{
  "version": 1,
  "topic": "RF07",
  "title": "Fit permeability from core flood pressure and rate data",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Local Porosity-Permeability Calibration Calculator",
    "premise": "Fixed controls: .",
    "formula": "yi = log10(ki in mD); fitted yi = a+b*porosity_percent; minimize sum(log residual^2); predicted k = 10^(a+b*porosity_percent) mD; log RMSE = sqrt(S/N).; R_squared = 1-S/sum((yi-mean_y)^2); multiplicative RMSE factor = 10^log_RMSE.",
    "variables": "Porosity for local prediction is varied within the supported input range. This is a locally calibrated empirical log-linear transform, not a universal porosity-to-permeability equation, Kozeny-Carman pore-geometry model, NMR transform or effective-permeability upscaling method. Porosity alone does not identify pore connectivity, grain size, diagenesis, fracture contribution or directional permeability.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Porosity for local prediction (%) = 20",
        "explanation": "Fixed controls: ."
      },
      {
        "label": "Select the relation",
        "calculation": "yi = log10(ki in mD); fitted yi = a+b*porosity_percent; minimize sum(log residual^2); predicted k = 10^(a+b*porosity_percent) mD; log RMSE = sqrt(S/N).; R_squared = 1-S/sum((yi-mean_y)^2); multiplicative RMSE factor = 10^log_RMSE.",
        "explanation": "Porosity for local prediction is varied within the supported input range. This is a locally calibrated empirical log-linear transform, not a universal porosity-to-permeability equation, Kozeny-Carman pore-geometry model, NMR transform or effective-permeability upscaling method. Porosity alone does not identify pore connectivity, grain size, diagenesis, fracture contribution or directional permeability."
      },
      {
        "label": "Substitute the values",
        "calculation": "phi = 20 %; with the fixed controls above, the workflow evaluates Predicted local permeability = 100 mD; Log10 intercept (k in mD) = -2 dimensionless",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Predicted local permeability (mD) = 100; Log10 intercept (k in mD) (dimensionless) = -2",
        "explanation": "Only measured positive permeabilities from 1e-9 to 1e6 mD and porosities from 0.01 to 60 percent are accepted. Zero values, below-detection-limit/censored data and missing fields are rejected, not replaced with tiny invented numbers. No outlier removal, weighting, uncertainty interval or facies classification is performed. Exponentiating the fitted log trend does not estimate arithmetic mean permeability without an additional statistical correction and assumptions. No such correction is silently applied. High training R-squared is not out-of-sample validation; reserve independent core data before applying the relationship to logs or reservoir simulation."
      }
    ],
    "result": "Predicted local permeability (mD) = 100; Log10 intercept (k in mD) (dimensionless) = -2",
    "interpretation": "Only measured positive permeabilities from 1e-9 to 1e6 mD and porosities from 0.01 to 60 percent are accepted. Zero values, below-detection-limit/censored data and missing fields are rejected, not replaced with tiny invented numbers. No outlier removal, weighting, uncertainty interval or facies classification is performed. Exponentiating the fitted log trend does not estimate arithmetic mean permeability without an additional statistical correction and assumptions. No such correction is silently applied. High training R-squared is not out-of-sample validation; reserve independent core data before applying the relationship to logs or reservoir simulation.",
    "control": "Porosity for local prediction (%)",
    "columns": [
      "Porosity for local prediction (%)",
      "Predicted local permeability (mD)",
      "Log10 intercept (k in mD) (dimensionless)"
    ],
    "rows": [
      [
        10,
        0.999999999999999,
        -2.000000000000001
      ],
      [
        15,
        9.999999999999995,
        -2.000000000000001
      ],
      [
        20,
        100,
        -2.000000000000001
      ],
      [
        25,
        1000,
        -2.000000000000001
      ],
      [
        30,
        10000,
        -2.000000000000001
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Porosity for local prediction (%) = 10. Predicted local permeability (mD): 1; Log10 intercept (k in mD) (dimensionless): -2.",
      "Porosity for local prediction (%) = 15. Predicted local permeability (mD): 10; Log10 intercept (k in mD) (dimensionless): -2.",
      "Porosity for local prediction (%) = 20. Predicted local permeability (mD): 100; Log10 intercept (k in mD) (dimensionless): -2.",
      "Porosity for local prediction (%) = 25. Predicted local permeability (mD): 1000; Log10 intercept (k in mD) (dimensionless): -2.",
      "Porosity for local prediction (%) = 30. Predicted local permeability (mD): 10000; Log10 intercept (k in mD) (dimensionless): -2."
