Transp Porous Med (2015) 109:433–453 DOI 10.1007/s11242-015-0527-4

Scaling Invariant Effects on the Permeability of Fractal Porous Media Y. Jin1,2 · Y. B. Zhu1,2 · X. Li1,2 · J. L. Zheng1,2 · J. B. Dong1,2

Received: 25 November 2014 / Accepted: 3 June 2015 / Published online: 17 June 2015 © Springer Science+Business Media Dordrecht 2015

Abstract Porous media are interconnected systems, in which the distribution of pore sizes might follow scaling invariant property and will affect the fluid flow through it significantly. Thus, except for a detailed understanding of the fundamental mechanism at pore scale, hard is it to determine the appropriate relationship between the permeability and the basic properties. In this study, in terms of the size distribution and spatial arrangement of the pores, we analytically derived a permeability model using series–parallel flow resistance mode firstly. And then, together with the scaling invariant characteristics of the porosity, specific area and hydraulic tortuosity, the analytical permeability model is reformulated into a fractal permeability–pore structure relationship. The results indicate that: (1) the square of the porosity (ϕ) is proportional to the permeability in a fractal porous media, not the cubic law described in Kozeny–Carman (KC) equation; (2) the hydraulic tortuosity is a power law model of the minimum particle size with the exponent (Df − d), where Df and d are the pore size fractal dimension and Euclidean space dimension, respectively, while  is a parameter characterizing the spatial arrangement of pores; (3) the KC numerical prefactor is not a constant in fractal porous media. Its value, however, increases linearly with the size ratio of the minimum to the maximum pores but decreases exponentially with Df . More importantly, it is found to be a parameter characterizing the difference of fluid flow in porous media from that in a straight tube described by the Poiseuille law. The performance of the new fractal permeability–pore model is verified by lattice Boltzmann simulations, and the numerical prefactor universality is examined as well.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 41102093, 41472128), and CBM Union Foundation of Shanxi Province of China (Grant No. 2012012002).

B

Y. Jin [email protected]

1

School of Resource and Environment, Henan Polytechnic University, Jiaozuo 454003, China

2

Collaborative Innovation Center of Coalbed Methane and Shale Gas for Central Plains Economic Region, Henan Province, Jiaozuo 454003, China

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Keywords Fractal porous media · Tortuosity–porosity model · Lattice Boltzmann method · Kozeny–Carman constant · Permeability–pore model

1 Introduction Investigation of fluid flow in porous media is of paramount importance in different areas. As one of the basic transport properties, permeability describes how easily the flow passes through the porous media (Jin et al. 2013; Matyka et al. 2008; Costa 2006), and is a dependent quantity on some fundamental and well-defined parameters determined solely by the geometry of porous media, such as the porosity, the specific surface area, the hydraulic tortuosity and the shape factor. In the past, a lot of semiempirical relationships between the permeability and the structure parameters were proposed experimentally or theoretically (Collins 1961; Duda et al. 2011; Bear 1972; Dullien 1991), among them the modified KC equation (Carman 1937, 1939; Kozeny 1927) should be the most well-known one for solid particles packing bed. Natural porous media microstructure might be disordered and complicated, with pores/particles distributed scaling invariantly (Adler and Thovert 1998; Smidt and Monro 1998; Thovert et al. 1990; Young and Crawford 1991; Krohn and Thompson 1986). Substantial difference has been observed in the measured permeability even when porous media share the same statistical quantity of fundamental properties (such as porosity), and the so-called KC constant in the modified KC equation is actually an empirical parameter, which will alter with pore structures (Costa 2006; Ahmadi et al. 2011; Cai et al. 2010; Panda and Lake 1994; Rahli et al. 1997; Xu and Yu 2008). Thus, one is hard to determine the appropriate relationship between the permeability and the basic properties because of the large number of related parameters, except for a detailed understanding of size distribution and spatial arrangement of pores and particles in porous media (Costa 2006). To improve the estimation accuracy, various approaches were employed to investigate the permeability–pore structure (short by permeability–pore later) relationship, approximately catalogued into three types: experiment-based analysis, analytical derivations and numerical simulations. The experimental results are usually influenced by testing techniques, scales, experimental environments, etc. Meanwhile, owing to the potential continuous assumptions underlying in experiments, the microstructural effects on fluid flows may be thus ignored (Costa 2006; Yu et al. 2003). That will lead to some unexplained items, always called semiempirical parameters. Following the analytical approaches, a mathematical framework can be established with clear physical meanings for a problem at hand by some simplifications, guiding us to adapt the inherent coefficients to suit experiments (Ahmadi et al. 2011). Pitchumani and Ramakrishnan (1999) proposed a permeability model for fractal porous media by assuming them as fractal tube bundles other than the idealized arrangements of tube bundles with same length (Kozeny 1927; Happel 1959; Sangani and Acrivos 1982). However, the model presented in Pitchumani and Ramakrishnan (1999) presents unreasonable results, interested readers may consult the comments by Yu (2001) for detail. And then, Yu et al. (2003, 2005) analytically derived a permeability–porosity relationship, assuming that the size distribution of pores follows fractal statistics as well as that the relationship between the diameter and the length of capillary tubes admits a fractal scaling law (Wheatcraft and Tyler 1988). Later, these authors developed a new form of permeability and KC constant, and demonstrated how KC constant alters with the range of pore size (Xu and Yu 2008). Other than the accumulation method based on the

