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Pressure transmission in yield stress fluids - An experimental analysis
Journal of Non-Newtonian Fluid Mechanics 261 (2018) 50–59
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Pressure transmission in yield stress fluids - An experimental analysis Rodrigo S. Mitishita, Gabriel M. Oliveira, Tainan G.M. Santos, Cezar O.R. Negrão∗ Federal University of Technology-Paraná – UTFPR, Academic Department of Mechanics – DAMEC, Post-graduate Program in Mechanical and Materials Engineering – PPGEM, Research Center for Rheology and Non-Newtonian Fluids – CERNN, Rua Deputado Heitor Alencar Furtado, 5000 - Bloco N - Ecoville - CEP 81280-340, Curitiba / PR - Brazil
a r t i c l e
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Keywords: Pressure transmission Yield stress fluid Laboratory-scale experiments
a b s t r a c t The current work presents an experimental investigation of the pressurization of a yield stress fluid contained in a closed pipeline under isothermal conditions. The tests were performed in a laboratory-scale flow loop placed inside a thermally controlled chamber. Sensors located along the pipeline measured fluid pressure and temperature. Differently from Newtonian fluids, experiments conducted with a viscoplastic fluid showed that the pressure imposed at one end of a closed pipeline was not fully transmitted to the other end, supporting prior mathematical model results. The results also revealed that the final pressure distribution was dependent not only on the fluid yield stress but also on the shear history the fluid underwent during pressurization and on the ratio between the pressure wave inertia and viscous dissipation. A comparison of the fluid yield stress obtained from rheometric measurements with the shear stress at the pipeline wall showed that they were of the same order of magnitude and that the higher the pressure wave inertia-viscous dissipation ratio the higher was the discrepancy between them.
1. Introduction The always-increasing demand for energy and the reduction of oil reserves have motivated the oil industry to drill deeper and deeper wells. These very long wells can negatively affect the pressure propagation in drilling fluids, which is essential for well control operations, mainly under static conditions. For example, completion valves installed at the drillpipe end, near the well bottom, are hydraulically opened by pressurizing the fluid at the well surface. Engineers have argued that the pressure imposed at the surface is not fully transmitted to the valve position, preventing its operation. A possible solution for the problem is the substitution of the drilling fluid by a Newtonian fluid, usually water, which allows a complete pressure transmission from the surface to the valve. Nevertheless, this expensive and time-consuming solution should be avoided. Drilling fluids are usually formulated as viscoplastic materials to inhibit flow below the fluid yield stress, precluding cuttings to drop to the well bottom under static conditions. On the other hand, some theoretical works in the literature [1,2] have demonstrated that the nontransmission of pressure is related to the fluid viscoplasticity. This pressure transmission problem in viscoplastic fluids confined in pipelines was firstly investigated mathematically by Oliveira et al. [1]. The authors showed that pressure was only transmitted in yield stress (or viscoplastic) materials if the shear stress at the pipeline wall, caused by pressure gradients, exceeded the fluid yield stress. The authors also con∗
cluded that the final pressure distribution along the pipeline was not only dependent on the fluid yield stress but also on the relation between the pressure wave inertia (product of fluid density and sound speed divided by pipe length) and the viscous dissipation (ratio of viscosity and square of pipe diameter). In another study developed by Oliveira et al. [2], they showed that the fluid compression rate could also affect the pressure propagation and consequently, the final pressure distribution along the pipeline. A similar situation of pressure propagation takes place in the wellknown water hammer problem that is caused by sudden valve closures during steady state flows in pipelines [3–5]. Despite this problem being extensively studied for many years, most publications have focused on Newtonian fluids [4,5]. Several authors [3,4,6–8] have proposed mathematical models to show the attenuation of pressure waves after a fast valve closure. Analogous to the pressure transmission problem in viscoplastic fluids, the pressure wave dissipation in water hammer is also dependent on the ratio between inertia and viscous forces. Differently from previous works, Wahba [9] and Oliveira et al. [10] studied the rapid valve closure in power law and Bingham fluid flows, respectively. While the first showed that the pressure wave attenuation increases with the power law index due to the higher viscous dissipation, the second verified a non-uniform pressure distribution along the pipeline after the complete pressure wave dissipation, similar to what was observed by [1] and [2]. Oliveira et al. [10] also demonstrated that the final pres-
Corresponding author. E-mail address: [email protected] (C.O.R. Negrão).
https://doi.org/10.1016/j.jnnfm.2018.08.007 Received 8 April 2018; Received in revised form 15 June 2018; Accepted 14 August 2018 Available online 18 August 2018 0377-0257/© 2018 Elsevier B.V. All rights reserved.
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Journal of Non-Newtonian Fluid Mechanics 261 (2018) 50–59
sure distribution was influenced by inertia, viscous dissipation and the yield stress magnitude. Additionally, pressure propagation in yield stress fluids during flow start-up has also been investigated mathematically by numerous authors [11–16]. While a constant pressure or flow rate is applied at the inlet during flow restarts, the outlet is open so that the fluid is displaced throughout the whole pipeline. Similar to what has been observed in problems where pressure is not transmitted, Negrão et al. [15] found out that the fluid flow did not start up if the pressure difference imposed to the pipeline was not enough to exceed the fluid yield stress. Numerous experimental works have also been conducted to deal with flow start-up of yield stress fluids [17–19]. Their focus was mainly on measuring the required time for a Newtonian fluid to expulse a gelled waxy crude oil from the pipeline. El Gendy et al. [18] also verified that the flow only started up if the wall shear stress surpassed the fluid yield stress. Rønningsen [20] reported good agreement between values of shear stress and shear rates for waxy crude oils measured in a flow loop and in a controlled stress rheometer. The deviations ranged from 15% to 20%. Similarly, Lee et al. [21] corroborated the flow restart pressures of a waxy crude oil by comparing rheometric data with measurements obtained from an experimental apparatus. Aqueous Carbopol solutions are commonly used as yield stress fluids in start-up experiments because they are transparent and also cheap and easy to formulate [22]. In spite of the slightly elastic behavior and weak thixotropy, Carbopol gels can be adequately described as a viscoplastic fluid [22,23]. Taghavi et al. [24], for instance, studied the displacement of a Carbopol solution by a Newtonian fluid and Alba et al. [25] extended the work of [24] for inclined pipelines. Sierra et al. [26] also evaluated the effect of the imposed pressure rate on the displacement of a Carbopol solution by a Newtonian fluid. Despite the interest of the petroleum industry in the pressure transmission problem, there is yet a lack of experimental studies in the area. For instance, the works of [1,2] stated that the pressure transmission could be affected not only by the yield stress, but also by the pressure wave inertia and viscous dissipation. However, the only experimental investigation found in the open literature [2] was performed in a fullscale drilling rig in which the operation was quite complex, the process variables were difficult to control, such as the fluid properties and temperature, and the results were not quite repeatable, so that further investigation is still needed. In order to fulfill this absence of results, the current work puts forward an experimental investigation of the pressure transmission problem in a long closed pipeline containing a viscoplastic fluid. For that purpose, a laboratory-scale flow loop was built and was employed to conduct transient and steady state experiments using a Carbopol solution as the working fluid.
(L1 ) away from P1 , P3 was 16 m (L2 ) from P2 and P4 was also 16 m (L3 ) from P3 . These three lengths summed up the total axial pipeline length of 48.3 m (L = L1 + L2 + L3 ). The 12 inch diaphragms of the pressure transducers were positioned nearly coincident with the inner wall of the pipe so as to avoid any potential measurement errors due to yield stress effects. The pipeline was built in a helical shape to reduce the space occupied by the system while keeping a long circuit length. A volume of 50 l of fluid was added to the system, out of which 17.3 l filled up the flow loop and the remaining was stored in the tank. Three electro-pneumatic valves installed at the inlet and outlet of the helical pipeline, V1 and V2 respectively, and in the bypass, V3 , controlled the fluid flow through the hydraulic circuit. Fluid was displaced exclusively to the helical pipeline (full black lines in Fig. 1) when the inlet and outlet valves were opened and the bypass valve was closed and flowed only through the bypass (dashed lines in Fig. 1) when V3 was opened and V1 and V2 were closed. A manual valve (MV) was also installed in the bypass pipeline downstream the automatic valve V3 . The purpose of this valve is explained in Section 3. A Coriolis flow meter, capable of measuring flow rates ranging from 0.015 to 1.53 l/s with 0.1 % uncertainty, was positioned downstream the outlet valve. This flow meter was also capable of measuring fluid density and temperature. The chamber temperature was controlled within 5 and 30°C by using a LabVIEW routine that actuated a refrigeration system and an electric heater. The controlled temperature was based on the average readings of eight type-T thermocouples installed along the helical pipeline walls. The oscillations of the controlled temperature during the experiments were not larger than ± 0.2 °C. The data acquisition-control system was responsible for measuring pressure, flow rates and fluid temperatures, for actuating the electro-pneumatic valves and for controlling the pump and the fluid mixer speeds. Fast pressure transients were detected by four pressure transducers with measuring range of 0 to 16 bar, accuracy of 0.1 % of the span, and frequency of 500 Hz. The pump speed was controlled from 0 to 105 rpm by a frequency inverter. 3. Experimental procedure The following steps describe the experimental protocol used for the pressure transmission experiments: •
•
•
2. Experimental setup A schematic representation and a photograph of the experimental apparatus are shown in Fig. 1. The experimental setup consisted of three main parts: hydraulic, temperature control and data acquisition systems. The hydraulic system installed inside a thermally isolated chamber was composed of a progressive cavity pump that sucked the working fluid from a 50 l storage tank (fluid reservoir) and delivered it either to a main pipeline that worked as the test section or to a bypass pipeline that prevented system overpressure. The pump operated within the pressure range of 0 to 12 bar and flow rates up to 0.37 l/s. The main pipeline was 48.3 m long, with an internal diameter of 20.45 mm, built in stainless steel and shaped in a helical form, while the bypass pipeline had an internal diameter of 13.8 mm. The helical diameter and pitch measured 727 mm and 52 mm, respectively. In order to homogenize the fluid mixture, a 1 hp electric agitator was installed on the top cover of the storage tank. Four diaphragm pressure transducers, indicated by P1 (inlet), P2 , P3 and P4 (outlet) in Fig. 1(a), were installed along the main pipeline walls to measure the fluid pressure inside the system. P2 was located 16.3 m
•
The chamber temperature was firstly controlled at the desired value during 90 minutes. After the temperature stabilization, the inlet valve (V1 ), the bypass valve (V3 ) and the manual bypass valve (initially completely open) were opened and the outlet valve (V2 ) was maintained closed. The tests were initiated by starting the data acquisition routine and 15 s later the pump was turned on to a desired flow rate. While the fluid flowed through the bypass, the pressure was increased within the main pipeline. Due to inertia, the pump rotor did not reach a constant speed instantaneously, and consequently the fluid was gradually pressurized to the desired final pressure. To further pressurize the fluid in the main pipeline, the manual bypass valve (MV) was partially closed to restrict fluid flow in the bypass. The pump was turned off after 90 s and the data acquisition was disabled after 150 s, ending the experiment.
