GEOMORPHIC MODELING AND ROUTING IMPROVEMENTS by Darius James

GEOMORPHIC MODELING  AND ROUTING IMPROVEMENTS by Darius James
GEOMORPHIC MODELING AND ROUTING IMPROVEMENTS
FOR GIS-BASED WATERSHED ASSESSMENT IN ARID REGIONS
by
Darius James Semmens
A Dissertation Submitted to the Faculty of the
SCHOOL OF RENEWABLE NATURAL RESOURCES
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
WITH A MAJOR IN WATERSHED MANAGEMENT
In the Graduate College
THE UNIVERSITY OF ARIZONA
2004
2
The University Of Arizona g
Graduate College
As members of the Final Examination Committee, we certify that we have read the
dissertation prepared by Darius James
Semmens
Geomorphic Modeling and Routing Improvements for GIS-Based Watershed
entitled
Assessment in Arid
Regions
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of
Doctor of Philosophy
Date
Waite R. Osterkamp
[
' (71
4"
Date
Richard H. Hawkins
151 01.1Date
icente Lopes
e6
3/ 510 f
-
Dat
D. Philip Guertin
3- cot
Date
Dat
David9> Qrodrich
Final approval and acceptance of this dissertation is contingent upon the
candidate's submission of the final copy of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
WELat1aCzei2
Dissertation Director
Wai e R. Osterkamp
Date
2 6: t 9
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an
advanced degree at The University of Arizona and is deposited in the University Library
to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgment of source is made. Requests for permission for
extended quotation from or reproduction of this manuscript in whole or in part may be
granted by the head of the major department or the Dean of the Graduate College when in
his or her judgment the proposed use of the material is in the interests of scholarship. In
all other instances, however, permission must be obtained from the author.
SIGNED:
APPROVAL BY DISSERTATION DIRECTOR
This thesis has been approved on the date shown below:
-7)40,te,4
Waite R. Osterkamp
Adjunct Professor of Watershed Management
z
2 e20
Date
4
ACKNOWLEDGEMENTS
This dissertation was inspired by my work at the USDA-ARS Southwest
Watershed Research Center (SWRC), and drew substantially on my training in
geomorphology at Northern Arizona University (NAU) and in watershed hydrology and
GIS at the University of Arizona (U of A). A great number of people were instrumental
to my efforts, and I am indebted to all of them.
At the SWRC I benefited greatly from the experience and guidance of my
supervisor, David Goodrich. Scott Miller was an invaluable source of knowledge on
many fronts, and our constant exchange of ideas was invaluable during the formative
stages of this research. Integrating the geomorphic and routing models into KINEROS2
could not have been accomplished without the generous assistance of Carl Unkrich, a
master problem solver. Ryan Miller, Mariano Hernandez, Lainie Levick, Soren Scott,
Averill Cate, Shea Burns, Ginger Paige, and Mary White all contributed significantly to a
very enjoyable and productive tenure at the ARS.
I came to the University of Arizona largely because of the encouragement of my
advisor, Waite Osterkamp. He was instrumental in getting me to Tucson, and providing
me with the funding to get me started. Although my research ultimately diverged from
our original plan, Waite was unwavering in his support, and his thoughtful comments on
this dissertation have improved it substantially.
Phil Guertin played a central role in this research. To him I am indebted for my
initial training in spatial analysis and watershed management, my position at the ARS,
helping shape my ideas about the geomorphic assessment tool, and his guidance during
its development and testing. Also at the U of A, Richard Hawkins, Craig Wissler, and
Vicente Lopes were key instructors and patient committee members.
Funding for this research came from the U.S. Environmental Protection Agency
Landscape Ecology Branch for the Automated Geospatial Watershed Assessment
(AGWA) project. That project was overseen by William Kepner, who has been a mentor
and a friend over the course of my involvement with it.
This research would not have been possible were it not for the love, support, and
patience my wife, Betsy. She is the center of my life before, during, and after work. I
cannot thank her enough.
5
DEDICATION
To Betsy and Kira.
You are the smile at the beginning and end of my everyday.
6
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS 8
LIST OF TABLES 13
ABSTRACT 14
1.0 INTRODUCTION 16
1.1 PROBLEM STATEMENT 1.1.1 Intermediate-Scale Watershed Runoff Models 1.1.2 Watershed Geomorphic Models 16
17
19
1.2 HYPOTHESES 1.3 APPROACH 21
21
1.3.1 Runoff Routing 1.3.2 Geomorphic Modeling 1.4 STUDY AREA 2.0 ROUTING MODEL 22
22
24
27
2.1 INTRODUCTION 2.2 SCALE-RELATED PROBLEMS IN WATERSHED MODELING 2.3 CHANNELIZED FLOW ROUTING IN WATERSHED MODELS - PREVIOUS WORK 2.3. 1 Evaluating Which Model is Applicable 2.3.2 Diffusion wave modeling with the Muskingum-Cunge scheme 2.3.3 Practical Considerations 27
28
30
34
45
50
2.4 ROUTING IN AGWA 2.5 MODEL DESCRIPTION 2.6 MVPMC4 INSTABILITY AT Low FLOW 51
52
3.0 ROUTING MODEL RESULTS 3.1 METHODOLOGY 3.2 DESIGN STORM RESULTS 3.2. 1 Komolgorov-Smirnov Comparison 3.2.2 Metric and Visual Comparisons 3.3 OBSERVED STORM RESULTS 3.4 DISCUSSION 4.0 GEOMORPHIC MODEL 4.1 CHANNEL-MORPHOLOGIC MODELS 4.2 CHANNEL NETWORK EVOLUTION (SURFICIAL PROCESS) MODELS 4.3 WATERSHED GEOMORPHIC MODELS 4.3 APPROACH 4.5 NUMERICAL MODEL 4.5.1 Channel adjustments — deposition 4.5.2 Channel adjustments — erosion 4.5.3 Erosion and Sediment Transport in K2G 4.6 GEOMORPHIC MODELING TOOL 4.6.1 Profile Smoothing 4.6.2 Results of Profile Smoothing 54
59
59
60
61
62
70
78
81
81
85
88
93
94
101
106
110
117
118
127
7
4.6.3 Mass Balance Calculations 4.7 PROBLEMS 4.8 LIMITATIONS 5.0 GEOMORPHIC MODEL RESULTS 129
131
134
137
5.1 METHODOLOGY 5.2 RESULTS 137
145
5.2.1 Graphical Output 5.2.2 Mass-Balance Calculations 5.2.3 A Perspective on Model Output 5.3 DISCUSSION 145
158
164
167
6.0 RELATIVE GEOMORPHIC CHANGE AND CHANNEL STABILITY 169
6.1 MODEL ERROR AND RELATIVE CHANGE ASSESSMENT 6.2 METHODOLOGY 6.3 RESULTS 169
171
172
6.3.1 Relative Differences for the Hydraulic-Geometry Channels 6.3.2 Relative Differences for the Observed Channel Geometries 6.3.3 Relative Differences in Runoff Transmission Loss, and Sediment Yield 6.4 DISCUSSION 172
174
176
179
7.0 SUMMARY AND CONCLUSIONS 7.1 WATERSHED RUNOFF ROUTING AND SCALE 7.1.1 Conclusions 7.1.2 Future Research 7.2 GEOMORPHIC MODELING AT THE WATERSHED SCALE 7.2.1 Conclusions 7.2.2 Future Research APPENDIX A: DESIGN AND OBSERVED EVENT HYDROGRAPHS A.1 DESIGN STORM HYDROGRAPHS A.2 OBSERVED EVENT HYDROGRAPHS APPENDIX B: DERIVATION OF CHANNEL GEOMETRY ADJUSTMENTS B.1 DEPOSITIONAL DEPTH CHANGES B.2 EROSIONAL DEPTH CHANGES B.3 BANK FAILURE RESULTING FROM EROSIONAL OVERSTEEPENING B.4 WIDTH CHANGES REFERENCES 181
181
182
182
183
184
187
190
190
190
204
204
205
207
208
209
8
LIST OF ILLUSTRATIONS
Figure 1.1. Map of the Walnut Gulch Experimental Watershed showing cultural features.
25
Figure 1.2. Map showing Walnut Gulch subwatersheds. Subwatersheds used in this
26
research were WG1, WG6, WG11, and LH104
characteristic
space-time
a
range
of
Graph
showing
hydrological
processes
at
Figure 2.1.
scales, and the superimposed spatio-temporal domains of large, small, and
intermediate-scale watershed models for arid regions. 30
Figure 2.2. Graphs derived using the Ponce et al. (1978) equation for determining when
the diffusion wave model yields reasonable results in comparison to the dynamic
wave model. (A) Minimum wave period vs. flow depth; (B) Minimum wave period
36
vs. channel slope.
Figure 2.3. Graphs illustrating the minimum wave period required for the kinematic
wave model to be 95% as accurate as the diffusion wave model. Values are shown
38
for a range of (A) flow depths, and (B) channel slopes.
Figure 2.4. Graph showing division of the slope/reference water depth space into four
regions characterized by different wave types for a channel length of 1000 meters
39
and a Froude number of 0.4.
Figure 2.5. Graphs showing river wave approximation zones obtained from the analysis
40
of the momentum equation for three values of i = 1, 8, and 20. dynamic-wave
results
that
of
kinematicand
Figure 2.6. Graph showing a comparison
42
demonstrating the propagation of errors in the kinematic wave solution. Figure 2.7. Graph showing the influence of varying the time and space step during finite43
difference solutions of the kinematic wave equation. 49
Figure 2.8. Space-time discretization of Ponce and Yevjevich (1978) Figure 3.1. K2 and K2MC simulated WG6 runoff hydrographs for design storms ranging
from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower
63
right).
Figure 3.2. Graph showing normalized differences in the time to peak discharge at the
65
watershed outlet between K2 and K2MC, plotted as a function of area. Figure 3.3. Graph showing normalized differences in the simulated peak discharge at the
66
watershed outlet between K2 and K2MC, plotted as a function of area. Figure 3.4. Graph showing relative error in the total discharge volume at the watershed
67
outlet between K2 and K2MC as a function of area. Figure 3.5. Graph showing the average trends in mass-balance error as a function of
69
watershed area Figure 3.6. Graph showing the best match of simulated and observed hydrographs for
72
WG1 Figure 3.7. Graph showing the best match of simulated and observed hydrographs for
72
WG6 Figure 3.8. Graph showing the best match of simulated and observed hydrographs for
74
WG11 9
Figure 3.9. Graph showing the best match of simulated and observed hydrographs for
LH104 74
Figure 3.10. Graph showing average model efficiency plotted as a function of watershed
area for K2 and K2MC 76
Figure 3.11. Graph showing average error (%) in the timing of flow onset at the
watershed outlet plotted as a function of watershed area for K2 and K2MC 77
Figure 3.12. Graph showing average error (%) in the timing of peak flows at the
watershed outlet plotted as a function of watershed area for K2 and K2MC 77
Figure 4.1. Schematic representation of erodible trapezoidal channels in CASC2D 90
Figure 4.2. Profile adjustment over 18 years in the Goodwin Creek Watershed 92
Figure 4.3. Conceptualization of a compound channel in K2G showing separate channel
and overbank areas. 96
Figure 4.4. Graph showing change in sediment storage for each computational node in a
reach 100
Figure 4.5. Sketch illustrating depositional width reduction for the K2G trapezoidal
channels 102
Figure 4.6. Sketch illustrating depositional depth reduction for the K2G trapezoidal
channels 103
Figure 4.7. Plots showing depositional adjustments at the upstream (A) and downstream
(B) ends of a 210-meter reach following a 52 mm rainfall event with a peak flow of
35.3 m 3 /s (61.9 mm/hr). 105
Figure 4.8. Sketch illustrating erosional width increase for the K2G trapezoidal channels
106
Figure 4.9. Sketch illustrating erosional depth increase for the K2G trapezoidal channels
108
Figure 4.10. Plots showing erosional adjustments at the upstream (A) and downstream
(B) ends of a 25-meter reach following a 52 mm rainfall event with a peak flow of
3.1 m 3/s (129.7 mm/hr). 109
Figure 4.11. Graph showing measured and simulated runoff for an event on the Catsop
catchment in South Limburg, Netherlands. From Smith et al. (1999). 115
Figure 4.12. Comparison of discharge (Qt), sediment transport rate (St), and
concentration (Ct) between measured rainfall simulation data (circles) and K2predicted values for three events on road plots. From Ziegler et al. (2001). 116
Figure 4.13. Graph showing the channel profile near the outlet of Walnut Gulch
Subwatershed 223 before and after a K2G batch simulation of five 10-year 1-hour
events and no profile smoothing between events 120
Figure 4.14. Diagrammatic representation of two-reach junction profile adjustments
121
showing variables used in derivation of the adjusted elevation, EA Figure 4.15. Diagrammatic representation of three-reach junction profile adjustments
124
showing variables used in derivation of the adjusted elevation, EA Figure 4.16. Walnut Gulch Subwatershed 223 profile near the outlet before and after a
K2G batch simulation of five 10-year 1-hour events with profile smoothing between
events
128
10
Figure 5.1. Maps showing the watershed discretization (A), DEM (B), SSURGO soils
(C), and rain gauge configuration (D) for Walnut Gulch subwatershed 11. 139
Figure 5.2. Maps showing the four land-cover scenarios used to derive model parameters
for the test simulations: (A) NALC 1973, (B) NALC 1997, (C) Partial urbanization,
and (D) Total urbanization. The legend for the NALC classification scheme is
shown for reference. Outlines of the discretized model elements are superimposed
on the land-cover grids
142
Figure 5.4. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_0G_73, 1977_0G_73, and 1978_0G_73 (top to bottom,
respectively).
150
Figure 5.5. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_HG_97, 1977_HG_97, and 1978_HG_97 (top to bottom,
respectively).
151
Figure 5.6. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_0G_97, 1977_0G_97, and 1978_0G_97 (top to bottom,
respectively).
152
Figure 5.7. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_HG_PU, 1977_HG_PU, and 1978_HG_PU (top to bottom,
respectively).
153
Figure 5.8. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_0G_PU, 1977_0G_PU, and 1978_0G_PU (top to bottom,
respectively).
154
Figure 5.9. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_HG_U, 1977_HG_U, and 1978_HG_U (top to bottom,
respectively).
155
Figure 5.10. Maps showing depth and width changes (left and right, respectively) for
simulations 1964_0G_U, 1977_0G_U, and 1978_0G_U (top to bottom,
respectively).
156
Figure 5.11. Cumulative runoff depth (mm) per unit contributing area for the three batch
simulations arranged in order of decreasing rainfall total from 1964 (top) to 1978
(bottom). Pictured results are for the hydraulic geometry simulations, but the spatial
patterns for the observed geometry simulations were virtually identical 157
Figure 5.12. Bar graph showing the model (red) and geometric (blue) net mass-balance
error (%) for 1964 batch simulations.
161
Figure 5.13. Bar graph showing the relative magnitude of channel erosion (kg) and the
mass equivalent of the simulated geometric adjustment for the 1964 batch
simulations.
162
Figure 5.14. Bar graph showing the model (red) and geometric (blue) net mass-balance
error (%) for 1977 batch simulations. 163
Figure 5.15. Bar graph showing the relative magnitude of channel erosion (kg) and the
mass equivalent of the simulated geometric adjustment for the 1977 batch
simulations.
164
Figure 6.1. Maps showing the difference (meters) in simulated average depth change
(left) and average width change (right) between the partial urbanization and NALC
11
1997 land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations
based on the HG channels. Shades of red (positive values) indicate that geomorphic
adjustments yielded relatively larger depths/widths for the PU scenario 173
Figure 6.2. Maps showing the difference (meters) in simulated average depth change
(left) and average width change (right) between the partial urbanization and NALC
1997 land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations
based on the OG channels. Shades of red (positive values) indicate that geomorphic
adjustments yielded relatively larger depths/widths for the PU scenario 175
Figure 6.3. Maps showing the relative differences in cumulative runoff (mm, top),
transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1964
simulations. Results for the HG and OG channels are pictured on the left (A-C), and
right (D-F), respectively.
177
Figure 6.4. Maps showing the relative differences in cumulative runoff (mm, top),
transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1977
simulations. Results for the HG and OG channels are pictured on the left (A-C), and
178
right (D-F), respectively.
Figure A.1. K2 and K2MC simulated LH104 hydrographs for design storms ranging
from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower
right). Plots are discharge in cubic meters per second as a function of time in
192
minutes.
Figure A.2. K2 and K2MC simulated WG11 hydrographs for design storms ranging
from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower
right). Plots are discharge in cubic meters per second as a function of time in
minutes.
193
Figure A.3. K2 and K2MC simulated WG6 runoff hydrographs for design storms
ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event
(lower right). Plots are discharge in cubic meters per second as a function of time in
194
minutes.
Figure A.4. K2 and K2MC simulated WG1 hydrographs for design storms ranging from
the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right).
Plots are discharge in cubic meters per second as a function of time in minutes 195
Figure A.5. Observed and simulated hydrographs for the 8/1/78 event on LH104. 196
Figure A.6. Observed and simulated hydrographs for the 7/28/81 event on LH104 196
Figure A.7. Observed and simulated hydrographs for the 9/10/83 event on LH104 196
Figure A.8. Observed and simulated hydrographs for the 8/10/86 event on LH104 197
Figure A.9. Observed and simulated hydrographs for the 8/29/86 event on LH104 197
Figure A.10. Observed and simulated hydrographs for the 7/27/76 event on WG11 197
Figure A.11. Observed and simulated hydrographs for the 8/1/78 event on WG11 198
Figure A.12. Observed and simulated hydrographs for the 8/4/80 event on WG11 198
Figure A.13. Observed and simulated hydrographs for the 8/29/86 event on WG11 198
Figure A.14. Observed and simulated hydrographs for the 8/1/90 event on WG11 199
Figure A.15. Observed and simulated hydrographs for the 7/27/76 event on WG6 199
Figure A.16. Observed and simulated hydrographs for the 8/1/78 event on WG6 199
Figure A.17. Observed and simulated hydrographs for the 8/4/80 event on WG6 200
12
Figure A.18. Observed and simulated hydrographs for the 8/29/86 event on WG6. 200
200
Figure A.19. Observed and simulated hydrographs for the 8/1/90 event on WG6 201
Figure A.20. Observed and simulated hydrographs for the 7/27/76 event on WG1 for
the
8/1/78
event
on
WG1
201
simulated
hydrographs
Figure A.21. Observed and
201
Figure A.22. Observed and simulated hydrographs for the 8/4/80 event on WG1 202
Figure A.23. Observed and simulated hydrographs for the 7/30/81 event on WG1 202
Figure A.24. Observed and simulated hydrographs for the 8/17/86 event on WG1 202
Figure A.25. Observed and simulated hydrographs for the 8/2986 event on WG1 203
Figure A-26. Observed and simulated hydrographs for the 8/1/90 event on WG1. Figure B.1. Sketch illustrating the depositional depth reduction calculations for K2G
205
trapezoidal channels Figure B.2. Sketch illustrating the erosional depth increase calculations for K2G
206
trapezoidal channels Figure B.3. Sketch illustrating the bank failure calculations for K2G trapezoidal channels
207
Figure B.4. Sketch illustrating depositional width reduction calculations for K2G
208
trapezoidal channels 1.3
LIST OF TABLES
Table 3.1. Results of the Komolgorov-Smirnov tests indicating whether the null
hypothesis was accepted or rejected. Results are shown for the six design-storm events
ranging from 5-year 30-minute (5 _ 30) to 100-year 1-hour (100_1) for each subwatershed.
62
Table 3.2. Difference between the timing of K2 and K2MC flow onset (mins) 64
Table 3.3. Mass balance error (%) computed from K2 and K2MC output files 68
Table 4.1. Sample output for one channel element from the geom.out file written during
118
a K2G simulation Table 4.2. Results of the volume-balance calculations for one 10-year 1-hour event with
(PS) and without (NS) profile smoothing, and five 10-year, 1-hour events with profile
smoothing. Insignificant digits are carried to prevent rounding error. 131
Batch simulations, and their associated inputs, used to evaluate the
Table 5.1.
139
geomorphic model. Table 5.2. Characterization of the precipitation records for the summer monsoon on
140
WG11 during 1964, 1977, and 1978 Table 5.3. Composition of land cover classes on WG11 for the four land-cover scenarios
143
used during model testing. Table 5.4. Hydrologic parameters used by AGWA during land cover and soils
143
parameterization Table 5.5. Sediment grain-size distributions as percentages of the dry sample weight for
145
the channel and overbank elements. Table 5.6. Sediment mass balance results for the entire channel network from each
simulation. Error is presented in percent, and everything else has the units of kilograms.
160
Table 5.7. Simulated cumulative outflow and sediment yield for the observed land-cover
165
scenarios Table 5.8. Summary of sediment accumulation on Walnut Gulch from Nichols and
166
Renard (2003). The italicized values are from a pond within WG11. 14
ABSTRACT
Watershed models have two significant shortcomings that limit their application
to management problems in arid and semi-arid regions. The first is that the performance
of event-based hydrologic models for ephemeral stream networks declines significantly
as watershed size increases. The second is that no single model is capable of simulating
runoff, erosion, and geomorphic response in the channel network for multiple
consecutive events.
A diffusion-wave routing subroutine was developed for the Kinematic Runoff and
Erosion Model (K_INEROS2) using a four-point iterative solution to the modified
variable-parameter Muskingum-Cunge (MVPMC4) technique. It was tested against
kinematic-wave routing at scales ranging from 0.05 to 150 km 2 on the Walnut Gulch
Experimental Watershed in southeastern Arizona. Analyses demonstrated that MVPMC4
routing significantly improves simulated outflow hydrographs for small to moderate
events on watersheds that are 95 km 2 and larger.
A geomorphic model was developed by modifying KINEROS2 to compute width,
depth and slope adjustments from computed changes in sediment storage at each time
step. Width and depth adjustments are determined by minimizing total stream power for
each reach. A GIS-based interface was developed for model parameterization,
coordinating multiple-event batch simulations, tracking cumulative geomorphic change,
computing the sediment mass balance, visualizing results, and comparing results from
different simulations.
15
Simulated geomorphic adjustments are particularly sensitive to the number and
magnitude of events in the rainfall record. Widespread erosion was predicted during the
wettest sequence of rainfall events, mixed erosion and deposition during intermediate
sequence, and predominantly deposition during the driest sequence.
Simulation results from before and after urbanization in part of the watershed
were compared for wet and intermediate rainfall records. Differences in computed
geomorphic change in the unaffected part of the watershed were approximately zero.
Erosion, primarily manifested as channel incision, increased within the urbanized area,
and decreased downstream of it during the wet year. For the intermediate year, relative
increases in deposition extended further upstream as transmission loss increased relative
to the runoff volume. The overall pattern of relative geomorphic response was very
similar regardless of the initial channel geometry, suggesting that the model can be used
for broad-scale management and planning in the absence of detailed channel-geometry
observations.
16
1.0 INTRODUCTION
1.1 Problem Statement
Land, water, and biological resource managers have realized for some time that
their mandates overlap substantially; the resources they manage are all parts of linked
climatic, hydrologic, and geomorphic systems. These systems are defined and linked by
the movement of water on the Earth's surface, and watersheds thus represent the most
convenient spatial entities within which they can be described. The scientific tools
available to identify, study, and remediate watershed problems, however, are often
severely handicapped in that they only account for parts of the larger system, or can only
be applied at very limited spatial and temporal scales.
While much progress has been made towards improving watershed assessment
tools in recent years, efforts have been primarily focused on humid environments. Arid
and semi-arid regions, hereafter referred to collectively as arid regions, are characterized
by processes operating at different spatial and temporal scales that cannot be adequately
represented with assessment tools designed for humid regions. Comprising 30% of the
global land surface, and housing roughly 20% of the human population, arid regions
represent an important geography that has been largely overlooked in the development of
landscape management and planning tools.
This dissertation addresses two important limitations of arid-region watershed
models that stand as obstacles to their widespread and interdisciplinary use as assessment
and planning tools. The first limitation is one of scale, and the need to expand the spatial
17
limits of event-based hydrologic models by improving their performance on intermediatescale arid-region watersheds. The second limitation is the inability of watershed models
to compute geomorphic change throughout a channel network in response to changing
climate and land cover/use.
1.1.1 Intermediate-Scale Watershed Runoff Models
A major difficulty in hydrologic modeling has been to bridge successfully the
spatio-temporal transition between small watersheds with ephemeral streams and large
basins with perennial flow. Numerous models are very successful on either end of this
spectrum, but are less effective as changing spatial and temporal scales cause conceptual
and numerical models to break down in the middle. This scale gap is generally described
as occurring between about 100 and 1000 km 2 , depending on the climatic region and
event magnitude (e.g. Goodrich, 1990; Goodrich et al., 1997; Syed, 1999).
In humid environments, where rainfall is relatively uniform and subsurface flow
buffers runoff hydrographs, large-scale watershed models operating on a daily time step
can be successfully applied to smaller areas. This is not the case in arid and semi-arid
environments where rainfall is highly spatially variable and runoff is generated almost
exclusively by Hortonian overland flow. To model runoff in intermediate-scale arid
watersheds it is thus necessary to use distributed precipitation input and a time step on the
order of minutes to adequately represent the characteristic spatial and temporal variability
of rainfall and runoff. This is most effectively accomplished by extending the range of
scales in which small-scale (event-based) watershed models can be applied.
18
Previous studies have demonstrated that model efficiencies decrease substantially
with increasing watershed size for event-based runoff models (e.g. Syed, 1999; Goodrich,
1990; Goodrich et al., 1997). In arid regions, characterized by influent channels, the
runoff to rainfall ratio decreases with increasing watershed scale, thereby magnifying the
effects of errors derived from the conceptual and numerical components of the model. A
variety of sources of model error have been investigated, and it is widely believed that the
largest source lies in the measurement of rainfall, both at a point and its distribution in
space (Michaud and Sorooshian, 1994). Much recent work has been aimed at reducing
rainfall error through the use of radar rainfall data (e.g. Ogden and Julien, 1993; Morin et
al., 2003). Another source of error results from model conceptualization of nonchannelized surface runoff. The implementation of microtopography on upland plane
elements (Smith et al., 1995), and the use of grid-based models (e.g. Ogden and Heilig,
2001) have been explored as a means of addressing this. Error resulting from the
geometric complexity of watershed discretization has been investigated by numerous
authors (e.g. Goodrich, 1990; Syed 1999; Miller, 2002). Errors resulting from land
use/cover misclassification and the spatial resolution of soil data were explored by Miller
(2002), and Bradley (2003), respectively.
Whereas there is substantial literature on channel routing, the author was unable
to identify any work that has specifically evaluated the merits of different routing
techniques in watershed channel networks with respect to scale. The conditions under
which various routing models are most suitable have been explored in detail, however,
and based on this research it is clear that kinematic wave routing can accumulate error
19
when applied to large channel networks. Kinematic wave routing has been widely used
in event-based watershed models because it is relatively easy to implement, and can be
applied in the absence of downstream boundary conditions (e.g. USACE, 1994; Singh,
2002). The latter is particularly important for the simulation of ephemeral flows.
Kinematic routing, however, cannot directly account for flood-wave diffusion, which
becomes increasingly important as travel time increases and channel slopes decrease in
the downstream direction.
1.1.2 Watershed Geomorphic Models
The vast majority of geomorphic models are designed to facilitate the
investigation of important engineering problems such as channel design and bridge scour.
These models are applied on a reach basis, and require detailed channel geometry and all
water and sediment inflows as input. As such, they cannot easily be used to investigate
the cumulative impacts of dispersed inputs at larger scales. Land-use and climatic
impacts cannot be simulated directly because the models do not contain upland routing
and/or erosion subroutines.
To investigate problems relating to spatially variable land management and
planning it is necessary to conduct geomorphic simulations on a watershed basis.
Changing conditions in one subwatershed can be negated or compounded as a result of
conditions elsewhere in the watershed. Thus, accurate simulation of sediment discharge
and geomorphic change requires simultaneous consideration of the upland areas and
channel network throughout the watershed. Benefits of this type of approach include
20
improved estimates of sediment discharge and the ability to simulate probable
geomorphic response to distributed land cover/use change. With a watershed-based
geomorphic model it also becomes possible to compare the impacts of multiple change
scenarios, and to identify problem areas (in the channel network and/or on the uplands)
for selective remediation.
Although the benefits of a watershed-based approach to geomorphic modeling
have been widely recognized (e.g. Schumm et al., 1987), few watershed models possess
the ability to track geomorphic change in the channel network over time, and only one of
these, CASC2D (Ogden, 1998) can be applied to watersheds characterized by ephemeral
flows. Over longer periods of time, however, it becomes increasingly necessary to
account for changes in channel width in addition to depth (and hence slope) to account
for changing sediment storage and delivery. No watershed models are available that
simulate width, depth, and slope changes in arid and/or semi-arid environments.
Despite the infrequent and highly variable nature of rainfall in arid regions, the
rate of geomorphic change can be high. For instance, Graf (1983a) computed that
approximately 6 million cubic meters of sediment were removed from the channel
network of a 150 square kilometer semi-arid rangeland watershed in SE Arizona (Walnut
Gulch) during a 15-year period. Similar erosional episodes were observed throughout the
American Southwest during the late 1800s and early 1900s, and although no single causal
factor can be identified, land use/cover change has commonly been interpreted as a
significant contributor (Elliot et al., 1999). Given that the nine fastest growing cities in
the United States (USCB, 2003) are in the arid and semi-arid Southwest, considerable
21
land cover/use change continues to occur in this region. In the absence of a watershed-
based geomorphic model specifically designed for application in arid environments it is
not possible to evaluate how cumulative landscape change impacts channel networks and
the invaluable riparian corridors they support.
1.2 Hypotheses
•
A significant source of error in the use of kinematic routing at intermediate scales
results from the inability to account directly for diffusion of the flood wave as it is
routed through the channel network.
