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. 2022 Aug 3;7(32):28675–28684. doi: 10.1021/acsomega.2c04185

Solution and Analysis of Wellbore Temperature and Pressure Field Coupling Model under Lost Circulation

Ao Yang †, Zhongxi Zhu †,*, Nan Zhang ‡, Yuchen Ye ‡
PMCID: PMC9386830  PMID: 35990465

Abstract

graphic file with name ao2c04185_0014.jpg

The fluid in the annulus is in a variable mass flow state under the lost circulation condition, significantly affecting wellbore pressure distribution and the heat transfer efficiency between wellbore and formation. Therefore, based on the hydrodynamics and energy conservation law, a coupling model of transient wellbore temperature and pressure field under lost circulation conditions is established, which fully considers the casing program, bottom hole assembly, and heat source generated in drilling, as well as the influence of the temperature and pressure coupling in wellbore on the physical parameters of drilling fluid. The calculation results of the model in this paper are in good agreement with the field measured data and the previous research results, which verifies the rationality and accuracy of the model in this paper. The effects of the loss rate and the lost circulation zone location on the wellbore temperature and pressure distribution are analyzed, and the variation rule of the physical parameters of drilling fluid under the lost circulation condition is studied. The numerical simulation results show that the density and viscosity of drilling fluid in the annulus and bottom hole pressure increase with the increase in the circulation time; the annulus temperature decreases gradually with the increase in cycle time and tends to become stable after 8 h of the cycle. The annulus temperature and bottom hole pressure decrease with the increase in loss rate and closeness of the loss zone to the well bottom. When the loss zone is in the upper open-hole section, an inflection point consistent with the location of the loss zone appears on the annular temperature difference curve between the loss circulation and the nonloss circulation, and the position of the loss zone can be judged according to the inflection point.

1. Introduction

With the increase in oil and gas exploration and development, the research frontier of the petroleum industry has gradually shifted to the development of land deep oil and gas reservoirs and deep-water drilling.1,2 Complex geological conditions result in drilling operations facing engineering problems such as abnormally high temperature, high pressure, and narrow safe density window, resulting in frequent downhole complex accidents such as lost circulation.3,4 In addition, borehole temperature is not only an important parameter for wellbore stability analysis and drilling design but also has a significant impact on drilling fluid properties and cementing quality.5−8 Moreover, under the loss condition, the fluid in the annulus is in an unsteady-state flow, and the leaked fluid also reduces the formation temperature near the wellbore and changes the original temperature and pressure distribution in the wellbore, which seriously affect the physical parameters such as drilling fluid density and viscosity, making the loss situation more complex.9−13 Therefore, studying the variation law of wellbore temperature and pressure in the process of lost circulation is of great significance for safe and efficient drilling.

In drilling engineering, the research methods of wellbore temperature field are mainly divided into the analytical method and numerical method. Holmes14 used the steady-state linear heat transfer approximation to replace the heat transfer between the annulus fluid and the formation and derived an analytical model to calculate the steady-state heat transfer between the fluid in the drill string and the fluid in the annulus. The calculation results show that the maximum temperature of the fluid in the annulus does not appear at the well bottom. Based on this research result, Kabir15 assumes that the heat exchange process between the fluid in the drill string and the fluid in the annulus belongs to steady-state heat transfer, and the heat exchange between the annulus fluid and the near wellbore area belongs to pseudo-steady-state heat transfer. Combined with the energy conservation law, a one-dimensional transient heat transfer model based on dimensionless time has been established. Based on the above model and considering the different construction environment, mode, and other factors, researchers have established different analytical models to analyze the temperature distribution law of wellbore and formation and the heat transfer efficiency between fluid and near wellbore area in the construction process of cementing16 and oil and gas production.17,18

