# Inertial Effects on Dynamics of Immiscible Viscous Fingering in Homogenous Porous Media

^{*}

## Abstract

**:**

## 1. Introduction

_{2}sequestration [2,3], Enhanced Oil Recovery (EOR) [4,5], and separation in chromatography [6,7]. Depending on the application, sometimes the fingering phenomenon is desirable to enhance process productivity, while sometimes it has a detrimental effect on the final yield. Due to its importance in a wide range of applications related to viscosity-mismatching fluid interactions, many experimental and numerical studies have been conducted to investigate the dynamics and suppression of the fingering phenomenon. Although researchers have focused on various aspects of fingering dynamics and its suppression, it is noticeable that during numerical modelling and their validation with experiments, they have relied on the empirically formulated Darcy’s law which was introduced in 1856 [8] and does not take into account inertial effects. While Darcy’s equation successfully describes the flow in many applications of porous media, the linear relation between the pressure gradient and the velocity is only valid in specific ranges of fluid velocity. The non-linear relationship between the fluid velocity and the pressure gradient for high speed flows in porous media was first highlighted by Forchheimer in 1901 [9]. In addition to the viscous force which was already considered in Darcy’s equation, he introduced the inertial force for post-Darcy’s zone with a newly-introduced empirical parameter β, where β is a non-Darcy proportionality coefficient which arises in the Forchheimer equation to accommodate inertial effects and has dimension (m

^{−}

^{1}). Before applying the Forchheimer or Darcy’s equation, it is important to distinguish between flow regimes for correct applicability of the respective equation. At this point, it is worth mentioning that unlike the pure fluid flow region, researchers have found it difficult to distinguish between the Darcy and post-Darcy zone on the basis of a non-dimensional number. Currently, there exist two different criteria to distinguish between the Darcy and post-Darcy zone. The first criterion—which is similar to non-porous media flow—is related to the evaluation of the Reynolds number and is considered to identify post-Darcy regime. To evaluate the Reynolds number, different studies define it in different ways and set a criterion for the post-Darcy region accordingly [10,11]. More recently, the second criterion for recognizing the post-Darcy zone was presented by Zeng et al. [12]. The criterion to consider inertial effects in porous media flow is given by dimensionless Forchheimer number, such that $Fo=\raisebox{1ex}{$k\beta \nu $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.\ge 0.11$. Although the value of the non-Darcy coefficient β is obtained through experiments, its nature in terms of variability for different systems has been widely discussed in various studies. Earlier it was considered that for a specific system, β remains constant for all flow rates. However, Baree et al. [13] showed that for the Forchheimer equation, β changes with flow speeds and does not remain constant, thus making it more challenging to incorporate inertial effects in porous media flow. Apart from the Forchheimer equation, several other models have been presented to incorporate the effect of inertial forces in porous media flow. One such model is the modified Hagen–Poiseuille model [14]. It is a macroscale model and is based on the hydraulic radius definition. However, just like previous models, it is also based on empirical constants, and the ultimate expression is entirely based on empirical results. After modifications by Burke Plummer [15], and then Ergun et al. [10], the flow equation took the form as:

## 2. Mathematical Modelling

_{1}and density ρ

_{1}enters through the inlet and displaces Fluid II which has dynamic viscosity and density of µ

_{2}and ρ

_{2}, respectively. It is assumed that µ

_{1}< µ

_{2}. Considering 2D flow in a Hele-Shaw cell, the system of equations for 2-phase flow can be given by,

_{ri}is relative permeability. For simplicity, we consider relative permeability as ${k}_{ri}={S}_{i}^{2}$ in this work. However, when we consider the inertial corrections, instead of using Darcy’s model, we use Quil’s model (called the modified Darcy’s model in current work), which in the case of immiscible multiphase flow is given by,

## 3. Results and Discussion

^{−4}–10

^{−6}[37]. However, inertial effects are more prominent at higher Reynolds numbers [19]; therefore, for the current work we perform numerical simulations for Reynolds number of orders of 10

^{−1}–10

^{−3}. The flow domain is considered to have a uniform porosity of $\varphi =0.5$ We change the properties of Fluid I and Fluid II to attain different Reynolds number for a comparative study between Darcy’s law and Quil’s modified Darcy’s law (to be called modified Darcy’s law henceforth). It should be emphasized here that the modified Darcy’s law encompasses expression of inertial force combined with convective acceleration, hence the results of the modified Darcy’s law should be construed in this context. We define Reynold’s number as $\left(Re=\raisebox{1ex}{$\rho Uh$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.\right)$ while ‘h’ is gap thickness between the plates and denotes characteristic length. Moreover, in order to generalize our results, we define another parameter viscosity ratio, R defined as $\left(R=\raisebox{1ex}{${\mu}_{2}$}\!\left/ \!\raisebox{-1ex}{${\mu}_{1}$}\right.\right)$, and present our results with a dimensionless time parameter $\left(T=\raisebox{1ex}{${U}_{inlet}t$}\!\left/ \!\raisebox{-1ex}{$h$}\right.\right)$. In this study, the fingering phenomenon is studied for values of R ranging from 10–1000. We present our results in two sections: The first section relates to qualitative analysis where we compare the evolution of fingers with and without inertial and convection acceleration effects alongside various combinations of Re and R. In the second section, we quantify effects of the modified Darcy’s law and present comparative analysis for Darcy’s and the modified Darcy’s models.

