Temperature model provides information for well control

Eirik Karstad, Bernt S. Aadnøy
Stavanger College
Stavanger

• Equations [20,865 bytes]
• Equations [20,094 bytes]

The misinterpretation of well bore signals, especially those concerning drilling-fluid temperatures and volumetric expansion, has been identified as a key problem when drilling high-pressure, high-temperature (HPHT) wells offshore Norway.

Several severe incidents have occurred over the past 8 years, the worst case being an underground blowout that took 1 year to control.

The aim of this article is to show how to derive exact temperature information from a well during drilling and circulating operations so that more-accurate information can be provided to drillers and engineers during well-control situations.

This article presents a field case demonstrating the applications of the model. The derived model is an important tool for interpreting mud ballooning effects so that actual bottom-hole pressures (BHP) can be more exactly determined.

Prior work

Temperature information is essential for modeling and is essentially related to effective mud density, forward predictions, and reservoir and borehole stability analysis.

A leading researcher on the subject, Aadnøy, has concluded that mud heating causes large variations in the effective BHP and that the pressure at the bottom of the hole significantly differs from the mud density measured at the surface,¹ an effect that is flow-rate dependent.

Expansion or contraction of the drilling fluid in the well has also been shown to cause variations in the effective BHP. This volumetric change gives the appearance of a gain or a loss in the mud pits. Adding to the above work, Gill has defined a phenomenon known as mud ballooning.^{2 3}

To further understand the volumetric mud behavior, numerous workers have investigated changes in open-hole annular, casing, and mud volumes caused by variations in temperatures and pressure inside the entire borehole.^{1 4}

The conclusion was that mud-volume variation contributed about 90% to the total volume variation. Over the last couple of years, measurement-while-drilling (MWD) systems have been used to measure down-hole pressures and temperatures, providing additional insights.^{5 6}

Interpretations of the data show that variations in temperatures and pressures were much larger than expected and that the measured data remained consistent with the results reported by Aadnøy.¹ Temperature effects were early identified as a major cause of variations in effective mud density and mud volume.

Temperature profiles

Fig. 1 [64,459 bytes] shows a number of temperature profiles that exist in a well at any given time. The "virgin" temperature profile of the formation, T _e, represents the temperature that exists in the formation before the well is drilled.

The term "virgin" refers to the mud and formation temperature that occurs at time zero, occurring instantaneously as the well is drilled. This temperature is not measured directly, but is an important parameter, both as a reference temperature and for analysis work related to hydrocarbon pressure-volume-temperature (PVT) behavior.

During the circulation of drilling fluids, the bottom part of the well is usually under a condition of cooling. The drilling fluid at bottom is heated in the process, resulting in a heat transfer to the upper part of the well.

As shown in the Equations box [20,865 bytes], Parts A-C, the drill pipe acts as a counter-current heat exchanger. The cold mud pumped down inside the drill pipe is heated by contact with the pipe wall (Fig. 2 [173,048 bytes]), while the return mud on the outside of the pipe is exposed both to the outside of the pipe and the annulus. For this reason, the inside and outside of the pipe have different temperatures.

Fig. 1 shows a typical temperature profile for the formation at the borehole wall (T_wb), which is different from the virgin temperature (T_e). This difference depends on the heat conduction properties of the rock formation and the open-hole time exposure.

At the borehole wall, there is a heat exchange between the formation and the drilling mud, resulting in a temperature profile in the mud in the annulus defined by T_a. Because the drill pipe is a counter-current heat exchanger, the mud temperature (T_t) inside the drill pipe is usually lower than T_a.

It has been observed that the total heat-exchange process results in four different temperature profiles.

As drilling is initiated, a temperature-measurement probe is placed somewhere in the BHA.^{5 6} Because this probe is located between the inside and outside of the drillstring, the measured temperature probably reflects some average between T_a and T_t.

Because of the thermal insulation on the inside of the drillstring, the probe may approximately read T_a. It is important to observe that a temperature measurement in the well may not directly apply to one of the curves T_a or T_t shown in Fig. 1.

Approach description

The physical problem is to determine the temperature profile throughout the well under various operational modes. These include both static conditions (pump kicked out) and dynamic conditions of forward and reverse circulation.

Several implicit solutions are found in the literature.^{7 8} However, if a numerical simulator is required, implicit solutions may have limited value for parametric studies. Equation box, Part A, defines an implicit solution.

By performing parametric studies, explicit solutions for the temperature behavior were derived. By simulating wells with various flow rates, times, and depths, several temperature profiles were generated.

By studying the form of these profiles, simple functions were defined that fitted the temperature profile within a given accuracy. In other words, these equations define the temperature system's behavior.

All analytical expressions of the constants in this section are provided in the Equation box, Part C.

