Question

The term \(\frac{∂\overline{u}}{∂t}\) in terms of coefficient of volume change and coefficient of permeability is given by ____________

(a) \(\frac{∂\overline{u}}{∂t}=\frac{k}{m_v γ_w}  \frac{∂\overline{u}}{∂z}\)

(b) \( \frac{∂\overline{u}}{∂t}=\frac{k}{m_v γ_w}  \frac{∂\overline{u}}{∂z}\)

(c) \( \frac{∂\overline{u}}{∂t}=\frac{k}{m_v γ_w} \frac{∂^2\overline{u}}{∂z^2}\)

(d) \( \frac{∂\overline{u}}{∂t}=\frac{k}{m_v γ_w}  \frac{∂\overline{u}}{∂z}\)

The question was asked by my college professor while I was bunking the class.

I'm obligated to ask this question of Terzaghi’s Theory of One Dimensional Consolidation in section One Dimensional Consolidation of Geotechnical Engineering I

Answer 1

Correct answer is (c) \( \frac{∂\overline{u}}{∂t}=\frac{k}{m_v γ_w} \frac{∂^2\overline{u}}{∂z^2}\)

The explanation: The decrease in volume is given by,

 ∆v=-mv V0 ∆σ’—————–(1)

also \(\frac{∂v}{∂z}=m_v  \frac{∂\overline{u}}{∂t}\)  ——————————-(2)

∴ from (1) and (2), we get,

\(\frac{∂\overline{u}}{∂t}=\frac{k}{m_v γ_w}\frac{∂^2 \overline{u}}{∂z^2}.\)