- Email: [email protected]
Frictional pressure loss of drilling fluids in a fully eccentric annulus
Accepted Manuscript Frictional pressure loss of drilling fluids in a fully eccentric annulus Oney Erge, Ali Karimi Vajargah, Mehmet Evren Ozbayoglu, Eric van Oort PII:
S1875-5100(15)30051-2
DOI:
10.1016/j.jngse.2015.07.030
Reference:
JNGSE 888
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 16 June 2015 Revised Date:
17 July 2015
Accepted Date: 18 July 2015
Please cite this article as: Erge, O., Vajargah, A.K., Ozbayoglu, M.E., van Oort, E., Frictional pressure loss of drilling fluids in a fully eccentric annulus, Journal of Natural Gas Science & Engineering (2015), doi: 10.1016/j.jngse.2015.07.030. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Frictional pressure loss of drilling fluids in a fully eccentric annulus Oney Erge1, Ali Karimi Vajargah2*, Mehmet Evren Ozbayoglu1, and Eric van Oort2
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*Corresponding Author: Ali Karimi Vajargah, Email: [email protected], phone: +1 (918) 924-6677 1: Tulsa University Drilling Research Projects, 2450 E Marshall St., Tulsa OK, 74110. 2: Department of Petroleum and Geo-System Engineering, The University of Texas at Austin, 200 E. Dean Keeton St., Stop C0300, Austin TX, 78712.
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Authors email: Oney Erge: [email protected] Ali Karimi Vajargah: [email protected] Mehmet Evren Ozbayoglu: [email protected] Eric van Oort: [email protected]
1. Abstract
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It is common practice when drilling oil and gas wells to assume that the drillstring is placed concentrically in the annular space with either the open hole or previous casing strings in order to predict annular frictional pressure losses. The assumption of such a concentric annulus is, however, a considerable simplification that may not properly reflect the majority of drilling applications in the field. In fact, with an increasing number of deviated / horizontal and extended reach wells being drilled, a fully eccentric annulus is actually present in a large section of the wellbore. In this study, we apply experimental, analytical, and numerical approaches to investigate the impact of drillpipe eccentricity on the annular pressure loss while circulating non-Newtonian drilling fluids. The length of the experimental section of a flow loop was 27.74 m (91’) and it consisted of 0.0245 m (1”) steel drillpipe and 0.0571 m (2.25”) acrylic casing with the inner diameter of 0.0508 m (2”). Drillpipe was placed at the bottom of the casing, thereby simulating a fully eccentric annulus. Four Yield Power Law (YPL) drilling fluids were tested in this flow loop. Annular pressure loss for a wide range of laminar flow rates was recorded for each fluid. A numerical model based on a finite difference approach was developed to estimate the annular pressure loss. Subsequently, the experimental data was compared with the proposed model and also with several other widely used analytical and numerical approaches previously reported in the literature. The obtained results show that in the laminar flow regime, the annular frictional pressure loss in a fully eccentric annulus is considerably less than a concentric annulus, on occasions by less than 50%. In general, all the applied models under-estimated the effect of eccentricity on pressure loss. However, the novel proposed model showed the least discrepancy with the experimental data. Furthermore, it was found that the difference between the estimated and experimental results increases with increasing fluid yield stress. This suggests that models and/or correlations that are developed to correct for the eccentricity effect for fluids with negligible yield stress (for instance Power Law fluids) are not suitable to estimate the pressure loss for YPL fluids with elevated yield stress.
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2. Introduction
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Non-Newtonian fluids are used extensively throughout the petroleum industry. One of the most important applications occurs during drilling of an oil and gas well when drilling fluid, also referred as “drilling mud” or simply “mud”, circulates through the drillstring, comes out the bit and carries cuttings to the surface through an annular space. Another example involves the use of non-Newtonian cementing fluids to isolate the annular space between casing strings and open hole formations. During fluid circulation (e.g. of mud or cement), the frictional pressure loss in the annular space between the drill string and wellbore/casing needs to be added to the hydrostatic pressure to obtain the precise annular pressure profile. This determines the ECD. During complex well drilling operations, the “mud window,” i.e. the difference between the fracture gradient and the pore pressure (or the mud pressure required to prevent shear failure at the wellbore wall, whichever of the two is higher), tends to be very narrow. Exceeding the boundaries of the mud window usually results in significant well trouble (e.g., well control incidents, lost circulation, borehole instability, stuck pipe, etc.) and associated trouble time and recovery costs (Ameen Rostami et al., 2015; Karimi Vajargah and van Oort, 2015a; Kinik et al., 2014). As a common practice when drilling oil and gas wells, a concentric annulus is assumed while predicting annular frictional losses. However, in deviated/horizontal and extended reach drilling, a fully eccentric annulus is present in a large section of the wellbore. It is known that eccentricity reduces the annular frictional pressure loss and results in a lower ECD than anticipated, which can potentially cause disastrous incidents such as kicks (Karimi Vajargah and van Oort, 2015a). Furthermore, accurate prediction of ECD plays a vital role in successful implementation of some innovative technologies such as the Constant Bottomhole Pressure (CBHP) Technique of Managed Pressure Drilling (Karimi Vajargah et al., 2014a; Kinik et al., 2015, Mammadov et al., 2015). Usually, two or three parameters are used to describe the non-Newtonian rheological behavior of drilling fluids. The three-parameter model proposed by Herschel and Bulkley (1926), which is also known as the Yield Power Law (YPL) model, is widely used through the industry to describe the shear stress behavior as a function of shear rate of drilling fluids and cements. It has been shown that the YPL model (three parameter) fits the data points much better than two parameter models (e.g., Power Law or Bingham Plastic) (Hemphill et al., 1993; Maglione and Ferrario, 1996; Kelessidis et al., 2005). Four and five parameter models have also been introduced as a further refinement in characterizing rheology. However, due to the complexity in calculating the flow parameters such as Reynolds number, velocity profile, and pressure loss these models are not widely accepted (Kelessidis et al., 2006). Many researchers have investigated flow of non-Newtonian fluids in concentric and eccentric annuli: Concentric annuli: Fredrickson and Bird (1958) conducted analytical studies of laminar flow of Bingham-Plastic and Power Law fluids in concentric annuli. No analytical solution exists for laminar or turbulent flow of YPL fluids in concentric annuli. To obtain a solution for laminar flow, a non-linear equation with four unknowns must be integrated numerically to obtain the pressure gradient at a given flow velocity. An analytical integration is not possible due to the yield stress term. A numerical integration and an iterative procedure are necessary to obtain a solution for YPL fluid flow in a concentric annulus (Erge, 2013). To avoid this complexity, a narrow-slot model approximation was proposed (e.g., Tao and Donovan, 1955) in which the annulus is presented by a rectangular slot. This approach is widely 2
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used in the petroleum industry to estimate the pressure loss of non-Newtonian fluids in the annulus. It is proven to be relatively accurate for practical estimation with diameter ratios (e.g., the ratio of diameter of the drillpipe to diameter of the wellbore) greater than 0.2. During regular drilling operations, the diameter ratio typically ranges from 0.4 to 0.6 (Ahmed and Miska, 2009). In some special drilling application such as casing drilling (larger diameter ratios) or coil tubing drilling (smaller diameter ratios), diameter ratios out of the regular range are observed. It should be noted that the Narrow Slot approach disregards the effect of eccentricity. Hanks (1979) was the first to propose an approximate solution for laminar axial flow of YPL fluids in concentric annuli. He presented a numerical and iterative procedure and provided tables and design charts for practical predictions. Fordham et al. (1991) presented a numerical solution and limited experimental data for laminar flow of YPL fluids. Gucuyener and Mehmetoglu (1992) presented a solution for the flow of Yield-Pseudo-Plastic fluids in a concentric annulus. Their solution is based on the Robertson-Stiff (1976) rheological model, which includes Newtonian, Power Law and Bingham Plastic fluids. Eccentric annuli: Piercy (1933) was the first to present an analytical solution for flow of Newtonian fluids through eccentric annuli. Kozicki et al. (1966) developed generalized hydraulic equations for laminar flow of a generalized fluid in ducts of arbitrary shape. Sestak et al. (2001) and Ahmed et al. (2006) evaluated the performance of this model with experimental measurements that were obtained from an eccentric annulus. This model will be referred to in the remainder of this paper as the Pipe Equivalent approach. For more accurate prediction of annular pressure losses, Computational Fluid Dynamics (CFD) approach can be applied (Karimi Vajargah et al., 2014b). Sorgun et al. (2012) evaluated laminar and turbulent axial flow of Newtonian fluids in concentric and fully eccentric annuli. Haciislamoglu (1989) numerically evaluated the laminar flow of non-Newtonian fluids in eccentric annuli. He transformed the equation of motion into a bipolar coordinate system and used a control-volume formulation as the discretization method. Haciislamoglu and Langlinais (1990) presented a correlation for flow of PL fluids in an eccentric annulus based on numerical simulation results:
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= 1 − 0.072 . − 1.5 . √ + 0.96 . √ . !".
(1)
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The proposed correlation is valid for 0 to 0.95 dimensionless eccentricity, diameter ratios of 0.3 to 0.9 and a flow behavior index of 0.4 to 1.0. It relates the pressure in an eccentric annulus to a concentric one. Kelessidis et al. (2011) applied this correlation to obtain the pressure loss of non-Newtonian fluids in a fully eccentric annulus. Ahmed and Miska (2009) proposed that Eq. 1 could be applied to predict the annular laminar pressure loss of YPL fluids by replacing the fluid behavior index, m, with the generalized flow behavior index, N (Eq. 2): = 1 − 0.072 . − 1.5 . √# + 0.96 . √# . # !".
3
(2)
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3. Theory
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Furthermore, several other CFD studies addressed the flow of YPL fluids in eccentric annuli. Luo and Peden (1990) provided a practical approximation for the flow through eccentric annuli. The eccentric annulus was treated by suggesting it is composed of infinite concentric annuli with variable outer radii. This model only considers the slices of a cross section of the annuli and therefore it neglects the circumferential shear stress variations. Hashemian Adariani (2005) developed a numerical model for the laminar flow of YPL fluids through eccentric annuli. A boundary-fitted coordinate system was used and the momentum equations were transformed into a computational domain and solved using finite differencing. His method uses a Cartesian grid network to define the annulus. In a more recent study, Mokhtari et al. (2012) the effect of eccentricity on both the pressure losses and velocity profiles. Several studies also investigated the effect of pipe rotation and drill string deflection on pressure loss (e.g. Escudier and Gouldson, 1995; Hansen and Sterri, 1995; Ahmed and Miska, 2008; Erge et al., 2013; Erge et al. 2014; Escudier et al., 2002; Hemphill, 2015). Note that the focus in this paper is restricted to the eccentricity effect, and that the effects of pipe rotation will be the topic of future work. Many of these prior studies indicated that eccentricity can dramatically reduce the annular pressure loss, and must be taken into account explicitly. Furthermore, the eccentricity effect is more prominent in laminar flow in comparison with turbulent flow (Karimi Vajargah and van Oort, 2015b). So far, most of the studies conducted to investigate the effect of eccentricity on annular pressure loss were not validated with relevant experimental data, particularly for drilling fluids. We remedy this situation in the present paper, which offers both an experimental investigation and a new theoretical model to account for the effect of drillpipe eccentricity on the annular pressure loss experience with YPL fluids in deviated/horizontal and extended reach wells. First, a theoretical model and a solution procedure is proposed. Then, the experimental data is presented and compared with this new theoretical model as well as other most widely used models in the drilling industry.
