{"id":92213,"date":"2026-04-17T07:00:29","date_gmt":"2026-04-17T07:00:29","guid":{"rendered":"https:\/\/gaeatech.com\/knowledge-center\/?p=92213"},"modified":"2026-04-13T22:59:14","modified_gmt":"2026-04-13T22:59:14","slug":"pollutev10-example-11-time-varying-source-chloride-diffusion","status":"publish","type":"post","link":"https:\/\/gaeatech.com\/knowledge-center\/pollutev10-example-11-time-varying-source-chloride-diffusion\/","title":{"rendered":"POLLUTEv10 Example 11: Time-Varying Source Concentration with Diffusion (Chloride in Clay)"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"h-modeling-transient-source-histories-in-landfill-systems\">Modeling Transient Source Histories in Landfill Systems<\/h2>\n\n\n\n<p><strong>POLLUTEv10 Example 11<\/strong> focuses on a critical real-world scenario: <strong>time-varying source concentration<\/strong> combined with <strong>diffusion-only transport<\/strong> in a clay medium.<\/p>\n\n\n\n<p>Unlike previous examples, this case highlights how <strong>changing contaminant inputs over time<\/strong> influence subsurface concentration profiles\u2014even in the absence of groundwater flow.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Problem Overview<\/h2>\n\n\n\n<p>This example simulates:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A <strong>hypothetical landfill cell<\/strong><\/li>\n\n\n\n<li>A <strong>conservative contaminant (chloride)<\/strong><\/li>\n\n\n\n<li><strong>No advection (Darcy velocity = 0)<\/strong><\/li>\n\n\n\n<li>A <strong>time-dependent source concentration history<\/strong><\/li>\n\n\n\n<li>A <strong>semi-infinite clay domain<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Initial Conditions<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Clay pore water contains <strong>chloride at 120 mg\/L<\/strong><\/li>\n\n\n\n<li>Landfill cell is filled with <strong>water (6 m depth)<\/strong><\/li>\n\n\n\n<li>No waste is present for the first <strong>7 years<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Conceptual Model<\/h2>\n\n\n\n<p>The system evolves through several stages:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Stage 1: 0\u20137 Years (Pre-Waste Phase)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Clean water in landfill cell<\/li>\n\n\n\n<li>Chloride diffuses <strong>from clay \u2192 into water<\/strong><\/li>\n\n\n\n<li>Gradient driven by initial clay concentration (120 mg\/L)<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Stage 2: 7\u201310 Years (Waste Placement)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Chloride concentration increases <strong>linearly to 2100 mg\/L<\/strong><\/li>\n\n\n\n<li>Gradient reverses direction<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Stage 3: 10\u201313 Years (Peak Phase)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Source concentration remains <strong>constant at 2100 mg\/L<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Stage 4: 13\u201315 Years (Decline Phase)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Concentration decreases to <strong>1180 mg\/L<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Stage 5: 15\u201319 Years (Stabilization)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Concentration remains constant again<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Key Modeling Objective<\/h2>\n\n\n\n<p>The goal is to:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\"><strong>Predict chloride concentration distribution with depth at year 19<\/strong><\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Input Parameters<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Property<\/th><th>Value<\/th><th>Units<\/th><\/tr><\/thead><tbody><tr><td>Darcy Velocity (va)<\/td><td>0.0<\/td><td>m\/a<\/td><\/tr><tr><td>Diffusion Coefficient (Dm)<\/td><td>0.00663<\/td><td>m\u00b2\/a<\/td><\/tr><tr><td>Distribution Coefficient<\/td><td>0.0<\/td><td>cm\u00b3\/g<\/td><\/tr><tr><td>Dispersivity<\/td><td>0.0<\/td><td>m<\/td><\/tr><tr><td>Soil Porosity (n)<\/td><td>0.37<\/td><td>&#8211;<\/td><\/tr><tr><td>Dry Density<\/td><td>1.6<\/td><td>g\/cm\u00b3<\/td><\/tr><tr><td>Thickness of Interest (H)<\/td><td>1.5<\/td><td>m<\/td><\/tr><tr><td>Sub-layers<\/td><td>15<\/td><td>&#8211;<\/td><\/tr><tr><td>Source Concentration<\/td><td>Variable<\/td><td>mg\/L<\/td><\/tr><tr><td>Leachate Height (Hr)<\/td><td>6.0<\/td><td>m<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Key Equation<\/h2>\n\n\n\n<p>Since there is <strong>no flow<\/strong>, transport is purely diffusive:<\/p>\n\n\n\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>D<\/mi><mo>=<\/mo><msub><mi>D<\/mi><mi>m<\/mi><\/msub><mo>+<\/mo><mi>\u03b1<\/mi><mfrac><msub><mi>v<\/mi><mi>a<\/mi><\/msub><mi>n<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">D = D_m + \\alpha \\frac{v_a}{n}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p>With:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>v<\/mi><mi>a<\/mi><\/msub><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">v_a = 0<\/annotation><\/semantics><\/math>va\u200b=0<\/li>\n\n\n\n<li><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>\u03b1<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\alpha = 0<\/annotation><\/semantics><\/math>\u03b1=0<\/li>\n<\/ul>\n\n\n\n<p>\u27a1\ufe0f Therefore:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>D = Dm<\/strong><\/li>\n\n\n\n<li>Dispersion effects are negligible<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Time-Varying Source Representation<\/h2>\n\n\n\n<p>A key modeling technique in this example:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>Reference Height of Leachate (Hr)<\/strong> is set very large during intervals<\/li>\n\n\n\n<li>This ensures the <strong>source concentration remains constant<\/strong> within each time step<\/li>\n<\/ul>\n\n\n\n<p>This allows POLLUTEv10 to simulate:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Stepwise or linear changes in concentration<\/li>\n\n\n\n<li>Complex source histories<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Transport Mechanisms<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1. Diffusion Only<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Driven by concentration gradients<\/li>\n\n\n\n<li>No advective transport<\/li>\n\n\n\n<li>Symmetric spreading behavior<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. No Sorption<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Chloride is conservative<\/li>\n\n\n\n<li>Moves freely with no retardation<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. Infinite Domain Assumption<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Clay thickness is effectively infinite<\/li>\n\n\n\n<li>No boundary effects at depth<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-graphical-output-concentration-vs-time\">Graphical Output: Concentration vs Time<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1007\" height=\"910\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-22.jpg\" alt=\"\" class=\"wp-image-92215\" srcset=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-22.jpg 1007w, https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-22-300x271.jpg 300w, https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-22-768x694.jpg 768w\" sizes=\"auto, (max-width: 1007px) 100vw, 1007px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-pdf-report\">PDF Report<\/h2>\n\n\n<div class=\"ead-preview\"><div class=\"ead-document\" style=\"position: relative;padding-top: 90%\"><div class=\"ead-iframe-wrapper\"><iframe src=\"\/\/docs.google.com\/viewer?url=https%3A%2F%2Fgaeatech.com%2Fknowledge-center%2Fwp-content%2Fuploads%2F2026%2F04%2Fpollute-example11.pdf&amp;embedded=true&amp;hl=en\" title=\"Embedded Document\" class=\"ead-iframe\" style=\"width: 100%;height: 100%;border: none;position: absolute;left: 0;top: 0;visibility: hidden;\"><\/iframe><\/div>\t\t\t<div class=\"ead-document-loading\" style=\"width:100%;height:100%;position:absolute;left:0;top:0;z-index:10\">\n\t\t\t\t<div class=\"ead-loading-wrap\">\n\t\t\t\t\t<div class=\"ead-loading-main\">\n\t\t\t\t\t\t<div class=\"ead-loading\">\n\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/loading.svg\" width=\"55\" height=\"55\" alt=\"Loader\">\n\t\t\t\t\t\t\t<span>Loading&#8230;<\/span>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"ead-loading-foot\">\n\t\t\t\t\t\t<div class=\"ead-loading-foot-title\">\n\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/EAD-logo.svg\" alt=\"EAD Logo\" width=\"36\" height=\"23\"\/>\n\t\t\t\t\t\t\t<span>Taking too long?