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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.

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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.

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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

      Flux_TSSequence of time series flux values for the link input by the modeller.  

fFlowLG(Image Added)	Function that returns a lateral loss/gain flux for a given index flow rate Image Added. Also referred to as the flow based flux function.	volume
fFlowLG(Image Added)	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(Image Added) where Image Added 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

the

Link

Storage Routing

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

 

Image Added

Division index flow rate, which is the index flow for the current time step. See

the

Link

Storage Routing

storage routing - SRG for more information.

volume/time

 

Image Added

Owner’s share of division’s index flow rate for the current time step.

volume/time

 

Image Added	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(Image Added).	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

storage routing - SRG for more information)

States of a division: live or dead

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A division is dead if x = 1 and  or if x ≠ 1 and the following is true:

Equation 1	Image Modified

Ownership of division storage

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The live storage in a division is the storage above dead storage, as shown in Figure 1, and is obtained from:

Equation 2	Image Modified

The total volume of water in a division is then:

Equation 3	Image Modified

Sharing dead storage

Ownership of the dead storage is shared by fixed ratio (Ratio_ds(o)) to all of the owners. Hence:

Equation 4	Image Modified

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

Equation 5	Image Modified

Also if the maximum total dead storage specified by the modeller is Storage_dsMAX then each owner’s share of this is:

Equation 6	Image Modified

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

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

Equation 7	Image Modified

Equation 8	Image Modified

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	Image Modified

Equation 10	Image Modified

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.

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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 expressionfunction), and
2. Proportional loss fluxes that are shared in proportion to the ownership of the water in the division.

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The total fixed loss, Loss_fixed, is distributed to owners based on the specified ratios, expressionfunction, or time series such that:

Equation 11	Image Modified

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 the section on Ownership Adjustments for 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 the Proportional  Proportional Routing section for an explanation of this approach).

Equation 12	Image Modified

Equation 13	Image Modified

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

Equation 14	Image Modified

Murray-style High Flow Loss Sharing

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It is assumed that ownership is conserved in every division of a reach. This is reflected in the ownership mass balance equation below.  This shows that the difference between an owner’s share of storage at the beginning and end of a time step should be the sum of their share of all the division’s fluxes.

Equation 15	Image Modified

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	Image Modified

Live division model, assumptions and equations

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

Equation 17	Image Modified

Anchor
Proportional routing Proportional routing

Proportional routing of ownership

Proportional routing is used to share the division’s active (live) storage between owners. This is based on the idea that ownership travels at the rate that each owners’ flow influences the flow in the division. If we consider a division and divide each owners’ inflow into a very large number of small pieces,  that each time one more of these slices is passed through, the increment in division storage (Storage) can be approximated as a linear function g of the increment in the index flow rate q‾ Image Added:

Equation 18	Image Modified

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	Image Modified

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	Image Modified

Owner's outflow formula (excludes high flow losses)

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To simplify later steps, r is defined as:

Equation 21	Image Modified

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

Equation 22	Image Modified

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

Equation 23	Image Modified

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.

Info
icon false
Note: If the routing division is using an attenuation factor x=0 (i.e. Image Modified), 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.

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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 q‾(o) is Image Addedis owner o’s current flow rate, Ratio_HFT(o) is their share of the high flow threshold, q‾_HFT is Image Addedis the high flow threshold,  the Image Addedthe total flow rate, and Loss_HF the total high flow loss, then owner o’s share of the high flow loss is:

Equation 24	Image Modified

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

Equation 25	Image Modified

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	Image Modified

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

 

Image Added

where:

j  Owners: Loss_HF(j) ≠ 0 is Image Added 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	Image Added

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

Equation 29	Image Added

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

Equation 30	Image Added

Rearranged:

Equation 31	Image Added

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	Image Added

Equation 33	Image Added

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

Equation 34	Image Added

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 rateImage Addedfrom equation (17) and rearranging for O(o):

Equation 35	Image Added

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

Equation 36	Image Added

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

Equation 37	Image Added

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	Image Added

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	Image Added

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 changeImage Added. 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 Image Addedand 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	Image Added

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	Image Added

Rearranging this:

Equation 42	Image Added

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

Equation 43	Image Added

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

Equation 44	Image Added

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

Equation 45	Image Added

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

Equation 46	Image Added

Anchor
Ownership adjustments Ownership adjustments

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	Image Added

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	Image Added

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	Image Added

Equation 50	Image Added

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	Image Added

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

Equation 52	Image Added

Equation 53	Image Added

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	Image Added

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	Image Added

Equation 56	Image Added

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 Image Added 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

Image Added

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(Image Added), 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, 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(Image Added).
   • 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	Image Added

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

        Equation 58	Image Added

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

     Equation 59	Image Added

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	Image Added

         • 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	Image Added

           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	Image Added

      2. Set the following

         Equation 63	Image Added

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

      Equation 64	Image Added

      Equation 65	Image Added

   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	Image Added

      • Owner’s groundwater flux: 

        Equation 67	Image Added

        Equation 68	Image Added

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	Image Added

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

      Equation 70	Image Added

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

      Equation 71	Image Added

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	Image Added

     Equation 73	Image Added

      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	Image Added

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	Image Added

        Equation 76	Image Added

     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	Image Added

        Equation 78	Image Added

10.  Calculate owner mass balances to report, based on equation (15) or (16), as appropriate, using adjusted values of O(o)=0.

    Equation 79	Image Added

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	Image Added

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	Image Added

   2. Set each owner’s surplus or deficit:

      Equation 82	Image Added

      Equation 83	Image Added

   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	Image Added

      Equation 85	Image Added

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	Image Added

   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	Image Added

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

Image Added

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	Image Added

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

      Equation 89	Image Added

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

      Equation 90	Image Added

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	Image Added

   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	Image Added

            (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	Image Added

            Equation 94	Image Added

            Equation 95	Image Added

         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	Image Added

         Equation 97	Image Added

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	Image Added

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	Image Added

   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	Image Added

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	Image Added

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	Image Added

   Equation 103	Image Added

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

   Equation 104	Image Added

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	Image Added

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: