Modeling of fracture concentration by Sel’kov fractional dynamic system

¹ Institute of Cosmophysical Research and Radio Wave Propagation of the Far Eastern Branch of Russian Academy of Science, 684034, Kamchatskiy kray, Paratunka, Mirnaya str. 7, Russia
² National University of Uzbekistan after named Mirzo Ulugbek, 100174, Universitetskaya St., 4. Tashkent. Republic of Uzbekistan
³ Institute of Mathematics named after V. I. Romanovskiy Academy of Sciences of Uzbekistan, Academy of Sciences of Uzbekistan, 100170, Mirzo Ulugbek str., 85, Tashkent, Republic of Uzbekistan

^★ e-mail: romanparovik@gmail.com
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Abstract

Microseismic phenomena are studied by a Sel’kov generalized nonlinear dynamic system. This system is mainly applied in biology to describe substrate and product glycolytic oscillations. Thus, Sel’kov dynamic system can also describe interaction of two types of fractures in an elastic-friable medium. The first type includes seed fractures with lower energy and the second type are large fractures which generate microseisms. The first type of fractures are triggers for the second type of fractures. However opposite transition is possible. For example, when large fractures lose their energy and partially become seed ones. After their concentration increase, the process repeats providing auto oscillation character of microseism sources. Generalization of Sel’kov dynamic system is its analogue which is based on hereditarity. Hereditarity is studied within hereditary mechanics and it shows that a dynamic system can “remember” for some time the impact which was made upon it. It is typical for viscoelastic and yielding mediums. The Sel’kov generalized dynamic system will be called Sel’kov fractional dynamic system as long as from the point of view of mathematical description, it can be represented in the form of a system of differential equations with fractional derivatives. Fractional derivative orders are associated with system hereditarity and are responsible for energy dissipation intensity emitted by first- and second-type fractures. In the paper, the Sel’kov fractional dynamic model was numerically solved by Adams-Bashforth-Moulton method. Oscillo-grams and phase trajectories were plotted. It was shown that fractional dynamic model may have relaxation and damped oscillations.