Engineering lesson
Ramey wellbore temperature profiles and heat transfer assumptions
A Ramey-style wellbore temperature model balances flowing-fluid heat capacity with heat transfer through the well and transient conduction into the formation. Its result depends on flow duration and the chosen thermal resistances.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Define the fluid and formation temperatures
The formation profile is commonly represented by a reference temperature plus a geothermal gradient. Inlet fluid temperature supplies a separate boundary condition.
The supported workflow is a liquid temperature model. It does not automatically include gas expansion cooling, phase change or a fully coupled multiphase energy balance.
Separate well resistance from formation response
The overall heat-transfer coefficient and the formation conductivity describe different parts of the thermal path. Thermal diffusivity and elapsed time determine the transient formation term.
The explorer varies flow duration using the same mass rate, heat capacity and geometry. It keeps the workflow's documented long-time approximation visible rather than using a hidden formation-history state.
Check the time-domain assumptions
A line-source long-time expression has a domain of usefulness. Very early times or complex completion histories may require another treatment.
Use the formation-history workflow for explicit heating, shut-in and restart memory. A fresh elapsed-time Ramey calculation is not automatically equivalent to that history.
Explore the example
Original recorded synthetic cases solved with the site's ramey-liquid-temperature-profile 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.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Elapsed steady injection or production time (day) = 30. Outlet liquid temperature (K): 303.87118; Effective conductance per unit length (W/(m.K)): 2.13754099.
- Outlet liquid temperature (K)
Worked example
Ramey Wellbore Liquid Temperature Profile Calculator
Fixed controls: Inlet liquid absolute temperature = 300 K; Undisturbed formation temperature at inlet = 290 K; Signed formation gradient along flow = 0.03 K/m; Distance along flow = 2000 m; Constant liquid mass flow rate = 5 kg/s; Liquid specific heat capacity = 4200 J/(kg.K); Tubing inside radius (U reference area) = 0.03 m; Outer radius at formation heat-transfer interface = 0.1 m; Overall wellbore coefficient excluding formation = 20 W/(m2.K); Formation thermal conductivity = 2.5 W/(m.K); Formation thermal diffusivity = 0.000001 m2/s.
Relation and unit basis
tD=alpha*t/rw^2, with t in seconds. F=0.5*ln(4*tD)-0.290. Resistance per length R'=1/(2*pi*ri*U)+F/(2*pi*k). Relaxation length A=mass_rate*cp*R'. U=0 is the insulated limit.; Earth temperature=Te0+G*s. dT/ds=(earth-T)/A. T=Te0+G*s-G*A+(T0-Te0+G*A)*exp(-s/A). Net liquid heat gain=mass_rate*cp*(T-T0); positive means heating. Heat to formation per length=(T-earth)/R'.
Elapsed steady injection or production time is varied within the supported input range. Ramey's approximate single-phase incompressible-liquid case at one fixed operating time, with constant flow rate, heat capacity, U and formation thermal properties. Radial formation conduction and quasi-steady wellbore heat transfer; linear geothermal background along one uniform interval. It does not solve an early-time thermal transient or arbitrary injection/shut-in history.
1. Define the baseline
Elapsed steady injection or production time (day) = 30
Fixed controls: Inlet liquid absolute temperature = 300 K; Undisturbed formation temperature at inlet = 290 K; Signed formation gradient along flow = 0.03 K/m; Distance along flow = 2000 m; Constant liquid mass flow rate = 5 kg/s; Liquid specific heat capacity = 4200 J/(kg.K); Tubing inside radius (U reference area) = 0.03 m; Outer radius at formation heat-transfer interface = 0.1 m; Overall wellbore coefficient excluding formation = 20 W/(m2.K); Formation thermal conductivity = 2.5 W/(m.K); Formation thermal diffusivity = 0.000001 m2/s.
2. Select the relation
tD=alpha*t/rw^2, with t in seconds. F=0.5*ln(4*tD)-0.290. Resistance per length R'=1/(2*pi*ri*U)+F/(2*pi*k). Relaxation length A=mass_rate*cp*R'. U=0 is the insulated limit.; Earth temperature=Te0+G*s. dT/ds=(earth-T)/A. T=Te0+G*s-G*A+(T0-Te0+G*A)*exp(-s/A). Net liquid heat gain=mass_rate*cp*(T-T0); positive means heating. Heat to formation per length=(T-earth)/R'.
Elapsed steady injection or production time is varied within the supported input range. Ramey's approximate single-phase incompressible-liquid case at one fixed operating time, with constant flow rate, heat capacity, U and formation thermal properties. Radial formation conduction and quasi-steady wellbore heat transfer; linear geothermal background along one uniform interval. It does not solve an early-time thermal transient or arbitrary injection/shut-in history.
