Pore-network modeling of particle dispersion in porous media

Colloids and Surfaces A 580 (2019) 123768

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Pore-network modeling of particle dispersion in porous media Xiaoyan Meng, Daoyong Yang

⁎

T

Petroleum Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

GRAPHICAL ABSTRACT

ARTICLE INFO

ABSTRACT

Keywords: Pore-network model Dispersion coefficient Particle size Size exclusion Distribution profile

Coupling with the random walk particle tracking method and pore-network modeling simulation, this work systematically analyzed dispersion of particles with different particle sizes in pore-network models with variable heterogeneity. The newly developed pore-network model has been verified against results from the published papers. Effect of particle size on dispersion coefficient varies and depends upon the heterogeneity of porous media. Dispersion coefficient of particles is found to be greatly affected by particle size in a homogeneous model; however, for a heterogeneous model, throat velocity difference caused by heterogeneity plays an important role on particle dispersion. Comparing particle dispersion coefficient with and without size-exclusion effect in a homogeneous model, for large particles (i.e., 5 × 10 7 m), size-exclusion effect needs to be considered, especially at late times. In other words, without size-exclusion, particle dispersion coefficient is overestimated, while the size-exclusion effect becomes more important as flow rate increases. In the heterogeneous models, however, sizeexclusion effect of particles (i.e., 5 × 10 7 m) can be neglected. Effects of the widely used uniform distribution and volumetric distribution profiles on particle dispersion have been compared in different pore-network models. The dispersion difference between volumetric and uniform distributions increases with particle size and heterogeneity of the pore-network model. For a homogeneous model, dispersion coefficient with uniform distribution leads to a larger value than that with volumetric distribution; however, as heterogeneity increases, dispersion coefficient with volumetric distribution shows a larger value.

1. Introduction Due to the importance of dispersive transport of solutes and

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Corresponding author. E-mail address: [email protected] (D. Yang).

https://doi.org/10.1016/j.colsurfa.2019.123768 Received 4 August 2019; Accepted 6 August 2019 Available online 12 August 2019 0927-7757/ © 2019 Elsevier B.V. All rights reserved.

particles in porous media in various areas of science and engineering, it is necessary to investigate dispersion behaviour of solutes and particles in porous media [1–3]. In a circular tube, dispersion is attributed to the

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Nomenclature

t tD

Notation

dp D Dm Dne L N Ni NPe p Qij r r R R

t uav v v¯ x y z

Particle diameter, L Dispersion coefficient, L2/t Molecular diffusion coefficient, L2/t Dispersion coefficient without size-exclusion effect, L2/t Throat length, L Normal distribution, dimensionless Pores connected to pore #i by throats, dimensionless ¯ Dm ), dimensionless Peclet number (NPe = vR Pore pressure, Pa Volumetric flux from pore #j to pore #i, L3/t Radial distance from the tube centerline, L Radial distance of injection location from the tube centerline, L Throat radius, L Mean value of R, L

Time, t Dimensionless time tD = t [(R 0.5d p )2 Dm], dimensionless Time step, t Average flow rate of pore-network model, L/t Local fluid velocity inside a throat, L/t Average flow velocity of throat, L/t Diffusive displacement in the x direction, L Diffusive displacement in the y direction, L Diffusive displacement in the z direction, L

Greek Letters 2

R

µ

combination of diffusive and convective transport caused by the velocity gradient over the tube cross-section [4–12]. For solute or particle transport in porous media, similarly, molecular diffusion and mechanical dispersion which is defined by spreading of a component due to microscopic variations in flow velocity contribute to the dispersion process [13]. Mechanisms including non-uniform velocity distribution in a pore, stream splitting, and tortuosity effect lead to the complex dispersion phenomenon in porous media [14]. This work focuses on passive particles and longitudinal dispersion parallel to the direction of bulk flow. Generally, two stages including pre-asymptotic and asymptotic stages are used to describe dispersion phenomenon in porous media. For the pre-asymptotic stage, once being injected into porous media, solutes or particles begin to spread and dispersion coefficient is a function of travel distance or time and increase with them. After a period of time when the heterogeneity at different scales has been sampled, the plume follows an asymptotic spreading known as the asymptotic stage [15,16]. Due to the complex fluid behaviour and inherent heterogeneity, it is difficult to predict transport properties in real systems. Pore-network models representing a porous medium with pores connected by throats have been widely used to study transport problems in porous media [15,17–28]. The detailed review about pore-network modeling can be found elsewhere [27,29]. For dispersion in porous media, based on twodimensional (2D) square networks with various throat radius distributions, Bruderer and Bernabé [30] simulated solute dispersion in porous media with different degrees of heterogeneity. It was found that transit time and asymptotic dispersion coefficient increases with heterogeneity, and, even for homogenous porous media, it takes at least 100,000 s to reach the asymptotic stage. Bijeljic et al. [15] studied solute dispersion in a 2D circular network with the random walk particle tracking (RWPT) method. Based on the property of Berea sandstone, throat length and throat size of the network were assigned and the calculated dispersion coefficients match well with experimental data for Peclet number NPe < 4000. It was concluded that hundreds to thousands pores need to be transversed to approach the asymptotic dispersion. Coupling with the continuous time random walk method, the same pore network model was applied by Bijeljic and Blunt [31] to describe solute dispersion in porous media. Simulation results show that 4000 pores or a distance on the order of meters are required to be 400 , and it is transversed to reach asymptotic dispersion for NPe suggested that, in a natural system, asymptotic dispersion will never be reached because of the large heterogeneity. Comparing solute dispersivity (related to dispersion coefficient through velocity) obtained from solving the mixing cell model and with the RWPT method on a

