Full Equations (FEQ) Model for the Solution of the Full, Dynamic Equations of Motion for One-Dimensional Unsteady Flow in Open Channels and Through Control Structures

U.S. GEOLOGICAL SURVEY WATER-RESOURCES INVESTIGATIONS REPORT 96-4240

13.6 Network-Matrix Control Block--Network-Matrix Table

Purpose: The equations relating the flow and water-surface height values at all the flow-path end nodes in the stream system are defined with this block. The input in this block specifies how the flow-path end nodes are connected with internal boundary conditions (at special features) and external boundary conditions to represent the system. Complete details on stream-network schematization are given in section 3; details on internal boundary conditions are presented in section 8.1.

Heading: One line of user-selected information. The suggested string is NETWORK MATRIX CONTROL.

The code/type combinations applied in FEQ simulation are the following:

Code 1: Branch

Code 2: Number of nodes at a junction

Code 3: Equality of water-surface elevation between nodes

Code 4: One-node head-discharge relation

Type 1: Flow over a weir

Type 2: Table relating discharge and head

Type 3: Channel control of flow

Type 4: Structure capacity as function of time

Type 5: Structure capacity varied dynamically

Type 6: Pump with capacity limited by tail water

Code 5: Two-node head-discharge relations

Type 1: Expansion or contraction with critical flow

Type 2: Bidirectional flow in tables, plus pumping

Type 3: Variable-speed variable-head pump

Type 4: Bridge with flow over roadway

Type 5: Abrupt expansion with inflow or outflow

Type 6: Two-dimensional tables

Type 7: Variable height weir

Type 8: Sluice gates at Stratton Dam at McHenry, Ill.

Type 9: Underflow gates tables

Code 6: Node with forced value of flow or elevation (Boundary conditions)

Code 7: Level-pool reservoir

Code 8: Critical depth

Code 9: Not in use

Code 10: Not in use

Code 11: Conservation of momentum or constant elevation

Code 12: Match average elevation at two nodes

Code 13: Conservation of momentum or energy

Code 14: Side-weir flow

Code 15: Dummy branch

Variable: HEAD

Format: A80

Example: CODE A B C D E F G H I J FA FB FC FD FE

Explanation: This is a user-selected heading for the lines of input to follow.

LINE 2 (repeated as many times as needed to input all the equations describing flow and water-surface elevation in the stream system including external boundary conditions)

Variables: CODE, N(1),. . .,N(10), F(1),. . .,F(5)

Format: I5, 10I4, 5F7.0

Example: The FEQ model and example inputs and outputs may be obtained by electronic retrieval from the World Wide Web (WWW) at http://water.usgs.gov/software/feq.html and by anonymous File Transfer Protocol (FTP) from water.usgs.gov in the pub/software/surface_water/feq/ directory.

Explanation: Line 2 is the information used to define an equation or equations in the matrix relating the flow-path end nodes and branches. The relation and the meaning of the remainder of the line is specified by the entry for CODE. All fields on the line are read whether needed or not. The following sections describe the meaning of each input variable for each code.

If CODE=1 (Branch equations):

N(1) is the branch number. Two equations in the matrix are generated for each branch.

If CODE=2 (number of nodes at a junction, sum of flows=0):

N(1) is the number of flow-path end nodes at the junction. Must be at least 2 and no more than
9. N(2), . . . ,N(N(1)+1) are the flow-path end nodes at the junction.

If CODE=3 (Equality of water-surface elevation between nodes):

N(1) is the first flow-path end node. N(2) is the second flow-path end node.

If CODE=4 (One-node head-discharge relation at a node):

N(1) is the type number for the relation, with 1 N(1) 6.

N(2) is the number of the flow-path end node at which the head is specified.
N(3) is the direction of positive flow; +1 means flow into the system, and - 1 means flow out of the system where the system is that part of the stream that includes the discharge node for the hydraulic control structure.
N(4) is the number of the flow-path end node at which the flow is specified.

