How is the St Venant Equation Solved for in the Dynamic Wave Solution of SWMM 5?

Subject:   How is the St Venant Equation Solved for in the Dynamic Wave Solution of SWMM 5?

How is the St Venant Equation Solved for in the Dynamic Wave Solution of SWMM 5? by dickinsonre Subject:   How is the St Venant Equation Solved for in the Dynamic Wave Solution of SWMM 5? An explanation of the four St. Venant Terms in SWMM 5 and how they change for Gravity Mains and Force Mains. The HGL is the water surface elevation in the upstream and downstream nodes of the link. The HGL for a full link goes from the pipe crown elevation up to the rim elevation of the node + the surcharge depth of the node.  The four terms are: dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) / Link Length or dq2 = Time Step * Awtd * (HGL) / Link Length Qnew = (Qold – dq2 + dq3 + dq4) / ( 1 + dq1) when the force main is full dq3 and dq4 are zero and Qnew = (Qold – dq2) / ( 1 + dq1) The dq4 term in dynamic.c uses the area upstream (a1) and area downstream (a2), the midpoint velocity, the sigma factor (a function of the link Froude number), the link length and the time step or dq4 = Time Step * Velocity * Velocity * (a2 – a1) / Link Length * Sigma the dq3 term in dynamic.c uses the current midpoint area (a function of the midpoint depth), the sigma factor and the midpoint velocity dq3 = 2 * Velocity * ( Amid(current iteration) – Amid (last time step) * Sigma dq1 = Time Step * RoughFactor / Rwtd^1.333 * |Velocity| The weighted area (Awtd) is used in the dq2 term of the St. Venant equation: dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) / Link Length The four terms change at each iteration and time step to determine the new flow (Figure 1) based on the two equations: Denom = 1 + dq1 + dq5 Q = [Qold – dq2 + dq3 + dq4] / Denom If you look at a table of the values you will see that the terms add up to zero when the flow is constant and to delta Q or the change in Q when the flow is NOT constant (Figure 2). Figure 1.  The four terms define the new flow at each iteration in the dynamic wave solution of SWMM5 Figure 2.   The magnitude of the four terms determine the flow at the new iteration and ultimately the new Time Step.  If the flow is constant then the value of the term is constant. via Blogger http://www.swmm5.net/2013/08/how-is-st-venant-equation-solved-for-in.htm

An Example of the Importance of the Term DQ4 in the SWMM 5 St Venant Solution

Subject:   An Example of the Importance of the Term DQ4 in the SWMM 5 St Venant Solution

An Example of the Importance of the Term DQ4 in the SWMM 5 St Venant Solution by dickinsonre Subject:   An Example of the Importance of the Term DQ4 in the SWMM 5 St Venant Solution The four terms are are used in the new flow for a time step of Qnew: Qnew = (Qold – dq2 + dq3 + dq4) / ( 1 + dq1) when the force main or gravity main is full dq3 and dq4 are zero and  Qnew = (Qold – dq2) / ( 1 + dq1) The dq4 term in dynamic.c uses the area upstream (a1) and area downstream (a2), the midpoint velocity, the sigma factor (a function of the link Froude number), the link length and the time step or dq4 = Time Step * Velocity * Velocity * (a2 – a1) / Link Length * Sigma where Sigma is a function of the Froude Number and the Keep, Dampen and Ignore Inertial Term Options.  Keep sets Sigma to 1 always and Dampen set Sigma based on the Froude number, Ignore sets Sigma to 0 all  of the time during the simulation. The value of dq4 increases when there is a significant difference in the cross sectional  area of the downstream end of the link and the upstream end of the link.  In this  example, the downstream storage node causes a backflow in the link.   The flow may look unstable in the link  flow time series but the change in flow is simply due to the water sloshing back and forth.  There is not continuity error as the term dq4 keeps the water in the link  in balance. via Blogger http://www.swmm5.net/2013/08/an-example-of-importance-of-term-dq4-in.html

Keep and Dampen options and their effect on the four main terms of the St Venant equation

Note:  The Keep and Dampen options and their effect on the four main terms of the St Venant equation. 

