How Mathematics Improves The World | Getting Clean Water to Every Tap at the Right Pressure
Turn a tap in a high-rise flat.
Water arrives.
Not too slowly.
Not with so much pressure that pipes and fittings are overstressed.
It should still arrive when thousands of neighbours are showering.
It should still arrive when a major pipe is isolated for repair.
It should still have acceptable water quality after travelling through kilometres of underground pipes, storage tanks, valves and pumping stations.
And the system has to do this every hour of every day while demand changes.
Morning peak.
Midday lull.
Evening peak.
Firefighting demand.
Pipe leak.
Pump outage.
Tank level falling.
The network does not have one operating condition.
It is a hydraulic system changing through time.
Mathematics is how engineers make that invisible system legible.
Quick Read
A drinking-water distribution network contains reservoirs, storage tanks, pumps, valves, junctions and pipes. Engineers need to know the flow in every pipe, pressure at every node, tank levels over time and sometimes the movement of disinfectant or contaminants through the network. The U.S. Environmental Protection Agency’s EPANET software is used worldwide for exactly these problems, modelling hydraulic and water-quality behaviour in pressurised pipe networks.
The Mathematics begins with conservation of mass: at a junction, inflow minus outflow must equal demand or storage change. Energy relationships connect pressure, elevation, velocity and pump head. Pipe friction creates head loss, described through equations such as Darcy–Weisbach or empirical formulas such as Hazen–Williams. Pumps add head. Valves remove or regulate it. Tanks change system head as their water level rises and falls.
The network equations are coupled and nonlinear. Change one valve and flows redistribute through many pipes. Increase demand in one district and pressures change elsewhere. Engineers therefore solve the entire network iteratively. They calibrate models against pressure and flow sensors, simulate emergencies, optimise pump schedules, identify likely leaks and plan pipe renewal.
Singapore provides a strong local example. PUB describes a potable-water pipe network of roughly 6,000 km and uses permanent acoustic leak-detection sensors, data analytics and condition assessment to identify at-risk pipes. Its smart-water initiatives monitor pressure, flow and water quality so network operations can be observed rather than guessed.
One-sentence answer: Mathematics improves the world by converting a hidden network of pipes, pumps and changing demand into a solvable hydraulic model, allowing water utilities to maintain usable pressure, reduce leakage, minimise energy use and keep clean water moving reliably from source to every tap.
Pressure Is Stored Energy Per Volume
Water pressure feels like force.
Open a tap and it pushes water out.
In hydraulics, pressure is one form of mechanical energy.
It can be expressed as pressure head:
hp = p / ρg
where p is pressure, ρ is water density and g is gravitational acceleration.
Pressure head has units of length.
This is convenient because elevation is also measured in length.
A water engineer can therefore compare pressure energy and gravitational elevation in the same unit: metres of water head.
A high reservoir can supply pressure without a pump because gravity converts elevation head into pressure head as water descends.
Bernoulli’s Equation: Track Mechanical Energy Along the Water
For ideal steady incompressible flow, Bernoulli’s equation relates elevation, pressure and velocity:
z + p/ρg + v²/2g = constant
Real pipe networks are not ideal.
Friction removes mechanical energy.
Pumps add energy.
Valves dissipate it.
So a more practical energy equation becomes:
head at A + pump head = head at B + friction losses + valve losses
Every pipe path is an energy accounting problem.
Conservation of Mass: Water Cannot Disappear at a Junction
Take a pipe junction with three incoming pipes and two outgoing pipes.
If the node has no local storage, mass conservation requires:
sum of inflows − sum of outflows = local demand
This looks simple.
Now apply it to tens of thousands of junctions simultaneously.
Each pipe connects two nodes.
Every pipe flow depends on the head difference between its endpoints.
Every node head depends on flows through connected pipes.
The equations lock together across the whole network.
A water utility is running a giant coupled system of simultaneous nonlinear equations underground.
Friction: Long Pipes Lose Pressure
Water rubbing against a pipe wall dissipates mechanical energy.
The Darcy–Weisbach equation expresses friction head loss as:
hf = f (L/D) (v²/2g)
where f is friction factor, L is pipe length, D is diameter and v is average velocity.
Several lessons appear immediately.
- Longer pipe means more loss.
- Smaller diameter increases loss strongly.
