Darcy's law and groundwater movement

Darcy's law states that the rate of flow of water through a saturated porous medium is directly proportional to the hydraulic gradient and to the cross-sectional area of flow.

$$Q = K\,A\,i = K\,A\,\frac{\Delta h}{L}$$

where $Q$ = discharge ($m^3/s$), $K$ = hydraulic conductivity ($m/s$), $A$ = total cross-sectional area ($m^2$), $i = \Delta h / L$ = hydraulic gradient (dimensionless), $\Delta h$ = head loss over flow length $L$.

Discharge velocity $v = Q/A = K\,i$. The actual seepage velocity is $v_s = v/n$, where $n$ is porosity.

Relation of gradient and conductivity

Since $v = K\,i$, for a given gradient a higher $K$ gives a larger flow, and for a given flow the gradient is inversely proportional to $K$ ($i = v/K$): a low-$K$ layer needs a steep gradient to pass the same flow. Water moves from high head to low head; the gradient is the driving force, and $K$ (depending on grain size, sorting, fractures and fluid viscosity) is the ease of flow. $K$ is large for gravel (about $10^{-2}$ m/s) and small for clay (about $10^{-9}$ m/s).

Describing groundwater movement

Groundwater moves slowly, in laminar flow, from areas of high hydraulic head (recharge) to low head (discharge) in the direction of the gradient. Darcy's law is valid for laminar flow (Reynolds number below about 1-10).

Example

$K = 2\times10^{-4}$ m/s, head difference 5 m over 50 m, flow area 20 $m^2$:

$$\begin{aligned} i &= \frac{5}{50} = 0.1 \\ Q &= 2\times10^{-4}\times 20\times 0.1 = 4\times10^{-4}\ m^3/s \end{aligned}$$

Answer: $Q = 4\times10^{-4}\ m^3/s$ ($\approx 34.6\ m^3$/day).

Importance

• Estimating groundwater yield of wells and aquifers.
• Seepage under dams and from reservoirs and canals.
• Design of dewatering and drainage systems.
• Calculating transmissivity ($T = K b$) and aquifer testing.
• Groundwater contamination and flow modelling.