What is the Kirpich formula?

Kirpich (1940) gives Tc in minutes as 0.0078 k (L / square root of S) to the power 0.77, with L the flow length in feet and S the slope in ft/ft. It was fitted to seven small agricultural watersheds in Tennessee of 0.4 to 45 ha with defined channels. Bare soil and natural channels take k = 1.0, grassed overland flow 2.0, paved overland flow 0.4 and concrete-lined channels 0.2. A 1 200 m path at 2 % gives 20.6 minutes.

Which time of concentration method should I use?

Match the method to the dominant flow regime. Use Kirpich or the USBR equation for natural catchments with a defined channel, NRCS lag for rural catchments under about 8 km² with a reliable curve number, FAA for small paved areas where C is known, and Kerby only for the overland part of the path up to 365 m. Use the FHWA segmental method when you can map where the flow changes character. Compute at least two and record why you chose one.

Why do Tc methods give such different answers?

Each equation was fitted to a different kind of catchment. In the worked example on this page one 1 200 m flow path at 2 % gives 21 minutes with Kirpich and 78 minutes with NRCS lag. Kirpich assumes most of the path is a channel; the NRCS lag and FAA equations treat it as overland flow. A spread of a factor of two or more usually means at least one method is outside its calibration range.

What is the maximum length for sheet flow?

In the FHWA and TR-55 segmental method, sheet flow is limited to about 90 m (300 ft), and the calculator accepts up to 100 m. Kerby's overland equation applies up to 365 m (1 200 ft). Beyond those lengths water gathers in rills, swales and gutters, so the rest of the path should be computed as shallow concentrated flow and then channel flow, each with its own velocity.

Is lag time the same as time of concentration?

No. Lag time runs from the centroid of rainfall excess to the peak of the runoff hydrograph, and the NRCS relation is lag = 0.6 Tc. The NRCS lag equation computes the lag, so the calculator divides by 0.6 to return Tc. Every result also reports the lag as 0.6 Tc, which is the value a unit hydrograph method needs.

How much does Tc change the peak flow?

In the Rational Method the storm duration equals Tc, so Tc decides which intensity is read from the IDF curve. A shorter Tc gives a higher intensity and a higher peak. A 20 % change in Tc typically moves the Rational peak by 10 to 20 %, depending on how steep the IDF curve is at that duration, which is why Tc deserves the same scrutiny as the runoff coefficient.

Time of Concentration Calculator: Kirpich, NRCS Lag, FHWA

Lag time and average velocity
Kirpich	lag 12.4 min · V 0.97m/s
NRCS/SCS Lag	lag 46.9 min · V 0.26m/s
FAA (Rational)	lag 35 min · V 0.34m/s
USBR / SANRAL (DM eq. 3.17)	lag 12.4 min · V 0.97m/s

Using the calculator above

Choose “Compare equations” to run Kirpich, NRCS lag, FAA, Kerby and USBR on one flow path, or “Sheet + shallow + channel” for the FHWA segmental method, then a unit system: SI (metres, mm) or US customary (feet, inches). Every method needs the flow length and the average slope along that path, entered as a percentage (a fall of 24 m over 1 200 m is 2 %). The other inputs depend on the method, and a method whose input is left blank is skipped:

Kirpich: the adjustment factor k (1.0 for bare soil and natural channels).
NRCS lag: the curve number CN of the catchment.
FAA: the Rational Method runoff coefficient C.
Kerby: the retardance coefficient r. The path must be overland flow and no longer than 365 m.
USBR (DM eq. 3.17): the length and average slope of the main watercourse.
FHWA segmental: one row per segment. Sheet flow takes Manning’s n and the 1:2-year, 24-hour rainfall depth P₂; shallow concentrated flow takes the land-cover coefficient k; channel flow takes Manning’s n and the hydraulic radius R.

The result is Tc in minutes and hours, the lag (0.6 Tc) and the average velocity along the path, which is a quick check that length and slope were entered in the right units.

The six methods

The calculator reports every method in minutes and hours. The classical forms of Kirpich, NRCS lag, FAA and Kerby are in US units; the calculator converts SI inputs before applying them.