- Higher velocity increases loss roughly with velocity squared through this relation.
- Rougher pipe walls increase friction factor.
A pipe that was adequate when new can become hydraulically worse as internal roughness grows through ageing, deposits or corrosion depending on material and water conditions.
Hazen–Williams: An Engineering Shortcut With Boundaries
Water utilities often use the empirical Hazen–Williams relation for pressurised water pipes.
It is convenient because it avoids explicit viscosity and Reynolds-number calculations.
But it is empirical.
It is not a universal fluid-mechanics law.
EPANET supports Hazen–Williams, Darcy–Weisbach and Chezy–Manning formulations for head loss.
The choice of formula should fit the engineering context and modelling assumptions.
A simpler equation is useful only when its domain of validity is understood.
Pipe Diameter Is Expensive and Powerful
A larger pipe costs more material, excavation and installation.
But a larger diameter dramatically reduces friction loss for the same flow.
This creates a life-cycle optimisation problem.
Small pipe:
- cheaper to build;
- higher friction;
- greater pumping energy;
- less spare capacity.
Large pipe:
- more expensive initially;
- lower friction;
- lower operating energy;
- more future capacity;
- possibly longer water age if oversized and demand is low.
There is no automatically best diameter.
The answer depends on demand, energy cost, future growth, water quality and reliability.
Pumps: Add Head Where Gravity Is Not Enough
A pump increases water head.
But the amount of head it adds depends on flow.
A pump curve shows this relationship.
At low flow, head may be high.
At higher flow, available head usually falls.
The pipe network itself has a system curve: the head required to overcome static elevation and friction at each flow rate.
The operating point occurs where pump curve and system curve intersect.
A pump cannot choose its own flow independently.
It negotiates with the network hydraulically.
Variable-Speed Pumps: Change the Curve Instead of Accepting It
Variable-speed drives let pump rotational speed change with demand.
Pump affinity laws give useful approximate scaling:
- flow ∝ speed;
- head ∝ speed²;
- power ∝ speed³.
Reduce speed modestly and power demand can fall sharply.
This makes variable-speed control valuable when demand varies through the day.
But pumps must still operate inside efficient and safe regions.
Optimisation cannot ignore cavitation, minimum flow or motor constraints.
Tanks: Store Water and Hydraulic Flexibility
A storage tank does more than hold water.
It decouples supply and demand in time.
Pumps can fill the tank during lower-demand or cheaper-electricity periods.
The tank can supply demand during peaks.
Its elevated water level contributes pressure.
It provides reserve for outages or firefighting depending on design.
But tank operation has a water-quality trade-off.
Leave water too long and disinfectant residual may decay while water age increases.
A well-operated tank balances hydraulic resilience with water turnover.
Valves: Sometimes the Network Needs Deliberate Resistance
Engineers normally try to reduce friction.
Valves deliberately create it.
Pressure-reducing valves protect low-elevation zones from excessive pressure.
Flow-control valves limit flow.
Check valves prevent reverse flow.
Isolation valves allow sections to be removed for maintenance.
Network design is not “make every path as easy as possible”.
It is “shape the energy so pressure and flow remain acceptable everywhere”.
High Pressure Is Not Automatically Better
Low pressure causes weak service and can make firefighting demand difficult to meet.
Excessive pressure creates its own problems.
Leaks flow faster.
Pipe bursts may become more likely in vulnerable assets.
Customer fixtures experience greater stress.
Water loss increases.
The objective is not maximum pressure.
It is sufficient pressure inside a safe operating envelope.
Elevation Changes Everything
Two houses on the same pipe can experience different pressure if one sits on a hill.
Every metre of elevation gain consumes roughly one metre of hydraulic head.
A low district may need pressure reduction.
A hilltop district may need pumping or a separate pressure zone.
Topography therefore partitions cities into hydraulic zones.
Maps become pressure maps.
Loops Make Networks More Resilient
A tree-like network has one path from source to customer.
Break that path and the customer loses supply.
A looped network provides multiple paths.
Close one pipe and flow can reroute.
That improves resilience.
It also makes the hydraulics harder because flows choose paths according to resistance rather than following a single predetermined branch.
Network redundancy creates mathematical coupling.
Flow Splitting: Water Chooses the Easier Route
Two parallel pipes connect the same nodes.
How much water flows through each?
The head loss between the common endpoints must be equal along both routes.
