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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 951298, 10 pages
Research Article

Numerical Simulation of Gas-Liquid-Solid Three-Phase Flow in Deep Wells

1National Engineering Laboratory for Pipeline Safety, Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum, Beijing 102249, China
2MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

Received 7 January 2013; Accepted 20 March 2013

Academic Editor: Yi Wang

Copyright © 2013 Jianyu Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A gas-liquid-solid flow model which considers the effect of the cuttings on the pressure drop is established for the annulus flow in the deep wells in this paper, based on which a numerical code is developed to calculate the thermal and flow quantities such as temperature and pressure distributions. The model is validated by field data, and its performance is compared with several commercial software. The effects of some important parameters, such as well depth, gas kick, cuttings, and drilling fluid properties, on the temperature and pressure distributions are studied.

1. Introduction

With the worldwide increasing energy consumption and the development of petroleum industry, deep well and ultradeep well will become an important approach to explore oil and gas in the future. When the wellbore pressure is below the formation pressure during the drilling process, minor incidents such as gas cut and gas kick may emerge to form air pocket, which may in the worst case lead to disastrous blowout to generate formidable pressure and shock waves to cause heavy casualties and extensive property damages, especially in deep well drilling process. Besides, the property parameters and rheological parameters of the multiphase fluid in the wellbore are the sensitive functions of the temperate. Therefore, In order to predict the pressure distribution and optimize the design and operation of the production of the oil well systems, the accurate calculation of the temperature distribution of the wellbore should be guaranteed. Therefore, accurate prediction of the pressure and temperature changes under the gas kick condition to guide the deep drilling operation has practical significance.

For pressure prediction, early researchers used the empirical correlations [13], which are still widely used in commercial software. For example, the Beggs-Brill correlation is used in MUDLITE, based on which the bottomhole pressure, liquid holdup, and so forth can be evaluated. However, the empirical correlations sometimes result in great deviations for limited valid ranges [4]. Recently, numerical models have been developed to predict the multiphase flow behaviors in the wellbore, for example, Hasan and Kabir’s two-phase flow model [5], Ping et al.’s modified model [6] based on Hasan and Kabir’s work, Bijleveld et al.’s unbalanced drilling model [7], and Perez-Tellez et al.’s unified unbalanced drilling model [8]. Comparison studies show that the solutions of those models are generally much more accurate than the empirical correlations [9].

As to temperature prediction, both analytical and numerical methods are used. With regard to analytical solutions, representative works can be referred to those of Ramey [10], Holmes and Swift [11], Kabir et al. [12], and Hasan et al. [13]. Since numerical solutions obtained with less simplified models are generally more accurate than the analytical solutions, numerical simulation methods are widely adopted nowadays. Raymond [14] neglected effects of energy source in the drilling system and presented the first numerical model that coupled unsteady-state heat transfer in the formation and the steady heat transfer in the wellbore. On this basis, Keller et al. [15] made a hypothesis that no axial heat transfer existed in the drilling fluid and developed a mathematical model for temperature distribution of circulating fluid in the wellbore. Later, Marshall and Bentsen [16] analyzed solution stability and velocity of the finite difference method on the basis of Keller et al.’s work.

The available studies have been carried out primarily for shallow wells with a depth around 3 kilometers, while very few studies have been reported for the flow and thermal characteristics of deep wells though they should have deserved much more attention. Based on this point, in this study we aim to develop a model, taking important parameters into account, to describe the fluid flow and thermal properties in the deep drilling and then apply the established model to simulate the temperature and pressure drop distribution under the influences of different factors.

2. Hydraulic Model

2.1. Flow Patterns Transition Criteria

The pressure distribution in the wellbore is greatly affected by flow patterns, which is generally determined by flow patterns transition criteria in the calculations. Flow patterns transition criteria were primarily proposed for circular pipe flow such as those proposed by Taitel et al. [17]. The flow patterns transition criteria for annulus flows firstly presented by Caetano [18] and Kelessidis and Dukler [19] in 1986 and improved later by many researchers as presented below.

2.1.1. Bubble-to-Slug Transition

Caetano [18] indicated that the bubble-to-slug transition occurs when the velocity of dispersed bubbles is greater than that of Taylor bubbles, with the mean gas volumetric fraction being around 0.2. The flow pattern transition formula is given by the critical superficial velocity of liquid phase as follows:

2.1.2. Slug-to-Churn Transition

In 1999, Tengesdal et al. [20] first built a slug-to-churn transition model based on the drift-flux model, and it was improved by Kaya et al. [21] in 2001, in which the gas volumetric fraction of the fluid was evaluated by

Kaya et al. [21] declared that slug-to-churn transition occurs at a critical void fraction of 0.78; later Yu et al. [22] argued that the critical value should be 0.64 based on their experimental data. In this paper, the value of 0.64 is adopted.

