Research Article  Open Access
Shichen Gao, Zhixin Tian, Bailu Teng, "Analytical Solutions for Characterizing Fluid Flow through SandPack in Pipes", Mathematical Problems in Engineering, vol. 2020, Article ID 3091402, 10 pages, 2020. https://doi.org/10.1155/2020/3091402
Analytical Solutions for Characterizing Fluid Flow through SandPack in Pipes
Abstract
When fluid flows through a pipe that is packed with sand particles, the fluid will bear the resistance from the sandpack, as well as the viscous shear from the pipe wall. If the viscous shear from the pipe wall can be neglected, the fluid flow will obey Darcy’s law, and one can think the equivalent permeability of the packedpipe equals the permeability of the sandpack. However, if the viscous shear from the pipe wall cannot be neglected, the fluid flow will obey the Brinkman equation, and the permeability of the packedpipe will be less than that of the sandpack due to the additional viscous drag. In this work, on the basis of the Brinkman equation, we derived a series of analytical solutions for characterizing the fluid flow in packedpipes. These solutions can be used to depict the velocity profiles, estimate the flux rate, and calculate the equivalent permeability of a packedpipe. On the basis of these analytical solutions, we found that Poiseuille’s law is a special form of the derived equivalent permeability solution. We further divided the fluid flow in a packedpipe into three regimes, including NS flow, Brinkman flow, and Darcy flow. During the regime of Brinkman flow, the dimensionless flow velocity at the pipe center is 1, and the dimensionless flow velocity is gradually decreased to 0 at the pipe wall. We also investigated the effects of sorting, sand particle size, and sandpack porosity on the packedpipe permeability. The calculated results show that a more uniform size of the sand particles or a smaller mean particle diameter can lead to lower packedpipe permeability. Compared to the sorting and mean particle diameter, the sandpack porosity exerts a more significant effect on the packedpipe permeability.
1. Introduction
The Navier–Stokes (NS) equation describes fluid flow on the microlevel and it accounts for the effect of viscous shear on the fluid flow [1, 2]. For an incompressible fluid, neglecting the inertial terms, the NS can be reduced towhere p is the pressure (MPa), μ is the viscosity (mPa·s), β_{1} and β_{2} are the unit conversion factors, is the velocity (m/d), and ϕ is the porosity. On the basis of the reduced NS equation, Hagenbach [3] derived an analytical solution for characterizing the flux rate in empty pipes (in this work, an empty pipe indicates a pipe that is filled only with fluid), which can be expressed aswhere q is the flux rate (m^{3}/d), R is the inner radius of the pipe (m), and L is the length of the pipe (m). Equation (2) is also widely known as the Poiseuille–Hagenbach equation. Figure 1(a) shows the velocity profile of fluid flow in an empty pipe. From Figure 1(a), one can find that the fluid exhibits small flow velocity at the pipe wall due to the effect of viscous shear. In addition, if we use equivalent permeability to characterize the ability of the empty pipes for transmitting the fluid, on the basis of Darcy’s law [4] we can havewhere k_{e} is the equivalent permeability (mD). Substituting equation (2) into equation (3) gives
(a)
(b)
(c)
Equation (4) describes the relationship between the equivalent permeability of the empty pipe and the inner radius of pipes. This equation is commonly called Poiseuille’s law.
Darcy’s law characterizes the fluid flow on the macrolevel, and it considers the resistance from porous media on the fluid flow. If fluid flows through a cylinder sandpack whose radius is R and whose length is L, the flux rate can also be calculated with Darcy’s law (i.e., Equation (3)). The only difference is that the equivalent permeability in Equation (3) should equal the sandpack permeability:where k_{s} is the permeability of sandpack (mD). Figure 1(b) illustrates the velocity profile of Darcy flow in the cylinder sandpack. On the macrolevel, one can think that the velocity of Darcy flow is uniform along a given crosssection.
According to the aforementioned arguments, one can find that we can use analytical methods to describe the fluid flow in empty pipes or sandpacks. However, how to calculate the flux rate through a pipe that is packed with sand (see Figure 1(c)) is investigated. It should be noted that, for the scenario of Figure 1(a), the fluid flow is only affected by the viscous shear from the pipe wall, whereas, for the scenarios of Figure 1(b), the fluid flow is only affected by the effect of the sandpack. Therefore, if fluid flows through a packedpipe, both the viscous shear and the sandpack will exert their effects on the fluid flow.
