A transport phase diagram for pore-level correlated porous media

Advances in Water Resources 92 (2016) 23–29

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A transport phase diagram for pore-level correlated porous media M. Babaei, V. Joekar-Niasar∗ School of Chemical Engineering and Analytical Science, University of Manchester, M13 9PL Manchester, UK

a r t i c l e

i n f o

Article history: Received 24 August 2015 Revised 21 March 2016 Accepted 25 March 2016 Available online 30 March 2016 JEL classification: 100: subsurface hydrology 100.040: pore-scale modelling 100.070: single-phase flow and transport 100.020: numerical model solution approaches Keywords: Transport phase diagram Pore-level correlation Pore-network modelling Advection Dispersion

a b s t r a c t Transport in porous media is often characterized by the advection–dispersion equation, with the dispersion coefficient as the most important parameter that links the hydrodynamics to the transport processes. Morphological properties of any porous medium, such as pore size distribution, network topology, and correlation length control transport. In this study we explore the impact of correlation length on transport regime using pore-network modelling. Earlier direct simulation studies of dispersion in carbonate and sandstone rocks showed larger dispersion compared to granular homogenous sandpacks. However, in these studies, isolation of the impact of correlation length on transport regime was not possible due to the fundamentally different pore morphologies and pore-size distributions. Against this limitation, we simulate advection–dispersion transport for a wide range of Péclet numbers in unstructured irregular networks with “different” correlation lengths but “identical” pore size distributions and pore morphologies. Our simulation results show an increase in the magnitudes of the estimated dispersion coefficients in correlated networks compared to uncorrelated ones in the advection-controlled regime. The range of the Péclet numbers which dictate mixed advection–diffusion regime considerably reduces in the correlated networks. The findings emphasize the critical role of correlation length which is depicted in a conceptual transport phase diagram and the importance of accounting for the micro-scale correlation lengths into predictive stochastic pore-scale modelling. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Geological media have structure on all length scales, and the scale of a few pores is no exception (Lindquist et al., 20 0 0; Blunt, 2001). For example, based on micro CT imaging Yao et al. (1997) reported that the red sandstone from Vosges Mountains in France exhibits correlation lengths of 17.8 μm, 17.5 μm and 20.6 μm in three orthogonal directions for the porosity measurements variations. For the same sandstone Mees et al. (2003) reported an average correlation length of 27.8 μm. Regarding this inherent nature, if one desires to quantitatively reproduce the trends of macroscopic experimental data, the pore structure characteristics must be accounted for and realistically embodied in the models. Numerous image analysis studies on natural porous media (see e.g., Wardlaw et al. 1987; Bryant et al., 1993a, 1993b; Knackstedt et al., 1998, 2001; Lindquist et al., 20 0 0; Al-Raoush and Willson, 2005) infer the message that spatial correlations exist at the pore scale between pore body and pore throat sizes. Furthermore, the importance of incorporating the spatial correlations at microscale

∗

Corresponding author. Tel.: +44 1613064867. E-mail address: [email protected] (V. Joekar-Niasar).

http://dx.doi.org/10.1016/j.advwatres.2016.03.014 0309-1708/© 2016 Elsevier Ltd. All rights reserved.

and their key impact on macroscopic flow properties have been pointed out in a large number of works (see e.g., Jerauld and Salter, 1990; Renault, 1991; Ferrand and Celia, 1992; Ferrand et al., 1994; Blunt, 1997; Rajaram et al., 1997; Knackstedt et al., 2001; Jang et al., 2011). First we briefly review the literature focused on the effects of correlated pore scale heterogeneity on transport properties and next we present the scope of this work. 1.1. Effects of spatial correlation on transport Transport in porous media is characterized by dispersion coefficient (DL ) for different pore velocities. Péclet number (Pe), which is the ratio of advective to diffusive transport, is utilized to identify the transport regime in a porous medium. Péclet number by definition is defined as Pe = vL/D, where L is the characteristic length scale, v denotes the pore velocity and D is the diffusion coefficient. Typically three different regimes are defined based on the trend between dispersion coefficient and Péclet number: (a) a diffusion-dominated regime for low Péclet numbers (Pe < 0.3), where the relationship DL /Dm = 1/F φ holds (Sahimi, 1993), F is the formation resistivity factor and φ is the porosity of the porous medium;

