Anisotropic permeability evolution model of rock in the process of deformation and failure

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2012,24(1):25-31 DOI: 10.1016/S1001-6058(11)60215-1

ANISOTROPIC PERMEABILITY EVOLUTION MODEL OF ROCK IN THE PROCESS OF DEFORMATION AND FAILURE* WANG Huan-ling Key Laboratory of Coastal Disaster and Defence, Ministry of Education, Hohai University, Nanjing 210098, China, E-mail: [email protected] CHU Wei-jiang Hydrochina Huadong Engineering Corporation, Hangzhou 310014, China HE Miao Jiangsu Transportation Research Institute CO., LTD, Nanjing 211112, China

(Received April 29, 2011, Revised September 30, 2011) Abstract: The rock permeability is an important parameter in the studies of seepage and stress coupling. The micro-cracks and pores can initiate and grow on a small scale and coalesce to form large-scale fractures and faults under compressive stresses, which would change the hydraulic conductivity of the rock, and therefore, the rock permeability. The rock permeability is, therefore, closely related with the micro-cracking growing, coalescence, and macro new fracture formation. This article proposes a conceptual model of rock permeability evolution and a micro kinematics mechanism of micro-cracking on the basis of the basic theory of micromechanics. The applicability of the established model is verified through numerical simulations of in situ tests and laboratory tests. The simulation results show that the model can accurately forecast the peak permeability evolution of brittle rock, and can well describe the macro-experimental phenomenon before the peak permeability evolution of brittle rock on a macro-scale. Key words: rock hydraulics, micro-mechanics, permeability, model validation

Introduction The stresses will be re-distributed in a rock mass, under external forces or other engineering situations, which in turn will significantly change the permeability within the rock mass. This process is now a hot issue in the study of the coupling between the seepage field and the stress field in rock masses. With the development of new methods and new equipment in rock mechanics tests, especially, the application of the seepage coupling experimental method in the stressstrain process, it is widely recognized that the permea* Project supported by the Natural National Science Foundation of China (Grant Nos. 51009052, 11172090), the Three Gorge Research Center for Geo-Hazards, the Ministry of Education (Grant No. TGRC201026), the Fundamental Research Funds for the Central Universities (Grant No. 2010B02514), and the Key Laboratory of Coastal Disasters and Defence of Ministry of Education, Hohai University (Grant No. 2010016). Biography: WANG Huan-ling (1976-), Female, Ph. D., Associate Professor

bility of rock is closely related with the micro-crack growth and coalescence, and the macro-fracture formation[1]. A large number of experimental curves related with stress-strain relations and permeability evolutions[2] show that the rock stress state and the deformation degree in different directions in the stressstrain process are not the same, therefore, the evolution of micro-crack opening, density, and connectivity are different and the permeability is not the same in different directions, that is, the permeability shows a significant anisotropy[3]. So the rock deformation not only changes the permeability value, but also brings about a significant anisotropy, which should be considered in rock engineering. Extensive researches were carried out on the impact of the load (the stress state) on the rock permeability, which may be summarized as follows: (1) The empirical formulas obtained through extensive experiments. Early studies focused on establishing the relationship between seepage and stress by using direct experiments or indirect analyses. Among them, the most important result is on the effect of the normal

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Fig.1 Conceptual model of permeability evolution caused by microscopic micro-cracks changes in the process of deformation of brittle rock

stress on the permeability of a single fracture. The representative formulas were obtained by Barton (1995) and Oda et al.[4], and the Barton’s formula is widely used at present. The studies of the impact of the shear stress on the permeability of a single fracture lagged behind and with fewer significant results. But it was confirmed by many studies that the shear stress has a significant effect on the rock permeability. The most influential results were obtained by Olsson and Barton[5], Lee and Cho[6], Makurat[7]. With the development of technology in the rock mechanics tests, the seepage characteristics of the natural fracture rock under the three-dimensional stress state were studied by Tang et al.[8], Tan et al.[9], Liu et al.[10]. Useful conclusions were obtained, which may serve as a guide for practical projects. (2) The theoretical formulas based on the conceptual model. Liu et al.[10] derived the permeability formula for fractured rock mass under complex conditions based on the relationships between the stress-seepage and the strain-seepage. In these studies, a common problem is a strong hypothesis towards the anisotropy of permeability. For instance, Liu et al.[10] assumed that the principal directions of seepage and stress were the same, which, strictly speaking, is unsubstantiated, and thus the practicability of the established model is limited. (3) Statistical laws based on a large number of numerical experiments. Many numerical experiments were carried out by using Discrete Fracture Network (DFN) and Discrete Element Model (DEM). The relationship between permeability and stress state, and loading path was obtained by statistical analyses of calculation results. This method is the most accurate in its consideration of anisotropy as compared with other methods, but so far they are limited to two-dimensional cases.

