| Problem of flow instability in porous media are important for applied fields like mining, water
supply, etc. There is a fundamental interest to mechanisms are influence on flow
too. E.g., a viscous fingering is typical phenomenon of displacement processes in
porous medium [1,2]. The instability of gas flow in liquid-saturated domain have no
wide studies but it can make significant influence on heat and mass transport. If
the one phase have a high saturation, the other phase will form the droplets are
break and captured within pores due to the capillary forces [2–4]. It is possible to
neglect the capillarity if the saturation of both fluids exceed a percolation thresholds
[5,6].
We consider an infinite flat layer of uniform porous medium is saturated with gas and
liquid have close saturation. Its upper boundary is impermeable for liquid phase and gas can
pass freely through the border, and the down boundary is permeable for both phases. The
temperature and pressure are fixed at the top while their gradients are fixed at the bottom side.
Neglecting the capillarity, gas solubility, liquid evaporation and any phase transitions, we
obtain a steady solution and study its’ stability. The governing parameter of the flow
is
α = αgAPe, αg = (ÏwCg )-(ÏsCs), A = Ïstatvstat
(1)
where Pe is the thermal Peclet number determines a ratio between convective and conductive
heat transfer, αg is ratio of thermal capacities of fluid and matrix, and A is determined by gas
density and velocity in the steady state.
Analyzing the perturbations, we found that a long-wave instability realizes in the system.
The critical value of parameter is:
αc = a1 + k2a2 + O(Ïg-Ïw),
(2)
where a1,a2 are positive coefficients are calculated using thermal perturbations
combinations and k is wave number along horizontal direction. The minimal αc equals
2.47, and it correspond the critical Peclet number near 200 in the methane–water
system. An error of the dependence is of order of gas to water density ratio at small
k.
The critical value of α is independent on system parameters in the first approximation.
Therefore, the main mechanism of instability development is only an interaction of
conductive and convective heat transfer in fluid flows unlike systems are described in [7,8].
The α is independent on average values of temperature and pressure. The fundamental
mechanism of long-wave instability of gas flow is not connected with capillarity effects,
phase transitions, etc. Thermal expansion of the liquid does not change the flow stability
threshold rapidly.
[1] Fedotov, S.P. and Mikhailova, N.A. Instability of a steady-state regime for the
combustion of petroleum in a porous medium. J. Eng. Phys., 55, 1244–51 (1988)
[2] Ho, C.K. and Webb, S.W. (Ed.) Gas transport in porous media. Springer
(2006)
[3] Lyubimov, D.V., and Shklyaev, S., and Lyubimova, T.P. and Zikanov, O.
Instability of a drop moving in a Brinkman porous medium. Phys. Fluids, 21, 014105
(2009)
[4] Bentsen, R.G. The role of capillarity in two-phase flow through porous media.
Transport Porous Med., 51, 103–12 (2003)
[5] Goldobin, D.S. et al. Non-fickian diffusion and the accumulation of methane bubbles
in deep-water sediments. arXiv:1011.6345 (2011)
[6] Shklovskii, B.I. and Efros, A.L. Percolation theory and conductivity of strongly
inhomogeneous media. Phys.–Usp., 18, 845–62 (1975)
[7] Bulgakova, G.T., and Kalyakin, L.A. and Khasanov, M.M. Stability of filtration of a
gas-liquid mixture. J. Appl. Mech. Tech. Phy., 41, 1029–35 (2000)
[8] Caro, F., and Saad, B. and Saad, M. Two-component two-compressible flow in a
porous medium. Acta Appl. Math., 117, 15–46 (2012) |