What is seepage through an earth dam?

Seepage through an earth dam is the slow movement of water through the dam body and its foundation due to the hydraulic head difference between the upstream and downstream sides. Controlled seepage is expected, but excessive seepage or concentrated flow paths can lead to internal erosion and potential failure.

Which method does this calculator use?

This calculator implements a simplified version of Casagrande’s method for homogeneous earth dams with a horizontal toe drain. It uses an approximate phreatic line and an equivalent flow net to estimate seepage discharge and exit gradient.

Is this calculator suitable for zoned or rockfill dams?

No. The calculator is intended for preliminary seepage checks on homogeneous earth dams with relatively simple geometry. Zoned embankments, rockfill dams, and dams with complex foundations require more advanced numerical analysis such as finite element seepage modeling.

What factor of safety against piping is acceptable?

Many design guidelines recommend a factor of safety against piping of at least 3.0 for new dams, although the required value depends on local codes, dam hazard classification, and soil investigation results. Always follow the requirements of the relevant dam safety authority.

Earth Dam Seepage Calculator

Estimate seepage discharge, phreatic line profile, exit gradient, and factor of safety against piping for a homogeneous earth dam using a simplified Casagrande method.

Input Data

Dam Geometry

Dam height, H (m)

Crest width, B (m)

Upstream slope (H:V)

Enter horizontal component m for slope m:1 (H:V).

Downstream slope (H:V)

Enter horizontal component n for slope n:1 (H:V).

Hydraulic Conditions

Upstream water depth above impervious base, h_u (m)

Downstream tailwater depth, h_d (m)

0 for no tailwater at downstream toe.

Coefficient of permeability, k (m/s)

Typical core soils: 10⁻⁷–10⁻⁵ m/s.

Dam length (perpendicular to section), L (m)

Safety Parameters

Critical hydraulic gradient, i_cr

Often 0.8–1.0 for sandy foundations (i_cr ≈ (G_s − 1)/(1 + e)).

Horizontal toe drain length, L_d (m)

0 if no drain. Drain reduces exit gradient.

Results

Seepage Discharge

Through dam body (per unit length and total).

q (per m length): – m³/s/m

Q (total for length L): – m³/s

Exit Gradient & Piping Safety

Exit gradient, i_exit: –

Factor of safety against piping, FS_piping: –

Phreatic Line (Approximate)

The plot shows an approximate phreatic line using Casagrande’s method (parabolic profile) from upstream water level to downstream toe or drain.

Engineering background: seepage through earth dams

All earth dams experience some seepage through the embankment and its foundation. As long as this seepage is small, uniform, and controlled, it is not a problem. However, excessive seepage or concentrated flow paths can cause:

Piping (internal erosion) at the downstream toe or through cracks and animal burrows.
Softening and sloughing of the downstream slope.
Instability of the foundation or abutments.

Dam safety guidelines from agencies such as the USACE, USBR, and state dam safety programs emphasize careful seepage analysis and the use of filters, drains, and cutoff measures to keep gradients within safe limits.

Method used in this calculator

This tool implements a simplified Casagrande method for homogeneous earth dams on an impervious base, with an optional horizontal toe drain. It is intended for preliminary design and teaching, not for final design of critical structures.

1. Geometry and hydraulic head

The dam is idealized as a homogeneous embankment of height \(H\) on an impervious base. The upstream water depth above the base is \(h_u\), and the downstream tailwater depth is \(h_d\). The net head causing seepage is:

\( H_{\text{net}} = h_u - h_d \)

The total horizontal distance between the upstream and downstream toes is:

\( L_{\text{dam}} = m \, H + B + n \, H \)
where:
\(m\) = upstream slope (H:V), \(n\) = downstream slope (H:V), \(B\) = crest width.

2. Approximate phreatic line (Casagrande)

Casagrande proposed a graphical construction for the phreatic line (top seepage line) in a homogeneous earth dam. In this calculator we approximate it as a parabola starting near the upstream water surface and ending at the downstream toe or at the end of a horizontal toe drain.

