On the use of small stress excurtions to investigate the mechanical behaviour of porous rocks

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 32, No. 1, pp. 93-99, 1995

Pergamon

0148-9062(94)00019-0

Copyright ((", 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/95 $9.50 + 0.00

Technical Note On the Use of Small Stress Excursions to Investigate the Mechanical Behaviour of Porous Rocks Y. BERNABl~t D. T. FRYER~

INTRODUCTION A standard hypothesis used in the mechanics of materials is that the deformation of a stressed body is the sum of elastic and non-elastic terms ([1]; for a historical review see [2]). Experimental procedures for accurately evaluating the elastic part of the strain during laboratory testing of rocks are therefore needed. By definition, elastic deformation must be instantaneous and reversible. But linearity, in contradiction to common practice, is not required. In fact, the presence of pores and grain contacts in elastic porous materials is known to produce non-linearity [3-5]. In most rocks the initial, non-linear, upwardly curved, portion of the stress-strain curves is associated with elastic closure of microcracks. It is usually followed by a linear segment traditionally considered elastic [6]. However, dynamically measured moduli along such linear segments are often found to be significantly greater than the static tangent moduli at the same points (similar observations were also made using static measurements [7]; also, see Figs 5 and 6 in [8]). This discrepancy is probably due to frictional sliding on crack surfaces or grain contacts [9] and shows that, despite the linearity, the behaviour along these segments is not purely elastic. In previous studies, small stress excursions towards the interior of the yield surface were used to measure the elastic moduli at discrete intervals along the main stress path during laboratory testing of rock samples ([10], [7]), From these values, the elastic part of the deformation can then be inferred [7]. Here, we want to illustrate the possibilities of this technique by showing how its application helped in improving the correlation of mechanical properties and microstructure in two porous carbonates. Rocks of this type are often oil bearing and, therefore, economically important. We limited this study to prefailure stresses which are often prevalent in situ. However, even in moderate stress conditions, accurately predicting the rock behaviour around wellbores may still be difficult because significant inelastic deformation can occur. For instance, acoustic emissions, a sure evidence of non-elastic behaviour, were observed to begin at tlnstitut de Physique du Globe de Strasbourg, 67084 Strasbourg Cedex, France. ++Chevron,U.S.A. 93

relatively low stresses (see Fig. 2 in [1 l], or Fig. 2 in [12]) and to increase afterwards at an accelerated rate as the failure stress was approached. Similarly, many studies in the literature focused on the failure and post-failure regions where strain localization occurs [13] and the elastic moduli exhibit a drastic drop (see [14] or [10]). But, if the ultimate aim is to forecast failure, the pre-failure region should logically be considered more relevant.

DESCRIPTION OF SAMPLES

In this work, two carbonate rocks denoted C6 and C l 1 were considered. The samples were precisely ground to a right-cylindrical shape, 2.54cm in diameter and 5.08 cm in length. They were cleaned of cutting oil and thoroughly dried in a vacuum oven. Their porosity, ~, permeability, k, and grain density, pg, were then measured using standard equipment (i.e. helium porosimeter, air-permeameter and precision scale). Thinsections were prepared from intact rock samples and examined in the optical and electron microscopes (Figs 1 and 2). C6 is a sparitic, oolitic limestone with abundant, well cemented, bioclastic elements including siliceous spicules ( ~ - - 1 8 . 0 % , k = 1 0 - 1 4 m 2, pg=2.69gcm-3). C11 is a micritic limestone with some fossils and sparitic areas dispersed in a matrix of very small calcite grains ( ~ = 1 4 . 5 % , k = 5 . 4 . 1 0 - ~ S m 2, pg=2.71gcm-3). The two rocks essentially differ in the way the bioclastic elements are cemented: C6's cement consists of solid, polycrystalline calcite with very few micropores and visible grain boundaries; whereas C ll's cement is a porous, microgranular calcite aggregate. EXPERIMENTAL SETTING The triaxial testing apparatus used here has already been described in [7] and we refer the reader to this paper for details on the experimental set-up. The triaxial cell was capable of confining pressures up to 50 MPa. The confining pressure, a 3, and the axial stress, a~, were digitally recorded and computer controlled to _ 0.03 MPa for o-3 and + 0.12 MPa for ~j. The axial and lateral strains, q and ~3, were measured by means of two

94

BERNAB]~ and FRYER:

