{
  "version": 1,
  "topic": "MB07",
  "title": "Buckley Leverett breakthrough pore volumes explained",
  "createdAt": "2026-10-24",
  "case": {
    "title": "A synthetic shock construction",
    "premise": "Symmetric Corey-style fractional flow fw = S²/[S² + (1 − S)²], Swi = 0. Shock saturation Sf = 1/√2.",
    "formula": "shock slope = fw(Sf)/Sf; breakthrough PV = 1/shock slope",
    "variables": "S is an idealized saturation fraction with zero residual endpoints; gravity, capillary diffusion and compressibility are excluded.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Water saturation = 0.707106781",
        "explanation": "Symmetric Corey-style fractional flow fw = S²/[S² + (1 − S)²], Swi = 0. Shock saturation Sf = 1/√2."
      },
      {
        "label": "Select the relation",
        "calculation": "shock slope = fw(Sf)/Sf; breakthrough PV = 1/shock slope",
        "explanation": "S is an idealized saturation fraction with zero residual endpoints; gravity, capillary diffusion and compressibility are excluded."
      },
      {
        "label": "Substitute the values",
        "calculation": "Sf = 0.707106781; fw(Sf) = 0.853553391; slope = 1.20710678; breakthrough = 0.828427125 PV",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Water fractional flow = 0.853553391; Shock tangent value = 0.853553391",
        "explanation": "The tangent touches at the shock state. Its reciprocal slope gives breakthrough pore volumes on this convention."
      }
    ],
    "result": "Water fractional flow = 0.853553391; Shock tangent value = 0.853553391",
    "interpretation": "The tangent touches at the shock state. Its reciprocal slope gives breakthrough pore volumes on this convention.",
    "control": "Water saturation",
    "columns": [
      "Water saturation",
      "Water fractional flow",
      "Shock tangent value"
    ],
    "rows": [
      [
        0.2,
        0.058823529411764705,
        0.24142135623730954
      ],
      [
        0.4,
        0.30769230769230776,
        0.4828427124746191
      ],
      [
        0.6,
        0.6923076923076923,
        0.7242640687119286
      ],
      [
        0.7071067811865475,
        0.8535533905932737,
        0.8535533905932738
      ],
      [
        0.9,
        0.9878048780487805,
        1.086396103067893
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Water saturation = 0.2. Water fractional flow: 0.0588235294; Shock tangent value: 0.241421356.",
      "Water saturation = 0.4. Water fractional flow: 0.307692308; Shock tangent value: 0.482842712.",
      "Water saturation = 0.6. Water fractional flow: 0.692307692; Shock tangent value: 0.724264069.",
      "Water saturation = 0.707106781. Water fractional flow: 0.853553391; Shock tangent value: 0.853553391.",
      "Water saturation = 0.9. Water fractional flow: 0.987804878; Shock tangent value: 1.0863961."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Water fractional flow without rounding intermediate values.",
        "answer": "Sf = 0.707106781; fw(Sf) = 0.853553391; slope = 1.20710678; breakthrough = 0.828427125 PV. Water fractional flow = 0.853553391. The tangent touches at the shock state. Its reciprocal slope gives breakthrough pore volumes on this convention."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Water saturation changes from 0.2 to 0.9. Other controls remain as stated in the premise. Water fractional flow changes from 0.0588235294 to 0.987804878."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. S is an idealized saturation fraction with zero residual endpoints; gravity, capillary diffusion and compressibility are excluded. The tangent touches at the shock state. Its reciprocal slope gives breakthrough pore volumes on this convention."
      }
    ]
  },
  "recorded": null
}