•
A continuous simulation event-based geomorphic model describing channel width
and depth changes can simulate the anticipated geomorphic response to landscape
change in semi-arid watersheds.
1.3 Approach
The general approach in this research was to build on previous research by
developing new components for the USDA-ARS Kinematic Runoff and Erosion Model,
KINEROS2 (Smith et al., 1995), hereafter referred to as K2. Component development,
testing, and evaluation varied significantly for the two objectives, however, and are
described separately below.
22
1.3.1 Runoff Routing
Alternative routing methods suitable for use in an intermediate-scale arid
watershed model were explored, and the Muskingum-Cunge method with variable
parameters, specifically the MVPMC4 methodology (Ponce and Chaganti, 1994), was
selected as most suitable. A routing subroutine was developed for K2 based on this
approach. The subroutine utilizes the same infiltration, rainfall, and sediment routing
components of K2, and requires no additional inputs.
Kinematic and MVPMC4 routing techniques were compared using a series of
design (uniform) and observed (spatially variable) rainfall events. Simulated
hydrographs were compared to determine if they were statistically different, and to define
how they differed. Simulated and observed hydrographs were compared to evaluate
model performance.
All analyses were conducted for four subwatersheds in the Walnut Gulch
Experimental Watershed: Lucky Hills 104 (LH104, 4.74 ha), Walnut Gulch 11 (WG11,
782 ha), Walnut Gulch 6 (WG6, 9,558 ha), and Walnut Gulch 1 (WG1, 14,664 ha).
Results are presented as functions of watershed size and event magnitude.
1.3.2 Geomorphic Modeling
A watershed geomorphic model was developed to compute width and depth
changes on a reach basis. The geomorphic model is a modification of K2 that computes
geomorphic change resulting from computed change in sediment storage following each
time step. Width and depth change calculations are based on the minimum energy
23
dissipation rate theory (Yang and Song, 1979), and the theory's special case, the
minimum stream power theory (e.g. Chang, 1980; Song and Yang, 1980).
A GIS-based interface was developed for the geomorphic model to facilitate
multiple-event simulations by tracking geomorphic change between simulations. The
interface is based on the Automated Geospatial Watershed Assessment (AGWA) tool
(Miller et al., 2002). The interface conducts most aspects of watershed delineation,
discretization, and parameter estimation for geomorphic simulations. In addition, it
provides tools for the spatial visualization of model results, and the comparison of
multiple simulation outputs for change assessment.
The geomorphic model was evaluated on subwatershed 11 of Walnut Gulch
(WG11). Sediment mass balance was evaluated on a reach basis and for the channel
network as a whole using two consecutive 10-year 1-hour design storm events. Model
performance was evaluated by conducting three multiple-event (batch) simulations using
all measured monsoonal events for wet (1977, 47 events), average (1964, 54 events), and
dry (1978, 41 events) summers. The three batch simulations were each conducted based
on parameter inputs derived from 1973 and 1997 land cover/use data sets, and a 100%
urbanized artificial land-cover map. These tests were designed to evaluate the relative
behavior of the model under different precipitation and land-use regimes in the absence
of observed geomorphic change.
Results from the geomorphic modeling are presented individually and relative to
each other. The former is used as a means of evaluating whether the model produced
physically realistic results in terms of the magnitude and spatial distribution of erosion in
24
the watershed. The latter is used to evaluate the model in terms of its ability to identify
those reaches most susceptible to geomorphic change.
1.4 Study Area
The study area for this research is the Walnut Gulch Experimental Watershed
maintained by the USDA-ARS Southwest Watershed Research Center. Walnut Gulch is
approximately 120 kilometers southeast of Tucson, and contains the town of Tombstone,
Arizona (figure 1.1). The watershed is representative of mixed grass-brush land (Renard
et al., 1993), and lies in the transition between the Sonoran and Chihuahuan deserts. The
average elevation of the watershed is -1500 m, and the topography consists of gently
rolling hills incised by alluvial channels of moderate slope. Hillslopes vary from 2 to
65%, and channel slopes are predominantly between 1 and 3%. Surface soil textures (0-5
cm) include gravelly and sandy barns containing an average of 30% rock and little
organic matter (Renard et al., 1993; Kustas and Goodrich, 1994).
Walnut Gulch receives 250 to 500 mm of rainfall annually, most occurring during
two relatively rainy seasons: winter and summer. Winter rains are predominantly from
relatively uniform frontal events of low intensity and long duration. Summer rains
account for approximately 2/3 of the annual total, and are most commonly associated
with highly non-uniform convective thunderstorms of relatively high intensity and short
duration. Rainfall is recorded in the Walnut Gulch Watershed through a network of 96
recording rain gauges, or approximately one gauge for every 1.5 km2.
25
# ARS Headquarters
Roads
Improved dirt
A/ Paved
/ State highway
•
Unimproved dirt
,A,„/ Utility access
Land Ownership
Bureau of Land Management
Private
State Trust
A/
2
0
2
4
6 Kilometers
Figure 1.1. Map of the Walnut Gulch Experimental Watershed showing cultural
features.
Walnut Gulch is divided into 12 primary subwatersheds varying in size from 7.85
to 148 km 2 (figure 1.2). Each of the subwatersheds is equipped with a pre-calibrated
critical-depth flume for runoff measurement. Twelve smaller subwatersheds, ranging in
size from 0.4 to 89 ha, are equipped with flumes or detention ponds and weirs for runoff
measurement. The present study concentrates on four subwatersheds that were selected
to cover the range of available sizes. These include: Walnut Gulch 1 (WG1, 148 ktn 2 ),
Walnut Gulch 6 (WG6, 93.6 km 2 ), WG11 (7.85 km2 ), and Lucky Hills 104 (LH104.
0.047 km2).
26
Watershed Configuration
nested subwatersheds and measunna
[...
Primary Watershed
Watershed
1 (WG1)
2
3
4
5
6 (WG6)
7
a
148
112
942
229
221
93.6
136
14.8
239
15.8
7.85
23.7
9
10
11 (WG11)
15
A
2
0
$ Stock Pond
Runoff Measuring Device
Recording Rain Gauge
sInactive
er
Active
Unit Area Subwatershed
Gaged Pond Subwatershed
I 1 Ungaged Pond Subwatershed
Primary Subwatershed
2
4 Kilometers
Map Created 9-21-01 by Scott Miller
Figure 1.2. Map showing Walnut Gulch subwatersheds. Subwatersheds used in this
research were WG1, WG6, WG11, and LH104.
27
2.0 ROUTING MODEL
2.1 Introduction
The Automated Geospatial Watershed Assessment (AGWA) tool (Miller et al.,
2002) is the most comprehensive and widely used GIS-based watershed assessment tool
available specifically for arid and semi-arid environments. A significant problem with
the two hydrologic models currently incorporated into AGWA, however, is that neither is
appropriate for simulating runoff/erosion in watersheds between about 100 and 1000 km 2 .
The Soil Water Assessment Tool, SWAT (Arnold et al., ) is designed to simulate large-,
or basin- scale watersheds larger than approximately 1000 km 2 . In contrast, the
Kinematic Runoff and Erosion Model (K2) performs best when applied to watersheds
less than about 5 km 2 , but can do reasonably well for larger events on watersheds up to
about 100 km 2 (Goodrich, 1990; Goodrich et al., 1997; Syed, 1999). Whereas this multiscale approach to watershed assessment is one of AGWA's greatest strengths, the
inability of either model to simulate runoff and erosion in intermediate-scale watersheds
represents a significant obstacle to the practical application of the tool.
A potential source of error in applying K2 at intermediate scales is theoretically
the inability of the kinematic wave routing to account directly for diffusion of the flood
wave as it is routed through the channel network. Alternative routing methods suitable
for use in an arid-region watershed model were explored, and the Muskingum-Cunge
method with variable parameters, specifically the MVPMC4 methodology (Ponce and
28
Chaganti, 1994), was adopted and tested against K2 and observed outflow hydrographs at
multiple scales.
2.2 Scale-Related Problems in Watershed Modeling
Watershed models are commonly associated with a scale at which they are most
successful at reproducing observed water and sediment discharges. Small-watershed
models concentrate on describing hillslope processes, but their predictions tend to break
down as watershed area increases and runoff hydrographs become dominated by channel
processes. Conversely, large-watershed or basin-scale models may employ more robust
channel routing components, but often do not adequately describe spatially variable
runoff generation on the hillslopes, particularly in arid and semi-arid environments where
runoff duration is short relative to the model time step, which is usually 24 hours.
An ideal watershed model would be physically based, would describe hillslope
and channel processes equally well, and would be capable of simulations for long periods
of time. In practice, however, the amount of input data, and the computational time
required to achieve these objectives are insurmountable obstacles to implementing such a
model. As a result, myriad modeling strategies have been developed, each with a specific
purpose or problem. In AGWA, for instance, SWAT was selected for its ability to model
large areas for long periods of time. To do this it simplifies the spatial resolution of its
input data requirements and the temporal resolution at which computations are made. In
contrast, KINEROS was selected for its ability to simulate event discharge in smaller
areas. In this situation input data requirements for a given area are more extensive than
29
those for SWAT, but computational costs are about the same because the model is
applied to smaller watersheds. Similarly, whereas the computational time step is much
smaller in KlNEROS, the length of the simulation is much shorter. Reducing the
duration of the simulation rather than the temporal resolution at which processes are
computed thus minimizes computational cost.
In addition to the size of the modeled area and the time length of the simulation,
scale dependence in watershed modeling results at least in part from the different spatial
and temporal scales at which watershed hydrologic processes themselves are active. In
figure 2.1, for example, it can be seen that overland-flow processes operate at smaller
spatial and temporal scales than do channel-flow processes. There is overlap between
them, however, and it is within this spatial scale (-1 km) that K2 and similar smallwatershed models perform best (Goodrich, 1990; Woolhiser, 1996). As watershed size
increases channel processes become increasingly dominant, and the accuracy of smallwatershed model predictions declines. Improving the numerical description of this
transition from a hydrograph dominated by hillslope processes to one dominated by
channel processes is the primary objective of the routing component of this research.
30
100 yrs
Annual Raintall, Snowmelt, Evapo
Unsaturated
Flow
1 min
10m
100m
Figure 2.1. Graph showing hydrological processes at a range of characteristic space-time
scales, and the superimposed spatio-temporal domains of large, small, and intermediatescale watershed models for arid regions. Modified from Bloschl and Sivapalan (1995).
2.3 Channelized Flow Routing in Watershed Models — Previous Work
The physical size of watersheds limits the complexity of channelized flow routing
procedures in watershed models to relatively simplistic one-dimensional representations.
One-dimensional models describe a uniform flow velocity parallel to the channel only,
and cannot therefore account for localized effects such flow separation, eddies, or helical
31
flow around channel bends. Because flow cannot vary perpendicularly to the flow
direction neither can sediment movement, and as a result channels are generally
described as a series of reaches within which boundary conditions vary uniformly or not
at all.
Jean-Claude Barre de Saint-Venant originally derived the continuity and
momentum equations for one-dimensional gradually varied unsteady flow in the late 19 th
century. In classical form, the Saint-Venant equations can be expressed as:
aA aQ r(x,t) Continuity
+—=
(2.1)
\ u
ay
au au
0 – S
f )-- r Momentum
— + u— + g — = gS
(
A
at ax ax
(2.2)
at ax
—
Variables are defined as
A - cross-sectional area of the flow
Q – discharge
r(x,t) - the rate of lateral inflow per unit length of the channel
u - flow velocity
g – acceleration due to gravity
So –bed slope
Sf friction, or water-surface slope
ur/A – momentum exchange between the lateral inflow, r, and the main channel flow, u.
-
In hydraulic modeling, the continuity and momentum equations are coupled to
form the nonlinear dynamic wave model. Solution of these equations yields results for
flows that vary with local and convective accelerations, pressure gradient, and friction
and bed slope. Obtaining solutions for the full dynamic wave model, however, is a
complex and difficult process for unsteady flows in a network of channels. Most authors
describe the full dynamic wave model as inappropriate for steep channels, initially dry
32
channels, transcritical flows, and complex channel networks owing to difficulties in
obtaining numerical solutions (e.g. Ogden, 1998; Wu and Vieira, 2000; Meselhe and
Holly, 1997; Singh, 1996). A variety of numerical techniques has been developed to
work around these problems, however, and at least one author has claimed to have
developed a full dynamic wave model for arid watersheds (El-Hames and Richards,
1998). Although no detailed information can be found about this model, the authors
admit that it is so computationally intensive that it cannot run on normal desktop
computers.
Owing to the computational expense of solving the full dynamic wave equations,
simplifications, or approximations, of the full equations have been derived for certain
circumstances to provide simpler but acceptable solutions. These equations can be
classified according to the terms of the momentum equation that they neglect or consider
as quasi-steady dynamic wave, diffusion wave, or kinematic wave.
The quasi-steady dynamic wave neglects only the local acceleration term
(atyat )
from the momentum equation, which is then coupled with the continuity equation to form
the nonlinear quasi-steady dynamic wave model (Yen, 1979).
au
ay
u—+g—=
ax ax
f
–s )-- 11 r
0
0
f
A
(2.3)
This model accounts for downstream backwater effects and permits distortion,
translation, and attenuation of the hydrograph peak. Upstream and downstream boundary
conditions must be specified to obtain a unique solution. Neglecting only the local
acceleration term, however, produces more error than the diffusion wave approximation
33
for which both local and convective terms are neglected, and it is not widely used as a
result (Yen, 1979).
For gradually varied unsteady flows, except in highly nonuniform channels, local
and convective accelerations are commonly small, of the same order of magnitude, and
have opposite signs (Yen, 1979). Assuming that they cancel each other, the momentum
equation can be expressed in terms of flow depth as:
ay
ax
=S -S
0
(2.4)
when lateral inflows are not present. Combined with the continuity equation this forms
the diffusion wave model. Inclusion of the pressure term substantially improves solution
accuracy over the kinematic wave model (Yen, 1979). Flow peaks can be attenuated, and
the hydrograph can be distorted and translated. A unique solution to the diffusion wave
equation, however, requires that upstream and downstream boundary conditions and
lateral inflows be specified. This requires a more complicated simultaneous or iterative
solution than does the kinematic wave model (Beven, 1993).
In kinematic wave theory flood waves are assumed to be long and flat so that the
friction slope is nearly equal to the bed slope. Considering only the two slope terms of
the momentum equation, the kinematic wave equation can be expressed as:
S„ = S f(2.5)
when lateral inflows are absent. Solution of this equation requires only the upstream
boundary condition, making it the easiest to implement for channel networks where flow
depths at the drainage divide are considered equal to zero. The kinematic wave equation
34
cannot predict backwater effects, dispersion, or attenuation of the flood peak. In practice,
however, numerical diffusion associated with approximation errors in the numerical
solution of the kinematic-wave equation can lead to some attenuation of the hydrograph,
which is most pronounced for large time and space steps (Beven, 1993; Yen, 1979).
Numerical dispersion, also a result of approximation errors, manifests itself as the
steepening or flattening of the of the rising limb of the calculated runoff hydrograph, and
is most pronounced when the Courant number (also a function of the time and space
steps) is substantially different from 1 (Ponce, 1991). In general, when used for
channelized flow routing the kinematic wave model predicts a steeper wave than actually
occurs, and the effect of the accumulation of errors shows that the approximations made
in the development of the kinematic-wave equations are not generally justified for most
flow-routing applications (Miller, 1984; Hromadka and DeViies, 1988).
2.3.1 Evaluating Which Model is Applicable
Channelized flow can be spatially generalized within a basin as conforming to one
of the three primary approximations of the momentum equation (excluding the quasisteady dynamic wave). Flow in the steepest upstream channels is best described by the
kinematic wave approximation (Beven, 1993). This is particularly convenient because
the kinematic wave approximation also works well for overland flow on hillslopes, and
can be used with only the upper boundary condition of zero flow across the drainage
divide. Proceeding downstream, flows in channels of intermediate and low slope are well
described by the diffusion wave approximation. For the lowest slopes, slowest velocities,
35
and greatest flow depths, local and convective accelerations are large enough to
necessitate use of the full dynamic wave model. Evaluating which model is applicable
for a given channel or watershed has been the subject of considerable research and
debate. In general, however, the problem can be approached from two directions:
identifying the various flow conditions characterized by kinematic, diffusive, and
dynamic waves; and weighing the former against known limitations of the available
solution techniques.
Ponce et al. (1978) evaluated downstream transitions among the three wave types
by using linear stability analysis. They found that the diffusion wave model yielded
reasonable results in comparison with the full dynamic model when
\ l/ 2
(
TpS0
g
-
30
(2.6)
\YN /
where Tp is the wave period of a sinusoidal perturbation of steady uniform flow, yN is the
steady, uniform flow depth, and So is the longitudinal slope of the channel bed. If this
equation is satisfied, the diffusion model will accurately approximate the unsteady flow.
Scenarios demonstrating the conditions under which this is true are presented in figure
2.2 for the range of conditions expected in the Walnut Gulch Experimental Watershed in
southeastern Arizona. From these it can be seen that for small flow depths and steep
slopes, a short wave period is required to get a 'reasonable' solution using the diffusionwave approximation. It appears reasonable that the diffusion-wave equations would be
acceptable for the range of flow depths and channel slopes that are typical of small- to
intermediately-sized semi-arid watersheds.
36
0.9
Slope = 1%
0.8
o
0.7
Diffusion
.c
Dynamic
0
2
0
4
6
12
10
8
Flow depth (m)
A
1.6
Flow depth = 0.3 m
E
0.6
Diffusion
E 0.4
C
5
0.2
Dynamic
•
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Channel Slope
Figure 2.2. Graphs derived using the Ponce et al. (1978) equation for determining when
the diffusion wave model yields reasonable results in comparison to the dynamic wave
model. (A) Minimum wave period vs. flow depth; (B) Minimum wave period vs.
channel slope.
37
Ponce et al. (1978) also evaluated where to transition between diffusion and
kinematic wave models, and found that if the kinematic wave model is to be 95% as
accurate as the diffusion model after one wave propagation period then
Tp . 171y N
So u N
(2.7)
where UN is the normal flow velocity, So is the channel bed slope and y N is the flow depth.
Figure 2.3 shows that the minimum wave period increases with increasing flow depth and
decreasing channel slope. Unlike the transition between diffusion- and dynamic-wave
equations, however, the transition between kinematic- and diffusion-wave equations will
almost certainly be observed for streams of small- to intermediate-sized watersheds.
Moramarco and Singh (2000) extended the work of Ponce et al. (1978) by
subjecting a simplified dimensionless form of the momentum equation to quantitative
linear analysis. Three dimensionless parameters were derived in terms of the Froude
number and geometric characteristics of the river that permited quantification of the
influence of inertia and pressure in the momentum equation for various flow conditions.
Their results agree closely with those presented above; dynamic and diffusion waves
occur on the lowest slopes and can be separated in terms of the Froude number, and
kinematic waves are found on slopes greater than about 0.01, therefore having a wide
range of application (figure 2.4). For the range of slopes (0.001 - 0.05) and flow depths
(< 3) that are likely in channel networks of intermediate-sized arid and semi-arid
watersheds, diffusive and kinematic waves are likely to occur. Dynamic waves may
occur under certain circumstances, but should not be expected.
38
25.00
Channel slope = 0.01, Flow velocity = 0.3 m/s
20.00
o
0
;
15.00
Kinematic
a.
o
co
>
10.00 —
E
Diffusion
'"E 5.00
0.00
0
1
2
3
4
5
Flow depth (m)
A
50.00
45.00 -
Flow depth = 0.3 m, Flow velocity = 0.3 m/s
a
-
o
40.00
35.00
'c 30.00 a)
a.
25.00
20.00 E 15.00 Kinematic
"E 10.00 5
5.00
Diffusion
0.00
0
0.005
0.01
0.015
0.02 0.025
0.03 0.035
0.04
0.045
Channel Slope
Figure 2.3. Graphs illustrating the minimum wave period required for the kinematic
wave model to be 95% as accurate as the diffusion wave model. Values are shown for a
range of (A) flow depths, and (B) channel slopes.
39
Figure 2.4. Graph showing division of the slope/reference water depth space into four
regions characterized by different wave types for a channel length of 1000 meters and a
Froude number of 0.4. From Moramarco and Singh (2000).
Moussa and Bocquillon (2000) evaluated the approximation zones of the SaintVenant equations given variable overbank flow. The terms from the momentum equation
were analyzed as a function of the balance between friction and inertia using linear
perturbation theory. Flood waves were expressed in terms of three non-dimensionalised
variables: the Froude number, a dimensionless wave number, and the ratio, n, between
the width of the flooded zone and the width of the main channel. As expected, large
flood-plain widths introduced convective and inertial accelerations that restricted the
domain of application of the diffusive and kinematic wave models (figure 2.5).
Consideration of flood plains is necessary for the simulation of geomorphic change,
strengthening the argument for having diffusive wave routing in K2.
40
Fo-F.A.ONINO1g.y.)
1000
Tv.W2A1/1=
t
100
Gravity ware
Full
10
Salnt-Venant
system
1,ticn1c0.1
Kinematic
WA 1/111
hacr2K0.1
hs/o21‘0.1
lukal<0.1
0.1
0.01 0.01
0.1
1
100
1000
Dimensionless wave period
Full
Salnt-Venant
system
1000
Figure 2.5. Graphs showing river wave approximation zones obtained from the analysis
of the momentum equation for three values of n = 1, 8, and 20. From Moussa and
Bocquillon (2000).
41
The above evaluations of the flow conditions under which the approximations of
the Saint-Venant equations are appropriate considers only the relative magnitude of the
various terms in the momentum equation. The manner in which the momentum and
continuity equations are solved, however, can alter the approximation zones significantly.
Using the kinematic wave model as an example, two case studies are particularly
illustrative of the impacts of solution techniques on the model results.
Miller (1984) compared kinematic and dynamic wave models for a sinusoidal
hydrograph in a hypothetical channel reach, and for flow conditions meeting the accuracy
criteria of Woolhiser and Ligget (1967), and Ponce et al. (1978) for kinematic routing.
The kinematic-wave model was solved using the method of characteristics, and the
dynamic-wave model was solved using a linear implicit finite-difference technique. A
hydrograph was routed downstream using both models for a distance of approximately
6.4 km to demonstrate that small errors incurred in the kinematic-wave approximation
continuously accumulate as the flood wave travels downstream. The results are shown in
figure 2.6, where it can be seen that the Ponce et al. (1978) criterion for 95% accuracy
after one wave period (equation 2.7), appears to be correct. As the hydrograph is routed
beyond one wave period, however, the error continues to accumulate until the solution
becomes unreasonable. In addition to pointing out that errors associated with the
kinematic wave approximation can be propagated downstream, this study highlights the
importance of the solution method. The method of characteristics used to solve the
kinematic wave equations for this study provides nearly exact solutions and therefore
does not account for attenuation of the hydrograph as a result of diffusion and dispersion.
42
800
700
600
Kinematic
wave model
500
400
300
I
200
0
5000
10,000
/
15,000
I
20,000
i
25,000
I
30,000
I
35,000
40,000
TIME, IN SECONDS
Figure 2.6. Graph showing a comparison of kinematic- and dynamic-wave results that
demonstrating the propagation of errors in the kinematic wave solution. From Miller
(1984).
In contrast to Miller (1984), Zoppou and 0-Neill (1982) successfully applied the
kinematic wave model in situations for which the acceptance criteria of Ponce et al.
(1978) were violated. A four-point fully implicit finite-difference scheme (Preissmann,
1961) was used to solve the dynamic, diffusion, and kinematic wave equations for a 33.2
km reach of the River Yarra, near Melbourne Australia. Two measured floods were
routed through this reach, both of which should have been characterized by diffusive
waves according to the Ponce et al. (1978) criteria. Despite this, however, the kinematic
43
wave model performed just as well as both the diffusion and dynamic wave models. The
authors attribute this good agreement to specific characteristics of the channel reach in
question, and suggest that the Ponce et al. (1978) criteria be reevaluated or abandoned for
natural channels. The analysis is flawed, however, in that the solution technique is
responsible for the good performance of the kinematic wave model in a situation that
should not be characterized by kinematic waves. Numerous authors have demonstrated
that finite difference methods introduce significant numerical diffusion and dispersion
into the kinematic wave solutions. Hromadka and DeVries (1988), for example,
demonstrated that solving the kinematic wave equations with finite difference methods
could introduce substantial errors depending on the space and time steps used in the
computations (figure 2.7).
INFLOW IlYtOMPIAM
LO
3.5
40
Figure 2.7. Graph showing the influence of varying the time and space step during finitedifference solutions of the kinematic wave equation. From Hromadka and DeVries
(1988).
44
Confusion over the difference between flow conditions characterized by
kinematic waves and model conditions in which the kinematic wave equations may be
applied was largely put to rest by Ponce (1991). He argued that the artificial numerical
effects of using finite difference solution techniques are substantial enough to represent
another model parameter that requires calibration. He went so far as to state that "...it
seems pointless to try to "calibrate" a kinematic wave model by varying a physical
parameter such as Manning's n in order to match calculated results and observed data.
This practice amounts to curve-fitting; at best it is good conceptual modeling, but it
should not be interpreted as deterministic modeling." (p. 515). Given the extensive
discussion following the article in which this appeared it probably could have been
worded more judiciously, but the point remains: numerical diffusion can produce correct
results for the wrong reasons. Physically-based flow routing with finite difference
solutions of the kinematic wave model is not possible. Instead, Ponce (1991) proposed
extending kinematic wave theory to encompass the related diffusion wave theory. Three
methods were suggested to solve the diffusion wave equation without introducing further
grid-dependent numerical diffusion:
1. Analytic solutions, leading to Hayami's (1951) diffusion-analogy solution for
flood waves.
2. Numerical schemes for parabolic equations, such as the Crank-Nicolson scheme
(Crandall, 1956).
3. Extending the finite difference solution of the kinematic wave to the realm of
diffusion waves by matching physical and numerical diffusivities (Cunge, 1969;
45
Dooge, 1973). When the Muskingum scheme is used in this manner to model the
kinematic wave, the extended method is the well-known Muskingum-Cunge
model.
The third method has been explored at length by Ponce and others, and represents the
most practical means of accounting for diffusive flow in ephemeral watershed models
such as K2. It does not require a downstream boundary condition, yet it can match
physical and numerical diffusivity to ensure that flood-wave diffusion is not simply a
function of the selected time and space steps. As a result, this method is discussed in
greater detail in the following section.
2.3.2 Diffusion wave modeling with the Muskingum-Cunge scheme
Previously described routing models can be broadly classified as hydraulic, or
process-type models; they are based on the solution of the partial differential equations of
unsteady open channel flow, which are also referred to as the St. Venant, or dynamic
wave equations. Muskingum-Cunge routing falls into another class of routing models
called hydrologic routing models. Hydrologic routing uses a conceptual or systems
approach by employing the continuity equation and an analytical or empirical relation
between storage within a reach and discharge at its outlet. The distinction between
hydraulic and hydrologic models is appropriate for well-defined regular channels, but for
natural rivers the distinction is less obvious. The complex physical properties of natural
river systems defy exact mathematical representation, and any form of modeling
therefore necessarily involves a conceptual element (Weinmann and Laurenson, 1979).
46
The Muskingum method is based on the assumption of a linear relation between
the inflow, I, the outflow, 0, and the reach storage, V, that can be defined as
V = K[X/ + (1– X)0]
(2.8)
where K and X are the parameters. In the original Muskingum method these parameters
were determined through calibration to measured inflow and outflow hydrographs. The
formula for the Muskingum method is
Q n++11 = Qin e2 Q
j
r; + I
+
e3 Q n+1
i
(2.9)
where the routing coefficients are defined as
At
—
CI =
+ 2X
(2.10)
K
\
— + 2(1– X)
At
— – 2X
C2 – "
J
— + 2(1
(2.11)
–
X)
\ At
2(1– X )– —
c3= At
(2.12)
—+2k1-
The Muskingum-Cunge channel routing technique (Cunge, 1969) improved
Muskingum routing by computing K and X based on physical properties of the reach. As
such, it can be generally described as a nonlinear coefficient method based on physical
channel properties. The parameters K and X are calculated as
(2.13)
47
1
X=—1
2
q
S o cAx i
(2.14)
where zlx= reach length; c = flood wave celerity; q = unit width discharge; and So=
channel bed slope. While this allowed K and X to be evaluated in the absence of
measured inflow and outflow hydrographs, the parameters were constant, and could not
vary with the stage and celerity of passing flows, or along the channel with changes in its
geometry. The assumption of constant parameters made the solution highly dependent on
the reference values used to evaluate them (Dooge, 1973).
Ponce and Yevjevich (1978) expressed the routing parameters of the MuskingumCunge method in terms of two physically and numerically meaningful values that could
vary in space and time: the Courant and cell Reynolds numbers. These were defined,
resepectivel y, as
cA t
C
L. =- -1
AX
(2.15)
q
S
D=
cdx
(2.16)
'
By substituting (2.13 and 2.14) into (2.10 — 2.12) and solving for the routing coefficients
in terms of C and D yields
Ci =
C2 =
C3 =
1+C —D
1+C+D
(2.17)
—1+C+D
1+C+D
(2.18)
1—C+D
1+C+D
(2.19)
48
When lateral inflows are present a fourth routing coefficient is needed
C, = 2C
1+C+D
(2.20)
To allow C and D to vary in time and space the values of c and q were
recomputed for each time and space step by solving
dQ
c=—
dA
(2.21)
q=
(2.22)
-
B
where Q= total discharge, A = flow area, and B= top width. Four methods of solving for
c and q were proposed, culminating in the modified iterative four-point variable-
parameter method (MVPMC4; Ponce and Chaganti, 1994). In this method the routing
parameters C and D are computed for each computational cell based on the average unitwidth discharge at four points in time and space as
(
qa
=
a n + n n +1 n n+1
"/./
4
,n++11)
(2.23)
where the space-time discretization is defined in figure 2.8. Iterative solution is required
because the unit-width discharge at time n+1 and space j+1 is unknown. By computing
the average celerity, ca , based on the discharge q a , the routing parameters C and D are
thus based on the properties of the flow at the time and space step in which they are
actually applied. This nonlinear solution method significantly enhanced the technique's
applicability to real-world routing problems.
