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Category Physics & Science

Hydraulic Conductivity (K)

Soil permeability coefficient

Soil Presets:

Cross-Sectional Area (A)

Area perpendicular to flow direction

Hydraulic Head Difference (Δh)

h₁ − h₂ (upstream minus downstream)

Flow Path Length (L)

Distance along flow direction

Effective Porosity (n_e)

dimensionless

Fraction of interconnected pore space (0 to 1)

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About

Darcy's Law quantifies fluid flow through porous media under a hydraulic gradient. The relationship Q = −K × A × (Δh / L) governs subsurface hydrology, contaminant transport modeling, and aquifer yield estimation. The negative sign indicates flow occurs from high to low hydraulic head. Errors in hydraulic conductivity selection propagate directly into flow predictions - misidentifying clay (K ≈ 10⁻⁹ m/s) as sand (K ≈ 10⁻⁴ m/s) yields five orders of magnitude error.

This calculator computes volumetric discharge, specific discharge (Darcy velocity), and actual seepage velocity accounting for effective porosity. Darcy's Law assumes laminar flow (Reynolds number < 1 - 10), saturated conditions, and homogeneous isotropic media. The law breaks down in fractured rock, karst systems, and near pumping wells where turbulent flow dominates.

Formulas

Darcy's Law describes laminar flow of fluid through saturated porous media under a hydraulic gradient. The volumetric flow rate is proportional to the hydraulic conductivity, cross-sectional area, and hydraulic gradient.

Q = −K × A × ΔhL

Where Q = volumetric flow rate (m³/s), K = hydraulic conductivity (m/s), A = cross-sectional area perpendicular to flow (m²), Δh = hydraulic head difference (m), and L = length of flow path (m). The negative sign indicates flow direction from high to low head.

The hydraulic gradient i is the dimensionless ratio:

i = ΔhL = h₁ − h₂L

Specific discharge (Darcy velocity) represents apparent velocity through the total cross-section:

q = QA = −K × i

Actual seepage velocity accounts for the fraction of area available for flow (effective porosity n_e):

v = qn_e = QA × n_e

The seepage velocity is always greater than Darcy velocity since n_e < 1. For contaminant transport calculations, seepage velocity determines actual travel time through aquifers.

Reference Data

Material	Hydraulic Conductivity K (m/s)	Hydraulic Conductivity K (m/day)	Effective Porosity n_e	Classification
Gravel (coarse)	10⁻² - 1	864 - 86,400	0.25 - 0.35	Very High Permeability
Gravel (fine)	10⁻³ - 10⁻²	86 - 864	0.20 - 0.30	High Permeability
Sand (coarse)	10⁻⁴ - 10⁻³	8.6 - 86	0.25 - 0.35	High Permeability
Sand (medium)	10⁻⁵ - 10⁻⁴	0.86 - 8.6	0.25 - 0.32	Moderate-High
Sand (fine)	10⁻⁶ - 10⁻⁵	0.086 - 0.86	0.20 - 0.30	Moderate Permeability
Silty Sand	10⁻⁷ - 10⁻⁵	0.0086 - 0.86	0.15 - 0.25	Low-Moderate
Silt	10⁻⁸ - 10⁻⁶	0.00086 - 0.086	0.01 - 0.20	Low Permeability
Sandy Clay	10⁻⁹ - 10⁻⁷	8.6×10⁻⁵ - 0.0086	0.05 - 0.15	Very Low
Clay	10⁻¹¹ - 10⁻⁹	8.6×10⁻⁷ - 8.6×10⁻⁵	0.01 - 0.10	Aquitard/Aquiclude
Glacial Till	10⁻¹² - 10⁻⁶	8.6×10⁻⁸ - 0.086	0.05 - 0.20	Variable
Sandstone	10⁻¹⁰ - 10⁻⁶	8.6×10⁻⁶ - 0.086	0.05 - 0.30	Variable (fractured)
Limestone (unfractured)	10⁻⁹ - 10⁻⁶	8.6×10⁻⁵ - 0.086	0.01 - 0.20	Low-Moderate
Limestone (karst)	10⁻⁶ - 10⁻²	0.086 - 864	0.05 - 0.50	High (conduit flow)
Fractured Basalt	10⁻⁸ - 10⁻⁴	0.00086 - 8.6	0.05 - 0.25	Variable
Granite (unfractured)	10⁻¹⁴ - 10⁻¹⁰	8.6×10⁻¹⁰ - 8.6×10⁻⁶	0.0001 - 0.01	Impermeable
Shale	10⁻¹³ - 10⁻⁹	8.6×10⁻⁹ - 8.6×10⁻⁵	0.005 - 0.05	Aquiclude
Peat	10⁻⁶ - 10⁻⁴	0.086 - 8.6	0.30 - 0.50	Moderate
Loess	10⁻⁷ - 10⁻⁵	0.0086 - 0.86	0.15 - 0.25	Low-Moderate
Alluvium	10⁻⁵ - 10⁻³	0.86 - 86	0.20 - 0.35	High
Weathered Rock	10⁻⁸ - 10⁻⁵	0.00086 - 0.86	0.10 - 0.30	Variable