    ],
    "provenance": "Original recorded synthetic cases solved with the site's permeability-calibration engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.",
    "questions": [
      {
        "question": "Reproduce the baseline Predicted local permeability (mD) without rounding intermediate values.",
        "answer": "phi = 20 %; with the fixed controls above, the workflow evaluates Predicted local permeability = 100 mD; Log10 intercept (k in mD) = -2 dimensionless. Predicted local permeability (mD) = 100. Only measured positive permeabilities from 1e-9 to 1e6 mD and porosities from 0.01 to 60 percent are accepted. Zero values, below-detection-limit/censored data and missing fields are rejected, not replaced with tiny invented numbers. No outlier removal, weighting, uncertainty interval or facies classification is performed. Exponentiating the fitted log trend does not estimate arithmetic mean permeability without an additional statistical correction and assumptions. No such correction is silently applied. High training R-squared is not out-of-sample validation; reserve independent core data before applying the relationship to logs or reservoir simulation."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Porosity for local prediction (%) changes from 10 to 30. Other controls remain as stated in the premise. Predicted local permeability (mD) changes from 1 to 10000."
      },
      {
        "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. Porosity for local prediction is varied within the supported input range. This is a locally calibrated empirical log-linear transform, not a universal porosity-to-permeability equation, Kozeny-Carman pore-geometry model, NMR transform or effective-permeability upscaling method. Porosity alone does not identify pore connectivity, grain size, diagenesis, fracture contribution or directional permeability. Only measured positive permeabilities from 1e-9 to 1e6 mD and porosities from 0.01 to 60 percent are accepted. Zero values, below-detection-limit/censored data and missing fields are rejected, not replaced with tiny invented numbers. No outlier removal, weighting, uncertainty interval or facies classification is performed. Exponentiating the fitted log trend does not estimate arithmetic mean permeability without an additional statistical correction and assumptions. No such correction is silently applied. High training R-squared is not out-of-sample validation; reserve independent core data before applying the relationship to logs or reservoir simulation."
      }
    ]
  },
  "recorded": {
    "kind": "workflow",
    "workflow": "permeability-calibration",
    "parameter": "phi",
    "runs": [
      {
        "inputs": {
          "phi": 10
        },
        "result": {
          "outputs": [
            {
              "label": "Predicted local permeability",
              "value": 0.999999999999999,
              "unit": "mD"
            },
            {
              "label": "Log10 intercept (k in mD)",
              "value": -2.000000000000001,
              "unit": "dimensionless"
            },
            {
              "label": "Log10 slope per porosity percentage point",
              "value": 0.20000000000000004,
              "unit": "1/%"
            },
            {
              "label": "Log-space RMSE",
              "value": 2.220446049250313e-16,
              "unit": "log10(mD)"
            },
            {
              "label": "Log-space R-squared",
              "value": 1,
              "unit": "dimensionless"
            },
            {
              "label": "Multiplicative RMSE factor",
              "value": 1.0000000000000004,
              "unit": "dimensionless"
            },
            {
              "label": "Measurement count",
              "value": 5,
              "unit": "points"
            }
          ],
          "columns": [
            "Porosity (%)",
            "Measured permeability (mD)",
            "Fitted permeability (mD)",
            "Log10 residual"
          ],
          "rows": [
            [
              10,
              1,
              0.999999999999999,
              4.440892098500626e-16
            ],
            [
              15,
              10,
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            ],
            [
              20,
              100,
              100,
              0
            ],
            [
              25,
              1000,
              1000,
              0
            ],
            [
              30,
              10000,
              10000,
              0
            ]
          ],
          "chart": {
            "xLabel": "Porosity (%)",
            "yLabel": "Log10 permeability (k in mD)",
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                "name": "Measured core",
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                ],
                "y": [
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                "mode": "markers"
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                  10,
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                  14,
                  14.4,
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                  16,
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                  18,
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              }
            ]
          },
          "warnings": [
            "Local log-space fit only. Predicted permeability is the back-transformed log trend, not an arithmetic-mean estimate or a universal rock correlation."
