j+1.p J' 7+1-p _ 11p
(Ii/11.4 +192. h1- +01. "1+1 —d1.

' j
K1+1/2

AZ1A31+V2

J
K 1—1/2

AZz'AZr-—1/2

(lj Z —- (‘27 : —
1. ’ “t

. . . 01+“)
J—P _ _ J J 1

b1 — (az- + Cl- ) + At

 

' j 7
014-14) . , K._ 9 — K.
_1.___ J+1,p—1 ,3 t l/.. 2+1/2

hi. + I'lz- +

At A22. At

0137+1,p—1 _ 01:]

l
(2.62)

We observe that al and e; do not change from iteration to iteration so the iteration

index p is dropped from the notation. The above system can be solved efficiently

using the Thomas algorithm [Press et al., 1997].

53

2.3.6. Saturated groundwater ﬂow model
The unconﬁned aquifer is conceptualized as a series of vertical layers. In each vertical

layer, we solve the 2-dimenional groundwater equation:

T[a_H
89

+2.
(91/

SﬂiziT

+R+W—D 2.3
8t 8:1: 1) (6)

8:1:

 

 

 

[a

 

 

 

Where S is the storativity (dimensionless), T is the transmissivity of the aquifer [L2/T],

T=Kb where K is the saturated hydraulic conductivity [L/T] and b is the saturated
thickness of the aquifer [L], H is hydraulic head [L], R is recharge or discharge (L/T ) ,
W is the source and sink term due to pumping or root extraction [L/T] (inﬂow as
positive) and Dp is percolation into deeper aquifers [UT]. The above equation is

discretized using standard backward-in-time, center-in-space ﬁnite difference scheme:

 

 

 

 

. n+1 _ n+1 n+1 _ 71+]
5221' 9+1 _ 71.41) = Tel/2n Hi—Lj Haj _ Tel/2.1 Haj H 141...
m w
n+1 n+1 +1 n+1
Tr, j—1/2 Hi,j—1 — Hr, j Ti,j+1/2 H1, j — Hi,j+1 n ,n .6
91,1 3’4141/2 921.1 yr.j+1/2 '
4)

Here I and j are y and x coordinate indices, respectively, n is the time level.
t
W". = ft”+1Wijdt is the integral source term throughout the time step.

in

We solve for the percolation term implicitly:

54

(2.65)

 

where K] is the hydraulic conductivity [L/T] of the aquitard beneath the current
aquifer, A21 is the thickness of the aquitard, H1" is the hydraulic head in the

aquifer below. Equation (2.64) becomes:

 

 

 

n+1 __ n+1 71+1_ n+1
31.1(H7141_ 1.41): TIA/2.1 Hi—Lj Hm _Ti+1/2.j Haj Hi+Lj
1] 1.}
n+1 n+1 n+1 n+1
+ T: '—1/2 Hi,j-1_Hi,j __T13.j+1/2 Hm —Hi.j+1
may“ A”Fuel/2 Afar Axi,j+1/2

+1
(H31 “5")

 

 

—Kl + R + 3 W2. .dz
2: z ’3
1 b
(2.66)
The x-direction interface transmissivity is taken as:
T22—1/2, j = 211.jTi—1j (2 67)
A . . , '
$er “1.1717 + Afr—1.77141

y direction interface transmissivity is deﬁned in a similar fashion. Re-arranging Eq
(2.64) we obtain:
n+1 n+1 n+1 n+1 n+1 _
“1.111 1—10' + waLH + 01.1” 21.1 + di.sz'+1,j + half-47H — far (2'68)

Where

55

 

 

 

 

a _ Ti—1/2.j _ Tia—U2
1.7 — — ’ 111' "‘ —
A91, jAyi—l/Q, j Axtn'AxiJ-l/Z’
d 7141/2. j _ Ti.j+1/2
f, ' — — a i] _ —
J A3/1,jA?/2:+1/2, j AztjA‘Tz’JH/Tl (2-69)
5. - K
_ _ _7_J_ _l
S. . KH"
_ _ﬂ n 771 l 1
fig]. _ At Hm- ”12.1 + R+ A21

This gives rise to a sparse, symmetric, penta-diagonal and positive deﬁnite system
which can be efficiently solved by many advanced matrix solvers. We employ the
Conjugate Gradient method (described in [Leveque, 2007]) here due to its proven

efﬁciency, stability and readily-available codes [Press et al., 1997].

The implementation of boundary condition is straightforward in the model. Cells
outside the watershed boundary are marked as inactive, and their conductivity values
are set to zero. Thus no ﬂow exists at the watershed boundaries. It is known that
groundwater divides don’t always coincide exactly with the watershed boundaries.
However unless we are provided with more data regarding groundwater ﬂow, this is

the best approach we can use.

The nonlinear drainable porosity unconﬁned aquifer ﬂow model
The above solver deals with the linear groundwater ﬂow equation, whereas in reality
the storage coefﬁcient of the unconﬁned aquifer is nonlinear. [Hilberts et al., 2005]

has studied the effect of nonlinear storage on the speciﬁc yield of the unconﬁned

56

aquifer. The storage coefficient of the unconﬁned aquifer was found to be not a
constant but a ﬁrnction of soil water retention characteristics and depth to water table.
The ﬁnding is also consistent with previous experimental studies (e.g. [Healy and
Cook, 2002]). The concept of drainable porosity is based on the hydrostatic
equilibrium state. This condition certainly deviates from reality because the soil is
constantly wetting and drying and hydrostatic equilibrium condition rarely exists, but
it represents a signiﬁcant improvement over the common Bousinessq approximation,
which assumes the speciﬁc yield as simply the porosity. The stored moisture in a soil
column can be obtained by integrating the moisture proﬁle under hydrostatic

equilibrium, using a modiﬁed van Genuchten formulation:

 

 

 

 

—1/71
5141:1241“)mew-211"] 44—414
, _ n+1 (2.70)
S'(H) = (6.9 —01,)1—[1+(01(H — 2))72] n
The ﬂow equation for the unconﬁned aquifer is then written as:
aST(H) __ 8 8H 6 ,. 8H

 

 

 

 

 

We see that equation (2.71) reverts back to the ordinary groundwater ﬂow equation
(2.63) if the function ST(H) is a linear function of H. However, when we use the
nonlinear storage function in (2.70), this equation becomes nonlinear and need to be
solved using an iterative approach such as the Newton iteration. One important

consideration on any method we choose is mass conservation. We enforce

57

conservation by demanding that the water storage/water table position curve must be
single valued and the solution to the PDE must fall on this curve. To do this we

discretize equation (2.71) in a conservative manner:

,"_+1 _ 71, n+1 n+1 r n+1 n+1
5(HM ) 8(HM) = Ti—1/2,j Hi—l.j _ Haj _ i+1/2,j Hij _ Hi+l.j
A A11".

At A371, j Aft—1&4 $1,)“ z+1/2.j
n+1 n+1 n+1 n+1
Try—U2 Hi,j—1 ‘ Hi.j _ YTJ'l'l/Q Haj ‘ Hi.j+1

n+1 71
Hij ’ HI )

Azl

 

 

 

 

 

 

71. 71

(2.72)

Re-arranging terms and writing in a minimization form:

 

3(anl) n+1 5(H71.) '
G(Hl’fi'l):__L+KlHi,j _ 1,] —R?1__W/_7li_K Hz"
1.7

+ 1,j—1/2 z,j-—1 t,j _ i.j+1/2 1..j+1

L Ayigj ( Ayuj—l/Q } Ayij I Ayiaj+1/2

 

 

l l
n+1 n+1] n+1 n+1 l
T‘ H' _Hi.j 113+l/2,j Hij _H'

_ 2—1/2.j 2—1,j _ 1+1~j
Al’mk Ami—l/Zj J A331,]:k ABM/23‘

 

 

 

 

 

 

 

 

(2.73)

The Newton iteration approach involves computing the Jocabian of G(H):

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30m 802;]. 0... 8G”.
8H“. aH1414 6H1.j+1
301414 30241.)“ 86141) 0 801414
8H1) aHi+1,j 8H1+2,j 8H1,j+1
J H =
l l (9le 8G”, (’3le (‘9le
__~‘ 0... ' ‘ —" 0...
aHl,k01 aHl—l}: dHlJc aHl+1Jr
8071.71 0 6671.71 0071.71
aHn.,n-—1 aHit—1,71 aHngJ
which can be evaluated as:
8027‘]. Ti—1/2.j
301.)- Tar/23
3H1+Lj AmajAmi-H/Zj
601.]: _ Try—U2
émtg—1 A3/-1',jA?/1.j—1/2
6013‘ __ Ti.j+1/2
aH-i,j+1 Ayi,jAyi,j+l/2
801.j_S,(H'Z~z/) Kl
8H”. _ At A21
_ 80“. + 80).]. + 8G“. + 8G“
aHi—m aHt+1.j aHiJ—l 8111.341

 

 

The Newton update is obtained by solving the linear system of equations:

59

 

(2.74)

(2.75)

 

 

 

 

p _ Tl _ 772
G..,j(H I‘ At 32, R14 ”2:4
6019' (HP .— P)+ a U HI” .—HP.)+
3112;] J z—LJ I J 8,1241 J z+LJ 2,]
z. z r. — r,
_6H1,j+1 '1 3 (”115—1 ‘J J J

 

 

(2.76)
where p is the iteration number. Since I is symmetrical and positive deﬁnite, it can be
solved, again, using the conjugate gradient method. We keep computing the solution
to equation (2.76) until H converges. Newton iteration converges quadratically when
the initial guess is close enough to the solution. Normally it converges in 2~4

iterations.

2.3.7. Snowpack model.

The UEB snowpack model [Luce et al., 1999; Luce and .Tarboton, 2004] is adapted as
the snow module of the proposed model. Some coding errors in the original package
have been corrected. The UEB model keeps track of snow water equivalent and

energy content of the snowpack:

 

dU

717%...+9..+Qp+Qg-Q..+9.+Qe-Qm

dSl/VE (2.56)
dt ’ 3 '

. . . . . -2 -1
In the energy balance equation (all per unit of horizontal area 1.e. in U m hr ) terms

are: Q5“, net shortwave radiation; Q“, incoming longwave radiation; Qp, advected

60

heat from precipitation; Qg, ground heat ﬂux; Q16, outgoing longwave radiation; Qh,
sensible heat ﬂux; QC, latent heat ﬂux due to sublimation/condensation; and Qm,
advected heat removed by meltwater. In the mass balance equation (all in m/hr of
water equivalence) terms are: SWE: snow water equivalent (m), P,, rainfall rate

(m/hr); PS, snowfall rate (m/hr); Mr, meltwater outﬂow from the snowpack (m/hr);

and E, sublimation from the snowpack (m/hr) [Luce and Tarboton, 2004].

2.4. Interactions between domains

The above sections (Sec. 2.3) complete the descriptions of individual solvers. The
hydrologic system is a fully coupled system such that problems in each domain
cannot be studied without interactions with other domains. In this section we discuss
how these different domains are coupled in the model. We will see that the vadose
zone plays a central role in the model by solving runoff and recharge (also ET,
although this has already been discussed in Sec. 2.3) together with soil moisture. We
will detail how we decompose the 3D subsurface ﬂow into a combination of a 2D and
an array of 1D problems, while efﬁciently obtaining physically consistent solutions.

This has been a major roadblock in physically-based hydrologic modeling.

2.4.1. Coupling of the Unsaturated Richards equation and the groundwater ﬂow
equaﬁon:

Water movement in the three-dimensional subsurface domain can be described by a

61

single three-dimensional Richards’ equation:

8h
(C(11) + SS)??? = W(h) (2 56)
8 8h 8 8h 8 8h '
+—K h—+1+—K.l—+——Kl.-—
82 z( ) 82 82: 1(035 8y y(2)8y]

 

 

 

 

 

 

 

 

 

where h is the pressure head [m], KX, Ky, K2 are hydraulic conductivities [md-l],
C(h)= 86 / 8h is the differential water capacity [ml], 83 is the speciﬁc storage [m-l],
W(h) is the volumetric source or sink term [d-l]. Due to the strong non-linearity of
unsaturated conductivity and differential water capacity, a large matrix resulting from
3-D discretization must be solved iteratively. Applying this approach on a watershed
scale is computationally unfeasible. There is some research going in the direction of
directly solving this 3D equation [Maxwell and Kollet, 2008]. We note that even with
the help of massively-parallel super computers, the solution to Eq (2.56) in forward
mode is still difficult, not to mention long-term simulations in the context of a

watershed hydrologic model and model calibrations.

To cope with this difficulty, we seek to lower the dimensionality of Eq. (2.56) by
separating it into the one dimensional Richard’s equation that governs the unsaturated

portion above the unconﬁned aquifer:

l

8” J+ W(h) (2.56)

—+1
82

 

1% a

C(h — 82:

K(h)

 

 

 

and the two-dimensional groundwater ﬂow equation that governs ﬂow in the saturated

unconﬁned aquifer:

62

5%:3
8t 8:1:

8

— + R + W — D (2.63)
01/

8H
T— +
[ml

To avoid confusion, here we use H to represent the hydraulic head (and thus also the

 

T(.<21:
(91/

 

 

 

 

location of the water table of the unconﬁned aquifer). Eq. (2.63) has been used
ubiquitously to describe ﬂow in unconﬁned aquifers [e.g., Modﬂow]. We want to ﬁnd
a method that solves the two above equations and couples their solutions to
approximate the 3D system.

Many previous studies have attempted such a simpliﬁcation. One approach is to solve
Equations (2.56), taking the solution to equation (2.63) from the last time step as the
lower boundary condition. Several examples of this approach are [Downer and Ogden,
2004b; Jia et al., 2001; Twarakavi et al., 2008]. This method implicitly assumes that
the head in the unsaturated zone does not change throughout the time step. However,
such a scheme is not accurate when recharge from the vadose zone or lateral ﬂow of
groundwater changes the location of the water table. When recharge/discharge is large
during one time step, the method may cause numerical instability because it neglects
the head change of the water table induced by recharge itself. In addition, a theoretical
inconsistency could occur as a head difference is always required to maintain recharge
to the aquifer. We will illustrate this in the next section in more detail.

Another method is to iteratively solve the two equations until their solutions converge.
[Stoppelenbmg et al., 2005; van Dam et all, 2008] coupled the two equations via the
deep percolation ﬂux at the bottom of the saturated domain and the vadose zone

model. When iteration is involved on the equation level, the method can be much

63

more expensive. Moreover, the coupling is ad-hoc as apparently, if an impervious
layer sits at the bottom and the deep percolation is zero, such a coupling is not

possible. Here we seek a method that does not use equation-level iterations.

Before we discuss the new coupling method, we introduce below a test case to better
illustrate the method. This case closely resembles the widely known experiment ﬁrst
reported in [Vauclin et al., 1979] and later frequently studied in the literature (e.g.
[Dogan and Motz, 2005; Twarakavi et al., 2008]). As shown in Fig 2.4, the ﬂow
domain consists of a rectangular soil slab with width L and height Z. Due to
homogeneity in the y direction, the problem reduces to a 2D problem with x and z
axes. z is O at the bottom of the unconﬁned aquifer (saturated ﬂow domain) and
positive upward. Initially the soil has established hydrostatic equilibrium with the
water level on the left and right boundaries. Then a constant inﬂow is applied at the
center of the top surface from XL to XR (XL=-XR). The rest of the top surface is
covered to prevent evaporation. Water level on the boundaries is maintained at a
steady height of H0. Aﬁer the wetting front reaches the water table, the pore spaces
are ﬁlled up and the water table starts to rise. We can consider the saturated zone as
being ‘recharged’. As the head difference is created, water ﬂows to the sides, brings
up the water table at locations away from the inﬂow zone and ﬁnally exits from the
boundaries. After some time, a steady state will be reached when the water table level

becomes static (if the inﬂow rate does not exceed a certain threshold). Analyzing this

64

steady state situation helps shed light into the problem. At steady state, we must have
inﬂow rate that is equal to the outﬂow rate. Denoting the water table height as HW(x),

we must have:

z=HW(r)

: Qm (2.77)

 

 

 

Keeping in mind that HW is a function of the location x, we will simply write HW in

the rest of the section for shorter notation.

 

III

   

 

 

 

 

(0.0) "

Figure 2.4. sketch of the Vauclin 1979 test problem. HW(x) is the water table location
at x, z is measured 0 at the bottom.

65

111111

111111

 

 

 

 

 

 

Figure 2.5. Illustration of Assumptions 1 and 2
The method is based on two assumptions describe below and illustrated in Fig 2.5.

The ﬁrst assumption is that water moves only vertically in the unsaturated zone.

Denoting the lateral ﬂow into a unit volume of the soil column as q,, with the unit
[T-l], this can be written as:
Assumption 1:

q(:1:, z) = 0, z>HW (2.78)
This assumption is used by many and can be regarded as simplifying Equation (2.56)

by replacing 03[Kx(h)g—h] +
:r :r

A

K (h)a—h- with q=0 in the unsaturated portion
8y 9 83;

of the ﬂow domain. The rationale is that in the unsaturated domain gravity dominates
over lateral moisture diffusion.

The next assumption is that the lateral ﬂow is uniformly distributed along the

66

saturated thickness of the unconﬁned aquifer:

Assumption 2:
aqﬂ H W(zr)
a; = 0 and bf q7.(:1:,z)dz .—_ DR(x), z<=HW (2.79)

 

where DR(x) is the integrated lateral drainage term (m/day), positive for inﬂow. With

respect to Eq. (2.63), we see that:

F

q

 

 

 

 

 

 

 

 

 

 

’ 8:1: 8:1: ‘ 8y 8y (2 80)
DR : .2. T 2111 + .2 T 6_H
1 8:1: 811:] 8y 83/
And we can denote the drainage ﬂux, simply solved from (2.79), as
DR .
qr(2)=HW,zSHW (2.81)

Assumption 2 is a natural result of the Dupuit-Forchheimer (D.F.) assumption, which
assumes the ﬂow lines to be parallel. This condition has been used by many models to
apply groundwater ﬂow terms in the Richards Equation, e.g. [van Dam et al., 2008].
However, the original D.F. assumption also states that the hydraulic gradient is zero in
the vertical direction of the saturated zone:

H (x,z) = H W , z<=HW (2.82)
The paradox of the DP. assumption is well known (e.g. [Kirk/1am, 1967]). Although
often providing acceptable solutions to unconﬁned aquifer ﬂow problems, Eq. (2.82)
creates a theoretical inconsistency for our coupling approach here and therefore needs
modifcation. To see this, we need only look at the mass balance of a unit volume at

steady state, using Eq. (2.56), (2.80) and assuming W=0:

67

 

 

 

8h 8 8H
Ch.+S.—=O=—K—+ .
( () 5)8t 82[ z 82 q (283)
Asaresult:
8 8H

Thus Eq. (2.82) cannot be directly applied. For the steady state case, consider deep

percolation (qbot, m/day, positive for downward percolation), the above equation

forms a second order Boundary Value Problem (BVP) that can be quite easily solved.

The boundary conditions are:

 

 

 

 

l (a, 2:0
1
Z \az 0
r z=HW
K 8h 1 — DR
. z 67” _ +110, (2.85)
1
h(rr,HW) = 0
And the solution is:
‘1 1 DR 2 2
hrc.z’=1——b—“t- HW—z +——(HW —z) 2,86
( I ) i K2 ]( ) 2 Kz ( )

where h(x,z)’ is the steady state pressure head (m), K2 is the vertical saturated

hydraulic conductivity (m/day), HW is the water table location(m), z is the vertical
coordinate and DR is the integrated lateral drainage ﬂux (m/day), positive for inﬂow.
As we can see, h is a quadratic function of HW when head loss due to drainage is
considered instead of the linear relationship assumed in Eq. (2.82). Figure 2.6 shows h

as a function of 2 at different DR values. Clealy h(x,z) is less than HW—z when DR is

68

negative (water draining away). Although (2.86) is derived from the steady state case,
we will still use it for the unsteady case to provide a functional relationship between
the pressure head at z and the water table location HW.

Previous studies have directly used Eq. (2.82) to provide a Dirichlet boundary
condition to the Richards equation. An important implication of this approach is that,
even at steady state, the saturated/unsaturated interface found in the soil proﬁle must
be higher than HW because a head difference must exist in the soil proﬁle to supply
the required recharge to the saturated zone. Thus it is almost certain that the moisture
proﬁle will be erroneous because the saturated/unsaturated interface should be placed
right at the groundwater table. We will show that Eq. (2.86) allows the moisture
distribution in the vadose zone to agree with the water table location and thus
provides an improvement over the previous studies.

There is yet another challenge facing the approximation of (2.56) with (2.56) and
(2.63) which is the storage term S in (2.63). The speciﬁc yield of the unconﬁned
aquifer is a loosely deﬁned concept. Non-existent in the full 3D Richards equation, its
use in the groundwater ﬂow equation is largely for practical purposes. The speciﬁc
yield of a Boussinesq aquifer in theory is the effective porosity, or drainable porosity.
However, aquifers often take exceedingly long time (sometimes several years) to
release water for this value to be meaningful [King, 1899]. Some interesting
discussions of the drainable porosity concept can be found in [Hilberts et al., 2005].

However, the value of the speciﬁc yield, if deﬁned as the water yield as a result of

69

unit decline of the head, is also dependent on the transient moisture conditions of the
unsaturated soil column above and the depth to water table. Since considerable
uncertainty remains as to the evaluation of S [Healy and Cook, 2002], the speciﬁc
yield is often kept as a calibration parameter in groundwater models to match
observations. We propose linking the speciﬁc yield to the drainable porosity deﬁned
in [Hilberts et al., 2005] as:

5y = of (2.87)
where f is the drainable porosity, because ﬁeld capacity represents the ability of the

soil to hold water under normal ﬁeld conditions.

 

 

 

 

 

1.4 I I
1.2 . - - - hydrostatic _
. ----- - DR<0
1 - \’\..‘ —DR>O -

A 0.8 - _

E.

N 0.6 - H
0.4 - -
0.2 - -

I "x I ‘s
0
0 0.5 1 1.5

h—pressure head (m)

Figure 2.6. h(x,z) as a function of z and DR values.

“With the above two assumptions and equation (2.86) we now detail our coupling

70

scheme. First, assumption 1 allows us to discretize the 3D unsaturated ﬂow domain
into an array of 1D columns connected to the 2D saturated ﬂow domain at the bottom,
as shown in Figure 2.5b. Each one of these columns can now be governed by Eq.
(2.56). As the groundwater ﬂuctuates, the water table may progress or retreat from
soil column cells, but must share at least one cell with the soil columns. Assuming
elastic storage SS is minor compared to the speciﬁc moisture capacity C(h), the 3

dimensional Richards equation (2.56) is re-written for the soil column as:

C(h)% - 3

_ K ]
8t 82 2“)

 

82

 

+ q,. + W(h) (2.88)

 

 

This equation is simply the 1D Richards Equation (2.56) with the addition of a

drainage ﬂux term qr, thus it is treated the same way as described in Section 2.3.5.
With Assumptions 1 and 2, we have:

0 z>HW

q,.(rc.z)= DR 2 (HW (2.89)
HW(x) _—

 

This last cell of the soil column serves as the linkage between the two ﬂow domains.
We use this cell to represent the unconﬁned aquifer and bring the dynamics of the
groundwater aquifer into the soil column. The upper boundary of this cell is marked

Zn. The center of this cell should always be below the water table. A mass balance

equation is written for this cell appears as shown below:

71

304:]:

 

 

 

8 8h 8 8h
— K —— +— K — d2 +
8t 821 1 82:] 8y[ y 83/
0 , O (2.90)
8h “u an 2,,
KWE-+1 4823241 +f0 Wdz

 

 

 

 

 

 

Applying Eq. (2.80) and re-arranging terms, this equation actually looks similar to Eq.

(2.63):

3'11
5% = K
8t 2

0
8h 8h
—+1 —K —+1
82 2[82 ]

 

 

 

 

Z 2
J—DR “ Wd 2.91
+ W + [0 z 1 )

By writing down this equation, it seems we have conceptualized the bottom cell as a
layer that has some storage capability. However, as we have neglected SS, the storage
is obviously 0 in this cell. The use of S is to mimic the reaction of the unconﬁned
aquifer, whose stage will rise after receiving recharge. Under this context, the RHS of
the equation (2.91) represents the net inﬂow into the unconﬁned aquifer. Thus we can
understand S in another way by linking to the speciﬁc yield of the unconﬁned aquifer:

dh dH W dH W dh
— = S = S
dt 3’ (if 3’ dh dt

 

(2.92)
dH W
db.

 

Thus, S = Sy

And dHW/dh can be obtained from differentiating Eq. (2.86).
The next step is to solve Eq. (2.91) for this cell together with the unsaturated zone Eq.

(2.88). Applying a semi-implicit temporal discretization to Eq. (2.91), we obtain:

72

hj+1_hj+1

S _( ,
7_12 hJ+1 _ h] )___ If 712—1 712 +1
712 712 712—1 2
At / A2 z71.2—1/2

 

7+1_ 7

h .
_K _7LZ__I+1+ Zu DR-f-IVj
1 A2 HW’ m

 

(2.93)
in which, nz is the index of the bottom cell, SnZ is the storativity (dimensionless) as in
Eq. (2.92), th is the pressure head of the conﬁned aquifer that is sitting beneath the
unconﬁned aquifer, K I is the hydraulic conductivity of the aquitard between the
unconﬁned and conﬁned aquifers. We treat th and DR explicitly because their
temporal changes tend to be much smaller than the rest of the terms. To be consistent
with Eq. (2.62), here we have used j to denote the time level. This equation is linked
to the state of the groundwater by the inclusion of kn]: , which we take from Eq.
(2.86):

(HW—z _1_DR(

MK

2

2
712 712 HW _ Zr2VZ ) (2'94)

lij = h($,2n,)' = [1 — 2119’—
“‘ K

 

Z

Note that both HW and lateral ﬂow term DR have already been computed in the
groundwater ﬂow model in (2.64), lagged by one time step. Thus Eq. (2.93) presents a

closure to the vadose zone model (2.61). Writing (2.93) in the same form as (2.62):

ai 7+177 7' 7+1p _ 7
a172h71.2- +b712h712 _ (1112
. KK_ . s . K
a] :_ "‘21/2,b;7z:_ﬂi_a1Jz~+_l_.
A2 ‘ At ‘“ A21 (2.95)

712—1/2

. S th Kl
d¢= “17+ -%Km_ —K)+ —DR+WJ
At 712 Azl 712 1/2 HW

 

 

712

as in (2.62), p denotes iteration number. Note the dimension of (2.95) is different from

73

(2.62). After this equation is solved, we calculate the recharge as the water leaving the

bottom of the soil column:

 

HW hj+1 _hj+1
_ 7 2~1 ’ 2
R _ f qr‘d" + I{712—1/2 11 AZ n +1 (2-96)
2 712—1

11

If the vadose zone model and the groundwater model share the same temporal step

size, Eq. (2.96) can be directly fed into (2.63). However since we allow adaptive time

steps for the vadose zone model, R needs to be integrated over time and divided by

the time step of the groundwater ﬂow model to be consistent in dimension. The

coupling procedure can be summarized as follows:

1. At the beginning of each time step, extract water table depth HW, and drainage
ﬂux DR from the groundwater ﬂow solution from last time step

2. Use Eq. (2.94) and (2.89) to calculate hnZ and qr.

3. Solve the system resulting from (2.88) and (2.91) as described in Section 2.3.5.

4. Calculate R with (2.96), integrate it over time, and divide the result by the time
step of the groundwater solver

5. With the predicted R, solve (2.63) as described in 2.3.6

6. Calculate DR

7. Go to step 1 and repeat

In essence, we are using the last cell (in some computational areas called the ghost

cell) to emulate the behavior of the groundwater system and make a prediction, albeit

a more reliable one, about the ﬂux between the vadose zone and the saturated zone.

