Pipe Network - Surface Flow Boundaries

Interactions between the pipe network and surface 1D channels, 2D areas, or storage areas fall into one of the three categories listed below.

Boundary Conditions

Users may define inflow, stage, or normal depth boundary conditions at pipe nodes. Because of assumptions in the computational geometry of the pipe network, specific types of boundary conditions are only allowed at certain node types. A graphical description of these node types and the allowable boundary conditions for each of them is given in Pipe Network - Geometry .

Flow

Prescribed boundary flows entering the pipe network at nodes are assumed to enter vertically through drop/side inlets or curb openings. As such, they enter the system as source flows without any initial momentum.

Stage

Stage boundaries are applied at External node types and set a hydraulic head (water surface elevation + pressure head) at the most downstream computational face of a conduit. The user-entered value may be overridden when it drops below a minimum depth in the pipe. This is done to prevent model instability. The minimum boundary depth is a function of the flow at the outlet and the slope of the pipe. It is calculated assuming the low elevation, user-defined stage boundary results in plunging flow at the outfall location (see Plunging Flow in Pipe Systems).

Pumping

At any node, users may define pump connections using the pipe network node as either the pump inlet or outlet. With this boundary condition water can be moved into, out of, or within the pipe system.

Flow through Top/Side Inlets

At each node cell where a top and/or side inlet is defined, water may move between the pipe network and the overlying surface water geometry. Flow into the pipe network is computed based on a family of rating curves developed to relate flow to water surface elevations in the pipe cell and the surface cell. The rating table is developed using both the weir and orifice equations.

Top Inlet

Top inlets are assumed to have horizontally oriented openings. Flow is determined using the following equations:

1)	$\begin{array}{l}\displaystyle Q_{\textrm{w}} = C_{\textrm{w}} \, L_{\textrm{in}} \Delta H^{3/2}\end{array}$

2)	$\begin{array}{l}\displaystyle Q_{\textrm{o}} = C_{\textrm{o}} \, A_{\textrm{in}} \sqrt{2g\Delta H}\end{array}$

$\begin{array}{l}\displaystyle Q_{\textrm{inlet}} = \textrm{sgn}(z_{\textrm{sfc}} - H_{\textrm{pipe}}) \min (Q_{\textrm{w}}, Q_{\textrm{o}})\end{array}$

where

$\begin{array}{l}Q_{\textrm{w}}\end{array}$ is weir flow [L³/T],
$\begin{array}{l}Q_{\textrm{o}}\end{array}$ is orifice flow [L³/T],
$\begin{array}{l}C_{\textrm{w}}\end{array}$ is the weir coefficient [-],
$\begin{array}{l}C_{\textrm{o}}\end{array}$ is the orifice coefficient [-],
$\begin{array}{l}L_{\textrm{in}}\end{array}$ is the inlet length [L],
$\begin{array}{l}A_{\textrm{in}}\end{array}$ is the inlet area, all specified for a given top inlet [L²],
$\begin{array}{l}z_{\textrm{sfc}}\end{array}$ is the water surface elevation of the surface water above the top inlet [L],
$\begin{array}{l}H_{\textrm{pipe}}\end{array}$ is the hydraulic head of the pipe node cell below the top inlet [L], and
$\begin{array}{l}\Delta H\end{array}$ is the head differential at the top inlet [L].

The head differential at the top inlet is calculated taking into account the pipe and surface water levels, as well as the top inlet crest elevation.

$\begin{array}{l}\displaystyle \Delta H = \max(z_{\textrm{sfc}}, H_{\textrm{pipe}}, z_{\textrm{crest}}) - \max(\min(z_{\textrm{sfc}}, H_{\textrm{pipe}}), z_{\textrm{crest}})\end{array}$

where $\begin{array}{l}z_{\textrm{crest}}\end{array}$ defines the minimum elevation at which flow can move between the pipe network and surface water elements. $\begin{array}{l}z_{\textrm{crest}}\end{array}$ is calculated as the maximum of the user-defined top inlet elevation and the minimum elevation of the surface computational element.

Side Inlet

Side inlets are assumed to have vertically oriented openings, which may be rectangular or circular in shape. Different methodologies are employed for each.

Rectangular side inlets use a computational approach similar to that described in Hydraulic Computations through Gated Spillways. When the headwater depth is less than the inlet crown, the weir flow equation 1) is used. When the ratio of the headwater depth to the height of the inlet opening is greater than 1.25, the orifice equation 2) is used. Between a headwater to depth ratio of 1.0 and 1.25, a transition region applies. Both flow values are computed and linear interpolation is used to transition between the weir flow result at 1.0 and the orifice flow result at 1.25.

One significant difference between side inlet and gated spillway computations involves tailwater submergence. For high tailwater submergence, gates use a table-based discharge reduction factor (see High Flow Computations Discharge Reduction). Side inlets instead use the Villemonte equation

3)	$\begin{array}{l}\displaystyle C_d = (1.0-SB^{1.5})^{0.385}\end{array}$

where $\begin{array}{l}C_d\end{array}$ is the discharge reduction factor and $\begin{array}{l}SB = \frac{\max \left(H_{ds} - z_{\textrm{crest}}, 0 \right)}{H_{us} - z_{\textrm{crest}}}\end{array}$ is the submergence factor, where $\begin{array}{l}H_{us}\end{array}$ and $\begin{array}{l}H_{ds}\end{array}$ are the upstream and downstream water surface elevation or hydraulic head values.

