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Improving basic relationships of pipe hydraulics
Flow Measurement and Instrumentation 72 (2020) 101698
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Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst
Improving basic relationships of pipe hydraulics �rbara Biosca d, Jesús Díaz-Curiel a, *, María J. Miguel b, Natalia Caparrini c, Ba a Lucía Ar�evalo-Lomas a
Department of Geological and Mining Engineering, School of Mines and Energy, Universidad Polit�ecnica de Madrid, Address: C/ Ríos Rosas 21, 28003, Madrid, Spain Spanish Ministry of Science, Innovation and Universities, Address: Paseo de la Castellana 162, 28046, Madrid, Spain c Department of Natural Resources and Environmental Engineering, School of Mining Engineering, Universidad de Vigo, Address: C/ Maxwell. Campus LagoasMarcosende, 36310, Vigo, Pontevedra, Spain d Department of Energy and Fuels Systems, School of Mines and Energy, Universidad Polit�ecnica de Madrid, Address: C/ Ríos Rosas 21, 28003, Madrid, Spain b
A R T I C L E I N F O
A B S T R A C T
Keywords: Well-log Laminar flow Turbulent flow Transition interval Velocity profile Velocity factor Friction factor
This communication presents the approaches set up for processing spinner flowmeter well logs in vertical wells with a single fluid phase, which is the most widely used in assessing wells productivity. These focus on improving the pipe hydraulics relationships so that the different fluid inputs throughout the well can be quantified. Since vertical flow inside wells varies with depth between laminar flows (very low Reynolds number, i.e. Re < 103) and turbulent (Re > 4⋅103) the aim has been to reduce the uncertainty in the transition interval. Starting from bibliographical data and/or well-known formulas for laminar and for turbulent flow, several continuous re lationships have been developed for any regime: 1) an expression for the radial distribution of velocity inside the pipeline (velocity profile) was developed. 2) A relationship between the average velocity and the velocity at the axis (velocity factor) was created. 3) A third equation was generated to obtain the friction factor in smooth pipes (and starting from this, a new explicit equation for rough pipes). The purpose has been to have a set of empirical expressions of easy and continuous application for any regime, as an alternative to the use of computer simulations.
1. Introduction Flowmeter well logging is used to determine variations in the flow velocity along the pipeline of a well, and serves to quantify the fluid inputs at different depths that contribute to the extraction rate. These quantities are useful in estimating changes in the hydraulic character istics with depth, enhance the rational exploitation of reservoirs and optimize the resources management. Flowmeter sondes were developed in the late 1950s for use in oil wells [1,2]. Although different types of sensors have been constructed, the most widely used in assessing the productivity of wells are spinner flowmeters (a type of turbine with lightweight propeller). To improve the signal provided by flowmeter sondes, industry has designed different types of sensors, of which the so-called ‘free type’ (sondes are not confined in any way inside the pipe) are the most frequently used. The revolutions per unit time, as measured by the equipment, the advance velocity of the sonde along the pipeline, and the necessary static and dynamic calibration curves are used to determine the velocity at the
radial distance from the spinner to the well axis. The results of these measurements are a function of the mass of the blades and their geo metric features (section, angle, curvature, edges, etc.), which do not need to be considered since they are implicit in the cited calibration curves of the sonde. The scope of the calibration of well flowmeters will be detailed in the discussion section. To obtain the average velocity starting from measurements at any distance from the axis of the well (as in flowmeter well logs), the radial distribution of velocity (named velocity profile) must be used. This depends on the regime of fluid movement characterized by the Reynolds number, Re. The relationships governing this profile is fully established for the laminar regime and widely accepted for the turbulent regime, but it is not yet sufficiently established in the transition interval between them. In this study, the gradual disappearing of the inflectional behaviour as increasing the presence of suspended particles (see for example [3]) has not been considered. For typical pumping rates of between 0.0002 and 0.02 m3 s 1, this transition interval occurs between the bottom of the well (approximately 102
* Corresponding author. E-mail addresses: [email protected] (J. Díaz-Curiel), [email protected] (M.J. Miguel), [email protected] (N. Caparrini), [email protected] (B. Biosca), [email protected] (L. Ar�evalo-Lomas). https://doi.org/10.1016/j.flowmeasinst.2020.101698 Received 30 August 2019; Received in revised form 13 December 2019; Accepted 22 January 2020 Available online 30 January 2020 0955-5986/© 2020 Elsevier Ltd. All rights reserved.
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Flow Measurement and Instrumentation 72 (2020) 101698
permeable layers (approximately 104
function of the value of Re, the range of which varies according to the author. Eckhardt [21] cites the following long-established values: laminar flow at Re < 1000; intermediate or transitional flow at 1000 < Re < 4000; and turbulent flow at Re > 4000. Since the flow velocity is the focus of this study, the Re value cannot be extracted directly from the value of the punctual velocity measured by the flowmeter and an iterative process is therefore performed. First, the velocity in the pipeline axis, Vmax, is obtained from the measured velocity and is then used as the flow velocity to calculate an initial Re value. Using the relationship between V(r), Vmax and, a new average velocity is obtained and a new Re value is then adopted. This process is repeated until the variation between consecutive values of Re is negligible (for a given convergence criterion). Thus, the first step uses a velocity law that is generally given by the relation between the velocity V(r) at a distance r from the axis and Vmax. Since this relationship depends on the diameter of the well, the normalized notation rD ¼ r/(D/2), which ranges between zero and one, is used here, where D represents the pipeline diameter. For laminar flow in smooth pipes, the most widely accepted expression governing this velocity ratio was set by Hagen-Poiseuille equation [9,10] as a function of rD: VðrD Þ ¼ 1 Vmax
� r2D ; laminar flow:
(1)
For turbulent flow, logarithmic relationships and potentials of the velocity, depending on the turbulence degree, have been historically considered [7,8,11,22,23]. These relationships are still being reformu lated [12,13,16]. In general, these laws have a slight discontinuity near the axis of the pipeline for the lowest values of Re. From the range of possible equations, the expression given by Schlichting [10] as a func tion of radial distance has been selected here: VðrD Þ ¼ ð1 Vmax
rD Þ1=n ; turbulent flow;
(2)
where n is an exponent related to the degree of turbulence, for which Schlichting [11] stated that the minimum value is n > 6 (where n is not necessarily an integer). Fig. 1 presents the curves of the measured values based on Nikuradse [23], the curves obtained using Eq. (1) for laminar flow, and the curves from Eq. (2) with n ¼ 6 and n ¼ 10.5 for turbulent flow. The extrapolated curves for exponents n ¼ 3.5 and n ¼ 2.0 are also added, showing that the values of Eq. (2) for n < 6 are clearly different from the curve cor responding to laminar flow, indicating the need for an expression that
2. Theoretical background 2.1. Determining flow velocity from flowmeter well logs In the conventional process for determining the flow velocity from flowmeter well logs, certain principles of pipe hydraulics have obviously great relevance [18,19]. As is well known, the movement of a fluid along a pipe does not occur at the same velocity across its section, but depends on the distance from the axis. The velocity V(r) of the fluid at a certain distance from the pipe axis depends on the flow regime (degree of turbulence), which is quantified by the Re number [20] obtained from the fluid density ρ (kg⋅m 3), the flow velocity (m⋅s 1), the pipeline diameter D (m) and the fluid viscosity μ (Pa⋅s). In general, three types of flow regimes can be distinguished as a
Fig. 1. Velocity profiles in circular pipelines. 2
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Flow Measurement and Instrumentation 72 (2020) 101698
can provide gradual values from laminar to turbulent flow. In order to follow the process proposed in [18], the second step is the application of a velocity factor based on the average velocity and the maximum velocity Vmax along the axis, Fvel ¼ /Vmax. The actual average velocity is obtained from the expression
For turbulent flow through smooth pipes, the friction factor is given by the Blasius equation [7], which is given in Eq. (6): fBlasius ¼
0:3164 ; for turbulent flow Re1=4
(6)
�rma �n [22] and Prandtl Later, Colebrook [27] combined the Von Ka [8] equations to quantify the effect of the relative roughness rr (the ratio between the height of the relief of the wall and the inner diameter of the pipe) of the pipe walls, giving Eq. (7) in which the friction factor is implicit: � � 1 2:51 pffiffi ¼ 2 log 3:71rr þ pffiffi (7) f Re f Fig. 3 shows the resulting curves from fH-P and fBlasius and the values obtained by iteratively solving the Colebrook equation, including those values extrapolated for laminar flow and the transitional interval. In addition to the above relationships, the inflectional behaviour shown by Nikuradse [28] in the transitional interval (also shown in data from the University of Oregon in [29] and in data from Princeton Uni versity in [30]) is considered in this study.
