Permeability and Seepage

Permeability

Permeability is the ease with which water flows through the interconnected voids of a soil. It controls:

• seepage through and under dams, levees and sheet piles, and uplift pressures,
• rate of consolidation settlement of clay layers,
• drainage of excavations, roads and retaining walls,
• yield of wells and dewatering design.

Darcy's law

For laminar flow through saturated soil:

$v$ = discharge (superficial) velocity — discharge divided by the total cross-sectional area; $i=h/L$ = hydraulic gradient; $k$ = coefficient of permeability (hydraulic conductivity), in m/s or cm/s.

The actual average velocity through the pores is the seepage velocity:

Darcy's law holds when flow is laminar — true for silts, clays and most sands; it may not hold for coarse gravels at high gradients (turbulent flow).

Typical permeability

Factors affecting permeability

The Kozeny–Carman expression shows the main influences:

1. Grain size — $k\propto D^2$ (Hazen uses $D_{10}$).
2. Void ratio — $k$ rises sharply with $e$ ($e^3/(1+e)$).
3. Properties of pore fluid — $k\propto {\gamma }_w/\mu$; warm water (lower viscosity) flows faster. Standard results are reported at 27 °C in Indian practice: $k_{27}=k_T{\mu }_T/{\mu }_{27}$.
4. Shape and arrangement of particles, soil structure — flocculated clays are more permeable than dispersed ones at the same void ratio.
5. Degree of saturation — entrapped air reduces $k$.
6. Adsorbed water in clays reduces the effective pore space.
7. Stratification — horizontal permeability usually far exceeds vertical.
8. Impurities / fines clog pores.

Hazen's empirical relation (clean uniform sands)

Laboratory tests

Constant-head test (coarse-grained soils)

Water flows under a steady head $h$ through a sample of length $L$ and area $A$; volume $Q$ is collected in time $t$:

Falling-head (variable-head) test (fine-grained soils)

Water from a standpipe of area $a$ falls from $h_1$ to $h_2$ in time $t$:

(Very low permeabilities are also found indirectly from consolidation test data: $k=c_v{m}_v{\gamma }_w$.)

Field pumping tests

Laboratory samples may not represent stratification and fissures; pumping-out tests measure average field permeability. Water is pumped at a steady rate $q$ and water levels are observed in two observation wells at radii $r_1$ and $r_2$.

Other field tests: pumping-in tests in bore holes, packer tests in rock.

Heads in soil water

At any point in flowing ground water:

• Elevation (position) head $z$ — height above a chosen datum.
• Pressure head $u/{\gamma }_w$ — height to which water would rise in a piezometer.
• Total head $h=z+\frac{u}{{\gamma }_w}$ (velocity head is negligible in soil).

Water flows from higher total head to lower total head. Head loss per unit length is the hydraulic gradient.

Seepage pressure and seepage force

As water flows through soil it loses head by friction with the grains and transfers that energy to the soil skeleton:

• Seepage pressure $=h{\gamma }_w$ over a flow path losing head $h$.
• Seepage force per unit volume $=i{\gamma }_w$, acting in the direction of flow.

Upward flow reduces effective stress (seepage force opposes gravity); downward flow increases it:

Laplace equation

For steady two-dimensional flow in a homogeneous, isotropic, saturated, incompressible soil obeying Darcy's law, continuity gives

Its solution is represented by two families of orthogonal curves — the flow net.

Flow nets

• Flow lines — paths followed by water particles.
• Equipotential lines — lines joining points of equal total head.
• Flow channel — the space between two adjacent flow lines; equipotential drop — head loss between adjacent equipotentials.

Methods of drawing: graphical trial sketching, electrical analogy, sand models, numerical methods.

Uses of a flow net

Uplift pressure on the base of a structure is obtained from the heads at the base points, and must be resisted by the weight of the floor.

Critical hydraulic gradient and quicksand

When water flows upward with a gradient large enough that the seepage force equals the submerged weight of the soil, the effective stress becomes zero:

For typical sands ($G$ ≈ 2.65, $e$ ≈ 0.65), $i_c\approx 1.0$.

At $i=i_c$, a cohesionless soil loses all strength and behaves like a liquid — the quick condition (quicksand). It is a hydraulic condition, not a soil type; it occurs in fine sands and silts (coarse soils need very large flows; clays have cohesion). Examples: excavations below water table with upward seepage, downstream of sheet piles and weirs.