The Effect of Thixotropy on Pressure Losses in a Pipe

energies

Article The Eﬀect of Thixotropy on Pressure Losses in a Pipe

Eric Cayeux * and Amare Leulseged

Norwegian Research Centre, 4021 Stavanger, Norway; [email protected] * Correspondence: [email protected]; Tel.: +47-47-501-787

Received: 23 October 2020; Accepted: 20 November 2020; Published: 24 November 2020

Abstract: Drilling fluids are designed to be shear-thinning for limiting pressure losses when subjected to high bulk velocities and yet be sufficiently viscous to transport solid material under low bulk velocity conditions. They also form a gel when left at rest, to keep weighting materials and drill-cuttings in suspension. Because of this design, they also have a thixotropic behavior. As the shear history influences the shear properties of thixotropic fluids, the pressure losses experienced in a tube, after a change in diameter, are influenced over a much longer distance than just what would be expected from solely entrance effects. In this paper, we consider several rheological behaviors that are relevant for characterizing drilling fluids: Collins–Graves, Herschel–Bulkley, Robertson–Stiff, Heinz–Casson, Carreau and Quemada. We develop a generic solution for modelling the viscous pressure gradient in a circular pipe under the influence of thixotropic effects and we apply this model to configurations with change in diameters. It is found that the choice of a rheological behavior should be guided by the actual response of the fluid, especially in a turbulent flow regime, and not chosen a priori. Furthermore, thixotropy may influence pressure gradients over long distances when there are changes of diameter in a hydraulic circuit. This fact is important to consider when designing pipe rheometers.

Keywords: non-Newtonian; thixotropy; pressure loss; pipe

1. Introduction Drilling fluids play multiple and important roles in drilling operations, such as maintaining an adequate pressure in the open hole section of the borehole to avoid formation fluid influx or wellbore instabilities, transporting drill-cuttings to the surface, and cooling down the downhole equipment. Drilling fluids need to be sufficiently viscous to transport drill-cuttings and yet the pressure drops caused by viscous flow should be limited to avoid excessive pump pressures. As drilling fluids are pumped into a drill-string and returned through an annulus offering a larger cross-sectional area than the drill-pipes, they are designed to be shear-thinning, therefore limiting viscous pressure drop at high bulk velocity while presenting a high effective viscosity at low bulk velocities. The downhole hydrostatic pressure is related to the density of the drilling fluid, which often results from the addition of weighting solid particles in the fluid mix. Drilling fluids are therefore designed to gel when left at rest in order to keep both weighting material and drill-cuttings in suspension, thus exhibiting a viscoelastic behavior. As a result of these design principles, their rheological properties depend on the shear history, which means that in practice they are thixotropic. The estimation of viscous pressure losses associated with the circulation of drilling fluids has been calculated for many years, but mostly without considering the impact of thixotropy [1]. Yet under pump acceleration, thixotropy must play a role on the time evolution of pressure along the hydraulic circuit [2,3]. Additionally, any change in diameter along the hydraulic circuit influences the shear history of the fluid and therefore the resulting viscous pressure losses [4,5].

Energies 2020, 13, 6165; doi:10.3390/en13236165 www.mdpi.com/journal/energies Energies 2020, 13, 6165 2 of 23

Energies 2020, 13, x FOR PEER REVIEW 2 of 22 2. Viscous Flow in a Pipe 2. ViscousThe flow Flow of ain fluid a Pipe in a cylindrical pipe is axisymmetric. We will consider the force balance, in the axial directionThe flow andof a influid laminar in a cylindrical flow conditions, pipe is on axisymmetric. a small shell element We will of consider radius r the(dimension force balance, [L] (m)), in widththe axialdr (dimension direction and [L] in (m)) laminar and length flow conditions,ds (dimension on [L]a small (m)) shell (see Figureelement1): of radius 𝑟 (dimension [L] (m)), width 𝑑𝑟 (dimension [L] (m)) and length 𝑑𝑠 (dimension [L] (m)) (see Figure 1): d(τr) dp = r , (1) dr =𝑟ds , (1) 1 2 1 2 wherewhere τ𝜏is is the the shear shear stress stress (dimension (dimension [ML [ML− −T1T−−2]] (Pa))(Pa)) andand p𝑝is is the the pressure pressure (dimension (dimension [ML [ML− −T1T−−2]] (Pa)).(Pa)).

FigureFigure 1.1. SchematicSchematic viewview ofof anan axisymmetricaxisymmetric shellshell elementelement ofof ﬂuidfluid insideinside aa pipe.pipe.

SinceSince thethe viscousviscous frictionfriction componentcomponent ofof pressurepressure doesdoes notnot dependdepend onon thethe radialradial positionposition inin aa cross-section,cross-section, wewe cancan integrateintegrate EquationEquation (1):(1): 𝜏𝑟r=dp +C , (2) τ(r) = + , (2) 2 ds r where 𝐶 is an integration constant. At 𝑟=0, is a finite value only when 𝐶=0, and therefore the C whereshear stressC is an at integrationa radial position constant. is simply: At r = 0, r is a ﬁnite value only when C = 0, and therefore the shear stress at a radial position is simply: 𝜏𝑟 = , (3) r dp and more particularly, at the pipe wall, we have:τ(r) = , (3) 2 ds 𝜏 = , (4) and more particularly, at the pipe wall, we have:

where 𝜏 is the shear stress at the wall and 𝑑 is the internal pipe diameter. dw dp Equation (3) means that the shear stressτ =is a linear, function of the radial position. However, (4) w 4 ds certain fluids may exhibit a yield stress (here noted 𝜏), i.e., a minimum stress for which viscous flow wherecannotτ happen.w is the shear For such stress fluids, at the the wall shear and stressdw is therema internalins constant pipe diameter. and equal to the yield stress when

the radialEquation position (3) means is smaller that the than shear a certain stress isvalue, a linear here function noted of𝑟 the. As radial it is possible position. to However, obtain different certain fluidsvalues may of the exhibit yield a stress yield stressas a function (here noted of theτγ ),measurement i.e., a minimum method, stress it for is whichdebated viscous whether flow the cannot yield happen.stress really For suchexists fluids, [6]. For the the shear sake stress of this remains paper, constant we will andnot take equal sides, to the and yield consider stress when both thecases, radial i.e., positionwith and is without smaller thanyield a stress. certain The value, viscous here notedshear rstpress. As is it isin possibleturn a function to obtain of di thefferent shear values rate, of𝛾 the(dimension yield stress [T−1 as] (s a− function1)), that characterizes of the measurement the slippage method, between it is debated two adjacent whether shells. theyield The shear stress rate really at existsthe wall [6]. is For noted the sake𝛾 , and of thisthe shear paper, rate we at will the not center take of sides, the andpipe consideror in the bothplugged cases, region, i.e., with i.e., and𝑟≤ . −1 1 without𝑟, is necessarily yield stress. zero. The The viscous local fluid shear velocity, stress is in𝑣 turn(dimension a function [LT of] the(m/s)), shear is rate,relatedγ (dimension to the shear [T rate− ] 1 (sby:− )),𝛾 that= . characterizes The fluid velocity the slippage at the wall between is equal two to zero adjacent as a result shells. of The the shear no slip rate at the at the wall wall hypothesis, is noted . γw, and the shear rate at the center of the pipe or in the plugged region, i.e., r rp, is necessarily zero. while the fluid velocity is maximum at the centerline or in the non-shear region.≤ .Figure 2 shows a The local fluid velocity, v (dimension [LT 1] (m/s)), is related to the shear rate by: γ = dv . The fluid schematic view of the shear stress, shear rate− and fluid velocity in a cross-section of the drtube. velocity at the wall is equal to zero as a result of the no slip at the wall hypothesis, while the fluid velocity is maximum at the centerline or in the non-shear region. Figure2 shows a schematic view of the shear stress, shear rate and fluid velocity in a cross-section of the tube.

Figure 2. Schematic view of the shear stress, shear rate and velocity profiles in a cross-section.

Energies 2020, 13, x FOR PEER REVIEW 2 of 22

2. Viscous Flow in a Pipe The flow of a fluid in a cylindrical pipe is axisymmetric. We will consider the force balance, in the axial direction and in laminar flow conditions, on a small shell element of radius 𝑟 (dimension [L] (m)), width 𝑑𝑟 (dimension [L] (m)) and length 𝑑𝑠 (dimension [L] (m)) (see Figure 1):

=𝑟 , (1) where 𝜏 is the shear stress (dimension [ML−1T−2] (Pa)) and 𝑝 is the pressure (dimension [ML−1T−2] (Pa)).

Figure 1. Schematic view of an axisymmetric shell element of fluid inside a pipe.

Since the viscous friction component of pressure does not depend on the radial position in a cross-section, we can integrate Equation (1): 𝜏𝑟 = + , (2) where 𝐶 is an integration constant. At 𝑟=0, is a finite value only when 𝐶=0, and therefore the shear stress at a radial position is simply: 𝜏𝑟 = , (3) and more particularly, at the pipe wall, we have: 𝜏 = , (4) where 𝜏 is the shear stress at the wall and 𝑑 is the internal pipe diameter. Equation (3) means that the shear stress is a linear function of the radial position. However, certain fluids may exhibit a yield stress (here noted 𝜏), i.e., a minimum stress for which viscous flow cannot happen. For such fluids, the shear stress remains constant and equal to the yield stress when the radial position is smaller than a certain value, here noted 𝑟. As it is possible to obtain different values of the yield stress as a function of the measurement method, it is debated whether the yield stress really exists [6]. For the sake of this paper, we will not take sides, and consider both cases, i.e., with and without yield stress. The viscous shear stress is in turn a function of the shear rate, 𝛾 (dimension [T−1] (s−1)), that characterizes the slippage between two adjacent shells. The shear rate at the wall is noted 𝛾 , and the shear rate at the center of the pipe or in the plugged region, i.e., 𝑟≤ −1 𝑟, is necessarily zero. The local fluid velocity, 𝑣 (dimension [LT ] (m/s)), is related to the shear rate by: 𝛾 = . The fluid velocity at the wall is equal to zero as a result of the no slip at the wall hypothesis, whileEnergies the2020 fluid, 13, 6165 velocity is maximum at the centerline or in the non-shear region. Figure 2 shows3 of 23a schematic view of the shear stress, shear rate and fluid velocity in a cross-section of the tube.

FigureFigure 2. 2. SchematicSchematic view view of of the the shear shear stress, stress, shear shear rate rate and and velocity velocity profiles proﬁles in a cross-section.