For plotting purposes we assume a simple parabolic profile in normalized coordinates:

\( y(x) = H_u \left[ 1 - \left( \dfrac{x}{L_{\text{eff}}} \right)^2 \right] \)
where \(0 \le x \le L_{\text{eff}}\) is the distance along the base from the upstream toe, and \(L_{\text{eff}}\) is the effective seepage length (dam base plus drain).

This is an idealization; in practice, engineers often use flow nets or finite element seepage models for more accurate phreatic lines.

3. Seepage discharge

For a homogeneous dam on an impervious base, the seepage discharge per unit length can be approximated as:

\( q' = k \, \dfrac{H_{\text{net}}^2}{L_{\text{eff}}} \)
where:
\(q'\) = seepage discharge per meter of dam length (m³/s/m),
\(k\) = coefficient of permeability (m/s),
\(L_{\text{eff}} = L_{\text{dam}} + L_d\) (if a horizontal drain of length \(L_d\) is present).

The total seepage discharge for a dam of length \(L\) (perpendicular to the cross-section) is:

\( Q = q' \times L \)

4. Exit gradient and factor of safety against piping

The exit gradient at the downstream toe is a key indicator of piping risk. In a simplified form, we estimate:

\( i_{\text{exit}} \approx \dfrac{H_{\text{net}}}{L_{\text{eff}}} \)

The factor of safety against piping is then:

\( FS_{\text{piping}} = \dfrac{i_{\text{cr}}}{i_{\text{exit}}} \)

where \(i_{\text{cr}}\) is the critical hydraulic gradient of the foundation soil, often estimated as:

\( i_{\text{cr}} \approx \dfrac{G_s - 1}{1 + e} \)

with \(G_s\) the specific gravity of soil solids and \(e\) the void ratio. For many sands and silty sands, \(i_{\text{cr}}\) is in the range 0.8–1.0.

How to interpret the results

Seepage discharge (Q): Higher permeability, higher head, and shorter seepage paths all increase discharge. Excessive seepage may require drains, filters, or cutoffs.
Exit gradient (i_exit): If this approaches or exceeds the critical gradient, there is a risk of boiling and piping at the downstream toe.
Factor of safety (FS_piping): Values > 3 are often targeted for new dams, but always follow your governing code or dam safety authority.
Effect of a toe drain: Increasing the horizontal drain length \(L_d\) increases \(L_{\text{eff}}\), reducing both seepage discharge and exit gradient.

Typical input values and checks

Permeability k: 10⁻⁷–10⁻⁹ m/s for clay cores; 10⁻⁵–10⁻⁶ m/s for compacted silty clays.
Slopes: Upstream 3:1 to 4:1 (H:V), downstream 2:1 to 3:1 are common for small dams.
Critical gradient: Use site-specific lab data where possible; otherwise use conservative estimates.

Limitations and professional use

This calculator is designed for preliminary analysis and education. It does not replace detailed seepage modeling or professional dam design. Important limitations include:

Assumes a homogeneous dam on an impervious base.
Assumes steady-state seepage and 2D conditions.
Uses a simplified phreatic line and flow net approximation.
Does not model cracks, animal burrows, or anisotropic permeability.

For real projects, always consult a qualified geotechnical or dam safety engineer and follow the requirements of your local dam safety authority.

FAQ

Can I use this for dams with a cutoff wall or grout curtain?

Not directly. Cutoffs and grout curtains change the seepage path and head distribution. This simple model does not explicitly include them. You can approximate their effect by increasing the effective seepage length, but detailed analysis should be done with a numerical seepage model.

What if there is significant seepage through the foundation?

This calculator assumes an impervious base. If the foundation is pervious, seepage may occur both through the dam and through the foundation. In that case, a separate foundation seepage analysis (e.g., flow net or finite element model) is required.

How accurate is the phreatic line shown?

The plotted phreatic line is an idealized parabola based on Casagrande’s method. It is useful for understanding trends and for quick checks, but it is not a substitute for a detailed flow net or numerical analysis.