Interpretations on how the macroscopic mechanical behavior of sandstone affected by microscopic properties – revealed by bonded-particle model

(1)

Interpretations on how the macroscopic mechanical behavior of sandstone affected

by microscopic properties

—Revealed by bonded-particle model

Yo-Ming Hsieh

a

, Hung-Hui Li

b

, Tsan-Hwei Huang

b

, Fu-Shu Jeng

b,

⁎

a

Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan b_{Department of Civil Engineering, National Taiwan University, Taipei, Taiwan}

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 August 2007

Received in revised form 8 January 2008 Accepted 23 January 2008

Available online 23 February 2008

Macroscopic mechanical properties of sandstones, such as uniaxial compressive strength and Young's modulus were found to be signiﬁcantly affected by their petrographic properties, e.g. the porosity n and the grain area rationGAR. The intricate relationship between the macroscopic properties of sandstones and their petrographic or microscopic properties necessitates further study in exploring how the microscopic properties inﬂuence the macroscopic mechanical behavior.

In this research, numerical analyses based on the bonded-particle model, the microscopic properties of which originated from the bonded strength and stiffness, were thus conducted as a systematic study aiming at unraveling these microscopic mechanisms. A series of tests was conducted, and the results were compared with the actual behavior of sandstone. A numerical model comprised of three types of particles, grain particles GP, matrix particles MP and porous matrix particles PP, was accordingly proposed to represent the sandstone. The results of analyses demonstrated how the petrographic parameterGAR and porosity n determined the proportions and the numbers ofGP, MP and PP. The strength and stiffness of these bonds were estimated based on back analyses. Accordingly, the results of parametric study indicate that matrix particles tend to have stronger bonding strength yet softer stiffness, when compared to the grain particles.

Sandstone is a detrital sedimentary rock composed of rock fragments and mineral grains. The macroscopic mechanical behavior of sand-stones, the uniaxial compressive strength (UCS) (ISRM, 1981) for instance, is inherently affected by the microscopic properties of the sandstone, e.g. the mineral types, porosity, bonding strength, etc. of the constituting particles. How do these microscopic properties of sand-stones affect the macroscopic mechanical behavior of these rocks had been focus of interests in a number of recent researches. The inﬂuence of microscopic factors on the macroscopic strength of rock, were studied by various researchers with regard to the following aspects:

a) Mineralogical composition—The quartz content may contribute positively to the uniaxial compressive strength of sandstones (Smart et al., 1982; Gunsallus and Kulhaway, 1984; Shakoor and Bonelli, 1991). Nevertheless, there are situations that the quartz content of sandstones has no effect on their strength (Bell, 1978; Barbour et al., 1979; Dobereiner and De Freitas, 1986).

b) Cement and matrix_{—The strengths of sandstones are closely} rela-ted to their cementation. Sandstones with higher cement content tend to have higher strength than those with lower cement con-tent (Clough et al., 1981; David et al., 1998). The type of cement also

affects the sandstones' strength. For instance, sandstones with silica or calcareous cement have higher strength than the ones with clay mineral cement (Vutukuri et al., 1974). The term“matrix” is often referred to thefine particles between the grains. In sedi-mentology, thesefine particles in fact include “matrix” deposited during sediment processes and“matrix” formed during diagenetic processes. In practice, especially when observing thin sections, it is difficult to distinguish the cement content within the fine particles. Hereinafter, the term “matrix” used in this work refers to the mixture offine particles including both matrix and cement. As such, the matrix may exhibit bonding strength owing to the ce-ment content within it.

c) Grain size—The effect of grain size on the strengths of sandstones may vary in accordance with the particular types of rocks studied. It has been found that increase in grain sizes results in decrease in uniaxial compressive strength for greywacke (Singh, 1988). While in other cases, there are no correlation between the grain size and the strength (Shakoor and Bonelli, 1991; Plachik, 1999).

d) Grain particle packing—The compactness of grain particle packing can be described by parameters such as the packing density, the grain contact, and the grain area ratio (GAR), which is deﬁned as the ratio between the grain area and the total area in the selected range of the specimen image (Ersoy and Waller, 1995).Dobereiner and De Freitas (1986)suggested the strength of sandstones has no obvious correlation with the packing density, but positively corre-lates to the grain contact.Bell (1978)andBell and Culshaw (1993),

⁎ Corresponding author. Tel.: +886 2 2363 0530; fax: +886 2 2364 5734. E-mail address:[email protected](F.-S. Jeng).

Contents lists available atScienceDirect

Engineering Geology

(2)

however, suggested higher packing density results in higher mate-rial strength.

Furthermore, other fundamental physical properties such as poro-sity, unit weight, and water content have been correlated to the uni-axial compressive strength of rocks (Dyke and Dobereiner, 1991; Hawkins and McConnell, 1992; Jeng et al., 1994; Hatzor and Plachik, 1997, 1998; Plachik, 1999; Jeng et al., 2002, 2004; Lin et al., 2005; Weng et al., 2005, 2008; Tsai et al., 2008). It was showed the strength decreases with increasing porosity and water content, and increases with increasing unit weight.

