Groundwater Flow
Theory: groundwater flowing to a well
Overview
Pumping from a well lowers the surrounding water table. Can we calculate how far and how much?
Yes. Knowing the aquifer's ability to transmit water (transmissivity) and how much water it releases as levels fall (storativity), the Theis equation gives drawdown versus time and distance. Continued pumping forms a cone of depression around the well that deepens and widens over time. Knowing the actual groundwater velocity also lets us estimate how long contaminated water takes to reach the well. Let's calculate for a specific aquifer.
The Theis equation
$s$ is drawdown, $Q$ pumping rate, $T$ transmissivity (conductivity × thickness), $S$ storativity. For small $u$ (near the well, long times) the logarithmic Cooper–Jacob approximation applies and drawdown grows with the log of time.
Velocity, travel time and radius of influence
A groundwater formula borrowed from heat conduction
In 1935 Theis of the US Geological Survey published an equation for how drawdown around a well evolves in time. Noticing that the groundwater flow equation has the same form as heat conduction, he applied, with help from the mathematician Lubin, a solution known from heat conduction. The equation let aquifer properties be inferred from pumping-test records and became the standard method of groundwater investigation.
Worked example: pumping a confined aquifer
Hydraulic conductivity 5×10⁻⁵ m/s, thickness 20 m (transmissivity 1×10⁻³ m²/s), storativity 1×10⁻⁴, pumping 10 L/s (36 m³/h).
Drawdown
| Time | 1 m from well | 10 m | 100 m | 1,000 m | Radius of influence |
|---|---|---|---|---|---|
| 1 h | 8.99 m | 5.33 m | 1.72 m | 0.00 m | 285 m |
| 1 day | 11.52 m | 7.86 m | 4.19 m | 0.74 m | 1.4 km |
| 10 days | 13.35 m | 9.69 m | 6.03 m | 2.38 m | 4.4 km |
The Cooper–Jacob approximation agreed with Theis within 1% wherever u < 0.01 (most points); at 1,000 m after 1 day (u = 0.29) it erred by 29%.
Groundwater velocity and travel time (porosity 0.25)
| Item | Value |
|---|---|
| Velocity at 10 m | 2.75 m/day |
| Velocity at 100 m | 0.28 m/day |
| From 50 m to the well | 45 days |
| From 100 m to the well | 182 days |
| From 200 m to the well | 727 days |
Drawdown reaches over 1 km in a day, yet the water itself takes half a year to travel 100 m.
Drawdown travels as a pressure change, so it spreads fast. In a confined aquifer water is released by slight compression of water and rock, so storativity is small and pressure changes propagate far quickly. The water particles themselves move only slowly through the pores. So nearby wells feel the effect at once, while contaminants take a long time to arrive. Travel time scales with distance squared — twice as far takes four times as long — and wellhead protection zones are often set from these travel times.
Analysis tips
- Fit pumping-test records with Theis or Cooper–Jacob to obtain transmissivity and storativity.
- Check confined or unconfined (unconfined thickness changes with drawdown).
- Nearby rivers and boundaries can be represented by superposing image wells.
- Solve multiple wells and heterogeneous aquifers numerically (FD/FE).
- For contaminant transport, include dispersion and sorption in arrival times.
“The neighbour's well ran dry”
When a factory began heavy pumping from a new deep well, a farmer's shallow well several hundred metres away saw its level drop and ran dry in summer. The factory had assumed the distance was enough, but in a confined aquifer pressure drops spread quickly and far. Re-estimating with Theis predicted several metres of drawdown even at the farm. An agreement was reached to adjust pumping by season and monitor levels in observation wells.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Darcy flux as velocity | Arrival time overestimated | Divide by porosity |
| Cooper–Jacob at large u | Large far-field error | Check u < 0.01 |
| Steady Thiem only | Time effects missed | Use Theis transiently |
| Ignoring boundaries | Wrong drawdown | Use image wells |
| Confined/unconfined mix-up | Storativity off by orders | Check geology |
I'd like to learn related topics.
The ground-source heat pump tool is also relevant. Related pages include Darcy flow, Forchheimer equation, Brinkman equation and packed-bed flow and heat transfer.
Related Topics
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