Flow through porous media — proportionality in Darcy’s law According to Darcy’s law for laminar flow through a saturated porous medium, the average (Darcy) velocity is directly proportional to which parameter?

Introduction / Context:
Darcy’s law is the cornerstone of groundwater flow and seepage analysis. It relates the specific discharge (Darcy velocity) to the hydraulic gradient through the medium’s permeability (hydraulic conductivity). Correctly identifying the proportionality helps distinguish driving forces from material properties and state variables in seepage problems.

Given Data / Assumptions:

• Saturated, steady, laminar flow (validity range of Darcy’s law).
• Homogeneous, isotropic porous medium for simplicity.
• Temperature (and thus viscosity) constant.

Concept / Approach:
Darcy’s law states: q = k * i, where q is specific discharge (Darcy velocity), k is hydraulic conductivity, and i is hydraulic gradient (head loss per unit length). Therefore, for a given k, the velocity is directly proportional to i. Other listed quantities (effective stress, cohesion, stability number) influence strength and stability, not flow rate according to Darcy’s empirical relation.

Step-by-Step Solution:
Write the law: q = k * (Δh / L) = k * i.For fixed k, q scales linearly with i.Conclude the proportional parameter is the hydraulic gradient.

Verification / Alternative check:
Dimensional analysis confirms that q has dimensions of velocity, and i is dimensionless, leaving k to carry the necessary dimensions. Laboratory constant-head/permeameter tests also demonstrate linear q–i behavior within Darcy’s regime.

Why Other Options Are Wrong:
Effective stress and cohesion are shear-strength parameters; stability number is a slope stability metric; none enter Darcy’s proportionality directly.

Common Pitfalls:

• Applying Darcy’s law beyond its laminar range (high Reynolds number in pores).
• Confusing seepage velocity (q/n) with Darcy velocity q.

Final Answer:
Hydraulic gradient