Q36: Based on Allen-Hazens experiments, the co-efficient of permeability k (cm/s) is related to the effective size of soil D10 (cm) as (where C is a constant with a value between 100 and 150):
📝 Detailed Explanation
This is the general form of Allen Hazen’s empirical permeability formula, already applied numerically in Q25 — here tested as a direct “identify the correct formula structure” recognition question.
✅ Why “k = C·D10²” Is Correct
Allen Hazen’s formula expresses permeability as k = C(D10)², where D10 is the effective size in cm, k is permeability in cm/s, and C is an empirical constant typically between 100 and 150, depending on the soil’s uniformity.
🔑 Key Point
Hazen’s formula squares the EFFECTIVE SIZE (D10), not the empirical constant (C) — and permeability increases with larger effective size (bigger particles, bigger pores), which correctly rules out any “inverse” variant of the formula.
💡 Key Concepts for Students
- See Q25 for a fully worked numeric example applying this exact formula, using D10 = 0.05 cm to find K = 0.25 cm/sec.
- The constant C (100 to 150) accounts for variations in grain shape and uniformity not directly captured by D10 alone — a rounder, more uniform sand tends toward the higher end of this range.
- This formula is only valid for a specific particle-size range (roughly 0.1 mm to 3 mm) — outside that range, the empirical fit breaks down and the formula becomes unreliable.
- See the Going Deeper section for a complete summary of every permeability-related formula covered across this whole topic, gathered as a single capstone reference.
📚 Going Deeper: Every Permeability Formula in This Topic, Gathered Together
As a fitting close to this topic, here is every major formula covered across all 36 questions, gathered into one consolidated reference:
| Formula | Purpose | Where Covered |
|---|---|---|
| v = K·i ; q = K·i·A | Darcy’s law — basic velocity/discharge relationship | Q1, Q7, Q8 |
| K = QL/(Aht) | Constant head lab test | Q9 |
| K = 2.303(aL/At)·log10(h1/h2) | Falling head lab test | Q3, Q22, Q24 |
| K = Cv·mv·γw | Indirect, from consolidation test | Q15 |
| Q = πK(h2²−h1²)/[2.303·log10(r2/r1)] | Unconfined aquifer field pumping test | Q20 |
| Q = 2πKD·Sw/ln(R/Rw) | Confined aquifer field pumping test | Q21 |
| k ∝ γw/μ (Kozeny-Carman) | Effect of pore fluid properties on k | Q11, Q28, Q35 |
| k = C(D10)² | Hazen’s empirical formula from effective size | Q25, this question |
| V = n·Vs | Discharge velocity to seepage velocity | Q23 |
| ic = (G−1)/(1+e) | Critical hydraulic gradient (quicksand) | Q33, Q34 |
| Q = K·H·(Nf/Nd) | Flow net discharge (isotropic soil) | Q31 |
Every one of these formulas ultimately traces back to the same single foundational idea introduced in Q1 and Q26: permeability (K) is the soil property connecting how much water moves through soil to the driving force (hydraulic gradient) behind that movement — everything else in this topic is either a way of MEASURING K (lab and field tests), a way of ESTIMATING it from simpler data (Hazen’s formula), or a way of APPLYING it to a specific engineering scenario (wells, flow nets, quicksand risk).