2.1.3. Bubble/Slug-to-Dispersed Bubble Transition

When the velocity of liquid phase is relatively large, bubbles are scattered in it, and then the bubble/slug to dispersed bubble transition appears. Based on Caetano’s model [18], the expression describing the transition curve can be given as below:

2.1.4. Churn-to-Annular Transition

With flux and superficial velocity of the gas phase being large enough, churn flow transforms to annular flow. In fully developed annular flow, the liquid phase moves upward along the pipe wall smoothly and continuously in the form of liquid film, with continuous air stream entrained in the pipe center. Taitel et al. [17] announced that annular flow occurs only when gas superficial velocity is high enough to lift the entrained liquid droplets and carry them along. On this basis, an equilibrium relationship exists between the gravity of a liquid droplet and the resistance imposed on it by the gas stream, according to which the expression of churn to annular transition can be deduced as

2.2. Pressure Drop Model in the Drill String

The drilling mud is assumed to be injected into the drill string as a single-phase fluid without any solid particles or gas, and thus, the total pressure drop in the drill string is composed of gravity component and friction component, which is given by with evaluated by Colebrook formula:

2.3. Pressure Drop Model in the Annulus

The total pressure drop in the annulus is composed of the contributions of gravity, friction drag, convective acceleration loss, and interaction between cuttings and fluid: For annular flow, two total pressure drop formulae can be deduced as follows:

The total pressure drop can be obtained by solving (8), and the details can be referred to [19]. For other flow patterns, the first three terms on the right hand side of (7) are calculated by the formulae listed in Table 1.

Table 1: Expressions of pressure drop components of various flow patterns.

2.3.1. Pressure Drop Caused by Cuttings

For simplification, the following assumptions are made in this paper when modeling the pressure drop caused by cuttings:(a)the cuttings elements are uniformly distributed in the liquid phase and have no effects on the flow pattern;(b)the properties of the cuttings elements are assumed the same;(c)the interaction among the discrete elements is neglected.And then the pressure drop caused by cuttings can be calculated by where, is given by [26]: and N can be calculated by with being

And the term is the velocity difference between the liquid phase and the cuttings entrained in the liquid stream, which can be figured out through Newton’s second law and velocity interpolation.

3. Thermal Model

In this study, the flow and heat transfer are assumed to be steady at a certain drilling step due to the very low rate of penetration. Several assumptions have been made to establish a steady-state thermal model to describe the heat transfer process:(a)the fluid flow and heat transfer are assumed to be symmetric as shown in Figure 1;(b)since heat convection is dominant both in the drilling string and in the annulus during the drilling process, heat conduction between the drilling muds is neglected;(c)the heat convection is simplified as a one-dimensional problem.

Figure 1: Physical model.

Based on the above assumptions, the heat transfer governing equations for the drilling muds in the drill string and annulus and that for the formation can be written as (13), (14), and (15), respectively, as follows:

The convective heat transfer coefficient correlations [27] used in the present study are listed in Table 2.

Table 2: Convective heat transfer coefficient correlations under different flow patterns.

For the dispersed bubble and slug/churn cases, the is the same in both the laminar state and the turbulent state.

4. Model Solving and Validation

The models are implemented in a FORTRAN computer program. The well is divided into 150 segments. Calculation starts from the surface, based on the pressure and temperature at the wellhead, proceeds down the drilling string to the bottomhole, then up through the wellbore, and then terminates in the ground surface. And at each increment, the pressure and temperature are solved alternately. The model and numerical code are validated by the field data of Muspac53 well in the Mexican field [32] with the main test conditions given in Table 3. Figure 2 shows the comparisons of the field data with the predicted wellbore pressures by the present model and the other four models, that is, LSU model by Perez-Tellez, MUDLITE V-2, MUDLITE V-3, and Neotec WELLFLO 7 in [9]. The absolute errors and relative errors between the five models and the field data are, respectively, listed in Table 4. It is seen clearly that the predicted pressure by the present model agrees well with the field data and agrees better than the other three empirical-correlations-based models for this case.

Table 3: Data of pseudosteady state conditions in Muspac53.
Table 4: Comparison of the accuracies of different models.
Figure 2: Wellbore pressures by different pressure models.

5. Results and Discussions

In this section, the influences of the well depth, the gas kick, the cuttings, and the physical properties of the drilling mud on the temperature and pressure distributions are studied by employing the validated model and code. The relationship between pressure and temperature of the gas phase per mole is given by

The density and viscosity of the drilling mud used in the calculations are modeled as the functions of temperature and pressure and given as follows: where a, b, c, A, and B are fitting constants and the unit of temperature is here.

