--- title: "Average permeability of layered soil" subject: "Soil Mechanics" chapter: 6 key: "semester_4/soil_mechanics:layered-soil-average-permeability" source: https://www.bctengineer.com/q/semester_4/soil_mechanics/layered-soil-average-permeability site: BCT Engineering --- # Average permeability of layered soil Soil Mechanics · Chapter 6: Principle of Effective Stress, Capillarity and Permeability - Asked in: 2076 Baisakh, 3 marks - Video: [Permeability of Stratified Soils Deposits](https://www.youtube.com/watch?v=VQg6GwaIVlQ) — Elementary Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/layered-soil-average-permeability - Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-6 ## Question When water flows through layered soils, the average permeability, $k_{avg}$, depends on the flow direction with respect to the bedding plane. Find the value of $k_{avg}$ for the composite soil shown in the figure when water flows in the vertical and horizontal directions. Here $k_1$, $k_2$ and $k_3$ are the coefficients of permeability of the respective soils. [Figure: a block of soil $k_1$ of length $L_1$ over the full height ($H_1 + H_2$), next to a block of length $L_2$ made of $k_2$ (upper, height $H_1$) above $k_3$ (lower, height $H_2$).] ## Short answer Horizontal flow: right block layers in parallel give k_h2 = (k2 H1 + k3 H2)/(H1+H2), then left and right blocks in series yield k_avg,H = (L1+L2)/(L1/k1 + L2/k_h2). Vertical flow: right block layers in series give k_v2 = (H1+H2)/(H1/k2 + H2/k3), then left and right blocks in parallel yield k_avg,V = (k1 L1 + k_v2 L2)/(L1+L2). ## Answer Treat each zone of the soil as layers in series or parallel. Let the left block (zone 1) have permeability $k_1$, length $L_1$ and height $H = H_1 + H_2$. The right block (zone 2) has length $L_2$ and is made of $k_2$ (upper, $H_1$) over $k_3$ (lower, $H_2$). ```text <------ L1 ------><------ L2 ------> +-----------------+-----------------+ ^ | | k2 H1 | | | k1 +-----------------+ H | | k3 H2 | | +-----------------+-----------------+ v ``` #### 1. Horizontal flow (left to right) Inside zone 2 the flow is along the layers, so the two layers act in parallel: $$ k_{h2} = \frac{k_2H_1 + k_3H_2}{H_1 + H_2} $$ Zone 1 and zone 2 are met one after the other, so they act in series (same flow, lengths add): $$ k_{avg,H} = \frac{L_1 + L_2}{\dfrac{L_1}{k_1} + \dfrac{L_2}{k_{h2}}} $$ #### 2. Vertical flow (top to bottom) Inside zone 2 the flow crosses the layers, so they act in series: $$ k_{v2} = \frac{H_1 + H_2}{\dfrac{H_1}{k_2} + \dfrac{H_2}{k_3}} $$ Zone 1 and zone 2 now carry flow side by side under the same head (parallel). Weighting by the widths: $$ k_{avg,V} = \frac{k_1 L_1 + k_{v2} L_2}{L_1 + L_2} $$ Substituting $k_{v2}$ gives the full expression: $$ k_{avg,V} = \frac{k_1L_1 + \dfrac{(H_1+H_2)\,L_2}{\dfrac{H_1}{k_2}+\dfrac{H_2}{k_3}}}{L_1+L_2} $$ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For flow parallel to the layers (zone 2 horizontal), the equivalent k is: A. (H1+H2)/(H1/k2 + H2/k3) B. (k2 H1 + k3 H2)/(H1 + H2) C. (k2 + k3)/2 weighted by length D. sqrt(k2 k3) Answer: B. Layers act in parallel: thickness-weighted average of k. (Key point: Flow along layers) 2. For vertical flow across the layers k2 over k3, the equivalent k is: A. (k2 H1 + k3 H2)/(H1+H2) B. (H1+H2)/(H1/k2 + H2/k3) C. k2 + k3 D. k2 k3 Answer: B. Layers act in series: total thickness divided by sum of H/k. (Key point: Flow across layers) 3. Overall for horizontal flow, zone 1 and zone 2 are combined as: A. Parallel, weighted by widths B. Series: (L1+L2)/(L1/k1 + L2/k_h2) C. Arithmetic mean of k1 and k_h2 D. Geometric mean Answer: B. Flow meets one zone after the other, so they are in series with lengths adding. (Key point: Horizontal combination) 4. Overall for vertical flow, the zones are combined as: A. Series with lengths adding B. Parallel: (k1 L1 + k_v2 L2)/(L1+L2) C. Harmonic mean D. k1 only Answer: B. Zones carry flow side by side under the same head, so a width-weighted average applies. 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