Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient |
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Authors: | Prabir Daripa G Pa?a |
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Institution: | (1) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;(2) Institute of Mathematics “Simion Stoillow” of Romanian Academy, Bucharest, 70700, Romania |
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Abstract: | In the presence of diffusion, stability of three-layer Hele-Shaw flows which models enhanced oil recovery processes by polymer
flooding is studied for the case of variable viscosity in the middle layer. This leads to the coupling of the momentum equation
and the species advection-diffusion equation the hydrodynamic stability study of which is presented in this paper.
Linear stability analysis of a potentially unstable three-layer rectilinear Hele-Shaw flow is used to examine the effects
of species diffusion on the stability of the flow. Using a weak formulation of the disturbance equations, upper bounds on
the growth rate of individual disturbances and on the maximal growth rate over all possible disturbances are found. Analytically,
it is shown that a short-wave disturbance if unstable can be stabilized by mild diffusion of species, where as an unstable
long-wave disturbance can always be stabilized by strong diffusion of species. Thus, an otherwise unstable three-layer Hele-Shaw
flow can be completely stabilized by a large enough diffusion, i.e., by increasing enough the magnitude of the species diffusion
coefficient. The magnitude of this diffusion coefficient required to completely stabilize the flow will depend on the magnitude
of interfacial viscosity jumps and the viscosity gradient of the basic viscous profile of the middle layer. |
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Keywords: | Three-layer Hele-Shaw flows Improved oil recovery Stability of flows Sturm-Liouville problem Upper bound |
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