It is worth mentioning that the system main control variable was the fluid pressure that depended not only on the flow rate controlled by the pump speed but also on the opening of the bypass manual valve. The experimental procedure in each condition was performed three times to assure repeatability and the measurements were carried out at 5 and 25°C to evaluate the temperature effect on the results. 4. Preliminary tests For the sake of comparison, pressure transmission experiments were firstly conducted with a Newtonian fluid and the results are discussed in Section 4.1. A summary of the test preparation and rheology of the 51
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Fig. 1. (a) Schematic representation and (b) photograph of the experimental apparatus.
Fig. 2. Time evolution of the pressure measured by sensors P1 and P4 during the pressurization of water to 3 bar at 5°C. Pressure readings were taken before and after closing V1 .
viscoplastic fluid, and also a comparison of the rheometer data with the experimental setup results are presented in the Section 4.2 and 4.3, respectively.
higher frequency oscillation taking place after the valve closure was possibly due to electrical noise. A comparison between frequencies of the pressure oscillations before closing valve V1 with different pump speeds showed that they were directly proportional, supporting the argument that the pressure oscillations were produced by the pump pulsations. It is also worth noting that after closing valve V1 the standard deviation of the pressure measurements was of the same order of magnitude of the transducer error band. As still shown in Fig. 2, the amplitude of the oscillations enlarges as the pressure increases from the start-up to the steady-state in 10 s. Other experiments performed at higher pressure levels showed that the magnitude of the oscillations increased with the pressure level once the pump ability of maintaining a constant flow rate was reduced as the pressure was magnified. Considering that the oscillations were caused by pump pulsations, the pressure transmission in the Newtonian fluid was verified by calculating the pressure differences between the average values measured at positions P1 , P2 , P3 and P4 during 10 s after the steady state was reached. Fig. 3 shows the pressure differences for three measurements performed at the same condition and also two dot-dashed lines that indicate the transducer error band (± 0.016 bar). As all the differences lied within
4.1. Pressure transmission in water In order to check if pressure was transmitted in Newtonian fluids, water was firstly employed as working fluid. Fig. 2 depicts the time evolution of the pressure measured at P1 and P4 positions during the pressurization of water in the closed helical pipeline. The pump was started at t = 0 s, and the pressure increased gradually to reach the steady-state in 10 s. As shown, the pressure readings of both P1 and P4 oscillated between the same average values, indicating that the pressure was instantaneously transmitted along the whole 48.3 m of the pipeline length. After 40 s approximately, the inlet valve V1 was closed isolating the fluid within the main pipeline. The pressure oscillations were thus reduced suggesting the pumping pulsations were mostly responsible for the oscillations. The steady state pressure measurements are zoomed in Fig. 2 to show the magnitude of oscillations before (inset 1) and after (inset 2) the inlet valve closure. Not only the magnitude of the oscillations decreased but also the frequency increased when the valve V1 was closed. The 52
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Fig. 3. Pressure differences along the main pipeline for three different pressure measurements taken at the same conditions.
Table 1 Carbopol solution properties (yield stress, consistency index, power-law index and density) for 5 and 25°C.
the transducer error bands, these results evinced not only the pressure transmission in Newtonian fluids but also the measurement repeatability. 4.2. Fluid preparation and rheology An ultrasound gel diluted in 30% of water was chosen as the working fluid for the current study. The gel manufactured by the Brazilian company RMC Gel Clínico was basically an aqueous Carbopol solution that was composed of carboxyvinyl polymer (Carbopol), imidazolidinyl urea, methylparaben, 2-amino-2-methyl-1-propanol and deionized water. For now on, this composition will be simply called as Carbopol solution. The ultrasound gel was chosen because it is more stable than the formulated working fluid, especially for the large quantities required. The density and the isothermal compressibility of the final solution were assumed equal to the corresponding water properties due to the low concentration of the polymer. According to [22,23], the flow curve of Carbopol solutions can be adequately represented by the Herschel-Bulkley equation: { 𝜏(𝛾) ̇ = 𝜏0 + 𝐾 𝛾̇ 𝑛 , if 𝜏 > 𝜏0 (1) 𝛾̇ = 0, if 𝜏 ≤ 𝜏0
Temperature [°C]
𝜏 0 [Pa]
k [Pa.sn ]
n [-]
𝜌 [kg/m3 ]
5 25
15.2 12.8
7.64 7.43
0.41 0.39
1000 999
The results shown in Fig. 4 and Table 1 were obtained soon after the solution had been prepared. In order to check for the solution degradation, a fresh sample was collected every week from the storage tank of the experimental apparatus and submitted to rheometric tests. It was observed that the fluid degradation occurred after two weeks. The results shown in this work were performed within a time interval where no significant fluid degradation was detected. 4.3. Comparison of the rheometric data with experimental setup results In order to verify the accuracy of the experimental apparatus, flow curves obtained from pressure differences and flow rates were compared to the rheometer flow curves. The flow curve tests were performed by maintaining all three valves, V1 , V2 and V3 , open in order to obtain a constant flow rate through the main pipeline. Different flow rates were imposed to the main pipeline by not only changing the pump speed but also setting up different positions to the manual bypass valve (MV). The rheometer experiments were performed at atmospheric pressure, while the pressures measured in the flow loop were gauge pressures. For each imposed flow rate measured by the Coriolis flow meter, a steady state pressure drop based on 10 s average values (Δ𝑃 = 𝑃1,𝑎𝑣𝑔 − 𝑃4,𝑎𝑣𝑔 )was obtained. The shear stress was then calculated from the pressure difference by using a force balance on the pipeline wall:
where 𝛾̇ is the shear rate, 𝜏(𝛾) ̇ is the shear stress as a function of 𝛾, ̇ K is the consistency index and n is the power-law index. It is worth noting that the elastic effects of the Carbopol solution were neglected as an approximation for the fluid characterization. Flow curves for the Carbopol solution based on the results of a controlled shear stress rheometer (DHR-3 manufactured by TA Instruments) were built. A 40 mm diameter parallel-plate with 0.8 mm gap and crosshatched surfaces was used to avoid wall slip. Nine shear rate steps, 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100 s−1 , were chosen for building up the flow curve. According to the results of [27], the minimum shear rate of 0.01 s−1 can be considered acceptable for good fit for the Herschel-Bulkley equation and for an extrapolation of the yield stress for Carbopol solutions. Initially, a sample of approximately 100 ml was mixed for 30 min using a magnetic mixer. After homogenization, the solution was introduced in the rheometer geometry and the temperature was maintained at either 5°C or 25°C. The fluid was then sheared at a fixed rate during a time interval of approximately 1∕𝛾̇ for all shear rate values. Finally, the constants of Eq. (1) were fit to the measured steady state values of 𝜏 [Pa] for each value of 𝛾̇ [s-1 ] and the results are presented in Fig. 4 for temperatures 5 and 25°C. The symbols shown in Fig. 4 correspond to the average of three measurements. The constants of Eq. (1) and the solution density are shown in Table 1 for 5 and 25°C. As shown, the yield stress, the consistency and the power-law indices decrease with the temperature increase.
𝐷Δ𝑃 (2) 4𝐿 where 𝜏 w is the shear stress at the pipeline wall, D is the pipeline internal diameter and L is the total pipeline length. The shear rate at the pipeline wall was calculated from the average flow rate by using the Weissenberg-Rabinowitsch equation, assuming no slip at the pipe walls, fully developed, laminar, steady, incompressible and axial flow [28]: ( ) 𝛾̇ 𝑑 ln 𝑄 𝛾̇ 𝑤 = 𝑎𝑤 3 + (3) 4 𝑑 ln 𝜏𝑤 𝜏𝑤 =
where 𝛾̇ 𝑎𝑤 = 4𝑄∕(𝜋(𝐷∕2)3 ) is the apparent shear rate, or the shear rate for a Newtonian fluid, at the pipeline wall. Although Carbopol solution is expected to slip on the smooth internal surface of the steel pipe, the flow loop experiments were performed at sufficiently high shear rates. 53
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Fig. 4. Flow curves obtained from the rheometer experiments at 5 and 25°C and Herschel-Bulkley equation data fit.
Fig. 5. Flow curves obtained from the rheometer and the experimental apparatus for the Carbopol solution at 5 and 25°C.
According to [29] and [30] that conducted rheometric measurements with rough and smooth surfaces, wall slip does not affect the results at high shear rates, for their particular cases, larger than 10 s−1 . Fig. 5 shows good agreement between the flow curves obtained from the rheometer and the experimental apparatus. The percentage differences between the measured shear stresses are within 13 %. As depicted, the experimental apparatus was unable to provide shear rates lower than 30 s−1 and shear stress higher than 100 Pa, which correspond to the limits of the flow loop. Despite the good agreement at high shear rates, discrepancies would be envisage at low shear rates because of the steel pipe wall slip and zero-shear rate extrapolation to obtain the yield stress from the flow loop results must be avoided. It is also important to discuss the influence of the helical geometry in the pressure measurements. According to [31], fluid flow in a curved pipeline is affected by centrifugal forces, which may affect pressure measurements. In order to check the influence of the helical shape on the results, the Dean number Dn [32,33] was computed: √ 𝐷𝑛 = Re
𝐷 𝐷𝑠𝑝𝑖𝑟𝑎𝑙
where Re is the Reynolds number and Dspiral is the diameter of the helix which is approximately 0.727 m. Prior laminar flow tests performed with glycerin, a Newtonian fluid, showed that deviations between the measured pressures and the calculated values for a straight pipeline are only significant when Dean number was larger than 50, corresponding to a Reynolds number of 238. The results displayed in Fig. 5 are for laminar flows with Dean numbers ranging from 0.1 to 12.5 so that centrifugal forces did not affected the Carbopol solution results. It is also worth mentioning that the fluid was almost stationary within the closed pipeline during the pressure transmission experiments, and consequently, the centrifugal forces were negligible.