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To predict the annular pressure losses in an annulus, a model is proposed based on simplifying the Navier-Stokes equations in cylindrical coordinates. Major simplifying assumptions include: • Steady-state, isothermal, fully developed incompressible flow • Gravitational force is neglected • No slip at the wall • The inner pipe is rotating about its own axis and outer pipe is stationary • Axisymmetric flow Fig. 1 illustrates the geometry of the problem. In Appendix A, it is shown that annular pressure loss can be obtained from Eq. 3: $ $ $*+, 1 $*+, = & () ./ + & ) / $% $' $' ' $'
(3)
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No analytical solution exists for this equation and hence an implicit finite difference scheme is applied. After implementing the finite difference approximation and further algebraic simplifications, we obtain (Erge, 2013):
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$ 1 ∆' = 1*)23 + )2 -4+, 23 5 − *)23 + 2)2 +)26 -4+, 2 5 + *)2 + )26 -4+, 26 57 $% 2 +, − +, 26 + )2 ∆' 23 2'
(4)
Non-Newtonian behavior of drilling fluids is accounted for by considering YPL fluid behavior, in which shear stress is related to shear rate by the following equation:
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(5)
where shear rate function including the effect of rotation is given by: $ +? $+,
@ + $' ' $'
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(6)
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The proposed model is valid for Newtonian and/or non-Newtonian fluids (including Bingham-Plastic, PL and YPL) for a concentric annulus with inner pipe rotation and any flow regime (laminar or turbulent). This model is easy to implement, fast and bypasses the complexity of using commercial CFD software without sacrificing accuracy. Furthermore, contrary to the Narrow Slot approximation, it considers drillpipe rotation and can be applied for unusual diameter ratios as well, which makes it suitable for special drilling applications such as coiled tubing. A CFD analysis was conducted by using commercial software in order to validate the proposed model, initially for the concentric annulus, and to allow comparison with other models in the literature. A very fine mesh size was used (0.4 mm) to obtain accurate velocity profiles. The meshing is illustrated in Fig. 2. Inputs for the CFD analysis and fluid properties are given in Table 1. The laminar velocity profile obtained from the proposed model was compared with results from the Narrow Slot approach for flow of an arbitrary YPL fluid through concentric annuli (Fig 3). Very good agreement between the commercial CFD software and the proposed model was observed. In addition, Fig. 3 shows that the Narrow Slot approach presents a reasonable accuracy for this geometry (diameter ratio = 0.5). Erge et al. (2015b) presented a stepwise procedure to obtain the velocity profile for the Narrow Slot approach. To incorporate the effect of eccentricity, the pressure loss value obtained for a concentric annulus is corrected for the eccentric annulus, using an appropriate method. For instance, in this study Eq. (2) is used to incorporate pipe eccentricity. It should be noted that Eq. (1) was initially developed for PL fluids and similar correlation has not yet been reported for YPL fluids. However, as proposed by Ahmed and Miska (2009), a generalized flow behavior index, N, can be used instead of fluid behavior index, m, for YPL fluids (see Appendix B).
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4. Experimental Setup
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Experimental data presented by Erge (2013) used in this study. The data has been collected from the outdoor dynamic testing facility located at the Tulsa University Drilling Research Projects (Fig. 4). The length of the experimental section is 27.74 m (91 ft) and it consists of 0.0245 m (1”) steel pipe and 0.0571 m (2.25”) acrylic casing with the inner diameter of 0.0508 m (2”). Drillpipe lies at the bottom of the casing due to its weight, thereby simulating a fully eccentric annulus. Properties of the experimental set up are shown in Table 2. Fig. 5 shows the schematic of the test facility. The pressure data was obtained by using both point pressure transducers and differential pressure transducers. There are three temperature transducers mounted on the test setup. Temperature was measured at the inlet of the supply line and at the inlet and outlet of the annular test section. The entrance and exit lengths were estimated based on empirical correlations from literature (Knudsen and Katz, 1958; White, 2011). A 454-lpm (20-hp) centrifugal pump was used for fluid circulation. Fluid volume for circulation was handled in a 227 liter (60-gal) reservoir tank. A Coriolis flow meter was installed at the inlet of the supply line, measuring fluid density and flow rate. Four non-Newtonian drilling fluids (labeled YPL-A, YPL-B, YPL-C, YPL-D) were used in this study. A sophisticated rheometer, which is very sensitive at low and high shear rates, was used to form the rheogram (i.e. the plot of shear stress vs. shear rate) for each fluid (Fig. 6). Based on the rheometer readings, the YPL model was used to characterize the rheological profile of these fluids (Fig. 7). Tables 3 and 4 present the composition, density, and rheological parameters of each fluid respectively. The fluid behavior index (m) varies from 0.42 to 0.57, consistency index (K) from 0.21 to 0.95 Pa.sm and yield stress from 2.38 to 14.85 Pa. By designing these test fluids, it has been attempted to cover a wide range of drilling fluid rheological properties routinely used in field practice. Prior to the experiments, calibration tests were conducted with water to verify the pressure loss readings. These test results were compared with the analytical solution for flow of Newtonian fluid (water) in an eccentric annulus (Ahmed and Miska, 2009). Excellent agreement was achieved between the analytical model and the experimental results for a fully eccentric annulus as shown in Fig. 8.
5. Results and discussion
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5.1 Extracting the laminar data points. Annular pressure loss vs. flow rate was measured for a wide range of flow rate; however, in this study, we are only interested in laminar data points. Hence, it is required to distinguish the transition from laminar to turbulent flow. This can be done by plotting wall shear stress (τw) vs. nominal Newtonian shear rate (8v/Dhyd) and careful examination of the trend of the data points for a sharp increase in wall shear stress as nominal Newtonian shear rate increases (Ahmed and Miska, 2009). Fig. 9 shows the wall shear stress vs. nominal Newtonian shear rate for YPL-B. The red line shows the transition from the laminar flow regime to the turbulent regime. A similar approach was applied to distinguish the laminar-turbulent transition for other fluids as well. 5.2 Model comparisons and error analysis. Proposed model in this study is compared with the experimental data (Erge, 2013) as well as other widely used theoretical and numerical models, including: Narrow Slot (Appendix B), corrected Narrow Slot (Appendix B), Luo and Peden (1990), Sestak et al. 6
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(2001), Pipe Equivalent Model (Ahmed et al., 2006) and Hashemian Adariani (2005). Since, the Narrow Slot approximation does not consider the effect of pipe eccentricity, the obtained pressure loss from this approach is corrected by Eq. 2 for a fully eccentric annulus and is denoted as the Corrected Narrow Slot approach (Appendix B).
* -DE. − * -FG. = × 100 * -FG.
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Figs. 10 through 13 show the laminar pressure loss vs. average velocity for four YPL fluids (A through D). These figures compare the obtained pressure loss from the proposed model with the experimental data and other models in the literature. The Narrow Slot approach and Sestak et al. (2001) model are not shown in these figures due to their poor predictions for the fully eccentric annulus and are included in the error analysis only. It should be noted that YPL-A is the least viscous drilling fluid with the lowest yield stress and YPL-D is the thickest fluid with the highest yield stress. Note that when yield stress is negligible or zero, the YPL model is the same as the PL model. Figs. 10 through 13 indicate that all models, including the proposed model in this study, underestimate the effect of pipe eccentricity on annular pressure loss. Values predicted by the Corrected Narrow Slot model are very close to the proposed model. It should be noted that the proposed model and Corrected Narrow Slot approach are initially derived for the situation of a concentric annulus and both use Eq. 2 to correct the pressure loss for a fully eccentric annulus ( = 1). Eq. 1 is developed for the PL fluids and ignores the fluid yield stress, while realistic drilling fluids typically show YPL behavior. Unfortunately, a similar correlation is not available for the YPL model. As a common practice, the fluid behavior index (m) is replaced with generalized flow behavior index, N for YPL fluids as presented in Eq. 2 (Ahmed and Miska, 2009). However, based on the presented experimental data, this approach seems to be ineffective for YPL fluids with high yield stress and a more accurate correlation is required. Table 5 present the Absolute Average Percent Error (AAPE) for the models applied in this study. Relative error in percent, Average Percent Error (APE), and Absolute Average Percent Error (AAPE) are defined according to Eqs. 7, 8 and 9:
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1 IJA'KLA NA'OAPQ R''S' = *T ABC,2 P 2V
(8) "
1 IWXSYQA IJA'KLA NA'OAPQ R''S' = *TZABC,2 ZP
(9)
2V
5.3 Discussion. Table 5 shows that the proposed model had the best performance among all the other models. It predicts the annular pressure loss in a fully eccentric annulus for the low yield stress fluids (YPL-A and YPL-B) with less than 5% error. Higher error (up to 13%) was observed for fluids with high yield stress (YPL-C and YPL-D) that can be due to strong shear thinning ability of these fluids. Also, all the models presented here disregard the possible flow instabilities and only consider a two dimensional space. 7
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According to the obtained results, using the pressure loss values obtained from the Narrow Slot approach for a fully eccentric annulus is not recommended as it may result in more than 100% error. However, results obtained from the Corrected Narrow Slot approach show reasonable accuracy for the presented geometry and in several cases performed even better than the simplified CFD models for eccentric annuli such as the models by Luo and Peden (1990) or Hashemian Adariani (2005). Theoretical models that are developed based on introducing geometric parameters such as the Pipe Equivalent approach (Ahmed et al., 2006) or the model by Sestak et al. (2001) yielded very poor predictions and greatly underestimated the effect of eccentricity on pressure loss in a fully eccentric annulus. The comparative study presented here provides valuable insight towards superior hydraulic planning and real-time equivalent circulating density (ECD) management in deviated, horizontal and extendedreach wells. Its beneficial impact on managing the annular pressure between the fracture and pore pressure in the field to prevent drilling problems such as kicks, mud loss and wellbore instability events is expected to be immediate. Its use will also be key to successful implementation of innovative drilling techniques such as managed pressure drilling and dual gradient drilling, which require highly accurate annular pressure prediction and management.
6. Conclusions
This study aims to better predict annular pressure loss of YPL drilling fluids in a fully eccentric annulus for the laminar flow regime. A new model is proposed and validated with commercial CFD software for a concentric annulus. The model considers the inner pipe rotation and is valid for both laminar and turbulent flow regime for any diameter ratio. The effect of eccentricity is incorporated by applying Eq. 2. The model results as well as those of other most widely used models for annular pressure calculation was compared to experimental data.