T E C H N I C A L NOTE

where the superscript (e) refers to purely elastic deformations. K denotes the bulk modulus, G the shear modulus and @, Sq, &p and (~Eqare the stress and strain increments (note that K and G may depend on p and q). However, coupling (i.e. dependence of 6Ep and (~q on both Sp and Sq) appears when the rock behaviour becomes inelastic. Shear enhanced compaction and dilatancy are common manifestations of this effect (see, among others, [16] and [7]). Under certain conditions, this coupling may be used to identify the onset of non-elastic deformation or yield point (caution must be exercised in the case of anisotropic rocks since similar coupling is also caused by anisotropy [7]). The tests began with a hydrostatic compression up to 28 MPa (this stress path is represented by a horizontal line in the {p, q } plane; and was called the p-path in [7]). We then followed a constant mean stress path (i.e. a vertical line in the {p,q} plane or q-path) up to q = 4 2 M P a for C6 and 52MPa for CII. In neither experiment was failure reached. Unloading was conducted along the paths identical to the loading paths but

undeformed

deformed Fig. 1. Thin sections of intact and deformed C6 (SEM in a backscattered electrons mode). The pictures are 2 m m high and 2.5 m m wide. Note areas of solid (i.e. sparitic) calcite cement. In the deformed rock, a number of cracks in a direction sub-parallel to at (i.e. vertical) can be seen in the sparitic cement.

undeformed

sets of two strain-gauges axially and circumferentially oriented and diametrically mounted on the central portions of the samples. The estimated uncertainty was ___5 microstrains, mainly due to temperature variations. The maximum strain measurable was 25,000 microstrains or 2.5%. EXPERIMENTAL

PROCEDURES

For porous materials which may undergo large changes in volume during straining, the most convenient stress variables are the mean stress, p = (al + 2a3)/3, and the deviator stress, p = a ~ - a3 [15]. The corresponding strain variables are the volumetric strain, cp = E1+ 2E3, and the distortion, Cq= 2(EI- e3)/3. In elastic, uniform, isotropic materials the following uncoupled relations are satisfied: ~E~pe ) =

~p/g

(1)

6E
Oq/3G

(2)

deformed Fig. 2. As for Fig. 1 but for C1 I. Note the microgranular, microporous (i.e. micritic) cement. Intact and deformed C l l appear almost identical.

BERNABI~ and FRYER: TECHNICAL NOTE

95

considered elastic during hydrostatic compression. However, the two rocks presented some differences: C6 appeared linear during the first hydrostatic loading cycle; whereas C l l was significantly non-linear [Fig. ~0 4(a)]. During the q-path, C6 displayed a greater amount of irreversible strain than C ll [Fig. 4(b) and (d)]. Increasing deviator stress at constant mean stress pro"7, duced a strong compaction in C6 while C11 was slightly ¢0 dilatant. Inelasticity is often associated with change in microstructure (i.e. damage) and, as a consequence, with variations of the elastic moduli. The linearity noticed in ,- .......... ~:"-."" C6 for the first p-path loading cycle was not present anymore during the unloading cycle [Fig. 4(a)]. AccordD i s t o r t i o n , Eq ingly, this may be attributed to damage occurring during Fig. 3. An illustration of the stress excursion technique. The elastic the preceding q-path. The shape of the ¢p vs p curve moduli are givenby the slope of the small unload-reloadcycles.When these cyclescoincidewith the main stress-strain curve the behaviour remained unchanged at the beginning of the second cycle is considered purely elastic (i.e. points C1 and C4). Note that the as should be expected since no damage should take place stress-strain curve represented here is not real and that the un- during the rest time between the two tests. Finally, load-reload cycleswere highlyexaggeratedto be more visible(in fact, increasing curvature in the last unloading cycle revealed the hysteresiswas negligible). that more damage had occurred during the second test. In C ll, to the contrary, the two tests were almost in the opposite direction. The entire cycle was performed identical and it can be concluded that damage was twice, first with dry samples, second with water-satuprobably insignificant. rated samples drained in the atmosphere. However, successive cycles produced almost identical results and Elastic constants the effect of water was therefore considered negligible in Figure 5 shows the values of K and 3G at discrete these experiments. points along the stress-strain curves for both rocks. In order to evaluate the elastic part of the deforProbably due to their high porosities, C6 and C11 mation, we interrupted the main stress path at discrete have Ks and 3Gs much lower than the theoretical points and performed small unload-reload cycles as values for a non-porous calcite aggregate (i.e. illustrated in Fig. 3. According to plasticity theory, K = 70,000-80,000 MPa, 3G = 75,000-105,000 MPa materials can be assumed to behave elastically along [17]). In C6, during the first p-path loading cycle, K and such stress excursions if they are small enough and 3G were constant (i.e. their variations were smaller than oriented toward the interior of the yield surface [1]. the expected experimental errors). Furthermore, K was Reversibility, the most direct evidence of elastic becomparable to the tangent modulus, K*=Op/OEp, haviour, was systematically checked and the data remeasured at the same points along the main stress path jected when non-reversible cycles occurred. If an elastic [Fig. 5(a)]. This confirms that C6 behaved elastically unload-reload cycle coincided with the stress-strain during hydrostatic loading. More interesting is the obcurve, the behaviour along the main stress path was servation that K and 3G also appeared essentially conconsidered purely elastic. Otherwise, the deformation along the unload-reload cycle was assumed identical to stant during the first q-path loading cycle although the behaviour was non-elastic (indeed, the tangent modulus, the elastic part of the strain at the same point along the 3G* = Oq/aEq, decreased sharply with q as soon as q was main stress-strain curve. The inelastic strain increments could then simply be evaluated as the difference between increased [see Fig. 5(b)]. Afterwards, both K and 3G total and elastic strain increments (i.e. &~) = &p - &~e), decreased dramatically as q was unloaded. This sharp &~) = &q - &~qe)).The unload-reload cycles were always drop of the elastic moduli continued during the hydroexecuted using regular triaxial paths (i.e. changing at static unloading cycle although the behaviour was again while keeping a 3 constant). Along this path both p and elastic (K ~ K*; in fact, K* was slightly greater than K q vary, allowing K and 3G to be measured simul- which suggests that C6 was mildly anisotropic [7]), During further cycles, this pattern did not change anytaneously in isotropic rocks. more. In C11, the comparison o f K a n d 3G with K* and 3G* also confirms that the behaviour was elastic during RESULTS hydrostatic compression and inelastic during the q-path Triaxial tests (yielding again began at very small qs). Unlike C6, Ci 1 Figure 4 shows the four stress-strain curves (i.e. Epand began non-linearly. Both K and 3G increased with Eqversus p and q) for both C6 and C11. It appears clearly increasing p, K also increased with increasing q, but 3G that, in both rocks, significant irreversible deformation was not affected by changes in deviator stress. Imporwas only produced during the q-path. Furthermore, tantly, this pattern did not vary with successive cycles during hydrostatic compression, the distortion Eqwas not confirming that straining did not produce damage in this affected by p [Fig. 4(c)]. Therefore, the behaviour can be rock. .

