Influence of Fluid Rheology on Fluid Flow in a Natural Fracture Network

This study demonstrates through Navier-Stokes simulations that non-Newtonian rheology, specifically yield stress and shear-thinning behavior, significantly alters flow connectivity, distribution, and pressure dynamics in natural fracture networks compared to Newtonian fluids, necessitating its inclusion in fracture-flow modeling.

The Gist

Imagine you are trying to pour a thick, sticky honey through a maze of cracks in a rock. Now, imagine that honey isn't just sticky; it has a mind of its own. It can turn into a solid block if you don't push hard enough, and if you push really fast, it suddenly becomes as slippery as water.

This is exactly what researchers Cuong Mai Bui and Stephan K. Matthäi investigated in their new study. They looked at how non-Newtonian fluids (like the thick polymer gels used in oil drilling) move through complex networks of natural cracks in the ground.

Here is the story of their discovery, broken down into simple concepts:

1. The Old Way vs. The New Reality

For a long time, scientists treated fluids like water. They assumed that if you push water through a crack, it flows smoothly and predictably, like cars on a highway. This is called Newtonian flow.

But in the real world (especially in oil fields), the fluids are often "smart" fluids. They are Non-Newtonian.

• The Honey Analogy: Think of ketchup. If you leave it alone, it sits there like a solid. You have to shake the bottle (apply force) to get it moving. Once it's moving, it flows easily.
• The Study's Twist: Previous studies only looked at single, straight cracks. But nature is messy. Rocks have a "spaghetti bowl" of intersecting cracks, dead ends, and sharp turns. The researchers wanted to see what happens when these "smart" fluids hit this messy maze.

2. The Two Superpowers of the Fluid

The fluids they studied (specifically Xanthan gum solutions, used to pump oil out of the ground) have two special powers:

• Power A: The "Sleeping Giant" (Yield Stress)
  Imagine a crowd of people trying to walk through a narrow hallway. If they aren't pushed hard enough, they just stand still and block the way. This is Yield Stress.

  • What happened in the study: At low speeds, the fluid acted like a solid. It formed "rigid blocks" that took up to 65% of the space in the cracks. These blocks didn't move; they just sat there, blocking the flow and trapping oil in dead-end cracks. It was like a traffic jam where the cars refused to start the engine.

• Power B: The "Shape-Shifter" (Shear-Thinning)
  Now, imagine that same crowd starts running. As they run faster, they get thinner and more aerodynamic, slipping past each other easily. This is Shear-Thinning.

  • What happened in the study: When the researchers pumped the fluid in fast, the fluid realized, "Hey, I'm moving fast! I can get thinner!" Its viscosity (thickness) dropped dramatically. This allowed it to zoom through the cracks, but it also created wild swirls and whirlpools at the intersections, like water rushing around a bend in a river.

3. The Surprising Results

The researchers ran computer simulations on a real map of a rock fracture network (from Norway) and found some counter-intuitive things:

• The "Blockage" Effect: When the fluid moved slowly, the "Sleeping Giant" took over. The fluid refused to go into side branches, sticking only to the main, straight paths. This meant a huge part of the rock network was completely ignored, leaving oil behind.
• The "Swirl" Effect: When the fluid moved fast, the "Shape-Shifter" took over. Because the fluid got so thin and fast, it started creating swirls and vortices at the intersections. Instead of just going straight, the fluid started spinning around.
• The Great Equalizer: Here is the most surprising part. In simple, single cracks, shear-thinning usually makes fluid stick to the center (channeling). But in this complex, messy network, shear-thinning actually helped the fluid spread out more evenly! By becoming thinner and faster, it could overcome the friction of the winding, tortuous paths and reach more branches than water (which stayed thick and slow) could.

4. Why This Matters

This isn't just a physics puzzle; it's a money and energy problem.

• Oil Recovery: If you are trying to pump oil out of a rock using these thick gels, you need to know how fast to pump.
  • Too slow: The gel turns into a solid block, clogs the cracks, and you can't reach the oil in the side branches.
  • Just right: The gel thins out, swirls around the corners, and sweeps the oil out of the entire network.
• The "Traffic" Lesson: The study shows that you can't just use simple math (like Darcy's Law) to predict how these fluids move. The relationship between how hard you push (pressure) and how fast it goes (flow) is non-linear. It's like driving a car where the engine behaves differently depending on whether you are in a parking lot or on a racetrack.

The Bottom Line

Nature is messy, and fluids are complex. This study proves that when you push a "smart" fluid through a messy rock network, it doesn't just flow; it transforms. It can turn into a solid wall or a slippery slide depending on how hard you push it. To get the most oil out of the ground, engineers need to treat these fluids not as simple water, but as dynamic, shape-shifting agents that react to the geometry of the rock itself.

Technical Summary

1. Problem Statement

Subsurface fluid flow in fractured rock is critical for applications such as Enhanced Oil Recovery (EOR), hydraulic fracturing, and geothermal energy. While non-Newtonian fluids (e.g., polymer solutions, heavy oils, foams) are widely used in these industries, most existing fracture-flow studies assume Newtonian behavior (constant viscosity).

• The Gap: Current models often rely on simplified 1D representations or single-fracture geometries, ignoring the complex interplay between fluid rheology (yield stress, shear-thinning), fracture network geometry (intersections, dead-ends, variable apertures), and fluid inertia.
• The Challenge: Non-Newtonian fluids exhibit complex behaviors like yield stress (requiring a minimum stress to flow) and shear-thinning (viscosity decreases with shear rate). These properties, combined with the high geometric heterogeneity of natural fracture networks, create flow regimes that standard Darcy's law or cubic laws cannot accurately predict.

