Formula sheet

Environmental Water Systems Formula Sheet

Environmental water formulas for water balance, runoff, infiltration, flow, storage, pollutant load, pipe hydraulics, pressure, pumping, and reliability.

Branch: Environmental Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.9 · reviewed

This formula sheet collects common first-pass calculations for environmental water, stormwater, wastewater, and distribution systems. Use it for screening, design review, troubleshooting, and model checks. Detailed design still requires local rainfall data, utility standards, verified field data, hydraulic modelling, treatment requirements, and regulatory criteria.

State the basis before calculating: design storm, dry-weather flow, wet-weather flow, average day, peak hour, service area, catchment boundary, pipe-full condition, treatment basis, or monitoring interval.

Water balance

General balance:

\text{accumulation}=\text{inflow}-\text{outflow}+\text{source}-\text{sink}

Storage balance:

\displaystyle \frac{dS}{dt}=Q_{in}-Q_{out}+P-E-I+G

where $S$ is stored water, $Q_{in}$ and $Q_{out}$ are hydraulic flows, $P$ is precipitation input, $E$ is evaporation or evapotranspiration, $I$ is infiltration or seepage, and $G$ is groundwater exchange under the selected sign convention.

Discrete time approximation:

S_{t+\Delta t}=S_t+(Q_{in}-Q_{out})\Delta t

Add rainfall, infiltration, evaporation, or groundwater terms when they cross the selected boundary.

Rainfall volume

Rainfall volume over a catchment:

V=P_dA

where $P_d$ is rainfall depth and $A$ is catchment area.

If only a runoff coefficient $C_r$ is used:

V_{runoff}=C_rP_dA

The coefficient is a simplification. It depends on imperviousness, soil condition, slope, vegetation, surface storage, rainfall intensity, and antecedent moisture.

Rational method peak runoff

A common screening equation for small catchments is:

Q_p=C_r i A

where $Q_p$ is peak runoff, $C_r$ is runoff coefficient, $i$ is rainfall intensity for the selected duration and return period, and $A$ is drainage area.

Use local unit conventions carefully. The method assumes the rainfall duration is at least the time of concentration and is not appropriate for every catchment scale or storage condition.

Infiltration volume

Approximate infiltration volume:

V_I=fA\Delta t

where $f$ is infiltration rate, $A$ is infiltration area, and $\Delta t$ is time.

If infiltration capacity varies with time, use:

V_I=\int_0^T f(t)A\,dt

Soil infiltration is not constant. It changes with moisture, compaction, vegetation, clogging, surface condition, and groundwater level.

Flow rate

Volumetric flow:

Q=vA

Mass flow:

\dot{m}=\rho Q

For a circular pipe:

\displaystyle A=\frac{\pi D^2}{4}

Average velocity:

\displaystyle v=\frac{Q}{A}

Velocity affects head loss, sediment transport, erosion, noise, water age, and transient pressure.

Continuity at junctions

For an incompressible steady-flow junction:

\displaystyle \sum Q_{in}=\sum Q_{out}

With storage:

\displaystyle \frac{dS}{dt}=\sum Q_{in}-\sum Q_{out}

Use this relation to check nodes in distribution systems, sewer networks, detention basins, tanks, and treatment units.

Pollutant loading

Mass loading rate:

\dot{M}=QC

where $Q$ is flow and $C$ is concentration.

Total load over time:

M=\int_0^T Q(t)C(t)\,dt

For discrete samples:

M\approx \sum_i Q_iC_i\Delta t_i

Concentration without flow does not define total pollutant load. Composite sampling and flow-weighted methods are often needed.

Detention and storage sizing

Required storage from inflow and outflow hydrographs:

S_{req}=\max \int_0^t \left(Q_{in}-Q_{out}\right)dt

For discrete data:

S_k=S_{k-1}+(Q_{in,k}-Q_{out,k})\Delta t

The required volume is the maximum positive storage during the event.

Available rectangular basin volume:

V=LWH

where $L$ is length, $W$ is width, and $H$ is usable water depth.

Freeboard, sediment storage, side slopes, outlet control, maintenance access, and emergency spillway capacity must be handled separately.

Hydrostatic pressure

Pressure increase with depth:

p=\rho gh

Force on a horizontal area at uniform pressure:

F=pA

Hydrostatic pressure matters for tanks, wet wells, retaining structures, buried chambers, pipe thrust, groundwater uplift, and inspection safety.

Reynolds number

Pipe Reynolds number:

\displaystyle Re=\frac{\rho vD}{\mu}

Common screening ranges:

Re<2300 \quad \text{laminar}

Re>4000 \quad \text{turbulent}

Most full-pipe water infrastructure operates turbulently at normal service flow, but low-flow, small-diameter, viscous, or treatment-process conditions can differ.

Head loss and pressure drop

Darcy-Weisbach form:

\displaystyle h_f=f\frac{L}{D}\frac{v^2}{2g}

Pressure drop from head loss:

\Delta p=\rho gh_f

Minor loss:

\displaystyle h_m=K\frac{v^2}{2g}

Total dynamic head for a simple pumped line:

\displaystyle H_{TDH}=H_{static}+h_f+\sum h_m+\frac{\Delta p_{required}}{\rho g}

Check pipe roughness, fittings, valves, entrance and exit losses, aging, fouling, air pockets, and operating range.

Orifice and outlet control

Idealized orifice discharge:

Q=C_dA\sqrt{2gH}

where $C_d$ is discharge coefficient, $A$ is orifice area, and $H$ is head across the opening.

This relation is useful for outlet controls, tanks, basins, and hydraulic structures, but coefficients depend on geometry, approach conditions, submergence, clogging, and maintenance.

Pumping power

Hydraulic power:

P_h=\rho gQH

Input power with pump and motor efficiency:

\displaystyle P_{in}=\frac{\rho gQH}{\eta}

where $\eta$ is combined efficiency.

Energy over operating time:

E=P_{in}t

Pump checks should include minimum flow, maximum flow, net positive suction conditions, cycling, standby capacity, controls, surge, and backup power.

Water age and residence time

Nominal residence time:

\displaystyle \tau=\frac{V}{Q}

For storage tanks, treatment basins, and pipes, residence time affects disinfection, settling, biological activity, odor, corrosion, water quality, and treatment performance.

Nominal residence time is not the same as residence-time distribution. Short-circuiting, dead zones, stratification, and intermittent operation can change performance.

Reliability screening

If independent component availability is $A_i$ , simple series availability is:

A_{series}=\prod_i A_i

For two independent redundant components in parallel:

A_{parallel}=1-(1-A_1)(1-A_2)

These are simplified checks. Real water systems often have common-cause failures such as flooding, shared electrical panels, blocked access, common controls, or the same maintenance error.

Risk priority number

A simple screening metric is:

RPN=SOD

where $S$ is severity, $O$ is occurrence, and $D$ is detection ranking.

Use this only as a prioritization aid. It should not replace hydraulic analysis, public-health risk assessment, regulatory compliance, or emergency planning.

Practical checklist

Use these formulas with a short review checklist:

Define boundary, design event, and operating condition.
Close the water balance before sizing equipment.
Separate average, peak, wet-weather, dry-weather, and emergency flow cases.
Check storage, outlet control, overflow routing, and drain-down time.
Check pipe velocity, head loss, pressure, transients, and pump power.
Pair concentration data with flow data for pollutant loading.
Include maintenance, clogging, sediment, sensor quality, and backup operation.
Confirm that failure paths protect people, property, and receiving waters.

These equations are screening tools. Environmental water systems also depend on field inspection, monitoring, maintenance, public-service reliability, and local regulatory requirements.

REF

Disciplines