Analytical and numerical mathematical models for the study of geothermal variations caused by environmental temperature variability during Olocene

A large number of geological phenomena may be linked to anomalies of the local underground thermal gradient. The appearance of permafrost levels, the karstification with a predominant horizontal growth and travertine deposition in limestones at specific geological times may be just a few consequences of geothermal anomalies. One of the causes of the anomalies can be attributed to the past variations of the temperatures. Various proposed numerical models (for example Dahl-Jensen et alii, 1998; Pasculli & Sciarra, 2005), would demonstrate the feasibility of the occurrence of local geothermal gradient inversion due to the environmental temperature variations. Thus the target of this paper was essentially to discuss, furthermore, the mathematical feasibility of inversion and its peculiarity in space and in time, not only by means of the elaboration and the solution of a finite element numerical model, but by means of an analytical mathematical model as well. This was applied to simplified geological systems, in areas characterized by low gradient values, in which it is possible to exclude the formation of permafrost also in the past. The permafrost is a system with remarkable �thermal inertia� which, consequently, causes a delay and a lowering of the �answer� to the external temperature variations. This characteristic implies that surface thermal oscillations cannot be strictly correlated with temperature profile below thick permafrost layer. As a consequence the reversals of the geothermal gradient occurrence would have been enhanced. Moreover the exclusion of the permafrost, beyond simplifying the elaboration of the analytical calculation, allows the study of the feasibility of reversals also in warmer areas and, therefore, less prone from this point of view. Furthermore, it is noteworthy to point out that, as it�s well known, an analytical approach, if possible, allows to obtain indications and informations more general than those that can be obtained from typically numerical results. Indeed, numerical ones are specific of the particular ensemble of values assigned to the parameters characterizing the problem. The implemented and discussed models consider temperature distribution in a semi-infinite soil slab. We took into account the earth surface temperature variation throughout the last 90,000 years, deduced from the rate of two oxygen isotopes (18O/16O) measured in Greenland glaciers by GRIP (Greenland Ice-core Project) group. From the experimental oxygen isotopes ratio curve, applying the FFT (Fast Fourier Transform) tool, we built up an equivalent and correlated temperature analytical curve, implemented as Dirichlet�s boundary condition. The resulting well known analytical �thermal waves� have been implemented in a mathematical time scanning model, in order to obtain the evolution of oscillation of the subsoil temperature distribution excluding, however, permafrost formation. Also this kind of simulation shows an actual feasibility of a local thermal gradient inversion due to environmental temperature variability, at a depth ranging from few dozens up to a few hundred meters, at different geological periods. This can be happened also in relatively warm regions and characterized by vanishing or null values of the pre-existing geothermal gradient. The finite element numerical model, briefly reported in this paper, take into account the eventual formation of permafrost, with its thermal inertia. Also in this case, geothermal reversals have been evidenced