Jeng et al. (2004)andWeng et al. (2005)performed mechanical experiments and petrographic analyses on 13 different sandstone layers collected from the Western Foothill Range in Taiwan. The voids between particles can be either empty or partlyﬁlled with matrix materials. It has been found that fracture surfaces tend to go through particle–particle contacts or in between matrix when the sandstone is dry or wet, respectively (Lin et al., 2005).

In addition, since grains, matrix and porosity constitute the whole sandstone; how these entities affect the macro behavior is of interest. Through petrographic analysis, the inﬂuences of microscopic para-meters (or petrographic parapara-meters) on the macroscopic mechanical behaviors had been studied. Two parameters, the porosityn and the grain area ratioGAR were found to be the key parameters (Jeng et al., 2004). As a result, the uniaxial compressive strength (UCS) can be related to porosityn and GAR as shown in Eq. (1).

UCS ¼ 133:7d e0:107n_ð₃_{:2 0:026 GAR}_Þ _ð1Þ

Unit: MPa

Furthermore, the deformability (Young's modulusE) was found to be related to porosityn and GAR as well (Jeng et al., 2004) and can be expressed as:

E ¼ 36:3d e0:106n_ð₀_{:354 þ 0:017 GAR}_Þ _ð2Þ

Unit: GPa

From Eq. (2), it indicates that the Young's modulus of sandstones decreases with increasing porosityn and with decreasing GAR. It should be noted that the porosityn typically ranges between 5% and 25%, andGAR ranges between 20% and 75% for the studied sandstones. These two empirical equations, Eqs. (1) and (2), represent how the macroscopic strength and deformability of natural sandstones are related to porosityn and GAR. These empirical equations have been compared to sandstone worldwide (Bell and Culshaw, 1993, 1998; Ulusay et al., 1994) and were found to be consistent with the behavior of sandstones studied.

These two equations are adopted as the key relations to be met when validating and modifying numerical models and later analyses. Some of the microscopic properties of rock can be directly mea-sured or observed. For instance, the type of mineral and packing of grains can be measured from observing the thin sections of rocks under microscope. However, some other properties, e.g. the strength of matrix or the bonding strength of the grains, are difficult to measure directly. Moreover, in studying the influence of a particular micro-scopic factor, it would be ideal if the factor of interest could be varied while the other factors remained unchanged. Unfortunately, when natural rocks are used, frequently more than one factor will vary from specimen to specimen, rendering it difficult to evaluate, on identical bases, the unbiased influence of a particular factor.

There is very little study on the effect of matrix strength and stiffness on the macroscopic strength of rock. Despite these dif ﬁ-culties, a numerical model that is capable of accounting for the dis-crete packing nature of grains and the bonding strength, can serve as convenient means in studying the inﬂuence of microscopic properties on the macroscopic behavior of rock. Among the existing discrete numerical models, bonded-particle model (BPM) possesses the

desirable requirements mentioned above, and this model wasﬁrstly tested and eventually adopted in this research. By using BPM, the bonding strength between the matrix and the grain can be system-atically varied and the corresponding variation of macroscopic mecha-nical behavior can be observed. As a result, clariﬁcation on the effects of the matrix on mechanical behaviors of sandstones is accordingly obtained.

2. Methodology—a modiﬁed bonded-particle model

The framework of this study is illustrated inFig. 1. Before adopting BPM for studying microscopic mechanisms, the numerical model must be tested and modified until it shows adequate macroscopic mechanical behavior. Therefore, as shown inFig. 1, uniaxial compress tests were simulated and compared (Steps 1 and 2) and the results were veri_{fied (Verification I).}

As such, the uniaxial compression wasﬁrst modeled using BPM, and it was revised iteratively until it yields reasonable macroscopic uniaxial compression behavior (similar uniaxial compressive strength and Young's modulus). This test also serves the purpose of determin-ing BPM's microscopic parameters. After the test, a 2-dimensional numerical model was found to be capable of yielding a reasonable macroscopic mechanical response and was thus adopted as the tool for further analyses.

2.1. Setup of the modiﬁed BPM

In this study, a bonded-particle model (Potyondy and Cundall, 2004), which enables assembly and bonding of discrete circle par-ticles, is considered for simulating the aggregation and cementation of the sandstone. Intuitively, the bonded-particle model differs from the natural sandstone in the following aspects: (1) the grains of nature

(3)

sandstone are not necessarily spherical; (2) the contact of natural grains may often show_{“line-contact” as observed in thin sections;} however, the contact of BPM is either“point contact” or a segment of line-contact in between two particles, the so-called“parallel bond”, as shown in Figs.2; and (3) the number of grains opted when conducting numerical analysis can be much less than the actual number of grains in the real rock. Therefore, it is necessary to test and perform nec-essary modiﬁcations on the BPM so that adequate mechanical be-haviors can be reproduced by the numerical model.

