The mechanical behaviour of synthetic, poorly consolidated granular rock under uniaxial compression

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Tectonophysics 370 (2003) 105–120

The mechanical behaviour of synthetic, poorly consolidated

granular rock under uniaxial compression

F. Saidi*, Y. Bernabe, T. Reuschle

Institut de Physique du Globe de Strasbourg, CNRS-Universite Louis Pasteur, 5 rue Rene Descartes, 67084 Strasbourg, France

Accepted 31 March 2003

Abstract

In order to isolate the effect of grain size and cementation on the mechanical behaviour of poorly consolidated granular rock,

we prepared synthetic rock samples in which these two parameters were varied independently. Various proportions of sand,

Portland cement and water were mixed and cast in a mold. The mixture was left pressure-free during curing, thus ensuring that

the final material was poorly consolidated. We used two natural well-sorted sands with grain sizes of 0.22 and 0.8 mm. The

samples were mechanically tested in a uniaxial press. Static Young’s modulus was measured during the tests by performing

small stress excursions at discrete intervals along the stress–strain curves. All the samples exhibited nonlinear elasticity, i.e.,

Young’s modulus increased with stress. As expected, we found that the uniaxial compressive strength increased with increasing

cement content. Furthermore, we observed a transition from grain size sensitivity of strength at cement content less than 20–

30% to grain size independence above this value. The measured values of Young’s modulus are well explained by models based

on rigid inclusions embedded in a soft matrix, at high cement content, and on cemented grain-to-grain contacts, at low cement

content. Both models predict grain size independence in well-sorted cemented sands. The observed grain size sensitivity at low

cement content is probably due to microstructural differences between fine- and coarse-grained materials caused by small

differences in grain sorting quality.

D 2003 Elsevier B.V. All rights reserved.

Keywords: Strength; Elastic properties; Granular rocks; Cementation; Grain size

1. Introduction

Poorly consolidated granular rocks are commonly

encountered in oil reservoirs, and often linked to

problems like earth surface subsidence, well–bore

instability or sanding (e.g., Maury and Sauzay,

0040-1951/03/$ - see front matter D 2003 Elsevier B.V. All rights reserve

doi:10.1016/S0040-1951(03)00180-X

* Corresponding author. Present address: Division de Mecani-

que Appliquee, Institut Franc�ais de Petrole, 1 a 4 Avenue de Bois-

Preau, 92506 Rueil-Malmaison Cedex, France.

E-mail address: [email protected] (F. Saidi).

1987; Smart et al., 1991; Fredrich et al., 2000; Zhang

et al., 2000). Better knowledge about the mechanical

properties of these rocks is required in order to

anticipate and minimise the technical difficulties men-

tioned above (Schutjens et al., 1995). This is difficult

to achieve, however, because of the near impossibility

to collect and characterise well-preserved samples of

poorly consolidated reservoir rock. An alternative

approach is to prepare synthetic materials with the

required specifications (e.g., Visser, 1988; Dass et al.,

1993; Holt et al., 1993; Wang et al., 1997; David et

al., 1998). Most importantly, this method allows

d.

F. Saidi et al. / Tectonophysics 370 (2003) 105–120106

varying the relevant structural parameters independ-

ently and hence isolating their effect.

The mechanical behaviour of granular materials is

primarily controlled by cementation (e.g., Clough et

al., 1981; Chang et al., 1990; Bruno and Nelson, 1991;

Dvorkin et al., 1991, 1994; Bernabe et al., 1992; Zang

and Wong, 1995). In particular, it was observed that

very small amounts of cement, if deposited at grain-to-

grain contacts, strongly increase the stiffness and

strength of granular materials. The importance of

cement is also illustrated by a recent study that showed

that artificially induced cementation in situ can greatly

improve well–bore stability (Mamora et al., 2000).

Another important parameter is grain size. In the

ductile metals, an inverse relationship was established

between plastic yield stress and the square root of grain

size (Hall, 1951; Petch, 1953). Similar effects were ob-

served in rocks deformed in the brittle and semibrittle

fields (Handin and Hager, 1957; Skinner, 1959; Brace,

1961; Olsson, 1973). Fredrich et al. (1990) found that

in calcite aggregates, brittle strength, low temperature

yield stress and the brittle–ductile transition pressure

depended on the inverse of grain size. Zhang et al.

(1990) observed the same kind of scale effect in sand-

stones and attributed it to scaling with grain size of the

flaws responsible for grain fracturing. In poorly con-

solidated rock, we do not expect predominant grain

fracturing but rather rupture of cemented bonds. The

effect of grain size may therefore be quite different.

In this work, we prepared synthetic granular rock

samples in which cement content and grain size were

varied independently. The samples were tested in a

uniaxial press. The paper is organised as follows. In

Section 2, we describe the technique of fabrication of

the synthetic materials and the testing procedure. The

results of the mechanical tests are then reported in

Section 3.We present a detailedmicrostructure study of

the undeformed and deformed materials in Section 4.

Finally, the results are modelled and discussed in

Section 5.

2. Experimental procedure

2.1. Preparation of the synthetic granular rocks

The samples were prepared as follows: (1) Desired

amounts of dry sand and Portland cement were

weighed and thoroughly mixed. (2) Water was added

to produce complete curing. We used the smallest

possible amount of water in order to avoid gravity

segregation and/or flushing of the cement. (3) The

mixture was cast into a 4-cm-long rectangular pris-

matic mold with a square cross-section 2 cm in sides.

(4) The mixture was allowed to dry inside the mold

for 1 day. We finally removed the hardened sample

from the mold and completed the curing process by

leaving it in direct contact with the atmosphere for

six additional hours. In a previous series of experi-

ments, we assessed the influence of curing time on

the mechanical properties of the prepared materials

(Saidi, 2002) and verified that precisely following the

procedure above allowed us to obtain consistent

results (i.e., the properties of the various materials

can be meaningfully compared). Notice that no

pressure was applied to the samples during curing

in order to ensure that the final material was poorly

consolidated. Since the samples were very weak and

friable, we did not machine their faces to a precisely

flat surface.

