---
title: "Muskingum Method | Technical Reference"
slug: "muskingum-method"
description: "Find the equations for the muskingum method. "
tags: ["1D river modelling", "1D urban modelling", "Muskingum routing"]
updated: 2022-11-07T13:40:27Z
published: 2022-11-07T13:40:27Z
canonical: "help.floodmodeller.com/muskingum-method"
stale: true
---

> ## Documentation Index
> Fetch the complete documentation index at: https://help.floodmodeller.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Muskingum Method

It can be shown that the outflow and storage are related in the form of an infinite series as follows:

|  | \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn1.gif) \| (1) \| \| --- \| --- \| where a, x and n are constants unknown at the outset. |
| --- | --- |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn1.gif) | (1) |

If this series is truncated in the second term, Equation (1) can be expressed as:

|  | \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn2.gif) \| (2) \| \| --- \| --- \| |
| --- | --- |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn2.gif) | (2) |

From this equation, dS/dt can be obtained and replaced in the continuity equation to obtain the following storage expression :

|  | \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn3.gif) \| (3) \| \| --- \| --- \| |
| --- | --- |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn3.gif) | (3) |

Replacing (1/a)(1/n) by k and taking n as unity we obtain the storage equation for the Muskingum method, that is:

|  | \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn4.gif) \| (4) \| \| --- \| --- \| where: k is a storage constant and expresses the ratio of storage to discharge and has the dimension of time. It can be approximated by considering the travel time through the reach the constant x considers the relative importance of inflow and outflow in determining storage. For many cases x lies between 0.0 and 0.3 |
| --- | --- |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn4.gif) | (4) |

If Equation (4) is rewritten as:

|  | \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn5.gif) \| (5) \| \| --- \| --- \| |
| --- | --- |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn5.gif) | (5) |

and substituted in the discretised storage equation (see [Storage Equation](/v1/docs/storage-equation) equation (2)) we obtain an explicit equation for the outflow at the end of the time step where all terms on the right-hand side are known:

|  | \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn6.gif) \| (6) \| \| --- \| --- \| where: \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn7.gif) \| (7) \| \| --- \| --- \| \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn8.gif) \| (8) \| \| --- \| --- \| \| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn9.gif) \| (9) \| \| --- \| --- \| |
| --- | --- |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn6.gif) | (6) |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn7.gif) | (7) |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn8.gif) | (8) |
| ![](https://cdn.document360.io/5074bc26-a9ef-4bf4-b1de-a5c6ce628aca/Images/Documentation/MuskingumEqn9.gif) | (9) |

Combining the equations for C0, C1 and C2 gives :

|  | \| C0 + C1 + C2 = 1 \| (10) \| \| --- \| --- \| |
| --- | --- |
| C0 + C1 + C2 = 1 | (10) |

Also, all the coefficients must be positive in order to obtain valid results.

It is obviously important to choose appropriate values for k and x. If data from measured flood hydrographs are available, equation (5) can be used to plot storage as a function of the weighted inflow and outflow by using different values of x as shown in [Storage Equation](/v1/docs/storage-equation) (Figure 1).

The value of x that gives the curve that is closest to a straight line is assumed to be the most appropriate value to use. The value of the parameter k comes from the slope of the straight line.