74

Thus iteration on the equation level is avoided. Although the head in this last cell is
updated while solving the Richards’ Equation in the vadose zone, its ﬁnal state is not
important. Only the ﬂux through the interface is used later to calculate recharge in the
groundwater model (2.64). The more accurate the emulation, the less the error would
be. Our advantage at hand is that the groundwater system is well-known to be steady,
low frequency one such that equation (2.80) calculated from last time step is usually a
good approximation to DR. This way we have decoupled two systems to enable large
scale simulation, while retaining the salient nature of interactions between the two
compartments.

Here we must comment that the coupled lD/2D system will not completely reproduce
the behavior of the 3D equations, especially in places where lateral diffusion of soil
moisture is important on the scale of interest. However, the coupled system is a viable
alternative in watershed scale modeling where fully 3D solvers are not plausible, even
in the foreseeable future. In later sections we validate this approach using an

experiment dateset.

2.4.2. Coupling of the vadose zone and surface ﬂow:
Similar to the lower boundary condition, we write a separate mass balance equation

for the upper boundary cell, which is the ground surface storage layer:

db h, 71,
—1= — _—
dt (P E1) K1/2 A21/2 +1

 

— Fg (2.97)

 

 

where, h 1 is the surface ponding depth, P is precipitation (m/day), E1 is the surface

75

evaporation (m/day), Fg is the surface runoff (m/day) and K1/2 is the surface

hydraulic conductivity, which is calculated as the geometric mean of the saturated
current-state unsaturated conductivity of the ﬁrst soil layer (layer 2), considering the
fraction of pervious area:

K1/2 : (1‘ fimp)\/K1KSI (2-98)
The surface runoff is the contribution from surface ponding to various ﬂow paths. If
we assume the total length of overland ﬂow paths in the cell as 1, similar to [Panday

and Huyakorn, 2004], the surface runoff contribution can be computed as

__ Q9 _ (ht Th0)“ll
F9011) — I-W (2”)

Where Qg is the contributing discharge from surface storage to overland ﬂow paths
(m/day), u] is the ﬂow velocity (m/s), A is the area of the cell (m2), ho is a minimum
depth of water for surface runoff to occur (m) and the constant 86400 is for unit
conversion from rn/s to m/day. Here we conceptualize the runoff contributing process

as flOW on a rectangular plane and ICI I be calculated as:
dl ( . )

where (11 is the average distance to the nearest ﬂow paths (m). We use the Manning’s

formula to compute ﬂow velocity and apply the kinematic wave concept (S0=Sf):
‘2 3
’ul 2 '1—(1’1 — I10) / 53/2 (2.101)
n

where S0 is the average slope of the cell and n is the manning’s roughness coefﬁcient.

Then:

76

Q 86400 .
F. <4>=r=7.1,—<1—7.1/3sr (2.102)

ho is the depth of surface depression storage, and may also be interpreted as part of

the initial abstractions. Eq. (2.97) is a nonlinear equation that can be solved iteratively
together with the rest of the soil proﬁle. In order to achieve faster convergence, we

partially linearize the Fg term and the above equation is re-written as:

. . 2/3 .
1.. 1. 7—1 I. —1
F (hj+1’p) ___ SN‘(hl]+ ,p _ h0)(hl]Jr J _h’O) ’ hi]+ ,p > ho
g

0’ [li]+1,p—1 S ho

_ 8640053/2
— ndl

SN

(2.103)

where j and p are time level and iteration number as discussed in (2.61), k0 is the

surface runoff coefficient. With this runoff formulation, Eq. (2.97) is discretized as:

7' 7+Lp 7' 741.79 _ 7'
blh’l +61% —d1

7
7=__K1i b7 =—1-—cj+k
’ l 1 0
A21/2 At
71' (2.104)
. — 11 A
(11] —PI/1+E— 1/2+koh’0
. 2/3
, __ +1, —1
7.0 _ SN . (7119 P — 710)

When saturation excess occurs, i.e., the soil column is saturated and Qin>0, the

groundwater and the surface water are connected. In this case, the model will solve

the coupled equations (2.97) and (2.91):

77

 

 

dhi hl _ hnz ZO
Tit—”(P E1)—KT ——Z]:—+1—Fg(lz.)+fzt W112
8h 711 — h. a], ’b 2,
S——"i=K ——"Z+1—K —+1 + .+ Wd
81 T AZT 2 OZ QD7 2b Z

 

 

 

 

 

(2.105)

where, AzT is the total depth of the vadose zone, KT is the total average
conductivity of the vadose zone, and other notations are explained with Eq. (2.97) and
(2.91). To improve efficiency, we assume that when saturation excess occurs, root

extractions can be instantaneously supplied by the excess water, thus the collective

source term in the soil proﬁle, f~0 Wdz , is lumped into the top cell equation. Using
Z
t

the same modeling approach discussed above, we have:

 

 

 

 

N+1 N
_} ) N+1 N+1 2—1
(hl l1 =(P—E)—K hi _h": +1 —F 7N+1 +1: A2W
At 1 T ZT g ‘1 . 2 1 1'
z:
N+1 N
(hnz ‘— 712) thH—hn’i“ 7131—72.,”
At =KT AZT +1-Kl———A:l——+1+DR+WW

(2.106)

By solving for hnZ from the second equation:

 

S N T N+1 K1 N ,

—7. +———7 +———h +K —K+ +11
N+1— At ’nz A27 h A21 1 T 1 QDr 712
712 _ S K K]

T
__+___ _—
At AzT A21

(2.107)

and substituting it back into the ﬁrst one, we obtain a nonlinear equation:

78

 

 

 

11" +1 : p1[p2 Fg (711” +1) (2.108)
With
1
p], : 1
A—t + 72 — 7274
th 712—1 N N
1’2 = Eff (P‘ E1)+ 2: Azz‘Wt _ KT + 74 71,1112 +2’3h1 + 75
i=2
5 KT K1 ’72
:——, :-—-” :——-’ : ,'_:K,+ +VV —K
’71 At ’72 AZT ’73 Azl 74 71 + 72 + 73 lo I’ QD7 712 l

K T = harmonicMean(K)

(2.109)
This equation can be quickly solved by either Picard or Newton iterative methods.
Convergence is so fast with Newton iteration that it seldom takes more than 3
iterations.
2.4.3. Interaction between overland ﬂow and channel ﬂow
A zoomed-in sketch helps illustrate the exchange between the river and the land
(Figure 2.7a). From the point of numerical accuracy, it is perhaps best to use an
unstructured grid for the overland ﬂow, enhancing the resolution and ﬁt to the river
orientation near the streams. However, many other concerns dictate over the
computational accuracy in the ﬂow model, e.g., hydrological response unit,
integration of different model components, input data resolution and model generality.
Therefore we choose to employ a structure grid for land discretization. Here we
denote a reach of a river that is completely inside a land cell as a ‘reach segment’ (5).

A segment enters from one edge of a land cell one and leaves from another (maybe

79

the same edge) as shown in (Figure 2.7a). Because we also allow signiﬁcant
ﬂexibility for the river discretization, the edges of river cells and segments usually do
not correspond.

Two transformation matrices are pre-computed and stored to facilitate the calculation
of exchange ﬂux. TM records the fraction of each reach segment that belongs to each

river cells; TN records the fraction of each river cell that belongs to each segment:

 

 

T111 M T1111, "3
TM = .
TIM" 7',1 ' ' ' T111717; 71.9
TNLl TNLm,
TN : (2.110)
TanJ ' " TN718,717'
TMM = 73-2819 6 (1,2,...,ns},j e {1,2,...,m~})
TNj,i = 31-7?“ ,(1' E {l,2,...,nr},j E {l,2,...,ns})

Where ri, i= are the river cells, nr is the number of river cells of the river, 5],
i=(1,. . .,ns) are the reach segments, ns is the number of reach segments, and ﬂ is the
spatial join operator. TM and TN are both sparse matrices. With these two matrices, it
is possible to project variables from river cell grid to the reach segment and vice versa.
The length-weighted river stages of the segments can be calculated from the stages
deﬁned on the river cells:

Zchl,Zc172,...,ZchnS] = [n1,772,...,77n_7, x TM (2,111)
Where Zch is the river stage of the reach segments, 11 are the free surface elevation of

the river cells. After the exchange ﬂux is computed for the reach segments using the

80

method described below, it can be distributed to the river cells by:

qoc,1 qoc,1
' =TMX 5 mun

qoc,m~ (100,713

in which th is the exchange ﬂux (m3/day) computed for the reach segments, q0C is
the lateral ﬂow into river from overland ﬂow for each river cell. An occasional
situation is that one river may have several segments inside one land cell as it
meanders in and out of the edges of the same cell. A correction pass is used to remove
the redundantly calculated ﬂux and ensure mass balance.

It has been proposed in [Panday and Huyakorn, 2004] that interaction between
overland ﬂow and channel can be modeled by the equations for ﬂow over a wide
rectangular weir. Similar to this approach, we developed an efficient and stable
procedure to compute river/land exchanges on a physical basis. The cross-sectional
and plane sketches of a river cell are given in Fig 2.7. Zbank is the elevation of the.
bank (m), ho is the depth of the overland flow (m), E is the average elevation of the
cell (m), Z0 is the average free surface elevation of the cell (m) (Z0 = E+h0), hC is the

channel ﬂow depth and Zch is the stage of the channel (m).

81

 

    

 

 

Riverbed

 

 

(a)

 

 

 

l

 

 

 

 

 

 

 

 

x——-x River Segment |-—-| River Cell
(b)

Figure 2.7. sketch of a river cell in the model. (a). cross-sectional view, (b) plane view

82

The exchange mass M (m3) between land and channel is computed with the following

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

procedure:
<1> ZO>Zch?
Yes I No
<2> ho>0? <3> Zch>zbank'AND:hc>O
Yes No Yes No
M=min(Mex,ME,Ma) M=0 M:Mim M=

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.8. Calculation of River/Land exchange
In the above diagram, the ﬁrst condition <l> is to determine the direction of the ﬂow.

If it is from the land to the river (Zo>Zch) and there is water on the land (condition

<2>), we ﬁrst attempt to compute the contribution explicitly using the diffusive wave

 

 

 

equaﬁon:
1/2 1/2
111 = sﬁn(ﬂ) __2LCAt (ls/3 £3.11 : ___2LCAt [157/3 ‘20 — Ina“K(chr' ZBarl/x')i
er: 0 (917 71 oc (9:1: n o l (Ax/2)
(2.113)

However, this ﬂux cannot exceed the amount of currently available water on the land
cell:
Ma = Ar-ho (2.114)

Also, there is an equilibrium state, at which the river stage will be the same as the

83

land free surface elevation. Denoting this stage as Z*, the mass transfer equation is
written as:
>1! II!
(2 .411)».ID = (20—2 )Ar (2.115)
Where Ar is the area of the land cell, Ar=Ax*Ay and Ab is the area of the river cell

that spans this land cell. Then we can ﬁnd the Z*:

(Ar + Ab)Z* = ZOAr + ZChAb
ZoAr + ZCIIAI) -Z : (ZO-ZCII)AIJ
(Ar + Ab) C“ b (1+Ab/Ar)

 

*
IVE : (Z -ZCII)Ab :

 

(2.116)

And thus the exchange mass will be the minimum of Mex, ME and Ma.

On the other hand, if the river stage rises higher than the land free surface elevation
(and also the bank elevation), ﬂooding would occur, which is solved using an implicit
approach to enhance stability. Again using the diffusive wave formulation, the mass

exchange function can be written in the form of two ordinary differential equations:

 

ATdZO =—]l[ :__2_£cht~Zo (Z —Z )5/3
dt 2"" n Arr: / 2 ch Bank (2 117)
dZCh dZo .

 

Ab 2 —Ar
111 (if
Again these two equations can be solved by Picard or Newton iteration. The resulting

scheme is very stable.

2.4.4. Interaction between groundwater and channel ﬂow

84

The interactions between groundwater ﬂow and channel ﬂow is solved immediately
after the channel ﬂow step in section 2.3.4. We use the concept of operator splitting to
couple the river ﬂow and exchange with groundwater. After the river ﬂow model is
solved explicitly with Runge-Kutta approach, an implicit step is solved to calculate
the exchange between streams and groundwater using the conductance concept

[Gundzrz and Aral, 2005]. The governing equation for the fractional step is written as:

F

 

 

 

h. —h
.9 r _
ﬁll = ,. AZb by > (2b AZb) (2 “8)
dt Kr (z, — A2,) —— h, h s (21 _ AZb)
AZb 9

in which, 2}, is the river bed elevation (m), A21, is the thickness of the river bed
material (111), Kr is the river bed conductivity (m/s) and hg is the groundwater table

elevation (m). The above equation can be discretized implicitly as:

N+1 _ =1: [1* _ hN+1
h___h_=Kr_9___ (2.119)
At AZb

h* is the computed from the solution to equation (2.51) and h* is max(hg, zb-Azb). We

N+1 . .
can solve h from this equatlon:

* *
K,(hg—h)

N +1
h =
A Zb + K (2.120)
At 7'
and then the lateral ﬂow from groundwater is computed as:
(70,, = w(hN+1—h*) (2.121)

qgc is the lateral inﬂow from groundwater (mZ/s), w is the river width (m).

85

2.5. Data preparation steps general to all watersheds

A list of required input to create and drive a model is given in Table A. 1. All of inputs
except for the groundwater data are readily available for most of the US and can be
processed by the program without much additional effort. Some data processing steps
are general to all watersheds. They are described in this section. Creating the
conceptual groundwater model (i.e. how many layers to model) requires some
knowledge of the geological conﬁguration. Required quantitative groundwater inputs
include conductivities and thickness of the aquifers, which are not available in any
readily-usable national database. Therefore the groundwater data may come from
different sources for different regions and the processing is slightly more ad-hoc. For
the model application in this dissertation, groundwater data source and processing are

given in Sec. 3.1.1.

2.5.1. Digital elevation data

86

x 105 NED - DEM (m)

 

1.45
1.4
1.35
1.3
1.25

 

 

 

 

3.98 3.99 4 4.01 4.02 4.03 6
X

Figure 2.9. Discrepancies between NED and DEM when aggregated into the same
grid

The digital elevation data are widely available in many formats and resolutions. In the
U.S., the most commonly used elevation data include Digital Elevation Model (DEM)
and National Elevation Dataset (NED). The DEM normally has a resolution of
approximately 90m. The 30m resolution NED is available for most of US. and in
some regions even with 10m resolution. Figure 2.9 shows the diﬁ'erences of the model
grid elevation as a result of aggregating the DEM and the NED data using the mean
method. As we can see, local discrepancies between the NED and DEM can be at
times large. A general pattern is that NED is higher on the highlands than the DEM

but comparatively lower on the lowlands.

87

2.5.2. River data processing

The River network is established from the National Hydrography Dataset (NHD). The
NHD contains geographic information of streams as a collection of stream segments.
Each segment is a short stream reach that records the spatial coordinates of its curve
vertices with an attribute table detailing information including the ‘from’ and ‘to’
node of each river segment, the stream order and stream names if applicable. However,
the dataset in its original form cannot be utilized by the model as all segments are
listed as a relational table, not grouped by actual streams and do not exactly follow
upstream-downstream sequence. So some effort is required to sort the data into a
usable format. First, some pre-processing is done to identify the rivers that are going
to be modeled, by using the stream order and stream name information. Unused river
segments are removed from the data. The stream segments with the same stream order
and same stream names are tagged with the same river id (RID). Then river segments
with the same RID are extracted and grouped for further processing. Within one group
of segments that belongs to a river, the segment that has no upstream reach is the head
of the river. After we ﬁnd this segment, we sequentially ﬁnd its discharging segments
using the information in the ‘to’ ﬁeld until a clean list of segments from upstream to
downstream is completed and the X-Y coordinates of the vertices are also sorted.
Next, the river network is established by going through all rivers and recording their
tributaries, downstream rivers, and locations at which they are joined by their

tributaries. Finally, we establish a list that sorts all the rivers ﬁ'om upstream to

88

downstream. This list is used by the watershed model so that it always runs the river
models in an upstream-to-downstream sequence.

River bed elevation is found by subtracting the bankful river depth from the local
elevation values (taken as the bank elevation). However, due to its relatively large
grid size, DEM often has difficulty capturing thin features like streams and creeks. It
is found that the elevation data extracted from the DEM tend be too high for the

evaluation of bank elevations.

The bankful river depth data are not available in most cases and therefore must be
established from empirical relations and ﬁeld measurements. For the Red Cedar River
we have measurements of water depth at bankful conditions from Acoustic Doppler
Current Proﬁler (ADCP) surveys at various locations, as well as operational USGS
instruments. These measurements are used to ﬁt to the empirical relation of [Bjerklie,

2007f

0.39
'11)
D = c

 

500.24 (2.122)

where w is the river width, S0 is the slope of the river and c is a ﬁtting parameter. c is
found to be 0.027 from the Red Cedar River measurements and applied to all reaches
of the rivers. While it is possible to use other approaches to estimate the bankful river
depth, the simple relation seems to work well with some veriﬁcation measurements

taken on Red Cedar River and its impact on the watershed modeling is limited.

89

Fig 2.10 shows the river bed elevations estimated based on the DEM and NED using
the above approach for the Red Cedar River, as well as the groundwater table
provided by wellhead observations from the wellogic database [G WIM, 2006; Simard,
2007], later described in the groundwater input data section. As we can see, the
NED-based estimates closely correspond to the groundwater heads while the DEM
based values generally tend to be much higher, especially at reaches with large
elevation drop. Close examination of the discrepancies reveals that the places where
the two estimates diﬁ’er the most are often places where the river feature is missing
ﬁom the DEM map due to its resolution. Therefore, we conclude that NED-based

values are much more accurate.

 

 

 

 

 

290 l l I I T
- ------ River Bed-DEM
230 — ‘ \ - - -River Bed—NED a
E “a, —Groundwater Head
2’ 270 —
.2
‘s’
.01 260 -
LU
250 h .
I I 1 I I
O 20 4O 60 80 100

Distance along stream from head (km)

Figure 2.10. the river bed elevations estimated based on the DEM and NED as
compared to the groundwater head

90

2.5.3. Soils
With the initial guess of the groundwater table, hydrostatic equilibrium is assumed as
the initial guess for soil moisture. At hydrostatic state, we must have

h(w, w) = z — H (w, 77) (2.123)
where h(x,y,z) is the pressure head at any coordinate (x,y,z) in the subsurface domain,
H(x,y) is the water table elevation of a grid cell at location (x,y). With this initial state
and the van Genuchten formulation (2.57), the initial soil moisture condition

anywhere in the watershed can be evaluated.

2.5.4. Climatic data

The Theissen polygon method is used to assign climatic data to cells. At the time of
discretization, each climatic station records the indices of cells that they control.
When a data record is missing, the ﬁrst approach is to borrow data from adjacent
stations. If no valid data exists on this day in all neighboring stations, the monthly
mean value of the region is used.

As the model is based on an hourly time step while climatic data are often available in
daily format, there must be ways to disaggregate the data into hourly time steps.
Where daily precipitation from the NCDC is used, historical rainfall pattern for the
Midwest is used to distribute the precipitation into hours according to [NRCS, 1986].

Michigan belongs to the type II (intensive rain) rainfall zone, and the rainfall

91

hyetograph is obtained in the form of cumulative rainfall distribution Pfrac as shown
in Figure 2.11a:
t
Pfrac = [0 P (1)11 (2.124)
where P(t) is the rainfall rate (m/day). The rainfall rate in hour is then evaluated as:

x Pfrac(h7' + 1) — Pfrac(h7‘)
“31 1 / 24

 

P(hr) = Pday x f = Pd (2.125)

where Pday is the precipitation in a day (m/day), P(hr) is the rainfall rate in the given

hour (m/day), :4 f (k) = 1 and is discretely valued as shown in Figure 2.11b.

 

 

1 I I I
@05- —
H—
0 I I I
0 5 10 15 20

 

 

 

 

0 5 10 15 20
Hours of day

Figure 2.11. Rainfall pattern of the Midwest from

92

As discussed previously, given maximum temperature me (°C) and minimum
temperature Tmn (°C) in a day, the sub-daily temperature is parsed using [Campbell,

1985], assuming highest temperature at 3PM local time:
T . .. — T
T = Tm, + 1L2—"yicos(0.2618(t — 15)) (2.126)

in which hr is the hour of the day, Tav is average temperature of the day (°C).

2.6. Model Flow Diagram

Figure 2.12 shows the ﬂow diagram for the proposed model. The model marches
using the calendar days. The hydrologic processes, including Evaporation,
transpiration, snowpack and unconﬁned aquifer are updated on an hourly basis. The
vadose zone model solves inﬁltration, runoff and depression storage together with soil
moisture. It uses an hour as a base time step but adjusts it depending on flow
condition and. convergence rate, similar to [van Dam and F eddes, 2000]. The overland
ﬂow and river network also have the ability to adaptively select time steps to ensure

stability and computational efﬁciency.

93

Daily Loop

 

Hourly Loop

 

Model Start

 

_

 

Weather I

%

Vad oze

  
    
 

 
 
   

Canopy
Interc . tion

Snow . ack
Eva o oration
Im o ervious

   
 

 

 

 
 

  
  

 

Zone

Gndwater

9%

 

Output

Model End

 

 

 

     
   

Rivers Loop

 
   

Solar Rad

 
 
      

I epressron
Stora ; e

Overlan 0
Exchange

     
 
  
 

 
   
 

" nanne
Flow

   
 

 
 

 

Gndwater
Exchan 1 e

 

 

   

Figure 2.12. Program Flow Chart for the proposed model

94

2.7. Test cases

Numerical codes must be carefully tested and veriﬁed before they can be put to use.
There are two levels of code testing. In the ﬁrst level, numerical code is compared to
available analytical solution to ensure there is no bug in the code and that the
numerical scheme solves the PDE with acceptable accuracy (within the range covered
by the test problem). In this level the match between numerical solution and analytical
solution should be very good otherwise numerical error may inﬂuence our
interpretation of the results. In the second level, the numerical code is compared
against experimental data to examine how much the idealized governing equation
approximates the real-world. Due to experimental error and simpliﬁcations of the
processes, this is the step where more deviations are commonly observed.

In this section, we present comparisons with analytical solutions for all of the four
ﬂow domains (overland ﬂow, channel ﬂow vadose zone and groundwater). Instead, it
is compared to ﬁeld measurements. The coupling between overland flow and channel
flow is tested together with overland and channel flow codes in the V-catchment test
problem. Whereas since analytical solution for 3D saturated/unsaturated ﬂow is not
available, the coupling between vadose zone and groundwater ﬂow is compared to

experimental data reported in [Vauclin et al., 1979].

Flow over an inclined plane

The first test case for the overland ﬂow model is ﬂow over an inclined plane. The

95

plane is of length 200 meters and has a slope of 0.001 in the longitudinal direction but
is level in the lateral direction. The Manning’s coefﬁcient n for the slope is 0.03. A Ax
of 10m and a At of 3 seconds are used. A uniform rainfall rate of 1/60 mrn/s for the
duration of 1 hour is applied starting from the beginning of the solution. The Orlanski
free outﬂow boundary condition (reference) is used. An analytical solution is
available for the kinematic wave (KW) equation but not for the Diffusive wave (DW)
and Dynamic wave (DyW) equations. We solve the DW with the RKFV scheme,
solve the DyW with the SISL scheme and compared them with the KW analytical
solution in Figure 2.13. The time of concentration (to) for the plane is 18965. This is
the time when outﬂow equals rainfall rate and the hydrograph reaches a plateau. For
this problem, there is a small difference between the solution of KW and DW on the
rising limb around the time of concentration. This is also observed in the comparison
reported in [Gottardi and Venutelli, 2008]. The DW has a more smoothed comer as
opposed to the sharp edge predicted by the KW. This is not fully attributed to
numerical dispersion but rather due to the fact that kinematic wave neglects the
backward pressure from downstream. Thus it over-predicts the outﬂow rate before
time of concentration as compared to the full DyW. The solutions from DW and DyW,

on the other hand, are almost indistinguishable.

96

 

    

 

   

1.5
ml"!
52
x
1'5 1— _
\
E.
3
O
E 0.5 — f Analytical—KW —
0 RKFV—DW
---SISL—DyW

  
 

J I I I I

1000 2000 3000 4000 5000 6000
Time (s)

 

 

Figure 2.13. Outﬂow hydrograph of the inclined plane test problem compared to the
analytical solution

V-catchment

The V-catchment problem is a standard test case for overland ﬂow models
[DiGiammarco et al., 1996; Stephenson and Meadows, 1986]. The domain consists of
two inclined planes draining into a sloped channel. The sketch of the problem is
shown in Figure 2.14. Both of the planes are 800m in the lateral direction and 1000m
in the longitudinal direction and the slope is 0.05. The channel has a slope of 0.02.
The Manning’s coefﬁcient n used for comparison with literature is 0.015 for the
planes and 0.15 for the channel. Due to symmetry, only half of the domain needs to be

modeled. The plane is solved using the 2D RKFV scheme and the channel is solved

97

on a separate channel grid, using the channel RKF V scheme. The solutions obtained
from using the SISL scheme are almost identical and thus will not be reported here. In
order to test not only the overland and river ﬂow models but also the land/river
exchange scheme of the proposed scheme, we use exactly the same procedure
described in section 2.4.3. The land grid uses a Ax of 50m and a At of 5 seconds.
Because the channel has a milder slope and large n, a At=lOs is used. The channel
outﬂow rate is shown in Figure 2.15a, as compared to the analytical solution and
Finite Element (FE) solution reported in [DiGiammarco et al., 1996]. The analytical
solution is not available for the declining limb of the hydrograph. We observe that the
RKFV scheme matches the analytical solution very well during the initial rising
period, slightly better than the FE solution. The plane-side outﬂow rate, the rate at
which land contributes to the river, is shown in Figure 2.15b. There is a very small
oscillation around t = 30 min due to the mass exchange scheme. This is also present in
FE solution, only stronger. In watershed-scale modeling, such a small feature hardly
has any impact just like its inﬂuence is not seen in Figure 2.15a. Also note the
declining limb is captured noticeably better by the RKFV. Considering the simplicity
and computational efﬁciency of the RKFV, these results make the scheme very

favorable.

98

 

Figure 2.14. Sketch of the V—Catchment test problem

Red Cedar River measured hydrodynamic data-

The river ﬂow code has also been tested by comparing with analytical solutions of the
inclined plane and V—catchment test problems. The solution is identical to the
overland ﬂow solver reported above and thus are not shown here. In order to test the
model structure and the applicability of the diffusive equation to local rivers, we test

out numerical model against ﬁeld collected data.

