## 1 Introduction  
A review of the previous research regarding the dryout of bottom-heated saturated porous media shows, by the large number of published papers, the remarkable interest of researchers in this phenomenon over recent decades. However, there is a lack of publications related to the dryout/replenishment phenomena in porous media with an overlying liquid layer limited by a free surface. Dryout following boiling in porous media occurs in many engineering applications, such as geothermal systems, heat pipes, and post-accident analyses of liquid-cooled nuclear reactors. Since uncertainties exist about many aspects of the boiling process, experimental studies intend to elucidate the heat transfer process from the heating surface to the porous medium. A thin liquid layer in contact with the heating surface, a vapor layer, and a layer with both regions--vapor and liquid--can exist during this process. Experimental studies report the formation of an isothermal two-phase zone above the bottom whenever the temperature there exceeds saturation. The dryout condition requires accurate knowledge of how the vapor condenses on the capillary surface of the two-phase region in the porous media, and how the interface of the two-phase plain liquid saturated regions moves upward. In this regard, the works of the researchers [PERSON] [1], [PERSON] and [PERSON] [2], [PERSON] and [PERSON] [3], [PERSON] [4], [PERSON] et al. [5], [PERSON] [6,7] and many others have contributed to better knowledge of the aforementioned phenomena. The transport of heat from the liquid-vapor countercurrent in a two-phase liquid layer region was studied by [PERSON] and [PERSON] [8]. When boiling occurs, an almost isothermal two-phase zone forms near the bottom. [PERSON] and [PERSON] [9] reported experimental observations of boiling in a vertical circular cylinder heated from below and cooled from above. A simple one-dimensional model is used to predict the high of this zone, the necessary condition for its formation, and the dryout heat flux. The effects of gravity, capillarity, multiphase flow, and phase change in a fluid-filled porous media were studied by [PERSON] [10; 11] by means of one-dimensional analysis considering steady-state conditions. If a critical heat flux is exceeded, a vapor zone will occur at the heated surface. In this study, [PERSON] established the criteria for determining the heat flux at the dryout. The dryout heat flux in particulate beds heated through the base has also been studied. [PERSON] et al. [12], in a laboratory study, discussed the role of capillary forces in bed dryout. Then, [PERSON] and [PERSON] [13] reviewed the interaction between boiling and convection phenomena in a fluid-saturated porous layer heated from below and cooled from above. They used a stability diagram to provide a qualitative understanding of the experimental results. With regards to model boiling and dryout in particle beds, and concurrent with the experimental research, significant efforts have been made to develop understanding in this area. The aim of most of these models is to predict the incipient dryout power, namely the necessary power level to cause the appearance of a dry area in the bed. [PERSON] and [PERSON] [14] presented a model based on their experiments. A step forward was made by [PERSON] and [PERSON] [15] when they considered the combined resistances of liquid and vapor flow and developed their dryout criterion based on the conservation laws for mass, momentum, and energy. [PERSON] and [PERSON] [16] extended the Hardee-Nilson model to include the effect of capillary forces and [PERSON] [17; 18] went further in the same direction, assuming a capillary force and using the relative permeability to express two-phase friction. For the dryout in channeled porous beds, the modeling efforts for determining the heat flux refer only to shallow beds. Most of the related studies, such as those by [PERSON] et al. [12], [PERSON] and [PERSON] [14], and [PERSON] and [PERSON] [19], involved bottom-heated cases.