Prove the one$-$dimensional consolidation governing equation from first principles and state your assumptions.

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}=c_v\frac{de{l}^{2}u}{del{z}^2},$ where $c_v=\frac{K}{{}_w{m}_v}$

where $u=$ excess pore water pressure; $t=$ time; $z=$ depth; $c_v=$ coefficient of consolidation; ${}_w=$ unit weight of water; $K=$ hydraulic conductivity; and $m_v=$ coefficient of volume compressibility.