Since the natural sandstone is composed of grains, matrix, cement and void, particles representing grains and matrix were accordingly introduced into the BPM. One of the major difficulties simulating natural sandstones is to model sandstone with high porosity. Natural sandstone typically shows porosity in the range of 10%–25%; however, the use of BPM to model sandstone with large porosity may often result in“floating particles”, i.e. some particles are not supported or in contact with other grains owing to the limited number of particles opted. If a wider range of grain size distribution, like natural sandstone may have, is used, it is expected that there would be much less or none floating particles in the model.

To overcome the aforementioned issue ofﬂoating particles in BPM, an additional type of particle representing the“porous matrix” was introduced. Therefore, this work allows modeling of high-porosity

sandstones without leading to unsupported,ﬂoating particles. As a result, the proposed model contains three types of particles repre-senting grain, matrix and porous matrix. These particle types are respectively denoted as GP, MP and PP. This modiﬁcation enables controlled variation of the_{GAR and porosity n of the model while} observing the macroscopic variation ofUCS and Young's modulus E.

Once the numerical procedure with desired control on both microscopic and index properties is validated, the mechanisms behind “wet softening” can then be studied. Some of the weak sandstones possess“wet softening” behavior, and were found to be related to GAR and porosity n as well as wet weakening of the matrix (Jeng et al., 2004; Lin et al., 2005). The modiﬁed BPM predicts macroscopic strength and deformability based on independent parameters includ-ing: (1)GAR and porosity; and (2) the strength and the deformability of the matrix particles and the porous matrix particles.

The three types of particles (GP, MP and PP) in the proposed model resulted in six different types of bonds between these three particle types (Fig. 2). Since the porous matrix particle PP should be the weakest material among the three types of particles, the bonding strength ofPP with other types of particles should be dominated by the weakest bonding strength ofPP. Therefore, the P–G and P–M bonds are replaced byP–P bonds. As a result, a total of four types of bonds were used in the proposed model (Fig. 2).

2.2. Properties of natural sandstone simulated

The natural sandstone simulated is the one studied byJeng et al. (2004), and itsUCS and E of are summarized inTable 1. The grain size distribution of this natural sandstone was obtained by petrographic studies, as shown inFig. 3(Jeng et al., 2004). Particle sizes of the model were determined based on the natural size distribution. As shown in

Fig. 3, the sizes of the particles in simulation are about 10 times greater

Table 1

Macroscopic properties of the sandstone studied and the simulated results yielded by the proposed numerical model

Items Value Unit

Natural sandstone studied

Uniaxial compressive strengthUCS 39.6 MPa

Young's modulusE 7.9 GPa

Prediction of the proposed model

Uniaxial compressive strengthUCS 41.6 MPa

Young's modulusE 8.4 GPa

Fig. 3. Grain size distribution of the sandstone studied and the grain size distribution adopted.

Fig. 2. Illustration of the three particles and the corresponding four contact types adopted in this research. Symbols:GP=grain particle, MP=matrix particle, PP=porous particle. The thick lines depict the width of the parallel bond between the particles.

(4)

than the actual grain size owing to the limitation of total number of particles limited by the available computing resource. The grain size distributions for various_{GAR are selected to be similar to the natural} one, as shown inFig. 3. The size of particles ranges between 0.5 mm and 6 mm, and particles with diameter less than or equal to 2 mm are selected as matrix particle_MP.

The material-genesis procedures is summarized and illustrated in

Fig. 4.Fig. 5a, b, and c shows the respective typical distributions ofPP, MP and GP particles based on the above-mentioned concepts. The

particles were randomly distributed inside a specimen of 5.5 cm in width and 13 cm in height. Fig. 5d shows the complete model specimen with_{GAR=55% and n=25%, and there are 8487 particles and} 17,123 contact points in this numerical model. The numbers ofG–G, M–G, M–M and P–P contacts are 170, 1564, 1587 and 13,802, re-spectively. The number of particles and contact points varies with the variation of petrographic parameters, e.g. GAR and n, and these parameters also affects the relative ratio of the four different types of contacts, as will be discussed in Section 4.

(5)

2.3. Input parameters required

The proposed modeling was carried out using PFC2D, which has the following features: (1) the particles are considered as homogenous rigid disks; (2) the interaction between particles is described as a soft contact, which occurs at the contact point between two particles; (3) the particles are allowed to slightly overlap each other at the contact points when subjected to compression; and (4) the slip condition between particles is governed by the Mohr–Coulomb friction (PFC2D, 2004). As a result, the microscopic parameters involved include:

(1) The stiffness of particle and parallel bond—the normal stiffness kn, shear stiffness ks, the parallel-bond normal stiffness knpand

the parallel-bond shear stiffness ksp;

(2) The surface friction coefﬁcient μ between particles;

(3) The strength of bonding—the normal bonding strength σ, the shear bonding strengthτ; and

(4) The width of bonding_{—the parallel-bond radius multiplier λ.} (5) In this work, a parallel bond in between two particles was

adopted to simulate the bonding behavior between particles. As shown inFig. 2, parallel-bond model assumes that two particles are bonded by a rectangle block with a width of 2λR− =2λ min (RA_{, R}B_{), in which the}_{λ is parallel-bond radius multiplier; R}A_,

RB_{are radii of adjacent particles A and B. Parallel bonds are}

capable of taking normal force, shear force and moment. When the normal stress or shear stress acting on the rectangle block exceeds the strength, the parallel bond fails and will be removed from the model.