The Keep and Dampen options and their effect on the four main terms of the St Venant equation in SWMM5 by dickinsonre Note:  The Keep and Dampen options and their effect on the four main terms of the St Venant equation.  The four terms are are used in the new flow for a time step of Qnew: Qnew = (Qold – dq2 + dq3 + dq4) / ( 1 + dq1) when the force main or gravity main is full dq3 and dq4 are zero and  Qnew = (Qold – dq2) / ( 1 + dq1) The dq4 term in dynamic.c uses the area upstream (a1) and area downstream (a2), the midpoint velocity, the sigma factor (a function of the link Froude number), the link length and the time step or dq4 = Time Step * Velocity * Velocity * (a2 – a1) / Link Length * Sigma where Sigma is a function of the Froude Number and the Keep, Dampen and Ignore Inertial Term Options.  Keep sets Sigma to 1 always and Dampen set Sigma based on the Froude number, Ignore sets Sigma to 0 all  of the time during the simulation the dq3 term in dynamic.c uses the current midpoint area (a function of the midpoint depth), the sigma factor and the midpoint velocity. dq3 = 2 * Velocity * ( Amid(current iteration) – Amid (last time step) * Sigma dq1 = Time Step * RoughFactor / Rwtd^1.333 * |Velocity| The weighted area (Awtd) is used in the dq2 term of the St. Venant equation: dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) / Link Length or dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) / Link Length Normally, dq1 (Friction Loss / Maroon in the Graph) balances dq2 (Water Surface Slope Term or Green in the Graph) but often for links with a large difference between upstream and  downstream depths dq4 (Red in the Graph) can have a significant value.  If dq4 or dq3 are important then the depth of water to increases to pass the same flow using the Keep option over the Ignore.   If you have a link with a Froude number near or over 1.0 (Supercritical) then using Keep or Dampen  for the Options may result in depth differences.   The effect of Keep is to increase the "loss" terms in the St VenantEquation.   The effect of Dampen and Ignore is to decrease the sum of the "loss" terms in the St. Venant Solution and lower the simulated depth. via Blogger http://www.swmm5.net/2013/08/the-keep-and-dampen-options-and-their.html

Surcharged Node and the Link Connection in SWMM 5

Subject:   Surcharged Node and the Link Connection in SWMM 5

Surcharged Node and the Link Connection in SWMM 5 by dickinsonre Subject:   Surcharged Node and the Link Connection in SWMM 5  A surcharged node in SWMM 5 uses this point iteration equation (Figure 1):  dY/dt = dQ / The sum of the Connecting Link values of  dQ/dH  where Y is the depth in the node, dt is the time step, H is the head across the link (downstream – upstream), dQ is the net inflow into the node and dQ/dHis the derivative with respect to H of the link  St Venant equation.  If you are trying to calibrate the surcharged node depth, the main calibration variables are the time step and the link  roughness:  1.   Mannings's N 2.   Hazen-Williams or 3.   Darcy-Weisbach  The link roughness is part of the term dq1 in the St Venant solution and the other loss terms are included in the term dq5.  You can adjust the roughness of the surcharged link  to affect the node surcharge depth.  Figure 1.  The Node Surcharge Equation is a function of the net inflow and the sum of the term dQ/dH in all connecting links. Generally, as you increase the roughness the value of dQ/dH increases and the denominator of the term dY/dt = dQ/dQdH increases.   Figure 2.  The value of dQ/dH in a link as the roughness of the link increases. via Blogger http://www.swmm5.net/2013/08/surcharged-node-and-link-connection-in.html

How to Search These Blogs for Information

Note:  How to Search These Blogs for Information

How to Search These Blogs for Information about SWMM5, InfoSWMM or InfoSewer by dickinsonre Note:  How to Search These Blogs for Information  In each of the blogs search  for a term or a set of terms using the search button.   For example, here is http://swmm5.blogspot.com with a search for venant   An equivalent Search in http://www.swmm2000.com    An equivalent Search in http://swmm5.wordpress.com/    via Blogger http://www.swmm5.net/2013/08/how-to-search-these-blogs-for.html dickinsonre | August 4, 2013 at 7:28 am | Tags: Blogger, H2oMAP SWMM, IFTTT, InfoSWMM,swmm5 | Categories: H2OMAP SWMM, InfoSWMM, swmm5 | URL: http://wp.me/pnGa9-2wI

SWMM 5 Loss Term Values for various velocities and K values

Subject:   SWMM 5 Loss Term Values for various velocities and K values

SWMM 5 has three loss terms available for each link:  Entrance, Exit and Other losses.  The Entrance loss uses the upstream link velocity, the  Other loss uses the center link velocity and the Exit loss uses the downstream link velocity.  The general form of the loss term in the St. Venant equation is K*V^2/2g Table 1 shows the loss in feet of head for various combinations of velocity and K value.  If you want to  simulate a little loss of head at each node then a small value of K should be used otherwise the cumulative loss in the whole networks will be many feet of head.