Variance of particle displacement, L Standard deviation, L Fluid viscosity, Pa s Random number between 0 and , dimensionless Random number between 0 and 2 , dimensionless

three-dimensional (3D) pore network model, Acharya et al. [32] concluded that a network size of 33 × 23 × 23 pore-units is required to reach the asymptotic stage. With 3D pore network models and continues-time random walk method, Rhodes et al. [33,34] proposed a methodology to simulate solute dispersion from pore-scale to field scale and showed that both results from 2D and 3D pore network models can accurately reproduce the experimental data. With the RWPT method and a 3D network model extracted from a dense random packing of spheres, Jha et al. [35] examined effect of diffusion on dispersion coefficient while the dispersion coefficient of solutes obtained from their model match well with experimental data. Vasilyev et al. [36] assumed Taylor dispersion of solutes and solved the advection-diffusion equation for solutes transporting in 3D pore-network models, showing that dispersivity is greatly affected by the mean network coordination number. With three different pore-network models extracted from core images and the RWPT method, Blunt et al. [37] compared and analyzed solute dispersion behaviour. It should be noted that this work focuses on passive particles, and thus publications on dispersion of reactive particles are not reviewed here. Since dispersion coefficient is usually assumed to be only a function of porous medium, dispersion coefficient of particles are commonly obtained based on solute experiments. However, solutes and particles disperse differently [38–41]. Considering the size-exclusion effects of particles, James and Chrysikopoulos [42] theoretically estimated particle dispersion coefficients in both a parallel-plate fracture and a circular tube with Poiseuille flow valid at long times. It was concluded that the finite size of a particle excludes it from the slowest moving portion of the velocity profile, and thus the effective particle velocity is increased and the overall particle dispersion is reduced [42]. Similar to particles in a tube or fracture, due to size exclusion, particle can only sample part of the velocity profile at a pore scale in porous media. Besides, particles cannot enter pore spaces with an opening smaller than particle size, which essentially leads to reduction of the volume of accessible void-space [20,21,24,43]. There exist great disagreements about the relationship between particle size and dispersion coefficient. With the equations for quantifying asymptotic dispersion in a tube, Sahimi and Jue [22] solved the one-dimensional (1D) convection diffusion equation to study dispersion in porous media represented by 2D square pore-network models. In the absence of flow, an equation was proposed by Sahimi and Jue [22] to describe the negative exponential dependence of dispersion coefficient on size ratio (i.e., particle size to tube radius). Sahimi [23] applied the same methodology to study particle dispersion in porous media and compared his results with experimental data in the literature. Using particles at micrometer scale 2

