Hi, I’m Kkoogongnam.
Today we’ll break down Darcy’s law in the simplest, most practical way — the way you actually use it in soil mechanics and groundwater flow. (vs seepage velocity)
If you understand just three things, you can solve most seepage/flow-through-soil problems:
- Total head (from Bernoulli’s concept)
- Hydraulic gradient (i = Δh / L)
- Darcy’s law (v = k i)
1) Bernoulli Equation → Total Head (Energy of Water)
From fluid mechanics, Bernoulli’s idea tells us the total head is composed of three parts:
In seepage through soil, flow inside pores is usually slow enough that the velocity head term is often negligible. So we commonly use the simplified total head expression:
Note: If water flow is strong enough to move soil particles (erosion/piping), velocity head may not be negligible.
2) Hydraulic Gradient (i) = Head Loss per Distance
The hydraulic gradient is simply the head difference per flow length. It is dimensionless (m/m or ft/ft). Here’s the standard visualization:
The head difference can be written as:
And the hydraulic gradient is:
L is the flow length (the distance water travels through the soil specimen or soil layer).
Flow regimes: why linearity matters
In many soils, seepage velocities are low, so flow is laminar and we observe a nearly linear relationship between velocity and hydraulic gradient. But in coarse gravel, fractured rock, or very high gradients, flow can transition toward non-linear behavior.
3) Darcy’s Law (v = k i) & Seepage Velocity
Now the main event: Darcy’s law expresses discharge (superficial) velocity through soil as:
- v = discharge velocity (a.k.a. Darcy velocity / specific discharge): flow rate per total cross-sectional area
- k = hydraulic conductivity (permeability coefficient)
- i = hydraulic gradient (Δh/L)
Important: v is NOT the actual pore-water velocity. Because water only flows through the void space, the real seepage velocity is higher.
Discharge velocity vs. seepage velocity (why v is “smaller”)
Let q be volumetric flow rate (m³/s or ft³/s). Then:
Where:
- vs = seepage velocity (actual velocity through voids)
- Av = void area in cross-section
- A = total area = Av + As
So seepage velocity becomes:
In practical terms, this is the key takeaway:
n is porosity. (If you prefer void ratio e, you can convert between them depending on soil mechanics context.)
4) Worked Example (Quick Calculation)
Let’s do a simple Darcy’s law calculation in the form engineers actually use.
- Hydraulic conductivity: k = 1×10-4 m/s
- Hydraulic gradient: i = 0.5
- Cross-sectional area: A = 0.02 m²
Step 1) Discharge velocity: v = k i = (1×10-4)×0.5 = 5×10-5 m/s
Step 2) Flow rate: q = vA = (5×10-5)×0.02 = 1×10-6 m³/s
If porosity is n = 0.35, then seepage velocity is vs = v/n ≈ 1.43×10-4 m/s.
5) Official / Authoritative References (Buttons)
If you want primary references (recommended for students and engineers), start here:
6) FAQ
1) What is Darcy’s law in simple terms?
Darcy’s law links flow through a porous material to hydraulic gradient: v = k i. More gradient → more flow, assuming laminar conditions.
2) Is Darcy velocity the real water velocity?
No. Darcy velocity (discharge velocity) uses total area. Real pore-water speed is higher: vs = v/n.
3) What does hydraulic conductivity (k) depend on?
It depends on soil structure (grain size, voids) and fluid properties. So k changes across soil types and even with temperature (viscosity effects).
4) When does Darcy’s law fail?
It becomes less accurate when flow is not laminar (e.g., very coarse materials, fractures, or high gradients causing non-linear behavior).
5) What’s the difference between permeability and hydraulic conductivity?
In many geotech contexts, “permeability coefficient” is used for hydraulic conductivity (k). Strictly, permeability can refer to intrinsic permeability (medium-only), while k includes fluid properties.
7) Wrap-up
That’s Darcy’s law in the cleanest practical workflow: Total head → hydraulic gradient → v = k i → seepage velocity vs = v/n.