  Premelting describes the confluence of phenomena that are responsible for the
stable existence of the liquid phase of matter in the solid region of its bulk
phase diagram. Here we develop a theoretical description of the premelting of
water ice contained in a porous matrix, made of a material with a melting
temperature substantially larger than ice itself, to predict the amount of
liquid water in the matrix at temperatures below its bulk freezing point. Our
theory combines the interfacial premelting of ice in contact with the matrix,
grain boundary melting in the ice, and impurity and curvature induced
premelting, the latter occurring in regions which force the ice-liquid
interface into a high curvature configuration. These regions are typically
found at points where the matrix surface is concave, along contact lines of a
grain boundary with the matrix, and in liquid veins. Both interfacial
premelting and curvature induced premelting depend on the concentration of
impurities in the liquid, which, due to the small segregation coefficient of
impurities in ice are treated as homogeneously distributed in the premelted
liquid. Our principal result is an equation for the fraction of liquid in the
porous medium as a function of the undercooling, which embodies the combined
effects of interfacial premelting, curvature induced premelting, and
impurities. The result is analyzed in detail and applied to a range of
experimentally relevant settings.
