Homework #9  2012, Earth Surface Processes, Humphrey

 

 

We have been talking about the variability of river discharge.  There are many ways of looking at how river discharge varies.  A useful type of plot is shown below, which nicely shows that the Laramie river typically has only one flood (snow melt dominated) in early summer.

 

 

For this exercise, we are going to look at something that is of major interest to many, the question of the likelihood of large floods.  Here is some recent data from the Laramie river at Laramie (this is the only recent data of which I am aware):

 

Flow measurements as reported on UPRR quarterly Discharge Monitoring Reports (DMR) for WDEQ Permit WY0032590,

Laramie Tie Plant.

Year

Oct       Nov      Dec      Jan       Feb      March   Apr       May      June     July      Aug      Sept

Streamflow (cfs)

1987     8.1        77         49                     54         65         62         157       46         22         7.8        4.5

1988     28         40         26         7.9        7.0       68         201       508       680       48         8.9        5.5

1989     17         44         34         7.1        2.0        61         nr         16         45         13         11         13

1990     36         75         60         27         30         189       36         30         351       50         20         10

1991     7.9        41         76         11         24        104       20         169       612       19         21         11

1992                                                                                                     231

1993     49         102       147       196       264       309       39         407       1016     115       11         36

1994     14         38         67         177       209       93         53         288       77         7.6       7.0        4.7

1995     45         55         53         86         65         68         6.8        111       1281     281      14         15

1996     51         62        88         77         68         79         107       552       1136     420       20         21

1997     115       128       115       55         62         82         101       503       1277     149       110       87

1998     34         55        63         70         75         87         100       305       425       139       66         29

1999     20         18         67         56         60         75         68         496      1126     267      20         17

2000     18         40         13         84         72         45         54         297       101       18         6          13

2001     56         20        54         77         90        75         21         177       66         11         18         26

2002     11         16         21         160       207       174       32         68         56

2003     18         49        68         28         22         22         20         230       556                               14

2004     131       122       113       68        83         72

2005     47         49         81         94         98         90         53         306       1115     95         80         16

2006     188       201       200       126       109       107       159       386       105       187       109       115

2007     143       104       68         200       186       281       146       549       378       78         109       34

2008                                         68         68         68         98         469       1715     316       180       169

 

The above table shows several problems with flood analysis.  Real data sets have missing and questionable data.  The above table lists measurements taken once a month, obviously it probably misses the actual flood peaks.  There are more subtle problems with stream data: even if you find more continuous data (e.g. USGS typically reports daily discharge), the actual discharge is not measured, but estimated from river depth.  Any bed or bank erosion or deposition will create errors in this depth based estimate, this is especially a problem at high flows when erosion/deposition is common.  The biggest problem with the Laramie river data, is that over time, various water projects have diverted water from the river.  As a result, the flood data is not ‘stationary’, in other words the flood data does not represent a sample from the same river over time.  The river has been changing, so that we can’t trust the old data to predict the future.

 

To continue our analysis we will use a better data set as shown below.

 

 

Grey River, at Dobson New Zealand

1968 to 2004

year

Max flow (m3/s)

 

 

1997

5950.8

 

 

1988

5840.4

 

 

1998

5670.0

 

 

1970

4899.1

 

 

1994

4844.5

 

 

1977

4841.4

 

 

1984

4814.3

 

 

1983

4228.2

 

 

1969

4203.4

 

 

1972

4125.6

 

 

1975

4117.8

 

 

1980

4039.4

 

 

1973

4012.5

 

 

1979

4000.9

 

 

1982

3975.2

 

 

1996

3866.7

 

 

2000

3809.5

 

 

1974

3771.7

 

 

1968

3678.3

 

 

2002

3517.9

 

 

1976

3463.4

 

 

1981

3448.9

 

 

1993

3422.3

 

 

2001

3342.7

 

 

1978

3302.9

 

 

2004

3224.6

 

 

2003

3221.9

 

 

1989

3217.1

 

 

1995

3185.8

 

 

1992

3177.6

 

 

1991

3091.4

 

 

1999

3070.0

 

 

1990

2806.8

 

 

1971

2420.5

 

 

1987

2385.4

 

 

1986

2364.9

 

 

1985

1794.8

 

 

 

Mean 3761.0

 

 

 

 

 

 

 

 

Question 1

a.       Calculate the recurrence intervals for floods on the Grey River.  There are a variety of methods to calculate recurrence interval, but probably the simplest is as follows: order your data from largest to smallest (this I have done for you).  Now apply the following formula to each datum:

Tr = (N+1)/n,  where N is the total number of observations, n is the ranking in the above list from top to bottom (eq the second from the top is n=2) and Tr is the recurrence interval in years.

b.      Plot the recurrence intervals on a semi log plot.  Use log time on the x-axis and discharge on the y-axis.  We will use a log axis, however there is considerable discussion in the literature about the expected shape of a recurrence interval curve, or more precisely, how floods should be distributed in time.  (If you would like to investigate this more, look up Gumbel Distribution on the web.)  We use a log plot since it is straightforward to plot, not because it is correct.

c.       Use your plot to estimate the 100year flood on the Grey river.  Comment on the accuracy of your prediction.

d.      The channel forming discharge for a meandering river is typically about the 2year flood.  What is the 2 year flood on the Grey River.

 

 

Question 2

This is a follow-up to last week’s homework, question 2, which most of you missed because you assumed that the depth of a river didn’t change, when I asked about the slope changing.  So a slightly different question:

a.       How does the basal shear stress in a river change if the river enters a section where the slope is doubled?  Assume width and discharge and sediment in transport stay the same.  You can approach the problem with the Manning’s eqn and discharge continuity.

b.      Since the size of the material on the bed will probably change, and Manning’s n will increase slightly: will the increase in ‘n’ increase or decrease the basal shear stress.

 

Question 3

We introduced Hydraulic Geometry in class.  Remember, these relationships have no physics in them, they are only compilations of real world data on how real river behave. I presented some equations for the downstream changes in the major stream variables.  By mistake, I actually gave you the ‘at a station’ equations, in other words the equations that describe how a river location changes as discharge changes.  So here we will investigate a little.

width = a Qwb,   depth = c Qwf,   velocity = k Qwm

where b = .26,  f = .4 and m = .34, and Qw is the water discharge.  The values of a, c, and k are dependent on each river and highly variable.

a.       Explain why  b+f+m should equal 1 ! (big hint: width * depth * velocity should equal Qw at any point of the river).  While you are thinking about this, explain also why a*c*k should also be 1?

b.      (needs careful thinking) Let us investigate whether the hydraulic relations match what we know about the behavior of rivers from our work with the more theoretically based equations such as Manning’s equation.  We know from our discussion in class that the slope of a river changes very slowly.  Assume it doesn’t change over a flood.  Also assume Manning’s ‘n’ is constant.  Now compare the results of the hydraulic geometry equations to a doubling of discharge, to the results you get from Manning’s equation.  (Hints: if it makes you more comfortable, you can assume values of a=2, c=.5, k=1, although they are not needed. Try substituting the hydraulic geometry equations into Manning’s equation.  Don’t forget to include the width in Manning’s equation for total discharge.  Use depth, not hydraulic radius in Manning’s)

c.       (puzzle) If you use the more correct hydraulic radius instead of depth in question b above, does it improve the match between hydraulic geometry and Manning’s formula?

c.