If TYPE=1 (Flow over a weir):

N(5) is the number of the table that includes the weir coefficient as a function of head.

F(1) is the elevation of the reference point used for defining head.
F(2) is the weir length.

If TYPE=2 (Table relating discharge and head):

N(5) is the number of the table specifying flow as a function of head.

F(1) is the elevation of the reference point used for defining head.

If TYPE=3 (Channel control of flow):

N(5) is the slope source for defining flow. If N(5)= - 1, then the slope is given in F(1); if N(5)=0, then the bottom slope of the stream channel is used; and if N(5)=1, then the water-surface slope at the previous time point is used.

F(1) is the value of slope if N(5)= - 1.

If TYPE=4 (Structure capacity given as a function of time):

N(5) is the number of the table specifying the maximum flow through the structure as a function of head.

N(6) is the number of the table specifying proportion of maximum flow as a function of time.
F(1) is the elevation of the reference point used for defining head.
It is assumed that time varying flow can be represented by the product of two functions. The first function is the proportion of the maximum flow rate as a function of time. The second function is the maximum flow rate as a function of head at the head node. These two functions are simple to define given records of the operation of the facility.

If TYPE=5 (Structure capacity varied dynamically in FEQ computations):

N(5) is the number of the table specifying maximum flow through the structure as a function of head.

N(6) is the number of the operation block controlling the operation of the structure. All operation blocks must be numbered consecutively, starting at 1. The operation table appears in the Operation of Control Structures Block (section 13.12).
F(1) is the elevation of the reference point used for defining head.

If TYPE=6 (Pump with capacity limited by tail water):

N(5) is the number of the table specifying the pumping rate as a function of upstream head only. Tail-water variations are assumed to have only a small effect on the effective pumping rate.

N(6) is the source for the controlling level in the tail water. If value is greater than zero, it represents a time-series table number. If value is less than zero, it represents the Fortran unit number (see appendix 3) for a point-time-series file. This file must be specified in the Input Files Block (section 13.10). The controlling level can be stage or flow.
N(7) is the number of the table in which the tail-water level is converted into a limiting flow for the pumping rate. This table includes an argument of stage or flow taken from the source in N(6), and the limiting pumping rate is specified in this table. The actual pumping rate is the lesser of the limiting value or the rate from the table in N(5).

F(1) is the elevation of the reference point used for defining head for the table in N(5).

If CODE=5 (Two-node head-discharge relations):

N(1) is the type of relation, where 1 N(1) 9.
N(2) is the upstream node of the two nodes in the relation.
N(3) is the downstream node of the two nodes in the relation.
N(4) is the node at which the flow through the structure is specified; it must be either N(2) or N(3).

If TYPE=1 (Expansion or contraction with critical flow):

N(5) is the sign of the transition. Sign=+1 if expansion in flow results when flow is from upstream node to downstream node; sign= -1 otherwise.

N(6) is the number of the table giving the hydraulic characteristics of the cross section where critical flow is computed.
F(1) is the loss factor on velocity-head change for flow from upstream node to downstream node.
F(2) is the loss factor on velocity head change for flow from downstream node to upstream node.
F(3) is the elevation for defining depth in the section of critical flow.

If TYPE=2 (Bidirectional flow given by tables, plus pumping):

N(5) is the number of the table specifying flow in the positive direction; that is, from upstream node to downstream node. If N(6)=0, then N(5) specifies the number of the table containing square root of conveyance as a function of water-surface height.

N(6) is the number of the table specifying submergence reduction for positive flow. If N(6)=0, the conveyance option is applied. The pumping parameters are ignored if the conveyance option is selected.
N(7) is the number of the table specifying flow in the negative direction.
N(8) is the number of the table specifying submergence reduction for negative flow.

