Question 1

What is the Kozeny parabola?

Accepted Answer

The Kozeny parabola is the theoretical phreatic line through a homogeneous earthfill dam discharging to a horizontal underdrain. It has the form y² = y₀² + 2·y₀·x, where y₀ is the focal distance and the origin is at the start of the toe drain. Casagrande's correction adds a small adjustment near the upstream entrance to account for the fact that real phreatic lines start tangent to the upstream water surface.

Question 2

What is the focal distance y₀?

Accepted Answer

y₀ is the distance from the upstream end of the toe drain to the focus of the Kozeny parabola. For a homogeneous embankment with horizontal toe drain, Casagrande gives y₀ = √(d² + h²) − d, where d is the horizontal distance from the upstream water surface entry point on the dam to the upstream end of the toe drain, and h is the head difference between reservoir and tailwater. y₀ also equals the unit-width seepage flow rate divided by permeability: Q = k·y₀.

Question 3

When does this method fail?

Accepted Answer

The Casagrande/Schaffernak method assumes the embankment is homogeneous and isotropic, the dam rests on an impervious foundation, and there is a horizontal underdrain at the toe. For zoned embankments (clay core + sand-gravel shells), anisotropic permeability (kx ≠ ky), or pervious foundations, you need a 2-D seepage analysis (SEEP/W, MIKE-SHE) — but this calc is still useful for screening and for a first-cut Q estimate.

Question 4

How does an underdrain change the seepage line?

Accepted Answer

Without an underdrain, the phreatic line exits on the downstream slope above the toe — a free-discharge face — and tends to saturate the lower downstream embankment, threatening slope stability and creating piping potential at the exit point. A horizontal toe drain captures the seepage internally and depresses the phreatic surface, so the downstream slope stays unsaturated. Most modern earthfill dams have a chimney drain + toe drain combination.

Question 5

What permeability values are typical?

Accepted Answer

Compacted clay core: k ≈ 1×10⁻⁷ to 1×10⁻⁹ cm/s. Compacted silty sand shell: k ≈ 1×10⁻⁵ to 1×10⁻³ cm/s. Sand and gravel shell: k ≈ 1×10⁻² to 1×10⁰ cm/s. Filter / drain: k ≥ 1×10⁻¹ cm/s. For dam-safety analysis use the upper bound of the design range — overestimating Q is conservative for filter/drain sizing.

Question 6

What is the exit gradient and why does it matter?

Accepted Answer

The exit gradient i_e is the hydraulic gradient at the point where seepage exits the embankment into the toe drain or downstream face. If i_e exceeds the critical gradient i_c = (Gs − 1)/(1 + e) ≈ 1.0 for typical soils, soil grains can be lifted out by seepage forces — piping initiates. USACE EM 1110-2-1901 requires factor of safety FS = i_c/i_e ≥ 4. The toe drain reduces i_e by capturing seepage internally before it reaches a free face.

Material	k (cm/s)	k (ft/day)
Clean gravel, GW/GP	10⁻¹ – 10¹	300 – 30,000
Sand and gravel, SW	10⁻³ – 10⁻¹	3 – 300
Silty sand, SM	10⁻⁵ – 10⁻³	0.03 – 3
Compacted silt, ML	10⁻⁶ – 10⁻⁴	0.003 – 0.3
Compacted clay (core), CL/CH	10⁻⁹ – 10⁻⁷	3×10⁻⁶ – 3×10⁻⁴
Drain / filter material	10⁻¹ – 10⁰	300 – 3,000

Embankment Seepage Calculator

Permeability of compacted earth-dam materials

Worked example

Example — small homogeneous earthfill dam

Limitations and when to upgrade to FE

Related tools