We used two varieties of natural sand from Hague-

nau, France. The grain size distributions were deter-

mined from scanning electron microscope (SEM)

images of polished sections of the cemented materials

prepared as explained above. We observed that the

smallest sand grains of the fine-grained variety were

comparable in size to the biggest cement grains.

Fortunately, the two types of grains can be distin-

guished because cement grains appear lighter in SEM

images and have more angular shapes than sand

grains. The grain size histograms are shown in Fig.

1. Both varieties appear relatively well sorted. The

fine-grain variety has a mean grain diameter d of 0.22

mm and a standard deviation rd of 0.12 mm, while

d = 0.8 mm and rd = 0.29 mm for the coarse-grain

sand. The normalised standard deviation rd/d is equal

to 0.55 in the case of the fine sand and 0.36 for the

coarse sand. Hence, the distribution of normalised

grain diameters is broader in the fine sand than in the

coarse sand. We also estimated the grain aspect ratio

ar and found the two varieties to be almost identical in

that respect (fine sand, ar = 0.66; coarse sand,

ar = 0.69; in both cases, the standard deviation ra

was 0.13).

Two suites of samples were prepared using fine

and coarse sand (noted F and C, respectively).

Table 1

Bulk density q, macroscopic porosity / and cement volume fraction

Ccem of the prepared materials as a function of Cdry

Cdry Fine sand Coarse sand

q(g cm� 3)

/ Ccem q(g cm� 3)

/ Ccem

0.049 1.521 0.417 0.044 1.493 0.427 0.043

0.070 1.580 0.390 0.065 1.546 0.403 0.064

0.090 1.639 0.363 0.087 1.579 0.386 0.083

0.130 1.666 0.344 0.126 1.632 0.358 0.123

0.167 1.717 0.317 0.165 1.703 0.322 0.164

0.230 1.832 0.257 0.239 1.823 0.261 0.238

0.285 1.896 0.220 0.303 1.880 0.226 0.300

0.333 1.900 0.208 0.351 1.917 0.201 0.354

0.375 1.902 0.198 0.391 1.980 0.165 0.408

0.444 1.940 0.168 0.465 2.010 0.138 0.482

0.600 2.121 0.057 0.664 2.131 0.053 0.668

0.800 2.098 0.028 0.840 2.103 0.026 0.842

1.000 2.081 0.000 1.000

Fig. 1. Grain size histograms for the two varieties of sand (F and C).

F. Saidi et al. / Tectonophysic

Samples of pure Portland cement were also fabri-

cated. We varied the cement content in the F and C

samples by adjusting Cdry, the ratio of dry cement

weight to total dry weight (i.e., measured before

adding water), to pre-set values (see Table 1). How-

ever, Cdry is useless as model input parameter. We

need to estimate the volume fraction of cement Ccem.

For that purpose, we measured the bulk volume of

cured cement obtained per gram of dry cement (0.6

cm3/g), the cured cement density (2.081 g cm� 3) and

the bulk density q of each prepared F or C sample

(see Table 1). Although the cured cement is micro-

granular and microporous (using a mineral density of

2.5 g cm� 3 we estimated the microporosity to be

about 17%; Ashby and Jones, 1980), we will here-

after consider it a continuous phase. Aside the

cement microporosity, the rest of the pore space

consists of voids larger than the grain size of cement.

It will therefore be referred to as macroporosity.

Thus, the F and C samples contain three phases,

namely sand grains (we assume the density of quartz,

i.e., 2.65 g cm� 3), microporous cement and macro-

porosity. The volume fraction of each phase (i.e.,

sand volume fraction Cg, cement volume fraction

Ccem and macroporosity /) can be easily calculated

from the quantities described above. Each sample

was duplicated three or four times to test the repro-

ducibility of the fabrication procedures. We observed

maximum differences in / and Ccem of less than 5%.

The average experimental values of /, Cg and Ccem

are given in Table 1.

2.2. Uniaxial compression tests

Samples were tested in a computer-controlled uni-

axial press using axial strain as control variable. They

were loaded at a constant strain rate of 4.15� 10� 3

s� 1. The axial shortening was measured using a

LVDT displacement transducer installed between the

sample holders (resolution of 0.01 Am). The press was

equipped with an axial load cell with a 10 N precision.

Because of the extremely weak and irregular structure

of the material, we could neither measure the lateral

deformation nor record acoustic emissions. Transmis-

sion of acoustic signals is very poor in this type of

material.

In order to gain more information on the mechan-

ical behaviour, we measured static Young’s modulus

of some of the samples and its evolution along the

loading path. Small unloading–reloading cycles were

performed at regular intervals along the stress–strain

curve before and after the peak-stress. According to

the plasticity theory, materials behave elastically along

such cycles if the cycles are oriented towards the

interior of the yield surface. This technique was

successfully applied to natural rocks in the past

although hysteresis was generally observed (e.g.,

Hilbert et al., 1994; Bernabe and Fryer, 1995; Plona

and Cook, 1995). To avoid hysteresis problems, we

measured the static Young’s modulus as the slope of

the unloading portion of the cycles. A typical stress–

s 370 (2003) 105–120 107

Fig. 2. (a) A typical stress–strain curve (Cdry = 28.5%). (b) The

corresponding evolution of Young’s modulus as a function of r1 inthe same sample.


strain curve with the unloading–reloading cycles is

shown in Fig. 2a. Notice that the initial portion of the

curve is nearly horizontal. This abnormally large

compressibility is related to irregularities of the end-

faces. As a consequence, it was difficult to determine

the point at which the loading piston first contacted

the sample and therefore the absolute values of the

axial strain are not accurately known.

