Applying the Muskingam Method | Civil Engineering Optional Notes for UPSC PDF Download

 Page 1


– 4
Page 2


– 4 Background– 6
BACKGROUND
The Muskingum routing method uses a conservation of mass approach to route an inflow 
hydrograph.  The Muskingum method can also account for “looped” storage vs. outflow 
relationships that commonly exist in most rivers (i.e. hysteresis).  As such, this method can simulate 
the commonly observed increased channel storage during the rising side and decreased channel 
storage during the falling side of a passing flood wave.  To do so, the total storage in a reach is 
conceptualized as the sum of prism (i.e. rectangle) and wedge (i.e. triangle) storage, as shown in 
Figure 1.  During rising stages on the leading edge of a flood wave, wedge storage is positive and 
added to the prism storage.  Conversely, during falling stages on the receding side of a flood wave, 
wedge storage is negative and subtracted from the prism storage.  Through the inclusion of a travel 
time for the reach and a weighting between the influence of inflow and outflow, it is possible to 
approximate attenuation. 
 Figure 1. Muskingum Representation of Channel Storage, reproduced from Linsley, Kohler, and Paulhus, 
1982
Parameters that are required to utilize this method within HEC-HMS include the initial condition, K
[hours], X, and the Number of Subreaches.
Page 3


– 4 Background– 6
BACKGROUND
The Muskingum routing method uses a conservation of mass approach to route an inflow 
hydrograph.  The Muskingum method can also account for “looped” storage vs. outflow 
relationships that commonly exist in most rivers (i.e. hysteresis).  As such, this method can simulate 
the commonly observed increased channel storage during the rising side and decreased channel 
storage during the falling side of a passing flood wave.  To do so, the total storage in a reach is 
conceptualized as the sum of prism (i.e. rectangle) and wedge (i.e. triangle) storage, as shown in 
Figure 1.  During rising stages on the leading edge of a flood wave, wedge storage is positive and 
added to the prism storage.  Conversely, during falling stages on the receding side of a flood wave, 
wedge storage is negative and subtracted from the prism storage.  Through the inclusion of a travel 
time for the reach and a weighting between the influence of inflow and outflow, it is possible to 
approximate attenuation. 
 Figure 1. Muskingum Representation of Channel Storage, reproduced from Linsley, Kohler, and Paulhus, 
1982
Parameters that are required to utilize this method within HEC-HMS include the initial condition, K
[hours], X, and the Number of Subreaches.
•
•
•
 ESTIMATE INITIAL PARAMETER VALUES
K
The Muskingum K parameter is equivalent to the travel time through the reach.  This parameter can 
be estimated in multiple ways including:
Using known hydrograph data.
Comparing flow length to a flood wave velocity.
Using regression equations which were developed from observed data in a similar region.
Using Known Hydrograph Data
The travel time of a flood wave moving through a reach can be estimated by taking the difference 
between "similar points" on known inflow and outflow hydrographs.  The similar points can be the 
peaks of either hydrograph, the centroid of the area underneath each hydrograph, or between some 
reference flow on the rising limb of either hydrograph.  The inflow and outflow hydrographs for this 
tutorial are shown and detailed within Figure 2.
 Figure 2. Hydrographs
Use Figure 1 to answer the following questions.
Question 1: Use the time of peak for both hydrographs shown in Figure 2 to estimate a 
representative travel time (in hours) for this event.
Page 4