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nonintersect assumption of the capillary tubes, another idea is to reformulate the classical KC equation directly, by inserting the scaling invariant relationships between fundamental parameters and the microstructure of porous media with special settings (Sahimi 1993; Rawls et al. 1993; Guarracino 2007; Nasta et al. 2013). Accordingly, Costa (2006) proposed a twoparameter permeability–porosity equation, and then Henderson et al. (2010) analytically derived a three-parameter permeability model. Jin et al. (2013) found that the permeability is a function of the size of the largest pore, the porosity, the pore size fractal dimension (Df ) and a semiempirical parameter. Based on fractal geometry and Poiseuille’s law, Wang et al. (2014) analytically derived a semiempirical fractal permeability model, indicating that the porosity to the power of (4 − Df )/(2 − Df ) is proportional to the permeability of a twodimensional porous medium. Obviously, direct investigations are more reliable than those obtained on the assumption of the capillary tubes apart from each other (Jin et al. 2013; Costa 2006; Wang et al. 2014). Except for the solutions mentioned above, more and more efforts are now devoted to direct numerical simulations because of their advantages in understanding the basic physics of a certain problem (Croce et al. 2007). In numerical experiments, one can easily select or neglect any relevant effects, and reduce the uncertainty from coupling effects as far as possible. Moreover, the computational fluid dynamics (CFD) models are not subjected to any experimental techniques, scales or environments. Among these CFD models, the newly developed lattice Boltzmann method (LBM) has drawn broad and special attention (Chen and Doolen 1998; Kandhai et al. 1999; Koponen et al. 1997; Ladd 1994; Succi 2001), and has been proven to be a powerful tool to explore the controlling mechanism of complex flow problems (Nithiarasu and Ravindran 1998; Degruyter et al. 2010; Vita et al. 2012). However, different relationships can be derived from the same numerical results if the background guiding models are different. Thus, to understand the complex fluid flow process in porous media mechanistically, we need to establish a mathematical framework in advance, and then investigate the basic physics by numerical simulations to reduce the uncertainty in it. Based on the analytical–numerical coupling solution, here we first review some classical permeability equations. Under the hypothesis of fractal pore-space geometry and using series–parallel flow resistance mode, we then analytically derive a permeability estimation model with clear physical background. With the help of LBM, we investigate the basic physics of the geometry effects on the fluid flow. Finally, we establish a permeability–pore relationship for fractal porous media, and demonstrate its validity via comparisons between the existing correlations and the numerical results.

2 Theories and Methodologies 2.1 The Fluid Flow Resistance Model of Porous Media Since the pioneer work by Kozeny (1927), great efforts have been devoted to establishing the estimation models of the permeability K of porous media. Carman (1937, 1939, 1956) modified the original Kozeny’s equation as follows: K =

ϕ3 , k(1 − ϕ)2 S 2

(1)

where ϕ is the porosity, S is the specific surface and k = Cf τ 2 is the KC constant, in which τ is the hydraulic tortuosity, and Cf is a constant shape factor dependent on the capillary pore

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shape. Assuming the porous media is deposited by monosized solid particles, substituting for S in Eq. (1) leads to K =

ϕ3 δ2 , C0 τ 2 (1 − ϕ)2

(2)

where δ is the particle size, e.g., the side length of a cubic block or the diameter of spherical particles. C0 has been considered to be a constant relative to the shape factor of the solid particles which are monosized filled in the porous media. As pointed out by Bear and Verruijt (1987), the flow rate of a porous media can be expressed as a function of its flow resistance: Q=

P , R

(3)

where Q is the total flow rate, R is the flow resistance and P denotes the pressure difference between the inlet and outlet boundary. Similar to Ohm’s law, R follows the parallel and series calculation models. Combining Darcy’s law with the definition of flow resistance [see Eq. (3)], the relationship R = μL/(AK ) is obtained, where μ is the fluid dynamic viscosity, A and L is the crosssectional area and sample length of the porous medium, respectively. By borrowing the concept of electrical resistivity, the fluid flow resistivity Rf can be defined as the quantity of flow resistance of certain porous media with unit sample length and cross-sectional area, which yields Rf = μ/K . Together with Eq. (2), the flow resistivity of porous media filled with monosized solid particles reads: Rf =

C0 (1 − ϕ)2 μ τ 2 , ϕ3 δ2

(4)

which implies that Rf is the function of τ and δ if porosity ϕ and the shape of solid particles are same.

2.2 Lattice Boltzmann Methods LBM is actually a mesoscopic description of microscopic physics, and has been used widely to study the microstructural effect on the fluid flow at pore scale. For simplicity and convenience, the model description and fluids simulations are presented using the classical lattice Bhatnagar–Gross–Krook (BGK) model over a D2Q9 lattice structure. In LBM framework, based on the principle of effective conversion between physical and lattice systems, the real pore space Dp is discretized in terms of a regular lattice with spacing δx , time t in terms of a time step δt and velocity space in terms of a small set of velocities ci to ensure that ci δt is a vector connecting two adjacent lattice sites (Succi 2001; Dünweg et al. 2007; Sukop and Thorne 2007). For the employed D2Q9 lattice structure (Qian et al. 1992), a simple-cubic lattice has a set of probability functions f i , representing the mass density of fluid particles going through one of the 9 discrete velocities ci . Thus, the local mass density ρ and velocity u at each lattice position x and time t are given by ρ(x, t) =

8  i=0

8 f i (x, t),

u(x, t) =

f i (x, t) . ρ(x, t)

i=0

(5)

By Chapman–Enskog expansion of the Boltzmann equation, the time evolution of fluid flow is described by the lattice Boltzmann equation (LBE) (Sukop and Thorne 2007), as:

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f i (x, t) + Ω i = f i (x + ci δ t , t + δ t ),

(6)

where Ω i is the collision operator modifying 8 the populations 8 at x according to the mass and momentum conservative requirements of i=0 Ω i = i=0 Ω i ci . According to BGK model, the collision operator takes the single-relaxation-time approximation (Chen and Doolen 1998; Succi 2001; Qian et al. 1992), Ωi =

δt  τlbm

 eq f i (x, t) − f i (x, t) ,

(7)

eq

where τ lbm is a dimensionless relaxation time, and f i (x, t) is a quasi-equilibrium distribution function. And then, its discrete velocities ci are defined as ⎧ i =0 ⎨ (0, 0), (cos θ, sin θ ), i = 1 : 4, θ = i−1 (8) ci = c × √ 2 π , ⎩ 2(cos θ, sin θ ) i = 5 : 8, θ = 2i−9 4 π eq

to recover the Navier–Stokes (NS) equation for the fluid flow, f i is constructed by Eq. (9), and the kinetic viscosity of the fluid (ν) is given by Eq. (10), respectively.

u · ci 9(u · ci )2 3u 2 eq , (9) − f i (x, t) = ωi ρ(x, t) 1 + 3 2 + c 2c4 2c2 ν = (τlbm − 0.5)

δ 2x . 3δ t

(10)

with c = δ x /δ t and lattice sound speed c2s = c2 /3 due to the constrains of conservation and isotropy. Equation (10) imposes a constraint on the choice of τlbm being greater than 0.5 for a physically correct viscosity (Sukop and Thorne 2007), here we choose τlbm = 1. And the weight assignments follow ω0 = 4/9, ω1−4 = 1/9, and ω5−8 = 1/36, to satisfy ωi = 1 for symmetry reasons. In the complex fractal porous media, there is almost no net fluid motion that exists (Succi 2001). The boundary condition at solid–fluid interfaces can, therefore, physically approximate the no-slip boundary condition (Jin et al. 2013; Wang et al. 2014; Chen et al. 2013). For simplicity and without loss of generality, the complete bounce-back scheme is adopted in our flow simulations.

3 Characteristics of Fractal Porous Media and Their Flow Resistance Porous media in nature always consist of numerous irregular pores of different sizes spanning several orders of magnitude in length scales, such as soil, sandstones in oil reservoir, matrix pores in coal, packed beds in chemical engineering, fabrics used in liquid composite molding and wicks in heat pipes. These media possess pore microstructure, including both pore sizes and pore interfaces, and exhibit fractal characteristics, thus called fractal porous media (Adler and Thovert 1998; Smidt and Monro 1998; Thovert et al. 1990; Krohn and Thompson 1986).

3.1 Fractal Characteristics of Porous Media In a fractal pore space, the cumulative distribution of pore size λ has been proven to follow fractal scaling law. Obviously, a natural porous medium is composed of one or more representative units of linear size L at which the fractal behavior starts. These representative

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units share the same physical properties, such as porosity, pore structure, pore size range and transport property as the porous medium in a statistical mean, but contain only one pore/solid of the largest size. And, the fractal dimension of the pore size distribution, Df , is a function of the porosity and the ratio of the lower limit to upper limit of self-similar regions (Yu and Li 2001):  r d−Df , (11) ϕ(λ ≥ r ) = L where r is the scale, d is the Euclidean space dimension, the fractal dimension Df is in the range of 0 < Df < 2 and 0 < Df < 3 in two and three dimensions in natural porous media (Yu and Li 2001), respectively. Recently, some authors indicated that Df could possess a much wide range, for more details one can refer to Ghanbarian-Alavijeh and Hunt (2012). In our previous study (Wang et al. 2014), two definitions have been proposed to describe the microscopic physical properties following the fractal theory (Mandelbrot 1983), denominated as Frequency of pore growth and Lacunarity of pore size, denoted by Fλ and Pλ , respectively. Fλ is the ratio of the pore number of size λi to that of size λi−1 , while Pλ is the ratio of λi−1 /λi , where λi and λi−1 are two successive pore sizes in a fractal porous media and λi−1 > λi . Consequently, the fractal dimension Df can be calculated by log(Fλ )/ log(Pλ ) because it is another form of the number-size relation. Meanwhile, in a fractal porous medium (for example the Sierpinski carpet or Menger sponge fractals), if the smallest pore size is r , the smallest particle size will be ξr in a pore fractal (Ghanbarian et al. 2013a), where ξ is a constant and ξ ≥ 1 (in standard Sierpinski carpet, ξ = 1). Thus, the number-size distributions of the pores and particles are power laws with identical exponent −Df in a fractal porous media (Perrier and Bird 2002). Consequently, after n times iterations, the accumulative pore–particle surface area in a representative unit of a fractal porous medium is then expressed by: As =

n 

−(i−1)

βs δid−1 N (δi ) , where N (δi ) = Fλi−1 , δi = δmax Pλ

,

(12)

i=1

where δi and N (δi ) represent the size and number of the minimum solid particles after i-time iteration, respectively. δmax is the largest size of solid particle in a representative unit, βs is the shape factor, for square solid particles βs = 2d (Mortensen et al. 2005). Obviously, the cumulative pore–particle surface As is the sum of a geometric series. Following the definitions of Pλ and Fλ , together with their description of Df , we obtain the relation δmax = δmin Pλn−1 . Instantaneously, Eq. (12) is rearranged into: d−1−Df

