## Description and rationale

Modelling of ownership on links is an essential component of modelling water ownership in Source, as it enables ownership to be tracked through links in Source models. The rationale for modelling water ownership, and the overall principles, are discussed in the Ownership Systems SRG entry.

## Scale

The concept of spatial scale in the context of ownership on links relates to the fact that it applies to link divisions and these have length, width and depth dimensions (even if they are represented as points for modelling). Ownership status can be updated as often as at every model time step, or less often if required.

## Principal developer

This version of modelling ownership on links has been developed by eWater CRC for Source.

## Scientific Provenance

Ownership on links has been modelled in predecessors to Source, such as IQQM and MSM, for many years. The concepts in these models have been updated and enhanced to suit the needs of Source.

## Version

Source v3.8.8.

## Dependencies

The requirement is that there should be at least two water users (as well as an ownership system) in the river system being modelled, in addition to at least one link.

## Definitions

Dead storage | The storage remaining in a division when the stream has ceased to flow. This storage is affected by fluxes which are independent of index flow in the division. See Link storage routing - SRG for more information. |

Division | In Source, a routing link represents a river reach, which is divided into one or more divisions of equal length. Ownership modelling takes place at the level of a division. |

Fixed flux | Loss fluxes whose ownership is known a priori because they are shared by fixed ratio or by some other means such as time-series or function. |

Flow based flux | Lateral flux in a division whose rate is a function of the division’s index flow rate. |

General purpose flow based flux | A modeller configured, piecewise monotonically increasing relationship between flux and index flow. See Link storage routing - SRG for more information. |

Groundwater flux | A function of head/water level which, in turn, is a function of flow. The flux calculated via a linked groundwater model. See Link storage routing - SRG for more information. |

Lateral flux | Flow into or from the division that is not from upstream or going downstream. In Source, this can consist of groundwater infiltration, evaporation, precipitation, time series flux (representing diversions etc.), or flow based flux (general purpose, could be used to represent overbank loss). See the Link Storage Routing SRG for more information. |

Live Storage | That part of the total storage in a division that is a function of the index flow rate (see the Link storage routing - SRG for more information). |

Murray-style loss | Method of sharing the loss (or gain) from a division due to high flow. Losses caused by flows in excess of the regulated flow range are shared to owners in proportion to how far each of them is above their fixed share of the regulated flow range. In Source, the losses to be shared in this way are represented by the flow based flux. |

Net evaporation | Evaporation less rainfall. |

Owner | An entity such as a state, country or water user group that has a defined share of water in the river system, where this share is managed completely separately from any other share. |

Ownership system | A component in Source used to track and manage the ownership of water in a defined section of a modelled river network. An ownership system has a set of owners that share water within the ownership system’s boundaries. Each of these owners may lend water surplus to their requirements to other owners with a deficit via the ownership system’s borrow and payback systems. Lending owners can be paid back some time later at any location within the ownership system boundary. |

Proportional flux | Loss fluxes that are shared in proportion to the ownership of the water in the division. |

Storage | Volume of water within a division at a defined point in time. |

Time series flux | A modeller specified time series used to represent known losses or gains of particular owners from a division. See Link storage routing - SRG for more information. |

Other definitions can be found in the eWater River Systems glossary.

## Assumptions and Limitations

#### Table 1. Assumptions and constraints applicable to modelling ownership on links

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. |

## 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*.

Figure 1 below shows a single routing division, its storage compartments and fluxes. Each owner’s share of these storage compartments and fluxes must be determined for every division. The principles on which these calculations are based are discussed in the **Theory** section and the sequencing of the calculations is described in the **Methodology** section.

#### 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 |

fFlow_{LG}() | Function that returns a lateral loss/gain flux for a given index flow rate . Also referred to as the flow based flux function. | volume |

fFlow_{LG}() | When Murray-style high flow losses are being modelled, the maximum volume of flow based flux that can be shared according to fixed ratio – calculated as fFlow where is the high flow threshold._{LG}() | volume |

Flux_{TS} | Sequence of time series flux values for the link input by the modeller. | |

Flux_{TS}(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 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 |

Loss_{fixed} | 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 |

Loss_{fixed}(o) | The (volume) share of fixed loss owned by owner o. | volume |

Loss_{fixedMAX}(o) | Maximum fixed loss that owner o has the capacity to sustain. | volume |

Loss_{HF} | Total high flow loss | volume |

Loss_{HF}(o) | High flow loss for an owner | volume |

Loss_{prop} | 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). | volume |

Loss_{prop}(o) | The (volume) share of proportional loss owned by owner o. | volume |

m | Mass of the sample taken | mass |

m_{i} | Mass of the substance in the sample | mass |

M | Total mass in a specified volume | mass |

M_{i} | Mass of substance in a specified volume | mass |

n_{o} | 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. SeeLink 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. | Time |