Circulation

Published models are all based on the assumption of a constant well depth and a constant mud-in temperature. ^7-10 For applications within production technology, this is usually sufficient (Equations box, Part A). However, because mud-pit temperature increases while drilling, operators are often faced with an increasing mud-in temperature. This article expands the model to handle both increasing well depth and variable mud-in temperatures.

The model developed in Part A, Equations A-10 and A-11, gives the fluid temperature profile in the tubing and in the annulus for a circulating well at constant depth. To model an increasing well depth, the model must be modified. The total well depth during drilling is given in Equation 1.

Also, by assuming the drilling process to be rapid as compared to the heat flow in the formation, it is assumed that the instant bottom-hole well bore temperature beneath the drill bit equals the virgin formation temperature,^{11 12} with T_wb = T_e. Note that this is equivalent to neglecting the time function, f(t_D)=0, in Equation A-8a.

Similar to the increasing well depth, the variable mud-in temperature is given in Equation 2. Since the temperatures at the drill bit and the mud return flow line are monitored while drilling, these are the most interesting depths to model.

The general solution for increasing well depth is obtained by inserting Equation 1 into Equations A-6 and A-11, resulting in Equations 3 and 4.

Using data from Holmes and Swift (Table 1 [36,609 bytes]),⁹ a number of simulations were used to derive explicit temperature models for this equation at different depths, z.

Mud-return temperature

When considering the return flow-line temperature while drilling, when the well depth is set z = 0, Equation 3 is reduced to Equation 5. Assuming a penetration rate of 5-25 m/hr, and a varying mud flow rate, the mud-return temperature was observed to obey Equation 6.

Because the out-temperature is usually higher than the in-temperature, the active mud pit will gradually heat up. This will again result in an increase in the mud-in temperature, and according to Equation 5, the out-temperature will further increase.

For offshore HPHT applications, the active mud pit temperature may increase from about 10° C. to 60-70° C. This temperature increase does not only depend on the return flow temperature, but also on the total heat and the system.

The model in this article only considers the return-flow temperature, leaving heat loss from the active mud pit, or the mud between the outlet and input, for later analysis.

Assuming a constant mud-in temperature, it was furthermore found that the mud-out temperature was nearly constant during drilling as shown in Equation 7.

Bottom-hole temperature

While drilling, Equation 8 depicts the bottom-hole temperature (BHT). Again, the data in Table 1 show the influence of drilling on the BHT of the mud. Performing a number of simulations, it was discovered that the modeled BHT showed a distinct linear trend with increasing depth. Equation 9 resulted from this analysis.

Studying the case of maintaining the mud-in temperature constant, the relationship as shown in Equation 10 was found. While drilling, viscous flow energy, rotational energy, and drill-bit energy are believed to have a strong effect on the BHT.

However, this will most likely result in a perturbed offset in Equation 9, but a constant temperature gradient, g_G. Thus, drilling with a constant penetration rate and constant mud-in temperature provides direct information on the local temperature gradient, g_G.

When drilling ceases, but circulation continues, the well bore undergoes cooling. This change in formation temperature affects the temperature of the circulating mud.

Flowing fluid temperature profiles may be calculated from Equations A-10 and A-11. However, these are again complex implicit functions of circulating time, and a simple explicit function of circulation time is desirable.

Also, because the expression of the temperature profile is independent of the form of the dimensionless temperature function, f(t_D), the most applicable solution would be to express the temperature as an explicit function of f(t_D). Then, different f(t_D) models may be applied. More information on the dimensionless time function is covered in the Equations box, Part D.

As for the drilling case, only surface and BHTs are modeled. To get numerical values, the data in Table 1 are used. The key factor in the application of the proposed model is the relative deviation of the temperature from its virgin value, denoted by the term d (Equations 11 and 12).

Note that these solutions are valid for both forward and reverse circulation cases. It is important to use the respective temperature expressions.

The d_bh was observed to be almost logarithmic. During simulation, a function of the form shown in Equation 13 provided satisfactory accuracy. The flow rate is known to have a strong effect on the temperature, so the model has to handle changes in the flow rate. By a proper determination of the model constants (Equation box, Part C) the relative error between T_bht⁰ (1-d_bh) and the implicit solution of T_bh was found to be less than 1.3%, for f(t_D) < 5, and flow rates below 5,000 l./min. Such a validity range covers all practical well operations.

Mud-return temperature

The temperature difference between in and out at surface is still governed by Equations 3 and 5. However, for this case, the time function, f(t _D), applies to Equation A-8a. The result of this is an out-temperature that increases over time relative to the in-temperature. This is seen in the d function of Equation 11, which by simulations was found to obey the form found in Equation 14. Also, the relative error between T_out⁰ (1-d_out) and the implicit solution of T_out was found to be less than 1.3% for f(t_D) < 5 and flow rates below 5,000 l./min.

