Transp Porous Med DOI 10.1007/s11242-012-0054-5
Effect of Mean Network Coordination Number on Dispersivity Characteristics Leonid Vasilyev · Amir Raoof · Jan M. Nordbotten
Received: 21 August 2011 / Accepted: 21 July 2012 © Springer Science+Business Media B.V. 2012
Abstract In this study, we investigate the role of topology on the macroscopic (centimeter scale) dispersion characteristics derived from pore-network models. We consider 3D random porous networks extracted from a regular cubic lattice with coordination number distributed in accordance with real porous structures. We use physically consistent rules including ideal mixing in pore bodies, molecular diffusion, and Taylor dispersion in pore throats to simulate transport at the pore-scale level. Fundamental properties of porous networks are based on statistical distributions of basic parameters. Theoretical calculations demonstrate strong correspondence with data obtained from numerical experiments. For low coordination numbers, we observe normal transport in porous networks. Anomalous effects expressed by tailing in concentration evolution are seen for higher coordination numbers. We find that the mean network coordination number has significant influence on averaged characteristics of porous networks such as geometric and hydraulic dispersivity, while other topological properties are of less significance. We give an explicit formula that describes the decrease of geometric dispersivity with growing mean coordination number. The results demonstrate the importance of network topology for models for flow and transport in porous media. Keywords
Dispersion · Porous media · Pore-network models · Coordination number
1 Introduction Transport phenomena of passive solutes in porous media are important in many problems of scientific and industrial interest, including transport of contaminants in soils and aquifers, extraction of oil and gas, geothermal energy, and carbon dioxide storage. Realistic models L. Vasilyev (B) · J. M. Nordbotten Department of Mathematics, University of Bergen, Joh. Brunsgate, 12, 5008 Bergen, Norway e-mail:
[email protected] A. Raoof Universiteit Utrecht, Budapestlaan 4, 3584 CD, Utrecht, The Netherlands e-mail:
[email protected]
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require accurate estimates of averaged parameters, where for transport, dispersion plays a key role. Pore-scale heterogeneities are present in all porous media, and lead to deviations from the mean flow. On a macroscale, this is described by the hydrodynamic dispersion. In porous media, hydrodynamic dispersion accounts for spreading phenomena that in general includes the effects of both mechanical dispersion and molecular diffusion. Dispersion as an interaction of two basic phenomena, i.e., advection and diffusion, acting on a macroscale, is usually described by Péclet number, which is defined as the ratio between the time needed for particles driven by diffusion to flow through a characteristic length L, and the time needed for particles to flow through the same length being driven by advection: Pe =
L 2 /Dm vL = , L/v Dm
(1.1)
where Dm is the molecular diffusion, and v is the average velocity. The relationship between the dispersion coefficient and Péclet number demonstrates the dependence of spreading phenomena on the flow regime. If Pe ≤ 1, diffusion dominates in fluid flow, while for high Péclet numbers advection plays more significant role in the flow process and diffusion can be neglected. In such cases dispersion accounts for spreading of the concentration profile. There are two basic physical effects included in the description of dispersion: natural heterogeneity of microstructure of real porous systems, and variation of velocity within a single pore throat, the latter caused by friction at the solid surfaces. If only the mean flow in pore throats is considered, leaving mixing to occur only at junctions, averaged dispersion is proportional to the first power of velocity, and as follows Péclet number (Scheidegger 1957). The coefficient of proportionality here, so-called geometric dispersivity, describes the geometric nature of porous media. Taylor (1954), in his one-dimensional analysis of flow in pipes suggested that dispersion, as a result of the parabolic velocity profile, should be proportional to the second power of Péclet number. In general, the dispersion coefficient is usually modeled as proportional to some power of Péclet number with exponent lying between 1 and 2. When