The numerical method is based on the one-dimensional numerical model established by Raymond19 to calculate the temperature distribution of annulus fluid in vertical wells under steady and pseudo-steady conditions. Keller20 added the inner heat source generated by the drill string rotation and the friction resistance of drilling fluid during drilling to the Raymond model, established the two-dimensional transient heat transfer model, and considered that these inner heat sources had a significant impact on the energy balance during drilling fluid circulation. In 1998, J. Romero21 first considered the influence of circulation time, displacement, and seawater temperature on wellbore temperature field during deep-water drilling and developed the corresponding calculation program. In 2013, Mou22 divided the drilling fluid in the wellbore into several grids in the radial direction, established the wellbore temperature field model considering the radial temperature gradient of the drilling fluid, discretized and solved the mathematical model using the finite difference method, and considered that the axial heat conduction of the drilling fluid had a weak influence on the wellbore temperature distribution. In 2014, Li23 pondered the heat source generated during horizontal well drilling, established a numerical model calculating the horizontal well temperature field, and analyzed the heat source distribution in the process. In 2016, Li24 considered the influence of bottom hole assembly and casing program on wellbore temperature distribution and established a wellbore-formation transient heat transfer model suitable for the process of drilling fluid circulation and well shut-in under overflow conditions. In 2019, Yang25 established a transient heat transfer model suitable for deep-water multigradient drilling based on the conservation equations of mass, momentum, and energy, aiming at the problem of variable mass heat transfer caused by the hollow glass spheres entering the annulus from the drill string during deep-water multigradient drilling, and analyzed the wellbore temperature distribution law in deep-water multigradient drilling through this model. Compared with the numerical method, the analytical method has the characteristics of high computational efficiency and unconditional convergence, but it is difficult to obtain the analytical solution of the nonlinear partial differential equations representing the wellbore temperature field and the influence of the thermophysical parameters of the wellbore fluid, heat sources, and different casing program on the wellbore temperature distribution cannot be fully considered. As a result, the analytical model is difficult to effectively complete the research on the wellbore temperature distribution during deep-water drilling and drilling into deep reservoirs. This also makes more and more scholars prefer to use numerical methods to study the transient heat transfer process between wellbore and formation.

Although the research on wellbore heat transfer of drilling fluid under no-loss circulation conditions has become mature, there are few studies on variable mass heat transfer of annulus fluid under loss circulation.26−38 In 2017, Chen et al.39 established the first variable mass heat transfer model in the wellbore under the lost circulation condition and also proposed a method to judge the position of the loss zone according to the temperature gradient curve. However, due to its failure to fully consider the heat source generated in the drilling process and the influence of the casing program on the wellbore temperature distribution, the model has a large error in the drilling process of deep and ultradeep wells. In 2020, Wang et al.,40 on the basis of the Chen model, supplemented the influence of the internal heat source generated in the drilling process and casing program on the temperature distribution in the wellbore and established the numerical model predicting the wellbore temperature profile under the lost circulation. In 2020, Zhang41 proposed a wellbore temperature distribution model based on two-dimensional bottom hole regional loss. It is considered that compared with one-dimensional regional loss, the loss of drilling fluid in a two-dimensional region has a more significant impact on the annulus temperature profile of drilling fluid; however, the location of the loss zone cannot be judged by this model.

The above models never consider the influence of temperature pressure coupling in the wellbore on the physical parameters of drilling fluid after loss circulation. Based on the Chen model and Wang model, first, considering the mutual effect of physical parameters of the drilling fluid and wellbore temperature and pressure, a coupling model of wellbore temperature and pressure field under loss circulation is established. The partial differential equation is discretized by the finite difference method, and the discrete equation is solved by the Gauss–Seidel algorithm. In addition, the effects of temperature pressure coupling in wellbore on physical parameters of drilling fluid, convective heat transfer coefficient, and circulating pressure loss under lost circulation are analyzed. Finally, the distribution law of temperature and pressure in wellbore under lost circulation is studied. The research results can provide theoretical guidance for an efficient leaking stoppage and safe drilling.