In recent years, various studies have been conducted to investigate the fluid flow in packedpipes. Chen et al. [5] developed a momentum equation for characterizing the fluid flow in a packed tube by combining the NS equation, Darcy’s law, and a superficial dispersion term due to phase discontinuity. Yuki et al. [6] utilized a matched refractiveindex method to perform a visualization investigation on the flow structure in a spherepackedpipe. Three flows are observed in their experiment, including bypass flow, secondary flow, and spouting flow. Siddiqui et al. [7] developed an analytical solution to characterize the fluid flow through the porous media between two coaxial cylinders. In their study, the inner cylinder is stationary, while the outer cylinder travels parallel to itself. Hashim and Kamel [8] proposed a new correlation method for calculating the pressure drop through a packedpipe. Their method is validated by comparing the results of their work to experimental data that are measured in a horizontal packedpipe with homogeneous spherical pellets. Wu et al. [9] studied the Taylor dispersion in a packedpipe by accounting for the effect of wall reaction. Their study is based on the method of Gill’s series solution. More studies about fluid flow in packedpipes can be found in Nazififard and Suh [10], Wu et al. [9], and Yang et al. [11]. Although these studies provided us with a comprehensive insight into the fluid flow in packedpipes, these works are mainly conducted with numerical methods, which are highly computationally demanding. Therefore, it is imperative to derive analytical solutions for characterizing such kind of flow.
In practice, one kind of utilizations of the packedpipes is to conduct sandpack permeability tests. In these tests, it is commonly assumed that the viscous shear from the pipe wall can be neglected. As such one can think that the sandpack permeability equals the measured permeability. However, if the permeability of the sandpack is sufficiently large or the diameter of the pipe is sufficiently small, the viscous shear from the pipe cannot be neglected. In such cases, it is not reasonable to approximate the sandpack permeability with the packedpipe permeability.
To account for the effect of viscous shear on the fluid flow in porous media, Brinkman [12] proposed a new equation by combining the reduced NS equation with Darcy’s law. This equation is subsequently named as Brinkman equation, which is given as
In equation (6), the fluid is incompressible, the flow is assumed to be laminar, and the inertial force is neglected. It is worth noting that equation (6) is different from the original Brinkman equation by considering the effect of volume fraction of solid in porous media. The original Brinkman equation ignored the effect of the volume fraction of solid; hence, it will induce significant error if the solid volume fraction is sufficiently high or ϕ is sufficiently small [13–16]; and [17]. For the sake of convenience, although equation (6) is a modified form of the original Brinkman Equation, in this work, we still call it Brinkman equation. With the aid of equation (6), we derived analytical solutions for characterizing the fluid flow in packedpipes. On the basis of the analytical solutions, we depicted the velocity profiles of the fluid flow in packedpipes. In addition, we conducted a comprehensive comparison between the equivalent permeability of the packedpipes and that of the sandpack.
2. Methodology
In order to obtain analytical solutions for characterizing the fluid flow through packedpipes, we made the following assumptions:(1)The fluid flow is laminar and is in steadystate(2)The properties of the fluid and sandpack are constant(3)The sandpack is homogeneous and the pipe has a uniform diameter(4)There is no slippage at the pipe wall(5)The effect of gravity is neglected
According to the above assumptions, the parameters of μ, ϕ, and k_{s} in equation (6) are constant. In addition, in a cylindrical coordinate system, equation (6) can be rewritten aswhere r is along the direction from the pipe center to the pipe wall (m). Rearranging equation (7) gives
Equation (8) is a generalized Bessel equation whose general solution iswhere I_{n} indicates the n^{th} order modified Bessel function of the first kind, K_{n} indicates the n^{th} order modified Bessel function of the second kind, and A and B are constants that require to be determined. At the pipe center, we have r = 0 and
Since the flow velocity should not have an infinite value at r = 0, the constant B should equal 0, and equation (9) can be reduced to
As we have assumed that there is no slippage at the pipe wall, we can have
Inserting equation (11) into equation (12) yields
Inserting equation (13) into equation (11) gives the equation characterizing the velocity profile:
Equation (14) is in the format of a vector. Rewriting equation (14) in the format of scalar, we can have
The flux rate q can be calculated as follows:
Comparing equation (16) to Darcy’s law (i.e., Equation (3)), we can obtain the relationship between the equivalent permeability of packedpipe and the permeability of sandpack:
Equations (15) through (17) are the equations that characterize the fluid flow in packedpipes.
