Permeability of Microcracked Porous Solids: Modeling from Homogenization to Percolation

Li, Le; Li, Kefei

doi:10.1061/(ASCE)EM.1943-7889.0002132

Open access

Technical Papers

Jun 21, 2022

Permeability of Microcracked Porous Solids: Modeling from Homogenization to Percolation

Authors: Le Li https://orcid.org/0000-0001-7798-7120 [email protected] and Kefei Li [email protected]Author Affiliations

Publication: Journal of Engineering Mechanics

Volume 148, Issue 9

https://doi.org/10.1061/(ASCE)EM.1943-7889.0002132

PDF

Abstract

This paper establishes a comprehensive modeling for permeability of porous solids incorporating cracks with clustering extent ranging from nonoverlapping to percolation. The geometry of cracks is characterized through the crack density, orientation, connectivity, and fractal dimension of crack networks. The interaction direct derivative (IDD) model from effective medium theory is developed to predict the permeability of solids with nonoverlapping cracks and further extended by a two-step scheme to account for cracks with local clustering. The continuum percolation theory is adopted to build the permeability scaling law for crack networks approaching percolation. The comprehensive model thus established was validated against numerical simulations for crack networks with different geometry characteristics. The obtained results show that (1) the classical IDD model can predict satisfactorily the permeability for nonoverlapping cracks and the crack opening has nearly no impact on overall permeability, (2) the extended IDD model considers successfully the local clustering of cracks, the prediction range is greatly enhanced for overall permeability, and the impact of crack opening is still limited, and (3) as crack networks approachi percolation, the overall permeability observes a scaling law highlighting the importance of crack connectivity, and the impact of crack opening assumes a power law with the exponent related to the fractal dimension of crack networks.

Introduction

Cracks are common defects in quasibrittle materials such as rocks and concretes (Adler et al. 2013). These cracks can have different origins and result from various actions, e.g., the residual stress arisen during manufacturing processes (Li et al. 2020), fatigue or thermal loadings during service (Nguyen et al. 2020; Liu et al. 2021), or the aging of materials under internal and/or external processes (Torrijos et al. 2010). In porous solids, cracks facilitate the flow of fluids (liquids and gas) by creating additional transport channels and enhancing the connectivity of the pore structure in the solid matrix (Adler et al. 2013). Individual cracks can propagate and intersect each other, forming a crack network that could further promote the fluid flow (Adler and Thovert 1999; Li and Li 2015). With certain degree of clustering, the cracks may serve as preferential flow paths for aggressive agents, accelerate the aging processes of materials, and, consequently, affect the service lives of structures using these materials (Li 2016). In other cases such as extracting geothermal energy from rocks, the maximized overall flow capacity is sought as a desirable attribute by generating artificial crack networks in porous rocks (Hu et al. 2021). Therefore, it is a primary issue is to describe the impact of cracks on transport properties of materials and further on the target performance of these materials for engineering applications (Timothy and Meschke 2017b).

The flow of fluids with low Reynolds number in a fully saturated porous solid can be described by Darcy’s law, which can also be derived by Navier-Stokes equation applied to steady flow on the scale of pores (Whitaker 1986). Introducing cracks into the porous matrix can have two consequences: physically, new channels, characterized by crack geometry, are added to pore channels for fluid flow (Wang et al. 2018), and chemically, a new interface (crack surface) is created, making possible the mass exchange between crack surface and flowing fluids such as the dissolution of solid phases or precipitation on the crack surface (Li et al. 2011). In this study, the chemical effect of a crack surface is neglected and only the physical effect is taken into account(Vinci et al. 2014; Mustapha et al. 2020). Thus, a cracked porous solid can be idealized in two phases: the cracks and the (homogeneous) porous matrix, and estimating its effective permeability requires averaging properly the permeabilities of the two phases, simultaneously taking into consideration the geometry of cracks (Guàuen et al. 1997). Two theories can help the estimation: effective medium theory (EMT) and percolation theory (PT).

The effective medium theory develops various schemes to estimate the effective permeability of the equivalent homogeneous medium by evaluating the perturbation induced by inclusions. From the view of EMT, cracks could be taken as inclusions with high local permeability embedded in a low-permeable matrix (Eshelby 1957). Through appropriate assumptions of the far-field boundary conditions, EMT can further consider the interactions among multiple inclusions with multiple schemes available (Guàuen et al. 1997). The dilute scheme takes cracks to be nonoverlapping and positioned in homogeneous matrix without any interaction among cracks (Shafiro and Kachanov 2000). The self-consistent (SC) approximation idealizes a single crack embedded in the sought homogenized medium (Lemarchand et al. 2010; Jeannin and Dormieux 2017), and the SC scheme has been successfully applied to deal with stress–permeability coupling (Dormieux et al. 2011; Wang et al. 2018).

The differential scheme assumes that cracks are positioned into the homogeneous matrix one by one, and the effective permeability obtained at each step is considered as the host matrix for the subsequent step (Norris 1985). The interaction direct derivative (IDD) scheme (Zheng and Du 2001) provides an explicit and simple structure for the prediction of permeability of cracked solids with low crack extent (Zhou et al. 2011; Li and Li 2019). The cascade continuum micromechanics (CCM) model considers the complexity of the embedded cracks through the hierarchization and the interaction of the cracks (Timothy 2016; Timothy and Meschke 2016). The multipole expansion method gives a complete solution to the boundary-value problem of the matrix–inclusion composites, comprising the continuous matrix and nonintersecting crack inclusions (Kushch 2020). The permeability upscaling scheme adapts Eshelby’s inhomogeneity problem for linear elasticity to the Darcy’s law (Dormieux and Kondo 2005), and the crack interaction is estimated by the SC scheme, which has been applied to cementitious materials (Pivonka et al. 2004; Damrongwiriyanupap et al. 2017) and bone tissue (Abdalrahman et al. 2015; Estermann and Scheiner 2018).

Effective medium theory estimates the effective permeability from averaging; thus, it has its limit, i.e., the EMT models apply to situations where the perturbance of cracks to the fluid flow is moderate. As cracks perturb the fluid flow to a larger extent, e.g., by higher crack density or important clustering, the EMT begins to lose its validity (Guàuen et al. 1997). This is the very starting point for percolation theory to intervene, taking the cracks no longer as homogeneous inclusions but as networks. A central task of the percolation theory is to predict the percolation threshold (limit) of fluid flow and estimate the overall permeability when the crack perturbance is large (Hunt and Ewing 2009).

The classical PT is established through lattice methods and has been applied to less regular or random networks through equivalent lattices (Langlois et al. 2018; Li et al. 2017). The continuum PT, more adapted to solids with random networks, has been used to evaluate the connectivity degree (Berkowitz 1995) and predict the effective permeability (Li and Li 2015). The percolation threshold and scaling exponent depend on the finite domain size and fractal morphology of random network (Liu and Regenauer-Lieb 2021). The correlation between the crack geometry and the permeability from percolation theory, lattice or continuum-based, has been reviewed by Li and Li (2015). PT intends to describe the flow behavior near the percolation threshold and cannot be applied to cases far below the threshold (Guàuen et al. 1997). Accordingly, the EMT and PT have their respective merits, suitable to address two characteristic ranges: the fluid flow is dominated by the solid matrix or the crack network.

On the basis of the aforementioned state of the art, this paper focuses on the comprehensive modeling of permeability of microcracked porous solids for the entire range of crack density, employing the EMT and PT models to estimate the overall permeability in their respective valid ranges. In the range of moderate crack perturbance, the IDD model from EMT was used and extended to consider the finite crack connectivity and crack opening. As crack pertubance becomes strong with the appearance of dominant/spanning clusters of cracks, a percolation model was established to estimate the effective permeability considering the permeability of porous matrix and the geometry parameters of crack clusters. Following this logic line, the paper is presented as follows: the geometry characterization of random crack networks is presented for two-dimensional (2D) cases; then the IDD model and percolation scaling law for overall permeability are established; after that, numerical analysis and validation are provided; then, further analysis of crack opening and experimental validation are given; and the concluding remarks are summarized finally.

Geometry of Random Crack Networks

The geometry of random crack networks includes single cracks and crack clusters (Zhou et al. 2012; Li and Li 2015). The single crack geometry refers to the length, opening aperture, and tortuosity of the crack surface, and the network geometry includes the crack extent (density), orientation, and the crack cluster size (Adler et al. 2013). These geometry parameters have different influences on the permeability evaluation (Akhavan et al. 2012; Li and Li 2018). Three local parameters, namely opening aperture, clustering extent, and connectivity degree, and two averaging parameters, namely crack density and orientation, are retained to describe the crack networks. The geometry parameters are defined as follows and they are applied to illustrate impact of the crack network on the permeability.