Equation (3) has been used in previous studies [19,24] to find the relationship between the flowmeter-measured velocity and the average velocity. In oil wells [25] Fvel is assumed to vary between 0.75 and 0.95, and when a single value is required, Fvel ¼ 0.83 is used. However, in wells Fvel ranges between 0.5 and 0.85, and hence using a value of 0.83 can produce a deviation of greater than 60%. The third step in the process of [18] is the application of the rela tionship between the two parameters Fvel and Re to determine the exponent of turbulence that corresponds to the average velocity, or the Re value for this flow. Although [18] used his own curve, we develop an analytical expression that fits the empirical data obtained by Nikuradse [23] and Song et al. [24], since such an expression is necessary for the processing of flowmeter well logs. The followed process is delineated in the flow chart in Fig. 2. As discussed further below, the main drawback of this process is the difference between the flow velocity and average velocity obtained from the velocity law.
3. Proposed relationships 3.1. Velocity law As stated in the introduction, for this conversion, a new expression for the velocity law must first be developed that considers the entire range between laminar and turbulent flow, since along wells, flows range from very low Reynolds numbers (Re~102 103) in the deepest part to high values (Re~105 106) in the upper part. The conditions imposed to generate this law are (i) that its results
2.2. Head loss The head loss (Δh) caused by the local factors of the wells, referred to in the introduction section, is normally added to the drawdown due to extraction. This decrease in hydraulic head is produced in the well during extraction, with the consequent gradient of pressures, and is what causes the flow of water from the permeable formations to the well for each flow used. The most widely accepted expression for calculating head loss due to the friction along the pipe’s length is the DarcyWeisbach equation [6,26]: Δh ¼ f
L < V>2 : D 2g
(4)
where Δh is the head loss (m), f is the friction factor (dimensionless), L is the pipe length (m), D is the internal diameter of the pipe (m), is the average velocity (m⋅s 1), and g is the acceleration due to gravity (m⋅s 2). The friction factor for laminar flow in smooth pipes is given by the Hagen-Poiseuille equation [9,10], which does not include the effect of the roughness of the pipe walls, and which is reduced to the expression in Eq. (5) after the Reynolds number is established: fH
P
¼
64 ; for laminar flow Re
Fig. 3. Friction factor for smooth pipes and as a function of the relative roughness from the Colebrook equation [27].
(5)
Fig. 2. Process flow chart. 3
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Flow Measurement and Instrumentation 72 (2020) 101698
must match those from Eq. (1) (Hagen-Poiseuille) for laminar flow, and (ii) that it must fit the data measured by Nikuradse [23] for the interval of turbulent flow, 4⋅103
3.2. Velocity factor For turbulent flow corresponding to pumping rates higher than 1⋅10 3 m3 s 1, variations in the velocity factor can generate errors in the average velocity value of up to 12%. However, variations of the fluid inputs determined in screens at greater depths, where the flow is not turbulent, can generate errors of approximately 40% from the actual value (as discussed above and shown in Eq. (3)). In contrast to the process described for the integration of Eq. (3) [11], Eq. (8) for V(r)/Vmax cannot be analytically integrated to obtain a relationship between the average velocity and the maximum velocity. The values resulting from a numerical integration of this equation were therefore fitted to an algebraic expression that is given in Eq. (9). Fvel ¼
rD
A :
(9)
Equation (9) can be considered a reliable expression of this integral for the values 1 < τ < 10, since its average deviation with respect to the values of the numerical integral is 0.12%. This adjustment is shown graphically in Fig. 5.
11=τ τþ2= τþ0:5 C
< V > τ þ 0:5 ¼ Vmax τ
(8)
3.3. τ(Re) and Fvel(Re) It is necessary to obtain one of these two relationships in order to know the regime of the flow. To generate suitable fitting functions, the empirical values of Patel and Head [32] in the range 400 < Re < 4000 and Nikuradse [23] for Re > 4000 are approximated by algebraic ex pressions. This fitting has typically been conducted separately for three regimes of laminar, transition and turbulent flow. However, in a similar way as for the velocity law, we develop a unique relationship that re flects the τ values for laminar and turbulent flow and the transition in terval in a continuous manner. Given that (τNIK,Re) data from Table 1 show an increasing gradient with Re (see Fig. 6), the generated fitting function for τ(Re) has been τ ¼
where τ denotes the new exponent of turbulence. In this new relation ship, which is valid for values 1 < τ < 10, the use of τ is a convenient notation to indicate that this exponent does not have the same values as the exponent n of the [11] expression (Eq. (2)). Fig. 4 shows the curves obtained from Eq. (8) for various exponents between τ ¼ 1 and τ ¼ 9.8; it also includes data from Nikuradse [23] and the curve corresponding to laminar flow according to Eq. (1) (Hagen- Poiseuille) [9,10]. Fig. 4 shows that the curve obtained for τ ¼ 1 using Eq. (8) is consistent with the result from Eq. (1). With regard to the Nikuradse data [23], the minimum deviation curves in Eq. (8) show a minor dif ference, although this is smaller (0.9%) than for the [11] function (1.4%). The τ values resulting from this comparison τNIK are shown in Table 1. The curves from Eq. (8) for τ ¼ 5.2 and τ ¼ 9.8 are consistent with the [11] curves shown in Fig. 1 with n ¼ 6.1 and n ¼ 10.5, respectively. Moreover, for 1 < τ < 5.2, Eq. (8) provides intermediate values be tween the [23] data and those corresponding to laminar flow.
ðRe=R1 Þt1þ1 , t2 ⋅ðRe=R1 Þt3þ1
(10):
τ¼
where t1, t2, t3, and R1 are fitting coefficients, obtaining Eq.
ðRe=2490Þ9:994 þ 1
(10)
0:2⋅ðRe=2409Þ9:9 þ 1
For values of Re > 4000, the mean relative deviation of τ(Re) with respect to τNIK is 0.2%. This exponent and the distance to the axis, which is equal to the radius of the well minus the outer radius of the spinner case, can be entered into Eq. (8) to obtain the velocity at the axis Vmax. The process can then be continued, since knowing τ allows the values of the velocity factor to be obtained using Eq. (9). However, we opted to use the methodology stated by [18] and to develop a fitting relationship for Fvel as a function of Re. Again, the experimental values of Nikuradse [23] and Patel and Head [32] were used. Since the average velocity should not exceed the maximum velocity, this expression must be generated so that for increasing Re values, its results tend asymptotically to the unit value. However, it was decided that the sign of the curvature should be the result of obtaining the minimum deviation. As for Eq. (10), a generalized sigmoidal algebraic function has been chosen, but in this case the initial value must be 0.5 which is the velocity factor for Re < 103 (1 and 2 have been taken as independent terms). The value of the Reynolds number in the inflection, the exponents that regulate the curvature before and after the inflection, and the factor that determines the amplitude of the sigmoid function, have been left as variables to obtain in the process of minimum devia φ1
2 Þ þ1 tion. The relationship developed is Fvel ¼ φ ðRe=R , where ⋅ðRe=R2 Þφ3 þ2 2
and R2 are fitting coefficients, obtaining Eq. (11):
Fig. 4. Velocity profiles in a smooth pipe determined by Eq. (8) for different τ exponents. 4
φ φ φ 1, 2, 3,
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Flow Measurement and Instrumentation 72 (2020) 101698
Table 1 Turbulence exponent of minimum deviation for Nikuradse data [23]. Re
4.00⋅103
6.10⋅103
9.20⋅103
1.67⋅104
2.33⋅104
4.34⋅104
1.05⋅105
2.05⋅105
τNIK
5.2 3.96⋅105
5.5 7.25⋅105
5.8 1.11⋅106
6.5 1.54⋅106
6.2 1.96⋅106
6.4 2.35⋅106
6.8 2.79⋅106
7.1 3.24⋅106
8.1
8.6
8.6
9.1
9.3
9.5
9.6
9.8
Re τNIK
a logarithmic function for Re > 4000 with an R2 of 0.9998. 3.4. Friction factor Bearing in mind that the Colebrook equation was established for the turbulent regime, its numerical solution when the flow tends towards laminar gives lower values for the friction factor than those of Eq. (5) (Hagen-Poiseuille) for smooth pipes. To obtain a new explicit relationship, a fitting function is developed for the friction factor values for smooth pipes resulting from the HagenPoiseuille [9,10] and Blasius equations, and the effect of roughness is then added; the first step in this process is simpler than that presented by [17]. This procedure has the advantages of giving the friction factor in the absence of significant roughness by a unique function, and it allows to approximate the roughness effect separately. The established function for the non-roughness friction factor, fsmooth ¼ 0:3164 Re
ρ
Reσ1 þR3 ρ, Reσ2 þR4
where σ1, σ2, R3, R4 and ρ are fitting coefficients,
was generated as the product of a function with the same gradient as the Hagen-Poiseuille equation [9,10] with a function that has a gradient that changes in the transition interval to that of the Blasius equation [7]. The obtained relationship after fitting is given by Eq. (12):
Fig. 5./Vmax versus τ turbulence exponent.
fsmooth ¼
0:3164 Re10:75 þ 485010 Re Re10 þ 285010
(12)
This equation has a relative mean deviation of 0.01% with respect to the original functions in the 102
Fig. 6. Velocity factor (left-axis) and turbulence exponent (right-axis) versus Reynolds number data.