96

BERNABI~ and FRYER:

TECHNICAL NOTE

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Energy dissipation

significant damage was generated in C6 during loading without immediately affecting the elastic moduli. The The strain energy expended in a sample during a test influence of damage was only felt during unloading. is given by: Finally, a puzzling observation can be made from E = SP &p + q &q (3) Fig. 6: a significant amount of dissipated energy apIf &p and &q can be decomposed into elastic and peared to be recovered during unloading of the q-path inelastic terms as was explained earlier, then, E can also in C6 (similar observation was made for sandstone in be written as a sum of recoverable (i.e. elastic) and [7]). This "negative" dissipation could not be reduced non-recoverable (i.e. inelastic) strain energy terms, E re) without changing the values of the elastic moduli used and E 0). Figure 6 shows E (~ (i.e. the energy dissipation) in the calculations by an amount much larger than the calculated for both samples. The results confirm pre- experimental error. vious conclusions that, in both rocks, the dissipation was negligible during hydrostatic compression and that Microstructure much more energy was dissipated during the q-paths in Thin sections of the undeformed and deformed maC6 than in C11. Also, we can see in Fig. 6 that most of terials were examined under an optical microscope and the dissipation took place during the loading cycles, a scanning electron microscope or SEM (see Figs 1 and From this and previous observations, it appears that 2). Using polarized transmitted light, we determined the

BERNABI~ and FRYER:

number of twins in each rock and their orientations with respect to ~h. The observed distributions for both deformed and undeformed rocks are shown in Fig. 7. In undeformed C6, we can note a large number of randomly oriented twins. Deformed C6 shows a significant increase of the twin density and, furthermore, a clear preferred orientation around 90 ° to trl. In C11, the twin density was always rather low but again showed a clear increase in the deformed sample perpendicularly to trl. It is likely that the twin density was underestimated in C11 because the small grain size in the micritic cement does not permit good microstructure observations. Using the SEM in a backscattered electron mode, images can be obtained with a sharp contrast between solid material and pore space. This allows easy detection of very thin microcracks. Quantitative stereology methods [18] can then be applied to quantify the amount of microcracks produced in the rock during the tests.

T E C H N I C A L NOTE

97

The classic method of determining the microcrack surface area [19, 20] was modified here because the rocks considered had too complex porous microstructures. Two 2 mm test lines, one parallel and the other perpendicular to a,, were positioned at random locations in the thin sections. The number of intersections of these lines with the pore-solid interface was then measured. This procedure was repeated 60 times for each thin section, yielding two quantities, noted P ~) and P [90},from which the specific internal surface area of the material, Sv, can be inferred:

(4) The difference, S ~)- S ~"), where (d) refers to deformed and (u) to undeformed rock, is a measure of the specific surface area of new microcracks and grain boundaries produced during straining.

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Fig. 5. The elastic moduli K and 3G as a function o f p and q for C6 and C1 ! (typical error bars are indicated; the wide grey arrows represent the tangent moduli K* and 3G*, see text for comments).

60

98

BERNABI~ and FRYER: TECHNICALNOTE

The results are the following: in C6, Stvu)= 62.2 mm -~ and S ~ ) = 9 7 . 4 m m - ~ ( P ~ = 3 2 . 2 + 9 . 8 m m - ~ and p~9o)=30.8_10.3mm-~ in intact rock; P ~ ) = 57.9+ 14.9mm -t and PtLg°)=46.2__ 13.2mm-' in deformed rock; note that for this type of measurement the standard deviation is more a measure of the heterogeneity of the material than of the experimental error [18]). The fact that both P~) and p~0) increased during straining and that only a weak anisotropy was produced, suggests that grain boundary cracking was important in this rock. The new internal surface area produced during straining was 35.2 mm -1. In C11: S~ ) = 123.5 mm -t and S~ ) = 127.7 mm -1 (P~) = 63.4 _ 17.5 mm -~ and p~0) = 61.3_15.2mm-1 in intact rock; P
deformed A

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Fig. 7. The twin densityobservedin intact and deformedC6 and CI i as a function of the angle to at (see text for comments). than in C6 and, probably, principally involved frictional sliding at grain contacts (microcracking was absent and twinning was low). This is again consistent with the granular character of the rock. DISCUSSION AND CONCLUSIONS

The behaviour of each rock appeared consistent with its cement microstructure: the limestone with sparitic cement was initially linear elastic (indeed, roughly spherical voids should produce less non-linearity than Hertzian contacts between grains). It underwent damage during straining (i.e. intragranular and grain boundary cracking accompanied by pore collapse) and, as a result, evolved into a more granular material (i.e. non-linear elastic). On the other hand, the micrite always behaved non-linearly because of the granular structure of its cement (this is consistent with previous observations [22]). The grains were probably densely packed and 60 dilatancy occurred as a result of rearrangement of the A m grains. Previous studies have shown the mechanical v ! importance of cement in sedimentary rocks [23, 24]. * ** **$ They focused on the influence of the composition (i.e. o . +:~0+ +÷ soft or rigid) and volume fraction (see also theoretical ¢ ÷ 30 ~n models [25] and [26]). Here, we demonstrated that the ÷ cement microstructure (i.e. solid or granular) can play a o~ considerable role as well. S A The stress excursion technique used here was crucial lU for successfully interpreting the data. A number of i e 3( interesting observations could not have been made without accurate knowledge of the elastic constants (recent Q. work has shown that static and dynamic moduli agree if) well with each other if precautions are taken to insure ° " i C6#2 + 8 purely elastic behaviour in the static measurements [27]). c 0 1 T 1 1 T For instance, although damage occurred in C6 during 0,1 0,2 0,3 0,4 0,5 0,6 [ 0,0 the q-path loading cycle, its effect was only felt during unloading (a possible explanation of this is that new energy dissipation (10e Jm "3) Fig. 6. The calculatedenergydissipationin C6 and CI 1 as a function microcracks remained closed during loading and opened only during unloading). Also, comparison of the elastic of p and q (see text for comments).

i

,

/

[.

,"

BERNABI~ and FRYER:

and tangent moduli revealed that yielding began at particularly small deviator stresses. This is not too surprising for C I 1 which behaves like a granular material (see the theory of Hertzian contact under shear [4]; also [28]). On the other hand, this is unexpected in C6. Finally, what should we think of the "negative" dissipation observed in C6? It could obviously be due to errors in the elastic constants but, as already stated, unrealistically large changes are needed to remove it. Another possibility is that the material was initially in a non-zero energy state (i.e. it contained residual stresses). We saw that the intact rock already had a significant density of twins suggesting that it had already been subjected to some relatively high stress levels. Lastly, we should mention that our basic assumption (i.e. total strain is the sum of two elastic and inelastic terms) might be incorrect. Acknowledgement--This work was partially performed at the former Chevron Oil Field Research Company. We are grateful to the management for permitting publication of the results. Accepted for publication 28 July 1994.

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