Permeability and Seepage

Permeability and Seepage

Permeability is the property of soil that allows water to flow through its interconnected voids. This is critical for predicting settlement rates, designing drainage systems, and assessing the stability of dams and excavations.

Darcy's Law

In 1856, Henry Darcy demonstrated that the flow rate through saturated soil is proportional to the hydraulic gradient.

Darcy's Equation

Darcy's Law (Velocity)

$$
v = k i
$$

Flow Rate ( $q$ ):

Darcy's Law (Flow Rate)

$$
q = v A = k i A
$$

Seepage Velocity ( $v_s$ )

Water only flows through the voids, not the solids. Thus, the actual velocity is higher than the discharge velocity.

Seepage Velocity

$$
v_s = \frac{v}{n} = \frac{v (1+e)}{e}
$$

Permeability in Stratified Soils

Natural soil deposits are typically layered horizontally. Because water always seeks the path of least resistance, the overall permeability of the deposit depends heavily on the direction of flow.

Equivalent Permeability

Horizontal Flow (Parallel to Layers): Water flows primarily through the most permeable layer. The equivalent horizontal permeability (

k_H

) is the weighted average based on layer thickness.

Equivalent Horizontal Permeability

$$
k_H = \frac{k_1 H_1 + k_2 H_2 + \dots + k_n H_n}{H_1 + H_2 + \dots + H_n}
$$

Vertical Flow (Perpendicular to Layers): Water is forced to flow through every layer sequentially. The equivalent vertical permeability (

k_V

) is governed by the least permeable layer (the bottleneck).

Equivalent Vertical Permeability

$$
k_V = \frac{H_1 + H_2 + \dots + H_n}{\frac{H_1}{k_1} + \frac{H_2}{k_2} + \dots + \frac{H_n}{k_n}}
$$

Laboratory Tests for Permeability

Constant Head Test

Used for coarse-grained soils (gravels, sands) with high permeability (

k > 10^{-3} cm/s

Constant Head Permeability

$$
k = \frac{Q L}{A h t}
$$

Falling Head Test

Used for fine-grained soils (silts, clays) with low permeability (

k < 10^{-3} cm/s

Falling Head Permeability

$$
k = 2.303 \frac{a L}{A t} \log_{10} \left( \frac{h_1}{h_2} \right)
$$

Seepage Analysis

For complex 2D flow problems (e.g., under a dam), we use Flow Nets.

Flow Nets

A graphical solution to the Laplace equation for steady-state flow.

Flow Lines ( $\psi$ ): Paths followed by water particles.
Equipotential Lines ( $\phi$ ): Lines of constant total head.
Flow lines and equipotential lines intersect at 90°.
Form "curvilinear squares".

Seepage Quantity ( $q$ ) per unit width:

Seepage Quantity from Flow Net

$$
q = k H \frac{N_f}{N_d}
$$

Key Takeaways

Darcy's Law ( $v = ki$ ) is the foundation for flow in porous media.
Permeability ( $k$ ) varies by orders of magnitude (Gravel > Sand > Silt > Clay).
In stratified soils, horizontal permeability ( $k_H$ ) is usually much larger than vertical permeability ( $k_V$ ) because vertical flow is bottlenecked by the least permeable layer.
Constant Head tests are for high $k$ ; Falling Head tests are for low $k$ .
Flow Nets allow calculation of seepage quantity ( $q$ ), exit gradient, and uplift pressure under structures.
Seepage Velocity ( $v_s$ ) is always greater than discharge velocity ( $v$ ) because water flows only through the voids.

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Darcy's Law