5. Results and discussion This section presents the results for the pressure transmission experiments performed with the Carbopol solution. The effect of the initial pressure distribution on the pressure transmission and the effects of the yield stress and of the pressure wave inertia on the final pressure distribution are also discussed in this section.
(4)
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Fig. 6. Measured pressures at P1 and P4 as a function of time for pressurization experiments of (a) water and (b) Carbopol solution to the average final value of 6.8 bar at 5°C.
5.1. Transmission of pressure in the carbopol solution
tion in water. While the pressure P1 reaches the final value in approximately 10 s, the pressure wave finds the outlet in approximately 0.03 s. Differently from the Newtonian fluid, the pressure distribution in the Carbopol solution is non-uniform even at the initial condition (Fig. 8(b)). The discussion about how this non-uniform initial condition was achieved is presented in Section 5.2. After the fluid pressurization at t = 2 s, the pressures P1 (0 m) and P2 (16.3 m) increase whilst the pressures P3 (32.3 m) and P4 (48.3 m) remain unaffected, indicating the pressure wave is dissipated before reaching the outlet. While the pressure P1 increases from the start-up to the steady state, the pressures at the other sensors rise from the inlet to the outlet. The pressures closer to the inlet (near the pump) increase faster than the pressures near the outlet because of the high viscous dissipation in comparison to the pressure wave inertia. Finally, the final steady state pressure distribution reduces approximately linearly from the inlet to the outlet, as the pressure is no longer able to propagate in the pipeline.
Fig. 6 presents the pressure P1 and P4 as a function of time for water and for the Carbopol solution at 5°C, during the pressurization of the closed pipeline. For the water results shown in Fig. 6(a), the average values of pressure P1 and P4 are very similar throughout the whole experiment, indicating an instantaneous, fully transmission of pressure to the outlet. As noted, pressure P4 oscillations are higher than those at transducer P1 located near the pump outlet. This is explained by the reflection and amplification of the pressure waves at the closed valve V2 , near the transducer P4 . Despite the pressure evolution for water and the carbopol solution at transducer P1 being quite similar, the pressure P4 for the carbopol solution is not only delayed but also not fully transmitted to the outlet, as shown in Fig. 6(b). While the pressure P1 after stabilization averages 6.8 bar, the pressure P4 stabilizes at an average value of 5.6 bar, depicting a pressure difference of 1.2 bar along the pipeline length. It is worth mentioning that the pipeline was kept pressurized for three hours and the average pressure difference between P1 and P4 remained approximately the same for this long period, indicating that the pressure was indeed not transmitted. According to [1] and [2], the sooner the wall shear stress produced by the pressure difference drops below the fluid yield point, propagation stops and consequently, the pressure is not transmitted anymore. On the contrary, pressure is always transmitted in a Newtonian fluid, although delayed in high viscous dissipating system. In order to demonstrate repeatability, the steady state pressures at the four pressure transducers (P1 , P2 , P3 and P4 ) for three different measurements are presented in Fig. 7 as a function of the pipeline axial length. The pressure values are averaged over 10 s after the steady state has been reached and the error bars represent the standard deviations of the pressure averages. As shown, not only the average pressure values are fairly close to one another with a maximum deviation of 0.07 bar, but also the slopes of each curve are very similar. Fig. 8 compares the time evolution of the pressure distribution along the pipeline length during pressurization tests for water (Fig. 8(a)) and for the Carbopol solution (Fig. 8(b)) at 5°C. The pressure distribution at time t = 0 s indicates the initial condition at the pump start-up, whereas the other curves represent the pressure evolution to the steady-state (t = SS). The water pressurization (Fig. 8(a)) yields an almost uniform pressure distribution from t = 0 s to the steady-state. In other words, the pressure across the pipeline length remains uniform throughout the entire experiment because the fluid is slowly compressed in comparison to the speed that the pressure propagates in water. The pressure wave propagates almost at sound speed (∼1500 m/s) due to small viscous dissipa-
5.2. Effect of the initial pressure distribution The effect of the initial condition on the time evolution of pressure and on the final pressure distribution is now discussed. The pressurization tests started from the two different initial pressure distributions shown in Fig. 9. Excepting the initial pressure distribution, the other problem parameters, such as temperature and valve configuration, were kept unchanged for both experiments. Hereafter, the first initial pressure distribution is referenced as IPD1 and the second initial pressure distribution as IPD2. The condition IPD1 was obtained by switching off the pump after the pressure P1 reached steady state at approximately 6.0 bar, whilst maintaining valve V1 and the bypass open and the outlet valve V2 closed. As the outlet valve was kept closed, depressurization occurred only through the bypass (see Fig. 1). Considering the sensor P1 (pipeline inlet) was the nearest to the bypass exit, the pressure P1 decreased to a value close to atmospheric pressure (0.0 bar). Differently from Newtonian fluids (see pressures at t = 0 s in Fig. 6(a)), the pressure at the other sensors did not decrease as much as the pressure P1 . The condition IPD2, on the other hand, was obtained by opening the outlet valve V2 after the condition IPD1 was reached. Although both inlet and outlet were depressurized by opening valve V2 , the pressures P2 and P3 did not reduce to zero as depicted in Fig. 9. Additionally, P4 did not completely decrease to zero because of the fluid contained within the bends, flow meter and the pipeline downstream of transducer P4 . Since a pressure transducer was not installed halfway the pipeline length, three distinct pressure gradients seem to exist in Fig. 9 for the condition IPD2. However, only two pressure gradients along the pipe were expected, with the maximum pressure close to the middle and pressures near zero at both the inlet (P1 ) and the outlet (P4 ). 55
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Fig. 7. Steady-state pressure values along the pipeline for three measurements experiments taken at 6.8 bar and 5°C.
Fig. 8. Time evolution of the pressure distributions along the pipeline length during the pressurization experiments for (a) water and (b) the Carbopol solution at 5°C.
Fig. 9. Two initial pressure distribution along the pipeline for the Carbopol solution at 5°C.
56
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Fig. 10. Time evolution of the pressure distributions along the pipeline length during the pressurization of the Carbopol solution to an average steady state value of 3.0 bar at the inlet, starting from two initial conditions: (a) IPD1 and (b) IPD2 at 5°C.
Fig. 11. Time evolution of the pressure distributions along the pipeline length during the pressurization of the Carbopol solution to an average steady state value of 6.9 bar at the inlet, starting from two initial conditions: (a) IPD1 and (b) IPD2 at 5°C.
Two pressurization experiments were performed from each initial pressure condition IPD1 and IPD2 at 5°C. In the first, the pressure at P1 reached an average steady state value of 3.0 bar and in the second, 6.9 bar. Fig. 10 and Fig. 11 show the time evolution of the pressure distribution along the pipeline for the two pressure levels, respectively. While t = 0 s and t = SS are the initial condition and the steady state, respectively, the other curves correspond to the pressure distribution at specific instants after the pump has been switched on. Excepting the steady state pressures that are averages, the other values in Fig. 10 and Fig. 11 are instantaneous pressure measurements. Fig. 10(a) shows that the initial pressure distribution for the condition IPD1 (t = 0 s) is approximately linear, varying from 0 bar at the inlet to 1.4 bar at the outlet. As soon as the pump is turned on, the pressure P1 (t = 2 s) increases, producing a wave that raises the pressure almost sequentially at positions P2 and P3 . As noted, the pressure P4 is affected only 4 s after the pump has been started. Curiously, the pressure P4 at 4 s is altered before the pressure P3 exceeds the pressure P4 . The pressure was expected to increase sequentially from the inlet to outlet for high dissipation viscous fluids, such as viscoplastic [1,2]. On the contrary, the pressure seems to propagate in a non-dissipating medium, indicating the fluid elasticity may play a role on the propagation. Finally, the steady state pressure distribution reduces linearly from P1 to P3 and pressure P4 is almost the same as P3 . Fig. 10(b) illustrates that the pressures at the initial condition IPD2 are different and smaller than the pressures at IPD1. Consequently, the pressure evolves differently from the start-up to the steady state in comparison to the results of Fig. 10(a). Despite the steady state results of Fig. 10(a) and (b) being similar from P1 to P3 , the pressures at P4 are different. While the final pressure P4 for condition IPD1 is 2.1 bar, its counterpart for condition IPD2 is only 1.7 bar. As noted in Fig. 10(a)
and (b), the pressure P1 has almost the same value for every instant of time and the pressure P4 is only affected 4 s after the start-up, meaning that the time for the pressure wave to reach the outlet is independent of the initial condition. According to [1], pressure propagation depends on two opposing factors: pressure wave inertia and viscous dissipation – the higher the first in comparison to the second the faster is the pressure wave – so that the initial condition should not affect pressure transmission. Additionally, elasticity must be equally present in both cases since the pressure is similarly transmitted to the outlet, although this is not evident for condition IPD2. Conclusively, the pressure P4 for condition IPD1 evolves to higher values because it has already started from a higher stored energy condition (higher initial pressure). Fig. 11, on the other hand, reveals that the final pressure distribution is independent on the initial condition when the position P1 is pressurized to 6.9 bar. The comparison of Fig. 11(a) and (b) shows that the pressures for conditions IPD1 and IPD2 evolve differently up to 8 s, and then they both increase almost linearly to the steady-state, indicating that the effect of the initial condition is mitigated after 8 s. Fig. 12 compares the steady state for both initial conditions, exhibiting the final pressure distribution is only affected by the initial condition for lower levels of pressurization. Therefore, the final pressure distribution is dependent not only on the initial pressure distribution, but also on the pressure level that may fade the effect of the pressure history. 5.3. Final pressure distribution The wall shear stresses obtained from the stabilized values of pressure along the pipeline are now compared with the fluid yield stress measured in the rheometer. The wall shear stress was computed by using Eq. (2), considering the average pressures of three measurements, 57
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Fig. 12. Effect of the initial pressure condition on the steady state pressure distribution along the pipeline after the pipe inlet being pressurized to (a) 3 bar and (b) 6.9 bar at 5°C.
Fig. 13. Dimensionless pressure distributions along the pipeline length for different pressure levels: 3 bar (l = 0.18; BR = 0.48), 6.8 bar (l = 0.93; BR = 0.21) and 9 bar (l = 1.4; BR = 0.16) at 5°C.