          ]
        }
      },
      {
        "inputs": {
          "phi": 15
        },
        "result": {
          "outputs": [
            {
              "label": "Predicted local permeability",
              "value": 9.999999999999995,
              "unit": "mD"
            },
            {
              "label": "Log10 intercept (k in mD)",
              "value": -2.000000000000001,
              "unit": "dimensionless"
            },
            {
              "label": "Log10 slope per porosity percentage point",
              "value": 0.20000000000000004,
              "unit": "1/%"
            },
            {
              "label": "Log-space RMSE",
              "value": 2.220446049250313e-16,
              "unit": "log10(mD)"
            },
            {
              "label": "Log-space R-squared",
              "value": 1,
              "unit": "dimensionless"
            },
            {
              "label": "Multiplicative RMSE factor",
              "value": 1.0000000000000004,
              "unit": "dimensionless"
            },
            {
              "label": "Measurement count",
              "value": 5,
              "unit": "points"
            }
          ],
          "columns": [
            "Porosity (%)",
            "Measured permeability (mD)",
            "Fitted permeability (mD)",
            "Log10 residual"
          ],
          "rows": [
            [
              10,
              1,
              0.999999999999999,
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            [
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            ],
            [
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              0
            ],
            [
              25,
              1000,
              1000,
              0
            ],
            [
              30,
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              10000,
              0
            ]
          ],
          "chart": {
            "xLabel": "Porosity (%)",
            "yLabel": "Log10 permeability (k in mD)",
            "series": [
              {
                "name": "Measured core",
                "x": [
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                  15,
                  20,
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                ],
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                ],
                "mode": "markers"
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              {
                "name": "Local log-linear fit",
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                  10.8,
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              }
            ]
          },
          "warnings": [
            "Local log-space fit only. Predicted permeability is the back-transformed log trend, not an arithmetic-mean estimate or a universal rock correlation."
          ]
        }
      },
      {
        "inputs": {
          "phi": 20
        },
        "result": {
          "outputs": [
            {
              "label": "Predicted local permeability",
              "value": 100,
              "unit": "mD"
            },
            {
              "label": "Log10 intercept (k in mD)",
              "value": -2.000000000000001,
              "unit": "dimensionless"
            },
            {
              "label": "Log10 slope per porosity percentage point",
              "value": 0.20000000000000004,
              "unit": "1/%"
            },
            {
              "label": "Log-space RMSE",
              "value": 2.220446049250313e-16,
              "unit": "log10(mD)"
            },
            {
              "label": "Log-space R-squared",
              "value": 1,
              "unit": "dimensionless"
            },
            {
              "label": "Multiplicative RMSE factor",
              "value": 1.0000000000000004,
              "unit": "dimensionless"
            },
            {
              "label": "Measurement count",
              "value": 5,
              "unit": "points"
            }
          ],
          "columns": [
            "Porosity (%)",
            "Measured permeability (mD)",
            "Fitted permeability (mD)",
            "Log10 residual"
          ],
          "rows": [
            [
              10,
              1,
              0.999999999999999,
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            ],
            [
              25,
              1000,
              1000,
              0
            ],
            [
              30,
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              10000,
              0
            ]
          ],
          "chart": {
            "xLabel": "Porosity (%)",
            "yLabel": "Log10 permeability (k in mD)",
            "series": [
              {
                "name": "Measured core",
                "x": [
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                  15,
                  20,
                  25,
                  30
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                  3,
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                "mode": "markers"
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              {
                "name": "Local log-linear fit",
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                  10,
                  10.4,
                  10.8,
                  11.2,
                  11.6,
                  12,
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                  14,
                  14.4,
                  14.8,
                  15.2,
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          },
          "warnings": [
            "Local log-space fit only. Predicted permeability is the back-transformed log trend, not an arithmetic-mean estimate or a universal rock correlation."
