A Novel Prediction Model of the Drag Coefficient of Irregular Particles in Power-Law Fluids

Abstract

The settlement drag coefficient of irregular particles in power-law fluids is a crucial parameter in the field of petroleum engineering. However, the irregular shape of the particle and the non-Newtonian rheological properties of the fluid make it challenging to predict the settlement drag coefficient. In this study, the spherical and irregular particle sedimentation processes in power-law fluids have been analyzed using a visual device and a high-speed camera system. A mechanical model dependent on the force balance of settlement particles was adopted to conduct a detailed statistical analysis of 114 spherical particle experimental results, and a prediction model of the drag coefficient of spherical particles in the power-law fluid was established with a mean relative error of 3.85%. On this basis, considering the influence of geometric shape on the law of particle sedimentation, a new irregular particle sedimentation resistance coefficient model in power-law fluid is established via the incorporation of the parameter circularity of 2D shape description c into the spherical particle sedimentation resistance coefficient predictive model. The parameters in the new irregular particle sedimentation resistance coefficient predictive model can be obtained via nonlinear data fitting of the 211 groups of irregular particles using experimental results in the power-law fluid. The model has high prediction accuracy for the drag coefficient of irregular particles in power-law fluid, with a mean relative error of 4.47, and expands the scope of engineering applications, which is of great significance for fracturing scheme design and wellbore cleaning.

1. Introduction

The sedimentation behavior of solid particles is common in the field of petroleum engineering [1,2,3]. For instance, during the drilling of extended-reach horizontal wells, due to gravity, the rock debris will be deposited at the lower edge of the annulus in highly deviated wells and horizontal wells to form a cutting bed, which will cause wellbore cleaning problems. On the other hand, in the process of hydraulic fracturing, the lack of sand-carrying capacity of the fracturing fluid will make the proppant quickly settle at the bottom of the fracture, resulting in the proppant not completely filling the fracture, and the stimulation effect is not obvious. Studying the settling velocity of particles in fluid can provide a basis for the performance evaluation and optimal design of drilling fluid and fracturing fluid [4,5]. In the case of the free settlement of a single particle in a static viscous fluid, it is ruled by three vertical forces, namely buoyancy F_B, gravity F_G, and drag force F_D (viscous resistance). Under laminar and turbulent flow conditions, Figure 1 shows the particle settlement process diagram.

F_D is the force from the fluid if a single particle moves in the fluid. It is the key coefficient describing the momentum and energy transfer between the fluid and the particle. It is one of the key representative coefficients of the mutual effect between the fluid and the particle. Its definition is as follows [6]:

$$F_D=\frac{1}{8}{\rho }_l{C}_D\pi d_{}^2{v}_s^2{}^{}$$

(1)

In the initial stage of settlement, the particles start sinking because of gravity F_G. However, with the increase in the settling velocity, the drag force F_D of the particles increases continuously until the drag force F_D, buoyancy F_B, and gravity F_G acting on the particles are balanced. At this time, the particles will settle at a constant speed, which is the terminal settling velocity of the particles v_s.

At this time, the resultant force of the particles in the vertical direction is 0:

$$F_G-F_B=F_D=F_{G\text{-}B}$$

(2)

Among them, the resultant force F_G-B of gravity and buoyancy is expressed with the following formula [6]:

$$F_{G\text{-}B}=\pi ({\rho }_s-{\rho }_l)g\frac{d^3}{6}$$

(3)

The terminal settling velocity of particles can be obtained by bringing Formulas (1) and (3) into Formula (2).

$$v_s=\sqrt{\frac{4}{3}\frac{gd}{C_D}\frac{{\rho }_s-{\rho }_l}{{\rho }_l}}$$

(4)

where ρ_l means the fluid density, kg/m³; vs. stands for the gravitational settling velocity of particles, m/s; d stands for the diameter of the particle, m; ρ_s is the density of particles, kg/m³; C_D means the resistance parameter, dimensionless; and g refers to the acceleration of gravity, m/s².