 1 − Pλ δδmax d−1 As (δ) = βs δ max 1 − PλDf +1−d

d−1−Df δ d−1 ≈ ζβs δ max , (13) δmax where ζ is a function of Pλ , a parameter relevant to pore structure (Wang et al. 2014). Actually, by substituting the relationships of δ = ξr and Pλ δmax /ξ = L into Eq. (13), the pore–particle surface area yields: 

d−1−Df 1 − Lr . (14) As (δ) = βs δ d−1 max 1 − PλDf +1−d

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By comparison, Eq. (14) is consistent with what proposed is in Yu et al. (2009) because these authors assumed that the size of the maximum pore λmax is equal to the linear size of the representative region/unit of the porous media, and the size of the minimum pores λmin is equal to the measuring scale r . Consequently, the specific area of fractal porous media is expressed by: As (δ) Ld

(d−1−Df )

δ 1 δmax d−1 = ζβs , L L δmax

S(δ) =

(15)

Equation (15) indicates that the pore–particle surface area admits a fractal scaling law with δ. Meanwhile, the number-size distributions of the pore–particle surface area and the solid particles are similar power laws with identical exponent because of the same scaling behaviors. Instantaneously, the fractal dimensions of pore–particle surface area, the pores and solid particles are the same in a fractal porous medium, being Df . According to the independent measuring results, some authors considered that the fractal behaviors of the pores, solid particles and pore–particle surface area in a fractal porous media are different (Perrier and Bird 2002; Perrier et al. 1999; Dathe and Thullner 2005). It must be noted that the fractal dimension is just a number by which to quantify the scaling invariant degree of a pattern in geometrical or statistical scenes, meaning that just by the fractal dimension, the geometries of fractal porous media could not be uniquely determined. Thus, to accurately describe the self-similar property of a porous media, except for the fractal dimension, the spatial or statistical pattern which scales must be accompanied, such as the generators of Menger sponge and the PSF model (Perrier et al. 1999). Otherwise, by measuring approaches ignoring the spatial arrangement of pores and particles completely, such as the box-counting method, one will obtain different fractal dimensions for the physical properties even if they possess the same scaling behaviors actually, as the results in Perrier and Bird (2002), Perrier et al. (1999), Dathe and Thullner (2005) and Zhou et al. (2010). When the porous medium is filled with monosized solid particles, there will be no fractal behavior of pore/particle sizes. In such a condition, Fλ → 0+ , thus Df will tend to be −∞ theoretically because of Df = log(Fλ )/ log(Pλ ). Obviously, this is consistent with what pointed out is in Ghanbarian-Alavijeh and Hunt (2012), in which the authors indicated that Df = −∞ represents a uniformly grain/pore size distributed porous medium. Denoting by S0 the specific area of such a medium, from Eq. (15) we get Eq. (16) instantaneously because of δ = δmax . S0 = ζβs

d−1 δmax . Ld

(16)

For a two-dimensional nonfractal porous medium, the boundary length of a solid particle d−1 , thus the specific area takes the form of Eq. (16). is equal to 4δmax , also βs δmax As aforementioned, at a measuring scale of pore size r , the minimum measurable size of solid particles δ takes the value ξr . Thus, together with Eq. (11) and with respect to the pore size, Eq. (15) is then rearranged into: S(λ ≥ r ) = S(δ) = ζβs ξ 1Df ξ d−1−Df

ϕ ϕ = Cs . r r

(17)

where ξ 1 = δmax /L, being a constant in a fractal porous media. For example, ξ1 = 1/3 in standard Sierpinski carpet.

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Fig. 1 Basic construction of the VmSqLnRl-type fractal porous media. The solid and void phases are denoted by black and white, respectively. In a space of Euclidean dimension d, the initiator (a) of linear size l defines the representative region/unit of a porous media, divided into m 2 equal parts. At the first iteration step, the generator (b) divides the m 2 parts into two sets of q 2 (solid) and m 2 − q 2 (void) subregions. At the next step, each of the void subregions is replaced by a reduced replicate of the generator

3.2 Construction of Porous Media with Arbitrary Fractal Dimension For simplicity and without loss of generality, our attention is focused on the permeability–pore relationship of fractal porous media in two-dimensional context. In our previous studies (Jin et al. 2013; Wang et al. 2014), an algorithm was proposed to construct a self-similar fractal object with arbitrary fractal behavior and without blind pores, denominated as VmSqLnRltype fractals. The modeling algorithm is briefly as following: (1) define the representative (R) region of linear size l as an initiator in a space of Euclidean dimension d (Fig. 1a). This region is then divided into m × m small subregions of linear size l/m and set to be void phase (V ); (2) solidify (S) the very central region with size q × l/m (Fig. 1b); (3) repeat step (2) in the reset void subregions for n times until the predefined porosity is achieved. The basic construction of a V5S3L2Rl-type porous media is demonstrated in Fig. 1, where Fig. 1a is an initiator or the representative region of a fractal porous media, Fig. 1b is the generator and Fig. 1c is a V5S3L2Rl-type fractal porous medium with Df = log 14/ log 5. It is noticeable that, with successive iterations, pores and solid particles keep their sizes reduced, while their number increased (Adler and Thovert 1993; Tarafdar et al. 2001). In a representative region of linear size l, λmax = l and λmin = l are satisfied at the very beginning. After one subdivision, λmin (also r ) turns into l/m, δmin = δmax = lq/m; after n times of iterations, λmin = lm −n , δmin = lq/m n . Hence, for a VmSqLnRl-type fractal porous medium, λmax and λmin are l and lm −n , respectively, while δmax = lq/m and δmin = qλmin . In these media, ξ and ξ1 are expressed by ξ = q, ξ 1 =

q , m

(18)

and the Frequency of pore growth (Fλ ) and the Lacunarity of pore size (Pλ ) are expressed by Pλ = m d ,

Fλ = m d − q d .

