# EXTRAN Theory - Dynamic Wave Solution

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**EXTRAN** (Dynamic Wave) is a hydraulic flow routing model for both open channel and closed conduits in dendritic and looped networks. EXTRAN receives hydrograph input at specific nodal locations by interface file from an upstream block (e.g., the Runoff Block) and/or by direct user input. The model performs dynamic routing of stormwater and sanitary flows throughout the major storm drainage system to the outfall points of the receiving water system. This method and the enhancements are documented here and are the default method in the Hydraulics mode and referred to in the software as the Dynamic Wave method.

Since the flow in sewers is usually non-uniform, turbulent, and subject to backwater and surcharge, a model is required that simulates all of the terms in the one dimensional dynamic flow equation. The EXTRAN Model will simulate branched or looped networks, backwater due to tidal or non-tidal conditions, free-surface flow, pressure or surcharge flow, flow reversals, flow transfer by weirs, orifices and pumping facilities, and pond or lake storage. Types of channels that can be simulated include circular, rectangular, horseshoe, eggshape, baskethandle pipes, trapezoidal, parabolic, natural (irregular) channels, circular and rectangular orifices, and arbitrary closed conduit shapes. Simulation output takes the form of water surface elevations and discharge at selected network locations.

## Introduction to EXTRAN

The specific function of EXTRAN is to route inlet hydrographs through the network of pipes, junctions, and flow diversion structures of the main storm water system to the treatment plant interceptors and receiving water outfalls.

Please note that the boundary between the Runoff (or TRANSPORT) and EXTRAN Blocks is dependent on the objectives of the simulation. EXTRAN must be used whenever it is important to represent severe backwater conditions and special flow devices such as weirs, orifices, pumps, storage basins, and tide gates. Normally, these conditions occur in the lower reaches of the drainage system when pipe diameters exceed roughly 20 inches (500 mm). The Runoff Block, on the other hand, is well suited for the simulation of overland and small pipe flow in the upper regions of the system where the nonlinear reservoir assumptions of uniform flow hold.

EXTRAN simulates the following elements: pipes, manholes (pipe junctions), weirs, orifices, pumps, storage basins, and outfall structures. These elements and their associated properties are summarized in Tables 14-1 and 14-2. Output from EXTRAN takes the form of:

- Discharge hydrographs and velocities in selected conduits in printed and plotted form, and
- Flow depths and water surface elevations at selected junctions in printed and plotted form. Hydrographs may be supplied to a subsequent block on the output interface file.

Storm sewers are usually designed to be dendritic or trellislike but over time, as more connections are built and the population grows larger, the sewer system may become a looped system. From a hydrodynamic viewpoint the flow in a storm sewer system is one of the most complicated and difficult problems [Pansic and Yen, 1981].

EXTRAN is based on the St. Venant equations for gradually varied one dimensional flow and handles dendritic and looped systems with backwater. However, some flow situations such as roll waves, bores, surges, supercritical flow and hydraulic jumps violate the assumption of gradually varied flow. EXTRAN allows the simulation of these exceptional flow situations by implementing the following procedures when they occur in the course of the simulation:

- The propensity for roll waves is tested by calculating the conduit Vedernikov number (French, 1986). Conduits with Vedernikov numbers < 1 are simulated using the kinematic wave equation.
- The simulation of supercritical flow in conduits is done by the kinematic wave equation. This is used when all the following conditions occur:
- Flow is positive
- Water surface slope is less than the conduit slope
- The flow calculated by the Manning equation is less thatn the flow calculated using dynamic flow equations.

- Surges or bores are simulated with some attenuation of the wave front using the full dynamic equation.
- A stationary hydraulic jump is simulated in EXTRAN through a combination of the kinematic wave equation for the supercritical conduit and the full dynamic equation for the critical conduit.

This version of EXTRAN is a combination of implicit and explicit finite difference formulations for solving the nodal continuity equation, combined conduit momentum and continuity equation, and the boundary conditions of the solved network. Briefly the difference between explicit and implicit methods of solution is as follows:

- An explicit method uses only known information to calculate the unknown value at the new time step, or y = f(x).
- An implicit method solves for the unknown value at the new time step based on known and unknown information, or f(x,y) = 0.

The nodal continuity equation is solved explicitly based on the last time step and value of the last iteration’s nodal depth and surface area. The boundary conditions are solved based on the last iteration value of flow in the outfall conduits. The flow in conduits is solved implicitly using the full dynamic flow equation. The kinematic wave flow is solved iteratively using the value of the last iteration’s upstream conduit cross sectional area and hydraulic radius.