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(b) an advection-dominated regime for large Péclet numbers Pe > Pecrit (representing the so-called mechanical dispersion). For this regime, we have DL /Dm ∼ Pe, where Pecrit (200– 40 0 0) depends on porous medium structure and increases with increasing heterogeneity (Bijeljic and Blunt, 2006); (c) a mixed diffusion–advection regime in between the two regimes (1 < Pe < Pecrit ), an approximate power-law regime with a supra-linear dependency as DL /Dm ∼ Peβ , holds for this regime. The power-law regime happens when there is an interplay between diffusion and advection. In this regime compared to the other two regimes, the relation between the dispersion and Péclet number is more sensitive to the morphological features of the pore space and heterogeneity. It is interesting to investigate the effects of geostatistical parameters of pore network on dispersion and transport regimes. The correlation length in pore size distribution is one of several geostatistical parameters of pore network models. The representation of spatially correlated fields by pore scale models to study macroscopic dispersion has been the focus of several studies. Using fractally distributed pore space with long range correlations, referred to as disordered porous media, Sahimi (1993) showed that large scale fluctuations in the velocity field due to the spatial variations of the permeability field is the principal cause of dispersion. He also argued that in order to have predictive models of porous media, the permeability of the throats should be assigned in a correlated pattern compatible in range and type with the available experimental data. Using fractional Brownian motion to represent macroscopic long-range correlations in permeability of rocks, Sahimi (1995) demonstrated that the interplay between the correlated nature of the structure of the media and the transport process can give rise to a rich variety of macroscopic transport regimes absent from uncorrelated systems. Numerous studies emphasized the impact of spatial correlations of “macroscopic parameters” on transport coefficients and solute transport (see e.g., Tsang and Neretnieks, 1998; Moreno and Tsang, 1994; Vogel, 20 0 0; Dentz et al., 2002; Knudby and Carrera, 2005, 2006; Renard and Allard, 2013). A few works due to Bernabé and Bruderer (1998), Makse et al. (20 0 0), Bruderer and Bernabé (2001), Bruderer-Weng et al. (2004) and Le Borgne et al. (2011) are dedicated to specifically target the effect of “pore-level correlations” on transport regimes. Bernabé and Bruderer (1998) investigated the influence of pore size variance on the transport in the 2D networks of cylindrical tubes inclined at 45° to the nominal flow direction. Bruderer-Weng et al. (2004) showed that in statistically isotropic, heterogeneous networks, flow channelling intensifies when normalized standard deviation and correlation length of the radii distribution of the pore throats are increased. Intensification of flow channelling increases the dispersion coefficient. Again for 2D models of pore networks, le Borgne et al. (2011) showed that correlation of the spatial velocity should be explicitly included to represent incomplete mixing in the pore throats. 1.2. The present work In spite of numerous pore-scale studies elucidating the effects of correlation length on transport properties, a systematic study using “3D, correlated, irregular and unstructured pore-network modelling” within the context of advective-dispersive transport with clearly defined correlation lengths is missing. Here, we isolate the effect of pore-scale correlation length on the macroscopic longitudinal dispersion coefficient (DL ) and identify transport regimes for different correlation lengths. As discussed in the previous subsection, Bruderer and Bernabé (2001) and Bruderer-Weng (2004) had