2.4. Simulation of uniaxial compression test

To simulate uniaxial compressive tests, the upper and lower boundaries were selected to be frictionless rigid plates, and lateral boundaries were unconstrained. Gravity effect was not considered in this study, as the specimen was small, and the gravity-induced stress gradient had negligible effect on the macroscopic behavior. The un-conﬁned uniaxial compression test was simulated by vertically moving both the upper and the lower plates toward the center of the specimen at a constant velocity of 10− 2m/s. This loading rate, conﬁrmed in this study and byIverson (2003), was the upper limit that the loading rate imposed little effects on theUCS.

Using the basic model shown inFig. 5, the_{GAR can be increased by} randomly changing some of theMP into GP. Models were setup with GAR of 35%, 55%, and 75%. Similarly, the porosity n can be decreased by randomly changing some ofPP into MP, and the resultant porosity used in this study ranges from 10% to 25%.

2.5. Validation of the proposed numerical model—Verification I As shown inFig. 1(Step 2), the validity of the proposed model was first checked by simulating a series of uniaxial compression tests and by comparing with the experimental results fromJeng et al. (2004) and Weng et al. (2005), as shown inFig. 6. During this process (Figs. 1 and 4), the simulated macroscopic deformational behavior and the strength were made consistent with natural ones by iteratively re-fining the microscopic model parameters. Until the resultant macro-scopic behavior compares well with the natural ones (Fig. 1, Step 3), the corresponding parameters are then determined.

These parameters were determined by matching the calculated macroscopic strength (UCS) and stiffness (Young's modulus E) with physically measured ones listed inTable 1. The model parameters were determined based on curve-ﬁtting method, and the values used in this study are listed inTables 2 and 3.

Fig. 6compares the stress-strain curves respectively obtained by experiment and by numerical simulation. It can be seen that these two curves shown inFig. 6are consistent in general and the peak strength is also similar. Similar to natural rocks and as shown inFig. 6, the bonded-particle model fractures at peak strength, followed by substantial drop of resisting stress after peak; such behavior repre-sents a typical brittle fracture. After the peak, a major, inclined fracture

Fig. 5. Distribution of the grain, matrix and porous matrix particles of the proposed numerical model, withn=25% and GAR=55%. The percentages of GAR and n are deﬁned as the ratios of grain area and void area over the total area, respectively. The addition of the two ratios of void (10%) and porous matrix (15%) altogether is taken as porosityn.

Fig. 6. Comparison of the experimental and simulated stress-strain curve whenn=20% andGAR=35%.

(6)

surface formed in the specimen upon subsequent loading, and eventually the original intact specimen is broken apart. This fracture pattern is also similar to the natural rock.

Accordingly, the set of parameters shown inTables 2 and 3was adopted as the base set of parameters used for further analysis. Since the strengths of shear and normal bonds were not actually measured, the values listed inTable 3are obtained based on back analyses. The strength of normal and shear bonds have been evaluated and it was found that, as long as the rock tends to have brittle failure under uniaxial compression tests, the relative strength of shear to normal bonds has little influence on the UCS (Potyondy and Cundall, 2004; Cho et al., 2007). As such, the ratio of shear to normal bonding strength was selected to be one. Back analyses have been conducted to evaluate this ratio of strength of shear and normal bonds for sand-stone in this research. The results indicate that: (1)UCS is controlled by normal bond and (2) the shear bond has little influence on the UCS. This result is similar to the results observed from granite. Therefore, the normal bonding strength was selected by best-_{fitting UCS' and the} shear bonding strength was selected to be identical to normal bond for convenience. However, it should be noted that the shear bonding strength selected in this research is not necessary the real or “repre-sentative” one.

Using BPM, it is possible to track the number of bonding failure between particles. Fig.6shows the number of failed parallel bonds in the course of a uniaxial compression test withn=20% and GAR=35%. The initial bonding breakage started when the axial strain reached 50% of the yield strain (the strain corresponds to the peak strength), and the number of breakage signiﬁcantly increased when the axial strain approached the yield strain. It was also observed that the breaking ofP–P and M–G bonds outnumbered the breaking of M–M andG–G bonds, as shown inFig. 6. The observation on the number as well as the type of bonding failure enables us toﬁnd the insights regarding the dominating failure mechanism of sandstones; this will be further discussed in Section 4.

2.6. Further validation—Veriﬁcation II

The validity of the proposed model was then further tested, as shown inFigs. 1 and 4(Step 4), by comparing the computed variation ofUCS with GAR and porosity n to the natural ones such as the observed correlation shown by Eqs. (1) and (2) andFigs. 7 and 8(Jeng et al., 2004). As will be discussed in later sections, the results of tests were found to be acceptable and the proposed model was thus adopted for further study of the sandstone.

