{"id":92302,"date":"2026-04-19T18:00:59","date_gmt":"2026-04-19T18:00:59","guid":{"rendered":"https:\/\/gaeatech.com\/knowledge-center\/?p=92302"},"modified":"2026-04-24T01:27:18","modified_gmt":"2026-04-24T01:27:18","slug":"migratev10-example-6-integration-parameters-negative-concentration","status":"publish","type":"post","link":"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-6-integration-parameters-negative-concentration\/","title":{"rendered":"MIGRATEv10 Example 6: Eliminating Negative Concentrations Through Improved Integration"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"h-introduction\">Introduction<\/h2>\n\n\n\n<p>MIGRATEv10 Example 6 builds directly on Example 5 by addressing a common numerical issue in contaminant transport modeling:<\/p>\n\n\n\n<p>\ud83d\udc49 <strong>Negative concentrations and flux values<\/strong><\/p>\n\n\n\n<p>These results are non-physical and indicate that <strong>numerical integration parameters need adjustment<\/strong>. This example demonstrates how to refine the solution by modifying key <strong>Talbot integration parameters<\/strong>, and optionally verifying results using <strong>Fourier integration<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-conceptual-overview\">Conceptual Overview<\/h2>\n\n\n\n<p>This example compares:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Unstable numerical results<\/strong> (Example 5)<\/li>\n\n\n\n<li><strong>Refined, stable solutions<\/strong> after adjusting integration parameters<\/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-problem-identified-in-example-5\">Problem Identified in Example 5<\/h2>\n\n\n\n<p>In Example 5, the model output showed:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u274c Negative concentrations<\/li>\n\n\n\n<li>\u274c Negative flux into the base<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-why-this-happens\">Why This Happens<\/h3>\n\n\n\n<p>These issues arise from:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Insufficient numerical integration resolution<\/strong><\/li>\n\n\n\n<li>Difficulty resolving:\n<ul class=\"wp-block-list\">\n<li>Very small concentrations<\/li>\n\n\n\n<li>Early-time or late-time behavior<\/li>\n<\/ul>\n<\/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-key-solution-adjust-talbot-integration-parameters\">Key Solution: Adjust Talbot Integration Parameters<\/h2>\n\n\n\n<p>The first parameters to examine are:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-1-n-number-of-terms\">1. N (Number of Terms)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Controls the <strong>resolution of the numerical inversion<\/strong><\/li>\n\n\n\n<li>Higher N \u2192 more accurate results<\/li>\n\n\n\n<li>But also \u2192 increased computation time<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-2-rnu-scaling-parameter\">2. RNU (Scaling Parameter)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Controls the <strong>contour used in Talbot inversion<\/strong><\/li>\n\n\n\n<li>Affects stability and convergence of the solution<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-important-note\">Important Note<\/h3>\n\n\n\n<p>Other Talbot parameters typically <strong>do not need adjustment<\/strong>.<\/p>\n\n\n\n<p>\ud83d\udc49 Focus first on <strong>N and RNU<\/strong> for most cases.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-optional-verification-fourier-integration\">Optional Verification: Fourier Integration<\/h2>\n\n\n\n<p>To confirm the accuracy of results, users can:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Run the same scenario using <strong>Fourier integration<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-why-this-helps\">Why This Helps<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Provides an <strong>independent numerical solution<\/strong><\/li>\n\n\n\n<li>Helps verify that results are:\n<ul class=\"wp-block-list\">\n<li>Stable<\/li>\n\n\n\n<li>Physically realistic<\/li>\n<\/ul>\n<\/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-modeling-approach-in-migratev10\">Modeling Approach in MIGRATEv10<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-step-1-review-output-from-example-5\">Step 1: Review Output from Example 5<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Identify:\n<ul class=\"wp-block-list\">\n<li>Negative concentrations<\/li>\n\n\n\n<li>Negative flux values<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-step-2-increase-talbot-parameter-n\">Step 2: Increase Talbot Parameter N<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Gradually increase number of terms<\/li>\n\n\n\n<li>Observe effect on solution stability<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-step-3-adjust-rnu-if-needed\">Step 3: Adjust RNU if Needed<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fine-tune