Crack Density

Crack density describes the damage extent of porous solids, defined as follows:

ρ = \frac{1}{A_{s}} \sum_{i = 1}^{N} {(\frac{l_{i}}{2})}^{2}

(1)

where

l_{i}

=

i

th crack length;

N

= total crack number; and

A_{s}

= cracked area. This area-based density definition is widely used in micromechanical models (Kachanov 1992) and also applied in effective properties estimate of cracked solids (Hasselman 1978). Fig. 1 illustrates three random crack networks with crack densities

ρ = 0.01

, 0.1, 1.0, respectively.

Fig. 1. Random crack networks with constant length and crack density of (a) 0.01; (b) 0.1; and (c) 1.0.

Orientation

Supposing

N

cracks with length

l_{i} (i = 1, 2, \dots, N)

in a finite domain, the orientation degree

ω

can be determined through an orientation factor

ι

ω = \frac{ι {(θ)}_{\min}}{ι {(θ)}_{\max}}, with ι (θ) = \frac{1}{A_{s}} \sum_{i = 1}^{N} {(\frac{l_{i}}{2})}^{2} \cos^{2} θ_{i}

(2)

where

θ_{i}

= angle between crack

l_{i}

and the flow direction; and

ι {(θ)}_{\min}

and

ι {(θ)}_{\max}

= minimum and maximum values of the orientation factor. The orientation degree

ω = 1

stands for a perfectly isotropic crack network, (Fig. 1) and

ω = 0

as all cracks are parallel to the transport direction.

Fractal Dimension

The fractal dimension is a measure of the spatial organization of the cracks, which is independent on the crack length distribution and limited by the number of topological dimensions of the domain (three for a volume and two for a surface) (Davy et al. 1990). A two-point correlation function was applied to account for the fractal characteristics (Darcel et al. 2003). The correlation function

C (r)

scales with

r

as follows:

C (r) = \frac{N (r)}{N_{T}} \propto r^{D_{c}}

(3)

where

N_{T}

= total crack number;

N (r)

= number of cracks for which the distance between crack centers is within

r

; and

D_{c}

= fractal dimension that can be regressed by

C (r) - r

on a logarithm scale. Fig. 2 illustrates three random crack networks with the same crack density

ρ = 0.5

and different fractal dimensions

D_{c} = 1.99

, 1.75, 1.62.

Fig. 2. Random crack networks with constant length, $ρ = 0.5$ , and fractal dimensions of (a) $D_{c} = 1.99$ ; (b) $D_{c} = 1.75$ ; and (c) $D_{c} = 1.62$ .

Connectivity

The connectivity measures the cracking extent to which the flow paths are formed in random crack networks, a critical parameter for permeability estimation. A connectivity factor,

f

, based on percolation concepts of Li and Li (2015) considering local clustering, was used here

f = \frac{ξ}{L} P^{(3 - D_{c})}, with f \in [0, 1]

(4)

where

ξ

= cluster correlation length;

L

= domain size;

D_{c}

= fractal dimension; and

P

= likelihood of a crack being connected to the spanning clusters. If all cracks connect with each other and form into one spanning cluster across the domain, i.e.,

ξ = L

and

P = 1

, the connectivity factor

f

achieves the maximum value of 1. The connectivity of crack networks in Fig. 2 is

5 \times 10^{- 5}

[Fig. 2(a)], 0.022 [Fig. 2(b)], and 0.089 [Fig. 2(c)]. When

L \to \infty

, the connectivity

f

near percolation threshold obeys the following scaling law:

f \propto {| ρ - ρ_{c} |}^{- ν + (3 - D_{c}) β}

(5)

where the percolation threshold,

ρ_{c}

, depends on the geometry characterization of crack clusters; beyond the threshold, a percolating crack flow path appears in statistical sense. The exponents

β

and

ν

are universal constants related to the topological dimension of the domain, i.e.,

β = 5 / 36

and

ν = 4 / 3

for the two-dimensional case and

β = 0.41

and

ν = 0.875

for the three-dimensional case (Aharony and Stauffer 2003).

Permeability of Cracked Solids

IDD Model for Nonoverlapping Cracks

For the matrix-inclusion morphology, the crack inclusions are separately distributed in a continuous matrix phase. Permeability is taken as the linear physical property to be studied. So the permeability fluctuation

H_{i}

of crack inclusion

i

can be expressed

H_{i} = K_{i} - K_{m}

(6)

where the permeability tensors

K_{i, m}

stand for crack inclusion and matrix phase, respectively.

The dilute estimate for the permeability increment

H_{i}^{d}

of an ellipsoidal matrix containing one ellipsoidal crack inclusion can be written (Shafiro and Kachanov 2000; Zhou et al. 2011)

H_{i}^{d} = c_{i} {(H_{i}^{- 1} + Ω_{i}^{0})}^{- 1} = c_{i} {(H_{i}^{- 1} + K_{m}^{- 1} J)}^{- 1}

(7)

where

c_{i}

= volumetric fraction of the

i

th crack inclusion;

Ω_{i}^{0}

is the eigenstiffness tensor characterizing the

i

th inclusion shape and orientation; superscript 0 = matrix; susperscript

d

= dilute solution; and

J

is the geometry tensor related to the size of ellipsoidal inclusion. Its components

J_{j = 1, 2, 3}

are written

J_{1} = \frac{a_{1} a_{2} a_{3}}{2} \int_{0}^{\infty} \frac{d u}{(a_{1}^{2} + u) \sqrt{(a_{1}^{2} + u) (a_{2}^{2} + u) (a_{3}^{2} + u)}}

(8)

with

J_{2}

and

J_{3}

obtained by cyclic permutations of all indices 1, 2, and 3. The terms

a_{1, 2, 3}

are semiaxes of the inclusion parallel to coordinate axes

x_{1, 2, 3}

, and

e_{1, 2, 3}

are unit vectors along them. For some special cases, e.g., two-dimensional analysis,

J_{j}

can be simplified as follows:

J_{1} = 0, J_{2} = \frac{a_{3}}{a_{2} + a_{3}}, J_{3} = \frac{a_{2}}{a_{2} + a_{3}}

(9)

The IDD estimate derives from the effective self-consistent (ESC) scheme; the root of them is the inclusion-matrix-composite model. The IDD estimate considers the inclusion spatial distributions as well as the interaction between inclusions and the immediate surrounding matrix (Zheng and Du 2001). This estimate has an explicit and simple structure for various inclusion shapes, inclusion volume fraction, as well as inclusion and matrix anisotropies. Explicit solutions with second-order accuracy can be provided by the IDD estimate in a statistical sense. The IDD scheme gives the permeability increment

H^{idd}

of a solid containing

N

groups of crack inclusions as follows (Zheng and Du 2001; Du and Zheng 2002):

H^{idd} = {(I - \sum_{i = 1}^{N} H_{i}^{d} \cdot Ω_{D i}^{0})}^{- 1} \cdot \sum_{i = 1}^{N} H_{i}^{d}

(10)

where

Ω_{D i}^{0}

is the eigenstiffness tensor of the matrix atmosphere for inclusion

i

. The mathematical formulations for the classical IDD model are provided in the Appendix, and in the following, we develop further this model to account for crack density and opening in permeability estimation.

The corresponding matrix atmosphere was selected as ellipsoidal for the three-dimensional case and elliptical for the two-dimensional case, with their principal axes aligned on the crack plane (Fig. 3). Considering the plane

e_{2} e_{3}

and letting

a_{1} \to \infty

for two-dimensional analysis, the terms

2 a_{2}^{0}

and

2 a_{3}^{0}

denote the axis lengths of the elliptical matrix atmosphere,

2 a_{2}

is the crack length,

2 a_{3}

is the crack opening aperture, and

a_{2}^{0} = a_{2} + a_{3}^{0}

. The crack density

ρ

in solid can be evaluated as follows:

ρ = \sum_{i = 1}^{N} \frac{{(a_{2})}_{i}^{2}}{A_{a}}

(11)

where

N

= number of cracks (inclusions); and

A_{a}

= domain area. To maintain the same density in the crack-atmosphere cell equal to that in the whole domain, the geometrical parameters can be evaluated as follows:

\frac{a_{2}^{2}}{π (a_{2} + a_{3}^{0}) a_{3}^{0}} = ρ \Rightarrow \frac{a_{3}^{0}}{a_{2}^{0}} = \frac{a_{3}^{0}}{a_{2} + a_{3}^{0}} = \frac{\sqrt{1 + \frac{4}{π ρ}} - 1}{\sqrt{1 + \frac{4}{π ρ}} + 1}

(12)

Fig. 3. IDD modeling of a porous matrix with crack inclusions.