3 1 3 Finally, the volumetric flowrate (Q, dimension [L T− ] (m /s)) is related to the fluid velocity field by: dw Z 2 Q = 2πv(r)rdr. (5) 0 In laminar flow, as everything else is imposed by the pipe dimension and physical relationships (e.g., relation between flowrate and velocity field, relation between shear rate and velocity, shear stress as a function of radial distance), the volumetric flowrate and the pressure gradient only depend on the relationship between the shear stress and the shear rate.

3. Pressure Gradients of Non-Thixotropic Fluid in a Pipe The simplest relationship between shear stress and shear rate is the Newtonian rheological behavior: . τ = µNγ, (6) 1 1 where µN is the viscosity (dimension [ML− T− ] (Pa.s)). All other relationships that are not a direct proportionality between the shear stress and the shear rate are called non-Newtonian. The relation between the shear stress and the shear rate can be measured with a rheometer. For instance, in a concentric rheometer configuration, the rotational speed is converted to a shear rate at the wall and the shear stress at the wall is deduced from the measured torque. However, the relationship between rotational speed and shear rate at the wall is constant only for Newtonian fluids. For non-Newtonian fluids, the relationship is more complex and requires the utilization of all the measurements of shear rate and shear stresses in order to be estimated [7]. Similarly, the ratio of the torque to the shear stress is constant only for a Newtonian fluid. For non-Newtonian ones, corrections for end-effects shall be applied [8]. In practice, drilling fluids are non-Newtonian, and several rheological behaviors are candidates to describe their viscous properties.

3.1. Non-Newtonian Rheological Behaviors A ﬁrst simple extension of the Newtonian rheological behavior is the Bingham plastic model:

. τ = τγ + µpγ, (7)

1 2 1 1 where τγ is the yield point [ML− T− ] (Pa) and µp is the plastic viscosity with the dimension [ML− T− ] τ 1 1 (Pa.s). The effective viscosity can be defined as: µ = . (dimension [ML T ] (Pa.s)). For the e f f γ − − Bingham plastic model, it is noticeable that the effective viscosity tends to infinity when the shear rate tends to zero. To eliminate the problem associated with an effective viscosity tending to infinity at zero shear rate, the Collins–Graves model utilizes a multiplicative term that ensures a finite effective viscosity at zero shear rate [9]: . . tCGγ τ = τγ + µpγ (1 e− ), (8) − where tCG is a time constant [T] (s). Energies 2020, 13, 6165 4 of 23

Another simple generalization of the Newtonian rheological behavior is the power law model:

. n τ = Kγ , (9)

1 n 2 n where K is the consistency index (dimension [ML− T − ] (Pa.s ) and n is the flow behavior index (dimensionless). It is somewhat disturbing that the physical quantity of the consistency index depends on n, as it may lead to believe that consistency indices cannot be compared with each other. This concern can easily be removed by normalizing the shear rate. Indeed, it is always possible to consider that . . . γ . γ the shear rate is expressed as γ = 1 . If we now introduce the dimensionless shear rate γˆ = . where γ1 . 1 γ1 = 1s− , Equation (9) can be written: . n τ = τKγˆ , (10) where τK has the dimension of stress. It has exactly the same numerical value as K but its physical dimension is now independent of the consistency index, therefore explaining why these values can be compared with each other for different fluids. The Herschel–Bulkley model combines both the Bingham plastic and the power law generalizations of the Newtonian rheological behavior [10]:

. n τ = τγ + τKγˆ . (11)

It reduces to the Bingham plastic model when n = 1 and to the power law model when τγ = 0. The Herschel–Bulkley model is also referred to as the yield power law (YPL) model. Robertson and Stiﬀ (1976) [11] proposed a model that has a smoother curvature than the YPL model:

. B τ = τA C + γˆ , (12) where τA has the dimension of stress, and B, C are dimensionless model parameters. In medicine, biology and the food industry, the Heinz–Casson model [12] is often used:

1 . n n n τ = τγ + τKγˆ , (13) as well as the Carreau model [13]:

n 1 τ . 2 −2 µe f f = . = µ + (µ0 µ )(1 + δγ ) , (14) γ ∞ − ∞

1 1 where µe f f is the effective viscosity (dimension [ML− T− ] (Pa.s)), µ0 is a Newtonian viscosity when the shear rate tends to zero, µ is a Newtonian viscosity when the shear rate tends to infinity, δ is a ∞ relaxation time (dimension [T] (s)) and n is a power index. Finally, Quemada developed a model [14] purely based on the physical principles established for the effects of deformable particles in suspension in a Newtonian fluid, which correspond rather well to typical drilling fluids where the viscosity behavior is caused by the level of entanglement of polymers or the assemblage of structures made of clay platelets:

 . n 2 γ  1 + .   γc  =   µe f f µ  q . n  , (15) ∞ µ γ   µ∞ + .  0 γc

. 1 1 where γc is a characteristic shear rate (dimension [T− ] (s− )). Energies 2020, 13, 6165 5 of 23

Each of these models can potentially describe the rheological behavior of drilling fluids. Yet, statistically, it can be shown that in a majority of cases, drilling fluids are better described by the Heinz–Casson, Quemada, Herschel–Bulkley, Robertson–Stiff and power law models in decreasing order, than by the Bingham plastic rheological behavior [15].

3.2. Pressure Losses in Laminar Flow The viscous pressure gradient of the laminar flow of Newtonian fluid in a tube is well known. For Bingham plastic fluid, one can use the Swamee–Aggarwal [16] equation of the Darcy–Weisbach friction factor. As the Collins–Graves model does not have a yield stress, we can use the Weissenberg–Rabinowitsch–Mooney–Schofield method to estimate its associated laminar viscous pressure gradient. A solution is provided in AppendixA. The laminar viscous pressure losses of a power law fluid in a pipe are also well-known [17]. The pressure loss of a Herschel–Bulkley fluid has been described by Kelessidis et al. (2006) [1], while the pressure loss of a Robertson–Stiff fluid is described in Beirute and Flumertfelt’s paper (1977) [18]. The viscous pressure loss gradient in laminar flow of a fluid following the Carreau rheological behavior can be efficiently calculated using the method described by Sochi [13], which is based on the Weissenberg–Rabinowitsch–Mooney–Schofield method. The same method can be used to estimate the laminar pressure gradient in a pipe of a fluid well-described by the Quemada rheological behavior. The derivation is given in AppendixB. The pressure losses in laminar flow of a Casson fluid in a pipe have been described by Lee et al. (2011) [19]. Unfortunately, to the best of our knowledge, an analytic expression of the laminar pressure loss of a Heinz–Casson fluid in a pipe is not yet known. However, it is possible to perform a numerical estimation. The magnitude of the gradient of the fluid velocity at a radial position is given by the shear rate: dv . = γ. (16) − dr As we know the shear rate at the wall, since the shear stress at the wall is imposed by the pressure gradient, it is possible to integrate the fluid velocity in the tube from the wall, also considering that the velocity at the wall is zero as a consequence of the no slip at the wall hypothesis:

. v + = v γ ∆r, (17) j 1 j − j where vj is the ﬂuid velocity at the radial position rj, vj+1 is the ﬂuid velocity at the radial position . r + = r ∆r, ∆r is a radial step, γ is the shear rate at the radial position r , and considering that at j 1 j − j j dw rj dp . r0 = 2 , v0 = 0. Since τj = 2 ds , we can calculate γj:

1 r !n ! n . 1 j dp n γj = τγ . (18) τK 2 ds −

If τj gets smaller than τγ, then the last value of the ﬂuid velocity is used, here noted vγ. The ﬂowrate can then be estimated: Xn Q πv (r2 r2 ). (19) ≈ j j − j+1 j=1

dp It is then simple to apply a Newton–Raphson method to ﬁnd the value of ds that gives the desired volumetric ﬂowrate. We have found that n = 100 gives approximations of the pressure gradient with an accuracy better than 0.1%. Energies 2020, 13, 6165 6 of 23

3.3. Pressure Gradient in a Tube in Transitional and Turbulent Flow The Rabinowitsch–Mooney equation for a pipe is [20]:    d ln 8v  . . 3 1 dw  . γw = γwN +  = fRMγwN, (20) 4 4 d ln τw 

. . 8v where γ is the actual shear rate at the wall, γ = is the Newtonian shear rate at the wall, τw is the w wN di actual stress at the wall, v is the bulk ﬂuid velocity and fRM is the Rabinowitsch–Mooney correction factor of the true shear rate at the wall compared to the Newtonian one. Note that for power–law and Herschel–Bulkley rheological behaviors, the Rabinowitsch–Mooney correction factor for pipe ﬂow is:

3 1 f = + n. (21) RM 4 4

The Dodge and Metzner relation [21] can be used to estimate the Fanning friction factor ( fF) in a pipe for power law rheological behavior:

1 4 1 n 0.395 = log R f − 2 . (22) p 0.75 10 e F 1.2 fF n − n

Following the method described by Founargiotakis et al. [22], the actual shear stress at the wall can be approximated by a power law rheological behavior such that it follows locally the rheological behavior of the ﬂuid around the given shear stress:

. n0 τw = K0 γwPL , (23) where K0 and n0 are, respectively, the locally ﬁtted consistency and ﬂow behavior indices of . the local power law rheological behavior and γwPL is the equivalent power law wall shear rate. . From Equations (20) and (21), the expression of γwPL is:

. 3 1 . γ = + n0 γ . (24) wPL 4 4 wN

The local power law ﬂow behavior index can be calculated as follows:

d ln τw n0 = . , (25) d ln γwN and the local power law consistency index is simply:

τw K0 = . (26) . n0 γwPL

It is then possible to deﬁne a local power law Reynolds number (Re0) as:

2 n n ρu − 0 d 0 R = w , (27) e0 n K 2 + 6 0 0 n0 R where ρ is the ﬂuid density. The local power law Reynolds number limits for laminar ( e0l ) and turbulent

(Re0t ) ﬂow are [21]: R0 = 3250 1150n0, (28) el − R0 = 4150 1150n0. (29) et − Energies 2020, 13, 6165 7 of 23

The Darcy–Weisbach friction factor at the laminar limit ( flDW) is:

64 flDW = , (30) Re0L while the Fanning friction factor at the limit of turbulent ﬂow ( ftF) is estimated using Equation (22) with Re0t . Note that the Darcy–Weisbach friction ( fDW) is simply related to the Fanning friction ( fF) by:

fDW = 4 fF. (31)

For transitional ﬂow, the Darcy–Weisbach friction ( ftrDW) is linearly interpolated between the laminar and turbulent limits as a function of the local power law Reynolds number:

R R e0 e0l ftrDW = f + − ( ftDW f ). (32) lDW R R − lDW e0t − e0l The pressure gradient for transitional and turbulent ﬂow is then calculated using the Darcy–Weisbach deﬁnition of the friction factor:

dp ρu2 = fDW . (33) ds 2dw

3.4. Comparison of Estimated Pressure Losses for Different Drilling Fluids In this section we will compare the influence of the matched rheological behavior with expected pressure losses in a circular pipe. All the examples are based on measurements made on real drilling fluids with a scientific rheometer and utilizing rotational speeds spread along a logarithmic scale in order to obtain a proper coverage at low shear rates. The example of Figure3 corresponds to a KCl /polymer water-based mud of density 1250 kg/m3 at 80 ◦C (composition in % by wt: fresh water 65.35, KCl 9.60, Soda ash 0.08, DuoTec NS 0.32, Trol FL 0.80, Glydril MC 2.40, barite 19.45). We can see that the best match is obtained with a power law rheological behavior, or equivalently, a Robertson–Stiff, Herschel–Bulkley or Heinz–Casson models as they can all degenerateEnergies 2020, to13, ax FOR power PEER law REVIEW formulation. 7 of 22

33 FigureFigure 3.3. For a KClKCl/Polymer/Polymer 1250 kgkg/m/m at 80 ◦°C,C, the best matching rheological behavior is the power lawlaw model.model.

