Surface water hydrology

This chapter covers the following:

• Hydrologic cycle and watersheds
• Storm/Flood frequency probabilities
• Hydrologic mass balance problems
• Rainfall computation and analysis
• Runoff computation and analysis
• Hydrograph and unit hydrograph

Hydrologic cycle and watersheds

The hydrologic cycle describes how water continuously moves through the atmosphere, across the land surface, and through the subsurface. Key processes include precipitation, evaporation, transpiration, infiltration, runoff, and groundwater flow.

A watershed sets the physical boundary for many hydrologic calculations. All surface runoff generated inside a watershed drains to a common outlet, so watershed properties strongly influence the hydrologic response. Important characteristics include area, topography, soil properties, and land use.

Hydrologic cycle

The hydrologic cycle represents the continuous movement of water through the Earth’s atmosphere, surface, and subsurface.

A common way to summarize the cycle is with a mass balance, which accounts for where precipitation goes over a chosen time period:

$$P=E+T+I+R+\Delta S$$

Where:

• $P$ = Precipitation
• $E$ = Evaporation
• $T$ = Transpiration
• $I$ = Infiltration
• $R$ = Runoff
• $\Delta S$ = Change in storage

Watersheds

A watershed is a geographic area that drains all precipitation to a common outlet.

Important features include:

• Area
• Shape
• Slope
• Land Use

These features affect how quickly runoff forms, how much runoff is generated, and how large peak flows can become.

Storm/flood frequency probabilities

Storm and flood frequency analysis uses historical records and statistical methods to estimate how likely extreme hydrologic events are, and how large they may be.

Results are often reported using a return period (also called a recurrence interval). The return period is an average time between events of a given magnitude, not a guarantee that events occur on a fixed schedule. These analyses support hydrologic design, flood risk assessment, and regulatory decisions.

General probability

$$p=\frac{n_i}{N}$$

Where:

• $p$ = probability of occurrence of a flood flow of class $i$ (variate $i$)
• $n_i$ = number of items in the $i$th class
• $N$ = total number of items in a series

Average recurrence interval (ARI)

$$p(x\ge x_T)=p=\frac{1}{T}$$

Where:

• $p$ = probability of a single occurrence in a given storm period
• $x$ = magnitude of event
• $x_T$ = design level event
• $T$ = storm/flood return period (years)

Risk or annual exceedance probability (AEP)

This expression gives the probability of exceeding the design level at least once over a multi-year period:

$$p(x\ge x_T\text{ at least once in }n\text{ years})=p=1-{\left(1-\frac{1}{T}\right)}^n$$

Where:

• $p$ = probability of exceeding a flow or intensity in a given period
• $n$ = number of years

Reliability (probability of nonexceedance)

This is the complementary idea: the probability that the design level is not exceeded in any year over the period:

$$p(x<x_T\text{ each year for }n\text{ years})=p={\left(1-\frac{1}{T}\right)}^n$$

Where:

• $p$ = probability of not exceeding a flow or intensity in a given period

Hydrologic mass balance problems

Hydrologic mass balance is based on conservation of mass: over a chosen time period, the difference between inflows and outflows must equal the change in storage within the system.

You can apply this idea to a lake, a soil column, a watershed, or a groundwater basin, under steady-state or transient conditions.

The mass balance equation applied to a control volume:

$$\text{Input}-\text{Output}=\Delta \text{Storage}$$

Mathematically:

$$P+Q_{in}-E-Q_{out}=\Delta S$$

Where:

• $Q_{in}$, $Q_{out}$ = Inflow and outflow (surface or subsurface)
• $\Delta S$ = Change in storage over time

Example:

Given:

• $P=80\text{mm}$
• $E=20\text{mm}$
• $Q_{in}=100\text{mm}$
• $Q_{out}=140\text{mm}$

Then:

$$\Delta S=P+Q_{in}-E-Q_{out}=80+100-20-140=20\text{mm}$$

Rainfall computation and analysis

Rainfall computation and analysis focuses on describing precipitation in ways that are useful for hydrologic modeling. Common descriptors include intensity, duration, depth, and temporal distribution.