99

 

  
    
   
   

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I
0.25 - '-
RKFV
a 0 Analytical _
M\ 0.2 ...... FE
E,
g 0.15
E.
5
a, 0.1
E
I
8 0.05
2
Q.
0 .
1
0 50 100 150
Time (min)
I I l
5 — —
"\.
_ 4 — ‘I -
V)
M\
E. .
3 3 \\
0 \.
g .
O 2 - _
5 ‘-
.2
a:
1 ._ _
30 35 40
0 ‘ I I J '-
0 50 100 150
Time [min]

Figure 2.15. Solution to the V-Catchment test problem: Comparison of River outﬂow
hydrograph with analytical solution and the ﬁnite element (FE) solution reported in
[DiGiammarco]. (a) River outﬂow (b) Plane side outﬂow

100

[Irfan, 2002] measured hydrodynamic data (Discharge, Stage and Velocity) over a
period of 8 days on the Red Cedar River (RCR). The rising limb and part of the
falling limb of a precipitation event is recorded during this period. The bathymetry of
the river reach, as well as the river width, is also provided. The data are fed to the
river ﬂow module of the proposed model. The comparisons between simulated and
observed data at the Library Bridge are shown in Figure 2.16. The model accurately
simulates the discharge and stage data. There is some discrepancy between the
numerical solution and the measured velocity data. The difference may be due to the
presence of a weir at the Library Bridge. In addition, the model assumed rectangular
channel geometry while .in reality the channel cross-section is slightly parabolic and
there are ﬂood plains at various locations of the study reach. The ﬂood plains are
expected to attenuate the ﬂood peak and thus reduce the peak ﬂow velocity. However,
incorporating the presence of local ﬂoodplains considerably complicate the model and
increases the data requirement and computational burden. Thus this is not considered

here

101

 

N
O

 

Discharge (m 3/s)
5'

 

 

 

 

 

 

 

 

 

c
1 2 3 4 5 6 7 8
1
E
w 0.5
U)
(U
a
o I I I I I I I
1 2 3 4 5 6 7 3
A 08 I I I I I I I
E 06 l- o O _
3 ° ° °
3 0.4 — -
G)
> 0.2 I I I I I I I
1 2 4 5 7 8
Time (day)

Figure 2.16. Simulated channel ﬂow vs measured data by [Irfan, 2002]. Red circle:
measured data; blue solid line: Simulated

Inﬁltration Test Problem
This widely-used test problem demonstrates the ability of the 1D Richards’ equation
to predict inﬁltration into very dry soils [Celia et al., 1990; Haverkamp et al., 1977].
It is one of the few analytical solutions available to the Richards equation. The soil
column has a total length of L=0.3m, with the initial condition:
h(z,0) = —10m (2.127)

And Dirichlet boundary conditions on both ends:

102

h(O, t) = —0.75m

h(L, t) = —10m (2'128)

The soil parameters for this test case are given in Table 2.5

Table 2.5. Soil Parameters for the test case: Inﬁltration into very dry soils

 

Parameter 9r (.) 98 (.) a (m- 1) n (-) KS (In/day) /l(-)

 

Value 0.102 0.368 3.35 2 7.9661 0.5

 

 

The solutions (pressure head) obtained with three different spatial sizes (Ax= 0.075,
0.25 and 0.6 cm) and the analytical solution at the ﬁnal simulation time t = 6hr are
presented in Figure 2.17. As noted by many other studies, the vertical resolution does
have an impact on the solution quality [Downer and Ogden, 2004a]. The solver
performs very well at Ax= 0.075cm and 0.25 spatial resolution. And the quality of the
solution at Ax= 0.25 is very similar to that reported in [Lehmann and Ackerer, 1998].
But we do observe some deviation when Ax gets larger. The upper boundary condition
serves as an inﬁnite supply of water. We observe a ‘piston’ type of wetting front that
is found in homogeneous porous media propagating without changing much of its
shape. A coarse spatial resolution is hard to maintain the sharp wetting front predicted
by the homogeneous Richards equation. One of the reasons is that the solver, as other
predominantly used schemes, is ﬁrst order accurate in space and time. More
importantly, at the wetting front the soil hydraulic conductivity changes dramatically

due to the strong nonlinearity of the soil water retention and hydraulic models.

103

Solving the Richards equation accurately using large spatial size remains a big
challenge. However, since in the environment there are much larger uncertainties, e.g.,
the existence of macropores and the parameterization of soil water properties, we
believe the numerical error with the solver will be dominated by other sources of

errors and uncertainties.

 

 

 

 

O -—I
0.50 "'
0.1 0 _I
g
.C —
«a 0 15
Q)
Q
0.20 _
0.25 -
0.30
0

 

Pressure Head (m)

Figure 2.17. Inﬁltration into very dry soil test problem at t = 6hr. circle: Analytical
solution; solid line: Ax= 0.075cm; dashed line: Ax= 0.25cm; dashed line with cross:
Ax= 0.6cm.

Puming near impervious wall
The classic Theis solution [Freeze and Cherry, 1979] is used to check the accuracy of

the groundwater solver. This test problem shows that when an impervious wall is

104

present, the drawdown in the conﬁned aquifer will be greater because water supply is
limited (Figure 2.18a). The analytical solution to this problem can be obtained by
modifying the classic Theis solution, considering the symmetry of the problem and
setting a ‘shadow’ well in the mirror position of the pumping well. The solution form

is given by [Freeze and Cherry, 1979]:

h.O — h(x, y, t) = ﬁlW (211) + W (11%. ))

i

 

 

 

 

 

 

(. — at +
Ur = . 4Tt ‘ (2.129)
(:1: + a.)2 + y?)
“’2' = * 4Tt :
OOe—"du
W(u) = L it

In which, h(x,y,t) is the groundwater hydraulic head (m) at location (x,y) and at time t
[T], ho is the initial head [L], S is the aquifer storage coefﬁcient H, T is the
transmissivity of the conﬁned aquifer [LZTJ], Q is the pumping rate [L3T_l] and a is
the distance of the pumping well from the impervious wall [L]. W(u) is the well
function which can be numerically integrated.

The problem setup is illustrated in Figure 2.18b. A well situated 47.5m from the
impervious wall and an observation well is located 50m away from it, further away
from the wall. Other related parameters are:

Pumping rate Q=1000 m3/day

Initial hydraulic head h0=20m

Conﬁned aquifer storativity: S = 0.0002

105

Aquifer Thickness b=20m
Hydraulic Conductivity K=50m/day

Ax=Ay=5m

The time-series comparison with analytical solution at the observation well location is
presented in Figure 2.19. As we can see, the solutions match perfectly.
Impervious

boundary
(0, 755)

  

:9- H.a'l"<_- vhug-;--_---yy-l--ﬁa-a.'-.u-'
- .. . :«I ragga. 441-: 'hm‘*s;= .1

(1000.0)

trait-anrrgrsggooo’ 755)

   

(0. 0)

 

[1:113:11 masses: earnest! In an: Ira-1

"I u
‘‘‘‘‘
I- '.'-‘.r--.'.= W ' ‘ ' -;- '.' '-lw-'.‘-‘.*e".“‘ ' 'n Ir 0' ﬁiTu- ‘ u---.. ., .-- .5‘1'.“'-‘;Ffir."""!. a -. --

" 3 , . _ 33“.": _ . s _ _ . _ ‘ a
nu...- -2- 4356-? .u "“§~.=“'L:‘u”"‘:‘l .Ngw-‘iﬁ‘. . a sl— 1 ~\n . bun: .MNQ .La art; I I. 1"": .,uidlr-‘ni nth-‘53.}.

I
I
I

 

 

Figure 2.18. Pumping near impervious wall test problem

106

 

 

 

 

 

20 I I
~ --Numerica|
19.95 - e 0 Analytical —
,... 19.9 L
3.5.
“D
(U
a, .
:1: 19.85 -
19.8 '-
19.75 ' '
0 50 100 150
Time (s)

Figure 2.19. Solution to the pumping near impervious wall test problem: transient
comparison;

Coupled Saturated/Unsaturated model

Previously, we described a novel approach to couple the unsaturated/saturated model,
but its eﬁectiveness to replace the 3-dimensional equation must be validated. As
analytical solution to such a coupled system is not available, we can only compare our
results to the results obtained from experimental study, or solutions from a
3-dimensional Richards model. Here we ﬁrst present a comparison with the
experimental results from [Vauclin et al., 1979]. This dataset has been employed by
many researchers to verify their 3D Richard’s equation code, e.g. [Dogan and Motz,

2005] [Clement et al., 1994]. The experiment consists of a soil slab 6.00 by 2.00

107

meters. Initially the soil slab has established hydrostatic equilibrium with the water
table at 0.65 meters from the bottom. At the center Im on the surface of the soil slab,
a constant ﬂux of 3.5 m/day ﬂux is applied. On the left and right boundary, water
level is maintained at 0.65m. As in other studies, we model only the right half of the
domain due to the symmetrical nature of the experiment setup. Thus no ﬂow
boundary is applied to the left boundary. The soil retention and conductivity data
published in the original data in [Vauclin et al., 1979] is ﬁtted to the Van Genuchten
formulation by [Clement et al., 1994; Dogan and Motz, 2005] and provided here in
Table 2.6. For easier discussion we name the soil columns that are directly receiving

inﬂow as recharge columns and the other columns as passage columns.

Table 2.6. Soil Parameters for the test case: Vauclin experiment

 

Parameter 9r H 65 H a (m- 1) n (-) KS (m/day) 4f")

 

Value 0.01 0.30 3.30 4.1 8.4 0.5

 

 

108

 

 

 

 

2 I I I I I n
-~---- Model
0 Vauclin et al (1979) Experimental Results
1.5 - '-
8hr
~-ﬁ mmm‘
.....° 4hr N'1) «a
1 Hjmw-M... __o 0““ we“ _.
3hr“ --~...._°"" ~01-..“ “guxk‘m
”vih m». ~~“NH‘O ~... wt: ““""~c.-__h ~i.‘\‘~\.\
r ‘ ““ ---« . “Mm-N- "“ ..
o ~--.~.-.. - ~- - ._ __
[mam _.__.......-___....._. - . ,_ .. .. . .. ..
0.5 -
O I I l I l I]
0 0.5 1 1.5 2 2.5 3

Figure 2.20. Comparison of the proposed model with experimental results from

The water table simulated using the proposed coupling method is shown in Figure
2.20. It is observed that the solution at hours 3, 4 and 8 ﬁt well to the experimental
data, whereas at hour 2 it has over-estimated the head at the left boundary. This
over-estimation can be explained by the deviation of the coupled system to the actual
3D system. At the beginning stages of recharge, the passage columns are dry as
compared to the recharge columns. Lateral moisture diffusion causes the inﬂow to be
re-distributed to the passage columns before it reaches the unconﬁned aquifer and the
unsaturated part stores a portion of the inﬂow. However, in the coupled system, water

can move laterally only after it enters the unconﬁned aquifer. Thus at early stages the

[Vauclin et al., 1979]

109

recharge has been over-estimated. However as the water table rises, the drainage term
DR increases, the unconﬁned aquifer is able to transport more and more water, and
the system gets closer and closers to steady state. The solution matches the
experimental results better at later stages. This diffusion effect is expected to be less
noticeable when the method is applied to a watershed-scale model, where the ratio of
lateral ﬂow ﬂux to recharge is much smaller as compared to this experimental

scenario.

Figure 2.21 shows a comparison of the soil moisture proﬁle at hour 8 between the
proposed coupling method and the traditional method, in which the hydraulic head of
the unconﬁned aquifer is provided as Dirichlet boundary condition to the soil column.
We notice that the soil moisture proﬁles of the recharge columns (x=0.025) are much
higher with the conventional method than the present approach. Whereas in one of the
passage columns (x=0.925) the soil moisture is lower than the steady state value. Both
deviations are explained by the formulation of Eq. (2.86). With the present approach,
the water table location (point at which soil moisture becomes saturated) agrees well

with the head in the unconﬁned aquifer (in Fig 2.20).

110

 

 

 

0-5 ' —x=0.025,EBC g; ‘
+x=0.925,EBC
0— ---x=0.025,DBC
- o — x=0.925,DBC

l I I I I I
0 0.05 0.1 0.15 0 0.2 0.25 0.3

 

 

 

 

 

 

Figure 2.21. Comparison of soil moisture proﬁle obtained using the approach in the
proposed and traditional method, 03=0.3, EBC=Equation Boundary Condition
(Present approach); DBC=DirichIet Boundary Condition (Conventional approach)

Snowpack model.

The UEB snowpack model [Luce et al., 1999; Luce and Tarboton, 2004] is adapted as
the snow module of the proposed model. This model has been applied in the
mountainous region of North East US. Here we apply the snow model using the

driving data ﬁom Michigan.

Mass Balance Properties of the model

111

All schemes used in this model are mass-conservative schemes. However since the
convergence of some iterative matrix solvers (e.g. the Conjugate Gradient method) is
decided when error is less than a certain threshold, it is possible to induce a small
amount of mass balance error. The mass balance error of the entire model and each
component is closely monitored. After 10 years of simulation, a total mass balance
error of 10-8 m is generated, which is insigniﬁcant compared to around 8m of rainfall

during the period of time.

2.8. Summary and Conclusions

We follow the Freeze and Harlan ( 1969) blueprint for physically-based hydrologic
modeling in that we believe each component of the model simulates a physical
domain and should be veriﬁable by itself. All flow components, including overland
ﬂow, channel ﬂow, vadose zone and groundwater ﬂow have been independently
veriﬁed by analytical solutions. Field measurements have been used to verify the
Channel ﬂow code. The novel coupling mechanism between vadose zone and
Saturated groundwater ﬂow has been applied to the Vauclin (1979) experimental data
and the results were found to be promising. This novel approach solves a
lOrig-standing bottleneck in PDE-based subsurface ﬂow modeling by removing the
Computational limitations while maintaining physically consistent solutions. Surface
ﬂow is solved by an efﬁcient Runge Kutta Finite Volume scheme. After testing

individual components of the model, we are ready to apply the whole model to a

112

real-world watershed to evaluate the performance of PAWS as an integrated modeling

system for the hydrologic cycle. This is accomplished in the next chapter.

113

Chapter 3. Development of the Hydrologic
Model: Model Applications and Comparisons

In this chapter, we apply the PAWS model to a medium-sized watershed in Michigan
and evaluate its performance as an integrated watershed model. The model outputs
including streamﬂow and groundwater predictions are compared with observations,
and average annual ﬂuxes are compared with that estimated by an accepted annual
budget simulator-the SWAT model. Through these comparisons we wish to establish
the credibility of the model. Then we employ the model to provide insights to the

dynamics of the study watershed and illustrate the interactions between the hydrologic

components in space and time.

3.1. The Red Cedar River watershed model

3. 1.1. Study site and input data

The Red Cedar River watershed is located in the Grand River watershed in Michigan
(Figure 3.1). The National Elevation Dataset (NED) for the watershed is shown in
Figure 3.2. The total area of the watershed amounts to ll69km2. The watershed has a
relatively low relief with the maximum elevation recorded as 324m and a minimum of
24 9m.

The watershed is characterized by the humid continental climate with ample

precipitation and distinct temperatures in different seasons. Figure 3.4 shows the

114

annual precipitation recorded for the watershed. Daily or sub-daily weather data are
obtained from various sources including National Data Climatic Center [NCDC, 2010]
and Michigan Automated Weather Network [ll/LAWN, 2010]. The locations of [the
stations are shown in Figure 3.3. The availability of data from these stations is given
in Table 3.1. Although the MAWN stations contain much more data ﬁelds, most
MAWN stations are operational only after 2006, with the exception of msuhort, which
started from 1996. Solar radiation data are available only from the MAWN site. Thus
the values for the NCDC sites are copied from nearest MAWN site when they are

available.

115

 

 

 

 

 

 

 

 

 

 

 

 

Legend

— Main Rivers W

[:1 Basin 0 3 6 12 km
84°30'0’W 84°15'0'W 84°d'o'w

Figure 3.1. Location of the Red Cedar River Watershed

116

42°45'0"N

42°30'0'N

 

 

  

 

 

. Legend
Elevation (111)
High:324
[__Iﬁ—I—jﬁﬁ—ﬂ
“ .Low:249 0 5 1° 20"“
84°30'0"w 84°Is'o"w s4°oio"w

Figure 3.2. Elevation map of the Red Cedar River watershed

117

42°45'0"N

42°30'0"N

 

  

 

 

 

Corona N 3"
- 8
So
V
bath
Lansing or
Airport _2
u 3
q-
E»:
v
Howell .cmc
Eaton z
Rapids Io
I 8
IV
V
Legend
Climatic Station)
' ' ' o MAWN Stations
0 4 8 16 km I NCDC Station:
[3 Buin
84°30'0'w 84°15'0'W 84°0'0'w 83°4'5'0'w

Figure 3.3. Locations of weather stations used for climatic input data

The MAWN stations also record relative humidity and wind velocity. Such data are
present only with station 204641 in the NCDC list, which is an airport station. A suite
of tools have been developed to process weather data from both types of sources and
parse them into the format required by the model. When a particular
precipitation/temperature record is missing, it is borrowed from the nearest available
station. For solar radiation, missing data are ﬁlled using the algorithm detailed in

Section 2.3.1.

118

 

1000— I I | l l | r l l l I _

200 —

    

 

 

1998 1999 2000 20012002 2003 2004 2005 2006 2007 2008

Figure 3.4. Basin-average Annual precipitation for the years from 1988 to 2008 (m)

Table 3.1. Climatic Data sources and data availability. The dates in YYYYMMDD
format are the dates when a station starts to have records; Frac is the fraction of the
RCR watershed that is controlled by this weather station; PRCP PCT is the
percentage of valid precipitation records from 1998 to 2007

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Station 99002 99003 20243 7 201818 209006 204641 203947
ID
NAME MSUHORT BATH Eaton Comma Williamston Lansing Howell
Rapids Airport
Type MAWN MAWN NCDC NCDC NCDC NCDC NCDC
Frac 32.9% 2.0% 2.4% 0.0% 41.2% 0.5% 21.1%
PRCP 19960801 20010801 19840101 20010401 19840701 19840101 19911001
TMAX 19960801 20010801 NA 20011201 NA 19840101 20040101
TMIN 19960801 20010801 NA 20011201 NA 19840101 20040101
WIND 19960801 20010801 NA NA NA 19840101 NA
RHMD 19960801 20010801 NA NA NA 19840210 NA
SRAD 19960801 20010801 NA NA NA NA NA
PRCP 100.0% 64.2% 88.4% 67.3% 83.6% 99.9% 99.9%
PCT

 

The National Land Cover Data (NLCD) are shown in Figure 3.5 and the percentages

of different land use classes are given in Table 3.2. The land use of the Red Cedar

119

 

 

 

 

 

 

River Will
10 5000 0
scattered
percent-.1;
are 0er1
deciduor
Natural .
Informati
Landsat '
3.20). Th
but dichr
knowledg
that the "l

l
or the 110
the MD,

 

River watershed is predominantly agricultural. Row crops plus forage crops stand up
to 56% of the watershed. Urban areas are in the Northwest while forested areas are
scattered throughout the watershed. According to NLCD, there is a signiﬁcant
percentage of wetlands present in the watershed (around 13%). However, these areas
are often classiﬁed as either low land shrub, herbaceous open land or lowland
deciduous forest in the land cover dataset provided by the Michigan Department of
Natural Resources (MDNR) retrieved from the Michigan Center for Geographic
Information [WNR, 2010b]. The MDNR data is derived from classiﬁcation of
Landsat Thematic MapperTM imagery and has a more detailed classiﬁcation (Table
3.2b). The MDNR and NLCD data agree well on the percentages of agricultural land
but differ quite signiﬁcantly on the wetland, forest and herbaceous percentages. Little
knowledge exists about which one of these two databases is more accurate. We found
that the ‘wetland’ areas in the NLCD are mostly in close neighborhood to the streams
or the ﬂood plain. Upon ﬁeld inspections of some conﬂicting areas we determine that
the MDNR database is to be used for the modeling. The data are reclassiﬁed into

model classes following the procedure outlined in section 2.2.1.

120

42°30'0"N

42°45'0"N

355.55 35.5 _ .Vw

3.559%

 

 

:32 m6 Ham 5

 

@5303 33838
venue? >eoo
3on >5
mumxaama
caning
£55
“modem coon
“modem 8233
“modem 30:28
was :05 ,
3.535 and
:35 3823 am.
95 38:85 5:60
58D 33:25 33
535
“Bowed

 

 

Figure 3.5. NLCD Land use/Land cover map for the RCR watershed

I21

Table 3.2. Land use percentages from the NLCD database and the MDNR databse

 

 

 

 

 

 

 

 

 

 

Land use Classes NCLD MDN R
Urban 18.17% 10.29%
Deciduous Forest 9.01% 15.02%
Evergreen Forest 0.63% 4.64%
Row Crops 35.44% 33.63%
Forage Crops 22.10% 22.14%
Herbaceous and Bushes 0.80% 11.97%
Bare 0.29% 0.19%

Wetland 12.99% 1.56%

Water 0.56% 0.55%

 

The watershed, as others in Michigan, is very rich in the number of rivers and has a

complex surface water system. The main rivers from the National Hydrography

Dataset (NHD) in the watershed are shown in Figure 3.6. Majors rivers of the

watershed are extracted from the NHD. These are the river objects that will be solved

in the proposed model. Other smaller creeks, ditches, furrows, rills, etc, are modeled

as the overland ﬂow domain as discussed in section 2.3.3. The Red Cedar River,

which is a tributary of the Grand River, is the major stream in the domain, followed
by the Sycamore creek. Due to the low relief of the terrain, the rivers generally have a
very mild slope.

The two stream gaging stations operating in the RCR are USGS 04112500 at fannlane,
East Lansing and 04111379 at Williamston. Since East Lansing is the downstream

gage, we focus our calibration at this gage. However, the comparison at Williamston

122

is also important as it shows if the model is working properly inside the watershed.

 

    

 

 

 

N 12
2°
In
v
0
‘l N
Williamston "
F
- Legend g
A USGS g
_ Red Cedar River v
— Other Modeled Rivers
— National Hydrography Dataset 0 4 8 16 km
[2] Basin
84°30’0"W 84°1sro"w 84°0;0"W

Figrn'e 3.6. River system of the RCR watershed with USGS ﬂow gages

Geology of Michigan including that of the present day Red Cedar River watershed is
the result of extensive glaciation during four major glacial periods including the
“Wisconsian, Illinoian, Nebraskan, and Kansian periods. Glaciation during the
Wisconsian period (the last of these periods, approximately 11,000 years ago) is
mainly responsible for the development of Michigan’s geology, soils, and topography.
In the RCR watershed the geology includes glacial till (poorly sorted material

including pebbles and boulders), glacial outwash (ﬁner material deposited by glacial

123

melt water), lacustrian material (ﬁne materials deposited in glacial meltwater) and
alluvial material (recently deposited material from rivers and streams) as shown in
Figure 3.7 [Corner et al., 1999; MDEQ, 2010; MDNR, 2010a]. These deposited
materials, as well as organic material, are the parent materials of the soils in the
watershed. Limestone, shale, sandstone, coal, and other sedimentary rocks arranged in
almost horizontal layers compose the bedrock surface of the Red Cedar Watershed.

These features contributed to the relatively high conductivities of soil materials.

124

 

 

‘ Legend
Land System

[El Ice-contact outwaah

1:: Lodge Till or Fine supraglaclal drift

E1] Proglacial outwth ﬂ_._.ﬁ_,_,_,ﬁ

E ice-marginal till 0 5 10 20 km

 

 

 

s4°30'*o"w 84°15’0"W s4°oio"w

 

. Legend
Bedrock Geology
— Bayport Limestone

- Coldwater Shale

- Grand River Formation

- Marshall Formation 71W

- Michigan Formation 0 4'5 9 18 km
[:1 Red Bed!

E] Saginaw Formation

 

 

 

 

34°45’0va 84°30’0"W 84°15’0"W 84°0‘o"w

Geology

125

42°45’0"N

42°30’0"N

42°45 ’0"N

42°30’0"N

Figure 3.7. Geology maps of the RCR watershed. a. Land Systems; b. Bedrock

(a)

(b)

As a result of the glacial geology, soils within the watershed are mostly sandy, loamy
0r muck soils (commonly classiﬁed as hydrologic soil group B) mostly share the same
common parent materials. The sand percentage map is shown in Fig 3.83. The soil

available water capacity and saturated conductivities (Ksat) of the ﬁrst soillayer as

reported in the STATSGO database are shown in Fig 3.8b and Fig 3.8c.

We show the STATSGO data here because it is much less noisy and thus easier to
visualize than the SSURGO database. However, for the model simulation, SSURGO
data is used as input. Ksat is presented here only to give us an impression of the
spatial variation of hydraulic properties of the soils. The Ksat values from SSURGO
or STATSGO cannot be directly used as it is believed that several factors during the
generation of these two databases have impacted the reported values. For example,

Ksat only takes a few unique values (28,34,...). This is clearly a result of unit

conversion from integer inch values to metric units. Personal communication with

SSURGO staff reveals that these Ksat valuesare not necessarily measured but may
also be inferred using a set of procedures established by the Natural Resource
Conservation Services (NRCS). Since such critical information as the sources of data

is obscure, we choose not to directly use the Ksat values in the SSURGO. Ksat is

estimated using empirical relations based on the sand, silt and clay [Rawls et al.,

1991].

126

Figure 3.8. Soil property maps of the RCR watershed as reported in the STATSGO
database. a. Sand percent; b. Soil Available Water Capacity; c. Saturated Hydraulic

Conductivity (Ksat)

 

42°45’0"N

   

 

 

 

 

   

 

 

 

(a)
E
‘ Legend 9
Sand Percent §
[1:1] 43.6% s;
43.7% - 44.3%
- 44.4% - 45% Wﬁ—r—r—I—‘I
- 45.1% - 55% 0 4.5 9 18 km
- 55.1% - 82.6%
84°30’0"W 84°15’0"W 84°0’0"W
N .z
’e
+ l?
°e
(b)
F
' Legend g
Available Water Capacity t;
1:] 10%
- 10.1% - 12%
- 12.1% - 14% r—I—I—I—r—I—I—I—I
_14.1°/. - 16% 0 4.5 9 18 km
- 16.1% - 35%
84°30’0"W 84°15’0"W 84°0’0"W

127

Figure 3.8 (cont’d)

 

   
  

 

 

 

 

.2
1°
I!)
<-
34
V
(C)

.2

Legend :°

] Ksat (cm/day) a

[:3 28 - 34 2;

1.9.1:. 34 - 45

m 45 ' 150 r—r—Tﬁ—I—I—I—I—I

-150-190 0 4.5 9 18km

- 190 - 29o

84°30’0"W 84°15’0"W 84°0"0"W
Groundwater data

Generally, some information should be available about the geological formation of the
study domain to employ the proposed model. Some ideas about the aquifer
conﬁgurations are necessary for constructing the conceptual model. For the case of
the RCR watershed, as discussed in section 3.1.1, we know a layer of more permeable
glacial deposit with a thickness ranging approximately from 5 meters to 30 meters
overlies the bedrock. Although the bedrock has very small permeability, the thickness
of the layer is great and the overall transmissivity is not low. This information allows

us to divide the groundwater model into two layers. The upper layer is the glacial

128

deposit, taken as the unconﬁned aquifer; the lower layer is the bedrock, taken as the
conﬁned aquifer. A layer of aquitard is assumed to exist between the two layers.
Besides qualitative descriptions, we also need to quantitatively know the thickness of
the aquifer layers and their conductivities. This is the difﬁcult step, as these types of
data are usually very expensive to collect. Data limitations have been the major
hamper of groundwater modeling. Conductivities are known to vary many orders of
magnitude over space whereas existing conventional data (e.g. ﬁeld survey) are
normally too scarce.