N(9): If N(9)=0, water-surface elevation is detected at the destination node for control of the pump. If N(9)=1, flow rate is detected at the destination node for control of the pump.
N(10) is the node to monitor if pump is to be switched off whenever flow at the flow-path end node given by N(10) is > 0.
F(1) is the elevation for the reference point used for computing head.
F(2) is the flow distance for the conveyance option.
F(3) is the pumping rating. If F(3)=0, then no pump is simulated. If F(3) > 0, then pumping from the upstream node to the downstream node is simulated. If F(3) < 0, then pumping from the downstream node to the upstream node is simulated. The node from which water is pumped is the source node and the node to which water is pumped is the destination node.
F(4) is the inlet elevation for the pump. The water-surface elevation at the source node must exceed the inlet elevation before the pump can be turned on. If the water-surface elevation at the source node is at least 0.1 ft below the inlet elevation, the pump will be turned off.
F(5) is the cutoff value at the destination node. F(5) is either flow or water-surface elevation as defined by the value of N(9). The cutoff value is defined by the user to turn the pump on or off depending on conditions at the destination node. If this value gives a cutoff elevation, then setting F(5) < F(1) prevents convergence problems resulting from the pump cycling in subsequent iterations of the computations for a solution at a time step.

The pump option is used to represent the simple on-off operation of a constant-flow pump. Water always moves in the direction given by the sign of the constant pumping rate defined in F(3). The pump is either on or off. If the pump is off, then it is turned on whenever the inlet elevation is exceeded by the water-surface elevation at the source node and the monitored value at the destination node is in the proper range. If the monitored value at the destination node is water-surface elevation, then the water-surface elevation at the destination node must be less than the cutoff value given in F(5) to be in the proper range for turning the pump on. If the monitored value at the destination node is flow rate, then the flow rate at the destination node must be positive and less than the cutoff value given in F(5) to be in the proper range for turning the pump on.

If the pump is on, some similar set of rules must be used to determine when to turn the pump off. The action of the pump will affect the monitored values at the destination node, and some tolerance region or null region must be provided to prevent endless on-off cycling of the pump. The pump is turned off if the water-surface elevation at the inlet is at least 0.1 ft below the pump-inlet elevation given in F(4). If the monitored value at the destination node is elevation, then the pump is turned off if the water-surface elevation at the destination node is greater than the elevation given in F(1). If the monitored value at the destination node is flow rate, then the pump is turned off if the flow at the destination node is greater than the turn-on value plus twice the absolute value of the pumping rate. Finally, if N(10) > 0, then the pump is turned off if the flow at the node given in N(10) is greater than zero.

If TYPE=3 (Variable-speed variable-head pump):

Update available for two-way pumps and time-varying control blocks

N(5) is the flow direction: 1 means pumping from upstream node to downstream node; - 1 means pumping from downstream node to upstream node. The node from which water is being drawn is called the source node, and the node to which water is being pumped is called the destination node.

