Reservoir simulation

A Nonisothermal Wellbore Model and Its Application in Well Testing

Although the wellbore is in a nonisothermal environment, heat transfer between the fluid in the wellbore and the formation is often ignored and temperature is usually assumed constant in data interpretation, which will lead to misunderstanding of the pressure profile.

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Although the wellbore is in a nonisothermal environment, heat transfer between the fluid in the wellbore and the formation is often ignored and temperature is usually assumed constant in data interpretation, which will lead to misunderstanding of the pressure profile. In this work, wavelet transform and a nonlinear-regression-analysis method were used to study the relationships between downhole temperature, pressure, and flow rate. A nonisothermal wellbore model was established that is capable of predicting temperature, pressure, flow rate, and liquid-fraction profiles under multirate and multiphase production scenarios.


In order to monitor the downhole production and injection conditions, as well as make decisions for well-performance optimization, the dynamic temperature data acquired from permanent downhole gauges (PDGs) and distributed-temperature sensors (DTSs) are often interpreted widely.

In this paper, first, the relationships among bottomhole-pressure (BHP), -temperature (BHT), and flow-rate data were analyzed briefly by use of wavelet transform and nonlinear regression. Next, a nonisothermal wellbore model with complex structure was derived theoretically step by step. Then, the wellbore model was coupled with an existing reservoir model through BHT. After that, several synthetic cases were simulated to verify the wellbore model. Finally, on the basis of the coupled wellbore/reservoir model, the transient temperature behavior during flowing and shut-down periods was drawn out for well-testing analysis and some typical thermodynamic parameters were estimated by use of a set of field data.

BHT, BHP, and Flow-Rate Relationships

It is worth noting that formation pressure and temperature emerged in many similar characteristics. First, temperature and pressure meet the same diffusion equation. Second, at the junction between formation and well, near-wellbore damage should be considered for pressure. At the same time, on the temperature side, heat loss should also be considered, located in the formation, casing, tubing, and cement. Third, wellbore storage is compared with heat-capacity coefficient in wells. Fourth, constant-pressure­ and impermeable boundaries have been set for pressure boundaries, but the temperature boundary is always regarded as infinite ground. Analyses of the measured PDG data show that the temperature can also be used to provide reservoir information.

Nonlinear-Regression Analysis. A nonlinear-regression-analysis method, which includes a calculation of optimal transformations from data sets, is applied for the purpose of demonstrating an existing relationship between temperature (response variable) and flow rate and pressure (predictor variables). The algorithm is known as the alternating-conditional-expectations (ACE) method.

The ACE method was used for a set of transient downhole data that were recorded by PDGs. Then, the maximal correlation and the set of optimal transformations that contains the transformed values in the same order as the inputs are produced.

Wavelet Transform. Wavelet transform is a mathematical method to analyze signals that have the characteristic of time variation. In other words, it is a multi­resolution frequency-analysis method. Compared with other signal-analysis techniques, such as the nonlinear signal-processing method, wavelet transform can provide more-accurate analysis results for certain classes of signals and images.

There are several kinds of wavelets. In this study, the simplest and most widely used method—the Haar wavelet—was selected to process the downhole transient pressure and temperature data acquired by PDGs.

If we treat the transient pressure and temperature data as signals of a drawdown period, the amplitude of well-test coefficients is positive and, for pressure buildup, the amplitude is negative. Therefore, the timing of the flow events can be identified with wavelet transform.

Theoretical Derivation of the Nonisothermal Wellbore Model

The nonisothermal wellbore model represented in this paper is capable of predicting the temperature, pressure, and flow-rate profiles under multirate and multiphase production scenarios. This numerical wellbore model calculates temperature and pressure from bottomhole to wellhead separately and iterates until the estimated and calculated values converge.