both advection and dispersion are present, there is a transition region to a power law D ∼ Peδ where δ ≈ 1.2 for experiments on bead packs and homogeneous sandstones (Sahimi 1995; Bijeljic et al. 2004). On a crossover to a purely advective, mechanical dispersion regime with Pe > 400, Sahimi (1995); Bijeljic et al. (2004) found that D ∼ Pe. For high Péclet numbers (Pe 1), accounting for advection-dominated regime, molecular diffusion is negligible and mechanical dispersion becomes an important mechanism for transport described as D ∼ Pe ln Pe (Saffman 1959a; Koch and Brady 1985). Saffman (1959b, 1960) in his works on flow through the networks of capillaries mentioned that for networks where D → 0 and Péclet number is very high: Pe > 2 × 104 , it is difficult to observe the logarithmic dependence. This is the case considered, herein, valid for pore networks with long thin capillaries and clogged channels, which have been studied in the context of porous filters (Kampel and Goldsztein 1988; Goldsztein 2008), consolidated rocks as idealized as networks of capillaries of random permeability (Charlaix and Gayvallet 1991), and some situations of granular medial flow (Gavrilov and Shirko 2010). In principle, it is possible to describe flow and transport by solving the governing equations at the pore scale. A popular approach to obtain a fast and approximate solution to this system are the so-called pore-network models. However, the application of network models still requires an appropriate representation of the topology of the real porous structure. Previous works have shown the importance of geometric structure of pores, in particular, pore size distributions, shape of pore throats and pore throat correlations (Larson et al. 1977, 1991; Øren and Pinczewski 1991, 1994; Blunt et al. 1992, 1994; Heiba et al. 1992; Øren
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et al. 1992; Ioannidis and Chatzis 1993; Paterson et al. 1996; Pereira et al. 1996; Knackstedt et al. 1998). Arns et al. (2004) has further shown that the connection patterns of pores or the topology of the network also has much influence on nonlinear processes such as two-phase flow. Our work contributes to these efforts of identifying the key topological measures that affect macroscopic parameters of transport in porous media. Pore-network topology can be expressed by parameters including mean network coordination number (mean number of connections to each pore body) and the distribution of coordination numbers. Early pore-network modeling studies in 3D considered only regular networks with coordination number z = 6, though it obviously does not express full complexity and randomness of real porous media. More recent studies demonstrated that rock samples have a broad distribution of coordination numbers (Ioannidis et al. 1997; Lindquist et al. 2000; Øren and Bakke 2003). Øren and Bakke (2003) observed that the coordination number ranges from 1 to 16 with an average of 4.45. Lindquist et al. (2000) have shown that coordination numbers in real porous media can be larger than 20 depending on the porosity of the medium. Arns et al. (2004) found that the equivalent network models for Fontainbleau sandstone samples have low mean coordination numbers (z < 4) with a wide distribution of node coordination numbers. They concluded that relative permeabilities for the equivalent sandstone networks cannot be reproduced be regular networks and stochastic networks with the same geometric properties, where the effect of topological disorder is small and therefore the use of realistic 3D topologies is important. In this paper, we consider the effect of topology on dispersivity characteristics of porous media computed from network models. We first discuss main parameters of network geometry, focusing especially on coordination number and distribution of pore size. We use a method, suggested by (Raoof and Hassanizadeh 2010), for generating 3D pore networks, representing real porous structures, which gives a wide distribution of coordination numbers while retaining logical simplicity. The assumed specific geometry of pore throats provides developing of Taylor dispersion. We find that averaged dispersion curves of porous network demonstrate strong dependence on Péclet number that can be expressed by the second-order polynomial in which coefficients are in good agreement with theoretical calculations. Finally, we investigate the influence of mean network coordination number on geometric dispersivity.
