Direct Runoff Hydrograph (DRH) Method

Open Design Flood Estimation

Compute a full design flood hydrograph — rising limb, peak, recession and total volume — by convolving an effective rainfall hyetograph with a unit hydrograph and routing the resulting inflow through the catchment’s channel system. This guide covers the DRH workflow, rainfall loss modelling, time-step selection, Muskingum routing, and a worked SA example.

Overview

The Direct Runoff Hydrograph (DRH) method is the detailed hydrograph-generation technique used in South African design practice and internationally in engineering software such as HEC-HMS. The approach dates to Bauer & Midgley (HRU Report 2/74) and is documented as the primary hydrograph method in the SANRAL Drainage Manual (6th ed., Chapter 3).

In essence, the DRH method builds the flood hydrograph by:

1. deriving an effective rainfall hyetograph (rainfall excess) from a design storm using a loss model,
2. convolving that hyetograph with a unit hydrograph (SCS dimensionless or HRU 1/72 synthetic) to produce a direct runoff hydrograph at the catchment outlet,
3. routing the hydrograph through the main channel reach (typically with Muskingum), and
4. adding baseflow to obtain the total discharge hydrograph.

The product is a complete flood hydrograph — peak, timing, volume, and shape — suitable for sizing reservoirs, attenuation ponds, culverts with storage effects, and floodplain inundation models.

DRH vs unit hydrograph

The Unit Hydrograph page covers the UH itself — what it is, how it is derived, and how to convolve. The DRH method is the end-to-end workflow that starts from a design storm and ends with a routed outflow hydrograph. DRH uses the UH as one of several building blocks.

The DRH workflow

The DRH method comprises five sequential steps:

1. Design storm selection — choose storm duration(s) and return period, and obtain a design rainfall depth from the Design Rainfall tool.
2. Design storm temporal distribution — distribute the storm depth over time using a design storm pattern (SCS Type II/III, Huff, alternating block, etc.).
3. Loss estimation — subtract infiltration and interception losses to obtain the effective rainfall hyetograph (rainfall excess).
4. Convolution — apply a unit hydrograph to convert the effective rainfall hyetograph into an unrouted direct runoff hydrograph.
5. Channel routing + baseflow — route through the channel (Muskingum) and add baseflow to obtain the total design flood hydrograph.

Effective rainfall

Effective rainfall (rainfall excess) is the portion of gross rainfall that becomes direct runoff after accounting for interception, depression storage, and infiltration. HydroDesign’s DRH module supports two loss models:

SCS Curve Number loss model

The default in SA practice. At any cumulative rainfall $P$, the cumulative rainfall excess is:

$$Q(P) \;=\; \frac{(P - 0.1\,S)^2}{P + 0.9\,S} \qquad \text{for } P > 0.1\,S$$

Cumulative SCS runoff (SA practice, λ = 0.1)

With $S = 25\,400/CN - 254$ (mm). Incremental effective rainfall in each time step is the difference of successive $Q(P)$ values — this captures the non-linear increase of runoff fraction as the soil wets up. See the SCS Method page for full CN tables and commentary on the $\lambda$ choice.

φ-index (constant loss rate)

A simpler model in which a constant loss rate $\phi$ (mm/hr) is subtracted from each rainfall block. $\phi$ is calibrated so that the total effective rainfall equals the expected runoff depth:

$$P_{e,j} \;=\; \max\bigl(\,P_j - \phi \cdot \Delta t,\;\; 0\,\bigr)$$

φ-index loss

The φ-index is useful when a CN is uncertain, or for calibrating to an observed storm. It does not capture the non-linear increase of runoff fraction with storm magnitude, so it is unsuitable for extrapolation to return periods much longer than the calibration event.

Green-Ampt (optional, advanced)

For research or forensic work, a Green-Ampt infiltration model can be substituted. It is physically based on soil hydraulic conductivity and suction head, but requires more input parameters than typical design work supports. Green-Ampt is not part of HydroDesign’s standard DRH workflow.

Time-step selection

The computational time step $\Delta t$ must be small enough to resolve the peak of the UH and the rising limb of the storm. HEC-HMS guidance and the SANRAL Drainage Manual recommend:

$$\Delta t \;\le\; 0.25 \cdot T_c$$

HEC/SANRAL time-step rule

For $T_c \approx 4$ hr, that’s $\Delta t \le 1$ hr. Smaller time steps are acceptable and often preferable (5-15 min) as long as the design storm and UH are both resolved at the same resolution. Choose the UH unit duration $D$ equal to $\Delta t$ to avoid S-curve conversion.

