Critical gradient; quick condition in sand

Expression for critical hydraulic gradient

Take a sand layer of thickness $L$ and saturated unit weight $\gamma_{sat}$ in a tank. Water flows upward under a net head $h$ (head loss over the sample).

   h ↑  head causing upward flow
  ---------------  water
  ======= sand  L
  ---------------  filter
     ↑ ↑ ↑ inflow

At the base of the sand layer:

$$\sigma = \gamma_{sat}L \quad (\text{ignoring water above the soil, it cancels in } \sigma')$$ $$u = \gamma_w (L + h)$$ $$\sigma' = \sigma - u = \gamma_{sat}L - \gamma_w L - \gamma_w h = \gamma' L - \gamma_w h$$

Quick condition begins when $\sigma' = 0$:

$$\gamma' L = \gamma_w h_c \quad \Rightarrow \quad i_c = \frac{h_c}{L} = \frac{\gamma'}{\gamma_w}$$

Using $\gamma' = \dfrac{(G-1)\gamma_w}{1+e}$:

$$i_c = \frac{G-1}{1+e}$$

For $G = 2.65$ and $e = 0.65$, $i_c = 1.0$.

Why sand more than clay

• Sand is cohesionless: once $\sigma' = 0$ the grains have no strength at all. Clay has cohesion, so it keeps some strength even when $\sigma'$ is zero.
• Sand is highly permeable, so a large hydraulic gradient can easily be developed and a large flow of water occurs at the exit surface. In clay, low permeability makes the flow very small, so the critical gradient is rarely reached.
• Clay particles are bound by electrochemical forces, which resist being lifted by seepage.