A review is given for the state of the art and future development of the geothermal power engineering. Different diagrams of GeoPP are shown for different levels of geofluid temperature. A special focus is made on challenges for petrothermal power production. The huge amount of heat from petrothermal resources can be harnessed through creating geothermal circulation systems within the dry rocks. Estimates for system lifetime and heat production capacity of circulation systems are given for different flow rates of coolant.

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The article reviews analytical methods of studying heat transfer processes running in the conditions of resource-saving underground coal processing, heat drying in processing facilities and in the course of heat accumulation and emission in caved rocks. The author considers problems with boundary conditions formulated at the moving boundary of the heat transfer domain. , processing and beneficiation sometimes needs to research unsteady-state conjugate heat transfer or the heat transfer processes in the course of filtration in heterogeneous media naturally, technologically and physicochemically complicated. A mining practice is an extremely complex multi-factor functionally connected system of operation conditions and results governed by the diversity of natural structure, mining-and-geological conditions and operational and performance and design philosophy. This dictates the variety of physical models, calculation schemes and problem solution techniques.Generally, a natural permeable reservoir is simulated by a homogeneous medium with effective properties, or by a heterogeneous medium with a periodic isotropic structure composed of classical geometry elements (plates, cylinders, balls) [1][2][3][4][5][6][7].In case of an infinitely large solid element of a heterogeneous medium, we arrive to the partial problems of unsteady filtration heat transfer. These are the problems of conjugate heat transfer when fluid flows in natural and artificial channels (strata) and fractures [1,6,8], and their analytical solutions are obtained based on the method borrowed from the classical theory of thermal conduction [9].The scale mathematical modeling, with comprehensive facilities offered by the modern computer science, is of interest at this time and by all means allows solution of many problems of mine thermophysics. The "perfect accurate" mathematical modeling requires sophisticated software products. The mathematical description of a process being most complete, the numerical analysis and experiment are restricted by specificity of application and input data, which always contains uncertainty and assumptiveness. Interpretation of the results can also be difficult.Simplification of an initial set of equations necessitates deviation from the real-life conditions of a mining practice. The simpler mathematical description and approximated solution methods require physical substantiation of the problem conventionalities and experimental checking of the results.Nevertheless, physically substantiated models, asymptotic or/and approximated solution methods produce simple analytical results suitable for including in engineering calculations, more complicated physical formulations and in the economical-mathematical modeling. 82 SMIRNOVA et al.

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