Critical storm duration

The critical storm duration — the duration that produces the maximum peak discharge — is typically close to the catchment’s time of concentration $T_c$. DRH runs should bracket the critical duration by computing hydrographs for $0.5\,T_c$, $T_c$ and $2\,T_c$, and reporting the largest peak. HydroDesign’s DRH implementation does this automatically.

Convolution

The direct runoff hydrograph at the catchment outlet is obtained by discrete convolution of the effective rainfall hyetograph with the unit hydrograph. For a storm discretised into $N$ blocks of effective rainfall depth $P_j$ (mm):

$$Q(n\,\Delta t) \;=\; \sum_{j=1}^{\min(n, N)} P_j \cdot u\bigl((n - j + 1)\,\Delta t\bigr)$$

DRH convolution

Where $u(k)$ is the $k$-th ordinate of the $\Delta t$-hour unit hydrograph (m³/s per mm). Tabulating this convolution row by row produces the full hydrograph — rising limb, peak, and recession. See the Unit Hydrograph page for the full derivation.

Muskingum channel routing

Once the direct runoff hydrograph has been computed at the upper end of a channel reach, it must be routed downstream to the study outlet. The Muskingum method is the standard for this in SA practice:

$$O_{n+1} \;=\; C_0\,I_{n+1} + C_1\,I_n + C_2\,O_n$$

Muskingum routing

With routing coefficients:

$$\begin{aligned} C_0 &= \frac{-KX + 0.5\,\Delta t}{K(1 - X) + 0.5\,\Delta t} \\[6pt] C_1 &= \frac{KX + 0.5\,\Delta t}{K(1 - X) + 0.5\,\Delta t} \\[6pt] C_2 &= \frac{K(1 - X) - 0.5\,\Delta t}{K(1 - X) + 0.5\,\Delta t} \end{aligned}$$

Muskingum coefficients

Where $K$ is the reach travel time (hr), $X$ is the weighting factor ($0 \le X \le 0.5$, typically 0.2 – 0.3), and $\Delta t$ is the time step. The coefficients sum to unity. In SA practice, $K$ is often estimated from:

$$K \;=\; a \cdot A^{\,0.318}$$

Muskingum K from catchment area

With $a$ a regional coefficient. HydroDesign defaults to $K = T_L$ (lag time) and $X = 0.25$, with overrides available.

Baseflow separation

Baseflow is the contribution from sustained groundwater discharge — typically a small fraction of the design flood peak but a significant fraction of total volume. Separation techniques:

• Constant baseflow. Add a constant value $Q_b$ (estimated from low-flow records or catchment area × specific yield) to the routed DRH. Adequate for design work.
• Straight-line separation. On an observed hydrograph, draw a straight line from the inflection point at the start of the storm to the recession inflection point; below the line is baseflow, above is direct runoff.
• Recession-constant method. Baseflow decays exponentially with a catchment-specific recession constant $k$: $Q_b(t) = Q_{b,0} \cdot e^{-t/k}$.

For design flood estimation the simple constant-baseflow assumption is almost always adequate, because baseflow is typically less than 5% of the design peak.

Peak and volume extraction

From the total discharge hydrograph $Q(t)$:

• Peak discharge: $Q_{\text{peak}} = \max_t Q(t)$.
• Time to peak: the $t$ at which the peak occurs, measured from the start of effective rainfall.
• Flood volume: $V = \sum_n Q(n\,\Delta t) \cdot \Delta t$.
• Time base: the duration over which $Q(t) >$ baseflow.

These quantities drive the design of culverts, detention ponds, floodplain maps and dam spillways.

Worked example

A 120 km² catchment in Mpumalanga (veld zone 8 — bushveld) is analysed for a 1:100 year design flood. The SCS CN has been estimated as 75; $T_c$ = 4.5 hr; UH parameters follow HRU 1/72 for zone 8.

Given information

• Catchment area: $A = 120$ km²
• Time of concentration: $T_c = 4.5$ hr
• Design storm: 1:100 yr, 6-hour, $P = 180$ mm, SCS Type II distribution
• SCS CN: 75 ($\lambda = 0.1$) → $S = 84.7$ mm
• UH (HRU 1/72 zone 8): $T_L = 2.7$ hr, $Q_p = 16.3$ m³/s/mm (1-hr UH)
• Muskingum: $K = 2.7$ hr, $X = 0.25$
• Baseflow: $Q_b = 3$ m³/s
• Time step: $\Delta t = 1$ hr (satisfies $\Delta t \le 0.25\,T_c$)

Step 1 — Hyetograph

The 180 mm SCS Type II storm is discretised into 1-hour blocks (typical Type II peaks in hours 3 – 4):

Hour $j$	1	2	3	4	5	6
$P_j$ (mm)	12	28	68	42	20	10

Step 2 — Effective rainfall (cumulative SCS)

At each cumulative rainfall $P_\text{cum}$, compute $Q(P)$ and take differences:

Hour	$P_\text{cum}$	$Q(P_\text{cum})$	$P_{e,j}$
1	12	0.3	0.3
2	40	6.9	6.6
3	108	49.6	42.7
4	150	82.0	32.4
5	170	98.0	16.0
6	180	106.1	8.1

Total effective rainfall $\approx$ 106 mm.