However, for the RCR watershed and Michigan in general, valuable groundwater
-related data have been collected, logged and ﬁled with the Michigan Department of
Environmental Quality (MDEQ) when water wells in the state of Michigan were
drilled. Reported information include depth to water table, pumping test results,
lithological layer descriptions, etc. From these records the local conductivities can be
estimated (e.g. [Mace, 1997]). The number of wells is so massive that it almost dense
covers the entire state of Michigan [Simard, 2007]. The large number of wells can
partially offset the data quality issue. If the correct statistical procedure is used to
remove the noises and outliers, the results are more reliable.

Several studies have attempted to employ this data source, e.g. [Hill-Rowley et al.,
2003; Reeves et al., 2003]. However, the difﬁculty is that, since these data are not
recorded by professional surveyors but common household drillers, the quality of the

data is mediocre and sometimes poor. Signiﬁcant noises, if not handled correctly,

129

exist in the data which can result in erroneous interpretations of the aquifers. More
information is to be found from [GW1M, 2006; Simard, 2007].

Fortunately, extensive studies have been underway to denoise and interpolate the data
at Michigan State University to produce a smooth, statistically coherent data source.
Kriging is identiﬁed as the best geostatistical approach to interpolate the data into a
spatial ﬁeld. Part of that research was reported in [Simard, 2007]. As a result,
groundwater head, conductivity and thickness of the glacial drift layer were processed
using the software IGW [Li and Liu, 2004; Li et al., 2006].

Nevertheless, at places where such kind of data is not available, the users should not
be daunted by the data limitation. Despite all the difﬁculties, some information can
always be obtained from either lithological data or wells pumping tests. Our
advantage is that groundwater varies smoothly over space and we do not have a
specially high requirement on the quality of the K data. Besides, some parameter

adjustment can be applied.

3.1.2. Model Calibration

Flow records at USGS gage 04112500, East Lansing from 2002/09/01 to 2005/12/31
were used to calibrate the model while the years prior to this period were used for
veriﬁcation (or validation). The reason for using water years 2002 to 2005 was that in
these three consecutive years the watershed has experienced high, normal and

unusually low precipitation periods. Auto-calibration of the model was done using the

130

Differential Evolution (DE) algorithm [Cha/t'raborty, 2008] with the computational
resources from the High Performance Computing Center at the College of
Engineering, Michigan State University [HPCC, 2009]. The original DE code has
been modiﬁed and implemented into a parallel computing code. Future versions of

PAWS will allow calibration on a single CPU.

3.2. SWAT Model for comparison

While developing the proposed model, we also created a Soil and Water Assessment
Tool (SWAT) model [Arnold and Allen, 1996; Arnold and Fohrer, 2005] to compare
with the proposed model. The SWAT model is a widely used spatially-distributed
parameter model for the assessment and management of water resources, soil
conservation, non-point source pollution and soil erosion, etc. It has been applied in
many countries for various applications. SWAT was designed as a long-term yield
model and is not designed to accurately simulate detailed, single-event ﬂood routing
[Neitsch et al., 2005]. In the past 5 years there have been over 400 publications related
to SWAT, showing its generally accepted validity, popularity and versatility. Many
papers refer to SWAT as a physically-based model while some regard it as a
semi-physically-based because there is still signiﬁcant amount of empiricism. For
example, for daily modeling the major inﬁltration scheme is the SCS curve number
method [Division, 1986]. The curve number method is built on many years of

accumulation of experience in runoff prediction [NRCS, 1986]. However, as an

131

empirical approach, it cannot distinguish the effects of different process, e.g.
inﬁltration excess or saturation excess. From the basis on which the SCS curve
number method is developed it is also unclear how much water has inﬁltrated or
retained on the ground. For another example, the lack of a physically-based
groundwater ﬂow model prevents SWAT from simulating regional groundwater ﬂow
and making reliable estimates of surface water-groundwater interactions. The
overland ﬂow and channel ﬂow modules, which are based on the reservoir concept,
are also regarded by some to be over-simpliﬁed and thus presents challenges to the
accurate description of transport problems. The empiricism makes the SWAT model
less suited to study the impact of human modiﬁcations (e.g. dam or other hydraulic
structures). Each model has its designed purpose and its own areas of strength, and we
seek not to arrive at the conclusions of which one is better. We are interested in
comparing the proposed model with the SWAT model primarily because it allows us

to see if the fully physically-based modeling approach does have its own advantages.

3.2.1. Brief summary of SWAT mathematical bases

We brieﬂy summarize some of the major SWAT modeling approaches here to
highlight the differences between SWAT and PAWS. The theories covered here can all
be found in the SWAT theoretical documentation [Neitsch et al., 2005]. The majority
of the literature use SWAT as a daily model, thus we only describe the daily modeling

options.

132

SWAT discretizes the study area into subbasins, each one with a main channel and a
tributary channel. The subbasin and the channel is the smallest unit with spatial
heterogeneity. Inside each subbasin, however, multiple Hydrologic Response Units
(HRUs) may be deﬁned. Each HRU is a unique combination of soil type, land
use/land cover type and slope (slope added in the SWAT2005 version).

The runoff volume is calculated using the SCS Curve number method. The curve
number for each HRU is looked up from tabulated empirical values corresponding to
common land use/soil combinations and adjusted for antecedent moisture content.
Once the curve number (CN) is determined, the runoff for the HRU is calculated as:

s = 25.4 1999—10
CN

2 (3.1)
_ (P — 0.23)
Q‘W — (P + 0.83)

where S is a retention parameter (mm), P is precipitation (mm) for the day and qurf

is the surface runoff. Due to the empiricism, the curve number approach cannot
distinguish the different runoff generation mechanisms (saturation excess, inﬁltration
excess) and does not intend to do so from the design [Arnold and Fohrer, 2005;
Borah and Bera, 2004].

SWAT allows water in the soil layers in excess of ﬁeld capacity to percolate to lower

layers:

SW = swly — qu swly > rely

ly,8$6688 0 SI/Vly S FCly (3-2)

133

2 SW

ly,e:rcess

 

'UJ
1001‘ch TT

 

1.]

 

1—exp[

pt‘l'C K

sat

 

where TT is the travel time for percolation (day), SW]y is the water content in the soil
layer (mm), SW1),,excesS is the drainable volume of water on a given day (mm), Ksat is
the saturated conductivity of the layer, FCly is the ﬁeld capacity of the soil layer (mm),

Wperc,ly is the volume of percolation of the day (mm), SAle is the amount of water in

the layer when it is fully saturated (mm). As we can see, these equations assume a
one-way percolation which does not take into consideration the pressure head (or
hydraulic head) of the soil water in the lower layer. Another mechanism termed
‘revap’ is used to account for groundwater entering the unsaturated soil zone from the
aquifer. However, that mechanism is also empirical and controlled by a revap
coefﬁcient. The revap coefﬁcient is ranked one of the least sensitive parameters in the
model [van Griensven et al., 2006].

The recharge to aquifers on a given day is in turn calculated by lagging the recharge

using the method proposed by [Sangrey et al., 1984] and [Venetis, 1969]:

1
u ulsecp + exp(_ )wl‘Chl'g,i—1

6

= 1— exp(-6—1—)
9w (3.3)

you

1 .
t‘chrg,z

 

 

wseep = wpechyzn + warhbtm

in which Wrchrg,i is the amount of total recharge to aquifers on a day (mm), wsegp is

the total amount of groundwater recharge on a given day (mm), 69w is the delay

time or drainage time of the overlying geologic formation (days), wpercjy=n is the

134

percolation from the lowest soil layer, 11, on the day (mm), Wcrk,btm is the amount of
water ﬂow past the lower boundary of the soil proﬁle due to bypass ﬂow on day i.
And the groundwater contribution to streams is calculated as:
ng’z. = qu,1,i—1 exp (—agﬁ,At) + wt'clu‘g.sh [I — exp(—agwAt) ,
if aq$h>aQShthnq (3'4)
Q9115;- = 0 a if aClsh‘<'_'aqshthr,q

where ngj is the groundwater ﬂow into the main channl on day I (mm), a Jis an

9'“
empirical baseﬂow recession constant, Wrchrg,sh is the recharge to the shallow aquifer,

aqsh is the amount of water stored in the shallow aquifer, and a‘Ishthr,q is the threshold
water level in the shallow aquifer for groundwater contribution to channel to occur
(mm). agw and 6 W are called, respectively, ALPHA_BF and GW_DELAY in the
SWAT model codes. In the SWAT literature these two are often used as calibration
parameters. Some papers use the hydrograph-baseﬂow separation technique [Arnald]
to identify agw.

SWAT routes runoff in a subbasin to its main stream using a simple lagging

relationship:

Qsm‘f : (qu-rf + Qsto7‘,i—l) '(3-5)

 

 

_ .1
1 __ exp[ 3m 09]
tCOTlC
where qurf is the amount of overland ﬂow contribution to the main channel on a
given day (mm), qurf, is the amount of surface runoff generated on the day (mm),

Qstor,i-l is the surface runoff stored or lagged from the previous day (mm), surlag is

the surface runoff lag coefficient and tconc is the time of concentration for the

135

subbasin (hours). This lagging relationship resembles the solution to a linear ordinary
differential equation. Surlag is usually used as a calibration parameter. From the
author’s experience, the proper value to use with surlag depends heavily on the size of
the delineated subbasins. The larger the subbasins are, the smaller surlag should be
used. However, the use of this relationship masks the surface ﬂow patterns in the
subbasin. For example, the overland ﬂow paths and water velocity are not known
using this approach.

SWAT routes water in the river using either the linear reservoir method or the
Muskingum Cunge method. For the RCR watershed SWAT model developed, we
have used the Muskingum method for river routing. The development of the
Muskingum routing method is rather lengthy and is avoided here. It can be found on

the water routing chapter in the SWAT theoretical documentation.

3.2.2. SWAT Model Setup

The SWAT model for the RCR watershed was created with the assistance of the
ArcSWAT GIS interface for SWAT2005 [Olivera et al., 2006]. Using a 90 meter
resolution DEM and the NHD stream network, 53 subbasins were delineated for the
RCR watershed (F igure3.9). The NLCD land use/land cover map and STATSGO soils
database (l:250,000) [USDA] was used for land use/soils overlay to obtain the
distribution of Hydrologic Response Unit (HRU). Each HRU is a unique combination

of land use, soil types and slope. As compared to SWAT2000, the 2005 version of

136

SWAT also calculates the slope for each HRU and overlays different slopes on top of
land use/soil combinations, thus creating signiﬁcantly more HRUs. According to
some recent studies, using local slope values for each HRU is of great importance to
correctly simulating hydrologic responses and nutrient output. We used three classes
for slope: slope<=3%, 3%<slope<=6% and slope>6%. Since the topographic relief of
the watershed is very gentle, only a small fraction of areas, mainly the hills, is larger
than 6%. To maintain the number of HRUs at a manageable level, SWAT uses
customizable thresholds to remove any land use or soil type found in a subbasin that
is below their respective thresholds, and assigns their areas proportionally to the
dominant classes. We employed a 20% threshold for land use, 10% for soil and 20%
for slope. The ﬁnal land use/soil distributions after these processing steps are given in

Table3.3. As a result, a total of 1936 HRUs were obtained. The median area of the

HRU is 0.069 m2.

137

 

42°45’0"N

 

 

 

 

 

.Z
I:
'5
("l
8:
V

Legend Itrrtttt.

1::1 Watershed 0 4 8 16 km

EjBasin

84°30’0"W 84°15’0"W 84°0;0"W

Figure 3.9. Watersheds delineated for the SWAT2005 model using the ArcSWAT GIS
interface

Table 3.3. SWAT Land use class percentages aﬁer apply thresholds. URLD: Low
Intensity Urban; URMD: Medium Intensity Urban; WETF: Forested Wetland; URHD:
High Intensity Urban; HAY: Hay/Forage Crops; AGRR: General Agriculture; F RSD:
Deciduous Forest

 

URLD* URMD WETF URHD HAY AGRR F RSD

 

5.07 5.86% 8.99% 1.55% 29.18% 47.45% 1.91%

 

*Notes: The SWAT land use codes:

138

The SWAT model is calibrated against the ﬂow data from USGS gage 04112500 at
farmlane, East Lansing using the Shufﬂed Complex Evolutionary algorithm [Duan et

al., 1992; Duan etal., 1994].

3.3. Results and Discussions

3.3.1. Model Evaluation

We ﬁrst evaluate the performance of the model by comparing it against observations
and other models. Then we use the model to shed insights into the hydrologic system
of the study watershed. Two forms of observations are included in the comparison.
One is the streamﬂow measurements at the two USGS gaging stations inside the
watershed, the other is the groundwater head data interpolated from the well records.
Output from the SWAT model which is an established annual budget simulator is also
used in the comparison. In addition, the places where saturation excess is produced is
compared to the Topographic Index proposed by Beven [Beven and Kirkby, 1979].
3.3.2. Model performance evaluations

Model performance is evaluated using the comparison with USGS ﬂow data,
groundwater wells records and average annual ﬂuxes reported by SWAT. The metrics
we use to measure the performance include coefficient of determination (R2), root
mean squared error (RMSE), the Nash-Sutcliffe model efﬁciency coefficient (NASH)

and the mean error (ME). The RMSE, NASH, RNASH and ME are deﬁned as:

139

 

RMSE = JZ(QS — an)2

t=1

 

 

 

 

T 2 T . 2
2(95 —Qf..) 5: 152—193..)
NASH =1— t=1 ,RNASH =1— :1 2 , (3.6)
2(Qg-Qo) z JEZTJEO
= t=1

T
[HE : Z<Qo — Qm)
i=1

where QO is the observed discharge, Qm is the modeled discharge, T is the number

of observations and 50- is the mean of the observed discharges. The RNASH is
simply the NASH formula applied to the square-root transformed data. This transform
reduces the relative importance of large

The daily observed and simulated ﬂows at USGS gage 04112500 (farmlane, East
Lansing) for the 10 years from 1998 to 2007 are compared in Figure 3.10. The
proposed model did a very good job at streamﬂow predictions, especially with the
baseﬂow/ low ﬂow periods. The performance metrics of the calibration, veriﬁcation
and entire periods are provided in Table 3.4. The Nash-Sutcliffe for the calibration
period is 0.694 while for the veriﬁcation period it is 0.585. A NASH greater than 0.60
for daily comparison is generally considered good performance for a watershed of this
size. By examining many models and, literature, it was found that hydrologic models
applied to larger-sized watersheds (>5000km2) tend to achieve higher NASH at more
downstream gages because input errors (e.g. error with precipitation records) and

model structural error may cancel out.

140

Fo\mo RENO 5506 no\ 5 5E0

 

_ _

AEEEc 880
385

.31 1144.4

535

   

   

4.4.1111.

 

 

        

 

 

  

V035 55m

 

om
ow
om
ow

O O O
to <1- N
(sun) aﬁieuastg

 

 

___—

 

 

mEmcﬂ “mam 65:52 ”comm, :5 mom:

Figure 3.10. Comparison of the observed and simulated daily ﬂow at USGS gage
04112500 at East Lansing. Upper: Entire period from 1998/09/01 to 2005/12/31

9

Middle and lower: Close-up look of the hydrograph

141

 

 

Discharge (cms)

 

I I | I I | I l

 

03/01 03/07 04/01 04/07 05/01 05/07 06/01 06/07
Dates (yy/mm)

Figure 3.11. Comparison of the observed and simulated daily ﬂow at USGS gage
0411 1379 at Williamston

Table 3.4. Performance metrics evaluating the model applied to the RCR watershed

 

 

 

 

 

NASH RMSE RNASH ME R2
SWTAT. 0.665 4.033 0.621 -0.698 0.675
Calibration
SWfAT. 0.675 4.261 0.561 -0.065 0.680
iEast Veriﬁcation
Lansing
PAWS. 0.694 3.855 0.693 -1.122 0.742
Calibration
PAWS. 0.585 4.816 0.661 -0.232 0.591
Verification

 

 

 

 

 

 

 

 

 

Close inspection of the hydrograph indicates that the mismatches are generally due to
the underestimation of streamﬂow peaks during snowmelt periods. The reason for the

underestimation is attributed to the UEB snowmelt module employed in the proposed

142

model not melting snow rapidly enough to produce large runoff that is apparent from
the hydrograph. Since the UEB model is intended as a model with no parameters for
adjustment, adjusting this aspect of the simulation has been difficult. Further in-depth
research is needed to understand the exact reason for this under-estimation and to
improve the results.

We note the NASH coefficient of PAWS ranks higher than SWAT during calibration
period but lOwer during veriﬁcation period. This reduction of NASH in veriﬁcation
period is almost entirely due to the under prediction of a single snowmelt peak in
2001. This is shown in Figure 3.12, in which the SWAT simulated ﬂow is also plotted.
Although the baseﬂow leading into the peak as well as the recession periods are
nicely captured by PAWS, the peak around 2001/02/ 12 is largely missed, whereas
SWAT does a better job at this peak, thus ranking higher in the veriﬁcation period.
The low ﬂow periods are better visualized in a log-transformed as in Figure 3.13. Due
to its simpliﬁed groundwater ﬂow structure, SWAT produces baseﬁow that very
different in shape from the observed data, while PAWS generally does a good job.
Study has shown that outliers can signiﬁcantly inﬂuence sample values of NASH
[McCuen et al., 2006]. Due to its mathematical formulation, NASH puts much more
weight on a few large peak values than the low ﬂow period. Low ﬂow periods are
when pollutant concentrations tend to be high and thus the risk to human health is
greater. The baseflow period is also very important to capture as it best describes the

subsurface dynamics of the watershed. From these viewpoints we argue that NASH is

143

not the entire story and it may be more appropriate to look at the indicator RNASH,
which put weight more evenly on high and low ﬂow periods. As we can see from
Table 3.4, PAWS consistently ranks higher than SWAT in terms of RNASH, both in
calibration and veriﬁcation period, indicating a more realistic subsurface

representation.

USGS 04112500: Farmlane, East Lansing
l I I I I l

 

 

 

   

 

80
60
40
7320
E
‘5 99 0 01 02 03 04 05
g“ Water year 2M
ﬁso ' ' -
'5 . —PAWS
6° ' ------ uses
40 - - -SWAT -
20 ' . ‘
01/01 Dates (W/mm) 01/07

Figure 3.12. Comparison of the simulated streamﬂow from PAWS and SWAT and
observed USGS streamﬂow data. Upper: from 1999 to 2005, Lower: Close-up look of
water year 2001

Figure 3.11 shows the hydrograph comparison at the inner gage Williamston. The

model performs equally well at this gage as East Lansing. These results give us more

conﬁdence that the model not only predicts ﬂow satisfactorily at a downstream gage,
but also captures the underlying hydrologic dynamics of the watershed.

The under-prediction of snowmelt peak may also be due to the freezing soil effect,
which reduces the conductivity of the frozen soil layers. Since at current stage there is
still not a soil temperature module, soil freezing cannot be physically-described. Since
there are some soil temperature measurement data from MAWN, it has been
attempted to use the measured soil temperature data to identify the periods of time
when soil is frozen. When the soil temperature was less than 0, the saturated
conductivity of the soil was reduced to 1/10. However, this attempt did not make any
material difference to the hydrograph because the measured soil temperature was only
less than 0 in a few very cold days, and in these days there was simply no snowmelt.
As such the soil freezing mechanism is not activated for the results presented in this
dissertation. Future research should consider including a soil temperature module and

a better parameterization of the freezing process.

145

 

|n(m3/s)]
J
5
a”: ' T
«v’
w
1’
,—
‘—
:.
~__." nu

 

 

I ._ .' ,
I " \ | ( . 'I :.
H100 \ 1 “J ‘9 V, l “’5’; ¢ :1".
A I ~. I ‘ '0‘ I
(U ' I n I II
g ----- uses 5': . 1
g _2 - - -SWAT . "5‘ ‘
£10 —PAW s l-
| I I I l I
01/01 01/07 02/01 02/07 03/01
Dates (yy/mm)

Figure 3.13. Comparison of the simulated streamﬂow from PAWS and SWAT and
observed USGS streamﬂow data in log-scale

The model was started in 1998 with a rough initial estimate of groundwater heads.
The initial guess was simply taken as the mean elevation of the ground surface and
bedrock. If the model works properly, inaccuracies in the initial head would be mostly
removed during the years prior to 2001. The resulting head is determined mostly by
internal and boundary forcing. The groundwater model should converge to the steady
state solution and ﬂuctuate around it. We examine if the ﬁnal state and the
time-averaged state is reasonably close to the observed values. The ‘observed’
groundwater heads are a groundwater head map interpolated ﬁom well records. Maps
of simulated groundwater heads (averaged from 2001 to 2005) are compared with the
heads interpolated from wellogic database in Fig 3.14. Since the groundwater table

ﬂuctuates continuously, the 5-yea.r average head is used for the comparison. The

statistics of the comparison is given in Table 3.5. Excellent results have been achieved
by the model to reproduce the observed data as evidenced by the high R2 value.
However, since the initial guess is bounded by the DEM, a large part of the variation
in the data may have already been explained by the initial guess. It is thus legitimate
to remain skeptical about the true ability of the model to ﬁnd the correct states. To
address this, we show the initial guess in Fig 3.14c and its comparison with observed
data in Fig 3.15b. The statistics are also given in Table 3.5. We note that the initial
data is negatively-biased as the mean error (ME) is -4.8 meters. Both the 5-year
average state and the ﬁnal state have much smaller bias. The R2 and NASH have
improved substantially after the simulation. We see that with rough initial estimates of
head as initial conditions, the model was able to evolve to a much more realistic state,
demonstrating that the internal and boundary forcings have been realistic.

From Figure 3.15 there seem to be some systematic under-estimation of groundwater
heads toward the higher range. This can be attributed to the implied no-ﬂow condition
at watershed boundaries in the model. Groundwater divide does not necessarily
coincide with watershed boundary, and surface water features on the other side of the
boundary such as lakes and rivers can have inﬂuence on the groundwater heads.

Table 3.5. Metric of Groundwater heads comparison to wellogic data

 

 

 

NASH RMSE ME R2
Initial Data 0.48 6.22 -4.81 0.83
Final Stage at 2005/12/31 0.94 2.07 -1 .00 0.96
5 year Average 0.97 1.59 -0.81 0.98

 

147

Figure 3.14. (a) 5 year averaged simulated groundwater, (b) observed heads
interpolated from wellogic database, (0) initial groundwater heads used to start the

 

model
x 105 5 year averaged groundwater head (m)
I I I l I

 

 

 

 

 

 

 

 

 

 

3.98 3.99 4 4.01 4.02 4.03 6
X 10
(a)
x 105 Groundwater head from Wellogic (m)
l l I l I l
1.45 '
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05 -I I I I I I -
3.98 3.99 4 4.01 4.02 4.03 6
X 10
(b)

148

Figure 3.14 (cont’d)

X 10 Groundwater head initial guess
I I I I I I

 

 

 

 

 

 

 

149

 

 

 

 

 

290 I I I
285 - , g: "
. 5114'-
a 280 - at}? -
E g 275 t
32 . <-
a’ a I‘ ( n‘
> ... 27o — .. .~ —
11! t0 '-’-‘\
h -.: . 3
~13, ’ . 3:. . _ ( )
g C 265 - 3, .5. _
'“ § 5793’
51260 - . Q 1,, ’ _
.Ej '”
255 — - ” -
25 ‘ I I I
950 260 270 280 290
30° I I I
290

N
W
O

(b)

InItIally Guessed
groundwater head (m)

N N

8 3‘

250.,

 

 

 

 

2 ' i I I I

4250 260 270 280 290
Groundwater heads interpolated from wellogic database
using Kriging (m)

Figure 3.15. Observed vs simulated groundwater heads. (a) Observed data vs 5 year
averaged value, (b) Observed data vs the rough initial guess. X-axis is the observed
data obtained from Kriging interpolation of the wellogic data

150

3.3.3. Additional model results and hydrologic system of the RCR watershed

Table 3.6. Annual Average Fluxes compared to SWAT

 

 

Fluxes SWAT SWAT PAWS PAWS
(mm) (% of prep) (mm) (% of prep)

PRCP 761.5 100.00% 760.3 100.00%

Sublimation 3.9 0.51% 9.8933 1.30%
Surface Runoff 87.94 11.55% 106.63 14.00%

Lateral Soil Q 0.63 0.08% 0 0.00%
Groundwater Q 78.61 10.32% 80.447 10.56%
Recharge 100.24 13.16% 82.634 10.85%
ET 552 72.49% 560.7 73.63%
Inﬁltration NA NA 760.3 100.00%

 

 

151

Sublimation

   
 
  
  

Surface Runoff

Groundwater Q
PAWS
Recharge
ET
Sublimation
Surface Runoff
Groundwater Q
SWAT

Recharge

Figure 3.16. Pie chart showing the comparison of the average annual ﬂuxes as a

percentage of total precipitation between PAWS and SWAT

Recharge for the proposed model is calculated from the vadose zone model using

equation (2.96). As discussed earlier, we take recharge as the ﬂux that leaves the

bottom of the soil layers and enters the saturated unconﬁned groundwater aquifer.

Recharge can be negative which means that groundwater is discharging.

152

The model-estimated annual average of hydrologic ﬂuxes for the years 1998 to 2005
are tabulated in Table 3.6 and graphically presented in Figure 3.16. There is a small
difference of the total amount of precipitation applied to the two models because the
discretization is different. We see that these ﬂuxes, especially percentages of the
precipitation, are in general similar to that estimated by the SWAT model. Inﬁltration
is not directly extractable from SWAT results when SCS curve number method is used
(when daily time step is employed) to calculate runoff. The proposed model reports
2% ET higher than the SWAT model while 2.3% less recharge and 2.5% more surface -
runoff. The SWAT model has been well-established as a long term hydrologic budget
simulator. These results give us the conﬁdence that the model is separating the -~

precipitation in a reasonable manner.

3.3.4. Understanding the hydrology
After these comparisons we are ﬁnally ready to use the model for our own purposes. .
Besides streamﬂow predictions, a physically-based model has the advantage of;
explicitly elucidating the interactions of components in space and time. With this
valuable tool we are able to gain insights of the inner workings of the hydrologic
system. With the following discussions we want to answer two questions: (1) Where
does the water in the rivers come from?*(2) What is the driving force behind the
annual cycle in the hydrograph? Why the hydrologic response to rainfall is so

different in different seasons?

153

 

 

 

03/10 03/11 03/12 04/01 04/02

 

 

 

04/05 04/06 04/07 04/08

Figure 3.17. Hydrograph partitioning into streamﬂow generation mechanisms:
Inﬁltration excess, saturation excess and groundwater contribution. X-axis are dates in
YY/MM/DD format. SatE: Saturation Excess, InfE: Inﬁltration Excess, Qgc:
Groundwater contribution, Qout: Red Cedar River outflow

154

Temporal patterns of hydrologic cycle

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E40 I I I I I I I _ro

9" 20

3

c

_>‘

c o

‘5

§
23 I I I I I I I I _a‘:"
10— ‘ —SatE
0

 

I I I l I I I
99 00 01 02 03 04 05 06
Dates (start of year)

cms

 

  

 

 

—0.05

 

04/01 04/04 04/07 04/ 10
Dates (YY/MM)
Figure 3.18. Temporal dynamics of ﬂuxes and state variables. Upper: Monthly mean
ﬂuxes (mm). Lower: Daily ﬂuxes (close up of 2004). Inf: Inﬁltration; ro: Overland
contribution to channel; Qgc: Groundwater contribution to channel; Rchrg: Recharge;
InfE: Inﬁltration Excess; SatE: Saturation Excess; Dperc: Deep Percolation

155

Figure 3.17 shows the streamﬂow partitioning into inﬁltration excess, saturation
excess and groundwater contribution. From the model, saturation excess is deﬁned as
runoff generated when the whole soil column is saturated. Inﬁltration excess is
calculated as the runoff that is not saturation excess. Due to the presence of other
processes (evaporation, overland ﬂow and river ﬂow routing, etc), the partitions do
not necessarily add up to be the ﬂow at the gage at any instant of time. However, it
stills gives us a good idea of the relative importance of the different processes. We
observe clearly that the main contributions to streamﬂow are saturation excess and
groundwater baseﬂow. The inﬁltration excess, on the other hand, occurs only at the
few extreme rain events. After entering the stream, the ﬂood wave is attenuated and
become more smooth. There is some loss of water from runoff generation to ﬁnal

outﬂow due to evaporation.