          ]
        }
      },
      {
        "inputs": {
          "phi": 25
        },
        "result": {
          "outputs": [
            {
              "label": "Predicted local permeability",
              "value": 1000,
              "unit": "mD"
            },
            {
              "label": "Log10 intercept (k in mD)",
              "value": -2.000000000000001,
              "unit": "dimensionless"
            },
            {
              "label": "Log10 slope per porosity percentage point",
              "value": 0.20000000000000004,
              "unit": "1/%"
            },
            {
              "label": "Log-space RMSE",
              "value": 2.220446049250313e-16,
              "unit": "log10(mD)"
            },
            {
              "label": "Log-space R-squared",
              "value": 1,
              "unit": "dimensionless"
            },
            {
              "label": "Multiplicative RMSE factor",
              "value": 1.0000000000000004,
              "unit": "dimensionless"
            },
            {
              "label": "Measurement count",
              "value": 5,
              "unit": "points"
            }
          ],
          "columns": [
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            "Measured permeability (mD)",
            "Fitted permeability (mD)",
            "Log10 residual"
          ],
          "rows": [
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              25,
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              1000,
              0
            ],
            [
              30,
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              10000,
              0
            ]
          ],
          "chart": {
            "xLabel": "Porosity (%)",
            "yLabel": "Log10 permeability (k in mD)",
            "series": [
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                "name": "Measured core",
                "x": [
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              {
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                  10.4,
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                  0.8799999999999999,
                  0.96,
                  1.0399999999999996,
                  1.1199999999999997,
                  1.1999999999999997,
                  1.2799999999999996,
                  1.3599999999999999,
                  1.4399999999999997,
                  1.5200000000000002,
                  1.5999999999999999,
                  1.6799999999999997,
                  1.76,
                  1.8399999999999999,
                  1.9200000000000004,
                  2,
                  2.0799999999999996,
                  2.16,
                  2.2399999999999998,
                  2.3200000000000003,
                  2.4,
                  2.48,
                  2.5600000000000005,
                  2.64,
                  2.7200000000000006,
                  2.8000000000000003,
                  2.88,
                  2.9600000000000004,
                  3.04,
                  3.1200000000000006,
                  3.2,
                  3.2800000000000002,
                  3.3600000000000003,
                  3.4400000000000004,
                  3.5200000000000005,
                  3.6000000000000005,
                  3.6799999999999997,
                  3.7600000000000007,
                  3.8400000000000003,
                  3.920000000000001,
                  4
                ]
              }
            ]
          },
          "warnings": [
            "Local log-space fit only. Predicted permeability is the back-transformed log trend, not an arithmetic-mean estimate or a universal rock correlation."
