{"id":92221,"date":"2026-04-16T18:00:58","date_gmt":"2026-04-16T18:00:58","guid":{"rendered":"https:\/\/gaeatech.com\/knowledge-center\/?p=92221"},"modified":"2026-04-24T01:06:22","modified_gmt":"2026-04-24T01:06:22","slug":"pollutev10-example-12-fractured-media-transport-analytical-solution","status":"publish","type":"post","link":"https:\/\/gaeatech.com\/knowledge-center\/pollutev10-example-12-fractured-media-transport-analytical-solution\/","title":{"rendered":"POLLUTEv10 Example 12: Fractured Media Transport vs Analytical Solution (Tang et al., 1981)"},"content":{"rendered":"\n

Validating Fracture Transport Modeling with Analytical Benchmarks<\/h2>\n\n\n\n

POLLUTEv10 Example 12<\/strong> is a benchmark validation case<\/strong> that compares numerical results from POLLUTEv10 with an analytical solution developed by Tang et al..<\/p>\n\n\n\n

This example focuses on transport in fractured porous media<\/strong>, where contaminant migration occurs rapidly along fractures and slowly into the surrounding rock matrix.<\/p>\n\n\n\n

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Problem Overview<\/h2>\n\n\n\n

The model simulates:<\/p>\n\n\n\n

    \n
  • A single fracture system<\/strong><\/li>\n\n\n\n
  • A conservative contaminant<\/strong> (no sorption in fracture)<\/li>\n\n\n\n
  • Advection and diffusion along fractures<\/strong><\/li>\n\n\n\n
  • Diffusion into the surrounding matrix<\/strong><\/li>\n\n\n\n
  • A constant source concentration<\/strong><\/li>\n<\/ul>\n\n\n\n

    Key Conditions<\/h3>\n\n\n\n
      \n
    • Source concentration (co<\/strong>) = 1.0 mg\/L<\/li>\n\n\n\n
    • Fracture spacing = 1 m<\/li>\n\n\n\n
    • Fracture aperture = 10 \u03bcm<\/li>\n\n\n\n
    • High groundwater velocity along fractures<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Conceptual Model<\/h2>\n\n\n\n

      The system consists of:<\/p>\n\n\n\n

        \n
      • Discrete fractures<\/strong> acting as fast-flow pathways<\/li>\n\n\n\n
      • A low-permeability rock matrix<\/strong> surrounding fractures<\/li>\n\n\n\n
      • Diffusion from fracture \u2192 matrix<\/strong><\/li>\n<\/ul>\n\n\n\n

        Key Insight<\/h3>\n\n\n\n

        Because matrix diffusion is extremely small:<\/p>\n\n\n\n

        \n

        There is effectively no interaction between adjacent fractures<\/strong><\/p>\n<\/blockquote>\n\n\n\n

        Thus:<\/p>\n\n\n\n

          \n
        • Results are independent of fracture spacing over the time scale considered<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          Key Calculations<\/h2>\n\n\n\n

          Darcy Velocity in Fractures<\/h3>\n\n\n\n

          v<\/mi>a<\/mi><\/msub>=<\/mo>(<\/mo>fractures per m<\/mtext>)<\/mo>\u00d7<\/mo>(<\/mo>fracture width<\/mtext>)<\/mo>\u00d7<\/mo>(<\/mo>seepage velocity<\/mtext>)<\/mo><\/mrow>v_a = (\\text{fractures per m}) \\times (\\text{fracture width}) \\times (\\text{seepage velocity})<\/annotation><\/semantics><\/math>v<\/mi>a<\/mi><\/msub>=<\/mo>10<\/mn>\u00d7<\/mo>10<\/mn>\u2212<\/mo>6<\/mn><\/mrow><\/msup>\u00d7<\/mo>1<\/mn>\u00d7<\/mo>730<\/mn>=<\/mo>0.73<\/mn>\u00d7<\/mo>10<\/mn>\u2212<\/mo>2<\/mn><\/mrow><\/msup>\u2009<\/mtext>m\/a<\/mtext><\/mrow>v_a = 10 \\times 10^{-6} \\times 1 \\times 730 = 0.73 \\times 10^{-2} \\, \\text{m\/a}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

