The Mechanical Behaviour of Synthetic
Sandstone with Varying Brittle Cement
ContentC. DAVID$
B. MENEÂ NDEZ$
Y. BERNABEÂ $
The purpose of this work was to investigate the in¯uence of cement on the
mechanical behaviour of granular rocks. Following the technique described
in den Brok et al. [den Brok, S. W. J., David, C. and BernabeÂ, Y., Prep-
aration of synthetic sandstones with variable cementation for studying the
physical properties of granular rocks. C. R. Acad. Sci., 1997, 325, 487±
492], two blocks of synthetic sandstones with di�erent cement content were
prepared for mechanical testing under hydrostatic and triaxial conditions.
The results of the mechanical tests show that the behaviour of the synthetic
rocks compares well with that of natural sandstones. Increasing the amount
of cement from 3 to 5% in volume had important consequences on the mech-
anical properties: the critical pressure, strength and elastic moduli were sig-
ni®cantly increased and the brittle-to-ductile transition was shifted towards
higher pressures. We compared our results to the models of Zhang et al.
[Zhang, J., Wong, T. -F. and Davis, D. M., Micromechanics of pressure-
induced grain crushing in porous rocks. J. Geophys. Res., 1990, 95, 341±
352] and Wong et al. [Wong, T. -F., David, C. and Zhu, W., The transition
from brittle faulting to cataclastic ¯ow in porous sandstones: Mechanical de-
formation. J. Geophys. Res., 1997, 102, 3009±3025]. We conclude that
Zhang et al.'s microstructural parameter fD (i.e. the product of porosity f
by grain size D) appeared to be a scaling parameter for both the failure
envelopes and the critical pressure as de®ned in these models. Intuitively, the
contact length L is expected to play a crucial role in the mechanical proper-
ties of granular materials. Accordingly, we made a statistical analysis of this
microstructural parameter in our synthetic materials and in a suite of natu-
ral sandstones. A positive correlation with Young's modulus and a negative
correlation with porosity were found. This last result gives a physical back-
ground for the use of parameter (fD) in theoretical models. We want to
emphasize that working on synthetic sandstones allows for a better control
of the structural parameters (grain size, sorting, cement content, etc.) which
appear to be so important for the mechanical properties of granular rocks.
# 1998 Elsevier Science Ltd.
INTRODUCTION
Understanding the compaction mechanisms in porous
rocks is of great importance in many geotechnical ap-
plications from both a geological and economical view-
point. For example, compaction of reservoir rocks
associated with the production of oil or natural gas
can produce severe subsidence of the Earth surface [4].
Sanding and well stability are other important pro-
blems in which a good knowledge of the mechanical
behaviour of granular rocks is required. Unfortunately
compaction of porous rocks is poorly known and it is
often di�cult to identify possible problems in a timely
fashion before the exploitation of a potential site is
started. There are two major mechanisms which can
explain the reduction of porosity in situ: mechanical
compaction and chemical compaction. The latter
includes long time scales processes such as pressure
solution. It should therefore not be relevant to
Int. J. Rock Mech. Min. Sci. Vol. 35, No. 6, pp. 759±770, 1998
# 1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
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759
geotechnical problems like those mentioned above.
Mechanical compaction, on the other hand, consists of
short term processes (i.e. rotation and sliding of grains,
pore collapse, grain fracturing, fragmentation and so
forth), all expected to play an important role in this
context. The present paper is only concerned with
mechanical compaction.
The mechanical behaviour of granular rocks has
been thoroughly investigated in the last decades, both
from an experimental and a theoretical viewpoint
(among others, see Refs. [3, 5±9]). One goal in these
studies was to identify the parameters controlling the
compaction mechanisms. Zhang et al. [2] emphasized
the role of porosity and grain size. According to their
model based on Hertz theory applied to the grain-to-
grain contacts, irreversible compaction induced by
grain fracturing and pore collapse occurs in hydro-
static conditions above the so-called critical pressure
Pcrit. The model predicts that Pcrit is proportional to
(fD)ÿ3/2, where f is the porosity and D the grain size.
This behaviour was indeed observed experimentally [10]
and the quantitative relationship above was found to
hold satisfactorily (Ref. [3] and references therein).
Interestingly, Hertzian contact mechanics predicts the
non-linearity of the elastic regime in granular rocks, as
is indeed observed in laboratory experiments [8, 11].
However, the description of grain contacts as idealized
point to point contacts is not supported by microstruc-
ture observations. Among others, Simmons et al. [12],
Fredrich et al. [13] and Mene ndez et al. [9] found that,
in sandstones, virtually all grain-to-grain contacts are
cemented, in which case the Hertz theory is not appro-
priate. One important consequence of the presence of
cement is that the sti�ness of cemented contacts should
be insensitive to pressure [14±16]. This apparent con-
tradiction between mechanical data and microstructure
observations needs to be elucidated. Recent experimen-
tal and theoretical studies have con®rmed the import-
ance of cement: in particular it was shown that even
minute amounts of cement, if situated at the grain-to-
grain contacts, can greatly increase the sti�ness and
strength of granular materials [17±20].
It is clear that more experimental data on the role
of cemented contacts are required, but this is di�cult
to do in natural rocks because of their extreme com-
plexity and the strong coupling between the di�erent
structural parameters. An attractive alternative
approach is to prepare synthetic rock analogues in
which only one parameter is allowed to vary. In our
case, this parameter was the cement content. Several
attempts were made before [18, 21±24], each of them
with di�erent choices for the starting granular material
and cement composition. We have developed an e�-
cient method for preparing synthetic sandstones in
which the amount of silica glass cement could be con-
trolled. In a previous paper [1], we described the prep-
aration technique in detail. In this paper we present
results of a suite of mechanical tests performed on
samples of our synthetic sandstones. We will particu-
larly focus on the following questions. How does
cement content a�ect the mechanical behaviour of the
rock analogues? How do the synthetic sandstones com-
pare with natural sandstones? Is it possible to identify
the key parameters at the microscopic scale controlling
the mechanical behaviour?