The scope of deriving temperature expressions as explicit functions of time provides a possibility to determine the model constants empirically.

If reasonable approximations of the model constants by measurements can be defined, the temperature behavior can be predicted without exact knowledge of the input parameters. In addition, it becomes possible to use the temperature measurements to gain more information concerning the downhole physical parameters.

Because the temperature is known to be strongly flow-rate dependent, a correlation between the model constants and the flow rate was sought. Using data in Table 1, all the model constants were approximated by the polynomial in Equation 15.

Table 2 [28,786 bytes] shows the experimental values of k₁, k₂, and k₃. For a maximum flow rate of 2,500 l./min, the maximum deviation between the polynomial approximation and the corresponding implicit solution is 2%.

Although these models are based on a single well, it is believed that the models describe the general functional behavior independent of well specifications. To verify this, acquisition of more field data is needed.

Field cases

A 1996 field case using data from a well drilled in the North Sea will be used to demonstrate the application of the models derived in this article. The BHT was regularly measured during drilling.

The temperature gradient prognosis for this well was an average of 3.6° C. /100 m. Thus, it was observed that the measured gradient locally deviates from the average trend.

The first case is based on downhole temperature measurements obtained during drilling. In this example, the mud-in temperature and the penetration rate remained constant during the time period of interest.

For this case, the simplified Equations 7 and 10 are valid. Inserting the data from Fig. 3 [106,382 bytes], Equation 10 becomes:

51° C. = 47° C. + g_G * 130 m (a)

or

g_G = 3.1 (°C./100 m) (b)

This is the virgin temperature gradient at the depth of 2,000 m.

For the second example, obtained in a deeper part of the same well, the mud-in temperature increased by approximately 0.8° C./hr while drilling. Therefore, the full solution of Equations 6 and 9 are required. The surface return temperatures are inserted into Equation 6 as follows:

36° C. = 30° C. - Bl1 * 0.8° C./hr * 5 hr (c)

or

Bl1 = 1.715 (d)

Similarly, inserting the data from Fig. 4 [120,197 bytes] into Equation 9, the BHT is given by:

54° C. = 47° C. + gG * 130 m + 0.8° C./hr * 5 hr * (1- l1/ l2)e 2,000l1 (e)

or

g_G = 6.63° C./100 m - 3.64° C./100 m * (1- l1/ l2)e 2,000l1 (f)

Clearly, the temperature gradient measured downhole is not identical to the virgin temperature gradient because of increasing mud-in temperature. To determine the virgin formation temperature gradient, gG, from the above equation, information about l1 and l2 is required.

Using the relationships between Equation A-12 and A-13:

l1/ l2 = AB l²₁ (g)

we get

(1 - l1/ l2) e ^2,000l1 = (1 + A/B(Bl1)²) e 2,000l1 (h)

Having well data to make qualified estimates of the ratio A/B and l1, it is possible to determine the virgin temperature gradient as follows:

A/B @ 2.5 (i) e 2,000l1 @ 0.1 (j)

Now, inserting Equations i and j into Equations g and h gives us:

g_G = 3.3° C./100 m (k)

Fig. 3 shows other effects. Three different rotary speeds were applied during the time of the plot. Decreasing the rotary speed at t = 1,610 hr leads to an almost immediate drop in the BHT. An increase at about 1,720 hr leads to an increase in BHT.

However, note that two separate temperature gradients must be determined at the upper and lower part of the plot. The change in rotary speed resulted in an offset in the temperature gradient plot.

This field case also demonstrates that the measured temperature gradient while drilling is only equal to the virgin formation-temperature gradient if the mud-in temperature remains constant. For varying mud-in temperature, a correction must be applied as demonstrated in the second example.

The changes in the temperature gradient reflect changes in thermal conductivity of the formations through which heat is flowing.^{13 14} Further analysis may show that transient drilling temperature profiles can be of practical use in estimating thermal conductivities for the surrounding formations.

Thus, if variations in the thermal conductivities can be detected, there is a possibility that shales, water sands, and hydrocarbon-bearing formations can be detected.¹⁴

The two field cases above demonstrate how the virgin temperature gradient can be determined while drilling, and therefore can become of immediate use for drilling-fluid and pore-pressure analysis.

Acquiring measurements from the periods without drilling, and combining these with the drilling data, complete temperature profiles may be generated. Due to a present lack of consistent data, this is not included in the present field case.

Acknowledgment

The authors wish to thank Dr. Chris Ward of Sperry Sun Drilling Services for permission to use the North Sea MWD data.

References

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The Authors