The lower-resistance path carries more flow.
Increase roughness in one pipe and flow redistributes automatically.
This is exactly why one ageing main can shift demand onto neighbours before anyone notices physically.
A calibrated network model can reveal that redistribution.
Demand Is Not Constant
People wake up.
Shower.
Cook.
Go to work.
Factories start.
Schools operate.
Demand changes over hours, days and seasons.
EPANET supports multiple demand categories and time patterns at nodes.
An extended-period simulation advances through time, updating demands, tank levels, pump states and network hydraulics repeatedly.
One static snapshot cannot tell whether the system survives the morning peak.
Demand Forecasting: Tomorrow’s Pump Schedule Depends on People
Utilities forecast water demand using historical consumption, weather, day of week, industrial schedules and other factors.
The forecast helps determine:
- which pumps run;
- when tanks fill;
- how much reserve is held;
- whether unusual demand indicates leakage.
A forecast is never exact.
Operations therefore require margin.
As with power grids, infrastructure should survive being slightly wrong about tomorrow.
EPANET: Turn the Underground City Into a Graph
A water network can be represented as a graph.
Nodes represent junctions, tanks and reservoirs.
Edges represent pipes, pumps and valves.
Each edge carries parameters.
Length.
Diameter.
Roughness.
Pump curve.
Valve setting.
Each node carries elevation and demand.
EPA’s EPANET solves this network repeatedly to compute flow, pressure, tank level, energy use and water-quality variables.
A city becomes a mathematical graph with physics on its edges.
Hydraulic Solver: Guess, Correct, Repeat
The equations are nonlinear because head loss depends nonlinearly on flow.
So solvers iterate.
Start with estimated flows.
Calculate continuity and energy residuals.
Update heads and flows.
Repeat until mismatch becomes acceptably small.
Convergence is necessary.
It is not enough.
A converged model using wrong pipe diameters, wrong demands and wrong valve states is still wrong.
Calibration: Make the Model Behave Like the Real Network
Engineers compare model predictions with measured pressure and flow.
If the model predicts 45 m head at a sensor and reality measures 31 m, something deserves investigation.
Possible causes:
- pipe roughness is wrong;
- a valve is partly closed;
- demand allocation is wrong;
- network records are inaccurate;
- a leak exists;
- sensor data is faulty.
Calibration adjusts uncertain parameters until the model reproduces observed behaviour within acceptable limits.
But over-calibration is dangerous.
A model can be forced to fit one day’s data by assigning physically absurd roughness values.
A calibrated parameter should remain physically credible.
Sensors: The Network Needs a Nervous System
Pressure sensors reveal hydraulic state.
Flow meters reveal movement.
Tank-level sensors reveal storage.
Water-quality sensors track conductivity, turbidity, disinfectant residual or other variables depending on deployment.
Acoustic sensors listen for leaks.
PUB has described smart-water systems that collect near-real-time pressure, flow and water-quality data, and in 2025 reported 1,500 permanent leak-detection sensors across its pipe network.
The model predicts.
The sensors answer.
That loop creates operational intelligence.
Leak Detection: Water Leaving the Network Without Being Metered
A small underground leak can run for months without surfacing.
Utilities look for discrepancies.
Night flow is unexpectedly high.
A district pressure falls.
Acoustic vibration appears on a pipe.
Measured flow differs from the hydraulic model.
Leakage becomes an inverse problem:
Which hidden leak location best explains the pressure and flow changes we observed?
Optimisation and statistical inference can rank candidate locations.
Acoustic Leak Detection: Listen for Water Escaping
Pressurised water escaping through a crack generates vibration and sound.
Place two acoustic sensors along a pipe.
The leak sound reaches each at a slightly different time.
Cross-correlate the signals.
Estimate time delay.
Combine it with wave speed and sensor spacing.
The leak position can be estimated.
PUB describes using acoustic data loggers, noise correlators and smartphone-based acoustic devices to locate leaks.
Sound becomes geometry underground.
Minimum Night Flow: The City Sleeps, Leaks Do Not
During deep night, legitimate customer demand falls.
Leakage continues approximately all the time, influenced by pressure.
Utilities measure minimum night flow in district-metered areas.
Subtract expected legitimate night use.
The remaining excess provides an estimate of leakage.
One quiet hour becomes a diagnostic window.