2. Mathematical Model

During the drilling operation, the drilling fluid enters the drill string at temperature Tin, flows to the well bottom through the drill string, then enters the annulus by the drill bit, and finally returns to the surface with rock cuttings. During this process, the fluid in the wellbore not only absorbs the heat generated by its flow and drill string rotation but also is heated by the lower high-temperature formation and returns to the ground at temperature Tout.4242

Different from the normal condition, the lost circulation makes the drilling fluid invade the formation, resulting in a decrease in the volume flow of the drilling fluid in the upper annulus. It is necessary to comprehensively consider the influence of variable mass flow on the wellbore heat transfer process under the loss circulation. The fluid flow and heat transfer model under loss circulation is shown in Figure 1.13

Figure 1.

Figure 1

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Schematic diagram of fluid flow and heat transfer under loss circulation.

In the process of establishing the model, according to the flow characteristics and heat transfer law of the fluid in the wellbore under the condition of loss circulation, the following assumptions are put forward:

  • (1)

    The heat exchange in the axial direction of the formation is ignored.

  • (2)

    The physical parameters (density, specific heat capacity, etc.) of the drill string, casing, and formation rock are not affected by temperature and pressure.

  • (3)

    The axial heat conduction and radial temperature gradient of drilling fluid is ignored.22

2.1. Circulating Pressure Loss

In the process of drilling fluid circulation, the circulating pressure loss is expressed as43

2.1. 1

where Pf is the circulating pressure loss of drilling fluid (Pa), ρm is the drilling fluid density (kg/m3), f is the friction coefficient (dimensionless), and d is the flow diameter (m).

The calculation of friction coefficient mainly depends on the flow state of fluid; however, in the process of lost circulation, the volume flow of annulus fluid in the upper part of the loss zone decreases, which changes the flow state of annulus fluid. Therefore, it is necessary to accurately calculate the flow state of fluid on each grid node. Equations 2 and 3 are the calculation methods of friction coefficient of drilling fluid under laminar flow and turbulent flow conditions, respectively.44

2.1. 2

where Re is the Reynolds number (dimensionless).

2.1. 3

where n is the flow-behavior index (dimensionless).

2.2. Wellbore-Formation Heat Transfer Model

In the process of drilling fluid flowing from the wellhead to the well bottom in the drill pipe, the energy change of fluid microelement is mainly reflected in (1) the convective heat transfer between the fluid in the drill string and the fluid in the annulus, (2) net heat in the axial direction, and (3) heat generated by fluid flow and rock breaking by the drill bit.

2.2. 4

where Qm is the drilling fluid displacement (m3/s), Tp is the fluid temperature in the drill string (°C), Ta is the fluid temperature in the annulus (°C), dpi is the inner diameter of the drill string (m), Cm is the specific heat capacity of the drilling fluid (J/(kg·°C)), h12 is the total convective heat transfer coefficient (W/(m2·°C)), Sm is heat generated by fluid flow (W/m), and Sb is heat generated by rock breaking (W/m).

Under the condition of no loss, the energy change of fluid microelement in the annulus is mainly reflected in (1) the convective heat transfer between the fluid in the annulus and the fluid in the drill string, (2) convective heat transfer between fluid in annulus and wellbore, (3) net heat in the axial direction, and (4) heat generated by fluid flow and drill string rotation.

2.2. 5

where Tf is the formation temperature (°C), dpo is the outer diameter of the drill string (m), dw is the wellbore radius (m), h3 is the convective heat transfer coefficient of the borehole wall (W/(m2·°C)), and St is the heat generated for drill string rotation (W/m).

Under the loss circulation, the energy change of fluid microelement in the annulus at the lower part of the loss zone is the same as that under the normal circulation condition, while the change in drilling fluid flow caused by fluid entering the formation needs to be considered in the upper part of the loss zone.

2.2. 6

where Ql is the loss rate (m3/s).