          Mix Diffusion Coefficient<\/h3>\n\n\n\n

          D<\/mi>m<\/mi><\/msub>=<\/mo>D<\/mi>f<\/mi><\/msub>\u00d7<\/mo>tortuosity<\/mtext><\/mrow>D_m = D_f \\times \\text{tortuosity}<\/annotation><\/semantics><\/math>D<\/mi>m<\/mi><\/msub>=<\/mo>0.077<\/mn>\u00d7<\/mo>0.0000983<\/mn>=<\/mo>7.57<\/mn>\u00d7<\/mo>10<\/mn>\u2212<\/mo>6<\/mn><\/mrow><\/msup>\u2009<\/mtext>m<\/mtext>2<\/mn><\/msup>\/<\/mi>a<\/mtext><\/mrow>D_m = 0.077 \\times 0.0000983 = 7.57 \\times 10^{-6} \\, \\text{m}^2\/\\text{a}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
          \n\n\n\n

          Input Parameters<\/h2>\n\n\n\n
          Property<\/th>Value<\/th>Units<\/th><\/tr><\/thead>
          Darcy Velocity (va)<\/td>0.73E-2<\/td>m\/a<\/td><\/tr>
          Soil Thickness (H)<\/td>400.0<\/td>m<\/td><\/tr>
          Sub-layers<\/td>4<\/td>–<\/td><\/tr>
          Fracture Spacing<\/td>1.0<\/td>m<\/td><\/tr>
          Fracture Opening<\/td>10E-6<\/td>m<\/td><\/tr>
          Fracture Diffusion Coefficient<\/td>0.077<\/td>m\u00b2\/a<\/td><\/tr>
          Fracture Distribution Coef.<\/td>0.0<\/td>cm\u00b3\/g<\/td><\/tr>
          Matrix Diffusion Coefficient<\/td>7.57E-6<\/td>m\u00b2\/a<\/td><\/tr>
          Matrix Distribution Coef.<\/td>1.0<\/td>cm\u00b3\/g<\/td><\/tr>
          Matrix Porosity (nm)<\/td>0.05<\/td>–<\/td><\/tr>
          Dry Density (Matrix)<\/td>0.0<\/td>g\/cm\u00b3<\/td><\/tr>
          Source Concentration<\/td>1.0<\/td>mg\/L<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
          \n\n\n\n

          Transport Mechanisms<\/h2>\n\n\n\n

          1. Advection in Fractures<\/h3>\n\n\n\n
            \n
          • Rapid contaminant movement<\/li>\n\n\n\n
          • Dominant transport pathway<\/li>\n<\/ul>\n\n\n\n

            2. Diffusion Along Fractures<\/h3>\n\n\n\n
              \n
            • Spreads contaminant longitudinally<\/li>\n\n\n\n
            • Controlled by Df = 0.077 m\u00b2\/a<\/strong><\/li>\n<\/ul>\n\n\n\n

              3. Matrix Diffusion<\/h3>\n\n\n\n
                \n
              • Very slow transfer into surrounding rock<\/li>\n\n\n\n
              • Controlled by low tortuosity<\/strong><\/li>\n<\/ul>\n\n\n\n

                4. No Dispersion<\/h3>\n\n\n\n
                  \n
                • Dispersivity = 0<\/li>\n\n\n\n
                • Simplifies comparison with analytical solution<\/li>\n<\/ul>\n\n\n\n
                  \n\n\n\n

                  Graphical Output: Depth vs Concentration<\/h2>\n\n\n\n
                  \"\"<\/figure>\n\n\n\n
                  \n\n\n\n

                  PDF Report<\/h2>\n\n\n