Stressed water-infiltrated silica rocks may deform by pervasive pressure solution transfer (PPST), which involves dissolution of the grain-to-grain contacts, transport by diffusion of the solute, and precipitation on the free surfaces of the grains. A fundamental question regarding this process is how to model rheological behavior at stresses and temperatures typical of the crust of the Earth. A Voigt-type poroviscoplastic model is modified by using a Cole-Cole distribution of relaxation times rather than a Dirac distribution used previously. The motivation of this choice is to account for the distribution of the grain size in the compaction of the porous aggregate assuming that this distribution obeys approximately a log normal distribution. This grain size distribution depends upon the initial grain size distribution and cataclasis in the early stage of compaction. We compared this modified viscoplastic model with the full set of experimental data obtained in various conditions of mean grain size, effective stress, and temperature by Niemeijer et al. (2002). These data provide tests of all aspects of the model, which can be considered to have no free parameters. We show the experiments of Niemeijer et al. (2002) on PPST are primarily diffusion-limited. The grain size distributions observed for three samples imply that the distribution of the relaxation time covers 5 orders of magnitude in grain size.