(19)

Thus, the fractal dimension of pore area and porosity are given by Eqs. (20) and (21), respectively.

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Fig. 2 Relationship between real specific area and that by Eq. (22) of different VmSqLnRl-type porous media, where the solid line is y = x for reference Table 1 Pore structure parameter ζ of some VmSqLn-type porous media by the best-fitted linear model between βs (q/m)d−1 (ϕ/λmin ) and the real specific area

Porous media type

d/Df

ζ

V3S1Ln

1.0566

1.6000

V4S2Ln

1.1158

1.5000

V5S3Ln

1.1611

1.4545

V7S5Ln

1.2246

1.4118

V9S3Ln

1.0275

1.1429

V11S3Ln

1.0164

1.1089

V11S5Ln

1.0507

1.1294

V11S7Ln

1.1214

1.1803

V25S3Ln

1.0023

1.0423

V45S7Ln

1.0032

1.0233

ln Fλ ln (m d − q d ) , = ln Pλ ln m

n  n λdmax − δ dmax Df −d ϕ= m = . λdmax

Df =

(20) (21)

Actually, following the PSF model (Perrier et al. 1999), Fλ is ranged in (0, m d −q d ]. When Fλ → 0+ , Df → −∞ theoretically (Ghanbarian-Alavijeh and Hunt 2012). According to Eq. (17), the specific area of VmSqLnRl-type porous media is then expressed by Eq. (22), which is validated from different VmSqLnRl-type porous media as shown in Fig. 2. S (λ ≥ λmin ) = ζβs

q d−1 ϕ . m Df λmin

(22)

Some pore structure parameter ζ are listed in Table 1. It is noted that ζ will approach to d/Df as Df increases.

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3.3 Flow Resistance of Fractal Porous Media According to Eq. (21), the porosity of a VmSqL1Rl-type porous medium is determined only by the parameters m and q, so is the hydraulic tortuosity due to the spatial arrangement of void and solid phase determined by the construction process in such type porous medium. And in a VmSqLnRl-type fractal porous medium, the space is filled with VmSqL1R(lm −n ) units and solid phase, as shown in Fig. 1. For clarity, the porosity and hydraulic tortuosity of VmSqL1Rl-type porous media are denoted by ϕ1 and τ1 , respectively. Taking them together, we can rewrite Eq. (4) into the form of Eq. (23) for certain fluid flow through VmSqL1Rl-type porous media with determined m and q. Rf = C 1

1 1 ∝ 2, δ2 δ

(23)

where ∝ stands for “proportional to,” δ = ξ 1 l for porous media of VmSqL1Rl-type, and constant C 1 reads C1 =

C0 (1 − ϕ 1 )2 μτ 21 ϕ13

.

(24)

Equation (23) implies that the spatial distribution of δ −2 satisfies the parallel and series mode. (d/i) For convenience, denote by R f the flow resistivity of the VmSqLiRl-type porous medium. Thus, when a VmSqL1Rl-type porous medium turns into the VmSqL2Rl-type after (d/1) (d/2) one iteration, its flow resistivity will change from R f to R f . According to Eq. (23), (d/1)

(d/2)

takes the value of C1 (qd/m)−2 , while R f can be calculated from Eq. (25) by the Rf parallel–series spatial integration in a two-dimensional space.

 q m−q (dm −1 )/1 (d/2) Rf + . (25) = Rf m m−q Replacing the expression (m − q)/m + q/(m − q) by f (m, q) for short, we can express the average flow resistivity of VmSqLnRl-type porous media by

n−1 (d/1) (d/n) Rf = m 2 f (m, q) Rf n−1 1

2 = C 1 m f (m, q) . (26) δ 2max Substituting δ max = q L/m into Eq. (26) yields

2 n m f (m, q) 1 (d/n) Rf = C1 2 . q f (m, q) L 2

(27)

where L is the linear size of the representative unit of a fractal porous medium, also the size of the largest pore. Although Eq. (27) is derived from the regular model, but it is applied to porous media with solid particles distributed randomly according to the parallel–series model. To explain intuitively, we demonstrate a VmSqL2Rl-type model with the first-level solid particle distributed randomly on porous media composed of VmSqL1Rlm−1 -type models, see Fig. 3. Denoting by R1 , R2 and R3 the flow resistivity of VmSqL1Rl-, VmSqL1Rl/m- and VmSqL2Rl-type model, respectively, by parallel–series mode, we can get the relationship R3 = R2 ((m − q)/m + q/(m − q)) for arbitrary a and b, same as Eq. (25) derived from

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Fig. 3 Demonstration of a VmSqL2Rl-type model with solid particle distributed randomly

regular model. So, the flow resistivity model of Eq. (27) is not only applied to regular fractal porous media, but also to the random ones on average. However, Ghanbarian-Alavijeh and Hunt (2012) have pointed out that the parallel–series approach does not model connectivity among pores, which are more descriptive rather than predictive. But we think that the randomness effect on the transport property is in the charge of the semiempirical parameter of C0 . Thus, according to the relationship between the permeability (K ) and flow resistivity (d/n) Rf and taking Eqs. (17), (21) and (27) all together, we obtain the permeability estimation model of VmSqLnRl-type porous media, reads K =

ζ 2 βs2 ϕ 1 ϕ 2 C0 S(λ)2 T 2

(28)



where T 2 = τ 21 f (m, q)n−1 is introduced as a geometrical parameter accounting for the hydraulic tortuosity of a fractal porous medium.