Kirpich (1940)

T_c = 0.0078\,k\left(\frac{L}{\sqrt{S}}\right)^{0.77}

$L$ is the flow length (ft), $S$ the average slope (ft/ft) and $k$ a surface adjustment factor. Kirpich fitted it to small agricultural watersheds in Tennessee with well-defined channels and slopes of roughly 3 to 10 %, so bare soil and natural channels are the $k = 1.0$ baseline.

NRCS lag

T_L = \frac{L^{0.8}\left(\dfrac{1000}{CN} - 9\right)^{0.7}}{1900\,\sqrt{Y}}, \qquad T_c = \frac{T_L}{0.6}

$T_L$ is the watershed lag (h), $L$ the hydraulic length (ft), $CN$ the curve number and $Y$ the average watershed slope in percent. It comes from the NRCS National Engineering Handbook Part 630, Chapter 15.

FAA

T_c = \frac{1.8\,(1.1 - C)\sqrt{L}}{S^{1/3}}

$C$ is the Rational Method runoff coefficient, $L$ the flow length (ft) and $S$ the slope in percent. It was written for airport pavements and similar small, smooth drainage areas, and it pairs naturally with a Rational calculation that already uses the same $C$ .

Kerby (1959)

T_c = 0.8268\left(\frac{L\,r}{\sqrt{S}}\right)^{0.467}

$L$ is the overland flow length (ft, at most 1 200 ft or 365 m), $r$ the retardance coefficient and $S$ the slope (ft/ft). Kerby describes sheet flow only.

USBR equation (SANRAL Drainage Manual eq. 3.17)

T_c = \left(\frac{0.87\,L^2}{1000\,S}\right)^{0.385}

Here $T_c$ is in hours, $L$ is the watercourse length (km) and $S$ the average slope of the main watercourse (m/m). The SANRAL Drainage Manual uses it for the defined-watercourse part of Tc in the Rational and SDF methods, with overland time added separately. Its exponents are the same as Kirpich’s, so with $k = 1$ the two give the same answer to within rounding.

FHWA segmental method (sheet, shallow concentrated and channel flow)

FHWA HDS-2 and NRCS TR-55 split the path into three regimes and add the travel times:

T_c = T_{sheet} + T_{shallow} + T_{channel}

T_{sheet} = \frac{0.42\,(n L)^{0.8}}{P_2^{0.5}\,S^{0.4}}

with $T_{sheet}$ in minutes, $L$ in ft (90 m / 300 ft at most) and $P_2$ the 1:2-year, 24-hour rainfall in inches. Shallow concentrated flow uses $V = k\sqrt{S_p}$ (m/s, with $S_p$ the slope in percent; multiply by 3.28 for ft/s). Channel flow uses Manning’s equation, $V = R^{2/3}S^{1/2}/n$ in SI. Each travel time is the segment length divided by its velocity.

Symbol	Meaning	Units
$T_c$	Time of concentration	min (h for USBR)
$L$	Flow or segment length	ft (km for USBR)
$S$	Slope along the path	ft/ft or m/m (percent where noted)
$k$	Kirpich factor, or shallow-flow land-cover coefficient	dimensionless
$CN$	NRCS curve number	30 to 100
$C$	Rational runoff coefficient	0 to 1
$r$	Kerby retardance coefficient	0.02 to 0.80
$n$	Manning’s roughness (sheet or channel)	s/m^(1/3)
$P_2$	1:2-year, 24-hour rainfall depth	in (mm in SI input)
$R$	Hydraulic radius of the channel	m or ft

Inputs and where they come from

Flow length is the longest hydraulic path from the divide to the outlet, measured on a DEM or a contour map. Watershed delineation returns it with the catchment boundary, and a watercourse profile gives the 10-85, equal-area and Taylor-Schwarz slopes of the main stream. Use the 10-85 slope for the USBR equation, and the average slope of each segment for the FHWA method. CN comes from land cover and hydrologic soil group (the TR-55 calculator page has the table). P₂ comes from a national rainfall atlas such as NOAA Atlas 14 or from a design rainfall tool. Read the hydraulic radius off a surveyed or typical channel section.