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(i.e., 2 µm -7 µm ), Auset and Keller [44] conducted experiments in micromodels saturated with water and observed dispersion increases as particle size decreases. Such a negative relationship between particle size and dispersion coefficient is attributed to the size-exclusion effect which large particles travel in the center streamlines leading to fewer detours [44]. Based on a series of experiments in silicon micromodels with carboxylated polystyrene microspheres (i.e., 0.5 µm and 1.0 µm ), it was found that flow rate, ionic strength, colloid size, colloid concentration, and colloid-matrix interactions jointly affect colloid dispersion [45]. They concluded that a smaller particle leads to a larger diffusion and accessibility to smaller pores, thus leading to a larger dispersion coefficient. Chrysikopoulos and Katzourakis [43] observed a positive correlation between particle size ranging from nanoscale to microscale and dispersivity in their experiments. With mathematical equations valid across the full-time scale, Meng and Yang [40] compared particle dispersion coefficient of different sizes ranging from nanoscale to microscale flowing in a circular tube. It was found that the relationship between particle size and dispersion coefficient varies with time and is affected by NPe . They are positively correlated if NPe is larger than its critical value; otherwise, they are negatively correlated. So far, no efforts have been extended to systematically investigate the effect of particle size and size exclusion on dispersion coefficient in porous media with different degrees of heterogeneity under various flow conditions. Throughout this study, particle size is used to differentiate solutes and particles. Particle size ranging from nanoscale to microscale has been considered. Only size-exclusion preventing particle from sampling the entire pore velocity profile is of prime interest in this study, and pore-plugging by particles is not considered. For dispersion in a parallel-plate fracture, it has been reported that source condition greatly affects dispersion coefficient especially at early times [39,46]. Generally, two types of distribution profile including uniform distribution and volumetric distribution are widely used in the literature. Uniform distribution which uniformly distributes particles across the throat cross-section has been used elsewhere [15,30,35,46]. Because, in a unit area, the number of influx particles is proportional to flux instead of area, and thus volumetric distribution which distributes particles after flow flux is reported to be more accurate and widely used [17–19,28,38–40,47–50]. However, no attempts have been made to compare those two distribution profiles, especially its effect on particle dispersion in porous media. In this study, 2D pore-network models with different degrees of heterogeneity are built to study particle dispersion in porous media. To be specific, the 2D pore-network model is firstly validated against results from the published papers. With the RWPT method, dispersion coefficients of particles with different particle sizes are then compared and analyzed once particles are instantaneously injected into the porenetwork models under various flow conditions. The size-exclusion effect on particle dispersion is also examined by comparing the RWPT results with and without size-exclusion term. Finally, different distribution profiles are compared to examine their effects on dispersion coefficient.

pressure of the network model subject to an externally applied pressure gradient can be calculated by solving a system of linear equations considering mass conservation at each pore. Then, the local fluid velocity inside each throat which is subsequently required to perform the RWPT simulations can be expressed as a function of location and average flow velocity of each throat v¯ . Subsequently, the RWPT method is applied to simulate solute and particle transport considering diffusion and advection at each time step once solutes and particles are injected into the systems. Finally, dispersion coefficient of solutes and particles can be calculated with the variance of particle displacement in network models. The pore-network model and RWPT method are detailed as follows. 2.1. Pore-network model A 2D diamond pore-network model validated well against experimental measurements was applied by Bijeljic et al. [15] to quantify solute dispersion in porous media. In this work, the same method is used to build 2D diamond pore-network models with a circular crosssection. The widely used assumption for pore-network model in the literature that pores are volumeless is also applied in this work [15,30,36,53]. For the network size, it has been reported that simulation on smaller (i.e., 40 × 40) and larger (i.e., 80 × 80) networks produce the same results [15]. Bruderer and Bernabé [30] also used 40 × 40 network model to simulate solute dispersion in porous media. Thus, considering simulation times, network size in this work is also fixed as 40 × 40. In order to study solute and particle dispersion in porous media with different degrees of heterogeneity, length of all throats is set to be constant as 100 μm and heterogeneity is generated by varying tube radius following the log-normal probability distribution. To be specific, heterogeneity is measured as the normalized standard deviation R R , where R and R are the mean value and the standard deviation, respectively. The minimum and maximum values of throat radius used by Fenwick and Blunt [54] to simulate three-phase imbibition and drainage in a 3D cubic pore-network model are adopted in this work. Parameters used to build pore-network models in this work are listed in Table 1. Fig. 1 shows a 2D 40 × 40 diamond network model consisting of 3121 pores and 6084 throats in total. As shown in Fig. 1, no-flow boundary condition is applied at both top and bottom boundaries, and periodic boundary condition is applied at the side faces. Particles and solutes are instantaneously injected into the system and initially there are no solutes or particles in the system. Considering the incompressible flow and volumeless pores in this study, the mass conservation equations of each pore can be written as [36]:

Qij = 0

(1a)

j Ni

Qij =

R4 (p 8µL i

pj )

(1b)

where Qij is the volumetric flux from pore #j to pore #i; Ni means all pores connected to pore #i by throats; R and L are throat radius and length, respectively; µ is fluid viscosity; and (pi pj ) means pressure gradient between pore #i and pore #j. Except for pores at inlet and outlet faces, pressure of each pore can

2. Methodology The following assumptions are made in this work: (1) Flow is incompressible, fully-developed, and laminar; (2) Dispersion is isothermal; (3) Molecular diffusion is independent of concentration [4–7,9,12,51,52]; (4) Particles are neutrally buoyant and travel at their centroid velocity; (5) No chemical reactions for passive particles [38–40,42]; and (6) Pores are volumeless [15,30,36,53]. Based on the above assumptions, several steps are involved to describe particle dispersion in porous media in this work. First, 2D porenetwork models with different degrees of heterogeneity are built. Because incompressible flow and passive particle are assumed, pore

Table 1 Parameters used to build pore network models.