Table 2 Comparison of the wall shear stress, 𝜏 w , averaged over three pressurization experiments, with the fluid yield stress, 𝜏 0 = 15.2 Pa, at 5°C. Pressure [bar]
𝜏 w1 [Pa]
(𝜏𝑤1 −𝜏0 ) 𝜏0
3.0 6.8 9.0
14.1 12.1 9.7
7% 20% 36%
× 100%
𝜏 w2
[ Pa ]
14.2 12.6 11.6
(𝜏𝑤2 −𝜏0 ) 𝜏0
stress may be the reason for the non-transmission of pressure, but it is now exclusively responsible for the final pressure distribution. As already mentioned, [1] reported that the final pressure distribution depends not only on the fluid yield stress, but also on the counterbalance between two opposing effects: the pressure wave inertia and viscous dissipation. To account for these effects, [1] defined a Bingham number:
× 100%
7% 17% 24%
𝐵𝑅 = 4𝐿𝜏0 ∕𝑃𝑅 𝐷
(5)
and a dimensionless parameter which is the ratio between pressure wave inertia and viscous dissipation for the Bingham fluid. This parameter is redefined here for the Herchel–Bulkley fluid: ( )1∕𝑛−1 ( ) 𝜌𝑐𝐷 ( 1 𝐷 )1∕𝑛 ( 4𝑛 ) P𝑅 𝜆= (6) 𝐾 8𝐾 4𝐿 3𝑛 + 1
assuming a linear pressure variation along the pipeline at 5°C and taking into account only cases in which the initial condition did not affect the final pressure distribution. Table 2 shows the comparison for three levels of pressurization between the yield stress and two values of wall shear stress: the first, 𝜏 w1 , was evaluated based on the pressure difference P1 P4 and the second, 𝜏 w2 , was calculated using the pressure difference P2 - P4 . As noted, 𝜏 w1 is closer to 𝜏 0 than 𝜏 w2 and the higher the pressurization level the higher is the difference between the wall shear and yield stress. The dimensionless pressure distributions for these three levels of pressurization are also presented in Fig. 13, depicting that the higher the pressure level the more distant is the pressure distribution from linearity. It is also worth noting that the pressures P2 , P3 and P4 are aligned on a straight line for the three level of pressurization, explaining the fact of 𝜏 w1 being closer to 𝜏 0 than 𝜏 w2 . These results evince that the shear
where PR is the inlet pipeline pressure and c is the pressure wave speed. [1] states that: i) for high dissipating cases, i.e. low values of 𝜆, the final dimensionless pressure distribution is linear, depends only on the Bingham number and the pressure gradient is equal to -BR ; ii) for low dissipating cases, i.e. high values of 𝜆,the final dimensionless pressure distribution deviates from linearity, because the pressure wave reflects at the outlet, and depends on both Bingham number and on 𝜆. The values of BR and 𝜆 computed by using Eqs. (4) and (5), respectively, are depicted in Fig. 13 for the three levels of pressurization. As seen, the re58
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sults of Fig. 13 are qualitatively similar to those presented and discussed in [1]. In other words, the pressure distribution is almost linear and the pressure gradient is quite closed to -BR for low values of 𝜆. Additionally, the pressure distribution deviates as much as from a straight line as 𝜆 increases, denoting pressure reflection at the outlet.
[7] B. Brunone, B.W. Karney, M. Mecarelli, M. Ferrante, Velocity profile and unsteady pipe friction, J. Water Resour. Plan. Manag. (2000) 236–244. [8] E.M. Wahba, Modelling the attenuation of laminar fluid transients in piping systems, Appl. Math. Model. 32 (2008) 2863–2871, doi:10.1016/j.apm.2007.10.004. [9] E.M. Wahba, Non-Newtonian fluid hammer in elastic circular pipes: Shear-thinning and shear-thickening effects, J. Nonnewton. Fluid Mech. 198 (2013) 24–30, doi:10.1016/j.jnnfm.2013.04.007. [10] G.M. Oliveira, A.T. Franco, C.O.R. Negrão, Mathematical Model for Viscoplastic Fluid Hammer, J. Fluids Eng. 138 (2016) 011301, doi:10.1115/1.4031001. [11] J. Sestak, M.E. Charles, M.G. Cawkwell, M. Houska, Start-up of gelled crude oil pipeline, J. Pipelines. 6 (1987) 15–24. [12] G. Vinay, A. Wachs, I. Frigaard, Start-up transients and efficient computation of isothermal waxy crude oil flows, J. Nonnewton. Fluid Mech. 143 (2007) 141–156, doi:10.1016/j.jnnfm.2007.02.008. [13] A. Wachs, G. Vinay, I. Frigaard, A 1.5D numerical model for the start up of weakly compressible flow of a viscoplastic and thixotropic fluid in pipelines, J. Nonnewton. Fluid Mech. 159 (2009) 81–94, doi:10.1016/j.jnnfm.2009.02.002. [14] G.M. de Oliveira, L.L.V.D. Rocha, A.T. Franco, C.O.R. Negrão, Numerical simulation of the start-up of Bingham fluid flows in pipelines, J. Nonnewton. Fluid Mech. 165 (2010) 1114–1128, doi:10.1016/j.jnnfm.2010.05.009. [15] C.O.R. Negrão, A.T. Franco, L.L.V Rocha, A weakly compressible flow model for the restart of thixotropic drilling fluids, J. Nonnewton. Fluid Mech. 166 (2011) 1369– 1381, doi:10.1016/j.jnnfm.2011.09.001. [16] G.M. de Oliveira, C.O.R. Negrão, The effect of compressibility on flow startup of waxy crude oils, J. Nonnewton. Fluid Mech. 220 (2015) 137–147, doi:10.1016/j.jnnfm.2014.12.010. [17] P.B. Smith, R.M.J. Ramsden, The Prediction Of Oil Gelation In Submarine Pipelines And The Pressure Required For Restarting Flow, SPE Eur. Pet. Conf. (1978) 283–290, doi:10.2118/8071-MS. [18] H. El-Gendy, M Alcoutlabi, M. Jemmett, M. Deo, J. Magda, R. Venkatesan, A. Montesi, The propagation of pressure in a gelled waxy oil pipeline as studied by particle imaging velocimetry, AIChE J. 58 (2012) 302–311, doi:10.1002/aic.12560. [19] L. Kumar, O. Skjæraasen, K. Hald, K. Paso, J. Sjöblom, Nonlinear rheology and pressure wave propagation in a thixotropic elasto-viscoplastic fluids, in the context of flow restart, J. Nonnewton. Fluid Mech. 231 (2016) 11–25, doi:10.1016/j.jnnfm.2016.01.013. [20] H.P. Rønningsen, Rheological Behavior of Gelled, Waxy North Sea Crude Oils, J. Pet. Sci. Eng. 7 (1992) 177–213. [21] H.S. Lee, P. Singh, W.H. Thomason, H.S. Fogler, Waxy oil gel breaking mechanisms: Adhesive versus cohesive failure, Energy Fuels 22 (2008) 480–487, doi:10.1021/ef700212v. [22] N.J. Balmforth, I.A. Frigaard, G. Ovarlez, Yielding to Stress: Recent Developments in Viscoplastic Fluid Mechanics, Annu. Rev. Fluid Mech. 46 (2013) 121–146, doi:10.1146/annurev-fluid-010313-141424. [23] J.M. Piau, Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges. Meso- and macroscopic properties, constitutive equations and scaling laws, J. Nonnewton. Fluid Mech. 144 (2007) 1–29, doi:10.1016/j.jnnfm.2007.02.011. [24] S.M. Taghavi, K. Alba, M. Moyers-Gonzalez, I.A. Frigaard, Incomplete fluid-fluid displacement of yield stress fluids in near-horizontal pipes: Experiments and theory, J. Nonnewton. Fluid Mech. 167–168 (2012) 59–74, doi:10.1016/j.jnnfm.2011.10.004. [25] K. Alba, S.M. Taghavi, J.R. de Bruyn, I.A. Frigaard, Incomplete fluid-fluid displacement of yield-stress fluids. Part 2: Highly inclined pipes, J. Nonnewton. Fluid Mech. 201 (2013) 80–93, doi:10.1016/j.jnnfm.2013.07.006. [26] A. Gaona Sierra, P. Ribeiro Varges, S. Santiago Ribeiro, Startup flow of elastoviscoplastic thixotropic materials in pipes, J. Pet. Sci. Eng. 147 (2016) 427–434, doi:10.1016/j.petrol.2016.09.003. [27] J. Pérez-González, J.J. López-Durán, B.M. Marín-Santibáñez, F. Rodríguez-González, Rheo-PIV of a yield-stress fluid in a capillary with slip at the wall, Rheol. Acta. 51 (2012) 937–946, doi:10.1007/s00397-012-0651-9. [28] H. Hu, T. Saga, T. Kobayashi, N. Taniguchi, Mixing process in a lobed jet flow, Wiley-VCH, New York, 2002, doi:10.2514/3.15201. [29] A. Poumaere, M. Moyers-González, C. Castelain, T. Burghelea, Unsteady laminar flows of a carbopol® gel in the presence of wall slip, J. Nonnewton. Fluid Mech. 205 (2014) 28–40, doi:10.1016/j.jnnfm.2014.01.003. [30] V. Bertola, F. Bertrand, H. Tabuteau, D. Bonn, P. Coussot, Wall slip and yielding in pasty materials, J. Rheol. (N. Y. N. Y) 47 (2003) 1211–1226, doi:10.1122/1.1595098. [31] M. Ghobadi, Y.S. Muzychka, A Review of Heat Transfer and Pressure Drop Correlations for Laminar Flow in Curved Circular Ducts, Heat Transf. Eng. 37 (2016) 815–839, doi:10.1080/01457632.2015.1089735. [32] W.R. Dean, LXXII. The stream-line motion of fluid in a curved pipe (Second paper), London, Edinburgh, Dublin Philos, Mag. J. Sci. 5 (1928) 673–695, doi:10.1080/14786440408564513. [33] W.R. Dean, XVI. Note on the motion of fluid in a curved pipe, London, Edinburgh, Dublin Philos. Mag. J. Sci. 4 (1927) 208–223, doi:10.1080/14786440708564324.