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Our investigations show that pipe eccentricity has a profound effect on annular pressure losses while circulating YPL drilling fluids. In some cases, more than 50% reduction in pressure loss was observed for a fully eccentric annulus. Therefore, drillpipe eccentricity must be taken explicitly into account during conducing hydraulic planning for horizontal and extended reach wells, in which a large fully(or near fully) eccentric section is anticipated. The proposed model showed the best performance among all the applied models in this study. It predicted the annular pressure loss of low yield stress drilling fluids with less than 5% error. For fluids with high yield stress, higher error was observed (<15%). This indicates using a generalized flow behavior index (N) instead of fluid behavior index (m) in Eq. 2 as proposed by Ahmed and Miska (2009) will not generate accurate results for high yield stress YPL fluids. A more comprehensive approach is still required.
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Using Narrow Slot approach, as is extensively done in the petroleum industry, for predicting annular pressure loss in a fully eccentric annulus is not recommended, since it yields large errors. Theoretical models that are based on introducing geometric parameters, such as Pipe Equivalent, generally delivered a poor result by greatly under-estimating the eccentricity effect. However, the Corrected Narrow Slot approach showed acceptable accuracy with comparison to other more sophisticated and
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computationally expensive CFD models (Luo and Peden, 1990; Hashemian Adariani, 2005) for the presented geometry.
7. Acknowledgments
Nomenclature geometric parameters, m diameter, m
m: N:
dP/dl:
frictional pressure loss gradient, Pa/m
Q:
dP
frictional pressure loss, Pa
r, R:
E:
offset distance, m
Re:
f:
friction factor
V:
h:
height of the slot, m
K:
consistency index, Pa ∙ s b
w:
Greek Letters shear stress, Pa τ: shear rate, 1/s ;:
density, kg/m viscosity, Pa ∙ s diameter ratio angular speed, rad./s velocity, m /s geometrical constant, m
radius, m
Reynolds number mean axial fluid velocity, m /s width of the slot, m
b:
bulk
d:
dimensionless inner hydraulic outer wall yield Yield Power Law
i: h,H: o: w: y: YPL:
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dimensionless eccentricity
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ρ: µ, η: κ: ω: υ: λ:
flow rate, m3 /X
Subscripts apparent app:
Table 1. Input Parameters for CFD (Erge, 2013)
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ε:
flow behavior index generalized flow behavior index
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Authors would like to thank Tulsa University Drilling Research Projects, Det norske oljeselskap ASA and Baker Hughes for their support and encouragement through this study. In addition, ANSYS is thanked for providing the commercial CFD software.
Geometry OD (mm)
50.8
ID (mm)
25.4
Length (m)
2.5 Fluid Herschel–
Model
Bulkley
Yield Stress (Pa) Consistency Index
(ef. g h )
Flow Behavior Index 3
Density (kg/m )
9
5 1 0.5 1000
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Velocity (m/s)
1
Incompressible and Isothermal Mesh and Solution Criteria 0.4
Max Face size (mm)
0.4
Max Size (mm)
0.4
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Min size (mm)
Side Face Meshing (mm)
3
RMS
1e
Nodes
-5
8,592,484
Table 2. Specifications of the test facility (Erge, 2015a) Properties of the Experimental Setup
Properties of Inner Pipe
31.70 m
Length of the test section
27.98 m
Length of the supply line
25.60 m
Outside diameter
25.4 mm
Acrylic
Thickness
0.889 mm
27.74 m
Material type
304L stainless steel
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Material type of the test pipe
Length
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Total length of the facility
Outside diameter of the test pipe
57.15 mm
Weight in the air
0.423 kg/m
Thickness of the test pipe
3.175 mm
Modulus of elasticity
2x10 MPa
227 liter
Moment of inertia
5.15x10 m
Volume of the reservoir tank Flow rate range
5
-8
4
0-454 lpm
RPM range
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0-120 c/min
Table 3. Composition and density of the test fluids (it should be noted 1 gr/350 cc = 1 lb/bbl) (Erge, 2013) Composition
YPL-A YPL-B YPL-C
Laponite (gr/350 cc)
0.2
5.33
1002.9
0.11
6.54
1005.5
---
9
1010
0.30
8
1008.6
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YPL-D
3
Density (kg/m )
PAC R (gr/350 cc )
EP
Test Fluid
Table 4. Rheological parameters of the test fluids Rheological Parameters m
K (Pa.S )
m
τy (Pa)
0.2145
0.5748
2.383
0.6585
0.4265
4.581
0.9462
0.4254
7.602
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0.4999
14.85
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0.769
Table 5. Error analysis for different models (AAPE is presented) YPL-B
Proposed model
4.10
Narrow Slot Corrected Narrow Slot by Eq. 2 Pipe Equivalent (Ahmed et al., 2006) Sestak et al. (2001) Hashemian Adariani (2005)
YPL-D
4.47
12.32
12.84
84.10
83.61
98.18
99.18
6.93
5.61
13.64
14.07
10.11
19.51
29.38
30.66
30.48
42.67
54.31
56.71
7.56
13.17
22.69
16.26
12.38
10.32
19.18
19.90
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Luo and Peden (1990)
YPL-C
SC
YPL-A
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Models (Error in %)
Fig. 1. Geometry of the problem (Erge, 2015b)
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Fig. 2. Mesh and geometrical representation of the CFD analysis (Erge, 2015b)
Fig. 3. Comparison the velocity profile from the proposed model with CFD analysis and the Narrow Slot approximation (concentric annulus, laminar flow regime) (Erge, 2013)
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Fig. 4. Outside dynamic testing facility at The University of Tulsa (Erge, 2013)
Fig. 5. Schematic for the outside dynamic testing facility (Erge, 2013)
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Fig. 6. Anton Paar (Physica MCR 301) rheometer is used to characterize the fluids (Erge, 2013)
Fig. 7. Rheometer readings and fitted YPL model to the data pints for test fluids
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9000 Theoretical Model 8000
Experimental Data
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Pressure loss (Pa/m)
7000 6000 5000 4000 3000
1000 0 0
1
2
3
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2000
4
5
6
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Velocity (m/s)
Fig. 8. Calibration test with water (annular pressure loss vs. velocity, fully eccentric annulus) (Erge, 2013)
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40
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Wall Shear Stress (Pa)
50
30
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20
Laminar
10
0 0
200
400
600
800
1000
1200
1400
1600
8v/Dhyd
Fig. 9. Plot of wall shear stress vs. nominal Newtonian shear rate to determine the transition from laminar to turbulent flow for YPL-B
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1050
Experimenta Data Pipe Equivalent Model Corrected Narrow Slot Model Proposed Model Luo and Peden Model Hashemian Model
1000
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Pressure loss (Pa/m)
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Fig. 10. Model comparisons with the experimental data for YPL-A
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Fig. 11. Model comparisons with the experimental data for YPL-B
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Experimental Data Pipe Equivalent Model Proposed Model Corrected Narrow Slot Model Luo and Peden Model Hashemian Model
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Fig. 13. Model comparisons with the experimental data for YPL-D
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Appendix A: Derivation of the proposed model In this section, derivation of the proposed model is presented. Continuity equation in the cylindrical coordinates is: $i 1 $*i'+B - 1 $*i+? - $*i+, + + + .=0 $Q ' $' ' $j
%$(A.1)
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(
Now, consider the Navier-Stokes (N-S) equations in cylindrical coordinates: In r- direction:
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$+B $+B +? $+B $+B +? + +B + + +, − . $Q $' ' $j $% '
k k lllllllll $ 1$ 1 $ $ 8?? i $*'+ B +B −> *'8BB - + 8?B + 8,B − @ + iLB − $' ' $' ' $j $% ' ' $' k k k k k k llllllllll llllllllll lllllllllll *+? +? i $*+B +? $*+B +, − −i +i ' $j $% '
=−
In θ- direction:
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$+? $+? +? $+? $+? +B +? i + +B + + +, + $Q $' ' $j $% ' 1 $ 1 $ 1 $ $ 8?B − 8B? =− − > *' 8B? - + 8?? + 8,? − @ + iL? ' $j ' $' ' $j $% ' k k k k k k lllllllllll llllllllll llllllllll llllllllll *+? k +B k i $*+ $*+ $*+ ? +? ? +B ? +, − −i −i − 2i ' $j $' $% '
In z- direction:
(A.2)
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i(
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$+, $+, +? $+, $+, i + +B + + +, $Q $' ' $j $% k k llllllllll $ 1$ 1 $ $ $*+ , +, =− −> *'8B, - + 8?, + 8,, @ + iL, − i $% ' $' ' $j $% $% k+ kk+ klllllllllll llllllllll i $*'+ i $*+ B , ? , − − ' $' '
%$(A.3)
(A.4)
where the stress tensors for cylindrical coordinates are defined as: 8BB = −) >2
$+B 2 @ + ) − *∇ ∙ n$' 3
(A.5)
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8,? = 8?, = −) > 8,B = 8B, = −) >
$ +? 1 $+B + @
$' ' ' $j
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8B? = 8?B = −) >'
(A.7)
1 $+, $+? + @ ' $j
%$$+B $+, + @ $% $'
1 $ 1 $+? $+, *'+B - + + ' $' ' $j
%$and the dissipation function:
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(A.8) (A.9) (A.10)
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8,, = −) >2
(A.6)
$+B 1 $+? +B $+, $ +? 1 $+B 1 $+, $+? Φp = 2 & + + + / + >'
+ @ +> + @ $' ' $j ' $% $' ' ' $j ' $j
%$(A.12)
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$+B $+, 2 1 $ 1 $+? $+, *'+B - + +> + @ − > + @ $% $' 3 ' $' ' $j
%$(A.11)
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The N-S equations presented here are valid for all fluids (Newtonian or non-Newtonian) and in any flow regime (laminar or turbulent). After applying the simplifying assumptions, presented in section 2, the following equation is obtained: k k lllllllllll $ 1$ $*+, i $*'+ B +, =& ()' ./ − $% ' $' $' ' $'
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(A.13)
Using the mixing length concept gives: $+, $+, llllllllllll *i+B k +, k - = i q q $' $'
(A.14)
$+, k k llllllllllll −*i+ q B +, - = )E q $'
(A.15)
$ 1$ $+, 1$ $+, => )' @ + > )E ' q q@ $% ' $' $' ' $' $'
(A.16)
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(A.17)
Instead of the viscosity term, an effective viscosity is introduced:
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) = *) + )E -
Therefore, the viscosity term is replaced with the Eq. A.18:
(3)
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$ $ $*+, 1 $*+, = & () ./ + & ) / $% $' $' ' $'
(A.18)
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− +, 26 $ 1 $*+, 1 $*+, 1 +, ∆' = v *)23 + )2 - ( − *)2 + )26 - ( . . x + ) 23 $% 2 $' 23w 2 $' 26w ' 2
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+, − +, 2 +, − +, 26 +, − +, 26 $ 1 ∆' = >*)23 + )2 - 23 − *)2 + )26 - 2 @ + ) 23 $% 2 ∆' ∆' 2'
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$ 1 ∆' = 1*)23 + )2 -4+, 23 5 − *)23 + 2)2 +)26 -4+, 2 5 + *)2 + )26 -4+, 26 57 $% 2 +, − +, 26 + )2 ∆' 23 2'
(A.19)
(A.20)
(A.21)
(A.22)
In order to incorporate the non-Newtonian behavior of drilling fluids, an apparent viscosity term is defined as: )yGG =
89 + :; =6 ;
(A.23)
where shear rate function including the effect of rotation given as: $ +? $+, ;< = >' @ + $' ' $'
(A.24)
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Appendix B: Laminar pressure loss of YPL fluids by using Narrow Slot approach z{
Eq. 4 represents the shear stress (8)-shear rate (zB = ;< ) relationship for YPL fluids.
