Link Area Types in SWMM 5

 Link Area Types in SWMM 5

Purpose of Link Flow Classification:

• Dynamic Wave Routing: SWMM 5's dynamic wave routing solves the full St. Venant equations, which require calculating the cross-sectional area of flow at various points along each link.
• Area Allocation: The classification system determines how the calculated flow area at the midpoint of a link (aMid) is distributed between the upstream and downstream nodes for use in the node continuity equation.
• Continuity Equation: The node continuity equation relies on knowing the surface area at each node. The area allocation from connected links contributes to this surface area calculation.

The Seven Link Flow Classification Classes:

Here's a more detailed look at each class:

0. Dry Conduit:

• Description: The link has no flow (it's completely dry).
• Area Assignment: Half of the link's area (which is zero in this case) is assigned to both the upstream and downstream nodes. This effectively means no area contribution from this link to the nodes.
• Implication: Dry links don't contribute to the surface area at connected nodes.

1. Upstream End is Dry:

• Description: The upstream end of the link has zero depth, but there is flow in the downstream portion of the link.
• Area Assignment: Half of the link's calculated flow area (aMid) is assigned to the downstream node. The upstream node gets no area contribution from this link.
• Implication: The upstream node's surface area calculation only considers other connected links or the node's intrinsic area (if it's a storage node, for example).

2. Downstream End is Dry:

• Description: The downstream end of the link has zero depth, but there is flow in the upstream portion of the link.
• Area Assignment: Half of the link's calculated flow area (aMid) is assigned to the upstream node. The downstream node gets no area contribution.
• Implication: Similar to class 1, but the area contribution goes to the upstream node.

3. Sub-critical Flow:

• Description: The flow in the link is subcritical, meaning the Froude number is less than 1 throughout the link. This is the most common flow regime in gradually varied flow.
• Area Assignment: Half of the link's calculated flow area (aMid) is assigned to both the upstream and downstream nodes.
• Implication: This is a balanced distribution of area, reflecting the relatively uniform flow conditions in subcritical flow.

4. Super-critical Flow:

• Description: The flow in the link is supercritical, meaning the Froude number is greater than 1 throughout the link. This occurs in steep channels or during rapid flow conditions.
• Area Assignment: Even though the flow is supercritical, SWMM 5 still assigns half of the link's calculated flow area to both the upstream and downstream nodes as it does in the Sub-critical flow. This may seem counterintuitive, as supercritical flow is controlled by upstream conditions. However, this assignment is necessary for maintaining numerical stability in the iterative solution of the St. Venant equations.
• Implication: The same as Sub-critical flow. The SWMM5 engine treats them both the same. Super-critical flow in SWMM5 is used for reporting purposes and shown in the Link Classification Table as you have described.

5. Free-fall at Upstream End (Entry Loss Condition):

• Description: The flow enters the link with a free overfall condition at the upstream end, meaning that the upstream water level is not controlling the flow in the conduit. This implies a loss of hydraulic connection between the node and the conduit.
• Area Assignment: Half of the link's calculated flow area (aMid) is assigned to the downstream node. The upstream node does not get area from this link.
• Implication: The upstream node's water level doesn't directly influence the flow in this link under this condition.

6. Free-fall at Downstream End (Exit Loss Condition):

• Description: The flow exits the link with a free overfall condition at the downstream end, meaning that the downstream water level is not controlling the flow in the conduit. This implies a loss of hydraulic connection between the node and the conduit.
• Area Assignment: Half of the link's calculated flow area (aMid) is assigned to the upstream node. The downstream node does not get area from this link.
• Implication: The downstream node's water level doesn't directly influence the flow in this link under this condition.

Important Notes:

• Froude Number: The Froude number (Fr) is a dimensionless number that describes the flow regime:
  • Fr < 1: Subcritical flow (tranquil flow)
  • Fr = 1: Critical flow
  • Fr > 1: Supercritical flow (rapid flow)
• Dynamic Classification: The flow classification of a link can change during the simulation as flow conditions change.
• Numerical Stability: The area assignment rules, even for supercritical flow, are designed to maintain numerical stability in the iterative solution of the St. Venant equations.
• Link Classification Table: As you mentioned, the Link Classification Table in the SWMM 5 report file (.RPT) provides a summary of how many times each link was classified into each of the seven categories during the simulation. This table is valuable for understanding the flow behavior in your model.

In summary, the seven link flow classification classes in SWMM 5 are essential for accurately allocating the link's flow area to the upstream and downstream nodes during dynamic wave routing. These classifications, based on the flow conditions within each link, ensure that the node continuity equation is correctly applied, leading to more accurate and stable simulations.

