Darcy's Law Calculator

Compute volumetric flow rate through porous media using Darcy's Law. Enhanced with detailed explanations, unit conversions, and interactive graph.

Darcy's Law (1856): Q = K · A · i

where Q = volumetric flow rate [m³/s], K = hydraulic conductivity [m/s], A = cross‑sectional area [m²], i = hydraulic gradient [dimensionless] (head loss per unit length).

m/s (or any consistent unit)
m²
Δh / L (dimensionless)
Sand (K=1e-3, A=10, i=0.01)
Silt (K=1e-4, A=5, i=0.02)
Clay (K=1e-5, A=20, i=0.005)
Gravel (K=0.01, A=2, i=0.1)
Fine sand (K=5e-4, A=50, i=0.008)
Computing...

Understanding Darcy's Law

Darcy's Law is a fundamental equation in fluid mechanics describing the flow of a fluid through a porous medium. It was formulated by Henry Darcy in 1856 based on experiments on water flow through sand beds.

Mathematical formulation:

Q = K · A · i

where:

  • Q – volumetric flow rate [m³/s] – total volume of fluid passing per unit time.
  • K – hydraulic conductivity [m/s] – a measure of the ease with which fluid moves through the pore space. It depends on both the porous medium (grain size, sorting, porosity) and fluid properties (density, viscosity).
  • A – cross‑sectional area perpendicular to flow [m²].
  • i – hydraulic gradient [dimensionless] = Δh / L, where Δh is the head difference (e.g., water level difference) over a length L along the flow direction. It represents the driving force per unit length.

In differential form: q = Q/A = -K (dh/dl), where q is the specific discharge (Darcy velocity) [m/s]. Note that Darcy velocity is an apparent velocity; the actual average pore velocity is v = q / n, where n is porosity.

Detailed Explanation of Parameters

1

Hydraulic conductivity (K): It combines properties of the fluid and the porous medium. For a given fluid, K = κ·ρg/μ, where κ is intrinsic permeability (m²) – a property only of the porous medium, ρ is fluid density, μ is dynamic viscosity, and g is gravity. Typical K values: gravel: 10⁻²–1 m/s; sand: 10⁻⁵–10⁻³ m/s; silt: 10⁻⁹–10⁻⁷ m/s; clay: <10⁻⁹ m/s.

2

Hydraulic gradient (i): Suppose two piezometers (wells) 50 m apart show water levels at 102 m and 100 m elevation. The head difference Δh = 2 m, so i = 2/50 = 0.04. A gradient of 0.01 means a 1 m drop over 100 m.

3

Cross‑sectional area (A): For a column of porous material, A is the total cross‑section (including both solids and pores). The flow rate is proportional to A: doubling the area doubles Q.

Worked Example

A sand aquifer has hydraulic conductivity K = 5×10⁻⁴ m/s. Groundwater flows through a vertical cross‑section of width 100 m and saturated thickness 20 m (so A = 2000 m²). The hydraulic gradient measured between two wells 300 m apart is 0.002. Find the flow rate.

Q = (5×10⁻⁴) × 2000 × 0.002 = 0.002 m³/s = 2 L/s = 172.8 m³/day

This matches the calculator's output for similar inputs.

Intrinsic Permeability and Fluid Dependence

Hydraulic conductivity K is not a pure medium property; it depends on fluid viscosity and density. The intrinsic permeability κ (unit m² or darcy) is solely a function of the pore structure. Conversion: 1 darcy ≈ 0.987×10⁻¹² m². The relation is:

K = κ · (ρg / μ)

For water at 20°C, ρg/μ ≈ 1×10⁷ (m·s)⁻¹, so κ (in m²) ≈ K / 10⁷. For example, K = 10⁻³ m/s corresponds to κ ≈ 10⁻¹⁰ m² ≈ 100 darcy.

Validity and Limitations

  • Darcy's law holds for laminar flow (Reynolds number based on grain diameter Re < 1–10). In coarse gravel or high gradients, inertial effects appear (Forchheimer's law).
  • Assumes a homogeneous, isotropic medium and a Newtonian fluid.
  • For unsaturated flow, K becomes a function of moisture content (Richards' equation).
  • For gases, K is pressure‑dependent; often a modified form using pressure squared is used.

Applications

  • Groundwater hydrology: estimating well yields, aquifer properties, contaminant transport.
  • Petroleum engineering: oil and gas flow in reservoir rock.
  • Civil engineering: seepage through dams, drainage design, slope stability.
  • Soil science: water movement in unsaturated soils.
  • Chemical engineering: flow in packed beds, filters.

Typical Hydraulic Conductivity Values

Material K (m/s) K (m/day)
Gravel 10⁻² – 1 864 – 86,400
Clean sand 10⁻⁵ – 10⁻³ 0.864 – 86.4
Silty sand 10⁻⁷ – 10⁻⁵ 0.00864 – 0.864
Silt 10⁻⁹ – 10⁻⁷ 8.64×10⁻⁵ – 0.00864
Clay < 10⁻⁹ < 8.64×10⁻⁵

Calculator note: All input values must be in consistent units (e.g., K in m/s, A in m², i dimensionless). The result Q will be in m³/s. The calculator also displays the flow rate in L/s and m³/day for convenience. You can also use other units as long as they are consistent (e.g., K in cm/s, A in cm², then Q in cm³/s).

Frequently Asked Questions

Hydraulic gradient i = Δh / L, where Δh is the head difference (e.g., water level difference) over a length L along the flow direction. It represents the driving force per unit length. For example, if water levels in two wells 100 m apart differ by 2 m, i = 0.02.

Yes, simply rearrange: K = Q / (A·i). This calculator currently solves for Q, but you can easily use it to verify K by inputting trial values. A future version may include a solver for any variable.

Yes, with modifications. For gases, the hydraulic conductivity depends on pressure, and often the equation is written in terms of pressure squared or using the ideal gas law. The calculator is intended for liquids (incompressible flow).

In hydrogeology, Q is often expressed in m³/s, L/s, or m³/day. 1 m³/s = 1000 L/s = 86,400 m³/day. Our calculator gives m³/s and also shows L/s and m³/day.

Darcy velocity (specific discharge) q = Q/A is the volumetric flux based on the total cross‑section. The actual average velocity of water molecules through the pores is v = q / n, where n is porosity (e.g., 0.3). Since water can only move through pore space, v is always greater than q.