POLLUTEv10 Example 12: Fractured Media Transport vs Analytical Solution (Tang et al., 1981)

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Validating Fracture Transport Modeling with Analytical Benchmarks

POLLUTEv10 Example 12 is a benchmark validation case that compares numerical results from POLLUTEv10 with an analytical solution developed by Tang et al..

This example focuses on transport in fractured porous media, where contaminant migration occurs rapidly along fractures and slowly into the surrounding rock matrix.

Problem Overview

The model simulates:

A single fracture system
A conservative contaminant (no sorption in fracture)
Advection and diffusion along fractures
Diffusion into the surrounding matrix
A constant source concentration

Key Conditions

Source concentration (co) = 1.0 mg/L
Fracture spacing = 1 m
Fracture aperture = 10 μm
High groundwater velocity along fractures

Conceptual Model

The system consists of:

Discrete fractures acting as fast-flow pathways
A low-permeability rock matrix surrounding fractures
Diffusion from fracture → matrix

Key Insight

Because matrix diffusion is extremely small:

There is effectively no interaction between adjacent fractures

Thus:

Results are independent of fracture spacing over the time scale considered

Key Calculations

Darcy Velocity in Fractures

$v_a = (\text{fractures per m}) \times (\text{fracture width}) \times (\text{seepage velocity})$ $v_a = 10 \times 10^{-6} \times 1 \times 730 = 0.73 \times 10^{-2} \, \text{m/a}$

Mix Diffusion Coefficient

$D_m = D_f \times \text{tortuosity}$ $D_m = 0.077 \times 0.0000983 = 7.57 \times 10^{-6} \, \text{m}^2/\text{a}$

Input Parameters

Property	Value	Units
Darcy Velocity (va)	0.73E-2	m/a
Soil Thickness (H)	400.0	m
Sub-layers	4	–
Fracture Spacing	1.0	m
Fracture Opening	10E-6	m
Fracture Diffusion Coefficient	0.077	m²/a
Fracture Distribution Coef.	0.0	cm³/g
Matrix Diffusion Coefficient	7.57E-6	m²/a
Matrix Distribution Coef.	1.0	cm³/g
Matrix Porosity (nm)	0.05	–
Dry Density (Matrix)	0.0	g/cm³
Source Concentration	1.0	mg/L

Transport Mechanisms

1. Advection in Fractures

Rapid contaminant movement
Dominant transport pathway

2. Diffusion Along Fractures

Spreads contaminant longitudinally
Controlled by Df = 0.077 m²/a

3. Matrix Diffusion

Very slow transfer into surrounding rock
Controlled by low tortuosity

4. No Dispersion

Dispersivity = 0
Simplifies comparison with analytical solution

Graphical Output: Depth vs Concentration

PDF Report

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Key Insights

Fractures dominate contaminant transport in rock
Matrix diffusion can be negligible depending on tortuosity
Analytical solutions are valuable for model validation
POLLUTEv10 accurately simulates dual-porosity systems

Importance of Model Setup

Even though only 4 sub-layers are used:

Large domain (400 m) simplifies gradients
Low matrix interaction reduces complexity

However:

More layers may be required for higher precision or stronger matrix interaction

Practical Applications

This example is critical for:

Fractured rock aquifer analysis
Nuclear waste repository studies
Contaminant transport in bedrock
Model verification and calibration

Conclusion

POLLUTEv10 Example 12 demonstrates the model’s ability to accurately simulate fracture–matrix transport systems and match established analytical solutions.

Key takeaways:

Fracture flow controls transport speed
Matrix diffusion may be minimal in some systems
Analytical comparisons are essential for validation

This example builds confidence in using POLLUTEv10 for complex fractured media problems.

Learn more about our Contaminant Transport Modeling Solutions

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