The discharge reduction factor is applied to flows below a ratio of headwater depth to inlet opening height of 1.25. Above this, the fully submerged orifice flow already incorporates the tailwater, if any, and is not reduced.

Circular side inlets use the standard orifice equation 2) for a headwater at or above the crown of the inlet. Below this, the semi-empirical method outlined in Guo and Stitt (2017) is used. The weir and orifice equations are normalized using the ratio of the headwater depth to the diameter of the inlet. Both a weir flow and a partially submerged orifice flow are calculated, and the smaller of the two flows is used.

$\begin{array}{l}Q_w^* = C_w \frac{2}{3} \sqrt{2} \left( \frac{y}{d} \right)^{1.87} \\ Q_o^* = C_o \sqrt{2} \frac{A_{\textrm{in}}}{d^2} \sqrt{\frac{y}{d}-\frac{Y_h}{d}}\end{array}$

where

$\begin{array}{l}Q_w^*\end{array}$ is normalized weir flow [-],
$\begin{array}{l}Q_o^*\end{array}$ is normalized orifice flow [-],
$\begin{array}{l}y\end{array}$ is the water depth above inlet invert [L],
$\begin{array}{l}d\end{array}$ is the diameter of the inlet [L], and
$\begin{array}{l}Y_h\end{array}$ is the depth from the water surface to the centroid of the flow area [L].

The normalized flow is minimum of the normalized weir flow and the normalized orifice flow

$\begin{array}{l}\displaystyle Q_{\textrm{inlet}}^* = \min(Q_w^*, Q_o^*)\end{array}$

and the unnormalized flow is computed

$\begin{array}{l}\displaystyle Q_{\textrm{inlet}} = Q_{\textrm{inlet}}^* d^{2.5} \sqrt{g}\end{array}$

If the tailwater is high enough to produce tailwater submergence, the Villemonte equation 3) is used to reduce the flow.

Rating Curves

Inlet rating curves are developed using the above equations and summing flow through the top and side inlet openings, if present.

An example set of ratings curves is shown below. Positive flow is defined as flow from the surface element to the pipe network. Negative flow indicates a surcharge situation, with flow from the pipe network to the surface area. Drop inlet flows enter the system as source flows without any initial momentum.

A limitation is placed on flow into the pipe network during conditions when the pipe network is flowing full. In these cases, the total flow is taken as the minimum of the rating table flow, $\begin{array}{l}Q_{\textrm{rt}}\end{array}$ , and the theoretical maximum inflow given the cross-sectional area of the conduits attached to the node, $\begin{array}{l}Q_{\textrm{Tor}}\end{array}$ .

$\begin{array}{l}\displaystyle Q_{\textrm{inlet}} = \min \left( Q_{\textrm{rt}}, Q_{\textrm{Tor}} \right)\end{array}$

The theoretical max flow is calculated using Torricelli's Law

$\begin{array}{l}\displaystyle Q_{\textrm{Tor}} = \sqrt{2g} \sum_i A_i^{\textrm{max}} \sqrt{\Delta H}\end{array}$

where $\begin{array}{l}A_i^{\textrm{max}}\end{array}$ is the maximum cross-sectional area of a conduit $\begin{array}{l}i\end{array}$ connected to the node.

Flow at Culvert Openings

For culvert opening type nodes, it is assumed that the conduit opens horizontally out to a modeled surface flow region.

Inflows to culverts are handled in a manner generally consistent with the existing Culvert Hydraulics computations and assumptions in HEC-RAS. Outlet control is assessed using the modeled hydraulic head immediately inside the conduit and the capacity of the conduit. Inlet control is computed assuming the flow at the inlet passes through critical, with entrance losses given by the user-specified entrance loss coefficient (instead of specifying inlet geometry chart and scale numbers). The minimum of the two flow controls is used to set the inflow.

One difference between the existing culvert calculations and pipe network computations occurs when an opening is partially buried. When the invert of a culvert is below the 2D cell invert (or storage area invert), a weir equation is used to limit flow into the pipe. This prevents the situation where a surface cell becomes wet and generates a high water surface slope between the surface cell and the pipe cell, which creates unrealistically large flows into the pipe and may drain the surface cell. In this case, the span of the pipe cross-section is used as the weir length, and the weir coefficient is set at 0.75, a typical value for a natural high ground barrier.

Culvert outflows are applied using stage boundary conditions. The stage at the culvert outflow is taken as the maximum of the surface cell water surface elevation and the free outfall depth, as described in Plunging Flow in Pipe Systems.

References

Guo, J.C.Y., Stitt, R.P. 2017. Flow through partially submerged orifice. Journal of Irrigation and Drainage Engineering 143(8).