Ratio_{ds}(o) | Owner o’s share/ratio of the dead storage volume. This value is specified by the modeller. | n/a |

Ratio_{live}(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 |

Ratio_{loss}(o) | Owner o’s share/ratio of losses. | |

Ratio_{HFT}(o) | Owner o’s share/ratio of the high flow threshold. This value is specified by the modeller. | n/a |

Ratio_{TS}(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. | |

Storage_{ds} | 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 Storage._{dsMAX} | volume |

Storage_{ds}(o) | Current dead storage in the division owned by owner o. | |

Storage_{dsMAX} | Maximum total dead storage in a division specified by the modeller. | volume |

Storage_{dsMAX}(o) | Owner o’s share of the maximum total dead storage in a division. | volume |

Storage_{exclude} | Total volume of water at current time step belonging to owners that are not contributing to the high flow loss. | volume |

Storage_{HFT} | Division storage threshold that corresponds to reach high flow rate threshold q , i.e. _{HFT}Storage = _{HFT}f._{Storage}() | volume |

Storage_{HF} | Total volume of current storage that contributes to high flow loss. | volume |

Storage_{live} | Current live or active storage volume in a division. | volume |

Storage_{live}(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 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. The general form of the continuity equation is discussed in the*Ownership Mass Balance*section, below (see equations 15 and 16).

OR

*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 (*Storage*) specified by the modeller._{dsMAX}

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 (*Ratio _{ds}(o)*) to all of the owners. Hence:

Equation 4 |
---|

Where the sum of all the ratios is equal to one. That is:

Equation 5 |
---|

Also if the maximum total dead storage specified by the modeller is *Storage _{dsMAX}* then each owner’s share of this is:

Equation 6 |
---|

Water is borrowed and lent between all of the owners so that the fixed ownership share given by equation (4) is always maintained.

#### Sharing live storage

Live storage is shared according to the owner’s share of the index flow rate () (see Proportional Routing, below, for an explanation of this approach). The live storage calculation is done after each owner’s outflow has been determined.

Equation 7 |
---|

Equation 8 |
---|

## Ownership of fluxes

#### Inflow flux

In the flow phase, the outflow flux from the division upstream becomes the inflow flux to the current division (or the outflow flux from the upstream node becomes the inflow to the division if the most upstream division in a link is being considered). The same is true for the inflow flux per owner (the ownership processing for the upstream division or node has already determined its outflow per owner).

Equation 9 |
---|

Equation 10 |
---|

#### Outflow flux

The total outflow volume is also known after the reach’s flow routing is complete. Ownership processing determines each owner’s share of this outflow flux. The owner’s share of this flux is calculated based on mass balance and proportional routing. The resultant equation (equation (23)) and its derivation are explained in the section on *Owner’s outflow formula (excludes high flow losses)* below.

#### Lateral loss fluxes

The volume of lateral loss fluxes from every link division is known from the routing calculations for each time step after the flow phase. These fluxes may be divided into two categories:

- Fixed loss fluxes whose ownership is known a priori (because they are shared by fixed ratio or by some other means such as time-series or function), and
- Proportional loss fluxes that are shared in proportion to the ownership of the water in the division.

Proportional fluxes may be further categorised into those based on total water in the division, and those above a specified threshold, as in the Murray-style losses. These Murray-style losses are also referred to as high flow losses.

__Fixed loss sharing__

The total fixed loss, *Loss _{fixed}*, is distributed to owners based on the specified ratios, function, or time series such that:

Equation 11 |
---|

An owner’s share of fixed losses is adjusted if that owner does not have sufficient water in the division to cater for the loss. In a live division, an owner’s highest possible fixed loss occurs when their outflow is zero. In a dead division, the owner’s fixed loss cannot be larger than their share of dead storage. The Borrow and Payback mechanism is used to adjust owner shares of fixed loss for these situations. See Ownership adjustments for more information.

__Proportional loss sharing__

If the division is live, owner shares of proportional losses are based on each owner’s index flow rate (see Proportional Routing for an explanation of this approach).

Equation 12 |
---|

Equation 13 |
---|

If the division is dead, proportional losses are shared according to the owner’s fixed share of dead storage:

Equation 14 |
---|

__Murray-style High Flow Loss Sharing__

In the Murray, losses caused by flows in excess of the regulated flow range are shared to owners in proportion to how far each of them is above their fixed share of the regulated flow range. When the modeller specifies Murray-style losses for an ownership system in Source, this method of sharing is applied only to the general purpose flux, which is a function of flow, when specified on links that fall within the ownership system. The calculation of each owner’s share of the high flow loss is explained in the section on *High flow loss formula (Murray-style)*, below.

