A certain varved clay consists of alternating horizontal layers of silt and clay. The silt layers are 5 mm thick and have k = 3×10^-4 cm/s; the clay layers are 20 mm thick and have k = 6×10^-7 cm/s. Compute kx and kz.

Question

A certain varved clay consists of alternating horizontal layers of silt and clay. The silt layers are 5 mm thick and have [latex]k = 3×10^{-4} cm/s[/latex]; the clay layers are 20 mm thick and have [latex]k = 6×10^{-7} cm/s[/latex]. Compute [latex]k_{x}[/latex] and [latex]k_{z}[/latex].

Accepted Answer

[latex]k_{x}=\frac{\sum{k_{i}H_{i}} }{\sum{H_{i}} } =\frac{\left(3 imes 10^{- 4} cm/s ight) \left(0.5 cm ight) + \left(6 imes 10^{7} cm/s ight) \left(2 cm ight) }{0.5 cm + 2 cm} =6 imes 10^{-5} cm/s[/latex]

[latex]k_{z}=\frac{\sum{H_{i}} }{\sum{\left\lgroup\frac{H_{i}}{k_{i}} ight group } } =\frac{0.5 cm + 2 cm}{\frac{0.5 cm}{3 imes 10^{-4} cm/s} + \frac{2 cm}{6 imes 10^{-7} cm/s} } =7 imes 10^{-7} cm/s[/latex]

Commentary

Water flowing horizontally moves through both types of soil in parallel. The ratio of thicknesses for the two strata is 20/5 = 4, but the ratio of [latex]k[/latex] values is [latex]3 imes 10^{-4} / 6 imes 10^{-7} = 500[/latex]. Thus, the silt layers dominate the horizontal flow, even though they represent only 20% of the total cross-sectional area of flow. This is why the value of [latex]k_{x}[/latex] is nearly equal to [latex]k[/latex] of the silt. However, water flowing vertically must pass through all of the soil layers, and thus is controlled by the ones with the lowest hydraulic conductivity. Therefore, [latex]k_{z}[/latex] is nearly equal to [latex]k[/latex] of the clay.

Question 7.4: A certain varved clay consists of alternating horizontal lay...

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