EXTRAN can thus be characterized as a point iterative method that utilizes the stability of an implicit method and the flexibility of an explicit method. Some of the advantages and power of explicit methods are:

- An explicit method eliminates convoluted numerical techniques for modeling boundary conditions required with implicit methods (Wubs, 1987).
- The flexibility afforded by eliminating the need for matrix solutions required by fully implicit models allows the creation of specialized conduit and diversion structures.
- The speed benefit from an implicit method is on the order of 5 to 10 based on the time step size, but not considering the added computational burden of implicit over the explicit methods and the accuracy limitations of implicit methods, which require time steps 1 to 2 times the Courant number. Thus the computational speed of an explicit model is comparable to an implicit model considering the computational overhead.
- Implicit methods are limited by the travel time a disturbance would take to traverse the conduit's computational length.

## Conceptualization of the EXTRAN Drainage System

EXTRAN uses a link-node description of the sewer network or system. This description in EXTRAN facilitates both the finite difference representation of the physical prototype and the mathematical solution of the gradually-varied unsteady flow (St. Venant) equations which form the mathematical basis of the model.

The conduit system is idealized as a series of links or conduits which are connected at nodes or junctions. Links and nodes have well-defined properties which, taken together, permit representation of the entire pipe network. Moreover, the link-node concept is very useful in representing flow control devices. The specific properties of links and nodes are summarized in Table 14-2.

Links transmit flow from node to node. The primary dependent variable in the links is the discharge, Q, which is calculated at the center of the conduit. Properties associated with the links are roughness, length, cross-sectional area, hydraulic radius, conduit depth, and surface width. Velocity, hydraulic radius, and the cross-sectional area of flow, or depth, are variable in the link and computed at the upstream and downstream ends of the conduit.

The cross sectional area, surface width, and hydraulic radius are functions of the instantaneous depth of flow at the upstream and downstream ends of the conduit. The solution uses the average flow in each link during one iteration to calculate the cross sectional area, surface width, and hydraulic radius.

**CONDUIT TYPES**

- Circular
- Rectangular
- Egg-shaped
- Horseshoe
- Gothic
- Catenary
- Semi-elliptic
- Basket-handle
- Semi-circular
- Modified basket-handle
- Rectangular, triangular bottom
- Rectangular, round bottom
- Trapezoidal
- Parabolic
- Power function
- Irregular (natural)
- Horizontal ellipse
- Vertical ellipse
- Arch
- Bridge
- User-defined
- Rating curve

**JUNCTION TYPES**

- Manhole
- Point Junction (confluence)

**DIVERSION STRUCTURES**

- Circular Orifice
- Pumps rated by depth in node
- Pumps rated by well volume
- Pumps rated by dynamic head
- Variable speed pumps
- Pumps rated by static head
- Pumps controlled by telemetry
- Regulators
- Hydro-brakes
- Check valves
- Inflatable weirs
- Bendable weirs
- V-notch weirs
- Compound trapezoidal weirs
- Transverse weirs
- Sideflow weirs
- User-defined weirs
- Bottom outlet orifice
- Side outlet orifice
- Circular orifice
- Rectangular orifice
- Time history orifice
- Gated orifice
- Orifice modeled as equivalent pipe
- Orifice modeled using orifice equation

**STORAGE STRUCTURES**

- Constant area storage
- Power function storage
- Stepwise-linear storage
- Storage starts from node invert
- Storage starts from ground surface

**OUTFALL STRUCTURES**

- Free outfall
- Fixed backwater outfall
- Tidal coefficients outfall
- Low/high tidal outfall
- Stage history outfall
- Flow history outfall
- Rating curve outfall

**Types of Elements Available in EXTRAN**

Element | Constraint | Properties computed at each time-step | Constant Properties |
---|---|---|---|

Nodes | Sum Q = Change in Storage | Volume, Surface Area | Invert, Crown and Ground Elevations |

Conduits | Qin = Qout | for Cross-sectional Area, Hydraulic Radius, Upstream and Downstream ends of conduit Surface Width, Flow Velocity for Cross-sectional Area, Hydraulic Radius, Mid-point of conduit Surface Width, Conduit Flow | Loss Coefficients, Conduit Shape, Conduit Length, Slope and Roughness |

Nodes are the storage elements of the system and correspond to manholes or pipe junctions in the physical system. The variables associated with a node are the nodal head and surface area. The nodal volume is related internally by the program to depth and surface area. The volume of the node at any time is equivalent to the water volume in the half conduit lengths connected to the node. Similarly, the surface area of the node at any time is equivalent to half of the water surface area of the connecting conduits.