  Loss Term units equals K * V^2/2g = ft/sec * ft/sec * sec^2/ft = ft

Table 1:  Loss in feet of head for various combinations of velocity and K values.

Velocity (ft/sec)	K	K	K	K	K	K
	0.050	0.100	0.250	0.500	0.750	1.000
1	0.001	0.002	0.004	0.008	0.012	0.016
2	0.003	0.006	0.016	0.031	0.047	0.062
3	0.007	0.014	0.035	0.070	0.105	0.140
4	0.012	0.025	0.062	0.124	0.186	0.248
5	0.019	0.039	0.097	0.194	0.291	0.388
6	0.028	0.056	0.140	0.280	0.419	0.559
7	0.038	0.076	0.190	0.380	0.571	0.761
8	0.050	0.099	0.248	0.497	0.745	0.994
8	0.050	0.099	0.248	0.497	0.745	0.994
9	0.063	0.126	0.314	0.629	0.943	1.258
10	0.078	0.155	0.388	0.776	1.165	1.553

SWMM 5 Culvert Data from FHWA, HDS No. 5, Hydraulic Design of Highway Culverts, 1985

Subject:  SWMM 5 Culvert Data from FHWA, HDS No. 5, Hydraulic Design of Highway Culverts, 1985

SWMM 5 Culvert Data from FHWA, HDS No. 5, Hydraulic Design of Highway Culverts, 1985 by dickinsonre Subject:  SWMM 5 Culvert Data from FHWA, HDS No. 5, Hydraulic Design of Highway Culverts, 1985 If you use the culvert option in later versions of SWMM 5 then when the inlet control equation flow is less than the computed St Venant flow then the FHWA equations will be used for the current iteration in the SWMM 5 Dynamic Wave Solution. via Blogger http://www.swmm5.net/2013/08/swmm-5-culvert-data-from-fhwa-hds-no-5.html

All Possible Culverts Example Model in SWMM5

Note:  Attached is an example SWMM 5 model that has all 57 culvert types possible in SWMM 5 in one model.  The culverts are 57 small individual networks consisting of an inflow node, an upstream open channel, upstream node for the culvert, culvert link with culvert code, downstream node of the culvert, downstream open channel and finally an outfall node.  The culvert code and the shape of the culvert determine which FHWA equation is used to determine the flow INTO the Culvert during the simulation:

 1.   The flow from the St Venant Equation or
2.   The flow from the FHWA equation
 The minimum flow is used by the program. 

all_culverts.inp Download this file

All Possible Culverts Example Model in SWMM5 by dickinsonre Note:  Attached is an example SWMM 5 model that has all 57 culvert types possible in SWMM 5 in one model.  The culverts are 57 small individual networks consisting of an inflow node, an upstream open channel, upstream node for the culvert, culvert link with culvert code, downstream node of the culvert, downstream open channel and finally an outfall node.  The culvert code and the shape of the culvert determine which FHWA equation is used to determine the flow INTO the Culvert during the simulation: 1.   The flow from the St Venant Equation or 2.   The flow from the FHWA equation The minimum flow is used by the program.  via Blogger http://www.swmm5.net/2013/08/all-possible-culverts-example-model-in.html

Link Simulated Parameters used in either the Normal Flow or St Venant Equation of SWMM 5

Subject:  Link Simulated Parameters used in either the Normal Flow or St Venant Equation of SWMM 5

St. Venant equation – this is the link attribute data used when the St. Venant Equation is used in SWMM 5.  Simulated Parameters from the upstream, midpoint and downstream sections of the link are used.

Normal Flow Equation – this is the link attribute data used when the Normal Flow Equation is used in SWMM 5. Only simulated parameters from the upstream end of the link are used if the normal flow equation is used for the time step.

Link Simulated Parameters used in either the Normal Flow or St Venant Equation of SWMM 5 by dickinsonre Subject:  Link Simulated Parameters used in either the Normal Flow or St Venant Equation of SWMM 5 St. Venant equation – this is the link attribute data used when the St. Venant Equation is used in SWMM 5.  Simulated Parameters from the upstream, midpoint and downstream sections of the link are used. Normal Flow Equation – this is the link attribute data used when the Normal Flow Equation is used in SWMM 5. Only simulated parameters from the upstream end of the link areused if the normal flow equation is used for the time step. via Blogger http://www.swmm5.net/2013/08/link-simulated-parameters-used-in.html

Non Linear Term in the Saint Venant Equation of SWMM 5

Non Linear Term in the Saint Venant Equation of SWMM 5The flow equation has six components that have to be in balance at each time step:

1. The unsteady flow term or dQ/dt

2. The friction loss term (normally based on Manning's equation except for full force mains),

3. The bed slope term or dz/dx

4. The water surface slope term or dy/dx,

5. The non linear term or d(Q^2/A)/dx and

6. The entrance, exit and other loss terms.

All of these terms have to add up to zero at each time step. If the water surface slope becomes zero or negative then the only way the equation can be balanced is for the flow to decrease. If the spike is due to a change in the downstream head versus the upstream head then typically you will a dip in the flow graph as the water surface slope term becomes flat or negative, followed by a rise in the flow as the upstream head increases versus the downstream head.