2 Description of the Pore-Network Model In pore-network modeling, the microstructure of porous rocks is represented by simple ideal elements that reflect the average transport process. The main advantage of synthetic porous networks is the ability to create a computationally efficient virtual laboratory for fluid flow in porous media with well-defined assumptions and simplifications, allowing for an exhaustive exploration of fundamental correlations in transport properties. In this study, we choose a method for generating 3D porous networks introduced by Raoof and Hassanizadeh (2010) as it gives more reliable representation of real homogeneous isotropic porous medium. The method is based on a regular cubic lattice which has two basic elements: pore bodies located at regular lattice points and pore throats that interconnect pore bodies. In multi-directional regular pore-network lattice, each pore body can be connected to a maximum of 26 other pore bodies, forming 26 connections, which defines the maximum coordination number. In reality, most pores have significantly lower coordination number, therefore some connections are eliminated, as described in (Raoof and Hassanizadeh 2010), until the desired distribution of coordination numbers is achieved. In this paper, we use the mean network coordination number as a fundamental property of porous media. Note that the
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average of all coordination numbers in the network may deviate from the mean coordination number due to decreasing number of connections at the boundaries. The construction outlined above leads to a skeleton of porous network with pore bodies interconnected by pore throats. We assume that pore bodies are non-volumetric objects where mixing of fluids from pore throats occurs. They play only a connecting role and therefore only pore throats form the void space of porous medium. We also assume that all pore throats have circular cross-section with specific radii. Following Joekar-Niasar et al. (2008), we choose truncated log-normal probability distribution to generate pore throat radius. Together with log-normal distribution, we also investigate flow process in the networks with truncated uniform distribution of radii, and all radii equal to some constant. These distributions are less realistic compared to log-normal distribution when applying to pore size, however, we use them in order to investigate the influence of pore size on fundamental parameters of porous media. For simplicity, we consider flow only along the principle direction which is defined by the longest edge of the network. Fluid is injected into the bodies located at the inlet boundary and leaks from the outlet boundary on the opposite side of the porous network, moving under the macroscopic pressure gradient. Following these assumptions, we consider the average transport in porous networks as essentially one-dimensional process, where flow across the principle direction is described as spreading of the concentration profile. Most commonly the porosity of porous networks is defined as the ratio between the volume of void space and the total volume of the sample. Though this definition is intuitive, it does not include the fact that due to network randomness dead pores (with relatively small velocities) or even dead blocks can appear. This means that fluid does not flow through these blocks or there is only a diffusive flux present. Therefore, in advection-dominated regime, this part of the porous medium does not play a significant role in transport and must be excluded. When calculating effective porosity, we consider only pores where the mean fluid profile moves with a significant velocity. In our calculations, we have chosen to exclude all pores with flow rates below machine precision.
3 Assumptions for Transport in Porous Networks Transport in porous network can be simulated in many different ways (Blunt et al. 1992; Bijeljic and Blunt 2007; Suchomel et al. 1998). A common characteristic is that pore throats are usually not discretized (Blunt et al. 1992; Bijeljic and Blunt 2007; Raoof and Hassanizadeh 2010), as fluid flow in pore throats is assumed to be fully advective, and thus can be resolved analytically from transport equation. This assumption breaks down when effects of molecular diffusion and Taylor dispersion in throats are significant as is the case in throats with slow velocities and dead pores (i.e., pores with blocked ends and zero fluxes). To include this aspect in our investigation, we have chosen to resolve the advection–dispersion equation (ADE) numerically in each pore. 3.1 Pressure Field in Porous Network Based on the description of porous network given above we assume that pore bodies are non-volumetric and void space is formed only by pore throats. This can be formulated in terms of mass conservation (equivalent to volume conservation, as only incompressible flow is considered) for node i