previously used 2D pore network modelling to exactly see the effect of heterogeneity and correlation length of pore networks on DL . However the first work was limited to coefficient of variation of the pore radii distribution and the second work did not study the effect of correlated networks for varying Péclet number regimes. Furthermore, the correlation structure in porosity and velocity has been studied at the pore-scale on micro-CT images of sandstones and carbonates (e.g., Bijeljic et al., 2013a,b; Porta et al., 2015), but as discussed correlation length was an integral part of the microCT images and could not be independently varied. To focus only on effects of correlation length, we generate networks with identical topology that resemble statistical and topological properties of natural systems. We also employ identical pore size distributions. Only the spatial distributions of pore sizes are assigned differently so as to have various correlation lengths in the networks. We use pore-network modelling because it allows us to simulate large domains at affordable computational costs. The computational efficiency in the proposed methodology and porescale analyses is superior to the direct-image simulation, and this is an important factor because multiple simulation scenarios and realizations for different correlation lengths may be required before a good characterization of transport is presented. In the following sections, first we present the network generation algorithm, and upscaling of the results from pore to REV (representative elementary volume) scale to obtain the macroscopic dispersion coefficient. Then, we present the results and discuss the effect of correlation length on the longitudinal dispersion at different Péclet numbers and we propose a conceptual transport phase diagram that shows variation of transport regime versus the correlation length and the Péclet number. 2. Pore network generation and simulation The first step in pore network modelling is to construct the topology of the network. We use Delaunay triangulation, Voronoi tessellation and a processing step to trim down the network from the excessive long throats such that a realistic mean coordination number is generated. For all simulations, the topology of the correlated and uncorrelated networks is identical, i.e. the pore body locations and the local coordination number assigned to each pore body are identical in all network realizations. The only difference between the networks is the spatial distribution of pore sizes. The correlated network takes a spatially correlated distribution of pore body sizes, whereas the pore bodies in the uncorrelated network are attributed with random values for pore radii chosen from exactly the same distribution of the correlated network. This algorithm allows us to conduct the direct comparison between the outcomes of the two networks based solely on the magnitude of pore size spatial correlation. An approach similar to the methodology developed by the authors in Leng (2013) is used here. The network generation is conducted in four steps: Step 1: We randomly populate the space with pore body centre points so that no two points are closer to each other than a minimum threshold value assigned by an input. Step 2: We generate the correlated fields for the pore body radii using the field generator developed by Nowak et al. (2008). We map the points generated in Step 1 to the field generated in Step 2 to assign the pore body radii. Nowak et al. (2008) used fast Fourier transform based on the spectral density estimation to estimate the spectral density function from a random autocorrelated field. We use anisotropic exponential variogram, γ (h ) = σY2 (1 − exp(−|h| )), for a second-order stationary field of Y = ln R, which R stands for the pore radii, σY2 is the variance of Y, and h[−−] is the (anisotropic) effective separation distance scaled by the correlation  length scales λi [L], i = x, y, z, such that h = ( 3i=1 h2i /λ2i )0.5 , and

M. Babaei, V. Joekar-Niasar / Advances in Water Resources 92 (2016) 23–29

hi [L] is the separation vector component in direction i. The mean of Y, μY , is added to the fast Fourier transform of the random autocorrelated field generated from γ (h) that has a zero mean. It is desirable to specify the mean and variance of the pore radii as the statistical parameters of the targeted distribution of the pore radii that we wish to generate. For this purpose, we use the relationship between the mean and variance of R (σY2 and μY ) and the mean and variance of the natural logarithm of R (σY2 and μY ):



μY = ln 

σ +μ 2 R

 



μ2R 2 R

,

σ = 2 Y

 σR2 ln +1 μ2R

(1)

qi j = Ki j Pi j ,

Ki j =

π ri4j 8 μl i j

(2)

where Kij is the conductivity of the throat connecting pore body i to j, μ is the dynamic viscosity, and lij and rij denote length and radius of the pore throat, respectively. Writing the conservation of mass for an incompressible flow for each pore body i, we can write:



qi j = 0

(3)

j∈Ni

where Ni is the set of pore bodies connected to pore body i. Setting up a linear system of equation (K P = B) where the coefficient matrix, K is composed of Kij , a symmetric sparse diagonal matrix is constructed with the dimension of n × n, where n is the total number of pore bodies, P is the pressure vector which is unknown and B is the right hand side vector which is all zero except for the pores located at the inlet where the constant positive pressure is assigned. The outlet pressure is set to zero. Solving this linear system of equations, the pressure field at each pore body will be known and the average velocity at each pore throat can be calculated as:

v¯ i j =

ri2j

8 μl i j

Pi j

pe =

Dm

where Dm [L2 T−1 ] is the molecular diffusion coefficient.