PREPARATION AND DESCRIPTION OF THE
SYNTHETIC SANDSTONES
The technique for preparing the synthetic sandstones
is described in detail in Ref. [1]. We will brie¯y recall
the main steps of the preparation procedures and the
basic properties of the material produced.
The starting material was Fontainebleau sand, an
almost pure quartz sand with a mean grain size of
130 mm (standard deviation 50 mm). After coating the
sand grains with an alkaline silica gel and burning the
solvent o�, hot isostatic pressing was applied in order
to produce a well consolidated aggregate of sand and
silica±glass cement. The P±T path during hot pressing
was as follow: ®rst the pressure was raised to 40 MPa,
then the temperature was increased to 8008C; after
holding these conditions for 1 h, pressure and tempera-
ture were slowly lowered down to room conditions.
The heating temperature was chosen to be higher than
the melting temperature of the silica glass, but lower
than that of quartz. At 8008C, the melted silica glass
migrated to the grain-to-grain contacts due to capillary
forces. After cooling, the silica glass appeared to be
quite similar to diagenetic cement in natural rocks.
Compared to other attempts [18, 19], our technique
has the advantage that the cement is brittle (helping
microstructure investigation of deformed materials)
and that hot-pressing allows for the preparation of
well-consolidated material with porosity comparable to
common sandstones. Depending on the volume of
silica gel mixed to Fontainebleau sand, it is possible to
produce sandstone-like material with di�erent amount
of cement. Furthermore, it is important to note that
cement content is the only signi®cantly variable par-
ameter: in particular mineralogical content and grain
sorting are unchanged in the process. We decided to
prepare two varieties of synthetic sandstone, hereafter
called SS3% and SS5% with cement contents corre-
sponding to, respectively, 3 and 5% volume fractions
(note that these are only rough estimates because it
was not possible in practice to control exactly the
volume fraction of cement). On thin sections, the syn-
thetic sandstones appeared quite similar to natural
Fontainebleau sandstone [1]. To verify that the melted
silica glass correctly migrated to the grain-to-grain
contacts, we used scanning electron microscopy and X-
ray analysis to ®nd out where potassium rich elements
were located. Since potassium is only present in the
composition of the silica glass, this technique allows to
map the spatial distribution of cement. An example is
shown in Fig. 1. On all the analysed micrographs, we
found evidence that silica glass is located preferentially
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE760
at grain-to-grain contacts and much less on free grain
surfaces.
From two blocks of SS3% and SS5%, we cored a
number of cylindrical samples for mechanical testing.
Prior to these tests, we made some simple benchtop
measurements in order to characterize each material.
The results are summarized in Table 1. We determined
the porosity, f, from measurements of the weight of
dry, water-saturated and water immersed samples. The
material with the largest amount of cement has a
lower porosity (fSS5%=18.7%) compared to the other
one (fSS3%=21.9%). Such a result is normal. Both
batches of coated grains were subjected to identical
conditions, therefore the volume between the grains
was approximately the same and a greater portion of
this volume was occupied by the silica glass cement in
SS5% than in SS3%. This is also in agreement with
the largest density r found for SS5% compared to
SS3% (Table 1). From these results, it is possible to
calculate the density of the solid phase rs=r/(1ÿ f):
for both materials, we found rs=2560 kg/m3, a value
lower than the density of pure quartz (2650 kg/m3).
This shows that the silica glass cement has a lower
density than quartz. This e�ect was possibly increased
by the presence of small gas bubbles in the silica glass
(den Brok, personal communication). Ultrasonic P-
wave velocities VP were measured at room conditions
using the experimental setup described in David et
al. [25]: we observed a 35% increase in velocity for
SS5% compared to SS3% (Table 1). This huge di�er-
ence can be explained by the fact that in granular ma-
terials acoustic velocities are mostly controlled by the
elastic sti�ness of grain contacts, which is strongly
dependent on cementation and contact
roughness [26, 27]. Anticipating on the results pre-
sented in Section 4, we reported in Table 1 our data
Fig. 1. Example of SEM analysis on a thin section. (a) SEM micrograph in backscattered mode. (b) Distribution of silicium(light grey) from X-ray analysis. (c) Distribution of potassium (light grey), only present in the glass cement: due to the lowconcentration, the image is quite noisy. (d) Interpretation: cement is located either at grain contacts, on free boundaries or
®lls transgranular cracks.
Table 1. Comparison of selected bulk properties for both syntheticsandstones
SS3% SS5%
Cement content 03% vol. 05% vol.Porosity 21.920.6% 18.720.4%Dry density 2000213 kg/m3 2113224 kg/m3
P-wave velocity 2.920.2 km/s 4.120.2 km/sYoung's modulus 7.320.5 GPa 14.721.4 GPaBulk modulus 9.321.3 GPa 12.021.6 GPa
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE 761
for the elastic moduli E (Young's modulus) and K
(bulk modulus) inferred from the triaxial and hydro-
static compression tests. Again, signi®cant di�erences
are observed, with material SS5% much sti�er than
SS3%. This con®rms that cementation has a strong in-
¯uence on elastic properties, as predicted by theoretical
models [19].
To summarize, these simple measurements show that
the two synthetic sandstones have very di�erent prop-
erties. The more cemented synthetic sandstone has a
lower porosity, a higher density and velocity and is
sti�er than the less cemented synthetic sandstone.
Since the only varying parameter in the preparation
procedure is the amount of cement added in the sand
mixture, these properties obviously are strongly depen-
dent on the cement content in each material. As was
pointed out in previous papers [16, 19], we think that
not only the volume of cement is important, but also
its location at grain-to-grain contacts.
EXPERIMENTAL PROCEDURES
The main goal of this study was to investigate the
in¯uence of cementation on the mechanical behaviour
of the synthetic sandstones. For this purpose, series of
mechanical tests were performed under various stress
conditions, on water-saturated core samples with a
nominal diameter of 20 mm and a length of 40 mm. In
all the experiments, pore pressure Pp was kept con-
stant at 10 MPa.