156

Soil Water Change from Inital State (mm)
I I I I I

 

 

 

 

 

 

 

 

(a)
99 00 01 02 03 O4 05 06
Snow Water Equivalent (mm)
0-03 I I I I I I I
0.02 - ._
(b)
0.01 — ‘.I A _
0 L J A
99 00 01 02 03 04 05 06

Figure 3.19. Basin average state variables (a) Soil Water Content (change from initial
state), (b) Snow Water Equivalent. X-axis marks the beginning of each year (YY).

Monthly basin-averaged ﬂuxes are presented in Figure 3.18. This ﬁgure well explains
the seasonal hydrologic cycle of the RCR watershed and the impact of energy input
on the system. Generally, starting in Spring, a number of warm days are responsible
for melting the snow that accumulated in Winter. The ﬁrst few precipitation events
can be very pronounced in terms of streamﬂow generation. Because ET has not
picked up and snow/ice is melting, the basin on average has high soil moisture content,
as shown in Figure 3.19 by the state variable SW. Aﬂer the effect of snowmelt has
waned, the increase in ET gradually takes water from the system and we observe that

baseﬂow starts to decline. From June to Aug, the large ET demand, created by heavy

157

input of solar radiation and high temperature, as well as the vegetation/crop at full
canopy, cannot be met by the precipitation and soil moisture. The suction of the soil
and vegetation roots start to tap water from the groundwater. We observe rapid decline
of soil moisture content in the Basin. Because soil is constantly dry in this period, it
can inﬁltrate much more than in spring. The effects of precipitation events in summer
are much more muted. In the fall, PET demand drops, the inﬁltrated water starts to
accumulate and groundwater level rises. On the hydrograph we observe the baseﬂow
begins to increase. The precipitation events induce much higher peaks than in summer.
In winter, precipitation comes in the form of snow and the streamﬂow is provided by
the groundwater. Since ET is very small in winter, inﬁltrated water is stored in the
system. The model results show that the RCR watershed is a primarily
energy-controlled watershed in terms of hydrologic cycle. Thus potential climate

change in the future may profoundly inﬂuence the hydrology in the region.

158

Daily Recharge vs
Groundwater contribution

 

— Qgc
- - - Recharge

0.6 —

(a)

 

 

 

(b)

  

o— I I ‘1“ I WE;

02/05 02/06 02/07 02/08 02/09 02/10

 

 

 

Figure 3.20. Daily ﬂuxes of recharge and groundwater baseﬂow. (a) Rising limb of
baseﬂow, (b) falling limb of baseﬂow. Again, dates are in YY/MM/DD format. Unit of
the ﬁgure is m

We get a peek at the recharge-baseﬂow relationship by looking at Figure 3.20. As
discussed previously, the geological conﬁgurations of the RCR watershed and its
medium size allows for a rather rapid response of the groundwater system. The time
lag between recharge and a response in groundwater baseﬂow is on the order of days.
Thus, on a monthly chart as in 3.18, such a time lag is almost not noticeable. In Figure
b, as it goes into summer, the recharge becomes increasingly small and falls below

Qgc. The groundwater compartment is thus losing mass.

159

Spatial patterns of ﬂuxes

Being a spatially distributed process-based model, the proposed model has the ability
to examine the spatial variation of hydrologic ﬂuxes due to different land uses,
climate input, soil properties and geological conﬁgurations. Here we examine a series
of spatial distributions of average annual ﬂuxes presented in the form of gridded
model output maps. The maps all possess the. same Geographic Coordinate System as
(GCS) (State Plane NAD1983, Michigan South)

Figure 3.21a shows the annual mean generation of saturation excess in the watershed.
This map is compared to the topographic index map in Fig 3.22. The topographic
index is a metric developed by [Beven and Kirkby, 1979] to evaluate the propensity of
areas in the catchment to reach saturation and produce saturation excess. Areas that
are prone to saturation tend to be near the stream channels or where groundwater
discharge occurs. They can grow in size during rainfall while shrink during dry
periods [Beven, 1978; Dunne and Black, 1970; Dunne et al., 1975]. These areas are
referred to as the variable contributing areas. [Hornberger et al., 1998]. The
TOPMODEL [Beven and Kirkby, 1979] attributes the majority runoff to saturation
excess and return ﬂow. The determination of contributing areas in a watershed is of
importance from water quality perspective because these are the areas that are most
prone to non-point source pollution generation.

The topographic index is calculated as:

T1 = ln(a/tanB) (3.7)

160

Where a is the upslope contributing area per unit contour length( A/c) and tanB is the
local slope (-). Higher TI values correspond to higher likelihood of saturation excess.
Although the two maps have different units, their similarity in spatial pattern can be
visually examined. Upon visual inspections of Fig 3.22 and Fig 3.21a we observe the
two maps agree well. Higher values of saturation excess are indeed found at areas
with higher TI. Comparing the two maps with the DEM, it was apparent that most of
the areas are cells surrounding the streams, which generally correspond to perennial
stream network and ﬂood plain. Local hillslope bottoms are also sources of saturation
excess. This ﬁnding is shared by [Ivanov et al., 2004a]. These areas are converging
regions of the groundwater ﬂow nets and receive inﬂow from their catchments.
However, high saturation excess is also found on the Southwest of the watershed
boundaries whose elevation is high. We ﬁnd the phenomenon is explained by the
thickness of the unconﬁned aquifer in this area (the glacial driﬂ layer), which is
signiﬁcantly smaller than that in the rest of the watershed. The shallow depth to the
bedrock limited the volume of water that soils can hold, thus this region gets saturated
more easily. This example shows the various factors that control the runoff generation
processes.

Inﬁltration excess, on the other hand, is more dependent on the land use type and soil
properties. Fig 3.21b shows the spatially-distributed annual inﬁltration excess. We can
see that the inﬁltration excess in agreement with the spatial distribution of saturated

conductivity in Figure 3.8. Relatively high relief areas also record higher inﬁltration

161

excess, a typical example being area (1) marked on the map because the large slope
values allow runoff to occur more rapidly. On the other hand, inﬁltration excess is
simulated consistently in the urban areas on North East corner of the watershed
because of the impervious areas.

The inﬁltration map in Figure 3.23a looks somewhat similar to the inverse of the
saturation excess map. The lower inﬁltration‘of the stream cells and hillslope bottoms
is explained by frequent groundwater discharging. The urban areas have low
inﬁltration as, intuitively, impervious cover is higher. Over the large area in the center
that is agricultural, inﬁltration is higher on the East than on the West. This is again
due to the spatial distribution of soil types.

Quantifying the amount of recharge that the aquifer receives and the spatial
distribution can be of great interest to water management and ecological planning.
Groundwater contribution to streams in summer has lower temperature than average
stream water temperature. It provides the prime habitat for many ﬁsh species. The
groundwater also serves as one of the major sources of water supply for human
consumption in many parts of the world. The annual average recharge map (Figure
3.23b) again conﬁrms the role of regional groundwater ﬂow plays in the hydrologic
cycle. The stream cells and hillslope bottoms serve as the exﬁltration point. We come
to the conclusion that recharge in the region, although highly dependent on both space

and time, is very much a topography-controlled phenomenon.

162

 

1.45
1.4
1.35
1.3

1.25

 

 

 

 

3.98 3.99 4 4.01 4.02 4.03 6

x 105 Topographic Index

 

 

 

 

 

3.98 3.99 4 4.01 4.02 4.03 6
X

Figure 3.21. Average annual runoff generation (mm) (a) Saturation Excess (b)
Topographic Index

163

 

1.45
1.4
1.35
1.3
1.25

 

 

 

 

'1 I l I I I '-
3.98 3.99 4 4.01 4.02 4.03 6
x

 

Figure 3.22. Average Annual Inﬁltration Excess

The total ET and the ground evaporation (EvapG) maps are shown in Figure3.24. The
ground evaporation (EvapG) is deﬁned as water that is evaporated from the ground
surface. Impervious areas have much higher EvapG than agriculture and forested
areas since water cannot inﬁltrate. The ET is also shown to vary signiﬁcantly in space,
ﬁ'om 140mm to 802 mm inside the watershed. And the variation is apparently related
to topography. At hillslope bottoms and lowland areas, regional groundwater ﬂow

supplies moisture to the soil and thus actual ET is higher at these areas.

5 Inﬁltration (mm/yr)
I I I I I

 

1.45
1.4
1.35
1.3
1.25

(a)

 

 

 

 

I I I l -
3.98 3.99 4 4.01 4.02 4.03 6

 

5 ' Recharge (mm/yr)

 

1.45
1.4
1.35
1.3
1.25

(b)

 

 

 

 

Figure 3.23. Annual Inﬁltration (a) and recharge map (b)

165

5 ET (mm/yr)
I

    
   

 

 

 

 

 

 

 

 

3.98 3.99 4 4.01 4.02

Figure 3.24. Annual ET (a) and EvapG (b) map. EvapG is evaporation from the
ground.

166

3.4. Summary and Conclusions

The newly developed model achieved high performance metrics in terms of
streamﬂow prediction not only during the calibration period (N ASH=0.68) but also in
the veriﬁcation period (NASH=0.614). Flow at an internal gaging station is also
described well. Starting from a rough initial estimate of the groundwater heads, the
model is able to evolve to a more realistic state, and achieve excellent comparisons
with the observed groundwater heads. The annual mean hydrologic ﬂuxes, including
ET, inﬁltration, recharge and groundwater contribution are close to those estimated
by the SWAT model. All these results indicate that the proposed model is a valid
representation of the hydrologic system of the watershed.
The model is able to explicitly detail the complex interactions of processes in the
watershed in space and time. The watershed is a subsurface-dominated system with
saturation excess found to be the main runoff generation process. The places that
produce saturation excess are cells near streams and groundwater discharging areas
which agree well with the Topographic Index map computed from method of [Beven].
Inﬁltration, recharge and ET are also found to be strongly related to topography and
groundwater ﬂow.
The geological characteristics of the watershed permit a fast-responding subsurface
hydrologic system. The large seasonal variation of energy input introduces a strong
annual cycle in the groundwater baseﬂow and markedly different responses of surface

runoff in different seasons. Responses to summer precipitation are often muted while

167

large rain in early spring tends to produce much higher peaks in the hydrograph.

3.5. Limitations and future research

From the above modeling practice and comparison with observed data, we can

summarize some limitations of the present model and some future development

considerations:

1. Soil moisture lateral ﬂow needs to be added. Although for the study watershed soil
lateral ﬂow seems to be a small component (0.08% of precipitation from SWAT),
it should be added for the completeness of the model. It may be more important
for other watersheds

2. Soil temperature and freezing module need to be added. At least a simple soil
temperature should be incorporated to allow the parameterization of soil freezing.

3. The vegetation module needs to be improved for better modeling of the interaction
of vegetation and the hydrologic cycle.

4. The model should. be applied to larger watersheds and watersheds with different
land use/ geology/ soil/ climate.

5. The effect of spatial resolution on the results should be carefully examined

3.6. Software package

A software package with Graphical User Interface (GUI) has been developed for the

present model to help interfacing with data and building up the models. Along with

168

this package are tools to visualize model results and calibrate the model.

Due to some iterative approaches used to solve the PDEs (e.g. conjugate gradient for
the groundwater ﬂow equation, the celia’s iteration for Richards’ equation), the
computational time can vary depending on the ﬂow conditions and soil properties. It
has been found that larger N values of the van Genuchten ﬁinction (Eq. (2.57)) can
lead to more iterations to converge, since the soils properties are more nonlinear with
larger N. Also at high ﬂow times the time steps of surface ﬂow modules are reduced
to meet stability criterion. Thus the computational time required to complete a model
cannot be completely predicted ahead of time. However, for the RCR watershed

model, the simulation time varies from 2.5 seconds to 3.5 seconds for each day of

simulation.

169

Chapter 4. Estimating Longitudinal Dispersion
and Surface Storage in Streams Using Acoustic

Doppler Current Proﬁlers

4.1. Introduction

The ability to accurately describe the transport of conservative as well as reactive
solutes in streams and rivers is fundamental to many branches of science and
engineering including stream ecology, geomorphology, river engineering, water
quality modeling, risk assessment etc. It is well known [Day, 1975; Thackson and
Schnelle, 1970] that the classical Taylor theory [Taylor, 1954], which leads to the
following advection dispersion equation (ADE), cannot adequately describe tracer

transport in natural streams.

2

4.1
at 3:1: 33:2 ( )

 

Here C is the solute concentration, uis the mean ﬂow velocity anda: and tdenote
space and time respectively. Extensive tailing and the persistence of skewness in the
observed tracer data (which cannot be adequately described using equation (4.1)) led
to the development of models based on the concept of transient storage (TS) [Bencala
and Walters, 1983; Thackson and Schnelle, 1970]. The TS models often use
conceptualizations based on two distinct zones: the main channel in which advection

and dispersion are the dominant processes and the storage zones that contribute to TS.

170

The governing equations for the TS model appear as follows:

60_ (200 18 00

q
_ -—--——-+——-[ADS +——L—(CL -—C) + a(Cs -— C)

 

 

797 Aug; Aaa 3a- A
(4.2)
30
sza—A—(C—C)
at A5 .

Here C and C S are the solute concentrations in the main channel and the storage
zones, A and AS are the sizes of the main channel and storage zones, and a is a
ﬁrst-order exchange rate between the main channel and the dead zones. DS denotes
the dispersion coefficient in the TS model. The subscript “S” is used to indicate the
fact that the dispersion coefficients estimated Using the ADE and TS models generally
tend to be different for the same river reach under similar ﬂow conditions. Since the
TS model has additional terms for the dead zones, the coefﬁcient D3 represents
only the shear-ﬂow contribution while the ADE estimates of dispersion tend to be
higher asD includes the effects of dead zones to some extent [Deng and Jung, 2009].
By comparing the dispersion coefficients obtained from the TS and ADE equations,
we will be able to assess the relative contribution of dead zones (or storage zones) in
different stream reaches and this idea will be explored later on in this chapter. Efforts
to interpret model parameters and to relate them to physical stream characteristics are
often confounded by the inability of current stream tracer techniques to separate TS
processes [Gooseﬂ et al., 2005] as well as mathematical difﬁculties associated with
parameter estimation, especially in the presence of competing parameters and

false/singular convergence [Runkel et al., 1998]. In this chapter, we propose two

171

 

 

approaches for independent, ﬁeld-based estimation of the longitudinal dispersion
coefﬁcient and the contribution of the total storage AS that is due to surface features
such as vegetation, eddies and pools, meander bends etc. This type of storage is called
surface storage to separate it from the hyporheic or sub-surface storage (exchange
with near-bed sediments or groundwater) found in many stream reaches. By
estimating the surface storage in different stream reaches, the importance of hyporheic
storage can be assessed. Both techniques reply on the measurement of high-resolution
velocity ﬁelds within the stream channel. The dispersion coefﬁcient can be estimated
by directly integrating the velocity ﬁelds using the shear-ﬂow dispersion theory. The
size of the surface storage zones can be estimated using wavelet decomposition of the
velocity data. Both estimates are expected to lead to improved model parameters and

solutions.

Wavelet analysis has been used extensively in the past decade to analyze both time
series and spatial data in geophysical and engineering applications. Examples include
the analysis of streamﬂows [Coulibaly and Burn, 2004], seismogram data [Lockwood
and Kanamori, 2006], longterm trends in geomagnetic activity [De Artigas et al.,
2006], and ocean waves [Pairaud and Auclair, 2005]. Wavelet analysis [Mallat,
1989]allows the original signal or image (in our case a two-dimension image of mean
velocity in the channel as a function of river depth and width) to be split into different

components so that each component can be studied with a resolution that is suitable

172

for its scale. In particular, the signal can be decomposed into slow changing (i.e.,
coarse or low frequency) features and rapidly varying (i.e., ﬁne or high frequency)
features using low-pass and high-pass wavelet ﬁlters at different levels (or scales) in a
multilevel decomposition. This feature of wavelet analysis is particularly attractive for
studying TS processes in streams since we are looking for fast and slow-ﬂowing
regions of the'river. Previous applications of two-dimensional wavelets include
characterization of permeability anisotropy [Neupauer et al., 2006], studies of
spatiotemporal dynamics of turbulence [Guan et al., 2003] and analysis of spatial

rainfall data [K umar and F oufoulageorgiou, 1993].

The longitudinal dispersion coefficient (D) is an important parameter that describes
the transport of solutes in streams and rivers. Accurate estimation of the dispersion
coefﬁcient is important from human health and public safety perspectives as the
parameter is needed to predict contaminant concentrations near drinking water intakes
and receiving water bodies such as lakes or oceans. Once a solute is released into the
stream and becomes vertically and laterally well-mixed, longitudinal dispersion is the
primary mechanism responsible for spreading the tracer plume and for reducing peak
concentrations. The one-dimensional transport of solutes following the initial period
of mixing can be described using the advection-dispersion equation (ADE) [F ischer,
1979; Rutherford, 1994]. The coefficient D in equation (4.1) can be estimated

using a number of empirical relations available in the literature [Deng et al., 2001;

173

Fischer, 1979; Sea and Cheong, 1998]; however, these estimates are generally known
to exhibit large variability. Dispersion estimates from tracer studies (which usually
involve ﬁtting the observed data to a solution of equation (4.1) for appropriate
boundary and initial conditions) are generally believed to be more reliable; however,
signiﬁcant time and resources are needed to conduct tracer studies, especially on large
rivers. An alternative approach, which is the main focus of this chapter, is to estimate

D from the theory of shear ﬂow dispersion [F ischer, 1979]:

 

D 18 ' I d y dy, III I "'l lid 11
_ —Z{u (301(9) .9! 0.1/1101')!“ (y My ) y (43)

where A is the channel cross-sectional area, y is the transverse coordinate which
varies from y = 0 at one bank to y = B at the other, h(y) is the depth of ﬂow at
a given y location, Dy is the transverse mixing coefficient, u’(y) = 17(3)) — (7 is
the deviation from the mean velocity, (7 is the cross-sectional mean velocity and
H(y) = 1;) 2 11(3), z)dz is the depth-averaged velocity at y. Eq. (4.3) is based on several
assumptions as described in [Carr and Rehmann, 2007; Fischer, 1979] including
one-dimensional ﬂow (no vertical and transverse gradients in concentration, that is
well-mixed). Eq. (4.3) Although the theory for estimating the dispersion coefﬁcient
from the velocity ﬁeld has been around for nearly four decades, the single most
limiting factor in the application of Eq. (4.3) in the past was the measurement of
detailed velocity distribution and depth across the river. Signiﬁcant time and resources

are needed when instruments designed to make point-measurements (such as the

Price-AA and Pygmy current meters) are used to measure the velocity ﬁelds in

174

streams. However, this situation has changed with the advent of Acoustic Doppler
Current Proﬁlers (ADCP). While the ADCP technology itself is not new, early models
of the instruments were geared towards oceanographic applications. Recent ADCP
models designed keeping rivers, streams and other shallow inland water bodies in
mind can be used to quickly make velocity and bathymetry measurements over long
distances with high accuracy and resolution. This new capability introduced the
opportunity to address important questions involving inland water bodies [Dinehart
and Burau, 2005; Garcia et al., 2007; Phanikumar et al., 2007] including the
calculation of D with relative ease using Eq. (4.3). However, there are several issues
that need to be examined before comparing tracer estimates with ADCP results based

on equation (4.3) and these are described below.

The dispersion coefﬁcient estimated from tracer data is often used as the “true” value
to evaluate other methods. The tracer estimate represents the bulk, reach-averaged
mixing strength in a given reach and is valid only for the particular stream reach and
the ﬂow conditions for which the experiment was conducted. The ADCP estimate, on
the other hand, represents a “point estimate” along the length of the river (valid for a
particular cross-section for which data was obtained). If the channel is fairly uniform
and the transect represents the conditions in the entire reach, then the point estimate
can be expected to represent reach-averaged conditions. If the river reach exhibits

signiﬁcant heterogeneity in properties, then a reach-averaged estimate can be

175

 

 

 

calculated using a distance-weighted average as suggested by [Rt/therford, 1994]:
_. 1 n
D = I: DAx. (4.4)

whereD is the reach-averaged dispersion coefﬁcient, Ari is the length of the
sub-reach containing the ADCP transect atz' and L is the total length of the river
reach. A number of mechanisms including surface storage in dead zones (e.g.,
vegetation, woody debris) and solute exchange and retention within the hyporheic
zone are known to produce an effect that is somewhat similar to the effects of
shear-ﬂow dispersion in streams. The transient storage (TS) model [Run/rel, 1998],
which describes the temporal trapping (and release) of solute particles in the dead
zones is often used to describe transport when dead zones play an important role and
the ADE cannot adequately describe the data. Ignoring lateral ﬂow contributions into
the channel due to groundwater, the TS model can be written as shown below

[Bencala and Walters, 1983; Runkel, 1998].

Fisher et al. [Fischer, 1979] numerically evaluated Eq. (4.2) for both laboratory
ﬂumes as well as natural streams and compared their results with tracer estimates (see
Table 5.3 in [Fischer, 1979]). Their velocity measurements were obtained using
standard current meters. Carr and Rehmann [Carr and Rehmann, 2007] recently
evaluated Eq. (4.2) using the velocity and bathymetry measurements obtained from
ADCPs and compared their estimates with tracer results for ten US rivers. Half of

their ADCP estimates are found to be within 50% of the values from tracer studies,

176

and 85% are within a factor of 3. They conclude that the ADCP method is at least as
accurate as the best empirical formula considered in their work. While these results
are encouraging, many questions remain unanswered. First, if the ADCP estimates are
only as good as the best empirical relations (which generally produce estimates within
an order of magnitude) then the ADCP method is not very attractive since it is easier
to use the empirical relations. Second, in many stream ecosystem studies the
primary interest is not in dispersion but in other processes (e.g., biogeochemical
processes, mortality/loss rates of bacteria or viruses, storage zone sizes and reaction
rates) but accurate estimates of dispersion are still needed to adequately describe these
processes. If the ADCP method has the potential to produce dispersion estimates that
are as accurate as the tracer method, then the result has important implications for
studies involving stream and river ecosystems. The aim of this paper is to explore this
question in more detail. The tracer data used in [Carr and Rehmann, 2007] spanned a
period of nearly three decades (1967 to 1991) while the ADCP measurements coved
the period from 2000 to 2004. Changes in both channel cross-section and (local) slope
are possible in the period following the tracer studies which could potentially
inﬂuence the results. In addition, different approaches were used to estimate D from
the tracer data - from a routing method [Rutherford 1994] to ﬁtting a line to the
variance obtained from the tracer breakthrough curve [F ischer, 1979] to modeling the
tracer response curve as a scalene triangle [Jobson, 1997]. These methods are known

to produce vastly different results when applied to the same river reach under similar

177

ﬂow conditions introducing additional sources of uncertainty into the comparison
between ADCP and tracer estimates of the dispersion coefficient. The aim of this
paper is to contribute additional tracer and ADCP datasets in support of the analysis
reported in [Carr and Rehmann, 2007] focusing mainly on datasets collected at the
same time. Recognizing that the tracer estimates of D have their own sources of
error and uncertainty, our aim is to understand if the ADCP method has the ability to
produce estimates that are comparable to the tracer method, especially when using
multiple/repeated transect data at the same site, a procedure that tends to average
errors involved in the data. Due to the randomness of the turbulent ﬂows involved,
raw ADCP data usually contain noise which depends on the river characteristics and
the operating conditions of the instrument. One of the aims of this paper is to
systematically examine the effects of different post-processing methods (e.g.,
smoothing) on the dispersion results. Finally we will examine the limitations of the
ADCP method in order to better understand conditions under which the method can

produce reliable dispersion estimates.

Tracer data from several mid-westem streams have been used to compare the ADCP
estimates of dispersion and surface storage with tracer-based estimates. Data for the
Grand River, Michigan is new and was published in [Shen et al., 2008]. Tracer studies
on the RCR (a tributary of the Grand) and the wavelet decomposition method for

estimating surface storage are described in [Phanikumar et al., 2007]. Tracer and

178

ADCP data collected by USGS staff on the Ohio River and the St. Clair River are

used for comparing the dispersion estimates (tracer versus ADCP).

4.2. Description of Sites

The Red Cedar River (RCR) is a fourth-order stream in south central Michigan,
United States that drains a landscape dominated by agriculture and urbanization. It
originates as an outﬂow from Cedar Lake, Michigan, and ﬂows into East Lansing and
Michigan State University (MSU). The river then connects with the Grand River in
Lansing, Michigan. The total stream length is approximately 70 km. The river and its
tributaries drain an area of about 1,230 km2, one fourth of which is drained by
Sycamore Creek (Figure 1). More general information is available in previous
chapters. Limestone, shale, sandstone, coal, and other sedimentary rocks arranged in
almost horizontal layers compose the bedrock surface of the Red Cedar Watershed.
The river is characterized by a meandering channel and low stream gradient. A USGS
gauging station (04112500) is located at the Farm Lane Bridge (Figure 1) and
measures runoff from about 75% of the basin. The study reach was bounded by
Hagadom Bridge on the East and the Kalamazoo Street Bridge on the West (4.1). The
RCR meanders through the MSU campus over a stretch of approximately 5 km (our
study reach in this paper) and has an average slope of 0.413 m/km. The river width
varies considerably, from 16.3 m to 40.4 m, with an average of 28.1 m (4.1). The

general trend is that the width decreases for the initial 2400 meters, increases over the

179

next 2100 meters and ﬁnally decreases for the remaining 600 meters.

 

  
   

   
 

 

 

 

 

 

  
  
   

  

 

 

 

Legend
—Stneams
Study Area Williamston lILeSub-Watersheds =2
7?" ' ‘ -Lakes 9
z
I
9
i. - °
.‘ “:1: 9
Creek . 0 5 10 20 $
Elﬁlometers
' I
84’35'0'1N 34°1o'o'w
l-—- C 'r B : A I
K°"°99 <— Flow :2
Kalamazoo Library _ g
Ha adorn 3’
Farm Lane Bogus g g
. . , _
Kilometers
I r
84°30'0"W 84°28'48"W 84'27"36"W

Figure 4.1. Red Cedar River and the sampling locations

The Grand River is a 420 km long tributary to Lake Michigan (4.2) and the tracer
study was conducted on a 40 km stretch of the river, starting from the city of Grand
Rapids and extending to Coopersville. Surﬁcial geology of the Grand River Basin is

dominated by rivers crisscrossing the moraines and outwash plains formed by

180

extensive glaciation during the Pleistocene (Supporting Information). Till plains,
moraines, kames, and eskers of the Port Huron system are the predominant surface
features. The Ann Street Bridge near downtown Grand Rapids was selected as the
injection point. This location is close to the combined sewage overﬂow (CSOs)
outfalls that serve the city of Grand Rapids. These CSOs are point sources of
pathogens discharging into the Grand River. One of the objectives of the current study
was to examine the potential health risks posed by the C80 discharges. Sampling was
carried out at four downstream sites; bridges (Wealthy Street, site 1; 28th Street, site 2;
Lake Michigan Drive, site 3; 68th Street, site 4). The study reach was sufﬁciently long
to make watershed inﬂuences important. A US. Geological Survey (USGS)
streamﬂow gaging station (04119000) is located 1.05 km upstream from the ﬁrst
sampling site. The discharge on the test date was on the end of a recession limb with
the values during the experiment gradually declining from 3230 to 3010 cubic feet per
second. The study reach is a perennial gaining stream. No precipitation event was
reported in the three days prior to the experiment. Contributions from the tributaries
and baseﬂow together, termed lateral inﬂow, are important to correctly describe the
downstream transport of both tracers. Although lateral inﬂow is often neglected in
tracer studies conducted on relatively short river reaches, it is not negligible in our

case.

