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# Variable Storage Routing Method

The variable storage routing method was developed by Williams (1969) and used in the HYMO (Williams and Hann, 1973) and ROTO (Arnold et al., 1995) models. 

        For a given reach segment, storage routing is based on the continuity equation:

                              $$V\_{in}-V\_{out}=\Delta V\_{stored}$$                                                                      7:1.3.1

where $$V\_{in}$$ is the volume of inflow during the time step (m$$^3$$ H$$*2$$O), $$V*{out}$$ is the volume of outflow during the time step (m$$^3$$ H$$*2$$O), and $$\Delta V*{stored}$$ is the change in volume of storage during the time step (m$$^3$$ H$$\_2$$O). This equation can be written as

               $$\Delta t\*(\frac{q\_{in,1}+q\_{in,2}}{2})-\Delta t\*(\frac{q\_{out,1}+q\_{out,2}}{2})=V\_{stored,2}-V\_{stored,1}$$                        7:1.3.2

where $$\Delta t$$ is the length of the time step (s), $$q\_{in,1}$$ is the inflow rate at the beginning of the time step (m$$^3$$/s), $$q\_{in,2}$$ is the inflow rate at the end of the time step (m$$^3$$/s),  $$q\_{out,1}$$ is the outflow rate at the beginning of the time step (m$$^3$$/s), $$q\_{out,2}$$ is the outflow rate at the end of the time step (m$$^3$$/s), $$V\_{stored,1}$$ is the storage volume at the beginning of the time step (m$$^3$$ H$$*2$$O), and $$V*{stored,2}$$ is the storage volume at the end of the time step (m$$^3$$ H$$\_2$$O). Rearranging equation 7:1.3.2 so that all known variables are on the left side of the equation,

                                  $$q\_{in,ave}+\frac{V\_{stored,1}}{\Delta t}-\frac{q\_{out,1}}{2}=\frac{V\_{stored,2}}{\Delta t}+\frac{q\_{out,2}}{2}$$                                              7:1.3.3

where $$q\_{in,ave}$$ is the average inflow rate during the time step: $$q\_{in,ave}=\frac{q\_{in,1}+q\_{in,2}}{2}$$**.**

                Travel time is computed by dividing the volume of water in the channel by the flow rate.

                                $$TT=\frac{V\_{stored}}{q\_{out}}=\frac{V\_{stored,1}}{q\_{out,1}}=\frac{V\_{stored,2}}{q\_{out,2}}$$                                                           7:1.3.4

where $$TT$$ is the travel time (s), $$V\_{stored}$$ is the storage volume (m$$^3$$ H$$*2$$O), and $$q*{out}$$ is the discharge rate (m$$^3$$/s).

           To obtain a relationship between travel time and the storage coefficient, equation 7:1.3.4 is substituted into equation 7:1.3.3:

                      $$q\_{in,ave}+\frac{V\_{stored,1}}{(\frac{\Delta t}{TT})*(\frac{V\_{stored,1}}{q\_{out,1}})}-\frac{q\_{out,1}}{2}=\frac{V\_{stored,2}}{(\frac{\Delta t}{TT})*(\frac{V\_{stored,2}}{q\_{out,2}})}+\frac{q\_{out,2}}{2}$$                                  7:1.3.5

which simplifies to

                      $$q\_{out,2}=(\frac{2\*\Delta t}{2*TT+\Delta t})*q\_{in,ave}+(1-\frac{2*\Delta t}{2*TT+ \Delta t})\*q\_{out,1}$$                                  7:1.3.6

This equation is similar to the coefficient method equation

                       $$q\_{out,2}=SC\*q\_{in,ave}+(1-SC)\*q\_{out,1}$$                                                    7:1.3.7

where $$SC$$ is the storage coefficient. Equation 7:1.3.7 is the basis for the SCS convex routing method (SCS, 1964) and the Muskingum method (Brakensiek, 1967; Overton, 1966). From equation 7:1.3.6, the storage coefficient in equation 7:1.3.7 is defined as 

                       $$SC=\frac{2\*\Delta t}{2\*TT+ \Delta t}$$                                                                                                7:1.3.8

It can be shown that

                        $$(1-SC)*q\_{out}=SC*\frac{V\_{stored}}{\Delta t}$$                                                                    7:1.3.9

Substituting this into equation 7:1.3.7 gives

                         $$q\_{out,2}=SC\*(q\_{in,ave}+\frac{V\_{stored,1}}{\Delta t })$$                                                                7:1.3.10

To express all values in units of volume, both sides of the equation are multiplied by the time step

                           $$V\_{out,2}=SC\*(V\_{in}+V\_{stored,1})$$                                                              7:1.3.11