Pressure Management: Reduce Leaks Without Starving Customers
Leak flow usually increases with pressure.
Reduce pressure and leakage falls.
But reduce it too much and service deteriorates.
Pressure-reducing valves and variable-speed pumps can keep pressure near a target envelope as demand changes.
This is feedback control.
Measure downstream pressure.
Adjust valve or pump.
Measure again.
Infrastructure reliability becomes a control problem.
Water Hammer: Flow Cannot Stop Instantly
Close a valve too quickly.
Moving water has momentum.
The sudden deceleration creates a pressure wave that travels through the pipe.
This is water hammer.
The pressure surge can be many times ordinary operating pressure.
Pipes can rupture.
Supports can fail.
Valves can be damaged.
Transient hydraulic models solve wave equations through time, often using the method of characteristics.
The steady-state network is not enough.
Fast events require dynamic Mathematics.
The Joukowsky Relation: Fast Velocity Change Creates Fast Pressure Change
A classic estimate for water-hammer pressure change is:
Δp = ρ a Δv
where a is pressure-wave speed and Δv is change in velocity.
A velocity change of only 1 m/s with wave speed near 1,000 m/s creates pressure change of order one megapascal.
That is roughly ten bar.
This is why pump trips and valve closures need controlled timing and surge protection.
Fire Flow: The Network Must Survive an Unusual Peak
Ordinary domestic demand may be modest.
Firefighting can require very large flow at one location.
Open hydrants.
Pressure drops.
Can the network still maintain a required residual pressure?
EPANET can perform fire-flow analysis under minimum pressure constraints.
Water network design therefore considers rare emergency demand, not only average household use.
Pressure-Driven Demand: When Low Pressure Means People Receive Less Water
Traditional hydraulic models sometimes treat nodal demand as fixed.
Even if pressure collapses, the model still forces the full demand to leave the node.
Real customers cannot draw full flow at zero pressure.
Pressure-driven analysis links delivered demand to available pressure.
EPA’s current EPANET supports pressure-dependent demands.
This becomes especially important during failures and emergency scenarios.
When the system is stressed, the simple model fails exactly where accurate prediction matters most.
Water Quality: Hydraulics Determines What Water Arrives Where
Water quality is not independent of pipe flow.
Flow direction determines where water travels.
Velocity determines travel time.
Tank mixing affects age.
Disinfectant residual decays through bulk and wall reactions.
Contaminants can spread along hydraulic paths.
EPANET models chemical concentration, water age and source tracing precisely because hydraulic and quality models are coupled.
Water Age: Too Much Storage Can Become Stale Infrastructure
A large tank improves emergency storage.
But if demand is low, some water remains for a long time.
Disinfectant residual may decay.
Temperature can rise.
Water quality risk can increase depending on system conditions.
Water age modelling tracks how long water has spent in the network.
The best hydraulic design is not automatically the best water-quality design.
Source Tracing: Which Reservoir Supplied This Tap?
A network may receive water from several treatment plants or reservoirs.
At one customer node, the supply could be 70% from Source A and 30% from Source B.
Change pump operation and the mixture changes.
Source-tracing models follow that blending through time.
This helps investigate water-quality events and plan operations.
The tap receives water.
Mathematics tells us where it came from.
Contamination Response: Reverse the Network
Suppose a sensor detects unusual water quality at one location.
Where could the contaminant have entered?
Which customers may be affected next?
Hydraulic and quality models can simulate candidate injection points and compare predicted sensor patterns with observed data.
This is another inverse problem.
Observation downstream.
Possible source upstream.
EPA notes that EPANET can support contamination-threat and resilience studies.
Sensor Placement: Where Should We Measure?
Sensors are expensive.
Put one at every junction and costs become enormous.
Put too few and important failures remain invisible.
Optimisation can choose sensor locations to maximise detection probability, minimise expected detection time or improve hydraulic observability.
A good sensor location is not always where failures are most likely.
It may be where many possible failures produce distinguishable signatures.
Measurement design becomes part of network design.
Pipe Renewal: Replace the Right Pipe Before It Breaks
A city cannot replace every old pipe at once.
Utilities prioritise.
Age.
Material.
Failure history.
Soil.
Pressure.
Criticality.
Customer consequence.
PUB describes using data analytics and condition assessment to identify at-risk pipes for targeted renewal and reported more than 330 km renewed since 2016 in its 2025 update.