WhenWhen applyingapplying thethe variousvarious rheological behaviors to pressure lossloss calculationscalculations in a 55 inin pipepipe forfor ﬂowratesflowrates rangingranging between 0 and 5000 LL/min,/min, we cancan seesee thatthat thethe expectedexpected pressurepressure gradientsgradients areare rather different for the Collins–Graves, Quemada and Heinz–Casson models than for the power law rheological behavior (see Figure 4). For the Quemada and Heinz–Casson models, the difference in pressure gradient (relative to the power law model) reaches 3% in laminar flow and 29% in turbulent flow, even though the flow curve error with the actual rheometer measurements does not exceed −2.5% on the under-evaluation side and 11% on the over-evaluation side.

Figure 4. Estimated pressure losses while circulating a KCl/polymer fluid (1250 kg/m3 at 80 °C) in a 5 in pipe for flowrates ranging between 0 and 5000 L/min.

Another KCl/polymer fluid (composition in % by wt: fresh water 50.92, KCl 8.00, soda ash 0.07, DuoTec NS 0.27, Trol FL 0.67, Glydril MC 2.00, barite 38.07) with a density of 1500 kg/m3 at 80 °C, is best described by the Herschel–Bulkley or the Robertson–Stiff model (see Figure 5). The 𝜒 = ∑ 𝜏̂ −𝜏̃ for the Herschel–Bulkley model is 0.013, while for the Robertson–Stiff model, it is 0.014. Note that here 𝜏̂ is the measured shear stress and 𝜏̃ is the modelled shear stress. The Robertson– Stiff model fits best at low shear rates, while the Herschel–Bulkley model is better at high shear rates. Other models such as Heinz–Casson, Quemada and Carreau are quite close as well (max percentage difference 7% compared to the YPL model). However, the power law and Collins–Graves models do not match the rheometer measurements so well. Yet, when simulating pumping the fluid in a 5 in pipe, the estimated pressure gradients are rather dispersed, with differences in the range of 2% in laminar flow, but exceeding 35% toward the end of the analyzed flowrates for the turbulent flow regime, each of the models giving a relatively different estimation of the pressure gradient as a function of the flowrate (see Figure 6).

Energies 2020, 13, x FOR PEER REVIEW 7 of 22

Figure 3. For a KCl/Polymer 1250 kg/m3 at 80 °C, the best matching rheological behavior is the power law model.

Energies 2020, 13, 6165 8 of 23 When applying the various rheological behaviors to pressure loss calculations in a 5 in pipe for flowrates ranging between 0 and 5000 L/min, we can see that the expected pressure gradients are rather didifferentfferent forfor thethe Collins–Graves,Collins–Graves, Quemada and Heinz–Casson models than for the power law rheological behavior (see Figure4 4).). ForFor thethe QuemadaQuemada andand Heinz–CassonHeinz–Casson models,models, thethe didifferencefference in pressure gradientgradient (relative(relative toto thethe power law model) reaches 3%3% in laminar flowflow and 29% in turbulent flow,flow, eveneven though though the the flow flow curve curve error error with with the actualthe actual rheometer rheometer measurements measurements does notdoes exceed not exceed2.5% − on−2.5% the on under-evaluation the under-evaluation side and side 11% and on 11% the on over-evaluation the over-evaluation side. side.

3 Figure 4. Estimated pressure losses while circulating a KClKCl/polymer/polymer ﬂuidfluid (1250(1250 kgkg/m/m 3 at 80 ◦°C)C) in a 5 in pipe forfor ﬂowratesflowrates rangingranging betweenbetween 00 andand 50005000 LL/min./min.

Another KCl/polymer fluid (composition in % by wt: fresh water 50.92, KCl 8.00, soda ash 0.07, Another KCl/polymer fluid (composition in % by wt: fresh water 50.92, KCl 8.00, soda ash 0.07, 3 DuoTec NS 0.27, Trol FL 0.67, Glydril MC 2.00, barite 38.07) with a density of 1500 kg/m at 803 ◦C, is best DuoTec NS 0.27, Trol FL 0.67, Glydril MC 2.00, barite 38.07) with a density of 1500 kg/m nat 80 °C, is 2 P 2 describedbest described by the by Herschel–Bulkley the Herschel–Bulkley or the Robertson–Stior the Robertson–Stiffff model (see model Figure (see5). TheFigureχ =5). The(τî 𝜒eτi)= i=1 − for∑ the𝜏̂ Herschel–Bulkley−𝜏̃ for the Herschel–Bulkley model is 0.013, model while foris 0.013, the Robertson–Sti while for theff Robertson–Stiffmodel, it is 0.014. model, Note it that is 0.014. here Note that here 𝜏̂ is the measured shear stress and 𝜏̃ is the modelled shear stress. The Robertson– τî is the measured shear stress andeτi is the modelled shear stress. The Robertson–Stiff model fits best at lowStiff shearmodel rates, fits best while at low the Herschel–Bulkleyshear rates, while modelthe Herschel–Bulkley is better at high model shear rates.is better Other at high models shear such rates. as Heinz–Casson,Other models such Quemada as Heinz–Casson, and Carreau Quemada are quite close and asCarreau well (max are quite percentage close as di ffwellerence (max 7% percentage compared todifference the YPL 7% model). compared However, to the the YPL power model). law Howeve and Collins–Gravesr, the power modelslaw and do Collins–Graves not match the models rheometer do measurementsnot match the rheometer so well. Yet, measurements when simulating so well. pumping Yet, when the fluid simulating in a 5 in pipe,pumping the estimated the fluid pressurein a 5 in gradientspipe, the estimated are rather dispersed,pressure gradie with dintsff erencesare rather in thedispersed, range of with 2% in differences laminar flow, in the but range exceeding of 2% 35% in towardlaminar theflow, end but of theexceeding analyzed 35% flowrates toward forthe the end turbulent of the analyzed flow regime, flowrates each offor the the models turbulent giving flow a regime, each of the models giving a relatively different estimation of the pressure gradient as a relativelyEnergies 2020 di, 13fferent, x FOR estimation PEER REVIEW of the pressure gradient as a function of the flowrate (see Figure6).8 of 22 function of the flowrate (see Figure 6).

3 FigureFigure 5.5. ForFor aa KClKCl/polymer/polymer 15001500 kgkg/m/m 3 atat 8080 ◦°C,C, the best match is obtainedobtained withwith aa Herschel–BulkleyHerschel–Bulkley rheologicalrheological behavior.behavior.

Figure 6. The estimated pressure gradients, when circulating a KCl/polymer, 1500 kg/m3 at 80 °C, differ substantially depending on the chosen rheological behavior, especially in turbulent flow regime.

In practice, depending on the fluid, any of the above-listed rheological behaviors may provide the best match with rheological measurements. For instance, a KCl/polymer with a density of 1750 kg/m3 at 50 °C (composition in % by wt: fresh water 39.18, KCl 6.86, Soda ash 0.06, DuoTec NS 0.23, Trol FL 0.57, Glydril MC 1.71, barite 51.39) may be best modeled by a Quemada model, while the Collins–Graves model may provide the best fit for a bentonite-based fluid as it can be seen in Figure 7 (composition in % by wt: fresh water 94.23, bentonite 5.49, caustic soda 0.27, soda ash 0.01).

Figure 7. Flow curves for a bentonite based water-based mud (WBM). Here the best fit is obtained with the Collins–Graves model.

Energies 2020, 13, x FOR PEER REVIEW 8 of 22 Energies 2020, 13, x FOR PEER REVIEW 8 of 22

Figure 5. For a KCl/polymer 1500 kg/m3 at 80 °C, the best match is obtained with a Herschel–Bulkley rheologicalFigure 5. For behavior. a KCl/polymer 1500 kg/m3 at 80 °C, the best match is obtained with a Herschel–Bulkley Energies 2020, 13, 6165 9 of 23 rheological behavior.

Figure 6. The estimated pressure gradients, when circulating a KCl/polymer, 1500 kg/m3 at 80 °C,

differ substantially depending on the chosen rheological behavior, especially in turbulent33 flow FigureFigure 6.6. TheThe estimatedestimated pressurepressure gradients,gradients, whenwhen circulatingcirculating aa KClKCl/polymer,/polymer, 15001500 kgkg/m/m atat 8080 ◦°C,C, diregime.differffer substantially substantially depending depending on on the the chosen chosen rheological rheological behavior, behavior, especially especially in turbulent in turbulent flow regime. flow regime. In practice,practice, dependingdepending on on the the fluid, fluid, any any of of the the above-listed above-listed rheological rheological behaviors behaviors may may provide provide the 3 bestthe best matchIn practice,match with with rheological depending rheological measurements. on themeasurements. fluid, any For of instance, Forthe instance,above-listed a KCl/ polymera KCl/polymer rheological with a density behaviorswith a ofdensity1750 may kg provideof/m 1750at 50kg/mthe◦ Cbest3(composition at 50match °C (composition with in rheological % by wt: in fresh% measurements. bywater wt: fresh 39.18, water KClFor 39.18, 6.86,instance, Soda KCl a ash6.86, KCl/polymer 0.06, Soda DuoTec ash with0.06, NS DuoTec 0.23,a density Trol NS FLof 0.23, 0.57,1750 GlydrilTrolkg/m FL3 at MC0.57, 50 1.71,°C Glydril (composition barite MC 51.39) 1.71, in may barite% by be wt: best51.39) fresh modeled ma watery be by best39.18, a Quemadamodeled KCl 6.86, by model, Soda a Quemada whileash 0.06, the model, DuoTec Collins–Graves while NS 0.23, the modelCollins–GravesTrol FL may 0.57, provide Glydril model the MC may best 1.71, provide fit for barite a bentonite-basedthe 51.39) best fitma fory bea fluidbentonite-based best asmodeled it can be by seenfluid a Quemada in as Figure it can7 bemodel, (composition seen inwhile Figure the in %7Collins–Graves (composition by wt: fresh waterin model % by 94.23, maywt: bentonitefresh provide water the 5.49, 94.23, best caustic fitbe ntonitefor soda a bentonite-based 0.27,5.49, sodacaustic ash soda 0.01).fluid 0.27, as it soda can ashbe seen 0.01). in Figure 7 (composition in % by wt: fresh water 94.23, bentonite 5.49, caustic soda 0.27, soda ash 0.01).