<\/span>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<p>\n\t\t\t\t\t\t\t<div class=\"ead-document-btn ead-reload-btn\" role=\"button\">\n\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/reload.svg\" alt=\"Reload\" width=\"12\" height=\"12\"\/> Reload document\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<span>|<\/span>\n\t\t\t\t\t\t\t<a href=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/pollute-example11.pdf\" class=\"ead-document-btn\" target=\"_blank\">\n\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/plugins\/embed-any-document\/images\/open.svg\" alt=\"Open\" width=\"12\" height=\"12\"\/> Open in new tab\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div><\/div>\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Key Insights<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Time-varying sources can significantly alter transport behavior<\/li>\n\n\n\n<li>Even without flow, <strong>diffusion redistributes contaminants over time<\/strong><\/li>\n\n\n\n<li>Early conditions can influence long-term profiles<\/li>\n\n\n\n<li>Conservative species respond directly to <strong>boundary condition changes<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Importance of Sub-Layer Resolution<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">Accuracy depends on the number of sub-layers when using Variable Properties.<\/p>\n<\/blockquote>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Time-dependent inputs require <strong>fine discretization<\/strong><\/li>\n\n\n\n<li>Recommended: <strong>\u2265 15 sub-layers<\/strong><\/li>\n\n\n\n<li>Ensures accurate capture of transient gradients<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Practical Applications<\/h2>\n\n\n\n<p>This example is useful for:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Landfill design and performance assessment<\/strong><\/li>\n\n\n\n<li><strong>Chloride migration studies<\/strong><\/li>\n\n\n\n<li><strong>Time-dependent contaminant modeling<\/strong><\/li>\n\n\n\n<li><strong>Calibration against monitoring data<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Important Disclaimer<\/h2>\n\n\n\n<p>This is a <strong>hypothetical example<\/strong> intended to demonstrate modeling techniques.<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">It is <strong>not a universal modeling approach<\/strong> for all landfill systems.<\/p>\n<\/blockquote>\n\n\n\n<p>Proper application requires:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Hydrogeologic expertise<\/li>\n\n\n\n<li>Site-specific parameter selection<\/li>\n\n\n\n<li>Engineering judgment<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Conclusion<\/h2>\n\n\n\n<p>POLLUTEv10 Example 11 demonstrates how <strong>time-varying source concentrations<\/strong> influence contaminant transport in diffusion-dominated systems.<\/p>\n\n\n\n<p>Key takeaways:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Diffusion alone can produce complex concentration profiles<\/li>\n\n\n\n<li>Source history plays a critical role in long-term outcomes<\/li>\n\n\n\n<li>Variable Properties enables realistic simulation of changing conditions<\/li>\n<\/ul>\n\n\n\n<p>This example is essential for understanding <strong>transient contaminant behavior in low-permeability soils<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Modeling Transient Source Histories in Landfill Systems POLLUTEv10 Example 11 focuses on a critical real-world scenario: time-varying source concentration combined with diffusion-only transport in a clay medium. Unlike previous examples, this case highlights how changing contaminant inputs over time influence subsurface concentration profiles\u2014even in the absence of groundwater flow. Problem Overview This example simulates: Initial [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":92217,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[858,1632],"tags":[1668,1663,501,1657,24,469,675,1636,1627,1667],"class_list":["post-92213","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-contaminant-transport-modeling","category-pollute-examples","tag-chloride-transport","tag-clay-soil","tag-contaminant-transport","tag-diffusion-modeling","tag-environmental-engineering","tag-groundwater-modeling","tag-landfill-modeling","tag-numerical-modeling","tag-pollutev10","tag-time-varying-source"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.4 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Time-Varying Chloride Diffusion in Clay Medium - 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