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Pressure drop models for gas-liquid two-phase flow and its application in underbalanced drilling*
Conference of Global Chinese Scholars on Hydrodynamics
PRESSURE DROP MODELS FOR GAS-LIQUID TWO-PHASE FLOW AND ITS APPLICATION IN UNDERBALANCED DRILLING* PING Li-qiu, WANG Zhi-ming, WEI Jian-guang Institute of Oil and Gas, Petroleum University of China, Beijing 102249, China; E-mail:[email protected] ABSTRACT: Hasan-Kabir’s model and modified Ansari’s model were used to analyze the pressure drop of two-phase flow in underbalanced drilling (UBD). The mathematical and physical models of UBD were established and solved numerically. The effects of injection rate of gas and liquid and, back pressure on bottom-hole pressure were analyzed. The bottom-hole pressure control procedures were concluded. The computer simulations were validated with data obtained from experiment performed in a full-scale well located at Louisiana State University. The prediction accuracy of two methods was analyzed. The results showed that the error with the Hasan-Kabir model is less than 10%, however the error with the modified Ansari model is less than 5%. Ansari’s model is superior to Hasan-Kabir’s model in predicting pressure. This supplies theoretical support for success in UBD. KEY WORDS: gas-liquid two-phase flow, pressure drop models, underbalanced drilling, pressure predictions, accuracy comparison
1. Introduction The emergence of UBD technology can be used to avoid complicated drilling problems, such as reservoir damage and circulation loss, etc.. In UBD operations, underbalanced pressure condition is needed to be maintained by bottom-hole pressure control according to specific operating conditions and actual status of fluid in wellbore. So the bottom-hole pressure must be accurately predicted. During the ordinary UBD operations with conventional rigs (jointed pipe drilling), drilling fluids (liquid or gasified liquid) are pumped down through the drill string, through the bit, and then up the annulus. Within the annulus, drilling fluids are mixed with rock cuttings and production fluids (gas, oil, or water). The underbalanced hydraulic circulating system is typically characterized by the complex flow of two or more phases (liquid mixture, gas mixture, and solid cuttings). Therefore, the foundation of UBD analysis and study is to establish multiphase flow mechanistic model to predict and control pressure according to formation-wellbore-ground response. In the past decade, many researchers[1-12] have studied this issue. Some flow models have been established with
homogeneous approach[1,2] and empirical correlation approach[3,4]. Some flow phenomena were explained. But there is some shortcomings in guiding actual operation and further study is needed. In order to solve this problem, Hasan-Kabir’s model and modified Ansari’s model were used to analyze gas-liquid two-phase flow in UBD. The authors are trying to find a kind of simple, practical and accurate pressure prediction method. 2. Gas-Liquid Two-Phase Flow Models in UBD Previous steady gas-liquid two-phase models in UBD operations fall into three categories. The first is the steady state computer programs that neglect slip between phases by assuming that aerated mud can be treated as a homogeneous mixture. The second is the steady state computer programs that used empirical correlations to take into account slip between phases and predict different flow patterns. The third is the steady state computer programs based on mechanistic models rather than empirical correlations to take into account slip between phases and predict different flow patterns. The steady models presented in this paper belong to the third category. Most studies on UBD models focused on the first and the second types of models in the recent years. But both two kinds of models’ validations show that lower prediction accuracy than the accuracy needed in practical operations. Since the mid-1970’s, a significant progress has been made in understanding the mechanism of two-phase flow in pipes and production systems. This progress has resulted in several two-phase flow mechanistic models to simulate pipelines and wells under steady state as well as transient conditions. The mechanistic or phenol-menological approach postulates the existence of different flow configurations and formulates separate models for each one of these flow patterns to predict the main parameters, such as gas fraction and wellbore pressure. Consequently, mechanistic models,
* Supported by program for Changjiang Scholars and Innovative Research Team in University (IRT0411). Biography :PING Li-qiu (1974-), Female, Ph. D. Student 405
mechanistic or phenomenological approach postulates the existence of different flow configurations and formulates separate models for each one of these flow patterns to predict the main parameters, such as gas fraction and wellbore pressure. Consequently, mechanistic models, rather than empirical correlations, have been used with increasing frequency for the design of multiphase production systems. Hasan-Kabir’s model and Ansari’s model are mechanistic models. Hasan-Kabir’s model [7][8][14] According to the hydrodynamic theory, Hasan and Kabir developed a mechanistic model to analyze the flow pattern transition mechanism in pipes. The methods for calculating flow pattern transition boundaries and pressure drop in pipes were presented and modified velocity distribution coefficient and hydraulic diameter were used to study flow characteristics in annulus. Hasan and kabir concluded that the flow patterns in annulus included four types: bubble flow, slug flow, churn flow, and annular flow according to the configurations of two-phase media’s distribution. Computational methods for gas void fraction and pressure drop were presented in each flow pattern respectively. However, for the slug flow, this represents a simplification that does not rigorously consider the difference in drift-flux between the liquid slug and the Taylor bubble. Hasan-Kabir’s model was described in detail as follows. (1) Bubble Flow Transition Criteria: vsg <0.429v sl +0.357 v∞ or 2.1
f g <0.52 and
vm
1.12
>4.68d
0.48
[ g ( ρ -ρ ) /σ ] (σ /ρ ) ( ρ 0.5
l
g
0.6
l
m
/μ l )
0.08
Void Fraction: f g =vsg / ( C0 vm +v∞ ) C0＝1.20＋0.371 ( d t /d c )
ρ m = (1-fg ) ρ l +fg ρ g
Pressure Drop: ( dp/dz ) = 2f m vm 2 ρ m /d +g ρ m +ρ m vm ( dvm /dz )
(
)
(2) Slug Flow Transition Criteria: vsg > 0.429vsl +0.357 v∞ and ρ g vsg < ⎡⎣17.1log10 ( ρ l vsl ) -23.2 ⎤⎦ if ρ l v sl 2 >50 2
2
ρ g vsg <0.00673 ( ρ l vsl ) ,if ρ l vsl 2 <50 2
1.7
Void Fraction: f g =vsg / ( C1vm +v∞T ) 406
C1＝1.182＋0.90 ( d t /d c )
Pressure Drop: ( dp/dz ) = ⎡⎣2f mvm2 ρm /d ⎤⎦ (1-fg ) +g ρm +ρmvm ( dvm /dz ) (3) Churn Flow Transition Criteria: vsg <3.1 ⎡⎣σ g ( ρ l -ρ g ) /ρ g ⎤⎦ 2
0.25
and ρ g vsg > ⎡⎣17.1log10 ( ρ l vsl ) -23.2 ⎤⎦ if ρ l vsl >50 2
2
2
ρ g vsg >0.00673 ( ρ l vsl ) if ρ l vsl 2 <50 1.7
2
Void Fraction: f g =vsg / ( C1vm +v∞T ) C1＝1.15＋0.90 ( d t /d c )
Pressure drop: as described in the slug flow (4) Annular Flow Transition Criteria: vsg >3.1 ⎡⎣σ g ( ρ l -ρ g ) /ρ g ⎤⎦ 2
(
Void Fraction: f g = 1+X X = ( ρ g /ρ l )
0.5
0.8
)
0.25
-0.378
[(1-x ) x ] ( μ /μ ) 0.9
l
0.1
g
ρ c = (vsg ρ g + Evsl ρ l ) /(vsg + Evsl )
Pressure Drop:
( dp/dz ) = ⎡⎣ ρ g + ( 2f ρ v c
c
c
2 g
/d