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Q i j = 0,
(3.1)
j∈Ni
where Q i j is the volumetric flux from pore body j to pore body i interconnected by throat s. Ni denotes the set of all bodies connected with pore body i (i.e., the coordination number). For laminar flow within a cylindrical tube, volumetric flux Q i j is related to the pressure gradient between two bodies via Poiseuille’s law Qi j =
π Rs4 ( pi − p j ). 8μL s
(3.2)
Here pressure gradient ( pi − p j ) is defined in the throat of radius Rs and length L s and μ is the fluid viscosity. Mass balance equation (3.1) for all mixing bodies together with (3.2) form a system of linear equations where pressures in the pore bodies are unknown variables. At the same time, boundary conditions specify pressures in the inlet and outlet bodies. Solution of such system returns the pressure field of the porous network as a function of the prescribed pressures at the bodies. 3.2 Transport in Pore Throats Pore throats in our network models are volumetric objects that carry fluid between the pore bodies. For simplicity, we assume them to have a circular cross-section. Solute mass transport in pore throats is driven by the pressure gradient. Molecular diffusion together with Taylor dispersion induce spreading of the concentration profile. Assuming 1D flow, the following equation describes transport of solute in pore throat s: ∂cs ∂qs =− , ∂t ∂x
(3.3)
where total flux qs is of the form qs = vs · cs − Ds ·
∂cs . ∂x
(3.4)
Here vs denotes the fluid velocity in the throat s, and Ds is the equivalent dispersion coefficient which is the combination of molecular diffusion and Taylor dispersion (Taylor 1954). For a given coefficient of molecular diffusion Dm , it takes a form Ds = Dm +
(vs · Rs )2 . 48 · Dm
(3.5)
In our simulations, we observe advection-dominated flow (Pe > 2 × 104 ), where Taylor dispersion plays a significant role. 3.3 Numerical discretization of Porous Network We use an implicit finite-difference upstream scheme to solve ADE in throats to avoid timestep constraints in pore throats (LeVeque 1992). Following these assumptions, the discretized form of ADE for pore throat s is
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3 2
C
i
1
N−1
N−2
2 3
Fig. 1 Connection of discretized throats to the mixing body
⎧ n n−1 n n n n n ⎨ ck −ck = −vs ck −ck−1 + Ds ck+1 −2ck2+ck−1 , for vs 0 ts xs x ⎩ ckn −ck
n−1
ts
= −vs
n −cn ck+1 k xs
s
+ Ds
n −2cn +cn ck+1 k k−1 , xs2
for vs < 0
(3.6)
Here the concentration in each pore throat is denoted by c (index s is omitted in (3.6)), while ckn is the concentration at the discretized space point k · xs at the time n · ts . The mean fluid velocity in the pore throat vs can be derived from Poiseuille’s law by dividing by the cross-sectional area: vs =
Rs2 ( pi − p j ). 8μL s
(3.7)
Using (3.6), we obtain the concentration profile in each throat. In the next step, we need to perform mixing in pore bodies. As it was mentioned above, pore bodies are non-volumetric and their function is only to connect the pore throats. We assume that all substances from the throats instantaneously mix in the pore body and an averaged concentration can be derived from mass conservation law: Fis = 0, (3.8) s∈Ni
where Fis is the total mass flux to throat s from pore body i, which connects the set of throats Ni . Mass flux Fis is driven by the advective-dispersive transport and can be expressed by Ci −cs (k) 2 s Fis = π Rs2 · (vs · Ci − Ds ∂c ∂ x ) ≈ π Rs · (vs · Ci − Ds xs ), for vs 0 (3.9) Ci −cs (k) 2 s Fis = π Rs2 · (vs · cs − Ds ∂c ∂ x ) ≈ π Rs · (vs · cs (k) − Ds xs ), for vs < 0.
Here, the space derivatives in (3.9) are replaced with their finite-difference approximation, the concentration in pore body i denoted with Ci . Discretization point k can be either k = 2 or N − 1 depending on which side of throat s is connected to pore throat i (Fig. 1 clarifies the connection to pore body i). Here, volume π Rs2 · x R fully belongs to throat s which is also in accord with (3.8). First, we solve (3.6) assigning concentrations in pore throats and constant fluxes as initialboundary conditions. Then we find concentrations in pore bodies, solving (3.8) and (3.9). After that we can proceed to the next time step. Both (3.6) and (3.9) are solved as the system of linear equations using Gaussian elimination algorithm.