6. Conclusions Pressure transmission in a yield stress fluid compressed in a long pipeline under isothermal conditions was studied experimentally. An ultrasound medical gel diluted in water (Carbopol solution) was used to represent a viscoplastic fluid. Results for water were obtained to verify the pressure transmission in a Newtonian fluid, to check the accuracy of the experimental setup and also to compare with the viscoplastic fluid results. The main conclusions of the pressurization experiments with the Carbopol solution can be summarized as: i. The pressure increases uniformly for water and sequentially from the inlet to the pipe outlet for the viscoplastic fluid during the pressurization. At the steady state, the pressure distribution remains uniform for the Newtonian fluid and decreases from the inlet to the outlet for the yield stress fluid. These results corroborate previous numerical works. ii. The time evolution of the pressure distribution depends on the initial condition (pressure history) for low levels of pressurization. Above a certain limit, the pressure distribution evolves independently of the pressure history. iii. The fluid elasticity seems to play a role when the pressure starts to propagate along the pipeline. As the time elapses, the elasticity fades and the pressure propagation evolves based on two concurrent effects: pressure wave inertia and viscous dissipation. iv. The final pressure distribution is linear and only a function of the yield stress for low values of 𝜆 (pressure wave inertia-viscous dissipation ratio) and deviates from linearity, depending on both yield stress and 𝜆 for high values of 𝜆. Acknowledgements We would like to acknowledge the financial support of PETROBRAS, FINEP (Funding Authority for Studies and Projects), CNPq (Brazilian Research Foundation) and CAPES (Brazilian Graduate Education Foundation). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jnnfm.2018.08.007. References [1] G.M. Oliveira, C.O.R. Negrão, A.T. Franco, Pressure transmission in Bingham fluids compressed within a closed pipe, J. Nonnewton. Fluid Mech. 169–170 (2012) 121– 125, doi:10.1016/j.jnnfm.2011.11.004. [2] G.M. de Oliveira, A.T. Franco, C.O.R. Negrão, A.L. Martins, R.A. Silva, Modeling and validation of pressure propagation in drilling fluids pumped into a closed well, J. Pet. Sci. Eng. 103 (2013) 61–71, doi:10.1016/j.petrol.2013.02.012. [3] E.B. Wylie, V.L. Streeter, L. Suo, Fluid Transients in Systems, Prentice Hall, New Jersey, 1993. [4] M.S. Ghidaoui, M. Zhao, D.A. McInnis, D.H. Axworthy, A review of water hammer theory and practice, Appl. Mech. Rev. 58 (2005) 49, doi:10.1115/1.1828050. [5] A. Bergant, A.R. Simpson, A.S. Tijsseling, Water hammer with column separation: A historical review, J. Fluids Struct 22 (2006) 135–171, doi:10.1016/j.jfluidstructs.2005.08.008. [6] B. Brunone, U.M. Golia, M. Greco, Effects of two-dimensionality on pipe transients modeling, J. Hydraul. Eng. 121 (1995) 906–912, doi:10.1061/(ASCE)0733-9429(1995)121:12(906).
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Pressure transmission in yield stress fluids - An experimental analysis Rodrigo S. Mitishita, Gabriel M. Oliveira, Tainan G.M. Santos, Cezar O.R. Negrão∗ Federal University of Technology-Paraná – UTFPR, Academic Department of Mechanics – DAMEC, Post-graduate Program in Mechanical and Materials Engineering – PPGEM, Research Center for Rheology and Non-Newtonian Fluids – CERNN, Rua Deputado Heitor Alencar Furtado, 5000 - Bloco N - Ecoville - CEP 81280-340, Curitiba / PR - Brazil
a r t i c l e
i n f o
Keywords: Pressure transmission Yield stress fluid Laboratory-scale experiments
a b s t r a c t The current work presents an experimental investigation of the pressurization of a yield stress fluid contained in a closed pipeline under isothermal conditions. The tests were performed in a laboratory-scale flow loop placed inside a thermally controlled chamber. Sensors located along the pipeline measured fluid pressure and temperature. Differently from Newtonian fluids, experiments conducted with a viscoplastic fluid showed that the pressure imposed at one end of a closed pipeline was not fully transmitted to the other end, supporting prior mathematical model results. The results also revealed that the final pressure distribution was dependent not only on the fluid yield stress but also on the shear history the fluid underwent during pressurization and on the ratio between the pressure wave inertia and viscous dissipation. A comparison of the fluid yield stress obtained from rheometric measurements with the shear stress at the pipeline wall showed that they were of the same order of magnitude and that the higher the pressure wave inertia-viscous dissipation ratio the higher was the discrepancy between them.
1. Introduction The always-increasing demand for energy and the reduction of oil reserves have motivated the oil industry to drill deeper and deeper wells. These very long wells can negatively affect the pressure propagation in drilling fluids, which is essential for well control operations, mainly under static conditions. For example, completion valves installed at the drillpipe end, near the well bottom, are hydraulically opened by pressurizing the fluid at the well surface. Engineers have argued that the pressure imposed at the surface is not fully transmitted to the valve position, preventing its operation. A possible solution for the problem is the substitution of the drilling fluid by a Newtonian fluid, usually water, which allows a complete pressure transmission from the surface to the valve. Nevertheless, this expensive and time-consuming solution should be avoided. Drilling fluids are usually formulated as viscoplastic materials to inhibit flow below the fluid yield stress, precluding cuttings to drop to the well bottom under static conditions. On the other hand, some theoretical works in the literature [1,2] have demonstrated that the nontransmission of pressure is related to the fluid viscoplasticity. This pressure transmission problem in viscoplastic fluids confined in pipelines was firstly investigated mathematically by Oliveira et al. [1]. The authors showed that pressure was only transmitted in yield stress (or viscoplastic) materials if the shear stress at the pipeline wall, caused by pressure gradients, exceeded the fluid yield stress. The authors also con∗
cluded that the final pressure distribution along the pipeline was not only dependent on the fluid yield stress but also on the relation between the pressure wave inertia (product of fluid density and sound speed divided by pipe length) and the viscous dissipation (ratio of viscosity and square of pipe diameter). In another study developed by Oliveira et al. [2], they showed that the fluid compression rate could also affect the pressure propagation and consequently, the final pressure distribution along the pipeline. A similar situation of pressure propagation takes place in the wellknown water hammer problem that is caused by sudden valve closures during steady state flows in pipelines [3–5]. Despite this problem being extensively studied for many years, most publications have focused on Newtonian fluids [4,5]. Several authors [3,4,6–8] have proposed mathematical models to show the attenuation of pressure waves after a fast valve closure. Analogous to the pressure transmission problem in viscoplastic fluids, the pressure wave dissipation in water hammer is also dependent on the ratio between inertia and viscous forces. Differently from previous works, Wahba [9] and Oliveira et al. [10] studied the rapid valve closure in power law and Bingham fluid flows, respectively. While the first showed that the pressure wave attenuation increases with the power law index due to the higher viscous dissipation, the second verified a non-uniform pressure distribution along the pipeline after the complete pressure wave dissipation, similar to what was observed by [1] and [2]. Oliveira et al. [10] also demonstrated that the final pres-
Corresponding author. E-mail address: [email protected] (C.O.R. Negrão).
https://doi.org/10.1016/j.jnnfm.2018.08.007 Received 8 April 2018; Received in revised form 15 June 2018; Accepted 14 August 2018 Available online 18 August 2018 0377-0257/© 2018 Elsevier B.V. All rights reserved.
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sure distribution was influenced by inertia, viscous dissipation and the yield stress magnitude. Additionally, pressure propagation in yield stress fluids during flow start-up has also been investigated mathematically by numerous authors [11–16]. While a constant pressure or flow rate is applied at the inlet during flow restarts, the outlet is open so that the fluid is displaced throughout the whole pipeline. Similar to what has been observed in problems where pressure is not transmitted, Negrão et al. [15] found out that the fluid flow did not start up if the pressure difference imposed to the pipeline was not enough to exceed the fluid yield stress. Numerous experimental works have also been conducted to deal with flow start-up of yield stress fluids [17–19]. Their focus was mainly on measuring the required time for a Newtonian fluid to expulse a gelled waxy crude oil from the pipeline. El Gendy et al. [18] also verified that the flow only started up if the wall shear stress surpassed the fluid yield stress. Rønningsen [20] reported good agreement between values of shear stress and shear rates for waxy crude oils measured in a flow loop and in a controlled stress rheometer. The deviations ranged from 15% to 20%. Similarly, Lee et al. [21] corroborated the flow restart pressures of a waxy crude oil by comparing rheometric data with measurements obtained from an experimental apparatus. Aqueous Carbopol solutions are commonly used as yield stress fluids in start-up experiments because they are transparent and also cheap and easy to formulate [22]. In spite of the slightly elastic behavior and weak thixotropy, Carbopol gels can be adequately described as a viscoplastic fluid [22,23]. Taghavi et al. [24], for instance, studied the displacement of a Carbopol solution by a Newtonian fluid and Alba et al. [25] extended the work of [24] for inclined pipelines. Sierra et al. [26] also evaluated the effect of the imposed pressure rate on the displacement of a Carbopol solution by a Newtonian fluid. Despite the interest of the petroleum industry in the pressure transmission problem, there is yet a lack of experimental studies in the area. For instance, the works of [1,2] stated that the pressure transmission could be affected not only by the yield stress, but also by the pressure wave inertia and viscous dissipation. However, the only experimental investigation found in the open literature [2] was performed in a fullscale drilling rig in which the operation was quite complex, the process variables were difficult to control, such as the fluid properties and temperature, and the results were not quite repeatable, so that further investigation is still needed. In order to fulfill this absence of results, the current work puts forward an experimental investigation of the pressure transmission problem in a long closed pipeline containing a viscoplastic fluid. For that purpose, a laboratory-scale flow loop was built and was employed to conduct transient and steady state experiments using a Carbopol solution as the working fluid.
