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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.

Dead storageThe 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.
DivisionIn 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 fluxLoss 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
function.
Flow based fluxLateral flux in a division whose rate is a function of the division’s index flow rate.
General purpose flow based fluxA modeller configured, piecewise monotonically increasing relationship between flux and index flow.  See Link storage routing - SRG for more information.
Groundwater fluxA 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 fluxFlow 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 lossMethod 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.
OwnerAn 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 systemA 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 fluxLoss 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 fluxA 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.

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No.Assumption/constraint
1Owners cannot have a negative share of water in storage or in transit.
2The sum of all owners’ shares of storage in a link equals the link’s total storage volume.
3The sum of all owners’ shares of flow in a link equals the link’s total flow.

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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SymbolDescriptionUnits
dtModel time-steptime
Deficit(o)Owner’s deficit to be made up using borrow and payback.volume
      
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
FluxTSSequence 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
gLinear function to translate between ratios: from change in storage/total storage into change in flow/total flow.n/a
jAn owner whose current storage contributes to the high flow loss, i.e. is greater than their share of the high flow threshold.n/a
IDivision inflow volumevolume
I(o)Division inflow volume for owner ovolume
LossTotal volume of loss from the division (negative if water is gained).volume
LossfixedTotal 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
LossHFTotal high flow lossvolume
LossHF(o)High flow loss for an owner volume
LosspropTotal 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
mMass of the sample takenmass
miMass of the substance in the samplemass
MTotal mass in a specified volumemass
MiMass of substance  in a specified volumemass
noNumber of ownersn/a
Net(o)Net volume of water that owner o has in storage (in a “dead” division)volume
ODivision outflow volume, including outflowing lateral fluxesvolume
O(o)Division outflow volume for owner ovolume
oOwner of water in the divisionn/a
 
Image Added
Division index flow rate, which is the index flow for the current time step. SeeLink 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
rSymbol 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. 
StorageCurrent 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. 
StoragedsCurrent 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. 
StoragedsMAXMaximum 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
StorageexcludeTotal 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
StorageHFTotal volume of current storage that contributes to high flow loss.volume
StorageliveCurrent 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
tTime-step indexn/a
t1Start time indexn/a
t2End time indexn/a
xMuskingum parameter (see Link storage routing - SRG for more information) 

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

Equation 1
Image Modified

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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
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The total volume of water in a division is then:

Equation 3
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Ownership of the dead storage is shared by fixed ratio (Ratiods(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
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Also if the maximum total dead storage specified by the modeller is StoragedsMAX 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 () (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
Image Modified
Equation 8
Image Modified

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

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, Lossfixed, 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 Ownership adjustments for more information.

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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
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

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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
Image Modified

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
Image Modified

It can also be assumed that go ≈ go-1 ; i.e. the ratio g is the same for each owner’s slice of water as it passes through the division.  After summing up all of the slices the following relationship is obtained:

Equation 19
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

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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.

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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 is owner o’s current flow rate, RatioHFT(o) is their share of the high flow threshold,is the high flow threshold,the total flow rate, and LossHF 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
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High flow losses are worn only by owners where Storage(o,t)-StoragedsMAX(o) is greater than their share of the high flow threshold (StorageHFTRatioHFT(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 LossHF=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 LossHF=0, a trial value of Storage(o,t) such that Storage(o,t) > StoragedsMAX(o) + StorageHFTRatioHFT(o)):

Equation 27
Image Modified

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. Storageexclude:

Equation 28
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Info
iconfalse
Note: in equation (28), j refers to owners whose current storage does not contribute to the high flow loss.

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

Equation 29
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Equation (26) can now be recast in terms of the total volume in the division:

Equation 30
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Rearranged:

Equation 31
Image Modified

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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 Modified
Equation 33
Image Modified

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

Equation 34
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Each owner’s storage Storage(o,t) - StoragedsMAX(o) is compared to StorageHFTRatioHFT(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.

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This solution is applied in divisions where Muskingum weighting x ≠ 1 to owners that have Storage(o,t) ≤ StoragedsMAX(o) + StorageHFTRatioHFT(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
Image Modified

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

Equation 36
Image Modified
Info
iconfalse
Note: Equation (36) does not work for the case where x = 1 as it would lead to an attempt to divide by zero.  This reflects the fact that in this case there is no relationship between the division’s outflow and live storage.

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

Equation 37
Image Modified

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) > StoragedsMAX(o) + StorageHFTRatioHFT(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 Modified

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 Modified


Potential issue with high flow loss calculation (Case of x ≠ 1): Outflow can be negative

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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 Modified

where:

M is the total mass in the volume;

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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 Modified

Rearranging this:

Equation 42
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Proportional losses in a dead division are shared in the proportions of the stored water, therefore:

Equation 43
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Reiterating equation (15), the mass balance of a routing division over a time step is:

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

Equation 45
Image Modified

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

Equation 46
Image Modified

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  • 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 the Borrow and Payback - SRG entry for  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.

