🎓 Lesson 3 D2

Computing Time of Concentration: Kirpich vs. SCS vs. FAA

Time of concentration is how long it takes for water from the farthest point in a watershed to reach the outlet — like counting seconds until rain runoff arrives at the drain.

🎯 Learning Objectives

✓ Calculate time of concentration using Kirpich, SCS (TR-55), and FAA methods for a given watershed geometry and surface condition
✓ Analyze differences in Tc results across the three methods and explain which is most appropriate for mining drainage applications
✓ Apply correction factors for disturbed surfaces (e.g., blasted rock cuts, haul roads) to adjust standard Tc formulas
✓ Explain how underestimating Tc leads to undersized culverts or erosion failures in mine site drainage systems

📖 Why This Matters

In mining operations, rapid and accurate estimation of time of concentration determines the design storm intensity used for sizing ditches, culverts, sediment ponds, and diversion structures. Underestimating Tc yields overly conservative (costly) designs; overestimating it risks catastrophic failure during intense storms — especially on steep, unvegetated slopes created by blasting and excavation. This lesson bridges hydrology theory with real-world mine drainage safety and regulatory compliance.

📘 Core Principles

Time of concentration reflects the longest travel time of surface runoff — not average flow time. It is dominated by the slowest segment: typically overland flow on bare, compacted, or rocky surfaces common in active mine sites. Kirpich assumes laminar overland flow on impervious surfaces and uses slope and length only. SCS (TR-55) adds flow regime classification (sheet, shallow concentrated, channel) and incorporates surface roughness via retardance class. FAA method was developed for airport drainage but widely adopted in mining due to its explicit treatment of pavement and gravel surfaces — critical for haul roads and blast debris zones. All three methods assume steady, uniform flow and neglect infiltration — a key limitation on porous spoil piles or weathered rock.

📐 Key Calculation

Each method uses distinct empirical relationships calibrated to field data. Kirpich is simplest and most conservative for steep, bare slopes; SCS offers flexibility for mixed land covers; FAA best represents engineered surfaces like crushed rock and graded haul roads. Selection depends on dominant flow path characteristics — not just watershed size.

Kirpich Equation

Tc = 0.01947 × L^0.77 × S^(−0.385)

Empirical equation for time of concentration on impervious, unchanneled surfaces (e.g., rock cuts, waste dumps).

Variables:

Symbol	Name	Unit	Description
Tc	Time of concentration	minutes	Time for runoff to travel from hydraulically farthest point to outlet
L	Hydraulic length	feet	Length of longest flow path along ground surface
S	Average slope	ft/ft (dimensionless)	Fall over run along hydraulic path

Typical Ranges:

Steep mine waste slopes (8–15% slope): 5 – 25 min

Gentle reclaimed areas (>2% slope): 20 – 60 min

💡 Worked Example

Problem: Given: watershed length (L) = 1,200 m, average slope (S) = 0.08 (8%), soil/rock type = blasted granite (impervious), no vegetation. Compute Tc using Kirpich.

1. Step 1: Convert L to feet: 1,200 m × 3.281 = 3,937 ft

2. Step 2: Apply Kirpich: Tc (min) = 0.01947 × L^0.77 × S^(−0.385)

3. Step 3: Compute: L^0.77 = 3,937^0.77 ≈ 326.5; S^(−0.385) = 0.08^(−0.385) ≈ 1.62; then Tc = 0.01947 × 326.5 × 1.62 ≈ 10.3 min

Answer: The result is 10.3 minutes, which falls within the typical range of 5–25 minutes for steep mine waste slopes.

🏗️ Real-World Application

At the Red Mesa Copper Mine (Arizona), engineers designed a sediment control basin for a 42-ha disturbed watershed with 12% average slope and extensive blast-rock overland flow paths. Initial Kirpich estimate gave Tc = 9.7 min; SCS TR-55 (retardance class C, shallow concentrated flow dominant) yielded 14.2 min; FAA (gravel surface, 2% slope in channelized section) gave 11.8 min. Field tracer tests confirmed 12.5 ± 1.1 min — validating FAA as most accurate for this mixed-flow-path, engineered-surface scenario. The final design storm duration was set to 15 minutes based on FAA + safety margin.