Flow Rate

Abstract: The aim of this paper is to analyse the flow rate of the Nolichucky River in Tennessee. Various mathematical models are compared so as to find the most appropriate one and then estimate some specific features such as the maximum, the mean and the rate of change.

The data used throughout this analysis originates from the Nolichucky River in Tennessee

between 27 October 2002 and 2 November 2002. The time is measured in hours past

midnight, starting at 00:00 on 27 October and the flow is measured in cubic feet per second

(cfs). Fig.1 shows the original data.

Fig.1

A scatter plot of the data illustrates the relationship between the points. The flow is

represented on the y-axis and time on the x-axis. This is shown in fig.21

It is important to know that a graph plotting flow rate over time is often called a Hydrograph.

1 Graph Plotted in Macintosh Microsoft Excel 2004.

1

Time 0 6 12 18 24 30 36 42 48 54 60 66 72Flow 440 450 480 570 680 800 980 1090 1520 1920 1670 1440 1380

Time 78 84 90 96 102 108 114 120 126 132 138 144Flow 1300 1150 1060 970 900 850 800 780 740 710 680 660

Fig.2

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160

Time (hours)

Flow (cfs)

The graph shows that the flow rate was less then 500 cfs2 when something caused it to

increase parabolically until reaching a maximum value 50 hours later. After this, the flow rate

decreased rapidly for about 20 hours and then steadily fell towards its initial value during the

next 60 hours.

Before finding the line of best fit, it is possible to estimate the rate of change of the data. This

is done using a numerical method and allows us to roughly illustrate the behaviour of the flow

rate. Considering the first two points of the graph:

€

(t = 0, flow = 440) and (t = 6, flow = 450) ,

the change

€

dx of the x values is

€

6 − 0 = 6 and the change

€

dy of the y values is

€

450 − 440 =10.

As a result, the rate of change of the data is

€

dy

dx=

10

6=1.66 . The procedure is repeated for all

points of data and the results are plotted in Fig.3. (Note that the time value is the average

between the two original points)

The rate of change of the flow rate is therefore positive and increasing during the first 52

hours and negative and decreasing for the rest of the time.

2 Cubic feet per second

2

Fig.3

-60

-40

-20

0

20

40

60

80

0 50 100 150

Time

dy/dx

In order to accurately process and analyse the data, it is necessary to find the lines of best fit,

or the mathematical relationship between the points on the graph. Best-fit lines are estimated

using technology in programs such as Microsoft Excel or the TI-84 GDC calculator.

A straightforward linear correlation would not be accurate because the data shows distinct

changes in patterns. Moreover, the graph does not resemble to any known polynomial

function except the quartic, which is still far from the real correlation as shows model 1

Model 1 shows the quartic best fit:

€

y = 5 ×10−5 x 4 − 0.0132x 3 + 0.807x 2 + 6.2022x + 346.64 .

Note that even if the line follows the trends to a certain extent, it is still very far from the original values. Moreover, the line increases again when time equals around 130 hours whereas the data shows a continuous decrease.

It seems that there is no single function that would show behaviour similar to the original

data. (i.e. increasing parabolically, reaching a maximum, decreasing parabolically and then

decreasing steadily.) It is therefore necessary to split the data into two parts in order to

produce more accurate best-fit lines.

3

Model 1

y = 5E-05x 4 - 0.0132x 3 + 0.8078x 2 + 6.2022x +

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160

Time (hours)

Flow (cfs)Best-Fit line

Estimating from the graph, the maximum value takes place when time equals 54 hours hence

points occurring before 54 hours belong to the increasing trend and points occurring after 54

hours belong to the decreasing trend. The composite function representing the original data

will be in the form of

€

Flow =f1(t) for 0 ≤ t ≤ 54

f2(t) for 144 ≥ t ≥ 54

⎫ ⎬ ⎭

⎧ ⎨ ⎩

with one function effective for

times before 54 hours and another for times after 54 hours (note that for more accuracy the

point

€

t = 54 belongs to both functions.) This technique enables us to use different types of

functions as best fit lines.

The first hypothesis was that each trend followed an exponential pattern as shown in Model 2

It is clear that an exponential correlation is not the best model for the river flow. Indeed, the

model’s greatest rate of flow is significantly bellow the data’s maximum value (

€

t = 54, flow =1920) while its shape does not closely follow the pattern. The “increasing

trend” best-fit is too steep at first and does not reflect the flow’s initial steady behaviour.

Moreover, its y-intercept is below the data’s lowest point. The “decreasing trend” line is also

inadequate since it shows a continuous decrease throughout the graph whereas the original

data decreases rapidly at first and then slower after a few hours.

