VPMC Cross Section
    • 23 Oct 2022
    • 9 Minutes to read
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    VPMC Cross Section

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    Article Summary

    The VPMC Cross Section uses the same method as the VPMC Routing Unit, but with the added ability to derive wavespeed and attenuation curves from cross-section details.


    Field in Data Entry Form
    Name in Datafile

    Section Label

    node label at discharge point



    First Lateral Inflow Node

    first lateral inflow node label



    Second Lateral Inflow Node

    second lateral inflow node label



    First Lateral Inflow

    First lateral inflow label



    Second Lateral Inflow

    Second lateral inflow label



    Third Lateral Inflow

    Third lateral inflow label



    Fourth Lateral Inflow

    Fourth lateral inflow label



    Distance to Next Section

    distance to next discharge point or cross section in metres (a zero specifies the end of the reach and for this last node the section data will not be used).



    Water Surface Slope

    average water surface slope over reach (must be positive). If this is unavailable, an average bed slope should be used



    Minimum Number of Subnodes

    minimum number of subnodes allowed (default and minimum 2, ie just the upstream and downstream nodes)



    Maximum Number of Subnodes 

    maximum number of subnodes (default and maximum 100)



    Low Flow Smoothing Factor

    low flow smoothing factor such that the wavespeed at Q=0 (c0) is limited below by c0fact × cmax, where cmax is the maximum in-bank wavespeed.




    number of ensuing data lines (can be zero if this is the last section of the reach and section data are not needed for velocity calculation)




    cross chainage [m]



    Bed Elevation

    elevation of bed [m]



    Manning’s N

    Manning's 'n' roughness coefficient



    Start of Panel

    character '*' denoting the first data pair in a panel




    relative path length (only following a '*' - optional, to take account of differences between floodplain and channel reach lengths. See RIVER SECTION for details)




    The words 'LEFT', 'RIGHT', or 'BED' specifying the left bank top, right bank top and thalweg. Currently, only the lower of LEFT and RIGHT bank is used, to determine bankful elevation.




    Easting coordinate - not used in calculations




    Northing coordinate - not used in calculations



    Specify Velocity Using

    For “VQ Section”, the velocity-flow relationship is calculated using the section geometry; for “VQ Power Law”, enter the coefficients a, b, V0 and Q0 to define the flow-velocity relationship; otherwise (VQ Rating), enter the relationship at discrete points in the supplied table.



    Constant a

    scaling parameter for velocity calculation (>0)


    Exponent b

    power law parameter for velocity calculation (>0)


    Lowest Permitted Velocity

    lowest permitted velocity (>0)


    Flow Below which Velocity is Set to Vo

    flow below which velocity is set to Vo (>0)


    Number of VQ Data Pairs

    number of subsequent V Q data pairs


    Velocity and Flow Table 

    a table of velocity and flow values used to interpolate velocity values for subsequent Flood Modeller Quality simulations.


     - these fields are only used when the results of the simulation are to be used for a subsequent water quality simulation. However, dummy data values, or the defaults in the 1D Simulation window, must always be specified unless VQ SECTION is set.

    Theory and Guidance

    The VPMC Cross Section (MUSK-XSEC) models the flow of water in natural and man-made open channels, using a Variable Parameter Muskingum-Cunge method to route the flow. Wavespeed and attenuation parameters are derived from section details supplied by the user, and the method is otherwise identical to the VPMC Routing (MUSK-VPMC) unit. Lateral inflows can be distributed along a reach between adjacent units.

    The VPMC Cross Section calculates the discharge within a river or channel reach given the inflow hydrograph at the upstream end, up to two flow boundaries as lateral inflows and up to four lateral inflow units.

    The VPMC Cross Section is based on the diffusion equation, and uses a numerical scheme which is similar to the Muskingum equation with variable parameters k and x. The method requires a relationship between discharge, wavespeed and attenuation, which is derived from cross-section details.

    In order to smooth out convergence problems at low flows, it is possible to impose a lower bound on the wavespeed as a proportion of the maximum in-bank wavespeed. The default value is 0.0 (i.e. no lower bound); a maximum value of 0.5 is suggested, and is that specified in the literature for the RIBAMAN software package (HR Wallingford, 1989). Left and/or right bank markers may be specified to denote the bank extents for this purpose.

    A minimum of two VPMC Cross Section or VPMC Routing nodes is required for each river or channel reach. The downstream section need not have any wavespeed/attenuation or section data, and such data are not used even if they are supplied. In general, it is only necessary to supply two nodes for any one reach, even if the distance between them is considerable.

    The program automatically selects a distance step based on the Courant condition at each timestep, but within user-defined constraints. The distance increment in any case cannot exceed the distance between adjacent MUSK sections. Intermediate nodes are generated internally, and so it is only necessary to provide a MUSK-VPMC or VPMC Cross Section at each end of a reach. Flows at the internal nodes cannot be examined directly, but the user can control how many such nodes are generated.

    Up to two boundary units can be connected as lateral inflows to each MUSK-VPMC unit. These can be flow/time boundaries or hydrological boundaries. The generated flow is distributed evenly along the reach between a pair of MUSK nodes. It is not possible to split a lateral inflow into more than one MUSK-VPMC or VPMC Cross Section, - one must attach the inflow boundary unit to a lateral inflow unit; the lateral inflow nodes (labels 4-7) within the VPMC Routing Section unit must then be cross-referenced in the lateral inflow unit.

    REPLICATE units cannot be used with VPMC Cross Sections.


