Towards enhanced mixing of a high viscous miscible blob in porous media

This study employs high-order numerical simulations to demonstrate that the deformation and mixing of a highly viscous miscible blob in porous media are non-ideally influenced by the Péclet number and log-mobility ratio due to initial curvature, revealing three distinct flow patterns and identifying optimal intermediate conditions for enhanced mixing with significant implications for industrial applications like oil recovery and CO₂ sequestration.

The Gist

Imagine you are pouring a thick, sticky syrup (like honey) into a fast-flowing river of water. Now, imagine that river isn't flowing through an open channel, but through a sponge. This is the basic setup of the research paper you provided.

The scientists, Mijanur Rahaman and his team, wanted to understand what happens when a blob of thick, "sticky" fluid gets pushed through a porous material (like sand or rock) by a thinner, "runny" fluid. This isn't just a physics puzzle; it's crucial for real-world problems like getting oil out of the ground, trapping carbon dioxide underground, or cleaning up pollution.

Here is a breakdown of their findings using simple analogies:

1. The Setup: The "Sponge Race"

Think of the porous medium as a giant, dense sponge.

• The Runner: A thin, fast fluid (like water) is injected to push everything forward.
• The Obstacle: A circular blob of thick, slow fluid (like honey) is sitting in the middle of the sponge.
• The Goal: See how the "honey" blob gets squished, stretched, and mixed as the "water" pushes it.

2. The Three Shapes: How the Blob Reacts

The researchers found that depending on how thick the honey is compared to the water, and how fast the water is pushing, the blob changes into three distinct shapes. They named them:

• The Comet (Low Viscosity Contrast):
  Imagine a comet with a round head and a long, thin tail. When the difference between the thick and thin fluids isn't too extreme, the water pushes the blob forward, but it doesn't break it apart. Instead, it stretches the blob out into a long, thin tail behind it. It's like dragging a heavy sled through snow; it leaves a long trail but stays mostly intact.

• The Lumpy Potato (Medium Viscosity Contrast):
  Now, imagine the fluids are more mismatched. The water pushes hard against the back of the blob, but the blob is too stubborn to let the water cut right through it. Instead of a clean tail, the blob gets squished into a weird, lumpy shape with a nose at the front and a tail at the back. It's like trying to push a wet, heavy ball of clay through a narrow tube; it bulges and deforms but doesn't split.

• The Viscous Fingers (High Viscosity Contrast):
  This is the most dramatic one. If the "honey" is much thicker than the "water," the water gets impatient. It doesn't just push the blob; it punches holes through it! The water creates finger-like spikes that invade the thick fluid, splitting it apart. This is called Viscous Fingering. It's like poking a thick pudding with a fork; the fork (water) creates channels that split the pudding (blob) into separate pieces.

3. The "Sweet Spot" for Mixing

The most exciting discovery in the paper is about mixing.

You might think that the more chaotic the situation (the more "fingers" formed), the better the mixing. But the researchers found a "Goldilocks" zone.

• If the fluids are too similar, they just slide past each other (Comet shape) and don't mix well.
• If the fluids are too different, the thick blob becomes so resistant that the water flows around it rather than through it (Lumpy shape), again limiting mixing.
• The Sweet Spot: There is a "just right" middle ground where the viscosity contrast is high enough to create those finger-like spikes, but not so high that the blob refuses to be penetrated. In this zone, the blob gets torn apart and mixed most efficiently.

4. Why This Matters

Why do we care about a blob of fluid in a sponge?

• Oil Recovery: Oil is often thick and stuck in rock. Water is pumped in to push it out. If we understand how to create the right "fingers," we can mix the water and oil better and get more oil out of the ground.
• Carbon Capture: When we pump CO2 underground, we want it to mix and spread out safely so it doesn't leak. Understanding these shapes helps us predict where the gas will go.
• Pollution Cleanup: If a thick chemical spill happens underground, knowing how it will deform helps us design better ways to wash it out.

5. The Computer Magic

To figure all this out, the team didn't just use beakers in a lab. They built a super-precise computer simulation.

• They used a mathematical "net" (a grid) that was incredibly fine, like a high-resolution camera sensor, to catch every tiny detail of how the fluid moved.
• They used a special math technique (called a "Compact Finite Difference Scheme") that is like a high-end lens: it captures the image clearly without needing a massive, slow computer. This allowed them to simulate scenarios that were too complex for older methods.

The Bottom Line

This paper tells us that when you push a thick blob through a sponge with a thin fluid, the blob doesn't just move; it dances. It can become a comet, a lumpy potato, or a spiky starfish. By finding the perfect balance of speed and thickness, we can make that blob mix perfectly, which is a huge win for industries trying to clean up our planet or find energy.

Technical Summary

1. Problem Statement

The study investigates the rectilinear displacement and deformation of a highly viscous, miscible circular blob as it is pushed by a less viscous fluid within a homogeneous porous medium.

