{"id":92170,"date":"2026-04-20T18:00:43","date_gmt":"2026-04-20T18:00:43","guid":{"rendered":"https:\/\/gaeatech.com\/knowledge-center\/?p=92170"},"modified":"2026-04-13T23:37:57","modified_gmt":"2026-04-13T23:37:57","slug":"pollutev10-example-3-advection-diffusion-aquifer-mixing","status":"publish","type":"post","link":"https:\/\/gaeatech.com\/knowledge-center\/pollutev10-example-3-advection-diffusion-aquifer-mixing\/","title":{"rendered":"POLLUTEv10 Example 3: Advection + Diffusion with Aquifer Mixing"},"content":{"rendered":"\n

This example builds directly on Example 2 by introducing advective transport<\/strong> and a permeable aquifer boundary<\/strong>. This scenario is much closer to real-world landfill hydrogeology, where both diffusion and groundwater flow<\/strong> control contaminant migration.<\/p>\n\n\n\n

\n\n\n\n

Overview of the Scenario<\/h2>\n\n\n\n

In this example, the system consists of:<\/p>\n\n\n\n

    \n
  • A 4 m thick aquitard (low permeability layer)<\/strong><\/li>\n\n\n\n
  • A constant contaminant source<\/strong> at the top (landfill)<\/li>\n\n\n\n
  • A 3 m thick active aquifer zone<\/strong> (upper portion of a 20 m aquifer)<\/li>\n\n\n\n
  • Downward flow (advection)<\/strong> through the aquitard<\/li>\n\n\n\n
  • Mixing in the aquifer<\/strong> controlled by flow continuity<\/li>\n<\/ul>\n\n\n\n

    Key Enhancements from Example 2:<\/h3>\n\n\n\n
      \n
    • \u2705 Advection included (va \u2260 0)<\/li>\n\n\n\n
    • \u2705 Aquifer explicitly represented as a boundary<\/li>\n\n\n\n
    • \u2705 Flow continuity used to calculate dilution<\/li>\n\n\n\n
    • \u274c Still no sorption (Kd = 0)<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

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

      The transport system includes:<\/p>\n\n\n\n

        \n
      1. Downward advection + diffusion<\/strong> through aquitard<\/li>\n\n\n\n
      2. Mass transfer into aquifer<\/strong> at 4 m depth<\/li>\n\n\n\n
      3. Dilution in flowing groundwater system<\/strong><\/li>\n<\/ol>\n\n\n\n

        The aquifer is treated as a well-mixed receptor zone<\/strong>, not a full 3D domain.<\/p>\n\n\n\n

        \n\n\n\n

        Input Parameters<\/h2>\n\n\n\n
        Property<\/th>Symbol<\/th>Value<\/th>Units<\/th><\/tr><\/thead>
        Darcy Velocity<\/td>va<\/td>0.1<\/td>m\/a<\/td><\/tr>
        Diffusion Coefficient<\/td>D<\/td>0.01<\/td>m\u00b2\/a<\/td><\/tr>
        Distribution Coefficient<\/td>Kd<\/td>0<\/td>cm\u00b3\/g<\/td><\/tr>
        Soil Porosity<\/td>n<\/td>0.4<\/td>–<\/td><\/tr>
        Dry Density<\/td>\u03c1d<\/td>1.5<\/td>g\/cm\u00b3<\/td><\/tr>
        Aquitard Thickness<\/td>H<\/td>4<\/td>m<\/td><\/tr>
        Sub-layers<\/td>–<\/td>4<\/td>–<\/td><\/tr>
        Source Concentration<\/td>c\u2080<\/td>1<\/td>g\/L<\/td><\/tr>
        Landfill Length<\/td>L<\/td>200<\/td>m<\/td><\/tr>
        Landfill Width<\/td>W<\/td>300<\/td>m<\/td><\/tr>
        Aquifer Thickness<\/td>h<\/td>3<\/td>m<\/td><\/tr>
        Aquifer Porosity<\/td>nb<\/td>0.3<\/td>–<\/td><\/tr>
        Base Outflow Velocity<\/td>vb<\/td>26.67<\/td>m\/a<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
        \n\n\n\n

        Governing Transport Equation<\/h2>\n\n\n\n

        With both advection and diffusion:<\/p>\n\n\n\n

        \u2202<\/mi>C<\/mi><\/mrow>\u2202<\/mi>t<\/mi><\/mrow><\/mfrac>+<\/mo>v<\/mi>a<\/mi><\/msub>\u2202<\/mi>C<\/mi><\/mrow>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac>=<\/mo>D<\/mi>\u2202<\/mi>2<\/mn><\/msup>C<\/mi><\/mrow>\u2202<\/mi>x<\/mi>2<\/mn><\/msup><\/mrow><\/mfrac><\/mrow>\\frac{\\partial C}{\\partial t} + v_a \\frac{\\partial C}{\\partial x} = D \\frac{\\partial^2 C}{\\partial x^2}<\/annotation><\/semantics><\/math>\u2202t\u2202C\u200b+va\u200b\u2202x\u2202C\u200b=D\u2202x2\u22022C\u200b<\/p>\n\n\n\n

