Engineering lesson
Steady state transient and pseudosteady state flow
Steady state has an unchanging pressure distribution; transient flow has an evolving pressure disturbance; pseudosteady state has an approximately stabilized spatial shape while average reservoir pressure declines.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
The outer boundary is a physical assumption
The steady radial calculator takes a maintained external pressure at a specified radius. It describes a fixed boundary supplied with fluid, not automatic depletion of a closed reservoir.
A transient disturbance initially samples only part of the reservoir. In a bounded producing system, later pressure decline can approach pseudosteady behavior under appropriate conditions.
Do not swap pressure labels without changing the equation
An external boundary pressure and a volume-average pressure are different quantities. Pseudosteady equations can contain different geometric constants because the reference pressure and boundary condition differ.
The worked case is deliberately steady state. Skin is retained as a local resistance term, and the same pressure boundary is used across the comparison.
Match the model to the data interval
A stable measured rate does not prove that the pressure field is steady. Use pressure history, test interpretation and boundary evidence to choose the regime.
The explorer changes permeability inside one steady model. It illustrates sensitivity after a boundary assumption has been selected; it does not diagnose the regime from production data.
Explore the example
Replayed with the same equation record and calculation engine as /calculators/steady-state-radial-liquid-flow-rate. Original synthetic inputs; engine outputs are saved in the downloadable record.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Permeability (mD) = 100. q_o (STB/day): 735.844574.
- q_o (STB/day)
Worked example
Steady-State Radial Liquid Flow Rate
Fixed controls: h = 50 ft; p_e = 3000 psi; p_wf = 2500 psi; mu_o = 2 cP; B_o = 1.2 bbl/STB; r_e = 1000 ft; r_w = 0.328 ft; s = 2 dimensionless. k is varied as shown.
Relation and unit basis
q_o = 0.00708 k h (p_e - p_wf) / (mu_o B_o (ln(r_e / r_w) + s))
k: Permeability (mD). h: Net Pay Thickness (ft). p_e: External Boundary Pressure (psi). p_wf: Flowing Bottom-Hole Pressure (psi). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). r_e: Drainage Radius (ft). r_w: Wellbore Radius (ft). s: Skin Factor (dimensionless).
1. Define the baseline
Permeability (mD) = 100
Fixed controls: h = 50 ft; p_e = 3000 psi; p_wf = 2500 psi; mu_o = 2 cP; B_o = 1.2 bbl/STB; r_e = 1000 ft; r_w = 0.328 ft; s = 2 dimensionless. k is varied as shown.
2. Select the relation
q_o = 0.00708 k h (p_e - p_wf) / (mu_o B_o (ln(r_e / r_w) + s))
k: Permeability (mD). h: Net Pay Thickness (ft). p_e: External Boundary Pressure (psi). p_wf: Flowing Bottom-Hole Pressure (psi). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). r_e: Drainage Radius (ft). r_w: Wellbore Radius (ft). s: Skin Factor (dimensionless).
3. Substitute the values
k = 100; h = 50; p_e = 3000; p_wf = 2500; mu_o = 2; B_o = 1.2; r_e = 1000; r_w = 0.328; s = 2 → q_o = 735.844574 STB/day
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
q_o (STB/day) = 735.844574
Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis.
Result and interpretation
q_o (STB/day) = 735.844574
Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis.
All rows retain a maintained outer pressure. Read the trend as steady-state sensitivity only.
Exact scenario data
Replayed with the same equation record and calculation engine as /calculators/steady-state-radial-liquid-flow-rate. Original synthetic inputs; engine outputs are saved in the downloadable record.
| Permeability (mD) | q_o (STB/day) |
|---|---|
| 25 | 183.9611 |
| 50 | 367.9223 |
| 100 | 735.8446 |
| 200 | 1,471.689 |
| 400 | 2,943.378 |
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. Can a constant production rate coexist with transient reservoir pressure?
Show the worked answer
Yes. The pressure disturbance can continue expanding even while rate is held constant.
2. Reproduce the baseline q_o (STB/day) without rounding intermediate values.
Show the worked answer
k = 100; h = 50; p_e = 3000; p_wf = 2500; mu_o = 2; B_o = 1.2; r_e = 1000; r_w = 0.328; s = 2 → q_o = 735.844574 STB/day. q_o (STB/day) = 735.844574. Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Permeability (mD) changes from 25 to 400. Other controls remain as stated in the premise. q_o (STB/day) changes from 183.961144 to 2943.3783.
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. k: Permeability (mD). h: Net Pay Thickness (ft). p_e: External Boundary Pressure (psi). p_wf: Flowing Bottom-Hole Pressure (psi). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). r_e: Drainage Radius (ft). r_w: Wellbore Radius (ft). s: Skin Factor (dimensionless). Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis.
Before using the result
- Name the pressure reference.
- Identify open versus closed boundary assumptions.
- Do not infer a flow regime from rate alone.
Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis.
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.