Hydraulic gradient definition in pipe/porous flow Hydraulic gradient (i) is defined as the drop in piezometric head between two points divided by which quantity?

Introduction / Context:
The hydraulic gradient governs head loss representation in pipes and seepage in soils. It appears in Darcy’s law and the energy equation, linking head drop to flow rate and resistance.

Given Data / Assumptions:

• Piezometric head h_p = p/γ + z (for incompressible flow).
• We consider two points along the same streamline or flow path.
• Local losses and elevation changes are captured in head differences.

Concept / Approach:

The hydraulic gradient i is the head loss per unit length along the flow path: i = Δh / L, where Δh is the drop in piezometric head (or total head if velocity heads are accounted consistently) and L is the distance measured along the flow direction between the two points.

Step-by-Step Solution:

Verification / Alternative check:

In Darcy’s law for porous media: v = k * i, reinforcing that i is a geometric ratio of head drop to flow length. In pipe flow, the slope of hydraulic grade line equals the hydraulic gradient.

Why Other Options Are Wrong:

(a) Time does not define i. (c) Vertical depth at a point is unrelated to two-point head difference ratio. (d) Pipe diameter is not part of head gradient definition. (e) Wetted perimeter affects resistance, not the definition of i.

Common Pitfalls:

Mixing total head and piezometric head inconsistently; measuring straight-line distance instead of the flow-path length in curved conduits or media.

Final Answer:

the distance measured along the flow path between the points