)⎤⎦ / ⎡⎣1- ( ρc vg 2 /p )⎤⎦
Modified Ansari’s model[10][13][15] Ansari presented the criterion for two-phase flow pattern transition based on preview studies of gas-liquid two-phase flow in 1990. He studied the mechanism and characteristics of flow patterns and established flow characteristics analysis models to describe bubble flow, slug flow and annular flow patterns. In comparison to Hasan-Kabir’s model, Ansari’s model improved prediction accuracy of slug flow by considering two possible conditions of slug flow: the fully developed Taylor bubble slug flow and the developing Taylor bubble slug flow. The fully developed Taylor bubble slug flow may occur where the bubble cap length is negligible as compared to the total Taylor bubble length and the free falling film thickness in the Taylor bubble reaches a terminal value. The developing Taylor bubble slug flow may occur which consists only of a cap bubble and film thickness varies continuously along the flow direction, rather than reaching a constant terminal value as in the fully developed Taylor bubble. The following is Ansari’s model. In order to improve prediction accuracy, the authors presented the acceleration pressure gradient in bubble flow and slug flow to modify Ansari’s model according to Fernandes’[10]
2.2
study results. (1) Flow-Pattern Prediction Bubble/Slug Transition vsg = 0.25v∞ + 0.333vsl
Dispersed Bubble Transition 0.5
0.5 ⎡ 1.6σ ⎤ ⎛ ρ ⎞0.6 ⎛ f ⎞0.4 ⎛ vsg ⎞ 1.2 l ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ vm = 0.725 + 4.15 ⎜ ⎟ ⎢⎣ ( ρl − ρ g ) g ⎥⎦ ⎝ σ ⎠ ⎝ 2d ⎠ ⎝ vm ⎠
Dispersed Bubble/Slug Transition vsg = 1.08vsl
Transition to Annular Flow vsg =3.1 ⎡⎣σ g ( ρ l -ρ g ) /ρ g ⎤⎦ 2
0.25
(2) Flow-Behavior Prediction Bubble Flow Model Void Fraction: (1 − f g )v∞ = vsg / f g − C0 vm Pressure Drop: ( dp/dz ) = 2fm vm 2 ρ m /d +gρ m +ρ m vm ( dvm /dz )
(
)
Slug Flow Model ( dp / dz ) = [(1 − β ) ρ mls + βρ mtb ] + 2 f ls ρ mls vm (1 − β ) / d 2
+ H lls ρl / LSU ( vlls + vltb
)(v
t
− vlls )
Annular Flow Model 2 (dp / dz) c = φc (dp / dz) sc + g ρ c 3. Flow Pattern Transitions and Algorithm for Pressure Prediction Hasan-Kabir’s method was widely used by many researchers[16] studying two-phase flow in annulus in UBD operations. However, the modified Ansari’s method has never been used in studying UBD two-phase flow characteristics. In order to compare the universality of the two methods, both methods were used to analyze upward gas-liquid two-phase flow phenomena in annulus in UBD operations. The algorithms implemented in a FORTRAN 90 computer program to simulate UBD operations. The following simplifications have been accepted for research object in order to establish flow model in UBD: (i)·It is a steady model and can only simulate an established situation; (2) The drilled cuttings are transported at the same velocity in the annulus as the liquid phase; (3) The cross section of the wellbore is circular and concentric with drillstring; (4) The gas resolution in drilling fluid is negligible and there is no chemical reaction; (5) The ·gas and liquid medium is in thermodynamics balanced status, and pressure and density is single valued function; (6) No fluid production from reservoirs are considered. In UBD operations, pressure along the wellbore length is affected by the gas and liquid injection flow rate, the flow pattern distribution and the back pressure at the wellhead. With the larger well depth, temperature and pressure in annulus increases
constantly which results in the varying gas and liquid superficial velocity and gas void fraction which determines flow pattern distribution and pressure. The pressure gradient predictions are conducted with a marching algorithm which allows calculating the flow parameters along the flow path after dividing it into cells. After dividing the well bore into axial increments or cells, the initial conditions of pressure and temperature existing at the wellhead, the gas and liquid injection flow rates, and a guessed total pressure drop across the axial increment are used to solve the system of equations to determine the flow pattern, the liquid holdup, and the total pressure gradient along the axial increment. The calculated pressure gradient and guessed pressure gradient are compared to ensure the accuracy needed. After that, the pressure and temperatures at the bottom of this cell can be estimated. These pressures and temperatures represent the initial conditions at the top of the next axial increment, which similarly are utilized to calculate the corresponding pressure and temperature at the bottom of this new cell. Following this procedure, the flow pattern, the liquid holdup, the two-phase flow parameters, and the wellbore pressure can be calculated along the well bore flow path. The mechanistic steady state model to calculate flow patterns, two-phase flow parameters, and well bore pressure along the flow path following the specific algorithm and procedure are described below. (1) Input gas and liquid flow rates, fluid properties, and well geometry. (2) Select the length of the axial increments. (3) Guess the total pressure drop corresponding to the length increment. (4) Use the surface temperature and geothermal gradient to, estimate the downstream temperature of the first axial increment. (5) Similarly, use the casing choke pressure and the guessed total pressure drop from step (3) to estimate the downstream pressure of the first axial increment. (6) Use the surface pressure and temperature and the downstream pressure and temperature previously estimated in steps (4) and (5), to calculate the average pressure and temperature corresponding to the axial increment. (7) Estimate surface liquid and gas velocities and fluid properties at average conditions. (8) Program the flow pattern prediction models given in Hasan-Kabir’s and modified Ansari’s methods. (9) After identifying the existing flow pattern, use the corresponding flow behavior prediction model to calculate hydrodynamic parameters and total pressure drop. (10) Compare the total pressure gradient calculated in step 9 against that guessed in step 3. If 407
the difference between them is less than a tolerance continue with the next step. Otherwise, substitute the total pressure gradient guessed in step (3) for that calculated in step (9) and repeat steps (3) through (10) until convergence. When that happens, the cell downstream pressure will be the actual wellbore pressure occurring at the end of the first axial increment for the existing flow conditions. 4. Sensibility Factors Analysis In certain engineering conditions, the effects on bottom-hole pressure of different back pressures at the wellhead, gas and liquid injection flow rate are simulated by use of Hasan-Kabir’s (simplified by H-B in following diagrams and tables) method and the modified Ansari’s method. 4.1 The effect of wellhead back pressure (choke pressure) on bottom-hole pressure The data and parameters input for simulationare as follows. This is a vertical well with 1793 m of depth and 0.219 m of inner diameter of casing and 0.107m of outer diameter of drillstring. Nitrogen was injected at the rate of 0.43 m3/s, mud of 1120 kg/ m3 with a plastic viscosity of 24 mpa·s was injected at the rate of 0.009 m3/s. The surface temperature was 285º K and geothermal gradient was 0.017 ºK/m. The choke pressure varies in a range of 0.1-1.5MPa.