Dei jf f = Dm 1 + κ pei j 2

Vi

(6)

C − Ci dCi = qi jCi + qi j C j + π Dei jf f ri2j j dt li j qi j >0

(7)

j∈Ni

This equation is solved explicitly after solving the pressure field. Initially the network is filled with water without solute. The solute is continuously injected from the top boundary of the network at constant concentration equal to 1. 3. Data analysis and upscaling Among different approaches to calculate the dispersion coefficient such as the method of moments (Jury et al., 1990) and continuous time random walk (CTRW) (Bijeljic et al., 2004), we propose fitting the analytical solution of the one-dimensional Advection–Dispersion Equation (ADE), given by Ogata and Banks (1961), to the volume-averaged concentration profiles resulted from the pore network simulations. This method or similar method have been used to evaluate the longitudinal dispersion coefficient in laboratory and field studies (e.g., Coats et al., 1964; Toride et al., 1995) and in pore network modelling (Acharya et al. 2007; Zaretskiy et al. 2010; Kӧhne et al. 2011; Oostrom et al., 2014). Acharya et al. (2007) used both methods of moments and a fitting procedure to upscale non-reactive tracer particles motion in a lattice network model. They condensed their results in Acharya et al. (2007, Fig. 4) arguing that only for Pe > 104 the method of moments is more reliable than the 1D ADE fitting and for smaller Péclet numbers the two methods produced similar results. Zaretskiy et al. (2010) used the same Ogata–Banks analytical solution of a 1D ADE to estimate the dispersion coefficient in a reconstructed 3D porous medium of 1.5 × 1.5 × 1.5 mm3 size. Finally, Kӧhne et al. (2011) used CXTFIT (Toride et al., 1995) to obtain transport parameters by fitting the ADE to both the experimental and the pore network modelling-predicted breakthrough curves. We follow a similar approach to estimate the dispersion coefficient. The following properties are estimated based on averaging: (1) Average concentration: To obtain the upscaled resident concentration Cˆ, we average the concentration over the whole network, weighted by the volumes of pores :

 ViCi Cˆ =  Vi

(8)

where Vi and Ci are the pore volumes and concentrations of pore i, respectively. (2) Average pore Péclet number: We calculate an average Péclet number, Pe, as the average of the local Péclet numbers over nth number of throats and the molecular diffusion Dm :



Pe =< pe >=

(5)



where κ is a correction factor analytically calculated for a cylindrical capillary tube equal to κ = 1/192. Finally, we can write the mass balance equation for the pore body i with the volume of Vi in molar units as:

(4)

Transport inside the pore network is controlled by the advection and diffusion inside the capillary tube. If the pore throat is sufficiently straight and long (rij <
v¯ i j ri j

As a result of parabolic velocity profile inside the capillary tube, following the Taylor–Aris dispersion, the effective diffusion coefficient will be given by:

qi j <0

Step 3: We generate the network connecting the points (pore bodies) using Delaunay triangulation. The vertices of the triangulations are the positions of pore bodies and the edges are the pore throats. Since Delaunay triangulation generates unrealistically high coordination number that cannot represent natural porous media (i.e. sandpacks or sandstones with average coordination number of 4), we simply identify and eliminate the unrealistically long pore throats. Step 4: Using the depth-first search algorithm (Gross and Yellen, 2004), the percolating network is identified and labelled, and the isolated pores are removed. We solve the advection–dispersion in the network under given conditions and assumptions: (1) the flow is laminar where the Hagen–Poiseuille equation is valid, (2) the flow is steady-state single phase, (3) pore throats have cylindrical shape and pore bodies are spherical, (4) compared to the pore throats and their radii, pore bodies’ resistance against the flow is negligible, and (5) complete mixing occurs in the pore bodies. For a cylindrical capillary tube, using the Hagen–Poiseuille flow, flow rate (qij ) from pore body i to j can be written as:

25

vi ri nth Dm

(9)

where vi and ri are the pore throat average velocity and radius of throat i, respectively. The pore velocity at each pore throat is calculated using Hagen–Poiseuille equation:

vi =

ri2 Pi 8 μl i

(10)

where li is the length pore throat i, and Pi denotes the pressure drop over the pore throat i.