4 Results and Discussions As a general function, Eq. (28) should be able to describe the permeability–pore relationship for porous media without fractal behavior. In such a condition, n = 1, ϕ1 = ϕ, and S takes the value of S0 . Then, Eq. (28) is reduced to K =

=

ϕ13 C0 τ12 ϕ13



2 δmax d 2

δmax L

2 δmax

C0 τ12 (1 − ϕ1 )2

(29)

which is equal to the modified KC function defined in Eq. (2) for porous media deposited by monosized particles. To make Eq. (28) practical, the parameter T must be described by the fundamental properties contained in Eq. (2). As aforementioned, the hydraulic tortuosity depends only on the spatial arrangement of solid particles, not on their sizes. That is, there is no effect on the hydraulic tortuosity for a determined spatial pattern when it is zoomed in or out. For

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Fig. 4 Relationships between ϕ1 and τ1 , ϕ1 and τ2 . The circles represent the experimental data, and the solid line its best power law-fitted result

example, in a VmSqLnRl-type porous medium, τ will alter with parameters m, q and n, but not with the linear size of the representative unit L. Meanwhile, a certain spatial pattern leads to a determined porosity; however, for a porous medium with a determined porosity, its hydraulic tortuosity will alter with the spatial arrangement of solid phase. As pointed out in Valdés-Parada et al. (2011), the hydraulic tortuosity cannot be a function of porosity only. To explore the tortuosity–porosity relationship, we calculate τ1 of different VmSqL1-type porous media by the method in Jin et al. (2015) based on the velocity fields simulated by the LBM. Meanwhile, inspired by the construction process of fractal porous media, we rewrite T 2 as τ 22 f (m, q)n , with τ 1 / f (m, q)0.5 denoted by τ 2 , and plot the relationship between ϕ 1 and τ1 , τ2 , as well as that between ϕ and f (m, q)n/2 (see Figs. 4, 5). In Fig. 4, one is hard to find a clear relation between ϕ1 and τ1 . But the best-fitted result indicates that τ2 and ϕ1 follows the relation τ2 = α1 ϕ1 after combining the spatial arrangement. Meanwhile, the relationship between f (m, q)n/2 and ϕ follows f (m, q)n/2 = α2 ϕ − , as shown in Fig. 5. α1 and α2 are the fitted coefficients of power law models and both approximate to 1.0; and the fitted exponents are almost the same, denoted by  and satisfying  = 0.71. As we all know, the tortuosity was first proposed as a fundamental parameter to predict the permeability (Carman 1937), as shown in Eq. (1). But, due to the ambiguous concept, the tortuosity has several definitions, some are based on geometrical approaches, while others prefer to the hydraulic ones, for details one can refer to the literatures published recently (Ghanbarian et al. 2013a, b). Vast investigations have shown that the tortuosity τ is related to the porosity, and one of the most invoked tortuosity–porosity models is given by: τ = 1 − pln (ϕ)

(30)

where p is a coefficient and will be semiempirically modified by the microstructure of porous media, as the estimations have been reported in various numerical or experimental results (Matyka et al. 2008; Comiti and Renaud 1989; Mauret and Renaud 1997).

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Fig. 5 Relationship between ϕ and f (m, q)n/2 . The circles represent the raw data, and the solid line the best power law-fitted result

Other investigations showed that the tortuosity–porosity relationship admits a power law model, which reads: τ = ϕ −β

(31)

where β is an empirical exponent. Mota et al. (2001) found β = 0.4 for binary mixtures of spherical particles when measuring the conductivity of porous media; Liu and Masliyah (1996a) reported that β = 0.5 for random packs of grains with porosity ϕ > 0.2. Recently, Ghanbarian et al. (2013b) found β = 0.378 in their geometrical tortuosity model for porous media whose pore size distribution is narrow. Actually, the hydraulic tortuosity cannot be a function of porosity only (also shown in Fig. 4 and pointed out in Valdés-Parada et al. 2011), and β cannot be universal (Ghanbarian et al. 2013a). More generally, Henderson et al. (2010) assumed a fractal scale law as follows: τ = Cτ ϕ −Dτ ,

(32)

where Cτ and Dτ are the fractal coefficient and fractal exponent of τ , respectively. For the consolidated porous media, Liu and Masliyah (1996b) found Cτ = 1.61 and Dτ = 1.15. By means of fractal geometry, together with the scale–measurement relation proposed by Wheatcraft and Tyler (1988) for a fractal capillary tube, Feng and Yu (2007) derived a geometrical tortuosity estimation model, approximately follows: τ≈

Df Df + DT − 1



L λmin

DT −1 (33)

where DT was defined as the tortuosity fractal dimension of the curve in Wheatcraft and Tyler (1988), and λmin is the diameter of the minimum capillary tube. Actually, Eq. (33) admits a fractal scaling law between tortuosity and porosity in terms of Eq. (11).