2. Methodology

The authors employed high-fidelity numerical simulations to investigate these interactions.

• Governing Equations: The study solved the incompressible Navier-Stokes equations using a Reynolds-Averaged Navier-Stokes (RANS) approach to capture inertial effects and turbulence.
• Constitutive Model: Non-Newtonian behavior was modeled using the Herschel-Bulkley-Papanastasiou (HBP) approach. This combines:
  • Yield Stress ($\tau_0$): Fluid behaves as a solid below a critical stress.
  • Shear-Thinning: Viscosity decreases as shear rate increases (Power-law index $n < 1$).
  • Regularization: The Papanastasiou scheme was used to smooth the discontinuity at the yield point for numerical stability.
• Fluids Simulated:
  • Real Fluids: Xanthan gum solutions (XG1500, XG3000, XG5000) with varying concentrations, representing EOR polymers.
  • Idealized Fluids: Newtonian, Power-law, and Bingham fluids were used for verification and isolating specific rheological effects.
• Geometric Models:
  • Single Intersection: An X-shaped fracture intersection (60° angle) used for mesh convergence and model verification against analytical solutions.
  • Natural Fracture Network: A Discrete Fracture Network (DFN) model derived from the Devonian Hornelen basin (Norway). The model is an 8m $\times$ 8m domain containing 431 intersections (Y, T, and X types) with variable apertures and dead-ends.
• Simulation Setup:
  • Solver: Ansys Fluent 2023R1 with User-Defined Functions (UDF) for the HBP constitutive relation.
  • Mesh: Triangular finite elements; ~15 million elements for the network model.
  • Boundary Conditions: Constant influx rates ranging from $10^{-5}$ to $10^{-2}$ m$^2$/s, simulating flow regimes from creeping to inertia-dominated.

3. Key Contributions

1. First Network-Scale Simulation: This is the first study to simulate non-Newtonian fluid flow in a realistic, complex natural fracture network, moving beyond single-fracture or single-intersection models.
2. Decoupling Rheological Effects: The study isolates and quantifies the distinct impacts of yield stress (blocking flow) versus shear-thinning (enhancing connectivity) within a complex geometry.
3. Inertia-Rheology Coupling: It demonstrates how shear-thinning amplifies fluid inertia effects at high flow rates, leading to complex secondary flows (vortices) that are absent in Newtonian simulations.
4. Validation of Reduced-Order Models: The results highlight the failure of 1D graph-based models to capture critical phenomena like rigid zone formation, flow separation, and recirculation in non-Newtonian flows.

4. Key Results

A. Low Injection Rates (Yield Stress Dominance)

• Rigid Zone Formation: At low velocities ($u_0 \le 10^{-3}$ m/s), the yield stress prevents flow in large portions of the network. Unyielded "rigid zones" occupy up to 65% of the network cross-sectional area.
• Flow Localization: These rigid zones act as blockages, confining flow to a few dominant pathways and deactivating many branches. This significantly reduces network connectivity compared to Newtonian fluids.
• Viscosity: The apparent viscosity remains extremely high (up to 10,000 times that of water) in low-shear regions, governed by the yield stress term.

B. High Injection Rates (Shear-Thinning & Inertia Dominance)

• Disappearance of Rigid Zones: As flow rates increase, shear rates become sufficient to yield the fluid throughout the network (except in dead-ends).
• Inertia Amplification: Shear-thinning reduces viscosity in high-shear zones (intersections), which amplifies inertial effects. This leads to the formation of large recirculation zones (vortices) at fracture intersections.
• Enhanced Connectivity: Contrary to single-fracture studies where shear-thinning causes channeling, in this complex network, shear-thinning promotes broader flow distribution. The reduced viscosity allows the fluid to overcome tortuous pathways and enter more branches than a Newtonian fluid would.
• Multimodal Velocity Distribution: The network exhibits a multimodal velocity distribution, reflecting the coexistence of stagnant rigid zones, slow-flowing regions, and high-velocity inertial jets.

C. Pressure Drop Relationships

• Non-Linearity: The relationship between pressure drop ($J$) and flow rate ($u_0$) is highly non-linear for non-Newtonian fluids, deviating significantly from Darcy's law.
• Low Rates (Pre-Darcy): Follows Izbash's law ($u \propto J^q$ with $q > 1$), driven by yield stress resistance.
• High Rates (Post-Darcy): Follows Forchheimer's law ($J = Au_0 + Bu_0^2$). The quadratic term (inertial losses) is significantly amplified by shear-thinning, contributing up to ~70% of the total pressure drop for certain fluids.

5. Significance and Implications

• Reservoir Management: Ignoring non-Newtonian rheology leads to severe underestimation of flow blockages (due to yield stress) and misprediction of sweep efficiency (due to shear-thinning). This is critical for designing polymer flooding and hydraulic fracturing operations.
• Modeling Accuracy: The study proves that reduced-order models (1D segments) are insufficient for non-Newtonian flows because they cannot resolve the spatial variability of shear rates, rigid zones, and inertial recirculation.
• Engineering Design: The findings suggest that while yield stress can trap fluids in dead zones (reducing recovery), shear-thinning can be leveraged to improve fluid distribution across complex, tortuous fracture networks at high injection rates.
• Future Directions: The authors note the need for 3D simulations, consideration of mechanical degradation of polymers at high shear, and time-dependent models to capture transient vortex structures.

In conclusion, this paper establishes that fluid rheology is a primary driver of flow dynamics in fracture networks, fundamentally altering connectivity, pressure losses, and velocity distributions in ways that Newtonian assumptions cannot capture.