– 4 Background– 6
BACKGROUND
The Muskingum routing method uses a conservation of mass approach to route an inflow 
hydrograph.  The Muskingum method can also account for “looped” storage vs. outflow 
relationships that commonly exist in most rivers (i.e. hysteresis).  As such, this method can simulate 
the commonly observed increased channel storage during the rising side and decreased channel 
storage during the falling side of a passing flood wave.  To do so, the total storage in a reach is 
conceptualized as the sum of prism (i.e. rectangle) and wedge (i.e. triangle) storage, as shown in 
Figure 1.  During rising stages on the leading edge of a flood wave, wedge storage is positive and 
added to the prism storage.  Conversely, during falling stages on the receding side of a flood wave, 
wedge storage is negative and subtracted from the prism storage.  Through the inclusion of a travel 
time for the reach and a weighting between the influence of inflow and outflow, it is possible to 
approximate attenuation. 
 Figure 1. Muskingum Representation of Channel Storage, reproduced from Linsley, Kohler, and Paulhus, 
1982
Parameters that are required to utilize this method within HEC-HMS include the initial condition, K
[hours], X, and the Number of Subreaches.
•
•
•
 ESTIMATE INITIAL PARAMETER VALUES
K
The Muskingum K parameter is equivalent to the travel time through the reach.  This parameter can 
be estimated in multiple ways including:
Using known hydrograph data.
Comparing flow length to a flood wave velocity.
Using regression equations which were developed from observed data in a similar region.
Using Known Hydrograph Data
The travel time of a flood wave moving through a reach can be estimated by taking the difference 
between "similar points" on known inflow and outflow hydrographs.  The similar points can be the 
peaks of either hydrograph, the centroid of the area underneath each hydrograph, or between some 
reference flow on the rising limb of either hydrograph.  The inflow and outflow hydrographs for this 
tutorial are shown and detailed within Figure 2.
 Figure 2. Hydrographs
Use Figure 1 to answer the following questions.
Question 1: Use the time of peak for both hydrographs shown in Figure 2 to estimate a 
representative travel time (in hours) for this event.
•
•
Answer
The inflow and outflow time of peak are 4/11/1994 20:00 and 4/12/1994 00:00, respectively.  The 
difference in time between these two points is 4 hours.
Question 2: Use the approximate centroid of the area beneath each hydrograph to estimate a 
representative travel time (in hours) for this event.
Answer
The centroid of the area beneath the inflow hydrograph lies at approximately 4/11/1994 21:00.  The 
centroid of the area beneath the outflow hydrograph lies at approximately 4/12/1994 00:00.  The 
difference in time between these two points is 3 hours.
Question 3: For a reference flow of 6500 cfs on the rising limb, estimate a representative travel 
time (in hours) for this event.
Answer
The inflow hydrograph reaches 6500 cfs at approximately 4/11/1994 17:00.  The outflow hydrograph 
reaches 6500 cfs at approximately 4/11/1994 19:30.  The difference in time between these two points 
is 2.5 hours.
Comparing Flow Length to a Flood Wave Velocity
The travel time, T, of a flood wave moving through a reach can also be estimated by dividing the 
length of the reach, L, by the flood wave velocity, V
w
:
To estimate a flood wave velocity, multiple approaches can be used including:
Manning's Equation (Manning, 1891)
Kleitz-Seddon Law (Seddon, 1900)
This tutorial will make use of the Kleitz-Seddon Law which can be written as:
where B = the top width of the water surface and  = slope of the flow-stage rating curve; both of 
these variables must be estimated for the flow rate in question and at a cross section that is 
representative of the routing reach.  The USGS maintains a flow-stage rating curve for the stream 
gage a y e n w a t u s x n u P t  and is shown in Figure 3.
Page 5