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4 Transport Equation for Averaged Flow in Porous Network Displacement of the average concentration in porous media can be described by the ADE: ∂c = −v · ∇c + ∇ · ((D + Dm T)∇c). ∂t
(4.1)
also known as the equation of hydrodynamic dispersion (Bear 1972). In (4.1), D is the coefficient of mechanical dispersion representing dispersive flux, Dm T is the coefficient of molecular diffusion in porous media, and T is the tortuosity tensor. Previous studies related dispersion to the velocity or Péclet number (Sahimi 1995; Bijeljic et al. 2004; Saffman 1959a; Koch and Brady 1985). We understand the coefficient of mechanical dispersion D as a combination of geometric dispersion (describing the network nature of porous media) proportional to average velocity v and dynamic dispersion (also known as Taylor dispersion (Taylor 1954), which takes into account distribution of velocities in pore throats), proportional to v 2 (Scheidegger 1957). In this study, we consider transport with essentially one-dimensional flow, therefore, (4.1) takes its 1D form ∂c ∂c ∂ ∂c = −v + (D + Dm T ) . (4.2) ∂t ∂x ∂x ∂x Consequently the coefficient of mechanical dispersion can be expressed in a form D = αG v + αT v 2 ,
(4.3)
where αG is the coefficient of the geometric dispersivity in the porous medium and αT denotes the dynamic dispersivity accounting for Taylor dispersion in throats. We denote the coefficient accounting for the second moment in ADE which includes the effects of both molecular diffusion and mechanical dispersion as D ∗ = Dm T + αG v + αT v 2 ,
(4.4)
Following the derivation in Bear (1972) and Mostaghimi et al. (2010), we assume the geometric dispersion to be related to the variance of some network parameters. Classical definition of dispersion, based on Gaussian probability distribution, gives σ 2 = 2Dt, where σ is the variance of the solute position, D is the geometric dispersion, and t the is time. Let us consider particles moving through the network of pores that have different size according to some distribution. While the concentration profile moves through the network, mixing occurs within the pore bodies of each cubic cluster. Therefore, we can assume that particles spend averagely t ∗ = d/v seconds between two mixings, where d is the size of the underlying lattice. Due to the differences in pore radii, some particles arrive at a pore body earlier or later. This causes deviation from the mean flow. The variance of the distance traveled by particles during time between two mixings is (vs t ∗ − vs t ∗ )2 . According to this, we define the geometric dispersion coefficient as (vs t ∗ − vs t ∗ )2 (vs − vs )2 d αG v = = . (4.5) 2t ∗ 2v Here “ ” denotes averaging over the pore-network model, i.e., over the set of connecting throats. As it was discussed in Sect. 2, not all pore throats play a significant role in transport process. Therefore, in (4.5) and (4.8), only pore throats with significant advective flow (i.e., that form part of the effective porosity) are considered.
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Tortuosity is another important physical parameter that influences transport in porous media. In (4.2), T describes the relation between effective molecular diffusion and diffusion in pore throats. Tortuosity is defined as the ratio between the length of the tortuous tube and the distance between its ends. In the other words, tortuosity shows how similar is the tube to a straight line. In terms of porous media, it is defined as the geometrical length of the system (i.e., the distance between the inlet and the outlet) divided by the average length of all travel paths that particles flow through (Carman 1937): L 2 T = , (4.6)
L s where L s is the length of the tortuous flow path. According to this definition, the tortuosity lies always between 0 and 1 as the flow path is longer than the geometrical distance. In this study, we consider the tortuosity by how it affects the average velocity (Carman 1937): √ (4.7) v = T vs , here v is the average (macroscale) velocity, and vs is the averaged velocity in a flow path denoted by s. In our simulations, we calculate vs as the average of all pore-scale velocities. This method also interprets how porous network affects pore-scale velocities. In (4.4), the tortuosity relates the pore-scale molecular diffusion with the macroscale molecular diffusion of a homogeneous porous medium (Bear 1972). In pore throats, molecular diffusion accounts for deviations from the average pore-scale velocity due to random motion of particles. In the same way, hydraulic dispersion describes the variation of the velocities in a single throat. Therefore, it is reasonable to assume that the average hydraulic dispersion is related with pore-scale hydraulic dispersions as DT∗ = DTs T , if variations of hydraulic properties of pore throats are small. Thus, we calculate the macroscale hydraulic dispersivity from:
(Rs vs )2 T αT = , (4.8) 48Dm v 2 where Rs is the radius, and vs is the average velocity in the throat denoted by s. Equations (4.5) and (4.8) give explicit expressions for the upscaled ADE based on the underlying network, with no free parameters.