Using the dilute estimate in Eq. (7), the permeability increment

H_{i}^{d}

in the plane

e_{2} e_{3}

can be calculated as follows:

H_{i}^{d} = c_{i} {[H_{i}^{- 1} + K_{m}^{- 1} {(\frac{a_{3}}{a_{2} + a_{3}} e_{2} e_{2} + \frac{a_{2}}{a_{2} + a_{3}} e_{3} e_{3})}_{i}]}^{- 1}

(13)

By using the definition in Eq. (6),

H_{i}^{d}

can be rearranged as follows:

H_{i}^{d} = \frac{π a_{2} a_{3}}{A_{a}} {[\frac{(K_{i} - K_{m}) \cdot K_{m} (a_{2} + a_{3})}{K_{m} a_{2} + K_{i} a_{3}} e_{2} e_{2} + \frac{(K_{i} - K_{m}) \cdot K_{m} (a_{2} + a_{3})}{K_{m} a_{3} + K_{i} a_{2}} e_{3} e_{3}]}_{i}

(14)

The two-dimensional (2D) eigenstiffness tensor of the elliptical matrix atmosphere is

Ω_{D i}^{0} = K_{m}^{- 1} \cdot {(\frac{a_{3}^{0}}{a_{2}^{0} + a_{3}^{0}} e_{2} e_{2} + \frac{a_{2}^{0}}{a_{2}^{0} + a_{3}^{0}} e_{3} e_{3})}_{i}

(15)

Combining Eqs. (14) and (15) with Eq. (10), the IDD estimate of permeability can be obtained.

Now consider a crack with opening aperture

w^{'}

embedded in a solid matrix of low permeability. The classical intrinsic permeability of the crack between two parallel impermeable surfaces is expressed as

K_{i} = w^{' 2} / 12

(Timothy and Meschke 2017b). Serving as a reference, the intrinsic permeability of a cement paste (matrix) is around

K_{m} = 10^{- 17} m^{2}

(Picandet et al. 2009) with crack permeability

K_{i} = 10^{- 12} m^{2}

with opening

w^{'} = 5 μm

, the lower limit for such cracks. Thus, it is rational to assume that

K_{i} ≫ K_{m}

for physically meaningful cracks in engineering porous materials like concretes and rocks (Li et al. 2011; Dormieux et al. 2011). Using this assumption and considering a single crack family characterized by the same aspect ratio

w = a_{3} / a_{2}

(Lemarchand et al. 2005), one can derive the relative permeability along the crack’s orientation for solids with parallelly arranged microcracks as follows:

\frac{K_{x}}{K_{m}} = \frac{1 + π ρ (1 + w) (1 - γ_{D})}{1 - π ρ (1 + w) γ_{D}}

(16)

and the effective permeability for solids with randomly oriented microcracks as follows:

\frac{K_{eff}}{K_{m}} = \frac{1 + \frac{π}{2} ρ (1 + w) [w + (1 + w) (1 - γ_{D})]}{1 - \frac{π}{2} ρ (1 + w) [- w + (1 + w) γ_{D}]}

(17)

where

γ_{D}

= geometrical parameter of an elliptical matrix atmosphere, expressed by

γ_{D} = \frac{a_{3}^{0}}{a_{2}^{0} + a_{3}^{0}} = \frac{\sqrt{1 + \frac{4}{π ρ}} - 1}{2 \sqrt{1 + \frac{4}{π ρ}}}

(18)

To investigate the influence of opening aperture

w

and crack density

ρ

in Eq. (17), the effective permeability

K_{eff} / K_{m}

was calculated for various crack opening aperture ratios (

w = 0.001

to 0.03 and crack densities (

ρ = 0

to 1) for nonoverlapping cracks (Fig. 4). The crack density contributes to the effective overall permeability in a monotone manner, and the crack opening does not change notably the effective permeability. It is rather reasonable according to the assumptions of the IDD estimate: the cracks with very high local permeability at large

w

are assumed positioned in a low permeable matrix, i.e., the cracks are separated and the matrix is continuous, serving as barriers for the flow.

Fig. 4. Effective permeability in terms of crack density $ρ$ and opening aperture ratio $w$ in Eq. (17) for networks with nonoverlapping cracks.

Extended IDD Model for Low and Medium Crack Clustering

For crack density far below the percolation threshold, there is not yet a backbone cluster of cracks dominating the permeability, but local clustering can occur to increase the flow perturbance of flow. In this section, the classical IDD model is extended to account for this local crack clustering. Fig. 5 shows a random network with constant length

l

, crack density

ρ = 0.2

, and domain size ratio

L / l = 5

. Assume that the projected length of the largest cluster on the transport direction equals to

ξ

. Supposing the crack network contains

N_{ξ}

clusters whose sizes are in the interval between

0.5 ξ

and

ξ

, the projected length of the

i

th cluster on the transport direction is denoted

ξ_{j} = ψ_{j} ξ, ψ_{j} \in [0.5, 1.0]

(19)

Fig. 5. Random network (a) with constant length, $D_{c} = 2$ , $ρ = 0.2$ , $L / l = 5$ ; (b) its small clusters; and (c) equivalent model for local clusters.

As two cracks interconnect with each other, it seems reasonable to arrange a parallel crack with length

ξ_{j}

on the transport direction to evaluate the connectivity impact on the effective permeability (Fig. 5). The equivalent density

ρ_{h}

is

ρ_{h} = \sum_{j = 1}^{N_{ξ}} \frac{ξ_{j}^{2}}{4 A} = ρ (\sum_{j = 1}^{N_{ξ}} ψ_{j}^{2} \cdot ξ^{2}) / (4 \cdot \sum_{i = 1}^{N} {(a_{2})}_{i}^{2})

(20)

which can further consider the connectivity definition in Eq. (4) and be rearranged as follows:

ρ_{h} = \sum_{j = 1}^{N_{ξ}} ψ_{j}^{2} {[4 f \cdot P^{- (3 - D_{c})}]}^{2}

(21)

As shown in Fig. 5, a two-step scheme can be adopted to extend the IDD model to account for the local clustering. First, nonoverlapping cracks are introduced into the matrix so that one can evaluate the overall permeability using classical IDD model. The effective permeability

K_{1}

obtained at this step is attributed to the host matrix for the second step. In this step, a parallel crack is added on transport direction at the same position to consider the amplification effect of flow by the local crack connectivity (clustering). The relative permeability in this step can be estimated by Eq. (16) as follows:

\frac{K_{x}}{K_{1}} = \frac{1 + π ρ_{h} (1 + w) (1 - γ_{D}^{x})}{1 - π ρ_{h} (1 + w) γ_{D}^{x}}

(22)

In total, the classical IDD model for effective permeability can be extended to crack networks with connected cracks as follows:

\frac{K_{eff}}{K_{m}} = \frac{1 + π ρ_{h} (1 + w) (1 - γ_{D}^{x})}{1 - π ρ_{h} (1 + w) γ_{D}^{x}} \cdot \frac{1 + \frac{π}{2} ρ (1 + w) [w + (1 + w) (1 - γ_{D})]}{1 - \frac{π}{2} ρ (1 + w) [- w + (1 + w) γ_{D}]}

(23)

where

ρ_{h}

= equivalent density related to the connectivity

f

; and coefficients

γ_{D}^{x}

and

γ_{D}

can be derived from crack densities

ρ_{h}

and

ρ

by Eq. (18). The quantification of

ρ_{h}

will be given in the “Numerical Analysis” section.

Percolation Model for High Crack Clustering

As clustering effects become dominant and the spanning cluster of cracks is about to occur, the permeability of a cracked solid is to increase significantly, and the primary controlling parameter in the permeability of a crack network is connectivity. The effective permeability of cracked solids can be expressed

K_{eff} = K_{eff} (K_{m}; f, τ, ρ, w)

(24)

where

K_{m}

= permeability of the homogeneous porous matrix;

f

= percolation-based connectivity in Eq. (5);

τ

= tortuosity of the backbone cluster; and

ρ

and

w

= crack density and crack opening aperture ratio, respectively. The continuum percolation theory allows the following scaling law for effective permeability:

K_{eff} = K_{0} (K_{m}, w) {| ρ - ρ_{c} |}^{μ} with μ = (3 - D_{c}) β - (2 - D_{τ}) ν

(25)

where exponent

μ

= connectivity and the tortuosity of dominating cluster;

D_{τ}

= fractal dimensions of the dominating cluster; and the term

K_{0}

= impact of crack opening and matrix permeability. Some further adaptations for a finite flow domain have been discussed by Li and Li (2015).