As long as the influence of petrographic factors on UCS and E obtained by empirical correlations can be simulated (Step 4 inFig. 1), the base set of microscopic parameters is determined in the meantime, as the ones summarized inTables 2 and 3. Parametric studies can then be performed to learn the influence of macroscopic properties, in-cluding bonding strengths in between different types of particles on the strength reduction owing to possible wet softening (Step 5 inFig.1). 3. Influence of GAR and n

3.1. Inﬂuence of GAR and n on UCS

Fig.7shows the variation ofUCS with GAR and porosity n obtained from the proposed model and the comparison to the empirical correlations founded byJeng et al. (2004). It can be seen that, for a given porosityn, increasing GAR decreases the UCS. This observation, combined with the fact that increasing ofGAR under a given porosity n reduces the matrix content, suggests the possibility that the bonding strength in between matrix particles (M–M) is greater than the bonding strength in between grain particles (G–G) (Jeng et al., 2004). On the other hand, under a particularGAR, the decrease of porosity n leads to an increase inUCS as shown inFig. 7. The increase of grains and matrix, i.e. the decrease of porosityn, tends to better support the applied load and accounts for the increase ofUCS.

When compared to the behavior of actual sandstone, the above-mentioned in_{ﬂuence of GAR and porosity n on UCS is consistent with} the empirical relations shown inFig. 7, except that, at lower porosity (e.g., n=10%), the empirical relation exhibits a somewhat greater range of variation upon varyingGAR.

Table 2

Microscopic properties of grain (GP), matrix (MP) and porous matrix particles (PP)

Items GP MP PP Unit

Particle density,ρ 2660.0 2660.0 2660.0 kg/m3

Young's modulus of particle 47.9 8.9 7.1 GPa

Particle normal/shear stiffness, kn/ks 2.4 2.4 2.4

Particle friction coefﬁcient, μ 0.5 0.5 0.5

Remarks: The parameters are determined based on theGAR=75%, n=15% and adopted as the base case.

Table 3

Parameters of parallel bonds forG–G, M–M, M–G and P–P contacts

Items Contact types Unit

G–G M–M M–G P–P

Parallel bond radius multiplier,λ 1.0 1.0 1.0 0.08

Young's modulus of parallel bond 47.9 8.9 7.5 7.1 GPa

Normal/shear stiffness of parallel bond, knp/ksp 2.4 2.4 2.4 2.4 Normal strength of parallel bond 35.8 143.7 50.0 143.7 MPa Shear strength of parallel bond 35.8 143.7 50.0 143.7 MPa Remarks: The parameters are determined based on theGAR=75%, n=15% and adopted as the base case.

Fig. 7. Empirical relation ofUCS with GAR and porosity n. The simulated UCS' are also plotted for comparison.

Fig. 8. Empirical relation of Young's modulusE with GAR and porosity n. The simulated Es are also plotted for comparison.

(7)

3.2. Inﬂuence of GAR and n on Young's modulus E

The effect ofGAR and porosity n on Young's modulus E is shown in

Fig. 8. It can be seen that, under a speciﬁc porosity n, increasing GAR increasesE. Since the increase of GAR under a constant n, which was modeled by replacing some of matrix particles with grain particles, leads to a greaterE, this observation suggests the stiffness in between grains is greater than that in between matrix particles.

On the other hand, under a constant GAR, a decrease in the porosityn increases the E as shown inFig. 8. Decreasing porosity means more matrix particles and less porous matrix particles and void and leads to a more consolidated packing of the particles and the matrix. This densely packed structure tends to give larger stiffness than loosely packed structures. Consequently, increased number of particles accounts for the increase inE.

The simulation of the in_{ﬂuence of GAR and porosity n on the UCS} andE enables further validation on the selected parameters for the adopted model. The base set of parameters was selected from a set of GAR and n, and simulations were conducted to yield variations of UCS and E. The simulated variations are then compared with those of natural sandstones under variousGAR and n as shown inFigs. 7 and 8. Fairly good agreement of the predictions with the actual behaviors shows that the proposed model is acceptable for use in further analyses.

4. Contact type distribution

As described in Section 2, the proposed model comprises three types of particles:GP, MP, and PP; and involves four types of contacts: G–G, M–M, M–G, and P–P. In the proposed model, the variations of porosityn and GAR change both the number of particles and the relative number of contacts amongst different contact types, as shown inTable 4andFig. 9.

Fig. 9a shows the proportions of four contact typesM–M, M–G, G–G, andP–P when GAR is 35%. It is seen inFig. 9a that, with increase in porosityn from 10% to 25%, the proportion of P–P signiﬁcantly in-creases and the proportions ofM–M and M–G contacts correspond-ingly decrease. The proportion ofG–G contact remains constant due to its tie toGAR. Similar changes to proportions of different contact types can be seen with variation inGAR in the range of 55% and 75% as shown inFig. 9b and c. Nevertheless, due to the increase ofGAR, the number of GP particles relatively increases, and the proportion of M–G and G–G contacts increases accordingly. It can also be seen inFig. 9that theG–G contact has a rather low percentage, not exceeding 2%, this rather low percentage of_{G–G is resulted from the relatively large diameter of the} grain particles (GP), when compared to other particle types in the adopted model.