3. Results

3.1. General observations

All the samples exhibited behaviours similar to that

shown in Fig. 2. Pre- and post-peak sections of the

stress–strain curve in Fig. 2a appear surprisingly

similar. Post-peak softening always proceeded regu-

larly down to stresses as low as we attempted to reach.

We never observed any instability (i.e., sudden stress

drop) after the peak stress, which suggests that dam-

age accumulation occurred progressively over the

entire post-peak domain (note that our testing machine

is only moderately stiff and therefore susceptible to

induce instability). Strain localisation was rarely

present at the sample scale. Only a few samples did

present visible axial fractures as is generally observed

in uniaxial compression experiments on harder rocks.

But we were not able to determine at which point on

the stress–strain curves these fractures formed. In Fig.

2a, we see that the small unloading–reloading cycles

did not coincide with the main stress–strain curves

even during the initial stages of deformation, implying

that the samples became nonelastic at very low

stresses. Fig. 2b shows the corresponding values of

Young’s modulus E as a function of axial stress r1. In

the pre-peak domain, E significantly increases with

r1, indicating nonlinear elastic behaviour. In the post-

peak domain, E decreases sharply as a result of

damage accumulation.

3.2. Effect of cementation and grain size on strength

Fig. 3 shows the uniaxial compression strength Su(i.e., peak-stress) of the F and C samples as a function

of cement content Ccem. The values plotted in Fig. 3

were obtained by averaging the results from three or

four experiments run in identical conditions. The

relative fluctuations were not larger than 20% (the

error bars in Fig. 3 represent plus or minus two stand-

ard deviations). Plotted on a linear scale in Fig. 3a the

curve of Su versus cement content has an S-shape with

an inflection point (i.e., maximum slope) at Ccemc20%. The observed sharp increase of Su at low cement

content is in a good agreement with recent theoretical

(Dvorkin et al., 1994; Bruno and Nelson, 1991) and

experimental work (Bernabe et al., 1992; Yin and

Dvorkin, 1994; David et al., 1998), which showed

that cement located at grain contacts is very effective

in strengthening granular materials. At high Ccem, Suincreases less dramatically because the additional

cement was deposited in less-effective locations.

One interesting observation is that the inflexion

point also corresponds to a transition from grain size

sensitivity of Su at Ccem < 20–30% to grain size inde-

Fig. 4. The evolution of Young’s modulus E with r1 in log– log

scale in two samples with different Cdry (symbols) and in pure

cement (thick line). The power law behaviour breakdowns at the

onset of damage accumulation (see text for more details).

Table 2

Average values of the pre-factor Eo and exponent n as function of

cement content


Eo

(GPa)

n Ccem Eo

(GPa)

n Ccem

0.090 2.29 0.70 0.087 2.05 0.76 0.083

0.130 2.34 0.77 0.126 2.09 0.75 0.123

0.167 3.19 0.59 0.165 2.29 0.73 0.164

0.230 3.13 0.73 0.239 3.17 0.65 0.238

0.285 2.71 0.73 0.303 2.57 0.74 0.300

0.375 3.28 0.69 0.391 3.08 0.80 0.408

0.600 3.78 0.61 0.664 4.06 0.59 0.668

0.800 2.92 0.74 0.840 3.41 0.68 0.842

1.000 2.59 0.62 1.000

Fig. 3. Uniaxial strength Su as a function of cement content Ccem in

a linear plot (a) and in a log– log plot (b). Variability is indicated by

the error bars (1 S.D.).

F. Saidi et al. / Tectonophysics 370 (2003) 105–120 109

pendence at Ccem>20–30%. This can be clearly seen in

log–log scale in Fig. 3b or by calculating the ratio of

the strength of fine-grain material (i.e., F samples) over

that of coarse-grain rock (i.e., C samples) qS = Su(F)/Su

(C).

For Ccem>20–30%, qS has a constant value near unity

indicating grain size independence, while it increases

up to about 2.5 for Ccem < 20–30%.

3.3. Effect of cementation and grain size on Young’s

modulus

It is impossible to represent all the Young’s mod-

ulus data for the complete set of F and C samples and

for all the stresses on a single plot. Fortunately, the

behaviour was similar in all the samples. Fig. 4 shows

log(E) versus log(r1), including the post-peak portion,

for C samples with Ccem equal to 60% and 9% (the data

for pure cement is also shown for comparison). We see

that, in the pre-peak domain, the log(E) versus log(r1)

curves are initially straight lines (i.e., power laws,

EcEo (r1/r0)n, where r0 = 1 MPa). As the peak-

stress is approached, the log(E) versus log(r1) curvesbecome more and more horizontal, eventually revers-

ing completely in the post-peak domain. Nonlinear

elasticity was recently recognised as an important

characteristic of rocks (e.g., Guyer et al., 1997; Ostrov-

sky et al., 2000). One possible cause of nonlinearity is

the presence of Hertzian contacts between grains,

leading to a power-law EcEo (r1/r0)n with n = 1/3.

The measured values of exponent n are listed in

Table 2. We see that n fluctuates between 0.6 and 0.8


independently of grain size. Given the scatter, n

remains quite close to 0.62, the value for pure cement,

independently of Ccem. As a consequence, it seems

reasonable to assume that the nonlinear elastic behav-

iour of the F and C materials is primarily caused by the

nonlinearity of the cement itself. Since n for pure

cement is much greater than 1/3, the nonlinearity of

cement cannot be explained by Hertzian contacts only.

The pre-factors Eo are also given in Table 2. Despite a

rather large scatter, we can observe that Eo first

increases when Ccem increases up to 60%, then

decreases to the value measured for pure cement (2.6

GPa). The grain size effect reported above for strength

is almost invisible in the scatter of the Young’s

modulus data. It merely appears that, for the F samples

with Ccem < 20–30%, Eo is higher than that of the C

samples by a rather moderate 1.2–1.4 factor.