scaling for improved convergence<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-step-4-re-run-simulation\">Step 4: Re-run Simulation<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Check if:\n<ul class=\"wp-block-list\">\n<li>Oscillations are removed<\/li>\n\n\n\n<li>Results are physically realistic<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-step-5-optional-cross-check\">Step 5: Optional Cross-Check<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Run using <strong>Fourier integration<\/strong><\/li>\n\n\n\n<li>Compare results<\/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-depth-vs-concentration\">Graphical Output: Depth vs Concentration<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"804\" height=\"881\" src=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-37.jpg\" alt=\"\" class=\"wp-image-92303\" srcset=\"https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-37.jpg 804w, https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-37-274x300.jpg 274w, https:\/\/gaeatech.com\/knowledge-center\/wp-content\/uploads\/2026\/04\/image-37-768x842.jpg 768w\" sizes=\"auto, (max-width: 804px) 100vw, 804px\" \/><\/figure>\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%2Fmigrate-example6.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\/migrate-example6.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\" id=\"h-interpretation-of-results\">Interpretation of Results<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-before-adjustment\">Before Adjustment<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Oscillating concentration curves<\/li>\n\n\n\n<li>Negative values (non-physical)<\/li>\n\n\n\n<li>Unstable flux calculations<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-after-adjustment\">After Adjustment<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Smooth concentration profiles<\/li>\n\n\n\n<li>Physically realistic values<\/li>\n\n\n\n<li>Stable flux behavior<\/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-when-to-increase-integration-parameters\">When to Increase Integration Parameters<\/h2>\n\n\n\n<p>You should refine integration when:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Results show <strong>non-physical behavior<\/strong><\/li>\n\n\n\n<li>Concentrations are <strong>very small<\/strong><\/li>\n\n\n\n<li>Simulation is:\n<ul class=\"wp-block-list\">\n<li>Far before peak concentration<\/li>\n\n\n\n<li>Far after peak concentration<\/li>\n<\/ul>\n<\/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-trade-off-accuracy-vs-computation-time\">Trade-Off: Accuracy vs Computation Time<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><th>Parameter Increase<\/th><th>Effect<\/th><\/tr><tr><td>Higher N<\/td><td>More accurate, slower<\/td><\/tr><tr><td>Adjusted RNU<\/td><td>More stable solution<\/td><\/tr><tr><td>Fourier check<\/td><td>More confidence, extra runtime<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\ud83d\udc49 The goal is to find a <strong>balance between accuracy and efficiency<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-key-takeaways\">Key Takeaways<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Negative concentrations are a <strong>numerical issue\u2014not a physical result<\/strong><\/li>\n\n\n\n<li><strong>N and RNU<\/strong> are the primary parameters for fixing instability<\/li>\n\n\n\n<li>Most cases do <strong>not require adjusting all Talbot parameters<\/strong><\/li>\n\n\n\n<li>Fourier integration is useful for <strong>validation<\/strong><\/li>\n\n\n\n<li>Always verify results with <strong>engineering judgment<\/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\" id=\"h-practical-tips\">Practical Tips<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Start with <strong>moderate increases in N<\/strong><\/li>\n\n\n\n<li>Avoid excessive computation unless needed<\/li>\n\n\n\n<li>Use <strong>parametric testing<\/strong> when results are uncertain<\/li>\n\n\n\n<li>Always inspect output files carefully<\/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-final-thoughts\">Final Thoughts<\/h2>\n\n\n\n<p>MIGRATEv10 Example 6 reinforces a critical lesson:<\/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>Accurate modeling requires both numerical understanding and engineering judgment<\/strong><\/p>\n<\/blockquote>\n\n\n\n<p>By refining integration parameters, users can eliminate non-physical results and ensure that model outputs are both:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Numerically stable<\/strong><\/li>\n\n\n\n<li><strong>Physically meaningful<\/strong><\/li>\n<\/ul>\n\n\n\n<p>This example is essential for anyone performing <strong>high-precision contaminant transport modeling<\/strong>, especially when working with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Low