3

Parameters

Value

Source

Network size Throat length (m) Minimum throat radius (m) Maximum throat radius (m)

40× 40 1.0E-04 1.0E-06 18.5E-06

Bruderer and Bernabé [30] Bijeljic et al. [15] Fenwick and Blunt [54]

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Fig. 1. A 2D 40 × 40 diamond pore-network model in this work. Fig. 3. Comparisons of solute dispersion coefficients simulated in this work and those provided by Bijeljic et al. [15].

Table 2 Parameters used to verify the newly built model for passive particles. Parameter

Value

Maximum flow rate (m/s) Particle diameter (m) Dynamic viscosity (Ns/m2) Temperature (oC) Tube radius (m) Time step t (s)

2.2. The RWPT method

Source −6

1 × 10 5 × 10−6 1.002 × 10−3 20 5 × 10−4 10

James and Chrysikopoulos [42] James and Chrysikopoulos [42] Zheng et al. [38] Zheng et al. [38] Taylor [4]

The RWPT method widely used to study transport problems in porous media is also used in this work to simulate solute and particle dispersion [15,35,50,55]. At each time step, convective displacement of particles equals to the product of the duration of time step and particle velocity inside the throat. For diffusive transport, the magnitude is a Gaussian distribution with a mean of 0 and standard deviation of 6Dm t where Dm is the diffusion coefficient and estimated by the Stokes-Einstein equation [50]. The convective transport parallels the throat axis; while, for diffusive transport, its direction is random. Thus, in the Cartesian coordinate system, diffusive displacement can be expressed as [50]:

be calculated by solving a set of linear equations with the preconditioned-conjugate-gradient method in Matlab software (version 2015a). With the known pore pressure, v¯ of the throat connecting pore #i and pore #j can be calculated as [36]:

R2 (p 8µL i

pj )

(2a)

and the local fluid velocity inside a throat is expressed as [35]:

v (r ) = 2v¯ 1

r R

(3a)

y = N (0, 6Dm t ) sin sin

(3b)

z = N (0, 6Dm t ) cos

(3c)

where N indicates a normal distribution with mean and variance in the argument; cos is uniformly distributed between −1 and 1; and is uniformly distributed between 0 and 2 . At the beginning of each simulation, 10,000 particles are simultaneously injected into the network model. The concept of “flow-biased probability” in the literature which means the greater the fractional flow rate into a throat or pore, the higher the probability that a particle chooses that entrance point, has also be used to determine initial throat location for the injected particles in this work [17–19,50,56,57]. Comparing to area-based distribution, flux-based distribution is more accurate because, in a unit area, the number of influx particles is proportional to flux instead of area [50]. Thus, for particle location in each throat, instead of uniformly assigning particles across the cross section of throats, particles are volumetrically distributed and the details can be found elsewhere [28,39,40]. Different methods have been used to choose particle outflow throat once it reaches a pore [15,26,50]. Comparing the streamline splitting method and flow-biased probability method, Yang and Balhoff [50] found differences between simulations are within 5%. Thus, considering computational efficiency, “flow-biased probability” relating the possibility that a throat selected as the outflowing path to its flow rate, is also applied to determine the outflow path here. Because this work focuses on passive particles, during each step if a particle hits the throat boundary it will bounce back into the throat. For periodic boundary conditions, particles are randomly redistributed into the throats across the inlet face after exiting the throats at the outlet face of

Fig. 2. Comparisons of dynamic dispersion coefficients of passive particles calculated by the equations proposed by Meng and Yang [40] (curves) and RWPT simulations (symbols) in this work for an instantaneous point source (r = 0 ) and volumetric planar source.

v¯ =

x = N (0, 6Dm t ) sin cos

2

(2b)

where r is the radial distance from the tube centerline. 4

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Fig. 4. Throat radius distribution of pore-network models with different degrees of heterogeneity (a)

5

R

R = 0.1; (b)

R

R = 0.5; and (c)

R

R = 1.0 .

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the network model. Because only a relatively small difference can be t = 0.0001 s and observed between simulations at time step t = 0.0010 s, t = 0.0010 s is selected in this work. Finally, based on particle travel distances, dispersion coefficient D can be calculated as [15]:

D=

1d 2 2 dt

where time.