4.2 The effect of gas injection flow rate on bottom-hole pressure Input data and parameters were the same as in Section 4.1. The choke pressure was maintained at 1.586 MPa and nitrogen injection rate varied from 0 to 2.0m3/s. Figure 2 shows that the bottomhole pressure decreased slowly with the increasing injection gas flow rate. However in Hasan-Kabir’s method, when gas injection rate was higher than 1.25m3/s, the bottom-hole pressure decreased quickly compared against smooth decrease in modified Ansari’s method. The analysis of flow pattern transition shows that when the gas injection rate is higher than 1.25m3/s in Hasan-Kabir’s method, the total flow patterns in annulus are churn and annular flows which represent sharply lower pressure drop compared against relatively smooth pressure drop of the slug flow in modified Ansari’s method. This shows that HasanKabir’s method had some limitation in practical engineering application.
Fig. 2 Bottom-hole pressure vs. gas injection rate
4.2 The effect of liquid injection flow rate on bottomhole pressure Input data and parameters were the same as in Section 4.1. The choke pressure was maintained at 1.586MPa and injection mud (liquid) flow rate varied from 0.005 to 0.7m3/s. Fig. 1 Bottom-hole pressure vs. choke pressure
As one can see from Fig. 1, the bottom-hole pressure increased with increasing choke pressure. When the choke pressure varied from 0.1 MPa to 1.5 MPa, both increments of bottom-hole pressure were about 2.0 MPa. The reason of this trend happened was that volumetric flow rate, gas void fraction and mixture density of multiphase-flow fluid in annulus controlled by state equation changed with the increasing back pressure at the wellhead, which resulted in the transitions of flow patterns, hydrostatic pressure, friction pressure drop, acceleration pressure drop and total bottom-hole pressure. 408
Fig. 3 Bottom-hole pressure vs. injection liquid flow rate
From Fig. 3 it could be found that the bottomhole pressure increased with increasing injection mud flow rate. Hence one way to enhance bottom-hole
pressure is to increase injection mud flow rate.
6.1 Validation of prediction experiments data
results
with
5. The Methods of Controling Bottom-Hole Pressure The methods to regulate bottom-hole pressure are choke pressure adjustment and injection flow rate regulation according to sensibility factors analysis described above. 5.1
Choke pressure adjustment method
The Choke pressure adjustment method is not only to regulate bottom-hole pressure but also to avoid washing wellhead and surface equipments by high velocity fluids and higher friction pressure drop when fluids returning to wellhead udergo exceedingly large expansion. However, the overtop back pressure will accelerate the gas liquid separation and nonessential gas injection. On the other hand, the choke pressure at the wellhead should be maintained moderate limited by injection equipments and rotation control head. 5.2
Injection flow rate regulation method
There are two reasons to determine injection flow rate regulation method to control bottom-hole pressure is a simple, prompt, cheap means. The first reason is to change gas or liquid flow rate only by simple calculation according to the need of pressure regulation. The second reason is to reduce time consumption to regulate mud density and corresponding manpower, resource and finance consumption. But one thing to care is to limit the flow rate within the rated flow scope of the injection equipment. 6.
Fig. 4 Wellbore pressure vs. well depth of the first experiment
Validation and Anylysis
Hasan-Kabir’s method and modified Ansari’s method were validated with data obtained from two experiments performed in a full-scale well located at Louisian State University. This is a vertical well with 1793 m of depth and 0.219 m of inner diameter of casing and 0.107 m of outer diameter of drill-string. During the first experiment, nitrogen was injected at the rate of 0.53 m3/s, mud at 1120 kg/ m3 with a plastic viscosity of 6 mpa·s was injected at 0.01 m3/s. The surface temperature was 297ºK and geothermal gradient was 0.012 ºK/m. The choke pressure was maintained at 0.972 MPa. Annular wellbore pressure was measured at the positions of 1186m and 1768m. During the second test, nitrogen was injcted at the rate of 0.43 m3/s, mud of 1120 kg/ m3 with a plastic viscosity of 24 mpa·s was injected at 0.009 m3/s. The surface temperature was 285 º K and geothermal gradient was 0.017 ºK/m. The choke pressure was maintained at 1.586 MPa. Annular wellbore pressure was measured at 1768m.