If we assume that power-law behaviour is the

signature of intrinsic nonlinear elasticity in our materi-

als, then we must interpret the growing deviations from

the initial power law as reflecting damage accumula-

tion (other observations such as presence of hysteresis

support this interpretation; Saidi, 2002). Following

damage mechanics practice (Lemaitre, 1996), we can

define a damage parameterD from the elastic moduli of

the intact and damaged materials, D = 1�E(r1)/

[Eo(r1/r0)n] (D is equal to zero as long as E has the

undamaged power-law form and approaches 1 when E

becomes strongly degraded owing to damage accumu-

lation). Typical examples of D as a function of normal-

ised axial strain e1/e1(peak) are shown in Fig. 5 and

Fig. 5. The damage parameter D for samples with different Cdry

(symbols) and in pure cement (thick line) as a function of

normalised strain (see text for more details).

compared to D for pure cement. In Fig. 5, we see that

the onset of damage accumulation is early (i.e., around

e1(peak)/2) in high Ccem samples and is progressively

retarded as the cement content is lowered. Notice,

however, that we observed several exceptions to this

trend (see for example the 80% C sample as shown in

Fig. 5).

4. Microstructure

Selected intact and deformed samples were

impregnated with epoxy. Planar sections were cut

using a diamond saw and polished. Note that polish-

ing extended patches of cement turned out to be

impossible because epoxy did not penetrate the

cement microporosity. As a consequence, samples

with Ccem>40% were not used in the microstructure

study. In intact samples, the sections were cut perpen-

dicular to the sample axis whereas in deformed

samples, we used axial sections, i.e., parallel to r1.The sections were examined in the scanning electron

microscope (SEM) in backscattered electron mode. To

facilitate comparison, we used magnifications such

that micrographs of F and C materials showed a

similar number of sand grains. This is illustrated in

Fig. 6 showing examples of F and C samples with

identical high and low Cdry (37.5 and 9%). F and C

samples appear fairly similar. At low Ccem, the cement

forms distinct patches around grain-to-grain contacts.

As cementation increases, the cement begins to fill the

pore space between the grains. Note that due to the

different magnifications, the microgranular nature of

the cement is more apparent in the micrographs of F

material than in the C samples.

4.1. Intact materials

The SEM micrographs are images with grey levels

between 0 (black) and 255 (white). Pores appear in dark

shades. To evaluate porosity, we need to determine

which pixels belong to the solid phase and which to the

pores. Here we simply applied thresholding to the gray

images to render a binary image. However, choosing a

threshold is somewhat arbitrary. Here the grey level

histograms appeared always roughly bimodal and we

picked the mid-point between the two modes as thresh-

old. Since the micrographs of the F samples had amuch

Fig. 7. Average values of macroporosity /* estimated from SEM

micrographs compared to / determined from density measurements.

Variability is indicated by the error bars (1 S.D.).

Fig. 6. Examples of SEM micrographs of the F and C materials: (a) F with Cdry = 9%. The scale bar represents 100 Am. (b) F with Cdry = 37.5%.

The scale is the same as in (a). (c) C with Cdry = 9%. The scale bar represents 1 mm. (d) C with Cdry = 37.5%. The scale is the same as in (c).


higher magnification than those of the C samples do,

the micropores inside the cement were resolved in

micrographs of F samples and therefore contributed

to the estimate of the macroporosity /*. We attempted

to correct this effect but did not come up with a

satisfactory procedure. The uncorrected results are

presented in Fig. 7 and compared to the values com-

puted from density in Table 1. Note that in Fig. 7 and in

the other microstructure figures hereafter, the error bars

represent an interval of F 1 standard deviation. They

illustrate the variability of the parameter considered

rather than the accuracy of the measurements. We see

that /* is lower in C samples than in F samples by

about 2 porosity units, a consequence of lower magni-

fication. At low Ccem, /* of F samples seems to be in

agreement with / but we know that /* includes some

cement microporosity and is therefore overestimated.

We conclude that our choice of threshold was probably

Fig. 9. Average ratio n of length of grain/cement interface to grain

perimeter. Variability is indicated by the error bars (1 S.D.).


not optimal and lead to a general underestimation of

/*. At high Ccem, /* is much smaller than / in both F

and C samples. This effect is too strong to be caused by

incorrect thresholding. The most likely explanation is

that epoxy had not penetrated some nearly isolated

macropores inside the cement, making them indistin-

guishable from the poorly polished cement back-

ground. Hence, we failed to identify them. Overall,

the conclusion of this section is that the F and C

samples did not appear different with respect to poros-

ity.

The packing topology is usually characterised by

the packing coordination number, i.e., the average

number of grain-to-grain contacts per grain. Since

the samples were not compressed during preparation,

it is likely that a layer of cement is always present

separating nearly touching grains. In that sense, there

are probably no actual grain-to-grain contacts and we

define the coordination number as the number of

cemented bonds per grain. Unfortunately, this param-

eter cannot be determined from two-dimensional

images. Instead, we counted the average number N

of close neighbours per grain. We defined a close

neighbour as a grain having at least one point at a

distance less than a quarter of the radius of the grain

considered. This threshold distance is somewhat arbi-

trary. We settled on it because it visually appeared to

be a good compromise. For each value of Ccem, a

minimum of 300 grains were analysed. The results are

shown in Fig. 8. We can see that N has a range

Fig. 8. Average number N of closest neighbours per grain as a

function of Ccem. Variability is indicated by the error bars (1 S.D.).

roughly between 3 and 4, with a tendency to decrease

as Ccem increases. Also, N is in general lower for C

material than for F, which is consistent with the

strength results reported earlier. Lastly, it is important

to note the very large variability of N as illustrated by

the error bars in Fig. 8.