181

 

- Urban

1:] Grassland
[:1 Barren

I: Open Water
— Wetland
- Forest

N

LakeMichiganDr. 1, l . _ ' a . -'
’ eahh SLIBﬂc16
8th St. Bride , . . -

:VY‘

[11.85.19 098

M ‘6 87‘ 07 098

   

Legend
I Injection Site 9
A USGS gage
0 Sampling Sites

—— Grand River

—— Tributaries

M..0.0€ 098

30 15 0 30 60
Deamhmems mm

 

 

 

I I U I
86° 2’24”W 85° 51’36”W 85° 40’48”W 85° 30’0”W

Figure 4.2. Red Cedar River and the study region showing the sampling locations of
the Grand River tracer study

4.3. Materials and Methods

Four tracer studies (slug additions) were conducted on the Red Cedar River in
summer 2002 on 17 May, 31 May, 6 June, and 21 June. Average discharge (Q)
recorded on these four days at the USGS gage near Farm Lane Bridge was 16.82,
14.41, 19.06 and 2.49 m3/s respectively. Discharge and hydrodynamic data were

collected from ADCP surveys during the period 2002—2006. ADCP data could not be

182

collected during the tracer tests in 2002 but on two occasions the discharge values
were similar to those during the tracer tests. ADCP data collected on 19 March 2006
(Q = 19.89 m3/s) and 29 September 2005 (Q = 2.0 m3/s) were used to compare with
results from the tracer studies conducted on 6 and 21 June 2002 respectively. Because
of low-ﬂow conditions (long travel times of the dye cloud), the tracer study on 21
June 2002 could not be completed. For this discharge, we present results for only one
sampling location. Tracer transport was described using the TS equations [Bencala
and Walters, 1983]. A one-dimensional hydrodynamic model based on the St. Venant
equations was used to examine two of the estimated parameters in the TS equations
(mean velocity and dispersion coefﬁcient). For all the four slug injections, the dye
was released at the Hagadom Bridge and samples were collected at the Farm Lane,
Kellogg and Kalamazoo Bridges, respectively (4.1). The distances to the three
sampling locations from the point of release are: 1400 m (Farm Lane), 3100 m
(Kellogg) and 5079 m (Kalamazoo). Flourescein dye was released in the middle 75%
of the channel to ensure conditions of instantaneous mixing. Sampling was done at the
middle of the cross section for each dye release. F luorescein solution with a
concentration of 179.06 g/L was used for all the tracer tests. The volume of the dye
used was based on a desired peak concentration range of 10—20 mg/L at the last
sampling point. Samples were analyzed using a TurnerlO-AU ﬁeld ﬂuorometer
(Turner Designs Inc.,Sunnyvale, CA). Observed tracer data was checked for mass

conservation. The mass of the tracer in the dye cloud as it passed a sampling point

183

was calculated using the relation

771(t) = Qf C(t)dt (4.5)
0

where C(t) is the dye concentration and Q is the discharge at the sampling point. The
fractional recovery was calculated by m/M where M is the mass injected. The
fractionalrecovery values for all the ﬂows (and at different sampling locations) varied
from 0.91 to 1.5 with a mean value of 1.29 and a standard deviation of 0.16. To
facilitate comparisons with mathematical models and to produce a data set that obeys
dye mass balance, fractional recovery corrections were applied to the individual
concentration values as reported by [Atkinson and Davis, 2000]. Point velocities and
river stages were measured on eight consecutive days in 2002 (4—11 April) to calibrate
the hydrodynamic model. Price AA current meters and a l6-MHz Sontek acoustic
Doppler velocimeter (ADV) (Sontek/YSI Inc., San Diego, CA) were used to measure
velocity proﬁles and discharge in the river. In addition, a vessel-mounted,
down-looking 1200 kHz Rio Grande ADCP (Teledyne-RD Instruments, Poway, CA)
was used to measure discharge and three-dimensional velocity ﬁelds in the river using
an Oceanscience2 Riverboat (a low-drag trimaran) equipped with radio modems for
real-time deployments. By towing the vessel-mounted ADCP across a river transect
(perpendicular to the ﬂow direction), we obtained snapshots of instantaneous velocity
ﬁelds in the river which were later used for wavelet analysis] (described below).

Typical boat speeds were in the 0.5—l m/s range. Several steps were involved before

184

analyzing the ADCP data. The speed at which the ADCP was towed across the river
depended on the discharge and the ADCP operating conditions. The ADCP data was
exported from WinRiver (the ADCP operating software) for further processing in
MATLAB including smoothing and removal of bad ensemble values. Details of
ADCP principles and processing are described by [Dinehart and Burau, 2005]. The
discharge values obtained from ADCP measurements at the Farm Lane Bridge were
compared with those from the USGS gauging station on several occasions over a
period of four years and an excellent agreement was obtained.

For the Grand River tracer study, Rodamine WT 20% (weight) solution was used. P22
was obtained from Samuel Farrah, University of Florida, Gainsville, Florida and was
maintained on the host Salmonella typhimurium LT-2 (ATCC 19585). P22 stock was
grown by inoculating 100 ml of log-phase S. typhimurium host with 1 mL of P22
stock (~ 1011 pfu/mL) and incubated at 37 °C for approximately 3—5 h. After
incubation, 0.01 g of lysozyme and 3 mL of 0.2 M sterile EDTA were added to the
ﬂask and mixed well. The culture was then centrifuged at 4000 rpm for 10—15 min,
and the supernatant was ﬁlter sterilized through a 0.45 ummembrane. P22 stock was
stored at 4 °Cuntil used. RWT and P22 solutions were injected into the river (Slug
release) from the Ann Street Bridge on May 8, 2006 at 7:00 am. A total of 8770 g of
RWT and 16 L of bacteriophage P22 (4X1011 PFU/ml)were released. At each station,
grab samples were collected from just below the surface using manual sampling. Two

samples were taken at the same time. One was stored in a dark cooler for RWT

185

analysis. A 5X trypticase soy broth (TSB) was added to the other sample to stabilize
the bacteriophage for P22 analysis. All samples were kept on ice and were analyzed
within 48 h in the laboratory. Meanwhile, water temperature, pH, suspended solids,
and weather data (i.e., ambient temperature, rainfall, wind, etc.) were noted during
sampling. A Turner Designs 10 AU ﬁeld ﬂuorometer (Turner Designs, Inc., Sunnyvale,
California) was used to initially detect the dye at the ﬁrst three sites. The sampling
frequency for both tracers was increased after receiving a RWT signal. RWT samples
were analyzed in the laboratory using the same 10 AU unit. Water samples were
assayed for P22 following the double agar layer procedure [Adams, 1959]. Samples
from site 1 were diluted to a 10-3 concentration, and between 1 and 2mL of each
sample in at least duplicate were assayed for the phage presence on tryptic soy agar.
The plates were incubated for 24 h at 37 oC. The detection limit of this method is less
than one plaque-forming unit per milliliter. Total suspended solids concentration was
determined according to Standard Method 2540-D—total suspended solids dried at
103—105 °C [Greenberg et al., 1992].The river is signiﬁcantly wider in comparison
with the numbers reported inmanyprevious tracer studies. Therefore, to obtain a better
idea about the lateral variability at each station, sampling was done at multiple
locations (left, right, and center) on each bridge except at site 3, where sampling was
done at two locations (left and right) due to the presence of an island in the middle of
the channel. Because of the long travel time to site 4, P22 data was not collected. In

addition to the manual sampling, a submersible ﬂuorometer (Turner Designs SCUFA)

186

equipped with a data logger and programmed to measure RWT concentrations every
10 s, was deployed at sampling sites 1, 3, and 4. Discharge at all the sampling
locations was measured using a Teledyne - RD Instruments (1200 kHz) Rio Grande
acoustic doppler current proﬁler (ADCP).

For estimating the dispersion coefﬁcient using the ADCP technology, Data from a
total of 505 ADCP transects collected from seven rivers in the states of Ohio, Indiana
and Michigan are used in the present study. In addition, seven tracer studies have been
conducted on some of the rivers. Details of the rivers are summarized in Table l and
maps of the sites showing the locations of the ADCP transects are shown in Figure 1.
Out of the seven rivers, simultaneous tracer and ADCP data are collected for three
rivers (Ohio River, Grand River and Burns Ditch). For one river (Red Cedar River)
tracer data was collected during summer 2002 while ADCP data was collected for
similar ﬂows between 2003 and 2006. No tracer data is available for three rivers
(Muskegon River, Thomapple River and St. Clair River) but estimates from the ADCP
method and empirical relations are Shown for comparison. Where conditions
warranted we ran multiple transects at the same location and at multiple locations
within the same river reach. Details of the tracer study and modeling for the Grand
River and Red Cedar River are available in [Shen et al., 2008]. Tracer and ADCP
data for the Ohio River (only ADCP data for the St. Clair River) were collected by ’
USGS staff and details are available in [Holtschlag and Koschik, 2003; Koltun et al.,

2006]. A continuous dye release was conducted on the Portage Burns waterway

187

(Burns Ditch) on June 21, 2008 using Rhodamine WT. Although the aim of the tracer
study is to understand nearshore processes in Lake Michigan, concentration —time
data collected within the stream are used to estimate a dispersion coefﬁcient by ﬁtting
an analytical solution for continuous release [Chapra, 2008] to the data. For the Ohio
River, Rhodamine-WT was released at one of the banks and the tracer did not
completely mix within the study reach. Breakthrough data is not available in the form
of concentration versus time data; however, concentration values are reported within
the channel cross-section (at different depths and distances from the bank) for several
different locations. We computed the cross-sectional average concentration at
different stations and ﬁtted the spatial data to the unsteady ADE to compute a
dispersion coefﬁcient. This method will likely introduce some error Since the tracer is
not fully mixed to justify the use of one-dimensional ADE; however, the estimated
dispersion coefﬁcient described the mean concentration values at different stations

accurately after the ﬁrst few sampling locations.

4.3.1. Transient Storage Modeling

Tracer transport in the RCR and GR was described using the TS equations (4.2) which
describe transport in the main channel and the storage zones respectively [Run/rel et
al., 1998]. If Eq. 4.2 applied on a reach basis, then the velocity (Q/A) in (2) is a
reach-averaged value and, in general, is not equal to the local (point) velocity u. The

TS model equations were solved using a fourth-order accurate compact numerical

188

scheme [Demuren et al., 2001]. Brieﬂy, the Spatial derivatives were approximated
using a fourthorder scheme with spectral-like resolution and a low-storage
fourth-order Runge-Kutta scheme was used for temporal differencing. The resulting
tridiagonal matrix system of equations was solved using the Thomas algorithm. A low
Courant number of 0.15 and a uniform grid of 1001 points were used for all model _
runs. The boundary and initial conditions for the model were as follows. The river
was assumed to be initially at zero tracer concentration. The upstream boundary was
modeled to simulate slug release into the main channel and the storage zone was
assumed to be initially solute-free. A no-ﬂux boundary condition was speciﬁed at the
downstream boundary for the transport equations. Parameters in the TS model (i.e., A,
AS, D, a) were estimated using a global optimization procedure, the Shufﬂed
Complex Evolution (SCE) algorithm [Duan et al., 1992] by minimizing the
root-mean-square error (RMSE) between the observed data and the model. This
algorithm found wide applications in the hydrologic research literature and was
shown to be robust and efﬁcient in ﬁnding the global minimum. For all the dye
studies, Optimal parameters were obtained in 3000—6000 iterations on a 512-core
Western Scientific Opteron Cluster computing system at MSU. The RCR is generally
a gaining stream; however, within the study reach the gain was not signiﬁcant enough
to change the TS parameters. After running two separate optimizations, with and
without qL, we decided to use the parameters obtained with qL = 0. The fourth-order

accurate compact scheme used to solve the TS equations in this paper was tested

189

extensively and was used to solve similar sets of equations in the past [Phanikumar
and McGuire, 2004]. To assess the accuracy of the compact scheme for solving the TS

equations, we compared our numerical solutions with the analytical solutions reported

 

 

 

by [De Smedt et al., 2005]:
t — , 2 1 (t— > '(t — )
C(cr,t) = fa+ 2:2 L —-————a J[ar,a T ]—aJ[g——T—,on'] Cl(a:,r)d'r
0 4D7'2 27 l3 3
a an n— —l Lm
J(a,b) =1—ebeeTAIO(2\/l;)d)l =1— e—a b25717
0 I=m Om

(4.6)

where 10 is the modiﬁed Bessel function of zero-th order, [3 = (As/A) and C1 (x,t) is
the classical solution to the advection-dispersion equation with the same initial and

boundary conditions [Chapra, 1997]:

2
_M (4.7)

M
> = ——exp
Axl47rDt 4Dt

Cl(a:,t

where M is the mass of tracer released. Comparisons with our numerical solutions
obtained using 200 grid points are presented in Figure 5 for different values of a. An ‘

excellent agreement is noted between the two solutions.

190

 

 

I I I

 

 

 

 

0.12 an,
P t .
(1 = 0 _9’? “'4 0 Analytical (De Smedt et a1. 2005)
0.1 ' f ’1 Numerical (Present Work) ‘
f . .,Q i
- I 5 1..-»-.. 01 = 0.0001 ~
am».
it . at it
I . 5¢
0.06 " if ' 4'7. \ ‘
it a I“ -,
l : 5° ‘ -
:1? 3‘ it» \ka
0'04 if ft 1" E0 = 0.001
03’ .

0 02 _ if 5;? ii2 ”is.

- «a - Ilia. _
0 ' A n " .".. .I:.H .... .. .'
_002 “I” “I” ”I” “I” --l-- “-1” --.

3000 3500 4000 4500 5000 5500 6000
Time (s)

Figure 4.3. Comparison of the numerical solution with the analytical solution of De

Smedt et al [2005]

4.3.2. Multi-resolution Wavelet Decomposition of ADCP Data

To examine the estimated parameters and to relate them to the physical characteristics

of the river, we used three-dimensional velocity data obtained from ADCP surveys.

Given the 3-D velocity ﬁeld in a river, the dispersion coefficient D can be computed

by numerically integrating the velocities [Fisher et al., 1979]. However, we did not

follow this approach as earlier studies found that dispersion estimates based on

time-dependent velocity ﬁelds tend to be highly sensitive to velocity ﬂuctuations

[Palancar et al., 2003]. In this paper we focus on the parameters AS and A. Repeated

191

ADCP surveys in the study reach clearly showed regions of high and low velocities
and acoustic backscatter (a well—known measure of suspended solids concentration,

SSC) for different cross sections and led us to test the hypothesis that in-stream TS
zones can be identiﬁed using the three-dimensional velocity ﬁelds obtained from
ADCP surveys. Earlier studies [Sukhodolov et al., 2004; Tipping et al., 1993] showed
that the concentration of suspended particulate matter reduces in the dead zones due
to sedimentation of faster sinking fractions of suspended matter in the decelerating
ﬂow. Therefore, by identifying regions of decelerating ﬂow or low SSC using ADCP
data we may be able to identify the relative importance of dead zones in a river reach.
[Engelhardt et al., 2004] noted a correspondence between SSC and mean velocity
vectors and this correspondence was also evident in our ADCP data. To identify
regions of relatively fast and slow moving water,we use multiresolution wavelet
analysis of the twodimensional (y, z) normalized mean velocity ﬁelds obtained from
ADCP surveys. The continuous wavelet transform (CWT) of a function f(y,z) for a
two-dimensional wavelet is deﬁned as the convolution with a scaled and shifted

version of the wavelet function \I/

 

 

b1 —b.
81) 2—J.1_—_-ffJZ y- Z 2]dde
(I,
“la? iii 01 2 (4.8)
a1 b1
a = ,b==
(‘2 b2

 

 

 

 

where a and b are the scale and translation vectors. We used the two-dimensional

discrete wavelet transform (DWT) for our analysis. Similar to the 1D wavelet

192

transform, the 2-D transform can decompose a given function into its slow changing
(or coarse) features (called approximations) and ﬁne (or rapidly changing) features

(called details). For the two-dimensional case, the details can be further decomposed

into horizontal, vertical and diagonal details. If (pyapz denote the scaling functions

and crux/2‘? the wavelet functions for the one-dimensional representation in the y and

2 directions respectively, then multidimensional wavelet bases can be constructed as

the tensor products of the one-dimensional wavelet bases as shown below.

F

3} z _ _
t"bmn'wmj’ a—H’3_H
fl .2 _ _
\paﬁ . :‘(pmjwnuj’a—L’ﬁ-H (49)
"1121.7 y 2 _ __ O
(Jaw/2W. a—H,ﬁ—L

 

{pyrnn' (pzmj’ Or : L’ﬂ : L
where m denotes the wavelet scale level and i and j denote the rows and columns of
the coefﬁcient matrices. Here L and H denote the low- and high-frequency content

(coarse and ﬁne features) corresponding to the different ﬁlter properties of the scaling

and wavelet functions in the spectral space. For the function f(y, z), the wavelet

mi J.“ (y, z) captures its coarse features at level m, while the remaining three

wavelets \I/ HL( LH( HH(y,z) contain the vertical,

y,Z),‘IJm’z-J y,2),\11m7,:,j

mm
horizontal, and diagonal details respectively. The function can be represented as the

sum of its approximations and details using the wavelet coefﬁcient matrices as shown

below.

193

N
.A LL v HL
le):ZW Nat.qu N.i.j(y’z)+ 22W Naxj‘p N.-i..j(y’z)
if

m=1 ij

N N
H LH D HH
+ Z 2W 1112‘,qu N,i.j(y’z) + E SW N,z‘,j‘I’ Maj-(31.2)

m=l ij m=1 ij
(4.10)

where

WAM = f f WM jly,21f(y.2)dydz
va -—— ff(1.21me,)f(,,z)d,,d,
WHNJJ' : ff‘IJLHN’i‘J-(y,z)f(y,z)dydz
WDN,i,j = ff\IIHHNJ.’J.(y,z)f(y,z)dydz

Here the W parameters are the wavelet coefﬁcient matrices, N denotes the number of
levels in the decomposition, and the superscripts A, V,H and D denote the
approximations (the ﬁrst term in equation (4.10)) and the vertical, horizontal and
diagonal details respectively (the last three terms in equation (4.10)). The
decomposition (4.10) allows us to examine the different components at multiple levels
or scales. Our aim was to extract the coarse features (the ﬁrst term in equation (4.10))
at different levels as they retain the essential features of the velocity ﬁeld (the
high-frequency content simply adds detail). Since our primary interest was in making
a distinction between regions of slow moving or stagnant water (dead zones) and the
main channel (two distinct scales), we used two-level decomposition (N = 2). After
plotting the single-level reconstructions based on the approximations at levels 1 and 2
in the physical space, (AS/A) was computed from the area of all the pixels greater

than a threshold value T (corresponding to the background pixel value in the two

194

images):

A—Sz ff_—f1—_(y, z)dydz
A ffﬁ2(y, z)dydz

1 iff WA .111” >T
1,3 = 2m 2H 4.11
f1(y ) [0 otherwise ( )
LL
112(1)”): 1 iff WA 111.1111 11.J.>T
0 otherwise

The Haar and Daubechies-12 wavelets [Daubechies, 1988] were used in our analysis;
however, other wavelets produced essentially similar results. Aﬁer performing the
wavelet decomposition and extracting the terms for N = l and N = 2 and plotting
them in the physical space (i.e., as a function of y and z to identify the channel and the

relative locations of the dead zones within the channel), functions in the MATLAB

wavelet processing toolbox were used to compute the areas and the ratio in equation

(4.11).

 

 

 

 

Figure 4.4. Schematic diagram illustrating the concept of discrete wavelet

195

decomposition. S denotes the original signal (the 2D image of the velocity ﬁeld
measured by ADCP). L and H denote the low-frequency approximations and
high-frequency details, respectively. Sufﬁxes denote wavelet scale levels

4.3.3. Estimating the Longitudinal Dispersion Coefﬁcient

Two ADCPS manufactured by Teledyne - RD Instruments, Poway, California (1200 or
600 kHz Workhorse Rio Grande) were used for this study. Data were collected by
mounting the ADCP on a trimaran (an OceanScience RiverboatTM with housing for
electronics and radio modems to communicate with a land-based laptop) and towing
the vessel across the river perpendicular to the direction of ﬂow either from a bridge
or behind a small motor boat. During an ADCP ping, the boat will travel a certain
distance along the cross-sectional transect and the corresponding water column is
called an ensemble. The width of this ensemble depends on the ping rate and the boat
velocity (typically 0.2 - 0.5m/s in this work). The ADCP measures three-dimensional
water velocities from vertical segments of the water column and each of the segments
is referred to as a bin. Simultaneously, the ADCP measures the bottom depth of the
river and the boat velocity relative to the river bed. After each transect is completed

we obtain a 2D ﬁeld of 3-dimensional water velocities for a given :r(longitudinal)

location 17(3),.2): (”NWE’UZ) where ”N and ”E are the North and East velocity
components in Earth coordinates and ”Z the vertical velocity component at
location (y, z) where y and 2 denote the transverse and vertical coordinates

respectively. The measurement of 53(y, z) by ADCP is equivalent to densely deploying

196

velocimeters throughout the cross-sectional area. The ADCP data is processed
carefully after collection including smoothing and removal of bad ensembles. General
principles of ADCP operation and post-processing for moving-vessel measurements
are described in Simpson [Simpson, 2001] and Muste et al. [Muste et al., 2004]. Apart
from bad ensembles, some bins in a good ensemble may also occasionally report bad
velocity readings (e.g., when data do not meet the echo intensity, correlation or other
thresholds or when readings from the four beams differ signiﬁcantly). In the ﬁnal
datasets, the bad ensembles and bad bins are replaced by using nearest neighbor
interpolation.

Noise is almost always present in the ADCP data due to the transient turbulent nature
of the ﬂow. In order to reduce the effect of random noise on the dispersion estimates,
we have considered smoothing the vertical velocity proﬁle before evaluating Eq. (4.3)
to produce a mean velocity ﬁeld in the river. Muste [Muste et al., 2004] discussed
several smoothing techniques including ﬁtting a power law or logarithmic proﬁles to

the data in an ensemble:
m
2 : a i (4.12)
27

where z is the vertical coordinate measured from the bottom, 2' is the location in
the boundary layer at which 11 = 0 , k is the von Karman constant, u is the
kinematic viscosity of water and a. , m ,B are ﬁtting parameters. We will assess the
effect of using these two smoothing formulations on the dispersion estimates in a later

section. In addition to proﬁle smoothing, we have considered two alternative

197

approaches for orienting the velocities before evaluating Eq. (4.3). In the ﬁrst
approach, which is used in [Carr and Rehmann, 2007], the water velocities are
projected onto the main direction of river ﬂow (streamwise direction). The main

direction of the ﬂow is calculated as:

_. n (“’07)
V=Z:§:6@ﬁ on)
121 j=tb(i)

where I7 : (VN’VE) is the cross-sectional average velocity vector, ill (2') and
db(i) are, respectively, the top good bin and bottom good bin of the ensemble 2'. The

velocities are then projected as:

Q1.

FF? MM)

 

u:

and the width of the ensemble, Ay , is Similarly projected to the diagonal direction

tol7. In the second approach, 11 is simply projected to the normal direction of the
transect trackji . This can be evaluated similar to the way the ADCP evaluates
discharge:

7"fAt
u:ar=2=“®%) @w)
0 Ag

 

Here ii is the unit vector normal to the transect, l: is the unit vector in the vertical

direction, qis the fractional discharge, 17b is the boat velocity vector, 0 is the
fractional area, At and Ag are the elapsed time and distance for the ensemble.
One potential advantage of the second formulation is that it can be used when the

channel width is changing along the stream. However, its actual performance will be

198

evaluated in the next section. It is clear from equation (4.3) that the success of the
ADCP method depends on accurate approximation of the transverse dispersion
coefﬁcient Dy. There is no general consensus about the estimation of Dy but an
approximate average based on experimental results is given by [F ischer, 1979]:

Dy z C’u*d (4.16)
where C’ is a constant, normally taken as 0.145, d is the depth of the channel and
11* is the shear velocity, which is computed as 11* =W where g is the
acceleration due to gravity, R is theihydraulic radius and S is the channel slope.
The other formula for Dy that was previously used in [Deng et al., 2001] and
[Perucca et al., 2009] is:

Dy = elf/1(3)) (4.17)

 

B
U* = fu*(y)dy (4.18)

where H is the cross-sectionally averaged depth and h(y) is the local depth. After

D1 is computed from either (4.16) or (4.17), Eq. (4.3) can be numerically integrated.

Repeated ADCP transects have been collected at various locations under different

ﬂow conditions in order to assess the variability in the ADCP method.

199

4.4. Results

4.4.4. Evaluation of Channel Features and Potential for Hyporheic Exchange in the

RCR

To estimate the TS parameters, the 5 km study reach was divided into three test
reaches as shown in Figures 4.1 and 4.2. Model parameters were estimated for all
three reaches for different ﬂow rates (tracer test dates). Comparison of observed and
simulated tracer concentrations is shown in Figure 4.5. Surﬁcial sediments in the
study reach consisted of a thin (5 cm or less) layer of sand and gravel underlain by a
heavily consolidated clay layer of depth 0.25 m or more [Uzarski et al., 2004]. Thus
there was limited potential for hyporheic exchange in the study reach although there
were differences between the test reaches. Reach A (between the Hagadom and Farm
Lane Bridges), a relatively straight section of the river, was free of alluvium and
surface storage was the primary mechanism contributing to TS in the reach. This
reach had extensive vegetation growing near the banks and dead trees within the
channel, particularly near the Hagadom Bridge. Reach B (between the Farm Lane and
Kellogg Bridges) was characterized by the presence of a large meander and
undifferentiated sand units (mostly ﬁne sand and silt) near the left bank over much of
its length. Previous studies indicated that meandering contributes to surface storage by
increasing eddies and pools in the reach. In addition, reach B has a weir located near

the Library Bridge. The backwaters of this weir extended upstream and created an

200

impoundment resulting in enhanced surface storage. The sand and silt unit hugs the
left bank in Reach B and extends all the way into reach C (between Kellogg and
Kalamazoo Bridges) and provides the opportunity for enhanced exchange of solutes
between the banks and the main channel compared to reach A [Puzio and Larson,
1982]. Thus reach B had the potential for both surface storage and hyporheic -
exchange. In reach C, the river comes down following a north-south course and runs
into the Mason esker [Leverett and Taylor, 1915], a well-deﬁned ridge of sand and
gravel oriented in the south-north direction. The grain size of the sediments decreases
radially outward from the centerline of the esker. Since the adjacent regions are
mainly loam and other ﬁner material, the gravel acts as a ledge and the river adjusts
its gradient producing a meandering channel. Reach C therefore has wide ﬂoodplains,
exhibits extensive meandering and is marked by the presence of wetlands and swamps
near the edge of the river. Meandering of the channel changed the ﬂoodplain alluvium
and created high-porosity sand and gravel deposits that provide conditions suitable for
hyporheic exchange. Variations in velocities and channel widths between the reaches
can be used to gain insight into how dispersion changes along the river.
Hydrodynamic modeling indicated that although there was a signiﬁcant variability in
the velocities and heads as a function of distance, each subreach can be considered a
fair approximation of a channel representing uniform hydraulic properties.
Examination of the spatial variations in ﬂows and velocities for the ﬁrst three dye

studies showed that there was a signiﬁcant difference in the average velocities in

201

reaches A and B but velocities in reaches B and C had similar values. For example, for
Q = 16.82 m3/s, the mean velocities for the three reaches were 0.44, 0.75 and 0.71
m/S respectively.