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8 = 89 + :*;< -=
(4)
It is assumed that flow is single-phase, axial, isothermal, and incompressible with constant rheological properties. The following stepwise procedure is adopted from Merlo et al. (1995) to calculate the annular pressure loss for the laminar flow regime:
1 + 2 12v 3 ! − 2
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1. Assume |y= 1 (Power Law fluid) and calculate shear rate at the wall, γ< ~ .
(B.1)
2. Then, calculate shear stress at the wall, 8~ .
8~ = 89 + :*γ< ~ -=
3. Knowing that = , Compute |y .
|y = 1 −
− 1+ 1+
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(B.3)
4. Now, calculate the new value for shear stress at the wall, 8~ .
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(B.4)
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5. Calculate the difference between 8~ and 8~ . If tolerance defined as
6
× 100 is less than a
certain value (for instance 1%), accept 8~ ; Otherwise, use 8~ and calculate the new value for , |y , 8~ until convergence is obtained. By knowing the shear stress at the wall for laminar flow, annular pressure loss can be obtained from: 4 8~ = ! − 2
(B.5)
In order to incorporate the pipe eccentricity (corrected Narrow Slot approximation in this study), calculate the generalized flow behavior index (N): 21
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N=
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(B.6)
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Then, replace N in Eq. 2 to correct the pressure loss obtained from Eq. B.5 for eccentricity (Ahmed and Miska, 2009). It should be noted that in PL fluids, 89 = = 0, |y = 1 , and N = m.
References
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Ahmed, R. and Miska, S. 2008. Experimental Study and Modeling of Yield Power-Law Fluid Flow in Annuli with Drillpipe Rotation. Paper SPE 112604 presented at the SPE/IADC Drilling Conference, Orlando, Florida, 4-6 March. Ahmed, R., Miska, S.Z., Miska, W.Z., 2006. Friction Pressure Loss Determination of Yield Power Law Fluid in Eccentric Annular Laminar Flow. Wiertnictwo Nafta Gaz 23 (1): 47-53. Ahmed, R., Miska, S.Z. Advanced wellbore hydraulics, chapter 4.1, (pp. 191-219). Advanced drilling and well technology. USA Society of Petroleum Engineers. Ed. Bernt S. Aadnoy. SPE, 2009. Ameen Rostami, S., Kinik, K., Gumus, G. 2015. Dynamic Calibration of the Empirical Pore Pressure Estimation Methods Using MPD Data. OTC-25953-MS, OTC 2015, Houston Texas. Erge, O. 2013. Effect of Free Drillstring Rotation on Frictional Pressure Losses. University of Tulsa, M.S. Thesis. Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2013. Effect of Drillstring Deflection and Rotary Speed on Annular Frictional Pressure Losses. Paper ASME 10448 presented at the 32nd International Conference on Ocean Offshore and Arctic Engineering, Nantes, France, 9-14 June; doi:10.1115/OMAE2013-10448. Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2014. Effect of Drillstring Deflection and Rotary Speed on Annular Frictional Pressure Losses. Journal of Energy Resources Technology 136 (4), 042909-042909-10; doi: 10.1115/1.4027565. Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2014. The Effects of Drillstring Eccentricity, Rotation and Buckling Configurations on Annular Frictional Pressure Losses While Circulating Yield Power Law Fluids. Paper SPE 167950 presented at the IADC/SPE Drilling Conference & Exhibition, 4-6 March, Fort Worth, Texas, USA; doi:10.2118/167950-MS Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2015. Laminar To Turbulent Transition of Yield Power Law Fluids in Annuli. Journal of Petroleum Science and Engineering 128, 128-139; doi: 10.1016/j.petrol.2015.02.007 Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2015. CFD Analysis and Model Comparison of Annular Frictional Pressure Losses While Circulating Yield Power Law Fluids. Paper SPE 173840 presented at the SPE Bergen One Day Seminar, 22 April, Bergen, Norway; doi:10.2118/173840-MS Escudier, M.P., Oliveira, P. J., Pinho, F. T. 2002. Fully Developed Laminar Flow of Purely Viscous Non-Newtonian Liquids Through Annuli, Including the Effects of Eccentricity And Inner-Cylinder Rotation. International Journal of Heat and Fluid Flow 23: 52-73. Escudier, M.P. and Gouldson, I.W. 1995. Concentric Annular Flow with Centerbody Rotation of a Newtonian and a Shear Thinning Liquid. International Journal of Heat and Fluid Flow 16 (3): 156-162. Founargiotakis, K., Kelessidis, V. C., Maglione, R. 2008. Laminar, Transitional and Turbulent Flow of HerschelBulkley Fluids in Concentric Annulus. Canadian Journal of Chemical Engineering 86: 676-683. Fredrickson, A. G. and R. B. Bird. 1958. Flow of Non-Newtonian Fluids in Annuli. Ind. Eng. Chem. 50, 347–352. Gucuyener, H.I., Mehmetoglu, T., 1992. Flow of Yield-Pseudoplastic Fluids through a Concentric Annulus. AIChE J. 38, 1139. Haciislamoglu, M. 1989. Non-Newtonian Fluid Flow in Eccentric Annuli and Its Application to Petroleum Engineering Problems. Louisiana State University, Ph.D. dissertation. Haciislamoglu, M. and Langlinais, J. 1990. Non-Newtonian Flow in Eccentric Annuli. J. Energ. Resour. 112: 163169. doi:10.1115/1.2905753. Hanks, R.W. 1979. The Axial Laminar Flow of Yield-Pseudoplastic Fluids in a Concentric Annulus. Ind. Eng. Chem. Process Des. Dev. 18: 488-493. doi: 10.1021/i260071a024.Hansen, S. A. and Sterri, N. 1995. Drill Pipe
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Rotation Effects on Frictional Pressure Losses in Slim Annuli. Paper SPE 30488 presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 22-25 October. Hashemian Adariani, Y. 2005. Numerical Simulation of Laminar Flow of Non-Newtonian Fluids in Eccentric Annuli. University of Tulsa, M.S. Thesis. Hemphill, T., Campos W., Pilehvari A. 1993. Yield-Power Law Model More Accurately Predicts Mud Rheology. Oil & Gas Journal 45. Hemphill, T. (2015, March 17). Advances in the Calculation of Circulating Pressure Drop with and without Drillpipe Rotation. Society of Petroleum Engineers. doi:10.2118/173054-MS Herschel, W.H., Bulkley, R., 1926. Konsistenzmessungen von Gummi-Benzollosungen. Kolloid-Z. 39, 291–300. Karimi Vajargah, A., Hoxha, B. B., van Oort, E. 2014. Automated Well Control Decision-Making during Managed Pressure Drilling Operations. Paper SPE 170324 presented at the SPE Deepwater Drilling and Completions Conference, 10-11 September; doi:10.2118/170324-MS Karimi Vajargah, A., Najafi Fard, F., Parsi, M., Buranaj Hoxha, B. 2014. Investigating the Impact of the “Tool Joint Effect” on Equivalent Circulating Density in Deep-Water Wells. Paper SPE 170294 presented at the SPE Deepwater Drilling and Completions Conference, 10-11 September, Galveston, Texas, USA. doi:10.2118/170294-MS Karimi Vajargah, A. and van. Oort, E. 2015. Automated Drilling Fluid Rheology Characterization with Downhole Pressure Sensor Data. Paper SPE 173085 presented at the SPE/IADC Drilling Conference and Exhibition, 17-19 March, London, England, UK; doi:10.2118/173085-MS. Karimi Vajargah, A. and van Oort, E. 2015. Determination of drilling fluid rheology under downhole conditions by using real-time distributed pressure data. Journal of Natural Gas Science and Engineering, 24, 400-411. Kinik, K., Wojtanowicz, A. K. and Gumus, F. 2014. Temperature-Induced Uncertainty of the Effective Fracture Pressures: Assessment and Control. Paper SPE 170316 presented at the SPE Deepwater Drilling and Completions Conference, Galveston, Texas, USA, 10-11 September. doi: 10.2118/170316-MS. Kinik, K., Gumus, F., & Osayande, N. (2015, April 1). Automated Dynamic Well Control With Managed-Pressure Drilling: A Case Study and Simulation Analysis. Society of Petroleum Engineers. doi:10.2118/168948-PA Kelessidis, V. C., A. Mihalakis and C. Tsamantaki. 2005. Rheology and Rheological Parameter Determination of Bentonite–Water and Bentonite–Lignite–Water Mixtures at Low and High Temperatures. Proceedings of the 7th World Congress of Chem. Engr., Glasgow. Kelessidis, V. C., R. Maglione, C. Tsamantaki and Y. Aspirtakis. 2006. Optimal Determination of Rheological Parameters for Herschel-Bulkley Drilling Fluids and Impact on Pressure Drop, Velocity Profiles and Penetration Rates during Drilling. J. Petrol. Sci. Eng. 53, 203–224 (2006). Kelessidis, Vassilios C., Panagiotis Dalamarinis, and Roberto Maglione. 2011. Experimental Study and Predictions of Pressure Losses of Fluids Modeled as Herschel–Bulkley in Concentric and Eccentric Annuli in Laminar, Transitional and Turbulent flows. Journal of Petroleum Science and Engineering 77 (3): 305-312. Knudsen, J.G., Katz, D.V. 1958. Fluid Dynamics and Heat Transfer. McGraw-Hill Series in Chemical Engineering. Kozicki, W., Chou, C. H. and Tiu, C. 1966. Non-Newtonian Flow in Ducts of Arbitrary Cross-Sectional Shape. Chemical Engineering Science 21: 665-679. Luo Y. and Peden, J.M. 1990. Flow of Drilling Fluids through Eccentric Annuli. SPE Production Engineering Journal 5 (1): 91-96. Maglione, R. and G. Ferrario. 1966. Equations Determine Flow States for Yield-Pseudoplastic Drilling Fluids. Oil Gas J. 94, 63–66. Mammadov, E., Kinik, K., Ameen Rostami, S., & Sephton, S. (2015, April 27). Case Study of Managed Pressure Tripping Operation through Abnormal Formations in West Canadian Sedimentary Basin. Society of Petroleum Engineers. doi:10.2118/174073-MS Mokhtari, M., Ermila, M., & Tutuncu, A. N. (2012, January 1). Accurate Bottomhole Pressure For Fracture Gradient Prediction And Drilling Fluid Pressure Program - Part I. American Rock Mechanics Association. Merlo, A., Maglione, R. and Piatti, C. 1995. An Innovative Model for Drilling Fluid Hydraulics. Paper SPE 29259 presented at SPE Asia Pacific Oil and Gas Conference, Kuala Lumpur, Malaysia, 20-22 March. doi: 10.2118/29259-MS. Piercy, N.A.V., Hooper, M.S. and Winny, H.F. 1933. Viscous Flow through Pipes with Core. London Edinburgh Dublin Phil. Mag. J. Sci. 15: 647-676. Pilehvari, A. and Serth, R. 2009. Generalized Hydraulic Calculation Method for Axial Flow of Non-Newtonian Fluids in Eccentric Annuli. SPE Drilling & Completion 24 (4): 553-563.