First, a geothermal gradient must be assumed and a timestep set, along with input for other wellbore/formation properties. Then, the Hagedorn-Brown model is modified and recoded to calculate pressure, flow-rate, and liquid-fraction profiles. After that, this pressure-calculation model is coupled with a heat-transfer model to calculate the temperature profile. Finally, the program will check the convergence of temperature profiles and continue to calculate for next timesteps by using the steady-state temperature and pressure profiles to replace the items in transient-state temperature-calculation equations. Because the pressure model and temperature model are not run simultaneously, the fluid parameters can be amended by iteration and the problem of having to enter either the exact pressure profile or the exact temperature profile in advance is solved.

Heat-Transfer Model. Thermodynamic behavior of the flowing fluid is one of the dominant factors that affect multiphase flow in the wellbore. In addition, PDGs usually are installed several hundred feet above the pay zones, so the heat transfer along that distance, which can lead to misunderstanding the pressure profile, should not be ignored. In order to study the principle of temperature well testing, first the temperature changes along the wellbore need to be figured out, including heat transfer by convection in the fluid and by conduction between wellbore and formation.

Pressure- and Flow-Rate-Profile Calculation Model. The Hagedorn-Brown model is one of the more successful wellbore-pressure-drop-calculation methods for multiphase steady-state flow. This correlation includes the effects of gas slippage and was generated by analyzing a wide range of experimental data, such as liquid rates, gas/liquid ratios, tubing sizes, and different fluid properties, obtained from a vertical well.

Although the Hagedorn-Brown method was developed for vertical wells, in this paper, a modification is made so that it suits inclined flow and even applies to horizontal wells. Fluid properties are also treated as functions of pressure and temperature rather than as constants. In addition, the Hagedorn-Brown method was found to be effective for slug-flow prediction in oil wells; however, on the basis of other existing correlations and assumptions, different flow regimes such as bubble flow are taken into account.

Application of the Nonisothermal Wellbore Model in Well Testing

The PDGs and DTSs are always some distance away from the sandface (approximately 400 ft); in fact, heat transformation occurring between the wellbore fluids and the surrounding formation will seriously affect the transient temperature recorded by the gauges. In order to use those downhole transient data for well-testing interpretation, it is necessary to couple the wellbore model with a reservoir model to simulate actual transient temperature behavior during production and shut-in periods. Then, some typical thermodynamic parameters can be estimated by matching the simulated results with the real data. In addition, the effect of wellbore storage on temperature can be simulated by the model, and it provides a new method to assist transient pressure analysis for well-testing interpretation.


  • The analyses for measured PDG data by wavelet transform and nonlinear regression show that temperature responds to changes in flow rate and pressure; therefore, temperature can be used to provide reservoir information.
  • A nonisothermal wellbore model was developed, and its reliability has been verified by comparing it with other published models.
  • The established wellbore model can calculate temperature profiles, integrate pressure data, and determine flow-rate profiles more accurately.
  • The transient temperature behavior at the gauges can be simulated by coupling the wellbore model with a reservoir model.
  • The Joule-Thomson coefficient, viscosity, permeability, and porosity are very sensitive to temperature changes, and, in accordance with an established nonisothermal well-testing model, these representative thermodynamic parameters can be obtained accurately.
  • For pressure-buildup tests, a wellbore-storage effect starts with wellbore-temperature increase and ends when the temperature decreases. The jump in temperature can be very small in the case of high permeability and small Joule-Thomson coefficients but can be relatively large when low porosity and large viscosity exist. Thus, the transient temperature data may be used as a fast and reliable diagnostic tool in well-testing analysis to detect the end of wellbore storage.

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper IPTC 18152, “A Nonisothermal Wellbore Model With Complex Structure and Its Application in Well Testing,” by Shiyi Zheng, SPE, London South Bank University, andYiqun Zhang, SPE, Heriot-Watt University, prepared for the 2014 International Petroleum Technology Conference, Kuala Lumpur, 10–12 December. The paper has not been peer reviewed. Copyright 2014 International Petroleum Technology Conference. Reproduced by permission.