2

(4) is the variance of particle displacement and t is the travel

3. Model verification The analytical solution valid across the full-time scale has been proposed by Meng and Yang [40] to calculate dynamic dispersion coefficient of particles flowing in a circular tube. Thus, parameters listed in Table 2 are used to verify the pore-network model built in this work. Fig. 2 compares dispersion coefficients of particles calculated by the equations proposed by Meng and Yang [40] and RWPT simulations (symbols) in this work for an instantaneous point source and volumetric planar source, respectively. Good agreements between the RWPT simulations in this work and theoretical calculations by Meng and Yang [40] have been achieved as shown in Fig. 2. It should be noted that, in order to verify this diamond pore-network model, dispersion coefficient paralleling the flow direction of a throat is calculated instead of the entire network model. Bijeljic et al. [15] built a diamond pore-network model with size 40 × 40 to simulate longitudinal dispersion of solutes in which throat length and radius distribution are similar to those of the Berea sandstone. The calculated dispersion coefficients match well with experimental data collected from literature for NPe < 4000. In order to further verify our pore-network model, Fig. 3 compares solute dispersion coefficient in the newly built pore-network model with normalized standard deviation = 0.1 and that from Bijeljic et al. [15]. The same solute diffusion coefficient is set as Dm = 10-9 m2/s through this work. It is not surprising that dispersion coefficients of solutes simulated in our network model are smaller than those given by Bijeljic et al. [15]. This is because our model almost represents a homogenous porous media with normalized standard deviation = 0.1; however, their pore-network model was built based on a real sandstone which is more heterogeneous. Except for that, simulation results in our work follow the same trend of Bijeljic et al. [15] and solute dispersion increases as NPe increases.

Fig. 5. Dispersion coefficients of particles with different particle sizes in the pore-network model (i.e., R R = 0.1) at uav = 0.11 cm/s.

that the x and y scales in Fig. 4 are different. Obviously, pore size distribution becomes broader as the normalized standard deviation increases. For convenience, comparing those three network models, models with R R = 0.1, R R = 0.5, and R R = 1.0 are respectively indicated as a homogeneous model, a moderately heterogeneous model, and a highly heterogeneous model in this paper afterwards. Considering different particle sizes, Fig. 5 compares dispersion coefficient of particles after being instantaneously injected into the homogeneous model at average flow rate uav = 0.11 cm/s. It should be noted that flow-biased probability is used to assign the initial throat for each particle, and particles are volumetrically distributed inside each throat. It can be seen that each curve can be described with preasymptotic and asymptotic stages. During the preasymptotic stage, dispersion coefficient increases with time and then remains constant during the asymptotic stage. Considering the small normalized standard deviation used to generate the throat-radius distribution which leads to small radius variations (see Fig. 4a), thus all particles approach the asymptotic stage quickly after being instantaneously injected into the system. Based on the simulation results, Blunt et al. [37] observed a Gaussian-like spread of solutes in the sandpack with a uniform flow field after 1 s. As shown in Fig. 5, at early times, dispersion coefficient of particles increases as particle size decreases; however, at late times, dispersion coefficient increases with particle size. Besides, at late times, great differences between dispersion coefficients of solutes and particles (i.e., d p = 10 7 m) can be observed. Berkowitz and Zhou [58] presented that, for solutes flowing in a parallel fracture the relationship between diffusion and dispersion is complex because longitudinal diffusion increases dispersion while transverse diffusion decreases dispersion. Considering the diamond pore-network model used in this work, radial diffusion increases dispersion if diffusion dominates. Thus, as shown in Fig. 5, at early time diffusion dominates, smaller particles corresponding to a larger diffusion coefficient leads to a larger dispersion coefficient. James and Chrysikopoulos [42] plotted average velocity of particles as a function of ratio of particle size to tube diameter for particles transporting in a circular tube at the asymptotic stage. As particle size increases, the average velocity of particles is reported to increase. Fig. 6 shows locations of solutes and particles (i.e., d p = 10 7 m) after 1 s. Compared to solutes, particles (i.e., d p = 10 7 m) travel faster and disperse broader at the asymptotic stage. As can be seen from Fig. 7, the average velocity of particles is larger than that of solutes. By changing the pressure difference across the pore-network model, Fig. 8a and 8b compare dispersion coefficient of particles with different