Table 4

Summation of the particle numbers, contact numbers and the relative contents of the adopted particles for the cases analyzed

GAR (%) n (%) Particle number Contact number G–G (%) M–M (%) M–G (%) P–P (%) (a) 35 10 12,113 23,856 0 82 18 0 15 0 61 15 24 20 0 43 12 45 25 0 27 9 63 (b) 55 10 8487 17,123 1 66 33 0 15 1 42 25 32 20 1 23 17 59 25 1 9 9 81 (c) 75 10 7606 15,328 2 51 47 0 15 2 28 34 36 20 2 12 21 64 25 2 3 9 87

Remarks: The numbers of bond breakings are plotted and compared inFig. 9.

Fig. 9. Variations of contents of four contact types with porosityn and GAR ranging from 0% ~ 25% and 35% ~ 75%, respectively. The numbers of contacts and bond breakings for three cases (GAR=35%, 55% and 75%) plotted in this ﬁgure are summarized inTable 4.

(8)

The observation on the changes in the proportion of different contact types and the failure of bonding provides us a sound basis to explore the micro-mechanism behind the macroscopic mechanical of sandstones, concerning the dominant contact type affecting theUCS under various combination of porosityn and GAR, especially.

Fig. 10a and b shows contours of _{UCS and E respectively under} various combination of porosityn and GAR, which are based on the actual behavior of natural sandstone (Jeng et al., 2004). Three cases of macroscopic failure, referred to as Cases A, B, and C, are discussed in the following. The corresponding bonding failure types when approaching particle failure for the three cases are summarized in

Table 5.

In transitioning from Case A to Case B, in whichGAR decreases under constant porosity, e.g.UCS increases and Young's modulus E decreases, as shown inFig. 10. Meanwhile, the major type of broken bonding transit from M–G and G–G dominant to M–G and M–M dominant, as shown inTable 5.

This phenomenon is readily interpreted by the fact that a decrease ofGAR reduces the number of GP and increases the number of MP. As

the number ofGP decreases, the number of M–G bonding decreases; as the number of MP increases, the number of M–M bonding increases. For theM–M bonding, its strength is higher than M–G bonds and its stiffness is lower thanM–G bonds, thus the macroscopic strengthUCS increases inFig. 10a from Case A to Case B; macroscopic Young's modulus_{E decreases from Case A to Case B in}Fig. 10b.

When transitioning from Case B to Case C, with porosity increasing from 10% to 25% under a constantGAR of 35%, as shown inFig. 10a and b, both macroscopicUCS and E decrease.

For Case C, due to the increase of porosity under a constantGAR, parts of MPs are replaced by PPs such that the number of P–P increases and the number ofM–M bond decreases in the meanwhile. As a result, the most failed bonds areP–P and M–G bonds. The P–P bond features relatively lower strength and stiffness, and therefore the moreP–P bonds leads to reductions of the macroscopic UCS and E as well.

5. Inﬂuence of bonding strength on the wet-softening behavior of sandstone

The wet-softening behavior of sandstones refers to the reduction of shear strength when the material is wetted by water. This wet-softening behavior is owing to the fact that the bonding strength of matrix materials is reduced when the sandstone is wetted ( Dober-einer and De Freitas, 1986; Lin et al., 2005). The degree ofUCS reduc-tion can be expressed in R, a measure of the so-called strength reduction ratio deﬁned as the ratio of the UCS of dry rock by the UCS of wet rock (Jeng et al., 2004; Weng et al., 2005). The relationship, proposed byJeng et al. (2004), betweenR and porosity n and GAR is shown in Fig.11. Under low porosityn and high GAR, there is little matrix content, thusR is close to unity and indicates little reduction of strength when soaked; on the contrary, with high porosity_{n and low} GAR, the matrix content is high, R is close to 0 and represents sig-niﬁcant wet-softening behavior.

Through mechanical regression analyses between laboratory ex-periments and thin-slice observations, the relationship betweenR and porosityn and GAR can be established. However, the inﬂuence of bonding on macroscopic strength is difﬁcult to study using the above-mentioned conventional methods. The proposed model offers a way to explore the effect of particle bonding on macroscopic mechanical behaviors.

Fig. 12shows the effect of bonding strength of each individual bonding type on the macroscopicUCS for Cases A, B, and C. InFig. 12, all the bonding strengths are normalized with respect to the corre-sponding dry strengths of Cases A, B, and C respectively. For Case A, as shown in Fig. 12a, with 10% of porosity and 75% of GAR, the reduction ofM–G bond most signiﬁcantly decreases the macroscopic UCS. Whereas, reductions of M–M and G–G bonding strength result

Table 5

Summation of the number of cracks yielded by Cases A, B and C

Case A Case B Case C

Contact type n = 10% n = 10% n = 10%

GAR = 75% GAR = 35% GAR = 75%

P–P 0% 0% 55%

M–M 0% 22% 1%

M–G 76% 75% 41%

G–G 24% 3% 3%

Total number of cracks 225 673 272

Fig. 11. Variations ofR corresponding to the GAR and n of Cases A, B and C.

(9)

in a moderateUCS reduction and P–P bonding has little effect on UCS.