If the permeability is not adequately considered in the coupling analysis of the seepage field and the stress field, the practicability of the established coupled model will be limited and the true coupling rules of rock can not be revealed. This study focuses on the anisotropy of the permeability evolution in the process of rock deformation from a micromechanics view, to develop a reasonable mathematical model of fluidsolid coupling. The applicability of the proposed anisotropy permeability model is validated by simulating the in situ experiments and the laboratory experiments.

1. Anisotropy permeability model 1.1 Conceptual model Extensive rock micro-experiments and theoretical analyses[11,12] show that the micro-evolution of rock micro-cracks in the process of deformation is the main influencing factor for the increase of permeability within the rock system, and micro-cracks are the main channels of water flow. Based on a great number of test results of the stress-strain curves and the permeability curves under the triaxial compression[13,14], a conceptual model of the permeability evolution in different stress and strain processes is established, and the permeability evolution involves the following four phases (Fig.1): (1) Elastic compression stage. The compression force tends to close the initial micro-cracks and the deformation of micro-cracks is reversible at this stage, that is, the deformation of micro-cracks disappears after unloading, and the permeability decreases at this stage. (2) Stable stage of compression. The initial opening of micro-cracks is the minimum at this stage.

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The micro-cracks do not grow or slip and the permeability at this stage changes very little and can be approximately considered as a constant. (3) Rapid increase stage for permeability. The micro-cracks slip and expand with the increase of deformation, the fracture opening increases, and the fracture network connectivity is enhanced at this stage. Meanwhile, the new fissures within the rock system begin to crop up. These factors increase the water conductivity of rock by 2-4 orders of magnitude. (4) Post-peak phase. The material becomes softened at this stage, and the rock permeability is changed closely according to the confining pressure during the test. When the confining pressure is high, the formed macro-cracks have a tendency of compaction, thus the permeability decreases, when the confining pressure is low, the permeability further increases with the increase of rock deformation, but the increase rate is very small. In order to study the macroscopic phenomenon of anisotropic permeability changes in the stress and strain processes using the established model, one must establish the micro-mechanical properties of microcracks based on the following rules and assumptions: (1) The change of the stress (deformation) state will change the opening, density, and connectivity of micro-cracks, and thus will change the rock permeability, that is, the change in rock permeability is related with the opening, density, and connectivity of microcracks. (2) The water flow within the micro-cracks obeys the generalized flow law. (3) The critical starting gradient is not considered. (4) The rock is regarded as a binary system composed of the rock matrix and micro-cracks. The rock permeability depends on the following two factors: the single fracture permeability, and the connectivity degree of the micro-cracks. (5) The rock mass has a Representative Element Volume (REV). (6) The shape of the rock fracture is assumed to follow the Baecher disk model. (7) The interaction between micro-cracks is neglected, and the mechanical effect of all micro-cracks is in line with the superposition principle. 1.2 Permeability of a single micro-crack The generalized flow law can be expressed as follows when the hydraulic opening of a single fracture is more than 50 Pm

U g Q = w b[ 'H 12 P f

(1)

where Q is the flow in unit time at the cross-section of micro-cracks, 'H is the hydraulic gradient, U w

is the density of water, g is the gravity acceleration, P is the viscosity of water, b is the mechanical opening of micro-cracks, [ is the index of mechanical opening of micro-cracks, and f is the roughness of the surface of micro-cracks. It can be seen from Eq.(1) that the generalized flow law would be turned into a cubic law when [ is equal to three. Equation (1) can be expressed in the form of Darcy’s law as c

v =

U w g [ 1 c b J 12 P f

(2)

c

where v is the actual velocity vector along the fracture, and the superscript c refers to micro-cracks, c

J is the hydraulic gradient along the micro-crack surfaces. Equation (2) can be expressed in the tensor form as vic =

U w g [ 1 b G ij  ni n j J cj 12 P f

(3)

where G ij is the unit vector, and ni and n j are normal unit vectors of fracture surfaces. Therefore, the permeability coefficient K of the single micro-cracks is expressed as

K=

U w g [ 1 b G  n … n 12 P f

(4)

where G is the second rank unit tensor. The relationship between the permeability coefficient and the permeability is as follows

K=

Uw g k P

(5)

Thus, the permeability k of a single microcrack can be written in the following form

k=

b[ 1 G n…n 12 f





(6)

By comparing Eq.(4) with Eq.(6), it can be seen that the permeability k is only related to the transport characteristics of fracture networks, while the permeability coefficient K is related to fluid properties in addition to the fracture network. The macro-permeability of a single group microcrack can be obtained by the homogenization method:

