8. As shown in Figure 1, the base $ABCD$ of the quadrilateral pyramid $F-ABCD$ is a rhombus, with diagonals $AC=2$, $BD=\sqrt{2}$, and $AE$, $CF$ are both perpendicular to the plane $ABCD$, $AE=1$, $CF=2$. Then the volume of the common part of the quadrilateral pyramids $E-ABCD$ and $F-ABCD$ is $\qquad$
8. $\frac{2 \sqrt{2}}{9}$.

As shown in Figure 2, let the line $A F$ intersect the line $C E$ at point $H$. Then the common part of the quadrilateral pyramid $E-A B C D$ and the quadrilateral pyramid $F-A B C D$ is the quadrilateral pyramid $H-A B C D$.
Draw $H P \perp$ plane $A B C D$, with $P$ being the foot of the perpendicular.
Since $E A \perp$ plane $A B C D$ and $F C \perp$ plane $A B C D$, it follows that plane $A C F E \perp$ plane $A B C D$.
Thus, point $P$ lies on $A C$, and $H P \perp A C$.
From $\frac{H P}{C F}+\frac{H P}{A E}=\frac{A P}{A C}+\frac{P C}{A C}=1$, we get $H P=\frac{2}{3}$.
Also, $S_{\text {base } A B C D}=\frac{1}{2} A C \cdot B D=\sqrt{2}$, so