49
•
At
Q;
•
•
Q in±i
Ax
Figure 2.8. Space-time discretization of Ponce and Yevjevich (1978).
With a nonlinear solution that matches physical and numerical diffusion, the
Muskingum-Cunge technique is essentially a diffusion wave model. Variable parameter
Muskingum-Cunge routing has been shown to compare well against the full unsteady
flow equations over a wide range of flow conditions (Ponce, 1981; Brunner, 1989). In
addition, it has been demonstrated that the technique is independent of the user-specified
computational interval, Ax (Ponce and Chaganti, 1994).
The major limitations of the MVPMC4 technique are that it cannot account for
backwater effects, and the method begins to diverge from the full unsteady flow solution
when very rapidly rising hydrographs are routed through channels with slopes of less
than about 0.00019 (USAGE, 1994). Neither of these conditions are prevalent, however,
in small- to intermediate-scale watersheds, particularly in arid and semi-arid regions.
50
2.3.3 Practical Considerations
Solving the numerical equations for overland and channel flow requires
simplification of complex, irregular watershed geometry. Grid and conceptual methods
are the two means of accomplishing this (Singh, 1996). Representing watershed
geometry using a regular grid can describe slope irregularities on uplands. At the
resolution of most hydrologic model grids, however, the small surface irregularities that
influence overland flow are not captured. Spatial variation of soil and vegetation
parameters is also better represented using the grid method, but again these are cellaveraged values, and commonly at a coarser resolution than the elevation grid.
Conceptual methods that divide a basin into model elements, such as the cascading planes
in K2, further average out spatial variability in watershed geometry and hydrologic
parameters, but are computationally much more efficient and therefore more easily
applied to large basins. Either of these geometric representations may be referred to as
quasi distributed, and it has not been shown that one performs better than the other.
Geometric simplification of the channels, for watershed models that contain a
channel routing component, is universally achieved by representing them with a series of
cross sections. If natural channels are represented by closely spaced variable crosssections (using breakpoint data) then irregularities in the channel geometry are more
likely to produce conditions where the acceleration terms are significant. For generalized
trapezoidal cross-sections that are placed further apart, however, the channel irregularities
are averaged, acceleration terms will be small and the full equations will perform no
better than a simplification of them. In watershed modeling it is often impractical to
51
gather detailed cross-section information, and most models use generalized trapezoidal
channel geometries. This does not diminish the need, however, to simulate diffusive
flood wave attenuation.
2.4 Routing in AGWA
Channel routing in K2 is based on a four-point finite difference solution of the
kinematic wave approximation of the one-dimensional continuity and momentum
equations for open channel flow. This routing method is ideal for overland flow, and
flow in short, steep channel networks, but model efficiency has been shown to decline
significantly when watershed size increases beyond about 100 km 2 (e.g. Syed, 1999).
Because runoff hydrographs from watersheds of this size are dominated by channel
processes, it seems likely that using routing algorithms that more completely describe
channel flow processes would improve model performance in larger areas. Upgrading
the channel routing algorithms may not eliminate the decline in model efficiency with
increasing watershed size, but it is likely that a model accounting for attenuation and
translation of flow hydrographs could be applicable to larger watersheds than one that
does not.
Channel routing in SWAT is accomplished through pseudo channels using the
variable storage routing, or Muskingum river routing methods. These are both
hydrologic methods, and provide less room for improving process representation. In
addition, the large time step used in SWAT (1 day) precludes a detailed representation of
flood hydrographs. Daily runoff totals are routed downstream as a volume using a
52
velocity calculated from Manning's equation. This works well for large areas where
discharge can be considered to be approximately uniform during each 24 hour period, but
without reducing the time step at which calculations are made it would be difficult to
improve hydrograph representation.
The objective of improving hydrograph representation in intermediate-scale
watersheds can be accomplished using either diffusion- or dynamic-wave routing, both of
which can account for hydrograph attenuation and translation. Based on the previous
discussion of the conditions in which each of these is appropriate, however, a diffusion
wave model seems adequate for small- and medium-sized watersheds, which are broadly
defined as having average sizes of 25 and 500 km 2 , respectively (Pilgrim and Cordery,
1993). Local accelerations that are accounted for only in the full dynamic-wave models
are less likely to be significant at these scales, which are characterized by relatively
steeper slopes and shallower flows than larger watersheds. In addition, the problems
associated with solving the full dynamic wave equations in channel networks with steep,
initially dry channel networks that may be characterized by transcritical flows make the
dynamic wave model less practical to implement in K2.
2.5 Model Description
Variable parameter Muskingum-Cunge (MVPMC4) was selected as the most
suitable method of implementing diffusion-wave routing within the K2 conceptual
model. By adopting the K2 conceptual model and numerical framework it was possible
to retain the well-established functionality of the model, and simply replace kinematic
53
with MVPMC4 routing for the channels. In addition, by holding everything else constant
it is possible to facilitate direct comparison of the two channel routing techniques at
multiple scales, which is the expressed objective of this exercise.
Implementation of MVPMC4 routing followed the same procedure and spacetime discretization as described in section 2.3.2. The K2 numerical framework contains
an outer time loop within which flow at each computational node is computed. Reaches
are divided into 20 spatial nodes at which routing calculations are excuted, regardless of
reach length. Flow at each spatial node is computed by the iterative 4-point technique,
for which iteration ceases when the 4-point average discharge is the same before and after
the routing calculation (i.e. Aga —> 0). To improve the convergence of the iterative
procedure, the discharge VI++; obtained from the 3-point method (known values) is used
in the first iteration.
Celerity and unit-width discharge are computed for each iteration to ensure that
they are representative of the flow. The top width of the flow is calculated based on the
normal depth computed by means of a separate iterative procedure retained from the
kinematic routing that determines the flow depth. It was necessary to retain this
calculation in the model because flow depth and wetted perimeter are used in the
infiltration and sediment routing subroutines.
As with kinematic routing, MVPMC4 routing is sensitive to the Courant number
and errors can occur if it gets too high. MVPMC4 routing works best when the Courant
number is kept as close to 1 as possible, or between 0.5 and 2 in practice. For this reason
the Courant time step adjustment loop from K2 was retained, and is used to adjust the
54
time step downwards when the Courant number exceeds 2. For initially dry channels,
however, it was not possible to meet the lower limit; for very small discharges the
Courant number is routinely less than 0.5. This may result in a small amount of
numerical dispersion for the smallest flows.
2.6 MVPMC4 Instability at Low Flow
Under certain conditions, generally smaller flows, the MVPMC4 routing develops
an instability problem that results in negative outflows and a wildly oscilating outflow
hydrograph. This problem was originally discovered when a calibrated parameter file for
Walnut Gulch was used to simulate design storm runoff hydrographs with both K2 and
the modified K2 with MVPMC4 routing (K2MC). Hydrographs from both models
predicted very low discharges, but those simulated by K2MC displayed wild and
sometimes growing oscillations. Although volume balances remained reasonable, and
the discharges were low enough to be inconsequential in the larger scheme of runoff
events, the underlying problem seemed significant enough to merit correction.
A single 4-element subcatchment (3 planes and a channel) was used to investigate
the problem further. This parameter file was run with a series of design storms from the
5-year 30-minute to the 100-year 1-hour. Oscillations developed under all but the 100year events with MVPMC4 routing, and never when kinematic routing was used. To
isolate the source of the oscillations the parameter file was simplified further; twoelement (1 upland plane and a channel) and three-element (2 lateral planes and a channel)
parameter files were developed and run with the series of design storm events.
55
Oscillations were not observed for any of the events using the two- and three-element
parameter files, so it was concluded that the problem somehow arose when hydrographs
from upstream and lateral inflows were combined in the channel.
Further investigation using the four-element model revealed that the problem
developed when either precipitation decreased or channel length (and transmission
losses) increased to the point where flows in the channel were very small — less than
about 1 m 3 s -1 . By comparing outflow hydrographs from K2 for the same events it
became obvious that under these conditions hydrographs from upstream and lateral
inflows separate in time to yield two distinct peaks in the outflow hydrograph. K2MC
was unable to resolve this separation and developed oscillations that increased
downstream until the initial dip between the two hydrographs had passed the last spatial
node of the reach. Generally, the receding limb of the hydrograph was unaffected.
Owing to the number of time and space steps involved in computing the routing
for a reach, it was difficult to isolate the source of the problem. Extensive use was made
of the Compaq array viewer provided with Visual Fortran 6 to visualize the flow profile
at each time step. By using this tool it was possible to see that the oscillations started at
the leading edge of the upstream inflows, and became amplified as the flood wave moved
downstream through time. These same initial oscillations show up in kinematic routing,
but are smoothed through time and eventually disappear. Upstream inflows, because they
are added at a single section, are significantly higher than the lateral inflows, which are
averaged over the reach. As a result, flow in node 1 could be two or more orders of
magnitude higher than in node 2, creating a major discontinuity in the flow profile.
56
Initial attempts to define further how and where the problem developed centered
around defining the Courant and cell Reynolds numbers, and comparing them (and other
derived quantities) with accuracy criteria published in the literature. Specifically,
accuracy criteria of Koussis (1976), Ponce and Theurer (1982), and Szel and Gaspar
(2000) were evaluated, as well as general recommendations by Ponce (1989; and SDSU,
written communication, 2003). Unfortunately, the results were more confusing than
diagnostic. Accuracy criteria were always met during the peak of the hydrograph, but
were violated during the rising and falling limbs. Increasing the time step was originally
considered (space step is fixed within a simulation) to bring up the Courant number
during low flow. However, further investigation demonstrated that accuracy criteria were
also violated during the larger events when oscillations were not observed. In addition,
increasing the time step during the rising limb when changes in discharge are large
seemed likely to introduce additional error as a result of inadequately representing the
hydrograph. It was concluded that if the accuracy criteria were really the source of the
numerical oscillations then the oscillations should be developing whenever the criteria
were violated, which was clearly not the case.
A hybrid kinematic-MVPMC4 routing model was developed in an attempt to
implement kinematic routing in situations where the accuracy criteria were violated.
Implementing kinematic routing when the maximum Courant number for a time step was
less than 0.5 did not resolve the problem. The code was then modified to implement
kinematic routing within a time step, when the Courant number at any given node
dropped below 0.5, but this too failed to prevent oscillations from developing.
57
Owing to a logic error in the development of a switching mechanism between
MVPMC4 and kinematic routing, it was discovered that the oscillations do not develop if
kinematic routing is commenced one spatial node downstream of where the problem
starts. With this logic error the routing became overwhelmingly kinematic, but the
results did indicate that if kinematic routing could be implemented in the node before the
discontinuity that the problem might be resolved. To accomplish this, the routing
subroutine was rewritten to allow a spatial backstep in the calculations for each time step.
Upon identification of the discontinuity in the MVPMC4 routing at a certain spatial node,
the program backs up a node and recomputes the profile using kinematic routing.
Whereas this did correct the problem it produced a hydrograph that was arguably
indistinguishable from pure kinematic routing.
A more acceptable methodology of preventing the numerical oscillations from
developing was ultimately discovered in the VPMC routing section of HEC-HMS code
provided by William Sharffenberg from the US Army Corps of Engineers' Hydrologic
Engineering Center. For steep hydrographs entering flat-sloped channels, the original
developers of the HMS Muskingum-Cunge routing algorithm observed the same
oscillations, and developed an adjustment to the C1 routing coefficient based on extensive
numerical experiments where the MC method was compared to the St. Venant solution.
Their solution involves modification of the routing coefficients when C1 is computed to
be less than 0 by the following:
C2 = C 2 + 0.5C1(2.24)
C3 = C3 + 0.5C1(2.25)
58
Following these adjustments to the C2 and C3 parameters CI is set to O. This adjustment
to the routing coefficients was found to be the best way to conserve mass and yield routed
hydrographs close to the St. Venant solution. That work was summarized for inclusion in
an official Corps of Engineers engineering manual. Later it was converted to an
appendix before being dropped entirely. It was never published.
When this simple fix was made it removed the numerical oscillations in the
hydrograph. It has the effect, however, of predicting slightly increased peak flows and
delaying the onset of the hydrograph, but the time to peak is approximately the same as
for kinematic routing. This effect is most pronounced for the smallest flows, which
require the most consistent modification of the routing coefficients, and decreases with
increasing discharge. Modifying the routing coefficients does not result in significant
volume balance errors, and the net effect on outflow hydrographs is negligible.
59
3.0 ROUTING MODEL RESULTS
3.1 Methodology
Modified variable parameter Muskingum-Cunge routing was compared with the
original kinematic routing of K2, hereafter referred to as the K2MC and K2 models,
respectively. Comparisons were based on a set of parameter files originally developed by
Syed (1999) for the Walnut Gulch subwatersheds 1, 6, 11, and Lucky Hills 104 (see
description in Chapter 1). These parameter files were selected primarily because they
had previously been used to investigate scale issues. Syed (1999) calibrated the WG11
watershed with marginal success, producing efficiencies of 0.46 and -1.1 for runoff
volume and peak flow, respectively, when applied to the validation data. Calibration for
WG1 was considerably less successful; efficiencies of 0 and -13.4 were reported. The
calibrations were carried out through the use of the K2 parameter multipliers that are
applied uniformly over the entire watershed, which did not modify the original values in
the parameter file. The WG6 and LH104 parameter files were not calibrated.
It was not deemed necessary to use the parameter multipliers of Syed (1999) for
WG1 and WG11, or calibrate the two remaining parameter files. The major objective of
the routing model comparison is to evaluate relative differences between the two methods
as a function of scale, for which calibration is not required. Indeed, if each parameter file
were calibrated separately for the two models, differences in computed hydrographs
could easily result from differences in the effectiveness of the calibration.
60
The comparison first involved running the parameter files with both K2 and
K2MC for a set of design storms ranging from the 5-year 30-minute to the 100-year 1hour events. This comparison was designed to establish if there were significant
differences between the simulated hydrographs as a function of scale. The second
comparison involved running both models for a series of rainfall events for which
observed runoff data were available. This comparison was designed to evaluate relative
performance of the routing techniques as a function of scale.
3.2 Design Storm Results
The four subwatersheds (WG1, WG6, WG11, and LH104) were subjected to six
design storm events: 5-year 30-minute, 5-year 1-hour, 10-year 30-minute, 10-year 1-
hour, 100-year 30-minute, and 100-year 1-hour. Design events represent uniform rainfall
over the entire watershed, so the impact of spatially-variable rainfall is removed. The
design storm events were derived using the AGWA design-storm generator based on
return-period depths observed at Walnut Gulch. AGWA reduces the depths based on an
area-reduction factor developed by Osborn et al. (1980). The design storms used in this
comparison, however, were not generated individually for each subwatershed. Instead,
the design events were generated for a watershed with an area of approximately 50
hectares, and the same events were used for all comparisons.
A variety of tests and metrics was used to evaluate differences (or lack thereof)
between the two simulated hydrographs for each event. The Komolgorov-Smimov (K-S)
test was used to determine if the hydrographs could be considered statistically different at
61
the 5% level of significance. The hydrographs were also compared in terms of runoff
volume, peak flow, hydrograph shape, and the timing of the onset of the hydrograph and
peak discharge.
3.2.1 Komolgorov-Smirnov Comparison
The K-S test is a nonparametric test (i.e. independent of sample distribution) that
is not affected by the magnitude of the values being compared. It is sensitive to
deviations in the tails of the distribution where frequencies are low, but the upper tail of a
cumulative plot of discharges represents the highest discharges, which also is important
from a hydrologic perspective.
The Komolgorov-Smirnov comparison was conducted using a two-tailed test and
a 5% level of significance. Simulated discharges from the K2 and K2MC outflow
hydrographs were first converted to cumulative form, normalized by the sample size.
The K-S statistic is the maximum difference between these two, and is compared to
critical values for a specific level of significance. The null hypothesis for a two-tailed KS test states that both samples have the same continuous distribution.
Results of the K-S testing show that runoff hydrographs simulated by kinematic
and MVPMC4 routing are statistically indistinguishable in most cases. Significant
differences between the outflow hydrographs were observed for the smallest events in the
largest watersheds (table 3.1). This is in keeping with the hypothesis that directly
accounting for flood-wave diffusion will improve hydrograph representation as
watershed size increases, although these results indicate only that outflow hydrographs
62
become statistically different as watershed size increases. The impact of event magnitude
can be explained by referring to figure 2-5. As discharge increases the nondimensionalised wave period increases, which pushes the system from the realm of
diffusion waves into that of kinematic waves.
Table 3.1. Results of the Komolgorov-Smirnov tests indicating whether the null
hypothesis was accepted or rejected. Results are shown for the six design-storm
events ranging from 5-year 30-minute (5_30) to 100-year 1-hour (100_1) for each
subwatershed.
10_1
100_30 100_1
10_30
5_1
5_30
Watershed Area (ha)
LH104
4.7
Accept Accept
Accept
Accept
Accept
Accept
WG11
782
Accept
Accept
Accept
Accept
Accept
Accept
WG6
9558
Reject
Reject
Accept
Accept
Accept
Accept
WG1
14664
Reject
Reject
Accept
Accept
Accept
Accept
3.2.2 Metric and Visual Comparisons
Visual inspection of the simulated hydrographs for the design storm events shows
that the greatest differences occur for the smallest events on the largest watersheds.
Hydrographs for LH104 are virtually indistinguishable, whereas those for WG1 look
significantly different. Simulated hydrographs for WG6, intermediate between these two
endpoints, shows a clear transition from very similar hydrographs for the largest events to
substantially different hydrographs for the smallest events (figure 3.1). Simulated
hydrographs for LH104, WG11, and WG1 are presented in Appendix A.
63
0.9
08
0.7
0
05
04
0.3
02
01
40
250 -
200
-.--
K2
K2MC
150
50
Figure 3.1. K2 and K2MC simulated WG6 runoff hydrographs for design storms ranging
from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right).
Plots are discharge in cubic meters per second as a function of time in minutes.
64
The timing of the onset and peak discharges, runoff volume, peak discharge, and
mass-balance error were computed for each design-event simulation. Where appropriate,
the normalized difference was computed using the following
(3.1)
(K2MC_value — K2_value) / K2_value
where 'value' refers to the metric in question.
Differences in the timing of the onset of flow are presented in table 3.2 as just the
difference in onset time computed using the numerator of equation 2.26. As the table
shows, the onset of flow is basically identical between the two models, with the
maximum difference at any scale or event magnitude being 1 minute. There is no
apparent pattern to the differences, and one would not be expected. Diffusion effects
increase with travel time and distance, but the velocity of the wave front is primarily
controlled by channel slope and roughness.
Table 3.2. Difference between the timing of K2 and K2MC flow onset (mins).
Watershed
Area (ha)
5_30
5_1
10_30
10_1
100_30
100_1
LH104
4.7
0
0
-1
0
0
0
WG11
782.0
0
0
0
0
-1
0
WG6
9557.6
0
-1
0
-1
0
-1
WG1
14664.3
-1
0
0
0
0
0
The timing of peak discharge was substantially more variable than the onset of
flow (figure 3.2). Normalized differences in the timing of peak flow almost universally
increased with scale, and the change was most pronounced for the smallest events. With
only one exception, the peak flow occurred at the same time or earlier in K2MC. The
exception was the 100-year 30-minute event on WG1, for which it appears that a very
65
sharp and asymmetric peak in the kinematic routing was smoothed in the MVPMC4
routing, potentially due to flood-wave diffusion, which caused the peak to shift forward
in time.
20
10
-10
— it—
5_30
5_1
10_30
10_1
— 3K—
100 30
100_1
50
so
-70
0
2000
4000
6000
8000
10000
12000
14000
Area (ha)
Figure 3.2. Graph showing normalized differences in the time to peak discharge at the
watershed outlet between K2 and K2MC, plotted as a function of area. Negative values
indicate that K2MC predicts an earlier peak discharge than K2, and visa versa.
Normalized differences in the peak discharge were also quite sensitive to
watershed size and event magnitude (figure 3.3). For most events K2MC predicted
slightly lower peak flows than K2. As watershed size increased, however, K2MC
became more likely to predict a larger peak flow. As with the other metrics, differences
were most pronounced for the smallest events, and there was very little difference for the
largest events. Visual inspection of the hydrographs reveals that the smaller peak flows
do not appear to be the result of flood-wave attenuation due to diffusion. It is extremely
66
difficult, however, to resolve the cumulative impacts of diffusion in a distributed channel
network, particularly when the amount of numerical diffusion in K2 varies with the grid
spacing in each reach.
100
-
so
5_30
NE--
5_1
3.(--
10_1
10_30
so
—NE—
100_30
—EN—
100_1
40
20
o
-20
0
2000
4000
6000
8000
10000
12000
14000
Area (ha)
Figure 3.3. Graph showing normalized differences in the simulated peak discharge at the
watershed outlet between K2 and K2MC, plotted as a function of area. Negative values
indicate that K2MC predicts a smaller peak discharge than K2, and visa versa. Values in
excess of 100% were reset to 100%.
Differences in the runoff volume (water yield) mirror those associated with peak
discharge (figure 3.4). In most instances water yield is slightly lower in K2MC, with the
exception of values computed for the smallest events in the largest watersheds. Again,
differences increase with increasing watershed size and decreasing event magnitude.
67
100
-•-- 5_30
80
5_1
10_30
--34--
1 0_1
--*-- 100_30
60
-4---100_1
40
t
Z
i
20
-
_
----, -.
-20
2000
0
4000
6000
8000
10000
12000
14000
Watershed Area (ha)
Figure 3.4. Graph showing relative error in the total discharge volume at the watershed
outlet between K2 and K2MC as a function of area. Negative values indicate that K2MC
predicts less total runoff than K2, and visa versa.
The parallel trends in peak discharge and runoff volume suggest a problem with
the MVPMC4 routing technique. Ponce and Chaganti (1994) describe a small but
persistent loss of mass associated with the variable parameter methods. The loss of mass
is smallest for the MVPMC4 method, but is present none-the-less. This effect can be
seen in the mass-balance error presented in table 3.3 for each simulation. In most cases
mass-balance error is greater in the K2MC simulations. Error generally decreases with
increasing watershed size, but there is no discernable trend associated with event
magnitude (figure 3.5). For the K2 simulations, error increases with increasing
watershed size and decreasing event magnitude. Differences are thus most pronounced
for the smallest watersheds and largest events, which also represent those simulations for
68
which K2MC predicts slightly lower peak discharge and water yield. For smaller events
and larger watersheds the difference in mass balance error decreases, and ultimately
changes sign as error becomes greater in K2 (figure 3.5, and italicized values in table
3.3).
The loss of mass observed by Ponce and Chaganti (1994) was evaluated for four
different flows routed through a reach 800 kilometers in length with no lateral inflows.
They speculated that this would represent the worst-case scenario since most practical
routing applications would not consider such a long reach without intervening lateral
inflows, which tend to mask the accuracy of the computation. Actual losses were shown
to increase with increases in the ratio of peak inflow to baseflow, from 0.44% for a ratio
of 4 to 2.43% for a ratio of 20.
Table 3.3. Mass balance error (%) computed from K2 and K2MC output files.
Events for which K2MC had less error are italicized. Positive values indicate that
inflows exceeded outflows and storage (loss of mass).
100_1
100_30
10_1
5_1
10_30
Model
5_30
WS
-0.10
-0.21
-0.40
-0.08
LH104
-0.22
-0.16
K2
2.88
3.17
2.44
2.56
2.17
LH104
K2MC
2.12
0.20
0.53
0.88
0.42
WG11
0.49
K2
1.39
1.67
2.35
2.48
1.88
2.25
1.87
WG11
K2MC
0.33
0.06
1.08
0.39
0.68
WG6
K2
1.75
1.35
1.09
0.82
1.52
1.66
WG6
K2MC
0.80
0.12
0.63
0.30
1.46
2.38
WG1
K2
1.57
0.78
0.64
0.64
1.35
WG1
1.19
K2MC
1.75
69
2.5 4
2
1.5
0.5
o
-0.5
0
2000
4000
6000
8000
10000
12000
14000
Area (ha)
Figure 3.5. Graph showing the average trends in mass-balance error (%) as a function of
watershed area.
In the present application the ratio of peak inflow to baseflow would seem to be
infinite because the channels are initially dry. However, Ponce and Chaganti (1994) only
considered upstream inflows relative to what was already in the channel. In K2MC
lateral inflows add some water to the channel before the upstream flow arrives, and
definitely before the peak. The addition of all upstream inflows at a point, however,
guarantees that they will always be considerably greater than the lateral inflows, which
are distributed along the length of the channel. This makes the inflow-to-baseflow ratio
very large, and is a likely source of mass-balance error.
Interestingly, the conditions producing the greatest loss of mass are the same as
those described in section 2.5.1 that were the source of the numerical oscillations
observed in smaller events. The solution to this problem was to modify the routing
70
coefficients in situations when the problem developed. Although the modifications were
"found to be the best way to conserve mass and still produce routed hydrographs close to
the St. Venant solution" (Scharffenberg, 2003, personal communication), their specific
effects on mass conservation have not been documented. In addition, the modifications
are utilized most during smaller events, for which the hydrograph differences are most
pronounced. As such, these modifications may be contributing to the relatively large
K2MC mass balance error in the smaller watersheds.
The reduction in K2MC mass balance error with increasing watershed size (figure
3.5) may occur because the ratio of inflow discharge to that already in the channel
decreases in the downstream direction as lateral inflows are afforded more time to fill the
channel before upstream flows arrive. Loss of mass would be most pronounced for small
watersheds with only one channel, but would tend to be averaged with increasing
watershed size and greater complexity of the channel network.
3.3 Observed Storm Results
The observed-event simulations were conducted to provide a comparison between
kinematic and MVPMC4 routing in terms of their performance relative to measured
flows as a function of scale. Simulated and observed hydrographs are compared in terms
of shape and the timing of onset and peak flows (Appendix A). The timing comparisons
were possible through the establishment of event time for the observed hydrographs by
setting the time of flow onset equal to the time difference (in minutes) between the first
recorded rainfall and the first recorded runoff. It should be noted that timing errors of
71
approximately 5 to 10 minutes are not uncommon in the observed data. In the absence of
calibrated models, the following analysis emphasizes relative performance of the models
rather than the actual performance.
For WG1 seven runoff events were selected for the routing comparison. Peak
observed discharges ranged from 1.2 to --29 m 3 s -i , for rainfall depths ranging from 7.85
to 17.81 mm over the watershed. For comparison, the 5-year 30-minute design storm
event used in section 3.2 was 16.77 mm. In general, simulated runoff was substantially
lower than the observed runoff, and showed a great deal less temporal variability than the
observed runoff. The best, although very poor, results for WG1 are shown in figure 3.6,
which shows that only the highest peaks were represented in the simulated hydrographs.
In keeping with the design storm results, which demonstrated greater runoff in K2MC for
small events, simulated runoff was higher in K2MC for all of the WG1 events.
Potentially as a result of this, K2MC was more likely to simulate multiple peaks in the
runoff hydrograph.
For WG6 five events were simulated, with peak observed discharges ranging from
15 to 65 M 3 S -1 for total rainfall depths ranging from 10.81 to 23.09 mm over WG6. For
reference, the 5-year 1-hour event had a rainfall depth of 26.65 mm. In general,
simulated hydrographs were again smaller than the observed hydrographs, but the
difference between simulated and observed values was much smaller than for WG1
(figure 3.7). Again, K2MC predicted discharges that were consistently higher than those
from K2. Observed hydrographs showed significantly less temporal variation (i.e. fewer
distinct peaks), and simulated hydrographs from both models were much more successful
72
30
25
Observed
—K2
K2 MC
5
0
0
200
400
600
800
1000
1200
Time (min)
Figure 3.6. Graph showing the best match of simulated and observed hydrographs for
WG1 (14664 ha).
30
25
--•—• K2
K2MC
Observed
5
I.
0
50
100
150
200
250
300
350
Time (min)
Figure 3.7. Graph showing the best match of simulated and observed hydrographs
for WG6 (9558 ha).
73
at reproducing it. Simulated hydrographs from K2MC, however, were more likely to
capture distinct peaks in the observed hydrograph.
For WG11, five events were simulated with observed peak discharges ranging
from 13 to 25 m 3 s -1 , for total rainfall depths ranging from 23.12 to 31.91 mm over WG11.
For reference, the 10-year 30-minute design event had a rainfall depth of 28.64 mm.
Simulated hydrographs for WG11 consistently predicted greater than observed runoff,
and results from both models were very comparable in terms of peak discharge (figure
3.8). K2 and K2MC were equally successful at reproducing the shape of the observed
hydrograph.
For LH104, five events were simulated, with observed peak discharges ranging
from 0.06 to 0.4 M 3 S 1 for total rainfall depths ranging from 12.86 to 37.57 mm over
LH104. For reference, the 10-year 1-hour design event had a rainfall depth of 33.31 mm.
Simulated runoff for LH104 consistently overpredicted observed values, and by a much
more substantial margin than for WG11 (figure 3.9). Observed hydrograph shapes were
best represented at this scale, and all significant peaks in the observed hydrographs were
represented by the models. K2 and K2MC predicted almost identical hydrographs for all
of the events.
74
25
fi-451
K2
20
K2fVC
Observed
5
0
ottmm
•
70
110
90
130
150
Time (min)
170
190
Figure 3.8. Graph showing the best match of simulated and observed hydrographs
for WGII (782 ha).