Model Summary

The permeability of porous solids incorporating random crack networks is expressed through two approaches: the IDD estimate and continuum percolation theory (Table 1). The established model covers three characteristic ranges of cracks: (1) for nonoverlapping cracks or cracks with low perturbance for fluid flow, the classical IDD model is employed, (2) for low and medium crack clustering, the extended IDD model is derived, and (3) for high clustering of cracks, the percolation scaling law is employed. The established model is clustering-sensitive and will be validated through numerical simulations for different clustering cases.

Table 1. Effective permeability models applied for the whole range of crack density and clustering

Crack clustering extent	$ρ < ρ_{c}$	$ρ \to ρ_{c}$	$ρ > ρ_{c}$
No clustering (nonoverlapped)	IDD model	IDD model	—
Low and medium clustering	Extended IDD model	—	—
High clustering	—	Percolation scaling law	—
Percolated crack networks	—	—	Percolated flow

Numerical Analysis

Numerical Generation of Random Crack Networks

A finite domain

L \times L

with random distributed cracks is established. The crack length follows a lognormal distribution from the geometry characterization on cracks of concrete damage under compressive loading (Zhou et al. 2012) (Fig. 6). The statistical analysis showed that the cracking opening aperture has a clear correlation with crack length. So the ratio of opening aperture to length,

w

, was taken as a basic variable ranging from 0.001 to 0.028 from the results in Fig. 6, when the mean crack length

l_{mean}

equals to 3.606 mm and the opening aperture ranges from 5 to 100 μm (Li and Li 2015). A length scale

χ = 5

, 10, 15, 20, 25, 30 was retained as the ratio between the finite domain size

L

and the mean crack length

l_{mean}

. The constant length distribution serves as comparative case in numerical analysis.

Fig. 6. Crack patterns of concrete specimens under compressive loading: (a) perpendicular to loading; (b) crack length distribution; and (c) parallel to loading. (Data from Zhou et al. 2012.)

A Monte-Carlo algorithm was designed to generate different crack networks. The crack density, crack opening aspect ratio, crack length distribution, and the average number of interaction per crack were preset before the simulation. According to the preset statistical parameters, one can generate random cracks and locate them one by one into the finite area. Once a random crack is placed, the total intersection number of all cracks is calculated; if the average interaction for each crack does not surpass the preset parameter, this crack will be adopted; otherwise, it will be deleted. Meanwhile, the isolated cracks and connected crack clusters are characterized and recorded on this step. These simulation steps are repeated until the preset parameters, i.e., crack density and average intersection number, are accomplished.

The geometry characterization for the isolated cracks and crack clusters in the generated crack network, including the fractal dimension

D_{c}

and the connectivity factor

f

, was derived. Controlling the intersection ratio per crack produces correlated networks with different fractal dimensions

D_{c} = 1.43

to 1.99; releasing the control of the intersection ratio results in an uncorrelated network with

D_{c} = 2.0

. Three correlated networks with the same density and domain size (

ρ = 0.5

and

χ = 20

) are generated in Fig. 2. The intersection ratio per crack was controlled as 0.0 (

D_{c} = 1.99

), 2.0 (

D_{c} = 1.75

), and 3.0 (

D_{c} = 1.62

), respectively. Fig. 7 shows two uncorrelated networks with the same crack density

ρ = 0.5

, domain size

χ = 20

, fractal dimension

D_{c} = 2.0

, and different length distributions.

Fig. 7. Uncorrelated crack networks with (a) constant; and (b) lognormal lengths, with $ρ = 0.5$ , $χ = 20$ , and $D_{c} = 2.0$ .

Permeability Simulation

The generated domain containing crack networks was retained for permeability analysis. The assumptions of incompressible and Newtonian fluid were adopted in the permeability simulation. The effective permeability

K_{eff}

can be calculated through Darcy’s law,

K_{eff} = - η q / \nabla p

, where

q

is the flow rate,

η

is the fluid viscosity, and

\nabla p

is the pressure gradient. In the numerical solution, along the flow direction, a pressure gradient was applied and perpendicular to the flow direction, null flux conditions were exerted. The finite-element method (FEM) was used to solve the pressure field

p (x, y)

: triangle elements were applied to mesh the matrix; line elements were prescribed to simulate the cracks; and the two nodes of line element as well as the intersection point of cracks were considered as shared nodes.

When the intrinsic permeability of the porous matrix takes

K_{m} = 10^{- 17} m^{2}

, the local crack permeability

K_{crack}

is eight magnitudes higher than the matrix through

K_{crack} = w^{' 2} / 12

with crack opening

w^{'}

of 50 μm.

Model Validation

IDD Model for Nonoverlapping Cracks

Crack networks of size

χ = 20

were constructed with randomly oriented and located cracks, which followed a constant length distribution. There was no intersection between generated cracks, shown in Fig. 2(a). The crack opening ratio

w = 0.0139

and the crack density ranged between 0 and 1 with an interval of 0.1. For each crack density level, 40 crack networks were generated, and the corresponding effective permeability

K_{eff}

was solved by FEM. The average values of

K_{eff}

are given in Fig. 8(a) together with the IDD estimate, dilute scheme, the Mori-Tanaka (MT) scheme (Timothy and Meschke 2018), and differential scheme (DS).

Fig. 8. IDD estimate of effective permeability compared with numerical results for crack networks with nonoverlapping cracks ( $χ = 20$ ): (a) impact of crack density at a fixed $w = 0.0139$ ; and (b) impact of crack opening ratio at a fixed $ρ = 1.0$ .

The IDD estimate showed good agreement with the numerical results for the whole range of crack densities, whereas the MT scheme and the dilute scheme lost their prediction capacity at crack densities higher than 0.15, and the DS overestimated the effective permeability at crack densities higher than 0.35. Fig. 8(b) presents the effective permeability in terms of the crack opening ratio

w

ranging from 0.0014 to 0.0277 with

ρ = 1.0

for both numerical simulations and IDD estimates. One can see that the overall permeability shows very little change for the simulated values of crack opening.

Extended IDD Model for Low and Medium Crack Clustering

To illustrate the extended IDD model and obtain the equivalent horizontal density

ρ_{h}

, numerical simulations were performed for two different crack patterns with constant length,

D_{c} = 2.0

, 1.75, shown in Figs. 2(b) and 7(a). The study domain size

χ = 20

, and crack opening ratio

w

was 0.0139. The crack density ranges from 0 to 1.4 (1.6 for

D_{c} = 1.75

), with an interval of 0.1, and 40 crack networks were generated for each crack density level. The equivalent horizontal density

ρ_{h}

can be obtained for each crack network. Figs. 9(a) and 10(a) show the average value of

ρ_{h}

in terms of crack density

ρ

for uncorrelated (

D_{c} = 2.0

) and correlated crack networks (

D_{c} = 1.75

), respectively.

Fig. 9. IDD extended model of effective permeability compared with numerical results for crack networks with connected cracks (constant length, $χ = 20$ , $w = 0.0139$ , and $D_{c} = 2.0$ ): (a) $ρ_{h}$ in terms of $ρ$ ; and (b) relative permeability in terms of $ρ$ .

Fig. 10. IDD extended model of effective permeability compared with numerical results for crack networks with connected cracks (constant length, $χ = 20$ , $w = 0.0139$ , and $D_{c} = 1.75$ ): (a) $ρ_{h}$ in terms of $ρ$ ; and (b) relative permeability in terms of $ρ$ .

From Fig. 9(a), it can be seen that the equivalent horizontal density

ρ_{h}

increased slowly with

ρ

until

ρ = 0.4

but almost linearly afterward. That is because the connectivity factor

f

is

ρ

-dependent and the

f

values are very small at a low crack density, e.g.,

f = 0.002

at

ρ = 0.2

.

The relative permeability was then evaluated by extended IDD model through Eq. (23), shown in Fig. 9(b). For

ρ < 0.4

, the relative permeability obtained by IDD estimation for nonoverlapping cracks also showed good agreement with numerical results. From Fig. 10(a), the equivalent horizontal density

ρ_{h}

increased almost linearly with

ρ

due to the generation procedure of correlated networks. By using the obtained

ρ_{h}

, the effective permeability was evaluated through Eq. (23) and is illustrated in Fig. 10(b).