#### Fill dead storage flux

The division’s dead storage is shared according to fixed ratios. As this rule dictates each owner’s share of part of the total storage, it necessitates changing other parameters of the ownership conservation (mass balance) equation when a dead division is transitioning to the live state. The volume required to fill the dead storage compartment is treated as an additional loss that is shared according to fixed ratios.

## Ownership mass balance

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 |
---|

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 |
---|

## Live division model, assumptions and equations

#### Index flow rate

The owner’s index flow rate () is determined from the inflow volume *I(o)*, outflow volume *O(0)* and the Muskingum parameter *x* (see Link storage routing - SRG* *for details):

Equation 17 |
---|

#### 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 :

Equation 18 |
---|

It can also be assumed that *g _{o} ≈ g_{o-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 |
---|

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 |
---|

#### Owner's outflow formula (excludes high flow losses)

As ownership is conserved, each owner’s outflow can be determined using the proportional routing, mass balance and division flow equations (note, this formula does not cater for ‘high flow losses’).

To simplify later steps, *r* is defined as:

Equation 21 |
---|

Combining mass balance equation (15) with equations (19) and (21) gives:

Equation 22 |
---|

Rewriting (22) by substituting equation (17) for , and rearranging in terms of owner outflow volume , gives:

Equation 23 |
---|

If high flow losses are not specified, the result of this outflow volume equation is used in the mass balance equation to determine division storage at the end of a time-step.

#### High flow loss formula (Murray-style)

In the Murray, losses caused by flows in excess of the regulated flow range are shared to owners in proportion to how far each of them is above their fixed share of this range. Hence, if is owner *o*’s current flow rate, *Ratio _{HFT}(o)* is their share of the high flow threshold,is the high flow threshold,the total flow rate, and

*Loss*the total high flow loss, then owner

_{HF}*o*’s share of the high flow loss is:

Equation 24 |
---|

Substituting equation (19) into equation (24) enables it to be re-written in terms of the division’s storages, as follows:

Equation 25 |
---|

High flow losses are worn only by owners where *Storage(o,t)-Storage _{dsMAX}(o)* is greater than their share of the high flow threshold (

*Storage*). Owners that will not be required to contribute the high flow loss are identified by calculating

_{HFT}Ratio_{HFT}(o)*Storage(o,t)*for each owner assuming that

*Loss*and finding those that fall short of their share of the high flow threshold.

_{HF}=0From equation (25), for those owners that __exceed their share of the threshold__:

Equation 26 |
---|

The denominator of equation (26) is the total volume above the high flow threshold of owners contributing to the high flow loss. The equations that follow are used to determine values required to find this.

Firstly, a modified high flow threshold is defined that applies to the owners, *j* (where *j* is defined in Table 2, and the relevant owners are those that have, with *Loss _{HF}=0*, a trial value of

*Storage(o,t)*such that

*Storage(o,t) > Storage*:

_{dsMAX}(o) + Storage_{HFT}Ratio_{HFT}(o))Equation 27 |
---|

where:

is set-builder notation to indicate that the modified high flow threshold is the sum of all the owners’ shares of the threshold where the high flow loss is not going to be zero.

Next, it is necessary to determine the total volume of water this time step belonging to owners that are not contributing to the high flow loss, i.e. *Storage _{exclude}*:

Equation 28 |
---|

The denominator of equation (26) (current live volume contributing to high flow loss) is then:

Equation 29 |
---|

Equation (26) can now be recast in terms of the total volume in the division:

Equation 30 |
---|

Rearranged:

Equation 31 |
---|

#### High flow loss solutions over a time-step

__Case of Muskingum weighting ____x = 1____ (Index flow rate = Inflow)__

In this case, the index flow rate is the same as the inflow rate, and an owner’s share of inflow will determine their share of active storage. The proportional routing formula (equation (19)) can be rewritten and applied to inflow as shown in equation (32), below and then rearranged to determine the owner’s share of storage.

Equation 32 |
---|

Equation 33 |
---|

The owner’s share of proportional lateral flux is then found by substituting into equation (20) and rearranging.

Equation 34 |
---|

Each owner’s storage *Storage(o,t) - Storage _{dsMAX(}o)* is compared to

*Storage*to determine whether a high flow loss applies. If it does, equation (24) is used to determine this value. Mass balance is applied to determine each owner’s share of outflow.