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Fig. 6 Relationship between ϕ 1 /ϕ and T , with T calculated by T =

Y. Jin et al.



τ 21 f (m, q)n−1 and ϕ1 , ϕ by

Eq. (21). The solid line is the best fitted power law model to the relationship represented by circles

Recently, based on the geometry of standard Sierpinski carpet named by V3S1-type fractal here, Li and Yu (2011) proposed a tortuosity–porosity model which yields τ = (19/18)ln ϕ/ ln (8/9) (Ghanbarian et al. 2013b). For a V 3S1-type porous medium, ϕ1 = 8/9 according to Eq. (21), thus the tortuosity model of Li and Yu (2011) can be rewritten into τ = (19/18)ln ϕ/ ln ϕ1 , indicating that the tortuosity is a function of ϕ, ϕ1 and the prefractal distribution of solid phase. Even though these models are either based on the highly idealized geometry or from special porous media, they do provide a mathematical framework to establish the tortuosity–porosity relationship. Of course, all these tortuosity–porosity models are validated in a certain range of porosity due to the percolation threshold, because at very low porosity, the system cannot even percolate that makes defining a tortuosity meaningless (Ghanbarian et al. 2013a). Obviously, the validity of porosity range is beyond our objective of the present work. Thus, in terms of the relationship between τ2 and ϕ1 , as well as that between f (m, q)n/2 and ϕ, it is reasonable to consider that T is a function of ϕ1 /ϕ, following the power law relationship expressed in Eq. (34), which is then validated by the best power law-fitted result in Fig. 6.  ϕ1 T = . (34) ϕ The difference between  and the exponents proposed before (Henderson et al. 2010; Mota et al. 2001; Liu and Masliyah 1996a, b) should be ascribed to the fractal behavior and the geometrical shape of solid particles. In terms of the characteristics of fractal porous media, the relationship between ϕ1 /ϕ and the size distribution of particles satisfies:

δ max d−Df ϕ1 = . (35) ϕ δ min

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Taking into account Eqs. (34) and (35), the hydraulic tortuosity for a fractal porous medium is then defined as:

δ max (d−Df ) τ = Cτ T = Cτ , (36) δ min where Cτ is introduced for the geometrical considerations and spatial arrangement of solid phase, with Cτ > 1. In porous media filled with monosized solid particles, the relationship Cτ = τ1 is satisfied. Equation (36) interrelates the tortuosity with determined parameters with clear physical meanings, such as ϕ1 , ϕ and tortuosity of the prefractal solid distribution τ1 in a power law model, which can be used directly for porous media with solid particles randomly distributed because the derivation process using parallel–series flow resistance mode without special assumption. And Eq. (36) is in accord with the real situation: 1. If a porous medium is filled with monosized solid particles, its hydraulic tortuosity will be mainly affected by the spatial arrangement of solid phase; 2. For the fractal porous media with the same Df , the larger the ratio between δmax and δmin , the higher the value of the hydraulic tortuosity; 3. For any porous media with Df → d, the hydraulic tortuosity τ → 1, because Df → d means that the pore space will be ultimately filled with void phase, consequently resulting in a straight-channel flow; 4. Because δmax /δmin ≥ 1 and (d − Df ) > 0, that hydraulic tortuosity satisfies τ ≥ 1 is always true, consistent with the tortuous nature of fluid flow through porous media. For visual comparison, the streamlines of some VmSq-type porous media are demonstrated in Fig. 7. In the LBM simulations, no-slip boundary conditions were imposed on the top and bottom walls and periodic boundary conditions were assumed at the inlet (left wall) and outlet (right wall). The flow was driven by an external force field whose magnitude was chosen so that the Reynolds number Re < 1 to ensure the Darcy’s flow. For short, the hydraulic tortuosity of fluid flow is denoted by TVmSqLnRl . In Fig. 7, the streamlines indicate that: (1) TV3S1L3Rl > TV3S1L2Rl > TV3S1L1Rl , which is due to that the ratio of δmax to δmin in V3S1L3Rl-type porous media is larger than that in V3S1L2Rl-type porous media, even if they share the same Df ; (2) TV3S1L2Rl > TV9S3L2Rl although the porosity of V3S1L2Rl- and V9S3L2Rl-type porous media is same. That is because the hydraulic tortuosity will alert with Df , and a large fractal dimension will result in a small hydraulic tortuosity, as aforementioned; (3) For a porous medium filled with monosized solid particles, as that in Fig. 7a, it could be denominated by any kind of V3xSxL1Rl-type porous media, such as V3S1L1Rl-, V6S2L1Rl- and V9S3LiRl-type as well. So, the hydraulic tortuosity of VmSqL1Rl-type porous media is determined by q/m, because q/m determines the spatial arrangement of solid phase. Obviously, Eq. (36) is consistent with the numerical result. Substituting Eqs. (17) and (36) into Eq. (28) gives: K =

1 ϕ13−2 ϕ 2 2 δmin C0 τ12 (1 − ϕ1 )2

(37)

According to Eq. (35), we can then rewrite Eq. (37) as K =

1 ϕ13−2 ϕ 2 C0 τ12 (1 − ϕ1 )2



ϕ1 ϕ



2 Df −d

2 δmax .

