Φ Index Method for Determining Losses from a Precipitation Event

Use the rainfall data below to determine the Φ index for a watershed of 0.875 square miles, where the runoffvolume is 228.7 acre-ft.

Note that in order to use the φ index method the volume of DRO must be known. This could be viewed as adisadvantage.

Index

losses

direct runoff

hyetograph

Intensity, in/hr

time, hr

The phi index is a “mythical loss function” whose value results in a volume of DRO equal to that measured. If the phi index is larger than the rainfall intensity during an interval of time then the computed DRO during that period is zero. For example the phi index is larger than the rainfall intensity during the interval t3 to t4, thus the DRO during that period is computed as zero

t1

t2 t3 t4

FULL FILENAME

C:\MYFILES\ARCHIVE 1\VISIO DRAWINGS\FLUID MECHANICS - OPEN CHANNEL HYDRAULICS, HYDROLOGY\PHI INDEX.VSD

It appears to me that the phi index method accounts for all losses occurring between the time precipitationoccurs and runoff is measured, be they initial abstractions, depression storage or infiltration

From Bedient, 3rd edition, " The φ index method assumes the loss is uniformly distributed across the rainfallevent. Sometimes the method is modified to include a greater initial loss or abstraction followed by aconstant loss for the event."

Example 1.7 Phi Index.mcdC:\myfiles\Mathcad application areas\Fluids-open channels-hydrology\Bedient and Huber 3rd edition\

7/27/20047:39 AM

1 of 4

From McCuen, 3rd edition "The phi index equals the rainfall intensity above which the depth of rainfall equalsthe depth of direct runoff. "

Compute the depth of runoff over the catchment depth228 acre⋅ ft⋅

.875 mi2⋅:= , depth 4.886 in=

The correct value of phi, Φ, applied to the rainfall hyetograph, will result in a volume equal to the volume ofrunoff, 4.9 in.

Enter the hyetograph coordinates in a matrix. Note the time value is the same for the vertical line segments

DATA

0

0

2

2

5

5

7

7

10

10

12

12

0

1.4

1.4

2.3

2.3

1.1

1.1

0.8

0.8

0.3

0.3

0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

:=

i 0 11..:=

extract and rename each column and provide proper units

timei DATAi 0, hr⋅:=

intensityi DATAi 1,inhr

⋅:=

Example 1.7 Phi Index.mcdC:\myfiles\Mathcad application areas\Fluids-open channels-hydrology\Bedient and Huber 3rd edition\

7/27/20047:39 AM

2 of 4

0 1 2 3 4 5 6 7 8 9 10 11 1200.30.60.91.21.51.82.12.42.7

3Rainfall Hyetograph

intensityiinhr

timei

hr

intensity1 1.4inhr

= intensity7 0.8inhr

=

intensity3 2.3inhr

= intensity9 0.3inhr

=

intensity5 1.1inhr

=

initial guess for phi index value Φ 1inhr

⋅:=

Now, the volume of runoff is equal to the difference between the rainfall intensity and the assumed phi indexmultiplied by the time interval over which that value of intensity occurred. If the phi index is LARGER thanthe rainfall intensity then the runoff volume during that period is computed as zero. I used conditionalstatements to do this.

2 hr⋅ if Φ intensity1> 0 ft⋅, intensity1 Φ−,( )( )⋅ 0.067 ft=

3 hr⋅ if Φ intensity3> 0 ft⋅, intensity3 Φ−,( )( )⋅ 0.325 ft=

2 hr⋅ if Φ intensity5> 0 ft⋅, intensity5 Φ−,( )( )⋅ 0.017 ft=

3 hr⋅ if Φ intensity7> 0 ft⋅, intensity7 Φ−,( )( )⋅ 0 ft=

2 hr⋅ if Φ intensity9> 0 ft⋅, intensity9 Φ−,( )( )⋅ 0 ft=

Example 1.7 Phi Index.mcdC:\myfiles\Mathcad application areas\Fluids-open channels-hydrology\Bedient and Huber 3rd edition\

7/27/20047:39 AM

3 of 4

0 1 2 3 4 5 6 7 8 9 10 11 1200.30.60.91.21.51.82.12.42.7

3Rainfall Hyetograph

1

intensityiinhr

timei

hr

Note that for the last 2 rainfall intensities, the phi index was larger and zero runoff was computed. Now, if wesum the computed runoff volumes the correct value of φ will give a value equal to the observed runoff volume.

Runoff_volume 2 hr if Φ intensity1> 0 ft, intensity1 Φ−,( )( ) 3 hr if Φ intensity3> 0 ft, intensity3 Φ−,( )( )+2 hr if Φ intensity5> 0 ft, intensity5 Φ−,( )( )⋅ 3 hr if Φ intensity7> 0 ft, intensity7 Φ−,( )( )⋅++

...

2 hr if Φ intensity9> 0 ft, intensity9 Φ−,( )( )⋅+...

:=

Runoff_volume 4.9 in= If this value does not equal the observed runoff, assume different values of φ until itdoes

phi index = 1 in/hr

Example 1.7 Phi Index.mcdC:\myfiles\Mathcad application areas\Fluids-open channels-hydrology\Bedient and Huber 3rd edition\

7/27/20047:39 AM

4 of 4