We also need to characterise the spatial distribution

of cement. We want to know if the cement is exclu-

sively located at the grain contacts (or near contacts)

or if it forms a continuous layer around the grains.

This distinction is particularly important at low

cement content but becomes less and less meaningful

as Ccem increases. For this purpose, we used classic

quantitative stereology techniques (Underwood, 1970)

and measured the ratio n of the cumulative length of

grain/cement interface by the sum of the grain perim-

eters. We counted the number of intersections of

grain/cement, grain/pore and cement/pore interfaces

(noted IGC, IGP and ICP, respectively) with a test line of

known length. Using quantitative stereology, it is easy

to see that n = IGC/(IGC + IGP). The results are shown inFig. 9. We observe that n is larger for C samples than

for F ones. As expected, the values in both cases tend

to increase with increasing Ccem. Indeed, at high Ccem,

there are no pores left and n should be close to 1. The

values at low Ccem are more interesting in that their

possible range is much wider (i.e., from 0.2–0.3 to 1).

We found 0.8 for C samples and 0.5 for F ones. For

comparison, we imagine a material in which cement is

deposited only at grain contacts. According to our

measurements of N, a section through a single grain of


radius R would intersect four cemented bonds, each

one having a length 2a. We will see later that a/

R = 0.25 is a reasonable value. We therefore estimate nas 4(2a)/(2pR) = 0.3, a value significantly smaller than

the observed ones. We conclude that the cement is not

deposited exclusively at the grain-to-grain contacts

but coats a large portion of the free surface of the sand

grains. The cement coating appears to be more com-

plete in C material than in F.

4.2. Deformed samples

We also examined polished sections of deformed

samples. An example is shown in Fig. 10. We

observed essentially the same small number of intra-

granular microcracks as in the intact sand, indicating

that grain fracturing did not occur. This is not surpris-

ing since the F and C materials are too weak to allow

sufficiently high stresses to arise around grain con-

tacts. On the other hand, we observed intergranular

tensile microcracks at grain–cement interfaces. As

expected, the tensile microcracks were generally sub-

parallel to r1. Sometimes, these tensile fissures

extended through the cement itself. These observa-

tions suggest that cement debonding was an important

damage mechanism. This is in good agreement with

the results of Bruno and Nelson’s (1991) simulations.

However, other undetectable mechanisms may also

Fig. 10. Example of SEM micrograph of deformed sample (C with

Cdry = 16.7%). Debonding cracks at the grain/cement interfaces are

indicated by A while B highlights tensile cracks in the cement. The

direction of r1 is shown. The scale is the same as in Fig. 6c.

have been significant. In particular, shear fracturing in

the cement or grain boundary sliding are very hard to

identify because of the absence of offset markers. We

also noted fewer and smaller tensile debonding cracks

at high Ccem than at low cement content, suggesting

that other mechanisms were indeed active in this case.

5. Modelling and discussion

5.1. Young’s modulus

In modelling, the elastic properties, the greatest

difficulty we have to face is the nonlinear elasticity of

the cement phase. In fact, a rigorous treatment is out

of the scope of the present paper. Instead, we assume

that the classic methods valid for linear elastic media

can still be meaningfully used here. We consider two

end-members: (a) at high Ccem, the sand grains act as

rigid inclusions in a soft matrix of cement and macro-

pores, and (b) at low Ccem, the grains are cemented to

each other at grain contacts (Dvorkin et al., 1991,

1994). In both cases, one basic assumption is that the

cement acts as a homogeneous, continuous phase, and

its elastic behaviour is identical to that observed in

pure cement samples. Notice that the effect of macro-

porosity / is automatically taken into account in case

(b) but not in case (a). One way to handle this problem

is to homogenise cement and macropores together in

our treatment of end-member (a). In order to simplify

the analysis, we additionally assume that the sand

grains are monodispersed spheres. This is justified

since we saw earlier that, in both varieties, the grains

have a narrowly distributed radius R and an aspect

ratio of 0.6 (above 0.5, grains can safely be assimi-

lated to spheres; Huang and Hu, 1995).

It is convenient to start with end-member (a). We

follow the self-consistent analysis of Luo and Weng

(1987). A composite sphere consisting of a sand grain

of radius R and a concentric coat of a mixture of

cement and macroporosity is embedded in an infinite

volume of a material having the effective properties of

the sample considered (see Fig. 11a). The advantage

of this self-consistent approach is that it is not

restricted to dilute concentrations of sand grains, i.e.,

to Ccem near unity. Note that the weak cement/macro-

porosity matrix is the only connected phase in this

system, implying that the effective material should be

Fig. 11. The geometric arrangements corresponding to (a) the

effective medium inclusion model, and (b) the cemented bond

model (arrangement 1 and arrangement 2).


weak as well. The outer radius RV of the composite

sphere is R(1�Ccem�/)� 1/3. According to Luo and

Weng (1987), the effective bulk modulus Keff and

shear modulus Geff are:

Keff ¼ K* 1þ ð1� Ccem � /ÞðKQ � K*ÞðCcem þ /Þa*ðKQ � K*Þ þ K*

� �ð1Þ

Geff ¼ G* 1þ ð1� Ccem � /ÞðGQ � G*ÞðCcem þ /Þb*ðGQ � G*Þ þ G*

� �ð2Þ

where KQ and GQ are the bulk and shear modulus of

quartz (36.6 and 45 GPa, respectively), K* and G* are

the effective bulk and shear modulus of the cement/

macroporosity mixture, and, the two parameters a*and b* can be calculated in terms of KQ, GQ, K* and

G*. Luo and Weng (1987) demonstrated that

a* = 3K*/(3K* + 4G*), i.e., Eq. (1) is identical to the

classic Hashin–Shtrikman lower bound. Their results

also show that with the values of KQ, GQ, K* and G*

considered here, b* = 1.2(K* + 2G*)/(3K* + 4G*), i.e.,Hashin–Shtrikman lower bound for shear modulus, is

an excellent approximation. These definitions of a*and b* will be used here.

