This research concerns fundamentals of spontaneous transport of saltwater (1 mol·dm-3 NaCl solution) in nanopores of calcium carbonates. A fully atomistic model was adopted to scrutinize the temperature dependence of flow regimes during solution transport under CaCO3 nanoconfinement. The early time of capillary filling is inertia-dominated, and the solution penetrates with a near-planar meniscus at constant velocity. Following a transition period, the meniscus angle falls to a stabilized value, characterizing the capillary-viscous advancement in the calcite channel. At this stage, brine displacement follows a parabolic relationship consistent with the classical Lucas-Washburn (LW) theory. Approaching the slit outlet, the meniscus contact lines spread widely on the solid substrate and brine leaves the channel at a constant rate, in oppose to the LW law. The brine imbibition rate considerably increases at higher temperatures as a result of lower viscosity and greater tendency to form wetting layers on slit walls. We also pointed out a longer primary inertial regime and delayed onset of the viscous-capillary regime at higher temperatures. Throughout the whole span of capillary displacement, transport of sodium and chloride ions is tied to dynamics and diffusion of the water phase, even at the mineral interface. The results presented in this study are of broad implications in diverse science and technological applications.