Permeability, Seepage and Effective Stress

Water flowing through soil controls seepage under dams, the stability of excavations and the strength of the soil itself through effective stress.

Darcy's law

v = k i, Q = k i A

• i = hydraulic gradient = head loss/length; k = coefficient of permeability (m/s).
• Typical k: gravel 10⁻² m/s, sand 10⁻⁴, silt 10⁻⁶, clay < 10⁻⁹.
• Seepage velocity v_s = v/n (water moves only through the voids).
• Hazen's formula for clean sands: k (cm/s) ≈ C D₁₀² (D₁₀ in cm, C ≈ 100).

Laboratory tests: constant-head test (coarse soils), falling-head test (fine soils): k = (aL/(At)) ln(h₁/h₂). Field: pumping-out tests.

Layered soils

For layers of thickness Hᵢ and permeability kᵢ:

• Flow parallel to the layers: k_h = Σkᵢ Hᵢ/ΣHᵢ (weighted average; dominated by the most permeable layer).
• Flow perpendicular: k_v = ΣHᵢ/Σ(Hᵢ/kᵢ) (dominated by the least permeable layer). Always k_h ≥ k_v.

Flow nets

Under a dam or sheet pile, seepage follows the Laplace equation. A flow net of flow lines and equipotential lines (forming curvilinear squares) gives:

q = k H (N_f/N_d) per unit length

N_f = number of flow channels, N_d = number of equipotential drops, H = total head loss. For anisotropic soils use k = √(k_h k_v) and transform the scale.

Effective stress (Terzaghi)

σ' = σ − u

Total stress σ is carried by the grains (effective stress σ') and the pore water (u). Strength and compression of soil depend only on σ'.

• Upward seepage reduces σ'; downward seepage increases it.
• Quick sand (boiling): σ' becomes zero when the upward gradient reaches the critical hydraulic gradient: i_c = (G − 1)/(1 + e) = γ'/γ_w (about 1 for most sands).
• Factor of safety against piping = i_c/i_exit.
• Capillary rise above the water table gives negative pore pressure (increases σ').

Worked example

A sand layer (γ_sat = 20 kN/m³) is 4 m thick with the water table at the surface (γ_w = 10).

• σ = 80 kPa, u = 40 kPa, σ' = 40 kPa at the base.

Practice