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We present a new conceptual approach for modeling of fluid flows in random porous media based on explicit exploration of the treelike geometry of complex capillary networks. Such patterns can be represented mathematically as ultrametric spaces and the dynamics of fluids by ultrametric diffusion. The images of p-adic fields, extracted from the real multiscale rock samples and from some reference images, are depicted. In this model the porous background is treated as the environment contributing to the coefficients of evolutionary equations. For the simplest trees, these equations are essentially less complicated than those with fractional differential operators which are commonly applied in geological studies looking for some fractional analogs to conventional Euclidean space but with anomalous scaling and diffusion properties. It is possible to solve the former equation analytically and, in particular, to find stationary solutions. The main aim of this paper is to attract the attention of researchers working on modeling of geological processes to the novel utrametric approach and to show some examples from the petroleum reservoir static and dynamic characterization, able to integrate the p-adic approach with multifractals, thermodynamics and scaling. We also present a non-mathematician friendly review of trees and ultrametric spaces and pseudo-differential operators on such spaces.Entropy 2016, 18, 249 2 of 28 process with a non-linear relationship to time of its dispersion, in contrast to the standard diffusion, in which the dispersion σ r 2 is a linear function of time). We remark that a number of successful models

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree-like behavior and is locally constant for some time periods. It is natural to apply this kind of models for the investigation of avalanche processes and punctuated equilibrium as well as fractal-like analysis of time series generated by measurement of pressure in oil wells.

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