Figure 7.7. FlowFlow curvescurves forfor a a bentonite bentonite based based water-based water-based mud mud (WBM). (WBM). Here Here the bestthe best fit is fit obtained is obtained with thewithFigure Collins–Graves the 7. Collins–Graves Flow curves model. for model. a bentonite based water-based mud (WBM). Here the best fit is obtained with the Collins–Graves model. From the above described examples, it seems that the predicted pressure losses, especially in turbulent flow, can differ significantly as a function of the chosen rheological model. Therefore, it should be a good practice to analyze which model is best suited for a fluid rather than choosing one model that should fit all possible fluids. This being said, it remains to be verified that finding the best rheological model that matches rheometer values helps improving the pressure gradient predictions.

3.5. Verification in a Laboratory Flow-Loop To verify whether the choice of the best fitting rheological behavior based on rheometer measurements is also discernable when comparing actual pressure gradients in a pipe and estimated values, as described in Sections 3.2 and 3.3, we recourse to measurements made in a laboratory flow-loop. A positive displacement pump circulates fluid stored in a thermo-regulated tank with internal circulation. First the fluid flows into a pipe with an internal diameter of 25 mm and then into a glass tube with an internal diameter of 15.5 mm. Along the glass tubes there are three differential pressure sensors. The distances between the ports of the differential pressure sensors are, respectively, Energies 2020, 13, x FOR PEER REVIEW 9 of 22 Energies 2020, 13, x FOR PEER REVIEW 9 of 22 From the above described examples, it seems that the predicted pressure losses, especially in From the above described examples, it seems that the predicted pressure losses, especially in turbulent flow, can differ significantly as a function of the chosen rheological model. Therefore, it turbulent flow, can differ significantly as a function of the chosen rheological model. Therefore, it should be a good practice to analyze which model is best suited for a fluid rather than choosing one should be a good practice to analyze which model is best suited for a fluid rather than choosing one model that should fit all possible fluids. This being said, it remains to be verified that finding the best model that should fit all possible fluids. This being said, it remains to be verified that finding the best rheological model that matches rheometer values helps improving the pressure gradient predictions. rheological model that matches rheometer values helps improving the pressure gradient predictions. 3.5. Verification in a Laboratory Flow-Loop 3.5. Verification in a Laboratory Flow-Loop To verify whether the choice of the best fitting rheological behavior based on rheometer To verify whether the choice of the best fitting rheological behavior based on rheometer measurements is also discernable when comparing actual pressure gradients in a pipe and estimated measurements is also discernable when comparing actual pressure gradients in a pipe and estimated values, as described in Sections 3.2 and 3.3, we recourse to measurements made in a laboratory flow- values, as described in Sections 3.2 and 3.3, we recourse to measurements made in a laboratory flow- loop. A positive displacement pump circulates fluid stored in a thermo-regulated tank with internal loop. A positive displacement pump circulates fluid stored in a thermo-regulated tank with internal circulation. First the fluid flows into a pipe with an internal diameter of 25 mm and then into a glass circulation. First the fluid flows into a pipe with an internal diameter of 25 mm and then into a glass tube with an internal diameter of 15.5 mm. Along the glass tubes there are three differential pressure tube with an internal diameter of 15.5 mm. Along the glass tubes there are three differential pressure Energiessensors.2020 The, 13 ,distances 6165 between the ports of the differential pressure sensors are, respectively,10 0.209 of 23 sensors. The distances between the ports of the differential pressure sensors are, respectively, 0.209 m, 0.212 m and 0.206 m, following the direction of the flow. The first sensor is placed 1.955 m away m, 0.212 m and 0.206 m, following the direction of the flow. The first sensor is placed 1.955 m away from the change in diameter from 25 mm to 15.5 mm. The second sensor is positioned 1.415 m apart 0.209from m,the 0.212 change m and in diameter 0.206 m, followingfrom 25 mm the to direction 15.5 mm. of theThe flow. second The sensor first sensor is positioned is placed 1.415 1.955 m awayapart from the first one and the last sensor is separated from the previous by 1.46 m (see Figure 8). There is fromfrom thethe changefirst one in and diameter the last from sensor 25 is mm separated to 15.5 mm. from The the secondprevious sensor by 1.46 is positionedm (see Figure 1.415 8). mThere apart is also a pressure reducing valve placed 1.5 m upstream from the change in diameter. The purpose of fromalso a the pressure first one reducing and the valve last sensor placed is separated1.5 m upstre fromam thefrom previous the change by 1.46 in diameter. m (see Figure The8 purpose). There isof this valve is to ensure that the fluid is fully sheared before it continues its journey inside the flow- alsothis avalve pressure is to reducing ensure that valve the placed fluid 1.5is fully m upstream sheared from before the it change continues in diameter. its journey The inside purpose the of flow- this loop. valveloop. is to ensure that the fluid is fully sheared before it continues its journey inside the flow-loop.

Figure 8. Schematic description of the laboratory flow-loop. FigureFigure 8.8. Schematic description of thethe laboratorylaboratory flow-loop.flow-loop. All the experiments were conducted at a controlled temperature of 25 °C. The temperature AllAll thethe experimentsexperiments werewere conductedconducted atat aa controlledcontrolled temperaturetemperature ofof 2525 ◦°C.C. TheThe temperaturetemperature variations during the experiments have not exceeded ±0.5 °C. The first measurements have been variationsvariations duringduring the the experiments experiment haves have not not exceeded exceeded0.5 ±0.5◦C. The°C. The first measurementsfirst measurements have beenhave madebeen made with an aqueous solution of glycerin (50 %vol)± having a density of 1115 kg/m3 3 at 25 °C. This is withmade an with aqueous an aqueous solution solution of glycerin of glycerin (50 %vol) (50 %vol) having having a density a density of 1115 of 1115 kg/ mkg/mat3 25at 25◦C. °C. This This is is a a Newtonian fluid with a viscosity of 0.00445 Pa.s (see Figure 9a). Newtoniana Newtonian fluid fluid with with a viscositya viscosity of of 0.00445 0.00445 Pa.s Pa.s (see (see Figure Figure9a). 9a).

Figure 9. (a) Rheogram of the aqueous solution of glycerin (50 %vol) measured at 25 ◦C with a scientiﬁc rheometer, (b) comparison between measured pressure gradients with the three diﬀerential pressure sensors and pressure gradients calculated with a Newtonian rheological behavior.

Figure9b shows the measured pressure gradients taken with the three di fferential pressure sensors. The measurements from the first differential pressure sensor have been filtered out for Reynolds number above 4000 as entrance effects influenced the readings. This corresponds to a turbulent flow regime for which entrance effects can perturb measurements at a larger distance than in laminar flow [23]. The calculated pressure gradient is shown as the solid brown curve. It is in good agreement for laminar and turbulent flow, but does not match well in the transitional flow regime. These results are essentially provided for verification purposes. The second experiment makes use of an aqueous solution of Carbopol with a low molecular weight at the concentration of 0.07%. The acidity of the solution is neutralized by adding sodium hydroxide in a concentration of 0.009%. The fluid is shear-thinning and well-described by a power law rheological behavior (τK = 0.019Pa.s, n = 0.848), seconded by the Quemada model Energies 2020, 13, x FOR PEER REVIEW 10 of 22

Figure 9. (a) Rheogram of the aqueous solution of glycerin (50 %vol) measured at 25 °C with a scientific rheometer, (b) comparison between measured pressure gradients with the three differential pressure sensors and pressure gradients calculated with a Newtonian rheological behavior.

Figure 9b shows the measured pressure gradients taken with the three differential pressure sensors. The measurements from the first differential pressure sensor have been filtered out for Reynolds number above 4000 as entrance effects influenced the readings. This corresponds to a turbulent flow regime for which entrance effects can perturb measurements at a larger distance than in laminar flow [23]. The calculated pressure gradient is shown as the solid brown curve. It is in good agreement for laminar and turbulent flow, but does not match well in the transitional flow regime. These results are essentially provided for verification purposes. The second experiment makes use of an aqueous solution of Carbopol with a low molecular weight at the concentration of 0.07%. The acidity of the solution is neutralized by adding sodium Energies 2020, 13, 6165 11 of 23 hydroxide in a concentration of 0.009%. The fluid is shear-thinning and well-described by a power law rheological behavior ( 𝜏 = 0.019𝑃𝑎. 𝑠, 𝑛 = 0.848 ), seconded by the Quemada model (𝜇 = . 1 (0.0053𝑃𝑎.µ = 0.0053 𝑠, 𝜇Pa=0.1𝑃𝑎.𝑠,𝛾.s, µ0 = 0.1 =Pa 14𝑠.s, γc ,𝑛= 14s =0.35− , n) =and0.35 the) and Heinz–Casson the Heinz–Casson rheological rheological behavior behavior (𝜏 = ∞ (0.08𝑃𝑎,τγ = 0.08 𝜏Pa=, 0.046𝑃𝑎,τK = 0.046Pa 𝑛, n = =0.50) (see0.50 )Figure (see Figure 10a). 10 Ina). Figure In Figure 10b, 10 web, we can can see see that that for for this this particular ﬂuid,fluid, the powerpower law rheologicalrheological behaviorbehavior provides more accurate pressure gradients for the wholewhole range of measurementsmeasurements thanthan thethe estimationsestimations mademade withwith thethe QuemadaQuemada andand Heinz–CassonHeinz–Casson models.models.

Figure 10.10. ((aa)) Rheogram Rheogram of of an aqueous solution of Carbopol with low molecular weight (0.07%) measured at 25 ◦°CC with a scientiﬁcscientific rheometer, (b)) comparisoncomparison betweenbetween measuredmeasured pressurepressure gradientsgradients with thethe three three di ﬀdifferentialerential pressure pressure sensors sensors and pressureand pressure gradients gradients calculated calculated with power with law, power Quemada law, andQuemada Heinz–Casson and Heinz–Casson rheological rheological behaviors. behaviors.