Fvel ¼
ðRe=2580Þ9:914 þ 1 1:265⋅ðRe=2580Þ9:9 þ 2
(11)
Fig. 6 shows the values provided by Eq. (11) with data extracted from the cited references. The mean relative deviation of the fitting of Eq. (11) from the experimental data of Nikuradse [23] for turbulent flow is 0.46%, which is comparatively low. The mean relative deviation is somewhat higher (2.5%) with respect to the data of Patel and Head [32] for the laminar and transitional regimes. Among other reasons, this deviation is related to the greater dispersion of the data compared with a continuous func tion; for low values of Re, the data of Patel and Head [32] present a similar mean deviation relative to the value of Fvel ¼ 0.5 established for laminar flow. Hence, Eq. (11) is considered to accurately reflect the value of the velocity factor. It is also interesting to note that Eq. (11) fits
Fig. 7. Friction factor for smooth pipes given by Eq. (12) (left-axis) and roughness component given by Eq. (13) (right-axis). 5
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Flow Measurement and Instrumentation 72 (2020) 101698
of the Blasius equation [7], to which the inflectional behaviour shown in the Nikuradse data [28] is added. The resulting function is given in Eq. (13): ) � Rebðrr Þ ⋅expð Rp5 Rep Þ 0:25 f ðr1 ; ReÞ ¼ (13a) logðr1 =3:71Þ2 Rebðrr Þ þ Aðrr Þbðrr Þ
down the sonde at different velocities, obtaining a calibration curve (see [19]). Hence, it does not take into account either the velocity profile or the different behaviour in the transitional regime as considered by D�zemi�c et al. [35]. Another problematic feature of the flowmeter well logs would be the effects of lateral fluid input in the screened section. Since most wells are not continuously screened, to minimise this effect the fluid input at each screened section is therefore obtained by subtracting the flow mea surement in the axial direction, before and after each section. Related to the relationship for the velocity profile, the transition interval is problematic since the distribution of the fluid velocity inside the pipe is particularly heterogeneous. In laminar flow, the fluid ad vances homogeneously along the length and width of the pipe, and when the flow is completely turbulent, the fluid becomes homogeneously disordered. Within the transition interval, heterogeneity arises in terms of space and time, and instantaneous images or punctual measurements can give different results. In the transition from laminar to turbulent flow, where turbulence is first encountered, it only occurs in localized patches, so called puffs, which are spatially separated by laminar flow [36]. The aim of the relationship developed here is to simulate the measurements of a spinner flowmeter by providing an expression that can generate the mean values of the local advance and temporal varia tions in this transition interval. At this scale, it is feasible to fit a mean behaviour to an empirical function within the transition interval [37], which scope must be taken in an engineering sense. About the expression for the velocity factor Fvel ¼/Vmax, data for the turbulent flow from Nikuradse [23] show a slight curvature in the results from Eq. (11) (see Fig. 6). Although the relative deviation from this equation is similar (0.46%) to the logarithmic law developed by Nikuradse [23] (0.35%), data provided for turbulent flow by other au thors, such as Zagarola and Smits [12] and George [13], show stronger alignment. However, the turbulence exponent τ improves the fit of the Schlichting relationship [11] (Eq. (2)) to the Nikuradse data [23], and the use of Eq. (2) to determine the velocity factor for each Re value gives a certain curvature that tends asymptotically to a value of one. This is expected behaviour for the flat profile, corresponding to a very high degree of turbulence. Hence, the slight curvature provided by Eq. (11) can be considered correct. For this reason, the choice of other fitting expressions whose deviation from a logarithmic behaviour in the tur bulence zone is obviously smaller, for example the relationship Fvel ¼ 0.5þ(0.0117⋅ln [Re]þ0.205)⋅Re10/(Re10þ264,010), has been ruled out. It should also be pointed out that the above fittings are based on the /Vmax data for each value of Re taken from Table 2 of Nikuradse [23]; however, the other values in Table 9 from the same work show a different trend, which is more similar to those revealed in later works (e. g. [12]). If these other values were considered, the coefficients of the relationships developed here would change slightly. The difference may lie that calculation of the mean velocity from the velocity profile is not necessarily equal to the flow velocity (total flow rate divided by cross-sectional area); it would be due to the presence of the viscous sublayer near the wall (see [37]), which produces results that are not considered in the velocity laws. Concerning the friction factor, although Joseph and Yang [17] developed a relationship that includes the slight increase measured by McKeon et al. [16] starting from Re > 3⋅105 in smooth pipes, with respect to the potential behaviour derived by the Blasius equation, the fsmooth value determined by Eq. (12) is sufficiently accurate for the goals of this work. For turbulent flow in rough pipes, after the charts elaborated by Rouse [38] and Moody [39] were published with the numerical solution of the Colebrook equation [27], many works published explicit expres sions for obtaining these friction factor curves (see [15,40–43]). Many of these relationships are similar to the expression f ¼ 0.25/[log (rr/3.7 þ 3.7/Re0.875)]2 (modified from Swamee and Jain [44]). However, the values provided for the laminar flow and the transition intervals are dissimilar from the expected values.
where R5 and p are fitting coefficients (R5 ¼ 2700 and p ¼ 6), and A (rr) y b (rr) are fitting sub-functions that depend on roughness, as given by the following expressions: ) Aðrr Þ ¼ 5000⋅ð1=rr Þ1:3 þ 800 � 0:1 (13b) bðrr Þ ¼ 0:23 rr Eq. (13a) was generated starting from the product of the Nikuradse relationship [28] for the friction factor in the fully developed turbulent flow, with a function that reflects the change in the friction factor from laminar flow (zero) to values established from this relationship, in such a way that the difference from the results of the Colebrook equation is minimized. A continuous expression for the friction factor for any value of roughness can be determined by the sum of Eqs. (12) and (13): f ¼ fsmooth þ f ðr1 ; ReÞ
(14)
The resulting curves are shown in Fig. 8, with the curves obtained from analytical fitting functions of the Colebrook equation. The mean relative deviation between both sets of curves for Re > 4000 is 1.6%. Given the very high accuracy of the approximations in Eq. (12) for smooth pipes, the somewhat reduced accuracy of Eq. (14) must be attributed to the use of Eq. (13). 4. Discussion With reference to the calibration process of well flowmeters, it is not a similar process than the precise method realised in laboratory (see for example [33] or [34]). One of the characteristics of flowmeter well logs is that measurements are made while sonde moves in depth to reduce the influence of the sensor threshold. Given the hard conditions of the well logs, it is advisable to repeat a calibration process just before to measure. The standard calibration process only consists in to displace up and
Fig. 8. Friction factor as a function of the relative roughness from the Cole brook equation and Eq. (14). 6
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For the interval 102
[6] [7]
[8]
[9]
[10]
[11] [12]
5. Conclusions
[13]
New expressions for the velocity law, the velocity factor and the friction factor have been developed and their validity is verified using the deviation analysis presented and graphically illustrated in Figs. 4, 6 and 8. The improvements in the processing of flowmeter well logs have been optimized with respect to the flow regimes (laminar, transition and turbulent) produced in wells. There are many specialized publications in fluid mechanics that deal with turbulent flow, and the expressions re ported in these works have similar or better accuracy; however, the new relationships developed here cover the entire flow range using a continuous approach. In future research, the influence of this reformulation on the result ing permeability values in wells with conventional relationships versus depth should be quantified.
[14]
Author statement
[20]
Jesús Díaz-Curiel: Conceptualization, Methodology, Formal anal ysis, Investigation, Writing - Original Draft, Writing Review & Editing Preparation, Visualization, María Jesús Miguel: Resources, Writing �rbara Biosca: Inves Original Draft, Natalia Caparrini: Validation, Ba tigation, Resources, Writing Review & Editing Preparation, Lucía �valo-Lomas: Investigation, Resources, Writing - Review & Editing Are Preparation.
[21]
Acknowledgments
[24]
[15] [16] [17] [18] [19]
[22]
[23]
Part of this study was funded through the CARESOIL-CM (S2013/ MAE-2739 & P2018/EMT-4317) research grant of the regional gov ernment of Madrid (Comunidad de Madrid, Spain) and the DENSOIL (CTM2016-77151-C2-2-R) research grant from the Government of Spain (Ministerio de Economía y Competitividad). The authors also express their gratitude to Professor Eduardo de Miguel for his valuable inputs and considerations that have proven very useful to fulfil this work.