Frequently Asked Questions

Darcy velocity (specific discharge, q) is the volumetric flow rate divided by total cross-sectional area - it represents an apparent velocity as if water flowed through the entire section. Seepage velocity (v) is the actual average velocity of water particles moving through pore spaces, calculated by dividing Darcy velocity by effective porosity. Since effective porosity n_e ranges from 0.01 to 0.35 for most materials, seepage velocity is 3 - 100 times greater than Darcy velocity. For contaminant transport modeling, seepage velocity determines actual travel time.

Darcy's Law assumes laminar flow where viscous forces dominate. It breaks down when Reynolds number exceeds approximately 1 - 10, occurring in: coarse gravel with high gradients, fractured rock with large apertures, karst conduits, near pumping wells at high extraction rates, and unsaturated conditions where two-phase flow occurs. In these cases, the Forchheimer equation adds a quadratic velocity term to account for inertial effects. Darcy's Law also assumes homogeneous isotropic media - anisotropic formations require a hydraulic conductivity tensor.

Hydraulic conductivity can be measured via: (1) laboratory permeameter tests on undisturbed samples - constant head for coarse materials (K > 10⁻⁶ m/s), falling head for fine materials; (2) field slug tests or pump tests for in-situ values; (3) empirical correlations using grain-size analysis (Hazen, Kozeny-Carman equations). Laboratory values typically underestimate field conductivity by factors of 2 - 10 due to preferential flow paths and macropores absent in small samples. For preliminary estimates, use the reference table values for your dominant soil type.

The negative sign indicates that flow occurs in the direction of decreasing hydraulic head (opposite to the gradient direction). In practical calculations where you define Δh as the head drop (high minus low) and want positive flow in the downstream direction, the sign is already accounted for. This calculator uses the convention where Δh = h_upstream − h_downstream (positive value), yielding positive flow rate without the explicit negative sign.

Hydraulic conductivity K is temperature-dependent because it incorporates fluid viscosity: K = kρg/μ, where k is intrinsic permeability (temperature-independent), ρ is fluid density, and μ is dynamic viscosity. Water viscosity at 5°C is approximately 1.5 times that at 25°C, so K decreases by 50% in cold groundwater. Published K values typically reference 20°C; apply correction factors for significantly different temperatures.

Yes, Darcy's Law applies to any flow direction. For vertical flow, include the gravitational component in the total head. The hydraulic head h = z + p/(ρg) combines elevation head and pressure head. For upward flow against gravity, the gradient is reduced; for downward flow, it increases. In vertical percolation through unsaturated zones (infiltration), use Richards' equation which extends Darcy's Law to variable saturation conditions.