  We present theoretical analysis on old white dwarf stars with \emph{some}
hydrogen, that possess a mass of surface hydrogen from
$1\times10^{-11}M_{\sun}$ to $1\times10^{-7}M_{\sun}$. The evolution of such
objects is complicated by convective mixing from surface convection zone to the
underlying helium layer. In this paper, we provide first self-consistent,
quantitative investigation on the subject of convective mixing. Numerical
cooling curves and chemical evolution curves are obtained as a function of
white dwarf mass and hydrogen content. Such results will be applied to the
investigation of the non-DA gap of \citet{1997ApJS..108..339B} in a later
paper.