0.3
0.25
00
K2
K2MC
Observed
0.2
1 015
a
0.1
0.05
o
90
110
150
130
170
190
Time (min)
Figure 3.9. Graph showing the best match of simulated and observed hydrographs for
LH 1 04 (4.7 ha).
75
Simulated and observed hydrograph shapes were compared using the NashSutcliffe statistic, or model efficiency (ME) calculation, which is essentially a measure of
goodness of fit between simulated and observed discharges. The Nash-Sutcliffe (1970)
statistic is calculated as
ME = 1
(3.2)
where
9. = observation at time i
= simulated value at time i
= mean of all observed values
A model efficiency of 1 indicates that the modeled hydrograph is in perfect agreement
with the observed hydrograph. If the value is negative, then the mean value of observed
discharge would be a better predictor of observed runoff than the predicted discharge.
Results of the uncalibrated K2 and K2MC model efficiency calculations show
that model efficiency increases with increasing watershed size for both models (figure
3.10). Without model calibration, however, the absolute values for ME are less important
than the average difference between model efficiencies at each scale. As can be seen in
figure 3.10, K2MC model efficiencies are greater (or less negative) at all scales, but the
magnitude of the difference shows no consistent relationship to scale.
76
0.5
0
—a-- K2
—a—
K2MC
3
-3.5
0
2000
4000
6000
8000
10000
12000
14000
Area (ha)
Figure 3.10. Graph showing average model efficiency plotted as a function of
watershed area for K2 and K2MC.
Error in the timing of the onset of flow consistently, and almost linearly,
increased with watershed area for K2. K2MC, however, showed a less consistent
relationship to scale (figure 3.11). Despite variation at the intermediate scales, error at
the smallest scale was approximately the same as at the largest scale; just the sign is
different. In addition, the difference in onset error increases from almost 0 for the
smallest watershed to a maximum of approximately 60% for the largest watershed.
Average error in the timing of the peak discharge increased for both models as a
function of scale (figure 3.12). Increases, however, were significantly more pronounced
for K2, which had a maximum error of 184% for WG1 versus a K2MC error of 24%.
The difference between K2 and K2MC peak error also increased with increasing
watershed size, from almost 0 for LH104 to 160% for WG1.
77
80
70
60
50
j 40
al
"Zil
cn
oc
30
20
10
0
-10
-20
0
2000
4000
6000
8000
10000
12000
14000
Area (ha)
Figure 3.11. Graph showing average error (%) in the timing of flow onset at the
watershed outlet plotted as a function of watershed area for K2 and K2MC.
100
90
—.—
80
K2
—s— K2MC
70
60
50
40
30
20
10
0
-10
,
0
2000
4000
6000
8000
10000
12000
14000
Area (ha)
Figure 3.12. Graph showing average error (%) in the timing of peak flows at the
watershed outlet plotted as a function of watershed area for K2 and K2MC.
Values in excess of 100% were reduced to 100%.
78
3.4 Discussion
A comparison of the K2 and K2MC models was conducted to evaluate their
general behavior and performance as a function of scale. Analysis of the design storm
simulations demonstrated that K2 and K2MC could only be statistically distinguished for
the smallest events on the largest watersheds (4 of the 24 simulations). Although it was
the 5-year return period events that were statistically different for WG6 and WG1, this
was an underestimation. No area-reduction was used when the design storms were
generated, so the rainfall depth associated with the 5-year return period events on WG6
and WG1 (the two largest watersheds) is comparable in magnitude to the 10-year events
when area reduction is accounted for.
A comparison of the peak discharge, its timing, and runoff volume showed the
greatest differences between K2 and K2MC under the same circumstances. Most
notable, however, was that K2MC consistently predicted earlier and larger peaks, and
greater runoff volumes than did K2.
An assessment of model mass-balance error showed opposite trends in model
performance as a function of scale. K2MC average mass balance error was greatest for
the smallest watersheds; it consistently decreased with increasing watershed area. K2
showed the opposite trend, with mass balance error increasing above that reported by
K2MC for the largest watershed (WG1).
Analysis of the observed events demonstrated that K2MC simulated hydrographs
are marginally superior to those from K2 at approximating the shape of observed
hydrographs at all scales. K2 demonstrated consistent increases in the error of its
79
hydrograph onset timing with increasing scale. In contrast, K2MC showed no consistent
relationship between onset timing error and watershed scale. Differences between K2
and K2MC onset timing error, however, increased with scale. Peak discharge timing was
most poorly simulated by both models at the largest scale. The relative difference
between K2 and K2MC time to peak error, however, increases with watershed size, with
K2MC yielding the largest improvement for WG1.
The overestimation of observed runoff for small watersheds and underestimation
for large watersheds by both K2 and K2MC demonstrates that the models do not describe
runoff processes equally well at all scales. This problem is analogous to the decrease in
runoff associated with watershed geometric simplification and information entropy loss
(e.g. Miller, 2002). A contributing (channel) source area (CSA) of 0.5% of the watershed
area was used during the discretization of all watersheds. The actual CSA thus varied
from 0.02 ha for LH104 to 75 ha for WG1. Miller (2002) demonstrated that runoff
volume decreased with increasing CSA due to significant information entropy loss
among model input parameters. Strong correlations were found between simulated
runoff decreases and information entropy loss associated with Manning's roughness,
saturated hydraulic conductivity, and slope as model element size increased.
Not considered in the analysis of Miller (2002) was the representation of specific
process relationships at different scales. At the smallest scales representation of
hillslopes as planar features with uniform overland flow seems reasonable. As element
size increases, however, concentrated (rill) flow becomes more important, and the
assumption of uniform planar flow is less suitable (e.g. Willgoose and Kuczera, 1995).
80
This observation was the primary motivation behind the development of a
microtopography feature in K2 to permit flow concentration in rills, but this feature has
not been tested against measured data. Flow concentrated in rills prior to its entering the
channel system reduces upland infiltration, and results in more water being conveyed to
the channel network and ultimately the watershed outlet. The combined effect of lower
average slopes for larger model elements, and the assumption of uniform planar flow
over those slopes thus is likely to be the main cause of runoff underestimation with
increasing watershed scale.
These complementary analyses demonstrate that:
(1) MVPMC4 routing yields significantly different results from those of kinematic
routing for the vast majority of all events on watersheds greater than about 100 km 2 .
(2) The difference between hydrographs simulated with MVPMC4 and kinematic routing
makes the MVPMC4 routing more successful at replicating observed hydrographs for
watersheds larger than about 100 km2.
81
4.0 GEOMORPHIC MODEL
4.1 Channel-Morphologic Models
A major objective of river and watershed managers is to simulate the morphologic
response of channels to disturbance. Knowledge of how a channel will adjust its width,
depth, and slope to changing water and sediment supply is crucial to the protection of
water, biological, and cultural resources. Countless models have been developed with
this purpose in mind, and many have been successful at predicting morphologic change
under specific circumstances. A problem common to most of these, however, has been
simulating changes in channel width.
Despite years of research on the physics of channel width adjustment, no
physically-based models of channel-width adjustment can account for overbank flow,
vegetation impacts, or channel constriction. In addition, river width adjustment models
are subject to the same types of limitations as are watershed models. Complex models
describing physical processes in detail have been the most successful in simulating
certain types of river width adjustment, but are so data intensive that they can only be
applied to isolated channel reaches. Models applicable to larger areas necessarily
simplify a system conceptually to reduce the data requirements and computational cost,
but model output can only be as good as the conceptual framework from which it is
derived. Generally, this means that river width adjustment is computed on a reach basis,
and that it cannot account for unidirectional width adjustment (i.e. on the outside of
meander bends).
82
Complex, multi-dimensional flow models cannot be realistically applied to
watersheds, so the opportunities for simulating river width adjustment are limited.
Simplified bank stability algorithms (e.g. Vieira and Wu, 2000) can be applied with onedimensional hydraulic flow models such as K2. These are designed to compute bank
stability in terms of a factor of safety which, when exceeded, results in bank failure and
channel widening. Bank failure algorithms are usually combined with an empirical
method derived by Arulanandan et al. (1980) for computing bank toe erosion in onedimensional flows. Bank toe erosion thus provides a means of oversteepening banks to
cause failure, and a method of removing failed bank material deposited at the bank toe.
The bank stability approach requires initial and simulated pore water pressure in the
banks, making it useful in continuous simulation models such as CCI-1E1D (Vieira and
Wu, 2000), which track soil moisture between events.
The only other method for simulating channel morphologic change in onedimensional routing models is the use of extremal hypotheses in an equilibrium approach.
Extremal hypotheses offer a third equation (in addition to the momentum and continuity
equations) that can be used to complete the set of equations needed to solve for channel
width, depth, and slope. In applying an extremal hypothesis it is assumed that a channel
achieves stability or equilibrium when a specified function of some combination of the
hydraulic variables (water discharge, sediment discharge, sediment size, and channel
width, depth, and slope) has an extremum (maximum or minimum). A variety of
extremal hypotheses has been proposed in the literature, including: minimum stream
power (Chang, 1980), minimum unit stream power (Yang and Song, 1979), maximum
83
friction factor (Davies and Sutherland, 1980), and maximum sediment transport rate
(White et al., 1982). Of these, the minimum stream power (MSP) theory, a special case
of the minimum rate of energy dissipation theory (Yang et al., 1981), has received the
most attention. It has been applied in two widely used river simulation models: the
Generalized Stream Tube model for Alluvial River Simulation model, GSTARS (e.g.
Yang and Simôes, 1998); and the FLUVIAL-12 Mathematical Model for Alluvial
Channels (e.g. Chang, 1982). The MSP hypothesis is stated as (Chang, 1980, p. 1445):
For an alluvial channel, the necessary and sufficient condition of equilibrium
occurs when the stream power per unit channel length yQS is a minimum subject
to given constraints. Hence, an alluvial channel with water discharge Q, and
sediment (discharge), Qs, as independent variables, tends to establish its width
(B), depth (D) and slope (S) such that yQS is a minimum. Since Q is a given
parameter, minimum yQS also means minimum channel slope.
The quantity yQS has the dimension of energy per unit time per unit length, in which y is
the specific weight of the fluid (water).
The equilibrium methods offer predictions of the magnitude, rather than the rate
of width adjustment, which is the principal conceptual objection to channel morphologic
models based on extremal hypotheses (ASCE, 1998). When applied in GSTARS,
however, channel geometry is adjusted at each time step to produce the lowest possible
stream power given the potential geometries that could result from the computed change
in sediment storage. The rate of width/depth adjustment is thus governed by the rate of
erosion/deposition in the channel during a given time step. Limits on the amount of
lateral change are also imposed.
Width change in GSTARS is limited by the computed change in sediment storage
in stream tubes bordering the channel banks (Yang and Simoes, 1998). Change in
84
sediment storage computed away from the banks can be used only for vertical change.
The quasi-two dimensional approach to flow routing thus provides a mechanism for
limiting erosion and deposition on the channel banks.
By accounting for channel curvature and associated transverse energy gradients,
Fluvial-12 simulates variable erosion/deposition along a cross-section despite its onedimensional model framework. As with GSTARS, this approach limits width change.
Width adjustment as a result of bank erosion is also limited by an empirical bank
erodibility factor that controls the rate of bank erosion.
A major criticism of extremal hypotheses in general is that they lead to
conclusions that are incompatible with observations when combined with conventional
sediment transport and flow resistance equations (Griffiths, 1984). Specifically, for wide,
straight, unconstrained alluvial reaches in equilibrium it can be shown that the
combination of hypotheses and equations leads to the result that the Einstein sediment
discharge, 41), and the Shields entrainment function, 1/w, are nearly constant. According
to Griffiths (1984), the results of flume and field observations do not agree with this
conclusion, but it is not clear what effect this has on computed equilibrium geometries.
The American Society of Civil Engineers' (ASCE, 1998) task committee on hydraulics,
bank mechanics, and modeling of river width adjustment did not raise this issue in its
review of river width adjustment models based on extremal hypotheses.
Despite the aforementioned criticism of the MSP technique, numerous case
studies demonstrating its utility have been reported in the literature. Chang (1982)
demonstrated that the concept of minimum stream power could be successfully used as a
85
physical principle governing the morphology of alluvial streams when applied within a
water and sediment routing model. Both channel profile and width changes were very
closely predicted during a 35-hour flood in 1978 that caused significant geomorphic
change in the simulated reach. Song et al. (1995) were equally successful at simulating
channel degradation and widening in the earthen spillway of the Lake Mescalero Dam
and Dike, New Mexico, during a flood in 1984. By conducting the same simulation with
and without stream power minimization Song et al. (1995) demonstrated that accounting
for lateral sediment transport improved estimates of total sediment yield.
4.2 Channel Network Evolution (Surficial Process) Models
Channel network evolution models represent another major type of geomorphic
model. Unlike the morphologic models that concentrate on describing channel
properties, the evolution models are designed to simulate evolution of the larger channel
network and surrounding uplands on geologic time scales. This type of model is aimed at
evaluating landscape evolution in response to changing climatic, geologic, and
biophysical conditions.
Distributed, physically-based models of drainage basin evolution have been
steadily improving since the late 1980s when Roth et al. (1989) simulated the erosional
development of drainage patterns. In their treatment, overland flow over a flat, tilted
surface is modeled two-dimensionally according to the normal flow hypothesis, which
perfectly balances the vector of flow resistance with the vector of gravitational force on
the surface. Deterministic equations of mass and momentum conservation are coupled
86
with random fluctuations of sediment entrainment and transportation to provide an active
source of channelization (Roth et al., 1989). This approach produces topographic lows
that connect in the downstream direction and have a similar appearance to the early
stages of drainage basin formation.
Willgoose et al. (1991a,b) developed a numerical model capable of simulating the
long-term evolution of drainage basins. Their treatment of sediment transport is similar
to that of Roth et al. (1989), but a channelization equation is used to govern the
development and extension of channels. This equation is adapted from one developed by
Meinhardt (1982) to differentiate leaf-vein formation, and is based on the
phenomenology of channel-head extension. The channelization equation provides a
physically based mechanism for channel initiation and growth, but is not based on the
controlling transport physics at the channel head.
Howard (1994) developed a purely erosional treatment of overland flow that
eliminates assumptions based on analogies to physically different problems. His model is
again based on flow and sediment transport equations similar to those used by Roth et al.
(1989), but it employs a much different method of transition between channelized and
unchannelized surfaces. A cell in the simulation matrix is converted from nonalluvial to
alluvial if the actual bedload rate exceeds the potential rate. This mechanism works
because the model assumes that the system is detachment-limited, and therefore each cell
is capable of transporting all of the sediment supplied to it. Howard (1994) justifies this
assumption for simulating fluvial erosion of natural slopes and headwater channels
because they are dominantly detachment-limited.
87
Although the temporal resolution of channel evolution models is inadequate to
determine relatively short-term impacts on water quality and riparian condition, they
possess important features that can benefit channel morphologic models when applied on
a watershed scale. In particular, channel evolution models have necessarily paid
significant attention to channel extension (headward erosion) and channel slope.
Channel extension through headward erosion can be an important geomorphic
process on relatively short time scales. Extensive gullying on Walnut Gulch, for
instance, continues following a 15-year erosional episode beginning in about 1930 (Graf,
1983a). Whereas the vast majority of sediment generated during the erosional episode
was derived from channels with large contributing areas, the effects of the change in base
level that resulted continue. Unchannelized swales have transformed into valleys with
small incised channels (gullies) as headcuts have migrated upstream. The cumulative
impacts of sediment erosion by this process are affecting present morphologic response
downstream, where sediment is now accumulating. Representation of channel extension
is thus an important component of evaluating morphologic response on a watershed scale.
Channel slope is a product of the delicate balance between sediment supply,
transport, and deposition throughout the channel network. It is significantly influenced
by sediment grain size distribution, channel roughness, geological and biological
controls, contributing area, climate, and host of other factors. Regardless of the extent to
which these are accurately represented in geomorphic models, their single most important
feature is the ability to adjust slope dynamically in response to variable water and
sediment discharges. Many watershed models proclaiming a geomorphic component
88
presume that the timescale over which channel geometry changes occur is orders of
magnitude greater than the timescale of event discharges, and that morphologic change
can thus be ignored during event simulations (e.g. K2). This assumption, however,
breaks down as event magnitude increases and flows become capable of substantially
modifying the channel morphology.
4.3 Watershed Geomorphic Models
To understand how an individual stream reach responds to external stresses it is
necessary to study the channel network as a whole (e.g. Schumm et al., 1987). This
revelation has been the primary motivation for the development of watershed geomorphic
models, which allow individual reaches to respond independently to flows within a
dynamic, linked system. As described in section 4.2, many of the watershed-based
geomorphic models have been developed using a surficial-processes approach for
application to research questions about drainage basin evolution over long periods of
time. The other approach has been to combine channel morphologic models with
watershed runoff and erosion models, where the later provide input to the former. This
hybrid approach carries the advantage of being able to simulate short-term changes in
channel morphology that incorporate the distributed impacts of upland and upstream
disturbance. The hybrid approach necessarily incorporates the use of geographic
information systems (GIS) for watershed and channel network delineation and to assist
with model parameterization.
89
Two examples of the hybrid approach have gained notoriety in recent years: the
U.S. Army Corps of Engineers CASC2D model (Ogden and Heilig, 2001), which has
been incorporated into the Watershed Modeling System (WMS) hydrologic model
interface; and the One-Dimensional Channel Network Model CCRE1D (Vieira and Wu,
2002), developed at the National Center for Computational Hydroscience and
Engineering.
CASC2D combines a two-dimensional grid-based overland-flow routing scheme
with one-dimensional grid-based channel routing based on the diffusion- or dynamicwave equations. The channel network in CASC2D is represented as a series of links and
nodes, where a link is a channel segment, or reach, between two or more computational
nodes, which are located a the center points of grid cells. Cross-sections, defined at links,
can only be represented by trapezoids when routing sand-sized sediment. No mention is
made of overbank routing, so this component of geomorphic systems is not considered.
The potential to compute channel morphologic change in CASC2D is limited.
Erosional and depositional changes in channel bed elevation modify channel slope, but
erosion can occur only on the channel bed (figure 4.1). It does not appear that channel
side slopes are modified as a result of vertical change, and there is no mechanism for
bank failure or channel widening.
90
Figure 4.1. Schematic representation of erodible trapezoidal channels in CASC2D.
Despite the capability of CASC2D to simulate continuously, it has not been used
to simulate cumulative geomorphic change. Indeed, the model seems to have included
sediment primarily for water-quality assessment, with minimal geomorphic adjustments
primarily to improve the accuracy of sediment-discharge calculations. The strength of
this model, however, lie in the robustness of the hydrologic model, particularly its
integration of two-dimensional overland flow and one-dimensional flow routing in the
channel network, and in the grid-based representation of the channel network. Twodimensional overland flow routing should improve the model's ability to represent
landscape change relative to models that represent uplands as planes, although the extent
to which this might be true has not been investigated. The diffusion- or dynamic-wave
channelized-flow routing improves the model's ability to simulate runoff from large
watersheds. The grid-based representation of the channel network yields numerous short
reaches that afford the model greater flexibility to adjust channel gradient in response to
localized deposition and erosion.
CCHElD is perhaps the most comprehensive watershed geomorphic model
available. It was developed as a detailed, continuous simulation channel-network model
91
with flow routing based on the diffusion- or dynamic-wave equations. The channel
network is represented as a series of links, where a link is either a hydraulic structure or a
single channel consisting of several reaches. Reaches are defined as the channel segment
between two cross-sections, or computational nodes. Channels are represented by
breakpoint data, where bank tops must be specified in the event of overbank flow.
Variable deposition/erosion along a cross-section is facilitated by variable flow
depths along the section, and through an empirical equation for computing bank toe
erosion (Arulanandan et al., 1980). Bank slopes can thus increase as a result of channel
degradation, or lateral erosion at the bank toe. Bank failure is computed when stability
criteria are exceeded, and sediment is deposited at the bank toe.
CCHElD was successfully used to predict channel profile adjustments over an
18-year period in the Goodwin Creek Watershed in Mississippi. Measured and observed
profile adjustments during the 18-year period following the installation of a measuring
flume are remarkably well matched (figure 4.2), with very good representation of erosion
at the upstream end of the reach and deposition downstream in the backwater behind the
measuring flume.
92
Measures] in he_ 1972
•Measured
Ln Feb., 1992
buaal newel in Calculaboa. ha.. 1971
74
CAL, Wu et al s Fartnaia. Feb.. 1992
CAL, SEDTRA Module, Fob., 1992
- - - - C. Engelund-Hanwn. Feb., 1992
72
70
68
66
0
100)15002(100
2500
3000
Distance Upstream of Flume No, I (in)
Figure 4.2. Profile adjustment over 18 years in the Goodwin Creek Watershed.
Simulated profiles are shown in colors other than blue. The final observed profile
is represented by orange dots. From http://hydra.cche.olemiss.edu/ccheld/.
A major drawback of CCHElD is that it is a channel-network model. The
watershed runoff and erosion model, AGNPS (Agricultural Non-Point Source Pollution
Model), or an equivalent must be used to generate water and sediment inflows at the
upstream end of each tributary in the network. Efforts to improve this cumbersome
procedure are underway, but ideally there would be seamless interaction between the
upland and channel network models as in CASC2D.
CCHElD is designed for use in humid regions characterized by perennial flows,
and requires a downstream boundary condition for the entire simulation period. As such,
it is not possible to apply the model in arid-region watersheds with ephemeral flows.
Another significant point is that approximately 5 years after its initial release CCHElD
3.0 is still available only for beta-testing to researchers and engineers who sign a Beta-
93
Testing Agreement with NCCHE, and only the computer server at NCCHE can be used
to perform simulations. Practical application of the tool is thus difficult.
4.3 Approach
To address the stated objective of developing a geomorphic model for arid-region
watersheds, components of both channel morphologic and channel network evolution
models were combined. Rather than adapting a channel-network model to incorporate
distributed inputs from a watershed model, as was done in CCHE1D, a distributed
watershed runoff and erosion model was adapted to compute channel morphologic
change in the channel network. The model selected for this purpose was the USDA-ARS
Kinematic Runoff and Erosion model (K2), which has been widely and successfully used
in small watersheds characterized predominantly by overland flow.
The numerical model, hereafter referred to as K2G, incorporated a variety of
modifications for the purpose of simulating channel morphologic change:
•
Computation of channel slope based on channel-bed elevation at the reach
endpoints to enable dynamic adjustment of slope in response to deposition/erosion
•
Tracking of available sediment up to a maximum erodible depth
•
Calculation and tracking of channel geometric change in response to changing
sediment storage
•
Calculation of width and depth changes
•
Enabling bank failure if banks are oversteepened
94
The theory of total stream power minimization was adopted to compute changes in
channel width in the absence of information about the lateral distribution of
deposition/erosion along a cross-section. In addition, this technique requires no
additional parameter information beyond that of the parent model, K2. More specifically,
information about pore-water pressure and channel-bank material is not required.
To facilitate the assessment of geomorphic change at the watershed scale, it was
necessary to develop a GIS-based component that could work in concert with K2G as
part of a larger assessment tool. This component was developed based on the Automated
Geospatial Watershed Assessment tool (AGWA), which was customized to work with the
geomorphic model. The GIS interface, hereafter referred to as AGWA-G, coordinates
watershed and channel-network delineation, model parameterization, model simulations,
spatial visualization of model results, and the comparison of multiple simulation results
based on different initial conditions. The following sections describe the K2-based
numerical model and the AG WA-based modeling tool.
4.5 Numerical Model
The K2G geomorphic model was developed within the framework of K2. The
channelized flow routing subroutine of K2 was modified to translate change in sediment
storage into channel geometric change that can be tracked from one simulation (event) to
the next. Because K2G channels are trapezoidal, changes in channel geometry are
limited. A collection of prescribed changes, however, was developed to maximize the
flexibility of geometric adjustment within the trapezoidal framework. Channels are
95
allowed to adjust by changing their width, depth, bank slopes, horizontal bank widths,
and channel-bed elevations at both ends of each reach. In addition, a maximum erodible
depth has been implemented below which the channel can no longer erode vertically.
Slope is uniform for any given reach, and is allowed to vary as a function of elevation
changes at either end of the reach. All other parameters are required as input and
computed independently at either end of the reach. With the exception of slope,
geometric parameters vary linearly along the reach from the values provided at the
upstream end to those downstream. Bank slopes may be different on either side of the
channel if data are provided by the user, but because flow is one-dimensional (i.e. no
lateral variation in flow), volumetric change on either side of the reach is uniform.
The original intent of this research was to incorporate the MVPMC4 diffusionwave routing described in Chapters 3 and 4 into the geomorphic model. It was not
possible to do so, however, without enabling flow in compound channels. Compound
channels, or channels with flood plains, are an essential component of the geomorphic
model for two reasons. First, flood plains represent reservoirs of stored sediment, and the
conceptual model would be incomplete if it failed to account for changes in flood-plain
sediment storage. Second, when flood plains are not present the hydrologic model
extends the channel banks vertically to contain all flows. This can cause erroneously
high stream power when flow depth exceeds the bank tops, which results in exaggerated
channel incision. Despite a concerted effort to implement compound channels in K2MC,
it could not be accomplished in the available time. The original kinematic-wave routing
of K2 was thus retained in K2G to permit the use of compound channels.
96
Overbank areas in K2G can be specified by the user by entering their width, side
slope, and a lateral slope (figure 4.3). As in K2, overbank areas can only be represented
on one side of the channel. Although not required, the use of compound channels is
highly recommended when there is any potential for overbank flow. If overbank
parameters are not provided K2G arbitrarily extends bank tops to contain the flow if its
depth exceeds the channel depth. This can lead to inflated estimates of total stream
power, and hence erroneous geomorphic adjustments. Overbank widths are adjusted in
proportion to the computed channel top-width change, and overbank elevation (slope) is
allowed to vary independently from channel slope.
SS I
Depth
Width
Width (overbank)
Figure 4.3. Conceptualization of a compound channel in K2G showing separate channel
and overbank areas.
Geomorphic change is computed following each computational time step a the
upstream and downstream ends of each reach. K2G computes stage, discharge, and
change in sediment storage for each of 20 computational nodes in a channel reach during
each time step. Given the computed volume change of sediment storage it is possible to
affect only a limited amount of geometric adjustment. To determine what portion of the
change in sediment storage should be applied to the bed and banks, the total stream
97
power is computed for a maximum of 100 different geometric permutations ranging from
all width change to all depth change.
Geometric adjustments commence by applying all of the available sediment
volume towards raising or lowering the channel bed. With each successive iteration the
volume applied to vertical change is decreased by 1% and the volume applied to
horizontal change is increased by the same amount until the maximum permissible
volume of sediment has been applied to horizontal change. For each iteration the flow
depth and velocity are recomputed based on the discharge at that time step and channel
geometry for that iteration. Using these values total stream power is computed for the
iteration by integrating stream power along the reach:
CDT
j
= •Wsdx
(4.1)
where OT represents the total stream power. This expression is discretized following
Chang (1982) as:
=E0.5r(asi+Qi+isi+JAxi
(4.2)
where N is the number of nodes/stations along the reach, dx, is the distance between
stations I and I + 1, y is the specific weight of water, and S is the energy gradient. The
rate of energy dissipation due to sediment movement is neglected. The energy gradient is
computed using the following:
S
= Z, + U12 ( Z, + U '2+ 1Ax
2g
2g \
(4.3)
98
where Z is the water-surface elevation, U is the average velocity, and g is the acceleration
due to gravity. The geometry that results in the minimum total stream power is adopted,
and geometric parameters are updated for the next time step.
Geometric change through total stream power minimization is computed only at
the second and second-to-last computational nodes in the reach (nodes 2 and 19). The
primary reason that nodes 2 and 19 are used instead of nodes 1 and 20 is that the energy
gradients at the first and last nodes of the reach cannot be evaluated given the K2G
conceptualization of the channel network as a series of connected but independent
reaches. Deposition is computed for the intervals between nodes and is reported for the
downstream node of each interval; change in sediment storage is not reported at node 1.
Geometric change is not computed in the middle of the reach to minimize the input
requirements and computational expense of applying the model to large areas. Instead,
all geometric parameters are interpolated linearly between the 2 nd and 19 th nodes and to
reach endpoints before each time step.
As a result of the linear interpolation of geometric parameters between the reach
endpoints, the computed volume of geometric change for a reach can be severely
distorted, relative to the modeled net change in sediment storage, if deposition at nodes 2
and 19 is considered representative of deposition along the entire reach. Because K2G is
a transport-capacity model, it contains no mechanism other than settling velocity to limit
the rate of deposition when the sediment load exceeds transport capacity. When transport
capacity is well below the inflowing sediment load, deposition thus predominantly
99
occurs at the upstream end of the reach, and decreases exponentially in the downstream
direction.
Figure 4.4 applies to a reach from Walnut Gulch Subwatershed 11 for which there
was approximately uniform erosion during the rising limb of a runoff event, followed by
deposition during the receding limb that was concentrated at the upstream end of the
reach. As a result, using change in sediment storage at the ends of the reach to compute
changes in channel geometry (that are interpolated linearly between the endpoints) causes
total geometric change that severely overestimates net deposition in the reach. To ensure
that the equivalent mass of geometric adjustments is equal to the net change in sediment
storage for the reach it is therefore necessary to compute an average trend in the change
in sediment storage along the reach.
The nonlinearity is always concentrated at the upstream end of the reach, so K2G
assumes that the change in sediment storage at the downstream end of the reach is
representative of the downstream change and this value is used to compute downstream
change in channel cross-sectional area. It is thus necessary to compute a change in cross-
sectional area at the upstream end of the reach that will yield a volumetric change, VG,
equal to the modeled change in sediment storage, Vm, when integrated along the reach.