It can be seen from Formula (4) that a key parameter is needed in the calculation of settling velocity v_s, namely the particle settlement resistance coefficient C_D, which means the proportion of the viscous force of the fluid to the kinetic energy of the particles during the settlement process. According to Formula (4), the resistance coefficient C_D can be calculated according to the properties of fluid and particles and the settling velocity, v_s. The expression is as follows [7]:

$$C_D=\frac{4}{3}\frac{g{d}_{}}{v^2}\frac{{\rho }_s-{\rho }_l}{{\rho }_l}$$

(5)

C_D mainly depends on factors such as the shape and Reynolds number of the particle Re_s. Re_s means another main dimensionless number needing to be measured to depict particle sedimentation features, whose definition is the ratio of particle inertial force to particle viscous force. For particles settling in one stationary Newtonian fluid, the expression of the Reynolds number of the particle Re_s-n is shown below [8]:

$$R{e}_{s\text{-}n}=\frac{{\rho }_l{v}_{s}d}{\mu }$$

(6)

When the particle settles in the power-law fluid, its Re_s-p is expressed as follows [9]:

$$R{e}_{s\text{-}p}=\frac{{\rho }_l{v}_s^{2-n}{d}_{}^n}{K}$$

(7)

where μ is the viscosity of Newtonian fluid, Pa·s; n is the dimensionless flow index of the fluid; and K is the consistency coefficient of the fluid, Pa·sⁿ.

Research on the resistance coefficient of particle sedimentation can be traced back to 1851. Stokes [10] solved the Navier–Stokes equation by ignoring the nonlinear inertial term in the laminar flow region and obtained the formula for the calculation of the drag force of the fluid when the sphere moves slowly in the viscous fluid (particle Reynolds number Re_s-n < 0.1), namely, the famous Stokes formula. The study of Stokes provides a basis for the establishment and development of particle sedimentation. However, when the particle settling Reynolds number is large (Re_s-n > 1), the particle settling velocity will be influenced by the turbulent vortex behind the particle, and the Stokes formula will no longer be applicable. At the same time, for a higher Reynolds number range, the drag parameter model using theoretical analysis cannot be obtained because it is hard to acquire the analytical solution using theoretical analysis. Therefore, the association between the resistance parameter C_D and the particle Reynolds number Re_s is mainly based on the experimental curve to obtain a semi-empirical formula. Since the pioneering work of Stokes, scholars at home and abroad have conducted a lot of research work on free spherical particle sedimentation in Newtonian fluid. Abraham [11] and Terfous [12] have built C_D-Re_s-n calculation models under varied Re_s-n. These C_D-Re_s-n calculation models have high accuracy in predicting the resistance coefficient and are widely used.

The viscosity of Newtonian fluid is not affected by the shear rate, so the settling velocity of the particles in Newtonian fluid is only related to the particle diameter. However, most of the fluids used in practical engineering are power-law fluids [13]. The viscosity of the non-Newtonian is affected by the shear rate, and an increase in the shear rate will reduce the viscosity and increase the settling rate of the particles. Therefore, the settling rate of particles in non-Newtonian fluids is affected not only by particle diameter but also by flow index n and consistency index K. In their early studies, Lali [14] and Chhabra [15] used apparent viscosity instead of the viscosity term in Newton’s correlation to predict the sedimentation resistance coefficient of a single sphere in a power-law fluid. However, with increasingly deepened research, Peden [16] and Reynolds [17] found that there is a certain correlation between the resistance coefficient of particle sedimentation in non-Newtonian fluid and the flow index n of non-Newtonian fluid. Therefore, based on the power-law fluid flow index, the researchers proposed different C_D-Re_s-p prediction models. Kelessidis [18] put forward a five-coefficient suggested model to characterize the C_D-Re_s-p calculation model using the nonlinear regression method. Shah [19] collected the experimental data from the published literature and established C_D-Re_s-p calculation models in the power-law fluid. Okesanya [20], Battistella [21], Mohammad [22], Shajahan [23], Ahonguio [24], and Xu [25] also proposed C_D-Re_s-p prediction models for spherical particles in power-law fluids, respectively.