(38)

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Fig. 7 (Color) The fluid flow streamlines of some VmSqLn-type porous media when the LBM simulations reached a stable condition. a–d are the flow streamlines of V3S1L1Rl-, V3S1L2Rl-, V3S1L3Rl- and V9S3L2Rltype porous media, respectively. The color represents the relative magnitude of the dimensionless velocity which was normalized by the max velocity of a flow

The correlation between the analytical permeability (K as ) by Eq. (38) without parameter C0 and that calculated from the LBM simulations is investigated for different VmSqLntype porous media of various linear sizes (K ns , calculated from the LBM simulations). The relationships between K ns and C0 K as all yield highly linear correlation(not shown here), but the coefficient C0 will alter with the pore structure characterized by m, q and n. Substituting the fitted C0 into Eq. (38), we note that the values of permeability found in LBM simulations are in excellent agreement with those estimated by Eq. (38), some small deviation should be ascribed to the numerical precision and calculation errors (Fig. 8). The corresponding parameters of the porous media for comparison in Fig. 8 are listed in Table 2. As aforementioned, parameter C0 has been considered to be a constant in the porous media filled by monosized particles. But in fractal porous media, C0 is found to alert with the porosity and pore structure, as shown in Fig. 9. The fitted result shows that C0 is a function of the porosity ϕ and Df , approximately follows: 1

C0 ≈ 32ϕ d−Df

123

(39)

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Fig. 8 (Color) Relationship between K as and K ns . The symbols represent the relationship between K as and K ns from different VmSqLn-type porous media, and the solid line serves as a reference, where y = x indicates that K as has an identical value to K ns . l.u. is the dimensionless lattice unit Table 2 The corresponding parameters of the VmSqLn-type porous media in Fig. 8 for validation of Eq. (38)

Porous media type

m

q

Df

λmax /λmin

n

V3S1

3

1

1.893

3–35

1–5 1–4

V4S2

4

2

1.792

4–44

V5S1

5

1

1.975

5–53

1–3

V6S2

6

2

1.934

6–63

1–3

1.946

9–92

1–2

V9S3

9

3

In a fractal porous medium, that Df = d means the Euclidean space is fully occupied by void phase. In such a condition, the flow rate Q in a circular capillary is usually represented by the classical H–P equation, reads: Q=

πλ4 P , 128μL

(40)

where λ is the capillary diameter. Meanwhile, according to the Darcy’s law, Q interrelates the permeability K by: Q=K

AP . μL

(41)

where A the total cross-sectional area. Substituting Eq. (40) into Eq. (41) yields: K =

πλ4 . 128A

(42)

Because that the Euclidean space is fully occupied by void space, thus A = πλ2 /4 and λ = L are satisfied. Consequently, Eq. (42) is rearranged into K = L 2 /32. For a VmSqLnRld−1 type porous medium, when Df = d, the specific area S = βs LL d , T = 1 (because τ1 = 1 and

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1

Fig. 9 Relationship between ϕ d−Df and the fitted coefficients C0 . Symbol markers represent the fitted C0 from different porous models, and the solid line is the best-fitted power law model

f (m, q) = 1), ζ 2 = 1 (see Table 1, and because Pλ → 0 when Df → d), and ϕ1 = ϕ = 1 are all satisfied. Together with Eq. (28), the relation C0 = 32 is obtained, which is obviously consistent with the fitted result that is approximately expressed by Eq. (39), where the small errors should be ascribed to the numerical precision. In the meantime, taking Eqs. (11) and (39) both into account, we can get C0 ∝ λmin /L, which is consistent with that pointed out by Xu and Yu (2008). Together with Eqs. (28), (34) and (35), the KC constant k approximately yields:

λmin ϕ1 2 k ≈ 32 L ϕ

1+2(Df −d) 2(Df −d) λmin = 32Pλ . (43) L Equation (43) indicates that the KC constant k is a function of the fractal dimension, range and the Lacunarity of pore size. 1. If Df → d, then k → 32, meaning the fluid flow in porous media is approaching a Poiseuille flow; 2. If the scaling characteristics is weak or there is no fractal behavior of the pore size distribution. Thus, ϕ → ϕ1 and then k → 32 λmin L ; L and Df ≤ d are always satisfied, 3. If the size range of pores is fixed, because Pλ < λmin meaning that k will decrease with the fractal dimension Df ; in terms of Eq. (11), the porosity increases with fractal dimension monotonically if the pore size range is fixed (Jin et al. 2013), thus the smaller the porosity is, the larger is the value of k. The variation characteristic indicates that the KC constant is not a constant, which characterizes the difference of fluid flow in a porous media between that in a straight tube. A large deviation of a fluid flow from the Poiseuille’s flow will result in a small k, vice versa.

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Even though the variation trend of the KC constant proposed here is consistent with that in Xu and Yu (2008), their estimation expressions are not the same. The main difference is that we use intersected pore space other than the nonintersecting assumption of fluid paths.

5 Conclusions In this study, we investigate the scaling invariant characteristics of the fractal porous media. Following the analytical–numerical coupling solution, the scaling effects of pores on the permeability, hydraulic tortuosity and the KC constant are fully analyzed, and some conclusions are drawn as follows: (1) In fractal porous media, the permeability–porosity relationship follows power law correlation with exponent being 2, not the cubic law described in KC equation; (2) The hydraulic tortuosity of the fractal porous medium is a function of the size range of solid particles, fractal dimension of pore size and the prefractal solid distribution; the tortuosity–porosity relation admits a fractal scaling law with tortuosity fractal dimension approximately being 0.71 in two-dimensional context; (3) The KC constant is not a constant, which characterizes the derivation of the fluid flow from the Poiseuille flow. If the pore size fractal dimension is deterministic, a large value of λmin /L will yield a small KC constant; and KC constant will decrease with the fractal dimension if the range of pore size is fixed; (4) When the Euclidean space is occupied fully by the void phase, the fluid flow in porous media will turn into the Poiseuille flow; while if there is no fractal behavior of pores size, the media’s permeability follows KC equation characterizing the fluid flow in the pore space filled by monosized particles. The shift of these two kinds of fluid flow is controlled by the fractal dimension and the range of pore size distribution.

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