To accomplish this, the change in modeled sediment storage for the time step is summed
for each node in the reach to obtain Vm. Because changes in geometric the geometric
parameters are linear along the reach, change in channel cross-sectional area can be
represented by a single value defined as
100
2.00E-01
—a—
150E-01
•
Channel Deposition
Overbank Deposition
Ts'
1.00E-01
Je
c n
5.00E-02
0
0.
&&&&&
a, 0.00E+00
•
•
-5.00E-02
-1.00E-01
4
10
7
13
16
19
Computational Node
Figure 4.4. Graph showing change in sediment storage for each
computational node in a reach (20 is the downstream end). Note that
change in sediment storage does not vary uniformly along the reach.
A Ave =
Au + A D
(4.4)
2
\
where Au and AD are the changes in upstream and downstream cross-sectional area. The
volumetric change resulting from geometric adjustment for the reach can thus be
expressed as
Vm =VG
=
AA„ •
L
(4.5)
where AA ve and L are the average change in channel cross-sectional area and reach length,
respectively. Solving (4.4) for Au yields
Au = 2. AA „, — AD(4.6)
By solving (4.5) for AA„ and substituting the result into (4.6) the change in upstream
cross-sectional area that will conserve mass for the reach is
101
Au =2
Vm
L
(4.7)
The area Au is used as a volume per unit length of channel to adjust channel crosssectional geometry at the upstream end of each reach. It should be noted that K2G
computes deposition as a mass (kg). This mass is converted to a volume in by assuming
an average particle density of 2.65 (g/cm 3 ) and a porosity of 0.40 (40%) for the deposited
material.
4.5.1 Channel adjustments — deposition
During deposition channel depth and/or width can decrease in proportion to the
volume of deposited sediment, resulting in the geometry that produces the minimum total
stream power. Derivations of the geometric adjustments to erosion are given in
Appendix B.
Channel width change as a result of bank deposition proceeds by simple bank
translation towards the center of the channel such that the volume of bank accretion
equals the amount allotted to producing width change (figure 4.5). The volumetric
change is computed such that half is added to each bank, which allows different bank side
slopes to be accommodated on either side of the channel. The change in channel width is
added to the overbank width to maintain the section width, which is defined as the sum of
the horizontal bank widths, the channel bed width, and the overbank width. For the same
reason, bank slopes and widths remain constant during channel narrowing.
102
Figure 4.5. Sketch illustrating depositional width reduction for the K2G
trapezoidal channels.
Deposition on the banks must occur along the entire bank height to maintain a
trapezoidal geometry. Bank deposition, however, is limited by the ratio of wetted bank
perimeter to wetted bed perimeter. This limitation is necessary to determine the
maximum portion of channel deposition computed by the model that can be deposited on
the banks if the resultant geometry causes a lower total stream power. It has the added
benefit of not allowing significant bank deposition until a significant portion of the banks
is submerged. The assumption embodied in this limitation is that bank deposition per
unit surface area, at a maximum, is equal to that in the channel bed. In reality bank
deposition is limited by the bank width and slope, which would tend to result in lower
deposition per unit surface area on the banks. Bank shear stresses, however, are also
likely to be lower than on the channel bed (on average), which would have the opposite
effect of resulting in greater deposition per unit surface area on the banks. When taken
together these two factors are assumed to offset each other, and provide a reasonable
maximum volume of sediment available for deposition on the banks.
Bank deposition (channel narrowing) is limited by a minimum width. Channels
are not allowed to narrow past 10% of the width computed as a function of contributing
103
area by means of a hydraulic-geometry relation developed by Miller (1995). This limit
ensures that channel width does not reach zero, but it was never reached during the
testing of the model.
Depth change as a result of deposition on the channel bed proceeds by reducing
the depth of the channel while keeping the bank widths constant (figure 4.6). As a result,
deposition reduces bank slopes, but channel bottom and top widths remain constant.
Figure 4.6. Sketch illustrating depositional depth reduction for the K2G
trapezoidal channels.
Deposition on the channel bed can proceed until a minimum depth of 10 cm is
reached. This depth does not vary in proportion to the channel contributing area, and is
based on the observation that defined channels are rarely much less than 10 cm deep.
Unlike the minimum width, the minimum-depth limit serves a major purpose in the
geomorphic model. Sediment available for vertical change after the minimum depth is
met is used to raise the elevation of the cross-section uniformly. This provides a means
of increasing channel slope, particularly for long channels, without causing model failure
because channel depth has reached zero. It also is intended to reproduce conditions
observed in depositional channels — channels become wide and shallow, or braided,
104
during extended periods of deposition. K2G mimics this process when used in
conjunction with compound channels; the minimum depth of 10 cm allows flows to
spread over a wide area while retaining a portion that is slightly deeper. This small inset
channel can contain smaller flows and continue to effect deposition on the channel banks
until the minimum width is reached. It also provides a focus for subsequent erosional
events and channel incision.
To illustrate the magnitude and manner of simulated depositional adjustments
during a single large event, adjustments were plotted for the upstream and downstream
ends of an individual reach (figure 4.7). The adjustments plotted in figure 4.7 resulted
from a 52-mm rainfall event on WG11 with a peak flow of 35.3 m 3 /s (61.9 mm/hr) for
the reach. They show non-uniform deposition for the reach, with approximately 0.04
meters of deposition upstream, and no significant change downstream. Channel width
decreased by approximately 0.05 meters upstream, and increased by approximately 0.003
meters downstream. Bank slopes decreased from 1.02 to 0.98 upstream as a result of the
depositional depth decrease, and did not change significantly downstream. Sediment was
deposited on the flood plain at both ends of the reach, but most significantly upstream.
All geomorphic adjustments represent net change; adjustments during an event are not
output by K2G.
105
1439.1
Upstream Adjustment
1439
1438.9
1438.8
1438.7
Final
1438.6
1438.5
1438.4
1438.3
1438.2
1438.1
0
2
4
6
10
8
12
A
1437.1
Downstream Adjustment
1437 1436.9 1436.8 1436.7 -
Final
1436.6 1436.5 1436.4 1436.3 1436.2
0
246
8
1012
Figure 4.7. Plots showing depositional adjustments at the upstream (A) and downstream
(B) ends of a 210-meter reach following a 52-mm rainfall event with a peak flow of 35.3
m 3 /s (61.9 mm/hr). Elevation (y-axis) and distance (x-axis) both have the units of meters.
106
4.5.2 Channel adjustments — erosion
During erosional events depth and/or width increases in proportion to the volume
of eroded sediment, resulting in the geometry that produces the minimum total stream
power. Derivations of the geometric adjustments to erosion are in Appendix B.
Channel-width change as a result of bank erosion proceeds by simple bank
translation outwards from the center of the channel such that the volume of material
removed from the banks equals the amount allotted to width change (figure 4.8). The
volumetric change is computed such that half of the volume is removed from each bank,
which allows different bank side slopes to be accommodated on either side of the
channel. The change in channel width is subtracted from the overbank width to maintain
section width. For the same reason, horizontal bank widths remain constant during
channel narrowing.
Figure 4.8. Sketch illustrating erosional width increase for the K2G
trapezoidal channels.
As with deposition, erosion on the banks must occur along the entire bank height
to maintain a trapezoidal geometry. Bank erosion, however, is limited by the ratio of
wetted bank perimeter to wetted bed perimeter. This limitation is necessary to
approximate the maximum portion of channel erosion computed by the model that can
come from the banks in the absence of local scour information. It has the added benefit
107
of not allowing significant bank erosion until a significant portion of the banks are
submerged. The assumption embodied in this limitation is that bank erosion per unit
surface area, at a maximum, is equal to that in the channel bed. In reality bank erosion is
limited by relatively higher shear strength of bank versus bed material, which results in
less erosion per unit surface area on the banks. In addition, average bank shear stresses
are lower than on the channel bed, which accentuates the preference for vertical as
opposed to lateral erosion. During erosional events, however, the tendency is to
minimize total stream power by incising the channel and reducing channel slope, so
lateral erosion is relatively insignificant except during the largest events when the banks
are overtopped.
Depth change resulting from erosion of the channel bed proceeds by increasing
the depth of the channel while keeping the bank widths constant (figure 4.9). As a result,
vertical erosion increases bank slopes, but channel bottom and top widths remain
constant. Allowing channel side slopes to vary in this fashion was also necessary
because without it vertical depth changes cause the channel bottom width to be
continually reduced until the section becomes a 'V'. This change in geometry causes
problems in the numerical model, and is poorly representative of natural channel incision.
If channel incision increases either or both channel side slopes beyond a certain
critical slope, then the bank(s) fail, top width is increased, and bank slope(s) are reduced.
The critical side slope is set to be 3.75 (-75'). If this slope is reached, the side slope is
reset at that section to 1.75 (-60°), which is the side slope associated with the "best
hydraulic section" described in basic hydraulics texts (e.g. Chang, 1988). The volume of
108
sediment removed from the banks is used to offset any volume that might have been
removed from the banks, and the remainder is used to raise the channel bed uniformly.
Figure 4.9. Sketch illustrating erosional depth increase for the K2G
trapezoidal channels.
Erosion of the channel bed can proceed until the maximum erodible depth (MED)
is reached. The MED defaults to 5 meters, but can be adjusted manually by the user by
adding it to the input parameter file. Once the MED is reached, further depth increases
are prevented. To limit the amount of erosion computed in the sediment routing
subroutine, a parameter (PAVE) is set to indicate that the portion of the channel
perimeter on the bed is not erodible. A limited amount of erosion can thus continue to
occur, and is manifested as channel widening.
To illustrate the magnitude and manner of simulated erosional adjustments during
a single large event, adjustments were plotted for the upstream and downstream ends of
an individual reach (figure 4.10). The adjustments plotted in figure 4.10 resulted from a
52-mm rainfall event on WG11 with a peak flow of 3.1 m 3 /s (129.7 mm/hr) for the
presented reach. They show approximately uniform erosion for the reach, with 0.067
109
1504 —
Upstream Adjustment
1503.9
1503.8
1503.7 -
1503.6
1503.5
1503.4
2
4
6
8
10
12
A
1502.3 -Downstream Adjustment
1502.2
1502.1
1502 -
1501.9 -
1501.8 -
1501.7
0
2
4
6
8
10
12
Figure 4.10. Plots showing erosional adjustments at the upstream (A) and downstream
(B) ends of a 25-meter reach following a 52 mm rainfall event with a peak flow of 3.1
m 3 /s (129.7 mm/hr). Elevation (y-axis) and distance (x-axis) both have the units of
meters.
110
meters of erosion upstream, and 0.065 meters downstream. Width changes are
approximately an order of magnitude smaller than the depth adjustments, and not readily
apparent. Bank slopes increased from 1.29 to 1.53 upstream, and from 1.06 to 1.29
downstream as the channel incised. No sediment was eroded or deposited on the flood
plain.
4.5.3 Erosion and Sediment Transport in K2G
The general equation used in K2, and K2G, to describe the sediment dynamics at
any point along a surface flow path is a mass-balance equation similar to that for
kinematic water flow (Bennett, 1974):
a(Ac
at
s
)
awc,
ax
= q s (x,t)
(4.8)
in which
= sediment concentration [L3112],
Q = water discharge rate [L3 /T],
A = cross sectional area of flow [L 2 ],
e = net rate of erosion of the soil bed [L2/1],
q s = rate of lateral sediment inflow for channels [L 3 /T/L].
For upland surfaces, e is assumed to result from two major components: soil
erosion by rain splash on bare soil; and hydraulic erosion (or deposition) due to the
interplay between the shearing force of water on the loose soil bed, and the tendency of
soil particles to settle under the force of gravity. Thus e may be positive (increasing
concentration in the water) or negative (decreasing concentration). The net-erosion rate
is a sum of the splash-erosion rate, es , and hydraulic-erosion rate, eh,
e = e, + eh
(4.9)
111
For a more detailed description of splash erosion in K2G, refer to Woolhiser et al. (1990),
from which much of this description was derived.
The hydraulic-erosion rate represents the rate of exchange of sediment between
the flowing water and the sediment over which it flows, and may be either positive or
negative. K2G assumes that any given flow condition (velocity, depth, etc.) has a
specific equilibrium concentration of sediment that can be carried if that flow continues
steadily. The hydraulic-erosion rate is estimated as being linearly dependent on the
difference between the equilibrium concentration and the current sediment concentration.
Hydraulic erosion/deposition is thus modeled as a kinetic-transfer process by
e h = c g—C )44
(4.10)
in which C„, is the concentration at equilibrium transport capacity, C, = C,(x,t) is the
current local sediment concentration, and cg is a transfer-rate coefficient [T -1 ]. When
deposition is occurring, cg is equal to the particle-settling velocity divided by the
hydraulic depth, h. For erosion conditions, cg is set by the user based on the properties of
the soil or channel sediment.
Transport capacity in K2G is computed using the Engelund and Hansen (1967)
total-load formula:
C„ = 5x104Suu
g 2 dh(S —1) 2
in which
S is specific gravity of the particles
u is velocity [L/T],
u * is shear velocity, defined as jghS , for which Sb is bed slope
(4.11)
112
g is acceleration due to gravity [LIT/TI
d is particle diameter [L],
h is water depth.[L]
This formula equates the work done by tractive forces to the potential energy gained by
the grains as they move up the stream face of bed dunes. The effective tractive force is
related to the total load using similarity considerations (Alonso et al., 1981). Although
the Engelund and Hansen formula was developed using limited flume data, experimental
work by Govers (1990) and others using shallow flows over soil has demonstrated
relations that are similar to the transport capacity relation of Engelund and Hansen
(1967). Alonso et al. (1981), in a broad comparison of transport formulas, found that the
Engelund-Hansen formula yielded the best results for concave field-plot tests. For
channel flow, Alonso et al. (1981) found that the Engelund-Hansen formula yielded
acceptable results without excessive scatter, although it tended to overestimate
systematically the field data and underestimate the flume data. For light-weight
materials, Alonso et al. (1981) found that the Engelund-Hansen formula was one of only
two formulas to replicate the data satisfactorily. For very fine soil particles, Alonso et al.
(1981) again found the Engelund-Hansen formula to be one of only two that gave
estimates close to the observed load.
Particle settling velocity is calculated from particle size and density, assuming the
particles have drag characteristics and terminal fall velocities similar to those of spheres
(Fair and Geyer, 1954). This relation is
2
v=
4 g(p —1)d
3
C E,
(4.12)
113
in which CD is the particle drag coefficient. The drag coefficient is a function of particle
Reynolds number,
24 3
C D =—++0.34
R,
(4.13)
in which R„ is the particle Reynolds number, defined as
=
d
(4.14)
where V is the kinematic viscosity of water [L2/1]. Settling velocity of a particle is found
by solving equations 4.12-14 for
The general approach to sediment-transport simulation for channels is nearly the
same as that for upland areas. The major difference in the equations is that splash erosion
(e s ) is neglected in channel flow, and the term q s becomes important in representing
lateral inflows. Equations 4.8 and 4.10 are equally applicable to either channel or
distributed surface flow.
The erosion computational scheme for any element uses the same time and space
steps employed by the numerical solution of the surface-water flow equations. In that
context, equations 4.8 and 4.10 are solved for Cs (x,t), starting at the first node below the
upstream boundary, and from the upstream conditions for channel elements. If there is
no inflow at the upper end of the channel, the transport capacity at the upper node is zero
and any lateral input of sediment is subject to deposition. The upper boundary condition
is then
114
Cs(0,t)=
qc
qs
+v s WB
(4.15)
where WB is the channel bottom width. A(x,t) and Q(x,t) are assumed known from the
surface-water solution.
Performance of the K2 sediment-routing model has been evaluated in several
studies, all of which found it to yield acceptable results. At the watershed scale, Smith et
al. (1999) tested K2 against a dataset from the 41.2 ha Catsop catchment in South
Limburg, Netherlands. Calibration of the model parameters was performed based on six
events, with regard to the temporal distribution of runoff and sediment rather than single
values such as total or peak rates. The authors concluded that the overall ability of the
model to reproduce the measured data was relatively good considering the limited data
regarding soil conditions (agricultural activity) between events. Results from one of the
validation runs for this study are plotted in figure 4.11. The hydrograph and sedigraph
were both well represented by K2. Although these results were obtained using a
calibrated model, they demonstrate that K2 represents well the runoff and sediment
erosion/transport process in small watersheds.
115
50
Event 22.01.93
— Rainratail
— Simulated Runoff Rate
Measured Runoff Rate
• • Simulated Sediment, right scale
Measured Sediment, right scale
!
*
!
50
150
100
Time from Start, min.
200
0
Figure 4.11. Graph showing measured and simulated runoff for an event on the
Catsop catchment in South Limburg, Netherlands. From Smith et al. (1999).
K2 has also been shown to perform well on a plot scale. Ziegler et al. (2001)
tested the performance of K2 on 3-m 2 unpaved road plots in Thailand using a rainfall
simulator. Following parameter optimization based on five events, they compared
simulated and observed (water) discharge, sediment-transport rate, and sediment
concentration for five additional events with variable plot slopes and initial soil-moisture
conditions. Average model efficiencies were 0.40 and 0.36 for sediment output and
concentration, respectively, for the validation runs. Results from three of the validation
events are presented graphically in figure 4.12. Based on these results, Ziegler et al.
(2001) concluded that although K2 performed well overall, it had the most difficulty in
predicting the time-dependent sediment outputs. Most notably, early flush peaks and the
temporal decay in sediment output were not predicted, owing to the inability of K2 to
model removal of a surface sediment layer of finite depth.
116
0 14 0 12
0 10*
r.
0.08
.
0.08
0.04 - 0 n
002 -
1.4.OE
ROACIS
c:: froaskre0
— XAEFIOrs2
iot)
80 r
O
KHOO
0 riamiut
KINE FK;r52
ROADS
fripWrIllitit
01404062
0
C
-40
s 15 25
25 45 S5 '55
lirra rrnrl
Comparison of discharge (Qt), sediment transport rate (St), and
Figure 4.12.
concentration (Ct) between measured rainfall simulation data (circles) and K2-predicted
values for three events on road plots. From Ziegler et al. (2001).
The results of Ziegler et al. (2001) demonstrate that K2 is capable of adequately
representing hydraulic and splash erosion processes at the plot scale, although they also
suggest that the model could benefit substantially from temporally variable surfaceerodibility parameters. For the purpose of the present study, the sediment-routing
component of K2 seems sufficiently robust to permit generalized comparisons of
geomorphic response to landscape change.
117
4.6 Geomorphic Modeling Tool
Model input parameters are developed through a customized GIS-based interface
developed specifically for this research. The interface is a modified version of the
Automated Geospatial Watershed Assessment tool (hereafter referred to as the AGWA
Geomorphic tool, AGWA-G) that develops input parameter sets and visualizes
distributed output specific to K2G. AGWA-G coordinates batch event simulations by
conducting separate, sequential simulations using each precipitation input (*.pre) file in a
specified directory and modifying input parameters between simulations based on
computed change from the previous simulation.
Channel geometric parameters are tracked between simulations by means of a
simple output text file containing the minimum output data needed to develop a modified
input parameter file for the next simulation. This output file is written to the directory
containing the K2G model executable. A sample element output from this file is shown
in table 4.1. The second column in table 4.1 contains parameter values for the upstream
end of the reach, and the third column contains parameter values for the downstream end
of the reach. Parameter names followed by an asterisk represent the minimum output
required to pass channel modifications to subsequent simulations. The remaining
parameters are used in AGWA-G to track and visualize cumulative change resulting from
multiple simulations. The last three rows are overbank parameters that are only recorded
for compound elements.
118
Table 4.1. Sample output for one channel element from the geom.out file
written during a K2G simulation. Parameter abbreviations are defined in
grey, and all units are in meters except slope, which is dimensionless. Four
significant digits are carried to minimize rounding error.
194
194
Elem* (c/cmcra /0)
1347.9685
1348.9463
Elev* (c/cvarion,
0.2313
dcpth
0.2337
Dpth*
1.1164
0.8760
Cwid* (chtlitn('l 11 0111)
/0/4
1.3602
/tic
.
n
bank
1.2990
1*
(
i
SS
5 S2* (ri,:;/// bank si ( h , ,/ op c )
1.3602
1.2990
0.0010
0.0008
TWA (rota/ wichh adji/Arnicia)
0.0162
0.0134
TDA (iota! cicp111
10.0000
10.0000
ED* (crodibie (/(7)///)
1.1144
Owid* (orci-b(///k 11 /(///i)
0.8744
0.0162
0.0134
chwincl (1(7)111
TCDA
.
,
,
,
.
inchtding (11 crb,1nk acpth
SOB (siminica ovcrbarik acpmition)
.
0.0000
0.0000
The reported depth and width adjustments (TWA, TDA, TCDA, and SOB) are
calculated at nodes 2 and 19, and do not necessarily equal the difference between input
and output widths and depths. The parameters required for subsequent simulations are
for the reach endpoints, and are extrapolated linearly from where they are computed at
nodes 2 and 19. It was not, however, deemed necessary to do this for the parameters not
required in subsequent simulations.
4.6.1 Profile Smoothing
The inherited conceptual model of KINEROS treats the channel network as a
series of connected but independent reaches. Outflows from upstream contributing
reaches are summed to determine inflow for a downstream reach, but that is the extent of
the interaction between reaches. As a result, flow at channel junctions, or confluences, is
poorly described by the model. Together with the KINEROS transport capacity relation
119
that exaggerates deposition or erosion immediately downstream of a channel junction,
this can result in substantially different elevation adjustments among the reaches
connecting at a junction. During a simulation the initially continuous profile thus can
become discontinuous, with abrupt drops or ridges at the channel junctions.
Whereas this poses no computational problems in the model, it results in
computed geomorphic change that is unreasonable when considering the entire channel
network. Figure 4.13, for example, shows the lower end of the channel profile in Walnut
Gulch subwatershed 223 before and after a batch simulation of five 10-year 1-hour events
without any external modification of the flow profile between simulations. It can be seen
that elevations diverge significantly between reaches as the slope is adjusted
independently for each reach in an attempt to convey the inflowing sediment.
To correct this problem an algorithm was developed to redistribute sediment
between the downstream ends of upstream reaches and the upstream ends of downstream
reaches, thereby ensuring that channel elevations are the same for all reaches connecting
at a junction. The algorithm is designed to determine the weighted average elevation that
produces a volume change downstream equal to the volume change in the upstream
contributing channel(s). For cases when downstream deposition has caused an abrupt
increase in elevation at the junction a volume of sediment is transferred from the
downstream reach to the upstream reaches such that the final elevations at the junction
are equal. Conversely, if erosion downstream of the junction produces a sharp drop in
the profile then a volume of sediment is transferred from the upstream reach(es) to the
downstream reach such that the final elevations at the junction are equal. The algorithm
120
1339.0
Original
1338.5
.
Final
1338.0
3: 1337.5
o
t 1337.0
1336.5
1336.0
1335.5
1300
1400
1350
1450
1500
Distance (m)
Figure 4.13. Graph showing the channel profile near the outlet of Walnut Gulch
Subwatershed 223 before and after a K2G batch simulation of five 10-year 1-hour
events and no profile smoothing between events. See text for discussion.
is applied to each reach for which the upstream elevation may need to be adjusted. First
order channels are not adjusted. Derivations for this volume adjustment are presented
below for the cases where there are one and two upstream contributing reaches,
respectively.
4.6.1.1 Two-Reach Junction Two-reach junctions are defined as junctions of one
-
upstream reach with one downstream reach (figure 4.14). This type of junction occurs
where a single long reach must be divided into two or more shorter reaches to accomplish
any of the following objectives:
• Model output is required for a location where the channel was not automatically
split, for instance at a gauging station
121
•
Reach characteristics vary significantly within a longer reach, and it must be split
into two or more shorter reaches with different geometric parameters to
adequately describe the variability.
•
A long reach must be divided into two shorter reaches to enhance the ability of
the model to adjust reach gradient through elevation changes at the endpoints
Figure 4.14 shows the initial endpoint elevations for an upstream reach (El & E2) and a
downstream reach (E3 E4) following a K2G simulation in which a lower slope in the
downstream reach resulted in preferential deposition at its upstream end (E3). The
objective of the profile-smoothing algorithm is to compute the weighted-average
elevation, EA, that results in equal volume changes in the upstream and downstream
reaches (i.e. VI = V2). This requirement preserves mass within the channel network
even though volume changes in individual reaches may result in a net change in mass.
Figure 4.14. Diagrammatic representation of two-reach junction profile adjustments
showing variables used in derivation of the adjusted elevation, EA. Elevations are
represented by symbols beginning with `E', volumes with 'V', and lengths with `L'.
122
Given upstream and downstream reach lengths of Li and L2, respectively, the
volumes can be calculated as:
V1= 0.5(W1)(L1)(EA — E2)
(4.16)
V2 = 0.5(W 2)(L2)(E3 — EA)
(4.17)
where WI and W2 are the average widths of the upstream and downstream reaches,
respectively. The average width for any given reach is determined by computing
summing the upstream and downstream widths, and dividing by two. Because erosion in
the downstream reach could produce the opposite situation, the volumes can be expressed
such that they are independently of sign by letting:
(4.18)
EC1=1EA— E21
EC2=1EA— E31
(4.19)
VI = 0.5(W1)(L1)(EC1)
(4.20)
V2 = 0.5(W 2)(L2)(EC2)
(4.21)
to get
By requiring the volume change upstream (V1) to be equal to the volume change
downstream (V2), equations 4.20 and 4.21 can be combined as
(W1)(L1)(EC1)= (W2)(L2)(EC2)
If ED is defined as
(4.23)
ED =1E3—E21
EC1+ EC2= ED
(4.22)
(4.24)
123
yielding two equations, 4.22 and 4.24, and two unknowns, EC1 and EC2. To solve for
EC I, 4.22 and 4.23 are combined to yield
(W1)(L1) EC2 ED—Ed ED
1
EC1
(W2)(L2) EC1
EC1
(4.25)
Let the ratio of widths and lengths, RWL be defined as
RWL=
(W 1)(L1)
(W 2)(L2)
(4.26)
ED
RWL+1
(4.27)
Substituting and solving (4.25) for EC1 yields
EC1=
Because EC1 is an absolute value,
EA= E2± EC1
(4.28)
To determine the value for the adjusted elevation, the two possibilities are compared to
the midpoint elevation between E2 and E3, which is computed as the algebraic average of
these two elevations. The value of EA that is closer to the average elevation is selected.
4.6.1.2 Three-Reach Junction - Three-reach junctions are defined as junctions of
two upstream reaches with one downstream reach. This type of junction is the most
common, and channels are always split at this type of junction. Whereas the case of three
upstream reaches is a possibility, and is permissible in the model, this type of junction is
almost never observed in nature, and is therefore not considered in this derivation.
Figure 4.15 shows the initial endpoint elevations for two upstream reaches (El
and E2) and a downstream reach (E3) following a K2G simulation in which a lower slope
124
in the downstream reach caused deposition at its upstream end (E3). The objective of the
profile-smoothing algorithm is to compute the weighted average elevation, EA, for which
the combined volume change as a result of elevation adjustments for the upstream
reaches equals the volume change as a result of elevation adjustment in the downstream
reach (i.e. V1 + V2 = V3). Again, this requirement is designed to preserve mass within
the larger channel network even though volume changes in individual reaches may result
in a net change in mass.
Figure 4.15. Diagrammatic representation of three-reach junction profile adjustments
showing variables used in derivation of the adjusted elevation, EA. Elevations are
represented by symbols beginning with `E', volumes with 'V', widths with 'W', and
lengths with `L'.
Following the same variable conventions as the two-reach junction scenario,
average width, length, and volume change are referred to as W, L, and V, respectively.
The two upstream reaches are denoted as reaches 1 and 2, and the downstream reach as 3.
125
As before, the volume adjustments can be calculated as
V1= 0.5(W1)(L1)(EC1)
(4.29)
V2 = 0.5(W2)(L2)(EC2)
(4.30)
V3 = 0.5(W3)(L3)(EC3)
(4.31)
where the elevation-change variables, EC, are absolute values of the difference between
the original and adjusted elevations. Given the preservation of mass requirement (VI +
V2 = V3), the three volumes can be rewritten as
(W1)(L1)(EC1)+ (W 2)(L2)(EC2) = (W3)(L3)(EC3) (4.32)
Equation 4.32 contains three unknowns, so two additional equations are needed to solve
it. From figure 4.15, EC1 and EC2 can be expressed in terms of the known original
elevation differences between the three points as
EC1= El — E31— EC3 = ED 1 _ 3 — EC3
EC 2 =1E2 — E31— EC3 = ED 2_3EG3
(4.33)
(4.34)
Substituting these into (4.32) yields
(W1)(L1)(ED 1 _ 3 — EC3)+ (W 2)(L2)(ED 2 _ 3 — EC3) = (W3)(L3)(EC3) (4.35)
Rearranging to get all EC3 values on one side of the equation yields
126
(W1)(L1)(ED,_ 3 )+ (W2)(L2)(ED 2 _ 3 ) = (W 1)(L1)(EC3)+ (W 2)(L2)(EC3)+ (W3)(L3)(EC3)
(4.36)
which can be simplified to
(W1)(L1)(ED 1 _ 3 )+ (W2)(L2)(ED 2 _ 3 ) = (EC3)[(W 1)(L1) + (W 2)(L2) + (W3)(L3)]
(4.37)
The final solution for EC3 is thus
EC3 =
(W1)(L1)(ED1 _ 3 )+ (W2)(L2)(ED 2 _ 3 )
1( 4 7 1)(L1) + (W 2)(L2) + (W 3)(L3)]
(4.38)
Because EC3 is an absolute value,
EA = E3 EC3
(4.39)
To determine the value for the adjusted elevation, the two possibilities are compared to
the algebraic average of the three elevations, El, E2, and E3. The value of EA that is
closer to the average elevation is selected.