Particularly, for an empty pipe, we can think it is packed with a sandpack whose permeability is infinite and whose porosity is 1. Thus, the term in equation (17) approaches 0. On the basis of Taylor series, expanding I_{1} and I_{0} at 0 and using the first two terms of the series to approximate the results, we can have
The truncation errors of the Taylor series of I_{1} and I_{0} are at the order of and , respectively. For empty pipes, we have . Inserting into equation (18), equation (18) will be reduced to equation (4). In addition, if the radius of the pipe is sufficiently large, we will have
This implies that both equations (4) and (5) are special forms of equation (17).
3. Validation
In order to validate the derived analytical solutions, we compared the permeability that is calculated with equation (17) to the permeability that is calculated from the commercial software [18]. The model that is used for validation is shown in Figure 2. Figure 2 indicates a pipe that is filled with porous media. The model built in COMSOL has the same assumptions to the model used in this work. These assumptions can be found in the section of methodology. The permeability of the porous media (k_{s}) is 1×10^{7} mD, the porosity of the porous media (ϕ) is 0.1, the radius of the pipe (R) is varied from 0.001 m to 0.1 m, the length of the pipe (L) is 0.1 m, the flow velocity () is set to be 8640 m/d, and the viscosity of the fluid (μ) is 1 mPa·s. The fluid flows into the pipe through the inlet surface and flows out of the pipe through the outlet surface. The equivalent permeability of the packedpipe can be calculated based on Darcy’s law if the pressure difference between the inlet surface and outlet surface is measured. With the aid of COMSOL, we measured the average pressure at the inlet surface and the outlet surface and subsequently calculated the equivalent permeability of the packedpipe with the measured data. The results are summarized in Table 1.

Taking the case of R = 0.001 m as an example, the equivalent permeability can be calculated as follows:
In addition, the equivalent permeability can be calculated with equation (17); thus, we can validate the proposed solution by comparing the results from this work to those from COMSOL. Figure 3 shows the comparison. In this figure, one can find that although the results from COMSOL are slightly larger than those from the proposed solution, these two plots show a similar trend. Actually, the larger values from COMSOL can be ascribed to numerical effects. In COMSOL, the modified Brinkman equation is solved with the finite element method, which will induce numerical errors during the computation. For example, for the case of R = 0.1 m in Table 1, the calculated equivalent permeability k_{e} is 10005636.16 mD, which is even larger than the permeability of the sandpack (k_{s} = 1×10^{7} mD). However, in practice, the equivalent permeability of the packedpipe should be less than the permeability of the sandpack due to the additional viscous shear from the pipe wall. This indicates that the permeability of the packedpipe has been overestimated by COMSOL because of the effect of numerical errors.
4. Results and Discussion
In this section, we recognized the flow regimes in packedpipes and depicted the velocity profiles of these flow regimes. In addition, we conducted a comprehensive sensitivity analysis to study the effects of sandpack properties on the packedpipe permeability. For the sake of convenience, we define the following dimensionless parameters:where r_{D} is the dimensionless radius, is the dimensionless velocity, D_{a} is the Darcy parameter, and k_{r} is the permeability ratio. Inserting equations (21) through (24) into equations (15) and (17) gives
4.1. Flow Regimes
Figure 4 shows the value of k_{r} with different values of D_{a} (D_{a} is varied from 1 to 1000) in a loglog plot. From this figure, one can find that the permeability ratio is increased as the Darcy parameter D_{a} is increased. At large value of D_{a,} the permeability ratio k_{r} approaches 1, which indicates Darcy flow. At small values of D_{a}, k_{r} can be much smaller than 1, which implies that the equivalent permeability of the packedpipe is much smaller than the permeability of the sandpack.
Figure 5 illustrates the equivalent permeability of the packedpipe with a constant value of sandpack permeability (k_{s} = 1×10^{7} mD), a constant value of sandpack porosity (ϕ = 0.1), and different values of pipe radius. In Figure 5, it can be observed that, at small value of pipe radius, the equivalent permeability approaches the results ofwhereas at large value of pipe radius, the equivalent permeability approaches the results of equation (5) (i.e., k_{e} = k_{s}). According to Figure 5, one can divide the fluid flow into three regimes, including NS flow, Brinkman flow, and Darcy flow.
NS flow: at small pipe radius, the fluid flow in a packedpipe is mainly influenced by the effect of the viscous shear from pipe wall, and the sandpack exerts its effect through its volume fraction. The equivalent permeability of packpipes approaches the results of equation (27) during this regime; Darcy flow: at large value of pipe radius, the viscous shear from pipe wall can be neglected, and the fluid flow is mainly influenced by the resistance from the sandpack. During this regime, the equivalent permeability of the packpipe approaches the permeability of sandpack; Brinkman flow: this flow regime is in a transition between NS flow and Darcy flow. During this flow regime, both the viscous shear from pipe wall and the resistance from the sandpack can significantly influence the fluid flow.
