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Permeameter Calculator

K equals Q times delta L over A times delta h

Solution

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Solve for Permeability Coefficient (K)

The permeability coefficient describes how easily water moves through a soil sample. It is the key output of a constant-head permeameter test.

K = Q × ΔL / (A × Δh)

Solve for Flow Rate (Q)

Given permeability, cross-sectional area, and hydraulic gradient, calculate the volumetric flow rate through the sample.

Q = K × A × Δh / ΔL

Solve for Cross-Sectional Area (A)

Determine the required sample area for a given flow rate and permeability.

A = Q × ΔL / (K × Δh)

Solve for Length Change (ΔL)

Find the sample length needed to produce the observed flow rate under the given conditions.

ΔL = K × A × Δh / Q

Solve for Pressure Head (Δh)

Calculate the hydraulic head difference required to drive the measured flow rate.

Δh = Q × ΔL / (K × A)

How It Works

A permeameter is a lab device that pushes water through a soil sample under a known hydraulic head. By measuring the flow rate, sample length, cross-sectional area, and head difference, you can calculate the permeability coefficient K using Darcy's law. K tells you how easily water moves through the material — critical for landfill liner design, well construction, and drainage engineering. The formula Q = K × A × Δh / ΔL can be rearranged to solve for any of the five variables.

Example Problem

Water flows at 0.0002 m³/s through a soil sample with cross-sectional area A = 0.05 m², sample length ΔL = 0.3 m, and head difference Δh = 1.2 m. What is the permeability coefficient K?

  1. Identify the formula: K = Q × ΔL / (A × Δh).
  2. Substitute the measured values: K = 0.0002 m³/s × 0.3 m / (0.05 m² × 1.2 m).
  3. Calculate the numerator: 0.0002 × 0.3 = 0.00006 m⁴/s.
  4. Calculate the denominator: 0.05 × 1.2 = 0.06 m³.
  5. Divide: K = 0.00006 / 0.06 = 0.001 m/s.
  6. Interpret: K ≈ 10⁻³ m/s corresponds to coarse sand or fine gravel — highly permeable material suitable for drainage but unsuitable for a clay liner.

Always report the test temperature — water viscosity (and thus K) changes measurably with temperature.

When to Use Each Variable

  • Solve for Permeability (K) — when you have lab measurements of flow rate, sample area, length, and head difference, e.g., characterizing soil from a borehole sample.
  • Solve for Flow Rate (Q) — when you know the soil permeability and want to predict seepage, e.g., estimating leakage through a dam embankment.
  • Solve for Area (A) — when designing a filter or drain and need to size the cross-section for a target flow rate.
  • Solve for Length (ΔL) — when determining the required thickness of a clay liner or filter layer to limit seepage to a target rate.
  • Solve for Head (Δh) — when you know the flow rate and soil properties and need to find the driving head, e.g., estimating water table elevation from observed seepage.

Key Concepts

Darcy's law assumes laminar flow through a homogeneous, isotropic porous medium. The permeability coefficient K is not a pure material property — it depends on both the soil structure and the fluid viscosity. For fine-grained soils a falling-head test is more accurate because flow rates are too slow to measure reliably under constant head. Field permeability frequently differs from lab permeability by orders of magnitude because of macro-pores, root channels, and soil layering.

Applications

  • Geotechnical engineering: designing landfill liners with K below regulatory thresholds (typically 10⁻⁹ m/s)
  • Groundwater hydrology: modeling aquifer yield and contaminant transport rates
  • Dam safety: estimating seepage through embankments, cutoff walls, and foundations
  • Agriculture: evaluating soil drainage capacity and irrigation infiltration rates
  • Construction dewatering: sizing wellpoint systems based on site permeability

Common Mistakes

  • Using a constant-head test on fine-grained soils — flow is too slow to measure accurately; use a falling-head permeameter instead
  • Ignoring temperature effects on water viscosity — K varies with temperature because viscosity changes; always report the test temperature
  • Assuming the lab sample represents field conditions — soil layering, root channels, and macro-pores in situ can make field K orders of magnitude different from lab K
  • Mixing head units — Δh is the difference in hydraulic head (meters of water), not a pressure in pascals, unless you convert using ρg

Frequently Asked Questions

How do you measure soil permeability in the lab?

Pack a disturbed or undisturbed soil sample into a cylindrical permeameter of known cross-sectional area A and length ΔL. Apply a known head difference Δh and measure the volumetric flow rate Q under steady state. Then K = Q × ΔL / (A × Δh). Coarse soils use a constant-head apparatus; fine soils use a falling-head apparatus where the head declines over time.

What's the difference between constant-head and falling-head permeameter tests?