First we conducted a series of hydrostatic compac-
tion tests, following the procedures described in
Refs. [2, 28]. Jacketed samples were subjected to e�ec-
tive pressures (de®ned here as PcÿPp) up to about
500 MPa, by incrementally increasing the con®ning
pressure Pc. At each pressure step, the pore volume re-
duction was determined from the amount of water
withdrawn from the sample in order to keep the pore
pressure constant. This was done by means of a press-
ure generator coupled to a displacement transducer [2].
Porosity reduction Df was calculated from the pore
volume reduction normalized to the bulk volume, with
an uncertainty of about20.1% [3]. In addition acous-
tic emissions (AE) were continuously recorded during
the tests. Recording acoustic emissions while deform-
ing rocks is a very e�cient method to estimate damage
accumulation in real time [29±31]. We can, thus, accu-
rately identify the stress state corresponding to the
onset of intense microcracking (i.e. sharp acceleration
in the AE activity) which, in general, is associated with
an in¯ection on the stress±strain curves (i.e. accelera-
tion of the deformation rate). The corresponding e�ec-
tive pressure is often referred to as the critical pressure
Pcrit [2].
We also performed triaxial compression experiments,
at di�erent values of the con®ning pressure, following
the procedure described in Wong et al. [3]. Rock
samples were loaded at a constant axial strain rate
(5� 10ÿ5 sÿ1) while maintaining pore pressure and con-
®ning pressure constant. The axial shortening was
measured with a displacement transducer and the axial
load with an external load cell, from which we inferred
the axial strain e1, the di�erential stress q = s1ÿs3(where s1 is the axial stress and s3=s2=Pc) and the
mean e�ective stress p = (s1+2s3)/3ÿ Pp. The same
technique as for the hydrostatic compression tests was
used to determine pore volume variations. Acoustic
emission activity was also recorded during these tests.
In granular rocks at high con®ning pressures, massive
AE activity is usually observed simultaneously with an
in¯ection on the stress±strain curve at a stress level
C*. The critical stress C* indicates the onset of shear-
enhanced compaction where rocks deform by distribu-
ted cataclastic ¯ow and continuous strain-hardening [3].
At low con®ning pressures, a sharp acceleration in AE
activity usually occurs slightly after the peak stress S
and is associated with shear localization, dilatancy and
strain softening. The parameters Pcrit, C* and S were
also observed and measured in our synthetic rocks:
comparing their values will give us valuable infor-
mation on the e�ect of cementation on the mechanical
properties of the synthetic sandstones.
RESULTS OF THE MECHANICAL TESTS
Hydrostatic compaction experiments
In Fig. 2 we plotted our results for the hydrostatic
compaction experiments. For both materials, three
di�erent samples were tested, in order to check the
reproducibility. The results for the samples tested up
to the maximum capability of the pressure vessel
(about 450 MPa e�ective pressure) are plotted with
solid lines and open symbols. In other experiments
(represented in dashed lines), the highest applied press-
ure was lower, due to leaks or other technical pro-
blems. Vertical bars in Fig. 2 correspond to acoustic
emission ``activity'' de®ned as the number of AE gen-
erated per pressure increment. The critical pressure
Pcrit could be identi®ed for synthetic sandstone SS3%:
in Fig. 2(a), a signi®cant increase in AE activity ac-
companied a weak acceleration in porosity reduction
at 380 MPa e�ective pressure. On the contrary, there is
no evidence on Fig. 2(b) that the critical pressure was
reached for SS5% at 450 MPa e�ective pressure: how-
ever looking at the slow but signi®cant increase of AE
activity, we suspect that Pcrit was probably slightly
higher than this value. Bulk moduli K were calculated
from the linear part of the compaction curves in Fig. 2:
the results are reported in Table 1. As expected we
found that SS5% is less compressible than SS3%.
Notice that reproducibility is not very good for SS3%,
which explains the large standard deviation on the de-
termination of K (about twice the value for SS5%). In
Fig. 2(c) we compared the compaction curves for the
synthetic sandstone to that of Berea sandstone, a natu-
ral granular rock of similar porosity [2]. On this plot,
the x-axis corresponds to the pressure dependent por-
osity f = f0ÿDf calculated from initial porosity f0
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE762
(=21% for Berea sandstone) and porosity reduction
Df. It is clear that the mechanical behaviour of our
synthetic sandstones under hydrostatic compression re-
semble that of natural rocks. Notice that the critical
pressure for Berea sandstone is the same as for SS3%
(380 MPa) [2], but is associated with a much larger re-
duction in porosity than occurring in SS3%. We sus-
pect that further increment in pressure would have
caused accelerated compaction in SS3% as observed in
Berea sandstone. Note also that a sharp in¯ection on
the compaction curve at the critical pressure is not sys-
tematically observed in natural rocks: David et al. [28]
reported for a Fontainebleau sandstone a sharp peak
in AE activity at Pcrit, with almost no noticeable conse-
quence on the porosity reduction. The important con-
clusion from the hydrostatic compaction tests is that
the critical pressure for SS3% is lower than for SS5%.
This result is in agreement with the model proposed by
Zhang et al. [2] and will be discussed in more detail in
Section 6.
Triaxial compression experiments
The results for the triaxial tests performed on the
synthetic sandstones are given on Fig. 3, with the top
diagram showing the di�erential stress±axial strain
curves and the bottom one showing the porosity re-
duction±mean e�ective stress curves. Each curve corre-
sponds to a sample tested at a ®xed e�ective con®ning
pressure s3ÿPp as indicated on the plots. To check the
reproducibility, two experiments were performed under
the same conditions, namely s3ÿPp=30 MPa. One can
see that reproducibility is fairly good for SS3%, but
rather poor for SS5%: this is opposite to what we
observed in the hydrostatic compaction tests. A poss-
ible cause for such a variability can be the location of
the cored samples within the block. Most probably the
Fig. 2. Results of hydrostatic compaction tests. (a) Compaction curves and acoustic emission activity (vertical bars) forSS3%. (b) Same for SS5%. (c) Comparison of the porosity evolution under hydrostatic stress for the synthetic sandstones
and Berea sandstone.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE 763
pressure and temperature were not uniform in the
blocks during hot-pressing, leading to variations of
porosity and mechanical properties. In particular, sig-
ni®cant di�erences are expected between the center of
the blocks and the vicinity of the canister walls. The
values of S, C* and the corresponding mean e�ective
stress are given in Table 2.
