DOI: 10.12732/ijam.v37i1.10
GENERALIZED RELAXATION
FRACTIONAL DIFFERENTIAL
MODEL OF FLUID FILTRATION
IN A POROUS MEDIUM
B. Khuzhayorov 1, T.O.
Djiyanov 2, M.S. Zokirov 3,§
1,2,3 Samarkand State University
Samarkand - 140100, UZBEKISTAN
Abstract. The process of anomalous filtration of a homogeneous liquid in a porous medium is modeled by differential equations with a fractional derivative. Fractional derivatives are used as defined by Caputo. The problem of filtration in a finite homogeneous reservoir is posed and numerically solved. The influence of process abnormality on filtration characteristics was estimated. It is shown that a decrease in the exponent of the derivative in the relaxation term with respect to pressure leads to the decrease of the pressure distribution up to a certain distance from the beginning of the medium, and then to an increase. Reducing the order of the derivative in the relaxation term with respect to the filtration velocity acts inversely. The corresponding dynamics
with decreasing orders of derivatives las the filtration velocity. As a special case, the case with the predominance of the filtration velocity relaxation time over the pressure relaxation time is singled out, in particular, when the latter is equal to zero. In this case, the solution of the filtration equation acquires a wave character. With an increase in the difference between relaxation times in terms of filtration velocity and pressure, the propagation velocity of pressure waves decreases.
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DOI: 10.12732/ijam.v37i1.10
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 1
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