Fig. 5 Wellbore pressure vs. well depth of experiment
the second
6.2 Analysis As one can see from Figs. 4 and 5 that both predicted pressure trends are the same and modified Ansari’s prediction accuracy is better than HasanKabir’s prediction results. For the sake of comparison, the absolute error and relative error are introduced in this paper. e = Calculated－Measured Calculated-Measured
× 100% Measured The absolute and relative errors are presented in Table 1. e%=
Table 1
Accuracy comparison of two models first experiment
M depth(m) 1186 M(MPa) 11.81 H-B C(MPa) 10.78 Ansari C(MPa) 11.48 H-B e (MPa) 1.03 Ansari e(MPa) 0.33 H-B e% 8.7% Ansari e% 2.8% Note: M-measured, C-calculated
1768 17.61 16.23 17.03 1.38 0.58 7.8% 3.6%
second experiment 1768 18.73 17.53 18.06 1.2 0.67 6.4% 3.6%
Table 1 shows that the relative error of Hasan-Kabir’s method is less than 10% and that of modified Ansari’s method is less than 5%. With the 409
increase of depth of the well, both absolute errors (e) and relative errors increase. Although the difference of relative errors is smaller, the difference of absolute errors is larger. When the well depth is 1768 m, the absolute error of Hasan-Kabir’s method is more than 1.0 MPa which is nearly close to the underbalanced pressure difference of 1.0-3.0 MPa. Hence, the maximum absolute error of 1.38 MPa (the absolute error of Hasan-Kabir’s method) may jeopardize the success of UBD operations and lose economic target of UBD. Prediction accuracy of modified Ansari’s method is better and the absolute value error of 0.67 MPa will meet the accuracy need of UBD design. So modified Ansari’s method is recommended in UBD design. The reason of better prediction accuracy of modified Ansari’s method over Hasan-Kabir’s method is the detailed description of slug flow characteristics: fully developed and developing Taylor bubble slug flow conditions. The evaluation results with the modified Ansari’s model show that when both fully developed and developing Taylor bubble slug flow conditions are taken into account, the model predictions are very good (the relative error is less than 5%). On the other hand, when only the simplified fully developed Taylor bubble slug flow model is used, the relative error is near 15%. 7.
Conclusions The study of this paper shows that mechanistic analysis pressure drop models are superior to experimental models in studying gas-liquid two-phase flow pressure drop in UBD. The progress in gas-liquid two-phase flow analysis makes its industrial application possible. The following conclusions can be drawn from the present study. (1) In the study of two-phase steady flow in annulus of UBD operations, convergent solutions can be obtained by Hasan-Kabir’s method and modified Ansari’s method, but there is difference in the accuracy of prediction. (2) The accuracy difference of Hasan-Kabir’s method and modified Ansari’s method shows that accurately describing slug flow parameters is important to improve the pressure prediction accuracy. (3) In order to guarantee the success of UBD operation, modified Ansari’s method is recommended in UBD design. Nomencalture
v =velocity, m/s f g =void fraction, dimensionless d =pipe diameter, m dt =tubing diameter, m 410
d c =casing diameter, m dp / dz =total pressure gradient, Pa/m (dp / dz )c =gas core pressure gradient, Pa/m (dp / dz ) sc =superficial friction pressure gradient in the core, Pa/m φc2 =dimensionless group
g =acceleration due to gravity, m/s2
ρ =density, kg/m3 μ =viscosity, kg/m•s
E=entrainment H=liquid holdup, dimensionless f=fanning friction factor, dimensionless x=mass fraction of gas phase, dimensionless X=Lockhart-Martinelli parameter β=relative bubble length parameter,dimensionless σ=surface tension Subscripts sg=superficial gas sl=superficial liquid ∞T =terminal rise velocity of a single Taylor bubble ∞= terminal bubble-rise velocity in bubble flow m=mixture l=liquid g=gas c=core liquid (in case of annular flow) ls=liquid slug References [1] GUO, B, HARELAND, G., and RAJTAR, J., Computer Simulation Predicts Unfavorable Mud Rate for Aerated Mud Drilling. SPE26892, 1993 [2] SHARMA, Y., KAMP, A., Yonezawa, T., RIVERO, L.M., Kobayashi, A., and Gonzalez, J., Simulating Aerated Drilling. SPE59424, 2000 [3] TIAN, Shifeng, MEDLEY, GEORGE H. and STONE, CHARLES R., Optimizing circulation while drilling underbalanced. World Oil, June 2000，pp48-55 [4] LIU, Gefei, MEDLEY Jr, GEORGE H, Houston, Foam computer model helps in analysis of underbalanced Drilling. Oil and Gas J. ,July 1,1996,pp18-27 [5] BIJLEVELD, A. F., KOPER, M., SAPONJA, J. , Development and Application of an Underbalanced Drilling Simulator. SPE39303,1998 [6] LAGE, A.C.V.M., TIME, R.W., An Expermental and Theoretical Investigation of Upward Two-Phase Flow in Annuli. SPE64525,2000 [7] HASAN, A. R. and KABIR, C. S., Two-phase flow in vertical and inclined annuli. Int. J. Multiphase Flow, Vol. 18, No. 2, 1992,pp279-293 [8] HASAN, A. R., Void fraction in bubbly and slug flow in downward vertical and inclined systems. SPE26522,1995 [9] LAGE, A.C.V.M. and TIME, R.W., Mechanistic Model for Upward Two-Phase Flow in Annuli. SPE63127,2000 [10] PEREZ-TELLEZ, C. Smith, J.R, EDWARDS, J. K., A New comprehensive mechanistic model for
[11] [12]
[13]
[14]
underbalanced drilling improves wellbore pressure predictions. SPE74426,2002 BEGGS, H.D. and BRILL, J.P., A study of two phase flow in inclined pipes. JPT, May 1973,pp607～617 ZHOU Kaiji, Multiphase-flow circulation system analysis in UBD operations[J]. Natural Gas Industry, 1998，18（5）：44～47 CHEN Jialang, CHEN Taoping, WEI Zhaosheng, Gas-Liquid Two-phase Flow of Rod Pump Well. Petroleum Industry Press, 1994，5，pp6-15 HASAN, A. R. and KABIR, C. S., A Study of
[15]
[16]
multiphase flow behavior in vertical wells. SPE15138,1988 ANSARI, A. M., SYLVESTER, N. D., SARICA, C., SHOHAM, O. and BRILL, J. P., A Comprehensive mechanistic model for upward two-phase flow in wellbores. SPE Production & Facilities, May 1994 LUO Shiying, A mechanistic model and simulation for underbalanced drilling. Ph.D Dissertation, Southwest Petroleum Institute, 1999,5 (in Chinese)
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PRESSURE DROP MODELS FOR GAS-LIQUID TWO-PHASE FLOW AND ITS APPLICATION IN UNDERBALANCED DRILLING* PING Li-qiu, WANG Zhi-ming, WEI Jian-guang Institute of Oil and Gas, Petroleum University of China, Beijing 102249, China; E-mail:[email protected] ABSTRACT: Hasan-Kabir’s model and modified Ansari’s model were used to analyze the pressure drop of two-phase flow in underbalanced drilling (UBD). The mathematical and physical models of UBD were established and solved numerically. The effects of injection rate of gas and liquid and, back pressure on bottom-hole pressure were analyzed. The bottom-hole pressure control procedures were concluded. The computer simulations were validated with data obtained from experiment performed in a full-scale well located at Louisiana State University. The prediction accuracy of two methods was analyzed. The results showed that the error with the Hasan-Kabir model is less than 10%, however the error with the modified Ansari model is less than 5%. Ansari’s model is superior to Hasan-Kabir’s model in predicting pressure. This supplies theoretical support for success in UBD. KEY WORDS: gas-liquid two-phase flow, pressure drop models, underbalanced drilling, pressure predictions, accuracy comparison