5 Simulations We conduct numerical experiments for random pore networks created in accordance with the method described in Sect. 2. All networks are constructed of 30 × 10 × 10 bodies so that the whole structure has physical dimensions of 58 × 18 × 18 mm, where the longest side is parallel to the principal direction of flow. Porosity is kept constant at 20 % for all cases. In order to compare the closed form expressions derived in Sect. 4 with the results from the network model, the results from the network model are represented by the solution of the ADE, where the parameters are chosen in order to create a best fit to the data. 5.1 Hydrodynamic Dispersion of Porous Networks In this section, we investigate the parabolic dependence described by (4.4) for random networks with log-normal distribution of radii. In order to explore the effect of
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3.5
x 10
z=4 z=5 z=6 z=7 z=8
Despersion, D/Dm
3 2.5 2 1.5 1 0.5 0
0
0.5
1
1.5
2
Peclet number
2.5
3 5 x 10
Fig. 2 Dispersion derived from fitting the experimental data plotted with theoretical dependence (4.4) (dashed lines). Dispersion coefficients are calculated from theoretical assumptions (4.5) and (4.8)
the average velocity, and hence the Péclet number, we apply various pressure gradients to the pore networks. In Fig. 2, the dashed line represents the theoretical dependence (4.4), where αG and αT were calculated from the statistical information about the network according to (4.8) and (4.5). Experimental points (markers on Fig. 2) were obtained from fitting the breakthrough curves (BTCs) with the ADE. The data is plotted in terms of normalized dispersion and Péclet number such that (4.4) turns into D αG αT Dm T =1+ Pe + Pe2 , Dm T L L2
(5.1)
where L is the length of the porous sample. As follows, the geometric and hydraulic dispersivity coefficients in (5.1) are replaced with their normalized values αLG and αT Dm T . L2 Numerical experiments confirm the parabolic dependence of the hydraulic dispersion on the average velocity. Furthermore, the hydraulic dispersion derived from the statistical properties of the porous network is a good approximation of the experimental results from the network simulation. Figure 3 represents the data on a log-scale. High Péclet numbers (Pe > 400) account for advective-dominated regimes of flow (Bijeljic and Blunt 2007). In principle, the dispersion curves can be approximated with D ∼ Pe1.2 with transition to D ∼ Pe ln Pe as pointed by Sahimi (1995), and Bijeljic et al. (2004). On Fig. 3, D ∼ Pe ln Pe is plotted as the dashed line. It is seen that this dependence works sufficiently well only for Pe < 2 × 104 . We conclude that parabolic dependence D = αG Pe + αT Pe2 is a better fit to the transport phenomena in the porous structures considered in this study. It was also mentioned by other authors that logarithmic dependence fails to describe the hydrodynamic dispersion in networks of capillaries where diffusion is negligible and Péclet number is very high (Goldsztein 2008; Saffman 1959b).
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Despersion, D/Dm
105
104
z=4 z=5 z=6 z=7 z=8
103
102 4 10
5
10
Peclet number Fig. 3 Dispersion versus Péclet number compared with Eq. (4.4). Dashed line denotes D ∼ Pe ln Pe as proposed by Saffman (1959a). High Péclet number accounts for advection-dominated regime
5.2 Geometric and Dynamic Dispersivity As it is discussed in Sect. 4, we understand the coefficient of hydrodynamic dispersion as a combination of molecular diffusion, geometric dispersion and dynamic dispersion. The second term in Eq. (4.4) accounts for distribution of velocities due to branching in porous networks, while the third term describes Taylor dispersion due to the velocity profile in pore throats. Geometric dispersivity αG represents network nature of a porous medium, and therefore, we consider it as a fundamental property that depends on the network topology. We investigate geometric dispersivity as a function of a mean coordination number for three types of porous networks: (A) random networks with truncated log-normal distribution of radii, (B) random networks with truncated uniform distribution of radii, (C) random networks where all radii are the same, We assume the first type to be the most realistic representation of the porous structure. For type (A), we produced three sets of networks rebuilding their structures from the beginning (construction steps described in Sect. 2) in order to confirm that obtained parameters are independent of the realization. We observed (not shown) that all networks demonstrate similar behavior with the deviation of parameters (dispersivities, permeability, tortuosity) less then 1 %. On Fig. 4, the geometric dispersivity calculated from the statistical properties of the porous network (markers) is smaller in the networks with higher mean coordination number. An approximate formula connecting the geometric dispersivity with the mean coordination number (plotted as dashed lines on Fig. 4) is introduced below. Note that the geometric dispersivity is constant for all pressure gradients and depends only on the properties of the network structure. The hydraulic dispersivity demonstrates similar behavior decreasing for higher mean coordination numbers (see Fig. 5).
