Flood routing by the muskingum method

Abstract The Muskingum flow routing method has been very well researched and established in the hydrological literature. Its modest data requirements make it attractive for practical use. The paper gives a general overview of the Flood routing concept and types, and then goes on to explain the Muskingum method in detail. Introduction to Flood Routing Flood routing is a technique which is used to determine the flow hydrograph characteristics like shape and movement along a water course, and how these are affected by various factors like system storage and system dynamics on the shape and movement of flow hydrographs along a watercourse.

In other words Flood routing can be described as a process of calculating outflow rates, reservoir stage and storage volume from a stream channel once inflows and channel characteristics are known. The process of flood routing is used for the hydrologic analysis in flood forecasting, flood protection, reservoir design and spillway design etc. The principle of routing is used here for predicting the temporal and spatial distribution of hydrograph, during the course of its travel through the various sections of a stream (Subramanya 2002).

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Basic Principles of Routing All hydrologic routing methods use a common continuity equation as their common base. According to this equation, the difference between inflow and outflow rates is equal to the rate of change of storage. Mathematically the equation can be written as below: (Gosh 1997 p. 67) In the above equation, I is the rate of inflow, and at any time the corresponding outflow is O.

dS is the storage that is accumulated during a very small duration of time dt. Figure below represents the pictorial relation between storage S and discharge Q:

(Flood Routing) The above equation considers the losses due to seepage, evaporation and direct accretion to storage, as small enough to be ignored. The equation can be written in integral form as below: (Watson 1983 p. 490) Flood routing methods aim to solve the above one-dimensional mass continuity equation. Flood routing methods classification There are many flood routing methods available like using St. Venant equations, Level Pool Routing, Goodrich Method, Pul’s Method, Kinematic Routing, Muskingum equation, Muskingum-Cunge routing etc.

These methods can be divided into the following two categories: 1. Hydrological routing – These methods mainly use the continuity equation 2. Hydraulic routing – These methods combine the equation of continuity with the equation of motion for unsteady flow. (Subramanya 2002 p. 271) Types of Flood Routing In all the hydrologic analysis applications mentioned above, two categories of routing can be clearly recognised: 1. Reservoir routing – In this type of routing, the effect of a flood wave entering a reservoir is studied.

This is done by determining the volume-elevation characteristic of a reservoir in addition to the outflow-elevation characteristic of the spillway and also other outlet structures present in the reservoirs (Chadwick & Morfett 1986 p318). The results are used to predict the variation of reservoir elevation and outflow discharge with respect to time. This type of routing is necessary for: a. Designing the capacity of the spillway and other outlet structures b. Determining the correct location and size of capacity of the reservoir pertaining to a particular requirement condition. 2.

Channel routing – In this type of routing, a study is made of the change in shape of a hydrograph as it travels down a channel. This done by considering a channel reach i. e. the specific length of the stream channel, and an input hydrograph at the upstream end of the stream. The results are used to predict the flood hydrograph at various sections of the reach (Chadwick & Morfett 1986 p. 322). The output data obtained using this method is, the information on the flood-peak attenuation and, the total duration of the high-water levels. This type of routing is considered very important for: a. Flood-forecasting operations b.

Flood-protection related work Hydrologic Channel Routing In case of reservoir routing, the storage is a function of output discharge, whereas in case of channel routing, the storage is a function of both inflow and outflow discharges. This is the main reason why entirely different routing methods are needed for Channel routing. When a river is in flood, the flow can be characterized as gradually varied unsteady flow. In a particular channel reach the water surface as expected is not parallel to the channel both. Additionally it also varies with time. At the time of flood, the total volume in storage can be divided into two categories:

1. Prism storage – This is defined as the volume that would exist in case there is uniform flow at the downstream depth. i. e. Prism storage = 2. Wedge storage – This term represents the wedge-like volume which is formed between the actual water surface profile and the prism storage surface i. e. Wedge storage = In the downstream section of a river reach, the prism storage is observed to be constant, when the depth is fixed. However, the wedge storage changes from positive to negative depending on the type of flood. The wedge storage is positive at the time of advancing flood, while it is negative in case of a receding flood.

(Subramanya 2002 p. 282-283) (Flow Routing 2) .Muskingum Method Introduction Flood routing in open channels can be determined using a variety of modeling procedures. These methods follow a wide range of methodologies, which can be categorized as: 1. Simple like Muskingum-type approximations – Which have modest data requirements 2. Complex like Muskingum–Cunge methods – Where the typically calibrated Muskingum routing parameters are related to physical and hydraulic characteristics such as reach length, flood wave celerity, unit width discharge and channel bed slope 3. Highly complicated like the solution of the full dynamic flow i. e.

St Venant equations – Which require surveyed cross-sectional channel profiles and flow resistance data Out of these the Muskingum and Muskingum–Cunge methods are well established in the hydrological literature, and the modest data requirements make these procedures attractive even though more rigorous hydraulic models are available for unsteady flow routing. (Birkhead & James 2002 p. 113) For Muskingum method which is a hydrologic method, the discharge measurements alone are sufficient for routing. This is because it is assumed that the parameters of the Muskingum model capture the combined flood-propagating characteristics of a river reach.

When the water resources schemes to be built are in their initial planning stages, the river gauging system may remain either underdeveloped or insufficient to give precise and rigorous measurements of flow depths and discharges. The Muskingum method is useful for predicting the preliminary outflow hydrographs required at the initial stages of planning spillway capacities. In addition, these outflow hydrographs can also aid the design of stream gauges for future use. Hence, the Muskingum model has a high significance for modern civil engineers (Das 2004 p. 130).

The Muskingum equation is frequently used for routing of floods in river channels. The Muskingum method for routing flood waves in rivers and channels has been widely used in applied hydrology, since its first use in connection with a flood control project in the Muskingum County of Ohio about fifty years ago. Since its development around 1934 by McCarthy, the Muskingum method has also been a subject of many investigations (Strupczewski, Napiorkowski & Dooge 2002 p. 235) The figure below shows the translational and storage processes in stream channel routing. (Gill 1979 22).