_{HFT}Ratio_{HFT}(o)__Determining whether an owner has a high flow loss when ____x ≠ 1__

If the division’s Muskingum weighting *x ≠ 1 *both inflow and outflow impact storage, equation (23) is used to determine each owner’s share of outflow O(o). From this a trial estimate of each owner’s storage (*Storage(o,t)*) is calculated by assuming *Loss _{HF }= 0*. It is then possible to determine if owners have a share of the high flow loss or not, i.e. whether

*Storage(o,t) - Storage*is less or more than their share of the high flow threshold:

_{dsMAX}(o)*Storage*.

_{HFT}Ratio_{HFT}(o)__Case of Muskingum weighting ____x ≠ 1____, owner without a high flow loss__

This solution is applied in divisions where Muskingum weighting *x ≠ 1 *to owners that have *Storage(o,t) ≤ Storage _{dsMAX(}o) + Storage_{HFT}Ratio_{HFT}(o)*. To solve mass balance, the outflow volume (

*O(o)*) is recast in terms of the division’s live storages. Recalling the index flow ratefrom equation (17) and rearranging for

*O(o)*:

Equation 35 |
---|

Equation (35) can be rewritten in terms of live storage using the proportional routing equation, equation (19), as follows:

Equation 36 |
---|

Combining the mass balance and proportional loss equations (i.e. equations (15) and (19)) with equation (36), and rearranging yields:

Equation 37 |
---|

__Case of Muskingum weighting ____x ≠ 1____, owner with a high flow loss__

This solution is applied in divisions where Muskingum weighting *x ≠ 1 *to owners that have *Storage(o,t) > Storage _{dsMAX(}o) + Storage_{HFT}Ratio_{HFT}(o)*. The approach is based on defining a modified proportional loss which is the remaining proportional loss not accounted for after considering those owners not contributing to the high flow loss:

Equation 38 |
---|

Combining the mass balance and proportional loss equations (i.e. equations (15) and (19)), the loss equation (31), equation (35), and equation (38) above, and rearranging yields:

Equation 39 |
---|

__Potential issue with high flow loss calculation (Case of ____x ≠ 1____): Outflow can be negative__

A problem may occur with equation (39) as it is possible for the modeller to configure a perverse case where an owner would be required to borrow from other owners to pay for their share of the high flow loss. In the case where *x = 1* this does not cause a problem as borrowing between owners does not affect the share of the division’s storage. For other cases, borrowing between owners will change the share of the division’s storage (*Storage(o,t)*) as changes in outflow will change. In theory this would indicate that high flow loss should be solved iteratively. However, as iterative solutions tend to impact performance, and the situation will only occur where outgoing lateral fluxes are so large as to reduce an owner’s outflow to less than zero, a solution that uses borrow and payback on division outflow is proposed. This means accepting in these cases a mismatch between and *Storage(o,t)*.

## Dead division model, assumptions and equations

#### Fully mixed (continuously stirred reactor) model

Proportional routing cannot be used to determine owner shares where there is no active storage in a division. In this situation, the fully mixed (continuous stirred reactor) model is used. This approach is based on the concept that ownership will travel as if it were a substance mixed uniformly throughout the routing storages. If a substance, *i*, is completely mixed throughout a volume and a sample from that volume is taken, the following relationship applies:

Equation 40 |
---|

where:

*M* is the total mass in the volume;

*M _{i}* is the mass of the substance in the volume;

*m* is the mass of the sample taken; and

*m _{i}* is the mass of the substance in the sample.

#### Owner’s storage formula

To calculate an owner’s storage volume (*Storage(o,t)*) in a dead division, the fully mixed principle expressed in equation (40) is applied to a stored volume of water (*Storage*), with outflow (due to fluxes) *O* as the sample, and ownership *o* as the substance of interest. The resultant relationship is:

Equation 41 |
---|

Rearranging this:

Equation 42 |
---|

Proportional losses in a dead division are shared in the proportions of the stored water, therefore:

Equation 43 |
---|

Reiterating equation (15), the mass balance of a routing division over a time step is:

Equation 44 |
---|

Substituting equations (42) and (43) into (44), and rearranging the unknowns to the left hand side yields:

Equation 45 |
---|

This can be rearranged to solve for an owner’s storage volume, *Storage(o,t)*, as follows:

Equation 46 |
---|

## Ownership adjustments

In some situations, adjustments need to be made to owners’ shares of a division’s mass balance equation so that the shares all add up to the correct total. An imbalance can occur when:

- The division is transitioning from dead to live, i.e. between the continuous stirred reactor and proportional routing models. In this case, owner fixed loss and the previous time step storage are adjusted.
- The fixed losses specified by the modeller exceed the owner’s ability to meet a lateral outflow flux requirement. When this occurs, owner shares of fixed losses are modified, and the changes tracked in the appropriate borrow-and-payback account balances. (Refer to Borrow and Payback - SRG for a description of these balances).
- High flow losses cause outflow to be negative. When this occurs, owner shares of outflow are modified using the borrow and payback mechanism.