(L1 ) away from P1 , P3 was 16 m (L2 ) from P2 and P4 was also 16 m (L3 ) from P3 . These three lengths summed up the total axial pipeline length of 48.3 m (L = L1 + L2 + L3 ). The 12 inch diaphragms of the pressure transducers were positioned nearly coincident with the inner wall of the pipe so as to avoid any potential measurement errors due to yield stress effects. The pipeline was built in a helical shape to reduce the space occupied by the system while keeping a long circuit length. A volume of 50 l of fluid was added to the system, out of which 17.3 l filled up the flow loop and the remaining was stored in the tank. Three electro-pneumatic valves installed at the inlet and outlet of the helical pipeline, V1 and V2 respectively, and in the bypass, V3 , controlled the fluid flow through the hydraulic circuit. Fluid was displaced exclusively to the helical pipeline (full black lines in Fig. 1) when the inlet and outlet valves were opened and the bypass valve was closed and flowed only through the bypass (dashed lines in Fig. 1) when V3 was opened and V1 and V2 were closed. A manual valve (MV) was also installed in the bypass pipeline downstream the automatic valve V3 . The purpose of this valve is explained in Section 3. A Coriolis flow meter, capable of measuring flow rates ranging from 0.015 to 1.53 l/s with 0.1 % uncertainty, was positioned downstream the outlet valve. This flow meter was also capable of measuring fluid density and temperature. The chamber temperature was controlled within 5 and 30°C by using a LabVIEW routine that actuated a refrigeration system and an electric heater. The controlled temperature was based on the average readings of eight type-T thermocouples installed along the helical pipeline walls. The oscillations of the controlled temperature during the experiments were not larger than ± 0.2 °C. The data acquisition-control system was responsible for measuring pressure, flow rates and fluid temperatures, for actuating the electro-pneumatic valves and for controlling the pump and the fluid mixer speeds. Fast pressure transients were detected by four pressure transducers with measuring range of 0 to 16 bar, accuracy of 0.1 % of the span, and frequency of 500 Hz. The pump speed was controlled from 0 to 105 rpm by a frequency inverter. 3. Experimental procedure The following steps describe the experimental protocol used for the pressure transmission experiments: •
•
•
2. Experimental setup A schematic representation and a photograph of the experimental apparatus are shown in Fig. 1. The experimental setup consisted of three main parts: hydraulic, temperature control and data acquisition systems. The hydraulic system installed inside a thermally isolated chamber was composed of a progressive cavity pump that sucked the working fluid from a 50 l storage tank (fluid reservoir) and delivered it either to a main pipeline that worked as the test section or to a bypass pipeline that prevented system overpressure. The pump operated within the pressure range of 0 to 12 bar and flow rates up to 0.37 l/s. The main pipeline was 48.3 m long, with an internal diameter of 20.45 mm, built in stainless steel and shaped in a helical form, while the bypass pipeline had an internal diameter of 13.8 mm. The helical diameter and pitch measured 727 mm and 52 mm, respectively. In order to homogenize the fluid mixture, a 1 hp electric agitator was installed on the top cover of the storage tank. Four diaphragm pressure transducers, indicated by P1 (inlet), P2 , P3 and P4 (outlet) in Fig. 1(a), were installed along the main pipeline walls to measure the fluid pressure inside the system. P2 was located 16.3 m
•
The chamber temperature was firstly controlled at the desired value during 90 minutes. After the temperature stabilization, the inlet valve (V1 ), the bypass valve (V3 ) and the manual bypass valve (initially completely open) were opened and the outlet valve (V2 ) was maintained closed. The tests were initiated by starting the data acquisition routine and 15 s later the pump was turned on to a desired flow rate. While the fluid flowed through the bypass, the pressure was increased within the main pipeline. Due to inertia, the pump rotor did not reach a constant speed instantaneously, and consequently the fluid was gradually pressurized to the desired final pressure. To further pressurize the fluid in the main pipeline, the manual bypass valve (MV) was partially closed to restrict fluid flow in the bypass. The pump was turned off after 90 s and the data acquisition was disabled after 150 s, ending the experiment.
It is worth mentioning that the system main control variable was the fluid pressure that depended not only on the flow rate controlled by the pump speed but also on the opening of the bypass manual valve. The experimental procedure in each condition was performed three times to assure repeatability and the measurements were carried out at 5 and 25°C to evaluate the temperature effect on the results. 4. Preliminary tests For the sake of comparison, pressure transmission experiments were firstly conducted with a Newtonian fluid and the results are discussed in Section 4.1. A summary of the test preparation and rheology of the 51
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Fig. 1. (a) Schematic representation and (b) photograph of the experimental apparatus.
Fig. 2. Time evolution of the pressure measured by sensors P1 and P4 during the pressurization of water to 3 bar at 5°C. Pressure readings were taken before and after closing V1 .
viscoplastic fluid, and also a comparison of the rheometer data with the experimental setup results are presented in the Section 4.2 and 4.3, respectively.
higher frequency oscillation taking place after the valve closure was possibly due to electrical noise. A comparison between frequencies of the pressure oscillations before closing valve V1 with different pump speeds showed that they were directly proportional, supporting the argument that the pressure oscillations were produced by the pump pulsations. It is also worth noting that after closing valve V1 the standard deviation of the pressure measurements was of the same order of magnitude of the transducer error band. As still shown in Fig. 2, the amplitude of the oscillations enlarges as the pressure increases from the start-up to the steady-state in 10 s. Other experiments performed at higher pressure levels showed that the magnitude of the oscillations increased with the pressure level once the pump ability of maintaining a constant flow rate was reduced as the pressure was magnified. Considering that the oscillations were caused by pump pulsations, the pressure transmission in the Newtonian fluid was verified by calculating the pressure differences between the average values measured at positions P1 , P2 , P3 and P4 during 10 s after the steady state was reached. Fig. 3 shows the pressure differences for three measurements performed at the same condition and also two dot-dashed lines that indicate the transducer error band (± 0.016 bar). As all the differences lied within
4.1. Pressure transmission in water In order to check if pressure was transmitted in Newtonian fluids, water was firstly employed as working fluid. Fig. 2 depicts the time evolution of the pressure measured at P1 and P4 positions during the pressurization of water in the closed helical pipeline. The pump was started at t = 0 s, and the pressure increased gradually to reach the steady-state in 10 s. As shown, the pressure readings of both P1 and P4 oscillated between the same average values, indicating that the pressure was instantaneously transmitted along the whole 48.3 m of the pipeline length. After 40 s approximately, the inlet valve V1 was closed isolating the fluid within the main pipeline. The pressure oscillations were thus reduced suggesting the pumping pulsations were mostly responsible for the oscillations. The steady state pressure measurements are zoomed in Fig. 2 to show the magnitude of oscillations before (inset 1) and after (inset 2) the inlet valve closure. Not only the magnitude of the oscillations decreased but also the frequency increased when the valve V1 was closed. The 52
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Fig. 3. Pressure differences along the main pipeline for three different pressure measurements taken at the same conditions.
Table 1 Carbopol solution properties (yield stress, consistency index, power-law index and density) for 5 and 25°C.
the transducer error bands, these results evinced not only the pressure transmission in Newtonian fluids but also the measurement repeatability. 4.2. Fluid preparation and rheology An ultrasound gel diluted in 30% of water was chosen as the working fluid for the current study. The gel manufactured by the Brazilian company RMC Gel Clínico was basically an aqueous Carbopol solution that was composed of carboxyvinyl polymer (Carbopol), imidazolidinyl urea, methylparaben, 2-amino-2-methyl-1-propanol and deionized water. For now on, this composition will be simply called as Carbopol solution. The ultrasound gel was chosen because it is more stable than the formulated working fluid, especially for the large quantities required. The density and the isothermal compressibility of the final solution were assumed equal to the corresponding water properties due to the low concentration of the polymer. According to [22,23], the flow curve of Carbopol solutions can be adequately represented by the Herschel-Bulkley equation: { 𝜏(𝛾) ̇ = 𝜏0 + 𝐾 𝛾̇ 𝑛 , if 𝜏 > 𝜏0 (1) 𝛾̇ = 0, if 𝜏 ≤ 𝜏0
Temperature [°C]
𝜏 0 [Pa]
k [Pa.sn ]
n [-]
𝜌 [kg/m3 ]
5 25
15.2 12.8
7.64 7.43
0.41 0.39
1000 999
The results shown in Fig. 4 and Table 1 were obtained soon after the solution had been prepared. In order to check for the solution degradation, a fresh sample was collected every week from the storage tank of the experimental apparatus and submitted to rheometric tests. It was observed that the fluid degradation occurred after two weeks. The results shown in this work were performed within a time interval where no significant fluid degradation was detected. 4.3. Comparison of the rheometric data with experimental setup results In order to verify the accuracy of the experimental apparatus, flow curves obtained from pressure differences and flow rates were compared to the rheometer flow curves. The flow curve tests were performed by maintaining all three valves, V1 , V2 and V3 , open in order to obtain a constant flow rate through the main pipeline. Different flow rates were imposed to the main pipeline by not only changing the pump speed but also setting up different positions to the manual bypass valve (MV). The rheometer experiments were performed at atmospheric pressure, while the pressures measured in the flow loop were gauge pressures. For each imposed flow rate measured by the Coriolis flow meter, a steady state pressure drop based on 10 s average values (Δ𝑃 = 𝑃1,𝑎𝑣𝑔 − 𝑃4,𝑎𝑣𝑔 )was obtained. The shear stress was then calculated from the pressure difference by using a force balance on the pipeline wall:
where 𝛾̇ is the shear rate, 𝜏(𝛾) ̇ is the shear stress as a function of 𝛾, ̇ K is the consistency index and n is the power-law index. It is worth noting that the elastic effects of the Carbopol solution were neglected as an approximation for the fluid characterization. Flow curves for the Carbopol solution based on the results of a controlled shear stress rheometer (DHR-3 manufactured by TA Instruments) were built. A 40 mm diameter parallel-plate with 0.8 mm gap and crosshatched surfaces was used to avoid wall slip. Nine shear rate steps, 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100 s−1 , were chosen for building up the flow curve. According to the results of [27], the minimum shear rate of 0.01 s−1 can be considered acceptable for good fit for the Herschel-Bulkley equation and for an extrapolation of the yield stress for Carbopol solutions. Initially, a sample of approximately 100 ml was mixed for 30 min using a magnetic mixer. After homogenization, the solution was introduced in the rheometer geometry and the temperature was maintained at either 5°C or 25°C. The fluid was then sheared at a fixed rate during a time interval of approximately 1∕𝛾̇ for all shear rate values. Finally, the constants of Eq. (1) were fit to the measured steady state values of 𝜏 [Pa] for each value of 𝛾̇ [s-1 ] and the results are presented in Fig. 4 for temperatures 5 and 25°C. The symbols shown in Fig. 4 correspond to the average of three measurements. The constants of Eq. (1) and the solution density are shown in Table 1 for 5 and 25°C. As shown, the yield stress, the consistency and the power-law indices decrease with the temperature increase.