Class	Description	Link Area Assignment
0      Dry conduit		1/2 Upstream and 1/2 Downstream Node
1      Upstream end is dry		1/2 Downstream Node
2      Downstream end is dry		1/2 Upstream Node
3      Sub-critical flow		1/2 Upstream and 1/2 Downstream Node
4      Super-critical flow		1/2 Upstream and 1/2 Downstream Node
5      Free-fall at upstream end		1/2 Downstream Node
6      Free-fall at downstream end		1/2 Upstream Node

Known and Unknown Variables in the Node Continuity Equation of SWMM5

 Known and Unknown Variables in the Node Continuity Equation of SWMM5

  Node Continuity Equation

The fundamental principle behind the node depth calculation is the conservation of mass (continuity):

• Change in Storage = Inflow - Outflow - Losses

In the context of a SWMM 5 node, this translates to:

• Change in Node Volume = (Sum of Inflows) - (Sum of Outflows) - (Losses due to Evaporation and Infiltration)

Equation Breakdown:

The equation you provided is a discretized form of the continuity equation, specifically using the average of the old and new inflows and outflows:

New Iteration Node Depth = Old Node Depth + [ ½ * (New Inflow – New Outflow) + ½ * (Old Inflow – Old Outflow) - Node Losses ] / New Surface Area * Time Step

• New Iteration Node Depth: The depth of water at the node at the current iteration (this is what we're trying to calculate). It is called the Iteration Node Depth because it will change on the next iteration until the Iteration Node Depth is within 0.005 feet of the previous Iteration Node Depth.
• Old Node Depth: The depth of water at the node at the previous time step.
• New Inflow: The sum of all inflows to the node at the current iteration. This includes external inflows (e.g., rainfall, user-defined inflows) and flows from connected links calculated in the current iteration.
• New Outflow: The sum of all outflows from the node at the current iteration. This includes flows to connected links calculated in the current iteration.
• Old Inflow: The sum of all inflows to the node at the previous time step.
• Old Outflow: The sum of all outflows from the node at the previous time step.
• Node Losses: The volume of water lost due to evaporation and infiltration at the current iteration.
• New Surface Area: The surface area of the node at the current iteration. This is calculated based on the node's geometry (e.g., depth-area curve for storage nodes, width and length for other nodes).
• Time Step: The simulation time step.

Known and Unknown Variables:

At the start of each time-step iteration:

• Known:

  • Old Node Depth
  • Old Inflow
  • Old Outflow
  • Time Step

• Calculated during the current iteration but based on values from the previous iteration:

  • New Inflow (Based on link flows from the previous iteration)
  • New Outflow (Based on link flows from the previous iteration)
  • New Surface Area (Based on the Iteration Node Depth from the previous iteration or on the first iteration it is based on the Old Node Depth)
  • Node Losses (Based on the Iteration Node Depth from the previous iteration or on the first iteration it is based on the Old Node Depth)

• Unknown:

  • New Iteration Node Depth (This is what the equation solves for). On the 2nd to 8th iteration the New Node Depth is found by averaging the New Iteration Node Depth and the Old Iteration Node Depth.

Iterative Process:

SWMM 5 uses an iterative process to solve for the New Node Depth because the New Inflow, New Outflow, and New Surface Area are dependent on the New Node Depth itself. This creates a circular dependency that requires iteration.

1. First Iteration:

   • An initial estimate for New Iteration Node Depth is made. It is based on the New Iteration Node Depth equation and uses the Old Node Depth, the average of the New and Old Inflow, the average of the New and Old Outflow, the Time Step, the Node Losses and the New Surface Area.
   • Based on this New Iteration Node Depth, the New Inflow, New Outflow, and New Surface Area are updated. The New Node Depth is set equal to the New Iteration Node Depth.

2. Second to Eighth Iterations:

   • The New Iteration Node Depth equation is used to calculate a new New Iteration Node Depth.
   • The New Inflow, New Outflow, and New Surface Area are updated based on the New Iteration Node Depth.
   • The New Node Depth is set equal to the average of the current New Iteration Node Depth and the previous iteration New Node Depth. This averaging helps to stabilize the solution and prevent oscillations.

3. Convergence Check:

   • After each iteration, the New Iteration Node Depth is compared to the New Node Depth from the previous iteration.
   • If the absolute difference is less than a tolerance (0.005 feet in your case), the solution is considered to have converged, and the iteration stops.
   • If the difference is greater than the tolerance, the iteration continues, up to a maximum of 8 iterations (in SWMM 5.0.020).

Why Iteration is Necessary:

• Nonlinearity: The relationships between node depth, inflow, outflow, and surface area are often nonlinear. For example, the flow through a link might be related to the square root of the head difference between two nodes.
• Interdependence: The flow in each link depends on the depths at the connected nodes, and the depths at the nodes depend on the flows in the connected links.

Benefits of this Approach:

• Accuracy: The iterative approach allows for a more accurate solution of the node continuity equation, especially in situations with rapidly changing flow conditions.
• Stability: The averaging of the New Node Depth in the second to eighth iterations helps to improve the stability of the numerical solution and prevent oscillations.