Worked example: one flow path, six answers

A rural catchment has a 1 200 m flow path at an average 2 % slope: about 30 m of grassed overland flow, 370 m of unpaved swale, then 800 m of natural channel. The calculator gives:

Method	Inputs	Tc (min)
Kirpich	k = 1.0	20.6
USBR (DM eq. 3.17)	L = 1.2 km, S = 2 %	20.7
FHWA segmental	sheet n = 0.15, P₂ = 55 mm; shallow k = 0.491; channel n = 0.035, R = 0.4 m	26.7
FAA	C = 0.30	71.7
NRCS lag	CN = 75	78.1
Kerby	30 m overland segment only, r = 0.40	11.5 for that segment

The FHWA run splits into 11.8 min of sheet flow, 8.8 min of shallow concentrated flow and 6.1 min of channel flow. The first 30 m, 2.5 % of the length, take 44 % of the time, and Kerby applied to the same 30 m agrees to within 3 %. Kirpich, USBR and FHWA land within 21 to 27 minutes because all three treat most of the path as channel flow. FAA and NRCS lag were derived for drainage areas where overland flow dominates, which this catchment is not, so they would be set aside for this site.

Coefficient tables

Kirpich adjustment factor k, after Chow, Maidment and Mays as tabulated by TxDOT:

Surface	k
Bare soil, roadside ditches, well-defined natural channel	1.0
Overland flow on grass	2.0
Overland flow on concrete or asphalt	0.4
Concrete-lined channel	0.2
Natural channel with vegetation (practitioner value)	1.0
Channel with heavy brush (practitioner value)	1.5

Kerby retardance coefficient r:

Surface	r
Smooth impervious (asphalt, concrete)	0.02
Smooth bare packed soil	0.10
Poor grass, cultivated row crops	0.20
Pasture or average grass	0.40
Deciduous forest	0.60
Dense grass, coniferous forest, deciduous forest with deep litter	0.80

Manning’s n for sheet flow (TR-55):

Surface	n
Smooth surfaces (concrete, asphalt)	0.011
Fallow, no residue	0.05
Cultivated soil, residue cover ≤ 20 %	0.06
Cultivated soil, residue cover > 20 %	0.17
Short prairie grass	0.15
Dense grass	0.24
Bermuda grass	0.41
Natural range	0.13
Woods, light underbrush	0.40
Woods, dense underbrush	0.80

Shallow concentrated flow land-cover coefficient k (HDS-2): forest with heavy litter or hay meadow 0.076; minimum tillage, contour cropping or woodland 0.152; short-grass pasture 0.213; straight-row cultivation 0.274; nearly bare ground or alluvial fans 0.305; grassed waterway 0.457; unpaved 0.491; paved areas and small upland gullies 0.619.

Limitations and when to use something else

Calibration range. Kirpich was fitted to watersheds up to about 45 ha and under-predicts on very flat, urban or sheet-flow-dominated catchments. NRCS lag was calibrated for catchments under about 8 km² with slopes of 0.5 to 30 % and over-predicts where flow is channelised. Kerby stops at 365 m.
Uniform conditions. The single-equation methods assume one slope and one surface along the whole path. Where that is plainly untrue, use the segmental method.
Empirical transfer. These equations were derived in specific regions. Applying them in very different terrain carries a bias nobody has measured, so check computed Tc against an observed lag wherever a gauge exists.
Large catchments. Tc assumes the whole catchment contributes at the peak. For large catchments use a hydrograph approach: the unit hydrograph generator, or for South African sites design flood estimation.

What a free Studio account adds

Results on this page are free. A free Studio account adds:

Saving: each Tc calculation is filed under a project with its inputs, so a reviewer can trace it.
Word and Excel reports with the equation, inputs and result.
Comparison mode: every applicable method on one set of inputs, with methods outside their range flagged.
Linked inputs: fill length and slope from a saved watershed delineation, or choose the 10-85 or another slope from a saved watercourse profile.
A minimum Tc floor that clamps very short results and says so on the result.
Send to: pass the Tc straight to the Rational Method calculator, the TR-55 calculator, the unit hydrograph generator or the design storm generator as the storm duration.

Time of concentration calculator