concentrations<\/li>\n\n\n\n<li>Long simulation times<\/li>\n\n\n\n<li>Sensitive boundary conditions<\/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-learn-more-about-our-contaminant-transport-modeling-solutions\">Learn more about our Contaminant Transport Modeling Solutions<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.gaeatech.com\/pollute.php\" target=\"_blank\" rel=\"noreferrer noopener\">POLLUTE and MIGRATE Contaminant Modeling and Landfill Design<\/a><\/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-migrate-examples\"> MIGRATE Examples<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-1-rcra-landfill-composite-liner\/\">MIGRATEv10 Example 1: Modeling a RCRA Subtitle D Landfill with a Composite Liner<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-2-double-liner-leachate-system\/\">MIGRATEv10 Example 2: Composite Liner System with Primary &amp; Secondary Leachate Collection<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-3-pure-diffusion-conservative-contaminant\/\">MIGRATEv10 Example 3: Pure Diffusion of a Conservative Contaminant<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-4-finite-mass-aquifer-mixing\/\">MIGRATEv10 Example 4: Finite Mass Source and Aquifer Mixing with Base Outflow<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-5-integration-accuracy-modeling\/\">MIGRATEv10 Example 5: Understanding Integration, Accuracy, and the Role of Engineering Judgment<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-7-fourier-integration-accuracy\/\">MIGRATEv10 Example 7: Improving Accuracy with User-Selected Fourier Integration<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-8-lateral-contaminant-transport\/\">MIGRATEv10 Example 8: Evaluating Contaminant Migration at Multiple Lateral Positions<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-9-tdast-comparison\/\">MIGRATEv10 Example 9: Comparison with the TDAST Analytical Solution<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-10-fractured-media-sorption\/\">MIGRATEv10 Example 10: Contaminant Transport in Fractured Media with Sorption<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-11-multiple-landfills-contaminant-transport\/\">MIGRATEv10 Example 11: Contaminant Migration from Two Adjacent Landfill Cells<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-12-time-dependent-source-landfills\/\">MIGRATEv10 Example 12: Modeling Time-Dependent Source Histories for Multiple Landfill Cells<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migratev10-example-13-leachate-collection-termination\/\">MIGRATEv10 Example 13: Termination of Primary Leachate Collection System<\/a><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-comparison-between-pollute-and-migrate\">Comparison between POLLUTE and  MIGRATE<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migrate-vs-pollute-pure-diffusion\/\">MIGRATEv10 vs POLLUTEv10: Pure Diffusion Comparison<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migrate-vs-pollute-advective-diffusive-transport\/\">MIGRATEv10 vs POLLUTEv10: Advective\u2013Diffusive Transport Comparison<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/wp-admin\/post.php?post=91707&amp;action=edit\">MIGRATEv10 vs POLLUTEv10: Finite Mass Source Comparison<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migrate-vs-pollute-hydraulic-trap\/\">MIGRATEv10 vs POLLUTEv10: Hydraulic Trap (Finite Mass Source) Comparison<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/gaeatech.com\/knowledge-center\/migrate-vs-pollute-fractured-layer-sorption\/\">MIGRATEv10 vs POLLUTEv10: Fractured Layer with Sorption Comparison<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introduction MIGRATEv10 Example 6 builds directly on Example 5 by addressing a common numerical issue in contaminant transport modeling: \ud83d\udc49 Negative concentrations and flux values These results are non-physical and indicate that numerical integration parameters need adjustment. This example demonstrates how to refine the solution by modifying key Talbot integration parameters, and optionally verifying results [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":92305,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[858,1690],"tags":[501,1699,469,1689,1702,1701,1700],"class_list":["post-92302","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-contaminant-transport-modeling","category-migrateexamples","tag-contaminant-transport","tag-fourier-integration","tag-groundwater-modeling","tag-migratev10","tag-model-accuracy","tag-numerical-integration","tag-talbot-method"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.5 (Yoast SEO v27.5) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>MIGRATEv10 Example 6 Integration Stability Basics - 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