6 - p-Adic and Ultrametric Models in Geophysics

Summary

The cooperation between the research groups of K. Oleschko (applied geophysics and petroleum research) and A. Khrennikov (p-adic mathematical physics) led to the creation of a new promising field of research [245], [367]: p-adic and more generally ultrametric modeling of the dynamics of flows (of, e.g., water, oil, and oil-in-water and water-in-oil droplets) in capillary networks in porous random media. The starting point of this project is the observation that tree-like capillary networks are very common geological structures. Fluids move through such trees of capillaries and, hence, it is natural to try to reduce the configuration space to these tree-like structures and the adequate mathematical model of such a configuration space is given by an ultrametric space.

The simplest tree-like structure of a capillary network can be modeled as the field of p-adic numbers on the ring of p-adic integers (or the ring of p-adic integers). In p-adic modeling the variable x ∈ and the real time variable t ∈ R are used. Here x is the “pore network coordinate,” meaning that each pathway of pore capillaries is encoded by a branch of the p-adic tree. The center of this tree is selected as an arbitrary branching point of the pore network. For the moment, it plays the role of the center of coordinates, i.e. it is a purely mathematical entity. Thus, by assigning the p-adic number x to a system, one gets to know in which pathway of capillaries it is located, nothing more. Hence, the p-adic model provides a fuzzy description of pore networks. In particular, the size of capillaries is not included in the geometry. It can be introduced into the model with the aid of the coefficients of the anomalous diffusion– reaction equation playing the role of the master equation. From the dynamics, one can know the concentration of fluid (oil, water, or emulsions and droplets) in capillaries. However, the model does not give the concentration of fluid at any precisely fixed point of Euclidean physical space.

This modeling heavily explores the theory of p-adic pseudodifferential equations, equations with fractional differential operators D^α (Vladimirov's operators), see e.g. [18], [259], [275], [286], [287], [302], [233], [232], [290], [470], and more general pseudodifferential operators.