However, for the laminar flowflow regime,regime, pressurepressure loss gradients estimated with didifferentfferent rheological models models do do not notdiffer di much.ffer much. Figure 11a Figure shows 11thea measured shows the rheological measured behavior rheological of an behavioraqueous solution of an of aqueous polyanionic solution cellulose of polyanionic(concentration cellulose 0.2%) measured (concentration at 25 °C 0.2%) with a measured scientific at 25 C with a scientific rheometer together with curve-fitted models for power law rheometer◦ together with curve-fitted models for power law (𝜏 = 0.070𝑃𝑎. 𝑠, 𝑛 = 0.728,), Heinz– (τK = 0.070 Pa.s, n = 0.728), Heinz–Casson (τγ = 0.19 Pa, τK = 0.069 Pa, n = 0.324) and Quemada Casson ( 𝜏 = 0.19𝑃𝑎, 𝜏 = 0.069𝑃𝑎, 𝑛 = 0.324 ) and Quemada ( 𝜇 = 0.00323𝑃𝑎. 𝑠, 𝜇 = . 1 (µ = 0.00323 Pa.s, µ0= 0.100 Pa.s, γc = 1845 s− , n = 0.35) models. As it can be seen in Figure 11b, 0.100𝑃𝑎.∞ 𝑠, 𝛾 = 1845𝑠 ,𝑛 =0.35) models. As it can be seen in Figure 11b, the estimated pressure thegradients estimated are almost pressure indistinguishable gradients are almost from each indistinguishable other. Note that from past each a volumetric other. Note flowrate that past of 21 a volumetricl/min,Energies the 2020 flow flowrate, 13, xregime FOR ofPEER 21is transitional.REVIEW l/min, the flow regime is transitional. 11 of 22

FigureFigure 11. 11.( a(a)) Rheogram Rheogram of an an aqueous aqueous solution solution of of polyan polyanionicionic cellulose cellulose (PAC) (PAC) (0.2%) (0.2%) measured measured at 25 at 25°C◦C with with a scientific a scientiﬁc rheometer, rheometer, (b) comparison (b) comparison between between measured measured pressure pressure gradients gradients with the with three the threedifferential diﬀerential pressure pressure sensors sensors and pressure and pressure gradients gradients calculated calculated with power with law, power Quemada law, Quemada and Heinz- and Heinz-CassonCasson rheological rheological behaviors. behaviors.

However, it should be noted that transitional flow is observable on the differential pressure measurements, by the appearance of spikes of pressures. Just past the upper limit of laminar flow, these spikes are only occasional and of low amplitudes, but the more the Reynolds number increases within the transitional flow regime, the more frequent the spikes are and the larger are their amplitudes (see Figure 12). This effect is not reflected through the use of an interpolation of the pressure gradient between the laminar and turbulent pressure losses as described by Equation (32), as the result is only deterministic while in reality the phenomenon is stochastic in nature. When fitting the two above examples with a Herschel-Bulkley or a Robertson-Stiff rheological behavior, the fitted value of the yield stress turns out to be zero. If we utilize a zero yield stress in the Heinz-Casson rheological behavior, the model reduces to the one of a Newtonian fluid: 𝜏= 0 +𝜏𝛾 =𝜏𝛾 and therefore is not capable to reproduce the shear-thinning tendency that is measured with the rheometer. Therefore, the fitted parameters of the Heinz-Casson model always utilize a non-zero yield stress as soon as there is a shear-thinning tendency, even though, other models like Herschel-Bulkley or Robertson-Stiff would indicate that the yield stress is zero.

Energies 2020, 13, 6165 12 of 23

However, it should be noted that transitional flow is observable on the differential pressure measurements, by the appearance of spikes of pressures. Just past the upper limit of laminar flow, these spikes are only occasional and of low amplitudes, but the more the Reynolds number increases within the transitional flow regime, the more frequent the spikes are and the larger are their amplitudes (see Figure 12). This effect is not reflected through the use of an interpolation of the pressure gradient between the laminar and turbulent pressure losses as described by Equation (32), as the result is only deterministicEnergies 2020, 13 while, x FOR inPEER reality REVIEW the phenomenon is stochastic in nature. 12 of 22

FigureFigure 12.12.This This graph graph shows shows how how the the measured measured diﬀ erentialdifferenti pressuresal pressures start tostart be moreto be andmore more and erratic, more theerratic, more the the more Reynolds the Reynolds number number overpasses over thepasses laminar the tolaminar transitional to transitional limit. limit.

WhenFigure fitting 13a shows the two the above measured examples rheogram with a Herschel-Bulkleyof an aqueous solution or a Robertson-Sti of Carbopolff rheological with high behavior,molecular the weight. fitted This value time of the fluid yield has stress a yield turns stress. out to The be Herschel-Bulkley zero. If we utilize parameters a zero yield are: stress 𝜏 = in1.21𝑃𝑎, the Heinz-Casson 𝜏 = 0.266𝑃𝑎, rheological 𝑛 = 0.642, the behavior, parameters the modelfor the reducesRobertson-Stiff to the model one of are: a Newtonian 𝜏 = 380𝑃𝑎, fluid: 𝐵= 1 . n n . 0.597,n 𝐶 = 10.458 , those of the Heinz-Casson model are: 𝜏 = 0.24𝑃𝑎, 𝜏 = 0.488𝑃𝑎, 𝑛 = 0.271 and τ = 0 + τKγˆ = τKγˆ and therefore is not capable to reproduce the shear-thinning tendency that the parameters of the Quemada model are: 𝜇 = 0.00604𝑃𝑎. 𝑠, 𝜇 = 10000𝑃𝑎. 𝑠, 𝛾 = 970𝑠 ,𝑛 = is measured with the rheometer. Therefore, the fitted parameters of the Heinz-Casson model always 0.364. The comparison between the estimated pressure gradients and the measured ones is shown in utilize a non-zero yield stress as soon as there is a shear-thinning tendency, even though, other models Figure 13b. There are no significant differences between the predicted pressure gradients made with like Herschel-Bulkley or Robertson-Stiff would indicate that the yield stress is zero. the different models. There is a good agreement between the measured pressure gradients and the Figure 13a shows the measured rheogram of an aqueous solution of Carbopol with high molecular estimated ones made with the rheological models calibrated with the rheometer measurements, until weight. This time the fluid has a yield stress. The Herschel-Bulkley parameters are: = 1.21 Pa, = approximatively 22 L/min. However, for the largest flowrates, there is a discrepancyτγ building up,τK as 0.266 Pa, n = 0.642, the parameters for the Robertson-Sti model are: = 380 Pa, B = 0.597, C = the measurements tend to indicate that the fluid becomesff less shear-thinningτA than what is expected = = n = 10.458from the, those rheometer of the measurements. Heinz-Casson model are: τγ 0.24 Pa, τK 0.488 Pa, 0.271 and the . 1 parameters of the Quemada model are: µ = 0.00604 Pa.s, µ0 = 10, 000 Pa.s, γc = 970 s− , n = 0.364. ∞ The comparison between the estimated pressure gradients and the measured ones is shown in Figure 13b. There are no significant differences between the predicted pressure gradients made with the different models. There is a good agreement between the measured pressure gradients and the estimated ones made with the rheological models calibrated with the rheometer measurements, until approximatively 22 L/min. However, for the largest flowrates, there is a discrepancy building up, as the measurements tend to indicate that the fluid becomes less shear-thinning than what is expected from the rheometer measurements.

Figure 13. (a) rheogram of an aqueous solution of Carbopol with high molecular weight measured at 25 °C with a scientific rheometer, (b) comparison between measured pressure gradients with the three

Energies 2020, 13, x FOR PEER REVIEW 12 of 22

Figure 12. This graph shows how the measured differential pressures start to be more and more erratic, the more the Reynolds number overpasses the laminar to transitional limit.

Figure 13a shows the measured rheogram of an aqueous solution of Carbopol with high

molecular weight. This time the fluid has a yield stress. The Herschel-Bulkley parameters are: 𝜏 = 1.21𝑃𝑎, 𝜏 = 0.266𝑃𝑎, 𝑛 = 0.642, the parameters for the Robertson-Stiff model are: 𝜏 = 380𝑃𝑎, 𝐵= 0.597, 𝐶 = 10.458, those of the Heinz-Casson model are: 𝜏 = 0.24𝑃𝑎, 𝜏 = 0.488𝑃𝑎, 𝑛 = 0.271 and the parameters of the Quemada model are: 𝜇 = 0.00604𝑃𝑎. 𝑠, 𝜇 = 10000𝑃𝑎. 𝑠, 𝛾 = 970𝑠 ,𝑛 = 0.364. The comparison between the estimated pressure gradients and the measured ones is shown in Figure 13b. There are no significant differences between the predicted pressure gradients made with the different models. There is a good agreement between the measured pressure gradients and the estimated ones made with the rheological models calibrated with the rheometer measurements, until approximatively 22 L/min. However, for the largest flowrates, there is a discrepancy building up, as the measurements tend to indicate that the fluid becomes less shear-thinning than what is expected Energies 2020, 13, 6165 13 of 23 from the rheometer measurements.

Energies 2020, 13, x FOR PEER REVIEW 13 of 22 FigureFigure 13. 13.( a()a) rheogram rheogram ofof anan aqueous solution solution of of Carbop Carbopolol with with high high molecular molecular weight weight measured measured at at25 25 °CC with with a ascientific scientific rheometer, rheometer, (b) (comparisonb) comparison between between measured measured pressure pressure gradients gradients with the with three the differential◦ pressure sensors and pressure gradients calculated with power law, Quemada and Heinz– three differential pressure sensors and pressure gradients calculated with power law, Quemada and Casson rheological behaviors. Heinz–Casson rheological behaviors. All the previous fluids are non-thixotropic. This can be verified by confirming the time All the previous fluids are non-thixotropic. This can be verified by confirming the time independence of the shear stress after step-changing the rheometer speed. Figure 14 shows a series independence of the shear stress after step-changing the rheometer speed. Figure 14 shows a of measurements made with a scientific rheometer where the Newtonian shear rate is changed from series of measurements made with a scientific rheometer where the Newtonian shear rate is changed 1020, 512, 341, 170, 102, 51, 10.2, 5.1, 10.2, 51, 102, 341, 512, 1020, consecutively. It is noticeable that the from 1020, 512, 341, 170, 102, 51, 10.2, 5.1, 10.2, 51, 102, 341, 512, 1020, consecutively. It is noticeable that polyanionic cellulose high viscosity (PAC-HV) 0.2% Carbopol with low molecular weight and the the polyanionic cellulose high viscosity (PAC-HV) 0.2% Carbopol with low molecular weight and the second aqueous solution of Carbopol with high molecular weight respond immediately to the change second aqueous solution of Carbopol with high molecular weight respond immediately to the change in shear rate. in shear rate.