N(6) is the number of the table specifying the flow through the pump for each head; that is, a pump-performance curve. The speed of the pump used to define this curve is the base speed to which all speeds are relative. The base speed should be the highest speed for the pump so that all other speeds are smaller. For example, if a pump has a speed range from 0 to 1,800 revolutions per minute (rpm), then 1,800 rpm should be used to define the head across the pump. A relative speed of 0.5 means that the pump is operating at 900 rpm. The relation between flow through the pump and head must be unique. There are pumps for which this is not true; that is, pumps with ranges of flow for which the head increases as the flow increases followed by a decreasing head with increasing flow. Thus, for a given head, two flows are possible. Operation of pumps with nonunique head-discharge relations could be unstable if applied for pumping in an open-channel network. Operation of these pumps also could prove unstable in FEQ simulation. Therefore, to represent the pump performance in FEQ, the pump curve must be modified for pump operation stability when applied in an open-channel network.
N(7) is the number of the table specifying the sum of the entrance losses at the inlet, friction losses in the inlet conduit, and friction losses in the outlet conduit. The sum is expressed in terms of head as a function of flow through the pump. In simulation, losses are assumed to increase with flow. This table is optional and if omitted, the losses are assumed to be included in the pump-performance curve table given in the table specified in N(6).
N(8) is the number of the table specifying the exit-loss coefficient on the velocity-head difference between the end of the outlet conduit and the destination node. This coefficient is given as a function of the depth of submergence of the outlet conduit. The loss coefficient must be 1.0 at zero submergence. If this table is omitted, then an exit-loss coefficient of 1.0 is applied for all levels of submergence.
N(9): The number of the table specifying the pump speed as a function of time is given if N(9) < 0. The operation block number (see section 13.12) controlling the pump is given if N(9) > 0. The pump is kept running at the base speed all the time if N(9)=0. Water will not be pumped if the water-surface elevation at the source node is below the inlet elevation.
N(10) is an optional name for the pump. The name can be four alphanumeric characters at most. The first character should be alphabetic. If a pump is given a name, its state may be printed in the Special-Output File by specifying the name in the field as for a flow-path end node in the Special-Output Locations Block (section 13.9). In this case, the state of the pump includes the relative speed and the nature of flow. NO H2O denotes that the pump is on but that the inlet is above the water surface. FP denotes free flow out of the pump outlet. SP denotes submerged flow out of the pump outlet. OFF denotes that the pump is currently not running.
F(1) is the elevation of the outlet conduit and the point of reference for submergence of the outlet. If the water-surface elevation at the destination node is below this elevation, then a loss equal to the exit velocity head from the outlet conduit is assumed. If the water-surface elevation at the destination node is greater than this elevation, then a loss coefficient is determined from the table given in N(8) if this table is specified; otherwise, the loss coefficient is specified as 1.0. The loss coefficient is used in the computation of the fraction of the velocity-head difference between the conduit exit and the destination node that is an exit loss. The argument for the table is the depth of submergence computed relative to the elevation of the outlet conduit.
F(2) is the area of the outlet conduit when flowing full. The outlet conduit is assumed to be flowing full at all pump speeds.
F(3) is the elevation of the inlet conduit. Pump flow is zero if the water-surface elevation is below the inlet.
F(4) is the factor on the velocity head at the source node, usually 0 or 1 to exclude or include, respectively, the velocity head at the source node. This factor is specified as 0 if the source node is a free node.
F(5) is the factor on velocity head at the destination node, usually 0 or 1 to exclude or include, respectively, the velocity head at the destination node. This factor is specified as 0 if the destination node is a free node.

If TYPE=4 (Bridge with flow over the roadway):

N(5) is the number of the table specifying the bridge-loss coefficient as a function of water-surface height at the bridge opening for positive flow.

N(6) is the number of the table specifying the bridge-loss coefficient as a function of water-surface height at the bridge opening for negative flow.
N(7) is the number of the table specifying the area of bridge opening as a function of water-surface height at the bridge opening.
N(8) is the number of the table specifying flow over the roadway as a function of head for positive flow.
N(9) is the number of the table specifying flow over the roadway as a function of head for negative flow.
N(10) is the number of the table specifying the submergence effect as a function of head ratio for flow over the roadway.
F(1) is the maximum flow area through the bridge.
F(2) is the elevation of the high point of the bridge opening. Two options are available for this elevation. The first is the actual elevation of the high point of the bridge opening, and the second is a value that is at least 1 ft higher. If the true value is given, a submerged-flow equation is applied in simulation if the upstream end of the bridge becomes submerged. Otherwise, the free-flow equation is used with the free-flow loss adjusted to match closely the submerged-flow loss. This later option is applied to avoid flow discontinuities during the transition from free flow to submerged flow. These discontinuities can result in severe computational problems.
F(3) is the submerged-flow discharge coefficient for the bridge.
F(4) is the elevation for computing head on the roadway. This value can be set so large that no water will flow over the roadway. This should be done to simulate bridges for which flow over the roadway is impossible instead of using the TYPE=3 option.