[25] [26] [27] [28]
References [29]
[1] R.C. Rumble, H.M. Buck, B.A. Peters, New subsurface flowmeter developed by Humble research engineers, Oil Gas J. (1959) 104–107. May. [2] H.L. Bryant, Production well logging techniques, Geophysics 25 (4) (1960) 905–927, https://doi.org/10.1190/1.1438771. [3] N. Agrawal, G.H. Choueiri, B. Hof, Transition to turbulence in particle laden flows, Phys. Rev. Lett. 122 (11) (2019) 114502. [4] P. Hufschmied, Estimation of three-dimensional statistically anisotropic hydraulic conductivity field by means of a single well pumping test combined with flowmeter measurements, Hydrogeologie 2 (1986) 163–174. [5] K.R. Rehfeldt, P. Hufschmied, L.W. Gelhar, M.E. Schaefer, Measuring Hydraulic Conductivity with the Borehole Flowmeter, Topical Report EN-6511, Electric
[30] [31] [32]
7
Power Research Institute (EPRI), Palo Alto, California, produced by Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1989. H. Darcy, Recherches exp�erimentales relatives au mouvement de l’eau dans les tuyaux [Experimental research on water movement in pipes], Mallet-Bachelier, Paris, 1857 (268 pages and atlas). H. Blasius, The law of similarity in friction processes in fluids (Das € Ahnlichkeitsgesetz bei Reibungsvorg€ angen in Flüssigkeiten), Forschungs-Arbeit des Ingenieur-Wesens 131 (1913), https://doi.org/10.1007/978-3-662-02239-9_1. Berlin. L. Prandtl, in: Neuere Ergebnisse der Turbulenzforschung. Zeitschrift des Vereines Deutscher lngenieure, vol. 77, Recent results of turbulence research. National Advisory Committee for Aeronautics (NACA) of USA, Technical Memorandum 720, 1933, pp. 105–114. G. Hagen, About The Movement of Water in Narrow Cylindrical Pipes (Über Die Bewegung des Wassers in engen zylindrischen R€ ohren), Poggendorfs Annalen der Physik und Chemie 16 (1839) 423–442, https://doi.org/10.1002/ andp.18391220304. J.L. Poiseuille, Experimental research on the movement of liquids in very small diameter tubes (Recherches exp� erimentales sur le mouvement des liquides dans les tubes de tr� es-petits diam� etres), Comptes rendus hebdomadaires des s� eances de l’Acad�emie des Sciences 12 (1841) 112–115. ISSN 0001-4036. H. Schlichting, Boundary-layer Theory, seventh ed., McGraw-Hill Book Co, 1979, ISBN 9783662529195, p. 150. Reference to Schlichting, H. 1932. Entstehung der Turbulenz in einem rotierenden Zylinder. Nachr. Ges. Wiss. G€ ottingen, 1932. M.V. Zagarola, A.J. Smits, Mean-flow scaling of turbulent pipe flow, J. Fluid Mech. 373 (1998) 33–79, https://doi.org/10.1017/S0022112098002419. W.K. George, Is there a universal log law for turbulent wall-bounded flows? Philos. Trans. Roy. Soc. A 365 (2007) 789–806, https://doi.org/10.1098/rsta.2008.0217. S.W. Churchill, Friction-factor equation spans all fluid flow regimes, Chem. Eng. 84 (24) (1977) 91–92. E. Romeo, C. Royo, A. Monz� on, Improved explicit equations for estimation of the friction factor in rough and smooth pipes, Chem. Eng. J. 86 (3) (2002) 369–374, https://doi.org/10.1016/S1385-8947(01)00254-6. B.J. McKeon, M.V. Zagarola, A.J. Smits, A new friction factor relationship for fully developed pipe flow, J. Fluid Mech. 538 (2005) 429–443, https://doi.org/ 10.1017/S0022112005005501. D.D. Joseph, B.H. Yang, Friction factor correlations for laminar, transition and turbulent flow in smooth pipes, Phys. Nonlinear Phenom. 239 (14) (2010) 1318–1328, https://doi.org/10.1016/j.physd.2009.09.026. Dresser Atlas, Interpretive methods for production well logs, in: Section 2: the Continuous Spinner Flowmeter: A Velocity Measuring Instrument, third ed., Dresser Industries, Inc, USA, 1982. J. Díaz-Curiel, D.A. Martín, A. Rozycki, A. Maldonado, Flowmeter Well logging in large diameter water wells (Testificaci� on de flowmeter en pozos de agua de gran di� ametro), in: Metodología y aplicaciones. Tecnología del Agua, vol. 129, Elsevier, Barcelona (Spain), 1994. O. Reynolds, On the dynamic theory of incompressible viscous fluids and the determination of the criterion, in: Philosophical Transactions of the Royal Society of London T 186 A, S. 123, 1895, https://doi.org/10.1098/rsta.1895.0004. B. Eckhardt, Introduction. Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds’ paper, Philos. Trans. Roy. Soc. A 367 (2009) 449–455. € T. Von K� arm� an, Mechanical Similarity and Turbulence (Mechanische Ahnlichkeit
und Turbulenz), in: C.W. Oseen, W. Weibull (Eds.), Proc. 3rd International Congress on Applied Mechanics (Stockholm, 1930), vol. I, AB Sveriges Lifografska Tryckenier, 1931, pp. 85–93, 1931. J. Nikuradse, Gesetzm€ assigkeiten der Turbulenten str€ omung in Glatten Rohren, Laws of Turbulent Flow in Smooth Pipes. 1966, 1-73, in: Forschung auf dem Gebiete des Ingenieurwesens 356, vol. 3National Aeronautic and Space Administration (NASA) of USA, 1932, pp. 1–76, https://doi.org/10.1007/ BF02716946. TT F-10, 359. S. Song, C. Jordan, D. Georgi, Theoretical and experimental studies on the determination of the average fluid velocity from spinner flowmeter responses, in: Transactions of 37th Annual Logging Symposium of SPWLA, Paper III, 1996, p. 10. Schlumberger, Fundamentals of Production Logging, Schlumberger, Briarpark Drive Houston, Texas, 2013, p. 77042, 13, 978-1-937949-07-5-3750. J. Weisbach, Textbook of Engineering and Machinery Mechanics (Lehrbuch der Ingenieur- und Maschinen-Mechanik), in: Theoretische Mechanik [Theoretical Mechanics], vol. 1, Vieweg und Sohn, Braunschweig, 1845, p. 535. C.F. Colebrook, Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws, J. Inst. Civ. Eng. (London) 11 (4) (1939) 133–156, 1938 and 1939. J. Nikuradse, Str€ omungsgesetze in rauhen Rohren, VDI-Forschungsheft, 361. Beilage zu Forschung auf dem Gebiete des Ingenieurwesens. Ausgabe B, Band 4, July-August. Laws of Flow in Rough Pipes, 1950, National Advisory Committee for Aeronautics (NACA), 1933. TM 1292. C.J. Swanson, B. Julian, G.G. Ihas, R.J. Donnelly, Pipe flow measurements over a wide range of Reynolds numbers using liquid helium and various gases, J. Fluid Mech. 461 (2002) 51–60. B.J. McKeon, C.J. Swanson, M.V. Zagarola, R.J. Donnelly, A.J. Smits, Friction factors for smooth pipe flow, J. Fluid Mech. 511 (2004) 41–44, https://doi.org/ 10.1017/S0022112004009796. F. Russo, N.T. Basse, Scaling of turbulence intensity for low-speed flow in smooth pipes, Flow Meas. Instrum. 52 (2016) 101–114. V.C. Patel, M.R. Head, Some observations on skin friction and velocity profiles in fully developed pipe and channel flows, J. Fluid Mech. 38 (1969) 181–201, https://doi.org/10.1017/S0022112069000115.
J. Díaz-Curiel et al.
Flow Measurement and Instrumentation 72 (2020) 101698 [41] M.T. Çoban, Error analysis of non-iterative friction factor formulas relative to Colebrook-White equation for the calculation of pressure drop in pipes, J. Naval Sci. Eng. 8 (1) (2012) 1–13. [42] A.R. Vatankhah, Approximate analytical solutions for the Colebrook equation, J. Hydraul. Eng. 144 (5) (2018), 06018007, https://doi.org/10.1061/(ASCE) HY.1943-7900.0001454. [43] D. Brki�c, P. Praks, Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function, Mathematics 7 (1) (2019) 34, https://doi.org/10.3390/math7010034. [44] P.K. Swamee, A.K. Jain, Explicit equations for pipe flow problems, J. Hydraul. Div. ASCE 102 (5) (1976) 657–664. [45] P.K. Swamee, Design of a submarine oil pipeline, J. Transport. Eng. 119 (1) (1993) 159–170, https://doi.org/10.1061/(ASCE)0733-947X(1993)119:1(159). [46] M. Ur�si�c, B. Kompare, Improvement of the hydraulic friction losses equations for flow under pressure in circular pipes, Acta Hydrotech. 21 (34) (2003) 57–74. ISSN 1581-0267. [47] N.S. Cheng, Formulas for friction factor in transitional regions, J. Hydraul. Eng., ASCE 134 (9) (2008) 1357–1362, https://doi.org/10.1061/(ASCE)0733-9429 (2008)134:9(1357). [48] D. Brki�c, P. Praks, Unified friction formulation from laminar to fully rough turbulent flow, Appl. Sci. 8 (2018) 2036, https://doi.org/10.3390/app8112036.