The other main common used parameters for calculations are given in Tables 5 and 6.

Table 5: Equipment parameters and formation parameters.
Table 6: Parameters of the gas phase, the liquid phase, and the cuttings.

Without special illustrations, the parameters given in Tables 5 and 6 are employed in the subsequent calculations. In Section 5.1, besides the well depth of 6000 m, the wells of 2500 m, 4000 m, and 5000 m are studied to analyze the effect of the well depth on the flow characteristics with the gas kicks at the bottomhole kept at 2.5 × 10−3 m3/s. In Section 5.2, four gas kicks of 0.1 m3/s, 0.5 m3/s, 1 m3/s, and 2 m3/s under standard condition are used. For the case of a gas kick of 0.5 m3/s, we studied the effect of cuttings on the pressure drop. In Section 5.3, drilling mud densities and viscosities are studied separately. Four densities at standard condition (900, 1000, 1100, and 1200 kg/m3) are employed, and four standard viscosities (0.03, 0.04, 0.05, and 0.06 Pa·s) are used.

5.1. Influence of the Well Depth on Deep Drilling

Figure 3 describes the relation between the temperatures of the drilling mud in the annulus with the depths in the four wells. It can be seen that at the same depth, the temperature of the drilling mud increases evidently with the increase of the well depth. The temperature in the deep well of 6000 m is highest, and it is 45°C higher than the bottomhole temperature in the 2500 m well. Such large temperature difference is because of the large amount of heat absorbed from the deeper formation when drilling mud goes upward.

Figure 3: Temperature versus depth in different wells.

When gas kicks at the bottomhole are the same, the holdups in the annulus differ obviously for the four wells as shown in Figure 4, with lower holdup existing in deeper well. It is also seen that though for all the wells, there are two identical flow patterns, slug flow and bubble flow, but the transition location is different. The deeper the well is, the lower the transition occurs. For the well of 6000 m, transition occurs 1500 m from the surface, while for the well of 2500 m, it is 1000 m. In addition, at the same depth the liquid holdup of the deeper well is smaller, because when the mud goes upward, the pressure becomes smaller and the gas volume becomes larger.

Figure 4: Liquid holdup versus depth in different wells.

Figure 5 compares the pressure distributions in the annulus of the four wells. It is seen that at the same depth, the pressure difference is small compared with the pressure itself. The maximum pressure difference among them is only 3.2 MPa at the depth of 2,500 m, which indirectly shows that pressure due to the gravity force plays the most important role. The pressure difference is mainly due to different holdups and flow patterns, and so forth.

Figure 5: Pressure versus depth in different wells.
5.2. Influence of the Gas Kick and Cuttings on Deep Drilling

The effects of gas kick on liquid holdup and pressure distributions in the annulus are shown in Figures 6 and 7, respectively. It is seen clearly in Figure 6 that with the increase of gas kick, more flow pattern exists along the wellbore. From the smallest gas kick volume 0.1 m3/s to the largest 2.0 m3/s, the number of flow patterns varies from 1 to 3. When the gas kick is 0.1 m3/s, only bubble flow occurs. When the gas kicks are 0.5 m3/s and 1.0 m3/s, bubble flow and slug flow coexist. When the gas kick is 2.0 m3/s, three flow patterns coexist with churn flow existing near the ground surface. With the increase of the gas kick volume, the transition of bubble flow to slug flow occurs at a deeper depth. Figure 7 shows that gas kick greatly affects the magnitude of pressure, the larger the gas kick volume, the smaller the pressure. The smaller pressure at the bottomhole at the larger gas kick volume is due to the smaller density of the drilling fluid leading to a smaller gravity force. It should be noted that the decreasing bottomhole pressure at large gas kick volume may further induce the increase of the gas kick due to the large pressure difference between the formation and the bottomhole of the well and may eventually destroy the normal operations and result in dangerous blowout.

Figure 6: Liquid holdup versus depth under different gas kicks.
Figure 7: Pressure versus depth under different gas kicks.

In most previous models, the effect of cuttings on pressure is ignored completely or treated as a correction factor; in the present study its effect is clarified by solving (9). Figure 8 presents the pressure distributions in the annulus with and without cuttings with the gas kick being 0.5 m3/s under the standard condition. It is seen that cuttings do have effects on the pressure, but the influence is not so large; for this case, the maximum increment of pressure is about 0.7 MPa at the bottomhole.

Figure 8: Pressure versus depth with and without cuttings.
5.3. Influences of Mud Properties on Deep Drilling

Since the physical properties of drilling mud may affect the drilling process greatly, in this section two most important parameters are studied, that is, density and viscosity. The gas kick of 0.5 m3/s under the standard condition is employed in the calculation.