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0,05
LogN Equal Uniform
0,04
0,03
0,02
0,01
0
2
4
6
8
10
12
14
16
Mean network coordination number Fig. 4 Geometric dispersivity of porous networks calculated from statistical properties. Equation (5.2) approximates the relation between geometric dispersivity and coordination number
x 10−7
T
Hydraulic dispersivity α Dm/L
2
2.5
2 LogN Equal Uniform
1.5
1
4
6
8
10
12
14
16
Mean network coordination number Fig. 5 Hydraulic dispersivity of porous networks calculated from statistical properties
5.3 Discussions The main goal of this study is to investigate the coefficient of geometric dispersivity, which accounts for the network nature of porous media. We consider porous networks constructed in the same manner with different coordination numbers. Increasing of the mean coordination number leads to increasing of the number of throats in a porous network. In order to have the same porosity in all structures we reduce the pore radii (by scaling appropriately the corresponding log-normal distribution) and consequently (according to Eq. 4.8) this cause decreasing of the hydraulic dispersivity as shown on Fig. 5.
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Tortuosity
0.62 0.6
LogN Equal Uniform
0.58 0.56 0.54 0.52 0.5
4
6
8
10
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14
16
14
16
Mean network coordination number Fig. 6 Tortuosity of random porous networks evaluated from statistical properties 0.198
0.196
Effective porosity
0.194
0.192
0.19
0.188
0.186
0.184
4
6
8
10
12
Mean network coordination number
Fig. 7 Effective porosity of random porous networks with log-normal distribution of radii
In order to explain the effect of lower dispersivity in networks with higher coordination number, described in Sect. 5.2, we look at other porous medium characteristics such as tortuosity and effective porosity. We calculate the medium’s tortuosity from (4.7), obtaining the average velocity from the injection rate found in the pore-network simulations. Pore velocities are given by the pressure field. We observe that the tortuosity grows with the mean coordination number (Fig. 6) which means that particle flow paths become similar to a straight line representing the principal direction. This leads to a smaller variance in velocities as all paths converge to the straight line. Increase in the effective porosity also confirms that particle flow paths converge to a straight line (Fig. 7). Low effective porosities represent high number of dead pores or dead
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Radii
α0
c
Random
8.5 × 10−4
3.3 × 10−3
Equal
8.2 × 10−4
2.6 × 10−3
Uniform
9.0 × 10−4
3.5 × 10−3
δ = 0.43
zones where pressure gradient is extremely low and transport in throats becomes diffusion dominated. These dead blocks of the porous network force particles to flow through more tortuous paths which also leads to lower tortuosity factor (see Eq. 4.6). Following these observations, we conclude that higher network coordination numbers produce smaller variance of the pore-network parameters and the velocities that cause falling of the geometric dispersivity described by (4.5). In Fig. 4, the dispersivity converges to some value specific for the pore radii distribution and grows exponentially for low coordination numbers. Based on these assumptions, we introduce an empirical formula for the geometric dispersivity coefficient: αG = α0 + c · e−δ(z−2) ,
(5.2)
where α0 is the geometric dispersivity at the highest mean network coordination number (26 for the considered pore-network model), c and δ are some constants and z is the mean network coordination number. Term z − 2 points out that low coordination numbers are not realistic (if ever possible). z = 2 in a case of regular network shows that all pore bodies have two connections, i.e., they connect only two pore throats. This case represents, for example, a set of separate flow channels with no cross-flow. Corresponding random networks produce a high fraction of dead pores and dead zones which makes a homogeneity assumption doubtful. This formula works sufficiently well for z ≥ 4 (Fig. 4). Estimated parameters of (5.2) for the different types of networks are summarized in Table 1. In our investigations, we observed the evolution of the concentration profile obtained at several cross-sections. Concentration data must agree with the solution of ADE, however, anomalous transport effects may cause