Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media

We develop coupled evolution equations for viscous fingering (VF) and phase separation in partially miscible systems by combining a simple double-well thermodynamic free energy and Korteweg force with a classical miscible VF model for a binary system. The VF pattern transition into a droplet formation pattern by the spinodal decomposition effect is demonstrated, and the simultaneous increases in the depth of the energy minimum, in the difference in the equilibrium concentrations, and in the Korteweg force, enhance the droplet growth. The pattern's interfacial length increases with the spinodal decomposition effects. These results match the corresponding experimental results.

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“…We next analyse the interfacial length I, which is defined as (Mishra, Martin & De Wit 2008;Pramanik & Mishra 2015a)…”

Section: Phase Separation Effects On Interfacial Lengthmentioning

confidence: 99%

Numerical study on topological change of viscous fingering induced by a phase separation with Korteweg force

et al. 2022

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“…(2.2) 3). It is noted that the pressure-like term Q(c) due to Korteweg stress, appearing in the Darcy-Korteweg equation, can be defined as (Joseph & Renardy 1992;Pramanik & Mishra 2013, 2015bSwernath et al 2010)…”

Section: Governing Equationsmentioning

confidence: 99%

“…The effect of EIT on miscible VF dynamics has been numerically studied (Chen, Wang & Meiburg 2001;Chen et al 2008;Pramanik & Mishra 2013, 2015bSwernath, Malengier & Pushpavanam 2010). Chen et al (2008) showed that the miscible VF patterns in a radial geometry with EIT can be made to resemble the more familiar morphologies of their immiscible counterparts by increasing the value of a Korteweg stress parameter.…”

Section: Introductionmentioning

confidence: 99%

“…Chen et al (2008) showed that the miscible VF patterns in a radial geometry with EIT can be made to resemble the more familiar morphologies of their immiscible counterparts by increasing the value of a Korteweg stress parameter. Swernath et al (2010) showed that Korteweg stress could stabilize miscible viscous fingers in a linear geometry and Pramanik & Mishra (2013, 2015b showed that Korteweg stress delays the onset of instability so that miscible viscous fingers are stabilized. In short, numerical studies have indicated that EIT can stabilize the Saffman-Taylor instability and that the fingers are wider due to the effect of EIT.…”

Section: Introductionmentioning

confidence: 99%

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Numerical study of the effect of Péclet number on miscible viscous fingering with effective interfacial tension

Deki,

Nagatsu,

Mishra

et al. 2023

J. Fluid Mech.

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Viscous fingering or Saffman–Taylor instability shows fingering interfacial patterns when a more mobile fluid displaces a less mobile one in porous media. The effective interfacial tension (EIT) is like capillary force, acting at the miscible interface on time scales shorter than interface relaxation. It has been numerically reported so far that the fingers are wider with EIT compared with those without EIT. A recent experiment observed finger widening with increasing flow rate in a miscible system with EIT, which is not observed in classical immiscible and miscible systems. In this study, we have numerically investigated the effect of $Pe$ (which corresponds to the injection flow rates in the experiment) on miscible viscous fingering with/without EIT. We have shown that the fingers are monotonically thinner with an increase in $Pe$ in the system without EIT, while finger widening with increasing $Pe$ is observed in the system with EIT. Furthermore, we have also examined a one-dimensional underlying concentration profile and EIT profile by using a one-dimensional diffusion–convection model because EIT is proportional to the squared concentration gradient. We have then shown that the concentration gradient is steeper and, thus, the EIT is larger as $Pe$ is larger. Therefore, this is the first numerical study that can theoretically verify finger widening with increasing flow rate, which occurs only in a miscible system with EIT to the extent of our targeting EIT values, and explain the mechanism by one-dimensional analysis.

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“…If the injected fluid is more viscous, the interface is stable and the fluid front moves uniformly, whereas if the injected fluid is less viscous the displacement front can be unstable and lead to complex fingering patterns. Miscible viscous fingering has been extensively studied using a number of different approaches (for example Tan & Homsy 1986 , 1987 , 1988 ; Zimmerman & Homsy 1991 , 1992 ; Jha, Cueto-Felgueroso & Juanes 2011 a , b ; Chui, De Anna & Juanes 2015 ; Pramanik & Mishra 2015 b ). Recently, Nijjer, Hewitt & Neufeld ( 2018 ) described the full lifecycle of miscible viscous fingering: they identified the different regimes through which the flow evolves and in each regime modelled the evolution of the flow.…”

Section: Introductionmentioning

confidence: 99%

Stable and unstable miscible displacements in layered porous media

2019

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The effect of permeability heterogeneities and viscosity variations on miscible displacement processes in porous media is examined using high-resolution numerical simulations and reduced theoretical modelling. The planar injection of one fluid into a fluid-saturated, two-dimensional porous medium with a permeability that varies perpendicular to the flow direction is studied. Three cases are considered, in which the injected fluid is equally viscous, more viscous or less viscous than the ambient fluid. In general it is found that the flow in each case evolves through three regimes. At early times, the flow exhibits the concentration evolves diffusively, independent of both the permeability structure and the viscosity ratio. At intermediate times, the flow exhibits different dynamics including channelling and fingering, depending on whether the injected fluid is more or less viscous than the ambient fluid, and depending on the relative magnitude of the viscosity and permeability variations. Finally, at late times, the flow becomes independent of the viscosity ratio and dominated by shear-enhanced (Taylor) dispersion. For each of the regimes identified above, we develop reduced-order models for the evolution of the transversely averaged concentration and compare them to the full numerical simulations.

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Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media

Cited by 17 publications

References 30 publications

Numerical study on topological change of viscous fingering induced by a phase separation with Korteweg force

Numerical study on topological change of viscous fingering induced by a phase separation with Korteweg force

Numerical study of the effect of Péclet number on miscible viscous fingering with effective interfacial tension

Stable and unstable miscible displacements in layered porous media

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