Unlike spherical particles, the shape of the cuttings or proppants is mostly irregular. For irregularly shaped particles, the drag parameter C_D acts as both the particle Reynolds number Re_s and the particle shape. Pettyjohn [26], Komar [27], and Dioguardi [28] found that the irregularity of particle shape will lead to higher resistance in the fluid, resulting in a decrease in particle settling velocity. How to accurately find a suitable shape factor to depict irregular particle irregularity is a challenge in the current study on sedimentation resistance coefficient. Many researchers have proposed various parameters or shape factors to characterize the shape characteristics of irregular particles. Waddel [29] proposed the concept of sphericity to characterize the shape of particles, which defines the proportion of the surface area of a sphere with the same volume as the actual particle to its actual surface area. Bouwman [30] evaluated nine different shape factors and proposed a combined shape element to characterize the shape and roughness of particles. Corey [31] proposed the Corey shape factor to illustrate the shape characteristics of particles for the three-axis diameter of particles, and Williams [32] proposed the concept of roughness for the surface of particles. Tran-Cong [33] used the circularity c to illustrate the shape of the particle. Among the many parameters employed for the description of the shape of particles, sphericity is the most frequently adopted shape description element, which is defined as the ratio of the surface area of the equivalent-volume sphere to the actual surface area of the particle. Based on the parameter of sphericity, Chien [34], Yang [35], Hölzer [36], Swamee [37], and Zhu [38] established C_D-Re_s calculation models with different Reynolds numbers. However, sphericity is a function of the particle surface area. For particles with highly irregular shapes, it is hard to realize accurate measurement of the actual particle surface area, as a result, the calculation error of C_D-Re_s model based on sphericity is large, and it cannot guide engineering application effectively [39,40]. Therefore, how to use simple parameters to depict the irregularity of irregular particles and establish the C_D-Re_s correlation in power-law fluids is a problem demanding prompt solutions in the engineering area.

The spherical and irregular particle sedimentation behavior in power-law fluid is here studied experimentally using a visual device and a high-speed camera system. Based on the trial data of spherical particles, the predictive model of the spherical particle sedimentation resistance coefficient in power-law fluid is established. On this basis, considering the article irregularity’s impact on the sedimentation velocity of particles, the calculation model of the irregular particle sedimentation resistance coefficient in power-law fluid is built up via incorporating parameter circularity for shape description c. The parameters in the new irregular particle sedimentation resistance coefficient predictive model can be obtained via nonlinear data fitting of the irregular particle experimental results. The new model can more accurately depict irregular particle settlement behavior such as proppants as well as debris in power-law fluids, and expand the scope of engineering application, which is of great significance for fracturing scheme design and wellbore cleaning.

2. Experimental Method

2.1. Experimental Apparatus and Materials

The particle sedimentation experiment was performed in a transparent cylinder made of plexiglass, whose inner diameter was 100 mm and whose height was 1600 mm, with one high-velocity camera employed for capturing its sedimentation process. The sedimentation experiment device is shown in Figure 2.

The fluids used in the experiment were different concentrations of a CarboxymethylCellulose aqueous solution. CMC aqueous solution refers to one classical power-law fluid with rheological characteristics capable of being expressed via the expression as follows [41]:

$$\tau =k{\gamma }^n$$

(8)

where τ denotes the shear stress, Pa; $\gamma$ denotes the shear rate, 1/s.

When the solution of different concentrations is configured, the solution needs to be mixed for over 8 h using the mixer for the purpose of complete dissolution. Then, the Anton Paar MCR 92 rheometer (Figure 3) is used to test configured CMC aqueous solution rheological characteristics, whereas the rheological parameters are obtained via fitting. The rheological parameters of CMC fluids with different concentrations are shown in

Table 1. The rheological property of the test fluid.