Figure 9 reveals that the density of the drilling mud can greatly affect the bottomhole pressure, whose value is only 55.9 MPa at density of 900 kg/m3 (standard condition) but increases to 75.5 MPa at the density of 1200 kg/m3. It is seen that the increase ratio of the pressure (35%) is almost linearly proportional to the increase ratio of the mud density (33.3%). This is due to that the gravity contribution to pressure is almost proportional to density. It is seen in Figure 10 that with the increase of the drilling mud density, the temperature decreases. The major reason is due to the increase of the heat capacity per mass.

Figure 9: Pressure versus depth under different mud densities.
Figure 10: Temperature versus depth under different mud densities.

Figures 11 and 12, respectively, present the pressure and temperature distributions in the annulus under different drilling mud viscosities. It is seen that the effect of mud viscosity on pressure is much smaller than that of mud density, showing again that frictional drag pressure loss is a small component in the deep well. However, the temperature is also apparently affected by the viscosity. When density at the standard condition increases from 0.03 Pa·s to 0.06 Pa·s, the bottomhole temperature drops from 117°C to 108°C. The increase of viscosity decreases the turbulent heat transfer, leads to less heat absorption from the formation, and results in a lower temperature.

Figure 11: Pressure versus well depth under different mud viscosities.
Figure 12: Temperature versus well depth under different mud viscosities.

6. Conclusions

In this study, a gas-liquid-solid model is proposed to study the heat and fluid flow characteristics in annulus of deep wells. The model has been validated by field data. Based on the validated model, the effect of cuttings, gas kick, well depth, and drilling mud physical properties are clarified. It is found that the effect of the cuttings on the total pressure drop is small; temperature and pressure are greatly affected by well depth, drilling mud density, and gas kick; though drilling mud viscosity weakly affects the total pressure, it may obviously affect the temperature distributions in the annulus. The quantitative data obtained in this study can give a reference to the deep well drilling practice.


Flow area
:Factor of drag force, dimensionless
:Volume concentration of the cuttings, dimensionless
:Specific heat capacity of the fluid (J/(kg·°C))
:Specific heat capacity of the fluid in the drill string and in the annulus (J/(kg·°C))
:Diameter (m)
:Inner and outer diameters of the tube, inner diameter of the casing (m)
d(T):Temperature gradient ()
: Fanning friction factor, dimensionless
:Acceleration due to gravity (m/s2)
:Convective heat transfer coefficient (W/(m2·°C))
:Convective heat transfer coefficient of the inner wall of the tube , that of the outer wall of the tube  , and that of the surface of the casing (W/(m2·°C))
:Liquid holdup, dimensionless
:Length (m)
:Number of cuttings elements, dimensionless
: Nusselt number, dimensionless
:Pressure (Pa without otherwise specified)
:Wellhead back pressure (MPa)
Pr: Prandtl number, dimensionless
:Flow rate (kg/s or m3/s)
:Radius (m)
:Perfect gas constant, which is equals 8.314 J/(mol·°C)
Re: Reynolds number, dimensionless
ROP:Rate of penetration (m/h)
:Wetted perimeters (m)
:Temperature (°C  without otherwise specified)
:Temperature of the drilling mud in the drill string and in the annulus
:Velocity (m/s)
:Mean velocity (m/s)
:Volume (m3)
:Depth (m)
:Compressibility factor.
Greek Symbols
:Gas volumetric fraction, dimensionless
:Error follows the units of the physical quantities, or dimensionless as relative errors
:Thermal conductivity (W/(m·°C))
:Dynamic viscosity (Pa·s)
:An irrational number equals 3.14  with two decimal places, dimensionless
:Density (kg/m3)
:Density of the mixture,
:Interfacial tension (N/m)
: Shear stress (N/m2)
:Form factor of the cuttings, dimensionless.
Acc:Convective acceleration
dm:Drilling mud
et:External wall of the coiled tubing
Fric:Friction drag
FT:Fluid temperature
:Gas phase
GSC:Gas under standard condition
:Liquid phase
LF:Liquid film
LLS:Liquid phase in liquid slug
LS:Liquid slug
LSC:Liquid under standard condition
:Mixture of the liquid and gas phase
:Earth’s surface
SG:Superficial gas
SL:Superficial liquid
SU:Slug unit
TB: Taylor bubble
WT:Well temperature.


The study is supported by the National Key Projects of Fundamental R/D of China (973 Project: 2010CB226704) and the Science Foundation of China University of Petroleum, Beijing (no. 2462011LLYJ33, no. 2462011LLYJ55, no. 2462012KYJJ0403, and no. 2462012KYJJ0404).


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