1. Introduction The emergence of UBD technology can be used to avoid complicated drilling problems, such as reservoir damage and circulation loss, etc.. In UBD operations, underbalanced pressure condition is needed to be maintained by bottom-hole pressure control according to specific operating conditions and actual status of fluid in wellbore. So the bottom-hole pressure must be accurately predicted. During the ordinary UBD operations with conventional rigs (jointed pipe drilling), drilling fluids (liquid or gasified liquid) are pumped down through the drill string, through the bit, and then up the annulus. Within the annulus, drilling fluids are mixed with rock cuttings and production fluids (gas, oil, or water). The underbalanced hydraulic circulating system is typically characterized by the complex flow of two or more phases (liquid mixture, gas mixture, and solid cuttings). Therefore, the foundation of UBD analysis and study is to establish multiphase flow mechanistic model to predict and control pressure according to formation-wellbore-ground response. In the past decade, many researchers[1-12] have studied this issue. Some flow models have been established with
homogeneous approach[1,2] and empirical correlation approach[3,4]. Some flow phenomena were explained. But there is some shortcomings in guiding actual operation and further study is needed. In order to solve this problem, Hasan-Kabir’s model and modified Ansari’s model were used to analyze gas-liquid two-phase flow in UBD. The authors are trying to find a kind of simple, practical and accurate pressure prediction method. 2. Gas-Liquid Two-Phase Flow Models in UBD Previous steady gas-liquid two-phase models in UBD operations fall into three categories. The first is the steady state computer programs that neglect slip between phases by assuming that aerated mud can be treated as a homogeneous mixture. The second is the steady state computer programs that used empirical correlations to take into account slip between phases and predict different flow patterns. The third is the steady state computer programs based on mechanistic models rather than empirical correlations to take into account slip between phases and predict different flow patterns. The steady models presented in this paper belong to the third category. Most studies on UBD models focused on the first and the second types of models in the recent years. But both two kinds of models’ validations show that lower prediction accuracy than the accuracy needed in practical operations. Since the mid-1970’s, a significant progress has been made in understanding the mechanism of two-phase flow in pipes and production systems. This progress has resulted in several two-phase flow mechanistic models to simulate pipelines and wells under steady state as well as transient conditions. The mechanistic or phenol-menological approach postulates the existence of different flow configurations and formulates separate models for each one of these flow patterns to predict the main parameters, such as gas fraction and wellbore pressure. Consequently, mechanistic models,
* Supported by program for Changjiang Scholars and Innovative Research Team in University (IRT0411). Biography :PING Li-qiu (1974-), Female, Ph. D. Student 405
mechanistic or phenomenological approach postulates the existence of different flow configurations and formulates separate models for each one of these flow patterns to predict the main parameters, such as gas fraction and wellbore pressure. Consequently, mechanistic models, rather than empirical correlations, have been used with increasing frequency for the design of multiphase production systems. Hasan-Kabir’s model and Ansari’s model are mechanistic models. Hasan-Kabir’s model [7][8][14] According to the hydrodynamic theory, Hasan and Kabir developed a mechanistic model to analyze the flow pattern transition mechanism in pipes. The methods for calculating flow pattern transition boundaries and pressure drop in pipes were presented and modified velocity distribution coefficient and hydraulic diameter were used to study flow characteristics in annulus. Hasan and kabir concluded that the flow patterns in annulus included four types: bubble flow, slug flow, churn flow, and annular flow according to the configurations of two-phase media’s distribution. Computational methods for gas void fraction and pressure drop were presented in each flow pattern respectively. However, for the slug flow, this represents a simplification that does not rigorously consider the difference in drift-flux between the liquid slug and the Taylor bubble. Hasan-Kabir’s model was described in detail as follows. (1) Bubble Flow Transition Criteria: vsg <0.429v sl +0.357 v∞ or 2.1
f g <0.52 and
vm
1.12
>4.68d
0.48
[ g ( ρ -ρ ) /σ ] (σ /ρ ) ( ρ 0.5
l
g
0.6
l
m
/μ l )
0.08
Void Fraction: f g =vsg / ( C0 vm +v∞ ) C0＝1.20＋0.371 ( d t /d c )
ρ m = (1-fg ) ρ l +fg ρ g
Pressure Drop: ( dp/dz ) = 2f m vm 2 ρ m /d +g ρ m +ρ m vm ( dvm /dz )
(
)
(2) Slug Flow Transition Criteria: vsg > 0.429vsl +0.357 v∞ and ρ g vsg < ⎡⎣17.1log10 ( ρ l vsl ) -23.2 ⎤⎦ if ρ l v sl 2 >50 2
2
ρ g vsg <0.00673 ( ρ l vsl ) ,if ρ l vsl 2 <50 2
1.7
Void Fraction: f g =vsg / ( C1vm +v∞T ) 406
C1＝1.182＋0.90 ( d t /d c )
Pressure Drop: ( dp/dz ) = ⎡⎣2f mvm2 ρm /d ⎤⎦ (1-fg ) +g ρm +ρmvm ( dvm /dz ) (3) Churn Flow Transition Criteria: vsg <3.1 ⎡⎣σ g ( ρ l -ρ g ) /ρ g ⎤⎦ 2
0.25
and ρ g vsg > ⎡⎣17.1log10 ( ρ l vsl ) -23.2 ⎤⎦ if ρ l vsl >50 2
2
2
ρ g vsg >0.00673 ( ρ l vsl ) if ρ l vsl 2 <50 1.7
2
Void Fraction: f g =vsg / ( C1vm +v∞T ) C1＝1.15＋0.90 ( d t /d c )
Pressure drop: as described in the slug flow (4) Annular Flow Transition Criteria: vsg >3.1 ⎡⎣σ g ( ρ l -ρ g ) /ρ g ⎤⎦ 2
(
Void Fraction: f g = 1+X X = ( ρ g /ρ l )
0.5
0.8
)
0.25
-0.378
[(1-x ) x ] ( μ /μ ) 0.9
l
0.1
g
ρ c = (vsg ρ g + Evsl ρ l ) /(vsg + Evsl )
Pressure Drop:
( dp/dz ) = ⎡⎣ ρ g + ( 2f ρ v c
c
c
2 g
/d
)⎤⎦ / ⎡⎣1- ( ρc vg 2 /p )⎤⎦