Fluid Type	Mass Concentration, %	Temperature, T °C	Density, kg/m3	Consistency Index, K	Flow Behavior Index, n
CMC1	0.25	23.0	1001.2	0.0187	0.9032
CMC2	0.5	22.0	1001.5	0.0805	0.7805
CMC3	0.75	22.5	1001.7	0.0642	0.7011
CMC4	1	21.5	1002	0.5104	0.6562
CMC5	1.25	22.0	1002.8	1.1247	0.5914
CMC6	1.5	23.2	1003	1.8019	0.5604
CMC7	1.75	22.7	1003.5	2.9412	0.5198
CMC8	2	22.1	1005	4.7646	0.4806

.

The solid particles employed here are zirconia, stainless steel, quartz sand, and glass, and their density range was 2500–7930 kg/m³. Among them, stainless steel, zirconia, and glass are used as spherical particles, and irregular particles adopt white quartz sand particles. Shape factor circularity c was used to characterize white quartz sand particle irregularity. The circularity c refers to the rate of particle maximum projection from the surface perimeter to the equivalent circle perimeter. It is a kind of 2D shape parameter, and its expression is as follows [33]:

$$c=\frac{\pi d_A}{P_p}$$

(9)

where d_A denotes the equivalent circle diameter of maximum particle projection surface, m; P_p denotes the maximum particle projection surface perimeter, m.

Measurement of circularity c requires the use of image processing commercial software ImageJ to process the image. The specific measurement process can be obtained from previous studies [42]. The particle properties used here refer to

Table 2. Test particle properties.

Material Type	Particle Size dA, mm	Density ρs, kg/m3	Circularity c
Steel	1,2,3,4,5	7930	1
Zirconia	1,2,3,4,5	6080	1
Glass	1,2,3,4,5	2500	1
Quartz sand particle	1.64–5.8	2600	0.49~1

.

2.2. Experimental Method

Before the experiment, the configured CMC aqueous solution was poured into an inner tube made of organic glass and stood for 12 h for gas to fully escape and to stabilize the internal flow field. The solid particles used in the experiment need to be immersed in the experimental fluid for more than 24 h in advance so as to eliminate the influence of the physical interaction between the particle surface and the experimental fluid in terms of particle sedimentation speed during the sedimentation experiment.

After the experimental materials were fully prepared, the solid particles were placed below the liquid level of the vertical glass tube and settled freely along the center of the glass tube. A high-speed camera was used for capturing the settling motion trajectory of particles in a circular tube, and multiple image frames of the particle motion trajectory at various times were acquired. It is necessary to stay for 5 min to guarantee the internal flow field stability of the experimental fluid so as to eliminate the influence of particle sedimentation on the next experiment in the previous set of experiments; three repeated measurements should be conducted for each particle under the same settlement conditions to ensure that the average deviation of the three measurement results is less than 5%. After the experiment, the displacement value Δs and the time interval Δt between the two frames of images are determined, and Δs/Δt is the settling velocity v of the particles. Figure 4 is an example of a set of single-particle sedimentation experiments.

3. Results and Discussion

3.1. Spherical Particle Settling in the Power-Law Fluid

Experimental data for 114 spherical particle groups were brought into Formulas (5) and (7), and the values of C_D and Re_s-p were calculated, respectively. The relationship between C_D and Re_s-p was drawn in logarithmic coordinate form, referring to Figure 5.

As seen in Figure 5, the distribution of spherical particle sedimentation resistance coefficient C_D in the power-law fluid under low Reynolds number conditions was observed to conform to the predicted trend using the Stokes equation. However, with the increase in the Reynolds number, there is a significant error in predicting the spherical particle settlement resistance coefficient in a power-law fluid using the Stokes equation. This shows that the rheological characteristics of power-law fluid badly influence the settling velocity of solid particles, and its relationship with Newtonian fluid cannot be used to compute the spherical particle settlement resistance coefficient in power-law fluid. Therefore, it is necessary to develop a novel model for accurate prediction of the spherical particle sedimentation resistance coefficient in power-law fluid.