4

Model 2

y = 370.33e0.0279x y = 3163.6e -

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160Time (hours)

Flow (cfs)

Increasing trend

Decreasing trend

In addition to the straightforward graphical analysis, it is possible to calculate the average

distance between the original data and the model. Let us consider the first point of the data,

which is situated on the y-axis. (

€

t = 0, flow = 440) If the model function is defined as

€

f (x) = 370.33e0.0279x , then its flow value is

€

f (0) = 370.33 cfs. The distance between the

model and the actual data is

€

f (x) − f (0) = 370.33− 440 = 70 when rounding off adequately.

Calculating the distance for all points is long and tedious however, computer software such as

Excel make the task much more efficient. Fig 4 shows the tables used to compute the average

distance for both increasing and decreasing patterns in the first model.

Fig.4

The average distance between the points and the model is 75 units for the increasing function

and 54 units for the decreasing function. The average distance for the entire model is 65 units.

5

Time (hours) Flow (cfs) F(t) F(t)-Flow Av. distance0 440 370 70 756 450 438 12

12 480 518 3818 570 612 4224 680 723 4330 800 855 5536 980 1011 3142 1090 1195 10548 1520 1413 10754 1920 1671 249

Time (hours) Flow (cfs) F(t) F(t)-Flow Av. distance54 1920 1691 229 5460 1670 1577 9366 1440 1471 3172 1380 1372 878 1300 1280 2084 1150 1194 4490 1060 1114 5496 970 1039 69

102 900 969 69108 850 904 54114 800 843 43120 780 786 6126 740 734 6132 710 684 26138 680 638 42144 660 595 65

Increasing trend

Decreasing trend

The second model involves two distinct quadratic functions. The local minima should follow

more accurately the original data because of the initial flatness and the rapidly increasing

gradient afterwards. This is shown in Model 3

Model 3 illustrates the graph of

€

y = 0.6271x 2 − 8.399x + 476.36 as the increasing function and

€

y = 0.1622x 2 − 44.772x + 3781.9 as the decreasing function.

This model is much better than the previous one. The increasing function cuts the y-axis very

close to the first point and remains flattened for about 12 hours. Its gradient then increases

rapidly, following the pattern of the river’s flow rate. The model’s maximum flow rate occurs

at the intersection of both functions which is just slightly bellow the point where

€

t = 54 . The

coordinates of the maximum are found by equating the two functions and solving for x using a

Graphic Display Calculator.

€

0.6271x 2 − 8.399x + 476.36 = 0.1622x 2 − 44.772x + 3781.9

x = 53.835266 and y =1841.6813

For simplicity, it is assumed that the actual maximum point has coordinates

€

(54,1920) hence

the distance between the two points is found using Pythagoras Theorem:

€

d = (x1 − x2)2 + (y1 − y2)2

d = (54 − 53.835266)2 + (1920 −1841.6813)2

d = 78 units

6

Model 3y = 0.6271x2 - 8.399x + y = 0.1622x2 - 44.772x +

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160Time (hours)

Flow (cfs)

Increasing trend

Decreasing trend

It is also useful to find the average distance between the model and the data so as to further

optimize the analysis. Once again, this can been done using Excel tables as shown in Fig.5

Fig.5

The average distance is 37 units for the increasing function and 26 units for the decreasing

function proving that this model is better than the previous one.

Although this third model seems quite reliable, it is possible to optimize it even more by

changing the nature of the decreasing best-fit line. Indeed, it was found that a power

correlation is closer to reality for

€

54 ≤ t ≤144

7

Time (hours) Flow (cfs) F(t) F(t)-Flow Av. distance0 440 476 36 376 450 449 1

12 480 466 1418 570 528 4224 680 636 4430 800 789 1136 980 987 742 1090 1230 14048 1520 1518 254 1920 1851 69

Time (hours) Flow (cfs) F(t) F(t)-Flow Av. distance54 1920 1837 83 2660 1670 1680 1066 1440 1533 9372 1380 1399 1978 1300 1277 2384 1150 1166 1690 1060 1066 696 970 979 9

102 900 903 3108 850 838 12114 800 786 14120 780 745 35126 740 716 24132 710 698 12138 680 692 12144 660 698 38

Increasing trend

Decreasing trend

Model 4 includes the same quadratic function for

€

0 ≤ t ≤ 54 but also a power function

€

y =147191x−1.095 for

€

54 ≤ t ≤144

This fourth model appears to be the most accurate of all. It contains both a quadratic and a

power function and its shape is very close to the original data. The maximum occurs when

€

0.6271x 2 − 8.399x + 476.36 =147191x−1.095

x = 54.149827 and y =1860.34

hence the distance between the data and the model at the maximum is

€

d = (54 − 54.149827)2 + (1920 −1860.34)2

d = 60 units

The average distance between the power function and the data is illustrated in fig.6 and equals

22units. Fig.6

8

Time (hours) Flow (cfs) F(t) F(t)-Flow Av. distance54 1920 1866 54 2260 1670 1663 766 1440 1498 5872 1380 1362 1878 1300 1247 5384 1150 1150 090 1060 1067 796 970 994 24

102 900 930 30108 850 874 24114 800 823 23120 780 778 2126 740 738 2132 710 701 9138 680 668 12144 660 637 23

Model 4y = 0.6271x 2 - 8.399x + y = 147191x -1.095

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160Time (hours)

Flow (cfs)

Increasing trend

Decreasing trend

The fourth and last model is therefore the most reliable since the maximum is closer to reality

and the average distance between the function and the original data smaller.