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according to assumption (6), and for describe the geometric properties of a group of micro-cracks, the folloW wing three variables are required, that is, n , 1 W , W

and aW , where n

is the normal vector of the ith

group microcrack ( W = 1, n ), and 1 W and aW denote the numbers and radii of micro-cracks in unit volume, respectively. Here, the fracture density can be expressed as d W = N W aW 3 . The volume integral is defined as follows for the flow velocity within all micro-cracks in REV, in order to obtain the average velocity within a group of micro-cracks W

v =

1

1

v d: = ³ : ³: : : i

c

vic d: c

(7)

where vi is the actual velocity vector of water within the rock mass, vic is the actual velocity vector of water flow along the fracture within the rock mass, and : c is the volume occupied by fractures. As for the group of micro-cracks with the unit normal vector of n , the average openings of microcracks in the same direction in the REV are assumed to have the same value, and thus : c can be written as

d: c = N W S  aW bW 2

concerned. The connected fractures would form a water flow channel, and disconnected fractures are not involved in the water flow and the cycle alternation and need not be considered in the seepage calculation. Therefore, it is necessary to introduce a parameter to describe the overall connectivity of micro fracture networks in the permeability tensor. We assume that  is the fracture connectivity factor with values in the range [0, 1] . O = 0 indicates that various micro-cracks are isolated and they cannot form a water flow network, while O = 1 indicates that the micro-cracks are fully developed, and each micro-crack is involved in the water flow. Obviously, the fracture connectivity is related to the fracture density: the greater the density, the greater the connectivity will be. Accordingly, the connectivity factor can be expressed as § d  d0 · ¸ © d ¹

O (d ) = O0 + ¨

2

(12)

where O0 is the initial connectivity factor of microcrack groups, d 0 is the initial density of micro-crack groups, and d is the average density of micro-crack groups under the current load.

(8)

Equations (3) and (8) are substituted into Eq.(7), which is then integrated on unit spheres, to obtain the following equation W

v =

Uw g S 1 J N W b[ (a) 2 G  n … n 12 P f 4S

Uw g J N W b[ (a)2 G  n … n 48P f

(9)

Fig.2 The test curves of permeability of Lac du Bonnet granite ( V c = 5 MPa )

Therefore, the macro-permeability and permeability coefficients of a group of micro-cracks can be expressed as: K

Wc

Wc

=

k =

Uw g W W [ W 2 N (b ) (a ) G  n … n 48P f



1 N W (bW )[ (aW )2 G  n … n 48 f



(10)

(11)

1.3 Permeability of groups of micro-cracks Generally, the fractures can be classified according to whether they are connected or disconnected as far as the permeable fractures containing water are

Fig.3 The test curves of permeability of Lac du Bonnet granite ( V c = 10 MPa )

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Table 1 Parameters of numerical simulation of Lac du Bonnet granite in seepage test K U vs c d 105 N Es K K b f 0

c

(1010Pa) 16.1 13.0

ni

(105J/m2) 0.4

(MPa)

2.3

25

8

31

0.001

1.85

1.5

2

(10 Pa)

(Pm)

0.3

0.4 0.3

In order to determine the superimposed effect of micro-cracks in all directions, the density of microcracks in all directions is integrated over the unit sphere S 2 = ^n, n = 1` , and then the average density

of micro-cracks in the macroscopic scale is obtained as d=

c

10

m 1 W W W d(n)(n … n)dS = ¦ wW d(n )(n … n ) ³ S 4S W =1

(13)

0.8

O0 0.05

b0

Confining pressure

(Pm)

(MPa)

0.4

10

0.8

5

2. Validation for the proposed model of permeability evolution To test whether the established evolution equation matches the real macro test characteristics of brittle rock, according to in situ and experimental data, the evolution law of the permeability tensor with the changes of stress and strain under load is numerically simulated.

where m denotes the number of integration points, and wW is the micro-crack integral weight in the n direction, which can be obtained by a look-up table using the integral method[15]. According to assumption (4), combining Eq.(10) with Eq.(12) and following the superposition principle, the permeability coefficient and the permeability of multiple groups of micro-cracks can be obtained, respectively, as: c

K =

c

k =

Uw g O (d )dN W (bW )[ (aW )2 G  n … n 48P f



1 O (d )dN W (bW )[ (aW )2 G  n … n 48 f



(14)

Fig.5 Numerical simulation of permeability behaviors of Lac du Bonnet granite ( V c = 10 MPa )

(15)

It can be seen from Eq.(15) that the permeability of rock is not a constant during the process of loading, but rather a variable related to the micro-crack state within the rock mass, the density, the connectivity, and other factors.

