Water resources engineering can be considered a broad field as it involves aspects of the physical sciences, engineering, biology, ecology, and even the social sciences. In this article, I will be focusing on the hydrologic and hydraulic aspects of water resources engineering. Hydrology is the study of how water moves across the land. Hydraulics is the study of how water moves through open channels (e.g., streams or engineered channels), pipes, and aquifers (groundwater).
This list is comprised of the most commonly used equations in water resources engineering (at least in my experience). Note that these equations are not listed in any particular order (i.e., importance).
1. Manning’s Equation
Manning’s equation is probably the most widely used equation in hydrology and hydraulics. It is an empirical equation, which means it is based on observations rather than the principles of physics. Manning’s equation was developed by Robert Manning in the late 19th century. Interestingly, Robert Manning did not have a background in fluid mechanics. He was actually an accountant by trade.
Manning’s equation relates average velocity to slope, channel geometry, and channel roughness. The equation is shown below.
where,
A = Flow area (sq ft)
R = Hydraulic Radius (A/P)
S = Slope (ft/ft)
n = Manning’s roughness coefficient
The following assumptions apply to Manning’s equation:
- Uniform flow
- Constant shear stress at the boundary
- Energy loss is due to friction
To determine flow, multiply velocity and flow area.
2. TR-55 Time of Concentration Equations
Chapter 3 of the United States Natural Resources Conservation Service (NRCS) Technical Release 55 (TR-55) presents a set of simple procedures to estimate time of concentration.
Time of concentration is defined as the time it takes a drop of water to travel from the most hydraulically distant part of the watershed to the outlet. It is an important hydraulic parameter because it impacts the shape and peak of the hydrograph for a particular watershed.
You calculate time of concentration by adding the travel time, which is the time it takes water to travel from one location to another, of various flow segments.
where,
Tc = Time of concentration
Tt = Travel time
You compute the travel time depending on whether a particular flow segment could be defined as sheet flow, shallow concentrated flow, or concentrated flow.
Sheet Flow
Sheet flow is defined as flow over a plane or surface. An example is shown below.
You should not use the following equation for watersheds more than 300 feet of sheet flow. This is because the TR-55 methodology assumes that shallow concentrated flow occurs after 300 feet of sheet flow. The following equation shows who to calculate the travel time for sheet flow. This equation is called Manning’s kinematic solution.
where,
Tt = Travel time (hrs)
n = Manning’s roughness coefficient
L = Length of flow (ft)
P2 = 2-year, 24-hour rainfall (in)
s = Slope (ft/ft)
Shallow Concentrated Flow
After a maximum of 300 feet of sheet flow, the flow will become shallow concentrated flow. The calculations for shallow concentrated flow depends on whether the surface is paved or unpaved.
To calculate sheet flow, use the following graph from TR-55 to determine the velocity of the shallow concentrated flow based on slope.
After calculating velocity, use the following equation to calculate travel time.
where,
Tt = Travel Time (hr)
L = Length of flow (ft)
V = Velocity (fps)
Concentrated Flow
Finally, water becomes concentrated in gutters, ditches, swales, streams, or engineered channels. The calculations for the travel time of concentrated flow is similar to shallow concentrated flow except velocity is computed using Manning’s equation.
3. Rational Method
The Rational Method is a simple equation used to estimate peak flow from small watersheds. It was developed by Emil Kuichling, a German water engineer, in 1889.
The United States Natural Resources Conservation Service (NRCS) states that the Rational Method is valid for watersheds up to 50 acres. It is expressed by the following equation:
Q = CiA
where,
Q = Flow (cfs)
i = Rainfall intensity (in/hr)
A = Area (acres)
You can find runoff coefficients in most water resources engineering textbooks or on the internet. For study areas located within the United States, precipitation intensity can be obtained from the NOAA Precipitation Data Frequency Server.
Water resources engineers use this equation to design smaller structures like pipes and stormwater best management practices (BMPs).
4. Froehlich (1995) Peak Flow Prediction
When performing a dam breach analysis, you typically determine a number of dam breach parameters. Then you compare the modeled results (using these dam breach parameters) to peak flow equations. These equations predict the peak outflow from a dam breach. The most famous peak flow equation is Froelich (1995).
Froelich developed this equation by performing a best-fit regression analysis using data from 22 case studies.