For SS5%, all the samples are representative of the
brittle regime, with the presence of a clear peak stress
on the di�erential stress±axial strain curve, a transition
from a compacting behaviour to a dilating behaviour
at a given di�erential stress level and localization of
deformation on a shear fracture. Slightly after the
peak stress we observe the usual increase in acoustic
emission rate (de®ned as the number of AE recorded
per second) probably corresponding to the creation of
a macroscopic fracture. Beyond this point, a sharp
stress drop and strain softening are observed. Notice
that, on the porosity reduction plot, one of the curves
(open diamonds) does not correspond to the general
trend. The discrepancy is mostly caused by an unu-
sually large porosity reduction at the very beginning of
the experiment, resulting in an important o�set. We
have no clear explanation for this initial o�set not
observed in other samples.
For SS3%, both brittle and ductile behaviour are
observed, depending on the value of the e�ective con-
®ning pressure. The experiments at 10 and 30 MPa are
representative of the brittle regime, with similar
characteristics as for SS5%. Notice however that there
is very little dilatancy close to the peak stress for both
experiments performed at 30 MPa e�ective con®ning
pressure. Such a behaviour (shear localization but little
dilatancy) was referred to as ``transitional'' by
Fig. 3. Results of triaxial tests with on top the axial strain±di�erential stress curves, and on bottom the porosity reduction±e�ective stress curves compared to the hydrostatic compaction curves. Notice on the top right the sharp increase of the
acoustic emission rate (vertical lines) at the peak stress for the experiment at 50 MPa e�ective con®ning pressure.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE764
Mene ndez et al. [9]. The sample tested at 50 MPa has
a di�erent type of failure, representative of the com-
pactive cataclastic ¯ow regime: neither shear localiz-
ation nor dilatancy were observed in this sample.
However a small stress drop is visible in Fig. 3.
Comparing the results for both synthetic sandstones,
clear di�erences were observed regarding the mechan-
ical behaviour under triaxial conditions. First the tran-
sition from brittle to ductile behaviour occurred at
lower pressures for the less cemented material (actually
the transition was not reached in the more cemented
material up to 50 MPa). Second, at any value of the
e�ective con®ning pressure the strength of SS5% is
systematically higher than that of SS3%. Third
Young's modulus calculated from the mean slope on
the di�erential stress±axial strain curves is larger for
SS5% compared to SS3%: the same is also true for
bulk moduli derived from the hydrostatic compaction
curves (Table 1). In addition to ``tangent'' Young's
modulus calculated from the slope of the linear part of
the loading curves, we also measured Young's modulus
at di�erent stress levels following the method described
in Refs. [8, 11]. Selected samples were subjected to
small stress loops which allows to determine elastic
modulus E as a function of applied mean e�ective
stress: such stress cycles are shown in Fig. 3 (exper-
iments at 30 MPa plotted with a continuous line). In
Fig. 4, one can see that these moduli for both synthetic
sandstones increase with mean e�ective stress: such a
behaviour is similar to what is observed in natural
sandstones and reveals non-linear elasticity [8]. For
comparison, the range of tangent Young's modulus for
all tested samples is also given in Fig. 4 (horizontal
dashed bars). Surprisingly, it seems that there is more
non-linearity (i.e. stronger pressure dependence) in the
more cemented synthetic sandstone compared to the
other one. However the uncertainty on these measure-
ments is quite large (about 20%, see Fig. 4) due on
one hand to O-ring friction and on the other hand to
the limited range investigated during each stress cycle.
To summarize, our results show that increasing the
amount of cement has strong consequences on the
mechanical behaviour of the synthetic sandstones, with
enhanced mechanical strength and sti�ness, as well as
higher critical pressure and brittle to ductile transition.
MICROSTRUCTURE OBSERVATIONS
To analyze the microstructure of our synthetic sand-
stones, both optical and scanning electron microscopes
were used. Optical micrographs were presented in a
previous paper (see Fig. 1 in Ref. [1]), which showed
that the rock analogue is visually very similar to gran-
ular rocks like Fontainebleau sandstone. SEM mi-
croscopy coupled with X-ray analysis revealed the
location of silica glass cement in the synthetic sand-
stones (Fig. 1): on the sections investigated, we
observed that silica glass cement is located preferen-
tially at grain contacts as expected. To go further in
the study, we decided to perform a quantitative analy-
sis of selected features observed at the microstructural
level. Presumably, the size of the cemented grain-to-
grain contacts, their number per grain, and also the
grain size should play a crucial role regarding the
mechanical behaviour of the synthetic sandstones. For
SS3% and SS5%, we measured the statistics of the
grain-to-grain contacts in two thin sections, each cov-
ering an area of 2� 1.4 mm. Figure 5 gives a graphical
de®nition of the parameters measured: for any single
grain in the section, we determined the length L of the
contacts between neighbouring grains and the ``grain
size'' D, calculated as the average of the major and
minor diameters across the grain. About 100 grains
Table 2. Results derived from the mechanical tests under triaxial conditions (see the text for the de®nitions)
E�ective con®ning pressure,s3ÿPp (MPa) Peak stress, S (MPa)
Onset of shear-enhancedcompaction, C* (MPa) Mean e�ective stress (MPa)
SS3% 10 102 ÿ 4430 148 ÿ 7930 146 ÿ 7950 ÿ 185 112
SS5% 30 173 ÿ 8830 222 ÿ 10450 243 ÿ 131
Fig. 4. Young's modulus values (with error bars) at di�erent stressesfor the synthetic sandstones. The results are compared to the tangentmodulus, derived from the average slope on the stress±strain curves.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE 765
were analysed on each thin section. We also counted
for each grain the number N of contacts between
neighbouring grains. However for the latter, a 2D
analysis does not give a good estimation of the 3D co-
ordination number, which is a topological parameter
rather than a morphological parameter [32]. The main
problem with parameter N arises from the fact that it
is virtually impossible to see point contacts (i.e. unce-
mented contacts) on 2D sections. In order to compare
with natural rocks, the same analysis was performed
on sections of Fontainebleau sandstones (FS1 to FS3),
Berea sandstone (BS) and some of the reservoir sand-
stones (RS1 to RS5) studied by Bernabe et al. [8].
