Modeling Fluid’s Dynamics with Master Equations in Ultrametric Spaces Representing the Treelike Structure of Capillary Networks

Abstract

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

P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)

“…We point out that, although the ultrametric pseudo-differential equations studied in [4][5][6] describe some distinguishing features of propagation of fluids in capillary networks, they did not explicitly reflect coupling with the physical parameters of such networks. Roughly speaking, they were designed on the basis of abstract reasoning about the natural mathematical form of fluid dynamics on the tree-like configuration space.…”

“…[4][5][6]. On the other hand, by proceeding from the discrete tree-like dynamics, for which the physical meaning of dynamics' parameters are well defined, to "continuous" pseudo-differential equations, we also clarify the physical meaning of their parameters.…”

“…The tree-like geometry of percolation networks was always of special interest of Mandelbrot, the father of fractal geometry. In the set of our previous publications [4][5][6] we have discussed the advantages of using the scaling, as well as hierarchical representations of several complex systems. Some simplified models of oil, oil-in-water, and water-in-oil emulsion propagation in capillary networks in random porous media based on their physically real tree-like geometry were proposed [4][5][6].…”

“…Section 3 is devoted to the derivation of discrete tree-like dynamical equations describing fluid flows in capillary networks in porous media. In Section 4 we present essentials of the theory of p-adic numbers (encoding branches of homogeneous trees) and the corresponding theory of the Fourier transform and pseudo-differential operators (this presentation is really very brief and the reader can find details, e.g., in our paper [4]). Finally, in Section 5 we transform the system of discrete tree-like dynamical equations into p-adic (for the simplest case p = 2) nonlinear pseudo-differential equations which can be treated as the tree-set analog of the Euclidean space-based system of the Navier-Stokes equations.…”

“…Mathematically, such tree-like fractals are represented as ultrametric spaces and there exist well developed theories of fractional (pseudo-)differential equations on such spaces [7][8][9][10][11][12][13][14][15], which were applied in [4][5][6] to represent the dynamics of fluids in these tree-like capillary networks. The class of homogeneous trees with the constant number of vertices, p > 1, leaving each vertex, the so called p-adic trees, is especially useful for theoretical modeling as representing a mathematically simple model which, at the same time, represents the basic features of fluids' propagation through capillary/fractures networks.…”