[33] B. Lee, R. Cheesewright, C. Clark, The dynamic response of small turbine flowmeters in liquid flows, Flow Meas. Instrum. 15 (2004) 239–248. [34] B. Wang, N. Zhang, Q. Cao, Y. Du, Evaluation approach to dynamic characteristic of turbine flowmeters considering calibration system response, Flow Meas. Instrum. 64 (2018) 126–132. � [35] Z. D�zemi�c, B. Sirok, B. Bizjan, Turbine flowmeter response to transitional flow regimes, Flow Meas. Instrum. 59 (2018) 18–22. [36] B. Hof, A. Juel, T. Mullin, Scaling of the turbulence transition threshold in a pipe, Phys. Rev. Lett. 91 (24) (2003) 244502. [37] G.I. Barenblatt, A.J. Chorin, Turbulence: an old challenge and new perspectives, Meccanica 33 (1998) 445–468, https://doi.org/10.1023/A:100431240. [38] H. Rouse, Evaluation of boundary roughness, in: Proceedings of the Second Hydraulic Conference, vol. 27, University of Iowa Bulletin, 1943. [39] L.F. Moody, Friction Factors for Pipe Flow, Transaction on Semi-annual Meeting of the American Society of Mechanical Engineers, Pittsburgh, Pensylvania, June 19–22, 1944. [40] F. Abdolahi, A. Mesbah, R.B. Boozarjomehry, W.Y. Svrcek, The effect of major parameters on simulation results of gas pipelines, Int. J. Mech. Sci. 49 (2007) 989–1000, https://doi.org/10.1016/j.ijmecsci.2006.12.001.
8
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst
Improving basic relationships of pipe hydraulics �rbara Biosca d, Jesús Díaz-Curiel a, *, María J. Miguel b, Natalia Caparrini c, Ba a Lucía Ar�evalo-Lomas a
Department of Geological and Mining Engineering, School of Mines and Energy, Universidad Polit�ecnica de Madrid, Address: C/ Ríos Rosas 21, 28003, Madrid, Spain Spanish Ministry of Science, Innovation and Universities, Address: Paseo de la Castellana 162, 28046, Madrid, Spain c Department of Natural Resources and Environmental Engineering, School of Mining Engineering, Universidad de Vigo, Address: C/ Maxwell. Campus LagoasMarcosende, 36310, Vigo, Pontevedra, Spain d Department of Energy and Fuels Systems, School of Mines and Energy, Universidad Polit�ecnica de Madrid, Address: C/ Ríos Rosas 21, 28003, Madrid, Spain b
A R T I C L E I N F O
A B S T R A C T
Keywords: Well-log Laminar flow Turbulent flow Transition interval Velocity profile Velocity factor Friction factor
This communication presents the approaches set up for processing spinner flowmeter well logs in vertical wells with a single fluid phase, which is the most widely used in assessing wells productivity. These focus on improving the pipe hydraulics relationships so that the different fluid inputs throughout the well can be quantified. Since vertical flow inside wells varies with depth between laminar flows (very low Reynolds number, i.e. Re < 103) and turbulent (Re > 4⋅103) the aim has been to reduce the uncertainty in the transition interval. Starting from bibliographical data and/or well-known formulas for laminar and for turbulent flow, several continuous re lationships have been developed for any regime: 1) an expression for the radial distribution of velocity inside the pipeline (velocity profile) was developed. 2) A relationship between the average velocity and the velocity at the axis (velocity factor) was created. 3) A third equation was generated to obtain the friction factor in smooth pipes (and starting from this, a new explicit equation for rough pipes). The purpose has been to have a set of empirical expressions of easy and continuous application for any regime, as an alternative to the use of computer simulations.
1. Introduction Flowmeter well logging is used to determine variations in the flow velocity along the pipeline of a well, and serves to quantify the fluid inputs at different depths that contribute to the extraction rate. These quantities are useful in estimating changes in the hydraulic character istics with depth, enhance the rational exploitation of reservoirs and optimize the resources management. Flowmeter sondes were developed in the late 1950s for use in oil wells [1,2]. Although different types of sensors have been constructed, the most widely used in assessing the productivity of wells are spinner flowmeters (a type of turbine with lightweight propeller). To improve the signal provided by flowmeter sondes, industry has designed different types of sensors, of which the so-called ‘free type’ (sondes are not confined in any way inside the pipe) are the most frequently used. The revolutions per unit time, as measured by the equipment, the advance velocity of the sonde along the pipeline, and the necessary static and dynamic calibration curves are used to determine the velocity at the
radial distance from the spinner to the well axis. The results of these measurements are a function of the mass of the blades and their geo metric features (section, angle, curvature, edges, etc.), which do not need to be considered since they are implicit in the cited calibration curves of the sonde. The scope of the calibration of well flowmeters will be detailed in the discussion section. To obtain the average velocity
* Corresponding author. E-mail addresses: [email protected] (J. Díaz-Curiel), [email protected] (M.J. Miguel), [email protected] (N. Caparrini), [email protected] (B. Biosca), [email protected] (L. Ar�evalo-Lomas). https://doi.org/10.1016/j.flowmeasinst.2020.101698 Received 30 August 2019; Received in revised form 13 December 2019; Accepted 22 January 2020 Available online 30 January 2020 0955-5986/© 2020 Elsevier Ltd. All rights reserved.
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Flow Measurement and Instrumentation 72 (2020) 101698
permeable layers (approximately 104
function of the value of Re, the range of which varies according to the author. Eckhardt [21] cites the following long-established values: laminar flow at Re < 1000; intermediate or transitional flow at 1000 < Re < 4000; and turbulent flow at Re > 4000. Since the flow velocity is the focus of this study, the Re value cannot be extracted directly from the value of the punctual velocity measured by the flowmeter and an iterative process is therefore performed. First, the velocity in the pipeline axis, Vmax, is obtained from the measured velocity and is then used as the flow velocity to calculate an initial Re value. Using the relationship between V(r), Vmax and
� r2D ; laminar flow:
(1)
For turbulent flow, logarithmic relationships and potentials of the velocity, depending on the turbulence degree, have been historically considered [7,8,11,22,23]. These relationships are still being reformu lated [12,13,16]. In general, these laws have a slight discontinuity near the axis of the pipeline for the lowest values of Re. From the range of possible equations, the expression given by Schlichting [10] as a func tion of radial distance has been selected here: VðrD Þ ¼ ð1 Vmax
rD Þ1=n ; turbulent flow;
(2)
where n is an exponent related to the degree of turbulence, for which Schlichting [11] stated that the minimum value is n > 6 (where n is not necessarily an integer). Fig. 1 presents the curves of the measured values based on Nikuradse [23], the curves obtained using Eq. (1) for laminar flow, and the curves from Eq. (2) with n ¼ 6 and n ¼ 10.5 for turbulent flow. The extrapolated curves for exponents n ¼ 3.5 and n ¼ 2.0 are also added, showing that the values of Eq. (2) for n < 6 are clearly different from the curve cor responding to laminar flow, indicating the need for an expression that
2. Theoretical background 2.1. Determining flow velocity from flowmeter well logs In the conventional process for determining the flow velocity from flowmeter well logs, certain principles of pipe hydraulics have obviously great relevance [18,19]. As is well known, the movement of a fluid along a pipe does not occur at the same velocity across its section, but depends on the distance from the axis. The velocity V(r) of the fluid at a certain distance from the pipe axis depends on the flow regime (degree of turbulence), which is quantified by the Re number [20] obtained from the fluid density ρ (kg⋅m 3), the flow velocity (m⋅s 1), the pipeline diameter D (m) and the fluid viscosity μ (Pa⋅s). In general, three types of flow regimes can be distinguished as a
Fig. 1. Velocity profiles in circular pipelines. 2
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Flow Measurement and Instrumentation 72 (2020) 101698
can provide gradual values from laminar to turbulent flow. In order to follow the process proposed in [18], the second step is the application of a velocity factor based on the average velocity
For turbulent flow through smooth pipes, the friction factor is given by the Blasius equation [7], which is given in Eq. (6): fBlasius ¼
0:3164 ; for turbulent flow Re1=4
(6)
�rma �n [22] and Prandtl Later, Colebrook [27] combined the Von Ka [8] equations to quantify the effect of the relative roughness rr (the ratio between the height of the relief of the wall and the inner diameter of the pipe) of the pipe walls, giving Eq. (7) in which the friction factor is implicit: � � 1 2:51 pffiffi ¼ 2 log 3:71rr þ pffiffi (7) f Re f Fig. 3 shows the resulting curves from fH-P and fBlasius and the values obtained by iteratively solving the Colebrook equation, including those values extrapolated for laminar flow and the transitional interval. In addition to the above relationships, the inflectional behaviour shown by Nikuradse [28] in the transitional interval (also shown in data from the University of Oregon in [29] and in data from Princeton Uni versity in [30]) is considered in this study.