Figure 6 presents the equivalent permeability of packed pipe as a function of sandpack permeability with a constant pipe radius of 5×10^{−3} m and a constant sandpack porosity of 0.1. In Figure 6, one can find that the fluid flow can also be divided into three flow regimes. These flow regimes can be recognized as Darcy flow, Brinkman flow, and NS flow as the sandpack permeability is increased.
It should be noted that, due to the fact that the NS flow and Darcy flow can be regarded as two extreme scenarios of Brinkman flow, the division of the flow regimes in Figures 5 and 6 is somehow qualitative rather than rigorously defined. The NS flow is Brinkman flow neglecting the resistance from the sandpack, whereas the Darcy flow is Brinkman flow neglecting the viscous shear from the pipe wall. In this work, to make a clear division between these three flow regimes, a threshold value of permeability relative difference is assigned:where K_{e} is the equivalent permeability from equation (27) or (5) and k_{e} is the equivalent permeability of the packedpipe that is calculated with equation (17). Inserting equations (5), (17), (21), (22), (23), and (27) into equation (28), if K_{e} is obtained with equation (27), we can haveand if K_{e} is obtained with equation (5), we can have
Taking 5% as the threshold value, in this work, if ≤ 5%, we can think the flow regime is NS flow or Darcy flow. Inserting ≤ 5% into equation (29) and (30) yieldsfor equation (29) andfor equation (30). This implies that the flow regime can be regarded as NS flow if D_{a} ≤ 0.5481, and as Darcy flow if D_{a} ≥ 41.3899.
4.2. Velocity Profile
In this section, we studied the velocity profiles in a packedpipe by varying the Darcy parameter D_{a} from 0.1 to 100. The velocity profiles of NS flow regime, Brinkman flow regime, and Darcy flow regime are presented in Figures 7–9, respectively. Figure 7 shows the velocity profiles of NS flow with D_{a} = 0.1, 0.2, 0.3, 0.4, and 0.5. As defined in equation (25), = 1 indicates Darcy flow. In this figure, one can find that, for NS flow regime, the dimensionless velocity has been significantly jeopardized due to the effect of viscous shear from pipe wall. The highest dimensionless velocity can be observed at the pipe center. As D_{a} is increased, the dimensionless velocity is also increased.
Figure 8 illustrates the dimensionless velocity profiles during the Brinkman flow regime in a packedpipe with D_{a} = 8, 16, 24, 32, and 40. In this figure, it can be observed that the dimensionless velocity approaches the value of 1 at the pipe center, which indicates the fluid flow approaches Darcy flow. A smooth transition can be found between r_{D} = 0 and r_{D} = 1. During this transition, the dimensionless velocity is decreased from = 1 to = 0 because of the effect of viscous shear from pipe wall. In Figure 8, one can find that the Brinkman flow is a combination of Darcy flow near the pipe center and the NS flow near the pipe wall. Figure 9 shows the dimensionless velocity profiles of Darcy flow regime with D_{a} = 60, 70, 80, 90, and 100. In this figure, the transition is very short, and the dimensionless velocity is rapidly decreased to 0 at the pipe wall. In these plots, = 1 accounts for a large proportion, and the fluid flow can be regarded to be Darcy flow.
4.3. Sensitivity Analysis
In this section, we conducted a thorough sensitivity analysis on the packedpipe permeability with different sandpack properties. Berg [19] built the relationship between the sandpack permeability and the sandpack properties, which is given aswhere is the sorting term in phi unit and D is the geometric mean diameter of the sand particle. Inserting equation (33) into equation (17) yields
Equation (34) characterizes the relationship between the packedpipe permeability and the sandpack properties. The benchmark values of the parameters that are used for conducting sensitivity analysis are as follows: = 0.8, D = 0.05 m, R = 0.01 m, and ϕ = 0.2.