4. Results and discussion In this section, based on the RWPT simulations in three pore-network models with variable heterogeneity, effect of particle size on dispersion coefficient is thoroughly analyzed under various flow conditions. Besides, size-exclusion effect on particle dispersion is examined by comparing the RWPT results with and without size-exclusion term. Additionally, different distribution profiles widely used in the literature are compared to examine its effect on particle dispersion coefficient. 4.1. Effect of particle size Using analytical equations, Meng and Yang [40] concluded that, for passive particles flowing in a circular tube, relationship between particle size and dispersion coefficient varies with time. There exists a critical NPe , if the corresponding NPe is larger than the critical value, dispersion coefficient and particle size are positively correlated; otherwise, they are negatively correlated. Fig. 4 shows throat radius distribution of three pore-network models with normalized standard deviation equals to 0.1, 0.5, and 1.0, respectively. It should be noted

6

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Fig. 6. Locations of (a) solutes and (b) particles (i.e., d p = 10

7

m) at t = 1 s after being instantaneously injected into the pore-network model (i.e.,

particle sizes at uav = 0.21 cm/s and uav = 0.52 cm/s, respectively. Compared to Fig. 5, at early times, dispersion coefficient and particle size is still negatively correlated at uav = 0.21 cm/s; however, at uav = 0.52 cm/s, relationship between particle size and dispersion coefficient is not obvious. This is reasonable because, as flow rate increases, convective transport becomes more important, and thus, at early times, dispersion coefficient increases slightly with particle size at uav = 0.21 cm/s as shown in Fig. 8a. In Fig. 8b, convective transport becomes dominant at uav = 0.52 cm/s and thus no clear relationship between particle size and dispersion coefficient can be observed at early times. Similarly, at late times, dispersion coefficient increases with particle size for both cases as shown in Fig. 8. Compared to curves at uav = 0.11 cm/s, curves at uav = 0.52 cm/s stay closer to each other. Fig. 9 compares ratio of particle dispersion coefficient (i.e., d p = 10 7 m) to solute dispersion coefficient at various uav . Considering the diffusion role, it is reasonable that generally the ratio increases with decreased uav as shown in Fig. 9. Besides, at late times, all values are

R

R = 0.1).

larger than 1. In other words, particle size greatly affects dispersion in this homogeneous network model. From Fig. 10, it can be seen that our model is capable to generate the same conclusion reported in the literature that dispersion coefficient increases with uav [43,44]. By increasing the normalized standard deviation, Fig. 11 shows dispersion coefficients of different particles flowing in the moderately heterogeneous model (i.e., R R = 0.5). Compared to particles in the homogeneous model as shown in Fig. 5, only preasymptotic stage exists and particle dispersion coefficient increases with time. This is reasonable because, as aforementioned, long times are required for particles to scan pore space heterogeneity to reach asymptotic stage in heterogeneous porous media. From Fig. 11, it can be seen that, at the beginning, dispersion coefficient increases as particle size decreases and then quickly increases with particle size. It is interesting that Figs. 5 and 11 are plotted at the same uav ; however, compared to Fig. 5, curves in Fig. 11 are closer to each other. As aforementioned, dispersion in porous media is caused by stream splitting within pores, non-uniform 7

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Fig. 7. (a) solute and (b) particle (i.e., d p = 10

7

m) velocity at t = 1 s after being instantaneously injected into the pore-network model (i.e.,

velocity profile within throats, and diffusion. Fig. 12 compares v¯ of throats for the homogenous and moderately heterogeneous models. Compared to the moderately heterogeneous model, v¯ of the homogeneous model shows fewer variations among different throats. It can be concluded that, for a homogeneous model, diffusion greatly affects dispersion and thus particles with different size disperse differently, leading to distinguishable curves as shown in Fig. 5; however, for a heterogeneous model, compared to particle size effect, throat velocity difference dominates, and thus, as shown in Fig. 11, curves stay closer to each other. Similarly, by changing the pressure difference, Fig. 13 shows dispersion coefficients of different particles at uav = 0.17 cm/s. It is reasonable that, as uav increases, diffusive transport becomes less important and thus curves in Fig. 13 stay much closer to each other than those shown in Fig. 11. Fig. 14 shows ratio of dispersion coefficient of particles (i.e., d p = 10 7 m) to solutes at various flow rates in the moderately heterogeneous model. Due to the larger velocity differences among throats,

R

R = 0.1).

compared to curves shown in Fig. 9, curves are less distinguishable and values are smaller in Fig. 14. Besides, no clear patterns can be observed. Same conclusions can be applied to the highly heterogeneous model in this work (i.e., R R = 1.0 ). Thus, effect of particle size on dispersion not only varies among different porous media, but also relates to the heterogeneity of porous media. As heterogeneity increases, velocity differences among throats increase, leading to a larger dispersion coefficient (see Fig. 15). 4.2. Size-exclusion effect Compared to solutes, early breakthrough of particles has been attributed to the size-exclusion effect [42,48,59–62]. Thus, in this part, the size-exclusion effect on particle dispersion coefficient is examined in pore-network models with different degrees of heterogeneity. With the proposed equations, Meng and Yang [40] theoretically analyzed size-exclusion effect on particle dispersion coefficient for 8

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Fig. 10. Comparisons of dispersion coefficient of particles (i.e., d p = 10 various uav in the pore-network mode (i.e., R R = 0.1).