4.6.1.3 Procedure — Junction-elevation adjustments are computed systematically
for each junction in the watershed. The algorithm iterates through each reach in the
channel network, and if the reach has at least one contributing reach then the adjustments
are carried out for the upstream end of the reach. Contributing reaches are queried from
the stream attribute table to get the parameters needed to compute the adjustment, and
adjustments to the downstream ends of the contributing channels are made at the same
time. The upstream ends of reaches with no upstream contributing reaches, and the
downstream end of the outlet reach are never adjusted during profile smoothing because
they are not located at junctions of 2 or more reaches.
127
In addition to adjusting elevations at the channel, or reach, junctions, several of
the other geometric parameters are adjusted to reflect the change in elevation. The
elevations represent channel-bed (thalweg) elevation, so reducing the channel-bed
elevation is translated into an increase in channel depth and a decrease in the erodible
depth by the same amount (as if the modeled infilling of the channel never happened).
Likewise, the following parameters used to track event and cumulative change in
AGWA-G are also modified by the same amount: channel depth adjustment (for the
simulation), cumulative channel depth adjustment (for the batch), total channel depth
adjustment (which includes overbank deposition/erosion for the simulation), and
cumulative total channel depth adjustment. To ensure that section widths do not change
(as in K2G), bank slopes are modified to reflect the new channel depth and a fixed
channel side width. Overbank parameters are not modified during the channel junction
elevation adjustments.
4.6.2 Results of Profile Smoothing
Adjusting reach junction elevations after and between simulations produces a
much more reasonable longitudinal profile than when elevations are not adjusted. Figure
4.16 shows the same section of the channel profile as figure 4.13 following a batch
simulation with the elevation adjustments (profile smoothing). The final profile
following this simulation is more reasonable from a geomorphic prospective, with profile
irregularities largely removed and a final profile with decreasing slope in the downstream
direction.
128
Profile smoothing causes significant sediment mass-balance errors in any given
reach. A portion of the sediment deposited in a downstream reach is transferred to the
upstream reach(es) to yield a common elevation at the junction. Whereas this seems
illogical, it takes sediment that is deposited in a very short section at the upstream end of
the downstream reach and redistributes a portion of it to the downstream end(s) of the
upstream reach(es). Mass is conserved during the adjustments, so there is no net effect
on the sediment balance for the channel network (see mass balance section below).
1339.0
Original
1338.5
-is
Final
1338.0
î 1337.5
o
LTI
1337.0
1336.5
1336.0
1335.5
1300
1350
1400
1450
1500
Distance (m)
Figure 4.16. Walnut Gulch Subwatershed 223 profile near the outlet before and
after a K2 0 batch simulation of five 10-year 1-hour events with profile smoothing
between events.
129
4.6.3 Mass Balance Calculations
At the end of each batch simulation AGWA-G computes the mass balance for
each channel, and for the entire channel network. This procedure involves (1) tracking
event and cumulative sediment output from each upland element (plane) in the
watershed, (2) computing change in sediment storage for each channel, and (3)
computing the sediment yield at the watershed outlet. To accomplish each of these tasks
it was necessary to create two additional database files for each batch simulation: one for
tracking event and cumulative outputs for the planes, and one in which the change in
sediment storage produced by geometric change can be calculated for each reach. These
files are added to the project with the same simulation name followed by "_batchp.dbf"
and "_volbal.dbr, respectively.
Whereas the batchp.dbf file tracks model output for each event, the volbal.dbf file
is used to compute the mass of sediment required for the calculated geometric changes in
each reach. This calculation is accomplished by computing the volume of sediment
produced by three types of geometric change: change in channel cross-sectional area,
overbank deposition/erosion, and section-wide deposition that may have occurred if the
channel filled to its minimum dimensions at any point during the simulations. These
types of geometric change are computed separately as VC1, VC2, and VC3, respectively,
so that the source of significant errors can be identified. The summed volumetric change
is converted to a mass by assuming an average particle density of 2.65 g/cm 3 , and a
porosity of 0.40 (40%). This mass is then compared to the value computed in K2G for
each reach, and the relative error is written to the "Error" field. During the volume-
130
balance calculations, the post-simulation section widths at the upstream and downstream
ends of each reach are computed and compared to the original values to ensure that no
unintended section width change has taken place. These values can be found in the
"SW1" and "SW2" fields, respectively.
Following calculations for each individual reach, the sediment mass balance is
computed for the entire channel network. A summary is written to the volbal.dbf table
comments, which can be accessed under the "Table" menu by selecting "Properties" in
the AGWA-G application. Sample results are shown below from a simulation of a single
10-year 1-hour event with profile smoothing.
Mass balance for profile smoothing run:
Network-Wide Mass-Balance Calculations for Batch Simulation:
Modeled plane erosion: -1299850 kg
Modeled channel erosion: -519410 kg
Geometric channel erosion: -503080 kg
Modeled sediment outflow: 1706310 kg
(myMociPlancOut)
(mvModChanDep)
(my(lcoChanDcp)
(niyMociSeclOut)
K2G sediment mass balance error: -6.62 (%)
Equation: (myModPlaneOut + myModSedOut + myModChanDep) * 100/
myModSedOut
Geometric adjustment mass balance error: -5.66 (`)/0)
Equation: (myModPlaneOut + myModSedOut + myGeoChanDep) * 100 /
myModSedOut
Error of geometric adjustment relative to modeled deposition: 3.14 (%)
Equation: (myModChanDep - myGeoChanDep) * 100 / myModChanDep
Volume-balance error was computed for simulations based on a single 10-year 1-
hour event with and without profile smoothing (table 4.2). As this output shows,
modeled change in network-wide sediment storage is very close to the mass required to
131
produce the computed geometric change. The difference between the two values can be
largely attributed to rounding errors that propagate through the many computations
involved in a simulation and the subsequent analysis. It is not clear why the difference is
greater for the simulation with no profile smoothing. Although the error for geometric
change in this simulation is less than the modeled error, it is likely that this could be
reversed for a depositional scenario.
To confirm that mass is not persistently lost during multiple consecutive
simulations, the model was run with profile smoothing for five consecutive 10-year 1-
hour events. Results show that the errors are almost identical to those from the singleevent run (table 4.2).
Table 4.2. Results of the volume-balance calculations for one 10-year 1-hour event with
(PS) and without (NS) profile smoothing, and five 10-year, 1-hour events with profile
smoothing. Insignificant digits are carried to prevent rounding error.
1 10-yr 1-hr, PS 1 10-yr 1-hr, NS 5 10-yr 1-hr, PS
-6499200
-1299800
-1299800
Modeled plane erosion (kg)
-2706500
-519400
-519400
Modeled channel erosion (kg)
-2619800
-453700
-503100
Geometric channel erosion (kg)
8632200
1706300
1706300
Modeled sediment outflow (kg)
-6.64
-6.62
-6.62
K2G volume-balance error (%)
-5.64
-2.77
-5.66
Geometric vol.-balance error (%)
3.21
12.64
3.14
K2G-geometric relative error (%)
4.7 Problems
Most problems that occurred during the development and testing of the model
have been eliminated. Most notable among those that remain, however, is the presence
of very short channel reaches that can occur when two tributary channels connect to the
132
main stem almost directly opposite each other. Very short channel reaches are subject to
massive shifts in the channel slope during the model run, as well as during the profile
smoothing. This can result in failure of the numerical model, and interruption of a batch
simulation. This problem is most pronounced when high-resolution DEMs are used
during the watershed discretization. In one instance a reach length of approximately 2
meters was created when using 1-meter Light Detection and Ranging (LIDAR) data for
the discretization. It was only possible to get the model to run by artificially lengthening
the channel to 15 meters.
Very long reaches can also cause problems in the numerical model, although not
as severe. Channel slope can only vary as a result of elevation change at the reach
endpoints, so very long reaches require a much greater combination of erosion and
deposition to have a significant impact on the channel slope. If the amount of deposition
required to increase the channel slope (and hence prevent continuing deposition) is
greater than the original channel depth, then unrealistic geometric change can result
because the channel will fill in to its minimum dimensions before the entire section can
be raised. Although this occurs in nature, by representing the natural system as a network
of longer reaches the model cannot adequately resolve where it happens. Computed
reach-average geometric change will therefore be less representative of on-the-ground
geomorphic change.
The parameter that channel adjustments are most sensitive to is slope, particularly
in relation to adjacent reaches. As a result, if reach slopes are derived from integer or
low-resolution DEMs computed geomorphic change throughout the channel can be
133
driven predominantly by error in the initial reach slopes. Narrow and short channels are
most likely to inherit errors from the elevation data, but all channels can be affected to
some degree depending on the resolution and vertical accuracy of the DEM. Experience
working with 30- and 10-meter DEMs, and 1-meter LIDAR has shown that as the
resolution increases problems with the computed slopes become less pronounced.
In the absence of a high-resolution DEM or field measurements of channel
elevation at the endpoints of each reach, it may be necessary to subject the model to an
initial series of flows to allow slopes to adjust. Unwanted width and depth changes are
commonly associated with these adjustments, so the 'primer' events only should be used
to update reach endpoint elevations. The desired simulation period should proceed with
the original width and depth values determined by the user.
Another problem is associated with the use of distributed precipitation files. For
five of the approximately 150 distributed precipitation files used in the testing of K2G,
massive deposition in certain reaches caused model failure. It appears that this occurs
when computed flow depth drops to approximately zero at the leading edge of the
primary flood wave, causing the majority of sediment at the leading edge of the flow to
deposit. The problem arises in the kinematic routing subroutine, but it is possible that the
geomorphic adjustments somehow interfere with the routing. Regardless, the volume of
deposited sediment results in very large adjustments to the channel geometry that
ultimately cause the model to fail. The problem is identified in K2G by comparing the
change in sediment storage for each time step to the area of the channel. If the change in
sediment storage is greater than the channel area, the model run is aborted and the
134
problem is reported. This error has not been observed with design (spatially uniform)
rainfall events.
4.8 Limitations
There is a variety of limitations associated with the K2G conceptual model and its
practical application that have been identified during its development. Perhaps the most
significant is that the model cannot be applied readily to large watersheds as was hoped.
This is largely due to automated procedures for discretizing the watershed in a GIS that
have not been able adequately to split long reaches into a series of shorter ones. In
addition, the evaluation of representative upstream and downstream geometries for each
reach is tedious and requires substantial time and effort. For finely discretized large
watersheds this will remain a daunting task until such time as reliable automated
procedures have been established, and high-resolution DEMs are widely available.
Representing the slope of each reach with a single value defined by the bed
elevation at each end of the reach is also limiting. This is necessary to allow dynamic
slope adjustment, but it has the effect of reducing the resolution of the channel profile
representation. As with any spatial averaging, a considerable amount of information is
lost, and there is an associated cost in terms of the accuracy of the computed results.
Considering short reaches helps minimize this problem, but as described above the more
reaches the greater the time and expense of developing parameter inputs.
Another limitation of the K2G model is the obvious problem of representing
highly variable natural channels as uniformly varying trapezoids.
This crude
135
simplification of channel geometry can be frustrating when attempting to define width
and depth from a measured cross-section. Gross generalizations must be made to
describe reach geometry in this fashion, and there is as yet no repeatable means of
accomplishing the task.
Finally, there are limitations inherited from the parent model, K2. Flow in
braided or anastomosing channels cannot be represented. Sediment grain-size
distributions on the channel bed and banks are the same and cannot vary through time,
thus precluding armoring. Strongly non-uniform flows and flows in channels with very
low slopes are not well represented. Model performance decreases with increasing
watershed size (particularly beyond about 100 km 2 ) and decreasing magnitude of the
rainfall input. Perhaps the largest source of error associated with K2 simulations is that
of rainfall input, and K2G simulations based on observed rainfall will not be widely
possible until AGWA-G has been updated to derive rainfall estimates from radar data.
From a sediment-transport modeling perspective, another important limitation of
K2G is the use of a transport-capacity relation to compute erosion and deposition. When
transport capacity is less than sediment load the rate of deposition is limited only by the
settling velocity of the particles. Thus if sand particles are being transported, the rate of
deposition can be quite high. When transport capacity exceeds the sediment load the rate
of erosion is limited by the transfer-rate coefficient, which can be high for noncohesive
sediment on channel beds. In gradually varied natural systems these erosion/deposition
rate limiters may represent the transfer of sediment between the flow and the channel bed
adequately. When they are applied in the K2/K2G conceptual model, however, they are
136
problematic because of abrupt changes in channel geometry, and hence transport
capacity, at reach junctions.
Additional general assumptions of the K2G sediment-transport calculations
include:
1) The fluid and sediment properties are steady and uniform
2) There is an infinite and continuous supply of sediment particle sizes represented
in the bed material (transport-limited conditions), unless specifically limited by
the user, or if the maximum erodible depth is reached
3) The particle-size distribution of the bed material does not vary as a result of
preferential transport of smaller particles
4) There is a specific relation between hydraulic and sedimentological parameters
and the rate at which the sediment load is transported
5) The sediment stored in a reach can be neglected
These assumptions are all used to simplify governing equations and numerical
procedures, and are frequently not well met in natural systems.
137
5.0 GEOMORPHIC MODEL RESULTS
5.1 Methodology
Despite the extensive network of automated hydrologic measuring devices on the
Walnut Gulch Experimental Watershed, there has been no systematic effort to document
channel morphology throughout the watershed. A variety of research has investigated
hillslope erosion and sediment yield, but only recently have efforts begun to monitor
channel morphologic change, and insufficient data are presently available to test the
performance of K2G in terms of its ability to reproduce observed morphologic change.
Testing of the model thus concentrated on evaluating relative model response to variable
land cover/use conditions, initial channel geometry, and climatic inputs.
Test simulations were performed on Walnut Gulch subwatershed 11, which was
discretized into model elements using a contributing (channel) source area, CSA, of 8.1
hectares, or 1% of the watershed area (figure 5.1). A 10-meter DEM was used during the
watershed delineation and discretization procedure. A large pond in the upper central
portion of the subwatershed contains all but the most extreme events, and was thus
removed from the model. The AGWA internal gauge feature was used to break up longer
reaches as much as possible, but problems with this functionality limited its success. The
average channel length is 329 m, and the maximum and minimum channel lengths are
927 and 25 m, respectively. Elevations at the upstream and downstream ends of each
reach were obtained from 1-meter LEDAR data to minimize channel-slope errors.
138
Twenty four batch simulations were conducted to test model performance for a
range of land use/cover and climatic (rainfall) inputs, as well as for two initial channel
geometry configurations. Table 5.1 lays out the initial conditions for each of these
simulations. Parameter inputs from the land cover and soil data were derived using
AGWA's default parameterization routines (Miller et al., 2002). The Soil Survey
Geographic Database (SSURGO) was used to derive soil-based parameter inputs for all
simulations.
Simulations were conducted based on distributed, observed precipitation records
from the summer monsoon seasons of three different years: 1964, 1977, and 1978. These
years were selected as representative of wet, average, and dry years based on
observations at the National Weather Service gauge in Tombstone, AZ. Unfortunately
there is no easy means of evaluating annual totals for any given subwatershed, and
because of the high spatial variability of rainfall on Walnut Gulch, the observations at
Tombstone were not representative of precipitation at WG11. After running the
simulations, which automatically interpolate rainfall depths across the watershed, it was
possible to characterize the distribution of events for each year based on the average
depth of rainfall over the watershed (Table 5.2). Based on this analysis, wet, dry, and
intermediate years were represented, but the rainfall totals for the intermediate year are
much closer to the wet year than the dry year. 1964 was the wettest year, and was
characterized by 52 events with an average depth of 9.29 mm. The intermediate year,
1977, was characterized by 47 events with an average depth of 7.86 mm. The dry year,
1978, was characterized by 41 events with an average depth of 1.02 mm.
139
•
•
•
Figure 5.1. Maps showing the watershed discretization (A), DEM (B), SSURGO
soils (C), and rain gauge configuration (D) for Walnut Gulch subwatershed 11.
Table 5.1.
Batch simulations, and their associated inputs, used to evaluate the
geomorphic model.
Simulation
Name
1964 HG 73
1964 HG 97
1964 HG PU
1964_HG_U
1964 OG 73
1964 OG 97
1964 OG PU
1964_0G_ U
1977 HG 73
1977_HG_97
1977_HG_PU
1977 HG U
Land Cover Input
Precipitation Input
Channel Geometry
NALC 1973
NALC 1997
Partially Urbanized
100% Urban
NALC 1973
NALC 1997
Partially Urbanized
100% Urban
NALC 1973
NALC 1997
Partially Urbanized
100% Urban
1964 Monsoon
1964 Monsoon
1964 Monsoon
1964 Monsoon
1964 Monsoon
1964 Monsoon
1964 Monsoon
1964 Monsoon
1977 Monsoon
1977 Monsoon
1977 Monsoon
1977 Monsoon
Hydraulic geometry
Hydraulic geometry
Hydraulic geometry
Hydraulic geometry
Observed
Observed
Observed
Observed
Hydraulic geometry
Hydraulic geometry
Hydraulic geometry
Hydraulic geometry
140
1977_0G_73
1977_0G_97
1977_0G_PU
1977_0G_U
1978_HG_73
1978_HG_97
1978_HG_PU
1978_HG_U
1978_0G_73
1978_0G_97
1978_0G_PU
1978_0G_U
NALC 1973
NALC 1997
Partially Urbanized
100% Urban
NALC 1973
NALC 1997
Partially Urbanized
100% Urban
NALC 1973
NALC 1997
Partially Urbanized
100% Urban
1977 Monsoon
1977 Monsoon
1977 Monsoon
1977 Monsoon
1978 Monsoon
1978 Monsoon
1978 Monsoon
1978 Monsoon
1978 Monsoon
1978 Monsoon
1978 Monsoon
1978 Monsoon
Observed
Observed
Observed
Observed
Hydraulic geometry
Hydraulic geometry
Hydraulic geometry
Hydraulic geometry
Observed
Observed
Observed
Observed
Table 5.2. Characterization of the precipitation records for the summer monsoon on
WG11 during 1964, 1977, and 1978.
1978
1977
1964
Year
Number of events
Total Precip. (mm)
Ave. event depth (mm)
Max. event depth (mm)
Standard deviation
52
483.3084
9.294393
51.92596
12.35652
47
369.1959
7.855232
40.39204
8.949365
41
41.92376
1.022531
1.765405
0.393512
The rainfall record may have differed substantially if rainfall intensity had been
used as a selection criteria in addition to the annual total rainfall depth. This would have
permitted the elimination of annual totals for which much of the precipitation was
derived from long-duration events of low intensity. In addition, it would have been
possible to characterize the rainfall record, perhaps more appropriately, in terms of its
erosive capacity. Runoff in K2G is generated as infiltration excess, so higher rainfall
intensities result in increased runoff. High-intensity rainfall has the greatest potential to
affect erosion on upland surfaces because of its potential for sediment detachment
through rain-splash erosion, and increased ability to transport that sediment because of
increased flow depths. Increased runoff, and hence stream power, in the channels
increases the transport capacity, and can result in increased erosion. Unfortunately,
141
rainfall intensity at a single gauge is not representative of intensity over a larger area, and
spatial averaging of measured rainfall intensities from a network of gauges for every
event in a year was not practical. For this reason, selecting the rainfall records in terms
of annual total depth was the most suitable means of obtaining records with variable
average and total rainfall depths.
Parameter estimates for the simulations were derived from four land-cover grids:
two observed, and two hypothetical. The land-cover grids (figure 5.2) included:
1) North American Landscape Characterization (NALC) project classified land
cover for 1973
2) 1997 Landsat TM imagery classified according to the NALC scheme.
3) A hypothetical 100% urbanized condition
4) A hypothetical partially-urbanized condition for which the 1997 land cover was
altered to convert part of the watershed to the urban class.
The 1973 land cover is characterized predominantly by grassland and desert-scrub
vegetation, with lesser amounts of mesquite woodlands (table 5.3). In 1997 the land
cover is characterized by increased mesquite woodlands and decreased grassland, but the
difference is less pronounced than would be ideal for comparative purposes. The 1973,
1997, and 100% urbanized land-cover grids are relatively homogenous, so the part-urban
land cover was designed to introduce substantial heterogeneity by converting the upland
area of the southernmost tributary to the urban class. The part- and all-urban grids
produce significantly different parameter sets than the other two. For the urban landcover class, AGWA assumes that 40% of the upland areas are impervious, and hence it
142
reduces saturated hydraulic conductivity estimates, derived from the underlying soil, by
40%. In addition to the saturated hydraulic conductivity reductions, erosion is reduced
by setting a parameter that designates the percentage, 40% in this case, of the surface
covered by pavement (i.e. non-eroding). Vegetative cover, and hence interception, is
reduced by the assumption that urban areas are characterized by 15% cover, relative to
20% for mesquite woodlands and 25% for grassland and desert scrub. Table 5.4
summarizes the parameter estimates used in AGWA for each land cover class.
I=
Forest
Oak Woodland
Mesquite Woodland
Grassland
Desertscrub
▪
Riparian
Agriculture
I= Urban
Water
Barren / Clouds
▪
1
MI
ill
Figure 5.2. Maps showing the four land-cover scenarios used to derive model parameters
for the test simulations: (A) NALC 1973, (B) NALC 1997, (C) Partial urbanization, and
(D) Total urbanization. The legend for the NALC classification scheme is shown for
reference. Outlines of the discretized model elements are superimposed on the landcover grids.
143
Table 5.3. Composition of land cover classes on WG11 for the four land-cover
scenarios used during model testing.
NALC 1973
NALC 1997
Part Urban 1997
All Urban
Mesquite Woodlands
0.0
4.9
1.2
0.0
Grassland
54.8
53.0
34.4
0.0
Desertscrub
45.2
42.1
28.9
0.0
Urban
0.0
0.0
35.5
100.0
Table 5.4. Hydrologic parameters used by AGWA during land
cover and soils parameterization.
Imperv
n
Cover
Int
Class Name
0
20
1.15 0.050
3 Mesquite Woodland
2.00 0.150
0
4 Grassland
25
25
3.00 0.055
0
5 Desert Scrub
0.10 0.015
40
15
8 Urban
All simulations were performed using compound channels, or channels with flood
plains. Two different methods were used, however, to determine the initial channel
geometries. The first method involved using hydraulic geometry relations derived for
Walnut Gulch by Miller (1995), which is the default method for estimating channel
geometry in AGWA. Miller (1995) derived relations for width and depth as:
W = 0.0724A °3377
(5.1)
D = 0.0502A" 523
(5.2)
where A is the contributing area in square meters. Hydraulic-geometry channels were
retained in this analysis to evaluate model behavior in the absence of observed-geometry
data. Flood-plain widths estimated by this method were three times the channel width.
Side slopes of the channel and overbank are set to 1.0 (rise/run), and the lateral slope of
the flood plain is set to 0.01.
144
The second method involved defining channel cross-sections from a high
resolution (1-meter) DEM that was developed from LIDAR data acquired in May, 2003.
Cross-sections were obtained for at least three locations on every reach, and the data were
imported into Excel where geometric parameters were computed. Translation of natural
channel cross-sections into trapezoidal sections for the model was done manually.
Placement of bank tops and toes was accomplished with guidance from the hydraulic-
geometry relation to improve estimation of channel area when multiple banks were
present. In general, however, the transformation was conducted by considering the most
appropriate means of representing the natural section with a compound trapezoidal one.
In many cases, for instance, channel areas were substantially larger than predicted by the
hydraulic-geometry relation because the channel locally was incised.
The maximum erodible depth of all reach endpoints was set to 5 meters for all
simulations in the absence of field observations. This assumption is probably not
justified for the majority of reaches on WG11, but the maximum simulated depth change
during the model tests was 0.475 meters.
The sediment grain-size distributions for the model runs were divided into five
size classes (table 5.5). These size classes are global parameters for the model, and
therefore apply to channel, overbank, and upland elements, although the percentage of
sediment in each size class can vary for each element. Distributions for the upland
elements are based on spatially averaged sand, silt, and clay fractions derived from the
soils data. For the sand class, 25% was split into the medium sand class with a median
diameter of 0.625 mm. The remaining 75% was left in the fine sand category, and the silt
145
and clay fractions were not modified. The coarse sand fraction was set to zero for all
upland elements. Grain sizes for all channel and overbank elements followed the
distributions laid out in table 5.5. Whereas the overbank distribution is based on the
AGWA default values, the channel distribution was derived from distributions measured
by Canfield (1998) on Walnut Gulch.
Table 5.5. Sediment grain-size distributions as percentages of the dry
sample weight for the channel and overbank elements.
3
0.625 0.15625 0.0332 0.002
Average particle size (mm)
4.1
13.7
22.7 59.5
0
Channel
Overbank
0
10
40
40
10
5.2 Results
5.2.1 Graphical Output
Results from the geomorphic model testing are presented in figures 5.3 to 5.10.
Each figure displays results for a particular land-cover scenario and initial channel
configuration (hydraulic geometry or observed). The six maps in each figure are
arranged such that average depth changes are presented on the left, and average width
changes on the right. It is important to note that the geomorphic model computes width
and depth changes at both the upstream and downstream ends of each reach. To reduce
the amount of data presented and simplify the analysis, these values were averaged.
The map pairs A & B, C & D, and E & F are arranged in order of decreasing
precipitation from top (1964) to bottom (1978). Color ramping of the results was
conducted by means of an automated procedure in AGWA-G, and is unique to each set of
146
displayed results. The legend for each map is thus displayed to serve as a reference for
the range of values in the simulation results. Negative values, shown in shades of blue,
represent simulated channel width or depth decreases (deposition). Positive values,
shown in shades of red, represent width or depth increases (erosion). For each map the
class containing the zero value (no change) is shown in white.
Figures 5.3 to 5.10 provide a glimpse of the enormously complex relations among
landscape change, climate, and geomorphic response. It is not possible to interpret
meaningfully the geomorphic response of any given reach to land changes in land cover
ancUor precipitation. The spatial and temporal variability of rainfall and erosion upstream
is too complex to lend itself to simple interpretation. Instead, only the broadest
observations are made of the various results.
The first and most obvious observation is that large, frequent precipitation events
are more likely to cause widespread erosion than are small, infrequent events. The 1964
monsoon caused the greatest and most widespread erosion throughout the watershed,
regardless of land cover. The 1978 monsoon produced the most widespread deposition,
but because the events were so small the amount of deposition was negligible. The 1977
monsoon, with more uniformly intermediate events, produced more mixed deposition and
erosion.
Differences in geomorphic response to the initial channel geometries are
significant. Here the magnitude of the response was less significant than the spatial
distribution of geomorphic changes. Changes were less uniform and more specific to the
individual reaches when observed geometries were used. Specifically, incised reaches
147
are likely to exhibit erosion, and wide reaches are likely to have deposition. In addition,
the wide channels, where present, promoted transmission loss, which in turn resulted in
lowered runoff volume (figure 5.11).
Observed geometries were highly variable, and alternating wide and narrow
sections were commonly observed within a single reach. The inability to split reaches at
these transitions severely limits the model's representation of the system. In addition,
when the geometry is adequately represented the absence of observed maximum erodible
depths likely results in exaggerated change for incised sections where flow velocities, and
hence transport capacity, are high.
Geomorphic response to different land-cover conditions is less pronounced than
to the different precipitation records or initial channel geometries. A minor mesquite
invasion between 1973 and 1997 produced a negligible impact on channel morphology,
and although a small increase in erosion can be observed it is most likely well within the
error of the simulations. The partial-urbanization scenario produced the most notable
differences in the spatial pattern of erosion/deposition, as would be expected given the
significant heterogeneity in its land cover. The specific response to partial urbanization,
however, was mixed. For the 1964 monsoon increased erosion was observed relative to
that for the NALC 1997 land cover, but for the 1977 monsoon the opposite was true.
These differences are discussed in more detail in the following chapter.
The all-urban land-cover scenario produced increased erosion in some cases,
increased deposition in others. The 1964 simulation based on the hydraulic geometry
channels produced a dramatic and widespread increase in erosion, but these results are
148
sufficiently different from all of the others that they are questionable (see the mass
balance discussion below for more details). The 1964 simulation based on the observed
channels produced very little geomorphic adjustment throughout the watershed, with the
exception of two short reaches in the lower portion of the watershed. For 1977, the allurban scenario produced widespread deposition for the hydraulic geometry channels, and
a wide range of erosion and deposition for the observed channels.
149
slid
A/ -0.088 - -0.071
slid
-0.012 - 0.017
0.017 - 0.046
0.046 - 0.074
A/ 0.074 - 0.103
0.103 -0.132
NO.132 - 0.16
No Data
wild
N
N
-0.071 --0.055
A/-0.055 - -0.038
A/ -0.038 - -0.022
-0.022 - -0.006
-0006-0.011
0.011 - 0.027
No Data
wild
511d
slid
-0.004 - 0.001
0.001 - 0.005
0.005 - 0.009
A/ 0.009 - 0.013
A/ 0.013 - 0.018
A/ 0.018 - 0.022
No Data
wi ld
A,
,Q‘
0-0
0-0
0 - 0.001
0.001 - 0.001
A/0.001 - 0.001
r.
N
Decreasing
Precipitation
A/ 0.001 - (loot
No Data
wild
slid
-0003--0002
002- -0.002
A/ -0002--0.001
A/ -0 001 -0.001
-0.001 0-0
0- 0.001
At• 0.001 - 0.001
/V000 1 -0002
NO.002 - 0.003
No Data
wild
4
Figure 5.3. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 HG 73, 1977 HG 73, and 1978 HG 73 (top to bottom,
respectively).
