{"id":92198,"date":"2026-04-19T07:00:07","date_gmt":"2026-04-19T07:00:07","guid":{"rendered":"https:\/\/gaeatech.com\/knowledge-center\/?p=92198"},"modified":"2026-04-13T23:35:51","modified_gmt":"2026-04-13T23:35:51","slug":"pollutev10-example-8-potassium-diffusion-clay","status":"publish","type":"post","link":"https:\/\/gaeatech.com\/knowledge-center\/pollutev10-example-8-potassium-diffusion-clay\/","title":{"rendered":"POLLUTEv10 Example 8: Laboratory Diffusion of Potassium in Clay"},"content":{"rendered":"\n

Laboratory diffusion testing is a cornerstone of contaminant transport analysis in low-permeability soils such as compacted clays. In POLLUTEv10 Example 8<\/strong>, the model is applied to simulate the diffusion of potassium (K\u207a)<\/strong> through a clay specimen under controlled laboratory conditions.<\/p>\n\n\n\n

This example is based on well-established experimental work by R. Kerry Rowe and colleagues, including Michael Caers and Franco Barone, whose studies in the late 1980s helped define diffusion behavior in clay liners used in environmental containment systems.<\/p>\n\n\n\n

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

The objective of this simulation is to reproduce and analyze the diffusive transport of potassium<\/strong> through a saturated clay layer, with the following characteristics:<\/p>\n\n\n\n

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  • Initial background concentration in clay:<\/strong> 10 mg\/L<\/li>\n\n\n\n
  • Source (leachate) concentration:<\/strong> 400 mg\/L<\/li>\n\n\n\n
  • Leachate head above specimen:<\/strong> 6 cm<\/li>\n\n\n\n
  • Boundary condition at base:<\/strong> Impermeable (zero flux)<\/li>\n\n\n\n
  • Transport mechanism:<\/strong> Pure diffusion (no advection)<\/li>\n<\/ul>\n\n\n\n

    This scenario represents a closed-bottom system<\/strong>, commonly used in laboratory column testing to isolate diffusion processes without interference from advective flow.<\/p>\n\n\n\n

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

    The model conceptualization includes:<\/p>\n\n\n\n

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    • A clay specimen<\/strong> of finite thickness<\/li>\n\n\n\n
    • A constant concentration source<\/strong> at the top boundary<\/li>\n\n\n\n
    • An impermeable boundary<\/strong> at the base (zero mass flux)<\/li>\n\n\n\n
    • An initial uniform background concentration<\/strong> throughout the clay<\/li>\n<\/ul>\n\n\n\n

      Since Darcy velocity is zero<\/strong>, transport is governed entirely by Fickian diffusion<\/strong>, driven by concentration gradients between the source and the clay.<\/p>\n\n\n\n

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      Input Parameters<\/h2>\n\n\n\n

      The following parameters are used in POLLUTEv10 for this example:<\/p>\n\n\n\n

      Property<\/th>Symbol<\/th>Value<\/th>Units<\/th><\/tr><\/thead>
      Darcy Velocity<\/td>va<\/td>0<\/td>m\/a<\/td><\/tr>
      Diffusion Coefficient<\/td>D<\/td>0.648<\/td>cm\u00b2\/d<\/td><\/tr>
      Distribution Coefficient<\/td>Kd<\/td>2.68<\/td>cm\u00b3\/g<\/td><\/tr>
      Soil Porosity<\/td>nm<\/td>0.39<\/td>–<\/td><\/tr>
      Dry Density<\/td>\u2014<\/td>1.68<\/td>g\/cm\u00b3<\/td><\/tr>
      Soil Layer Thickness<\/td>H<\/td>4.5<\/td>cm<\/td><\/tr>
      Number of Sub-layers<\/td>\u2014<\/td>10<\/td>–<\/td><\/tr>
      Source Concentration<\/td>co<\/td>400<\/td>mg\/L<\/td><\/tr>
      Leachate Reference Height<\/td>Hr<\/td>6<\/td>cm<\/td><\/tr>
      Background Concentration<\/td>\u2014<\/td>10<\/td>mg\/L<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
      \n\n\n\n

      Layer Discretization Strategy<\/h2>\n\n\n\n

      When modeling initial concentration profiles<\/strong>, proper discretization is critical for numerical stability and accuracy.<\/p>\n\n\n\n

      In this example:<\/p>\n\n\n\n

        \n
      • Layer 1 (top):<\/strong> 0.1 cm<\/li>\n\n\n\n
      • Layer 2 (main body):<\/strong> 4.3 cm<\/li>\n\n\n\n
      • Layer 3 (bottom):<\/strong> 0.1 cm<\/li>\n<\/ul>\n\n\n\n

        This ensures:<\/p>\n\n\n\n

          \n
        • Accurate representation of boundary conditions<\/strong><\/li>\n\n\n\n
        • Stability in modeling concentration gradients<\/strong><\/li>\n\n\n\n
        • Proper handling of initial background concentration<\/strong><\/li>\n<\/ul>\n\n\n\n

          The thin top and bottom layers act as buffer zones<\/strong>, allowing the model to better capture steep gradients near boundaries.<\/p>\n\n\n\n

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          Key Processes Simulated<\/h2>\n\n\n\n

          1. Diffusion<\/h3>\n\n\n\n

          The dominant process is molecular diffusion<\/strong>, described by Fick\u2019s Law. Since there is no flow:<\/p>\n\n\n\n

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          • Transport is driven solely by concentration differences<\/strong><\/li>\n\n\n\n
          • Movement occurs from high concentration (400 mg\/L)<\/strong> to low concentration (10 mg\/L)<\/strong><\/li>\n<\/ul>\n\n\n\n

            2. Sorption<\/h3>\n\n\n\n

            The distribution coefficient (Kd = 2.68 cm\u00b3\/g)<\/strong> indicates that potassium:<\/p>\n\n\n\n

              \n
            • Partially sorbs onto clay particles<\/li>\n\n\n\n
            • Experiences retardation<\/strong>, slowing its migration<\/li>\n<\/ul>\n\n\n\n

              This reflects realistic behavior in clay liners, where cation exchange plays a significant role.<\/p>\n\n\n\n

              3. Boundary Conditions<\/h3>\n\n\n\n
                \n
              • Top boundary:<\/strong> \u10db\u10e3\u10d3\u10db constant concentration source<\/li>\n\n\n\n
              • Bottom boundary:<\/strong> Zero flux (impermeable)<\/li>\n<\/ul>\n\n\n\n

                This creates a one-directional diffusion system<\/strong> with accumulation over time.<\/p>\n\n\n\n

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

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