Note that for the latter, if clays were present, we delib-
erately ignored them in the analysis: the reason for
doing so is that lumps of clays are very deformable
and may not play an important role in the mechanical
behaviour.
In Fig. 6, we plotted the histograms for the contact
length L for the synthetic and the Fontainebleau sand-
stones. One can see that the maximum of probability
for L is consistently shifted towards larger values when
porosity decreases (i.e. cementation increases), both for
the synthetic and Fontainebleau sandstones. Such a
dependence of contact length with porosity is intui-
tively expected. The mean values and standard devi-
ations (normalized to the mean values) for L and D
are reported in Table 3 (note that these are all 2D
measurements, except for porosity and Young's mod-
ulus). The grain size values measured here for the
reservoir sandstones are di�erent from those given in
Ref. [8] which were only rough estimates. The mean
contact length hLi was 54.5 mm for SS5% and 37.7 mm
for SS3%. In both rocks, the standard deviations sLwere very large (i.e. 39.5 and 30.6 mm, respectively).
Normalizing to the mean values, we found that sL/hLi
was equal to 0.72 for SS5% and 0.81 for SS3%. In
other words, the relative degree of heterogeneity was
slightly lower for the most cemented rock than for the
least cemented one. Interestingly, these values are very
similar to those obtained for the Fontainebleau sand-
stone (Table 3). This observation suggests that, in
nature also, the degree of heterogeneity of granular
rocks increases as the amount of cement decreases. A
larger variability in the microstructure parameters is
observed for the reservoir sandstones.
In Fig. 7(a), we plotted the normalized standard de-
viations for parameter L vs porosity. We found a fairly
good linear evolution for the synthetic sandstones and
the Fontainebleau sandstones. The reservoir sand-
stones follow roughly the same trend, but the results
are more scattered which should be expected since
these rocks cover a very wide range of microstructures.
In contrast with this, the Fontainebleau sandstones
and our synthetic rocks are all aggregates formed from
the same pure sand and a single cement (quartz for
Fontainebleau sandstones and silica glass for the syn-
thetic rocks). For the statistics of N (not shown here),
a weak positive correlation with porosity was found.
However there is a bias in such a result as the prob-
ability for counting grain contacts on 2D sections is
larger for more cemented materials. In fact, we believe
that the true coordination number should be the same
for both our synthetic sandstones, because the mech-
anical packing and forming conditions were exactly
the same. Our statistical results show that the higher
the porosity, the larger the degree of heterogeneity.
This is a preliminary result which needs to be con-
®rmed on a larger set of data for di�erent granular
rocks. Therefore, any de®nitive conclusion on the
relationship between the variability of the degree of
heterogeneity and the mechanical behaviour would be
premature. However, it is worth to note that our
observations are in excellent agreement with recent
Fig. 5. Graphical de®nition of microstructure parameters L (lengthof contact), N (number of contacts per grain) and D (grain size).
Fig. 6. Histograms of the contact lengths for the synthetic sandstones (SS) and three Fontainebleau sandstones (FS) withdi�erent porosity f.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE766
numerical simulations. Reuschle [33] clearly showed in
his work based on a network approach to fracture in
rocks that the larger the distribution of local strength
(to be associated in our case to each grain-to-grain
contact) the more ``ductile'' the mechanical behaviour
(i.e. less fracture localization, less mechanical instabil-
ity): in other words, one would expect a lower brittle
to ductile transition when the distribution of local
grain-to-grain geometry is more heterogeneous. This is
exactly what we observed for the synthetic sandstones.
This observation is potentially very useful: from a
thorough microstructure analysis, valuable information
can be obtained on the mechanical behaviour of gran-
ular rocks.
In Fig. 7(b) we plotted the mean contact length hLi
normalized to the mean grain size hDi (as de®ned in
Fig. 5) vs porosity f. There is a good linear corre-
lation between these parameters, with hLi/hDi decreas-
ing with increasing porosity. Interestingly, the
correlation is better than when using hLi alone. The
fact that normalized contact length increases with
decreasing porosity has a strong consequence on the
mechanical properties of granular rocks: given such an
evolution, the stress concentration at grain contacts
will be smaller and the material will be stronger and
more rigid. This is quite consistent with our mechan-
ical data.
DISCUSSION
The experimental data presented in this paper show
a striking similarity between our synthetic sandstones
and natural sandstones from the mechanical behaviour
viewpoint. Qualitatively, the stress±strain and compac-
tion curves are comparable and features like brittle±
ductile transition, critical pressure and non-linear elas-
ticity are reproduced quite well (Figs 2±4).
