# Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media

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## Abstract

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_{2}sequestration. The viscosity contrasts between heavy oil and water is several orders of magnitude larger than typical viscosity contrasts considered in the majority of the literature. We use the finite element method (FEM)-based COMSOL Multiphysics simulator to simulate miscible displacements in homogeneous porous media with very large viscosity contrasts. Our numerical model is suitable for a wide range of viscosity contrasts covering chromatographic separation as well as heavy oil recovery. We have successfully captured some interesting and previously unexplored dynamics of miscible blobs with very large viscosity contrasts in homogeneous porous media. We study the effect of viscosity contrast on the spreading and the degree of mixing of the blob. Spreading (variance of transversely averaged concentration) follows the power law ${t}^{3.34}$ for the blobs with viscosity $\sim \mathcal{O}({10}^{2})$ and higher, while degree of mixing is found to vary non-monotonically with log-mobility ratio. Moreover, in the limit of very large viscosity contrast, the circular blob behaves like an erodible solid body and the degree of mixing approaches the viscosity-matched case.

## 1. Introduction

_{2}sequestration [2], chromatographic separation [3,4], contaminant transport in aquifers [5], mixing in low-Reynolds number flow [6], intrinsic characteristics of nonlinear dynamics and pattern formation have fascinated active theoretical, numerical [6,7,8,9,10,11,12,13], and experimental [14] researchers for more than half a decade. Both rectilinear and radial displacements have been investigated rigorously and have their own significance that can be investigated independently as well as comparatively. Here, we are interested in rectilinear displacements in porous media. The majority of the studies are focused on the case when the defending fluid is separated from the invading fluid by a flat interface. In many situations, e.g., aquifers, the contaminant can be of an arbitrary shape and miscible to the ambient fluid [15]. When the underlying fluids are miscible to each other, the finger-like patterns that develop at the fluid–fluid interface depend, to a great extent, on the shape of the interface [13], viscosity contrast [1], and diffusion rate [16]. As a consequence, mixing, spreading, and breakthrough time alter greatly [6,13,15]. Pramanik et al. used the Fourier pseudospectral method to capture the dynamics of a more viscous blob in a homogeneous porous medium [13], and the results are summarized in the R-Pe parameter plane. Here, R is the log-mobility ratio and Pe is the Péclet number. Depending upon the blob dynamics, R-Pe plane was demarcated into three regions: lump-, comet-shaped deformations, and VF. Their studies were restricted to $R\le 2.5$, and Pe $\le 2500$, due to the limitations of the numerical convergence of the semi-implicit time integration of the pseudo-spectral method. The novelty of their work was identifying the re-entrance into the comet deformation region for large R and fixed Pe, which is in strong contrast with the planar interface cases.

## 2. Mathematical Formulation and Numerical Methods of Solutions

#### 2.1. Physical Description of the Problem

#### 2.2. COMSOL Multiphsysics Simulations

## 3. Results

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Spatial distribution of concentration at t = 40, 70, 100 seconds from top to bottom (in each subfigure), for Pe $=1000$ and (

**a**) $R=0.5$, (

**b**) $R=1.25$, (

**c**) $R=1.5$; (

**d**) $R=1.75$; (

**e**) $R=2$; (

**f**) $R=7$.

**Figure 3.**Transversely averaged concentration for different R (mentioned in each subfigure) in fixed reference frame at time $t=0$ (gray), 20 (red), 60 (blue), and 100 (black) seconds.

**Figure 4.**Streamline distribution in the vicinity of a circular blob for $R=10$ at (

**a**) $t=40,\phantom{\rule{3.33333pt}{0ex}}$ (

**b**) $t=70$ and (

**c**) $t=100$ seconds. The red lines are the concentration contours at concentration levels $0.05,\phantom{\rule{0.166667em}{0ex}}0.08$ and $0.5$.

**Figure 5.**Temporal evolution of the variance of transversely averaged concentration, ${\sigma}_{x}^{2}\left(t\right)$.

**Figure 6.**Temporal evolution of axial variance ${\sigma}_{x}^{2}(t)$ for viscosity ratio in the range $\mathcal{O}\left({10}^{2}\right)$–$\mathcal{O}({10}^{6})$.

**Figure 7.**(

**a**) Variation of the skewness $a\left(t\right)$ with time, for different R; (

**b**) $a\left(t\right)$ at fixed t showing a non-monotonicity with respect to R.

**Table 1.**Parameter values used for the COMSOL simulations in this paper. All of the quantities are in S.I. units.

$({\mathit{L}}_{\mathit{x}},{\mathit{L}}_{\mathit{y}})\times {10}^{2}$ | $\mathit{D}\times {10}^{9}$ | Number of Elements | $\mathbf{\Delta}\mathit{e}\times {10}^{2}$ | Pe = $\frac{\mathit{Ud}}{\mathit{D}}$ | $({\mathit{x}}_{0},{\mathit{y}}_{0})$ |
---|---|---|---|---|---|

($12,5$) | 5 | $216,000$ | $1/60$ | 1000 | ($1/90,1/40$) |

($20,5$) | 5 | $360,000$ | $1/60$ | 1000 | ($1/90,1/40$) |

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**MDPI and ACS Style**

Sharma, V.; Pramanik, S.; Mishra, M.
Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media. *Fluids* **2017**, *2*, 32.
https://doi.org/10.3390/fluids2020032

**AMA Style**

Sharma V, Pramanik S, Mishra M.
Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media. *Fluids*. 2017; 2(2):32.
https://doi.org/10.3390/fluids2020032

**Chicago/Turabian Style**

Sharma, Vandita, Satyajit Pramanik, and Manoranjan Mishra.
2017. "Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media" *Fluids* 2, no. 2: 32.
https://doi.org/10.3390/fluids2020032