The aim of this article is to present a one-dimensional finite volume modeling of mass transport phenomena in partially saturated open porous media. The finite volume model is based on a staggered solution strategy which permits to solve the equations sequentially. This appropriate partitioning reduces thus the size of the system to be solved at each time step. Therefore, two iterative algorithms are proposed to solve the discretized problem. Moreover, an original treatment of the convective boundary condition, based on a suitable linearization to derive the algebraic form of this condition, is also presented. The computational model is then specifically used for studying the drying process of a cementitious material. For this purpose, the two proposed algorithms are compared in terms of computational efficiency. Thereafter, the convective boundary condition is analyzed with regard to the drying kinetics. Finally, a one-dimensional drying experimental test is simulated.