𝐷Δ𝑃 (2) 4𝐿 where 𝜏 w is the shear stress at the pipeline wall, D is the pipeline internal diameter and L is the total pipeline length. The shear rate at the pipeline wall was calculated from the average flow rate by using the Weissenberg-Rabinowitsch equation, assuming no slip at the pipe walls, fully developed, laminar, steady, incompressible and axial flow [28]: ( ) 𝛾̇ 𝑑 ln 𝑄 𝛾̇ 𝑤 = 𝑎𝑤 3 + (3) 4 𝑑 ln 𝜏𝑤 𝜏𝑤 =
where 𝛾̇ 𝑎𝑤 = 4𝑄∕(𝜋(𝐷∕2)3 ) is the apparent shear rate, or the shear rate for a Newtonian fluid, at the pipeline wall. Although Carbopol solution is expected to slip on the smooth internal surface of the steel pipe, the flow loop experiments were performed at sufficiently high shear rates. 53
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Fig. 4. Flow curves obtained from the rheometer experiments at 5 and 25°C and Herschel-Bulkley equation data fit.
Fig. 5. Flow curves obtained from the rheometer and the experimental apparatus for the Carbopol solution at 5 and 25°C.
According to [29] and [30] that conducted rheometric measurements with rough and smooth surfaces, wall slip does not affect the results at high shear rates, for their particular cases, larger than 10 s−1 . Fig. 5 shows good agreement between the flow curves obtained from the rheometer and the experimental apparatus. The percentage differences between the measured shear stresses are within 13 %. As depicted, the experimental apparatus was unable to provide shear rates lower than 30 s−1 and shear stress higher than 100 Pa, which correspond to the limits of the flow loop. Despite the good agreement at high shear rates, discrepancies would be envisage at low shear rates because of the steel pipe wall slip and zero-shear rate extrapolation to obtain the yield stress from the flow loop results must be avoided. It is also important to discuss the influence of the helical geometry in the pressure measurements. According to [31], fluid flow in a curved pipeline is affected by centrifugal forces, which may affect pressure measurements. In order to check the influence of the helical shape on the results, the Dean number Dn [32,33] was computed: √ 𝐷𝑛 = Re
𝐷 𝐷𝑠𝑝𝑖𝑟𝑎𝑙
where Re is the Reynolds number and Dspiral is the diameter of the helix which is approximately 0.727 m. Prior laminar flow tests performed with glycerin, a Newtonian fluid, showed that deviations between the measured pressures and the calculated values for a straight pipeline are only significant when Dean number was larger than 50, corresponding to a Reynolds number of 238. The results displayed in Fig. 5 are for laminar flows with Dean numbers ranging from 0.1 to 12.5 so that centrifugal forces did not affected the Carbopol solution results. It is also worth mentioning that the fluid was almost stationary within the closed pipeline during the pressure transmission experiments, and consequently, the centrifugal forces were negligible.
5. Results and discussion This section presents the results for the pressure transmission experiments performed with the Carbopol solution. The effect of the initial pressure distribution on the pressure transmission and the effects of the yield stress and of the pressure wave inertia on the final pressure distribution are also discussed in this section.
(4)
54
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Fig. 6. Measured pressures at P1 and P4 as a function of time for pressurization experiments of (a) water and (b) Carbopol solution to the average final value of 6.8 bar at 5°C.
5.1. Transmission of pressure in the carbopol solution
tion in water. While the pressure P1 reaches the final value in approximately 10 s, the pressure wave finds the outlet in approximately 0.03 s. Differently from the Newtonian fluid, the pressure distribution in the Carbopol solution is non-uniform even at the initial condition (Fig. 8(b)). The discussion about how this non-uniform initial condition was achieved is presented in Section 5.2. After the fluid pressurization at t = 2 s, the pressures P1 (0 m) and P2 (16.3 m) increase whilst the pressures P3 (32.3 m) and P4 (48.3 m) remain unaffected, indicating the pressure wave is dissipated before reaching the outlet. While the pressure P1 increases from the start-up to the steady state, the pressures at the other sensors rise from the inlet to the outlet. The pressures closer to the inlet (near the pump) increase faster than the pressures near the outlet because of the high viscous dissipation in comparison to the pressure wave inertia. Finally, the final steady state pressure distribution reduces approximately linearly from the inlet to the outlet, as the pressure is no longer able to propagate in the pipeline.
Fig. 6 presents the pressure P1 and P4 as a function of time for water and for the Carbopol solution at 5°C, during the pressurization of the closed pipeline. For the water results shown in Fig. 6(a), the average values of pressure P1 and P4 are very similar throughout the whole experiment, indicating an instantaneous, fully transmission of pressure to the outlet. As noted, pressure P4 oscillations are higher than those at transducer P1 located near the pump outlet. This is explained by the reflection and amplification of the pressure waves at the closed valve V2 , near the transducer P4 . Despite the pressure evolution for water and the carbopol solution at transducer P1 being quite similar, the pressure P4 for the carbopol solution is not only delayed but also not fully transmitted to the outlet, as shown in Fig. 6(b). While the pressure P1 after stabilization averages 6.8 bar, the pressure P4 stabilizes at an average value of 5.6 bar, depicting a pressure difference of 1.2 bar along the pipeline length. It is worth mentioning that the pipeline was kept pressurized for three hours and the average pressure difference between P1 and P4 remained approximately the same for this long period, indicating that the pressure was indeed not transmitted. According to [1] and [2], the sooner the wall shear stress produced by the pressure difference drops below the fluid yield point, propagation stops and consequently, the pressure is not transmitted anymore. On the contrary, pressure is always transmitted in a Newtonian fluid, although delayed in high viscous dissipating system. In order to demonstrate repeatability, the steady state pressures at the four pressure transducers (P1 , P2 , P3 and P4 ) for three different measurements are presented in Fig. 7 as a function of the pipeline axial length. The pressure values are averaged over 10 s after the steady state has been reached and the error bars represent the standard deviations of the pressure averages. As shown, not only the average pressure values are fairly close to one another with a maximum deviation of 0.07 bar, but also the slopes of each curve are very similar. Fig. 8 compares the time evolution of the pressure distribution along the pipeline length during pressurization tests for water (Fig. 8(a)) and for the Carbopol solution (Fig. 8(b)) at 5°C. The pressure distribution at time t = 0 s indicates the initial condition at the pump start-up, whereas the other curves represent the pressure evolution to the steady-state (t = SS). The water pressurization (Fig. 8(a)) yields an almost uniform pressure distribution from t = 0 s to the steady-state. In other words, the pressure across the pipeline length remains uniform throughout the entire experiment because the fluid is slowly compressed in comparison to the speed that the pressure propagates in water. The pressure wave propagates almost at sound speed (∼1500 m/s) due to small viscous dissipa-
5.2. Effect of the initial pressure distribution The effect of the initial condition on the time evolution of pressure and on the final pressure distribution is now discussed. The pressurization tests started from the two different initial pressure distributions shown in Fig. 9. Excepting the initial pressure distribution, the other problem parameters, such as temperature and valve configuration, were kept unchanged for both experiments. Hereafter, the first initial pressure distribution is referenced as IPD1 and the second initial pressure distribution as IPD2. The condition IPD1 was obtained by switching off the pump after the pressure P1 reached steady state at approximately 6.0 bar, whilst maintaining valve V1 and the bypass open and the outlet valve V2 closed. As the outlet valve was kept closed, depressurization occurred only through the bypass (see Fig. 1). Considering the sensor P1 (pipeline inlet) was the nearest to the bypass exit, the pressure P1 decreased to a value close to atmospheric pressure (0.0 bar). Differently from Newtonian fluids (see pressures at t = 0 s in Fig. 6(a)), the pressure at the other sensors did not decrease as much as the pressure P1 . The condition IPD2, on the other hand, was obtained by opening the outlet valve V2 after the condition IPD1 was reached. Although both inlet and outlet were depressurized by opening valve V2 , the pressures P2 and P3 did not reduce to zero as depicted in Fig. 9. Additionally, P4 did not completely decrease to zero because of the fluid contained within the bends, flow meter and the pipeline downstream of transducer P4 . Since a pressure transducer was not installed halfway the pipeline length, three distinct pressure gradients seem to exist in Fig. 9 for the condition IPD2. However, only two pressure gradients along the pipe were expected, with the maximum pressure close to the middle and pressures near zero at both the inlet (P1 ) and the outlet (P4 ). 55
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Fig. 7. Steady-state pressure values along the pipeline for three measurements experiments taken at 6.8 bar and 5°C.
Fig. 8. Time evolution of the pressure distributions along the pipeline length during the pressurization experiments for (a) water and (b) the Carbopol solution at 5°C.
Fig. 9. Two initial pressure distribution along the pipeline for the Carbopol solution at 5°C.
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Fig. 10. Time evolution of the pressure distributions along the pipeline length during the pressurization of the Carbopol solution to an average steady state value of 3.0 bar at the inlet, starting from two initial conditions: (a) IPD1 and (b) IPD2 at 5°C.
Fig. 11. Time evolution of the pressure distributions along the pipeline length during the pressurization of the Carbopol solution to an average steady state value of 6.9 bar at the inlet, starting from two initial conditions: (a) IPD1 and (b) IPD2 at 5°C.