3. Substitute the values
days = 30 day; with the fixed controls above, the workflow evaluates Outlet liquid temperature = 303.87118 K; Effective conductance per unit length = 2.13754099 W/(m.K)
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
Outlet liquid temperature (K) = 303.87118; Effective conductance per unit length (W/(m.K)) = 2.13754099
No gas/steam, phase change, pressure profile, Joule-Thomson effect, viscosity update, frictional heating, axial conduction, multiple coupled strings, thermal recovery reservoir simulation or hydraulic feedback is included. Verify that the fluid remains liquid at actual local pressure and temperature separately. The thermal and hydraulic workflows are not automatically coupled. Software limits: inlet and entire undisturbed formation interval 200-600 K, |G| <= 0.1 K/m, time >= 7 days and tD >= 100, positive conductivity/diffusivity/heat capacity and outer thermal radius greater than tubing inside radius. The line-source expression uses Ramey's rounded constant 0.290; it is not an exact early-time cylindrical solution or a field-accuracy guarantee.
Result and interpretation
Outlet liquid temperature (K) = 303.87118; Effective conductance per unit length (W/(m.K)) = 2.13754099
No gas/steam, phase change, pressure profile, Joule-Thomson effect, viscosity update, frictional heating, axial conduction, multiple coupled strings, thermal recovery reservoir simulation or hydraulic feedback is included. Verify that the fluid remains liquid at actual local pressure and temperature separately. The thermal and hydraulic workflows are not automatically coupled. Software limits: inlet and entire undisturbed formation interval 200-600 K, |G| <= 0.1 K/m, time >= 7 days and tD >= 100, positive conductivity/diffusivity/heat capacity and outer thermal radius greater than tubing inside radius. The line-source expression uses Ramey's rounded constant 0.290; it is not an exact early-time cylindrical solution or a field-accuracy guarantee.
Flow duration changes while the liquid inlet, geothermal profile and thermal properties remain fixed.
Exact scenario data
Original recorded synthetic cases solved with the site's ramey-liquid-temperature-profile 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.
| Elapsed steady injection or production time (day) | Outlet liquid temperature (K) | Effective conductance per unit length (W/(m.K)) |
|---|---|---|
| 14 | 304.0718 | 2.254448 |
| 21 | 303.9625 | 2.190705 |
| 30 | 303.8712 | 2.137541 |
| 60 | 303.7052 | 2.041271 |
| 120 | 303.5528 | 1.953299 |
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. Does this liquid Ramey model calculate hydrate onset?
Show the worked answer
No. It predicts a temperature profile under its thermal assumptions; hydrate equilibrium needs a separate fluid and phase model.
2. Reproduce the baseline Outlet liquid temperature (K) without rounding intermediate values.
Show the worked answer
days = 30 day; with the fixed controls above, the workflow evaluates Outlet liquid temperature = 303.87118 K; Effective conductance per unit length = 2.13754099 W/(m.K). Outlet liquid temperature (K) = 303.87118. No gas/steam, phase change, pressure profile, Joule-Thomson effect, viscosity update, frictional heating, axial conduction, multiple coupled strings, thermal recovery reservoir simulation or hydraulic feedback is included. Verify that the fluid remains liquid at actual local pressure and temperature separately. The thermal and hydraulic workflows are not automatically coupled. Software limits: inlet and entire undisturbed formation interval 200-600 K, |G| <= 0.1 K/m, time >= 7 days and tD >= 100, positive conductivity/diffusivity/heat capacity and outer thermal radius greater than tubing inside radius. The line-source expression uses Ramey's rounded constant 0.290; it is not an exact early-time cylindrical solution or a field-accuracy guarantee.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Elapsed steady injection or production time (day) changes from 14 to 120. Other controls remain as stated in the premise. Outlet liquid temperature (K) changes from 304.071757 to 303.552802.
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. Elapsed steady injection or production time is varied within the supported input range. Ramey's approximate single-phase incompressible-liquid case at one fixed operating time, with constant flow rate, heat capacity, U and formation thermal properties. Radial formation conduction and quasi-steady wellbore heat transfer; linear geothermal background along one uniform interval. It does not solve an early-time thermal transient or arbitrary injection/shut-in history. No gas/steam, phase change, pressure profile, Joule-Thomson effect, viscosity update, frictional heating, axial conduction, multiple coupled strings, thermal recovery reservoir simulation or hydraulic feedback is included. Verify that the fluid remains liquid at actual local pressure and temperature separately. The thermal and hydraulic workflows are not automatically coupled. Software limits: inlet and entire undisturbed formation interval 200-600 K, |G| <= 0.1 K/m, time >= 7 days and tD >= 100, positive conductivity/diffusivity/heat capacity and outer thermal radius greater than tubing inside radius. The line-source expression uses Ramey's rounded constant 0.290; it is not an exact early-time cylindrical solution or a field-accuracy guarantee.
Before using the result
- Keep liquid-model scope.
- Record inlet and formation boundaries.
- Check the long-time formation approximation.
No gas/steam, phase change, pressure profile, Joule-Thomson effect, viscosity update, frictional heating, axial conduction, multiple coupled strings, thermal recovery reservoir simulation or hydraulic feedback is included. Verify that the fluid remains liquid at actual local pressure and temperature separately. The thermal and hydraulic workflows are not automatically coupled. Software limits: inlet and entire undisturbed formation interval 200-600 K, |G| <= 0.1 K/m, time >= 7 days and tD >= 100, positive conductivity/diffusivity/heat capacity and outer thermal radius greater than tubing inside radius. The line-source expression uses Ramey's rounded constant 0.290; it is not an exact early-time cylindrical solution or a field-accuracy guarantee.
Continue learning
Start with the broader workflow 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.