Fig. 8. Dispersion coefficients of particles with different particle sizes in a porenetwork model (i.e., R R = 0.1) at (a) uav = 0.21 cm/s and (b) uav = 0.52 cm/ s.

7

m) at

Fig. 11. Dispersion coefficients of particles with different particle sizes in the pore-network model (i.e., R R = 0.5) at uav = 0.11 cm/s.

Similarly, in this work, different particle sizes have been used to examine size-exclusion effect on particle dispersion coefficient. As shown in Fig. 16a, particle dispersion coefficient with and without size sizeexclusion effect are almost identical for particles with d p = 10 7 m. It should be noted that same results are obtained for smaller particles (i.e., d p < 10 7 m) and that, due to the limited space, similar figures for small particles are not shown. However, for larger particles (i.e., d p = 5 × 10 7 m) curves overlap each other at early times and then show large differences at late times (see Fig. 16b). Considering the volumetric distribution, this is reasonable that most particles stay around throat center at the beginning, and thus at early times sizeexclusion effect is not obvious; however, at late times, due to diffusive transport, more particles reach throat boundaries and thus leading to large dispersion difference as shown in Fig. 16b. Therefore, it can be concluded that, for the homogenous model used in this work, at uav = 0.11 cm/s, size-exclusion effect can be neglected for particles smaller than d p = 10 7 m; however, for larger particles (i.e., d p = 5 × 10 7 m), it needs to be considered, especially at late times. Similarly, it can be seen that, without size-exclusion effect, particle dispersion coefficient is overestimated. In order to further examine size-exclusion effect, Fig. 17 compares the ratio of particle dispersion coefficient (i.e., d p = 5 × 10 7 m) with and without size-exclusion effect in the homogeneous network model at various uav . It is clear that, for particles with d p = 5 × 10 7 m, size-

Fig. 9. Comparisons of ratio of particle dispersion coefficient (i.e., d p = 10 7 m) to solute dispersion coefficient at different uav in the pore-network mode (i.e., R R = 0.1).

particles flowing in a circular tube. As the ratio of particle size to tube diameter increases, size-exclusion effect becomes more important. Besides, ratio of particle dispersion coefficient with size-exclusion effect to that without size-exclusion effect varies with time and is smaller than 1. 9

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Fig. 14. Comparisons of ratio of particle dispersion coefficient (i.e., d p = 10 7 m) to solute dispersion coefficient at different uav in the pore-network mode (i.e., R R = 0.5 ).

exclusion in the moderately heterogeneous model at uav = 0.11 cm/s. As can be seen, dispersion coefficients are almost identical, which means size-exclusion effect important in the homogeneous network model can be neglected in this heterogeneous model for particles with d p = 5 × 10 7 m. Same results can be obtained for particles (i.e., d p = 5 × 10 7 m) flowing in this moderately heterogeneous model at increased uav . Due to limited space, figures at increased velocities are not shown here. For the highly heterogeneous model, same conclusions can be obtained. Therefore, size-exclusion effect which is important in the homogeneous model can be neglected in heterogeneous models. In other words, size-exclusion effect on particle dispersion also depends upon the heterogeneity of the network model. 4.3. Effect of distribution profile

Fig. 12. Comparisons of average velocities of throats for pore-network models at (a) R R = 0.1 and (b) R R = 0.5.

In this part, volumetric and uniform distributions are used to distribute particles in each throat across the cross section, while their effects on dispersion are systematically compared and analyzed in different pore-network models. Fig. 19 plots dispersion coefficients of solutes and particles (i.e., d p = 10 7 m) with uniform and volumetric distributions in the homogeneous model at uav = 0.11 cm/s. It can be seen that for solutes, only subtle difference can be observed for curves with volumetric distribution and uniform distribution. Meng and Yang [39] compared particle dispersion coefficient under different source conditions in a 2D fracture and found that, at early times, particles originating from uniform line source result in a larger dispersion coefficient than those from the volumetric line source. At late times, source effect on dispersion coefficient is negligible. Because Fig. 19 results from a homogeneous model, similar as particles flowing in a 2D fracture, for particles (i.e., d p = 10 7 m) at early times, curve with uniform distribution leads to a larger dispersion coefficient than that with volumetric distribution; and, at late times, those two curves generate almost identical values (see Fig. 19b). The same phenomenon can be observed for particles at increased uav , and thus similar figures are not provided here. Compared to a homogeneous model, however, for both solutes and particles (i.e., d p = 10 7 m), dispersion coefficient with volumetric distribution has a larger value than that with uniform distribution in a moderately homogeneous model (Fig. 20). For the homogeneous network model, mechanisms are similar as that in a circular tube where velocity gradient decreases from tube boundary to the center, and thus, at early times, more particles stay close to the tube boundary leading to a larger dispersion coefficient. Thus, as shown in Fig. 19b, compared to volumetric distribution, particle dispersion coefficient with uniform