Figure 10(a) shows the packedpack permeability and sandpack permeability with different values of . The value of normally ranges from 0.7 to 1, and a larger value of indicates that the particle size is more uniform. In Figure 10(a), one can find that both the packedpack permeability and sandpack permeability are decreased as the value of is increased. This manifests that a more uniform size of the sand particle can lead to a less packedpipe and sandpack permeability. Figure 10(b) shows the permeability ratio k_{r} (see equation (24)) that is calculated with the results in Figure 10(a). A larger value of k_{r} indicates a smaller relative difference between the packedpipe permeability and the sandpack permeability. As shown in Figure 10(b), the permeability ratio is increased as the value of is increased. However, it is worth noting that as is varied from 0.7 to 1, k_{r} is only slightly increased from 0.9936 to 0.9948. This implies that the sorting of the sand particles will not induce a large difference between the packpipe permeability and the sandpack permeability.
(a)
(b)
Figure 11 presents the packedpipe permeability, sandpack permeability, and permeability ratio as a function of mean particle diameter. In Figure 11(a), it can be observed that the packedpipe permeability and sandpack permeability are increased as the mean particle diameter is increased, whereas, in Figure 11(b), the permeability ratio is decreased as the mean particle diameter is increased. In Figure 11(a), the packedpipe permeability exhibits very similar values to those of the sandpack permeability. In Figure 11(b), the permeability ratio is slightly less than 1 although it is decreased from 0.9970 to 0.9940.
(a)
(b)
Figure 12(a) describes the packedpipe permeability and sandpack permeability as a function of sandpack porosity. As shown in this figure, the packedpipe permeability and the sandpack permeability are increased rapidly as the sandpack porosity is increased. Figure 12(b) shows the change of the permeability ratio with different sandpack porosity. It can be observed that the permeability ratio k_{r} is around 0.9 with ϕ = 0.8, indicating that the difference between and packedpipe permeability and sandpack permeability can be significant with large sandpack porosity.
(a)
(b)
Comparing the results shown in Figures 10 through 12, one can find that the sandpack porosity exerts a more significant effect on the packedpipe permeability and permeability ratio. This is because the sandpack porosity is highly related to the effective space in the porous media for transmitting the fluid. A higher porosity indicates a larger effective space; hence, the packed pipe permeability expresses a larger permeability with high sandpack porosity.
5. Conclusions
In this work, the authors derived a series of analytical solutions for characterizing the fluid flow through pipes that are packed with sand. These solutions can be used to depict the velocity profiles in a packedpipe (equation (15)), estimate the flux rate through a packedpipe (equation (16)), and calculate the equivalent permeability of a packedpipe (equation (17)). In real applications, one can use these solutions to correct the results of packedpipe permeability test and characterize the fluid flow in the pore volume of underground reservoirs. For example, in gas hydrate reservoirs, as the gas hydrate is partially decomposed into gas and water in the pore volume, the fluid flow in the space between the undecomposed gas hydrate within the pore volume can be characterized with the proposed solutions. With the aid of these equations, the authors recognized the flow regimes, investigated the velocity profiles, and conducted sensitivity analysis. On the basis of the calculated results, we can form the following conclusions: Poiseuille’s law (i.e., equation (4)) is a special form of the derived equivalent permeability solution (i.e., equation (17)); three flow regimes can be observed in a packedpipe with different values of Darcy parameter, including NS flow regime, Brinkman flow regime, and Darcy flow regime; a more uniform size of the sand particles can lead to a lower packedpipe permeability, while a higher permeability ratio and a larger mean particle diameter can lead to a higher packedpipe permeability and a lower permeability ratio; as the sandpack porosity is increased, the packedpipe permeability is increased, while the permeability ratio is decreased. In comparison to the sorting and mean particle diameter, the sandpack porosity can exert a more significant effect on the packedpipe permeability and permeability ratio.
Nomenclature
D_{a}:  Darcy parameter 
k_{e}:  Equivalent permeability, mD 
k_{r}:  Permeability ratio 
k_{s}:  Permeability of the sandpack, mD 
L:  Length of the pipe, m 
p:  Pressure, MPa 
p_{in}:  Average pressure at the inlet surface, MPa 
p_{out}:  Average pressure at the outlet surface, MPa 
q:  Flux rate, m^{3}/d 
r:  Cylindrical coordinate, which is along the direction from pipe center to pipe wall 
R:  Inner radius of the pipe, m 
r_{D}:  Dimensionless radius 
:  Dimensionless velocity 
β_{1}:  Unit conversion factors, 0.0853 
β_{2}:  Unit conversion factors, 1.01 × 10^{15} 
μ:  Viscosity, mPa·s 
:  Velocity, m/d 
:  Porosity 
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
References
 Q.H. Nguyen, P.T. Nguyen, and B. Q. Tang, “Energy equalities for compressible NavierStokes equations,” Nonlinearity, vol. 32, no. 11, pp. 4206–4231, 2019. View at: Publisher Site  Google Scholar
 Y.H. Yu and S. A. Kinnas, “Roll response of various hull sectional shapes using a NavierStokes solver,” International Society of Offshore and Polar Engineers, vol. 19, no. 1, pp. 46–51, 2009. View at: Google Scholar
 E. Hagenbach, “Ueber die Bestimmung der Zähigkeit einer Flüssigkeit durch den Ausfluss aus Röhren,” Annalen der Physik und Chemie, vol. 185, no. 3, pp. 385–426, 1860. View at: Publisher Site  Google Scholar