More details on the first two points are discussed in the following sections.

#### Division transitioning from dead to live

If the division has started flowing again in the current model time step - that is, it has gone from being dead to being live - a correction is required if there was airspace in the dead storage last time step (i.e. if dead storage was not full: *Storage(t-1) < Storage _{dsMAX}*). Firstly the airspace volume to be filled is calculated:

Equation 47 |
---|

Secondly, each owner’s fixed loss for the current model time step is increased to represent their contribution to filling the airspace storage volume, and is equivalent to reducing the volume available to contribute to filling live storage by the requisite amount. The relevant equation is:

Equation 48 |
---|

This is also consistent with the fully mixed principle discussed in the section on *Owner’s storage formula*, above. The final step is to adjust the value of the storage for last time step, so it is the appropriate value to use in calculations for a live division in the current time step (note, this adjustment occurs after outputs for the last time step are recorded). That is:

Equation 49 |
---|

Equation 50 |
---|

#### Borrow and payback

If an owner’s share of fixed losses is greater than their available capacity to meet a lateral outflow flux requirement, then an adjustment is made which entails borrowing from other owners that have surplus capacity available, with later payback. Different methods are used to determine whether each owner has a capacity deficit or has surplus capacity, depending on whether the division is live or dead (see the sub-sections below). The borrow and payback options available are:

- Fixed loss borrow and payback: Owners with a surplus that lend to others have their fixed loss
*Loss*increased by the amount loaned, and those with a deficit that borrow have their_{fixed}(o)*Loss*decreased by the amount borrowed._{fixed}(o) - Outflow borrow and payback: Owners with a surplus that lend to others have their outflow volume
*O(o)*decreased by the amount loaned, and those with a deficit that borrow have their*O(o)*increased by the amount borrowed.

__Live Division – Fixed Loss Borrow and Payback__

In a live division, the maximum fixed loss an owner could meet is that which would occur when their outflow is equal to zero (i.e. when *O(o) = 0*). The value of *Loss _{fixed}(o)* that will result in

*O(o) = 0*can be found from equation (23), re-expressed as follows:

Equation 51 |
---|

For each owner, the values of surplus and deficit for borrow and payback are therefore:

Equation 52 |
---|

Equation 53 |
---|

__Live Division – Outflow Borrow and Payback__

For each owner, the limiting surplus and deficit for potential borrow and payback is their share of the outflow volume.

__Dead Division__

In a dead division, determining whether each owner has a surplus or a deficit, and the magnitude, is based on the mass balance equation (equation (15)), re-expressed as follows:

Equation 54 |
---|

A positive net volume is a surplus, and a negative net volume is a deficit. Hence the values for borrow and payback are:

Equation 55 |
---|

Equation 56 |
---|

## Methodology

#### Model Phase: Configuration

Link ownership features are specified as input data by the modeller at the level of the ownership system and the individual link.

- At the ownership system level the modeller specifies for all links in the ownership system:
- Whether ownership of the time series flux is shared according to a fixed ratio or proportional to owner flow/storage in the link. (This setting can be overridden at the link level where the modeller can input a time series flux per owner if required).
- Whether ownership of other link lateral fluxes is shared according to a fixed ratio or proportional to owner flow/storage in the link.
- Whether the Murray style high-flow loss method is to be used for the flow based flux.

- Owner percentages of dead storage:
*Ratio*_{ds}(o) - Owner percentages of time series fluxes (for the fixed ratio option):
*Ratio*_{TS}(o) - Owner percentages of other lateral fluxes (for the fixed ratio option):
*Ratio*_{loss}(o) - Owner percentages of the high flow threshold (if applicable):
*Ratio*_{HFT}(o)

- Owner percentages of dead storage:
- For each individual link, the following are input:
- Initial owner shares of flow or active storage
- Whether the time series flux is to be input per owner (if so, this overrides the ownership system method of sharing time series flux), or not.
- If the time series flux is to be modelled by owner,
*Flux*, otherwise_{TS}(o)*Flux*_{TS} - If Murray style losses are to be modelled, the high flow threshold above which high flow conditions apply.

## Model Phase: Initialisation

When the model is initialised at the start of the scenario run, for a given link the ownership processing is initialised using the same values for all divisions in the link. Each owner’s share of the initial storage volume is calculated. In addition, if Murray-style losses are being modelled, the threshold storage volume and each owner’s initial share of this are determined, and the initial value of the volume of flow based flux caused by flow at the high flow threshold is also determined.