It is now possible to further investigate on the river’s flow rate using this mathematical model.

Let the model be defined by the function

€

Flow =0.6271t 2 − 8.399t + 476.36 for 0 ≤ t ≤ 54

147191t−1.095 for 144 ≥ t ≥ 54

⎫ ⎬ ⎭

⎧ ⎨ ⎩

Its derivative is

€

d( flow)

dt=

1.2542t − 8.399 for 0 ≤ t ≤ 54

−161174.145t−2.095 for 144 ≥ t ≥ 54

⎧ ⎨ ⎩

⎫ ⎬ ⎭

The amount of flowing water therefore increases when the gradient of the graph is positive,

i.e. when the first derivative has positive values. Plotting the graph of the derivative helps

visualising the situation.

Fig.7 illustrates the first derivative of both function. Note that the blue dotted line occurs

€

for 0 ≤ x ≤ 54 and the full red line

€

for 54 ≤ x ≤144

9

€

y =1.2542x − 8.399

€

y = −161174 x−2.095

Fig.7

The flow rate therefore increases when the first derivative

€

y =1.2542t − 8.399 is positive in

the given domain. ( between the point where the derivative crosses the x-axis and the model’s

maximum value

€

t = 54.1 )

€

0 =1.2542t − 8.399

t ≈ 6.70 hours

Rounding off to the nearest hour for consistency, the amount of flowing water increased from

07:00 on 27 October to 06:00 on 29 October.

Since the river is controlled by a series of dams, it is also useful to estimate the amount of

water that flowed in a given period of time. This could help understanding the river better and

allow a more efficient use of resources. The volume of water that flowed past the measuring

station between 00:00 on 28 October and 00:00 on 29 October is found by integrating the

function between

€

t = 24 and t = 48 . This is

€

(0.6271t 2 − 8.399t + 476.36)24

48

∫ dt =0.6271t 3

3−

8.399t 2

2+ 476.36t

⎡

⎣ ⎢

⎤

⎦ ⎥24

48

= 24 404 cubic feet

10

€

y = 0.6271t 2 − 8.399t + 476.36

€

y =147191t−1.095

Water flowed past

Fig.8

Fig 8 shows the volume of water that flowed between 00:00 on 28 October and 00:00 on 29 October

Another important feature of the original data that can be found using the model is the

average flow rate. At first glance and without any calculation, one would estimate the average

flow rate to be around 1000 cfs. This exact rate would also occur twice, first during the

increase of flow at

€

t ≈ 36 hours and then during the decrease at

€

t ≈ 95 .

The most reliable way of finding the average of a function in a given domain is to use the

Mean Value Theorem that takes the form:

€

mean =f (b) − f (a)

b − a. Because the model is defined

by two distinct functions, it is necessary to find the mean for each function separately and

then the average of the results. The average flow rate from 00:00 on 28 October to 00:00 on 2

November is found as an example.

Let “00:00 on 28 October” be defined as

€

t = 24 , “00:00 on 2 November” as

€

t =144 and the

“maximum value” of the model occur at

€

t = 54 , then the mean value of the increasing function

is:

€

mean increa sin g( ) =f (54) − f (24)

54 − 24=

0.6271(54)2 − 8.399(54) + 476.36[ ] − 0.6271(24)2 − 8.399(24) + 476.36[ ]

30

=1851.438 − 635.9936

30 =1215.4444

≈1215 cfs

and the mean of the decreasing function is:

€

mean decrea sin g( ) =f (144) − f (54)

144 − 54=

147191(54)−1.095[ ] − 147191(144)−1.095

[ ]

90

=1228.503 cfs

≈1228

Hence the average flow rate from 00:00 on 28 October to 00:00 on 2 November is

€

1215.444 +1228.503

2≈1222 cfs

This exact rate also occurs twice, once when the quadratic and the power functions equals the

mean value:

€

1222 = 0.6271t 2 − 8.399t + 476.36

t = 41.82

t ≈ 42

€

1222 =147191t−1.095

t = 79.48

t ≈ 80

11

Hence the average flow between those two dates occurs at 18:00 on 28 October and 08:00 on

29 October.

Using the information extracted from the mathematical model of the data, we can suggest a

possible weather patter that would account for the shape of the graph. Because the river’s

average flow rate between the 28th and the 2nd was significantly higher than the initial and

final rates, the total volume of water that flowed seems to be greater than usual. This could be

caused by rainfall in the area, which would then be collected in the river. Moreover, the model

shows a steep increase in flow rate that reaches its maximum value within a few hours. A

strong and sudden precipitation is therefore likely to have caused the shape of the graph.