If TYPE=5 (Abrupt expansion with inflow or outflow):

N(5) is the number of the table specifying the critical flow as a function of depth at the upstream node for the abrupt expansion. A table of type 2, type 4, type 22, or type 25 may be used. Types 22 or 25 are preferred, but types 2 and 4 are retained for consistency with earlier versions of FEQ.

Discussion: An abrupt expansion can be any feature that results in an abrupt increase in the area available for flow at all water-surface heights. Thus, a hydraulic drop is an abrupt expansion, as is a sudden increase in channel size. Critical depth is possible only at the upstream node. The user must supply a critical-flow table so that this condition can be detected. Inflow-outflow of water may result between the upstream and downstream nodes of the abrupt expansion. The direction of this flow is taken to be at right angles to the direction of flow from the upstream node to the downstream node. These features make it possible to represent structures used to divert water from the side of a channel. Frequently, an abrupt expansion is used to reduce the velocity of the water so that the performance of the side discharge is easier to evaluate and more efficient in diverting water.

The Type=5 option has several restrictions. First, the nodes must be on branches in their natural condition; that is, the upstream node for the abrupt expansion must be the downstream node on a branch upstream, and the downstream node for the abrupt expansion must be the upstream node for the branch downstream. Second, the discharge node for the abrupt expansion must be the upstream node of the abrupt expansion. Third, the flow cannot reverse at the discharge node. (Flow may reverse at the downstream node of the abrupt expansion.) Fourth, gravity and friction forces are ignored in the momentum balance.

If TYPE=6 (Two-dimensional tables):

N(5) is the number of the table specifying flow from the upstream node to the downstream node.

N(6) is the number of the table specifying flow from the downstream node to the upstream node.
N(7) is the number of an optional table specifying a multiplying factor to apply to the values derived from the tables given in N(5) and N(6). If left blank, the multiplying factor is taken to be 1.0.
N(8) is the number of an optional table specifying the elevation for computing heads as a function of time. If left blank, the elevation for computing heads remains fixed at the value given in F(1).
N(10) is the continuation field. If N(10) > 0, then the next line of input gives another set of values N(5) through F(1) so that more than one set of flow relations is available between the two nodes. The end of the input is defined by a value of N(10)=0. The number of sets of tables input is limited only by the memory allocated for the storage of the Network-Matrix Control Input.
F(1) is the elevation for computing heads.

With tables of type 14, a drop in water-surface elevation in the direction of flow is not required; however, only one flow path can be defined between a pair of flow-path end nodes. If more than one flow path is required, then additional branches or dummy branches must be added so that only one table exists for each flow path between nodes. This restriction results because type 14 tables include the downstream head and flow as arguments, and the upstream head is determined from these arguments. Thus, the values from these tables are not additive. Conversely, tables of type 6 and 13 include the upstream and downstream head as arguments, and flow values, which are additive, are determined from these arguments.

If TYPE=7 (Variable-height weir):

N(5): If N(5) > 0, it is the operation block number (see section 13.12); if N(5) < 0, it is the number of the table specifying the opening fraction as a function of time.

N(6) is the number of the table specifying the elevation of the weir crest as a function of the opening fraction.
N(7) is the number of the table specifying the weir coefficient for flow from the upstream node to the downstream node (positive flow) as a function of the opening fraction.
N(8) is the number of the table specifying the weir coefficient for flow from the downstream node to the upstream node (negative flow) as a function of the opening fraction.
N(9) is the number of the table specifying the submergence correction for flow over the weir.
N(10) is the optional name for the weir. The name can be four alphanumeric characters at most. The first character should be alphabetic. If the weir is given a name, its state may be printed in the
Special-Output File by specifying the name in the field as for a flow-path end node in the Special-Output Locations Block (section 13.9). In this case, the state of the weir includes the crest elevation and the nature of flow. NO FLOW denotes absence of flow over the weir. FW denotes flow over the weir free of downstream effects (free flow). SW denotes submerged flow over the weir.
F(1) is the weir length.
F(2) is the factor to apply to velocity head when computing the total head applicable to the weir equation.