To go further, we analysed our data according to
the model of Zhang et al. [2] and compared the results
to the prediction of that model for a large set of di�er-
ent sandstones. As said before, this model predicts that
the critical pressure Pcrit which marks the onset of
grain crushing and pore collapse in hydrostatic com-
paction experiments should scale as (fD)ÿ3/2, where f
is the porosity and D is the mean grain size. Implicit
in the model is that the rock can be described by a
packing of spheres, that the contacts between the
spheres obey Hertzian fracture theory and that the
Table 3. Statistics of the microstructural parameters de®ned in the text and Fig. 5, for the synthetic sandstones, three Fontainebleau sand-stones (FS), Berea sandstone (BS) and ®ve reservoir sandstones (RS) studied in Ref. [8]
Porosity (%)Young's modulus
(GPa)Grain size, hDi
(mm) sD/hDi
Contact length, hLi(mm) sL/hLi
SS3% 21.9 7.3 124 0.48 37.7 0.81SS5% 18.7 14.7 134 0.47 54.5 0.72FS1 26.7 ÿ 129 0.37 31.6 0.94FS2 22.7 ÿ 146 0.35 43.0 0.80FS3 18.1 ÿ 148 0.27 54.3 0.71BS 17.9 14.0 86 0.51 29.2 0.93RS1 28.1 11.5 66 0.49 13.2 0.98RS2 27.5 14.5 130 0.76 28.4 1.20RS3 20.5 23.0 473 0.60 187.6 0.72RS4 19.1 9.5 128 0.71 36.8 0.96RS5 18.0 8.5 70 0.52 19.2 0.89
Fig. 7. (a) Heterogeneity parameter (de®ned as the variance normalized to the mean value) vs bulk porosity for contactlength L. The analysis was done on the synthetic sandstones (open symbols), Fontainebleau sandstones (solid circles) and asuite of reservoir sandstones (solid triangles) studied by Bernabe et al. [8]. (b) Ratio hLi/hDi vs porosity f (h i is for statisti-
cal average).
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE 767
fracture nucleation sites scale as the grain size. Figure
8 is a log±log plot of the critical pressure Pcrit vs (fD)
for the synthetic rocks and other granular materials
studied by Wong et al. (Ref. [3] and references
therein). Figure 8 clearly shows that our synthetic
sandstones follow the general trend predicted by the
model of Zhang et al. [2]. Notice however that SS5%
and SS3% are located below the theoretical line with
slope ÿ2/3, which means that the model overestimates
the critical pressure of the synthetic sandstones by at
least a factor 2. This is a puzzling result, in the sense
that the model of Zhang et al. [2] does not include the
e�ect of cement, and one would expect the model to
underestimate rather than overestimate the critical
pressure. Note that in general the observed scatter is
important on the log±log plot. Perhaps (fD) is only a
®rst-order controlling parameter and second order par-
ameters like the geometry of contacts or the grain
shape may also have a signi®cant in¯uence.
We also investigated the failure envelope, comparing
our results to those reported by Bernabe and Brace [34]
and Wong et al. [3] for a suite of natural sandstones.
Figure 9 represents the failure envelopes in the p±q
coordinates, where p is the e�ective mean stress and q
the di�erential stress. Dark symbols indicate the stress
state at the peak stress S for rock samples which failed
in the brittle regime, whereas open symbols show the
stress state at the failure stress C* for rock samples
which failed in the cataclastic ¯ow regime. The data
on a such a plot approximately map out an elliptical
cap which amplitude is strongly controlled by the criti-
cal pressure [3]. We can see in Fig. 9 that our results
for the synthetic sandstones are in good agreement
with the data for natural rocks. In their recent paper,
Wong et al. [3] propose that the critical pressure Pcrit
acts as a scaling parameter for the failure envelopes:
indeed, when plotting the mechanical data for di�erent
sandstones with coordinates p/Pcrit and q/Pcrit, these
authors show that, within a reasonable scatter, a single
failure envelope can be ®tted to the normalized data.
Since, to ®rst order, the critical pressure is controlled
by (fD), one should expect that the failure envelope
for di�erent granular materials should also scale as
(fD). The work done by Wong et al. [3] uni®es in a
single model several mechanical properties: strength,
critical pressure, brittle to ductile transition and relates
those properties to parameter (fD). To check on this
idea, we have constructed a three-dimensional diagram
where (fD) was introduced as an additional parameter
(Fig. 10). The advantage of such a plot is that it
includes both descriptions of the mechanical data in
term of macroscopic failure envelopes and microstruc-
ture parameters. For the ®ve sandstones in Fig. 9, we
®rst ®tted a quadratic function f = Ap2+Bp on each
failure envelope. In general, the ®t was reasonably
Fig. 8. Critical pressure vs parameter fD (porosity times grain size)for the synthetic sandstones and a set of granular materials studiedby Wong et al. [3]. The line with slope ÿ2/3 is the prediction of the
model proposed by Zhang et al. [2].
Fig. 9. Failure data points and envelopes for the synthetic sandstonesand a suite of natural sandstones studied by Wong et al. [3]. Darksymbols correspond to the brittle regime, open symbols correspond
to the cataclastic ¯ow regime.
Fig. 10. Three dimensional plot showing the scaling of the failureenvelopes with parameter (fD). The failure envelopes correspond toparabolic regression curves on the data from Wong et al. [3], respect-ively for Boise (front), Rothbach, Kayenta, Darley Dale and Berea(back) sandstones. Grid lines correspond to interpolation on thewhole data set. For comparison our data for SS3% (open squares)
and SS5% (open circles) are also plotted.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE768
good. The values of the ®tting parameters A and B are
listed in Table 4. Notice that Wong et al. [3] used a
parabolic envelope only in the dilating brittle regime,
and an elliptical cap for the compacting ductile regime:
for our purpose, it was enough to use a single quadra-
tic ®t. In a second step, the parabolic envelopes were
plotted in Fig. 10 for each sandstone, using (fD) as
third coordinate axis. One can see that (fD) is a good
scaling parameter: the size of the parabolic failure
envelopes decreases regularly with increasing values of
(fD). To see how the synthetic sandstones ®t into this
description, the same analysis was done for SS3% and
SS5%. The experimental results are plotted as open
symbols (the short segments show the distance of the
points to the fD= 0 plane). Although only few data
points were available, we could also model the failure
envelope by a quadratic function and the results are
comparable to those of the natural sandstones
(Table 4). This new kind of analysis combining mech-
anical and microstructure data is potentially very useful
for people who try to model the mechanical behaviour
of granular materials. Our results also con®rm the
capability of our synthetic material to reproduce accu-
rately the mechanical behaviour of natural rocks.