In order to evaluate K* and G*, we apply the same

approach to macropores embedded in a cement

matrix:

K* ¼ Kcem 1� /Ccem þ / � aCcem

� �ð3Þ

G* ¼ Gcem 1� // þ Ccem � bCcem

� �ð4Þ

where Kcem and Gcem are the cement bulk and

shear moduli, and, a = 3Kcem/(3Kcem + 4Gcem) and

b = 1.2(Kcem + 2Gcem)/(3Kcem + 4Gcem). Note that

Eqs. (3) and (4) are identical to Hashin–Shtrikman

higher bounds. In a previous uniaxial test on a pure

cement sample, we measured the Young’s modulus

Ecem and Poisson’s ratio mcem of cement. We found

that mcem was approximately equal to 0.2 independ-

ently on stress. We then have Kcem =Ecem/3(1�2mcem) and Gcem =Ecem/2(1 + mcem), which implies

that Kcem and Gcem have the same stress depend-

ence as Ecem (i.e., Ko(r1/r0)n and Go(r1/r0)

n, where

Ko =Eo/3(1� 2mcem) and Go =Eo/2(1 + mcem)).Concerning the nonlinear elasticity, the classic

treatment (Suquet, 1995) is to replace nonlinear

phases by linear materials with adequately chosen

properties, i.e., corresponding to the average stresses

and strains. Here we assume the average stress in the

cement to be nearly equal to the remotely applied

stress. This assumption may not be too bad because

of the rather weak stress dependence of the cement

(i.e., the exponent n is lower than 1). Accordingly,

we simply replace Kcem and Gcem in Eqs. (3) and (4)

by Ko(r1/r0)n and Go(r1/r0)

n, where Ko and Go are

the experimentally measured pre-factors for pure

cement and r1 is the remotely applied stress. We

can then evaluate the stress dependence of Keff, Geff

and Eeff = 9KeffGeff/(3Keff +Geff) by performing the

above calculations for the values of / and Ccem in

Table 1, using a broad range of r1 (i.e., 0.1–100

MPa). We found that the calculated Eeff very closely

obeyed a power law Eeff =Eo(r1/r0)n. In all cases,

the exponent n was only slightly lower than n for

pure cement (i.e., 0.55–0.58 instead of 0.62). Fig.

Fig. 12. Experimental and theoretical values of Eo in a linear plot (a)

and in a log– log plot (b). The thick lines correspond to the

inclusion model and the thin ones to the cemented bond model. The

solid lines and symbols refer to F materials whereas the dotted lines

and open symbols correspond to C.


12a,b shows the calculated effective pre-factors Eo

and compares them to the observed ones. The strik-

ing result is that the agreement is good not only at

high cement content (i.e., Ccem>60%) where the rigid

inclusion model is expected to be adequate but also

at all Ccem down to 30%. In particular, the maximum

at Ccem = 60% and the decline of Eo for Ccem

between 60% and 30% are correctly predicted. As

a test, we tried another model in which the homog-

enisation steps described above were interchanged.

Namely, we homogenised grains and cement first

(i.e., Eqs. (1) and (2) with the appropriate constants),

and then mixed the resulting effective medium with

macropores (i.e., Eqs. (3) and (4)). The resulting Eo

values were always larger than the ones presented

above. The discrepancy was small only for

Ccem = 80% and regularly increased with decreasing

cement content. In fact, Eo did not show a maxi-

mum. It is intuitively clear that the alternative model

should be appropriate in the case of large voids (i.e.,

several grain sizes) embedded in a relatively strong

sand/cement mixture. To the contrary, our micro-

structure observations suggest that the macropores

were relatively small and embedded inside the

cement matrix. Clearly, the primary effect of such

macropores is to weaken the cement. This leads to a

weak effective medium because the cement/macro-

porosity matrix is the connected phase.

Below 30%, the inclusion model strongly under-

predicts Eo. This is due to the fact that, contrary to

what is assumed in the model, cement is, for the most

part, optimally deposited at the grain contacts. Its

strengthening effect is therefore enhanced. In order

to take this into account, we used Dvorkin et al.’s

(1994) cemented bond model. The normal and tan-

gential stiffnesses, SN and ST, of an individual

cemented bond are given by:

SN ¼ � 4pRGcemð1� mcemÞ1� 2mcem

kN

Do

ð5Þ

ST ¼ �2pRGcem

kT

Do

ð6Þ

where R is the grain radius and Do is an arbitrary

nonzero dimensionless factor. The dimensionless

parameters kN and kT are obtained by calculating the

following integrals:

kN ¼Z a

0

HNðtÞtdte þ t2=2

ð7Þ

kT ¼Z a

0

HTðtÞtdte þ t2=2

ð8Þ

where a = a/R is the normalised radius and e = h/R the

normalised thickness of the cemented contact (see Fig.

