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| No. | Assumption/constraint |
|---|---|
| 1 | Owners cannot have a negative share of water in storage or in transit. |
| 2 | The sum of all owners’ shares of storage in a link equals the link’s total storage volume. |
| 3 | The sum of all owners’ shares of flow in a link equals the link’s total flow. |
Theory
Background
In Source, components that are physically or logically connected are joined using a link. If the connection is significant enough to have an effect on the time that water would take to pass along it then the link is modelled as a routing link. Each routing link is subdivided into one or more divisions.
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Figure 1. Division ownership conceptual model
Definition of terms used in equations
Table 2. Symbols used in equations
| Symbol | Description | Units |
|---|---|---|
| dt | Model time-step | time |
| Deficit(o) | Owner’s deficit to be made up using borrow and payback. | volume |
| FluxTS | Sequence of time series flux values for the link input by the modeller. | |
| FluxTS(o) | Sequence of time series flux values for the link for owner o, input by the modeller. | |
| Storage routing function used to determine the live storage volume in a link division. See the Link Storage Routing SRG for more information. | volume | |
| g | Linear function to translate between ratios: from change in storage/total storage into change in flow/total flow. | n/a |
| j | An owner whose current storage contributes to the high flow loss, i.e. is greater than their share of the high flow threshold. | n/a |
| I | Division inflow volume | volume |
| I(o) | Division inflow volume for owner o | volume |
| Loss | Total volume of loss from the division (negative if water is gained). | volume |
| Lossfixed | Total volume of loss that is shared in a predetermined way. It is assumed to have been adjusted to reflect any shortfall in volume to meet it during the flow phase (trying to pump a division dry for example). | volume |
| Lossfixed(o) | The (volume) share of fixed loss owned by owner o. | volume |
| LossfixedMAX(o) | Maximum fixed loss that owner o has the capacity to sustain. | volume |
| LossHF | Total high flow loss | volume |
| LossHF(o) | High flow loss for an owner | volume |
| Lossprop | Total volume of loss that should be shared in proportion to the ownership of the water in the division. This volume is assumed to have been adjusted to reflect any shortfall that occurred in the flow phase (such as a division with non-zero area at empty trying to evaporate water from an empty division). | |
| Lossprop(o) | The (volume) share of proportional loss owned by owner o. | volume |
| m | Mass of the sample taken | mass |
| mi | Mass of the substance in the sample | mass |
| M | Total mass in a specified volume | mass |
| Mi | Mass of substance in a specified volume | mass |
| no | Number of owners | n/a |
| Net(o) | Net volume of water that owner o has in storage (in a “dead” division) | volume |
| O | Division outflow volume, including outflowing lateral fluxes | volume |
| O(o) | Division outflow volume for owner o | volume |
| o | Owner of water in the division | n/a |
| Division index flow rate, which is the index flow for the current time step. See the Link Storage Routing SRG for more information. | volume/time | |
| Owner’s share of division’s index flow rate for the current time step. | volume/time | |
| Threshold for high flow/upper limit to regulated flow (used to determine high flow losses). | volume/time | |
| r | Symbol used to simplify mass balance equations. | |
| Ratiods(o) | Owner o’s share/ratio of the dead storage volume. This value is specified by the modeller. | n/a |
| Ratiolive(o) | Owner o’s ratio of index flow rate to total index flow rate. Used to calculate their share of active storage and proportional losses. | n/a |
| Ratioloss(o) | Owner o’s share/ratio of losses. | |
| RatioHFT(o) | Owner o’s share/ratio of the high flow threshold. This value is specified by the modeller. | n/a |
| RatioTS(o) | Owner o’s share/ratio of time series flux. | |
| Storage | Current total volume of water stored in the division. | volume |
| Storage(o,t) | The total volume of water stored in the division at time step owned by owner o. | volume |
| Storage(o,t-1) | Total volume of water stored in the division at the previous time-step (t-1) owned by owner o. | |
| Storage(t) | Total volume of water stored in the division at time-step t. | volume |
| Storage(t-1) | Total volume of water stored in the division at the previous time-step t-1. | |
| Storageds | Current dead storage in the division. If the division is dead, this is the total division storage, Storage(t). If the division is live this is StoragedsMAX. | volume |
| Storageds(o) | Current dead storage in the division owned by owner o. | |
| StoragedsMAX | Maximum total dead storage in a division specified by the modeller. | volume |
| StoragedsMAX(o) | Owner o’s share of the maximum total dead storage in a division. | volume |
| Storageexclude | Total volume of water at current time step belonging to owners that are not contributing to the high flow loss. | volume |
| StorageHFT | Division storage threshold that corresponds to reach high flow rate threshold qHFT, i.e. . | volume |
| StorageHF | Total volume of current storage that contributes to high flow loss. | volume |
| Storagelive | Current live or active storage volume in a division. | volume |
| Storagelive(o) | Current live or active storage volume in a division owned by owner o. | volume |
| Surplus(o) | Owner’s surplus that can be shared using borrow and payback. | volume |
| t | Time-step index | n/a |
| t1 | Start time index | n/a |
| t2 | End time index | n/a |
| x | Muskingum parameter (see the Link Storage Routing SRG for more information) |
States of a division: live or dead
At any time-step a routing link division can be classed as being either:
- Live: A division is described as being live if a solution for the time step can be found that satisfies both the continuity and storage equations.
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- The general form of the continuity equation is discussed in the Ownership Mass Balance section, below (see equations 15 and 16).
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- Dead: If the only solution that can be found satisfies the continuity equation but not the storage equation, then the division is described as being dead (i.e. it has ceased to flow).
A division is dead if x = 1 and or if x ≠ 1 and the following is true:
| Equation 1 |
|---|
Ownership of division storage
At the start of division ownership processing, the last time-step’s storage (Storage(t-1)) and each owner’s share of it (Storage(o,t-1)) are known from the flow phase routing calculations. The volume of water in a routing link division can be divided into:
- Live Storage: If the division is live then the live storage is the result of evaluating the storage equation for the current inflow and outflow.
- Dead Storage: If the division is dead then the dead storage is the total volume of water in the division. It is possible for the dead storage to be larger than the maximum specified by the modeller as the division may be, for example, in the process of transitioning from dead to live but there is not enough water in the division to satisfy both the continuity and storage equations. If the division is live then the dead storage is equal to the maximum volume (StoragedsMAX) specified by the modeller.
The live storage in a division is the storage above dead storage, as shown in Figure 1, and is obtained from:
| Equation 2 |
|---|
The total volume of water in a division is then:
| Equation 3 |
|---|
Sharing dead storage
Ownership of the dead storage is shared by fixed ratio (Ratiods(o)) to all of the owners. Hence:
| Equation 4 |
|---|