The radial heat transfer in the formation is

2.2. 7

where kf is the thermal conductivity of rock (W/(m·°C)), ρf is the rock density (kg/m3), and Cf is the specific heat capacity of rock (J/(kg·°C)).

2.3. Auxiliary Equation

2.3.1. Convective Heat Transfer Coefficient

To simplify the calculation, the total convective heat transfer coefficient h12 is used to calculate the transient heat transfer between the drilling fluid in the drill string and the drilling fluid in the annulus.45Table 1 shows the calculation method of convective heat transfer coefficient.46

2.3.1. 8

where h1 and h2 are the convective heat transfer coefficients of the drill string inner wall and the drill string outer wall, respectively, (W/(m2·°C)), km is the thermal conductivity of drilling fluid (W/(m·°C)), and kp is the thermal conductivity of the drill string (W/(m·°C)).

Table 1. Calculation Formula of Convective Heat Transfer Coefficient.
region flow pattern calculation formula
inner wall of the drill string laminar flow
graphic file with name ao2c04185_m019.jpg
turbulent flow
graphic file with name ao2c04185_m020.jpg
outer wall of the drill string laminar flow
graphic file with name ao2c04185_m021.jpg
turbulent flow
graphic file with name ao2c04185_m022.jpg
well wall laminar flow
graphic file with name ao2c04185_m023.jpg
turbulent flow
graphic file with name ao2c04185_m024.jpg
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2.3.2. Heat Source During Drilling

Equations 11–13 represent the heat generated by fluid flow, drill string rotation, and rock breaking by the drill.23

2.3.2. 9
2.3.2. 10
2.3.2. 11

where M is the axial drill string torque (N·m), ω is the drill string rotary speed (r/s), Eb is the rock breaking efficiency of the bit (dimensionless), and Mb is the torque at the bit (N·m).

2.3.3. Physical Parameters of Drilling Fluid

Considering that the thermophysical parameters of drilling fluid change with temperature and pressure during drilling fluid circulation, eqs 11 and 12 represent the functional relationship of density and viscosity of water-based drilling fluid under the coupling action of temperature and pressure, respectively.47

2.3.3. 12
2.3.3. 13

where ρm0 is the density of surface drilling fluid (kg/m3), P0 is the surface pressure (MPa), T0 is the surface temperature (°C), μm0 is the viscosity of surface drilling fluid (mPa·s), A is equal to 4.9224 × 10–10, B is equal to −9.6877 × 10–19, C is equal to −3.1296 × 10–4, D is equal to −1.7432 × 10–6, E is equal to 4.9186 × 10–13, F is equal to 2.18 × 10–9, G is equal to −9.32 × 10–3, and H is equal to 1.09 × 10–5.

3. Model Solving

3.1. Initial and Boundary Conditions

  • (1)
    Known drilling fluid injection temperature
    graphic file with name ao2c04185_m014.jpg 14)
  • (2)
    The temperature of drilling fluid in the drill string at the well bottom is equal to that in the annulus
    graphic file with name ao2c04185_m015.jpg 15)
  • (3)
    The formation temperature changes linearly at an infinite distance from the wellbore
    graphic file with name ao2c04185_m016.jpg 16)
    where Ts is the surface temperature (°C), H is the well depth (m), and Tg is the formation temperature gradient (°C/m).

3.2. Model Solving Method

Figure 2a shows that the wellbore and formation are divided into n equally spaced grids in the axial direction, respectively. Figure 2b shows that m variable spacing grids are divided radially for the wellbore and formation with the drill string as the center.48 In this case, r0 is the inner diameter of the drill string, r1 is the outer diameter of the drill string, r2 is the well diameter, when j > 2, rj = r2e(j–2) α, where α is an equal ratio factor.

Figure 2.

Figure 2

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Schematic diagram of the wellbore and formation grid division.