Equation (3) has been used in previous studies [19,24] to find the relationship between the flowmeter-measured velocity and the average velocity. In oil wells [25] Fvel is assumed to vary between 0.75 and 0.95, and when a single value is required, Fvel ¼ 0.83 is used. However, in wells Fvel ranges between 0.5 and 0.85, and hence using a value of 0.83 can produce a deviation of greater than 60%. The third step in the process of [18] is the application of the rela tionship between the two parameters Fvel and Re to determine the exponent of turbulence that corresponds to the average velocity, or the Re value for this flow. Although [18] used his own curve, we develop an analytical expression that fits the empirical data obtained by Nikuradse [23] and Song et al. [24], since such an expression is necessary for the processing of flowmeter well logs. The followed process is delineated in the flow chart in Fig. 2. As discussed further below, the main drawback of this process is the difference between the flow velocity and average velocity obtained from the velocity law.
3. Proposed relationships 3.1. Velocity law As stated in the introduction, for this conversion, a new expression for the velocity law must first be developed that considers the entire range between laminar and turbulent flow, since along wells, flows range from very low Reynolds numbers (Re~102 103) in the deepest part to high values (Re~105 106) in the upper part. The conditions imposed to generate this law are (i) that its results
2.2. Head loss The head loss (Δh) caused by the local factors of the wells, referred to in the introduction section, is normally added to the drawdown due to extraction. This decrease in hydraulic head is produced in the well during extraction, with the consequent gradient of pressures, and is what causes the flow of water from the permeable formations to the well for each flow used. The most widely accepted expression for calculating head loss due to the friction along the pipe’s length is the DarcyWeisbach equation [6,26]: Δh ¼ f
L < V>2 : D 2g
(4)
where Δh is the head loss (m), f is the friction factor (dimensionless), L is the pipe length (m), D is the internal diameter of the pipe (m),
P
¼
64 ; for laminar flow Re
Fig. 3. Friction factor for smooth pipes and as a function of the relative roughness from the Colebrook equation [27].
(5)
Fig. 2. Process flow chart. 3
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Flow Measurement and Instrumentation 72 (2020) 101698
must match those from Eq. (1) (Hagen-Poiseuille) for laminar flow, and (ii) that it must fit the data measured by Nikuradse [23] for the interval of turbulent flow, 4⋅103
3.2. Velocity factor For turbulent flow corresponding to pumping rates higher than 1⋅10 3 m3 s 1, variations in the velocity factor can generate errors in the average velocity value of up to 12%. However, variations of the fluid inputs determined in screens at greater depths, where the flow is not turbulent, can generate errors of approximately 40% from the actual value (as discussed above and shown in Eq. (3)). In contrast to the process described for the integration of Eq. (3) [11], Eq. (8) for V(r)/Vmax cannot be analytically integrated to obtain a relationship between the average velocity and the maximum velocity. The values resulting from a numerical integration of this equation were therefore fitted to an algebraic expression that is given in Eq. (9). Fvel ¼
rD
A :
(9)
Equation (9) can be considered a reliable expression of this integral for the values 1 < τ < 10, since its average deviation with respect to the values of the numerical integral is 0.12%. This adjustment is shown graphically in Fig. 5.
11=τ τþ2= τþ0:5 C
< V > τ þ 0:5 ¼ Vmax τ
(8)
3.3. τ(Re) and Fvel(Re) It is necessary to obtain one of these two relationships in order to know the regime of the flow. To generate suitable fitting functions, the empirical values of Patel and Head [32] in the range 400 < Re < 4000 and Nikuradse [23] for Re > 4000 are approximated by algebraic ex pressions. This fitting has typically been conducted separately for three regimes of laminar, transition and turbulent flow. However, in a similar way as for the velocity law, we develop a unique relationship that re flects the τ values for laminar and turbulent flow and the transition in terval in a continuous manner. Given that (τNIK,Re) data from Table 1 show an increasing gradient with Re (see Fig. 6), the generated fitting function for τ(Re) has been τ ¼
where τ denotes the new exponent of turbulence. In this new relation ship, which is valid for values 1 < τ < 10, the use of τ is a convenient notation to indicate that this exponent does not have the same values as the exponent n of the [11] expression (Eq. (2)). Fig. 4 shows the curves obtained from Eq. (8) for various exponents between τ ¼ 1 and τ ¼ 9.8; it also includes data from Nikuradse [23] and the curve corresponding to laminar flow according to Eq. (1) (Hagen- Poiseuille) [9,10]. Fig. 4 shows that the curve obtained for τ ¼ 1 using Eq. (8) is consistent with the result from Eq. (1). With regard to the Nikuradse data [23], the minimum deviation curves in Eq. (8) show a minor dif ference, although this is smaller (0.9%) than for the [11] function (1.4%). The τ values resulting from this comparison τNIK are shown in Table 1. The curves from Eq. (8) for τ ¼ 5.2 and τ ¼ 9.8 are consistent with the [11] curves shown in Fig. 1 with n ¼ 6.1 and n ¼ 10.5, respectively. Moreover, for 1 < τ < 5.2, Eq. (8) provides intermediate values be tween the [23] data and those corresponding to laminar flow.
ðRe=R1 Þt1þ1 , t2 ⋅ðRe=R1 Þt3þ1
(10):
τ¼
where t1, t2, t3, and R1 are fitting coefficients, obtaining Eq.
ðRe=2490Þ9:994 þ 1
(10)
0:2⋅ðRe=2409Þ9:9 þ 1
For values of Re > 4000, the mean relative deviation of τ(Re) with respect to τNIK is 0.2%. This exponent and the distance to the axis, which is equal to the radius of the well minus the outer radius of the spinner case, can be entered into Eq. (8) to obtain the velocity at the axis Vmax. The process can then be continued, since knowing τ allows the values of the velocity factor to be obtained using Eq. (9). However, we opted to use the methodology stated by [18] and to develop a fitting relationship for Fvel as a function of Re. Again, the experimental values of Nikuradse [23] and Patel and Head [32] were used. Since the average velocity should not exceed the maximum velocity, this expression must be generated so that for increasing Re values, its results tend asymptotically to the unit value. However, it was decided that the sign of the curvature should be the result of obtaining the minimum deviation. As for Eq. (10), a generalized sigmoidal algebraic function has been chosen, but in this case the initial value must be 0.5 which is the velocity factor for Re < 103 (1 and 2 have been taken as independent terms). The value of the Reynolds number in the inflection, the exponents that regulate the curvature before and after the inflection, and the factor that determines the amplitude of the sigmoid function, have been left as variables to obtain in the process of minimum devia φ1
2 Þ þ1 tion. The relationship developed is Fvel ¼ φ ðRe=R , where ⋅ðRe=R2 Þφ3 þ2 2
and R2 are fitting coefficients, obtaining Eq. (11):
Fig. 4. Velocity profiles in a smooth pipe determined by Eq. (8) for different τ exponents. 4
φ φ φ 1, 2, 3,
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Flow Measurement and Instrumentation 72 (2020) 101698
Table 1 Turbulence exponent of minimum deviation for Nikuradse data [23]. Re
4.00⋅103
6.10⋅103
9.20⋅103
1.67⋅104
2.33⋅104
4.34⋅104
1.05⋅105
2.05⋅105
τNIK
5.2 3.96⋅105
5.5 7.25⋅105
5.8 1.11⋅106
6.5 1.54⋅106
6.2 1.96⋅106
6.4 2.35⋅106
6.8 2.79⋅106
7.1 3.24⋅106
8.1
8.6
8.6
9.1
9.3
9.5
9.6
9.8
Re τNIK
a logarithmic function for Re > 4000 with an R2 of 0.9998. 3.4. Friction factor Bearing in mind that the Colebrook equation was established for the turbulent regime, its numerical solution when the flow tends towards laminar gives lower values for the friction factor than those of Eq. (5) (Hagen-Poiseuille) for smooth pipes. To obtain a new explicit relationship, a fitting function is developed for the friction factor values for smooth pipes resulting from the HagenPoiseuille [9,10] and Blasius equations, and the effect of roughness is then added; the first step in this process is simpler than that presented by [17]. This procedure has the advantages of giving the friction factor in the absence of significant roughness by a unique function, and it allows to approximate the roughness effect separately. The established function for the non-roughness friction factor, fsmooth ¼ 0:3164 Re
ρ
Reσ1 þR3 ρ, Reσ2 þR4
where σ1, σ2, R3, R4 and ρ are fitting coefficients,
was generated as the product of a function with the same gradient as the Hagen-Poiseuille equation [9,10] with a function that has a gradient that changes in the transition interval to that of the Blasius equation [7]. The obtained relationship after fitting is given by Eq. (12):
Fig. 5.
fsmooth ¼
0:3164 Re10:75 þ 485010 Re Re10 þ 285010
(12)
This equation has a relative mean deviation of 0.01% with respect to the original functions in the 102
Fig. 6. Velocity factor (left-axis) and turbulence exponent (right-axis) versus Reynolds number data.