4.4.5. Evaluation of TS Model Parameters for RCR

The estimated parameters of the TS model for the RCR are summarized in Table 6.1.
Examination of the estimated parameters showed that, for all the dye studies (with the
exception of l-C in Table 1), the size of the TS zone increased in the downstream
direction. This is in agreement with descriptions of channel features and surﬁcial
geology presented in the previous section. The size of the TS zones also increased
with discharge, from 1.88 m2 for a low ﬂow of 2.49 m3/s to 7.28 m2 for Q = 19.06
m3/s. This is in contrast to the results of [Morrice et al., 1997], who found that the
size of the TS zone decreased with increasing discharge in a ﬁrst-order mountain
stream. In the RCR, stream cross-sectional area (A) increased with discharge, an
observation also made by [Morrice et al., 1997]; however, at higher discharges the
adjacent low-lying areas near the banks were ﬁlled with relatively stagnant water
which provided additional surface storage that was not available at low discharges.
Results from our ADCP surveys (presented in the next section) support this

explanation.

202

Figure 4.5. Comparison of observed and simulated tracer concentrations for four slug

releases conducted during summer 2002. (a) Q=2.49 m3/s. Sampling locations at

x=0.87km (Bogue Street Bridge) (b) Q = 14.41 m3/s, sampling locations are, namely,
Farm Lane Bridge (x = 1.40km). Kellogg Bridge (x = 3.10km) and the Kalamazoo

Bridge (x=5.08km), respectively. (c) Q = 16.82km3/s, ((1) Q: 19.06m3/s

 

80 I I I I I I

3 0 Observed
- = .4 0 -
(a) Q 2 9 m /S Simulated

p- .-

\l
O

 

O\
O

.. - .l

Ur
O

h d

O
I
O
I

(qdd) uonenuaouog
N w A
O O

.—
O
I
I

l l l

O
‘b
t

 

 

T I I I I I r

0 Observed .
Simulated

.1

at
O
I
A
G
v
0
H
H
P
.h
H
B
\u
m

 

Vt
O
1

d

N b)
O O
I f
u”;

(qdd) uouenuaouog
A
O

u—r
O

 

 

 

o

I
‘01—” -
" o

 

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Time after release (hours)

203

(Figure 4.5 contd.)

 

N
kn

N
O

).

._‘
LII

.—
O

(qdd) uonenuaouog

Us

0

l)

I
2
I

I

I I I I I I

(d) Q=16.82 m3/s

  

 

Observed
Simulated -

 

l
1
l
_1

 

\1
0

05
O

m
0
T"

40=

(qdd) uonenuaouog

 

O
l
I

g"
.3

I I I I I

(c) Q=19.06 m3/s

 

1.5 2 2.5 3 3.5 4
Time after release (hours)

204

 

Observed
. Simulated l

 

 

Table 4.1. Parameters in the Transient Storage model estimated for four different ﬂow

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rates
Dye Reach
Study- Length Q A As D a As/A Q/A RMSE T Fmed Fmed200
Reach (km) (m3/S) (m2) (m2) (mZ/S) (/S) (-) (m/Sl (-) (S) (-) (-)
I-A 1.40 16.82 39.93 6.38 0.75 6.26E-04 0.16 2.64 1.59 1598 3.89 0.64
I-B 1.70 16.82 33.73 6.70 0.79 6.58E-04 0.20 2.51 0.36 1521 5.95 0.85
l-C 1.98 16.82 30.48 5.48 0.79 4.75E-04 0.18 3.07 0.45 2106 4.02 0.46
2-A 1.40 14.41 38.69 5.38 0.58 3.54E-04 0.14 2.68 2.00 2824 2.06 0.32
2-B 1.70 14.41 33.05 5.28 1.45 3.24E-04 0.16 2.73 0.98 3084 2.52 0.32
2-C 1.98 14.41 28.94 5.66 1.45 3225-04 0.20 2.54 0.80 3103 3.63 0.41
3-A 1.40 19.06 42.52 6.62 0.69 5.20E-04 0.16 2.88 2.10 1925 3.01 0.48
3-B 1.70 19.06 36.36 7.28 1.19 5.50E-04 0.20 2.62 0.39 1819 5.01 0.69
3-C 1.98 19.06 32.19 7.22 1.19 5095-04 0.22 2.64 0.50 1964 5.82 0.69
4-D 0.865 2.49 15.47 1.88 0.31 1.54E-04 0.12 0.16 4.41 6489 6.11 1.89

 

 

 

 

 

 

 

 

 

 

 

 

 

The rate of exchange a between the main channel and the storage zones increased

with discharge Q. A positive relation between a and Q was also noted by [D 'Angelo et

al., 1993] who attributed it to an increased availability of solute per unit time. We

calculated the TS zone residence times as (t8 = (AS/A)/a) [Harvey et al., 1996].

Residence times ranged from 255 seconds (a = 6.26 x 10-45-1) for Q = 16.82 m3/s to

790 seconds (a = 1.5 x 10-451) for Q = 2.49 m3/s for the RCR. For all dye studies,

residence times increased with distance in the downstream direction and generally

decreased with discharge Q. We attribute the increase in tS with downstream distance

205

 

to the presence of sand and gravel deposits in the last two reaches compared to reach
A. Our estimated a values are in the same range as the values reported by [Salehin et
al., 2003] for the vegetated reach of an agricultural stream in Sweden (a = 6.1 x
10-4s—I). [Harvey and Ryan, 2004] also reported similar range of values (a = 4.7 x
10.4 and 5.6 x 10451) for a heavily vegetated stream in Arizona. The relatively high
tS values determined for reach C are attributed to the alluvium storage and the
sediment characteristics (gravel and coarse sand) in this reach. Results from our
ADCP surveys (and wavelet decomposition) showed that surface storage in this reach
was relatively small indicating that hyporheic exchange was the primary mechanism
that contributed to TS in this reach.

The cumulative effect of TS on downstream transport and reach-scale retention of
water depends on the parameters AS, a and the ﬂow velocity in the main channel. It is
well-known that efforts to interpret TS model parameters often lead to misleading
conclusions about the relative importance of TS processes compared to other
processes [Run/tel, 2002] as existing metrics such as the storage zone residence time
do not describe the overall effect of the TS parameters described above. [Run/rel, 2002]
proposed the use of a new metric (Fmed) which is the fraction (expressed as percent)

of the median reach traveltime that is due to TS.

AS’
x —=—— (4.19)

F = [I — (La/u
ncd A + As

7

 

Stream reaches that substantially inﬂuence the downstream transport of solute mass

due to TS will have higher values of Fmed and vice versa. Since reach lengths (L)

206

vary signiﬁcantly in different studies, a Fmed value obtained using a standard reach
length of 200 m (Fmed 200) was proposed as a metric to facilitate direct comparison
with other streams. F med and F med 200 values for the Red Cedar River indicate that
the importance of TS increases in the downstream direction (4.1). Comparison of the
F med 200 values with estimates for other streams [Runkel, 2002] showed that values
for the RCR were at the lower end of the range and were comparable to those for the
Snake River, an acidic and metalrich mountain stream in Colorado [Bencala et al.,
1990]. Although the RCR is a bigger stream in comparison, the limited potential for
hyporheic exchange at this site (due to the consolidated clay layer) was noted by other
researchers [Uzarski et al., 2004]. The estimated (AS/A) values and exchange
coefﬁcients (a) in several subreaches are comparable for the two streams.

4.4.6. Results from ADCP Surveys and Wavelet Analysis

Figure 4.6 Shows the observed bathymetry, channel cross sections and the mean
velocity ﬁelds at the Hagadom and Farm Lane Bridges for two different ﬂow rates, Q
= 5.49 m3/s (8 November 2003) and Q = 19.89 m3/s (19 March 2006). On both days,
the ﬂow near the Hagadom Bridge was highly nonuniform and became 'relatively
uniform with distance in the downstream direction. In addition, channel cross section
was W shaped at the Hagadom Bridge as opposed to the U-shaped cross sections
(which favor uniform conditions) at the other bridges. As discharge increased, the
river became wider and was marked by the presence of relatively stagnant water near

the banks. The relative extent of the low-velocity or stagnant zones decreased in the

207

downstream direction as the ﬂow became more uniform between Hagadom and Farm
Lane Bridges. Since the tracer data was used to estimate reach-averaged values for
AS/A, ADCP data collected at multiple stations within a reach can be averaged to
compare with the tracer results. If in-channel processes were primarily responsible for
storage within a reach, then we expect AS/A estimates from ADCP and tracer data to
agree. On the other hand, stream reaches dominated by hyporheic exchange are
expected to produce widely different estimates of AS/A from ADCP and tracer data.
The approximations to the original velocity ﬁelds based on two-level decomposition
allowed us to identify TS locations with the channel (4.7). The images marked L1 and
L2 show the ﬁrst term in equation (4.3) for level 1 and level 2 decompositions
respectively. The relative locations of the dead zones within the channel given by the
wavelet decomposition (L2 approximation in Figure 11) agreed with our observations
of relatively stagnant water during our ﬁeld work. The ratios of the areas (AS/A)
calculated based on this decomposition are shown in Table 2 for reach A for two

discharge values (2.0 and 19.8 m3/s).

208

 

 

 

Depth (m)

Ni—‘ONv-‘ONr-‘ONt-‘O.

 

 

 

 

 

 

 

 

 

E
*2.
I.)
Q
' L l I. l 1 4 .J- l l l 1 .l- I I I J A 1 I “I l l J l
5 10 15 20 25
'Wldth (m)

Figure 4.6. Observed mean velocity ﬁelds at two different stations in reach A in the

Red Cedar River obtained using a 1200 kHz ADCP (a and b) Q = 5.49m3/s (8

November 2003). (c and (I) Q = 19.89 m3/s (19 March 2006). Note that during the

high event the adjacent low lying areas near the banks were ﬁlled with relatively
stagnant water which were not available during low discharge

The average values obtained ﬁom the ADCP data were based on four transects and
were found to be in good agreement with results from tracer data. Using more
transects will likely improve the ADCP estimates but data from other transects (e.g.,
at the Bogue Street Bridge) showed that conditions were similar to those at the Farm
Lane Bridge. Since our primary focus was on ascertaining whether the sizes of the TS
zones independently estimated using the ADCP and tracer data were of a comparable

magnitude, we believe that the estimates in Table 4.2 are adequate. In ﬁrst-order

209

tropical headwater streams differing in channel morphology and hydraulic
characteristics, [Gucker and Boechat, 2004] compared the sizes of TS zones for
different stream morphotypes including straight run, meandering, step-pool, and
swamp reaches. They concluded that their (AS/A) estimates were lowest for straight
run reaches and highest for swamp reaches. In our case, reach A, predominantly a run
reach, had the lowest (AS/A) for all the discharge values. Since parameter values
estimated based on the TS model were comparable to those estimated based on the
ADCP data for similar discharge values, we conclude that TS was primarily
controlled by in-stream processes in reach A. We could not obtain (AS/A) estimates
from the ADCP data for reaches B and C as the stream was too shallow to operate our
1200 kHz instrument and obtain good transect data. Since (AS/A) values were highest
in reach C (which was consistent with our observations, e.g., the presence of swamps,
a meandering channel, wide ﬂoodplain), we wanted to test the hypothesis that
hyporheic exchange was important in this reach. If this was indeed true, then we
expect the (AS/A) estimates from ADCP data to be relatively small compared to the
estimates from tracer data. We were successful in obtaining several good transects at
the Kalamazoo Bridge (our last sampling point) on 15 October 2006 (Q = 3.35 m3/s).

Data from one such transect is shown in Figure 4.5.

210

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1' ‘ 1 ', l". J}-
.t_ 1. 4111“
f .
g. hi] I I “Lamb mud-Ann 1W if Ila/1"
Q I 1 .~,l 1m {inf
H“ (d) Ll
Width
I 5 v i ‘ . t I I i
.1: III" i . . . JI 1 I; y
H .1. 1}. Law . ‘. ”lo-b
8 ii] i “L “a u- I“ I“.
Q Vl‘m "I (c) L2
Width
.-=.. IMIII111.1111 1 l
a. . g
5 tilt.“ ‘ i . . viii“ 'I
“1‘ ch MUS III-i]; 0)) L1
Width
.101 , 1.1111
5 II .II ‘ F l
g li'tialﬂi ,. 1| 1.. '1 II ‘III UIIVI uh]. .- L2
AM .11 - b LIWIL‘IIo I» IJ’: (a)
Width

Figure 4.7. Multiresolution wavelet approximations for the images shown in Figure
4.6. After completing the wavelet analysis, the low-frequency content at wavelet
levels 1 and 2 (denoted by L1 and L2) was plotted in the physical space. (a and b)
Hagadom Bridge. (0 and (1) Farm Lane Bridge.

The channel was relatively narrow at this station and showed cross-sectional
uniformity in velocity which was an indication that surface storage may be relatively
unimportant. Average value obtained from wavelet analysis based on three transects at
this site gave (AS/A) = 0.05. Although we do not have estimates from tracer data to

compare with this value, our fourth dye study was conducted under similar low-ﬂow

211

conditions (Q = 2.49 m3/s). We obtained (AS/A) = 0.12 for reach D between the
Hagadom and Bogue Street Bridges in Table 1. All other dye studies showed that
(AS/A) increased in the downstream direction and with discharge. It is therefore
highly probable that the (AS/A) value for reach C on 15 October 2006 was
signiﬁcantly higher compared to the number 0.05 estimated from the ADCP data. An
important parameter in assessing the role of storage zones is the Damko"hler index
(Da) calculated from observed data. The parameter Da reﬂects the relative importance
of downstream processes in relation to TS and is computed as the ratio of the time
needed for the downstream tracer transport for a certain reach length to the mean
tracer residence time in the storage zones [Harvey and Wagner, 2000]. [Schmid, 2004]
analyzed slug release data and concluded that very close or nearly identical results are
obtained by the AD model and TS model if Da < 0.6 or Da > 60.0. In our case, the
calculated Da based on the estimated parameters of the TS model ranged between 1.5
and 7.6 for all cases, which conﬁrmed the preferred use of TS model over the AD

model.

212

Table 4.2. Comparison of the Relative Sizes of the Transient storage zones estimated
using tracer data and independently using the ADCP data for Reach A

 

 

 

 

 

 

 

 

 

 

 

 

Q Tracer Q ADCP As/A As/A Comments
(m3/s) (m3/s)

19.06 19.8 0.16 0.16 Farm Lane-1
19.06 19.8 0.16 0.12 Farm Lane-2
19.06 19.8 0.16 0.19 Hagadom-l
19.06 19.8 0.16 0.21 Hagadom-2
19.06 19.8 0.16 0.17 Average Value*
2.49 2 0.12 0.11 Farm Lane-1
2.49 2 0.12 0.14 Farm Lane-2
2.49 2 0.12 0.11 Hagadom-1
2.49 2 0.12 0.17 Hagadom-2
2.49 2 O. 12 0.12 Hagadom-3
2.49 2 O. 12 0.13 Average Value“

 

 

 

 

 

 

 

By examining the low-frequency contributions (the coarse features) at successive
levels in multilevel wavelet decomposition of ADCP data, we were able to identify
the relatively stagnant regions in the ﬂow ﬁeld. The observed velocity ﬁelds
contained both the mean and the highly oscillatory components of flow. By taking the
average of several transects, we were able to quantify the relative magnitudes of
surface storage and hyporheic exchange in different test reaches using wavelet
analysis. The decomposed states are plotted in the physical space as shown in Figure
11 and image processing was used to estimate the ratio of the two cross-sectional
areas in the images. In equation (4.10), the main channel cross-sectional area A can be
computed from either the level 0 (i.e., the original image) or the level 1 approximation
as shown in Figure 4.7 (they produced identical values for our data sets). The

highfrequency subbands in the wavelet decomposition (e.g., the horizontal, vertical

213

and diagonal details) were not shown as they did not contain information useful for
our present analysis. In a study of transient storage and hyporheic ﬂow along the
Willamette River in Oregon, [Fernald et al., 2001] indicated that they used a
boat-mounted ADCP to measure discharges and main channel cross-sectional areas (A)
at their sampling locations, although they did not show any comparisons between
observed (ADCP) and estimated cross-sectional areas. In the present work, we did not
directly compare the cross-sectional areas measured using the ADCP to the A
estimated from our TS modeling for the following reason. Depending on the mode of
operation and due to the time of delay required to transmit and receive acoustic
signals, ADCP data usually have a “blank distance” close to the transducer in which
velocity measurements are not available. In addition, there are difficulties in making
measurements close to the banks. These limitations become more pronounced in
shallow environments. Recent ADCP models specially designed for shallow
environments may be more suitable for the type of applications described in this paper.
Instead of directly using the cross-sectional areas (A) obtained from the ADCP, we
decided to focus on the ratio of the areas (AS/A) as errors involved in the
approximations of the areas may cancel out when ratios are involved. The assumption
here is that the (AS/A) values estimated from ADCP measurements (and wavelet
analysis) are representative of the entire cross section including the areas that could
not be reached using an ADCP. For the data reported in this paper, we were able to

make measurements close to the banks, therefore this assumption is unlikely to affect

214

our results and conclusions but it is not hard to imagine situations where a signiﬁcant
fraction of storage zones are in shallow areas and remain inaccessible to an ADCP.
The success of studies aimed at understanding functional relationships between
nutrient uptake and storage area depends critically on our ability to separate surface
storage from hyporheic exchange [Runkel et al., 2003; Salehin et al., 2003]. This
paper presents one approach for achieving this separation. We demonstrated that in
one of our test reaches (reach A), the tracer data and the ADCP estimates of (AS/A)
were in good agreement for both high and low discharge values. Additional data sets
and analyses are required to test the strengths and weaknesses of this approach. As
noted by [Shields et al., 2003] and [Dinehart and Burau, 2005], the study of river
reaches using ADCPs is hampered by the lack of custom software for data analysis.
For this study, we created software for extracting ADCP data, smoothing, correction,
visualization, wavelet analysis, image processing etc. The availability of standardized
software will make it easier to study river reaches using ADCPs on a routine basis.
4.4.7. Estimating Longitudinal Dispersion in Rivers

Before computing the dispersion coefficient using Eq. (4.3), we examined the ADCP
datasets to identify potential issues that could lead to a violation of the assumptions
involved in Eq. (4.3). We examined the depth-averaged velocity proﬁles in the
transverse direction to identify recirculating or secondary ﬂow regions and the
number of bad ensembles as a per cent of the total number of ensembles. Datasets

with a large percent of bad ensembles are not used for estimating the dispersion

215

coefﬁcient as explained later. Results from typical ADCP surveys are shown in

 

.0

Figure 2 in which the variable plotted is the mean velocity V = \ﬁtz + v2 + 2112 as a

 

 

function of the depth and width of the channel. Channel characteristics in all stream
reaches are such that the width-to-depth ratio(B / H ) > 10 , therefore equation (4.16)
can be expected to provide a reasonable approximation of the transverse dispersion
coefﬁcient [Deng et al., 2001]; however, unless otherwise stated, we have used
equation (4.17) to compute Dy. The raw velocity data obtained from the ADCPs is
conditioned by orienting (rotating) the velocities and smoothing the vertical proﬁles
to obtain a consistent mean velocity ﬁeld. Figure 3 shows the vertical velocity proﬁles
in two large rivers (Ohio and St. Clair). Power law and logarithmic proﬁles are ﬁtted
to the raw ADCP data. We notice that the logarithmic relation describes the data better
in large rivers such as the St. Clair River (especially closer to the bottom boundary
layer) and that smoothing produces much better conditioned data for further analysis.
For evaluating the dispersion coefﬁcient, however, we ﬁnd that the two proﬁles make
little difference. The reason is that during the evaluation of Eq. (4.3) only the mean
velocity in the water column is used (which is relatively insensitive to the smoothing
technique used). As described earlier, we have examined two approaches for
orienting the velocities (rotating the velocities in the streamwise direction and
projecting them along a direction that is normal to the transect track). These two
methods together with the three proﬁle smoothing methods (power law, log law and

no smoothing) yield a total of six cases. Assuming that the tracer estimate of the

216

dispersion coefﬁcient in a given reach represents a reasonable averaged measure of
dispersion in that reach, the relative “error” in the ADCP estimate for individual
transects within the same river reach is calculated for all six cases for the Grand River
datasets and the results are displayed as box plots in Figure 4.10. Results indicate that
the method of orienting the velocities has a relatively larger inﬂuence on the
dispersion results than the method used for proﬁle smoothing. For the Grand River
datasets used to generate Figure , log law smoothing with velocity projection in the
streamwise direction gave the best (closest to the tracer) results. For the relatively
shallow rivers such as Burns Ditch (not shown in Figure 4.10) the power law

smoothing gave slightly better results.

217

Ohio River VCIOCity (m/s)

  

 
   

l
A 2 as
S
v 4 0.6
“)g 6 .1 i 04
a 8 “L"... ' a 3‘ {.l (6.2

0 50 100 150 200 250 300 350 400
St. ChairRi er

vow-NU.)

Or-‘NW

‘OHNUJ-F

0 10 20 30 40 50 60

n—Nw-ﬁm

 

099999

0‘ 5 10‘ 15 20 25 30 35 40 45
Red Cedar River
7 "r " H r.- . vi
. ‘ J i ,"

“(a a.

 

Distance (m)

Figure 4.8. Sample ADCP transect data used for computing the longitudinal
dispersion coefficients.

Since one of the objectives of this paper is to quantify the uncertainty in the

218

dispersion estimates from the ADCP method by collecting repeated transect data at
the same location (repeating this procedure for several different locations in a given
reach), we plotted all the ADCP results against tracer data in Figure 5(a). The
variability in the ADCP estimates within a given reach is shown using box plots and
colors indicate different rivers. For the Red Cedar River, ADCP and tracer data are
shown for reach A, a 1.4 km reach as described in [Phanikumar et al., 2007]. Tracer

data was collected for ﬂow rates 19.06, 16.82, 14.41 and 2.49 m3/s and tracer values

of dispersion are marked in Figure 4.5 for all the four ﬂow rates. Several ADCP
datasets were collected within the same reach for ﬂow rates 19.98, 3.6 and 4.7 m3/s
and the dispersion values are shown using box plots (The X-axis labels mark the
locations of the box plots). For the Grand River, tracer and ADCP data are shown for
reaches 2 and 3 as described in [Shen et al., 2008]. The combined reach is
approximately 23.8 km long which explains the larger variability in the ADCP
estimates shown in Figure 4.5. Tracer values for Grand River plotted in the ﬁgure
represent average values for reaches 2 and 3. For Ohio and St. Clair Rivers all the
transect data reported in [Holtschlag and Koschik, 2003; Koltun et al., 2006] are

included to generate the box plots.

219

 

14liﬁr11

(a) Ohio River
12 -

Depth(m)
on
I

o ADCP Raw Data
— Power Law 2 *'
— Log Law

 

 

 

l I l I

o l I
O 0.4 0.8 1 .2
Longitudinal Velocity (m/s)

1 U I I I V '

 

25

 

(b) St. Clair River
20L

Depth(m)
8 G
r T

 

 

 

 

O l L l l l 4 l l I
10'1 10°
Longitudinal Velocity (m/s)

 

Figure 4.9. Vertical velocity proﬁles in the Ohio and St. Clair Rivers showing the effect of
ﬁtting a power law (black lines) and logarithmic proﬁles (red lines). The raw data from the
ADCP is shown using symbols

220

Two different tracer estimates of dispersion are plotted in Figure 4.11a — in the ﬁrst
approach transient storage modeling (Eq. 4.2) is used to estimate the dispersion
coefﬁcient. In the second approach, the tracer data is ﬁtted to an analytical solution of
the ADE to obtain the dispersion coefﬁcient as described in the earlier section. There
are additional details associated with the tracer methods that are important to
understand the comparison shown in Figure 4.11. Observed tracer data can be
modeled in two different ways. In the ﬁrst approach (method A), tracer mass is
injected into the stream at a: = Oand parameters in the analytical or numerical
solution are estimated by minimizing the deviation between simulated and observed
concentrations at each of the downstream sampling stations. In this method
parameters (e.g., D) estimated for reach 1 represent average conditions between the
injection site and the ﬁrst sampling location, however, parameters estimated for reach
2 represent conditions for both reaches 1 and 2 and so on. In the second approach
(method B), upstream conditions in the form of concentration versus time data
observed at the ﬁrst sampling location are speciﬁed at the beginning of reach 2,
therefore parameters estimated for reach 2 represent conditions in reach 2 only. For
relatively small rivers and using the TS model Eq. (4.2), our experience indicates that
the two approaches produce almost identical results [Phanikumar et al., 2007].
However, this situation is different while using the ADE for large rivers. For this case,
we found that methods A and B produce widely different results. This is not

surprising since the ADE does not have a separate term for the dead zones. Therefore,

221

the dispersion term in the ADE tends to capture the effects of the dead zones as well.
As a result, if method A is used with the ADE in large rivers, the cumulative effects of
dead zones as travel time increases could produce dispersion estimates that are
unreasonably high compared to the local/point estimates from the ADCP. For the
Grand River, for example, the TS estimates of the dispersion coefﬁcient in the four
reaches are 2.16, 1.6, 4.2 and 1.39 mz/s respectively while the values based on the

ADE using method A are 3.5, 27.72, 56.54 and 112.3 mz/s respectively. Results

obtained using method B in the same reaches (using the same initial mass) are 3.5,
13.05, 15.02, and 4.92 respectively. The ADE estimates shown in Figure 4.11 a are
obtained using method B in which parameters estimated for a reach represent
conditions only in that reach. We notice that the ADCP and tracer estimates are in
good agreement as the ﬂow rate changes over four orders of magnitude. In addition,
the tracer estimate is closer to the median value of the dispersion coefﬁcients obtained
from the ADCP method indicating that it is beneﬁcial to obtain multiple datasets at
the same location. These results establish the ADCP method as a reliable alternative to
the tracer method. From Figure 4.11a it appears that the difference between the TS
and ADE estimates increases with ﬂow indicating that dead zones play an important
role at high ﬂows. This was observed clearly for the Red Cedar River and the Grand
River during our ﬁeld studies (e.g., Figure 10 in [Phanikumar et al., 2007]). At high
ﬂows, the low lying areas near the banks of the river are ﬁlled with stagnant water

that contributed to additional storage. This additional storage is not available during

222

low flow conditions. For low flows, the difference between the two models (TS and
ADE) is not as high for the river sites considered in this paper and the dispersion
coefﬁcient from the TS model is in good agreement with the ADCP estimates. These
comparisons indicate that the ADCP estimates of dispersion include the effects of
dead zones as well. The ADCP measures velocities in both fast and slow moving
regions of the river. There are no guidelines in the literature on what constitutes a
dead zone (e. g., regions where velocities fall below a certain threshold value). In an
earlier paper [Phanikumar et al., 2007] we explored the idea of separating the ﬂow
ﬁeld measured using an ADCP into relatively fast and slow moving regions using
wavelet decomposition. We were successful in estimating the size of surface storage
zones (AS / A)based on ADCP data and estimates compared favorably with results
from a tracer-based method (TS modeling) for both high and low flows. These results
support the fact that the dispersion coefﬁcient estimated by the ADCP method
includes contributions from dead zones.

