Seepage and superficial velocity relation

Superficial (discharge) velocity $v$ is based on the total area; seepage velocity $v_s$ is the real velocity in the voids.

Derivation

Consider a soil of gross cross-section $A$ and length $L$. Let $A_v$ be the area of voids on the cross-section and $V_v$, $V$ the volumes of voids and total.

$$n = \frac{V_v}{V} = \frac{A_v L}{A L} = \frac{A_v}{A}$$

Let the discharge be $Q$. It passes through the gross area $A$ at velocity $v$ and through the void area $A_v$ at velocity $v_s$:

$$Q = v A = v_s A_v$$ $$v_s = v\,\frac{A}{A_v} = \frac{v}{n}$$

Since $v = k i$:

$$v_s = \frac{k\,i}{n}$$

Since $n<1$, $v_s > v$. In terms of void ratio, $n = \dfrac{e}{1+e}$ so $v_s = v\,\dfrac{1+e}{e}$.