Based on the finite difference method, the wellbore transient temperature control equation is discretized and fully implicit. Among them, the first-order space derivative in the governing equation adopts the first-order upwind scheme, and the first-order time derivative adopts the two-point backward difference. The second spatial derivative adopts the three-point central difference.49Equation 17 can be expressed as a difference scheme of any node.

3.2. 17

The equations of all control bodies are expressed in the matrix form of eq 18 and solved by the Gauss–Seidel algorithm. The temperature Ti,jt0 of each control body at t0 can be obtained. Then, the drilling fluid physical parameters at t0 can be obtained according to the current time pressure Pi,j. All the above can be used as the initial parameters of drilling fluid physical parameters at t = t0 + Δt time to solve the temperature and pressure distribution at each node at t. The specific solution process is shown in Figure 3.50

3.2. 18

Figure 3.

Figure 3

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Solution flow chart of temperature and pressure field coupling model under loss circulation.

4. Model Validation

To verify the accuracy of the model, the drilling data of Xinjiang Oilfield in Table 2 are substituted into this model and another classical model, and the calculation results are compared. The annulus temperature distribution curve is shown in Figure 4. The simulation results show that since the Chen model does not consider the influence of heat source on annulus drilling fluid temperature, the bottom hole temperature is 13.5 °C lower than that of the actual model and 10.7 °C lower than that of the Wang model. The calculation results of this model are close to those of the Wang model. However, the Wang model considers that the physical parameters of drilling fluid are constant, which leads to the calculation result being 2.8 °C lower than that of this model. In addition, the error between this model and the Wang model is no more than 5%, which shows that the calculation results of this model are reasonable and accurate.

Table 2. Drilling Data and Thermophysical Parameters.

parameters numerical value parameters numerical value
well depth (m) 5590 drilling fluid density (g/cm3) 1.44
casing length (m) 3600 drilling fluid viscosity (mPa·s) 45
length of the drill collar (m) 250 specific heat capacity of drilling fluid [J/(kg·°C)] 1600
drill pipe outer diameter (mm) 127 thermal conductivity of drilling fluid [W/(m·°C)] 1.73
drill pipe inner diameter (mm) 108 rock density (g/cm3) 2.64
drill collar inner diameter (mm) 57 specific heat capacity of rock [J/(kg·°C)] 837
drill collar outer diameter (mm) 158.75 thermal conductivity of rock [W/(m·°C)] 2.25
bit diameter (mm) 215.9 drill string density (g/cm3) 8
casing inner diameter (mm) 220.5 specific heat capacity of drill string [W/(m·°C)] 400
displacement (L/s) 32 thermal conductivity of drill string [W/(m·°C)] 43.75
surface temperature (°C) 20 drilling fluid injection temperature (°C) 46
bit nozzle combination (mm3) 12 + 12 + 14 formation temperature gradient (°C/m) 0.021
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Figure 4.

Figure 4

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Model validation.

Table 3 shows the comparison between the calculated values of drilling fluid outlet temperature and circulating pressure loss and the measured values on the construction site. The calculation results are in good agreement with the field measurement results. The maximum error of wellhead temperature is 3.9% and the maximum error of circulating pressure loss is 4.6%. The accuracy of the model in this paper meets the needs of practical engineering and the actual situation of the construction site.

Table 3. Simulation Results of Drilling Fluid Outlet Temperature and Circulating Pressure Loss of Well XX.

      outlet temperature (°C)
 circulating pressure loss (MPa)
well depth (m) displacement (L/s) drilling fluid injection temperature (°C) measured value calculated value error (%) measured value calculated value error (%)
5400 20.8 49.6 50.4 52.4 3.9 13.95 14.6 4.6
5450 26.2 59 59.8 59.4 0.6 18.13 18.8 3.7
5530 25.2 60.9 60.7 60.9 0.3 17.79 18.5 3.9
5570 21.8 59.1 58 57.5 0.8 14.62 14.5 0.8
5590 21.7 61.4 59.9 58.6 2.1 15.4 15.1 1.9
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5. Results and Discussion