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Robertson, R.E., Stiff Jr., H.A., 1976. An Improved Mathematical Model for Relating Shear Stress to Shear Rate in Drilling Fluids and Cement Slurries. SPE J. 16, 31–36. Sestak, J., Zitny, R., Ondrusova J. and Filip, V. 2001. Axial Flow of Purely Viscous Fluids In Eccentric Annuli: Geometric Parameters for Most Frequently Used Approximate Procedures. 3rd Pacific Rim Conference on Rheology, Canadian Group of Rheology, Montreal. Sorgun M., Ozbayoglu M. and Aydin I. 2010. Modeling and Experimental Study of Newtonian Fluid Flow in Annulus. J. Energy Resour. Technol. 132 (3): 033102-033102-6. doi: 10.1115/1.4002243. Tao, L. N., and W. F. Donovan. 1955. Through-flow In Concentric and Eccentric Annuli of Fine Clearance with and without Relative Motion of The Boundaries. Trans. ASME 77.8: 1291-1301. White, F.M., 2011. Fluid Mechanics. 7th Edition, McGraw-Hill Series in Mechanical Engineering.
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Highlights We investigated annular pressure losses experimentally, analytically and numerically
•
A novel numerical model for Yield Power Law drilling fluids is presented
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The presented model is compared with the experimental data and published models
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The presented numerical model better predicts the annular pressure losses
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Pressure losses reduced significantly with increasing eccentricity
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S1875-5100(15)30051-2
DOI:
10.1016/j.jngse.2015.07.030
Reference:
JNGSE 888
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 16 June 2015 Revised Date:
17 July 2015
Accepted Date: 18 July 2015
Please cite this article as: Erge, O., Vajargah, A.K., Ozbayoglu, M.E., van Oort, E., Frictional pressure loss of drilling fluids in a fully eccentric annulus, Journal of Natural Gas Science & Engineering (2015), doi: 10.1016/j.jngse.2015.07.030. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Frictional pressure loss of drilling fluids in a fully eccentric annulus Oney Erge1, Ali Karimi Vajargah2*, Mehmet Evren Ozbayoglu1, and Eric van Oort2
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*Corresponding Author: Ali Karimi Vajargah, Email: [email protected], phone: +1 (918) 924-6677 1: Tulsa University Drilling Research Projects, 2450 E Marshall St., Tulsa OK, 74110. 2: Department of Petroleum and Geo-System Engineering, The University of Texas at Austin, 200 E. Dean Keeton St., Stop C0300, Austin TX, 78712.
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Authors email: Oney Erge: [email protected] Ali Karimi Vajargah: [email protected] Mehmet Evren Ozbayoglu: [email protected] Eric van Oort: [email protected]
1. Abstract
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It is common practice when drilling oil and gas wells to assume that the drillstring is placed concentrically in the annular space with either the open hole or previous casing strings in order to predict annular frictional pressure losses. The assumption of such a concentric annulus is, however, a considerable simplification that may not properly reflect the majority of drilling applications in the field. In fact, with an increasing number of deviated / horizontal and extended reach wells being drilled, a fully eccentric annulus is actually present in a large section of the wellbore. In this study, we apply experimental, analytical, and numerical approaches to investigate the impact of drillpipe eccentricity on the annular pressure loss while circulating non-Newtonian drilling fluids. The length of the experimental section of a flow loop was 27.74 m (91’) and it consisted of 0.0245 m (1”) steel drillpipe and 0.0571 m (2.25”) acrylic casing with the inner diameter of 0.0508 m (2”). Drillpipe was placed at the bottom of the casing, thereby simulating a fully eccentric annulus. Four Yield Power Law (YPL) drilling fluids were tested in this flow loop. Annular pressure loss for a wide range of laminar flow rates was recorded for each fluid. A numerical model based on a finite difference approach was developed to estimate the annular pressure loss. Subsequently, the experimental data was compared with the proposed model and also with several other widely used analytical and numerical approaches previously reported in the literature. The obtained results show that in the laminar flow regime, the annular frictional pressure loss in a fully eccentric annulus is considerably less than a concentric annulus, on occasions by less than 50%. In general, all the applied models under-estimated the effect of eccentricity on pressure loss. However, the novel proposed model showed the least discrepancy with the experimental data. Furthermore, it was found that the difference between the estimated and experimental results increases with increasing fluid yield stress. This suggests that models and/or correlations that are developed to correct for the eccentricity effect for fluids with negligible yield stress (for instance Power Law fluids) are not suitable to estimate the pressure loss for YPL fluids with elevated yield stress.
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2. Introduction
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Non-Newtonian fluids are used extensively throughout the petroleum industry. One of the most important applications occurs during drilling of an oil and gas well when drilling fluid, also referred as “drilling mud” or simply “mud”, circulates through the drillstring, comes out the bit and carries cuttings to the surface through an annular space. Another example involves the use of non-Newtonian cementing fluids to isolate the annular space between casing strings and open hole formations. During fluid circulation (e.g. of mud or cement), the frictional pressure loss in the annular space between the drill string and wellbore/casing needs to be added to the hydrostatic pressure to obtain the precise annular pressure profile. This determines the ECD. During complex well drilling operations, the “mud window,” i.e. the difference between the fracture gradient and the pore pressure (or the mud pressure required to prevent shear failure at the wellbore wall, whichever of the two is higher), tends to be very narrow. Exceeding the boundaries of the mud window usually results in significant well trouble (e.g., well control incidents, lost circulation, borehole instability, stuck pipe, etc.) and associated trouble time and recovery costs (Ameen Rostami et al., 2015; Karimi Vajargah and van Oort, 2015a; Kinik et al., 2014). As a common practice when drilling oil and gas wells, a concentric annulus is assumed while predicting annular frictional losses. However, in deviated/horizontal and extended reach drilling, a fully eccentric annulus is present in a large section of the wellbore. It is known that eccentricity reduces the annular frictional pressure loss and results in a lower ECD than anticipated, which can potentially cause disastrous incidents such as kicks (Karimi Vajargah and van Oort, 2015a). Furthermore, accurate prediction of ECD plays a vital role in successful implementation of some innovative technologies such as the Constant Bottomhole Pressure (CBHP) Technique of Managed Pressure Drilling (Karimi Vajargah et al., 2014a; Kinik et al., 2015, Mammadov et al., 2015). Usually, two or three parameters are used to describe the non-Newtonian rheological behavior of drilling fluids. The three-parameter model proposed by Herschel and Bulkley (1926), which is also known as the Yield Power Law (YPL) model, is widely used through the industry to describe the shear stress behavior as a function of shear rate of drilling fluids and cements. It has been shown that the YPL model (three parameter) fits the data points much better than two parameter models (e.g., Power Law or Bingham Plastic) (Hemphill et al., 1993; Maglione and Ferrario, 1996; Kelessidis et al., 2005). Four and five parameter models have also been introduced as a further refinement in characterizing rheology. However, due to the complexity in calculating the flow parameters such as Reynolds number, velocity profile, and pressure loss these models are not widely accepted (Kelessidis et al., 2006). Many researchers have investigated flow of non-Newtonian fluids in concentric and eccentric annuli: Concentric annuli: Fredrickson and Bird (1958) conducted analytical studies of laminar flow of Bingham-Plastic and Power Law fluids in concentric annuli. No analytical solution exists for laminar or turbulent flow of YPL fluids in concentric annuli. To obtain a solution for laminar flow, a non-linear equation with four unknowns must be integrated numerically to obtain the pressure gradient at a given flow velocity. An analytical integration is not possible due to the yield stress term. A numerical integration and an iterative procedure are necessary to obtain a solution for YPL fluid flow in a concentric annulus (Erge, 2013). To avoid this complexity, a narrow-slot model approximation was proposed (e.g., Tao and Donovan, 1955) in which the annulus is presented by a rectangular slot. This approach is widely 2
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used in the petroleum industry to estimate the pressure loss of non-Newtonian fluids in the annulus. It is proven to be relatively accurate for practical estimation with diameter ratios (e.g., the ratio of diameter of the drillpipe to diameter of the wellbore) greater than 0.2. During regular drilling operations, the diameter ratio typically ranges from 0.4 to 0.6 (Ahmed and Miska, 2009). In some special drilling application such as casing drilling (larger diameter ratios) or coil tubing drilling (smaller diameter ratios), diameter ratios out of the regular range are observed. It should be noted that the Narrow Slot approach disregards the effect of eccentricity. Hanks (1979) was the first to propose an approximate solution for laminar axial flow of YPL fluids in concentric annuli. He presented a numerical and iterative procedure and provided tables and design charts for practical predictions. Fordham et al. (1991) presented a numerical solution and limited experimental data for laminar flow of YPL fluids. Gucuyener and Mehmetoglu (1992) presented a solution for the flow of Yield-Pseudo-Plastic fluids in a concentric annulus. Their solution is based on the Robertson-Stiff (1976) rheological model, which includes Newtonian, Power Law and Bingham Plastic fluids. Eccentric annuli: Piercy (1933) was the first to present an analytical solution for flow of Newtonian fluids through eccentric annuli. Kozicki et al. (1966) developed generalized hydraulic equations for laminar flow of a generalized fluid in ducts of arbitrary shape. Sestak et al. (2001) and Ahmed et al. (2006) evaluated the performance of this model with experimental measurements that were obtained from an eccentric annulus. This model will be referred to in the remainder of this paper as the Pipe Equivalent approach. For more accurate prediction of annular pressure losses, Computational Fluid Dynamics (CFD) approach can be applied (Karimi Vajargah et al., 2014b). Sorgun et al. (2012) evaluated laminar and turbulent axial flow of Newtonian fluids in concentric and fully eccentric annuli. Haciislamoglu (1989) numerically evaluated the laminar flow of non-Newtonian fluids in eccentric annuli. He transformed the equation of motion into a bipolar coordinate system and used a control-volume formulation as the discretization method. Haciislamoglu and Langlinais (1990) presented a correlation for flow of PL fluids in an eccentric annulus based on numerical simulation results:
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= 1 − 0.072 . − 1.5 . √ + 0.96 . √ . !".