At present, the Khan and Richadson model [43], the Kelessidis and Mpandelis model [18], and the Cheng model [44] are widely used and are representative of C_D-Re_s-p relations for power-law fluids. Specific forms in each model are referred to in

Table 3. Some representative C_D-Re_s-p correlations for spherical particles settling in the power-law fluid.

Author	CD-Res-p Empirical Correlations	Equation
Khan and Richadson	$C_D={(2.25R{e}_{{}_{s\text{-}p}}^{-0.31}+0.36R{e}_{{}_{s\text{-}p}}^{0.06})}^{3.45}$	(10)
Kelessidis and Mpandelis	$C_D=\frac{24}{R{e}_{s\text{-}p}}(1+0.1407R{e}_{s\text{-}p}^{0.6018})+\frac{0.2118}{1+0.4215/R{e}_{s\text{-}p}}$	(11)
Cheng	$C_D=\frac{24}{R{e}_{s\text{-}p}}{(1+0.27R{e}_{s\text{-}p})}^{0.43}+0.47\left[1-\exp (-0.04R{e}_{s\text{-}p}^{0.38})\right]$	(12)

.

On the basis of trial data on spherical particle sedimentation, C_D-Re_s-p models of the above different forms are fitted. The five-parameter model of Kelessidis and Mpandelisl was observed to have the best goodness of fit, whose expression is as follows [18]:

$$C_D=\frac{24}{R{e}_{s\text{-}p}}(1+AR{e}_{s\text{-}p}{}^B)+\frac{C}{1+D/R{e}_{s\text{-}p}{}^E}$$

(13)

where A, B, C, D, and E denote all correlation coefficients and are dimensionless.

By fitting the experimental data, the parameters in Equation (13) are finally obtained via regression (

Table 4. Fitting results of Equation (13).

A	B	C	D	E
0.16	0.498	0.422	0.129	0.072

).

Therefore, the predictive model of the spherical particle sedimentation resistance coefficient C_D-Re_s-p in power-law fluid is shown in Equation (14).

$$C_D=\frac{24}{R{e}_{s\text{-}p}}(1+0.16R{e}_{s\text{-}p}^{0.498})+\frac{0.422}{1+0.129/R{e}_{s\text{-}p}^{0.072}}$$

(14)

We use the maximum MRE (MAX MRE), mean relative error (MRE), as well as root mean square logarithm error (RMSLE) to evaluate the distinction between trial values and the foreseen values of each model. The expressions of MRE and RMSLE are as follows [7]:

$$MRE=\frac{1}{N}{\displaystyle \sum _{i=1}^N\frac{\left|C_{DP,i}-C_{DM,i}\right|}{C_{DM,i}}\times 100\%}$$

(15)

$$RMSLE=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(\ln C_{DP,i}-\ln C_{DM,i}\right)}^2}$$

(16)

where N is the number of samples; C_DP is the sedimentation resistance coefficient computed with the model, dimensionless; and C_DM denotes the experimental measured sedimentation resistance coefficient, dimensionless.

The experimental values are compared with the predicted results of the C_D-Re_s-p model in Table 3 and the calculation results in this study. The comparison results are shown in Figure 6, Figure 7, Figure 8 and Figure 9 and

Table 5. Error statistics of drag coefficient of sphere settling in power-law fluid.

Res-p	Author	Error
		MRE	RMSLE	Max MRE
0.001 < Res-p < 373	Khan and Richadson	9.66%	11.27%	26.08%
	Kelessidis and Mpandelis	8.09%	9.64%	17.61%
	Cheng	9.37%	11.17%	29.69%
	This chapter (Equation (14))	3.85%	4.88%	14.05%

.

It can be seen from Figure 6, Figure 7, Figure 8 and Figure 9 that the prediction results of the Khan and Richadson model, the Kelessidis and Mpandelis model, the Cheng model, and the model established in this study are consistent with the experiment. As per the error analysis listed in Table 5, the MREs between the foreseen and experimental values of the resistance coefficients of the four models are 9.66%, 8.09%, 9.37%, and 3.85%, respectively. The prediction accuracy of the spherical particle sedimentation resistance coefficient model in power-law fluid here is the highest, and the three error analysis parameters are the smallest.

