We present theoretical analysis on old white dwarf stars with intermediate amount of hydrogen. 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 of convective mixing in white dwarf stars. 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 Bergeron et al. in a later paper.