The analysis of the mechanical data showed that
(fD) has a strong in¯uence on the mechanical beha-
viour of both natural and synthetic sandstones. To
understand why this is so, one has to remember that
in granular rocks the mechanical properties are
strongly a�ected by the geometry of contacts between
neighbouring grains [16, 19]. In particular the contact
length L and number of contacts per grain N must
play an important role. The existence of correlations
between L, N and porosity or grain size would give a
justi®cation for the scaling observed in Fig. 10.
Unfortunately, we don't have the microstructure data
for the sandstones presented on that ®gure. However,
we can try to use the statistics available for the
Fontainebleau, reservoir and synthetic sandstones in
Fig. 7. Indeed the results in Fig. 7 show that a good
correlation exist between microstructure parameters
hLi/hDi and porosity for this limited set of granular
materials. This gives a physical background for the
scaling observed for the failure envelopes and the criti-
cal pressure with (fD). To go further, we also investi-
gated more quantitatively the interplay between the
microstructure parameters and the mechanical proper-
ties: this was done on the analysis of Young's mod-
ulus. For the reservoir and the synthetic sandstones,
we plotted in Fig. 11 Young's modulus E vs parameter
hLi/hDi and found an overall increase between both
parameters. Notice however that the linear ®t is not
very good. Again we used hLi/hDi instead of hLi:
doing so, our results show that for a ®xed grain size,
the granular rock will be sti�er when the contact
length is larger. Note also that contact length normal-
ized to grain size is often used in models for the elastic
properties of granular materials [35]. The next step
would be to develop a cementation model to quantify
the e�ect of contact length, porosity and coordination
number on the mechanical properties of granular
rocks: this is beyond the scope of this paper.
Obviously more data are required to con®rm the
conclusions drawn from our analysis on a limited set
of synthetic and natural sandstones. Nevertheless, the
correlations that we found show at least that the stat-
istics of the geometrical properties at grain contacts
has a signi®cant in¯uence on the mechanical properties
of granular materials. In this regard, our approach has
the advantage that it allows to produce synthetic ma-
terial with controlled amount of cement, all other
structural parameters being held constant: therefore it
is possible to work on a series of synthetic rocks cover-
ing a broad range of cement content to further check
the ideas exposed in this preliminary work.
CONCLUSION
We presented a set of mechanical and microstruc-
tural data for two varieties of synthetic sandstones pre-
pared following the technique described in Ref. [1] and
compared the results to those obtained on several
natural sandstones. Hydrostatic and triaxial exper-
iments showed that increasing the amount of cement
from 3% in volume (SS3%) to 5% in volume (SS5%)
results in larger values for critical pressure, strength
and sti�ness. The brittle-to-ductile transition is also
shifted towards higher pressures (in other words the
more cemented material tends to be more brittle). Our
results are quantitatively in agreement with the model
of Zhang et al. [2] which links the critical pressure to
the parameter (fD) and with the description in terms
Table 4. Fitting parameters A and B (see text) for the synthetic andnatural sandstones. Note that stresses need to be in MPa in the
quadratic function
A B
Boise ÿ0.0698 3.163Rothbach ÿ0.0107 2.633Kayenta ÿ0.00785 2.456Darley Dale ÿ0.00678 2.470Berea ÿ0.00604 2.439SS3% ÿ0.00625 2.373SS5% ÿ0.00583 2.624
Fig. 11. Correlation between Young's modulus and contact lengthnormalized to mean grain size for the synthetic (open symbols) and
reservoir (solid triangles) sandstones.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE 769
of failure envelopes proposed by Wong et al. [3]. In
order to better understand the importance of (fD), we
investigated the statistics of L, the length of the con-
tacts between adjacent grains and D, the mean grain
size. We found for our synthetic sandstones and a
suite of Fontainebleau and reservoir sandstones a
negative correlation between hLi/hDi and porosity.
This result gives a physical justi®cation for the scaling
observed for both the failure envelopes and critical
pressure with parameter (fD). We also found that the
lower the cement content, the higher the degree of het-
erogeneity for the local grain-to-grain geometry: het-
erogeneity may play an important role for the
mechanical behaviour of granular rocks, as suggested
by recent simulations [33].
We found a good similarity between the properties
of the synthetic sandstones and those of natural rocks:
therefore this gives us some con®dence in applying the
conclusions of our work to natural rocks. The
approach of preparing rock analogue is useful in the
sense that each structural parameter can be investi-
gated independently (grain size, porosity, sorting, etc.),
while maintaining the others unchanged. In addition,
not only the amount of cement can be modi®ed, but
also its composition. The study presented here can also
be extended to other rock properties, like acoustic or
transport properties, with the general idea that work-
ing on rock analogues like those used in this paper is
easier than in natural rocks.
AcknowledgementsÐWe thank Yves Gueguen for his commentswhich helped to improve the paper. Praeme Chopra kindly gave ushis gel recipe for the preparation of the synthetic sandstones. Thisstudy could not have been possible without the expertise of Bas denBrok who was in charge of preparing the synthetic sandstones. Thehot-pressing was done at ETH, Zurich: we thank Dave Olgaard andDavid Bruhn for their assistance. The mechanical tests were per-formed during short stays of two of the authors (C. D. and Y. B.) atSUNY Stony Brook, with the help of Teng-fong Wong and WenluZhu. NATO funded part of the travel expenses to SUNY. Thisresearch was funded by the GDR Ge ome canique des RochesProfondes.
Accepted for publication 30 December 1997
REFERENCES
1. den Brok, S. W. J., David, C. and Bernabe , Y., Preparation ofsynthetic sandstones with variable cementation for studying thephysical properties of granular rocks. C. R. Acad. Sci., 1997,325, 487±492.
2. Zhang, J., Wong, T.-F. and Davis, D. M., Micromechanics ofpressure-induced grain crushing in porous rocks. J. Geophys.Res., 1990, 95, 341±352.