The primary node dependent variable is the head (H), or the node invert elevation plus water depth. The head is assumed to be changing over time but constant throughout any one node. (Note: A plot of head versus distance along the sewer network yields the hydraulic grade line, HGL, which can be plotted using the XSECT program invoked using Plot from the Extran menu).

Inflows, such as inlet hydrographs, and outflows, such as weir diversions, take place at the nodes of the network. The node volume changes over time due to surface inflow, and the balance of flow entering and leaving the conduit. This change in nodal volume during a given time step, Ðt, forms the basis of head and discharge calculations in EXTRAN. The derivation and the solution of these connected node and conduit equations are discussed in **Basic Equations of EXTRAN** .

Basic Equations of EXTRAN

The basic differential equations for solving sewer flow in EXTRAN and other open channel unsteady flow problems are derived from the gradually varied, one-dimensional, unsteady flow equations for open channels, otherwise known as the *St. Venant* or shallow water equations (Lai, 1986). They are non-linear hyperbolic partial differential equations and analytical solutions are unknown or unwieldy except in simplified situations. Numerical methods must be used to solve the equations since no general analytical solution exists. In addition to a numerical solution the equations require that upstream and downstream boundary conditions (BC) and initial conditions (IC) be defined by the user. The St. Venant equations are valid as long as the flow is a gradually varied one-dimensional flow; vertical acceleration is neglected; hydrostatic pressure; and the frictional resistance is the same as for steady flow.

This release of Extran has an updated solution to the gradually varied, one-dimensional unsteady flow equations for open channels. The solution is implicit for conduit flow and explicit for junction depth. The program solves for the new time step flow implicitly using a point iteration scheme and combines the best points of the EXTRAN Version 3 and EXTRAN Version 4 solutions. The solution is iterative since it uses the conduit values at the nth and n+1st time steps to calculate the new time flow and head differential equations.

**Partial Differential Equations (PDE) **

**Significant Differences Between EXTRAN Versions 3, 4 and 5 **

**Special Conduit Flow Conditions **

**Flow and Head Computation During Surcharge and Flooding **

**Flow Control or Diversion Devices**

**Storage Devices, Ponds and Lakes **** **

**Oscillations During a Hydraulic Jump**

**Kinematic and Diffusion Wave Equations**

**Special Finite Difference Approximations **** **

## SWMM Interface File Format

Documentation of the SWMM routines may be found in the EPA SWMM User’s Manual (Huber and Dickinson, 1988). The output is either in U.S. customary units or metric units depending on the value of parameter METRIC on data group B2.

Hydrographs stored on the SWMM interface file may be accessed through a program written by the user or by conversion to an ASCII/text file by the Combine Block. The structure of this file is described in Appendix B and in Section 2 of the EPA SWMM User’s Manual (Huber and Dickinson, 1988). This file structure must be followed if the user wishes to create an interface file containing input hydrographs generated by a program external to SWMM or if you wish to use another program to access the Interface File generated by SWMM.

The Interface File is a binary Unformatted FORTRAN file. Unfortunately the construction of the file header for binary files is not consistent across compilers and therefore a binary file created by one brand of compiler cannot be read by any other brand of compiler. XP-SWMM uses MICROSOFT (or Digital, or Compaq) VISUAL FORTRAN to generate its interface files and therefore any external program creating or reading an interface file must also be compiled using MICROSOFT (or Digital, or Compaq) VISUAL FORTRAN or MICROSOFT C (Or MICROSOFT VISUAL C++).

Subroutine INFACE.FOR reads or writes the header information for a SWMM 5.0 or EXTRAN 5.0 interface file. Note that the time step, Dt, or DELT is single not double precision.