If TYPE=8 (Sluice gates at Stratton Dam at McHenry, Ill.):

N(5): If N(5) > 0, it is the operation block number (see section 13.12); if N(5) < 0, it is the number of the table specifying the gate opening as a function of time.

F(1) is the sill elevation for the sluice gates (731.15 ft).

F(2) is the Maximum gate opening, in feet. A value of 8 ft is about the limit of the fitted relations representing the flow through the gates.

F(3) is the factor to apply to the flows to represent change in width from the standard width at McHenry of 68.75 ft (five gates at 13.75 ft each). The default value is 1.0 if left blank. If the gates simulated differ in width from those at Stratton Dam on the Fox River at McHenry, Ill., then the ratio of widths can be used for this factor. Adjustments for differing approach conditions also may have to be considered.

If TYPE=9 (Underflow gates tables):

N(5): If N(5) > 0, it is the operation block number (see section 13.12); if N(5) < 0, it is the number of the table specifying the gate opening as a function of time. This is the opening in the same units used for the maximum gate opening in F(2).

N(6) is the name for the gate. The name can be four alphanumeric characters at most. The first character should be alphabetic. If a name is given for the gate, its state may be printed in the Special-Output File by specifying the name in the field for a flow-path end node in the Special-Output Locations Block (section 13.9). In this case, the state of the gate includes the opening and the nature of flow. FW denotes free-weir flow. SW denotes submerged-weir flow. FO denotes free-orifice flow. SO denotes submerged-orifice flow. OR denotes orifice-flow conditions such that it cannot be determined in the table lookup process whether the flow is free or submerged, only that the flow is orifice flow. NO FLOW denotes that either the gate is closed or the water-surface level is below the gate opening.
N(7) is the number of a type 15 table for flow from the upstream node to the downstream node.
N(8) is the number of a type 15 table for flow from the downstream node to the upstream node.
F(1) is the sill elevation for the sluice gates.
F(2) is the maximum gate opening in feet. This value is multiplied by the fractional gate opening generated in an Operation of Control Structures Block (section 13.12) to convert to an actual gate opening for table lookup in the table given in N(7) or N(8).

N(1) is TYPE of the forced value. TYPE=1 if flow as a function of time is forced at the node, and TYPE=2 if elevation as a function of time is forced at the node.

N(2) is the number of the flow-path end node where forcing takes place.

N(3) is the direction of positive flow at the node. Direction is given as 1 if a positive value of flow from the flow source is into the stream system and - 1 if out of the system.

N(4) is the source for the forcing values. If N(4) > 0, then the source is the table given by the number in N(4). If N(4)=0, then the source is the constant value given in F(1). If N(4) < 0, then the source is the file referenced by the Fortran unit number (see appendix 3) specified by the absolute value of N(4).

F(1) is the value of constant flow if N(4)=0 and TYPE=1. This value is the minimum value if the source for the forcing values (N(4)) is nonzero. This minimum value applies to both flow and elevation and to values from a table and from a file. If applied with tidal data including values possibly less than zero, F(1) should be less than the minimum tide expected. Otherwise, tide levels below this value will be truncated.

F(2) is the multiplier on the value given by the source for flows and elevation. The value of F(2) is specified by the user. F(2) is typically used to check the sensitivity of FEQ output to source flows and elevations without making major changes in model input. If left blank, a default value of 1.0 is applied.

If CODE=7 (Level-pool reservoir):

N(1) is the node number of the reservoir.

N(2) is the number of the table specifying storage as a function of elevation for the reservoir.

N(3), the number of inflow nodes, must be 1. Every level-pool reservoir must have one inflow node.

N(4) is the inflow node number.

F(1) is the slope factor for making the elevation at the reservoir node and the inflow node slightly different. F(1) defaults to a value of 110-8 if left blank.