Asset management is a risk-ranking problem.
Criticality: The Pipe That Fails Least Often Can Matter Most
A small local pipe may fail frequently and affect twenty homes.
A large transmission main may fail rarely and affect an entire district.
Risk is not failure probability alone.
risk ≈ probability × consequence
Criticality analysis simulates pipe outages and measures how many customers lose pressure, how much demand becomes unmet and whether alternative paths exist.
Network structure changes the consequence of failure.
N-1 Thinking: Can the System Lose One Important Component?
Power grids use contingency analysis.
Water networks can use similar resilience thinking.
Close one major pipe.
Turn off one pump.
Take one reservoir out of service.
Recalculate.
Do pressure violations appear?
Does a tank drain too quickly?
Which valve changes restore service?
A network becomes resilient when failure scenarios are tested before failure.
Emergency Valve Sequencing: The Order Matters
A pipe bursts.
Operators isolate it.
But closing the wrong valve first can create a pressure surge or cut supply to too many customers.
Network simulators can evaluate valve-operation sequences before operators execute them.
PUB has described simulator capabilities for contingency planning and valve operating sequences in its smart-water programmes.
Order is part of the solution.
Pump Scheduling: Electricity Price Becomes a Hydraulic Variable
A pump can run now or later.
If a tank has enough storage, utilities can shift pumping toward lower-cost electricity periods.
But tank levels must remain within limits.
Pressure must remain acceptable.
Water age must not become excessive.
Pumps have efficiency curves.
Frequent starts cause wear.
Pump scheduling becomes constrained optimisation over time.
Energy economics enters hydraulics.
Optimisation: Cheapest Is Not the Same as Safest
An optimiser could minimise electricity cost by letting tanks run almost empty.
That may reduce emergency reserve.
It could minimise pressure to reduce leakage.
That may reduce firefighting capability.
It could maximise turnover to reduce water age.
That may require expensive pumping.
The objective function is therefore multi-objective.
- energy;
- pressure;
- leakage;
- reliability;
- water quality;
- asset wear;
- emergency reserve.
Mathematics reveals the trade-off surface.
Water utility policy chooses where to operate on it.
Digital Twins: A Hydraulic Model That Keeps Listening
A static hydraulic model represents the network at one calibrated state.
A digital twin aims to remain connected to live operational data.
Sensor readings update demands.
Valve states update topology.
Tank levels update storage.
The model predicts near-future pressure and flow.
An anomaly triggers investigation.
The value is not a 3D picture of pipes.
The value is a continuously corrected model of hydraulic state.
Machine Learning: Predict Failures, But Keep Physics
Machine learning can predict demand, classify leak sounds, rank pipes for renewal and detect anomalies.
A neural network may learn relationships too complex for hand-built rules.
But a purely statistical model can violate hydraulics.
It may predict more flow leaving a node than enters.
It may extrapolate poorly during a major burst never seen in training.
Hybrid physics-informed systems combine data-driven learning with mass and energy constraints.
The best algorithm does not replace conservation laws.
It learns inside them.
A Classroom Thought Experiment: The Three-Tap Network
Draw one elevated tank connected to a pipe that splits into three taps.
Give each branch a different resistance.
Open one tap.
Pressure remains high.
Open all three.
Flow increases and head loss rises.
Now make one branch narrower.
Its flow falls.
Now lower the tank.
All pressures fall.
The child discovers:
- conservation;
- resistance;
- pressure;
- network interaction;
- the importance of elevation.
Primary Mathematics: Water Distribution Begins With Measurement
Primary students already learn:
- volume;
- capacity;
- rate;
- time;
- length;
- graphs;
- ratio;
- measurement.
Litres per minute is a flow rate.
Tank volume is storage.
A graph of household demand across a day is already an operations problem.
The sophistication comes later.
The foundation is ordinary measurement done carefully.
Secondary Mathematics: The Pipe Network Becomes Algebra and Graphs
Secondary students add:
- simultaneous equations;
- graphs and networks;
- quadratic relationships;
- functions;
- statistics;
- probability;
- optimisation.
Flow balance becomes simultaneous equations.
Friction becomes nonlinear functions.
Pipe connections become a graph.
Demand forecasts become statistics.
Leak location becomes inference.