Within the performed simulation ranges, namely

ρ < 1.1

,

ρ_{h} < 0.335

, and

f < 0.067

for

D_{c} = 2.0

and

ρ < 1.3

,

ρ_{h} < 0.356

, and

f < 0.168

for

D_{c} = 1.75

, the extended IDD estimate was validated and showed good accuracy for crack networks with low and medium clustering. For higher crack density and notable crack clustering extents, the IDD estimate diverged from the numerical results because the clustering effect becomes dominating, and this effect is beyond the prediction capacity of the extended IDD model.

Percolation Model for High Crack Clustering

The high crack clustering effect on permeability was quantified through the percolation scaling law. In a forgoing study (Li and Li 2015), for uncorrelated networks,

D_{c} = 2.0

, the percolation thresholds

ρ_{c}

were 1.43 and 1.67 for constant and lognormal length distributions. For correlated networks,

ρ_{c} = 1.68

and 1.78 for constant lengths with

D_{c} = 1.75

and 1.62, respectively. As crack density

ρ \to ρ_{c}

, the scaling law in Eq. (25) is valid, i.e.,

| ρ - ρ_{c} | \leq 0.33

or 0.38 for uncorrelated and correlated networks. The scaling exponent

μ

of permeability is calculated as

- 0.922

(constant length) and

- 0.919

(lognormal length) for

D_{c} = 2.0

,

- 1.122

(constant length) for

D_{c} = 1.75

, and

- 1.257

(constant length) for

D_{c} = 1.62

by linear regression between the effective permeability and

| ρ - ρ_{c} |

on a logarithm scale. The regressed parameters are presented in Table 2.

Table 2. Scaling parameters of permeability for different

D_{c}

Crack length distribution	$D_{c}$	$υ$	$λ$	$μ$
Constant	2.00	11.57	0.199	$- 0.922$
Constant	1.75	56.26	0.477	$- 1.122$
Constant	1.62	92.85	0.554	$- 1.257$
Lognormal	2.00	13.79	0.223	$- 0.919$

The effective permeability from the percolation scaling law was compared with results from the cascade continuum micromechanics (CCM) model proposed by Timothy and Meschke (2017a). The CCM model predicts the effective permeability through the Eshelby mean-field homogenization method. At the self-similar limit level [described as

n \to \infty

by Timothy and Meschke (2017a)], the CCM model degenerates to the self-consistent scheme. For low matrix permeability,

K_{m} = 10^{- 18} m^{2}

, local crack permeability

K_{crack}

is eight magnitudes higher than

K_{m}

. The CCM model displays a sharp phase transition as the crack density increases. The order of magnitude of the permeability increases beyond a critical threshold of the crack density parameter, depending on the value of

K_{crack} / K_{m}

(Dormieux and Kondo 2004).

An interesting connection between CCM model and the percolation model can be found. Although these two models were established based on two totally different assumptions, the effective permeability (

K_{eff} / K_{m}

) of uncorrelated crack networks with random distributed cracks, derived by the percolation scaling law in this study, is in good agreement with the effective permeability calculated by the CCM model (short-range interaction).

The CCM model can predict the effect of the crack opening aspect ratio on the features of the conversion from a nonconnected solid to a completely connected solid, i.e., the percolation threshold depends on the crack opening aspect ratio (Leonhart et al. 2015, 2017). The percolation scaling law deals with the crack clustering effect and takes the cracks no longer as homogeneous inclusions but as networks, i.e., the percolation threshold relies on the local crack clustering effect and the connectivity factor.

Further Analysis

Impact of Crack Opening

Fig. 11 presents the effective permeability evolution

K_{eff}

as the crack density ranges between [

0, ρ_{c}

] for cgs (

w = 0.001

to 0.028) for uncorrelated crack networks (

D_{c} = 2.0

) with domain size

χ = 20

. The influence of opening aperture

w

on

K_{eff}

is less important as

ρ ≪ ρ_{c}

and becomes remarkable as

ρ \to ρ_{c}

. Approaching the percolation threshold, a flow path is almost formed, and the flow rate is directly influenced by the crack opening aperture.

Fig. 11. Effective permeability evolution $K_{eff}$ versus crack density $ρ$ , ranging from 0 to $ρ_{c}$ , for different crack openings $w$ (constant length, $χ = 20$ , and $D_{c} = 2.0$ ).

To obtain the impact of crack opening

w

on

K_{eff}

in Eq. (25) as

ρ \to ρ_{c}

, the interested density range is

ρ \in [ρ_{c} - b, ρ_{c})

, with

b = 0.33

or 0.38 for uncorrelated and correlated crack networks, respectively. Figs. 12(a–d) illustrate the effective permeability

K_{eff}

in terms of crack openings

w

for correlated/uncorrelated networks with constant and lognormal crack length distributions. The effective permeability

K_{eff}

always increases with

w = 0.001

to 0.004 at a fixed crack density but keeps rather steady as

w

scales from 0.007 to 0.028. It is reasonable because for large

w

values, the local crack permeability becomes very high.

Fig. 12. (a, c, e, and g) Effective permeability $K_{eff}$ ; and (b, d, f, and h) $K_{0}$ in terms of opening aperture $w$ : (a and b) $D_{c} = 2.0$ and constant length; (c and d) $D_{c} = 1.75$ and constant length; (e and f) $D_{c} = 1.62$ and constant length; and (g and h) $D_{c} = 2.0$ and lognormal length.

Figs. 12(a–h) present the logarithmic plots of

K_{0} / K_{m}

to

w

for various

ρ

and

D_{c}

. On the logarithmic scale

K_{0} / K_{m}

is a linear function of

w

for the range

0.001 \leq w \leq 0.004

, i.e., as follows:

K_{0} \propto w^{λ} for 0.001 \leq w \leq 0.004

(26)

The

λ

exponents are, respectively, 0.199, 0.477, 0.554, and 0.223 for

D_{c} = 2.0

(constant length),

D_{c} = 1.75

(constant length),

D_{c} = 1.62

(constant length), and

D_{c} = 2.0

(lognormal length). The exponent

λ

versus

D_{c}

is given in the following equation:

λ = - 0.937 D_{c} + 2.09

(27)

Substituting Eq. (26) into Eq. (25) gives

\frac{K_{eff}}{K_{m}} = \frac{K_{0}}{K_{m}} {| ρ - ρ_{c} |}^{μ} = {\begin{cases} υ w^{- 0.937 D_{c} + 2.09} {| ρ - ρ_{c} |}^{μ} & 0.001 \leq w \leq 0.004 \\ 4 {| ρ - ρ_{c} |}^{μ} & w > 0.004 \end{cases}

(28)

where

υ

= fitting parameter, given in Table 2. Due to the amplification of local flow caused by clustering effects,

υ

increases for correlated crack networks (

D_{c} < 2.0

). The expression in Eq. (28) reveals that

K_{eff}

depends on the crack opening

w

via a power law through the term

K_{0}

.

With the specific parameters retained, the permeability of solids containing percolated crack networks presents an increase of several orders of magnitude compared with the unpercolated cases. When cracks intersect to form a percolated path across the boundaries of the domain, the effective permeability will be significantly altered (Leonhart et al. 2017; Renshaw et al. 2020; Benkemoun et al. 2018). After onset of percolation, the effective permeability is generally stable [Fig. 13(a)], and more likely to be determined by the total length of the percolated path. Although the permeability becomes higher when the crack density is larger, the overall impact of crack density on the permeability of percolated networks is limited. However, for the percolated crack networks, the effective permeability

K_{eff}

exhibits a power law with respect to the crack opening

w

. In Fig. 13(b), the exponent of power law is regressed as 2.968, which is in good agreement with the results reported in literature (Li et al. 2019; Reinhardt and Jooss 2003; Picandet et al. 2009).

Fig. 13. Effective permeability evolution after percolation ( $ρ > ρ_{c}$ , constant length, $χ = 20$ , and $D_{c} = 2.0$ ): (a) $K_{eff} - ρ$ for different crack openings $w$ ; and (b) $K_{eff} - w$ at a fixed $ρ = 2.0$ .

Experimental Data

The correlation between a two-dimensional (2D) crack network and the effective permeability of cement-based materials was investigated by Li and Li (2018). The actual crack networks were extracted, and the geometry parameters including crack density, crack orientation, connectivity factor, and crack opening aperture were evaluated. The effective permeability was obtained indirectly using moisture transport modeling and drying tests. The cracked specimens had density in the range of 0.044–0.399 and relative permeability

K_{eff} / K_{m}

of 1–4.3. The experimental values

K_{eff} / K_{m}

and

ρ

were employed to compare with the permeability evolution derived from the model in Fig. 14. From the extracted actual crack geometry, i.e.,

ρ < 0.4

, the established model was vibrated against the experimental results. The missing clear comparison for

ρ > 0.4

is due the limited crack density extent.