Modified Ansari’s model[10][13][15] Ansari presented the criterion for two-phase flow pattern transition based on preview studies of gas-liquid two-phase flow in 1990. He studied the mechanism and characteristics of flow patterns and established flow characteristics analysis models to describe bubble flow, slug flow and annular flow patterns. In comparison to Hasan-Kabir’s model, Ansari’s model improved prediction accuracy of slug flow by considering two possible conditions of slug flow: the fully developed Taylor bubble slug flow and the developing Taylor bubble slug flow. The fully developed Taylor bubble slug flow may occur where the bubble cap length is negligible as compared to the total Taylor bubble length and the free falling film thickness in the Taylor bubble reaches a terminal value. The developing Taylor bubble slug flow may occur which consists only of a cap bubble and film thickness varies continuously along the flow direction, rather than reaching a constant terminal value as in the fully developed Taylor bubble. The following is Ansari’s model. In order to improve prediction accuracy, the authors presented the acceleration pressure gradient in bubble flow and slug flow to modify Ansari’s model according to Fernandes’[10]
2.2
study results. (1) Flow-Pattern Prediction Bubble/Slug Transition vsg = 0.25v∞ + 0.333vsl
Dispersed Bubble Transition 0.5
0.5 ⎡ 1.6σ ⎤ ⎛ ρ ⎞0.6 ⎛ f ⎞0.4 ⎛ vsg ⎞ 1.2 l ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ vm = 0.725 + 4.15 ⎜ ⎟ ⎢⎣ ( ρl − ρ g ) g ⎥⎦ ⎝ σ ⎠ ⎝ 2d ⎠ ⎝ vm ⎠
Dispersed Bubble/Slug Transition vsg = 1.08vsl
Transition to Annular Flow vsg =3.1 ⎡⎣σ g ( ρ l -ρ g ) /ρ g ⎤⎦ 2
0.25
(2) Flow-Behavior Prediction Bubble Flow Model Void Fraction: (1 − f g )v∞ = vsg / f g − C0 vm Pressure Drop: ( dp/dz ) = 2fm vm 2 ρ m /d +gρ m +ρ m vm ( dvm /dz )
(
)
Slug Flow Model ( dp / dz ) = [(1 − β ) ρ mls + βρ mtb ] + 2 f ls ρ mls vm (1 − β ) / d 2
+ H lls ρl / LSU ( vlls + vltb
)(v
t
− vlls )
Annular Flow Model 2 (dp / dz) c = φc (dp / dz) sc + g ρ c 3. Flow Pattern Transitions and Algorithm for Pressure Prediction Hasan-Kabir’s method was widely used by many researchers[16] studying two-phase flow in annulus in UBD operations. However, the modified Ansari’s method has never been used in studying UBD two-phase flow characteristics. In order to compare the universality of the two methods, both methods were used to analyze upward gas-liquid two-phase flow phenomena in annulus in UBD operations. The algorithms implemented in a FORTRAN 90 computer program to simulate UBD operations. The following simplifications have been accepted for research object in order to establish flow model in UBD: (i)·It is a steady model and can only simulate an established situation; (2) The drilled cuttings are transported at the same velocity in the annulus as the liquid phase; (3) The cross section of the wellbore is circular and concentric with drillstring; (4) The gas resolution in drilling fluid is negligible and there is no chemical reaction; (5) The ·gas and liquid medium is in thermodynamics balanced status, and pressure and density is single valued function; (6) No fluid production from reservoirs are considered. In UBD operations, pressure along the wellbore length is affected by the gas and liquid injection flow rate, the flow pattern distribution and the back pressure at the wellhead. With the larger well depth, temperature and pressure in annulus increases
constantly which results in the varying gas and liquid superficial velocity and gas void fraction which determines flow pattern distribution and pressure. The pressure gradient predictions are conducted with a marching algorithm which allows calculating the flow parameters along the flow path after dividing it into cells. After dividing the well bore into axial increments or cells, the initial conditions of pressure and temperature existing at the wellhead, the gas and liquid injection flow rates, and a guessed total pressure drop across the axial increment are used to solve the system of equations to determine the flow pattern, the liquid holdup, and the total pressure gradient along the axial increment. The calculated pressure gradient and guessed pressure gradient are compared to ensure the accuracy needed. After that, the pressure and temperatures at the bottom of this cell can be estimated. These pressures and temperatures represent the initial conditions at the top of the next axial increment, which similarly are utilized to calculate the corresponding pressure and temperature at the bottom of this new cell. Following this procedure, the flow pattern, the liquid holdup, the two-phase flow parameters, and the wellbore pressure can be calculated along the well bore flow path. The mechanistic steady state model to calculate flow patterns, two-phase flow parameters, and well bore pressure along the flow path following the specific algorithm and procedure are described below. (1) Input gas and liquid flow rates, fluid properties, and well geometry. (2) Select the length of the axial increments. (3) Guess the total pressure drop corresponding to the length increment. (4) Use the surface temperature and geothermal gradient to, estimate the downstream temperature of the first axial increment. (5) Similarly, use the casing choke pressure and the guessed total pressure drop from step (3) to estimate the downstream pressure of the first axial increment. (6) Use the surface pressure and temperature and the downstream pressure and temperature previously estimated in steps (4) and (5), to calculate the average pressure and temperature corresponding to the axial increment. (7) Estimate surface liquid and gas velocities and fluid properties at average conditions. (8) Program the flow pattern prediction models given in Hasan-Kabir’s and modified Ansari’s methods. (9) After identifying the existing flow pattern, use the corresponding flow behavior prediction model to calculate hydrodynamic parameters and total pressure drop. (10) Compare the total pressure gradient calculated in step 9 against that guessed in step 3. If 407
the difference between them is less than a tolerance continue with the next step. Otherwise, substitute the total pressure gradient guessed in step (3) for that calculated in step (9) and repeat steps (3) through (10) until convergence. When that happens, the cell downstream pressure will be the actual wellbore pressure occurring at the end of the first axial increment for the existing flow conditions. 4. Sensibility Factors Analysis In certain engineering conditions, the effects on bottom-hole pressure of different back pressures at the wellhead, gas and liquid injection flow rate are simulated by use of Hasan-Kabir’s (simplified by H-B in following diagrams and tables) method and the modified Ansari’s method. 4.1 The effect of wellhead back pressure (choke pressure) on bottom-hole pressure The data and parameters input for simulationare as follows. This is a vertical well with 1793 m of depth and 0.219 m of inner diameter of casing and 0.107m of outer diameter of drillstring. Nitrogen was injected at the rate of 0.43 m3/s, mud of 1120 kg/ m3 with a plastic viscosity of 24 mpa·s was injected at the rate of 0.009 m3/s. The surface temperature was 285º K and geothermal gradient was 0.017 ºK/m. The choke pressure varies in a range of 0.1-1.5MPa.