## Model Phase: Ordering

No ownership calculations are involved in the ordering phase.

## Model Phase: Flow distribution

In this phase, ownership processing for a link is performed after other link processing is completed. Computational steps are summarised in Figure 2.

**Figure 2. Flowchart- of main steps in ownership processing for a division in a time step**

For every division, to determine owner volumes of inflow, lateral flux, outflow, storage and mass balance, the steps for each model time step are as follows:

- Initialise temporary parameters.
Set each owner’s upstream inflow

*I(o**)*to equal that owner’s share of the upstream component’s outflow using equation (10). (The upstream component is the next division upstream, or if it is the first division in the link, the upstream node).- Determine the volume of the total flow based flux,
*fFlow*, for this time step to be shared according to fixed ratio, function or time series specified by the modeller. Where relevant, also determine the volume,_{LG}()*Loss*, that is to be shared using high flow rules (i.e. Murray-style losses), which is the volume that is in excess of the value of_{HF}*fFlow*._{LG}()- If the ownership system uses Murray-style losses:
The flux based on flow up to the high flow threshold is shared according to fixed ratio. This volume is:

Equation 57 The flux based on flow above the high flow threshold is shared according to high-flow rules. This volume is:

Equation 58

Otherwise, all the flux is shared using fixed shares, function or time series specified by the modeller:

Equation 59

- If the ownership system uses Murray-style losses:
- Determine every owner’s fixed flux total for this time step (
*Loss*), where the modeller has specified which of the link’s lateral fluxes (time series, general purpose flow based, groundwater and net evaporation) are fixed at the ownership system level (noting these must satisfy equation (11))._{fixed}(o)- Time series flux:
- Determine the owner’s share of input time series flux(es)
*Flux*:_{TS}(o)If the time series flux for the current link is not input per owner, and for the ownership system time series flux sharing is by fixed ratio, apply the specified owner’s ratio to the total time series flux,

*Flux*, that was calculated prior to ownership processing (_{TS}*Flux*has been adjusted to ensure total loss does not exceed the amount of water in the division)._{TS }Equation 60 If the time series flux for the current link is input per owner, get the owner’s flux for this time step from the input time series:

If*LossGain*the flux is a gain, this needs no adjustment:_{TS}< 0Equation 61 Otherwise, the flux is a loss that may need to be scaled down (This is to allow for the case where the time series has pumped the division dry. The overall time series loss

*Flux*has already been adjusted by link processing to ensure losses do not exceed water in the link. Input owner time series losses are scaled down so their sum does not exceed the total adjusted for any owner time series gains.)._{TS}Equation 62

Set the following

Equation 63

- Determine the owner’s share of input time series flux(es)
Determine the owner’s fixed share of the flow based flux when the ownership system uses Murray-style losses

Equation 64 Equation 65 If the ownership system uses fixed ratio to share other lateral fluxes: Apply the owner’s configured

*Ratio*to the remaining lateral fluxes and add their share of these to their fixed flux total:_{loss}(o)Owner’s net evaporation flux:

Equation 66 Owner’s groundwater flux:

Equation 67 Equation 68

- Time series flux:
- Determine the total proportional flux,
*Loss*._{prop}- Initially
*Loss*_{prop}= 0 If time series flux sharing is

*‘**Proportional’*, a total time series was input, so:Equation 69 If Murray-style losses are not being modelled and other lateral flux sharing is

*‘**Proportional’*:Equation 70 If other lateral flux sharing is

*‘**Proportional’*:Equation 71

- Initially
- Determine the division’s state (‘dead’ or ‘live’).
- If Muskingum weighting factor and there is no inflow (
*I = 0*), it is ‘dead’:*State = Dead* Otherwise, compare the result of the storage function to the mass balance equation to determine division state (if they are within

*maxError*, the division is ‘live’):Equation 72 Equation 73 then

*State = Live*Otherwise

*State = Dead*

- If Muskingum weighting factor and there is no inflow (
- Find the division’s storage using a method appropriate to the state of the division (i.e. whether ‘dead’ or ‘live’). These methods are summarised in the two following sections.
Calculate each owner’s outflow,

*O(o)*, using equations (15) or (16), as appropriate, rearranged so the term*O(o)*is on the left hand side of the equation.Equation 74 - Adjust owner shares of outflow as necessary to ensure that none are negative. The principles are discussed in Ownership Adjustments, above. In summary, if
*O(o) < 0*, the owner concerned borrows from other owners to ensure*O(o)=0*.- If any owner’s outflow is negative then:
Calculate surplus/deficit for each owner.