(1)
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The proposed correlation is valid for 0 to 0.95 dimensionless eccentricity, diameter ratios of 0.3 to 0.9 and a flow behavior index of 0.4 to 1.0. It relates the pressure in an eccentric annulus to a concentric one. Kelessidis et al. (2011) applied this correlation to obtain the pressure loss of non-Newtonian fluids in a fully eccentric annulus. Ahmed and Miska (2009) proposed that Eq. 1 could be applied to predict the annular laminar pressure loss of YPL fluids by replacing the fluid behavior index, m, with the generalized flow behavior index, N (Eq. 2): = 1 − 0.072 . − 1.5 . √# + 0.96 . √# . # !".
3
(2)
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3. Theory
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Furthermore, several other CFD studies addressed the flow of YPL fluids in eccentric annuli. Luo and Peden (1990) provided a practical approximation for the flow through eccentric annuli. The eccentric annulus was treated by suggesting it is composed of infinite concentric annuli with variable outer radii. This model only considers the slices of a cross section of the annuli and therefore it neglects the circumferential shear stress variations. Hashemian Adariani (2005) developed a numerical model for the laminar flow of YPL fluids through eccentric annuli. A boundary-fitted coordinate system was used and the momentum equations were transformed into a computational domain and solved using finite differencing. His method uses a Cartesian grid network to define the annulus. In a more recent study, Mokhtari et al. (2012) the effect of eccentricity on both the pressure losses and velocity profiles. Several studies also investigated the effect of pipe rotation and drill string deflection on pressure loss (e.g. Escudier and Gouldson, 1995; Hansen and Sterri, 1995; Ahmed and Miska, 2008; Erge et al., 2013; Erge et al. 2014; Escudier et al., 2002; Hemphill, 2015). Note that the focus in this paper is restricted to the eccentricity effect, and that the effects of pipe rotation will be the topic of future work. Many of these prior studies indicated that eccentricity can dramatically reduce the annular pressure loss, and must be taken into account explicitly. Furthermore, the eccentricity effect is more prominent in laminar flow in comparison with turbulent flow (Karimi Vajargah and van Oort, 2015b). So far, most of the studies conducted to investigate the effect of eccentricity on annular pressure loss were not validated with relevant experimental data, particularly for drilling fluids. We remedy this situation in the present paper, which offers both an experimental investigation and a new theoretical model to account for the effect of drillpipe eccentricity on the annular pressure loss experience with YPL fluids in deviated/horizontal and extended reach wells. First, a theoretical model and a solution procedure is proposed. Then, the experimental data is presented and compared with this new theoretical model as well as other most widely used models in the drilling industry.
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To predict the annular pressure losses in an annulus, a model is proposed based on simplifying the Navier-Stokes equations in cylindrical coordinates. Major simplifying assumptions include: • Steady-state, isothermal, fully developed incompressible flow • Gravitational force is neglected • No slip at the wall • The inner pipe is rotating about its own axis and outer pipe is stationary • Axisymmetric flow Fig. 1 illustrates the geometry of the problem. In Appendix A, it is shown that annular pressure loss can be obtained from Eq. 3: $ $ $*+, 1 $*+, = & () ./ + & ) / $% $' $' ' $'
(3)
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No analytical solution exists for this equation and hence an implicit finite difference scheme is applied. After implementing the finite difference approximation and further algebraic simplifications, we obtain (Erge, 2013):
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$ 1 ∆' = 1*)23 + )2 -4+, 23 5 − *)23 + 2)2 +)26 -4+, 2 5 + *)2 + )26 -4+, 26 57 $% 2 +, − +, 26 + )2 ∆' 23 2'
(4)
Non-Newtonian behavior of drilling fluids is accounted for by considering YPL fluid behavior, in which shear stress is related to shear rate by the following equation:
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8 = 89 + :*;< -=
(5)
where shear rate function including the effect of rotation is given by: $ +? $+,
@ + $' ' $'
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;< = >'
(6)
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The proposed model is valid for Newtonian and/or non-Newtonian fluids (including Bingham-Plastic, PL and YPL) for a concentric annulus with inner pipe rotation and any flow regime (laminar or turbulent). This model is easy to implement, fast and bypasses the complexity of using commercial CFD software without sacrificing accuracy. Furthermore, contrary to the Narrow Slot approximation, it considers drillpipe rotation and can be applied for unusual diameter ratios as well, which makes it suitable for special drilling applications such as coiled tubing. A CFD analysis was conducted by using commercial software in order to validate the proposed model, initially for the concentric annulus, and to allow comparison with other models in the literature. A very fine mesh size was used (0.4 mm) to obtain accurate velocity profiles. The meshing is illustrated in Fig. 2. Inputs for the CFD analysis and fluid properties are given in Table 1. The laminar velocity profile obtained from the proposed model was compared with results from the Narrow Slot approach for flow of an arbitrary YPL fluid through concentric annuli (Fig 3). Very good agreement between the commercial CFD software and the proposed model was observed. In addition, Fig. 3 shows that the Narrow Slot approach presents a reasonable accuracy for this geometry (diameter ratio = 0.5). Erge et al. (2015b) presented a stepwise procedure to obtain the velocity profile for the Narrow Slot approach. To incorporate the effect of eccentricity, the pressure loss value obtained for a concentric annulus is corrected for the eccentric annulus, using an appropriate method. For instance, in this study Eq. (2) is used to incorporate pipe eccentricity. It should be noted that Eq. (1) was initially developed for PL fluids and similar correlation has not yet been reported for YPL fluids. However, as proposed by Ahmed and Miska (2009), a generalized flow behavior index, N, can be used instead of fluid behavior index, m, for YPL fluids (see Appendix B).
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4. Experimental Setup
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Experimental data presented by Erge (2013) used in this study. The data has been collected from the outdoor dynamic testing facility located at the Tulsa University Drilling Research Projects (Fig. 4). The length of the experimental section is 27.74 m (91 ft) and it consists of 0.0245 m (1”) steel pipe and 0.0571 m (2.25”) acrylic casing with the inner diameter of 0.0508 m (2”). Drillpipe lies at the bottom of the casing due to its weight, thereby simulating a fully eccentric annulus. Properties of the experimental set up are shown in Table 2. Fig. 5 shows the schematic of the test facility. The pressure data was obtained by using both point pressure transducers and differential pressure transducers. There are three temperature transducers mounted on the test setup. Temperature was measured at the inlet of the supply line and at the inlet and outlet of the annular test section. The entrance and exit lengths were estimated based on empirical correlations from literature (Knudsen and Katz, 1958; White, 2011). A 454-lpm (20-hp) centrifugal pump was used for fluid circulation. Fluid volume for circulation was handled in a 227 liter (60-gal) reservoir tank. A Coriolis flow meter was installed at the inlet of the supply line, measuring fluid density and flow rate. Four non-Newtonian drilling fluids (labeled YPL-A, YPL-B, YPL-C, YPL-D) were used in this study. A sophisticated rheometer, which is very sensitive at low and high shear rates, was used to form the rheogram (i.e. the plot of shear stress vs. shear rate) for each fluid (Fig. 6). Based on the rheometer readings, the YPL model was used to characterize the rheological profile of these fluids (Fig. 7). Tables 3 and 4 present the composition, density, and rheological parameters of each fluid respectively. The fluid behavior index (m) varies from 0.42 to 0.57, consistency index (K) from 0.21 to 0.95 Pa.sm and yield stress from 2.38 to 14.85 Pa. By designing these test fluids, it has been attempted to cover a wide range of drilling fluid rheological properties routinely used in field practice. Prior to the experiments, calibration tests were conducted with water to verify the pressure loss readings. These test results were compared with the analytical solution for flow of Newtonian fluid (water) in an eccentric annulus (Ahmed and Miska, 2009). Excellent agreement was achieved between the analytical model and the experimental results for a fully eccentric annulus as shown in Fig. 8.
5. Results and discussion
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5.1 Extracting the laminar data points. Annular pressure loss vs. flow rate was measured for a wide range of flow rate; however, in this study, we are only interested in laminar data points. Hence, it is required to distinguish the transition from laminar to turbulent flow. This can be done by plotting wall shear stress (τw) vs. nominal Newtonian shear rate (8v/Dhyd) and careful examination of the trend of the data points for a sharp increase in wall shear stress as nominal Newtonian shear rate increases (Ahmed and Miska, 2009). Fig. 9 shows the wall shear stress vs. nominal Newtonian shear rate for YPL-B. The red line shows the transition from the laminar flow regime to the turbulent regime. A similar approach was applied to distinguish the laminar-turbulent transition for other fluids as well. 5.2 Model comparisons and error analysis. Proposed model in this study is compared with the experimental data (Erge, 2013) as well as other widely used theoretical and numerical models, including: Narrow Slot (Appendix B), corrected Narrow Slot (Appendix B), Luo and Peden (1990), Sestak et al. 6
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(2001), Pipe Equivalent Model (Ahmed et al., 2006) and Hashemian Adariani (2005). Since, the Narrow Slot approximation does not consider the effect of pipe eccentricity, the obtained pressure loss from this approach is corrected by Eq. 2 for a fully eccentric annulus and is denoted as the Corrected Narrow Slot approach (Appendix B).
* -DE. − * -FG. = × 100 * -FG.
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Figs. 10 through 13 show the laminar pressure loss vs. average velocity for four YPL fluids (A through D). These figures compare the obtained pressure loss from the proposed model with the experimental data and other models in the literature. The Narrow Slot approach and Sestak et al. (2001) model are not shown in these figures due to their poor predictions for the fully eccentric annulus and are included in the error analysis only. It should be noted that YPL-A is the least viscous drilling fluid with the lowest yield stress and YPL-D is the thickest fluid with the highest yield stress. Note that when yield stress is negligible or zero, the YPL model is the same as the PL model. Figs. 10 through 13 indicate that all models, including the proposed model in this study, underestimate the effect of pipe eccentricity on annular pressure loss. Values predicted by the Corrected Narrow Slot model are very close to the proposed model. It should be noted that the proposed model and Corrected Narrow Slot approach are initially derived for the situation of a concentric annulus and both use Eq. 2 to correct the pressure loss for a fully eccentric annulus ( = 1). Eq. 1 is developed for the PL fluids and ignores the fluid yield stress, while realistic drilling fluids typically show YPL behavior. Unfortunately, a similar correlation is not available for the YPL model. As a common practice, the fluid behavior index (m) is replaced with generalized flow behavior index, N for YPL fluids as presented in Eq. 2 (Ahmed and Miska, 2009). However, based on the presented experimental data, this approach seems to be ineffective for YPL fluids with high yield stress and a more accurate correlation is required. Table 5 present the Absolute Average Percent Error (AAPE) for the models applied in this study. Relative error in percent, Average Percent Error (APE), and Absolute Average Percent Error (AAPE) are defined according to Eqs. 7, 8 and 9:
"
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1 IJA'KLA NA'OAPQ R''S' = *T ABC,2 P 2V
(8) "
1 IWXSYQA IJA'KLA NA'OAPQ R''S' = *TZABC,2 ZP
(9)
2V
5.3 Discussion. Table 5 shows that the proposed model had the best performance among all the other models. It predicts the annular pressure loss in a fully eccentric annulus for the low yield stress fluids (YPL-A and YPL-B) with less than 5% error. Higher error (up to 13%) was observed for fluids with high yield stress (YPL-C and YPL-D) that can be due to strong shear thinning ability of these fluids. Also, all the models presented here disregard the possible flow instabilities and only consider a two dimensional space. 7
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According to the obtained results, using the pressure loss values obtained from the Narrow Slot approach for a fully eccentric annulus is not recommended as it may result in more than 100% error. However, results obtained from the Corrected Narrow Slot approach show reasonable accuracy for the presented geometry and in several cases performed even better than the simplified CFD models for eccentric annuli such as the models by Luo and Peden (1990) or Hashemian Adariani (2005). Theoretical models that are developed based on introducing geometric parameters such as the Pipe Equivalent approach (Ahmed et al., 2006) or the model by Sestak et al. (2001) yielded very poor predictions and greatly underestimated the effect of eccentricity on pressure loss in a fully eccentric annulus. The comparative study presented here provides valuable insight towards superior hydraulic planning and real-time equivalent circulating density (ECD) management in deviated, horizontal and extendedreach wells. Its beneficial impact on managing the annular pressure between the fracture and pore pressure in the field to prevent drilling problems such as kicks, mud loss and wellbore instability events is expected to be immediate. Its use will also be key to successful implementation of innovative drilling techniques such as managed pressure drilling and dual gradient drilling, which require highly accurate annular pressure prediction and management.