#### 3.1. Qualitative Analysis

#### 3.2. Quantitative Analysis

_{f}) provides a meticulous capability to study morphology and pattern formations of the fingering phenomenon [38,39]. In the current work, quantification of fingers morphology and pattern formation was carried out using dimensionless fractal number analysis. As part of the post-processing route, we acquired color images of fingering patterns from COMSOL and processed them to convert to 8-bit binary images. The fractal dimensions of binary images were computed by the box-counting method using FracLac (a plugin of a freeware program ImageJ [40]). Overall, we observed that an increase in the Reynolds number increases the fractal number, irrespective of inclusion of inertial and convective acceleration effects. Moreover, the fractal number also increases with the evolution of fingers. This conforms the results by Pons et al. [38]. However, it is evident that the modification in Darcy’s law tends to decrease the fractal number at the high Reynolds number. Since high values of (D

_{f}) signify the capability of fingers to split into similar shapes, we observe that inertial forces combined with convective acceleration tend to produce fingers of shape which are less similar to the parent fingers. Similar findings without quantification were reported by Miranda et al. where they refer to these non-similar fingers as ‘interfacial lobes’ branching out sideward [41].

^{-5}where contact area plays a vital role in total oil recovery. In such cases, application of Darcy’s law can potentially misjudge the total oil recovery.

## 4. Conclusions

- The trend of saturation distribution remains the same, both for Darcy’s and the modified Darcy’s law with inertial corrections and convective acceleration. High saturation levels of invading fluid stay near the inlet while levels with low saturation travel far from the inlet.
- While considering the modification in Darcy’s law, the fractal number is found to decrease as the Reynolds number increases.
- The interfacial area is observed to be underestimated by Darcy’s law, especially at low Reynold’s numbers.
- Modification in Darcy’s law affects fingers morphology. At low viscosity ratios, fingers tend to widen while this effect is relatively small at high viscosity ratios.
- Length and growth of fingers are affected mostly at later stages of the fingers evolution, while the growth rate of fingers decreases with an increase in inertial effects and convective acceleration.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of radial flow domain where less viscous fluid (red) displaces more viscous fluid (blue).

**Figure 2.**Comparison of numerical model (dashed-curve) with the experiment by Ramachandaran [34].

**Figure 3.**Mesh sensitivity analysis for comparison of numerical model with the experiment by Ramachandran [34].

**Figure 4.**Fingering phenomenon with Fluid I saturation contour plots for (

**a**) Darcy’s and (

**b**) the modified Darcy’s models for $Re=0.0219$ and $R=1000$ at $T=454.5$.

**Figure 5.**Comparison of propagation of fingering for fluids with different viscosities for (

**a**) Darcy and (

**b**) the modified Darcy’s model for contours of saturation ${S}_{1}=0.02$ are compared for fluids of viscosity $R=100$ (blue) and $R=1000$ (red). Both contours are captured for the same inlet velocity at time $T=454.4$.

**Figure 6.**Comparison of the evolution of fingering for (

**a**) Darcy’s and (

**b**) the modified Darcy’s model for contours of saturation ${S}_{1}=0.02$ for fluids of viscosity ratio $R=100$ and Reynolds number $Re=2.19\times {10}^{-3}$ for time $T=18$ (blue), $T=181$ (red), $T=363$ (green), $T=727$ (grey), $T=1181$ (orange) and $T=1818$ (light blue).

**Figure 7.**Comparison of the evolution of fingering for (

**a**) Darcy’s and (

**b**) the modified Darcy’s model contours of saturation ${S}_{1}=0.02$ for fluids of viscosity ratio $R=1000$ and Reynolds number of $Re=2.19\times {10}^{-3}$ for time $T=18$ (blue), $T=181$ (red) and $T=363$ (green).

**Figure 8.**Comparison of the evolution of Fluid I saturation contour plots for Darcy’s (top) and the modified Darcy’s model (bottom) for $R=1000$ and $Re=2.12\times {10}^{-2}$.

**Figure 9.**Comparison of Darcy’s model with the modified Darcy’s Model at $R=1000$ for (

**a**) fractal number, (

**b**) interfacial length, (

**c**) number of fingers, (

**d**) growth rate and (

**e**) finger length at different Re.

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**MDPI and ACS Style**

Rabbani, S.; Abderrahmane, H.; Sassi, M.
Inertial Effects on Dynamics of Immiscible Viscous Fingering in Homogenous Porous Media. *Fluids* **2019**, *4*, 79.
https://doi.org/10.3390/fluids4020079

**AMA Style**

Rabbani S, Abderrahmane H, Sassi M.
Inertial Effects on Dynamics of Immiscible Viscous Fingering in Homogenous Porous Media. *Fluids*. 2019; 4(2):79.
https://doi.org/10.3390/fluids4020079

**Chicago/Turabian Style**

Rabbani, Shahid, Hamid Abderrahmane, and Mohamed Sassi.
2019. "Inertial Effects on Dynamics of Immiscible Viscous Fingering in Homogenous Porous Media" *Fluids* 4, no. 2: 79.
https://doi.org/10.3390/fluids4020079