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Seepage in inclined permeable layer

Short answer

Hydraulic gradient i = sin 5° = 0.0872. Flow per metre width q = k i H cosα = 4.8×10⁻⁵ × 0.0872 × 3×cos5° ≈ 1.25×10⁻⁵ m³/s.

As shown in the figure, an inclined permeable soil layer is underlain by an impervious layer. The coefficient of permeability of the permeable soil layer is equal to 4.8×10−54.8 \times 10^{-5} m/sec. If seepage of water in this soil layer occurs in the direction shown in the figure, then calculate (i) hydraulic gradient and (ii) rate of water flow (seepage) for that soil layer. Take the thickness of soil layer H = 3 m and the angle of inclination of that soil layer, α\alpha = 5°. Assume any other necessary conditions. [Figure: inclined permeable layer of thickness H = 3 m on an impervious layer sloping at α\alpha = 5°; ground water table parallel to the ground surface; seepage direction down the slope.]

Assumptions

The water table is parallel to the ground surface and the impervious layer beneath, so flow is parallel to the slope. The thickness H=3H = 3 m is measured vertically, so the flow area normal to the flow per metre width is Hcos⁡αH\cos\alpha.

        ground / water table (parallel)
    \ ~~~~~~~~~~~~~~~~
     \   flow  ->  H=3 m      
      \  (down the slope)    α = 5°
       \_________________ impervious

(i) Hydraulic gradient

Over a length LL along the slope, the head drop equals the fall in elevation Lsin⁡αL\sin\alpha:

i=Lsin⁡αL=sin⁡α=sin⁡5∘=0.0872i = \frac{L\sin\alpha}{L} = \sin\alpha = \sin 5^\circ = 0.0872

(ii) Rate of flow per metre width

q=k i A=k sin⁡α (Hcos⁡α)×1q = k\,i\,A = k\,\sin\alpha\,(H\cos\alpha)\times 1 q=4.8×10−5×0.0872×(3×cos⁡5∘)=4.8×10−5×0.0872×2.988q = 4.8\times10^{-5} \times 0.0872 \times (3 \times \cos 5^\circ) = 4.8\times10^{-5} \times 0.0872 \times 2.988 q=1.25×10−5 m3/s per metre width  (≈1.08 m3/day per m)q = 1.25\times10^{-5}\ \text{m}^3/\text{s per metre width} \; (\approx 1.08\ \text{m}^3/\text{day per m})

Answer: i=0.0872i = 0.0872; q=1.25×10−5q = 1.25\times10^{-5} m³/s per m width.

✓ Answer checked by Claude Opus 5.5 (high effort)

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