In this paper we present a review on our recent experimental investigations into the phase behavior of the deeply cooled water confined in a nanoporous silica material, MCM-41, with elastic neutron scattering technique. Under such strong confinement, the homogeneous nucleation process of water is avoided, which allows the confined water to keep its liquid state at temperatures and pressures that are inaccessible to the bulk water. By measuring the average density of the confined heavy water, we observe a likely first-order low-density liquid (SLDL) to high-density liquid (HDL) transition in the deeply cooled region of the confined heavy water. The phase separation starts from 1.12 +/- 0.17 kbar and 215 +/- 1K and extends to higher pressures and lower temperatures in the phase diagram. This starting point could be the liquid-liquid critical point of the confined water. The locus of the Widom line is also estimated. The observation of the liquid-liquid transition in the confined water has potential to explain the mysterious behaviors of water at low temperatures. In addition, it may also have impacts on other disciplines, because the confined water system represents many biological and geological systems in which water resides in nanoscopic pores or in the vicinity of hydrophilic or hydrophobic surfaces.