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Fig. 1. The pore networks for different correlation lengths of (a) λx, y, z = 1 μm, (b) λx, y, z = 10 μm, (c) λx, y, z = 20 μm, and (d) λx, y, z = 30 μm. Note that all networks have the same topology and pore size distribution.

(3) Since the average resident concentration with time is known, by fitting the analytical solution of one-dimensional advection–dispersion equation with a continuous injection boundary condition to the pore-network simulation results, the average longitudinal dispersion coefficient and average pore velocity are estimated. The analytical solution provides spatial and temporal distribution of concentration in a one dimensional system. In our study we fit the analytical solution to the concentration time series calculated by averaging the concentration over the whole network at each time step.

4. Results To be able to use curve fitting to calculate dispersion coefficient, we need to make sure that network is large enough to be REV. Therefore, we calculate porosity and permeability of the networks with similar geostatistical parameters but different sizes. The results show that for the largest correlation porosity and permeability will not change with size, for networks having more than 43,0 0 0 pores. We selected a large network with almost 87,0 0 0 pores to make sure that this size can represent an REV. We generate pore network models of Lx = 70 0 0 μm (along the flow), Ly = 4600 μm and Lz = 4600 μm in dimension. The number of pore bodies in the space is 87,750, with the minimum distance between each pair of points (body centres) being 100 μm. This allows a maximum pore radius of 50 μm. The average coordination number of the generated network is 5.5. After randomly populating the space with pores, the mean of the distances between the centres of the pore bodies is 119 μm. This distance is often used as the characteristic length of the pore network models (Bijeljic et al., 2004) as opposed to the grain size in the experimental works or the ratio of the volume to pore–grain area ratio used in direct image simulations. The value obtained in our model is in good agreement with Berea sandstone with the characteristic length of 131 μm and also with the measurement range of 100–

150 μm obtained in modelling of Øren and Bakke (2003), Bijeljic et al. (2004) and Mostaghimi et al. (2012). Based on the values of correlation length for sandstone rocks discussed in the literature, we generate pore size distributions with λx, y, z = 1, 10, 20 and 30 μm, providing a range from an uncorrelated to a strongly correlated fields of pore radii. In all the models we have μR = 16.65 μm, σ R = 9.57 μm and R ≤ 50 μm. The 3D representations of the four networks for different correlation lengths are shown in Fig. 1. The difference in spatial distributions of pores over the network for the same statistical distributions is visually clear when one compares an uncorrelated network, Fig. 1(a), with correlated networks, as shown in Fig. 1(b)–(d). The advective–dispersive transport of the solute is simulated for each pore network model at increasing global pressure gradients of p = 1, 5, 10, 50, 100, 500, …, 5 × 106 pa. Having Lx = 7×10−3 m, μ = 0.001 cp, Dm = 10−9 m2 /s, and calculating absolute permeability in order of 0.005 × 10−12 m2 for λx, y, z = 1 μm network, 0.009 × 10−12 m2 for λx, y, z = 10 μm network, 0.011 × 10−12 m2 for λx, y, z = 20 μm network, and 0.013 × 10−12 m2 for λx, y, z = 30 μm network, a range of average pore Péclet numbers from ≈10−1.7 to ≈104.2 is obtained by Eq. (9). The range requires 14 simulations for each network. Only for illustration purposes we have simulated higher average pore Péclet numbers of up to ≈105 . Fig. 2 shows snapshots of the simulations for networks with λx, y, z = 1 μm and λx, y, z = 30 μm for Pe ≈ 103 and Pe ≈ 105 , for the same average concentration values. As it can be clearly seen by comparing Fig. 2(a) with (b) for the same uncorrelated network (λx, y, z = 1 μm) with increase of injection rate, the advection of solute becomes stronger and the concentration front destabilizes, which initiates the microfingering of solutes. However, this effect for the same flow rate has been already happening at the lower Péclet number for Fig. 2(c) compared to Fig. 2(d) for the same correlated network (λx, y, z = 30 μm). This means that in correlated systems, the transition from fully diffusion-controlled regimes to advection controlled regime happens at lower Péclet numbers. In other words, in uncorrelated