          ]
        }
      },
      {
        "inputs": {
          "phi": 30
        },
        "result": {
          "outputs": [
            {
              "label": "Predicted local permeability",
              "value": 10000,
              "unit": "mD"
            },
            {
              "label": "Log10 intercept (k in mD)",
              "value": -2.000000000000001,
              "unit": "dimensionless"
            },
            {
              "label": "Log10 slope per porosity percentage point",
              "value": 0.20000000000000004,
              "unit": "1/%"
            },
            {
              "label": "Log-space RMSE",
              "value": 2.220446049250313e-16,
              "unit": "log10(mD)"
            },
            {
              "label": "Log-space R-squared",
              "value": 1,
              "unit": "dimensionless"
            },
            {
              "label": "Multiplicative RMSE factor",
              "value": 1.0000000000000004,
              "unit": "dimensionless"
            },
            {
              "label": "Measurement count",
              "value": 5,
              "unit": "points"
            }
          ],
          "columns": [
            "Porosity (%)",
            "Measured permeability (mD)",
            "Fitted permeability (mD)",
            "Log10 residual"
          ],
          "rows": [
            [
              10,
              1,
              0.999999999999999,
              4.440892098500626e-16
            ],
            [
              15,
              10,
              9.999999999999995,
              2.220446049250313e-16
            ],
            [
              20,
              100,
              100,
              0
            ],
            [
              25,
              1000,
              1000,
              0
            ],
            [
              30,
              10000,
              10000,
              0
            ]
          ],
          "chart": {
            "xLabel": "Porosity (%)",
            "yLabel": "Log10 permeability (k in mD)",
            "series": [
              {
                "name": "Measured core",
                "x": [
                  10,
                  15,
                  20,
                  25,
                  30
                ],
                "y": [
                  0,
                  1,
                  2,
                  3,
                  4
                ],
                "mode": "markers"
              },
              {
                "name": "Local log-linear fit",
                "x": [
                  10,
                  10.4,
                  10.8,
                  11.2,
                  11.6,
                  12,
                  12.4,
                  12.8,
                  13.2,
                  13.6,
                  14,
                  14.4,
                  14.8,
                  15.2,
                  15.6,
                  16,
                  16.4,
                  16.8,
                  17.2,
                  17.6,
                  18,
                  18.4,
                  18.8,
                  19.2,
                  19.6,
                  20,
                  20.4,
                  20.8,
                  21.2,
                  21.6,
                  22,
                  22.4,
                  22.8,
                  23.2,
                  23.6,
                  24,
                  24.4,
                  24.8,
                  25.2,
                  25.6,
                  26,
                  26.4,
                  26.8,
                  27.2,
                  27.6,
                  28,
                  28.4,
                  28.8,
                  29.2,
                  29.6,
                  30
                ],
                "y": [
                  -4.440892098500626e-16,
                  0.07999999999999963,
                  0.1599999999999997,
                  0.23999999999999955,
                  0.3199999999999996,
                  0.3999999999999997,
                  0.47999999999999976,
                  0.5599999999999998,
                  0.6399999999999997,
                  0.7199999999999998,
                  0.7999999999999998,
                  0.8799999999999999,
                  0.96,
                  1.0399999999999996,
                  1.1199999999999997,
                  1.1999999999999997,
                  1.2799999999999996,
                  1.3599999999999999,
                  1.4399999999999997,
                  1.5200000000000002,
                  1.5999999999999999,
                  1.6799999999999997,
                  1.76,
                  1.8399999999999999,
                  1.9200000000000004,
                  2,
                  2.0799999999999996,
                  2.16,
                  2.2399999999999998,
                  2.3200000000000003,
                  2.4,
                  2.48,
                  2.5600000000000005,
                  2.64,
                  2.7200000000000006,
                  2.8000000000000003,
                  2.88,
                  2.9600000000000004,
                  3.04,
                  3.1200000000000006,
                  3.2,
                  3.2800000000000002,
                  3.3600000000000003,
                  3.4400000000000004,
                  3.5200000000000005,
                  3.6000000000000005,
                  3.6799999999999997,
                  3.7600000000000007,
                  3.8400000000000003,
                  3.920000000000001,
                  4
                ]
              }
            ]
          },
          "warnings": [
            "Local log-space fit only. Predicted permeability is the back-transformed log trend, not an arithmetic-mean estimate or a universal rock correlation."
          ]
        }
      }
    ]
  }
}