When heavy rainfall occurs on land that cannot absorb and store the water because it is

urbanised, saturated or too compact, it flows on the surface and reaches the river within a

short time3. This will then lead to a temporary but rapid increase in flow rate and sometimes

floods. If the rain comes to an end, the surface runoff disappears and the flow of the river

returns to its normal level. This “falling limb” is always more gentle than the rising limb

because the water infiltrated in the ground starts reaching the river and continues contributing

to its flow. The graph of the original data is therefore an example of a “storm hydrograph”

and can be compared to the typical one in Fig.94

3 http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/hydrosphere/surface_water.html4 Ecole Polytechnique Federale de Lausanne: http://hydram.epfl.ch

12

Fig.9 shows the sketch of a typical storm hydrograph. The increase in flow is the rising limb, the peak is the maximum and the decrease is the falling limb, or flood recession. Note that subsurface flow arises after the peak because of water infiltration and the normal flow rate is the baseflow.

Model 4 shows the graph and model of the original data.

The model appears to be close to reality confirming our hypothesis. Nevertheless, there are

still several limitations that will affect the results.

The first and most important limitation is the transition between the rising and falling limb.

The model shows it as an abrupt change in gradient meaning the flow rises rapidly and then

suddenly falls with not intermediate stage in between. Most models, however, contain a

somewhat flattened peak since the runoff water may flow through different path and takes

some time to disappear completely. One way to solve this problem is to divide the model’s

composite function further and use another quadratic polynomial for the 3 highest points. On

the other hand, transitions between the various new functions could be even more ambiguous

and inaccurate hence this is a possible limit of technology.

Secondly, the model does not take into account the slight change in direction of the rising

limb at the 8th point of the graph. This point could have a great importance in a complex and

technical analysis as it could show, for example, that subsurface flow takes less time than

expected to reach the river and therefore adds to the surface runoff. Taking measurements of

13

Model 4y = 0.6271x2 - 8.399x + y = 147191x-1.095

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160Time (hours)

Flow (cfs)

Increasing trend

Decreasing trend

the flow at smaller time intervals is a possible solution to this limitation and it would greatly

improve the accuracy and reliability of the model.

Finally, we may attempt to apply a similar model and reasoning to a different set of data in

order to verify the hypothesis and analysis made in the discussion. The new data originates

from the Nolichucky River in Tennessee between 1st May 2007 and 20th May 2007. The flow

rate values are the averages of each day and are shown below5.

Note that some parameters of the data are changed. Here, 24 hours pass between each

measurement. The rise and fall of the flow rate also takes place over a much longer period of

time. (20 days)

Applying the same model to this set of data produces the graph in Model 5.

Model 5y = 105.07x2 - 276.53x + 880 y = 6698.6x -

0

500

1000

1500

2000

2500

0 5 10 15 20

Day, May 2007

Flow rate /cfs

5 http://nwis.waterdata.usgs.gov/tn/nwis/dv?cb_00060=on&format=html&begin_date=2007-04-27&end_date=2007-05-20&site_no=03465500&referred_module=sw

14

Day 1 2 3 4 5 6 7 8 9 10Flow 756 729 766 1780 2000 1670 1370 1170 1060 984

Day 11 12 13 14 15 16 17 18 19 20Flow 921 870 842 783 726 707 715 676 639 613

It appears that the power function for the falling limb fits reality closely. The rising trend,

however, is visibly inaccurate since the data shows a smooth peak whose gradient changes

slowly. Our initial mathematical model therefore cannot be effectively applied to this

situation.

Surprisingly, the fact that the model did not fit the new data reinforces our confidence in its

reliability. Indeed, the new data was taken over a much longer period of time meaning that the

weather pattern was different. The new data might have occurred in a month where it rained

often but less strongly. The water could then infiltrate more into the ground leading to less

surface runoff but more subsurface flow, explaining why it took about 4 days for the flow rate

to reach its maximum level. In this case, we our not dealing with a short and violent storm so

we cannot apply the same model. We may therefore contest the first limitation of the model

that said that the maximum flow rate should show a smoother pattern. Indeed, it appears that

such behaviour is likely to be caused by underground flow, not surface runoff.

Sources and references

www.water.usgs.gov, last visited 2007-12-11

http://hydram.epfl.ch, last visited 2007-12-11

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