Fig. 14. IDD model and IDD extended model of effective permeability compared with experimental results. (Data from Li and Li 2018.)

Conclusions

This paper established a comprehensive modeling for microcracked porous solids incorporating cracks with clustering extent from nonoverlapping to percolation. Both EMT and percolation models were employed to construct the modeling for overall permeability estimate and these models are validated against numerical simulations for cracks with different geometry characteristics. From the obtained results, the following conclusions can be reached:

•

Once porous solids contain internal cracks, the cracks are to promote the fluid flow and the overall permeability is increased. The classical IDD model from EMT can predict rather satisfactorily the overall permeability of porous solids containing nonoverlapped cracks. The pertinent geometry parameter is crack density, and the overall permeability changes gradually with the crack density. Because the perturbance of cracks on fluid flow is rather limited, the impact of the crack opening’s aperture is not important.

•

For cracks with local clustering, the fluid flow is notably influenced by the geometry characteristics of cracks, and pertinent parameters include the crack density and the crack connectivity. The classical IDD model was extended to consider this local clustering through a two-step scheme in which the local clustering is addressed by a supplementary crack density parallel to the flow direction. From the numerical simulations, this supplementary crack density was regressed for specific crack networks, and the extended-IDD model showed good prediction capacity for overall permeability in the crack density range of

ρ \in (0, ρ_{c} - b)

with

b = 0.33

or 0.38 for uncorrelated and correlated crack networks, respectively. The prediction range of the IDD model was greatly enhanced. Again, the impact of crack opening aperture is not important.

•

For crack clustering approaching the percolation threshold, the scaling law of the effective permeability of cracked solids was established through percolation theory as

K_{0} {| ρ - ρ_{c} |}^{μ}

. The crack opening now comes to exert remarkable influence on the overall permeability. The term

K_{0}

reveals the contribution of crack opening as well as the permeable matrix, and the power-law term reveals the contribution of network geometry characteristics through crack density, crack connectivity, and cluster tortuosity. Before the onset of percolation, the term

K_{0}

scales to the crack opening aperture through a power law with the exponent correlated to the fractal dimension of crack network; after the percolation, the power law is stable with an exponent regressed to 2.968.

Notation

The following symbols are used in this paper:

$A_{s, a}$: surface/domain area;
$a_{1, 2, 3}$: semi axes of inclusion;
$a_{1, 2, 3}^{0}$: semi axes of ellipsoidal matrix;
$c_{i}$: inclusion volume fraction;
$D_{c, τ}$: fractal dimension;
$F$: flux tensor;
$f$: connectivity factor;
$H, H_{i}$: permeability increment;
$H^{d}, H^{idd}$: dilute/IDD estimate of permeability;
$J$: geometry tensor of ellipsoidal inclusion;
$K, K_{i, m}$: permeability tensor;
$K_{eff}$: effective permeability;
$K_{i}$: crack permeability;
$K_{m}$: matrix permeability;
$K_{0}$: permeability contributed by $w$ and $K_{m}$ ;
$L, χ$: domain size/domain size ratio;
$l_{i}$: $i$ th crack length;
$l_{mean}$: mean crack length;
$N$: crack number;
$N_{ξ}$: cluster number;
$P$: pressure tensor;
$q$: flow rate;
$w$: crack opening aperture;
$β, ν, μ$: percolation exponents;
$γ_{D}$: geometrical parameter of matrix;
$\nabla p$: pressure gradient;
$η$: fluid viscosity;
$λ$: permeability exponent;
$ξ$: correlation length of clusters;
$ξ_{i}$: projected length of $i$ th cluster;
$ρ$: crack density;
$ρ_{c}$: percolation threshold;
$ρ_{h}$: equivalent horizontal density;
$ι (θ)$: orientation factor;
$υ, b$: scaling parameters;
$Ω_{D i}^{0}, Ω_{i}^{0}$: eigenstiffness tensor; and
$ω$: orientation degree.

Appendix. Basis for IDD Scheme

This section is derived from the following studies: Zheng and Du (2001), Du and Zheng (2002), Shafiro and Kachanov (2000), and Zhou et al. (2011). The linear permeation problem in cracked solids can be expressed

F = K \cdot P

(29)

where

F

= liquid flux;

P

= pressure gradient; and

K

is the permeability tensor. In the following, the permeability tensors

K_{m}

,

K_{i}

, and

K

stand for the matrix, the

i

th inclusion, and the effective medium, respectively. The tensors

H_{i}

and

H

refer to the permeability increment of the

i

th inclusion and effective medium compared with the matrix, described as follows:

H = K - K_{m}, H_{i} = K_{i} - K_{m}

(30)

For the inclusion-matrix morphology, the IDD scheme takes into account the interaction between inclusions and their surrounding matrix, giving explicit solutions for linear physical properties (Zheng and Du 2001). The dilute estimation of the permeability increment

H^{d}

for a solid containing

N

groups of inclusions is expressed

H^{d} = \sum_{i = 1}^{N} H_{i}^{d} = \sum_{i = 1}^{N} c_{i} {(H_{i}^{- 1} + Ω_{i}^{0})}^{- 1}

(31)

where

H_{i}^{d}

and

c_{i}

= dilute estimation and volumetric fraction of inclusion

i

, respectively;

Ω_{i}^{0}

is the eigenstiffness tensor, which characterizes the shape and orientation of inclusion

i

; superscript 0 = dependency on the matrix; and

d

= dilute.

By considering the interaction between inclusions, the permeability

H

obtained through IDD method can be written (Du and Zheng 2002)

H^{idd} = {(I - \sum_{i = 1}^{N} H_{i}^{d} \cdot Ω_{D i}^{0})}^{- 1} H^{d}

(32)

where

Ω_{D i}^{0}

is the eigenstiffness tensor of matrix atmosphere; and

H^{idd}

is directly calculated by

H_{i}^{d}

,

Ω_{D i}^{0}

, and

H^{d}

. These quantities all have definite physical meanings and can be quantitatively expressed. If all inclusion and matrix atmospheres have same shape and orientation,

Ω_{D i}^{0} = Ω_{D}^{0}

, and Eq. (32) degenerates to

H^{idd} = {({(H^{d})}^{- 1} - Ω_{D}^{0})}^{- 1}

(33)

Following the development of Shafiro and Kachanov (2000) and Zhou et al. (2011), considering an ellipsoidal matrix atmosphere with permeability

K_{m}

containing one ellipsoidal inclusion, the effective permeability estimated by dilute scheme is described

H^{d} = \sum_{i = 1}^{N} H_{i}^{d}, H_{i}^{d} = c_{i} {(H_{i}^{- 1} + K_{m}^{- 1} J)}^{- 1}

(34)

where

J

is the geometry tensor related to the size of ellipsoidal inclusion. Assuming that

a_{1, 2, 3}

are semi axes of the inclusion and

a_{1} > a_{2} > a_{3}

, the components

J_{j = 1, 2, 3}

are given

J_{1} = \frac{a_{1} a_{2} a_{3}}{(a_{1}^{2} - a_{2}^{2}) \sqrt{a_{1}^{2} - a_{3}^{2}}} [F (θ, κ) - E (θ, κ)] J_{3} = \frac{a_{1} a_{2} a_{3}}{(a_{2}^{2} - a_{3}^{2}) \sqrt{a_{1}^{2} - a_{3}^{2}}} [\frac{a_{2} \sqrt{a_{1}^{2} - a_{3}^{2}}}{a_{1} a_{3}} - E (θ, κ)] J_{2} = 1 - J_{1} - J_{3}

(35)

with

E (θ, κ) = \int_{0}^{θ} \sqrt{1 - κ^{2} \sin^{2} w} d w F (θ, κ) = \int_{0}^{θ} 1 / \sqrt{1 - κ^{2} \sin^{2} w} d w

(36)

and

θ = \arcsin \sqrt{1 - {(a_{3} / a_{1})}^{2}}, κ = \frac{\sqrt{a_{1}^{2} - a_{2}^{2}}}{\sqrt{a_{1}^{2} - a_{3}^{2}}}

(37)

By equating Eqs. (31) and (34), the eigenstiffness tensor of inclusion

i

can be written

Ω_{i}^{0} = K_{m}^{- 1} \cdot J

(38)

The eigenstiffness tensor of matrix atmosphere

Ω_{D i}^{0}

can also be obtained by Eq. (38) with the known geometry parameters.

Data Availability Statement

All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

The authors acknowledge the support of Natural Science Foundation of China (Nos. 52038004 and 51808218) and Natural Science Foundation of Jiangxi Province (No. 20202ACBL214018).