Fvel ¼
ðRe=2580Þ9:914 þ 1 1:265⋅ðRe=2580Þ9:9 þ 2
(11)
Fig. 6 shows the values provided by Eq. (11) with data extracted from the cited references. The mean relative deviation of the fitting of Eq. (11) from the experimental data of Nikuradse [23] for turbulent flow is 0.46%, which is comparatively low. The mean relative deviation is somewhat higher (2.5%) with respect to the data of Patel and Head [32] for the laminar and transitional regimes. Among other reasons, this deviation is related to the greater dispersion of the data compared with a continuous func tion; for low values of Re, the data of Patel and Head [32] present a similar mean deviation relative to the value of Fvel ¼ 0.5 established for laminar flow. Hence, Eq. (11) is considered to accurately reflect the value of the velocity factor. It is also interesting to note that Eq. (11) fits
Fig. 7. Friction factor for smooth pipes given by Eq. (12) (left-axis) and roughness component given by Eq. (13) (right-axis). 5
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Flow Measurement and Instrumentation 72 (2020) 101698
of the Blasius equation [7], to which the inflectional behaviour shown in the Nikuradse data [28] is added. The resulting function is given in Eq. (13): ) � Rebðrr Þ ⋅expð Rp5 Rep Þ 0:25 f ðr1 ; ReÞ ¼ (13a) logðr1 =3:71Þ2 Rebðrr Þ þ Aðrr Þbðrr Þ
down the sonde at different velocities, obtaining a calibration curve (see [19]). Hence, it does not take into account either the velocity profile or the different behaviour in the transitional regime as considered by D�zemi�c et al. [35]. Another problematic feature of the flowmeter well logs would be the effects of lateral fluid input in the screened section. Since most wells are not continuously screened, to minimise this effect the fluid input at each screened section is therefore obtained by subtracting the flow mea surement in the axial direction, before and after each section. Related to the relationship for the velocity profile, the transition interval is problematic since the distribution of the fluid velocity inside the pipe is particularly heterogeneous. In laminar flow, the fluid ad vances homogeneously along the length and width of the pipe, and when the flow is completely turbulent, the fluid becomes homogeneously disordered. Within the transition interval, heterogeneity arises in terms of space and time, and instantaneous images or punctual measurements can give different results. In the transition from laminar to turbulent flow, where turbulence is first encountered, it only occurs in localized patches, so called puffs, which are spatially separated by laminar flow [36]. The aim of the relationship developed here is to simulate the measurements of a spinner flowmeter by providing an expression that can generate the mean values of the local advance and temporal varia tions in this transition interval. At this scale, it is feasible to fit a mean behaviour to an empirical function within the transition interval [37], which scope must be taken in an engineering sense. About the expression for the velocity factor Fvel ¼
where R5 and p are fitting coefficients (R5 ¼ 2700 and p ¼ 6), and A (rr) y b (rr) are fitting sub-functions that depend on roughness, as given by the following expressions: ) Aðrr Þ ¼ 5000⋅ð1=rr Þ1:3 þ 800 � 0:1 (13b) bðrr Þ ¼ 0:23 rr Eq. (13a) was generated starting from the product of the Nikuradse relationship [28] for the friction factor in the fully developed turbulent flow, with a function that reflects the change in the friction factor from laminar flow (zero) to values established from this relationship, in such a way that the difference from the results of the Colebrook equation is minimized. A continuous expression for the friction factor for any value of roughness can be determined by the sum of Eqs. (12) and (13): f ¼ fsmooth þ f ðr1 ; ReÞ
(14)
The resulting curves are shown in Fig. 8, with the curves obtained from analytical fitting functions of the Colebrook equation. The mean relative deviation between both sets of curves for Re > 4000 is 1.6%. Given the very high accuracy of the approximations in Eq. (12) for smooth pipes, the somewhat reduced accuracy of Eq. (14) must be attributed to the use of Eq. (13). 4. Discussion With reference to the calibration process of well flowmeters, it is not a similar process than the precise method realised in laboratory (see for example [33] or [34]). One of the characteristics of flowmeter well logs is that measurements are made while sonde moves in depth to reduce the influence of the sensor threshold. Given the hard conditions of the well logs, it is advisable to repeat a calibration process just before to measure. The standard calibration process only consists in to displace up and
Fig. 8. Friction factor as a function of the relative roughness from the Cole brook equation and Eq. (14). 6
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Flow Measurement and Instrumentation 72 (2020) 101698
For the interval 102
[6] [7]
[8]
[9]
[10]
[11] [12]
5. Conclusions
[13]
New expressions for the velocity law, the velocity factor and the friction factor have been developed and their validity is verified using the deviation analysis presented and graphically illustrated in Figs. 4, 6 and 8. The improvements in the processing of flowmeter well logs have been optimized with respect to the flow regimes (laminar, transition and turbulent) produced in wells. There are many specialized publications in fluid mechanics that deal with turbulent flow, and the expressions re ported in these works have similar or better accuracy; however, the new relationships developed here cover the entire flow range using a continuous approach. In future research, the influence of this reformulation on the result ing permeability values in wells with conventional relationships versus depth should be quantified.
[14]
Author statement
[20]
Jesús Díaz-Curiel: Conceptualization, Methodology, Formal anal ysis, Investigation, Writing - Original Draft, Writing Review & Editing Preparation, Visualization, María Jesús Miguel: Resources, Writing �rbara Biosca: Inves Original Draft, Natalia Caparrini: Validation, Ba tigation, Resources, Writing Review & Editing Preparation, Lucía �valo-Lomas: Investigation, Resources, Writing - Review & Editing Are Preparation.
[21]
Acknowledgments
[24]
[15] [16] [17] [18] [19]
[22]
[23]
Part of this study was funded through the CARESOIL-CM (S2013/ MAE-2739 & P2018/EMT-4317) research grant of the regional gov ernment of Madrid (Comunidad de Madrid, Spain) and the DENSOIL (CTM2016-77151-C2-2-R) research grant from the Government of Spain (Ministerio de Economía y Competitividad). The authors also express their gratitude to Professor Eduardo de Miguel for his valuable inputs and considerations that have proven very useful to fulfil this work.