Equation 75 Equation 76 Pass owner surpluses and deficits to the Borrow method. This will return how much owners borrowed or lent (

*OwnerBorrowed(o), OwnerLent(o)*).Adjust the outflows to account for borrowing:

Equation 77 Equation 78

- If any owner’s outflow is negative then:
Calculate owner mass balances to report, based on equation (15) or (16), as appropriate, using adjusted values of

*O(o)=0*.Equation 79

**Dead division**

The steps involved in processing a dead division are as follows:

Calculate the final storage for each owner,

*Storage(o, t)*, using equation (14):Equation 80 - Fixed losses are adjusted so that mass balance can be achieved when the owner has the configured ratio of dead storage, using equations (54), (55), and (56).
Calculate how much each owner has in excess of the final storage:

Equation 81 Set each owner’s surplus or deficit:

Equation 82 Equation 83 Pass owner surpluses and deficits to the Borrow method (Refer to Borrow and Payback - SRG for more information). This will return how much owners borrowed or lent (

*OwnerBorrowed(o), OwnerLent(o)*).Adjust the fix losses to account for borrowing:

Equation 84 Equation 85

- Calculate each owner’s share of the proportional lateral flux (
*Loss*) using equation (43)._{prop}(o)If the time series flux is specified to be shared proportionally:

Equation 86 If other lateral fluxes are to be shared proportionally:

For each of groundwater, net evaporation fluxes and, if Murray-style losses are not being modelled, flow based flux (*Flux*):_{GW}, Flux_{NE}, Flux_{flow}Equation 87

**Live division**

Figure 3 provides an overview of the procedure. The steps are explained in more detail below.

**Figure 3. Flowchart- of main steps in processing a live division in a time step**

The steps involved in processing a live division are as follows:

- If the division is transitioning from dead to live (i.e.
*State*) and_{t-1}= Dead*Storage(t - 1) < Storage*, and the fixed lateral flux for each owner,_{dsMAX}, Storage(t - 1)*Loss*, are adjusted as described in the_{fixed}(o)*Division transitioning from dead to live*section under Ownership adjustments (equations (47), (48), (49) and (50) above). The steps are:Adjust each owner’s fixed lateral flux to include the volume to fill dead storage

Equation 88 Set the apparent storage for last time step to the maximum dead storage.

Equation 89 Set the apparent storage for each owner at the last time step to their share of the maximum dead storage.

Equation 90

- Calculate the total ‘live’ storage for reporting (and use in further calculations), where
*Storage*as per equation (20)._{live}= Storage(t) - Storage_{dsMAX} - Calculate trial storage volumes for each owner (that assume no high flow loss):
If Muskingum

*x = 1*, outflow and storage are not interdependent, so the proportional routing model can be applied directly:Equation 91 - Otherwise outflow and storage interact, so a more complex method is needed:
Adjust owner fixed lateral fluxes to ensure no owner has a negative outflow:

To determine the ratio

*r*, use equation (21)Equation 92 (Note that

*Storage*_{live}= Storage(t) - Storage_{dsMAX})Adjust owner fixed lateral fluxes to ensure no owner has a negative outflow. This involves finding the maximum fixed loss that each owner can sustain with non-negative outflow and the resultant surplus/deficit using equations (51), (52) and (53).

Equation 93 Equation 94 Equation 95 Pass owner surpluses and deficits to the Borrow method (Refer to Borrow and Payback - SRG for more information). This will return how much owners borrowed or lent (

*OwnerBorrowed(o), OwnerLent(o)*).For each owner: Add amount lent and subtract amount borrowed from the fixed loss - refer to equations 84 and 85.

Use the proportional routing model equation (37) to determine trial volumes for each owner (assuming no high flow losses at this stage):

Equation 96 Equation 97

- If there is a high flow loss (
*Loss*> 0):_{HF}If Muskingum

*x ≠ 1*high flow losses will impact storage volumes. The approach described in the section on*High flow loss formula (Murray-style)*, under**Live division model, assumptions and equations**above, is used to determine the final storage volume for each owner.Calculate owner shares of the high flow loss using equation (26):

Equation 98

- Calculate each owner’s share of each proportional lateral flux for reporting using equation (20), after rearranging so the term
*Loss*is on the right hand side (i.e. by rearranging to match the form of equation (13)):_{prop}If the time series flux is configured to be shared proportionally

Equation 99 If other lateral fluxes are configured to be shared proportionally:

For each of groundwater, net evaporation fluxes and, if Murray-style losses are not being modelled, flow based flux (

*Flux*):_{GW}, Flux_{NE}, Flux_{flow}Equation 100

**Adjust Owner Storage Volumes For High Flow Loss**

In a live division when there is a high flow loss (*Loss _{HF} > 0*) and the Muskingum