5.1. Influence of Loss Circulation on Convective Heat Transfer Coefficient

In the process of drilling fluid circulation, the flow condition is mainly affected by bottom hole assembly, casing program, and drilling fluid volume flow. Therefore, the effect of loss circulation on the convective heat transfer coefficient is more significant. Figure 5 shows the calculation results of the annular convective heat transfer coefficient when the location of the loss zone is located at 4000 and 5590 m and the loss rate is 20% of the displacement. It can be seen from Figure 5 that during the lost circulation process, the fluid volume flow in the annulus above the loss zone decreases and the flow state changes, resulting in the convection heat transfer coefficient of the drill string outer wall and borehole wall being lower than that in the normal circulation.

Figure 5.

Figure 5

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Annular convective heat transfer coefficients.

5.2. Annulus Temperature Profile under Loss Circulation

When loss occurs at the well bottom (5590 m) and the loss rate is 25% of the displacement, the annulus temperature profile under the lost circulation condition is shown in Figure 6. The simulation results show that with the increase in well depth, the annulus temperature increases first and then decreases. The highest temperature is about 800–1000 m from the well bottom. This is because when the local formation temperature is higher than the wellbore temperature, the drilling fluid absorbs heat from the formation; otherwise, it releases heat into the formation. In addition, with the increase in circulation time, the bottom hole temperature gradually decreases. When the circulation time reaches 8 h, the temperature change in the annulus gradually tends to be stable.

Figure 6.

Figure 6

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Annulus temperature profile during lost circulation.

5.3. Influence of Loss Circulation on Wellbore Pressure

When loss occurs at the bottom of the well (5590 m) and the loss rate is 25% of the displacement, the simulation results of circulating pressure loss and pressure distribution in the wellbore are shown in Figure 7. The simulation results show that in the case of lost circulation, the circulating pressure loss in the annulus is reduced by 0.85 MPa. This is because the loss circulation reduces the flow of drilling fluid above the loss zone so that the circulating pressure loss in the wellbore under the loss circulation condition is lower than that under the normal circulation of drilling fluid, resulting in the reduction of the pressure in the wellbore.

Figure 7.

Figure 7

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Simulation results of circulating pressure loss and wellbore pressure distribution during loss circulation.

5.4. Drilling Fluid Physical Parameter Profile

Figure 8 shows the simulation results of drilling fluid density and viscosity in the annulus during the lost circulation. The simulation results show that with the increase in well depth, the density and viscosity of drilling fluid in the annulus decrease first and then increase. This is because under the coupling effect of pressure and temperature, the annulus temperature gradually increases with the increase in well depth and the drilling fluid expands, resulting in a decrease in density and viscosity. On the contrary, when the well depth exceeds the highest point of the annulus temperature, the annulus temperature gradually decreases and the annulus pressure gradually increases, resulting in the compression of the drilling fluid and the increase in the density and viscosity of the drilling fluid.

Figure 8.

Figure 8

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Profile of drilling fluid physical parameters when the loss rate is 25%.

5.5. Effect of Loss Rate on Annulus Temperature

Figure 9 shows the effect of different loss rates on the annulus temperature distribution when the bottom hole (5590 m) has lost and the circulation time is 8 h. The simulation results show that compared with normal circulation, the fluid temperature in the annulus decreases significantly when loss occurs. On the one hand, when lost circulation occurs, the drilling fluid enters and reduces the temperature of the formation near the loss zone, reducing the temperature difference between the fluid in the annulus and the formation. On the other hand, the lost circulation also leads to a decrease in fluid volume flow in the upper part of the lost circulation zone and a reduction in the heat generated by fluid friction resistance. At the same time, the flow velocity of the fluid decreases, resulting in the highest point of annulus temperature moving up. In addition, with the increase in loss rate, more drilling fluid enters the formation, resulting in a further reduction in fluid temperature in the annulus.