        This equation shows:<\/p>\n\n\n\n

          \n
        • Advection term (va \u2202C\/\u2202x)<\/strong> \u2192 dominates transport<\/li>\n\n\n\n
        • Diffusion term (D \u2202\u00b2C\/\u2202x\u00b2)<\/strong> \u2192 smooths gradients<\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          Flow Continuity Calculations<\/h2>\n\n\n\n

          A key part of this example is determining the aquifer dilution<\/strong> using flow balance.<\/p>\n\n\n\n

          1. Inflow to Aquifer (Upgradient)<\/h3>\n\n\n\n

          q<\/mi>i<\/mi>n<\/mi><\/mrow><\/msub>=<\/mo>v<\/mi>i<\/mi>n<\/mi><\/mrow><\/msub>\u22c5<\/mo>h<\/mi>\u22c5<\/mo>W<\/mi>=<\/mo>20<\/mn>\u22c5<\/mo>3<\/mn>\u22c5<\/mo>300<\/mn>=<\/mo>18000<\/mn>\u2009<\/mtext>m<\/mi>3<\/mn><\/msup>\/<\/mi>a<\/mi><\/mrow>q_{in} = v_{in} \\cdot h \\cdot W = 20 \\cdot 3 \\cdot 300 = 18000 \\, m^3\/a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

          2. Flow from Landfill (Recharge)<\/h3>\n\n\n\n

          q<\/mi>a<\/mi><\/msub>=<\/mo>v<\/mi>a<\/mi><\/msub>\u22c5<\/mo>L<\/mi>\u22c5<\/mo>W<\/mi>=<\/mo>0.1<\/mn>\u22c5<\/mo>200<\/mn>\u22c5<\/mo>300<\/mn>=<\/mo>6000<\/mn>\u2009<\/mtext>m<\/mi>3<\/mn><\/msup>\/<\/mi>a<\/mi><\/mrow>q_a = v_a \\cdot L \\cdot W = 0.1 \\cdot 200 \\cdot 300 = 6000 \\, m^3\/a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

          3. Total Outflow<\/h3>\n\n\n\n

          q<\/mi>o<\/mi>u<\/mi>t<\/mi><\/mrow><\/msub>=<\/mo>q<\/mi>i<\/mi>n<\/mi><\/mrow><\/msub>+<\/mo>q<\/mi>a<\/mi><\/msub>=<\/mo>18000<\/mn>+<\/mo>6000<\/mn>=<\/mo>24000<\/mn>\u2009<\/mtext>m<\/mi>3<\/mn><\/msup>\/<\/mi>a<\/mi><\/mrow>q_{out} = q_{in} + q_a = 18000 + 6000 = 24000 \\, m^3\/a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

          4. Base Outflow Velocity<\/h3>\n\n\n\n

          v<\/mi>b<\/mi><\/msub>=<\/mo>q<\/mi>o<\/mi>u<\/mi>t<\/mi><\/mrow><\/msub>W<\/mi>\u22c5<\/mo>h<\/mi><\/mrow><\/mfrac>=<\/mo>24000<\/mn>300<\/mn>\u22c5<\/mo>3<\/mn><\/mrow><\/mfrac>=<\/mo>26.67<\/mn>\u2009<\/mtext>m<\/mi>\/<\/mi>a<\/mi><\/mrow>v_b = \\frac{q_{out}}{W \\cdot h} = \\frac{24000}{300 \\cdot 3} = 26.67 \\, m\/a<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
          \n\n\n\n

          Peak Concentration (Hand Calculation)<\/h2>\n\n\n\n

          Because advection dominates<\/strong>, the peak aquifer concentration can be estimated:<\/p>\n\n\n\n

          c<\/mi>m<\/mi>a<\/mi>x<\/mi><\/mrow><\/msub>=<\/mo>q<\/mi>a<\/mi><\/msub>\u22c5<\/mo>c<\/mi>0<\/mn><\/msub><\/mrow>q<\/mi>o<\/mi>u<\/mi>t<\/mi><\/mrow><\/msub><\/mfrac><\/mrow>c_{max} = \\frac{q_a \\cdot c_0}{q_{out}}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

          c<\/mi>m<\/mi>a<\/mi>x<\/mi><\/mrow><\/msub>=<\/mo>6000<\/mn>\u22c5<\/mo>1<\/mn><\/mrow>24000<\/mn><\/mfrac>=<\/mo>0.25<\/mn>\u2009<\/mtext>g<\/mi>\/<\/mi>L<\/mi><\/mrow>c_{max} = \\frac{6000 \\cdot 1}{24000} = 0.25 \\, g\/L<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

          \u2714 Key Insight:<\/h3>\n\n\n\n
            \n
          • The aquifer concentration is controlled by dilution<\/strong>, not diffusion<\/li>\n\n\n\n
          • This provides a quick validation check<\/strong> against model results<\/li>\n<\/ul>\n\n\n\n
            \n\n\n\n

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

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