4.2 The effect of gas injection flow rate on bottom-hole pressure Input data and parameters were the same as in Section 4.1. The choke pressure was maintained at 1.586 MPa and nitrogen injection rate varied from 0 to 2.0m3/s. Figure 2 shows that the bottomhole pressure decreased slowly with the increasing injection gas flow rate. However in Hasan-Kabir’s method, when gas injection rate was higher than 1.25m3/s, the bottom-hole pressure decreased quickly compared against smooth decrease in modified Ansari’s method. The analysis of flow pattern transition shows that when the gas injection rate is higher than 1.25m3/s in Hasan-Kabir’s method, the total flow patterns in annulus are churn and annular flows which represent sharply lower pressure drop compared against relatively smooth pressure drop of the slug flow in modified Ansari’s method. This shows that HasanKabir’s method had some limitation in practical engineering application.
Fig. 2 Bottom-hole pressure vs. gas injection rate
4.2 The effect of liquid injection flow rate on bottomhole pressure Input data and parameters were the same as in Section 4.1. The choke pressure was maintained at 1.586MPa and injection mud (liquid) flow rate varied from 0.005 to 0.7m3/s. Fig. 1 Bottom-hole pressure vs. choke pressure
As one can see from Fig. 1, the bottom-hole pressure increased with increasing choke pressure. When the choke pressure varied from 0.1 MPa to 1.5 MPa, both increments of bottom-hole pressure were about 2.0 MPa. The reason of this trend happened was that volumetric flow rate, gas void fraction and mixture density of multiphase-flow fluid in annulus controlled by state equation changed with the increasing back pressure at the wellhead, which resulted in the transitions of flow patterns, hydrostatic pressure, friction pressure drop, acceleration pressure drop and total bottom-hole pressure. 408
Fig. 3 Bottom-hole pressure vs. injection liquid flow rate
From Fig. 3 it could be found that the bottomhole pressure increased with increasing injection mud flow rate. Hence one way to enhance bottom-hole
pressure is to increase injection mud flow rate.
6.1 Validation of prediction experiments data
results
with
5. The Methods of Controling Bottom-Hole Pressure The methods to regulate bottom-hole pressure are choke pressure adjustment and injection flow rate regulation according to sensibility factors analysis described above. 5.1
Choke pressure adjustment method
The Choke pressure adjustment method is not only to regulate bottom-hole pressure but also to avoid washing wellhead and surface equipments by high velocity fluids and higher friction pressure drop when fluids returning to wellhead udergo exceedingly large expansion. However, the overtop back pressure will accelerate the gas liquid separation and nonessential gas injection. On the other hand, the choke pressure at the wellhead should be maintained moderate limited by injection equipments and rotation control head. 5.2
Injection flow rate regulation method
There are two reasons to determine injection flow rate regulation method to control bottom-hole pressure is a simple, prompt, cheap means. The first reason is to change gas or liquid flow rate only by simple calculation according to the need of pressure regulation. The second reason is to reduce time consumption to regulate mud density and corresponding manpower, resource and finance consumption. But one thing to care is to limit the flow rate within the rated flow scope of the injection equipment. 6.
Fig. 4 Wellbore pressure vs. well depth of the first experiment
Validation and Anylysis
Hasan-Kabir’s method and modified Ansari’s method were validated with data obtained from two experiments performed in a full-scale well located at Louisian State University. This is a vertical well with 1793 m of depth and 0.219 m of inner diameter of casing and 0.107 m of outer diameter of drill-string. During the first experiment, nitrogen was injected at the rate of 0.53 m3/s, mud at 1120 kg/ m3 with a plastic viscosity of 6 mpa·s was injected at 0.01 m3/s. The surface temperature was 297ºK and geothermal gradient was 0.012 ºK/m. The choke pressure was maintained at 0.972 MPa. Annular wellbore pressure was measured at the positions of 1186m and 1768m. During the second test, nitrogen was injcted at the rate of 0.43 m3/s, mud of 1120 kg/ m3 with a plastic viscosity of 24 mpa·s was injected at 0.009 m3/s. The surface temperature was 285 º K and geothermal gradient was 0.017 ºK/m. The choke pressure was maintained at 1.586 MPa. Annular wellbore pressure was measured at 1768m.
