Engineering lesson
Darcy versus Forchheimer when inertia changes the equation
Darcy–Forchheimer adds a velocity-squared inertial resistance to the viscous Darcy term. At low velocity it approaches Darcy behavior; at higher velocity a linear Darcy extrapolation overpredicts flow for a fixed gradient.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Write the pressure-gradient balance
The implemented form is G = μv/k + βρv² for nonnegative gradient magnitude and superficial velocity. Here β has units 1/m; it is not a dimensionless inertial coefficient.
The solver uses a stable positive root to avoid subtracting nearly equal large numbers at small G. Gravity is excluded from G and must be handled separately if relevant.
Use a ratio to compare the terms
The inertial-to-viscous ratio is βρvk/μ. A value much smaller than one indicates a small inertial contribution in this model; a larger value means it cannot be ignored.
That ratio is a diagnostic, not a universal transition criterion for every porous medium. β and k must represent the same geometry and measurement conditions.
Keep superficial and pore velocity distinct
Superficial velocity is rate divided by bulk cross-sectional area. An average connected-pore velocity requires an additional porosity or flow-area definition.
At G = 1,000,000 Pa/m with the stated controls, Darcy gives kG/μ = 0.1 m/s, while Forchheimer gives approximately 0.003113 m/s. The explorer sweeps G for the Forchheimer model; it does not infer β from data or substitute a pipe turbulent friction law.
Explore the example
Replayed with the same equation record and calculation engine as /calculators/darcy-forchheimer-superficial-velocity. Original synthetic inputs; engine outputs are saved in the downloadable record.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 100000. v (m/s): 0.00095124922.
- v (m/s)
Worked example
Darcy-Forchheimer Superficial Velocity
Fixed controls: mu = 0.001 Pa.s; k = 1e-10 m^2; beta = 100000000 1/m; rho = 1000 kg/m3. G is varied as shown.
Relation and unit basis
G: Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m). mu: Dynamic Viscosity (Pa.s). k: Intrinsic Permeability (m^2). beta: Inertial Resistance Coefficient (Not Dimensionless c_F) (1/m). rho: Fluid Density (kg/m3).
1. Define the baseline
Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) = 100000
Fixed controls: mu = 0.001 Pa.s; k = 1e-10 m^2; beta = 100000000 1/m; rho = 1000 kg/m3. G is varied as shown.
2. Select the relation
G: Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m). mu: Dynamic Viscosity (Pa.s). k: Intrinsic Permeability (m^2). beta: Inertial Resistance Coefficient (Not Dimensionless c_F) (1/m). rho: Fluid Density (kg/m3).
3. Substitute the values
G = 100000; mu = 0.001; k = 1e-10; beta = 100000000; rho = 1000 → v = 0.00095124922 m/s
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
v (m/s) = 0.00095124922
The inertial term limits superficial velocity relative to the linear Darcy extrapolation.
Result and interpretation
v (m/s) = 0.00095124922
The inertial term limits superficial velocity relative to the linear Darcy extrapolation.
The pressure-gradient sweep holds μ, k, β and density fixed; compare the two predicted superficial velocities.
Exact scenario data
Replayed with the same equation record and calculation engine as /calculators/darcy-forchheimer-superficial-velocity. Original synthetic inputs; engine outputs are saved in the downloadable record.
| Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) | v (m/s) |
|---|---|
| 1,000 | 6.1803e-5 |
| 10,000 | 0.0002701562 |
| 100,000 | 0.0009512492 |
| 1,000,000 | 0.003112673 |
| 1.0000e+7 | 0.009950125 |
Scroll the table to see all columns. The highlighted row is the worked baseline. Downloads retain full numerical precision.
Test your understanding
Work through the question before revealing its answer. Use the stated assumptions and include units in every numerical step.
1. What should happen as β approaches zero?
Show the worked answer
The inertial term disappears and v approaches kG/μ, the Darcy result.
2. Reproduce the baseline v (m/s) without rounding intermediate values.
Show the worked answer
G = 100000; mu = 0.001; k = 1e-10; beta = 100000000; rho = 1000 → v = 0.00095124922 m/s. v (m/s) = 0.00095124922. The inertial term limits superficial velocity relative to the linear Darcy extrapolation.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m) changes from 1000 to 10000000. Other controls remain as stated in the premise. v (m/s) changes from 0.0000618033989 to 0.009950125.
4. What would be needed to apply this example to a different physical system?
Show the worked answer
Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. G: Nonnegative Driving Pressure Gradient Magnitude (Gravity Excluded) (Pa/m). mu: Dynamic Viscosity (Pa.s). k: Intrinsic Permeability (m^2). beta: Inertial Resistance Coefficient (Not Dimensionless c_F) (1/m). rho: Fluid Density (kg/m3). The inertial term limits superficial velocity relative to the linear Darcy extrapolation.
Before using the result
- Use β in 1/m.
- Exclude gravity from the supplied driving gradient.
- Compare superficial velocities on the same area basis.
The inertial term limits superficial velocity relative to the linear Darcy extrapolation.
Continue learning
Start with the broader workflow guide · Related method guide
Sources and method
The explanation and synthetic examples are original PetroCalcHub teaching material. The references below support the definitions and model context. They do not supply field validation for the synthetic scenarios.