• Physical Context: The scenario models the Saffman-Taylor instability (viscous fingering) where a less viscous fluid displaces a more viscous one. Unlike previous studies focusing on infinite slices or radial geometries, this work focuses on a finite circular blob with physically realistic no-flux boundaries (impermeable walls), which is crucial for applications like oil recovery, CO2 sequestration, and chromatography.
• Governing Physics: The flow is governed by Darcy's law coupled with an advection-diffusion equation for solute concentration. The viscosity depends exponentially on concentration ($\mu = e^{Rc}$), creating a log-mobility ratio ($R$) that drives instability.
• Key Challenge: Previous numerical studies using pseudo-spectral methods were limited by the requirement for periodic boundary conditions. This study aims to overcome these limitations to explore a broader parameter space and understand how the blob's initial curvature and boundary conditions affect mixing and deformation.

2. Methodology

The authors developed a robust numerical framework to solve the coupled non-linear partial differential equations (PDEs).

• Mathematical Formulation:

  • The system uses a streamfunction-vorticity formulation to eliminate pressure.
  • The governing equations are non-dimensionalized using the Péclet number ($Pe = UL_c/D$) and the log-mobility ratio ($R = \ln(\mu_2/\mu_1)$).
  • Boundary conditions include no-flux for concentration and specific conditions for the streamfunction (zero-flux at longitudinal boundaries, Dirichlet zero at transverse boundaries).

• Numerical Scheme:

  • Spatial Discretization: A fourth-order accurate Higher-Order Compact (HOC) finite difference scheme is employed. This scheme uses the original differential equations to substitute truncation error terms, allowing for high accuracy without the periodic boundary restrictions of spectral methods.
  • Temporal Discretization: The second-order Crank-Nicolson (CN) method is used for time integration, ensuring stability and accuracy.
  • Solver: The resulting linear systems are solved using the BiCGStab iterative solver with an Incomplete LU (ILU) preconditioner.
  • Grid Independence: Simulations were validated using a grid size of $3751 \times 2501$ with a time step of $\Delta t = 10^{-2}$, ensuring mass conservation and grid independence.

3. Key Contributions

1. Novel Numerical Approach: The application of a HOC finite difference scheme with Crank-Nicolson time stepping to miscible viscous fingering in porous media with no-flux boundaries. This allows for simulations that are not restricted to periodic domains.
2. Extended Parameter Space: The study explores a significantly wider range of parameters than previous literature:
   • Péclet number: $500 \le Pe \le 3000$
   • Log-mobility ratio: $0 \le R \le 7$
3. Discovery of New Physics: The authors identified a non-monotonic relationship between viscosity contrast and mixing efficiency, revealing an "optimal mixing condition" that depends on the competition between fingering instability and tail formation.

4. Results and Findings

The simulations revealed three distinct pattern formations based on the interplay between $Pe$ and $R$:

• Pattern Regimes:

  1. Comet Shape (Region I): Occurs at low $R$ or very high $R$. The displacing fluid penetrates the blob slightly, creating a downstream tail, but the blob remains largely intact without significant fingering.
  2. Lump Shape (Region II): Occurs at intermediate $R$ (outside the fingering window). The blob deforms into a rounded, upstream-nosed shape with a tail, but the displacing fluid fails to break through the core.
  3. Viscous Fingering (Region III): Occurs within a critical "window" of $R$ ($R^l_{VF} < R < R^u_{VF}$). The displacing fluid penetrates the blob, forming finger-like structures.

• Critical Windows:

  • The study defined empirical best-fit equations for the lower ($R^l_{VF}$) and upper ($R^u_{VF}$) bounds of the fingering instability window as functions of $Pe$.
  • As $Pe$ increases, the fingering window expands (e.g., at $Pe=3000$, the window is $0.2 \le R \le 3.4$).

• Dynamics and Mixing:

  • Tip Splitting: The study captured unique tip-splitting phenomena where fingers split into multiple branches, a feature not previously reported for circular blobs in this specific configuration.
  • Non-Monotonic Mixing: Mixing efficiency (quantified by global variance $\chi$) does not increase linearly with $R$.
    • Optimal Mixing: Maximum mixing occurs at intermediate values of $R$ (within the fingering window).
    • Suppressed Mixing: At very high $R$, the blob deforms into a "lump" or "comet" with a thin tail. While longitudinal spreading increases, transverse spreading is suppressed, leading to lower overall mixing compared to the fingering regime.
  • Variance Behavior: Longitudinal variance ($\sigma_x^2$) increases with tail length, while transverse variance ($\sigma_y^2$) is maximized during fingering and minimized during lump/comet deformation.

5. Significance and Implications

• Optimization of Industrial Processes: The findings suggest that simply increasing the viscosity contrast ($R$) does not guarantee better mixing. Instead, there is an optimal operating window for $Pe$ and $R$ that maximizes mixing efficiency. This is critical for:
  • Enhanced Oil Recovery (EOR): Maximizing the sweep efficiency of injected fluids.
  • CO2 Sequestration: Ensuring efficient trapping and mixing of injected CO2.
  • Pollution Remediation: Optimizing the spread of remediation agents in groundwater.
• Methodological Advancement: The successful implementation of the HOC scheme with no-flux boundaries provides a reliable tool for future studies of complex flow geometries where periodic assumptions are invalid.

In conclusion, the paper demonstrates that the deformation and mixing of a viscous blob in porous media are governed by a complex competition between viscous fingering and tail formation, with an optimal mixing state achievable only at specific intermediate parameter values.