Structural barriers to complete homogenization and wormholing in dissolving porous and fractured rocks

This study utilizes three network models and a unified flow focusing metric to demonstrate that structural heterogeneity inherent in porous and fractured rock topologies imposes a fundamental limit on flow homogenization, preventing complete uniform dissolution regardless of system conditions.

The Gist

Imagine you have a block of rock, but instead of being solid stone, it's like a giant, complex sponge or a tangled web of pipes hidden inside. Now, imagine pumping a strong acid through this rock. The acid eats away at the rock, making the holes bigger.

This paper is a scientific investigation into what happens to the paths the acid takes as it eats its way through different types of "sponges." The researchers wanted to know: Can the acid eventually make the flow perfectly smooth and even, or will it always get stuck in a few favorite paths?

Here is the breakdown using simple analogies:

1. The Three Types of "Sponges"

The researchers tested three different kinds of rock structures to see how they react to the acid:

• The Perfect Grid (Regular Pore Network): Imagine a Lego structure where every pipe is exactly the same length, but some are slightly wider than others. It's very orderly.
• The Messy Web (Disordered Pore Network): Imagine a spiderweb where the strings are all different lengths and angles, and the holes are different sizes. It's chaotic and random.
• The Cracked Rock (Discrete Fracture Network): Imagine a block of concrete that has been smashed. It has huge, long cracks running through it, connecting in complex ways. This is like real-world fractured rock.

2. The Three Ways the Acid Eats

Depending on how fast the acid flows and how strong the reaction is, the rock changes in three distinct ways:

• Uniform Eating (The Sponge Soaking): The acid spreads out evenly. Every little hole gets slightly bigger at the same time. It's like a sponge soaking up water evenly.
• Channeling (The Highway Expansion): The acid finds the slightly wider paths and just makes those bigger. It's like a city where everyone decides to drive on the same main road, so that road gets wider and wider, while the side streets stay empty.
• Wormholing (The Tunnel Dig): The acid gets greedy. It finds one tiny weak spot and digs a single, super-fast tunnel straight through the rock, ignoring everything else. It's like a mole digging a straight line through a garden.

3. The Big Discovery: The "Structural Trap"

The most important finding of this paper is about why you can never make the flow perfectly smooth, even if you dissolve the rock for a long time.

Think of the rock structure like a city map:

• The "Pipe Size" (Aperture): This is the width of the streets. If you widen a narrow street, traffic flows better.
• The "Road Length & Connections" (Topology): This is how long the streets are and how they connect to each other.

The Analogy:
Imagine you have a city with a messy layout (The Disordered Web). Some streets are dead ends; some are long loops; some are short cuts.

• If you widen every street by the same amount (Uniform Dissolution), the traffic does get smoother.
• BUT, the traffic will never be perfectly even. Why? Because the city map itself is still messy! You still have those long, winding dead-end streets that no one uses, and those short, direct highways that everyone rushes toward.

The Conclusion:
The researchers found that in messy rock (like the Disordered Web or the Cracked Rock), the shape of the network (the map) creates a permanent "traffic jam" or "traffic rush" that the acid cannot fix.

• In the Perfect Grid, the acid can eventually make everything perfectly smooth because the map was simple to begin with.
• In the Messy Web and Cracked Rock, the acid eats away the "width" differences, but it cannot change the "length" and "connectivity" differences. The rock is still structurally messy, so the fluid will always prefer certain paths.

4. Why Does This Matter?

This is huge for real-world problems like:

• Storing Carbon Dioxide: We pump CO2 underground to store it. If we think the acid (or fluid) will spread out evenly, we might be wrong. It might get stuck in a few "wormholes," leaving huge parts of the rock untouched.
• Geothermal Energy: We pump hot water through rocks to generate power. If the water only flows through a few cracks, we might not get enough heat.
• Oil & Gas: When we use acid to clean up oil wells, we need to know if the acid will create a single tunnel (good for flow, bad for coverage) or spread out.

The Takeaway:
You can't just look at how big the holes are to predict how fluid will move. You have to look at the entire map of the rock. Even if you dissolve the rock until the holes are all the same size, the "shape" of the rock will always force the fluid to choose favorites. It's like trying to make traffic flow perfectly evenly in a city with a terrible road layout—widening the streets helps, but it won't fix the bad design.

Technical Summary

1. Problem Statement

The dissolution of reactive fluids through porous and fractured rocks fundamentally alters both fluid chemistry and solid matrix properties. This process creates complex feedback loops between advection, diffusion, surface reactions, and the initial heterogeneity of the medium. While previous research has extensively characterized dissolution regimes (compact, wormholing, channeling, and uniform) in idealized porous media, there is a critical gap in understanding how structural heterogeneity—specifically network topology, segment lengths, and connectivity—limits the ability of dissolution to homogenize flow.

Current continuum-scale models often assume that uniform dissolution can eliminate heterogeneity by enlarging pores or fractures uniformly. However, this paper argues that in natural systems, structural disorder (quenched disorder) acts as a fundamental barrier to complete flow homogenization, a factor often overlooked when upscaling from pore/fracture scales to reservoir scales.

2. Methodology

The authors employ a comparative network modeling approach to quantify dissolution dynamics across three distinct network geometries, utilizing a unified metric: the Flow Focusing Profile ($f_{50\%}$).

A. Network Models

Three contrasting network types were simulated to isolate different scales of heterogeneity:

1. Regular Pore Network: A diamond lattice with identical edge lengths. Heterogeneity arises only from a log-normal distribution of pore diameters (conduit-scale heterogeneity).
2. Disordered Pore Network: A Delaunay triangulation with randomly positioned nodes. This introduces heterogeneity in both pore diameters and pore lengths/intersection angles (conduit- and segment-scale heterogeneity).
3. Discrete Fracture Network (DFN): A semi-generic model based on the Pietrasecca Fault (Italy). It includes three fracture families with varying orientations, lengths, and apertures, introducing heterogeneity across conduit, segment, and network scales (connectivity).