3. Wong, T.-F., David, C. and Zhu, W., The transition from brittlefaulting to cataclastic ¯ow in porous sandstones: Mechanical de-formation. J. Geophys. Res., 1997, 102, 3009±3025.
4. Martin, J. C. and Serdengecti, S., Subsidence over oil and gas®elds. Geol. Soc. Am. Rev. Eng. Geol., 1984, 6, 23±34.
5. Handin, J.et al., Experimental deformation of sedimentary rocksunder con®ning pressure: Pore pressure tests. Am. Assoc. Petrol.Geol. Bull., 1963, 47, 717±755.
6. Gallagher, J. J.et al., Experimental studies relating to microfrac-ture in sandstone. Tectonophysics, 1974, 21, 203±247.
7. Zoback, M. D. and Byerlee, J. D., E�ect of high-pressure defor-mation on permeability of Ottawa sand. Am. Assoc. Petrol. Geol.Bull., 1976, 60, 1531±1541.
8. Bernabe , Y., Fryer, D. T. and Shively, R. M., Experimental ob-servations of the elastic and inelastic behaviour of porous sand-stones. Geophys. J. Int., 1994, 117, 403±418.
9. Mene ndez, B., Zhu, W. and Wong, T. F., Micromechanics ofbrittle faulting and cataclastic ¯ow in Berea sandstone. J. Struct.Geol., 1996, 18(1), 1±16.
10. Brace, W. F., Volume changes during fracture and frictional slid-ing: A review. PAGEOPH, 1978, 116, 603±614.
11. Bernabe , Y. and Fryer, D. T., On the use of small stress excur-sions to investigate the mechanical behaviour of porous rocks.Int. J. Rock Mech. Geomech. Abstr., 1995, 32, 93±99.
12. Simmons, G. et al., Physical Properties and Microstructures of aSet of Sandstones, Secondary, eds. G. Simmons et al.Schlumberger-Doll Research Center, 1982.
13. Fredrich, J. T., Mene ndez, B. and Wong, T. -F., Imaging thepore structure of geomaterials. Science, 1995, 268, 276±279.
14. Dvorkin, J., Mavko, G. and Nur, A., The e�ect of cementationon the elastic properties of granular materials. Mech. Mater.,1991, 12, 207±217.
15. Dvorkin, J., Yin, J. and Nur, A., E�ective properties of cemen-ted granular materials. Mech. Mater., 1994, 18, 351±366.
16. Zang, A. and Wong, T. -F., Elastic sti�ness and stress concen-tration in cemented granular material. Int. J. Rock Mech.Geomech. Abstr., 1995, 32, 563±574.
17. Bruno, M. S. and Nelson, R. B., Microstructural analysis of theinelastic behaviour of sedimentary rock. Mech. Mater., 1991, 12,95±118.
18. Bernabe , Y., Fryer, D. T. and Hayes, J. A., The e�ect of cementon the strength of granular rocks. Geophys. Res. Lett., 1992, 19,1511±1514.
19. Yin, J. and Dvorkin, J., Strength of cemented grains. Geophys.Res. Lett., 1994, 21, 903±906.
20. Wong, T. -F. and Wu, L. -C., Tensile stress concentration andcompresive failure in cemented granular material. Geophys. Res.Lett., 1995, 13, 1649±1652.
21. Almossawi, H. I. H., Physical properties of synthetic sandstonerocks. Geophys. Prosp., 1988, 36, 689±699.
22. Visser, R., Acoustic Measurements on Real and SyntheticReservoir Rock, Secondary, ed. R. Visser. Proefschrift,Technische Universiteit Delft, 1988.
23. Holt, R. M., Unander, T. E. and Kenter, C. J., Constitutivemechanical behaviour of synthetic sandstone formed understress. Int. J. Rock Mech. Geomech. Abstr., 1993, 30, 719±722.
24. Dass, R. N.et al., Tensile stress±strain behaviour of lightlycemented sand. Int. J. Rock Mech. Geomech. Abstr., 1993, 7,
711±714.25. David, C., Mene ndez, B. and Darot, M., In¯uence of stress-
induced and thermal cracking on physical properties and micro-structure of La Peyratte Granite. Int. J. Rock Mech. andGeomech. Abstr., in preparation.
26. Winkler, K. W., Contact sti�ness in granular porous material:Comparison between theory and experiment. Geophys. Res.Lett., 1983, 10, 1073±1076.
27. Palciauskas, V. V., Compressional to shear wave velocity ratioof granular rocks: Role of rough grain contacts. Geophys. Res.Lett., 1992, 19, 1683±1686.
28. David, C.et al., Laboratory measurement of compaction-inducedpermeability change in porous rocks: Implications for the gener-ation and maintenance of pore pressure excess in the crust.PAGEOPH±3, 1994, 143(1), 425±456.
29. Zhang, J.et al., Pressure-induced microcracking and grain crush-ing in Berea and Boise sandstones: Acoustic emission and quan-titative microscopy measurements. Mech. Mater., 1990, 9, 1±15.
30. Read, M. D.et al., Microcracking during triaxial deformation ofporous rocks monitored by changes in rock physical properties.II. Pore volumometry and acoustic emission measurements onwater-saturated rocks. Tectonophysics, 1995, 245, 223±235.
31. Zang, A., Wagner, C. F. and Dresen, G., Acoustic emission,microstructure and damage model of dry and wet sandstonestressed to failure. J. Geophys. Res., 1996, 101, 17507±17521.
32. Dodds, J. and Leitzlement, M., In Physics of the Finely DividedMatter, eds. N. Boccara and M. Daoud. Springer, 1985.
33. Reuschle , T., A network approach to fracture. PAGEOPH,1997, submitted.
34. Bernabe , Y. and Brace, W. F., Deformation and fracture ofBerea sandstone. Am. Geophys. Monogr., 1990, 56, 91±101.
35. Digby, P. J., The e�ective elastic moduli of porous granularrocks. J. Appl. Mech., 1981, 48, 803±808.
DAVID et al.: MECHANICAL BEHAVIOUR OF SYNTHETIC SANDSTONE770