Table 3

Normalised grain separation e assuming hexagonal packing and

normalised cemented bond radius a1 and a2 using Dvorkin et al.’s

(1994) arrangements 1 and 2


e a1 a2 e a1 a2

0.049 0.078 0.408 0.091 0.397

0.070 0.070 0.466 0.083 0.085 0.456

0.090 0.064 0.512 0.198 0.089 0.492 0.104

0.130 0.090 0.558 0.254 0.104 0.548 0.218

0.167 0.106 0.598 0.320 0.111 0.595 0.311

0.230 0.126 0.660 0.443 0.129 0.659 0.438

0.285 0.163 0.700 0.524 0.170 0.696 0.510

0.333 0.221 0.721 0.560 0.214 0.724 0.572

0.375 0.276 0.739 0.589 0.246 0.752 0.648

0.444 0.363 0.775 0.674 0.335 0.785 0.732

0.600 0.589 0.585

0.800 1.321 1.318


11b). The dimensionless functions HN and HT are

solutions to the integral equations described in Appen-

dix A. The effective bulk and shear moduli are then

given by:

Keff ¼Ncð1� Ccem � /ÞSN

12pRð9Þ

Geff ¼Ncð1� Ccem � /Þ

20pRSN þ 3

2ST

� �ð10Þ

where Nc is the average contact coordination number,

i.e., the mean number of cemented bonds per grain.

The cemented bond model of Dvorkin et al.’s

(1994) is more complicated to use than the inclusion

model because we have to evaluate three additional

parameters, Nc, a and e. None of these parameters can

be obtained directly from the microstructure observa-

tions reported in Section 4. However, the microstruc-

ture study provides some useful information. We saw

that N, a two-dimensional equivalent of Nc, varied

between 3 and 4. In three dimensions, we assume that

2N gives a reasonable estimate of Nc, which would

thus range between 6 and 8. Although we know that

the actual sand packing was random, we need to

choose a regular sphere packing in order to proceed.

Nc = 6 corresponds to the simple cubic packing. We

must reject it because the maximum volume fraction

of the spheres (i.e., Cg) is p/6 = 0.524, a value smaller

than those yielded by the experimental data of Table 1

(between 0.551 and 0.529 for the first few F and C

samples). Assuming simple cubic packing in these

cases would imply that the sand grains penetrated

each other, which we know is impossible. Nc = 8

corresponds to hexagonal packing, i.e., two-dimen-

sional hexagonal layers of spheres, stacked on top of

each others. The maximum Cg is now p/3M3 = 0.605,

a value larger than any Cg calculated from the data of

Table 1. We can then calculate the normalised sepa-

ration e of the sand grains assuming hexagonal pack-

ing and using the values of Table 1. We simply have

to solve / +Ccem = 1� (1 + e/2)� 3p/3M3. The calcu-

lated values are given in Table 3. We lack sufficiently

detailed information on the spatial distribution of

cement to calculate the normalised radius a = a/R of

the cemented bonds directly. However, we know that

a must be bounded by a1 and a2, two values that we

can calculate using the two arrangements proposed by

Dvorkin et al. (1994) and illustrated in Fig. 11b.

Arrangement 1 assumes that the cement is all depos-

ited at the grain contacts, whereas, in arrangement 2,

the cement forms a continuous layer around the

grains. Tedious but simple volume calculations lead

to the following equations:

Ccem

Cg

¼ C

43a21 1þ e=2ð Þ � 2þ 2 1� a21

� �3=2h ið11Þ

Ccem

Cg

¼ ð1� C=2Þ½ð1þ e=2Þ2 þ a22�3=2 � 1

þ C

2ð1þ e=2Þ3 þ 3Ca22

4ð1þ e=2Þ ð12Þ

where a1 and a2 refer to arrangements 1 and 2,

respectively. These equations have at most one root

between 0 and 1. The results are given in Table 3. We

see that no solutions were found for a2 in the samples

with the lowest Ccem. Indeed, we verified that, in these

cases, the normalised thickness of the cement layer

was less than e/2, implying a2 = 0. Now, the simplest

is to take a equal to the mean of a1 and a2. Notice,however, that the cemented bonds overlap signifi-

cantly for a larger than about 0.65, implying that the

cemented bond model becomes inadequate in such

cases. Accordingly, only the cases with a< 0.65 were

considered in the following.

Finally, we proceeded with Dvorkin et al.’s (1994)

model as with the inclusion model: we replaced the


nonlinear elastic cement by a linear material with

elastic properties chosen as before, i.e., corresponding

to the remotely applied stress. The calculated Eeff did

not display excellent power law behaviour (i.e.,

Eeff =Eo(r1/r0)n as before. Limiting r1 to the exper-

imental range 1–15 MPa, we obtained n between 0.42

and 0.50 (smaller than the pure cement exponent) and

values of Eo closer to the experimental data than the

results of the inclusion model (see Fig. 12a,b). How-

ever, the estimated Eo were still significantly smaller

than the experimental values, maybe owing to a

systematic underestimation of a or to nonlinear elas-

ticity effects (i.e., underestimation of stress concen-

tration in cemented bonds and, therefore, elastic

properties of cement). Interestingly we obtained 30–

50% smaller Eo in the C samples than in F (see Fig.

12b). We argue that this is probably a consequence of

our method of calculating e and a (the smaller

/ +Ccem, the smaller e and the larger a, finally leading

to a strong enhancement of Eo). If our goal was

merely to fit the experimental data, we could obvi-

ously obtain a much better match by defining a as an

adjustable linear combination of a1 and a2. But thiswould be quite arbitrary since we do not have any

independent constraint on a. We can nevertheless

confidently conclude that, at Ccem < 20–30%, the

cemented bond model is a clear improvement with

respect to the inclusion model.

5.2. Uniaxial strength

As explained above, damage was exclusively

located in the cement. Hence, it is clear that the

strength of the F and C materials is controlled by

the state of stress of the cemented regions between the

sand grains. The elastic models discussed above are

grain size independent, implying that the stress field

and therefore strength should also be insensitive to

grain size. This prediction agrees with the experimen-

tal results at Ccem>20–30% but not below this value.

We conclude that at least one important assumption

used in our implementation of Dvorkin et al.’s (1994)

cemented bond model was differently satisfied in the

F and C materials.

