Macroscopic Model of Two-Phase Compressible Flow in Double Porosity Media
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oscopic Model of Two-Phase Compressible Flow in Double Porosity Media M. B. Panfilova,b,*, Zh. D. Baishemirovc,d, and A. S. Berdyshevc,d a Institut
Elie Cartan – Université de Lorraine, Nancy, France Institut d’Alembert – Sorbonne Université, Paris, France c Abai Кazakh National Pedagogical University, Almaty, Kazakhstan d Institute of Information and Computational Technologies CS MES RK, Almaty, Kazakhstan *e-mail: [email protected] b
Received June 27, 2019; revised September 20, 2019; accepted November 13, 2019
Abstract—A macroscopic model of two-phase flow of compressible liquids in a compressible double porosity medium is developed and used to analyse various qualitative mechanisms of the occurrence of memory (delay). The two main mechanisms are non-instantaneous capillary redistribution of liquids, and non-instantaneous relaxation of pressure. In addition, cross effects of memory arise, caused by asymmetric extrusion of liquids from pores due to phase expansion and pore compaction, as well as nonlinear overlap of compressibility and capillarity (non-linear extrusion). To construct the model, the asymptotic method of two-scale averaging in the variational formulation is applied. Complete averaging has been achieved due to the separation of nonlocality and nonlinearity at different levels of the asymptotic expansion. All delay times are explicitly defined as functions of saturation and pressure. Keywords: double porosity, averaging, two-phase flow, compressible fluid, nonlocality, memory, delay, nonequilibrium DOI: 10.1134/S001546282007006X
INTRODUCTION A double porosity medium (it is also called fractured-porous medium) is a classical model for studying the effects of the occurrence of memory in fluid flow after changing the scale of consideration. Such a medium consists of weakly permeable blocks and highly permeable fractures. In the case of a single-phase compressible flow, the propagation of a perturbation in blocks is much slower than in fractures, so that the average pressure in the blocks is delayed and depends on the entire history of the variation of the average pressure in the fractures, which is a memory effect. Since this effect is observed in terms of average pressures, it is necessary to construct equations averaged over inhomogeneities. For single-phase flow, which is described by linear equations, it is possible to obtain a completely averaged model and explicit expression for the memory operator. The first mathematical model of such a process was proposed in [1]. Later, in the early 1990s, it was shown by averaging [2, 3] that high difference in the permeability of blocks and fractures leads not only to delay, but to long memory, described by an integro-differential equation. In the case of two-phase flow, another type of memory arises, which is generated by capillarity. Capillary forces push the wetting phase from the fractures into blocks, while displacing the non-wetting phase from the blocks into fractures. This phenomenon is known as capillary countercurrent imbibition. In
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