– 4 Background– 6
BACKGROUND
The Muskingum routing method uses a conservation of mass approach to route an inflow 
hydrograph.  The Muskingum method can also account for “looped” storage vs. outflow 
relationships that commonly exist in most rivers (i.e. hysteresis).  As such, this method can simulate 
the commonly observed increased channel storage during the rising side and decreased channel 
storage during the falling side of a passing flood wave.  To do so, the total storage in a reach is 
conceptualized as the sum of prism (i.e. rectangle) and wedge (i.e. triangle) storage, as shown in 
Figure 1.  During rising stages on the leading edge of a flood wave, wedge storage is positive and 
added to the prism storage.  Conversely, during falling stages on the receding side of a flood wave, 
wedge storage is negative and subtracted from the prism storage.  Through the inclusion of a travel 
time for the reach and a weighting between the influence of inflow and outflow, it is possible to 
approximate attenuation. 
 Figure 1. Muskingum Representation of Channel Storage, reproduced from Linsley, Kohler, and Paulhus, 
1982
Parameters that are required to utilize this method within HEC-HMS include the initial condition, K
[hours], X, and the Number of Subreaches.
•
•
•
 ESTIMATE INITIAL PARAMETER VALUES
K
The Muskingum K parameter is equivalent to the travel time through the reach.  This parameter can 
be estimated in multiple ways including:
Using known hydrograph data.
Comparing flow length to a flood wave velocity.
Using regression equations which were developed from observed data in a similar region.
Using Known Hydrograph Data
The travel time of a flood wave moving through a reach can be estimated by taking the difference 
between "similar points" on known inflow and outflow hydrographs.  The similar points can be the 
peaks of either hydrograph, the centroid of the area underneath each hydrograph, or between some 
reference flow on the rising limb of either hydrograph.  The inflow and outflow hydrographs for this 
tutorial are shown and detailed within Figure 2.
 Figure 2. Hydrographs
Use Figure 1 to answer the following questions.
Question 1: Use the time of peak for both hydrographs shown in Figure 2 to estimate a 
representative travel time (in hours) for this event.
•
•
Answer
The inflow and outflow time of peak are 4/11/1994 20:00 and 4/12/1994 00:00, respectively.  The 
difference in time between these two points is 4 hours.
Question 2: Use the approximate centroid of the area beneath each hydrograph to estimate a 
representative travel time (in hours) for this event.
Answer
The centroid of the area beneath the inflow hydrograph lies at approximately 4/11/1994 21:00.  The 
centroid of the area beneath the outflow hydrograph lies at approximately 4/12/1994 00:00.  The 
difference in time between these two points is 3 hours.
Question 3: For a reference flow of 6500 cfs on the rising limb, estimate a representative travel 
time (in hours) for this event.
Answer
The inflow hydrograph reaches 6500 cfs at approximately 4/11/1994 17:00.  The outflow hydrograph 
reaches 6500 cfs at approximately 4/11/1994 19:30.  The difference in time between these two points 
is 2.5 hours.
Comparing Flow Length to a Flood Wave Velocity
The travel time, T, of a flood wave moving through a reach can also be estimated by dividing the 
length of the reach, L, by the flood wave velocity, V
w
:
To estimate a flood wave velocity, multiple approaches can be used including:
Manning's Equation (Manning, 1891)
Kleitz-Seddon Law (Seddon, 1900)
This tutorial will make use of the Kleitz-Seddon Law which can be written as:
where B = the top width of the water surface and  = slope of the flow-stage rating curve; both of 
these variables must be estimated for the flow rate in question and at a cross section that is 
representative of the routing reach.  The USGS maintains a flow-stage rating curve for the stream 
gage a y e n w a t u s x n u P t  and is shown in Figure 3.
•
•
•
 Figure 3. Flow-Stage Rating Curve at Punxsutawney
Use the previously-shown equations and Figure 3 to answer the following questions.  Assume the 
following:
The length of the reach within this example is 68860 feet.
The slope of the rating curve shown in Figure 2 at a reference flow of 6500 cfs is approximately 1316 cfs / ft.
The top width of Mahoning Creek in the vicinity of the Punxsutawney gage at a reference flow of 6500 cfs is 
approximately 200 ft.
Question 4: For a reference flow of 6500 cfs, estimate a representative flood wave velocity (in 
ft/s) using the Kleitz-Seddon Law.
Answer
Question 5: Using the previously-computed representative flood wave velocity, estimate a 
representative travel time (in hours) for this event.
Answer