6. Conclusions
This study aims to better predict annular pressure loss of YPL drilling fluids in a fully eccentric annulus for the laminar flow regime. A new model is proposed and validated with commercial CFD software for a concentric annulus. The model considers the inner pipe rotation and is valid for both laminar and turbulent flow regime for any diameter ratio. The effect of eccentricity is incorporated by applying Eq. 2. The model results as well as those of other most widely used models for annular pressure calculation was compared to experimental data.
•
Our investigations show that pipe eccentricity has a profound effect on annular pressure losses while circulating YPL drilling fluids. In some cases, more than 50% reduction in pressure loss was observed for a fully eccentric annulus. Therefore, drillpipe eccentricity must be taken explicitly into account during conducing hydraulic planning for horizontal and extended reach wells, in which a large fully(or near fully) eccentric section is anticipated. The proposed model showed the best performance among all the applied models in this study. It predicted the annular pressure loss of low yield stress drilling fluids with less than 5% error. For fluids with high yield stress, higher error was observed (<15%). This indicates using a generalized flow behavior index (N) instead of fluid behavior index (m) in Eq. 2 as proposed by Ahmed and Miska (2009) will not generate accurate results for high yield stress YPL fluids. A more comprehensive approach is still required.
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Using Narrow Slot approach, as is extensively done in the petroleum industry, for predicting annular pressure loss in a fully eccentric annulus is not recommended, since it yields large errors. Theoretical models that are based on introducing geometric parameters, such as Pipe Equivalent, generally delivered a poor result by greatly under-estimating the eccentricity effect. However, the Corrected Narrow Slot approach showed acceptable accuracy with comparison to other more sophisticated and
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computationally expensive CFD models (Luo and Peden, 1990; Hashemian Adariani, 2005) for the presented geometry.
7. Acknowledgments
Nomenclature geometric parameters, m diameter, m
m: N:
dP/dl:
frictional pressure loss gradient, Pa/m
Q:
dP
frictional pressure loss, Pa
r, R:
E:
offset distance, m
Re:
f:
friction factor
V:
h:
height of the slot, m
K:
consistency index, Pa ∙ s b
w:
Greek Letters shear stress, Pa τ: shear rate, 1/s ;:
density, kg/m viscosity, Pa ∙ s diameter ratio angular speed, rad./s velocity, m /s geometrical constant, m
radius, m
Reynolds number mean axial fluid velocity, m /s width of the slot, m
b:
bulk
d:
dimensionless inner hydraulic outer wall yield Yield Power Law
i: h,H: o: w: y: YPL:
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dimensionless eccentricity
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ρ: µ, η: κ: ω: υ: λ:
flow rate, m3 /X
Subscripts apparent app:
Table 1. Input Parameters for CFD (Erge, 2013)
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ε:
flow behavior index generalized flow behavior index
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a, b D:
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Authors would like to thank Tulsa University Drilling Research Projects, Det norske oljeselskap ASA and Baker Hughes for their support and encouragement through this study. In addition, ANSYS is thanked for providing the commercial CFD software.
Geometry OD (mm)
50.8
ID (mm)
25.4
Length (m)
2.5 Fluid Herschel–
Model
Bulkley
Yield Stress (Pa) Consistency Index
(ef. g h )
Flow Behavior Index 3
Density (kg/m )
9
5 1 0.5 1000
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Velocity (m/s)
1
Incompressible and Isothermal Mesh and Solution Criteria 0.4
Max Face size (mm)
0.4
Max Size (mm)
0.4
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Min size (mm)
Side Face Meshing (mm)
3
RMS
1e
Nodes
-5
8,592,484
Table 2. Specifications of the test facility (Erge, 2015a) Properties of the Experimental Setup
Properties of Inner Pipe
31.70 m
Length of the test section
27.98 m
Length of the supply line
25.60 m
Outside diameter
25.4 mm
Acrylic
Thickness
0.889 mm
27.74 m
Material type
304L stainless steel
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Material type of the test pipe
Length
SC
Total length of the facility
Outside diameter of the test pipe
57.15 mm
Weight in the air
0.423 kg/m
Thickness of the test pipe
3.175 mm
Modulus of elasticity
2x10 MPa
227 liter
Moment of inertia
5.15x10 m
Volume of the reservoir tank Flow rate range
5
-8
4
0-454 lpm
RPM range
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0-120 c/min
Table 3. Composition and density of the test fluids (it should be noted 1 gr/350 cc = 1 lb/bbl) (Erge, 2013) Composition
YPL-A YPL-B YPL-C
Laponite (gr/350 cc)
0.2
5.33
1002.9
0.11
6.54
1005.5
---
9
1010
0.30
8
1008.6
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YPL-D
3
Density (kg/m )
PAC R (gr/350 cc )
EP
Test Fluid
Table 4. Rheological parameters of the test fluids Rheological Parameters m
K (Pa.S )
m
τy (Pa)
0.2145
0.5748
2.383
0.6585
0.4265
4.581
0.9462
0.4254
7.602
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0.4999
14.85
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0.769
Table 5. Error analysis for different models (AAPE is presented) YPL-B
Proposed model
4.10
Narrow Slot Corrected Narrow Slot by Eq. 2 Pipe Equivalent (Ahmed et al., 2006) Sestak et al. (2001) Hashemian Adariani (2005)
YPL-D
4.47
12.32
12.84
84.10
83.61
98.18
99.18
6.93
5.61
13.64
14.07
10.11
19.51
29.38
30.66
30.48
42.67
54.31
56.71
7.56
13.17
22.69
16.26
12.38
10.32
19.18
19.90
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Luo and Peden (1990)
YPL-C
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YPL-A
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Models (Error in %)
Fig. 1. Geometry of the problem (Erge, 2015b)
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Fig. 2. Mesh and geometrical representation of the CFD analysis (Erge, 2015b)
Fig. 3. Comparison the velocity profile from the proposed model with CFD analysis and the Narrow Slot approximation (concentric annulus, laminar flow regime) (Erge, 2013)
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Fig. 4. Outside dynamic testing facility at The University of Tulsa (Erge, 2013)
Fig. 5. Schematic for the outside dynamic testing facility (Erge, 2013)
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Fig. 6. Anton Paar (Physica MCR 301) rheometer is used to characterize the fluids (Erge, 2013)
Fig. 7. Rheometer readings and fitted YPL model to the data pints for test fluids
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9000 Theoretical Model 8000
Experimental Data
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Pressure loss (Pa/m)
7000 6000 5000 4000 3000
1000 0 0
1
2
3
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2000
4
5
6
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Velocity (m/s)
Fig. 8. Calibration test with water (annular pressure loss vs. velocity, fully eccentric annulus) (Erge, 2013)
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Wall Shear Stress (Pa)
50
30
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Laminar
10
0 0
200
400
600
800
1000
1200
1400
1600
8v/Dhyd
Fig. 9. Plot of wall shear stress vs. nominal Newtonian shear rate to determine the transition from laminar to turbulent flow for YPL-B
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1050
Experimenta Data Pipe Equivalent Model Corrected Narrow Slot Model Proposed Model Luo and Peden Model Hashemian Model
1000
900 850
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Pressure loss (Pa/m)
950
800 750 700
600 0.2
0.3
0.4
0.5
0.6
Velocity (m/s)
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650
0.7
0.8
0.9
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Fig. 10. Model comparisons with the experimental data for YPL-A
1400
1200
Experimental Data Pipe Equivalent Model Corrected Narrow Slot Model Proposed Model Luo and Peden Model Hashemian Model
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1600
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1800
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1000
800
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Velocity (m/s)
Fig. 11. Model comparisons with the experimental data for YPL-B
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2900 2700
2300
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Pressure loss (Pa/m)
2500
2100 1900
Experimental Data Pipe Equivalent Model Proposed Model Corrected Narrow Slot Model Luo and Peden Model Hashemian Model
1500 1300
0.5
1 Velocity (m/s)
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1.5
2
Fig. 12. Model comparisons with the experimental data for YPL-C
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3400
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3900
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2900
Experimental Data Pipe Equivalent Model Corrected Narrow Slot Model Proposed Model Luo and Peden Model Hashemian Model
2400
1900
0
0.5
1
1.5
Velocity (m/s)
Fig. 13. Model comparisons with the experimental data for YPL-D
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Appendix A: Derivation of the proposed model In this section, derivation of the proposed model is presented. Continuity equation in the cylindrical coordinates is: $i 1 $*i'+B - 1 $*i+? - $*i+, + + + .=0 $Q ' $' ' $j
%$(A.1)
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(
Now, consider the Navier-Stokes (N-S) equations in cylindrical coordinates: In r- direction:
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$+B $+B +? $+B $+B +? + +B + + +, − . $Q $' ' $j $% '
k k lllllllll $ 1$ 1 $ $ 8?? i $*'+ B +B −> *'8BB - + 8?B + 8,B − @ + iLB − $' ' $' ' $j $% ' ' $' k k k k k k llllllllll llllllllll lllllllllll *+? +? i $*+B +? $*+B +, − −i +i ' $j $% '
=−
In θ- direction:
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$+? $+? +? $+? $+? +B +? i + +B + + +, + $Q $' ' $j $% ' 1 $ 1 $ 1 $ $ 8?B − 8B? =− − > *' 8B? - + 8?? + 8,? − @ + iL? ' $j ' $' ' $j $% ' k k k k k k lllllllllll llllllllll llllllllll llllllllll *+? k +B k i $*+ $*+ $*+ ? +? ? +B ? +, − −i −i − 2i ' $j $' $% '