N(1) is the number of the branch-end node where cross-sectional hydraulic characteristics are defined.

Several restrictions apply to the option. The cross sections at both nodes must be identical and must be given by the same table number. Furthermore, the bottom elevations must be identical. There also should be a change in flow rate; otherwise, CODE=3 is a better option for this situation.

If CODE=12 (Match average elevation at two nodes):

N(1) is the upstream source node for averaging.
N(2) is the downstream source node for averaging.
N(3) is the node at which average elevation of nodes N(1) and N(2) will be forced.
F(1) is the weight to be used in computing the average. The weight is applied to N(1), and the
complement of the weight is applied to N(2). The weight, W, must satisfy 0 less than or equal to W less than or equal to 1.

If CODE=13 (Conservation of momentum or energy):

Update available for the explicit specification of one or two nodes for inflow

N(1) is the upstream node.
N(2) is the downstream node.
F(1) is the loss coefficient to apply to the energy equation. The coefficient is a factor applied to the change in velocity head that results when water is taken from the channel. The loss coefficient must be nonnegative and less than 1. The energy losses when water is discharged from a channel are generally small. Care must be applied in setting this value because computational failure can result if it is too large.

If CODE=14 (Side-weir flow):

N(1) is the upstream node on the source channel.
N(2) is the downstream node on the source channel.
N(3) is the node used to represent the outflow or inflow to the source channel.
N(4) is the flow table of type 13 defining the outflow over the side weir. This flow is computed as if the weir were a normal weir. Adjustments are made, on the basis of flow conditions, for the side-weir flow.
N(5) is a flow table of type 13 representing inflow over the side weir into the source channel. This flow is taken to be for a normal weir; no adjustment is made.
F(1) is the weight coefficient used in computing the average water-surface height, elevation, and flow rate in the source channel. The weight is applied to node N(1), and the complement of the weight is applied to node N(2). The value of the weight must satisfy 0 less than or equal to W less than or equal to 1.
F(2) is the elevation from which the heads included in the flow tables are measured.

Side weirs are a potential source of computational problems in FEQ simulation. If too much flow is diverted, then the water upstream from the weir in the source channel may approach critical depth or even become slightly supercritical. This may result in computational failure; most side weirs do not function well with a hydraulic jump at some point along the weir. Careful thought must be given to sizing the source channel and the weir so that supercritical flow will not result.

If CODE=15 (Dummy branch):

N(1) is the upstream node for the dummy branch.
N(2) is the downstream node for the dummy branch.
F(1) is the slope factor for assigning the elevation at the downstream and upstream nodes slightly different values. F(1) defaults to 110-8 if left blank.
F(2) is the surface area to use in routing flows through the dummy branch. F(2) defaults to 2.0 if left blank.

If CODE = - 1 (Input for network-matrix table is complete).

Variable: NAME1, BNODE

Format: A6, A5

Example: BNODE= U25

Explanation:

NAME1 is the identifying character string "BNODE=." This string must appear even if no boundary node is specified to initialize the matrix development. The name must be in upper case and must be given exactly as shown, beginning in column 1. Otherwise, a message will be issued that the boundary node for starting the network-matrix construction is missing.

BNODE is the boundary node to use for initiating the development of the network matrix. In order to develop the network matrix, a beginning flow-path end node on the boundary of the stream system is required in simulation. This must be a boundary node but must not be a boundary node at which elevation as a function of time is given. If elevation as a function of time is specified at all boundary nodes, then a dummy branch must be added to the model so that one of the other two boundary types can be specified. If the node field is left blank, then a matrix will be developed from each valid boundary node in the model and the node resulting in the smallest matrix length will be selected. It is best to leave the field blank and let the optimum starting boundary node be selected in simulation. Specifying this node will save only a fraction of a second of microcomputer time even when simulating a complex stream system. As the model is changed, the optimum boundary node also may change.