We saw in Section 2.1 that one important differ-

ence between the F and C sands was the width of their

grain size distributions as measured by rd/d. As a

consequence, N (i.e., a two-dimensional equivalent of

Nc) was 20% larger in F samples than in C (see

Section 4.1). A decrease of Nc implies a general

increase of the stress concentrations in the cemented

bonds and, hence, a decrease in strength. In addition,

the microstructure study showed that n (see Section

4.1) was larger in C materials by about 30%. An

increase of n indicates an increase of the amount of

cement inefficiently deposited away from the grain

contacts. This implies smaller a and, therefore,

smaller strength. Contrary to these observations, we

assumed in the cemented bond model that Nc was

equal to 8 and we used the same procedure to estimate

a in both F and C materials. Are these subtle differ-

ences in packing and cement distribution a plausible

explanation to the strength differences observed at

low cement content? If yes, why do they have a much

smaller effect on the elastic properties? Why does the

effect disappear at Ccem>30%?

The answer to the last question is easy. At high

Ccem, these differences in Nc and n tend to decrease

(the microstructure observations show the beginning

of this trend). Even if substantial differences re-

mained, at Ccem>30% the sand grains act as inclu-

sions. In this case, packing and cement distribution

become essentially irrelevant. As to the second ques-

tion, we can see readily that Nc appears linearly in the

cemented bond model. Hence, a 20% change in Nc

must produce a 20% change in the elastic properties.

We numerically verified that a similar change in ayielded a change in Young’s modulus of no more than

30%. Finally, in order to answer the first question, we

consider the strength S* of individual cemented bonds

and recognise that failure is not controlled by the

mean S* but by its minimum value, i.e., the weakest

link (see the numerical simulations of Reuschle, 1998;

Tang et al., 2000). A similar line of reasoning can be

followed using the magnitude of stress concentration

at individual cemented bonds. If all bonds are equally

resistant, failure is obviously controlled by the most

stressed ones. It is possible that, in our materials, a

small decrease in the mean value of Nc corresponded

to much greater fluctuations of the magnitude of stress

concentrations in cemented bonds. Likewise, a small

decrease of the average n may have induced much

larger fluctuations of S*. We conclude that the subtle

differences in packing and cement distribution

observed are indeed a plausible explanation to the

strength differences observed at low cement content.


As discussed above, there is a strong relationship

between the mechanical properties and the spatial

distribution of cement, which, of course, depends on

how the mixture was prepared, specifically, how the

water was introduced. The concentration of cement at

the grain contacts generally observed here suggests a

predominance of capillary forces. In this respect, our

fabrication procedure was similar to that of Holt et al.

(1993), who introduced the reactive fluid, CO2, using

two-phase flow conditions. In nature, analogues to our

materials might therefore be found in settings charac-

terised by unsaturated flow.

6. Conclusion

1. Poorly consolidated granular materials are signifi-

cantly strengthened by cementation. Uniaxial

strength strongly increased with increasing cement

content. The static Young’s modulus had a more

complex behaviour because the cement used in this

work was a weak, nonlinear elastic phase.

2. We observed a transition from grain size sensitivity

of strength at cement contents lower than 20–30%

to grain size independence above this value. To the

contrary, static Young’s modulus appeared rather

unaffected by grain size.

3. We were able to model the variations of static

Young’s modulus with cement content using two

very different approaches. At high Ccem, the sand

grains were treated as hard inclusions embedded in

a soft cement/macroporosity matrix. At low Ccem,

we used Dvorkin et al.’s (1994) cemented bonds

model. One striking result was that the inclusion

model remained valid in an unexpectedly wide

range of cement content, i.e., down to Ccem = 30%.

4. Because the sands used here are well sorted, the

models above predict that the elastic properties

should be unaffected by grain size. Theoretically,

this should be also true for strength. However, the

relative grain size distribution was narrower in F

sand than C, resulting in small differences in

packing and spatial distribution of cement. We

argue that strength is strongly affected by fluctua-

tions of the microstructural parameters while

elastic moduli are not. Consequently, it is plausible

that these subtle differences in packing and spatial

distribution of cement had a much greater impact

on strength than on the elastic properties of the F

and C materials.

Acknowledgements

We are grateful to A. Aydin and W. Olsson for

reviewing the manuscript. Their comments were very

helpful.

Appendix A

Here, we only need to concentrate on the integral

equation for HN:

Do þ HNðtÞ ¼ �KN

Z p

0

d/Z tcos/þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2�t2sin2/

p

0

� HNðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 þ s2 � 2tscos/

pÞ

e þ 1=2ðt2 þ s2 � 2tscos/Þ ds

ðA1Þ

where KN=(2Gcem/pGQ)(1 � mcem)(1 � mQ)/(1 �2mcem). Numerically solving this equation requires

using a sufficiently accurate numerical integration

method.

Consider the following integral I:

I ¼Z a

0

f ðsÞds ðA2Þ

I can be approximated by the following sum:

IcXNi¼0

a

Npifi ðA3Þ

where fi= f(ia/N) and pi are weights that depend on the

method used (in this work, we used the extended

Simpson’s rule; Press et al., 1987). Applying this

method twice, Eq. (A1) can be approximated by:

Do þ HN

kaM

� �¼ �KN

XMj¼0

ppjXMi¼0

abjkpi

� HNðxijka=MÞ

e þa2x2ijk2M 2

ðA4Þ


where

bjk ¼k

Mcos

jpM

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

M 2sin2

jpM

� �s

and

xijk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ i2b2jk � 2kibjkcos

jpM

� �s:

By interpolation, we can express HN(xijka/M) as a

function of the HN(ka/M). We thus obtain an easily

solved system of M+ 1 linear equations with M + 1

unknowns, i.e., HN(ka/M). Notice that system (A4) is

singular for e = 0. In that case, a more sophisticated

method is necessary.

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The mechanical behaviour of synthetic, poorly consolidated granular rock under uniaxial compression

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