Step 3 — Convolution with 1-hour UH

Using the HRU 1/72 UH with $Q_p = 16.3$ m³/s/mm and $T_L = 2.7$ hr (triangular UH, $t_p = 3.2$ hr, $t_b = 8.5$ hr), the 1-hour UH ordinates are approximately:

$t$ (hr)	0	1	2	3	3.2	4	5	6	7	8	8.5
$u$	0	5.1	10.2	15.3	16.3	13.6	10.5	7.4	4.3	1.2	0

Convolving with the effective rainfall blocks gives a direct runoff hydrograph peaking at about 920 m³/s around $t = 6$ hr.

Step 4 — Muskingum routing

Routing with $K = 2.7$ hr, $X = 0.25$, $\Delta t = 1$ hr:

$$C_0 = 0.077,\quad C_1 = 0.247,\quad C_2 = 0.676$$

Routed peak is attenuated to approximately 880 m³/s, shifted by one time step.

Step 5 — Total flood hydrograph

Adding baseflow of 3 m³/s, the design peak is ~883 m³/s at $t \approx 7$ hr. Total flood volume is approximately 12.7 × 10⁶ m³.

Report the full hydrograph

For pond and spillway design, the peak alone is not enough — the total volume above a threshold matters. The DRH method’s strength is that it gives you both peak and volume from the same computation. Always plot the hydrograph in your report.

Limitations

• All UH limitations carry through. Linearity, time-invariance, spatial uniformity — see Unit Hydrograph for detail.
• Loss model sensitivity. The choice of CN (or φ) and $\lambda$ can shift the peak by a factor of two or more. Document the loss model clearly.
• Time-step sensitivity. Too coarse a $\Delta t$ under-resolves the peak; too fine introduces numerical noise in convolution if the UH is not smooth.
• Muskingum is linear. For steep, flashy reaches with significant storage (e.g. floodplains), kinematic-wave or dynamic-wave routing is more appropriate.
• No distributed rainfall. The standard DRH workflow assumes rainfall is uniform over the catchment. For catchments > ~500 km², consider a semi-distributed approach.
• Antecedent conditions are assumed, not simulated. ARC II is the default; explicit simulation of soil moisture requires a continuous model (ACRU, HEC-HMS continuous, SWAT).

Related tools and guides

• Rational Method — peak-only method for small catchments
• SCS Method — Curve Number method for deriving rainfall excess
• Unit Hydrograph — UH theory, synthetic UHs, S-curve
• SDF Method — regional deterministic method (SA)
• Tc Calculator — time of concentration methods
• Design Rainfall — design storm depths
• Design Storm — temporal distributions (Type II, Huff, alternating block)
• TR-55 Calculator — graphical SCS peak discharge

References

• Bauer, S.W. & Midgley, D.C. (1974). A simple procedure for synthesizing direct runoff hydrographs, HRU Report 2/74. Hydrological Research Unit, University of the Witwatersrand, Johannesburg. (Foundational SA DRH reference.)
• Midgley, D.C. & Pitman, W.V. (1972). Surface Water Resources of South Africa, HRU Report 1/72. Hydrological Research Unit.
• SANRAL. (2013). Drainage Manual (6th ed.). South African National Roads Agency, Pretoria. Chapter 3 — Hydrology; Section 3.6: Synthetic Unit Hydrograph and Direct Runoff Hydrograph Methods.
• Chow, V.T., Maidment, D.R. & Mays, L.W. (1988). Applied Hydrology. McGraw-Hill, New York. Chapter 7 — Unit Hydrograph; Chapter 9 — Flow Routing.
• USACE. (1994). Engineering and Design — Flood-Runoff Analysis, Engineer Manual EM 1110-2-1417. US Army Corps of Engineers, Washington DC.
• USACE. (2000). Hydrologic Modeling System HEC-HMS Technical Reference Manual. US Army Corps of Engineers, Hydrologic Engineering Center, Davis CA.
• USDA-NRCS. (2007). National Engineering Handbook, Part 630, Chapters 15 – 17.

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Open Design Flood Estimation