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It is assumed that ownership is conserved in every division of a reach. This is reflected in the ownership mass balance equation below. This shows that the difference between an owner’s share of storage at the beginning and end of a time step should be the sum of their share of all the division’s fluxes.
| Equation 15 |
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When Murray-style high flow losses are specified, these are separated from other proportional losses, and the mass balance equation takes the form:
| Equation 16 |
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Live division model, assumptions and equations
Index flow rate
The owner’s index flow rate (q‾(o)) is determined from the inflow volume I(o), outflow volume O(0) and the Muskingum parameter x (see the Link Storage Routing SRGfor details):
| Equation 17 |
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Proportional routing of ownership
Proportional routing is used to share the division’s active (live) storage between owners. This is based on the idea that ownership travels at the rate that each owners’ flow influences the flow in the division. If we consider a division and divide each owners’ inflow into a very large number of small pieces, that each time one more of these slices is passed through, the increment in division storage (Storage) can be approximated as a linear function g of the increment in the index flow rate q‾:
| Equation 18 |
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It can also be assumed that go ≈ go-1 ; i.e. the ratio g is the same for each owner’s slice of water as it passes through the division. After summing up all of the slices the following relationship is obtained:
| Equation 19 |
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From this, lateral loss fluxes that are proportionally shared can also be shared in proportion to the ownership ratios in live storage, as this is the same as sharing in proportion to each owner’s index flow rate. Hence, substituting from equation (19) into equation (13) and rearranging yields:
| Equation 20 |
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