M. Babaei, V. Joekar-Niasar / Advances in Water Resources 92 (2016) 23–29

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Fig. 2. The concentration profiles of the simulation at the time of the simulation that Cˆ/C0 is approximately 0.5 in each system (a) λx, y, z = 1 μm, Pe ≈ 103 , (b) λx, y, z = 1 μm, Pe ≈ 105 , (c) λx, y, z = 30 μm, Pe ≈ 103 , and (d) λx, y, z = 30 μm, Pe ≈ 105 .

fields as there is no distinct fast and slow flow paths, the advective transport can be damped by the randomized flow field at lower Péclet numbers. Our analysis for the range of average pore Péclet numbers from ≈10−1.7 to ≈104.2 shows good match between the 1D ADE and averaged concentration results for λx, y, z = 1, 10, and 20 μm (see Supplementary information). However, for the case of λx, y, z = 30 μm the fitting result is not satisfactory. This is due to the limited physical dimension of the network compared to the correlation length that leads to a non-Fickian temporal concentration profile. Consequently we are leaving out λx, y, z = 30 μm from our further analysis. The values of the longitudinal dispersion coefficients with respect to the increasing Péclet numbers are shown in Fig. 3. Also shown are the experimental data for unconsolidated sandpacks and beadpacks by Pfannkuch (1962), Seymour and Callaghan (1997), Kandhai et al. (2002), Khrapitchev and Callaghan (2003), and Stöhr (2003). We have also compared our results with the direct image simulation of Bijeljic et al. (2011) for carbonate rocks and the pore network modelling of Bruderer and Bernabé (2001). The pore network modelling of Bruderer and Bernabé (2001) are comparably limited in the range of Péclet numbers used in the study. Also because the authors have defined the dimensionless parameter σ R /μR without discriminating between the variably correlated networks, all of our networks would have had the same DL /Dm − −Pe relationship, which is a wrong conclusion based on our findings. The following points can be read from the figure. 1. The results of the uncorrelated network simulations have been satisfactorily validated against experimental data for unconsolidated sandpack and beadpack. 2. Whereas we observe a clear power law region for the uncorrelated case for log(Pe) < 0.5 (supra-linear region above 45°), the values of the DL /Dm for the three correlated cases exhibit a transition from diffusion-dominated system to the

Fig. 3. The longitudinal dispersion coefficient scaled by molecular diffusion versus Péclet number for various cases of correlation lengths, and compared with experimental data for unconsolidated sandpacks and beadpacks (Pfannkuch, 1962; Seymour and Callaghan, 1997; Kandhai et al., 2002; Khrapitchev and Callaghan, 2003; Stöhr, 2003), the dispersion coefficients of the sandpack and carbonate rocks obtained by direct image simulation (Bijeljic et al., 2011), and pore network modelling of heterogeneously distributed network (Bruderer and Bernabé, 2001) with specific coefficient of variation of pore throat radius of 0.6 corresponding to networks of this study.

advection-dominated system that spans over only a short range of Péclet numbers. For log(Pe ) > −0.5 in the correlated networks, a linear relationship more closely describes the trend of log(Pe)–DL /Dm than a power law relationship.

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In practice for enhanced oil recovery techniques in petroleum engineering and for clean-up technologies in soil remediation engineering, a correct estimate of dispersion and resident time is essential, as they influence the economy of projects as well as the efficiency of the technologies. The transport phase diagram infers the important message that characterization of the rocks and special core analysis (SCAL) should not be only restricted to statistical properties of the rocks and important information about the geostatistical properties (e.g. correlation length) should be estimated. Based on SCAL and micro-CT image processing, it is possible to evaluate the correlation length of different rock materials. Better understanding of the correlation length will definitely improve the predictive modelling of chemical transport in subsurface and efficient designs of chemical flooding and clean-up plans where resident time of chemicals and their spreading are important. 5. Conclusions Fig. 4. A transport phase diagram illustrating the effects of the correlation length on the transport regimes, namely diffusion-controlled, transitional mixed advection–diffusion, and advection-controlled. The crosses are the approximate locations of the bound of the power law region for one uncorrelated and 5 correlated networks considered in this study.