The median dispersion values from repeated transect data shown in Figure 4.11a are
plotted against the tracer values in Figure 4.11b. We also plotted values reported in the
literature for comparison including data from Carr and Rehmann [Carr and Rehmann,
2007] and Fisher et al. [F ischer, 1979]. The lower end of the tracer values shown in
Figure 4.11b come from either laboratory flume data reported by Fisher et al. [F ischer,
1979] or relatively smaller rivers such as the Burns Ditch or the Red Cedar River

(present work). Results from individual datasets for all rivers are plotted in Figure

223

4.12(a) against some of the well known empirical relations available in the literature
including the relations of Fisher et al. [Fischer, 1979], Seo and Cheong [Sea and
Cheong, 1998] and Deng et al. [Deng et al., 2001]. The ADCP method generally
produces estimates that are comparable to the results from the empirical relations;
however ADCP and tracer values are generally lower. The deviation
(AD / D ADCP)where AD 2 (DADCP — DEmpiI.ical)between the ADCP values and
those from empirical relations is displayed using box plots in Figure 6(b) for all three
empirical relations considered. We ﬁnd that the Fisher et al. [F ischer, 1979] relation
matches closely with our ADCP estimates. A Kruskal-Wallis one-way ANOVA test on
ranks indicated that the three groups had statistically signiﬁcant differences
(p 3 0.001) among their median values. Further analysis using Tukey’s multiple
pairwise comparison procedure indicated that the relation of Fischer et a1. [F ischer,
1979] is responsible for the observed difference and that differences in the results
based on the [Sea and Cheong, 1998] and Deng et al [Deng et al., 2001] relations is

not statistically signiﬁcant.

224

1 2 3 4 5 6
100° I I I I I I

 

Grand River

1” r
100 H
10 i +

— Log-law (normal to transect)

M Log-law (streamwise direction)
0,1 — Power-law (normal to transect)
I: Power-law (streamwise direction)
- No smoothing (normal to transect)

:11: No smoothing (streamwise direction)

Figure 4.10. Effects of smoothing (logarithmic, power-law or no-smoothing) and
velocity projection methods (rotating velocities in the streamwise direction or
projecting them in a direction normal to the transect track) on the dispersion estimates
from ADCP

IIIIILLIL LllLLLLl]

 

 

% Error Relative to Tracer Estimate

—-I
lllllLLL|_l llllLLIJ

 

 

 

 

Examination of our ADCP data indicated that there is a correlation between the
quality of the dispersion estimates and the per cent bad ensembles in the transect data.
Estimated dispersion numbers were found to be unrealistically high or low when the
per cent bad ensembles exceeded about 12%. Bad ensembles can occur when
communication is interrupted, when aquatic vegetation or large debris enters the ﬁeld
of the transducer beams or when a change in the ADCP operating conditions is
warranted. A large number of bad ensembles could potentially inﬂuence the discharge

measurement which can be a problem in itself. Figure 4.13 shows typical

225

depth-averaged velocity proﬁles (Figures 4.13 a, b and c) and the computed discharge
and the dispersion coefﬁcient as a function of per cent had ensembles within the same
reach for Grand River. Symbols show the raw data and trend lines based on LOESS
smoothing are also plotted (no attempt was made to ﬁt the proﬁle to satisfy the no-slip
condition at the two banks). Examination of the velocity proﬁles can help isolate
datasets that could potentially lead to a violation of the assumptions involved in Eq.

(4.3).

226

Discharge, Q (ms/S)

 

 

 

10° 10I 102 103

: ﬁrnrnnl I I lllllll I I 111111] I ll:

103 e (a) L:
g 102 a” E
E — _
2101 er ‘9
_g : :
U ._ _
E 100 —E—' E i - Red Cedar River ’2
U E 0 12:1 BurnsDitch E
g E X [III Grand River I
'5 10-1 ? - OhioRlver q:
51, E I m SLClaIrRlver E
5 I I: Muskegoanver I
10-2 5— G Thomapple River —g

E X TracerEstimateﬂ'S) E

: l I I l l l E TmaBﬁTatemDBl :

 

._- «Sniff 3: we

 

 

 

Discharge, Q (ms/S)
104 E I llllllll lllllllll I |||l|lll lllllllll lllllllll lllll
— d”
(b) x9
103 g? V
I 0
° 102 =— ﬂ
.- E . C
E E o o
7’ 1 r .
I$10 E 0
é : 8
— 0
10° 5—
E 0 Fisher at al. (1979)
10.] =_ e Carr and Rehmann (2007)
E O PresentWork
_ 1 1111111] l llllllll l llllllll 1 1111111] lllllllll lLlJll,
10'1 100 101 102 103 104
TracerEstimate

Figure 4.11. Comparisons between ADCP and tracer estimates of the dispersion
coefﬁcient: (a) Box plots denote the variability in D estimated using the ADCP
method within a given river reach. Tracer estimates based on the ADE and the

transient storage modeling are shown using different symbols. (b) Comparisons

between ADCP and tracer estimates plotted on top of similar results reported in the
literature.

227

 

      

 

 

 

 

 

   

10 1 1111111] l 1111111[ 1 1111111 1 1111111[ llllfl_
: (a) E
3
a 10 E .
NE : 2.1'
v 2 r .. —=
:5; 10 E 3 ' i E
.g : + ‘ Z
“9: 10‘ E 0 8 E
o : :
.5 10" E" E
g : o ADCP Estimate :
g 10-1 g- + Fischer (1975) _E‘
D E o Deng et al. (2001) E
10-2 g— A Seo&Cheong (1998) ——5
§ V Tracer Estimate (ADE) E
_3l_ 11111111'1 1 1111111 1 1111111 1 1111111“4
10 . 10 10 10
Discharge, Q (ma/s)
3 =
10 E (b) 1 1 E
103 E s EE
1 E— rm E
[ AD ] 1° E LEI E
pm. 100 E— i E
10'1 E E _E
E 8 E
10-2 E— — Fischer (1975) 1:
E Seo & Cheong(1998)§
_3 - Deng etal. (2001) _
10 E’ 1 I l E

 

 

 

1 2 3 4

Figure 4.12. Comparison of ADCP estimates of the dispersion coeﬁicient with tracer
estimates and results from empirical relations. (b) Box plots showing the deviation
between estimates of D based on the ADCP method and empirical relations

228

 

0:65.29, D=19.14, Bad Ensembles = 3.7% Q=65.70, D=23.92, Bad Ensembles = 3.6 96
0.8 l 1 I 1 l

l ' l I l

. l

(a) (b) '

  
    

 

 

 

 

0.6
0.4
O
0.2
04
LOESS—smoothed proﬁle
.0 7 1 1 J 9.
0 20 40 60 0 1 O 20 30 40 50

0:67.73, D=15.82, Bad Ensembles = 7.8% 0:58.32, D=213.98, Bad Ensembles = 25.5 96
l l l l
1 _ (c) 1 2 (d) . _

 

 

Depth-Averaged Longitudinal Velocity (m/s)

 

 

 

. o u . ‘1
o
'00... . IO. . —l
. a o

0. .. o ° '0 -

I ' o 9

0 o ' o

. o. . o '-

. ‘. 1 . s

 

 

20 30 _ 4o 60
Distance, y (m) Distance, y (m)

Figure 4.13. Depth-averaged longitudinal (u) velocity proﬁles plotted as a function of
the transverse coordinate y .

4.5. Conclusions

As improved understanding of stream solute transport processes leads to better
models and new approaches [Boano et al., 2007; Deng and Jung, 2009], there is an
imperative need for independent, ﬁeld-based estimates of the dispersion coefﬁcient to
constrain models. The ADCP method of estimating the dispersion coeﬁ'lcient appears
to be an excellent alternative to the tracer approach if care is taken to identify spurious
data and repeated transects are used to estimate 5(or another appropriate measure

that represents average conditions within the stream reach)- Our r CSUItS indicate that a

229

measure of D based on repeated transects is more reliable than individual estimates.
For the river reaches in our work, the median value of the dispersion coefﬁcients
obtained from multiple datasets is found to be closer to the tracer estimate based on
the ADE (using method B as described earlier). Our comparisons indicate that the
ADCP method measures the inﬂuence of dead zones on the dispersion coefﬁcient as
does the estimate from the ADE. The ADCP method has its share of limitations
including the inability to make measurements close to the banks and in shallow
stream reaches. In addition, the method is not suitable for stream reaches dominated
by meander bends, recirculating regions or secondary ﬂows. Recent ADCP models
(e.g., the Sontek 55 or M9) using smaller sensor heads, small (~ 5 cm) blanking
distances and bin sizes and multiple acoustic frequencies (with features such as
frequency hopping) have the potential to further improve our ability to estimate
dispersion in streams and rivers. Additional datasets (including simultaneous tracer
and ADCP data) and analyses are needed to further assess the relative strengths of this
approach, especially for large rivers. The new ADCP models are also expected to
improve estimates of surface storage zones by reducing the blanking distance.

In conclusion, we demonstrated how useful insights into stream solute transport
processes can be obtained using high-resolution, three-dimensional hydrodynamic
data obtained from ADCPs. Coupled with traditional tracer studies, these approaches
have the potential to constrain TS modeling by providing independent estimates of the

dispersion coefficient as well as surface storage in different reaches. The methods

230

described in this chapter can be extended to obtain other types of useful information
(e.g., residence times in different reaches, exchange rates with surface storage zones,

lateral and vertical dispersion coefﬁcients to name a few).

231

Appendix A . Supplemental Tables

Table A. 1. List of input data to the watershed model and format

 

 

 

 

 

 

 

 

 

Data Common Data Source F ormat/Comments
DEM Either DEM or NED ASCII grid
from USGS
LULC map NLCD from US EPA ASCII integer grid
or local agencies.
Remotely sensed data
LULC Customized A variable called lulc saved in .mat
mapping table format that contains two ﬁelds: M for the
transformation matrix as listed in Table
A2; G for the group information for each
model RPT
Watershed Local agencies ESRI shapeﬁle: Polygon
Extent delineated watershed
boundaries
River NHD from USGS ESRI shapeﬁle: Polyline
Hydrography Need to be processed. A river needs to be
Dataset coded for its river ID. Currently, scattered
observations of river width and type
Soils Type SSURGO mapunit ASCII integer gn'd
map
SSURGO soils NRCS SSURGO A .mat ﬁle containing processed database
database database information
Weather NCDC and other ESRI shapeﬁle: Point
Station climatic data network
Locations

 

Climatic Data

NCDC and other
climatic data network

NCDC downloaded format or MAWN
downloaded format

 

 

 

 

 

 

Groundwater Local agencies or ASCII grid

Hydraulic inferred from

Conductivity geological information

Groundwater Local agencies or ASCII grid

Aquifer Layer inferred from The thickness (or elevation) of each layer
information geological information

Initial Local agencies or ASCII grid

Groundwater inferred from

Head geological information

 

232

 

Table A2. An example transformation matrix from MDNR dataset to model classes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LU/ Database Water Imperv Pine Oak Shrub Grass Corn Bare Alfalf
LC deﬁnition -ious -a
Low
1 , 0.4 0.2 0.4
Intensuy
Urban
High
2 _ 0.8 0.1 0.1
Intensrty
Urban
Airport
3 1
Road/
4 . 1
Parking Lot
Non-vegetat
5 0.2 0.8
ed Farmland
Row Crops
6 l
Forage
7 1
Crops
Orchards
9 l
Herbaceous
10 l
Openland
(N/A)
11
U land
12 p
Shrub
Parks/Golf
13 0.3 0.7
Courses
Northern
l4 1
Hardwood
Association
Oak
15 . . l
Assoc1at10n
As en
16 p , , 1
Assoc1at10n
Other
17 1
Upland
Deciduous

 

 

 

 

 

 

 

 

 

 

 

233

 

 

Table A2. (cont’d)

 

18

Mixed
Upland

Deciduous

 

19

Pines

 

20

Other
Upland

Conifers

 

21

Mixed
Upland

Conifers

 

22

Upland
Mixed

Forest

0.5

0.5

 

23

Water

 

24

Lowland
Deciduous

Forest

 

25

Lowland
Coniferous

Forest

 

26

Lowland
Mixed

Forest

0.5

0.5

 

27

Floating
Aquatic

 

28

Lowland
Shrub

 

29

Emergent
Wetland

 

30

Mixed
Non-Forest
Wetland

 

31

Sand / Soil

 

 

32

 

Exposed
Rock

 

 

 

 

 

234

 

 

 

 

 

 

 

Table A2. (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

Mud Flats

33 0.2 0.8
Other Bare /

34 0.2 0.8
Sparsely
Vegetated

 

235

 

 

Table A3. Summary of several watershed-scale hydrologic models

 

 

 

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237

Appendix B. User’s Manual for the model

Graphical User Interface

B.1. Creating and Running the model in interactive mode

The proposed model is implemented in a mixed environment of Matlab and Fortran.
Computationally intensive subroutines (mainly PDE solvers) are written in Fortran
and linked to the Matlab main program via the Matlab mex interface. The main
program is written in Matlab due to its efficiency in development and versatility in
data handling. It is ‘open-ended’, meaning data is all conveniently available for
inspection during run time. This is a big advantage for researchers interested in
advancing the model further. The data interfacing capability of Matlab and efﬁciency
is also important because it allows a researcher to spend less time writing auxiliary
subroutines and can instead focus on the scientiﬁc part of modeling. The development
of this model, along with the complete software package is impossible to be done by

the author himself had the model not been written in this fashion.

The model is packaged in a folder that can be run under mainstream computational
environments including Windows (32 and 64 bit), Linux (32 and 64) and Macintosh.
The model have been compiled, run and tested on all of these platforms. Ensuring the
compatibility and portability on all these platforms is not a trivial task. Such effort is

spent mainly because the calibrated results shown in this dissertation is done on the

238

Linux environment at the High Performance Computing Center (HPCC) at MSU.

A Graphical User Interface has been developed to assist interfacing with data and
setting up the model. The GUI has 7 core capabilities, including, loading data input,
specifying grid parameters, discretizing data into model, specifying runtime
parameters (Component solvers, Model Start Time, Model End Time, etc), running
the model, saving/loading model and displaying the results. The GUI is mainly used

during the setting-up stage. The model can be run with or without the GUI.

B. l .1. Installing and starting the model

The model can be run in the interactive mode (in Matlab) or the compiled mode (after

compilation by the mcc compiler). Running the model in the interactive mode is the

same as running any other Matlab programs. To create a model in the interactive
mode with the GUI:

1. Start up Matlab and browse to the root directly of the model (SMROOT).

2. When Matlab path is under SMROOT, enter ‘gpath’. This command adds the
relevant directories into the Matlab paths and also set the values of some
environmental variables (Env in matlab workspace)

3. Enter the command ‘mygui’. This command brings up the model main GUI
session. Fig B] shows the GUI window after it is started. (Depending on the

Operating System and the Matlab version, the look of the windows may be

239

trivially diﬂ‘erent.)

 

 

Figure B.1. GUI after the model is started

B.l.2. Loading the data

To load the raw data for discretization, click on the ‘Data’ button on the main GUI.
This brings up the data GUI (Figure B.2a). This window allows the user to specify the
input ﬁles. The data items listed in this window is summarized in Appendix A. For
each of these items, click on the item’s checkbox and then use the ﬁle browser to load
the ﬁle(s). For the ‘Weather Data File Folder’ and ‘Ground Water Files Folder’, a
directory that contains relevant ﬁles should be loaded The ‘Soils Map Files’ can
accept multiple ASCII raster data ﬁles because normally SSURGO data is organized

in county we may span several counties in our study domain. All other input boxes

240

expect one input ﬁle. After a ﬁle is loaded, its path is shown in the Edit box below the

checkbox (Figure B.2b).

241

was

 

Figure 3.2. Data GUI. (a) before loading data (b) after loading data ﬁles

Another way to quickly load the data ﬁles and avoiding much of the human actions is

242

to create a GUI list ﬁle. Using this GUI list ﬁle is the same as manually setting all the
ﬁelds. A GUI list ﬁle used for the GUIs in this model always has the format:
Field_id:input

where F ield_id is the identiﬁcation flag of the input ﬁeld. An example GUI list ﬁle

looks like:

 

wtrshd_ﬁle:C:\Work\PRISM\data\Shapeﬁles\Wtrshd_RCR_Union.shp
dem_ﬁle:C:\Work\PRISM\data\dem_lulc\ned_rcr.txt
ned_ﬁle:C:\Work\PRISM\data\dem_lulc\ned_rcr.txt
riv_ﬁle:C:\Work\PRISM\data\Shapeﬁles\RCRModelRivers.shp
lulc_ﬁle:C:\Work\PRISM\data\dem_lulc\lulc_rcr_bigger.txt
lulcTB_ﬁle:C:\Work\PRlSM\data\dem_lulc\lulc_mat_rcr. mat
soilsMap_ﬁle:C:\Work\PRISM\data\soils\ingham.txt;C:\Work\PRISM\data\soils\li
vingston.txt;

wea_ﬁle:C:\Work\PRlSM\data\Shapeﬁles\Stations_RCR.shp

 

 

 

To load this ﬁle, Click on the ‘Load Input’ button on the data GUI window, and then
select the GUI list ﬁle. If the GUI list ﬁle is successfully loaded, the edit boxes on the
data GUI will be ﬁlled with the correct records.

Click the ‘Apply’ button on the data GUI to close the window._

B.1.3. Setting up the grid

243

 

Figure 8.3. GUI after the model is started

Press the ‘Grid’ button on the main GUI will open up the grid GUI (Figure B3). This
window allows the user to specify discretization information. In the box ‘nx’ and ‘ny’,
user needs to input the number of cells in x and y direction. In ‘dx’ and ‘dy’, spatial
step size in meter should be input. The boxes origin_x and origin _y stand for the
location of the lower left corner of the grid. Origin_x and origin _y information is
automatically loaded for a given DEM raster grid. Aﬂer ﬁlling in these ﬁelds, the user
needs to click one of the options listed on the right to indicate what method should be
used to aggregate DEM data in a grid cell. Because DEM maps usually have ﬁner
resolution than the computational grid, there can be many DEM point values inside
each grid cell. ‘Area average’ means the elevation data inside one grid cell is averaged
to obtain the grid cell elevation. ‘Linear’, ‘Nearest Neighbor’, ‘Spline’ and ‘Inverse

Distance Weighting are different methods of interpolation. These options will take an

244

interpolated point value as the elevation for the grid cell. The ‘Area Average’ method
is best supported and tested right now and the user for the moment should generally
choose this option.

Similar to the data GUI, the grid GUI can also be ﬁlled by loading a GUI list ﬁle to

save the manual input time:

 

row:50

col:67

dx3900

dy:900
origin_x:3978025.6696

origin_y: 103832.3393

 

 

 

This GUI list ﬁle can be written by ﬁrst ﬁlling the information in the GUI and the
using the ‘Save’ button. Then it can be loaded by using the ‘Load’ button. After the
blanks are properly ﬁlled, hit the ‘Apply’ button to ﬁnish the grid set-up process. Note
that the ‘Apply’ button is available only after one of the ‘interpolation method’

options on the right is selected.

B. l .4. Discretization
When grid information step is done, the ‘Discretize’ button on the main GUI is
enabled (Figure 8.5). Here we can discretize one, several, or all of the components of

the watershed model. In this window the user should select the components that need

245

to be discretized. If a new watershed model is being created, all the boxes should be
checked. There are also a few edit boxes that needed to be ﬁlled out. The ‘DX array’
next to the ‘River’ check box states the spatial step-size for the discretization of the
rivers. The program will evaluate the content of the box and to get an array (in
Matlab), whose i-th element correspond to the i-th river dx. The nRPT next the
‘LULC’ checkbox is the number of RPT that are going to be modeled in the domain
(see section *** for the meaning of nRPT). Other edit boxes should be left untouched
at this moment.

Once the user clicks the ‘Discretize!’ button, the program will take a couple of
minutes to go through the discretization steps for all the components. New data
storages will be allocated in memory. Data will be discretized onto the computational
grid previously speciﬁed. Any data in the memory will be cleaned if the box of that

component is discretized.

r‘ " " dstaui

 

 

input either number at variable name

'V Watershed

'9‘ Weather

[7‘ DEM

v‘ HWB.’ ox array '(L‘OCLgerc' River Spec FIN-0318. ‘
59‘ LULC nRPT_ I“ 3' Thresnom Threshold
V Sou:- nwLayer I 20 I

«37‘ GW traKLayer‘ {Hi 77

: Discretize.l .

Figure B.4. Discretization GUI

246

B. 1 .5. Setting up solution schemes and time steps

The model is written in such a ﬂexible way that different solution schemes to ﬂow
domains can replace the default value with great ease. After clicking on the ‘Solvers’
button the main the solvers GUI is bought up (Figure B5). The solvers GUI contains
two sub-buttons and the ‘DT-speciﬁcation’ section where the user speciﬁes the
temporal time step that should be used for each component. First we should use the
‘Solver Functions’ sub-button to open the solver functions GUI. In this new window
(Fig B.6a), the left column ‘Field’ shows the components. And the user is expected to
ﬁll in the middle column under ‘Value’ to indicate which solvers they want to choose
for each component. These are expected to be Matlab function handles which accepts
input in certain format. For the results published in this work, the settings can be
directly loaded without any typing. On the new window, click on ‘F ileéOpen’ and
select the ﬁle ‘SMROOT/data/solversmat’. After clicking ‘Yes’ on the conﬁrmation
page (Figure B.6b), we see that some of the ﬁelds have been ﬁlled. (Figure B.6c). The
content in the ﬁlled ﬁelds are the Matlab function handles which correspond to a
matlab .m ﬁle in the model package. Each of these ﬁles can be opened by typing ‘edit
(FILENAME)’ under Matlab command prompt. For example, type the command ‘edit
GW_sol’ in Matlab command prompt will open the ﬁle ‘GW¥sol.m’. Some ﬁelds are
still ‘void’ because their solvers may have been combined in other functions. Hit ‘OK’

to close this window.

247

The next sub-button ‘Exchange Functions’ is the place to enter functions that calculate
interactions among domains. For the current model structure, the user only need to
input ‘@F_oc_dw’ for the ‘OC’ ﬁeld.

Then we are ready to specify the time steps. The unit of the input should be ‘day’. A
quick way to load the current setting is to click the ‘Load’ button on the solvers GUI
and load the ﬁle ‘$MROOT/data/dt.txt’. This will automatically ﬁll in the relevant

ﬁelds. (Fig B.5b). Hit ‘Apply’ to save the settings.

 

(a) (b)

Figure 8.5. Solvers GUI. (a) aﬁer it is opened (b) after dt ﬁle is loaded

248

 

 

 

  

7%“ zest

”eat/ram ‘ we- ___5: j

 

' ‘Eé’unace

 

 

 

 

 

   

Are you stire to load? You WIN notmodify'Cun'ent Data i‘you don’t hit
Appty or Save in the new Wndow

 

 

 

 

 

 

(b)

Figure 8.6. Loading solvers data into solvers functions GUI
B.l.6. Final model set up

249

Before model can be run, a few steps remain to be done. In the ‘Run Model’ section
of the main GUI, the user is expected to set the model start time and model end time
in ‘YYYY-MM-DD HH:MM:SS’ format. Clicking on the ‘Final Set Up’ button will
enter the settings into the model. It will also execute a function called ‘modelSetUp’
the do some preparation steps for the model to be ready to run. It is recommended that
the user save the model at this moment so the model can be run directly from here

without having to re-discretize the model.

B. l .7. Saving and Loading model

At any stage of model preparation or model running, the model can be saved and can
be later loaded to resume the previous operations by using the ‘Save’ and ‘Load’
buttons on the main GUI. This is a big advantage of building up the model in the
Matlab environment as it would not be a trivial task to write such a utility in a
completely Fortran-based program. The ‘Save’ and ‘Load’ ﬁinctions have proven to
be hugely convenient during the model development stage.

The ‘Clear All’ button can erase any model data from the current workspace and start
a new main GUI. It is recommended to use ‘Clear All’ function before loading or

creating a new model to avoid any potential memory issue.

B.l.8. Running the model

After the ﬁnal model setup is done, the model is ready to be run and the ‘Run Model’

250

button is now enabled. Clicking on this button will start to run the model from the
time speciﬁed in the ‘ModelStart’ box. The simulation can be paused by clicking on
the ‘Pause! ’ button and resumed by clicking on the ‘Run Model’ button again.

When the model is running, the Matlab command window will be displaying some
time usage information (in seconds) after each day of simulation. The items are,
sequentially:

Year, Julian Day, T-vadoze zone, T-overland ﬂow, T-river flow, T-groundwater,
T-total.

This will tell the user that the simulation is proceeding properly.

An example run-time information is given below:

 

>> run('nlST.mat',0,'example_Run')

Driver control ﬁle:

mat initalization :nlST.mat

Model Project Name: example_Run

Start Running Model at wallclock: 23-Nov-2009 16:48:02
File Output to: example_Run.txt & example_Run_result.txt
Model Start Time: 01-Sep-2001

Model End Time: 3 l-Dec-2005

2001 244 1.1549 1.6202 1.8675
0.35064 4.9932

2001 245 0.83704 1.0274 0.89959
0.11235 2.8764

2001 246 0.8243 0.98791 0.86801
0.11384 2.7941

2001 247 0.76272 0.95858 0.86146
0.10863 2.6914

 

 

 

251

B.2. Running the model in non-interactive mode

For model calibration or long term simulation, the model can be run in non-interactive
mode (without the GUI). The user can either run the model in Matlab with a
command or run a compiled version of the model without invoking Matlab at all.

To run the model in non-interactive mode, the .mat ﬁle saved after step B.1.6 must be
accessible. At Matlab prompt, the model can be run using the following command:
Run(matﬁle, par, prj)

The marﬁle is the ﬁlename of the .mat that isj'saved after step 8.1.6. It should be a
string variable. par is a Matlab structure‘array "for parameter adjustment information.
If no parameter is to be changed, put number,0 at this argument location. prj is the
project name for the simulation (a string). This. name will be used to write output.

The model has also been compiled on into staiidalone executables that can be run on
Windows/Linux/Macintosh platforms. However the Linux executable may not run on
a random Linux environment due to the numerously different Linux systems and
machine architectures. Normally a Linux program needs to be compiled from source.
Assuming the compiled version can work properly. The command can be run at the
command prompt of the operating system:

PRISM driverﬁle

driverﬁle is the ﬁlename of a text input driver ﬁle. This driver ﬁle contains control
information for the model simulation. It has ﬁve lines, with explanations in the

parenthesis:

252

 

 

project_name (Simulation name that is used to save output)

matﬁle (Mat ﬁle that contains the data)

ModelStartﬁ'me (YYYYMMDD, or 0 if ModelStartTime saved in the mat ﬁle is used)
ModelEndTime (YYYYMMDD, or 0 if ModelEndTime saved in the mat ﬁle is used)

Parameter__changer_/ile (a parameter changer control ﬁle, leave blank if none)

 

253

 

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