In z- direction:
(A.2)
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i(
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$+, $+, +? $+, $+, i + +B + + +, $Q $' ' $j $% k k llllllllll $ 1$ 1 $ $ $*+ , +, =− −> *'8B, - + 8?, + 8,, @ + iL, − i $% ' $' ' $j $% $% k+ kk+ klllllllllll llllllllll i $*'+ i $*+ B , ? , − − ' $' '
%$(A.3)
(A.4)
where the stress tensors for cylindrical coordinates are defined as: 8BB = −) >2
$+B 2 @ + ) − *∇ ∙ n$' 3
(A.5)
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8,? = 8?, = −) > 8,B = 8B, = −) >
$ +? 1 $+B + @
$' ' ' $j
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8B? = 8?B = −) >'
(A.7)
1 $+, $+? + @ ' $j
%$$+B $+, + @ $% $'
1 $ 1 $+? $+, *'+B - + + ' $' ' $j
%$and the dissipation function:
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(A.8) (A.9) (A.10)
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8,, = −) >2
(A.6)
$+B 1 $+? +B $+, $ +? 1 $+B 1 $+, $+? Φp = 2 & + + + / + >'
+ @ +> + @ $' ' $j ' $% $' ' ' $j ' $j
%$(A.12)
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$+B $+, 2 1 $ 1 $+? $+, *'+B - + +> + @ − > + @ $% $' 3 ' $' ' $j
%$(A.11)
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The N-S equations presented here are valid for all fluids (Newtonian or non-Newtonian) and in any flow regime (laminar or turbulent). After applying the simplifying assumptions, presented in section 2, the following equation is obtained: k k lllllllllll $ 1$ $*+, i $*'+ B +, =& ()' ./ − $% ' $' $' ' $'
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(A.13)
Using the mixing length concept gives: $+, $+, llllllllllll *i+B k +, k - = i q q $' $'
(A.14)
$+, k k llllllllllll −*i+ q B +, - = )E q $'
(A.15)
$ 1$ $+, 1$ $+, => )' @ + > )E ' q q@ $% ' $' $' ' $' $'
(A.16)
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(A.17)
Instead of the viscosity term, an effective viscosity is introduced:
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) = *) + )E -
Therefore, the viscosity term is replaced with the Eq. A.18:
(3)
SC
$ $ $*+, 1 $*+, = & () ./ + & ) / $% $' $' ' $'
(A.18)
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No analytical solution is available for Eq. 3. By applying an implicit finite difference scheme and further algebraic simplifications, Eq. 4 is obtained: − +, 26 $ $*+, $*+, 1 +, ∆' = v() − () . . x + ) 23 $% $' 23w $' 26w ' 2
− +, 26 $ 1 $*+, 1 $*+, 1 +, ∆' = v *)23 + )2 - ( − *)2 + )26 - ( . . x + ) 23 $% 2 $' 23w 2 $' 26w ' 2
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+, − +, 2 +, − +, 26 +, − +, 26 $ 1 ∆' = >*)23 + )2 - 23 − *)2 + )26 - 2 @ + ) 23 $% 2 ∆' ∆' 2'
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$ 1 ∆' = 1*)23 + )2 -4+, 23 5 − *)23 + 2)2 +)26 -4+, 2 5 + *)2 + )26 -4+, 26 57 $% 2 +, − +, 26 + )2 ∆' 23 2'
(A.19)
(A.20)
(A.21)
(A.22)
In order to incorporate the non-Newtonian behavior of drilling fluids, an apparent viscosity term is defined as: )yGG =
89 + :; =6 ;
(A.23)
where shear rate function including the effect of rotation given as: $ +? $+, ;< = >' @ + $' ' $'
(A.24)
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Appendix B: Laminar pressure loss of YPL fluids by using Narrow Slot approach z{
Eq. 4 represents the shear stress (8)-shear rate (zB = ;< ) relationship for YPL fluids.
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8 = 89 + :*;< -=
(4)
It is assumed that flow is single-phase, axial, isothermal, and incompressible with constant rheological properties. The following stepwise procedure is adopted from Merlo et al. (1995) to calculate the annular pressure loss for the laminar flow regime:
1 + 2 12v 3 ! − 2
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γ< ~ =
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1. Assume |y= 1 (Power Law fluid) and calculate shear rate at the wall, γ< ~ .
(B.1)
2. Then, calculate shear stress at the wall, 8~ .
8~ = 89 + :*γ< ~ -=
3. Knowing that = , Compute |y .
|y = 1 −
− 1+ 1+
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(B.2)
(B.3)
4. Now, calculate the new value for shear stress at the wall, 8~ .
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1 1 + 2 12J = 8~ = 89 + : >* @ |y 3 ! − 2
(B.4)
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5. Calculate the difference between 8~ and 8~ . If tolerance defined as
6
× 100 is less than a
certain value (for instance 1%), accept 8~ ; Otherwise, use 8~ and calculate the new value for , |y , 8~ until convergence is obtained. By knowing the shear stress at the wall for laminar flow, annular pressure loss can be obtained from: 4 8~ = ! − 2
(B.5)
In order to incorporate the pipe eccentricity (corrected Narrow Slot approximation in this study), calculate the generalized flow behavior index (N): 21
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N=
|y 1 + 2 *1 − |y -
(B.6)
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Then, replace N in Eq. 2 to correct the pressure loss obtained from Eq. B.5 for eccentricity (Ahmed and Miska, 2009). It should be noted that in PL fluids, 89 = = 0, |y = 1 , and N = m.
References
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Ahmed, R. and Miska, S. 2008. Experimental Study and Modeling of Yield Power-Law Fluid Flow in Annuli with Drillpipe Rotation. Paper SPE 112604 presented at the SPE/IADC Drilling Conference, Orlando, Florida, 4-6 March. Ahmed, R., Miska, S.Z., Miska, W.Z., 2006. Friction Pressure Loss Determination of Yield Power Law Fluid in Eccentric Annular Laminar Flow. Wiertnictwo Nafta Gaz 23 (1): 47-53. Ahmed, R., Miska, S.Z. Advanced wellbore hydraulics, chapter 4.1, (pp. 191-219). Advanced drilling and well technology. USA Society of Petroleum Engineers. Ed. Bernt S. Aadnoy. SPE, 2009. Ameen Rostami, S., Kinik, K., Gumus, G. 2015. Dynamic Calibration of the Empirical Pore Pressure Estimation Methods Using MPD Data. OTC-25953-MS, OTC 2015, Houston Texas. Erge, O. 2013. Effect of Free Drillstring Rotation on Frictional Pressure Losses. University of Tulsa, M.S. Thesis. Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2013. Effect of Drillstring Deflection and Rotary Speed on Annular Frictional Pressure Losses. Paper ASME 10448 presented at the 32nd International Conference on Ocean Offshore and Arctic Engineering, Nantes, France, 9-14 June; doi:10.1115/OMAE2013-10448. Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2014. Effect of Drillstring Deflection and Rotary Speed on Annular Frictional Pressure Losses. Journal of Energy Resources Technology 136 (4), 042909-042909-10; doi: 10.1115/1.4027565. Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2014. The Effects of Drillstring Eccentricity, Rotation and Buckling Configurations on Annular Frictional Pressure Losses While Circulating Yield Power Law Fluids. Paper SPE 167950 presented at the IADC/SPE Drilling Conference & Exhibition, 4-6 March, Fort Worth, Texas, USA; doi:10.2118/167950-MS Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2015. Laminar To Turbulent Transition of Yield Power Law Fluids in Annuli. Journal of Petroleum Science and Engineering 128, 128-139; doi: 10.1016/j.petrol.2015.02.007 Erge, O., Ozbayoglu, E. M., Miska, S. Z., Yu, M., Takach, N., Saasen, A., May, R. 2015. CFD Analysis and Model Comparison of Annular Frictional Pressure Losses While Circulating Yield Power Law Fluids. Paper SPE 173840 presented at the SPE Bergen One Day Seminar, 22 April, Bergen, Norway; doi:10.2118/173840-MS Escudier, M.P., Oliveira, P. J., Pinho, F. T. 2002. Fully Developed Laminar Flow of Purely Viscous Non-Newtonian Liquids Through Annuli, Including the Effects of Eccentricity And Inner-Cylinder Rotation. International Journal of Heat and Fluid Flow 23: 52-73. Escudier, M.P. and Gouldson, I.W. 1995. Concentric Annular Flow with Centerbody Rotation of a Newtonian and a Shear Thinning Liquid. International Journal of Heat and Fluid Flow 16 (3): 156-162. Founargiotakis, K., Kelessidis, V. C., Maglione, R. 2008. Laminar, Transitional and Turbulent Flow of HerschelBulkley Fluids in Concentric Annulus. Canadian Journal of Chemical Engineering 86: 676-683. Fredrickson, A. G. and R. B. Bird. 1958. Flow of Non-Newtonian Fluids in Annuli. Ind. Eng. Chem. 50, 347–352. Gucuyener, H.I., Mehmetoglu, T., 1992. Flow of Yield-Pseudoplastic Fluids through a Concentric Annulus. AIChE J. 38, 1139. Haciislamoglu, M. 1989. Non-Newtonian Fluid Flow in Eccentric Annuli and Its Application to Petroleum Engineering Problems. Louisiana State University, Ph.D. dissertation. Haciislamoglu, M. and Langlinais, J. 1990. Non-Newtonian Flow in Eccentric Annuli. J. Energ. Resour. 112: 163169. doi:10.1115/1.2905753. Hanks, R.W. 1979. The Axial Laminar Flow of Yield-Pseudoplastic Fluids in a Concentric Annulus. Ind. Eng. Chem. Process Des. Dev. 18: 488-493. doi: 10.1021/i260071a024.Hansen, S. A. and Sterri, N. 1995. Drill Pipe
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Robertson, R.E., Stiff Jr., H.A., 1976. An Improved Mathematical Model for Relating Shear Stress to Shear Rate in Drilling Fluids and Cement Slurries. SPE J. 16, 31–36. Sestak, J., Zitny, R., Ondrusova J. and Filip, V. 2001. Axial Flow of Purely Viscous Fluids In Eccentric Annuli: Geometric Parameters for Most Frequently Used Approximate Procedures. 3rd Pacific Rim Conference on Rheology, Canadian Group of Rheology, Montreal. Sorgun M., Ozbayoglu M. and Aydin I. 2010. Modeling and Experimental Study of Newtonian Fluid Flow in Annulus. J. Energy Resour. Technol. 132 (3): 033102-033102-6. doi: 10.1115/1.4002243. Tao, L. N., and W. F. Donovan. 1955. Through-flow In Concentric and Eccentric Annuli of Fine Clearance with and without Relative Motion of The Boundaries. Trans. ASME 77.8: 1291-1301. White, F.M., 2011. Fluid Mechanics. 7th Edition, McGraw-Hill Series in Mechanical Engineering.
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Highlights We investigated annular pressure losses experimentally, analytically and numerically
•
A novel numerical model for Yield Power Law drilling fluids is presented
•
The presented model is compared with the experimental data and published models
•
The presented numerical model better predicts the annular pressure losses
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Pressure losses reduced significantly with increasing eccentricity
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