You get more than the normal flow based on the head difference because in addition to the head difference you also get a push from the non linear terms or dq3 and dq4 in this graph.

If you use the Keep Inertial term option then you use all 5 terms at all times, if you use the Dampen option you use all five terms until the Froude number of the link greater than 1 and then you do not use terms dq3 and dq4, if you use the Ignore option then the dq3 and dq4 terms are not used at all for the calculation of the flow in the link.

Hysteresis Effect in the Link Flow versus Depth Relationship in SWMM 5

Subject: Hysteresis Effect in the Link Flow versus Depth Relationship in SWMM 5
You can often get a hysteresis effect for the Flow versus Depth relationship in SWMM 5 due to the five component St. Venant equation used to simulate the flows (http://swmm5.blogspot.com/2010/12/what-are-units-for-five-st-venant-flow.html) . A hysteresis effect is having two or more flow values for the same depth value in the link. For example, this image shows how the link 8100 has a different flow for the same depth in the rising and falling limb’s of the hydrograph. This is due to the different values for the upstream and downstream head, hydraulic radius and cross sectional area during the falling and rising hydrograph, respectively.

What are the Units for the five St. Venant Flow Terms in SWMM 5 and InfoSWMM?

Subject: What are the Units for the five St. Venant Flow Terms in SWMM 5 and InfoSWMM?

This is how the flow is calculated in a link in InfoSWMM.  It uses the
 ·         Upstream and downstream head,
·         The user input length,
·         The weighted cross sectional area and hydraulic radius as I explained in the previous email,
·         The Center velocity,
·         The Center Cross sectional area, and
·         The Upstream and Downstream Cross sectional area.

The slope as listed in the output file is more for reference and is actually not used in the St. Venant Solution.   The way the program usually works is that the friction slope lags the water surface head slope with the difference made up by the change in flow.  The two non linear terms are usually small and only affect the flow during reverse or backwater events.

The new flow (Q) calculated at during each iteration of time step as

(1)Q for the new iteration = (Q at the Old Time Step – DQ2 + DQ3 + DQ4 ) / ( 1.0 + DQ1 + DQ5)
In which DQ2, DQ3 and DQ4 all have units of flow (note internally SWMM 5 has units of CFS and the flows are converted to the user units in the output file, graphs and tables of SWMM 5).

The equations and units for DQ2, DQ3 and DQ4 are:

(2)Units of DQ2 = DT * GRAVITY * aWtd * ( H2 – H1) / Length = second * feet/second^2 * feet^2 * feet / feet = feet^3/second = CFS

(3)Units of DQ3 = 2 * Velocity * ( aMid – aOld) * Sigma = feet/second * feet^2 = feet^3/second = CFS

(4)Units of DQ4 = DT * Velocity * Velocity * ( aDownstream – aUpstream) * Sigma / Length = second * feet/second * feet/second * feet^2 / feet = feet^3/second = CFS


The equations and units for DQ1 and DQ5 are:

(5)Units of DQ1 = DT * GRAVITY * (n/PHI)^2 * Velocity / Hydraulic Radius^1.333 = second * feet/second^2 * second^2 * feet^1/3 * feet/second / feet^1.33 = Dimensionless

(6)Units of DQ5 = K * Q / Area / 2 / Length * DT = feet^3/second * 1/feet^2 * 1/feet * second = Dimensionless

The five components calculated at the each time step and at each iteration during a time step and together predict the new Link Flow (Q) in SWMM 5. The value of the different components can be seen over time in Figure 1 and as a component percentage in Figure 2 and 3.

Figure 1: The Five St. Venant Components over time.


Figure 2: The relative magnitude of the St Venant terms over time for the same for the same link as in Figure 1.

Figure 3: The relative magnitude of the St Venant terms over time for the same for the same link as in Figure 1 shown in an area chart normalized to 100 percent. Normally the DQ1 and DQ2 terms balance each other except for backwater conditions or reverse flow in which the terms DQ3 and DQ4 can dominate.