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)

Defines the discharge velocity of water through saturated soil as a function of permeability and hydraulic gradient.

v = k i

Variables

Symbol	Description	Unit
$v$	Discharge velocity (approach velocity)	cm/s
$k$	Coefficient of permeability (hydraulic conductivity)	cm/s
$i$	Hydraulic gradient (\Delta h / L)	-

Flow Rate ( $q$ ):

Darcy's Law (Flow Rate)

Calculates the total volumetric flow rate through a soil cross-section using Darcy's Law.

q = v A = k i A

Variables

Symbol	Description	Unit
$q$	Total flow rate (discharge)	-
$A$	Cross-sectional area perpendicular to flow	-
$v$	Discharge velocity	-
$k$	Coefficient of permeability	-
$i$	Hydraulic gradient	-

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

The actual velocity of water through the void channels; always greater than the macroscopic discharge velocity.

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

Variables

Symbol	Description	Unit
$v_{s}$	Seepage velocity	-
$v$	Discharge velocity	-
$n$	Porosity (V_v / V_t)	-
$e$	Void ratio	-

Since $n < 1$ , $v_s > v$ .

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

Weighted average permeability for flow parallel to layered strata; controlled by the most permeable layer.

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

Variables

Symbol	Description	Unit
$k_{H}$	Equivalent horizontal permeability	-
$k_{i}$	Permeability of layer i	-
$H_{i}$	Thickness of layer i	-

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

Equivalent permeability for flow perpendicular to layered strata; controlled by the least permeable layer.

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}}

Variables

Symbol	Description	Unit
$k_{V}$	Equivalent vertical permeability	-
$k_{i}$	Permeability of layer i	-
$H_{i}$	Thickness of layer i	-

Generally, natural deposits are strongly anisotropic, meaning $k_H$ is typically significantly greater than $k_V$ .

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

Laboratory formula for determining permeability of coarse-grained soils under a constant hydraulic head.

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

Variables

Symbol	Description	Unit
$k$	Coefficient of permeability	-
$Q$	Volume of water collected	cm³
$L$	Length of specimen	cm
$A$	Area of specimen	cm²
$h$	Constant head difference	cm
$t$	Time	s

Falling Head Test

Used for fine-grained soils (silts, clays) with low permeability ( $k < 10^{-3} cm/s$ ).

Falling Head Permeability

Laboratory formula for measuring the low permeability of fine-grained soils using a falling head standpipe.

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

Variables

Symbol	Description	Unit
$k$	Coefficient of permeability	-
$a$	Area of standpipe	cm²
$L$	Length of specimen	cm
$A$	Area of specimen	cm²
$t$	Time interval	s
$h_{1}$	Initial head in standpipe	-
$h_{2}$	Final head in standpipe	-

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

Calculates the total seepage flow per unit width through a soil mass using a graphically constructed flow net.

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

Variables

Symbol	Description	Unit
$q$	Total seepage quantity per unit width	-
$k$	Permeability of the soil	-
$H$	Total head loss (h_{upstream} - h_{downstream})	-
$N_{f}$	Number of flow channels (spaces between flow lines)	-
$N_{d}$	Number of potential drops (spaces between equipotential lines)	-

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

Darcy's Equation

Darcy's Law (Velocity)

Darcy's Law (Flow Rate)

Seepage Velocity (vsv_svs​)

Seepage Velocity

Permeability in Stratified Soils

Equivalent Permeability

Equivalent Horizontal Permeability

Equivalent Vertical Permeability

Laboratory Tests for Permeability

Constant Head Test

Constant Head Permeability

Falling Head Test

Falling Head Permeability

Seepage Analysis

Flow Nets

Seepage Quantity from Flow Net

Seepage Velocity ( $v_{s}$ )