Ownership in Links - SRG

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

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

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
fFlowLG()	Function that returns a lateral loss/gain flux for a given index flow rate . Also referred to as the flow based flux function.	volume
fFlowLG()	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 fFlowLG() where  is the high flow threshold.	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 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).	volume
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. 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
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. StorageHFT = fStorage().	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 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_dsMAX) 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 (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:

1. 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 expression), and
2. 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, expression, 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.

Note: If the routing division is using an attenuation factor x=0 (i.e. ), equation (23) can be rearranged to the continuous stirred reactor equation (equation (46)). This means that for fully forward weighted routing schemes, proportional routing is the same as fully-mixed.

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_HF the total high flow loss, then owner 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_HFTRatio_HFT(o)).  Owners that will not be required to contribute the high flow loss are identified by calculating Storage(o,t) for each owner assuming that Loss_HF=0 and finding those that fall short of their share of the high flow threshold.

From 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_HFTRatio_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_HFTRatio_HFT(o) 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.

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_dsMAX(o) is less or more than their share of the high flow threshold: Storage_HFTRatio_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_HFTRatio_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_HFTRatio_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_fixed(o) increased by the amount loaned, and those with a deficit that borrow have their Loss_fixed(o) decreased by the amount borrowed.
• 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)
• 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_TS(o), otherwise 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:

1. Initialise temporary parameters.

2. 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).

3. Determine the volume of the total flow based flux, fFlow_LG(), for this time step to be shared according to fixed ratio, expression or time series specified by the modeller.  Where relevant, also determine the volume, Loss_HF, 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 fFlow_LG().
   • If the ownership system uses Murray-style losses:

     1. The flux based on flow up to the high flow threshold is shared according to fixed ratio. This volume is:

        Equation 57

     2. 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, expression or time series specified by the modeller:

     Equation 59

4. Determine every owner’s fixed flux total for this time step (Loss_fixed(o)), 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)).
   1. Time series flux:
      1. 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_TS, that was calculated prior to ownership processing (Flux_TS has been adjusted to ensure total loss does not exceed the amount of water in the division).

           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_TS < 0 the flux is a gain, this needs no adjustment:

           Equation 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_TS 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.).

           Equation 62

      2. Set the following

         Equation 63

   2. Determine the owner’s fixed share of the flow based flux when the ownership system uses Murray-style losses or it uses fixed ratio to share other lateral fluxes:

      Equation 64

      Equation 65

   3. If the ownership system uses fixed ratio to share other lateral fluxes: Apply the owner’s configured Ratio_loss(o) to the remaining lateral fluxes and add their share of these to their fixed flux total:

      • Owner’s net evaporation flux: 

        Equation 66

      • Owner’s groundwater flux: 

        Equation 67

        Equation 68

5. Determine the total proportional flux, Loss_prop.
   1. Initially Loss_prop = 0

   2. If time series flux sharing is ‘Proportional’, a total time series was input, so:

      Equation 69

   3. If Murray-style losses are not being modelled and other lateral flux sharing is ‘Proportional’:

      Equation 70

   4. If other lateral flux sharing is ‘Proportional’:

      Equation 71

6. 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

7. 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.

8. 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

9. 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:

     1. Calculate surplus/deficit for each owner.

        Equation 75

        Equation 76

     2. Pass owner surpluses and deficits to the Borrow method. This will return how much owners borrowed or lent (OwnerBorrowed(o), OwnerLent(o)).

     3. Adjust the outflows to account for borrowing:

        Equation 77

        Equation 78

10.  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:

1. Calculate the final storage for each owner, Storage(o, t), using equation (14):

   Equation 80

2. 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).

   1. Calculate how much each owner has in excess of the final storage:

      Equation 81

   2. Set each owner’s surplus or deficit:

      Equation 82

      Equation 83

   3. 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)).

   4. Adjust the fix losses to account for borrowing:

      Equation 84

      Equation 85

3. Calculate each owner’s share of the proportional lateral flux (Loss_prop(o)) using equation (43).

   1. If the time series flux is specified to be shared proportionally:

      Equation 86

   2. 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:

1. If the division is transitioning from dead to live (i.e. State_t-1 = Dead) and Storage(t - 1) < Storage_dsMAX, Storage(t - 1), and the fixed lateral flux for each owner, Loss_fixed(o), are adjusted as described in the Division transitioning from dead to live section under Ownership adjustments (equations (47), (48), (49) and (50) above).  The steps are:

   1. Adjust each owner’s fixed lateral flux to include the volume to fill dead storage

      Equation 88

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

      Equation 89

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

      Equation 90

2. Calculate the total ‘live’ storage for reporting (and use in further calculations), where Storage_live = Storage(t) - Storage_dsMAX as per equation (20).
3. Calculate trial storage volumes for each owner (that assume no high flow loss):

   1. If Muskingum  x = 1, outflow and storage are not interdependent, so the proportional routing model can be applied directly:  

      Equation 91

   2. Otherwise outflow and storage interact, so a more complex method is needed:

      1. Adjust owner fixed lateral fluxes to ensure no owner has a negative outflow:

         1. To determine the ratio r, use equation (21)

            Equation 92

            (Note that Storage_live = Storage(t) - Storage_dsMAX)

         2. 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

         3. 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)).

         4. For each owner: Add amount lent and subtract amount borrowed from the fixed loss - refer to equations 84 and 85.

      2. 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

4. If there is a high flow loss (Loss_HF > 0):

   1. 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.

   2. Calculate owner shares of the high flow loss using equation (26): 

      Equation 98

5. Calculate each owner’s share of each proportional lateral flux for reporting using equation (20), after rearranging so the term Loss_prop is on the right hand side (i.e. by rearranging to match the form of equation (13)):

   1. If the time series flux is configured to be shared proportionally 

      Equation 99

   2. If other lateral fluxes are configured to be shared proportionally:

   3. 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:

1. Create a list of HighFlowOwners whose trial storage exceeds their share of the high flow threshold volume (Storage(o, t) > Storage_HFT(o) + Storage_dsMAX(o)), and another list of the remaining ExcludedOwners (that have Storage(o, t) ≤ Storage_HFT(o) + Storage_dsMAX(o)).

2. Find the modified high flow storage threshold that applies to HighFlowOwners:

   Equation 101

3. 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

4. Find the total division storage volume contributing to high flow losses this time step: 

   Equation 104

5. Calculate for each owner in HighFlowOwners their final storage for this time step using equation (39) with Storage_live substituted for 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

Output data

Output potentially available is summarised in Table 4.

Table 4. Link Ownership Output Variables

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: