Cemented Sandstone

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

0148-9062/98 $19.00+0.00PII: S0148-9062(98)00003-5

{Ecole et Observatoire des Sciences de la Terre, UniversiteÂ LouisPasteur, CNRS UMR 7516, 5, rue ReneÂ Descartes, 67084Strasbourg, France.

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 MeneÂ 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


(=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.


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.


MeneÂ 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.


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 BernabeÂ 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.


numerical simulations. ReuschleÂ [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 BernabeÂ et al. [8]. (b) Ratio hLi/hDi vs porosity f (h i is for statisti-

cal average).


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 BernabeÂ 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.


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.


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 GeÂ omeÂ canique des RochesProfondes.

Accepted for publication 30 December 1997

REFERENCES

1. den Brok, S. W. J., David, C. and BernabeÂ , 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. BernabeÂ , 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. MeneÂ 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. BernabeÂ , 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., MeneÂ 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. BernabeÂ , 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., MeneÂ 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. ReuschleÂ , T., A network approach to fracture. PAGEOPH,1997, submitted.

34. BernabeÂ , 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.


Cemented Sandstone

Documents

Transcript of Cemented Sandstone