7.2 Darcy's law & groundwater flow

Hydraulic head and gradient

Groundwater flows in response to differences in hydraulic head (h), the mechanical energy of water per unit weight, expressed as a length (e.g., meters). Total head is the sum of elevation head (z) and pressure head (ψ):

h = z + ψ

Velocity head is negligible in porous media. Water always moves from high head to low head, regardless of elevation alone — which is why pressurized confined water can flow upward. The hydraulic gradient (i) is the change in head over the flow distance between two points:

i = dh / dl = (h₁ − h₂) / L

The gradient is dimensionless (length divided by length).

Darcy's law

In 1856 Henry Darcy, studying sand filters in Dijon, France, showed that discharge is proportional to the head loss and cross-sectional area and inversely proportional to flow length. The law is written:

Q = K · i · A

where:

• Q = volumetric flow rate (L³/T, e.g., m³/day)
• K = hydraulic conductivity (L/T, e.g., m/day)
• i = hydraulic gradient (dimensionless)
• A = cross-sectional area perpendicular to flow (L²)

Dividing by area gives the specific discharge, or Darcy velocity (q):

q = Q / A = K · i

The Darcy velocity is a flux — the discharge per unit bulk area of aquifer (solids plus voids). It is not the true speed of a water molecule, because water can only travel through the pore space, not through the solid grains.

Hydraulic conductivity versus intrinsic permeability

K combines the medium and the fluid:

K = k · ρg / μ

where k is intrinsic permeability (L²), ρ is fluid density, g is gravitational acceleration, and μ is dynamic viscosity. Warmer, less-viscous water moves faster through the same medium, so K increases with temperature even though k is unchanged.

Average linear (seepage) velocity

Because flow is restricted to interconnected pores, the actual velocity of the water — and of any dissolved contaminant it carries — is faster than the Darcy velocity. The average linear velocity (v), also called seepage velocity, divides the Darcy velocity by the effective porosity (n_e):

v = q / n_e = K · i / n_e

Effective porosity (the interconnected, flow-contributing voids) is smaller than total porosity, so v is always greater than q. This velocity governs contaminant travel time and is the value a hydrogeologist uses to estimate when a plume will reach a receptor.

Fully worked Darcy calculation

Problem. A confined sand aquifer is 8 m thick and 200 m wide. Two monitoring wells 500 m apart along the flow path have heads of 62.0 m and 57.0 m. Hydraulic conductivity K = 15 m/day and effective porosity n_e = 0.25. Find (a) the hydraulic gradient, (b) the Darcy velocity, (c) the discharge through the aquifer cross-section, and (d) the average linear velocity and travel time over the 500 m.

(a) Gradient i = (62.0 − 57.0) / 500 = 5.0 / 500 = 0.010 (dimensionless)

(b) Darcy velocity q = K · i = 15 m/day × 0.010 = 0.15 m/day

(c) Discharge Cross-sectional area A = thickness × width = 8 m × 200 m = 1,600 m² Q = K · i · A = 0.15 m/day × 1,600 m² = 240 m³/day

(d) Average linear velocity and travel time v = q / n_e = 0.15 / 0.25 = 0.60 m/day Travel time t = L / v = 500 m ÷ 0.60 m/day ≈ 833 days ≈ 2.3 years

Note that v (0.60 m/day) is four times q (0.15 m/day) because 1 / n_e = 1 / 0.25 = 4. A contaminant released at the upgradient well needs about 2.3 years to reach the downgradient well — not the roughly 9 years the Darcy velocity alone would wrongly imply.

Flow nets and equipotentials

Groundwater flow can be mapped with a flow net, a grid of two families of curves: equipotential lines connecting points of equal head, and flow lines (streamlines) tracing flow paths. In an isotropic medium the two families intersect at right angles, and flow always runs from high to low head, perpendicular to the equipotentials. Where equipotentials are closely spaced the gradient is steep and flow is fast; where they spread apart the gradient is gentle. Real aquifers are often anisotropic (K differs with direction) and heterogeneous (K varies with position), so field-scale K is a directional, scale-dependent property rather than a single fixed constant.

Assumptions and limits

Darcy's law is valid for laminar flow through porous media, which describes nearly all natural groundwater. It breaks down where flow becomes turbulent — immediately around high-capacity pumping wells, in large solution channels (karst), and in very coarse open gravels — where the Reynolds number exceeds roughly 1–10 and discharge is no longer linearly proportional to gradient. It also assumes a representative continuum and should be applied to fractured rock only through an equivalent-porous-medium approximation.

Quick reference

Remember for the exam: q = Ki is the bulk flux, v = Ki/n_e is the real water speed, and v is always greater than q.