 H. Darcy, Les Fontaines Publiques De La Ville De Dijon, Dalmont, Paris, France, 1856.
 G. Q. Chen, L. Zeng, and D. Taylor, “Taylor dispersion in a packed tube,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2215–2221, 2009. View at: Publisher Site  Google Scholar
 K. Yuki, M. Okumura, H. Hashizume, S. Toda, N. B. Morley, and A. Sagara, “Flow visualization and heat transfer characteristics for spherepacked pipes,” Journal of Thermophysics and Heat Transfer, vol. 22, no. 4, pp. 632–648, 2008. View at: Publisher Site  Google Scholar
 A. M. Siddiqui, S. Sadiq, and T. Haroon, “Macroscopic analysis of flow through a porous medium between two circular cylinders,” Journal of Porous Media, vol. 14, no. 9, pp. 751–760, 2011. View at: Publisher Site  Google Scholar
 E. T. Hashim and R. S. Karmel, “Pressure drop in a horizontal packed pipe,” Petroleum Science and Technology, vol. 31, no. 1–4, pp. 1091–6466, 2013. View at: Publisher Site  Google Scholar
 Z. Wu, X. Fu, and G. Wang, “Complete spatial concentration distribution for taylor dispersion in packed tube flow,” International Journal of Heat and Mass Transfer, vol. 92, pp. 987–994, 2016. View at: Publisher Site  Google Scholar
 M. Nazififard and K. Y. Suh, “CFD analysis of a spherepacked pipe for potential application in the molten salt blanket system,” AtwInternational Journal for Nuclear Power, vol. 61, no. 89, pp. 543–547, 2016. View at: Google Scholar
 J. Yang, J. Wu, L. Zhou, and Q. Wang, “Computational study of fluid flow and heat transfer in composite packed beds of spheres with low tube to particle diameter ratio,” Nuclear Engineering and Design, vol. 300, pp. 85–96, 2016. View at: Publisher Site  Google Scholar
 H. C. Brinkman, “A calculation of the viscous force exerted by a following fluid on a dense swarm of particles,” Applied Scientific Research, vol. A1, pp. 27–34, 1949. View at: Google Scholar
 L. Durlofsky and J. F. Brady, “Analysis of the brinkman equation as a model for flow in porous media,” Physics of Fluids, vol. 30, no. 11, pp. 3329–3341, 1987. View at: Publisher Site  Google Scholar
 E. J. Hinch, “An averagedequation Approach to particle interactions in a fluid suspension,” Journal of Fluid Mechanics, vol. 83, no. 4, pp. 695–720, 1977. View at: Publisher Site  Google Scholar
 I. D. Howells, “Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects,” Journal of Fluid Mechanics, vol. 64, no. 3, pp. 449–476, 1974. View at: Publisher Site  Google Scholar
 M. Muthukumar and K. F. Freed, “On the Stokes problem for a suspension of spheres at nonzero concentrations. II. Calculations for effective medium theory,” The Journal of Chemical Physics, vol. 70, no. 12, pp. 5875–5887, 1979. View at: Publisher Site  Google Scholar
 H. A. Belhaj, K. R. Agha, S. D. Butt, and M. R. Islam, “A comprehensive numerical simulation model for nonDarcy flow including viscous, inertial and convective contributions,” in Proceedings of Nigeria Annual International Conference and Exhibition, pp. 4–6, Society of Petroleum Engineers, Abuja, Nigeria, August 2003. View at: Publisher Site  Google Scholar
 Comsol, COMSOL, Comsol, Inc, Burlington, MA, USA, 2019.
 R. R. Berg, “Method for determining permeability from reservoir rock properties,” Transactions of Gulf Coast Association of Geological Societies, vol. 20, pp. 303–335, 1970. View at: Google Scholar
Copyright
Copyright © 2020 Shichen Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.