  We discuss two conditions needed for correct computation of $2\nu \beta\beta$
nuclear matrix-elements within the realistic shell-model framework. An
algorithm in which intermediate states are treated based on Whitehead's moment
method is inspected, by taking examples of the double GT$^+$ transitions
$\mbox{$^{36}$Ar}\rightarrow\mbox{$^{36}$S}$,
$\mbox{$^{54}$Fe}\rightarrow\mbox{$^{54}$Cr}$ and $\mbox{$^{58}$Ni}
\rightarrow\mbox{$^{58}$Fe}$. This algorithm yields rapid convergence on the
$2\nu\beta\beta$ matrix-elements, even when neither relevant GT$^+$ nor GT$^-$
strength distribution is convergent. A significant role of the shell structure
is pointed out, which makes the $2\nu\beta \beta$ matrix-elements highly
dominated by the low-lying intermediate states. Experimental information of the
low-lying GT$^\pm$ strengths is strongly desired. Half-lives of
$T^{2\nu}_{1/2}({\rm EC}/{\rm EC};
\mbox{$^{36}$Ar}\rightarrow\mbox{$^{36}$S})=1.7\times 10^{29}\mbox{yr}$,
$T^{2\nu}_{1/2}({\rm EC}/{\rm EC};\mbox{$^{54}$Fe}\rightarrow
\mbox{$^{54}$Cr})=1.5\times 10^{27}\mbox{yr}$,$T^{2\nu}_{1/2}({\rm EC} /{\rm
EC};\mbox{$^{58}$Ni}\rightarrow\mbox{$^{58}$Fe})=6.1\times 10^{24}\mbox{yr}$and
$T^{2\nu}_{1/2}(\beta^+/{\rm EC};\mbox{$^{58}$Ni}
\rightarrow\mbox{$^{58}$Fe})=8.6\times 10^{25}\mbox{yr}$ are obtained from the
present realistic shell-model calculation of the nuclear matrix-elements.