Froehlich (1995) is considered the most accurate empirical peak flow equation available for dam breach analyses. This means you would, in most cases, pick the set of dam breach parameters that result in a peak flow that is closest to the Froelich (1995) equation.
where,
Qp = Peak outflow (m3/s)
Vw = Volume of water above the breach invert at the time of failure (m3)
hw = Hydraulic depth of water above the breach bottom at the time of failure (m)
5. SCS Curve Number Method
The SCS Curve Number Method is another very widely used method in water resources. It is a simple and efficient way to estimate stormwater runoff. It is defined by the equations shown below.
where,
Q = Runoff (in)
P = Rainfall (in)
S = Potential maximum retention after runoff begins (in)
CN = Curve Number
The curve number is a function of hydrologic soil group and land use. There are a number of curve number tables available in water resources engineering textbooks and on the internet. You can determine hydrologic soil group by searching for your site on the NRCS Web Soil Survey.
6. Water Balance
This is more of a conceptual equation, but the water balance is the foundation of hydrology. A water balance can be applied to a variety of hydrologic systems such as a drainage basin or a column of soil.
R = P – ET – I – ΔS
where,
R = Runoff
P = Precipitation
ET = Evapotranspiration
ΔS = Change in storage
7. Horton’s Equation
Horton’s equation is an empirical formula used to determine infiltration. It assumes that infiltration starts at a constant rate. Then, if rainfall exceeds infiltration capacity, infiltration will decrease in an exponential manner. The equation is shown below.
where,
fp is the infiltration rate (L/T) at time t
fc is the steady-state infiltration rate (L/T) at large times
f0 is the initial infiltration rate (L/T)
k is the decay constant for a given soil and initial condition t-1
t is the time (t)
8. Penman-Monteith Equation
The Penman-Monteith equation is used to estimate reference evapotranspiration (ET). Reference ET is defined as the water use of a hypothetical reference crop such as well-watered alfalfa or turfgrass. There are many other empirical methods that have been developed to calculate reference ET. Some are temperature-based while others are based on radiation or wind speed. The Penman-Monteith equation depends on a variety of factors as shown below.
where,
ETo = reference evapotranspiration [mm day-1]
Rn = net radiation at the crop surface [MJ m-2 day-1]
G = soil heat flux density [MJ m-2 day-1]
T = mean daily air temperature at 2 m height [°C]
u2 = wind speed at 2 m height [m s-1]
es = saturation vapour pressure [kPa]
ea = actual vapour pressure [kPa]
es – ea saturation vapour pressure deficit [kPa]
Δ slope vapor pressure curve [kPa °C-1]
γ psychrometric constant [kPa °C-1]
This is a somewhat complex equation. Luckily, there are a variety of calculators available online to calculate reference ET using the Penman-Monteith method.
After calculating reference ET, you can calculate crop ET using crop coefficients using the equation below. Crop ET is the water use of the plant of interest. This is the value you would put in a water balance.
ETc = ETO × kc
9. Darcy’s Law
Darcy’s Law is an equation that calculates how flow moves through a porous medium like soil. This equation was developed by Henry Darcy, a French engineer, in the 1800’s. The equation is shown below.
where,
Q is flow rate in cfs
K is the coefficient of permeability (ft/s)
i is the hydraulic gradient (ft/ft)
∆H is height (ft)
L is length (ft)
µ is absolute viscosity (lb-s/ft2)
k is intrinsic permeability (ft2)
γ is specific weight (lb/ft3)
Note that you should think about Q as specific discharge, which is discharge per area. This term is referred to as Darcy velocity.
There are a number of assumptions to keep in mind when using Darcy’s Law. These assumptions are as follows:
- The soil is saturated.
- Flow through soil is laminar.
- Flow through the soil is continuous and steady.
10. Bernoulli Equation
The Bernoulli Equation is a conservation of energy equation for two points along a horizontal streamline. In fluid mechanics, the Bernoulli Equation states that an increase in flow velocity indicates a decrease in static pressure or fluid’s potential energy. The Bernoulli Equation is named after Daniel Bernoulli who was a Swiss physicist. He published this equation in his book Hydrodynamica in 1738. The Bernoulli Equation is shown below.
The Bernoulli Equation operates on the following assumptions:
- The fluid is non-viscous
- The fluid is incompressible
- Flow is steady
11. Continuity Equation
The continuity equation is fundamental to your understanding of fluid mechanics. It is simple but powerful in the sense that it can be applied in many ways. The continuity equation states that the mass entering a system is equal to the mass leaving the system plus any accumulation of mass within the system (i.e., water storage in a pond). This is shown in equation form below.
12. Froude Number
The Froude Number (Fr) is a dimensionless parameter used to classify open channel flow into different flow regimes (e.g., subcritical, critical, and supercritical). The equation for the Froude Number is shown below. As you can see, it is the ratio of inertial forces to gravitational forces.