Advanced Mathematics: The Water Network as a Dynamic System
Modern distribution engineering draws on:
- nonlinear algebra;
- graph theory;
- numerical analysis;
- optimisation;
- probability;
- control theory;
- inverse problems;
- partial differential equations for transients;
- statistical estimation;
- machine learning.
Steady hydraulics solves a nonlinear network.
Water hammer solves wave propagation.
Leak detection solves an inverse problem.
Pump scheduling solves constrained optimisation.
Sensor placement solves observability.
A tap is connected to almost every branch of applied Mathematics.
Why This Improves the World
1. It keeps pressure usable across an entire city
Hydraulic models show where pressure will be too high or too low under changing demand.
2. It finds leaks sooner
Pressure, flow and acoustic data can be compared with model expectations to locate hidden losses.
3. It reduces energy use
Pump scheduling and variable-speed control can meet demand with less wasted head and electricity.
4. It protects water quality
Water-age, source-tracing and disinfectant models connect hydraulic operation with what customers actually receive.
5. It makes emergency scenarios testable
Utilities can simulate pipe failures, pump outages and fire flows before the real network is stressed.
6. It turns maintenance into prioritised risk reduction
Asset models rank pipes by failure likelihood and consequence rather than replacing infrastructure blindly by age alone.
What Mathematics Does Not Do
A hydraulic model does not repair a broken pipe.
It does not make inaccurate asset records correct.
It does not guarantee sensor data is trustworthy.
It does not replace water-quality laboratory testing.
It does not decide how much redundancy society should pay for.
It does not make every pump schedule equally robust to failure.
And a perfectly converged EPANET model can still be wrong if the physical network represented inside it is wrong.
Frequently Asked Questions
What is EPANET?
EPANET is public-domain software developed by the U.S. Environmental Protection Agency for modelling hydraulic and water-quality behaviour in pressurised drinking-water distribution networks containing pipes, pumps, valves, tanks and reservoirs.
Why does water pressure fall when many taps open?
Higher total flow increases friction losses through pipes and fittings. Unless pumps, tank levels or other sources provide enough head, pressure at customer nodes falls.
Why are water tanks elevated?
Elevation creates gravitational head, helping maintain pressure without continuous pumping. Tanks also provide storage that separates production from short-term demand.
How are underground leaks found?
Utilities use several methods including minimum-night-flow analysis, pressure and flow modelling, acoustic sensors, correlators, field inspection and data analytics. Combining independent methods improves confidence.
What is water hammer?
Water hammer is a transient pressure wave created when water velocity changes rapidly, such as during a sudden valve closure or pump trip. Surge analysis helps engineers design protective equipment and safe operating sequences.
Why not keep pressure very high everywhere?
Excessive pressure increases leakage, stresses infrastructure and wastes pumping energy. Utilities aim for sufficient pressure within defined operating limits rather than maximum pressure.
Sources and Further Reading
- U.S. Environmental Protection Agency, EPANET, describing hydraulic and water-quality modelling of drinking-water distribution systems.
- U.S. Environmental Protection Agency, EPANET 2.2 User Manual, including pipes, pumps, valves, head-loss equations, pressure-dependent demand and water-quality simulation.
- PUB, Singapore’s National Water Agency, Keeping Singapore’s Potable Water Pipe Network in Good Order, on pipe renewal, acoustic leak detection and network monitoring.
- PUB, Digitalising Water — Sharing Singapore’s Experience, describing smart-water-grid monitoring and operational analytics.
Continue Through eduKateSG
Continue with How Mathematics Works. Compare this article with Keeping Electricity Flowing Through a Changing Grid: both are infrastructure networks whose invisible state must be estimated, constrained and corrected continuously. Also compare it with Testing a Structure Before Reality Has To, where simulation lets failure scenarios happen mathematically before they happen physically.
Final Thought: The Tap Is the End of a Calculation You Never See
A reservoir level changes.
A pump starts.
A valve regulates pressure.
A tank fills.
A district wakes up.
Demand rises.
Flow redistributes.
One pressure sensor reads low.
A leak-detection model becomes suspicious.
A crew finds the pipe before a small leak becomes a large burst.
And upstairs, someone fills a glass.
The ordinary act works because an invisible network has been measured, modelled, balanced and repaired for decades.
Mathematics is not the water.
It is one of the reasons the water arrives.