These data support tools such as intensity-duration-frequency (IDF) relationships, hyetographs, and design storms, which are then used as inputs for runoff estimation and flood prediction.

Rainfall estimation methods

• Arithmetic Mean Method:

$$P=\frac{1}{n}\sum _{i=1}^n{P}_i$$

• Thiessen Polygon Method:

$$P=\sum _{i=1}^n{w}_i{P}_i$$

• Isohyetal Method: Use area-weighted average between isohyets.

Rainfall intensity (IDF relationship)

These equations relate design rainfall intensity to storm duration and (in some forms) return period.

Equation 1

$$i=\frac{c}{T_d^e+f}$$

Equation 2

$$i=\frac{c{T}^m}{T_d^e+f}$$

Equation 3

$$i=\frac{c{T}^m}{T_d^e+f}$$

Where:

• $i$ = design rainfall intensity (in./hr)
• $T_d$ = duration of event (min)
• $m$ = rainfall coefficient
• $T$ = return period (years)
• $c,e,f$ = return-period coefficients

Runoff computation and analysis

Runoff computation transforms precipitation into surface runoff by accounting for hydrologic losses such as infiltration, interception, evaporation, and surface storage.

Depending on watershed size and available data, you may use analytical methods, empirical methods, or a combination. Runoff analysis is used to estimate peak discharge, runoff volume, and the timing of flow.

Rational method

$$Q=CIA$$

Where:

• $Q$ = peak discharge (ft³/sec)
• $C$ = runoff coefficient
• $I$ = rainfall intensity from an IDF curve for a duration of $t_c$ (in./hr)
• $A$ = watershed area (acres)

Time of concentration

• $t_c$ = time of concentration: time required for runoff to travel from the hydraulically most distant point of the watershed to the point of interest (min). Multiple watersheds may require summation of times to determine an overall time for the system.

Conversion factor from acre-in/hr to ft³/sec is approximately 1.0.

Weighted composite runoff coefficient

Use this when the drainage area includes multiple land covers (and therefore multiple runoff coefficients):

$$C_w=\frac{A_1{C}_1+A_2{C}_2+\dots +A_n{C}_n}{A_1+A_2+\dots +A_n}$$

Where:

• $C_w$ = weighted/composite runoff coefficient for whole drainage area

Example:

Given: $C=0.75$, $i=60\text{mm/hr}$, $A=2\text{ha}$

Then:

$$Q=0.75\cdot 60\cdot 2=\frac{90}{360}=0.25m^3/s$$

SCS curve number (CN) method

Used to estimate direct runoff from rainfall.

$$Q=\frac{(P-0.2S{)}^2}{P+0.8S},\text{for }P>0.2S$$

Where:

$$S=\frac{25400}{CN}-254$$

• $CN$ = Curve Number (dimensionless)
• $P$ = Precipitation (mm)
• $S$ = Potential max retention after runoff begins

Hydrograph and unit hydrograph

A hydrograph shows how discharge at a specific location changes over time, often in response to a storm.

A unit hydrograph is a standardized way to represent watershed response: it describes the direct runoff hydrograph produced by a unit depth of effective rainfall applied uniformly over the watershed for a specified duration. Both tools are widely used in flood analysis and watershed modeling.

Hydrograph

A hydrograph shows variation of stream discharge with time. Key parts include:

• Rising limb
• Peak flow
• Falling limb
• Baseflow

Unit hydrograph (UH)

A Unit Hydrograph is the direct runoff hydrograph resulting from 1 cm of effective rainfall uniformly distributed over the watershed for a specified duration.

Assumes:

• Linearity
• Time invariance

Example:

3-hour unit hydrograph:

Time (hr)	UH (m³/s)
0	0
3	10
6	25
9	20
12	5
15	0

If 2 cm of effective rainfall:

Time (hr)	Direct runoff (m³/s)
0	0
3	20
6	50
9	40
12	10
15	0