*x ≠ 1*(meaning that outflow and storage interact), owner storage volumes need to be adjusted for the high flow loss. The steps are as follows:

- Create a list of
*HighFlowOwners*whose trial storage exceeds their share of the high flow threshold volume (*Storage(o, t) > Storage*), and another list of the remaining_{HFT}(o) + Storage_{dsMAX}(o)*ExcludedOwners*(that have*Storage(o, t) ≤*.*Storage*)_{HFT}(o) + Storage_{dsMAX}(o) Find the modified high flow storage threshold that applies to

*HighFlowOwners*:Equation 101 Find the total storage and the total proportional flux volumes owned by the (whose storage volume does not contribute to the high flow loss):

Equation 102 Equation 103 Find the total division storage volume contributing to high flow losses this time step:

Equation 104 Calculate for each owner in

*HighFlowOwners*their final storage for this time step using equation (39) with*Storage*substituted for_{live}*Storage(t) - Storage*:_{dsMAX}Equation 105

## Data

#### Input data

Details on data requirements are provided in the Source User Guide.

#### Parameters or settings

Input parameters and settings for ownership specified at the level of the link (as distinct from input parameters and settings at the ownership system level which apply to all links, and nodes, in the ownership system) are summarised in Table 3.

#### Table 3. Link ownership parameters

Parameter name | Parameter description | Unit type | No. of values | Allowable values & validation rules | Default value(s) |
---|---|---|---|---|---|

Ownership system | Name of the ownership system the link sits within. | n/a | 1 | Read only | Link's ownership system |

Configure time series flux per owner | Indicates whether each owner's time series flux is to be configured on the link (if so, overrides ownership system sharing method) | n/a | 1 | Yes or no | No |

Murray-style losses: High flow threshold | For Hurray-style high flow losses: Threshold above which high flow losses occur | Volume/time | multiple | Real ≥ 0 | Owner in link's ownership system |

Owner | Name of the owner the row's share parameters apply to | % | One per owner | Read only | Value for link's ownership system |

Dead storage % | Owner's percentage of dead storage in the link | % | One per owner | Read only | Value for link's ownership system |

Time series flux % | Owner's percentage of link time series fluxes that are shared according to fixed ratio | % | One per owner | Read only | Value for link's ownership system |

Other lateral flux % | Owner's percentage of other link lateral fluxes that are shared according to fixed ratio | % | One per owner | Read only | Value for link's ownership system |

Initial flow or live storage % | Owner's percentage of initial flow or live storage in the link | % | One per owner | Rea, 0 - 100 Total of all owners = 100%y | Equal value per owner |

#### Output data

Output potentially available is summarised in Table 4.

#### Table 4. Link Ownership Output Variables

Model element | Parameter | Units | Freq. | Display format |
---|---|---|---|---|

Division + owner | Upstream flow | Volume/time | Time-step | Displayed as: Graph, Table, Statistics (min, max average over the modelled time period. |

Upstream flow volume | Volume | |||

Downstream flow | Volume/time | |||

Downstream flow volume | Volume | |||

Storage volume | Volume | |||

Live storage volume | ||||

Dead storage volume | ||||

Lateral flow volume | ||||

Lateral flow | Volume/time | |||

Groundwater flux | ||||

Net evaporation | ||||

Flow based flux | ||||

Time series flux | ||||

High flow loss(if applicable) | ||||

Mass balance | Volume | |||

Borrow balance | Volume | See Borrow and Payback - SRG | ||

Net borrow | ||||

Link + owner | Same as above. Link upstream inflow = first division's upstream inflow, link downstream outflow = last division's downstream outflow. Storage is total for all link divisions. |

## References

Border Rivers Commission Border Rivers Bulk Water Sharing Plan

Commonwealth of Australia (2007) Water Act 2007 (Act No 137 of 2007 as amended, including amendments up to Act No. 46 of 2011 and SLI 2008 No. 106 (as amended by SLI 2011 No. 117)). Part 1A - The Murray‑Darling Basin Agreement. Available at www.comlaw.gov.au/Details/C2011C00621/Download

NSW and Queensland Governments (2008) New South Wales – Queensland Border Rivers Intergovernmental Agreement 2008. Available at www.derm.qld.gov.au/wrp/pdf/border/intergovt_agreement_2008.pdf and at www.water.nsw.gov.au/Water-management/Law-and-policy/Intergovernmental-agreements/Intergovernmental-agreements/default.aspx

i. The flux based on flow above the high flow threshold is shared according to high-flow rules. This volume is:

## 1 Comment

## Andrew Davidson

Note need to add catchment inflow to the list of lateral flows. Intention is to share based on fixed ratios.