Fig. 5 Wellbore pressure vs. well depth of experiment
the second
6.2 Analysis As one can see from Figs. 4 and 5 that both predicted pressure trends are the same and modified Ansari’s prediction accuracy is better than HasanKabir’s prediction results. For the sake of comparison, the absolute error and relative error are introduced in this paper. e = Calculated－Measured Calculated-Measured
× 100% Measured The absolute and relative errors are presented in Table 1. e%=
Table 1
Accuracy comparison of two models first experiment
M depth(m) 1186 M(MPa) 11.81 H-B C(MPa) 10.78 Ansari C(MPa) 11.48 H-B e (MPa) 1.03 Ansari e(MPa) 0.33 H-B e% 8.7% Ansari e% 2.8% Note: M-measured, C-calculated
1768 17.61 16.23 17.03 1.38 0.58 7.8% 3.6%
second experiment 1768 18.73 17.53 18.06 1.2 0.67 6.4% 3.6%
Table 1 shows that the relative error of Hasan-Kabir’s method is less than 10% and that of modified Ansari’s method is less than 5%. With the 409
increase of depth of the well, both absolute errors (e) and relative errors increase. Although the difference of relative errors is smaller, the difference of absolute errors is larger. When the well depth is 1768 m, the absolute error of Hasan-Kabir’s method is more than 1.0 MPa which is nearly close to the underbalanced pressure difference of 1.0-3.0 MPa. Hence, the maximum absolute error of 1.38 MPa (the absolute error of Hasan-Kabir’s method) may jeopardize the success of UBD operations and lose economic target of UBD. Prediction accuracy of modified Ansari’s method is better and the absolute value error of 0.67 MPa will meet the accuracy need of UBD design. So modified Ansari’s method is recommended in UBD design. The reason of better prediction accuracy of modified Ansari’s method over Hasan-Kabir’s method is the detailed description of slug flow characteristics: fully developed and developing Taylor bubble slug flow conditions. The evaluation results with the modified Ansari’s model show that when both fully developed and developing Taylor bubble slug flow conditions are taken into account, the model predictions are very good (the relative error is less than 5%). On the other hand, when only the simplified fully developed Taylor bubble slug flow model is used, the relative error is near 15%. 7.
Conclusions The study of this paper shows that mechanistic analysis pressure drop models are superior to experimental models in studying gas-liquid two-phase flow pressure drop in UBD. The progress in gas-liquid two-phase flow analysis makes its industrial application possible. The following conclusions can be drawn from the present study. (1) In the study of two-phase steady flow in annulus of UBD operations, convergent solutions can be obtained by Hasan-Kabir’s method and modified Ansari’s method, but there is difference in the accuracy of prediction. (2) The accuracy difference of Hasan-Kabir’s method and modified Ansari’s method shows that accurately describing slug flow parameters is important to improve the pressure prediction accuracy. (3) In order to guarantee the success of UBD operation, modified Ansari’s method is recommended in UBD design. Nomencalture
v =velocity, m/s f g =void fraction, dimensionless d =pipe diameter, m dt =tubing diameter, m 410
d c =casing diameter, m dp / dz =total pressure gradient, Pa/m (dp / dz )c =gas core pressure gradient, Pa/m (dp / dz ) sc =superficial friction pressure gradient in the core, Pa/m φc2 =dimensionless group
g =acceleration due to gravity, m/s2
ρ =density, kg/m3 μ =viscosity, kg/m•s
E=entrainment H=liquid holdup, dimensionless f=fanning friction factor, dimensionless x=mass fraction of gas phase, dimensionless X=Lockhart-Martinelli parameter β=relative bubble length parameter,dimensionless σ=surface tension Subscripts sg=superficial gas sl=superficial liquid ∞T =terminal rise velocity of a single Taylor bubble ∞= terminal bubble-rise velocity in bubble flow m=mixture l=liquid g=gas c=core liquid (in case of annular flow) ls=liquid slug References [1] GUO, B, HARELAND, G., and RAJTAR, J., Computer Simulation Predicts Unfavorable Mud Rate for Aerated Mud Drilling. SPE26892, 1993 [2] SHARMA, Y., KAMP, A., Yonezawa, T., RIVERO, L.M., Kobayashi, A., and Gonzalez, J., Simulating Aerated Drilling. SPE59424, 2000 [3] TIAN, Shifeng, MEDLEY, GEORGE H. and STONE, CHARLES R., Optimizing circulation while drilling underbalanced. World Oil, June 2000，pp48-55 [4] LIU, Gefei, MEDLEY Jr, GEORGE H, Houston, Foam computer model helps in analysis of underbalanced Drilling. Oil and Gas J. ,July 1,1996,pp18-27 [5] BIJLEVELD, A. F., KOPER, M., SAPONJA, J. , Development and Application of an Underbalanced Drilling Simulator. SPE39303,1998 [6] LAGE, A.C.V.M., TIME, R.W., An Expermental and Theoretical Investigation of Upward Two-Phase Flow in Annuli. SPE64525,2000 [7] HASAN, A. R. and KABIR, C. S., Two-phase flow in vertical and inclined annuli. Int. J. Multiphase Flow, Vol. 18, No. 2, 1992,pp279-293 [8] HASAN, A. R., Void fraction in bubbly and slug flow in downward vertical and inclined systems. SPE26522,1995 [9] LAGE, A.C.V.M. and TIME, R.W., Mechanistic Model for Upward Two-Phase Flow in Annuli. SPE63127,2000 [10] PEREZ-TELLEZ, C. Smith, J.R, EDWARDS, J. K., A New comprehensive mechanistic model for
[11] [12]
[13]
[14]
underbalanced drilling improves wellbore pressure predictions. SPE74426,2002 BEGGS, H.D. and BRILL, J.P., A study of two phase flow in inclined pipes. JPT, May 1973,pp607～617 ZHOU Kaiji, Multiphase-flow circulation system analysis in UBD operations[J]. Natural Gas Industry, 1998，18（5）：44～47 CHEN Jialang, CHEN Taoping, WEI Zhaosheng, Gas-Liquid Two-phase Flow of Rod Pump Well. Petroleum Industry Press, 1994，5，pp6-15 HASAN, A. R. and KABIR, C. S., A Study of
[15]
[16]
multiphase flow behavior in vertical wells. SPE15138,1988 ANSARI, A. M., SYLVESTER, N. D., SARICA, C., SHOHAM, O. and BRILL, J. P., A Comprehensive mechanistic model for upward two-phase flow in wellbores. SPE Production & Facilities, May 1994 LUO Shiying, A mechanistic model and simulation for underbalanced drilling. Ph.D Dissertation, Southwest Petroleum Institute, 1999,5 (in Chinese)
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