For a lowerGAR under a constant porosity, such as Case B with 10% of porosity and 35% ofGAR, reduction of M–G and M–M bonding strength leads to signi_{ﬁcant reduction of UCS, and the strength} re-duction ofG–G and P–P bonding have an insigniﬁcant effect on UCS (Fig. 12b).

Fig. 12c depicts, Case C with 25% of porosity and 35% ofGAR, P–P bonding strength reduction renders signiﬁcant reduction of UCS, M– M, M–G, and G–G bonding strength reductions have lesser and lesser effect onUCS reduction.

On one hand, the globalGAR or porosity n determines the numbers of each contact type. In turn, the speciﬁc distribution of the contact types, the three cases studied for instance, dominates not only_UCS andE but also the degree of wet softening of sandstone R.

6. Conclusion

In this study, how the microscopic factors, e.g. particle types and the corresponding bonding strength and stiffness, affect the macro-scopicUCS, E and R of sandstones was explored.

A model based on bonded-particle method was proposed to simu-late the macroscopic mechanical behaviors of sandstones. The pro-posed numerical model contained three types of particles: grain particles (GP), matrix particles (MP), and porous matrix particles (PP). These two additional types of particles introduced in this study,MP and_{PP, greatly facilitated the study on the effects of petrographic} parameters GAR and porosity n on the macroscopic mechanical behaviors. The model had been veriﬁed for modeling sandstones and showed good capability in simulating/predicting the macroscopic mechanical behavior of sandstones. Meanwhile, the porosity was taken into consideration, and porous matrix particles were intro-duced. These particles enabled simulations with a wide range of porosity, from 10% to 25%, and facilitated the corresponding observa-tion on the variaobserva-tions ofUCS and Young's modulus E with varying porosity.

Overall, this study revealed reasonable explanations on why the UCS and R vary according to GAR and porosity n as follows:

(1) As long as theM–M bond is stronger than G–G bond, a decrease inGAR will lead to the increase of UCS in the macroscopic manner under a constant porosity (Fig. 10a);

(2) On the other hand, provided thatM–M bond is softer than G–G orM–G bonds and under a constant porosity, a decrease in GAR will lead to the decrease of E under a constant porosity (Fig. 10b).

(3) An increase inP–P bonds leads to reductions of UCS and E (Change from Case B to Case C, shown inFig. 10); and (4) When a sandstone possesses a greater number ofP–P bonds or

M–M bonds, this sandstone tends to have a greater wetting reduction ofUCS, as shown inFigs. 11 and 12.

Given the methodology and the results presented in previous sections, this research has its limitations and further researches are of interest:

(1) The analysis in this study is 2-dimensional, and the true geometrical condition of specimens in unconﬁned uniaxial compression tests cannot be fully represented. Three-dimen-sional analyses are inherently more representative of the three-dimensional nature of rock and can be used to further check the validity of this work.

(2) The void ratio is 10%; however, other types of rocks may have much lower void ratio. Reduction of void ratio means more particles in the computer model are needed, which may not be feasible using standard single-processor computers.

(3) The types of grains considered are mainly based on sedimen-tary rocks. Such types of grains may not represent other rock types (including igneous rocks, metamorphic rocks and others). More studies should be carried out toﬁnd adequate models for different rock types.

(4) The type of grain contacts considered is point contact. As such, the in_{ﬂuence of grain-contact types on the macroscopic} be-havior awaits to be further studied. More simulation techniques should be developed to represent other types of rocks. (5) The used of porous particle facilitates simulation of natural

sandstone; however, it does pose limitations. The stiffness and bonding strength are determined based on back analyses,

(10)

which are not physical properties than can be obtained through laboratory experiments since the “porous matrix” is a con-ceptual model rather than an actual material.

Acknowledgments

The research is partly supported by the National Science Council of Taiwan, grant numbers 002-261 and NSC-95-2221-E-002-263. Authors thank Mr. Christopher Fong for his help and advices in improving the paper.

References

Barbour, T.G., Atkinson, R.H., Ko, H.Y., 1979. Relationship of Mechanical Index and Mineralogic Properties of Coal Measure Rock: 20th Symposium of Rock Mechanics, Austin, Texas, pp. 189–198.

Bell, F.G., 1978. Physical and mechanical properties of the fell sandstones, Northumber-land, England. Engineering Geology 12 (1), 1–29.

Bell, F.G., Culshaw, M.G., 1993. A survey of the geotechnical properties of some relatively weak Triassic sandstones. The Engineering Geology of Weak Rock: Proceedings of the 26th Annual Conference of the Engineering Group of the Geological Society, Leeds, United Kingdom. Balkema, Rotterdam, pp. 139–148.

Bell, F.G., Culshaw, M.G., 1998. Petrographic and engineering properties of sandstones from the Sneinton Formation, Nottinghamshire, England. Quarterly Journal of Engineering Geology 31, 5–19.

Clough, G.W., Sitar, N., Bachus, R.C., 1981. Cemented sands under static loading. Journal of Geotechanical Engineering Division 107 (6), 799–817.

Cho, N., Martin, C.D., Sego, D.C., 2007. A clumped particle model for rock. International Journal of Rock Mechanics and Mining Sciences 44, 997–1010.