[25] [26] [27] [28]
References [29]
[1] R.C. Rumble, H.M. Buck, B.A. Peters, New subsurface flowmeter developed by Humble research engineers, Oil Gas J. (1959) 104–107. May. [2] H.L. Bryant, Production well logging techniques, Geophysics 25 (4) (1960) 905–927, https://doi.org/10.1190/1.1438771. [3] N. Agrawal, G.H. Choueiri, B. Hof, Transition to turbulence in particle laden flows, Phys. Rev. Lett. 122 (11) (2019) 114502. [4] P. Hufschmied, Estimation of three-dimensional statistically anisotropic hydraulic conductivity field by means of a single well pumping test combined with flowmeter measurements, Hydrogeologie 2 (1986) 163–174. [5] K.R. Rehfeldt, P. Hufschmied, L.W. Gelhar, M.E. Schaefer, Measuring Hydraulic Conductivity with the Borehole Flowmeter, Topical Report EN-6511, Electric
[30] [31] [32]
7
Power Research Institute (EPRI), Palo Alto, California, produced by Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1989. H. Darcy, Recherches exp�erimentales relatives au mouvement de l’eau dans les tuyaux [Experimental research on water movement in pipes], Mallet-Bachelier, Paris, 1857 (268 pages and atlas). H. Blasius, The law of similarity in friction processes in fluids (Das € Ahnlichkeitsgesetz bei Reibungsvorg€ angen in Flüssigkeiten), Forschungs-Arbeit des Ingenieur-Wesens 131 (1913), https://doi.org/10.1007/978-3-662-02239-9_1. Berlin. L. Prandtl, in: Neuere Ergebnisse der Turbulenzforschung. Zeitschrift des Vereines Deutscher lngenieure, vol. 77, Recent results of turbulence research. National Advisory Committee for Aeronautics (NACA) of USA, Technical Memorandum 720, 1933, pp. 105–114. G. Hagen, About The Movement of Water in Narrow Cylindrical Pipes (Über Die Bewegung des Wassers in engen zylindrischen R€ ohren), Poggendorfs Annalen der Physik und Chemie 16 (1839) 423–442, https://doi.org/10.1002/ andp.18391220304. J.L. Poiseuille, Experimental research on the movement of liquids in very small diameter tubes (Recherches exp� erimentales sur le mouvement des liquides dans les tubes de tr� es-petits diam� etres), Comptes rendus hebdomadaires des s� eances de l’Acad�emie des Sciences 12 (1841) 112–115. ISSN 0001-4036. H. Schlichting, Boundary-layer Theory, seventh ed., McGraw-Hill Book Co, 1979, ISBN 9783662529195, p. 150. Reference to Schlichting, H. 1932. Entstehung der Turbulenz in einem rotierenden Zylinder. Nachr. Ges. Wiss. G€ ottingen, 1932. M.V. Zagarola, A.J. Smits, Mean-flow scaling of turbulent pipe flow, J. Fluid Mech. 373 (1998) 33–79, https://doi.org/10.1017/S0022112098002419. W.K. George, Is there a universal log law for turbulent wall-bounded flows? Philos. Trans. Roy. Soc. A 365 (2007) 789–806, https://doi.org/10.1098/rsta.2008.0217. S.W. Churchill, Friction-factor equation spans all fluid flow regimes, Chem. Eng. 84 (24) (1977) 91–92. E. Romeo, C. Royo, A. Monz� on, Improved explicit equations for estimation of the friction factor in rough and smooth pipes, Chem. Eng. J. 86 (3) (2002) 369–374, https://doi.org/10.1016/S1385-8947(01)00254-6. B.J. McKeon, M.V. Zagarola, A.J. Smits, A new friction factor relationship for fully developed pipe flow, J. Fluid Mech. 538 (2005) 429–443, https://doi.org/ 10.1017/S0022112005005501. D.D. Joseph, B.H. Yang, Friction factor correlations for laminar, transition and turbulent flow in smooth pipes, Phys. Nonlinear Phenom. 239 (14) (2010) 1318–1328, https://doi.org/10.1016/j.physd.2009.09.026. Dresser Atlas, Interpretive methods for production well logs, in: Section 2: the Continuous Spinner Flowmeter: A Velocity Measuring Instrument, third ed., Dresser Industries, Inc, USA, 1982. J. Díaz-Curiel, D.A. Martín, A. Rozycki, A. Maldonado, Flowmeter Well logging in large diameter water wells (Testificaci� on de flowmeter en pozos de agua de gran di� ametro), in: Metodología y aplicaciones. Tecnología del Agua, vol. 129, Elsevier, Barcelona (Spain), 1994. O. Reynolds, On the dynamic theory of incompressible viscous fluids and the determination of the criterion, in: Philosophical Transactions of the Royal Society of London T 186 A, S. 123, 1895, https://doi.org/10.1098/rsta.1895.0004. B. Eckhardt, Introduction. Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds’ paper, Philos. Trans. Roy. Soc. A 367 (2009) 449–455. € T. Von K� arm� an, Mechanical Similarity and Turbulence (Mechanische Ahnlichkeit
und Turbulenz), in: C.W. Oseen, W. Weibull (Eds.), Proc. 3rd International Congress on Applied Mechanics (Stockholm, 1930), vol. I, AB Sveriges Lifografska Tryckenier, 1931, pp. 85–93, 1931. J. Nikuradse, Gesetzm€ assigkeiten der Turbulenten str€ omung in Glatten Rohren, Laws of Turbulent Flow in Smooth Pipes. 1966, 1-73, in: Forschung auf dem Gebiete des Ingenieurwesens 356, vol. 3National Aeronautic and Space Administration (NASA) of USA, 1932, pp. 1–76, https://doi.org/10.1007/ BF02716946. TT F-10, 359. S. Song, C. Jordan, D. Georgi, Theoretical and experimental studies on the determination of the average fluid velocity from spinner flowmeter responses, in: Transactions of 37th Annual Logging Symposium of SPWLA, Paper III, 1996, p. 10. Schlumberger, Fundamentals of Production Logging, Schlumberger, Briarpark Drive Houston, Texas, 2013, p. 77042, 13, 978-1-937949-07-5-3750. J. Weisbach, Textbook of Engineering and Machinery Mechanics (Lehrbuch der Ingenieur- und Maschinen-Mechanik), in: Theoretische Mechanik [Theoretical Mechanics], vol. 1, Vieweg und Sohn, Braunschweig, 1845, p. 535. C.F. Colebrook, Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws, J. Inst. Civ. Eng. (London) 11 (4) (1939) 133–156, 1938 and 1939. J. Nikuradse, Str€ omungsgesetze in rauhen Rohren, VDI-Forschungsheft, 361. Beilage zu Forschung auf dem Gebiete des Ingenieurwesens. Ausgabe B, Band 4, July-August. Laws of Flow in Rough Pipes, 1950, National Advisory Committee for Aeronautics (NACA), 1933. TM 1292. C.J. Swanson, B. Julian, G.G. Ihas, R.J. Donnelly, Pipe flow measurements over a wide range of Reynolds numbers using liquid helium and various gases, J. Fluid Mech. 461 (2002) 51–60. B.J. McKeon, C.J. Swanson, M.V. Zagarola, R.J. Donnelly, A.J. Smits, Friction factors for smooth pipe flow, J. Fluid Mech. 511 (2004) 41–44, https://doi.org/ 10.1017/S0022112004009796. F. Russo, N.T. Basse, Scaling of turbulence intensity for low-speed flow in smooth pipes, Flow Meas. Instrum. 52 (2016) 101–114. V.C. Patel, M.R. Head, Some observations on skin friction and velocity profiles in fully developed pipe and channel flows, J. Fluid Mech. 38 (1969) 181–201, https://doi.org/10.1017/S0022112069000115.
J. Díaz-Curiel et al.
Flow Measurement and Instrumentation 72 (2020) 101698 [41] M.T. Çoban, Error analysis of non-iterative friction factor formulas relative to Colebrook-White equation for the calculation of pressure drop in pipes, J. Naval Sci. Eng. 8 (1) (2012) 1–13. [42] A.R. Vatankhah, Approximate analytical solutions for the Colebrook equation, J. Hydraul. Eng. 144 (5) (2018), 06018007, https://doi.org/10.1061/(ASCE) HY.1943-7900.0001454. [43] D. Brki�c, P. Praks, Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright ω-function, Mathematics 7 (1) (2019) 34, https://doi.org/10.3390/math7010034. [44] P.K. Swamee, A.K. Jain, Explicit equations for pipe flow problems, J. Hydraul. Div. ASCE 102 (5) (1976) 657–664. [45] P.K. Swamee, Design of a submarine oil pipeline, J. Transport. Eng. 119 (1) (1993) 159–170, https://doi.org/10.1061/(ASCE)0733-947X(1993)119:1(159). [46] M. Ur�si�c, B. Kompare, Improvement of the hydraulic friction losses equations for flow under pressure in circular pipes, Acta Hydrotech. 21 (34) (2003) 57–74. ISSN 1581-0267. [47] N.S. Cheng, Formulas for friction factor in transitional regions, J. Hydraul. Eng., ASCE 134 (9) (2008) 1357–1362, https://doi.org/10.1061/(ASCE)0733-9429 (2008)134:9(1357). [48] D. Brki�c, P. Praks, Unified friction formulation from laminar to fully rough turbulent flow, Appl. Sci. 8 (2018) 2036, https://doi.org/10.3390/app8112036.
[33] B. Lee, R. Cheesewright, C. Clark, The dynamic response of small turbine flowmeters in liquid flows, Flow Meas. Instrum. 15 (2004) 239–248. [34] B. Wang, N. Zhang, Q. Cao, Y. Du, Evaluation approach to dynamic characteristic of turbine flowmeters considering calibration system response, Flow Meas. Instrum. 64 (2018) 126–132. � [35] Z. D�zemi�c, B. Sirok, B. Bizjan, Turbine flowmeter response to transitional flow regimes, Flow Meas. Instrum. 59 (2018) 18–22. [36] B. Hof, A. Juel, T. Mullin, Scaling of the turbulence transition threshold in a pipe, Phys. Rev. Lett. 91 (24) (2003) 244502. [37] G.I. Barenblatt, A.J. Chorin, Turbulence: an old challenge and new perspectives, Meccanica 33 (1998) 445–468, https://doi.org/10.1023/A:100431240. [38] H. Rouse, Evaluation of boundary roughness, in: Proceedings of the Second Hydraulic Conference, vol. 27, University of Iowa Bulletin, 1943. [39] L.F. Moody, Friction Factors for Pipe Flow, Transaction on Semi-annual Meeting of the American Society of Mechanical Engineers, Pittsburgh, Pensylvania, June 19–22, 1944. [40] F. Abdolahi, A. Mesbah, R.B. Boozarjomehry, W.Y. Svrcek, The effect of major parameters on simulation results of gas pipelines, Int. J. Mech. Sci. 49 (2007) 989–1000, https://doi.org/10.1016/j.ijmecsci.2006.12.001.
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