A constant-head permeameter maintains a steady head Δh across the sample and measures the resulting flow rate — suitable for coarse, highly permeable soils (sands, gravels) where Q is easy to measure. A falling-head permeameter lets the water level in a standpipe drop through the sample over time — suitable for fine-grained soils (silts, clays) where flow is too slow for direct volume measurement. ASTM D2434 covers the constant-head method; ASTM D5084 covers falling-head with a flexible-wall cell.

What is Darcy's law used for in environmental engineering?

Darcy's law describes steady laminar flow through porous media. Environmental engineers use it for well yield estimation, contaminant plume transport, landfill leachate analysis, wetland hydrology, and sizing of French drains, sand filters, and leach fields.

What are typical permeability values for different soils?

Gravel: 10⁻² to 1 m/s. Clean sand: 10⁻⁵ to 10⁻² m/s. Silt: 10⁻⁹ to 10⁻⁵ m/s. Clay: 10⁻¹² to 10⁻⁹ m/s. Compacted clay liners for landfills must have K below 10⁻⁹ m/s (≈1 cm/year) to meet regulatory standards.

Why does soil permeability matter for landfill liner design?

Landfill liners must slow the leakage of leachate (contaminated water) into groundwater. Federal RCRA Subtitle D rules require a compacted-clay component with K ≤ 10⁻⁷ cm/s (10⁻⁹ m/s) or an engineered composite liner of equivalent performance. A permeameter test on each batch of clay verifies compliance.

How does temperature affect the permeability coefficient?

Water viscosity decreases about 2.5 % per °C rise, so measured K rises proportionally. Lab tests are typically conducted at 20 °C and corrected to a standard temperature using K₂₀ = K_T × (μ_T / μ_₂₀). Always record and report the test temperature.

What is the difference between permeability and hydraulic conductivity?

In groundwater hydrology the two terms are often used interchangeably for K (units m/s). Strictly, intrinsic permeability k (units m²) depends only on the soil skeleton, while hydraulic conductivity K = k × ρg / μ also includes the fluid's density and viscosity. Use k when comparing permeability across different fluids; use K for water-flow engineering.

Permeameter Formula

Darcy's law, rearranged for lab permeameter measurements, gives the hydraulic conductivity of a soil sample:

K = Q × ΔL / (A × Δh)

Where:

  • K — permeability (hydraulic conductivity), measured in meters per second (m/s)
  • Q — volumetric flow rate through the sample, in cubic meters per second (m³/s)
  • ΔL — length of the soil sample along the flow path, in meters (m)
  • A — cross-sectional area perpendicular to flow, in square meters (m²)
  • Δh — head difference driving the flow (inlet head minus outlet head), in meters (m)

The formula assumes steady-state, laminar, one-dimensional flow through a saturated, homogeneous, isotropic porous medium. These assumptions are valid for most soils under lab conditions.

Worked Examples

Geotechnical Engineering

What is the permeability of a foundation soil sample tested in a constant-head permeameter?

A soil sample from a building foundation investigation has A = 0.008 m² and ΔL = 0.15 m. Under a head of Δh = 0.5 m the flow rate stabilizes at Q = 5 × 10⁻⁶ m³/s. What is K?

  • K = Q × ΔL / (A × Δh)
  • K = 5×10⁻⁶ × 0.15 / (0.008 × 0.5)
  • K = 7.5×10⁻⁷ / 0.004
  • K ≈ 1.88 × 10⁻⁴ m/s

This value is typical of fine sand — drainable but not a candidate for a compacted-clay liner.

Environmental Engineering

Does a compacted-clay landfill liner meet the 10⁻⁹ m/s permeability target?

A 0.01 m² falling-head cell with ΔL = 0.1 m is run under Δh = 1.0 m. The measured flow rate is Q = 9 × 10⁻¹² m³/s. Calculate K and compare to the regulatory maximum.

  • K = Q × ΔL / (A × Δh) = 9×10⁻¹² × 0.1 / (0.01 × 1.0)
  • K = 9×10⁻¹³ / 0.01
  • K ≈ 9 × 10⁻¹¹ m/s

Result is well below the 10⁻⁹ m/s RCRA Subtitle D threshold — the liner passes.

Agricultural Engineering

What infiltration rate can an irrigated field soil sustain?

A tilled-soil core sample with A = 0.02 m² and ΔL = 0.12 m has measured permeability K = 1 × 10⁻⁵ m/s. Under a ponded head of Δh = 0.05 m, what flow rate can the soil accept?

  • Q = K × A × Δh / ΔL = 1×10⁻⁵ × 0.02 × 0.05 / 0.12
  • Q = 1×10⁻⁸ / 0.12
  • Q ≈ 8.33 × 10⁻⁸ m³/s (≈ 7.2 L/day through that core)

Scaled to a field, this gives the sustainable irrigation application rate before ponding or runoff occurs.

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