David, C., Menendez, B., Bernabe, Y., 1998. The mechanical behaviour of synthetic sandstones with varying brittle cement content. International Journal of Rock Mechanics and Mining Sciences 35 (6), 759–770.

Dobereiner, L., De Freitas, M.H., 1986. Geotechnical properties of weak sandstone. Geotechnique 36 (1), 79–94.

Dyke, C.G., Dobereiner, L., 1991. Evaluating the strength and deformability of sandstones. Quarterly Journal of Engineering Geology 24, 123–134.

Ersoy, A., Waller, M.D., 1995. Textural characterisation pf rock. Engineering Geology 39, 123–136.

Gunsallus, K.L., Kulhaway, F.H., 1984. Comparative evaluation of rock strength measurements. International Journal Rock Mechanics and Mining Sciences & Geomechanics Abstracts 21 (5), 233–248.

Hatzor, Y.H., Plachik, V., 1997. The inﬂuence of grain size and porosity on crack initiation stress and criticalﬂaw length in dolomites. International Journal of Rock Mechanics and Mining Sciences 34 (4), 805–816.

Hatzor, Y.H., Plachik, V., 1998. A microstructure-based failure criterion for Aminadav dolomites. International Journal of Rock Mechanics and Mining Sciences 35 (6), 797–805.

Hawkins, A.B., McConnell, B.J., 1992. Sensitivity of sandstone strength and deformability to changes in moisture content. Quarterly Journal of Engineering Geology 25, 115–130.

ISRM, 1981. In: Brown, E.T. (Ed.), Rock Characterization, Testing and Monitoring—ISRM Suggested Methods. Pergamon, New York. 211 pp.

Iverson, S.R., 2003. Investigation of bulk solids engineering properties and application of PFC2D to ore passﬂow problems. In: Konietzky, H. (Ed.), Numerical Modeling in Micromechanics via Particle Methods. Proceedings of the 1st International PFC Symposium, Gelsenkirchen, Germany. Balkema, Rotterdam, pp. 252–258. Netherlands.

Jeng, F.S., Ju, G.T., Huang, T.H., 1994. Properties of Some Weak Rock in Taiwan: Proceedings of The 1994 Taiwan Rock Engineering Symposium. National Central University, Chungli, Taiwan, pp. 259–267.

Jeng, F.S., Weng, W.C., Huang, T.H., Lin, M.L., 2002. Deformational characteristics of weak sandstone and impact to tunnel deformation. Tunnelling and Underground Space Technology 17, 263–264.

Jeng, F.S., Weng, M.C., Lin, M.L., Huang, T.H., 2004. Inﬂuence of petrographic parameters on geotechnical properties of Tertiary sandstones from Taiwan. Engineering Geology 73, 71–91.

Lin, M.L., Jeng, F.S., Tsai, L.S., Huang, T.H., 2005. Wetting weakening of tertiary sandstones—microscopic mechanism. Environmental Geology 48, 265–275. Shakoor, A., Bonelli, R.E., 1991. Relationship between petrographic characteristics,

engineering index properties, and mechanical properties of selected sandstone. Bulletin of the Association of Engineering Geologists 28, 55–71.

Singh, S.K., 1988. Relationship among fatigue strength, mean grain size and compressive strength of a rock. Rock Mechanics and Rock Engineering 21, 271–276.

Smart, B.D., Rowlands, N., Isaac, A.K., 1982. Progress towards establishing relationships between the mineralogy and physical properties of coal measures rocks. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 19, 81–89.

Tsai, L.S., Hsieh, Y.M., Weng, M.C., Huang, T.H., Jeng, F.S., 2008. Time-dependent Deformation Behaviors of Weak Sandstones. International Journal Rock Mechanics and Mining Sciences 45, 144–154.

PFC2D (Particle Flow Code in 2 Dimension) Version 3.1, 2004. Itasca Cons Group, Minneapolis.

Plachik, V., 1999. Inﬂuence of porosity and elastic modulus on uniaxial compressive strength in soft brittle porous sandstones. Rock Mechanics and Rock Engineering 32, 303–309.

Potyondy, D.O., Cundall, P.A., 2004. A bonded-particle model for rock. International Journal of Rock Mechanics and Mining Sciences 41, 1329–1364.

Ulusay, R., Tureli, K., Ider, M.H., 1994. Prediction of engineering properties of selected litharenite sandstone from its petrographic characteristics using correlation and multivariate statistical techniques. Engineering Geology 37, 135–157.

Vutukuri, V.S., Lama, R.D., Saluja, S.S., 1974. Handbook on Mechanical Properties of Rocks.

Weng, M.C., Jeng, F.S., Huang, T.H., Lin, M.L., 2005. Characterizing the deformation behavior of tertiary sandstones. International Journal of Rock Mechanics and Mining Sciences 42, 388–401.

Weng, M.C., Jeng, F.S., Hsieh, T.M., Huang, T.H., 2008. A simple model for stress-induced anisotropic softening of weak sandstones. International Journal of Rock Mechanics and Mining Sciences 45, 155–166.