B. Reactive Transport Model

• Physics: The model couples steady-state flow (Hagen–Poiseuille for pores, Reynolds equation for fractures) with 1D advection-reaction equations.
• Reaction Kinetics: A linear rate law ($R = k c_w$) is used, incorporating transverse diffusion effects via an effective reaction rate ($k_{eff}$). Axial diffusion is neglected (valid for high Péclet numbers).
• Dissolution Mechanism: Mineral dissolution enlarges conduit diameters (or fracture apertures) uniformly along the length of the conduit based on the total volume dissolved, preserving mass conservation.
• Dimensionless Parameters:
  • Effective Damköhler Number ($Da_{eff}$): Ratio of advective to reactive time scales.
  • Reaction-Diffusion Parameter ($G$): Characterizes hindrance due to transverse diffusion.
  • Dimensionless Time ($T$): Normalized cumulative dissolved volume.

C. Key Metric: Flow Focusing Profile

The authors utilize the flow focusing index, $f_{50\%}$, defined as the fraction of conduits required to carry 50% of the total flow in a given cross-section.
$$f_{50\%} = \frac{n_{X/2} - n_{50\%}}{n_{X/2}}$$
Tracking this index along the flow direction ($x$) creates a profile that reveals the evolution of flow paths:

• Decrease in profile: Indicates uniform dissolution (homogenization).
• Increase in profile: Indicates channeling or wormholing (flow focusing).
• Sigmoidal shape: Characteristic of a propagating wormhole front.

3. Key Contributions

1. Unified Metric Application: Successfully applies the flow focusing profile to compare porous media and discrete fracture networks (DFNs) on a single quantitative scale, revealing shared regimes and geometry-specific behaviors.
2. Identification of Structural Barriers: Demonstrates that while uniform dissolution can eliminate heterogeneity caused by conduit width (diameter/aperture), it cannot eliminate heterogeneity rooted in network topology (path lengths and connectivity).
3. Regime Differentiation: Provides a detailed characterization of how the three network types respond to dissolution across uniform, channeling, and wormholing regimes, highlighting that DFNs require significantly lower $Da_{eff}$ values to transition between regimes due to their high initial connectivity.

4. Key Results

A. Homogenization Limits

• Regular Networks: Uniform dissolution successfully eliminates diameter-based heterogeneity, leading to a nearly delta-function velocity distribution and complete flow homogenization ($f_{50\%} \to 0$).
• Disordered Networks: Even after uniform dissolution, a residual flow focusing index ($f_{50\%} \approx 0.25$) persists. The velocity distribution remains broad because flow is constrained by the disordered geometry (e.g., perpendicular edges that carry little flux).
• DFNs: Complete homogenization is unattainable. Even with uniform aperture growth, the flow focusing index remains high ($f_{50\%} \approx 0.7$) because the network is dominated by long-range connectivity and specific fracture families. The "quenched disorder" of the network structure sets a hard limit on homogenization.

B. Dissolution Regimes

• Uniform Regime:
  • Regular: Rapid homogenization.
  • Disordered/DFN: Slower homogenization; flow focusing decreases but plateaus at a non-zero value.
• Channeling Regime:
  • Regular: Flow focuses along pre-existing paths; slight inlet-outlet asymmetry.
  • Disordered: Inlet region homogenizes significantly before a single path dominates; growth is slower than in regular networks due to branching.
  • DFN: Minimal change in focusing as the system is already highly channelized.
• Wormholing Regime:
  • Regular: A single dominant wormhole propagates rapidly; high $G$ accelerates growth by pushing reactant to the tip.
  • Disordered: Multiple channels compete; growth is slower due to branching and screening effects.
  • DFN: High $G$ slows wormhole advance (opposite to porous media) because diffusion limitations restrict dissolution in the main channel, allowing smaller paths to share the flux.

C. Velocity Distributions

Analysis of velocity histograms confirms that in disordered and fractured systems, the post-dissolution velocity distribution closely matches that of an idealized network with uniform diameters but disordered geometry. This proves that the remaining heterogeneity is structural, not geometric (size-based).

5. Significance and Implications

• Limitations of Continuum Models: Standard Darcy-scale finite-difference models, which rely on evolving porosity/permeability fields, incorrectly predict that uniform dissolution leads to complete flow homogenization. This paper shows that such models fail to capture the persistent preferential flow paths caused by network topology.
• Upscaling Challenges: Extrapolating laboratory results (often based on regular or core-scale porous media) to field-scale fractured reservoirs is risky. The structural heterogeneity in DFNs fundamentally alters transport behavior, leading to higher degrees of channeling than predicted by pore-scale experiments.
• Engineering Applications: For processes like acidizing, geothermal stimulation, and geological carbon storage, assuming that dissolution will fully homogenize the reservoir is dangerous. Models must account for structural disorder (path lengths and connectivity) to accurately predict injectivity evolution, stimulation efficiency, and the risk of early breakthrough or clogging.
• Methodological Advance: The flow focusing profile serves as a robust, unified tool for diagnosing dissolution regimes and quantifying the impact of structural heterogeneity across diverse geological media.

In conclusion, the paper establishes that structural heterogeneity is a fundamental, irreducible barrier to flow homogenization in dissolving rocks. This "quenched disorder" dictates the ultimate transport behavior of the system, regardless of the degree of uniform dissolution achieved.