References

Abdalrahman, T., S. Scheiner, and C. Hellmich. 2015. “Is trabecular bone permeability governed by molecular ordering-induced fluid viscosity gain? Arguments from re-evaluation of experimental data in the framework of homogenization theory.” J. Theor. Biol. 365 (Jan): 433–444. https://doi.org/10.1016/j.jtbi.2014.10.011.

Google Scholar

Adler, P. M., and J. F. Thovert. 1999. Fracture and fracture networks. Dordrecht, Netherlands: Kluwer.

Crossref

Google Scholar

Adler, P. M., J. F. Thovert, and V. V. Mourzenko. 2013. Fractured porous media. London: Oxford University Press.

Google Scholar

Aharony, A., and D. Stauffer. 2003. Introduction to percolation theory. New York: Taylor & Francis.

Google Scholar

Akhavan, A., S. M. H. Shafaatian, and F. Rajabipour. 2012. “Quantifying the effects of crack width, tortuosity, and roughness on water permeability of cracked mortars.” Cem. Concr. Res. 42 (2): 313–320. https://doi.org/10.1016/j.cemconres.2011.10.002.

Google Scholar

Benkemoun, N., H. Alkhazraji, P. Poullain, M. Choinska, and A. Khelidj. 2018. “3-D mesoscale simulation of crack-permeability coupling in the Brazilian splitting test.” Int. J. Numer. Anal. Methods Geomech. 42 (3): 449–468. https://doi.org/10.1002/nag.2749.

Google Scholar

Berkowitz, B. 1995. “Analysis of fracture network connectivity using percolation theory.” Math. Geol. 27 (4): 467–483. https://doi.org/10.1007/BF02084422.

Google Scholar

Damrongwiriyanupap, N., S. Scheiner, B. Pichler, and C. Hellmich. 2017. “Self-consistent channel approach for upscaling chloride diffusivity in cement pastes.” Transp. Porous Media 118 (3): 495–518. https://doi.org/10.1007/s11242-017-0867-3.

Google Scholar

Darcel, C., O. Bour, P. Davy, and J. R. de Dreuzy. 2003. “Connectivity properties of two-dimensional fracture networks with stochastic fractal correlation.” Water Resour. Res. 39 (10): 1272. https://doi.org/10.1029/2002WR001628.

Google Scholar

Davy, P., A. Sornette, and D. Sornette. 1990. “Some consequences of a proposed fractal nature of continental faulting.” Nature 348 (6296): 56–58. https://doi.org/10.1038/348056a0.

Google Scholar

Dormieux, L., L. Jeannin, and N. Gland. 2011. “Homogenized models of stress-sensitive reservoir rocks.” Int. J. Eng. Sci. 49 (5): 386–396. https://doi.org/10.1016/j.ijengsci.2010.12.010.

Google Scholar

Dormieux, L., and D. Kondo. 2004. “Coupling between permeability and damage: A micromechanical approach.” In Proc., XXI Int. Congress of Theoretical and Applied Mechanics. New York: Springer.

Google Scholar

Dormieux, L., and D. Kondo. 2005. “Diffusive transport in disordered media: Application to the determination of the tortuosity and the permeability of cracked materials.” In Applied micromechanics of porous materials, edited by L. Dormieux and F.-J. Ulm, 83–106. New York: Springer.

Crossref

Google Scholar

Du, D. X., and Q. S. Zheng. 2002. “A further exploration of the interaction direct derivative (IDD) estimate for the effective properties of multiphase composites taking into account inclusion distribution.” Acta Mech. 157 (1): 61–80. https://doi.org/10.1007/BF01182155.

Google Scholar

Eshelby, J. D. 1957. “The determination of the elastic field of an ellipsoidal inclusion, and related problems.” Proc. R. Soc. London, Ser. A 241 (1226): 376–396. https://doi.org/10.1098/rspa.1957.0133.

Google Scholar

Estermann, S., and S. Scheiner. 2018. “Multiscale modeling provides differentiated insights to fluid flow-driven stimulation of bone cellular activities.” Front. Phys. 6 (Sep): 76. https://doi.org/10.3389/fphy.2018.00076.

Google Scholar

Guéguen, Y., T. Chelidze, and M. Le Ravalec. 1997. “Microstructures, percolation thresholds, and rock physical properties.” Tectonophysics 279 (1): 23–35. https://doi.org/10.1016/S0040-1951(97)00132-7.

Google Scholar

Hasselman, D. P. H. 1978. “Effect of cracks on thermal conductivity.” J. Compos. Mater. 12 (4): 403–407. https://doi.org/10.1177/002199837801200405.

Google Scholar

Hu, X., X. Song, K. Wu, Y. Shi, and G. Li. 2021. “Effect of proppant treatment on heat extraction performance in enhanced geothermal system.” J. Pet. Sci. Eng. 207 (Dec): 109094. https://doi.org/10.1016/j.petrol.2021.109094.

Google Scholar

Hunt, A., and R. Ewing. 2009. Percolation theory for flow in porous media. Cham, Switzerland: Springer.

Google Scholar

Jeannin, L., and L. Dormieux. 2017. “Poroelastic behaviour of granular media with poroelastic interfaces.” Mech. Res. Commun. 83 (Jul): 27–31. https://doi.org/10.1016/j.mechrescom.2017.04.002.

Google Scholar

Kachanov, M. 1992. “Effective elastic properties of cracked solids: Critical review of some basic concepts.” Appl. Mech. Rev. 45 (8): 304–335. https://doi.org/10.1115/1.3119761.

Google Scholar

Kushch, V. 2020. Micromechanics of composites: Multipole expansion approach. Oxford, UK: Butterworth-Heinemann.

Google Scholar

Langlois, V., V. H. Trinh, C. Lusso, C. Perrot, X. Chateau, Y. Khidas, and O. Pitois. 2018. “Permeability of solid foam: Effect of pore connections.” Phys. Rev. E 97 (5): 053111. https://doi.org/10.1103/PhysRevE.97.053111.

Google Scholar

Lemarchand, E., C. A. Davy, L. Dormieux, and F. Skoczylas. 2010. “Tortuosity effects in coupled advective transport and mechanical properties of fractured geomaterials.” Transp. Porous Media 84 (1): 1–19. https://doi.org/10.1007/s11242-009-9481-3.

Google Scholar

Lemarchand, E., L. Dormieux, and F.-J. Ulm. 2005. “Micromechanics investigation of expansive reactions in chemoelastic concrete.” Philos. Trans. R. Soc. London, Ser. A 363 (1836): 2581. https://doi.org/10.1098/rsta.2005.1588.

Google Scholar

Leonhart, D., J. J. Timothy, and G. Meschke. 2015. “Determination of effective transport properties of fractured rocks using the extended finite element method and micromechanics.” In Proc., ISRM Regional Symp.-EUROCK 2015. Austin, TX: OnePetro.

Google Scholar

Leonhart, D., J. J. Timothy, and G. Meschke. 2017. “Cascade continuum micromechanics model for the effective permeability of solids with distributed microcracks: Comparison with numerical homogenization.” Mech. Mater. 115 (Dec): 64–75. https://doi.org/10.1016/j.mechmat.2017.09.001.

Google Scholar

Li, K. 2016. Durability design of concrete structure: Phenomena, modelling and practice. New York: Wiley.

Crossref

Google Scholar

Li, K., and L. Li. 2019. “Crack-altered durability properties and performance of structural concretes.” Cem. Concr. Res. 124 (Oct): 105811. https://doi.org/10.1016/j.cemconres.2019.105811.

Google Scholar

Li, K., M. Ma, and X. Wang. 2011. “Experimental study of water flow behaviour in narrow fractures of cementitious materials.” Cem. Concr. Compos. 33 (10): 1009–1013. https://doi.org/10.1016/j.cemconcomp.2011.08.005.

Google Scholar

Li, L., and K. Li. 2015. “Permeability of microcracked solids with random crack networks: Role of connectivity and opening aperture.” Transp. Porous Media 109 (1): 217–237. https://doi.org/10.1007/s11242-015-0510-0.

Google Scholar

Li, L., and K. Li. 2018. “Experimental investigation on transport properties of cement-based materials incorporating 2d crack networks.” Transp. Porous Media 122 (3): 647–671. https://doi.org/10.1007/s11242-018-1019-0.

Google Scholar

Li, L., W. Liu, Q. You, M. Chen, and Q. Zeng. 2020. “Waste ceramic powder as a pozzolanic supplementary filler of cement for developing sustainable building materials.” J. Cleaner Prod. 259 (Jun): 120853. https://doi.org/10.1016/j.jclepro.2020.120853.

