Fourteen inches or twenty-eight: three riprap methods at the end of one barrel

Last issue ended at the outfall of a 24-inch barrel running full at 13.4 ft/s. I promised three riprap sizing methods that disagree about what that jet needs. They do, by a factor of two. But two of the three turn out to be the same equation, and most of the disagreement comes down to one input that's usually guessed: the tailwater.

The jet

The outlet is the one from issue 010: a 4 ft × 4 ft riser draining through 80 ft of 24-inch RCP at 1 percent onto a free outfall. At pond stage 107 the barrel is in control and passes 42.2 cfs. Flowing full, that's

$$V = \frac{Q}{A} = \frac{42.2}{\pi (2.0)^2/4} = 13.4\ \text{ft/s}$$

That's fast enough to pull apart most soils within a few storms. The question is how big the stone at the outfall must be.

Method 1: the HEC-14 apron equation

FHWA HEC-14 (Hydraulic Design of Energy Dissipators for Culverts and Channels, 3rd ed.) sizes a riprap apron at a circular outlet with

$$D_{50} = 0.2\,D\left(\frac{Q}{\sqrt{g}\,D^{2.5}}\right)^{4/3}\frac{D}{TW}$$

with D the barrel diameter and TW the tailwater depth. When the tailwater is unknown or below 0.4D, the manuals that adopt it use TW = 0.4D. A free outfall has no tailwater to speak of, so TW = 0.8 ft:

$$D_{50} = 0.2(2.0)\left(\frac{42.2}{5.67 \times 5.66}\right)^{4/3}\frac{2.0}{0.8} = 1.44\ \text{ft} = 17.3\ \text{in}$$

Method 2: the "0.02 / TW" formula

Several state erosion-control standards (New Jersey's and DelDOT's, for two) write the apron equation as

$$d_{50} = \frac{0.02}{TW}\left(\frac{Q}{D}\right)^{4/3}\qquad \text{(ft, cfs)}$$

which gives 0.02/0.8 × (21.1)^4/3 = 1.46 ft = 17.5 in. Expand the HEC-14 form and you get 0.2/g^2/3 × (Q/D)^4/3/TW. With g = 32.2, the coefficient is 0.0198. So methods 1 and 2 are the same equation: one keeps the units in the formula and the other folds them into the coefficient. The 0.2 in-difference is just 0.02 against 0.0198.

That's worth knowing when a reviewer asks why your calc doesn't follow the local manual. If the manual uses the 0.02 form, it agrees with HEC-14.

Method 3: Isbash on the jet velocity

Isbash's relation sizes a stone that a stream of given velocity won't move:

$$D = \frac{V^2}{2g\,C^2\,(S_s - 1)}$$

with S_s = 2.65 for typical stone. It comes with two coefficients: C = 1.20 for stones embedded in a bed with low turbulence, and C = 0.86 for exposed stones in high turbulence. Here D is read as D₅₀, as most manuals do.

The jet coming out of a barrel onto a flat apron is close to the definition of high turbulence, at least for the first few diameters. If you take Isbash at face value, the 0.86 coefficient governs and the stone is nearly twice the HEC-14 size.

Why they disagree

The apron equation is empirical. It was fitted to laboratory tests of riprap aprons downstream of culverts, so the spreading of the jet over the apron is built into the coefficient. Isbash knows nothing about spreading: it takes the full outlet velocity as the velocity on the stone. For the first stones at the barrel lip that's roughly right. Two or three diameters downstream it's conservative. So the two methods answer different questions. HEC-14 sizes the apron as a whole, and Isbash with C = 0.86 sizes the stone right at the jet.

The bigger lever is the tailwater, which sits in the denominator of the apron equation:

Going from no tailwater to a full pipe's depth of it cuts the stone by a factor of 2.5. On a real job the tailwater is the normal depth in the receiving channel at the same flow, and it can be computed. A channel with that normal depth needs about 7-in stone. The same outfall dumping onto a flat field needs 17-in stone, or 28-in by Isbash. That's the actual design decision. The choice of manual barely matters next to it.

What I'd put on the plans

• Compute the tailwater. Use normal depth in the receiving channel at the barrel flow, or 0.4D if there's no channel. Don't assume a convenient value.
• Size the apron with HEC-14 (or the local 0.02 form, which gives the same answer), then check the stone at the lip with Isbash. If the jet is high-velocity and the tailwater is low, as here, a plunge pool or a riprap basin (also in HEC-14) is usually a better answer than a wider apron of very large stone.
• Size it at the right flow. The barrel caps the flow at 42 cfs at stage 107 and 35 cfs at the handoff stage. Size the stone for the highest flow the barrel can deliver before the emergency spillway takes over. Don't use the riser rating, which, as issue 010 showed, overstates what the pipe can carry.
• Don't skip the filter. Stone of any size placed directly on soil loses fines through the voids and settles into the hole it was meant to prevent.

Running it yourself

• Free: the riprap sizing calculator on pe-calc (Isbash and USACE EM 1110-2-1601) and the culvert calculator for the barrel flow and outlet velocity.
• The whole outlet: HydroComplete's culvert outlet rates the barrel with HDS-5 inlet and outlet control, so the riprap check starts from the flow the barrel actually delivers rather than the riser rating.

— Michael Flynn, PE