Abstract
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.
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