FigureFigure 14. 14. WithWith non-thixotropicnon-thixotropic ﬂuids,fluids, the the shear shear stress stress reaches reaches astable a stable value value almost almost immediately immediately after aftereach each change change in rheometer in rheometer speed. speed.

Figure 15 shows that with an aqueous solution of KCl and xanthan gum, the shear stress stabilizes to a constant value only after several tens of seconds after the shear rate has changed. The KCl/polymer fluid is therefore thixotropic.

Energies 2020, 13, 6165 14 of 23

Figure 15 shows that with an aqueous solution of KCl and xanthan gum, the shear stress stabilizes Energiesto a constant 2020, 13, value x FOR onlyPEER afterREVIEW several tens of seconds after the shear rate has changed. The KCl/polymer14 of 22

ﬂuidEnergies is 20 therefore20, 13, x FOR thixotropic. PEER REVIEW 14 of 22

Figure 15. With a thixotropic fluid, it is noticeable that the shear stress stabilizes to a steady state value after several tens of seconds each time the rheometer speed is changed. Figure 15. WithWith a thixotropic thixotropic fluid, fluid, it is noticeable that the shear stress stabilizes to a steady state value after several tenstens of seconds each timetime the rheometer speedspeed isis changed.changed. Figure 16a shows the steady state rheogram measured with a KCl/xanthan gum fluid while Figure 16b shows that the three differential pressure sensors measure different pressure gradients Figure 16 a shows the steady state rheogram measured with a KClKCl//xanthanxanthan gumgum fluidfluid while until a flowrate of 25 L/min, in contradiction with what has been observed when utilizing non- Figure 1616bb showsshows that that the the three three di ffdifferentialerential pressure pressure sensors sensors measure measure different different pressure pressure gradients gradients until a thixotropic fluids. flowrateuntil a flowrate of 25 L/min, of 2 in5 contradictionL/min, in contradiction with what has with been what observed has been when observed utilizing non-thixotropic when utilizing fluids. non- thixotropic fluids.

FigureFigure 16. 16. ((aa)) Rheogram Rheogram of of an an aqueous aqueous solution solution of of xant xanthanhan gum gum and and KCl KCl measured measured at at 25 25 °C◦C with with a a scientificscientific rheometer, rheometer, ( (bb)) comparison comparison between between measured measured pressure pressure gradients gradients with with the the three three differential differential Figure 16. (a) Rheogram of an aqueous solution of xanthan gum and KCl measured at 25 °C with a pressurepressure sensors sensors and and pressure pressure gradients gradients calcul calculatedated with with power power law, law, Quemada Quemada and and Heinz–Casson Heinz–Casson scientific rheometer, (b) comparison between measured pressure gradients with the three differential rheologicalrheological behaviors. behaviors. pressure sensors and pressure gradients calculated with power law, Quemada and Heinz–Casson rheological behaviors. AsAs it it seems seems that that thixotropic effects effects influenceinfluence thethe pressurepressure gradientsgradients atat didifferentfferent positions positions along along a apipe pipe after after a a change change in in diameter, diameter, we we will will analyze, analyze, in in the the next next section, section, how how to to incorporate incorporate thixotropy thixotropy in As it seems that thixotropic effects influence the pressure gradients at different positions along inpressure pressure loss loss calculations calculations for for flow flow in in circular circular pipes. pipes. a pipe after a change in diameter, we will analyze, in the next section, how to incorporate thixotropy in pressure loss calculations for flow in circular pipes. 4. Laminar Flow Pressure Losses of a Thixotropic Fluid in a Hydraulic Circuit with Change in Diameters 4. Laminar Flow Pressure Losses of a Thixotropic Fluid in a Hydraulic Circuit with Change in Diameters

Energies 2020, 13, 6165 15 of 23

4. Laminar Flow Pressure Losses of a Thixotropic Fluid in a Hydraulic Circuit with Change in Diameters The thixotropic response of a fluid is described by a dimensionless variable called the structure parameter (λ). The structure parameter corresponds to the degree of entanglement of floccules in the fluid, such as that generated by the buildup of structures by polymers or clay platelets. Mewis and Wagner (2009) [24] propose a general description of the time evolution of the structure parameter through the following ordinary differential equation:

dλ . a . c = k γ (1 λ)b k γ λd, (34) dt 1 − − 2 where k1 and k2 are, respectively, the coefficients that characterize the breakdown and the buildup of floccules, and a, b, c and d are either explicitly specified by the model or fitted to a series of observations. During the derivation of his model, Quemada [14] utilizes a = 0, b = 1, c = 1 and d = 1 and we will follow the same assumption. The structure parameter is then used to modify the flow curve. Here, we will utilize a simple first order function of the structure parameter:

.  λ λγ,  . − ∞ .  τγ,λ = 1 + α . τγ, λγ, , (35)  ∞ . ∞  dλ = k (1 λ) k γλ dt 1 − − 2 . . where α is a scaling factor, λγ, is the value of the structure parameter for the shear rate γ in steady ∞ state conditions, i.e., when t , and τ . is the best match rheological behavior in steady state → ∞ γ, conditions, e.g., power law, Collins–Graves, Herschel–Bulkley,∞ Robertson–Stiff, Heinz–Casson, Carreau, Quemada. The integration of the ordinary differential equation of the structure parameter at constant shear rate gives: . (k +k γ)tstep k1 λ = βe− 1 2 + . , (36) k1 + k2γ where β is an integration constant and tstep is the time since the shear rate changed. If we assume a continuity of the structure parameter when the shear rate changes and if we consider that the origin of time is set at the moment when the shear rate is modified, then:

k1 k1 tstep = 0, λ0 = . + β β = λ0 . , (37) k1 + k2γ ⇔ − k1 + k2γ

. denoting λ0 the value of the structure parameter when the shear rate changed. The value of λγ, can then be calculated: ∞ k . 1 λγ, = lim λ = . . (38) tstep + ∞ →∞ k1 k2γ When the flowrate is constant, there is no acceleration in the portions of pipes with a constant diameter and, therefore, Equation (1) and consequently Equation (3) still yield, regardless of the thixotropic behavior. The hydraulic circuit is now discretized in short sections of length ∆s (dimension [L] (m)) and each of these sections is indexed by the subscript i. The cross-section is discretized in n d = wi small shells of radial thickness ∆ri 2n (dimension [L] (m)), where dwi is the internal diameter at along pipe position i. The radial position is indexed with the subscript j, starting at zero at the outer position in the cross-section. In laminar flow, we assume that the fluid contained in the shell i 1, j moves to vi 1,j − − the shell i, j after an elapsed time s , where vi 1,j is the fluid velocity at along hole and radial position ∆ − i 1, j (see Figure 17). It may be that the diameter of the pipe changes from along tube position i 1 to − − Energies 2020, 13, 6165 16 of 23

i from a diameter dwi 1 to a diameter dwi . Assuming that the ﬂuid is incompressible within the working − pressure, the mass conservation equation can simply be expressed as a volume conservation:

2 2! 2 dwi 1 dwi 1 dwi 1 dwi 1 dwi dwi vi 1,j 2− j 2n− 2− (j + 1) 2n− = vi,j 2 j 2n − − − − − − 2! (39) dw dw i (j + 1) i , 2 − 2n and therefore, we can express the ﬂuid velocity at the entrance of the shell with axial index i: Energies 2020, 13, x FOR PEER REVIEW 16 of 22 2 2! dwi 1 dwi 1 dwi 1 dwi 1 vi 1 2− j 2n− 2− (j + 1) 2n− − − − − vi,j = . (40) 2 2! dwi dwi dwi dwi 𝑣, = ( + ) . (40) 2 j2n 2 j 1 2n − − −

Figure 17. TheThe conditions conditions on on a ashell shell at at abscissa abscissa position position 𝑖 i andand radial radial position position 𝑗j are influencedinfluenced by the conditions at the preceding shell at position i 1 and j. conditions at the preceding shell at position 𝑖−1− and 𝑗. It is then possible to perform a numerical integration along the hydraulic circuit, considering that It is then possible to perform a numerical integration along the hydraulic circuit, considering the initial conditions of the structure parameter at each radial position are imposed by the fact that that the initial conditions of the structure parameter at each radial position are imposed by the fact the fluid has been flowing through a long section of pipe with a constant diameter and therefore the that the fluid has been flowing through a long section of pipe with a constant diameter and therefore . structure parameter is equal to λγ, anywhere in a cross-section upstream from the change in diameter. the structure parameter is equal to 𝜆 , anywhere in a cross-section upstream from the change in The same methodology is followed∞ as for solving the pressure losses with a Heinz–Casson fluid, except diameter. The same methodology is followed as for solving the pressure losses with a Heinz–Casson that we need to keep track of the time evolution of the structure parameter for each radial position and fluid, except that we need to keep track of the time evolution of the structure parameter for each at each axial position. radial position and at each axial position. In the laboratory flow-loop, when pumping at a constant volumetric flowrate of 6 L/min in the In the laboratory flow-loop, when pumping at a constant volumetric flowrate of 6 L/min in the example flow-loop depicted in Figure8, with a thixotropic fluid where k = 0.003, k = 0.001, α = 0.05, example flow-loop depicted in Figure 8, with a thixotropic fluid where1 𝑘 = 0.0032, 𝑘 = 0.001, 𝛼= the structure parameter is impacted by the change in pipe diameter from 25 to 15.5 mm (see Figure 18). 0.05, the structure parameter is impacted by the change in pipe diameter from 25 to 15.5 mm (see It is mostly the structure parameter closest to the pipe wall which is affected by the change in geometry. Figure 18). It is mostly the structure parameter closest to the pipe wall which is affected by the change In the central region of the pipe, there is an unsheared region corresponding to the yield stress of the in geometry. In the central region of the pipe, there is an unsheared region corresponding to the yield fluid. This part is not much influenced after the transition from the larger to the smaller diameter. stress of the fluid. This part is not much influenced after the transition from the larger to the smaller The axial evolution of the structure parameter impacts in turn the relationship between the shear diameter. stress and shear rate. As the shear stress needs to be a linear function of the radial position (to the limit of yield stress), this impacts the acceptable shear rate and consequently how the structure parameter evolves in the axial direction. Figure 19 shows the estimated values of the shear rate in both the axial and radial directions. We can see that the change in diameter continues to impact the shear rate over a long distance after the change in diameter.

Figure 18. Representation of the structure parameter as a function of the normalized radial position and axial distance after the pressure relief valve. The color scale is gradient based.

The axial evolution of the structure parameter impacts in turn the relationship between the shear stress and shear rate. As the shear stress needs to be a linear function of the radial position (to the

Energies 2020, 13, x FOR PEER REVIEW 16 of 22

𝑣, = . (40)

Figure 17. The conditions on a shell at abscissa position 𝑖 and radial position 𝑗 are influenced by the conditions at the preceding shell at position 𝑖−1 and 𝑗.