Two pressurization experiments were performed from each initial pressure condition IPD1 and IPD2 at 5°C. In the first, the pressure at P1 reached an average steady state value of 3.0 bar and in the second, 6.9 bar. Fig. 10 and Fig. 11 show the time evolution of the pressure distribution along the pipeline for the two pressure levels, respectively. While t = 0 s and t = SS are the initial condition and the steady state, respectively, the other curves correspond to the pressure distribution at specific instants after the pump has been switched on. Excepting the steady state pressures that are averages, the other values in Fig. 10 and Fig. 11 are instantaneous pressure measurements. Fig. 10(a) shows that the initial pressure distribution for the condition IPD1 (t = 0 s) is approximately linear, varying from 0 bar at the inlet to 1.4 bar at the outlet. As soon as the pump is turned on, the pressure P1 (t = 2 s) increases, producing a wave that raises the pressure almost sequentially at positions P2 and P3 . As noted, the pressure P4 is affected only 4 s after the pump has been started. Curiously, the pressure P4 at 4 s is altered before the pressure P3 exceeds the pressure P4 . The pressure was expected to increase sequentially from the inlet to outlet for high dissipation viscous fluids, such as viscoplastic [1,2]. On the contrary, the pressure seems to propagate in a non-dissipating medium, indicating the fluid elasticity may play a role on the propagation. Finally, the steady state pressure distribution reduces linearly from P1 to P3 and pressure P4 is almost the same as P3 . Fig. 10(b) illustrates that the pressures at the initial condition IPD2 are different and smaller than the pressures at IPD1. Consequently, the pressure evolves differently from the start-up to the steady state in comparison to the results of Fig. 10(a). Despite the steady state results of Fig. 10(a) and (b) being similar from P1 to P3 , the pressures at P4 are different. While the final pressure P4 for condition IPD1 is 2.1 bar, its counterpart for condition IPD2 is only 1.7 bar. As noted in Fig. 10(a)
and (b), the pressure P1 has almost the same value for every instant of time and the pressure P4 is only affected 4 s after the start-up, meaning that the time for the pressure wave to reach the outlet is independent of the initial condition. According to [1], pressure propagation depends on two opposing factors: pressure wave inertia and viscous dissipation – the higher the first in comparison to the second the faster is the pressure wave – so that the initial condition should not affect pressure transmission. Additionally, elasticity must be equally present in both cases since the pressure is similarly transmitted to the outlet, although this is not evident for condition IPD2. Conclusively, the pressure P4 for condition IPD1 evolves to higher values because it has already started from a higher stored energy condition (higher initial pressure). Fig. 11, on the other hand, reveals that the final pressure distribution is independent on the initial condition when the position P1 is pressurized to 6.9 bar. The comparison of Fig. 11(a) and (b) shows that the pressures for conditions IPD1 and IPD2 evolve differently up to 8 s, and then they both increase almost linearly to the steady-state, indicating that the effect of the initial condition is mitigated after 8 s. Fig. 12 compares the steady state for both initial conditions, exhibiting the final pressure distribution is only affected by the initial condition for lower levels of pressurization. Therefore, the final pressure distribution is dependent not only on the initial pressure distribution, but also on the pressure level that may fade the effect of the pressure history. 5.3. Final pressure distribution The wall shear stresses obtained from the stabilized values of pressure along the pipeline are now compared with the fluid yield stress measured in the rheometer. The wall shear stress was computed by using Eq. (2), considering the average pressures of three measurements, 57
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Fig. 12. Effect of the initial pressure condition on the steady state pressure distribution along the pipeline after the pipe inlet being pressurized to (a) 3 bar and (b) 6.9 bar at 5°C.
Fig. 13. Dimensionless pressure distributions along the pipeline length for different pressure levels: 3 bar (l = 0.18; BR = 0.48), 6.8 bar (l = 0.93; BR = 0.21) and 9 bar (l = 1.4; BR = 0.16) at 5°C.
Table 2 Comparison of the wall shear stress, 𝜏 w , averaged over three pressurization experiments, with the fluid yield stress, 𝜏 0 = 15.2 Pa, at 5°C. Pressure [bar]
𝜏 w1 [Pa]
(𝜏𝑤1 −𝜏0 ) 𝜏0
3.0 6.8 9.0
14.1 12.1 9.7
7% 20% 36%
× 100%
𝜏 w2
[ Pa ]
14.2 12.6 11.6
(𝜏𝑤2 −𝜏0 ) 𝜏0
stress may be the reason for the non-transmission of pressure, but it is now exclusively responsible for the final pressure distribution. As already mentioned, [1] reported that the final pressure distribution depends not only on the fluid yield stress, but also on the counterbalance between two opposing effects: the pressure wave inertia and viscous dissipation. To account for these effects, [1] defined a Bingham number:
× 100%
7% 17% 24%
𝐵𝑅 = 4𝐿𝜏0 ∕𝑃𝑅 𝐷
(5)
and a dimensionless parameter which is the ratio between pressure wave inertia and viscous dissipation for the Bingham fluid. This parameter is redefined here for the Herchel–Bulkley fluid: ( )1∕𝑛−1 ( ) 𝜌𝑐𝐷 ( 1 𝐷 )1∕𝑛 ( 4𝑛 ) P𝑅 𝜆= (6) 𝐾 8𝐾 4𝐿 3𝑛 + 1
assuming a linear pressure variation along the pipeline at 5°C and taking into account only cases in which the initial condition did not affect the final pressure distribution. Table 2 shows the comparison for three levels of pressurization between the yield stress and two values of wall shear stress: the first, 𝜏 w1 , was evaluated based on the pressure difference P1 P4 and the second, 𝜏 w2 , was calculated using the pressure difference P2 - P4 . As noted, 𝜏 w1 is closer to 𝜏 0 than 𝜏 w2 and the higher the pressurization level the higher is the difference between the wall shear and yield stress. The dimensionless pressure distributions for these three levels of pressurization are also presented in Fig. 13, depicting that the higher the pressure level the more distant is the pressure distribution from linearity. It is also worth noting that the pressures P2 , P3 and P4 are aligned on a straight line for the three level of pressurization, explaining the fact of 𝜏 w1 being closer to 𝜏 0 than 𝜏 w2 . These results evince that the shear
where PR is the inlet pipeline pressure and c is the pressure wave speed. [1] states that: i) for high dissipating cases, i.e. low values of 𝜆, the final dimensionless pressure distribution is linear, depends only on the Bingham number and the pressure gradient is equal to -BR ; ii) for low dissipating cases, i.e. high values of 𝜆,the final dimensionless pressure distribution deviates from linearity, because the pressure wave reflects at the outlet, and depends on both Bingham number and on 𝜆. The values of BR and 𝜆 computed by using Eqs. (4) and (5), respectively, are depicted in Fig. 13 for the three levels of pressurization. As seen, the re58
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sults of Fig. 13 are qualitatively similar to those presented and discussed in [1]. In other words, the pressure distribution is almost linear and the pressure gradient is quite closed to -BR for low values of 𝜆. Additionally, the pressure distribution deviates as much as from a straight line as 𝜆 increases, denoting pressure reflection at the outlet.
[7] B. Brunone, B.W. Karney, M. Mecarelli, M. Ferrante, Velocity profile and unsteady pipe friction, J. Water Resour. Plan. Manag. (2000) 236–244. [8] E.M. Wahba, Modelling the attenuation of laminar fluid transients in piping systems, Appl. Math. Model. 32 (2008) 2863–2871, doi:10.1016/j.apm.2007.10.004. [9] E.M. Wahba, Non-Newtonian fluid hammer in elastic circular pipes: Shear-thinning and shear-thickening effects, J. Nonnewton. Fluid Mech. 198 (2013) 24–30, doi:10.1016/j.jnnfm.2013.04.007. [10] G.M. Oliveira, A.T. Franco, C.O.R. Negrão, Mathematical Model for Viscoplastic Fluid Hammer, J. Fluids Eng. 138 (2016) 011301, doi:10.1115/1.4031001. [11] J. Sestak, M.E. Charles, M.G. Cawkwell, M. Houska, Start-up of gelled crude oil pipeline, J. Pipelines. 6 (1987) 15–24. [12] G. Vinay, A. Wachs, I. Frigaard, Start-up transients and efficient computation of isothermal waxy crude oil flows, J. Nonnewton. Fluid Mech. 143 (2007) 141–156, doi:10.1016/j.jnnfm.2007.02.008. [13] A. Wachs, G. 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6. Conclusions Pressure transmission in a yield stress fluid compressed in a long pipeline under isothermal conditions was studied experimentally. An ultrasound medical gel diluted in water (Carbopol solution) was used to represent a viscoplastic fluid. Results for water were obtained to verify the pressure transmission in a Newtonian fluid, to check the accuracy of the experimental setup and also to compare with the viscoplastic fluid results. The main conclusions of the pressurization experiments with the Carbopol solution can be summarized as: i. The pressure increases uniformly for water and sequentially from the inlet to the pipe outlet for the viscoplastic fluid during the pressurization. At the steady state, the pressure distribution remains uniform for the Newtonian fluid and decreases from the inlet to the outlet for the yield stress fluid. These results corroborate previous numerical works. ii. The time evolution of the pressure distribution depends on the initial condition (pressure history) for low levels of pressurization. Above a certain limit, the pressure distribution evolves independently of the pressure history. iii. The fluid elasticity seems to play a role when the pressure starts to propagate along the pipeline. As the time elapses, the elasticity fades and the pressure propagation evolves based on two concurrent effects: pressure wave inertia and viscous dissipation. iv. The final pressure distribution is linear and only a function of the yield stress for low values of 𝜆 (pressure wave inertia-viscous dissipation ratio) and deviates from linearity, depending on both yield stress and 𝜆 for high values of 𝜆. Acknowledgements We would like to acknowledge the financial support of PETROBRAS, FINEP (Funding Authority for Studies and Projects), CNPq (Brazilian Research Foundation) and CAPES (Brazilian Graduate Education Foundation). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jnnfm.2018.08.007. References [1] G.M. Oliveira, C.O.R. Negrão, A.T. Franco, Pressure transmission in Bingham fluids compressed within a closed pipe, J. Nonnewton. Fluid Mech. 169–170 (2012) 121– 125, doi:10.1016/j.jnnfm.2011.11.004. [2] G.M. de Oliveira, A.T. Franco, C.O.R. Negrão, A.L. Martins, R.A. Silva, Modeling and validation of pressure propagation in drilling fluids pumped into a closed well, J. Pet. Sci. Eng. 103 (2013) 61–71, doi:10.1016/j.petrol.2013.02.012. [3] E.B. Wylie, V.L. Streeter, L. Suo, Fluid Transients in Systems, Prentice Hall, New Jersey, 1993. [4] M.S. Ghidaoui, M. Zhao, D.A. McInnis, D.H. Axworthy, A review of water hammer theory and practice, Appl. Mech. Rev. 58 (2005) 49, doi:10.1115/1.1828050. [5] A. Bergant, A.R. Simpson, A.S. Tijsseling, Water hammer with column separation: A historical review, J. Fluids Struct 22 (2006) 135–171, doi:10.1016/j.jfluidstructs.2005.08.008. [6] B. Brunone, U.M. Golia, M. Greco, Effects of two-dimensionality on pipe transients modeling, J. Hydraul. Eng. 121 (1995) 906–912, doi:10.1061/(ASCE)0733-9429(1995)121:12(906).
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