150
slid
slid
N-0_159 -0.123
N-0123--0088
irve -0 088 -0.052
/V -0 052 - -0.016
-0016-0019
0 019 - 0.055
0 055 - 0.091
/\/0091 -0.127
NO 127 - 0.162
0 162 - 0 196
A/ No Data
11111 wild
/\./ -1.288 - -1.04
A/-1.04 - -0.792
/\,/ -0.792 - -0.544
A/ -0.544 - -0.296
/ -0.296 - -0.048
-0.048 - 0.2
0.2 - 0.447
No Data
r
mi
slid
slid
-0003- -0.002
-0.002 - 0
0- 0.002
0.002 - 0.004
A/ 0.004 - 0.006
0.006 - 0.008
0.008 - 0.01
No Data
wild
A/
4
Al-0.017 -0.014
A/ -0.014 -0.01
- -0.007
A/ -0.007 - -0.004
Decreasing
Precipitation
-0.004 - -0.001
-0.001 - 0.002
0.002 - 0.005
,. 0.005 - 0.009
A./ 0.009 - 0.012
No Data
wild
.Q‘
slid
No Data
slid
No Data
wl ld
1111 wild
Figure 5.4. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 OG 73, 1977 OG 73, and 1978 0G73 (top to bottom,
respectively).
151
slid
N -0.073 - -0.059
slid
-0.012-0.017
0_017 - 0.046
0.046 - 0.075
A/ 0.075 - 0.104
0.104 - 0.133
0.133 - 0 162
Q‘ No Data
/V
/V
wild
slid
-0.004 - 0.001
0.001 - 0.005
0.005 -0.01
001 - 0.014
A/ 0014- 0.019
A/ 0 019 - 0.023
No Data
wild
IV-0.059 - -0.045
N-0.045 - -0.031
-0.031 - -0.018
-0.018 - -0.004
N
r
-0 004 - 0.01
0.01 - 0 024
No Data
MN wild
slid
/ 0 - 0
0-0
0-0
0-0
- 0
NO - 0.001
N 0 .001 - 0.001
No Data
wild
Decreasing
Precipitation
N
slid
-0003- -0.002
-0002--0.002
-0002- -0.001
A/-0 001 - -0.001
0-0
0 - 0.001
Ai 0.001 - 0.001
0 001 -0002
A/ 0 002 - 0.002
No Data
wild
Figure 5.5. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 HG 97, 1977 HG 97, and 1978 HG 97 (top to bottom,
respectively).
152
slid
A/ -0.107 - -0.076
A/ -0.076 - -0.045
-0.045 -0.015
-0015- 0.016
0016- 0.047
0.047 - 0.078
0.078 - 0.108
A/ 0.108 - 0.139
A/ 0.139 - 0.17
No Data
wild
slid
A/ -0.004- -0.002
'
-0.002-0
0- 0.001
0.001 - 0.003
0.003 - 0.004
A/ 0.004 - 0.006
A/ 0.006 - 0.008
A/0.008
-
0.009
slid
/\/-1.506 - -1.228
/\/-1.228 - -0.951
N-0.951 - -0.673
A/ -0.673 - -0.396
-0.396--0.118
-0.118- 0.159
0 159 - 0 437
A No Data
MI wild
X;
014--0.011
N -0.011 --0008
N -0.008 - -0.006
-0006--0.003
-0003 - 0
0 - 0.003
0.003 - 0005
0.005 - 0008
Decreasing
Precipitation
A/ 0.008 -0.011
A/0.011 -0014
slid
0 - 0.001
0.001 - 0.002
A/ 0.002 - 0.003
0.003 - 0.004
A/13.004 - 0.005
0.005 - 0.006
No Data
wild
A/
A/
Figure 5.6. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 OG 97, 1977 OG 97, and 1978_0G_97 (top to bottom,
respectively).
153
slid
slid
/V
A/
A/
-0.012 -0025
0 025 - 0.062
0.062 - 0.098
0.098 - 0.135
0.135 - 0.172
0.172 - 0.208
No Data
wild
--0056
IV-°°82
A/ -0.066 --0051
A/ -0.051 --0 035
,&J-0.035 - -0.019
-0.019--0004
-0.004 - 0.012
0.012 - 0.027
0.027 - 0.043
No Data
wild
slid
slid
-0.023- -0.011
-0.011-0
0- 0.012
0.012 - 0 023
0023- 0 034
0.034 - 0.046
0046- 0.057
No Data
MI wild
A/
A/
A/
A/
-0.001 - 0
0 - 0.002
/0.002 - 0.003
A/ 0.003 - 0.005
A/ 0.005 - 0.006
A/ 0.006 - 0.007
Q‘ No Data
wild
Decreasing
Precipitation
slid
Al -0003- 0.002
-0002--0002
A/ -0002--0.001
-
A/ -0.001 -0.001
-0 001 0-0
0 - 0.001
/1v 0.001 - 0.001
0.031 - 0.002
0.002 - 0.002
A/
A/
No Data
wlid
Figure 5.7. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 HG PU, 1977 HG PU, and 1978 HG PU (top to bottom,
respectively).
154
slid
Al-0.144 -0.116
A/ -0.116- -0.087
A/-0.087 --0059
A/ -0.059 - -0.031
-0.031 --0.003
-C)003- 0.026
0.026 - 0.054
0 054 -0082
A/ 0.082 -011
No Data
wild
slid
A/-1.194 - -0.974
A/ -0974--0754
A/ -0.754 - -0.534
-0534 - -0.313
-0.313 - -0.093
-0093-0127
0.127 - 0.348
0.348 - 0 568
A/ 0.568 - 0.788
No Data
wild
A/
slid
A/ -0018--0013
A/ -0013--0008
-0.008 - -0.003
-0.003 - 0.002
0.002 - 0.007
Ar - 0.007 - 0.012
A/ 0.012 - 0.017
A/ 0.017- 0.022
A/ 0.022 - 0.027
No Data
wild
slid
-0039--0028
A/ -0.028 - -0.017
-0.017--0007
-0.007 - 0 004
0.004 - 0.015
0.015 - 0.026
A/ 0026-0.036
0.036 - 0.047
A/0.047 - 0.058
No Data
wild
A.
A/
Decreasing
Precipitation
4mief .rneff
slid
-0.001 -0
0- 0.001
0.001 - 0.002
0.002 - 0.003
A/ 0 003 - 0.005
N 0.005 - 0.006
No Data
wild
Figure 5.8. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 OG_PU, 1977 OG PU, and 1978_0G_PU (top to bottom,
respectively).
155
slid
/V -0.184-0.151
slid
-0.032 - 0.008
0 008 - 0.048
0 048 - 0.089
"./ 0 089 - 0.129
A/ 0 129 - 0.169
A/ 0 169 - 0.21
No Data
wild
A/ -0.151 -0.118
A/ - -0.085
A/ -0.085 - -0.052
-0.052--0.02
-0.02- 0.013
0.013- 0 046
" No Data
EN wild
slid
-0065- -0053
Af -0.053- 4104
Al -004- -0.028
A/ -0.028 - -0_016
-0.016 - -0.003
-0 003 - 0.009
0.009 - 0.021
A,• 0.021 - 0.034
/\/0.034 - 0 . 0 46
N 0 .046 - 0.058
No Data
wild
slid
A/ -0.141 - -0.117
/V -0.117--0.093
A/ -0.093 - -0.068
Decreasing
Precipitation
A/-0.068 --0044
-0.044 - -0.02
-0.02 - 0.005
No Data
wild
slid
Al0.003 - -0.002
-0002--0.002
-0002--0.001
-0001 --0001
-0.001 -0
0-0
0 - 0.001
0.001 - 0.002
N 0 .002 - 0.002
,AV 0.002 - 0.003
No Data
wild
Figure 5.9. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964 HG U, 1977 _ HG _U, and 1978_HG U (top to bottom,
respectively).
156
slid
A/ -2.824 - -2.339
slid
-0088-0.006
0.006 - 0 1
0.1 -0.194
A/ 0.194 - 0.288
/\/0.288 - 0.382
A/ 0.382 -0475
,Q1‘
No Data
wild
N -2.339 - -1.853
//.1.853-- -1.368
A/ -1.368 - -0.882
7 -0.882 - -0.397
-0.397 - 0.089
0.089 - 0.575
A No Data
Ili wild
slid
-0041--0.033
A/ -0033--0.026
A/ -0 026 --0.018
A/
A/
-0.018--0.011
-0.011 - -0.003
-0003-0005
0 005 - 0.012
0 012 - 0.02
0 02 - 0 028
0 028 - 0.035
No Data
w1 1c1
slid
A/ -1 .285 - -1.041
/\/-1.041 - -0.796
A/ -0.796 - -0.552
N -0.552 - -0.308
Decreasing
Precipitation
-0.308 - -0.063
-0.063 - 0.181
No Data
111111 wild
slid
-0.001 -0
0 - 0.001
0.001 - 0.002
0.002 - 0.003
A/ 0.003 - 0.005
A/ 0.005 - 0.006
N
No Data
wild
Figure 5.10. Maps showing depth and width changes (left and right, respectively) in
meters for simulations 1964
_ OG U 1977 OG U, and 1978 OG U (top to bottom,
respectively).
157
slid
56.99 - 68.071
68.071 - 79.153
/V 79.153 - 90.234
/\J90.234 - 101.316
A/ 101.316 - 112.397
No Data
wild
slid
1.293 - 2.835
2.835 - 4.377
4.377 - 5.919
A/ 5.919 - 7.46
A/ 7.46 - 9.002
/V No Data
MI wild
slid
0-0
0-0
\/, 0 - 0
A/ 0 - 0.001
A/ 0.001 -0.001
/V No Data
wil d
Figure 5.11. Cumulative runoff depth (mm) per unit contributing area for the three
batch simulations arranged in order of decreasing rainfall total from 1964 (top) to
1978 (bottom). Pictured results are for the hydraulic geometry simulations, but the
spatial patterns for the observed geometry simulations were virtually identical.
158
5.2.2 Mass-Balance Calculations
Cumulative changes in sediment storage and geometric adjustments were
computed for each batch simulation to facilitate net mass-balance calculations for the
channel network (see section 4.6.3). Results of these calculations are presented in table
5.6. Sediment inflows to the channel network are derived from erosion on the uplands
(planes). Sediment outflow is the sediment leaving the outlet channel. The change in
sediment storage is equivalent to the net amount of erosion and/or deposition in the
channels, plus the amount of sediment remaining in suspension at the last time step.
Mass-balance error (%) was calculated as:
(Inflows - Outflows +- AStorage) / Outflows * 100 (5.3)
where storage is added or subtracted depending on its sign.
Computing the percent error by dividing by the outflow is inappropriate in the
absence of outflow, which is frequently the case in arid regions. As such, the error
estimates for 1978, for which sediment outflow was universally zero, were computed as:
(Inflows — AStorage) / Inflows * 100 (5.4)
Table 5.6 presents both the modeled change in sediment storage and the
equivalent mass required to produce the simulated geometric adjustments. The modeled
change in sediment storage is that computed in the K2G sediment routing, which was not
altered from K2. Error associated with the modeled change in sediment storage is
referred to as model error. The equivalent mass of sediment required to produce the
simulated geometric change is computed at the end of the batch simulation by comparing
the final geometry with the original geometry. As such, it incorporates geometric
159
adjustments computed in K2G, geometric adjustments during profile smoothing in
AGWA-G, and cumulative rounding error from the calculations in each simulation. Error
associated with the equivalent mass of geometric adjustment is referred to as geometric
error.
Model and geometric error consistently increased with decreasing precipitation
for all land-cover scenarios (table 5.6). This effect reflects errors in the KINEROS
sediment-routing calculations, for which the same trend is observed. The difference
between computed and observed changes in sediment storage in KINEROS is roughly the
same regardless of event magnitude and thus increases in proportion to the volume of
routed sediment for small events. Errors in excess of 50% were common for the smallest
events.
Graphical presentation of the mass balance error presented in table 5.6 indicates
significant trends in the simulation output. For the 1964 monsoon (figure 5.12) model
error was highest for the partially urbanized land cover, and lowest for the NALC 1973
land cover. Geometric error was consistently lower than the model error when the initial
channel geometries were defined by the hydraulic-geometry relation. When observed
channel geometries are used as the initial condition, geometric adjustments were
consistently larger than the modeled change in sediment storage.
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161
20
15
10
5
0
-5
-10
o Model Error
• Geom Error
-15
-20
-25
64hg73
64hg97
64hgpu
64hgu
64og73
64og97
64ogpu
64ogu
Figure 5.12. Bar graph showing the model (red) and geometric (blue) net mass-balance
error (%) for 1964 batch simulations.
Comparison of the cumulative modeled change in sediment storage and the mass
equivalent of geometric adjustments reveals a disappointing trend in the performance of
the geomorphic model. For all 1964 batch simulations geometric adjustment exceeds the
computed change in sediment storage (figure 5.13). This effect is most pronounced when
the observed channel geometries are used as a starting point. Geometric mass-balance
error is consistently less than the model error for simulations based on the hydraulicgeometry channels, but this trend is reversed when observed geometries are used.
Channel geometric change and modeled change in sediment storage yield
different results. Modeled erosion decreased for the two urban scenarios, which is not
162
unreasonable because sediment production on the uplands increased substantially despite
the presence of erosion pavement. In addition, modeled erosion decreased when the
observed geometries were used and the presence of wide, shallow reaches enabled more
sediment to be captured and retained within the channel network. Geometric change,
however, failed to reflect these same trends.
64hg73
64hg97
64hgpu
64hgu
64og73
64og97
64ogpu
64ogu
1000000
0
-1000000
-2000000
-3000000
-4000000
-5000000
• Channel Eros
o Geometric Adjust
-6000000
Figure 5.13. Bar graph showing the relative magnitude of channel erosion (kg) and the
mass equivalent of the simulated geometric adjustment for the 1964 batch simulations.
For the 1977 monsoon the 1-1G U batch simulation computed sediment yield was
an order of magnitude different from the other seven simulations. This is thought to
result from problem discussed in section 4.7 that produces excessive deposition in a
single time step when flow depth drops precipitously prior to the arrival of the primary
163
flood wave. Results from the HG _U simulation were thus excluded from the following
analysis.
No consistent relation was observed between mass-balance error (model or
geometric) and land-cover scenario for the 1977 simulations (figure 5.14). Results from
the HG_U simulation were omitted for clarity. Geometric error was larger than model
error for all but one of the simulations. The relation between model and geometric massbalance error was more irregular when observed channel geometries were used.
200
o Model Error
• Geom Error
150
100
50
0
-50
77hg73
77hg97
77hgpu
77og73
77og97
77ogpu
77ogu
Figure 5.14. Bar graph showing the model (red) and geometric (blue) net mass-balance
error (%) for 1977 batch simulations.
For all but one of the 1977 batch simulations geometric adjustment exceeds the
computed change in sediment storage (figure 5.15). This effect is more pronounced for
164
the two urbanized scenarios, and most pronounced when the observed channel geometries
are used as starting points. Unlike the 1964 simulations, however, differences for 1977
are more consistent, and the overall trend of increased deposition with urbanization is
reasonably well represented by the geometric adjustments.
1900000
• Channel Eros.
o Geometric Adjust.
1400000
900000
400000
-100000
77hg73
77hg97
77hgpu
77og73
77og97
77ogpu
77ogu
Figure 5.15. Bar graph showing the relative magnitude of channel erosion (kg) and the
mass equivalent of the simulated geometric adjustment for the 1977 batch simulations.
5.2.3 A Perspective on Model Output
To put the simulated sediment yields into perspective, cumulative water and
sediment outflows from the observed land-cover scenarios were compiled for comparison
with observations on Walnut Gulch. The results, presented in table 5.7, are for the entire
165
watershed (WG11), which has a contributing area of 633 ha. The calculation of sediment
yield as a volume assumed a particle density of 2.65 g/cm 3 , and a porosity of 0.3.
Table 5.7. Simulated cumulative outflow and sediment
yield for the observed land-cover scenarios.
Simulation Runoff (mm) Sed. Yield (m 3/ha)
sv64hg73
63.3
11.2
sv77hg73
1.5
0.1
sv78hg73
0
0.0
sv64hg97
63.6
11.3
sv77hg97
1.5
0.1
sv78hg97
0
0.0
Both runoff and sediment yield were highest for the 1964 monsoon, which
yielded an order of magnitude more runoff, and two orders of magnitude more sediment
than did the 1977 (intermediate) monsoon. The 1978 monsoon yielded no net runoff or
sediment yield at the watershed outlet. Given the characterization of the rainfall records
presented in table 5.2, the model output suggests that both runoff and sediment yield from
a series of events increase exponentially as average event depth and the standard
deviation of event depths increase.
In comparison, Nichols and Renard (2003) computed sediment yield for several
small watersheds on Walnut Gulch through repeat surveys of stock tanks. The results of
their analysis is summarized in table 5.8. Although these values are not directly
comparable to sediment yield from the much larger (633 ha) WG11, they indicate that the
range of simulated sediment-yield values in K2G is reasonable. The 1964 monsoon was
characterized by two consecutive large runoff events that transported most of the
166
sediment outflow of that year. Nichols and Renard (2003) reported only two single-year
records, with computed sediment yields of 6.0 and 3.3 m 3 /ha for a watershed of 43.8 ha.
These records, however, were from approximately average rainfall years, indicating that
high sediment yields are likely to be substantially higher during wet years.
Table 5.8. Summary of sediment accumulation on Walnut Gulch from Nichols
and Renard (2003). The italicized values are from a pond within WG11.
Contributing area (ha) Years in Record Sediment Yield (m 3/ha/yr)
0.4
11
92.2
35.2
18
2.3
84.2
34
1.7
43.8
46
2.8
The unit rate of sediment yield generally decreases as contributing area increases
(e.g. Branson et al., 1981). This results from a host of factors, most notably including:
increasing sediment storage within the channel network, decreasing average event rainfall
depth, and increasing transmission loss with increasing watershed size. Given this, it is
possible that K2G has overestimated sediment yield for 1964. The fact that only a partial
year was simulated would accentuate that possibility, although most sediment discharge
occurs during the monsoon season. However, the 1964 monsoon was exceptional, and it
was selected for that reason. Sediment yield from the moderate 1977 monsoon was much
closer to what one would expect based on the results of Nichols and Renard (2003).
167
5.3 Discussion
Results from the test simulations highlight the importance of rainfall frequency
and magnitude in governing geomorphic response. The influence of transmission losses
is demonstrated by widespread deposition during intermediate and small events. When
observed channel geometries were used, deposition was greatest in the lowermost
channels for which flow reductions were most pronounced (figure 5.11). Graf (1983b)
observed a similar response in the Henry Mountains of Utah, where periods of
aggradation were associated with stream power reductions as a result of transmission
losses in the downstream direction. The larger events observed during the 1964 monsoon
produced less significant downstream decreases in runoff, and were associated with
widespread channel scour.
Test results indicate that the model is particularly sensitive to the initial channel
geometry. This is not surprising, because the minimum stream power theory that was
used to govern the channel adjustments is based on the hypothesis that a channel adjusts
its geometry to minimize the total stream power. Hydraulic geometry channels fail to
account for channel slope when they are used to assign width and depth values, and thus
are not necessarily representative of well-adjusted sections. Observed geometries may be
more representative of the reach and slope at the points where they were measured, but
cannot account for inter-reach variability. Regardless of the initial geometries, variable
adjustment was observed throughout the watershed, and those adjustments were different
depending on the initial conditions.
168
The land-cover scenarios produced unexpected results. Urbanization did increase
runoff as expected, but slightly larger increases in erosion resulted in increased
deposition during the 1977 monsoon, and decreased erosion during the 1964 monsoon.
Widespread channel incision was the expected result, but the reason it was not observed
is likely due to the way AGWA-K2G treats urban uplands. By reducing saturated
hydraulic conductivity, increased runoff and sediment entrainment are generated on the
uplands. Increasing the PAVE parameter helps offset this by reducing computed erosion
by the fraction it is set to, but it does not limit access of the runoff to sediment. In reality,
urban areas concentrate runoff in paved channels and culverts to prevent erosion, and
much of the additional runoff gets to the channels without a significant sediment load.
The persistent overestimation of deposition by the geometric adjustments during
the 1977 simulations is most likely a result of the profile-smoothing calculations between
each simulation. Profile smoothing and mass-balance calculations were developed and
debugged based on test simulations using design storms and initial channel geometries
defined by the hydraulic-geometry relation. As such, differences between the model and
geometric changes in sediment storage are least pronounced for the hydraulic geometry
batch simulations. What the profile smoothing calculations fail to account for is
significantly different bank slopes for the upstream and downstream channels at each
junction. Elevation adjustments computed at each junction may therefore fail to conserve
mass in the transfer of sediment between reaches.
169
6.0 RELATIVE GEOMORPHIC CHANGE
AND CHANNEL STABILITY
6.1 Model Error and Relative Change Assessment
Error in numerical models of watershed runoff and erosion can result from
numerous sources. Most notable among these is how faithfully a model describes the
real-world processes, both conceptually and numerically. All models are simplified
abstractions of real-world systems, and assumptions are necessarily made to accomplish
this. Although the simplifying assumptions made during model development are
designed to minimize model error, they cannot eliminate it, and all models are thus
characterized by a certain amount of inherent error.
For K2, Syed (1999) obtained Nash-Sutcliffe model-efficiency values of 0.87 and
0.83 for runoff volume and peak discharge, respectively, during calibration runs on
WG11 (perfect agreement of observed and simulated values produces an efficiency of 1).
When the calibrated model was applied to the validation data set, however, efficiency
values were reduced to 0.43 and -1.1. Syed (1999) observed that smaller events were
difficult to model because the runoff to rainfall ratios are low, and input rainfall errors (as
well as numerical errors) become a large percentage of the overall model output. In other
words, the model did not represent rainfall-runoff processes equally well at all scales.
Modeled upland- and channel-erosion predictions have been shown to represent
poorly observations. Risse et al. (1993), for example, reported an overall Nash-Sutcliffe
model efficiency of 0.58 for predictions of annual sediment yield from runoff plots using
the Universal Soil Loss Equation (USLE). They noted that UST F. over-predicted soil loss
170
when erosion rates were low, and under-predicted when erosion rates were high. This
same observation was made by Nearing (1998) in a broad analysis of erosion-model
performance.
Bravo-Espinosa et al. (2003) compared observed bedload discharge with
predictions from seven common bedload equations under a variety of conditions.
Inequality coefficients computed for each of these comparisons ranged from 0.2222 to
0.9999, and the vast majority were greater than 0.8 (perfect agreement of simulated and
observed values produces an inequality coefficient of 0).
Whereas these examples are not necessarily representative of conditions on
Walnut Gulch, or error specifically associated with K2, they demonstrate that even
carefully calibrated models cannot reliably predict water and sediment yields for any
given event. For multiple events errors can be compounded, or reduced, depending on
the magnitude and initial conditions of each event. As such, the simulated geomorphic
changes presented in chapter 5 are useful in their ability to display where changes are
likely to be greatest; the magnitude of the changes should be considered unreliable.
Another way to derive useful information from the test simulation results is to
compare the computed geomorphic change between two simulations for which only one
initial condition varied. For instance, if the same simulation is repeated using two
different land-cover grids to generate input parameters for the upland elements, then the
difference between the two simulation results can be attributed to the different land-cover
parameters. By holding everything else constant, repeatable errors associated with
erosion and routing calculations are negated. Consistent over- or under-estimation of
171
erosion still can influence the magnitude of the differences, but their relative magnitudes
are unlikely to be significantly affected. In other words, it is possible to identify where in
the watershed the impacts of land-cover change are likely to be significant.
6.2 Methodology
The partially urbanized (PU) and NALC 1997 (97) land-cover scenarios were
selected for the relative-change assessment for two reasons. The non-urban component
of partially-urbanized scenario is the same as the NALC 1997 land cover, so differences
in geomorphic response can be attributed exclusively to the presence of the urban landcover class. In addition, results for these two scenarios were significantly different and
spatially varied.
Relative differences were computed for the batch simulations performed for the
1964 and 1977 land-cover scenarios, with initial channel geometries derived from
observations (OG) and the hydraulic-geometry relation (HG). Insufficient runoff from
the 1978 events yielded irregular, very small changes, so the 1978 results were not
included in this analysis. Simulation outputs were compared using the AGWA-G
differencing feature for batch geomorphic simulations. This feature compares all output
parameters for each simulation, and writes the differences (as absolute values or in
percent) to a new output file for visualization in the GIS.
172
6.3 Results
Relative differences between the PU and 97 simulations were visualized for depth
and width changes, and are presented in figures 6.1 and 6.2. As in the chapter 5 figures,
maps are in order of decreasing precipitation from top to bottom, depth changes are on
the left, and width changes are on the right. All differences are presented in meters,
which are computed as:
Absolute Difference = PU - 97
where PU represents an output from the partially-urbanized scenario, and 97 represents
the corresponding output for the NALC 1997 land cover. Increased erosion or decreased
deposition is shown in shades of red (positive values), and reduced erosion or increased
deposition is shown in shades of blue (negative values). Another way to look at this is
that positive values indicate relatively large widths and/or depths (greater width and/or
depth increases, or smaller width and/or depth decreases), and negative values indicate
relatively smalle widths and/or depths (greater width and/or depth decreases, or smaller
width and/or depth increases) for the PU scenario.
6.3.1 Relative Differences for the Hydraulic-Geometry Channels
Results for the HG channels (figure 6.1) indicate that the largest changes occurred
during the wet year (1964). All channels in the urbanized tributary were characterized by
increased erosion or decreased deposition (relatively large channels) for the PU scenario.
Relative differences were of approximately the same magnitude for both width and depth
changes, but did not always have the same sign. Downstream of the urbanized area
173
channel depth decreased, but widths increased for the PU scenario. This pattern of depth
decreases and width increases is commonly associated with channel aggradation, and
suggests that the model is yielding reasonable results.
slid
A/ -0.028 - -0.02
/V -0.02 - -0.012
slid
A,. -0.012--0.003
-0.003 - 0.005
0.005 - 0 013
0.013 - 0.021
NO.021 -0.03
0.03 - 0 038
Ao• 0.038 - 0 046
No Data
tv'
/V
A/
A/
A/
'
No Data
Wild
wild
slid
/\," -0 02 - -0.013
-0.013 - -0.006
-0.006 - 0 001
0 001 - 0.007
0 007 - 0.014
A/ 0 014 - 0.021
A/ 0.021 - 0.028
A/ 0 028 - 0.034
No Data
wild
-0.009 - -0.004
-0.004 - 0.001
0.001 - 0.006
0.006 - 0.011
0.011 - 0.016
0.016 - 0.021
0.021 - 0.026
slid
-0.001 -0
0- 0.001
A, 0.001 - 0.003
A/ 0.003 - 0.004
A/ 0.004 - 0.005
A/ 0.005 - 0.007
No Data
wild
Figure 6.1. Maps showing the difference (meters) in simulated average depth change
(left) and average width change (right) between the partial urbanization and NALC 1997
land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations based on the
HG channels. Shades of red (positive values) indicate that geomorphic adjustments
yielded relatively larger depths/widths for the PU scenario.
Most channels in the non-urbanized area of the watershed showed little or no
change, but there are some width differences where there should be none. Maximum
174
differences in the non-urbanized section of the watershed, however, are on the order of
millimeters.
For the 1977 batch simulations, relative-depth changes were mixed. The PU
scenario yielded both increases and decreases in channel depth relative to the 97 scenario
within the urbanized area. Channels downstream of the urbanized area were again
characterized by depth decreases or no change. Width changes were very small, and only
occurred in the uppermost channels within the urbanized area. No significant width or
depth changes were reported for the non-urbanized area.
6.3.2 Relative Differences for the Observed Channel Geometries
Relative differences between the PU and 97 scenarios for the 1964 batch
simulations based on the observed channels (figure 6.2, top) are notable in that significant
depth changes were not reported within the urbanized area. For each reach downstream
of the urban area, however, deposition in the PU scenario reduced channel depths.
Simulated widths for the PU scenario were larger or the same as those for the 97 scenario
within the urbanized area and downstream of it. Relative width changes in the nonurbanized area were small, and less prevalent than for the simulations based on the
hydraulic-geometry channels.
For the 1997 simulations, relative width and depth changes again were mixed.
Increased and decreased widths and depths can be observed both within and downstream
of the urbanized area. The patterns of width and depth changes appear unrelated
spatially. No changes were simulated for the non-urbanized area.
175
slid
A/ -0.137 -0.111
A/ -0.111 - -0.085
/\/-0.085 - -0.059
N -0.059 - -0.032
,
-0.032 - -0.006
-0.006 - 0.003
0.003 - 0.02
No Data
wild
slid
A/-0.015 - -0.012
A/-0.012 --0009
-0.009--0005
-0.005 - -0.002
-0.002 - 0.001
0.001 - 0.005
Cv 0.005 - 0.008
A/0.008 - 0.011
A/0.011 -0014
/V0.014 - 0.018
No Data
w11
slid
-0045- 0.043
0.043 - 0.13
0.13 - 0.218
A/ 0.218 - 0.305
0.305 - 0.392
A/ 0.392 - 0.48
No Data
wi ld
N
slid
A/ -0.034 - -0 025
/V -OE025 -0.016
-0.016 - -0.006
-0.006 - 0.003
0.003 - 0.012
0.012 - 0.021
0.021 - 0.03
0.03 - 0.04
1V0.04 -0049
/\/
No Data
Wi
Figure 6.2. Maps showing the difference (meters) in simulated average depth change
(left) and average width change (right) between the partial urbanization and NALC 1997
land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations based on the
OG channels. Shades of red (positive values) indicate that geomorphic adjustments
yielded relatively larger depths/widths for the PU scenario.
176
6.3.3 Relative Differences in Runoff Transmission Loss, and Sediment Yield
To help explain the patterns of relative geomorphic response shown in figures 6.1
and 6.2, relative differences in cumulative runoff, transmission loss, and sediment yield
were mapped for the 1964 and 1977 simulations (figures 6.3 and 6.4, respectively). For
the 1964 simulations, differences in cumulative runoff (mm over the contributing area)
declined in the downstream direction along the southern tributary for the HG channels.