**Sample Subroutine INFACE.FOR**Expand source

C======================================================================= SUBROUTINE INFACE(IDO,NTAPE) INCLUDE 'TAPES.INC' INCLUDE 'INTER.INC' CHARACTER KJN*10 DIMENSION JN(NIE),KJN(NIE) C======================================================================= C Read interface headers. C======================================================================= IF(IDO.EQ.0) THEN REWIND NTAPE READ(NTAPE,ERR=999) TITLE(1) READ(NTAPE,ERR=999) TITLE(2) READ(NTAPE,ERR=999) IDATEZ,TZERO READ(NTAPE,ERR=999) TITLE(3) READ(NTAPE,ERR=999) TITLE(4) READ(NTAPE,ERR=999) SOURCE,LOCATS,NQUAL,TRIBA READ(NTAPE,ERR=999)(KAN(I),I=1,LOCATS) IF(NQUAL.GT.0) THEN READ(NTAPE,ERR=999) (PNAME(J),J=1,NQUAL) READ(NTAPE,ERR=999) (PUNIT(J),J=1,NQUAL) READ(NTAPE,ERR=999) (NDIM(J),J=1,NQUAL) ENDIF READ(NTAPE,ERR=999) QCONV ENDIF C======================================================================= C Read and write interface headers. C======================================================================= IF(IDO.EQ.1) THEN REWIND NTAPE READ(NTAPE,ERR=999) TITLE(1) READ(NTAPE,ERR=999) TITLE(2) READ(NTAPE,ERR=999) IDATEZ,TZERO READ(NTAPE,ERR=999) TITLE(3) READ(NTAPE,ERR=999) TITLE(4) READ(NTAPE,ERR=999) SOURCE,LOCATS,NQUAL,TRIBA WRITE(N6,1) TITLE(1),TITLE(2) WRITE(N6,2) TITLE(3),TITLE(4) WRITE(N6,3) SOURCE WRITE(N6,4) IDATEZ,TZERO WRITE(N6,5) LOCATS,NQUAL,TRIBA C======================================================================= C Read sequence of location numbers. C======================================================================= READ(NTAPE,ERR=999) (KAN(I),I=1,LOCATS) WRITE(N6,66) (KAN(I),I=1,LOCATS) IF(NQUAL.GT.0) THEN READ(NTAPE,ERR=999) (PNAME(J),J=1,NQUAL) READ(NTAPE,ERR=999) (PUNIT(J),J=1,NQUAL) READ(NTAPE,ERR=999) (NDIM(J),J=1,NQUAL) WRITE(N6,7) (J,PNAME(J),PUNIT(J), + NDIM(J),J=1,NQUAL) ENDIF READ(NTAPE,ERR=999) QCONV WRITE(N6,8) QCONV ENDIF C======================================================================= C Write interface headers. C======================================================================= IF(IDO.EQ.2) THEN REWIND NTAPE READ(NTAPE,ERR=999) NEWOUT,NPOLL IF(JCE.EQ.0) READ(NTAPE,ERR=999) (JN(I),I=1,NEWOUT) IF(JCE.EQ.1) READ(NTAPE,ERR=999) (KJN(I),I=1,NEWOUT) REWIND NTAPE WRITE(NTAPE,ERR=998) TITLE(1) WRITE(NTAPE,ERR=998) TITLE(2) WRITE(NTAPE,ERR=998) IDATEZ,TZERO WRITE(NTAPE,ERR=998) TITLE(3) WRITE(NTAPE,ERR=998) TITLE(4) WRITE(NTAPE,ERR=998) SOURCE,NEWOUT,NPOLL,TRIBA,JCE IF(JCE.EQ.0) WRITE(NTAPE,ERR=998) (JN(I),I=1,NEWOUT) IF(JCE.EQ.1) WRITE(NTAPE,ERR=998) (KJN(I),I=1,NEWOUT) IF(NPOLL.GT.0) THEN WRITE(NTAPE,ERR=998) (PNAME(J),J=1,NPOLL) WRITE(NTAPE,ERR=998) (PUNIT(J),J=1,NPOLL) WRITE(NTAPE,ERR=998) (NDIM(J),J=1,NPOLL) ENDIF WRITE(NTAPE,ERR=998) QCONV ENDIF RETURN END

__Section Pages__

- Partial Differential Equations
- Finite Difference Equations
- Implicit Time Weighting
- Significant Differences between EXTRAN Versions 3, 4 and 5
- Finite Difference Solution
- Convergence in EXTRAN
- Automatic Time Step Selection
- Special Conduit Flow Conditions
- Flow and Head Computation during Surcharge and Flooding
- Ground and Invert Elevations
- Flow Control or Diversion Devices
- Storage Devices, Ponds, and Lakes
- Orifices
- Weirs
- Weirs with Tide Gates
- Pump Stations
- Outfall Structures
- Boundary Conditions
- Initial Conditions
- Pit or Junction Losses
- Oscillations during a Hydraulic Jump
- Irregular Closed Conduits
- Irregular Open Channels
- Kinematic and Diffusion Wave Equations
- Special Finite Difference Approximations
- Conduit Depth Calculations