Fr = V/(gy)0.5
where,
Fr = Froude Number
V = Velocity
y = Hydraulic depth
g = Acceleration of gravity
The table below outlines how the Froude Number is related to flow regime.
| Fr < 1 | Subcritical |
| Fr = 1 | Critical |
| Fr > 1 | Supercritical |
13. Weir Equation (Broad Crested)
Another commonly used hydraulic equation is the weir equation. Typically, flow from a weir is determined when routing flow from a pond or lake.
where,
Q is discharge (cfs)
H is height of water (ft)
b is the width of the base (ft)
v is velocity (ft/s)
Note that when modeling flow over a dam, the approach velocity can be assumed to be zero.
It is also important to note that the computations for weir flow differ by the type weir. The equation above is only applicable to broad crested weirs.
14. Orifice Equation
Orifice is basically a fancy word for a hole. The orifice equation is used to determine flow through an orifice (hole). Like the weir equation, the orifice equation is used in routing exercises.
where,
Q = flow
Cd = orifice coefficient
d = diameter
g = acceleration of gravity
h = distance between the free surface and the center of the orifice
Keep in mind that this equation is only applicable to circular orifices.
15. Hazen-Williams Equation
The Hazen-Williams Equation is an empirical equation used to calculate head loss in pipes. The equation describes the relationship between the flow in a pipe and the pressure drop caused by friction. The Hazen-Williams Equation is shown below.
where,
hf = head loss due to friction (ft)
L = length of pipe (ft)
Q = volumetric flow rate (gpm)
C = Roughness Coefficient
d = diameter (ft)
This equation is used to design the pipes in water supply systems and irrigation systems. Typical C factors used in these designs are shown below.
| Pipe Material | C |
| Concrete | 130 |
| Cast Iron (New) | 130 |
| Cast Iron (20 years old) | 100 |
| Cast Iron (50 years old) | 120 |
| Welded Steel (New) | 120 |
| Wood Stave | 120 |
| Vitrified Clay | 110 |
| Riveted Steel (New) | 110 |
| Brick Sewers | 100 |
| Asbestos Cement | 140 |
| Plastic | 150 |
16. TMDL
A total maximum daily load (TMDL), as defined by the United States Environmental Protection Agency (EPA), is the maximum amount of a pollutant permitted to enter a waterbody.
Typically, a TMDL is developed for each waterbody/pollutant combination. For example, if a waterbody is impaired by two different pollutants, two TMDLs will be developed. The following equation is used to develop TMDLs.
TMDL = ΣWLA + ΣLA + MOS
where,
WLA = wasteload allocation (point sources)
LA = load allocation (nonpoint sources)
MOS = Margin of Safety
The MOS accounts for the uncertainty in predicting how reducing the pollutant loads will result in meeting water quality standards.
According to the Clean Water Act, each state must develop TMDLs for every waterbody listed on Section 303(d) list of impaired waters.
17. Reynold’s Number
The Reynolds Number is a dimensionless parameter and is the ratio of inertial forces to viscous forces. This number is used to classify flow as laminar or turbulent. A low Reynolds Number indicates laminar flow where viscous forces are dominant. In contrast, a high Reynolds Number indicates turbulent flow where inertial forces are dominant. Turbulent flow tends to include eddies, vortices, and other flow instabilities.
where,
R is Reynolds Number
D is internal pipe diameter
v is fluid velocity
μ is absolute viscosity
v is kinematic viscosity
ρ is density
The characteristic length depends on the shape of the hydraulic structure in question. For example, the hydraulic depth is used as L when calculating the Reynolds Number for a pipe.
18. Stoke’s Law
Stoke’s Law is used to determine the settling velocity of a particle. This equation is used in water/wastewater design. Stoke’s Law is named for George Stokes, an Irish physicist.
where,
vs is settling velocity in ft/s
SGparticle is the specific gravity of the particle
Dft is the diameter of the sphere in ft
v is kinematic viscosity
g is the acceleration due to gravity
19. Gutter Flow
The following equation is used to calculate gutter flow. This equation is derived from Manning’s equation.
where,
K = Gutter flow constant = 0.56 ft3/(s-ft)
z = inverse of the cross slope of the gutter (decimal)
n = Manning’s roughness coefficient
s = slope of the gutter (decimal)
y = water depth at the curb (ft)
20. Important Conversions
There are also a number of important conversions you should consider memorizing. This will allow you to complete engineering calculations more quickly.
1 acre = 43,560 sf
1 sq mi = 640 ac
1 CY = 27 cf
1 mile = 5,280 ft
1 feet = 0.3048 m
1 cfs = 314.18 gpm