Fig.6 The test curve of permeability-strain of Los ancient basalt

Fig.4 Numerical simulation of permeability behaviors of Lac du Bonnet granite ( V c = 5 MPa )

2.1 Validation through in situ test In order to evaluate the simulation effectiveness of the proposed model under more complex conditions, the classic and universally accepted in situ test results are used to validate the permeability evolution model. For example, the Lac du Bounet granite, stored in the Nuclear Waste Underground Research Laboratory (URL) in Canada, was extensively studied in literature[13]. The permeability evolution curves of two groups of Lac du Bonnet granite samples, taken from

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Table 2 Parameters of numerical simulation of Los ancient basalt in seepage test K U vs c d Es K K f 0

c

10

5

(10 Pa) 1.52

2

(10 J/m ) 0.23

3

45

0.2

0.1

1.1

a depth of 240 m underground in the Canadian URL, are shown in Figs.2 and 3. In figures, the (V 3  V 1 ) at the vertical axis indicates the deviatoric stress. Because the two groups of rock samples are taken from deep underground, most of the initial micro-cracks within the rock are in the closed state, as verified by images of electron microscopy. Thus the initial permeability of the rock is very low, within a range from 10 m2-18 m2 to 10 m2-19 m2. It can be seen from Figs.2 and 3 that the permeability decreases by 1-2 magnitude in the elastic compressive stage and increases by 2-3 magnitude at the stage of slip and expansion of the micro-cracks. Because the initial permeability and the changes of permeability of the two groups of rock samples are quite different, two sets of parameters are used to simulate the permeability curves, as shown in Table 1. In Table 1, E s and v s are the elastic modulus and Poisson’s ratio of the rock matrix, respectively, K is a material constant used to describe the mate-

ni

bc

(MPa)

10

(10 Pa)

(Pm)

2

5.5

20

105 N

O0

b0 (Pm)

950

0.03

90

the same basalt, their physical properties are similar and the mechanical laws they follow are relatively uniform. The parameters used in the numerical simulation are shown in Table 2 and the simulation results are shown in Fig.7 and Fig.8.

Fig.7 Numerical simulations of permeability behaviors of Los ancient basalt ( V c = 5 MPa )

rial’s expansibility for resisting the tensile stress, kc is used to describe the material’s surface energy when the micro-cracks are in the states of compression and shear, c is the rock cohesion, U is the critical friction of the micro-crack surface when the rock is in the states of compression and shear, f is used to describe the roughness of the fracture surface, K ni is the initial compressibility when the micro-cracks are in the state of elastic compression, b0 and bc are the initial opening and the maximum closure opening of the micro-cracks, respectively, and O 0 is the initial connectivity rate of micro-crack groups. The simulation results are shown in Figs.4 and 5. From Fig.4 and Fig.5, it is shown that the curves of the numerical simulation agree well with the test data with a clear indications of the four test stages of elastic compression, stable stage of permeability, dramatic increase of permeability, and slowing down of increase of permeability. 2.2 Validation through laboratory test According to the permeability test under loading for the Los ancient basalt samples in Liangshan, Sichuan Province, the results are shown in Fig.6 and the test was carried out on MTS815.02 servo testing machine. Because the two rock samples are selected from

Fig.8 Numerical simulations of permeability behaviors of Los ancient basalt ( V c = 10 MPa )

From Fig.7 and Fig.8, it is shown that the proposed model can well describe the permeability evolution behaviors before the peak stage of Los ancient basalt, and also the relation between permeability evolution and confining pressure, but the post-peak stage of the permeability evolution behaviors can not be well described by this model, because the changes of permeability at the post-peak stage are very complex, which is not adequately reflected in the proposed model. In fact, the micro to macro-break phenomena of rock under loading are difficult to be described by the continuum mechanics alone, other angles should be considered.

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3. Conclusion The seepage field and the stress field coupling in the rock mass is an important issue in rock mechanics. The established mathematical models of seepagestress coupling mainly involve the relationships between the permeability and the stress or strain, without a due consideration of the anisotropy and nonhomogeneity of the permeability, which has restricted their applicability. In this study, based on the micromechanics theory and the evolution law of microcrack opening, density, connectivity, and the loading process, an anisotropic permeability evolution model is established and is simultaneously homogenized to the macroscopic scale by mathematical methods, and then the established evolution equation is validated by experimental data. Numerical simulation results show that the proposed model can well describe the permeability evolution law of different brittle rocks under different confining pressures, and especially, the before-peak macro-test phenomenon of permeability evolution behaviors on the macro-scale.

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