It is then possible to perform a numerical integration along the hydraulic circuit, considering that the initial conditions of the structure parameter at each radial position are imposed by the fact that the fluid has been flowing through a long section of pipe with a constant diameter and therefore the structure parameter is equal to 𝜆 , anywhere in a cross-section upstream from the change in diameter. The same methodology is followed as for solving the pressure losses with a Heinz–Casson fluid, except that we need to keep track of the time evolution of the structure parameter for each radial position and at each axial position. In the laboratory flow-loop, when pumping at a constant volumetric flowrate of 6 L/min in the example flow-loop depicted in Figure 8, with a thixotropic fluid where 𝑘 = 0.003, 𝑘 = 0.001, 𝛼= 0.05, the structure parameter is impacted by the change in pipe diameter from 25 to 15.5 mm (see Figure 18). It is mostly the structure parameter closest to the pipe wall which is affected by the change in geometry. In the central region of the pipe, there is an unsheared region corresponding to the yield stress of the fluid. This part is not much influenced after the transition from the larger to the smaller Energies 2020, 13, 6165 17 of 23 diameter.

Energies 2020, 13, x FOR PEER REVIEW 17 of 22

limit of yield stress), this impacts the acceptable shear rate and consequently how the structure parameter evolves in the axial direction. Figure 19 shows the estimated values of the shear rate in both the axial and radial directions. We can see that the change in diameter continues to impact the Figure 18.Figure Representation 18. Representation of the of structure the structure parameter parameter as aa function function of theof the normalized normalized radial positionradial position shear rate overand axiala long distance distance after the after pressure the reliefchange valve. in The diameter. color scale is gradient based. and axial distance after the pressure relief valve. The color scale is gradient based.

Figure 19.Figure The 19.resultThe of result the oftime the evolution time evolution of the of thestructur structuree parameter parameter at didifferentfferent radial radial positions positions after after the change in diameter is axial evolution of the shear rate (here normalized). The color scale is the change in diameter is axial evolution of the shear rate (here normalized). The color scale is gradient gradient based. based. As a consequence, the pressure gradient after the change in diameter is larger than if it were As estimateda consequence, with a non-thixotropic the pressure fluid gradient having after a similar the rheological change in behavior. diameter The is pressure larger gradient than if it were decays to the one corresponding to steady state conditions at infinite distance to any change in estimated with a non-thixotropic fluid having a similar rheological behavior. The pressure gradient geometrical dimensions (see Figure 20). decays to the one corresponding to steady state conditions at infinite distance to any change in geometrical dimensions (see Figure 20).

Figure 20. (a) Pressure gradients along the flow loop for non-thixotropic and thixotropic fluids having the same steady state rheological behavior, (b) structure parameter in various cross-sections along the flow-loop, (c) shear rate in various cross-sections along the flow-loop.

The difference of the pressure gradient estimated with a thixotropic fluid compared to a non- thixotropic fluid with equivalent rheological behavior in steady state conditions increases with the volumetric flowrate (see Figure 21). The difference is of course largest the closest to the change in diameter, but it is estimated to still have effects more than 5.5 m after the change in diameter in the context of the laboratory flow-loop. With that particular fluid and with the described flow-loop dimensions, it is necessary to position a differential pressure sensor at least 7.7 m downstream from the change in pipe diameter to be within 1% of the pressure gradients for which thixotropic effects would be negligible.

Energies 2020, 13, x FOR PEER REVIEW 17 of 22 limit of yield stress), this impacts the acceptable shear rate and consequently how the structure parameter evolves in the axial direction. Figure 19 shows the estimated values of the shear rate in both the axial and radial directions. We can see that the change in diameter continues to impact the shear rate over a long distance after the change in diameter.

Figure 19. The result of the time evolution of the structure parameter at different radial positions after the change in diameter is axial evolution of the shear rate (here normalized). The color scale is gradient based.

As a consequence, the pressure gradient after the change in diameter is larger than if it were estimated with a non-thixotropic fluid having a similar rheological behavior. The pressure gradient decays to the one corresponding to steady state conditions at infinite distance to any change in Energies 2020, 13, 6165 18 of 23 geometrical dimensions (see Figure 20).

FigureFigure 20. (a) Pressure Pressure gradients along the flow flow loop for non-thixotropic and thix thixotropicotropic fluidsfluids having thethe samesame steadysteady state rheological behavior, ((bb)) structurestructure parameterparameter inin variousvarious cross-sectionscross-sections alongalong thethe flow-loop,flow-loop, ((cc)) shearshear raterate inin variousvarious cross-sectionscross-sections alongalong thethe flow-loop.flow-loop.

TheThe difference difference of of the the pressure pressure gradient gradient estima estimatedted with with a thixotropic a thixotropic fluid fluid compared compared to a non- to a non-thixotropicthixotropic fluid fluid with with equivalent equivalent rheological rheological behavior behavior in steady in steady state state conditions conditions increases increases with with the thevolumetric volumetric flowrate flowrate (see (see Figure Figure 21). 21 The). The difference difference is of is course of course largest largest the the closest closest to tothe the change change in indiameter, diameter, but but it is it estimated is estimated to still to still have have effects effects more more than than 5.5 5.5m after m after the thechange change in diameter in diameter in the in thecontext context of the of the laboratory laboratory flow-loop. flow-loop. With With that that pa particularrticular fluid fluid and and with with the the described flow-loopflow-loop dimensions,dimensions, itit isis necessarynecessary to to position position a a di differentialfferential pressure pressure sensor sensor at leastat least 7.7 7.7 m downstreamm downstream from from the changethe change in pipe in pipe diameter diameter to be to within be within 1% of 1% the of pressure the pressure gradients gradients for which for which thixotropic thixotropic effects wouldeffects Energiesbewould negligible. 2020 be negligible., 13, x FOR PEER REVIEW 18 of 22

Figure 21. 21. (a()a )Estimated Estimated pressure pressure gradients gradients at the at the position position of the of sensors the sensors along along the flow-loop the flow-loop for non- for thixotropicnon-thixotropic and thixotropic and thixotropic fluid fluid having having the same the same steady steady state state rheological rheological behavior, behavior, (b) (percentageb) percentage of theof the pressure pressure gradient gradient difference difference for for the the thixotropic thixotropic fluid fluid with with regards regards to tothe the non-thixotropic non-thixotropic fluid. fluid.

5. Discussion Discussion The estimation of pressure losses for drilling operat operationsions can basically be split into the estimation estimation of pressurepressure losseslosses inside inside the the string string and and in the in annulus.the annulus. The paperThe paper focuses focuses on the estimationon the estimation of pressure of pressurelosses in circularlosses in pipes circular and, pipes therefore, and, istherefore, applicable is applicable to pressure to losses pressure inside losses the string. inside Such the string. estimations Such estimationsare important are when important determining when determining the maximum the ﬂowrate maximum possible flowrate to use possib whenlebeing to use limited when bybeing the limitedmaximum by pressurethe maximum rating pressure of the surface rating installation of the surface pumping installation system. pumping Such pressure system. gradients Such pressure are also gradientsimportant are when also estimating important thewhen dynamic estimating response the dynamic of the drilling response system of the to drilling pump accelerations system to pump and accelerationsmore generally and when more solving generally the Navier–Stokes when solving equations the Navier–Stokes for the complete equations hydraulic for circuit.the complete hydraulicIn practice, circuit. it is not uncommon that the estimated pump pressures made with real-time hydraulic calculationsIn practice, are o ﬀitby isseveral not uncommon tens of bars. that There the may estimated be many pump reasons pressures for such inaccuracies,made with including,real-time hydraulic calculations are off by several tens of bars. There may be many reasons for such inaccuracies, including, but not limited to, the lack of precision of rheometer measurements made at the rig site, the accuracy of pressure and temperature extrapolation models for the fluid rheological behavior, uncertainty related to the actual pipe internal diameter, unprecise modelling of the effect of change in diameters at the level of tool-joints. However, as presented above, simply the choice of the rheological model may substantially influence the estimated pressure losses in the turbulent flow regime. Indeed, most of the time, the flow regime is turbulent inside the pipes during normal drilling operations. Yet, there is no special information available that may help decide which rheological model is the most pertinent to use for the turbulent flow regime. This problem could be resolved if inline pipe rheometers performed measurements in the turbulent flow regime. Such pipe rheometers could indicate which rheological behavior is best suited to estimate pressure losses in pipes when subjected to turbulent flow. At the same time, they could provide the parameters of the best-suited rheological model. The calibration of these parameters requires one to apply low flowrates in order to accurately estimate the yield stress, if any, of the drilling fluid. Yet, we have seen in Section 4 the importance of the change in diameters on the estimated pressure gradients in a pipe rheometer when utilizing thixotropic fluids. The estimated parameters of the rheological model may become biased by the impact of structuration and de- structuration of floccules after passing a restriction or an enlargement. It is only past long distances that these effects become negligible and therefore the size of a pipe rheometer could turn out to be unreasonably large to eliminate those effects. Another solution would be to position several differential pressure sensors, as shown in Figure 8, in order to collect information about thixotropic effects and thereafter calibrate the thixotropic model described by Equation (35). As a continuation of this work, we will, in the near future, investigate the impact of time dependent changes on the boundary conditions, for instance oscillations of the volumetric flowrate, when applied to thixotropic fluids. Figure 22 illustrates initial measurements that have been made in the laboratory flow-loop. It seems that the time dependence of the rheological properties of the thixotropic fluid influences the rapidity by which the fluid can respond to quick changes in flow

Energies 2020, 13, 6165 19 of 23 but not limited to, the lack of precision of rheometer measurements made at the rig site, the accuracy of pressure and temperature extrapolation models for the fluid rheological behavior, uncertainty related to the actual pipe internal diameter, unprecise modelling of the effect of change in diameters at the level of tool-joints. However, as presented above, simply the choice of the rheological model may substantially influence the estimated pressure losses in the turbulent flow regime. Indeed, most of the time, the flow regime is turbulent inside the pipes during normal drilling operations. Yet, there is no special information available that may help decide which rheological model is the most pertinent to use for the turbulent flow regime. This problem could be resolved if inline pipe rheometers performed measurements in the turbulent flow regime. Such pipe rheometers could indicate which rheological behavior is best suited to estimate pressure losses in pipes when subjected to turbulent flow. At the same time, they could provide the parameters of the best-suited rheological model. The calibration of these parameters requires one to apply low flowrates in order to accurately estimate the yield stress, if any, of the drilling fluid. Yet, we have seen in Section4 the importance of the change in diameters on the estimated pressure gradients in a pipe rheometer when utilizing thixotropic fluids. The estimated parameters of the rheological model may become biased by the impact of structuration and de-structuration of floccules after passing a restriction or an enlargement. It is only past long distances that these effects become negligible and therefore the size of a pipe rheometer could turn out to be unreasonably large to eliminate those effects. Another solution would be to position several differential pressure sensors, as shown in Figure8, in order to collect information about thixotropic effects and thereafter calibrate the thixotropic model described by Equation (35). As a continuation of this work, we will, in the near future, investigate the impact of time dependent changes on the boundary conditions, for instance oscillations of the volumetric flowrate, when applied Energiesto thixotropic 2020, 13, x fluids. FOR PEER Figure REVIEW 22 illustrates initial measurements that have been made in the laboratory19 of 22 flow-loop. It seems that the time dependence of the rheological properties of the thixotropic fluid conditions.influences the This rapidity is an byimportant which the step fluid to canstudy respond before to investigating quick changes the in flowimpact conditions. of thixotropy This is on an pressureimportant losses step toin studyan annulus before under investigating the influence the impact of a moving of thixotropy inner onpipe, pressure both axially, losses in radially an annulus and laterally.under the influence of a moving inner pipe, both axially, radially and laterally.