The same trend is observed for the OG channels, although the specific pattern varies
slightly. Relative differences in transmission loss (m 3/1(m) and sediment yield (kg)
increased in the downstream direction for simulations based on both the HG and OG
channels. The specific pattern of differences varies between the HG and OG simulations
for both variables, but their overall trends are similar.
For the 1977 simulations, differences in cumulative runoff declined in the
downstream direction along the southern tributary for both the HG and OG channels
(figure 6.4). In addition, the specific pattern of downstream runoff declines is almost
identical, regardless of the initial channel geometry. Relative differences in cumulative
transmission loss and sediment yield initially increased in the downstream direction, and
then decreased slightly for the 1977 simulations. This trend was observed regardless of
the initial channel geometry, although specific patterns differed.
177
slid
slid
0 - 5 686
5686- 11.373
/ 11.373 - 17.059
17.059 - 22.745
A/ 22.745 - 28.432
28.432 - 34.118
No Data
wild
-0.178-5427
5.427- 11.032
/ -N., 11.032 - 16.636
A/ 16.636 - 22.241
A/ 22.241 - 27.846
A/ 27.846 - 33.451
No Data
wild
A/
A/
4isof
slid
-8446- 1573.799
1573.799 - 3156.044
3156.044
- 4738.289
A/
4738.289 - 6320.535
6320.535 - 7902.78
7902.78 - 9485.025
A/
N
A/
No Data
wild
slid
A/
A/
4
-6.804 - 1678.208
1678.208 - 3363.22
3363 22 - 5048.232
5048.232 - 6733.244
6733.244 - 8418.256
8418 256 - 10103.268
No Data
vv11c1
slid
slid
-9661.55- 631152.795
631152795- 1271967.14
" 1271967 14- 1912781.485
1912781.485 - 2553595.83
2553595.83- 3194410.175
/\,/3194410.175 - 3835224.52
A/
A/
A/ No Data
wild
A/
N
A/
-111836.51 - 533047.232
533047.232 - 1177930.973
1177930.973- 1822814.715
1822814.715- 2467698 457
2467698.457 - 3112582.198
3112582.198 - 3757465.94
No Data
wild
Figure 6.3. Maps showing the relative differences in cumulative runoff (mm, top),
transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1964
simulations. Results for the HG and OG channels are pictured on the left (A-C), and
right (D-F), respectively.
178
slid
slid
0- 3.154
3.154 - 6.307
6.307 - 9.461
9.461 - 12.614
12 614 - 15 768
No Data
wild
-0.039 - 2.588
2.588 - 5.214
5.214 - 7.841
7.84
1
- 10.468
A/ 10.468 - 13 095
13.095 - 15.722
No Data
wild
N
N
A/
A/
slid
slid
-12.061 - 1159.779
1159 779 - 2331.619
/, 2331 619 - 3503 459
Al 3503 459 - 4675.299
A/4675 299- 5847.139
A/ 5847.139 - 7018.979
No Data
wild
-11.094 - 1230.359
1230.359 - 2471.811
2471.811 -3713,264
3713.264 - 4954.717
A/ 4954.717 - 6196.169
A/ 6196.169 - 7437.622
/
slid
0 - 144626.338
144626.338 - 289252.676
/ 289252.676 - 433879.014
A/ 433879.014 - 578505.352
A/ 578505.352 - 723131.69
No Data
wild
A/
il
wild
slid
-796.85 - 103452.793
103452.793 - 207702.437
207702.437 - 311952.08
311952.08- 416201.723
A/ 416201.723 - 520451.367
A/ 520451.367 -624701.01
No Data
wild
Figure 6.4. Maps showing the relative differences in cumulative runoff (mm, top),
transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1977
simulations. Results for the HG and OG channels are pictured on the left (A-C), and
right (D-F), respectively.
179
6.4 Discussion
Differences in the geomorphic response of the channel network were mapped for
the PU land-cover scenario relative to the 97 land cover for simulations based on the
observed channel geometries and channel geometries defined by the AGWA-G default
hydraulic-geometry relation. The differences thus represent relative responses given the
same precipitation input when part of the watershed is urbanized. They do not represent
pre- and post-urbanization response.
Results indicate that geomorphic response for any given reach depends largely on
the initial channel geometry. The general pattern of channel adjustment, however, is
surprisingly similar for the two initial channel geometries. Both yielded decreased depths
and increased widths downstream of the urbanized area for the 1964 monsoon. The
mixed pattern of depth changes for the 1977 simulations was almost identical for the
observed and hydraulic-geometry channels. Relative width changes for the 1977
simulations, however, were more pronounced, and significant depth increases were not
observed in the urbanized area for the 1964 simulations when the observed geometries
were used.
Similarities in relative channel response between simulations based on the two
initial geometries can be attributed largely to similar spatial patterns of relative runoff,
transmission loss, and sediment yield. Although values vary slightly in location and
magnitude, the spatial pattern of these relative outputs was always the same between OG
and HG simulations. In the absence of observed channel geometries, a suitable
180
hydraulic-geometry relation may be sufficient to provide some indication of how
channels might respond to landscape change.
Another important relation revealed by the relative assessment is that geomorphic
response to landscape change is highly dependent on the magnitude of precipitation.
Given two series of precipitation events that likely represent only a fraction of the
broader spectrum of potential monsoon rainfall, it is apparent that the pattern of channel
adjustments can vary significantly. The specific patterns presented above are likely to be
influenced by limitations in the conceptual representation of erosion in urbanized areas
(as described in chapter 5), but the spatial pattern of erosion and deposition within the
channel network would be just as likely to vary with event- and seasonal-rainfall
magnitude.
The relevance of spatial characteristics of stream power to erosion, sediment
transport, and sediment storage has been evaluated in several studies (e.g. Graf 1982,
1983a, 1983b, 1990; Magilligan, 1992; Lecce, 1997; Fonstad, 2003). All of these have
reported nonlinear variation of stream power and associated variability in sediment
transport and storage in the downstream direction. Although stream power was
regrettably not reported in the output for these analyses, its variability can be recognized
by the presented spatial trends in runoff and transmission loss. It can furthermore be
deduced that spatial patterns of stream-power variation change with event- and seasonalrainfall magnitude, and that the simulated pattern of geomorphic adjustments is a direct
result.
181
7.0 SUMMARY AND CONCLUSIONS
Prior to listing the major conclusions that can be drawn from this research, it is
appropriate to restate briefly the objectives. This study was designed to improve
watershed assessment tools for arid and semi-arid regions. To that end, specific goals
were identified: improving the upper watershed size limit to which event-based runoff
models can be applied confidently; and developing a geomorphic model capable of
simulating width, depth, and slope adjustments from multiple runoff events in ephemeral
stream networks.
7.1 Watershed Runoff Routing and Scale
A routing subroutine was developed for the Kinematic Runoff and Erosion Model
(Smith et al., 1995) using an iterative four-point solution of the modified variableparameter Muskingum Cunge (MVPMC4) technique (Ponce and Chaganti, 1994).
Analyses were designed to test the hypothesis that by accounting for flood-wave
diffusion the MVPMC4 technique could improve model performance on intermediatesized watersheds relative to kinematic routing. All analyses were conducted for the
Walnut Gulch Experimental Watershed (see section 1.4), and, as such, caution should be
used in extrapolating the conclusions to other areas. Although Walnut Gulch is assumed
typical of arid-region watersheds in many respects, the scale at which flood-wave
diffusion effects become significant is significantly influenced by channel slopes.
182
7.1.1 Conclusions
Major conclusions drawn from the comparison of kinematic and MVPMC4
routing at multiple scales included:
1. Hydrographs simulated by kinematic and MVPMC4 routing were statistically
different at the 5% level of confidence for events with a 5-year return period or
smaller on watersheds of 9558 ha (95.6 km 2 ) and larger. For larger events and/or
smaller watersheds, the two techniques yield outflow hydrographs that are
statistically indistinguishable.
2. Model mass-balance error decreases with increasing watershed area when
MVPMC4 routing is used. Conversely, mass-balance error increases with
increasing watershed area when kinematic routing is used.
3. Outflow hydrographs simulated with MVPMC4 routing more closely represent
observed hydrographs than those simulated with kinematic routing at all scales,
and performance gains increase with increasing watershed size.
4. Matched physical and numerical diffusion in the MVPMC4 routing technique
improves its performance relative to kinematic routing under conditions where
flood-wave diffusion is most pronounced: for smaller events in larger watersheds.
7.1.2 Future Research
A comparative analysis of routing performance for compound channels was not
attempted in this research. Hydrograph attenuation resulting from overbank flows
increases the importance of flood-wave diffusion, which has the effect of making
183
diffusive effects important on smaller watersheds when flood-plain spillage occurs than
for an equivalent channel network without flood plains. Flood plains represent important
reservoirs for sediment storage, and are thus crucial in geomorphic modeling. The
geomorphic model developed in this research thus requires MVPMC4 routing for
compound channels before it can be applied to larger watersheds.
The variable-parameter Muskingum-Cunge technique has been implemented for
compound channels in the hydrologic Drainage Network Channel Flow Routing model,
DNCFR, (Garbrecht and Brunner, 1991). DNCFR, however, is a continuous simulation
model for large watersheds with perennial flow. Variable-parameter Muskingum-Cunge
routing has not been applied previously to channel networks in event-based ephemeral
flow models.
7.2 Geomorphic Modeling at the Watershed Scale
A geomorphic model, K2G, was developed based on the Kinematic Runoff and
Erosion Model to compute width, depth, and slope adjustments from simulated changes
in sediment storage. Width and depth adjustments are computed based on the minimum
rate of energy dissipation theory (Yang and Song, 1986), and specifically the theory's
special case, the minimum stream-power theory (Chang, 1980). A GIS-based interface
was developed to coordinate model parameterization, multiple-event simulations, and the
visualization of model output.
In the absence of observed data, testing of the geomorphic model was designed to
accomplish the following:
184
•
Demonstrate that the model conserves mass
•
Confirm that the model behaves reasonably given different land use/cover
conditions and precipitation inputs
•
Evaluate the importance of utilizing observed channel-geometry information
7.2.1 Conclusions
Conclusions drawn from the individual batch simulation results presented in
Chapter 5:
1. The model conserved mass reasonably well when initial channel geometry was
defined by a hydraulic-geometry relation and bank slopes defaulted to 1.0. When
observed geometries with variable bank slopes were used, however, the mass
imbalance was significant.
2. Erosion was greatest and most widespread for the wettest precipitation record.
Mixed erosion and deposition was simulated for the intermediate record, and
deposition was most widespread for the driest record. This trend was expected,
and strongly indicated that a sediment balance could be achieved if the model was
run for a long enough period of time.
3. Simulated channel adjustments were particularly sensitive to the initial channel
geometry. The pattern of geomorphic adjustments for each land-cover scenario
varied significantly when observed geometries were used, and was much more
uniform when the default hydraulic geometry channels were used. Use of the
default hydraulic-geometry relation for model parameterization is not
185
recommended when estimates of channel adjustment are needed for a specific
reach or reaches.
Conclusions drawn from the differencing of simulation results presented in
Chapter 6 for the partially urbanized and NALC 1997 land-cover inputs (relative change
assessment):
4. The relative assessment of partial urbanization and un-urbanized scenarios did an
excellent job of removing background change. Results of the scenario-output
differencing show the concentration of impacts within and downstream of the
urbanized area, and no significant changes in the unaffected areas.
5. Partial urbanization of the upland area resulted in increased channel incision
within the urbanized area and increased deposition downstream for the wettest
precipitation record (1964). Increases were less pronounced than expected due to
contemporary increases in upland erosion that resulted from the lack of direct
hydraulic connectivity between impervious surfaces and the channels.
6. Increased aggradation downstream of the urbanized area for the 1964 simulations
was characterized by depth reductions and width increases. This pattern
commonly is observed in natural systems during periods of aggradation, and
provides a strong indication that the model is behaving properly.
7. Urbanization of the upland areas resulted in mixed increases in erosion and
deposition within the urbanized area, and increased deposition downstream for the
intermediate precipitation record (1977). The difference in relative response
186
between the 1964 and 1977 simulations indicates that any assessment of stability,
or vulnerability to degradation, requires consideration of a range of return-period
rainfall.
8. Spatial patterns of geomorphic adjustment to urbanization were closely linked to
variability in cumulative runoff depth and transmission loss. Runoff initially
increases as flows coalesce in the headwaters, and then declines as transmission
losses increase in the downstream direction. Sediment yield is highest at this
transition and begins to decline downstream, which closely corresponds with
where increased deposition was predicted by the model. The specific location of
the transition shifts upstream for simulations with small average event discharges
for which transmission losses can cause significant reductions in sedimenttransport capacity. When runoff volumes are large relative to transmission losses,
the locus of deposition shifts downstream.
9. The large-scale patterns of relative geomorphic response to urbanization were
strikingly similar regardless of the initial channel geometry. This suggests that
channel slopes and lengths are the most important parameters controlling
watershed-scale geomorphic response, even though specific adjustments may vary
from reach to reach. Watershed-scale assessments of geomorphic response to
landscape change can thus be carried out with reasonable confidence in the
absence of detailed channel-geometry information.
187
7.2.2 Future Research
Much work remains to be done before K2G/AGWA-G is ready for broad
distribution as an assessment tool. In particular, the profile-smoothing calculations need
to be corrected to account for variable widths, depths, and bank slopes within each reach
and between reaches connecting at a junction. With this correction, mass conservation
should be greatly improved (or at least be as good as it is in the K2 sediment-transport
calculations), particularly when observed sections are used.
Rigorous testing of the model must be conducted to demonstrate that simulated
geomorphic adjustments are representative of observed adjustments, and to quantify error
in the simulated adjustments over a range of conditions. Data for this type of testing
rarely is available on a watershed scale, but repeat LlDAR over-flights promise to be a
useful source. Some problems with vertical and horizontal offset errors have occurred
with surface matching for repeat LIDAR data (Crowell et al., 2003), but the errors can be
largely removed. In addition, LIDAR data, as opposed to at-a-point field measurements,
provides the ability to derive much more detailed information about the spatial variability
of channel response.
In its present form, the K2G/AGWA-G geomorphic model can only be applied to
areas smaller than about 10 km 2 . For larger areas, channel lengths are too long to
represent adequately channel-slope variability, and hence geomorphic response. In
addition, longer reaches limit the model's ability to adjust reach slope because bedelevation adjustments become vanishingly small relative to the length of the channel. To
increase the upper size limit of the geomorphic model, automated, GIS-based procedures
188
for channel network discretization need to be developed for which a maximum reach
length can be specified. Ideally, the channel-network discretization would identify the
most significant changes in channel slope. Another improvement to the modeling tool
that would increase the scale at which it can be applied is the addition of MVPMC4
routing described above.
The results presented in chapters 5 and 6 provide a tantalizing glimpse of the
many questions that can be addressed with a model such as K2G that can represent
relations between rainfall, transmission losses, stream power, and geomorphic response.
In particular, the geomorphic model holds great promise in its ability to address the issue
of short-term channel stability in response to landscape and climate change. Before this
can be undertaken, however, a good deal of work needs to be conducted to ascertain
model behavior under the conditions in which channel destabilization has been observed.
For instance, spatial patterns of geomorphic adjustment need to be mapped over a broader
range of precipitation records and over longer periods of time (years to decades). In
addition, the model must be pushed to evaluate its response to major disturbances in the
channel network and on the uplands, and whether it can recover its initial geometry after
the disturbance is removed. In other words, it is necessary to determine if externally
imposed disturbances result in the crossing of an extrinsic threshold to a new process
regime, which is characterized as responsive behavior (Werritty, 1997), and whether the
new regime is persistent (transitive) or short-lived (intransitive) using the terminology of
Chappell (1983).
189
When and if the assessment of channel stability can be accomplished, it would
represent a significant advancement in surficial-process modeling for arid regions. Used
in coordination with an alternative futures scenario-generation tool developed for AGWA
(Levick et al., 2003), the geomorphic modeling tool would provide the first
comprehensive assessment and planning tool for arid-region watersheds and their stream
networks.
190
APPENDIX A: DESIGN AND OBSERVED
EVENT HYDROGRAPHS
A.I Design Storm Hydro graphs
Hydrographs from both K2 and K2MC are plotted side by side in figures A.1-4
for six design events in each of the four Walnut Gulch subwatersheds: LH104 (4.74 ha),
WG11 (782 ha), WG6 (9558 ha), and WG1 (14664 ha). Each figure contains all six
design events in the same order; from upper left to lower right these are the 5-year 30-
minute, 5-year 1-hour, 10-year 30-minute, 10-year 1-hour, 100-year 30-minute, and 100year 1-hour. To facilitate visual comparison, the hydrographs are presented as discharge
in cubic meters per second as a function of time in minutes.
A.2 Observed Event Hydrographs
Simulated hydrographs from K2 and K2MC are plotted together with observed
runoff hydrographs from the Walnut Gulch event database in figures A.5-26. The timing
of simulated and observed hydrographs was synchronized by adding the difference in
time between the observed start of rainfall and the observed start of runoff to the
observed hydrograph times. A minimum of five events was simulated for each of the
four subwatersheds (LH104, WG11, WG6, and WG1). Events were selected to provide a
range of peak discharge values, but very small events were ignored because of poor
model performance for the smallest events.
All graphs for the observed events plot discharge in cubic meters per second as a
function of event time in minutes. Data points were omitted from the WG1 plots for
191
clarity; the relatively small discharge values for K2 and K2MC were very difficult to see
when data points were plotted. The time step for the simulation models was always one
minute. The time step for the observed hydrographs is variable; data points are collected
when changes in discharge are detected (breakpoint data).
0.7
06
r
192
-.-
i
0.5
04
.
K2
K2MC
‘
,
03
02
\.
1
0.1
i
__.
ç
0
10
30
50
70
90
0.9
08
K2
•
07
K2MC
06
05
0.4
03
0.2
0.1
0 10
30
50
70
90
1.4
1.2
K2
•
K2MC
0.8
0.6
0.4
0.2
0
10
30
50
70
90
Figure A.1. K2 and K2MC simulated LH104 hydrographs for design storms ranging
from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right).
Plots are discharge in cubic meters per second as a function of time in minutes.
193
6
30
5
25
•
A
K2
• K2MC
20
I
15
i
10
5
0
0
100
50
150
200
•
100
K2
K2MC
BO
Y
60
40
20
0
0
50
100
150
200
Figure A.2. K2 and K2MC simulated WG11 hydrographs for design storms ranging
from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right).
Plots are discharge in cubic meters per second as a function of time in minutes.
194
114
12
-4--
K2
• K2MC
10
8
4
2
-«- K2
K2MC
250
-
t
‘
200
150
100
0
50
100
150
200
250
300
Figure A.3. K2 and K2MC simulated WG6 runoff hydrographs for design storms
ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower
right). Plots are discharge in cubic meters per second as a function of time in minutes.
195
20
18
16
K2
• K2MC
14
12
10
0
100
150
200
250
300
350
0
400
50
100
150
200
250
900
350
400
4
180
160
250
140
K2
K,2 M C
K2
• K2MC
200
120
100
150
100
50
20
0
50
100
150200
250
300
350
400
4
CI
50
100
150
200
250
300
350
400
4501
Figure A.4. K2 and K2MC simulated WG1 hydrographs for design storms ranging from
the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots
are discharge in cubic meters per second as a function of time in minutes.
196
0.3
0.25
—4,-- K2
.4t.
-
Ig
K2MC
0.2
Observed
0.15
I
0.1
0.05
0 simm.**90
101101PPP•Pni.c
.
110
100
120
130
140
150
170
160
180
19
Figure A.5. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on
LH104 (discharge in cubic meters per second plotted as a function of time in minutes).
0.3
0.25
—4—
K2
K2MC
0.2
Observed
0.15
0.1
0.05
o
50
70
110
90
130
150
170
190
230
210
25
Figure A.6. Observed and simulated (uncalibrated) hydrographs for the 7/28/81 event on
LH104 (discharge in cubic meters per second plotted as a function of time in minutes).
0.8
f
0.7
0.6
.1
0.5
--•— K2
Observed
,..
\,..
J
0.4
0.3
"1".•....
0.2
-.
0.1
0
K2 MC
•
* \*
-41...,.”"
. 4.
45
4.6
or
•
r
55
65
75
85
1P ,P 111411.0.4.
95
105
Figure A.7. Observed and simulated (uncalibrated) hydrographs for the 9/10/83 event on
LH104 (discharge in cubic meters per second plotted as a function of time in minutes).
197
1
MC
Observed
0.7
0.6
\..,
0.5
.T
0.4
0.3
lO•
1
0.2
%
#
0.1
0
•
20
%.11414141'744‘'--"IRP liT 'low n vim.ord. -
—
0
10
30
40
50
60
70
90
80
1..
Figure A.8. Observed and simulated (uncalibrated) hydrographs for the 8/10/86 event on
LH104 (discharge in cubic meters per second plotted as a function of time in minutes).
0.35
a** ••••
0.3
K2
K2MC
0.25
Observed
0.2
0.15
0.1
0.05
90
100
110
120
130
140
150
160
170
180
19
Figure A.9. Observed and simulated (uncalibrated) hydrographs for the 8/29/86 event on
LH104 (discharge in cubic meters per second plotted as a function of time in minutes).
Figure A.10. Observed and simulated (uncalibrated) hydrographs for the 7/27/76 event
on WG11 (discharge in cubic meters per second plotted as a function of time in minutes).
198
25
20
—10—
K2
K2 MC
Observed
15
10
••••••n• •
•=,
190
170
150
130
110
90
70
Figure A.11. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on
WG11 (discharge in cubic meters per second plotted as a function of time in minutes).
35
30
—4.--
K2
- K2 MC
25
Observed
20
15
10
5
4411111411"41461111"1001 01011V1411111010rmwo NIMIllemsrmdadadam.
0
70
50
110
90
130
150
190
170
Figure A.12. Observed and simulated (uncalibrated) hydrographs for the 8/4/80 event on
WG11 (discharge in cubic meters per second plotted as a function of time in minutes).
40
35
—4.--
30
a
K2
K2 MC
Observed
25
20
15
10
5
•nnn •nnn•••"Ilt
100
120
140
160
180
200
220
240
260
280
30
Figure A.13. Observed and simulated (uncalibrated) hydrographs for the 8/29/86 event
on WG11 (discharge in cubic meters per second plotted as a function of time in minutes).
199
-
K2
-
K2MC
Observed
_ 3IFJP.PIW 11•••••nn•••_
0
40
20
60
80
100
140
120
180
160
Figure A.14. Observed and simulated (uncalibrated) hydrographs for the 8/1/90 event on
WG11 (discharge in cubic meters per second plotted as a function of time in minutes).
45 40
—4.—
35
•
K2
K2 MC
Observed
30
25
20
15
10
5
0
50
150
100
300
250
200
Figure A.15. Observed and simulated (uncalibrated) hydrographs for the 7/27/76 event
on WG6 (discharge in cubic meters per second plotted as a function of time in minutes).
70
60
K2
K2 MC
50
Obs erved
40
30
20
10
o
1111111. 1111011. 1111mor- 111WArar Immoirrier
100
150
200
250
300
Figure A.16. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event
on WG6 (discharge in cubic meters per second plotted as a function of time in
minutes).
200
16
14
12
--411--
K2
--MI—
K2 MC
Observed
10
8
6
4
2
o•• —•••n•!
—46-44aglimells—limilmint_asmor_pww_wr
150
100
350
300
250
200
Figure A.17. Observed and simulated (uncalibrated) hydrographs for the 8/4/80 event on
WG6 (discharge in cubic meters per second plotted as a function of time in minutes).
25
20
K2
K2MC
Observed
15
10
o
11, mummer majn
V
100
250
200
150
300
350
450
400
50
Figure A.18. Observed and simulated (uncalibrated) hydrographs for the 8/29/86 event
on WG6 (discharge in cubic meters per second plotted as a function of time in minutes).
30 25
—4--
K2
K2MC
20
Observed
15
10
5
0
50
100
150
200
250
Figure A.19. Observed and simulated (uncalibrated) hydrographs for the 8/1/90 event on
WG6 (discharge in cubic meters per second plotted as a function of time in minutes).
201
25
K2
20
- K2IVIC
Observed
15
10
0
50
100
150
200
250
300
350
400
450
Figure A.20. Observed and simulated (uncalibrated) hydrographs for the 7/27/76 event
on WGI (discharge in cubic meters per second plotted as a function of time in minutes).
35
30
Observed
-K2
25
- K2MC
20
15
10
o
150
200
250
300
350
400
450
Figure A.21. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on
WG1 (discharge in cubic meters per second plotted as a function of time in minutes).
1.2
Observed
1
-K2
K2 MC
0.8
0.6
0.4
0.2
0
250
300
350
400
450
500
Figure A.22. Observed and simulated (uncalibrated) hydrographs for the 8/4/80 event on
WG I (discharge in cubic meters per second plotted as a function of time in minutes).
202
30
Observed
25
-
20
K2
K2IVIC
15
10
5
0
o
200
600
400
1000
800
1200
Figure A.23. Observed and simulated (uncalibrated) hydrographs for the 7/30/81 event
on WG1 (discharge in cubic meters per second plotted as a function of time in minutes).
12
Observe
10
-K2
MC
8
K2
6
4
2
0 ---
0
,
-r----
100
200
300
-r-
,
400
500
600
700
80
Figure A.24. Observed and simulated (uncalibrated) hydrographs for the 8/17/86 event
on WGI (discharge in cubic meters per second plotted as a function of time in minutes).
20 Observed
16
K2
K2 MC
12
8
4
0
300
350
400
450
500
550
600
650
Figure A.25. Observed and simulated (uncalibrated) hydrographs for the 8/2986 event on
WG1 (discharge in cubic meters per second plotted as a function of time in minutes).
203
1
Observed
-K2
K2 MC
0 150
200
250
300
350
400
45
Figure A-26. Observed and simulated (uncalibrated) hydrographs for the 8/1/90 event on
WGI (discharge in cubic meters per second plotted as a function of time in minutes).
204
APPENDIX B: DERIVATION OF
CHANNEL GEOMETRY ADJUSTMENTS
Channel-depth adjustments in K2G are derived for depositional and erosional
scenarios. Channel bottom width is preserved during both adjustments, but bank side
slopes are allowed to vary. Both calculations compare the original channel crosssectional area (A 0 , volume per unit length of channel) with the sum of the original area
and the area change, IV
B.1 Depositional Depth Changes
To compute depth decreases as a result of deposition, the adjusted cross-sectional
area, A a , is computed first as:
= A0 —
(B.1)
The adjusted cross-sectional area can also be written as:
A, = d a • w+ 0.5.d, • swl + 0.5. d a • sw2
(B.2)
where da is the adjusted channel depth, and sw/ and sw/ are the bank widths on either
side of the channel (figure B.1). Bank widths are computed from the original geometric
parameters, and are only adjusted in the case of bank failure. Solving equation B.2 for
the adjusted channel depth yields
205
da =
A,
(B.3)
[w+ 0.5(swl + sw2)]
for which everything on the right hand side is known. Following the computation of
adjusted channel depth, the bank slopes (rise/run) are adjusted by
zl =
d
(B.4)
swl
and
z2 =
•
d
(B.5)
sw2
04
04
sw2
sw 1
Sketch illustrating the depositional depth reduction
Figure B.1.
calculations for K2G trapezoidal channels.
B.2 Erosional Depth Changes
To compute depth increases as a result of erosion, the adjusted cross-sectional
area, A a , is computed first as:
= A, + A,
The adjusted cross-sectional area can also be written as:
(B.6)
206
A,
= d a • w + 0.5 d a • swl + 0.5 d a • sw2 (B.7)
where da is the adjusted channel depth, and sw/ and sw/ are the bank widths on either
side of the channel (figure B.2). Bank widths are computed from the original geometric
parameters, and are only adjusted in the case of bank failure. Solving equation B.2 for
the adjusted channel depth yields
da =
Aa
[w + 0.5(swl + sw2)]
(B.8)
for which everything on the right hand side is known. Following the computation of
adjusted channel depth, the bank slopes (rise/run) are adjusted by
d
zl = a
swl
(B.9)
z 2= d '
(B.10)
and
sw2
swl
w
sw2
Figure B.2. Sketch illustrating the erosional depth increase calculations
for K2G trapezoidal channels.
207
B.3 Bank Failure Resulting from Erosional Oversteepening
Bank failure occurs when channel incision increases bank slope above 3.75, and
bank slope is automatically reduced to 1.75. The volume of failed material per unit
length of channel is computed by subtracting the channel area above the new bank from
that over the original (figure B.3). Since these are two right triangles the calculation
becomes
Vf
= 0.5. d • sw f 0.5. d • sw o(B.11)
where d is the channel depth, sw o is the original channel bank width (known), and swf is
defined as
SW f
=
d
1.75
because the final bank slope of 1.75 is known.
d
Figure B.3. Sketch illustrating the bank failure calculations for K2G
trapezoidal channels.
(B.12)
208
B.4 Width Changes
Depositional and erosional width changes in K2G are accomplished simply by
translating the channel banks towards or away from the center of the channel,
respectively. As a result, the volume per unit length of channel is computed as the area
of a parallelogram, which is defined as the base length times the height (perpendicular
distance between the top and bottom). To calculate this area, consider the parallelogram
on its side such that the base width is equal to the width change, and the height is equal to
the depth of the channel (figure B.4). The area of the parallelogram is known in this case,
and is defined as one half of the volume per unit length of channel that is available to
produce width changes. Width change, wc, (the base width) can thus be computed as
wc --
Ap
d
(B.13)
where Ap is the area of the parallelogram. The other half of the volume available for
width change is applied to the other side of the channel.
Figure B.4. Sketch illustrating depositional width reduction calculations
for K2G trapezoidal channels. Width reduction is pictured here, but the
variables and procedure is identical for width increases.
209
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