    The equations used in the VPMC Cross Section are identical to those for the MUSK-VPMC unit, and the user is referred to VPMC Routing for details.

    For a VPMC Cross Section, the wavespeed c and attenuation parameter a are derived from section properties according to the equations:








    a = attenuation parameter [1/m]

    B = flow surface width [m]

    h = stage [m]

    S0 = average reach slope [m/m].

    The derived curves can be inspected prior to a full unsteady run by using boundary mode. They are written to the unit file (extension .zzu) at the beginning of the run. They can then be modified and input as user-supplied data to a MUSK-VPMC unit if desired.

    For information on space increment determination, timestep considerations and attenuation parameter constraints, please refer to MUSK-VPMC.

    Water Quality Simulations

    If the flood routing model is to be used to generate output for a future water quality simulation, an approximate method must be used to calculate the nodal velocities required. See Routing Velocity Calculation for more information.


    Calculation of the wavespeed and attenuation curves involves derivation of a conveyance table as for RIVER sections, with the added requirement of an average reach slope.

    The calculated curves are saved to the Unit Characteristics file (extension .zzu) if the model is run in Boundary mode.

    The curves are calculated directly form the section data supplied by the use, and so may exhibit features resulting from sudden changes in cross-section shape. In particular, 'spikes' may appear on the wavespeed curve at around bankfull due to the rapid increase in conveyance and flow width. If this occurs then it is recommended to run the model in Boundary mode first, which writes the wavespeed and attenuation curves to the Unit Characteristics file (extension .zzu). These curves can then be smoothed manually to produce a more physically realistic shape, and used as the input data for a MUSK-VPMC unit.

    Automatic smoothing at low flows can be effected by specifying a lower bound on the wavespeed as a proportion of the maximum in-bank wavespeed. A value of less than 0.5 is recommended. The sole use of left and/or right bank markers in this unit is to denote the bank extents for this purpose.

    The wavespeed and attenuation curves are output as a VPMC Routing (MUSK-VPMC) unit, which can be imported directly into VPMC unit, or alternatively, applied as an “Event Data” file to a simulation, as long as it comprises a complete reach or reaches. It may be necessary to remove some data points, however, as a maximum of 100 points are permitted in the wavespeed/attenuation curves for a MUSK-VPMC unit.

    If Flood Modeller is used then Muskingum units can be preceded or followed by normal Flood Modeller river reaches. Muskingum and river sections cannot be mixed within a reach; reaches of the two types can be joined by JUNCTION units (with only two nodes if necessary). The three types of Muskingum unit (MUSKINGUM, MUSK-VPMC and MUSK-XSEC) can be mixed within a reach.

    The normal connectivity rules apply for Muskingum units and therefore an appropriate downstream boundary (e.g. Head-Time Boundary or Flow-Head Boundary is required at the downstream end of Muskingum routing reaches which are not connected to a structure, although this will not have any impact on the routing calculations.

    A normal depth calculation is carried out using Manning's equation, to give an approximation to the water level in the section which would be achieved if normal depth conditions applied. This can be overridden by supplying a QRATING unit with the same node label as the MUSK-XSEC unit. For the last MUSK-XSEC section in a reach, the unit immediately downstream determines the level and the section data are not used.

    REPLICATE units cannot be used to replicate MUSK-VPMC or MUSK-XSEC units. MUSK-XSEC units also cannot be used in conjunction with INTERPOLATE units.

    In very rare circumstances, it is possible in large models for the data storage space to become full. If an error message indicates that this has happened, then the storage required for each MUSK-XSEC unit can be reduced by decreasing the value of maxsub and allowing fewer internal subnodes. This may also decrease run times.

    The variation of the number of space increments may occasionally lead to the loss of resolution on the flood wave or poor volume conservation. This is because the space increment is based on the discharge at the upstream end of the reach, and so the number of calculation subnodes may drop dramatically while the floodwave is still within the reach. If this becomes a problem, the user should fix the parameters minsub and maxsub to give a suitable number of subnodes; the recommended number is that corresponding to the maximum inflow into the reach. This can be obtained by carrying out a full unsteady run and inspecting the resulting inflows and subnode numbers (which are stored in the unit state for the upstream section). The model can then be run again with the new fixed space increment.

    RIVER units should not be directly connected to Conduit units. Users can connect CONDUIT and RIVER reaches using a Junction if no head loss occurs at the join. Alternatively, the specialised Culvert Inlet and Culvert Outlet units can be used to model the losses associated with transitions from open channel to culverts and vice versa. Bernoulli Loss units are also available to model more generalised losses.

    Datafile Format

    Line 1 - keyword 'RIVER' [comment]

    Line 2 - keyword 'MUSK-XSEC'

    Line 3 - Label1 [, Label2 [, Label3]] [, Label4, Label5, Label6, Label7]

    Line 4 - dx, z, slope, minsub, maxsub, qmax, c0fact

    Line 5 - keyword 'CROSS SECTION'

    Line 6 - n1

    Line 7 to Line 6+n1 - xi, yi, ni, [, *, ri, LRB, easting, northing]

    Line 7+n1 - keyword 'VQ POWER LAW' or 'VQ RATING' or 'VQ SECTION'

    The remaining data must not be supplied if VQ SECTION is specified.

    Line 8+n1 - Vo, Qo, a, b, (if 'VQ POWER LAW') or ndat (if 'VQ RATING')

    Line 9+n1 to Line 8+n1+ndat - V, Q (if 'VQ RATING')

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