Fields of Filtration Rates in Layered Heterogeneous Beds

About the authors:

Aleksandr I. Filippov, Dr. Sci. (Tech.), Professor, Chief Researcher, Sterlitamak Branch of Ufa University of Science and Technology, Sterlitamak, Russia; filippovai1949@mail.ru, https://orcid.org/0000-0002-0964-9805

Oksana V. Akhmetova, Dr. Sci. (Phys.-Math.), Professor, Department of General and Theoretical Physics, Sterlitamak branch of the Bashkir State University; eLibrary AuthorID, ahoksana@yandex.ru

Aleksei A. Kovalsky, Cand. Sci. (Phys.-Math.), Director, Sterlitamak Branch of the Bashkir State University; eLibrary AuthorID, aakov68@mail.ru

Marat R. Gubaydullin, Junior Researcher, Sterlitamak Branch of the Bashkir State University; eLibrary AuthorID, Web of Science ResearcherID, fir_bmf@mail.ru

Abstract:

This article proposes an interpretation model of well logging for layered heterogeneous orthotropic formations with an arbitrary permeability distribution in the perforated layer. This model relies on the problem of determining the velocity field in the well-layer system. This problem is reduced to the problem of the pressure field in a layered inhomogeneous formation system with an arbitrary distribution of permeability components of the perforated layer k_zd(z_d), k_xd(z_d). The analytical solution of this problem was obtained on the basis of the asymptotic method of “stepwise spatial averaging”. It was then employed to reconstruct the velocity field using the Darcy law; it was realized using modern mathematical software packages.

Based on the analysis of the obtained analytical expressions and the computational experiment, the authors show that the velocity field is essentially determined by the permeability distribution over the thickness of the perforated layer, even though in the zero approximation, the pressure profile in the perforated layer does not depend on the vertical coordinate. In contrast to the known models, the zero approximation obtained takes into account the contribution of vertical interstitial overflows from the unperforated zone of the formation. Nevertheless, it does not allow estimating the effect of interplastic overflows in the perforated layer and their contribution to the solution of the pressure field problem in the first asymptotic approximation.

The results of calculations of the velocity profile at the exit from a porous medium for various practically important model distributions of permeability are presented. New regularities and features of the formation of the velocity profile are established. The developed model opens up prospects for solving inverse problems that can be used to develop methods for interpreting well logs.

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