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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) < StoragedsMAX).  Firstly the airspace volume to be filled is calculated:

Equation 47
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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
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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 Modified
Equation 50
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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:

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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 Lossfixed(o) that will result in O(o) = 0 can be found from equation (23), re-expressed as follows:

Equation 51
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For each owner, the values of surplus and deficit for borrow and payback are therefore:

Equation 52
Image Modified
Equation 53
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Live Division – Outflow Borrow and Payback

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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
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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 Modified
Equation 56
Image Modified

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

Image Modified

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, fFlowLG(), for this time step to be shared according to fixed ratio, expression function or time series specified by the modeller.  Where relevant, also determine the volume, LossHF, 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 fFlowLG().
    • 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 Modified
      2. The flux based on flow above the high flow threshold is shared according to high-flow rules. This volume is:

        Equation 58
        Image Modified
    • Otherwise, all the flux is shared using fixed shares, expression function or time series specified by the modeller:

      Equation 59
      Image Modified
  4. Determine every owner’s fixed flux total for this time step (Lossfixed(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) FluxTS(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, FluxTS, that was calculated prior to ownership processing (FluxTS has been adjusted to ensure total loss does not exceed the amount of water in the division).

          Equation 60
          Image Modified
        • 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 LossGainTS < 0 the flux is a gain, this needs no adjustment:

          Equation 61
          Image Modified

          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 FluxTS 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 Modified
      2. Set the following

        Equation 63
        Image Modified
    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 Modified
      Equation 65
      Image Modified
    3. If the ownership system uses fixed ratio to share other lateral fluxes: Apply the owner’s configured Ratioloss(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 Modified
      • Owner’s groundwater flux: 

        Equation 67
        Image Modified
        Equation 68
        Image Modified
  5. Determine the total proportional flux, Lossprop.
    1. Initially Lossprop = 0
    2. If time series flux sharing is Proportional’, a total time series was input, so:

      Equation 69
      Image Modified
    3. If Murray-style losses are not being modelled and other lateral flux sharing isProportional’:

      Equation 70
      Image Modified
    4. If other lateral flux sharing isProportional’:

      Equation 71
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  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 Modified
      Equation 73
      Image Modified

       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 Modified
  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 Modified
        Equation 76
        Image Modified
      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 Modified
        Equation 78
        Image Modified
  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 Modified

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 Modified
  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 Modified
    2. Set each owner’s surplus or deficit:

      Equation 82
      Image Modified
      Equation 83
      Image Modified
    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 Modified
      Equation 85
      Image Modified
  3. Calculate each owner’s share of the proportional lateral flux (Lossprop(o)) using equation (43).
    1. If the time series flux is specified to be shared proportionally:

      Equation 86
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    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 (FluxGW, FluxNE, Fluxflow ):  

      Equation 87
      Image Modified

Live division

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

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  1. If the division is transitioning from dead to live (i.e. Statet-1 = Dead) and Storage(t - 1) < StoragedsMAX, Storage(t - 1), and the fixed lateral flux for each owner, Lossfixed(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 Modified
    2. Set the apparent storage for last time step to the maximum dead storage.

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

      Equation 90
      Image Modified
  2. Calculate the total ‘live’ storage for reporting (and use in further calculations), where Storagelive = Storage(t) - StoragedsMAX 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 Modified
    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 Modified

          (Note that Storagelive = Storage(t) - StoragedsMAX)

        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
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        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
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        Equation 97
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  4. If there is a high flow loss (LossHF > 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 Lossprop 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:

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

...

    1. For each of groundwater, net evaporation fluxes and, if Murray-style losses are not being modelled, flow based flux (FluxGW, FluxNE, Fluxflow): 

      Equation 100
      Image Added

 

Adjust Owner Storage Volumes For High Flow Loss

 

In a live division when there is a high flow loss (LossHF > 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) > StorageHFT(o) + StoragedsMAX(o)), and another list of the remaining ExcludedOwners (that have Storage(o, t) ≤ StorageHFT(o) + StoragedsMAX(o)).
  2. Find the modified high flow storage threshold that applies to HighFlowOwners:

    Equation 101
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  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
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    Equation 103
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  4. Find the total division storage volume contributing to high flow losses this time step: 

    Equation 104
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  5. Calculate for each owner in HighFlowOwners their final storage for this time step using equation (39) with Storagelive substituted for Storage(t) - StoragedsMAX:

    Equation 105
    Image Added
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Note: For the owners in ExcludedOwners their final storage is equal to their trial storage.


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

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.

Parameter nameParameter descriptionUnit typeNo. of valuesAllowable values & validation rulesDefault value(s)
Ownership systemName of the ownership system the link sits within.n/a1Read onlyLink's ownership system
Configure time series flux per ownerIndicates whether each owner's time series flux is to be configured on the link (if so, overrides ownership system sharing method)n/a1Yes or noNo
Murray-style losses: High flow thresholdFor Hurray-style high flow losses: Threshold above which high flow losses occurVolume/timemultipleReal ≥ 0Owner in link's ownership system
OwnerName of the owner the row's share parameters apply to%One per ownerRead onlyValue for link's ownership system
Dead storage %Owner's percentage of dead storage in the link%One per ownerRead onlyValue 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 ownerRead onlyValue 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 ownerRead onlyValue 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 potentially available is summarised in Table 4.

Model elementParameterUnitsFreq.Display format
Division + ownerUpstream flowVolume/timeTime-step

Displayed as:

Graph,

Table,

Statistics (min, max average over the modelled time period.


 
Upstream flow volumeVolume 
Downstream flowVolume/time
Downstream flow volumeVolume
Storage volumeVolume


Live storage volume
Dead storage volume
Lateral flow volume
Lateral flowVolume/time
Groundwater flux
Net evaporation
Flow based flux
Time series flux
High flow loss(if applicable)
Mass balanceVolume
Borrow balanceVolumeSee Borrow and Payback - SRG
Net borrow
Link + ownerSame 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.

Border Rivers Commission (xxxx) Border Rivers Bulk Water Sharing Plan

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