Google Scholar

Li, M., Y. B. Tang, Y. Bernabé, J. Z. Zhao, X. F. Li, and T. Li. 2017. “Percolation connectivity, pore size, and gas apparent permeability: Network simulations and comparison to experimental data.” J. Geophys. Res.: Solid Earth 122 (7): 4918–4930. https://doi.org/10.1002/2016JB013710.

Google Scholar

Li, X., D. Li, and Y. Xu. 2019. “Modeling the effects of microcracks on water permeability of concrete using 3D discrete crack network.” Compos. Struct. 210 (Feb): 262–273. https://doi.org/10.1016/j.compstruct.2018.11.034.

Google Scholar

Liu, J., J. Huang, K. Liu, and K. Regenauer-Lieb. 2021. “Identification, segregation, and characterization of individual cracks in three dimensions.” Int. J. Rock Mech. Min. Sci. 138 (Feb): 104615. https://doi.org/10.1016/j.ijrmms.2021.104615.

Google Scholar

Liu, J., and K. Regenauer-Lieb. 2021. “Application of percolation theory to microtomography of rocks.” Earth Sci. Rev. 214 (Mar): 103519. https://doi.org/10.1016/j.earscirev.2021.103519.

Google Scholar

Mustapha, H., K. Makromallis, and A. Cominelli. 2020. “An efficient hybrid-grid crossflow equilibrium model for field-scale fractured reservoir simulation.” Comput. Geosci. 24 (2): 477–492. https://doi.org/10.1007/s10596-019-09838-3.

Google Scholar

Nguyen, T. T. N., M. N. Vu, N. H. Tran, N. H. Dao, and D. T. Pham. 2020. “Stress induced permeability changes in brittle fractured porous rock.” Int. J. Rock Mech. Min. Sci. 127 (Mar): 104224. https://doi.org/10.1016/j.ijrmms.2020.104224.

Google Scholar

Norris, A. N. 1985. “A differential scheme for the effective moduli of composites.” Mech. Mater. 4 (1): 1–16. https://doi.org/10.1016/0167-6636(85)90002-X.

Google Scholar

Picandet, V., A. Khelidj, and H. Bellegou. 2009. “Crack effects on gas and water permeability of concretes.” Cem. Concr. Res. 39 (6): 537–547. https://doi.org/10.1016/j.cemconres.2009.03.009.

Google Scholar

Pivonka, P., C. Hellmich, and D. Smith. 2004. “Microscopic effects on chloride diffusivity of cement pastes—A scale-transition analysis.” Cem. Concr. Res. 34 (12): 2251–2260. https://doi.org/10.1016/j.cemconres.2004.04.010.

Google Scholar

Reinhardt, H. W., and M. Jooss. 2003. “Permeability and self-healing of cracked concrete as a function of temperature and crack width.” Cem. Concr. Res. 33 (7): 981–985. https://doi.org/10.1016/S0008-8846(02)01099-2.

Google Scholar

Renshaw, C. E., E. M. Schulson, D. Iliescu, and A. Murzda. 2020. “Increased fractured rock permeability after percolation despite limited crack growth.” J. Geophys. Res.: Solid Earth 125 (8): e2019JB019240. https://doi.org/10.1029/2019JB019240.

Google Scholar

Shafiro, B., and M. Kachanov. 2000. “Anisotropic effective conductivity of materials with nonrandomly oriented inclusions of diverse ellipsoidal shapes.” J. Appl. Phys. 87 (12): 8561–8569. https://doi.org/10.1063/1.373579.

Google Scholar

Timothy, J. J. 2016. “Analytical and computational models for the effective properties of disordered microcracked porous materials.” Ph.D. thesis, Faculty of Civil and Environmental Engineering, Ruhr-Univ. Bochum.

Google Scholar

Timothy, J. J., and G. Meschke. 2016. “Cascade lattice micromechanics model for the effective permeability of materials with microcracks.” J. Nanomech. Micromech. 6 (4): 04016009. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000113.

Google Scholar

Timothy, J. J., and G. Meschke. 2017a. “Cascade continuum micromechanics model for the effective permeability of solids with distributed microcracks: Self-similar mean-field homogenization and image analysis.” Mech. Mater. 104 (Jan): 60–72. https://doi.org/10.1016/j.mechmat.2016.10.005.

Google Scholar

Timothy, J. J., and G. Meschke. 2017b. “The intrinsic permeability of microcracks in porous solids: Analytical models and Lattice Boltzmann simulations.” Int. J. Numer. Anal. Methods Geomech. 41 (8): 1138–1154. https://doi.org/10.1002/nag.2673.

Google Scholar

Timothy, J. J., and G. Meschke. 2018. “Effective diffusivity of porous materials with microcracks: Self-similar mean-field homogenization and pixel finite element simulations.” Transp. Porous Media 125 (3): 413–434. https://doi.org/10.1007/s11242-018-1126-y.

Google Scholar

Torrijos, M. C., G. Giaccio, and R. Zerbino. 2010. “Internal cracking and transport properties in damaged concretes.” Supplement, Mater. Struct. 43 (S1): 109–121. https://doi.org/10.1617/s11527-010-9602-z.

Google Scholar

Vinci, C., J. Renner, and H. Steeb. 2014. “A hybrid-dimensional approach for an efficient numerical modeling of the hydromechanics of fractures.” Water Resour. Res. 50 (2): 1616–1635. https://doi.org/10.1002/2013WR014154.

Google Scholar

Wang, Y., L. Jeannin, F. Agostini, L. Dormieux, and E. Portier. 2018. “Experimental study and micromechanical interpretation of the poroelastic behaviour and permeability of a tight sandstone.” Int. J. Rock Mech. Min. Sci. 103 (Mar): 89–95. https://doi.org/10.1016/j.ijrmms.2018.01.007.

Google Scholar

Whitaker, S. 1986. “Flow in porous media I: A theoretical derivation of Darcy’s law.” Transp. Porous Media 1 (1): 3–25. https://doi.org/10.1007/BF01036523.

Google Scholar

Zheng, Q. S., and D. X. Du. 2001. “An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution.” J. Mech. Phys. Solids 49 (11): 2765–2788. https://doi.org/10.1016/S0022-5096(01)00078-3.

Google Scholar

Zhou, C., K. Li, and X. Pang. 2011. “Effect of crack density and connectivity on the permeability of microcracked solids.” Mech. Mater. 43 (12): 969–978. https://doi.org/10.1016/j.mechmat.2011.08.011.

Google Scholar

Zhou, C., K. Li, and X. Pang. 2012. “Geometry of crack network and its impact on transport properties of concrete.” Cem. Concr. Res. 42 (9): 1261–1272. https://doi.org/10.1016/j.cemconres.2012.05.017.

Google Scholar

Information & Authors

Information

Published In

Journal of Engineering Mechanics

Volume 148 • Issue 9 • September 2022

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Oct 8, 2021

Accepted: Apr 13, 2022

Published online: Jun 21, 2022

Published in print: Sep 1, 2022

Discussion open until: Nov 21, 2022

Authors

Affiliations

Le Li https://orcid.org/0000-0001-7798-7120 [email protected]

Assistant Researcher, Dept. of Civil Engineering, Tsinghua Univ., Beijing 100084, PR China. ORCID: https://orcid.org/0000-0001-7798-7120. Email: [email protected]

View all articles by this author

Kefei Li [email protected]

Professor, Dept. of Civil Engineering, Tsinghua Univ., Beijing 100084, PR China (corresponding author). Email: [email protected]

View all articles by this author

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Abstract

Introduction

Geometry of Random Crack Networks

Crack Density

Orientation

Fractal Dimension

Connectivity

Permeability of Cracked Solids

IDD Model for Nonoverlapping Cracks

Extended IDD Model for Low and Medium Crack Clustering

Percolation Model for High Crack Clustering

Model Summary

Numerical Analysis

Numerical Generation of Random Crack Networks

Permeability Simulation

Model Validation

IDD Model for Nonoverlapping Cracks

Extended IDD Model for Low and Medium Crack Clustering

Percolation Model for High Crack Clustering

Further Analysis

Impact of Crack Opening

Experimental Data

Conclusions

Notation

Appendix. Basis for IDD Scheme

Data Availability Statement

Acknowledgments

References

Information

Published In

Copyright

History

ASCE Technical Topics:

Authors

Affiliations

Metrics

Citations

Download citation

Figures

Other

Share

Copy the content Link

Share with email

Share

Request Username

Create a new account

Change Password

Password Changed Successfully

Verify Phone

Congrats!