Darcy's Law and Permeability

The Darcy Equation

Darcy's Law in its simplest form is:

Q = K × i × A

• Q = volumetric flow rate (ft³/day, m³/s, or GPM)
• K = hydraulic conductivity (ft/day, m/s, or cm/s)
• i = hydraulic gradient (dimensionless, ft/ft or m/m)
• A = cross-sectional area perpendicular to flow (ft² or m²)

The hydraulic gradient is the change in hydraulic head per unit distance in the direction of flow: i = Δh / L, where Δh is the head difference and L is the flow path length. A gradient of 0.01 means the water level drops 1 foot for every 100 feet of horizontal distance.

Darcy's Law is valid for laminar flow (low Reynolds numbers) through porous media. It breaks down in very high-velocity flow (turbulent flow through coarse gravel or fractures), in very low-permeability materials where non-Darcy effects occur (clay, tight shales), and in unsaturated conditions where two-phase flow complicates the relationship.

Hydraulic Conductivity vs. Intrinsic Permeability

Hydraulic conductivity (K) is a combined property of both the porous medium and the fluid. It depends on the grain size, sorting, porosity, and packing of the medium, plus the density and viscosity of the fluid. Units are velocity (ft/day, m/s, cm/s).

Intrinsic permeability (k) is a property of the porous medium alone, independent of the fluid. It has units of area (darcy, millidarcy, or m²). The relationship between them is:

K = k × ρg / μ

Where ρ is fluid density, g is gravitational acceleration, and μ is dynamic viscosity. For water at standard conditions, 1 darcy corresponds to approximately K = 0.83 m/day = 2.7 ft/day. In petroleum engineering, permeability is always reported in millidarcys (md); in hydrogeology, hydraulic conductivity in ft/day or m/s is standard.

Typical hydraulic conductivity ranges:

• Gravel: 100–10,000 ft/day
• Clean sand: 1–500 ft/day
• Silty sand: 0.1–10 ft/day
• Silt: 0.001–1 ft/day
• Clay: 0.00001–0.01 ft/day
• Fractured rock: 0.01–100 ft/day (highly variable)

Aquifer Testing Basics

Aquifer tests (pump tests) are field methods for determining hydraulic conductivity, transmissivity, and storativity of aquifer systems. The basic procedure involves pumping water from a well at a constant rate and measuring the drawdown (decline in water level) in the pumping well and nearby observation wells over time.

Transmissivity (T) = K × b, where b is the aquifer thickness. It represents the ability of the full aquifer thickness to transmit water. Units are ft²/day or m²/day. A transmissivity of 1,000 ft²/day is considered a good aquifer for water supply.

Storativity (S) is the volume of water released from storage per unit area of aquifer per unit decline in head. For confined aquifers, S is typically 0.0001–0.001 (very small, water released by compression). For unconfined aquifers, S approximates the specific yield (typically 0.1–0.3).

The Theis equation and Cooper-Jacob approximation are the standard methods for analyzing pump test data. The Theis solution uses a type curve matching technique. The Cooper-Jacob straight-line method (valid for late-time data where u < 0.05) plots drawdown vs. log of time and extracts T and S from the slope and intercept.