Figure 9.

Figure 9

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Effect of loss rate on temperature distribution of annulus drilling fluid.

5.6. Influence of Loss Zone on Annulus Temperature

Figure 10 shows the effect of loss zone on annulus fluid temperature distribution when the circulation time is 8h. When there is a loss zone (4300m), the loss rate is 40% of the displacement; when there are two loss zones (4000m and 5590m), the loss rate is 20% of the displacement respectively. The results show that the closer the lost circulation zone is to the well bottom, the lower the annulus temperature is.

Figure 10.

Figure 10

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Effect of lost circulation zone on annulus temperature distribution.

Figure 11 shows the annulus temperature difference between the two drilling conditions. The simulation results show that when the loss occurs in the upper open-hole section, the annulus temperature at the lost circulation zone decreases greatly, resulting in the appearance of an inflection point consistent with the location of the loss zone on the temperature difference curve. In the case of lost circulation, the position of the lost circulation zone can be judged according to the inflection point by predicting the annulus temperature.

Figure 11.

Figure 11

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Temperature difference between non lost circulation and lost circulation.

5.7. Analysis of Factors Influencing Bottom Hole Pressure

Figure 12 shows the variation law of bottom hole pressure with cycle time under loss circulation. The simulation results show that the bottom hole pressure increases with the increase in the drilling fluid circulation time. When the drilling fluid circulation time is close to 8 h, the bottom hole pressure gradually tends to be stable. In addition, with the increase in loss rate and the gradual downward movement of lost circulation zone, the bottom hole pressure decreases accordingly. When the loss rate increases from 10 to 40%, the bottom hole pressure decreases by 0.4 MPa; when the position of the lost circulation layer moves down from 4300 to 5300 m, the bottom hole pressure decreases by only 0.1 MPa. It can be seen that the effect of loss rate on bottom hole pressure is more significant.

Figure 12.

Figure 12

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Influence of loss rate and lost circulation zone on bottom hole pressure.

6. Conclusions

According to the hydrodynamics and energy conservation law, a coupling model of wellbore temperature and pressure field is established. The influence of temperature pressure coupling in wellbore on drilling fluid physical parameters and convective heat transfer coefficient under loss circulation is analyzed, and the main factors affecting annulus temperature and pressure distribution are researched. The conclusions are as follows:

  • (1)

    Under the loss circulation, the density and viscosity of drilling fluid decrease first and then increase with the increase in well depth. To improve the accuracy of calculation, the physical parameters of drilling fluid need to be considered as a function of temperature and pressure.

  • (2)

    Different from the normal circulation condition of drilling fluid, the lost circulation reduces the volume flow in the upper part of the lost circulation zone, the annulus pressure loss, and convective heat transfer coefficient, resulting in a large decrease in the annulus temperature near the lost zone. Also, when the loss occurs in the upper open-hole section, the temperature difference curve has an inflection point at the near loss zone, and the location of the loss zone can be judged by referring to the inflection point.

  • (3)

    Compared with the position of loss zone, the influence of loss rate on annulus temperature and bottom hole pressure is more significant. The temperature and pressure distribution in the wellbore can be calculated using this model according to the real-time loss rate at the construction site to appropriately adjust the drilling fluid density and ensure safe drilling.

Acknowledgments

The authors wish to acknowledge the China Scholarship Council and National Engineering Laboratory of Petroleum Drilling Technology, Leak Resistance, and Sealing Technology Research Department.

Author Contributions

N.Z. and Y.Y. curated the data. Z.Z. and A.Y. performed the methodology and numerical simulation and wrote the original draft. Y.Y. supervised the experiment. Z.Z. and A.Y. wrote, reviewed, and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

This research was funded by the National Natural Science Foundation of China (Grant No. 51204024) and the Hubei Provincial University Outstanding Young and Middle-aged Science and Technology Innovation Team Plan (Grant No. T201804).

The authors declare no competing financial interest.

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