Fig. 13. Dispersion coefficients of particles with different particle sizes in the pore-network model (i.e., R R = 0.5) at uav = 0.17 cm/s.

exclusion effect cannot be neglected at various flow rates. At early times, curves separate from each other and values decrease as uav increases, and then become indistinguishable at late times. Based on above results, particles with d p = 5 × 10 7 m have been further analyzed in the moderately and highly heterogeneous models. Fig. 18 plots particle dispersion coefficient with and without size 10

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Fig. 16. Comparisons of dispersion coefficient of particles (a) d p = 10 7 m and (b) d p = 5 × 10 7 m with and without size exclusion in the pore-network model (i.e., R R = 0.1) at uav = 0.11 cm/s.

Fig. 15. Comparisons of particle (i.e., d p = 10 7 m) locations at t = 0.1 s after being instantaneously injected into the (a) homogeneous (i.e., R R = 0.1); (b) moderately heterogeneous (i.e., R R = 0.5); and (c) highly heterogeneous (i.e., R R = 1.0 ) pore-network models.

Fig. 17. Comparisons of the ratio of particle dispersion coefficient (i.e., d p = 5 × 10 7 m) with and without size-exclusion effect at various uav in the pore-network model (i.e., R R = 0.1).

distribution is larger. As aforementioned, however, for a heterogeneous network model, velocity difference among throats plays an important role and thus volumetric distribution which assigns more particles around the throat center leads to a larger dispersion coefficient. This conclusion has been further demonstrated as shown in Fig. 21 that for both solutes and particles, larger dispersion differences exist between

curves with uniform and volumetric distributions than those shown in Fig. 20. As heterogeneity increases, dispersion difference between volumetric and uniform distributions increases. Effect of uniform and volumetric distributions on particle dispersion coefficient is found to 11

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Fig. 18. Comparisons of dispersion coefficient of particles (i.e., d p = 5 × 10 7 m) with and without size exclusion at uav = 0.11 cm/s in the moderately heterogeneous pore-network model (i.e., R R = 0.5).

Fig. 20. Comparisons of dispersion coefficient of (a) solutes and (b) particles (i.e., d p = 10 7 m) with uniform distribution and volumetric distribution at uav = 0.11 cm/s in the moderately heterogeneous pore-network model (i.e., R R = 0.5 ).

5. Conclusions 2D diamond pore-network models with variable heterogeneity have been built and verified to describe particle dispersion in porous media under various flow conditions. For particles flowing in a homogeneous model, at early times, dispersion coefficient of particles is found to increase as particle size decreases; however, at late times, dispersion coefficient increases with particle size. In heterogeneous models, curves of particle dispersion coefficient become indistinguishable, and the relationship between particle size and dispersion coefficient is less obvious. Dispersion coefficient of particles is greatly affected by particle size in a homogeneous model; however, in the heterogeneous model, throat velocity difference caused by heterogeneity plays an important role on particle dispersion. It can be found that, in the homogeneous model without size-exclusion effect (i.e., d p = 5 × 10 7 m), particle dispersion coefficient is overestimated, while size-exclusion effect becomes more important as uav increases. However, in the heterogeneous models, size-exclusion effect of particles with d p = 5 × 10 7 m can be neglected. The widely used uniform and volumetric distribution profiles have been compared in different models. The dispersion difference between volumetric and uniform distributions increases with particle size. Effect of uniform and volumetric distributions on particle dispersion is found to depend upon the property of porous media. As heterogeneity increases, dispersion difference between volumetric and uniform distributions also increases. For the homogeneous model,

Fig. 19. Comparisons of dispersion coefficient of (a) solutes and (b) particles (i.e., d p = 10 7 m) with uniform distribution and volumetric distribution at uav = 0.11 cm/s in the homogeneous pore-network model (i.e., R R = 0.1).

depend upon the property of porous media. For the homogeneous media, dispersion coefficient with uniform distribution is larger; however, as heterogeneity increases, dispersion coefficient with volumetric distribution has a large value and the difference increases with particle size. 12

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