Figure 22. ((aa)) The The application application of of volumetric volumetric flowrate flowrate step step changes changes on on a a non-thixotropic non-thixotropic fluid fluid (here (here an an aqueous solutionsolution of of PAC) PAC) results results in a pressurein a pressure gradient gradient evolution evolution that is only that influenced is only influenced by momentum by momentumeffects, (b) aeffects, thixotropic (b) a fluid thixotropic (here a fluid KCl xanthan(here a KCl gum xanthan aqueous gum solution) aqueous responds solution) diff erentlyresponds to differentlysimilar volumetric to similar flowrates volumetric as aflowrates consequence as a ofconsequence the time necessary of the time for necessary the structure for parameterthe structure to parameterreach equilibrium. to reach equilibrium.

6. Conclusions We have derived semi-analytical solutions for the estimation of pressure gradients in cylindrical pipes with the Collins–Graves and Quemada rheological behaviors in laminar flow conditions. A method to numerically calculate the pressure gradients with the Heinz–Casson rheological behavior has also been described. It has served as a basis for estimating pressure gradients with thixotropic fluids. A general method has been explained to numerically calculate pressure gradients in turbulent flow. We have seen that different rheological behaviors, fitting accurately with high precision rheometer measurements, give similar pressure gradients in the laminar flow regime. However, the differences may be substantial when estimating the pressure gradients in turbulent flow. We have also seen that for thixotropic fluids, the passage through a change in diameter may influence the pressure gradients over substantial distances. We have exposed a method to estimate the pressure gradients in steady state conditions for thixotropic fluids circulating in a hydraulic circuit containing diameter variations. The results have been confirmed by measurements made with a laboratory scale flow-loop.

Funding: This research was funded by the Norwegian Research Council, Equinor and Sekal grant number 308826.

Acknowledgments: The authors would like to thank Hans Joakim Skadsem from the University of Stavanger for valuable inputs during the preparation of this work.

Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Energies 2020, 13, 6165 20 of 23

6. Conclusions We have derived semi-analytical solutions for the estimation of pressure gradients in cylindrical pipes with the Collins–Graves and Quemada rheological behaviors in laminar flow conditions. A method to numerically calculate the pressure gradients with the Heinz–Casson rheological behavior has also been described. It has served as a basis for estimating pressure gradients with thixotropic fluids. A general method has been explained to numerically calculate pressure gradients in turbulent flow. We have seen that different rheological behaviors, fitting accurately with high precision rheometer measurements, give similar pressure gradients in the laminar flow regime. However, the differences may be substantial when estimating the pressure gradients in turbulent flow. We have also seen that for thixotropic fluids, the passage through a change in diameter may influence the pressure gradients over substantial distances. We have exposed a method to estimate the pressure gradients in steady state conditions for thixotropic fluids circulating in a hydraulic circuit containing diameter variations. The results have been confirmed by measurements made with a laboratory scale flow-loop.

Author Contributions: Conceptualization, E.C.; methodology, E.C.; software, E.C.; validation, E.C.; formal analysis, E.C.; investigation, E.C.; resources, A.L.; data curation, A.L.; writing—original draft preparation, E.C.; writing—review and editing, E.C. and A.L.; visualization, E.C.; supervision, E.C.; project administration, E.C.; funding acquisition, E.C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Norwegian Research Council, Equinor and Sekal grant number 308826. Acknowledgments: The authors would like to thank Hans Joakim Skadsem from the University of Stavanger for valuable inputs during the preparation of this work. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A Weissenberg–Rabinowitsch–Mooney–Schofield have demonstrated that in laminar flow, the flow-rate in a pipe of a generalized Newtonian fluid is [13]:

3 Z τ 3 πd w . πd Q = w γτ2dτ = w I, (A1) 3 3 8τw 0 8τw

R τ . I = w 2d where 0 γτ τ. For the Collins–Graves rheological behavior, the expression of this integral is:

Z τ w . . 2 . 2 tCGγ I = γ τγ + µpγ (1 e− ) dτ. (A2) 0 − . Now, we will change the integration variable of this integral from τ to γ, by using the fact that the 1st derivative of Equation (8) relatively to the shear rate is:

d . . . τ tCGγ tCGγ . = tCG τγ + µpγ e− + µp(1 e− ). (A3) dγ −

Therefore, the integral I, for a Collins–Graves ﬂuid, is:

Z γ w . . 2 . 2 . . . . tCGγ tCGγ tCGγ I = γ τγ + µpγ (1 e− ) tCG τγ + µpγ e− + µp(1 e− )dγ. (A4) 0 − −

The integration result is: . A Be 3tCGγ I = − , (A5) − 4 648tGC Energies 2020, 13, 6165 21 of 23

P3 P3 . = 3 + 2 + 2 2 + 3 3 = i + 3 itCGγ + where A 3661µp 3450tCGτγµp 1530tGCτγµp 396tGCτγ and B aiτγ biµpe i=1 i=1 P4 P4 . P3 . P2 . 3 i i = 2 itCGγ = itCGγ = itCGγ = 4 4 ciµpγwtGC, with a1 a1,iµpe , a2 a2,iµpe , a3 a3,ie , b1 648γwtGC i=0 i=0 i=0 i=0 − − 3 3 2 2 4 4 3 3 2 2 324γ t 486γ t 486γwt 243, b = 648γ t + 648γ t + 1944γ t + 388γwt + 3888, w GC − w GC − CG − 2 w GC w GC w GC CG = 4 4 ======3 4 + b3 162γwtGC, c0 16, c1 48, c2 72, c3 72, c4 216, a1,0 648γwtGC 2 3 + 2 + = 3 4 2 3 2 = 3 4 + 216γwtGC 144γwtGC 48tCG, a1,1 1944γwtGC 972γwtGC 972γwtGC 486tCG, a1,2 1944γwtGC 2 3 2 − −3 4 − 2 4 − 3 2 1944γ t + 3888γwt + 3888t , a = 432γ t , a = 648γ t + 216γwt + 72t , a = w GC GC CG 1,3 − w GC 2,0 w GC GC GC 2,1 2 4 3 2 = 2 4 + 3 + 2 = 2 4 1944γwtGC 972γwtGC 486tGC, a2,2 1944γwtGC 1944γwtGC 1944tGC, a2,3 324γwtGC, − −4 3 − 4 3 4 3 − a = 216γwt + 72t , a = 648γwt 324t , a = 648γwt + 648t . 3,0 GC GC 3,1 − GC − GC 3,2 GC GC To determine the wall shear rate, one shall solve the following equation:

d dp . . d dp w tCGγ w τw = τγ + µpγ (1 e− ) = . (A6) 4 ds ⇔ − 4 ds Equation (A6) can be solved numerically using a Newton–Raphson method, starting with the . Newtonian wall shear rate (γwN) as an initial value:

. 8v γwN = , (A7) dw

Appendix B

q µ Let us introduce a shorthand notation of the Quemada rheological behavior by noting χ = ∞ µ0 . n and Γ = γc , i.e.,  . n 2 .  Γ + γ  τ = γµ  . n  , (A8) ∞ Γχ + γ  Following the Weissenberg–Rabinowitsch–Mooney–Schoﬁeld method, the expression of the integral I of Equation (A1) for a Quemada ﬂuid is:

. 4 Z τw  n  . 3  Γ + γ  I = 2   d γ µ  . n  τ. (A9) 0 ∞ Γχ + γ

From Equation (A8), we have:

. n2 . n . n . n . n2 µ Γ + γ + 2nµ γ Γ + γ 2nµ γ Γ + γ dτ ∞ ∞ ∞ . = , (A10) . n2 . n3 dγ Γχ + γ − Γχ + γ which we can substitute in Equation (A9), after changing the integration limits:

.  . n2 . n . n . n . n2  Z γ  . n 4 w . Γ + γ µ Γ + γ + 2nµ γ Γ + γ 2nµ γ Γ + γ  . 3 2    ∞ ∞ ∞  I = γ µ  .   dγ. (A11)  n   . n2 . n3  0 ∞ Γχ + γ  Γχ + γ − Γχ + γ 

. n If Γχ γ , then Equation (A11) simpliﬁes to: | |

.  . n2 . n . n . n . n2  Z γ  . n 4 w . Γ + γ µ Γ + γ + 2nµ γ Γ + γ 2nµ γ Γ + γ  . 3 2    ∞ ∞ ∞  I = γ µ    . dγ. (A12)  . n   . 2n n  0 ∞ γ  γ − γ3  Energies 2020, 13, 6165 22 of 23

The result of the integration Equation (A12) is:

 6 . 4 in  X − 3  αiγw  I = µ  , (A13) ∞ 4 in  i=0 −

2 3 4 5 where α0 = 1, α1 = 2Γ(3 n), α2 = 5Γ (3 2n), α3 = 20Γ (1 n), α4 = 5Γ (3 4n), α5 = 2Γ (3 5n), 6 − − − − − α6 = Γ (1 2n). − . n Otherwise, if Γχ cannot be neglected compared to γ , there is no analytic form for the integral | | and Equation (A11) shall be integrated numerically, for example by using Simpson’s method. It has been found that the precision of the integral with Simpson’s method is better than 0.1% past 100 discretization steps. To determine the wall shear rate, one shall solve the following equation:

 . n 2 + dw dp .  Γ γw  dw dp τw = γwµ  . n  = . (A14) 4 ds ⇔ ∞  4 ds Γχ + γw

Equation (A14) can be solved numerically using a Newton–Raphson method, starting with the . Newtonian wall shear rate (γwN) as an initial value.

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