3.2. Irregular Particle Resistance Coefficient

For irregular particles such as rock debris and proppants, their shape will irregularly and badly influence the drag force exerted by the fluid on them, thus changing their final settling velocity. The circularity c is more sensitive to particle profile irregularities and is easier to determine. Therefore, considering the influence of particle shape on particle sedimentation, the circularity c is introduced into Equation (14) to establish the predictive model of the resistance coefficient of irregular-shaped particles. In order to achieve this goal, the paper draws on Yan’s method [1] and uses Formula (17) to depict the functional relationship between the particle roundness c, the experimental value C_D-N of the irregular particle settlement resistance coefficient, and the calculated value C_D-S of Formula (14) under the same conditions. The expression is as follows:

$$C_{D\text{-}N}=C_{D\text{-}S}\exp [\alpha {Re}_{s\text{-}p}^{\beta }{(1-c)}^{\lambda }]$$

(17)

where the relation of f(c) is determined using Equation (18), as follows:

$$f(c)=\alpha R{e}_{s\text{-}p}^{\beta }{(1-c)}^{\lambda }$$

(18)

where α, β, and λ denote the correlation coefficients and dimensionless numbers.

The parameters in Equation (18) can be obtained via nonlinear data fitting of the sedimentation data of 211 groups of gravel in power-law fluid, as shown in

Table 6. Fitting results of Equation (18).

α	β	λ
1.35	0.028	0.002

.

Therefore, the relationship between irregular particles C_D-N-Re_s-p in power-law fluid is as follows:

$$C_{D\text{-}N}=C_{D\text{-}S}\exp \left[1.35R{e}_{s\text{-}p}{}^{0.028}{(1-c)}^{0.002}\right](0.01<R{e}_{s\text{-}p}<373)$$

(19)

The foreseen value using the model is compared with the trail value. The distinction of the foreseen sand sedimentation resistance coefficient value and the measured resistance coefficient value are referred to in Figure 10 and

Table 7. Error statistics of drag coefficient of Irregular Particles settling in power-law fluid.

Res-p	Error
	MRE	RMSLE	Max MRE
0.001 < Res-p < 373	4.47%	5.83%	13.72%

. As shown in the diagram, the resistance coefficient predictive model prediction accuracy with the particle shape description parameter c is higher, and the mean relative error is 4.47%, which meets the engineering requirements.

4. Conclusions

The conclusions of this study can be summarized as follows:

(1)

A model for predicting the settling drag coefficient of spherical particles in power-law fluid is established by modifying the correlation proposed by Kelessidis and Mpandelis. The mean relative error is 3.85%;

(2)

For irregular particles, based on the spherical particle prediction model, using image analysis technology, and introducing two-dimensional shape description parameters, a resistance coefficient model for predicting the settlement of irregular particles in power-law fluid is established. The results show that the model has good prediction ability, with a mean relative error of 4.47. This work can provide a valuable reference for those interested in fracturing scheme design and wellbore cleaning;

(3)

The existing C_D-Re_s prediction models are mostly for Newtonian fluid, power-law fluid, and Heba fluid. However, there are relatively few studies on C_D-Re_s prediction models for other non-Newtonian fluids, such as Bingham fluid and viscoelastic fluid, and the existing models are difficult to use in actual field operations. Therefore, it is the main research direction in the future to study the particle sedimentation law in other types of fluids via experimental means and establish the corresponding C_D-Re_s correlation;

(4)

At present, the single particle sedimentation method based on the Stokes theory formula is used to measure the particle sedimentation velocity in the laboratory. It is to observe and measure the time required for single particles to settle in the fracturing fluid to the bottom of the container in a static state and calculate the sedimentation rate. However, the single-particle sedimentation method cannot measure the sedimentation velocity of multiple particles. Therefore, it is urgent to develop new devices and new experimental methods to further explore the sedimentation mechanism of multi-particles.