This earlier achieved linearity, which has also been discussed in the description of Fig. 2, is attributed to the contribution of localized velocity variations in the correlated systems that enhance the dispersion even for the low Péclet number regimes. This has been observed in direct simulation of dispersion in carbonate rocks (Bijeljic et al. 2013a, Fig. 4). 3. For a given Péclet number larger than one, the values of DL /Dm for the correlated cases are around an order of magnitude larger than the uncorrelated case. Changing the correlation length from 1 μm to 10 μm makes a large impact on dispersion. It is important to examine the pore velocity variation with respect to increasing the Péclet number and the spatial correlation length. We calculate the coefficient of variation of the pore velocity, i.e., the ratio of the standard deviation of the pore velocity over the mean of pore velocity, CVv = σ (v )/v. Our analysis shows that for a given correlation length, CVv remains constant for all the different Péclet numbers used in the paper. For different correlation lengths following results have been obtained: CVv for λx, y, z = 1 μm is 1.05, and CVv for λx, y, z = 10 μm, 20 μm are 1.26 and 1.34, respectively. These numbers show a regime-independent increase in the coefficient of variation from uncorrelated network to correlated networks. This regime-independent increase in velocity variation can explain the enhanced dispersion in the correlated pore network models compared to the uncorrelated cases for the power-law region. To establish a relationship between the width of the power law region (in terms of Pe number values) with respect to the correlation length of the pore networks, we have simulated two more networks with λx, y, z = 5 μm and λx, y, z = 7 μm. As shown in detail in the Supplementary information, the width of the power law region shrinks with the increase of correlation length. To summarize our findings on the effects of correlation length on the transport regimes, we use a conceptual transport phase diagram. Assisted by the superposition of approximate locations of the transition zones between the diffusion controlled and advection controlled regimes in Pe − λx,y,x space, Fig. 4 shows that by increase of correlation length, the transition regime (mixed advection–diffusion) narrows down. The figure infers this message that for a given domain size, the Péclet number magnitude required to trigger an advection dominant regime is reduced for large correlation lengths.

This study focused on the impact of micro-scale correlation length on the single phase transport regimes and thereby derivation of an advection–dispersion transport phase diagram. Unstructured irregular networks with identical topologies and pore-size distributions but different correlation lengths have been generated. After solving the pore-scale advection–dispersion equation for different Péclet numbers in such networks and averaging the results, effects of Péclet number and correction length on dispersion were studied. The comparison between correlated and uncorrelated results showed three major features: (a) With increase of correlation length, the range of transition (mixed advection–dispersion) regime is getting smaller and less sensitive to the Péclet number. This suggests that for a given domain size, with increase in the correlation length the transport regimes can shift from a trimodal system to an almost bimodal system where only diffusion-controlled and advection-controlled regimes are distinguishable. (b) An increase of coefficient of variation of pore-scale velocity distributions can be seen from an uncorrelated network to a correlated one. This in turn leads to an order of magnitude increase of the longitudinal dispersion coefficient from an uncorrelated system compared to a correlated one. (c) Although the correlation length and tortuosity of porous media influence the effective diffusion coefficients, the variation of the diffusion-controlled regime is not so much influenced by the correlation length compared to the advectioncontrolled regime. Acknowledgement The authors thank the anonymous reviewers for their invaluable and constructive comments. Supplementary Materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.advwatres.2016.03. 014. References Acharya, R.C., van Dijke, M.I.J., Sorbie, K.S., van der Zee, S.E.A.T.M, Leijnse, A., 2007. Quantification of longitudinal dispersion by upscaling Brownian motion of tracer displacement in a 3D pore-scale network model. Adv. Water Res. 30 (2), 199–213. Al-Raoush, R., Willson, C., 2005. Extraction of physically realistic pore network properties from three-dimensional synchrotron X-ray microtomography images of unconsolidated porous media systems. J. Hydrol. 300 (1), 44–64.

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