significant changes in the profiles. While the low concentration part fits the ADE quite well, higher values form long tails (Fig. 8) cased by anomalous transport effects in porous networks known as tailing effect (Kennedy and Lennox 2001). This effect appears more significantly at high mean network coordination numbers which is noticeable by growing standard error in fitted curves. In Fig. 9, we plotted the standard deviation between concentration data and corresponding ADEs with theoretical coefficients. The theoretical ADEs describe the BTCs much better in comparison with fitted ADEs, which try to fit the long tails thus increasing the error in average transport. This error is caused also by numerical inaccuracy in obtaining concentration curves as well as derivation of theoretical parameters of porous networks. However it becomes more significant for high coordination numbers as a result of anomalous transport. Kennedy and Lennox (2001) concluded that one reason for tailing effect is the skewed density functions of underlying stochastic processes. The distribution of particles always converges to the normal distribution according to the central limit theorem which is seen as the long tails on concentration curves. However for highly skewed distributions, slower convergence is expected. We observed that the mean of the velocity distribution together with the coefficient of variations falls while coordination number grows (Fig. 10). This is also confirmed by the decreasing average velocity and geometric dispersivity (4.5). At the same
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Relative concentration
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
3.5
Pore volume injected Fig. 8 Concentration BTCs measured at points from 1.3 cm to 5.3 cm with 0.5 cm interval (dashed lines) and solutions of theoretical ADEs (solid lines) with coefficients obtained from Eqs. (4.4), (4.8), and (4.5). Mean network coordination number is 15
0.012
Standard deviation
0.01
0.008
0.006
0.004
0.002
0
4
6
8
10
12
14
16
Mean network coordination number Fig. 9 Calculated standard deviation shows the difference between concentration data and corresponding ADE with theoretical coefficients
time skewness increases which affects the rate of convergence to the central limit theorem and leads to a stronger effect of anomalous transport. On Fig. 10, variance and skewness of the velocity distributions are scaled by the mean value since it gives better representation for properties of distributions in terms of porous networks characteristics.
6 Conclusions We have investigated the role of topology on dispersion characteristics derived from 3D pore-network models. Using the algorithm introduced by Raoof and Hassanizadeh (2010)
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Network Topology on Dispersivity x 10 −3
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1100 1000 900 800 700
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Mean network coordination number Fig. 10 Mean, scaled variance, and scaled skewness of velocities distribution
to construct a suite of synthetic replica of real porous structures, we have explored the role of mean network coordination number. The synthetic porous structures include long pore throats with significant Taylor dispersion. We calculated the theoretical characteristics of the networks based on statistical properties and observed good correspondence when comparing to numerical simulations. Using dispersion as a combination of several physical and statistical processes, as confirmed by the experiments, improves our understanding of transport in porous networks. The results show that network topology, as represented by mean coordination number, has a significant influence on the dispersivity of the porous network. In particular, we observe that dispersivity is decreasing when the coordination number is growing, which affects the spreading of the concentration profile. For the porous structures studied, we have suggested an explicit empirical formula that describes the correlation between geometric dispersivity and coordination number for pore-network models. Finally, we also observed the appearance of anomalous transport effects for high coordination numbers (z ≥ 6) as evidenced by long tails of concentration curves, emphasizing the need for more advanced models to describe transport in highly connected media.
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