Groundwater Flow and Darcy's Law

Flow through rock and soil, not through a pipe

Water moving underground does not flow through an open channel — it squeezes through the connected pore spaces and cracks of an aquifer, a saturated layer of rock or sediment permeable enough to yield usable water. Henry Darcy discovered the governing relationship in 1856 while studying sand filters for the water supply of Dijon: the flow rate through a porous medium is proportional to the pressure (or elevation head) difference driving it, and inversely proportional to the distance the water has to travel.

live demo · a cone of depression forming around a pumped well● LIVE

Darcy's law

Q = -K · A · (dh/dl)

Q  = volumetric flow rate
K  = hydraulic conductivity (property of the rock/soil AND the fluid)
A  = cross-sectional area flow passes through
dh/dl = hydraulic gradient — change in head per unit distance

The minus sign just says water flows from high head to low head, same as heat flows from hot to cold. Hydraulic conductivity K spans an enormous range — clean gravel can be K ≈ 1 m/s, dense clay can be K ≈ 10⁻¹¹ m/s, ten orders of magnitude apart — which is why some aquifers can supply a city's wells and others barely seep.

Two kinds of aquifer, two different responses

A confined aquifer sits sandwiched between impermeable layers and is under pressure; pumping it lowers the potentiometric surface (the pressure head) without the aquifer itself draining — water is released almost entirely by the small compressibility of the rock and water. An unconfined (water-table) aquifer has no upper confining layer; pumping physically drains pore water and lowers the water table itself, releasing far more water per unit drawdown because it is emptying actual pore space rather than just relieving pressure. That difference in storativity — how much water a given drop in head actually yields — is why confined aquifers respond to pumping almost instantly across large distances, while unconfined aquifers respond more slowly but yield much more water for the same drawdown near the well.

The cone of depression

Pump a well continuously and the head drops fastest right at the well and less so with distance, carving out a roughly conical depression in the water table or potentiometric surface, radially symmetric around the well. For steady radial flow into a well fully penetrating a confined aquifer, integrating Darcy's law over concentric cylinders around the well gives the Thiem equation, relating drawdown at two radii to the pumping rate and the aquifer's transmissivity T = K·b (conductivity times aquifer thickness):

s1 - s2 = ( Q / (2π T) ) · ln(r2 / r1)

s1, s2 = drawdown at radii r1, r2 from the well
Q = pumping rate,  T = transmissivity = K·b

The logarithmic term means drawdown falls off slowly with distance close to the well and much faster far away — the classic funnel shape, steepest right at the borehole and flattening into a gentle, wide depression that can still be measurable a kilometre or more from a heavily pumped well in a very transmissive aquifer.

Why steady state is only half the story

The Thiem equation describes the final, steady-state cone after pumping has run long enough for the drawdown to stop changing — but real wells are switched on and the cone grows outward and deepens over time before it stabilises (if it ever fully does). Charles Theis solved the transient version in 1935 by borrowing the mathematics of heat conduction, introducing storativity S (the fraction of aquifer volume that yields water per unit drop in head) to describe exactly how fast that growth happens. Confined aquifers have very low S (roughly 10⁻⁵ to 10⁻³) and their cones expand outward rapidly with barely any local drawdown storage to slow them; unconfined aquifers have S closer to their drainable porosity (0.1 to 0.3) and their cones grow far more slowly because so much locally stored water has to physically drain before the depression can spread — the same physical distinction between confined and unconfined behaviour, now expressed as how fast rather than how much.

Frequently asked questions

What is the difference between hydraulic conductivity and permeability?

Permeability is a property of the rock or soil alone, describing how easily any fluid can move through its pore structure. Hydraulic conductivity, the K in Darcy's law, folds in the properties of the specific fluid too — its viscosity and density — so it specifically describes how easily water moves through that material, which is why K is the more directly useful quantity for groundwater calculations.

Why does pumping a confined aquifer lower the water level without draining any pore space?

Because a confined aquifer stays fully saturated under pressure the whole time. Pumping lowers the potentiometric surface, the pressure head, and water is released mainly through the small compressibility of the water and the slight compression of the aquifer skeleton, not by air moving in to replace drained pore water the way it does in an unconfined aquifer.

Why does the cone of depression widen faster in a confined aquifer than an unconfined one?

Confined aquifers have a very low storativity, meaning only a tiny amount of water is released for a given drop in head, so the pressure disturbance from pumping has little local storage to draw down and spreads outward quickly. Unconfined aquifers have a much higher storativity close to their drainable porosity, so far more water has to physically drain from the pores near the well before the depression can extend outward, which slows its spread.

Try it live

Everything above runs in your browser — open Groundwater Flow and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Groundwater Flow simulation

Groundwater Flow: Darcy's Law and the Cone of Depression