Abstract
P-adic numbers serve as the simplest ultrametric model for the tree-like structures arising in various physical and biological phenomena. Recently p-adic dynamical equations started to be applied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oil emulsion) in capillary networks in porous random media. In particular, a p-adic analog of the Navier–Stokes equation was derived starting with a system of differential equations respecting the hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium. This equation describes propagation of fluid’s flow through Geo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillary networks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomian decomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equation in term of series in general form. This solution may help researchers to come closer and find more facts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from the analogous aspects of the real world phenomena.
Highlights
The last decades have witnessed great use of Fourier and more generally wavelet analysis over the p-adic fields, and its various physical applications in physics, biology and cognitive science, and recently in geophysics
P-adic dynamical equations started to be applied to geophysics, to model propagation of fluids in capillary networks in porous random media
In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium
Summary
The last decades have witnessed great use of Fourier and more generally wavelet analysis over the p-adic fields, and its various physical applications in physics, biology and cognitive science, and recently in geophysics. The solvability of the p-adic analog of the Navier–Stokes equation via the wavelet theory is discussed by the example of real world problem: the precise modeling of fluid flow in highly heterogeneous, multiscale, and anisotropic porous media with strongly hierarchical architecture. This problem is recognized among key technical challenges of Petroleum Industry looking for new analytical solutions of classical mathematical analogue and new type of computing perspectives, more closed to the pure science. This paper demonstrates that, for fluids’ propagation through capillary networks in porous disordered media, p-adic linear models developed and investigated in our previous works [35,36] can be successfully generalized (at the mathematical level of rigorousness) to nonlinear phenomena
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